This series of articles showcases a variety of applications around the theme of inversion, which constitutes a strategic way of manipulating configurations of circles and lines in the plane. This article includes the Riemann sphere, rings of four tangent circles, and inverting the Sierpinski sieve.

The story of inversion, propelled by the study of conic sections and stereographic projection, is quite complex [1]. Although Vietá (1540–1603) already spoke of mutually inverted points, the proper development started with a plethora of notable geometers including L’Huilier (1750–1840), Dandelin (1794–1847), Quetelet (1796–1874), Steiner (1796–1863), Magnus (1790–1861), Plücker (1801–1868), Bellavitis (1803–1880), and Simpson (1849–1924). Steiner, in a manuscript published in 1913, is considered to have been the first to formulate inversion as a method to systematically simplify the study of complex geometric figures where circles play a prominent role. This work culminated in 1855 with the studies of Möbius (1790–1868) (hence the choice of the letter for the inversive circle in this article) [2]. Peaucellier (1832–1913) also applied inversion to his famous linkage [3], and Lord Kelvin (1824–1907) applied inversion to elasticity.

Let denote either the line or the segment from to , depending on the context. The length of the segment is denoted by . The circle with center and radius is denoted by . Denote by the inverse of in the current circle of inversion , where is either a symbolic point or often ; in `Manipulate` results, is often shown as a red dashed circle.

The following functions encapsulate the basic properties of inversive geometry developed in the first part of this series [4].

The function `squareDistance` computes the square of the Euclidean distance between two given points. (It is more convenient to use the following definition than the built-in Mathematica function `SquaredEuclideanDistance`.)

The function `collinearQ` tests whether three given points are collinear. When exactly two of them are equal, it gives `True`, and when all three are equal, it gives `False`, because there is no unique line through them.

The function computes the circle passing through the points , , and . If the points are collinear, it gives the line through them; if all three points are the same, it returns an error message, as there is no meaningful definition of inversion in a circle of zero radius.

The function computes the inverse of in the circle . The object can be a point (including the special point that inverts to the center of ), a circle, or a line (specified by two points).

Here is an example.

Since a circle can be inverted into a line, define a generalized circle to be either an ordinary circle or a line, as in the first article in this series. A consequence of theorem 2 in the next section is that the set of generalized circles is closed under inversion in a circle. The following `Manipulate` shows that as , the circle tends to the line , and the inverse of the circle tends to its reflection in the line , the circle .

Therefore, it makes sense to define inversion in a line to be reflection.

Here is an example.

The function `squareDistanceToLine` computes the square of the distance of a given point to a given line.

The function `exCircles` computes the four circles tangent to the lines forming the sides of a triangle having given vertices.

The function `inCircle` computes the incircle of a triangle having given vertices as the excircle of smallest radius.

The function `redPoint` is used to mark a given point in red.

The function draws an arc of radius `r` and center `b` from the line `ab` to the line `bc`.

The function `intersections` computes the possible points of intersection of a given line and circle. (There may be zero, one, or two points.)

The following `Manipulate` shows several of the functions introduced so far.

The concept of orthogonality between circles plays an important role in inversive geometry. Two intersecting circles are orthogonal if they have perpendicular tangents at either point of intersection. It is rather amusing to consider orthogonality of circles without mentioning any notion of perpendicularity [3, 4], as is done in the following definition.

**Definition**

Two intersecting circles and are orthogonal if one of them (say ) belongs to a set of three mutually tangent circles that touch each other at three distinct points of .

The following `Manipulate` shows the central circle is orthogonal to the black one by showing the other two circles (blue). Some values of the positions of the black circle or the radii of the blue circles make it impossible to close the three circles, so adjustments to close them are done on the fly.

Theorems 1–11 review the basic properties of inversion and introduce some of its remarkable properties. Consider all inversions to be with respect to the circle .

**Theorem 1**

The following `Manipulate` illustrates this property and shows that triangles and are similar. The gray circle is orthogonal to .

The following interesting variation of theorem 1 was presented at the Swiss Mathematical Contest

in 1999 [5, 6]:

Two circles intersect in points and . An arbitrary point of the first circle, which is not or , is joined to and , and the straight lines and intersect the second circle again in the points and . Prove that the tangent to the first circle at is parallel to the straight line .

**Theorem 2**

The inverse of a circle not passing through is also a circle not passing through . (In this case, the images of concyclic points are concyclic.) The inverse of a circle passing through is a straight line not passing through . (In this case, the images of concyclic points are collinear.)

For instance, invert the circle in .

The set of circles and lines, called *generalized circles*, is therefore closed under inversion.

**Theorem 3**

A straight line passing through is self-inverse. The inverse of a straight line not through is a circle through .

**Theorem 4**

Any circle through a pair of inverse points is orthogonal to the circle of inversion. Conversely, any circle cutting the circle of inversion orthogonally and passing through a point also passes through the inverse [7].

**Theorem 5**

(The effect of inversion on angles.) The angle of intersection of two circles is the same as that of their inverse circles. This is also true for any intersecting curve; that is, inversion is conformal.

**Theorem 6**

**Proof**

By the definition of inversion, . Then . Since , the result follows. □

**Theorem 7**

(The effect of inversion on the size of circles.) If the circle inverts into a circle of radius , then

To verify this result, notice that in the following expression (the condition implies does not pass through and thus indeed inverts into a circle).

**Theorem 8**

Two circles of radii and touching externally at the point invert into two parallel lines separated by the distance .

Subtracting the first points of the following lines obtained by inversion gives the result.

**Theorem 9**

where is the length of the tangent from to [8].

The following `Manipulate` shows the terms mentioned in theorem 9.

**Proof**

Let and be the ends of the diameter of shown in the figure above. The point is outside the segment , as otherwise would not be defined. Then

and the result follows. □

For instance, consider inverting in .

Then , which we verify as follows.

In the case of theorem 7, this is the corresponding comparison.

In order to verify this result, assume that and , with and (otherwise would not be a circle).

From this theorem, it follows that the product of the lengths of the tangents from to and

must be .

**Theorem 10**

(Ptolemy) Let , , , and be arbitrary points. Then , with equality if and only if the points are on a generalized circle.

**Proof**

Invert , , and in the circle . Then and (by theorem 6). Substituting this and similar results for , , , and in the inequality gives

which reduces to ; this always applies in the triangle . Equality holds if and only if the points , , and are collinear, that is, if and only if points , , , and are concyclic or collinear. □

Ptolemy (circa 127 AD) compiled much of what today is the pseudoscience of astrology. His Earth-centered universe held sway for 1500 years. In the words of Carl Sagan, in the first episode of his TV series *Cosmos*, “… showing that intellectual brilliance is no guarantee against being dead wrong.” You can find fascinating applications of Ptolemy’s theorem in [9, 10].

**Theorem 11**

(A fixed-point theorem.) Suppose two nonintersecting circles have centers and . Any circle that cuts those two circles orthogonally passes through the same two points on the line [10].

This result is useful when inverting to concentric circles elsewhere. In order to verify it, assume without loss of generality that the two circles are and , with , and that the orthogonal circle is . Then the following quantity is independent of , and it is equal to the position of one of the two fixed points mentioned.

The following provides a framework with which to interpret inversion. Consider a sphere of unit radius centered at the origin. Draw a line from the north pole to a point on the sphere. The point at which intersects the - plane defines a one-to-one correspondence between points on the sphere and points in the plane; is called *stereographic projection*. The image of the south pole is the origin. The image of the north pole is not defined, but is introduced as a new point to serve as the image of ; this makes the mapping continuous and one-to-one. In the context of stereographic projection, the sphere is referred to as the *Riemann sphere* after the prominent mathematician Bernhard Riemann (1826–1866), who studied under Steiner and earned his PhD degree under Gauss [11, 12].

Similar triangles give

By using these formulas, it is possible to prove that the stereographic projection of a circle on is a generalized circle in the plane. (Circles through go to lines and other circles go to circles.) So inversion in the unit circle in the plane induces a map from circles to circles on .

This defines stereographic projection.

An inversive pair of points maps to points reflected in the - plane.

Moreover, as the following `Manipulate` shows, an inversive pair of generalized circles maps to circles on that are reflections across the - plane. You can vary the radius and center of the control circle in the - plane. The circle inverts to in the unit circle, which is the equator of . You can show the stereographic cone joining to through its stereographic projection and the stereographic cone joining to through its stereographic projection , which is the reflection of in the - plane [13].

Amusing applications related to stereographic projection are found in [14, 15], and interesting Demonstrations in [16–18].

Consider a ring of four cyclically externally tangent circles as in the following `Manipulate`. The four points of tangency are concyclic (pink circle), even when some of the circles are internally tangent. You can drag the four points around this circle to alter the shape of the arrangement. To see many other similar patterns, see [19]. In contrast to the case of three tangent circles, the circle through the points of tangency is not necessarily orthogonal to the other four circles.

The function `perturb` slightly varies three points that are coincident or collinear.

Let us apply inversion to deduce the concyclic property. Invert in a circle centered at one of the tangent points, for instance, the red dashed circle in the following `Manipulate`. That inversion transforms the circles centered at and into two parallel lines and two circles in between these lines. These four inversions are sequentially tangent in three points. The problem then reduces to show that the three points of tangency lie on the same line (shown in blue).

Let us take a closer look at the arrangement formed by the inversion of the chain of circles, starting from a different point of view. The following `Manipulate` shows an arrangement of two parallel lines, one slanted line, and two circles with the tangencies indicated. The two orange disks indicate adjustable tangency points and of the circles and the parallel lines. It is easy to show that the slanted line always passes through the tangency point of the circles regardless of the positions of and and the value of one of the radii, say, of the lower circle. In fact, the sum of the radii remains constant for fixed positions of and . For some values of the radii there exists a third circle also tangent to the parallel lines and the circles. Selecting the appropriate option in the `Manipulate`, the radii are adjusted according to the position of the slanted line for this third (blue) circle to exist. When the slanted line is vertical and the circles are congruent, three of those third circles exist.

The next `Manipulate` inverts the previous arrangement. It shows that the tangent points of the set of (blue) pairwise tangent circles are concyclic. It also shows that there exist up to three circles tangent to the circles of , and sets of up to four circles orthogonal to the circles of . You can vary the center of the inversive circle. Similar examples can be found at [20, 21].

The quadrilateral joining the centers of the circles forming the ring has an inscribed circle. This is easily seen using Pitot’s theorem (1695–1771): a convex quadrilateral with consecutive side lengths , , , is *tangential*, that is, it has an inscribed circle, if and only if . Oddly enough, in the case of the quadrilateral, its incircle does not necessarily coincide with the circle passing through the four points of tangency, as the following result shows.

The *Sierpinski sieve* (or *Sierpinski gasket*, or *Sierpinski triangle*) [5, 6], named after the prolific Polish mathematician Waclaw Sierpinski (1882–1969), is a self-similar subdivision of a triangle. The function `Sierpinski` constructs the iteration corresponding to the recursive definition of this subdivision.

Inverting the vertices of the triangles from the first four iterations in the unit circle gives rise to stunning patterns.

The following `Manipulate` lets you vary the radius and center of the inverting circle, the number of iterations, and the type of transformation applied to the vertices forming the triangles forming the Sierpinski sieve.

I would like to thank the anonymous referee whose thoughtful and detailed comments on this article greatly improved its presentation, and also Queretaro’s Institute of Technology, which provided essential support for the completion of this second part.

[1] | H. W. Eves, A Survey of Geometry, Boston: Allyn and Bacon, 1972. |

[2] | B. C. Patterson, “The Origins of the Geometric Principle of Inversion,” Isis, 19(1), 1933pp. 154–180. www.jstor.org/stable/225190. |

[3] | J. Rangel-Mondragón. “Inversive Geometry II: The Peaucellier Inversor Mechanism” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/InversiveGeometryIIThePeaucellierInversorMechanism. |

[4] | J. Rangel-Mondragón, “Inversive Geometry: Part 1. Inverting Generalized Circles, Ellipses, Polygons, and Tilings,” The Mathematica Journal, 15, 2013. doi:10.3888/tmj.15-7. |

[5] | R. Todev, Geometry Problems and Solutions from Mathematical Olympiads, MathOlymps, 2010. |

[6] | D. Djukić, V. Janković, I. Matić, and N. Petrović, The IMO Compendium, 2nd ed., New York: Springer, 2011. |

[7] | D. E. Blair, Inversion Theory and Conformal Mapping, Providence, RI: The American Mathematical Society, 2000. |

[8] | H. Fukagawa and D. Pedoe, Japanese Temple Geometry Problems: San Gaku, Winnipeg, Canada: The Charles Babbage Research Centre, 1989. |

[9] | A. S. Posamentier, Advanced Euclidean Geometry, Emeryville, CA: Key College Publishing, 2002. |

[10] | E. W. Weisstein. “Ptolemy’s Theorem” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/PtolemysTheorem.html. |

[11] | D. Pedoe, Geometry, A Comprehensive Course, New York: Dover Publications Inc., 1988. |

[12] | E. W. Weisstein. “Riemann Sphere” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/RiemannSphere.html. |

[13] | T. Needham, Visual Complex Analysis, Oxford: Claredon Press, 1997. |

[14] | D. Gehrig, “The Orloj,” The Mathematica Journal, 7(4), 2000.www.mathematica-journal.com/issue/v7i4/features/gehrig. |

[15] | P. W. Kuchel, “Spatial Inversion: Reflective Anamorphograms,” The Mathematical Journal, 9(2), 2004. www.mathematica-journal.com/issue/v9i2/SpatialInversion.html. |

[16] | J. Rangel-Mondragón. “A General Cone” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/AGeneralCone. |

[17] | P. W. Kuchel, “Anamorphoscopes: A Visual Aid for Circle Inversion,” Mathematical Gazette, 63(424), 1979 pp. 82–89. |

[18] | E. Mahieu. “Inverse Stereographic Projection of Simple Geometric Shapes” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/InverseStereographicProjectionOfSimpleGeometricShapes. |

[19] | A. Akopyan, Geometry in Figures, CreateSpace Independent Publishing Platform, 2011. |

[20] | J. Rangel-Mondragon. “Problems on Circles IV: Circles Tangent to Two Others with a Given Radius” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ProblemsOnCirclesIVCirclesTangentToTwoOthersWithAGivenRadius. |

[21] | J. Rangel-Mondragon. “Problems on Circles X: Tangent Circles Generate Ellipses” from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/ProblemsOnCirclesXTangentCirclesGenerateEllipses. |

J. Rangel-Mondragón, “Inversive Geometry,” The Mathematica Journal, 2016. dx.doi.org/doi:10.3888/tmj.18-5. |

Jaime Rangel-Mondragón received M.Sc. and Ph.D. degrees in Applied Mathematics and Computation from the School of Mathematics and Computer Science at the University College of North Wales in Bangor, UK. He was a visiting scholar at Wolfram Research, Inc. and held positions in the Faculty of Informatics at UCNW, the Center of Literary and Linguistic Studies at the College of Mexico, the Department of Electrical Engineering at the Center of Research and Advanced Studies, the Center of Computational Engineering (of which he was director) at the Monterrey Institute of Technology, the Department of Mechatronics at the Queretaro Institute of Technology, and the Autonomous University of Queretaro in Mexico, where he was a member of the Department of Informatics and in charge of the Academic Body of Algorithms, Computation, and Networks. His research included combinatorics, the theory of computing, computational geometry, and recreational mathematics. Jaime Rangel-Mondragón died in 2015.

]]>We describe efficient algorithms for working with subgroups of . Operations discussed include join and meet, congruence testing, congruence closure, subgroup testing, cusp enumeration, supergroup lattice, generators and coset enumeration, and constructing a group from a list of generators.

The set of linear fractional transformations of the form

(1) |

known as Möbius transformations, has several interesting properties. First, the composition of functions of this form is still of the same form, as

Since the coefficients appearing in the composition are exactly those of the product of the two matrices and , it is most convenient to represent transformations of the form (1) by the matrix . A matrix and any nonzero scalar multiple of itself represent the same Möbius transformation, so we can consider only matrices with determinant 1 without loss of generality. Since the product of two matrices with determinant 1 also has determinant 1, such a set of matrices (or Möbius transformations) forms a group, where the group operation is matrix multiplication (or composition). If we further restrict the coefficients , , , and in (1) to be integers, the resulting group is known as . Now, if the matrix is in , then is also in and represents the same Möbius transformation. For this reason, we consider , known as the modular group, which is with each matrix identified with its negative. That is,

It is possible to show that every transformation in the modular group can be obtained as a combination of the two fundamental transformations

with corresponding matrices and . Another way of stating this fact is that is generated by and .

For example, the matrix may be obtained as the product .

The modular group is important because of the existence of modular functions, which are functions that have simple transformation laws under the action of the modular group. A prototypical modular function is the modular discriminant function, which may be defined for by

Since this product has zeros at every rational number, the real axis becomes a natural boundary of the domain of . Using the methods of analysis, it is possible to show that

(2) |

for every matrix in the modular group. Two observations are in order concerning the transformation formula (2). First, as

we see that the transformed value of still has a positive imaginary part, so it still lies in the domain of . Second, the values of where ranges over the whole upper half-plane are related to the values of where is restricted to the region shaded blue here.

This region is known as the fundamental domain for . This is because the transformations and can be used to bring any point in into this region, and no two points inside it differ by a Möbius transformation in . The transformation pairs the left edge with the right edge, while the transformation pairs the arc from to with the arc from to .

In the theory of modular functions one often wants to know what transformations leave a given function unchanged. For example,

will not be unchanged by all the transformations in , since the numerator does not have a transformation formula under all elements of . However, if is in and is divisible by 5, then

Thus, we are naturally led to the subgroup of given by

The package ModularSubgroups.m addresses the computational problem of working with such subgroups of the modular group. However, only certain subgroups of can be identified by congruence conditions on their entries, as is the case with . Such subgroups are called congruence subgroups and are discussed more later. For this reason, we need a better way to represent subgroups. The key to this lies in the matrices and . Since and generate

, and are also generators. However, it can be shown that is the free product of and ; that is, every matrix in can be written uniquely as a word in and as long as no two consecutive ’s appear and no three consecutive ’s appear in the word. This last condition is necessary because of the relations , where (the identity matrix) should be thought of as the empty word.

A subgroup of the modular group is said to have finite index (in ) if can be written as a disjoint union

of left cosets , where the left coset is defined as . In this way, the group is partitioned into several “copies” of , and the number of copies of that fit inside is called the index . If is a finite index subgroup of with index and left cosets (with a distinguished coset ), the matrices and permute the left cosets when acting by multiplication on the left; that is, we have equations

where and should be viewed as some permutations of the set , that is, elements of the symmetric group .

This identification of the matrices and as permutations gives rise to the permutation representation of , which we use to represent any subgroup of with finite index. Specifically, a subgroup is identified by: (1) its index ; (2) the permutation ; and (3) the permutation . The permutations and are not arbitrary. The following two conditions are necessary and sufficient for a given to appear as the representation of some group .

(1) in , where is the identity in . This condition arises from the fact that as matrices, we have .

(2) and generate a transitive subgroup of , or equivalently, the Schreier cosets graph discussed later is connected. This condition arises because the matrix sends the coset to the coset , hence the action of on the left cosets “connects” all of the cosets together.

If these two conditions are satisfied, the group may be identified as , where the condition needs to be evaluated after thinking of as a permutation by converting and into their corresponding permutations.

Since our representation of by two permutations and involves an arbitrary ordering of the nontrivial cosets , two different representations and represent the same group precisely when there is a relabeling of the indices in the permutations of that simultaneously converts into and into . For, example the two representations

represent the same group, as the relabeling converts to .

Another important combinatorial object attached to a subgroup of finite index is the Farey symbol for , as described in [1]. This symbol directly encodes a fundamental domain for as well as the edge-pairing matrices for this fundamental domain. Equivalently, it encodes independent generators for . However, since the equivalence of two representations of by two different Farey symbols is not as straightforward to detect, the permutation representation as described was chosen for the underlying representation for .

A subgroup of is called a congruence subgroup if it contains the principal congruence subgroup of level ,

for some natural number . If this is the case, we can describe as those matrices whose entries satisfy certain congruences modulo . For example, two families of congruence subgroups are and , which are defined as

Recently in [2], Hsu gave a simple test for determining if a given subgroup of is a congruence subgroup, based on a presentation for . This algorithm is implemented here and generalized to compute the congruence closure of a subgroup , which is the smallest congruence subgroup that contains .

The Schreier cosets graph of is of fundamental importance to several of the algorithms in the package. Given a subgroup with index and permutations and , the Schreier coset graph is the connected graph with vertices and labeled edges

Such a graph has the property of being folded. A graph is said to be folded if every vertex has at most one edge of a given orientation and label incident with it. If there is a vertex with two or more edges of the same label and orientation, then the graph is said to be unfolded. One property of the Schreier cosets graph is that the subgroup of the modular group consists of all words in and such that, when starting at vertex 1, the path that follows word must terminate at vertex 1. For example, take the subgroup with the following Schreier cosets graph.

The word , which corresponds to the matrix

is in as the path traced out by is given by ( must be read right to left since we are dealing with left cosets). Since the graph is folded, the process of tracing a path given by a word in and is deterministic. The group corresponding to this Schreier cosets graph turns out to be a congruence subgroup, and its defining congruences are given in Example 1.

All of these examples were tested in Mathematica 10.

Set the directory to be able to load the package and then evaluate the `Needs`.

Subgroups of the modular group are maintained in the container , and the names of the functions that operate on these groups start with a lower case “m” in order to avoid possible conflicts with built-in symbols. The matrices `mS`, `mO`, `mT`, `mR` of the package are set as follows.

Here is the group from the section on Schreier cosets graphs. The permutations are listed so that and , where and are the last two arguments of the `mGroup` container.

This group turns out to be a congruence subgroup of level 3, and it consists of those matrices that are congruent modulo 3 to one of the following matrices.

So, for example, the group has the description

The group generated by and is of finite index only for . Similarly, the group generated by and is of finite index only for .

There are two conjugacy classes of congruence subgroups of index 7, which we define here by

their generators.

In the printed form of the group :

- is the index of in .
- is the genus of the compact surface .
- is the number of cusps of , namely, the size of .
- is the number of fixed points of order 2 of in .
- is the number of fixed points of order 3 of in .

These numbers are related by .

They are indeed not conjugate.

The intersection of these two groups turns out to have index 28, while the group generated by these two groups turns out to be the full modular group.

A fundamental domain may be obtained with `mDomain`.

The edge pairings for this fundamental domain are given in the Farey symbol. Edges between two rational numbers with the same integer label are paired together, while edges with the label ● or ○ are paired with themselves.

The matrices returned by `mGenerators` are the matrices responsible for pairing the edges in this way.

The congruence closure of a subgroup is the smallest congruence subgroup that contains . We first start with the congruence subgroup of index 7 from the previous example and a non-congruence subgroup of index 9.

Its congruence closure is the theta subgroup.

One can also compute the congruence closure of a group by joining it with the principal congruence subgroup of the same level, but the package uses a much more efficient method. Membership in the groups , , and can be tested with , , and , respectively. The function `mFromMember` constructs the internal representation of the group given the group’s membership function.

We also have the following property, since `g` itself is congruence.

Next, we compute the congruence closure for the group given in Example 1.1 of [2]. It turns out to be the full modular group.

First get the principal congruence subgroup of level 5, which has index 60.

Generators may be computed quickly from this permutation representation, and we can also efficiently reconstruct the group from a list of generators.

We will graph the supergroup lattice for the principal congruence subgroup of level 4. The `mSupergroups` function is used to make the supergroup lattice. The index of each group (in )

is displayed in the lattice, and the actual group is displayed as a tooltip. Every subgroup of whose matrices can be described by congruence conditions modulo 4 appears somewhere on this lattice.

If is a subgroup of the modular group, then every matrix in acts on (the set of rational numbers with included) and partitions into equivalence classes. We say that two rational numbers and are equivalent under if there is an element of that sends to . The set of equivalence classes of under the action of is known as the cusps for , and there are finitely many cusps if has finite index in the modular group. The width of a cusp with respect to is defined to be the index of the stabilizer of inside the stabilizer of .

Let `g` and `h` be the subgroups

Here is a list of inequivalent cusps of `h` and their widths.

Here we reduce a list of random cusps to one of these four. The frequency of each cusp in the list should be proportional to its width.

The intersection of `g` and `h` may be computed.

Of course, the implementation in the package is more efficient.

If denotes the total number of subgroups of the modular group of index , then with so that it is possible to show that

and that

Since the radius of convergence of the power series is zero, this differential equation must be treated formally as a recurrence relation for the coefficients of . See Section 1 of [3] for details.

Caranica [4, Table 3.1] has computed the conjugacy classes of non-congruence subgroups of index 9. However, this table incorrectly claims that there are 11 conjugacy classes. In fact, there are 108 non-congruence subgroups of index 9 and 12 conjugacy classes. Vidal [5] has given a formula for the generating function of the total number of conjugacy classes of subgroups (congruence or not) of a given index.

Fortunately, the Farey symbols for the claimed groups are provided by Caranica, so we can recover the source of the error. First we verify that there are indeed 11 conjugacy classes of non-congruence subgroups in the table.

This is the group that is missing from the table.

The Mathematica built-in function `KleinInvariantJ` is invariant under and takes each complex value exactly once inside a fundamental domain for . A plot of this function and an outline of its fundamental domain are shown.

The upper half-plane, which is parameterized by , has been mapped into the unit disk, which is parameterized by , by the relation .

The hue of the color plotted at a point in the disk is the argument of the complex number , where is the point in the upper half-plane corresponding to .

Similarly, the built-in function `DedekindEta` can be used to construct such a function for , which is a congruence subgroup of of index 3. A plot of this function and a fundamental domain for are shown next.

Here is a similar construction for . This plots the fundamental domain for in the half-plane.

In the disk, such a region becomes a diamond shape. A univalent function on must be invariant under the generators of ; that is, , and such a function is provided by the built-in function `ModularLambda`.

In this case, the zero of occurs on this diamond where the colors coalesce. In the previous example, the zeros of are not visible in this way because they occur on the boundary of the

disk.

Many operations on subgroups of the modular group depend on operations on graphs. Several of the algorithms used here encounter unfolded graphs, and we use the efficient folding algorithm described in [6] to implement Stallings’s folding process, which converts any graph to a folded graph.

As described in the introduction, it is straightforward to convert a group described by the permutations and to the Schreier cosets graph. In order to reverse this procedure, it is necessary that each edge labeled either have the same initial and terminal vertex or be part of a three-cycle. Similarly, each edge labeled needs to occur in either a loop or a two-cycle. All of the graphs used here have this property. However, when building a group from a list of a generators, we may encounter a folded graph in which some vertex does not have valence four. Such graphs do not correspond to subgroups of of finite index. Let us illustrate the folding procedure on the following graph.

Such a graph represents the subgroup that is generated by the two words and , since, except for the trivial cycles induced by the relations , these are the only cycles in the graph. Whenever there are two edges and incident at the same vertex with the same orientation and label, causing the graph to be unfolded, the edge may be deleted and the vertex may be merged with without changing the subgroup represented by the graph. The progression of the graph folding procedure shown is left to right, top to bottom, with the edges to be deleted shown between graphs.

The subgroup of represented by the final folded graph has index three and is determined by the permutations and . This is also known as the theta subgroup. The reader is urged to work through the folding process for the group generated by and to see that it does not have finite index in . The starting graph is shown here.

It is useful to have a notion of a standard representation (in terms of the permutations and ) of a group whereby two groups are the same if and only if their permutations and are identical. This can be accomplished by visiting the coset first (denoted by the index 1 in the permutations). Once we have visited a coset , we then recursively visit the coset (assuming this has not been visited yet), and once this trip has returned to the coset , we visit the coset (also assuming this coset has not been visited yet). The standard labels for the indices for the nontrivial cosets may then be determined by the order in which that coset was visited.

In the case of testing a matrix for membership in a group , write as a word in and , then set and check if . Specifically, for a given matrix whose entries in the left column are non-negative, multiply on the left by the matrices

until the left column contains a zero. The variable holds the current coset, so every time is multiplied by , for example, needs to be updated to .

Given the membership function on matrices for a group, we construct the group coset by coset. Assume that has index at least three in . If , start with the four cosets ; otherwise, start with the three cosets . Proceed by adding either one or three cosets to at a time. If is such that:

- and , then add the coset to .
- and , then add the cosets , , to .

Where there is no such coset satisfying either of these conditions, we have found all of the cosets of . A naive implementation of this procedure would have worst-case running time , where is the index of the resulting group. The worst-case running time may be reduced to by keeping track of which cosets actually need to be checked.

We may compute coset representatives, generators, and a Farey symbol in operations for a subgroup of index . This works as follows. Let be the graph corresponding to a subgroup of index . First, the graph is cut into a tree so that the coset labeled is given by the resulting unique path from the vertex 1 to the vertex . Any time a cut is made or a fixed point is encountered, the corresponding matrix is added to the list of generators. Finally, after the cosets and generators are computed and the cuts have been recorded, the Farey symbol is computed by a clockwise traversal of the tree.

The cusps of a given subgroup are also important. The action of on the upper half-plane is given by

and this action extends to . The equivalence classes of under the action of , namely , are finite, and we may choose a representative for each one as follows. The stabilizer of in is generated by (or ). Therefore, any two cusps (say and for ) are equivalent under whenever there is an integer such that ; that is, and belong to the same cycle of the permutation . The width of a cusp can then be defined as the length of the cycle (of ) that contains the coset .

Joining and intersecting two groups is surprisingly simple. To compute the group that is generated by and , we can form the graph for . Then, for each generator of , merge the vertices in corresponding to the cosets and . This will in general result in a unfolded graph, which we can then fold and convert back to a permutation representation. In order to compute the permutation representations for the intersection of and , first find the orbit of under the action of and in terms of cosets of the form . A permutation representation of may then be obtained by the action of and on the cosets in this orbit of .

It is also straightforward to check if two groups are the same or conjugate, or if one group is contained in another. To test if two groups and are the same, we employ a strategy similar to the process for standardizing the representation. The cosets of and are visited simultaneously, starting with the pair . If we are currently visiting the pair , then we visit the pairs and as described in the standardization process. If the two paths ever become out of sync, that is, if cosets are visited in a different order, then we know the groups are not the same; otherwise the two paths will return back to and we know that and are the same. Checking if and are conjugate can be accomplished by the same procedure. We need to check if the path stays in sync when starting at some pair for .

The congruence functions use the list of relations of Hsu [2]. Recall that the congruence closure of a group is the smallest congruence subgroup that contains . We compute the congruence closure of as follows. Hsu gives a list of relations that are satisfied if and only if is a congruence subgroup. Let be the list of the permutations where is a relation in Hsu’s list. If contains a non-identity permutation , this represents an obstacle to being a congruence subgroup. Let denote the level of , which is defined as the order of the permutation . As it is known that contains , the set of relations for is also satisfied by . Let be any permutation in and an index of any coset in . Since must act trivially on the cosets of and is a subgroup of , the group obtained from by merging cosets and must also be contained in . Therefore, merging the cosets and of for all and must give .

We have described an efficient package for manipulating and constructing subgroups of the modular group. It is hoped that this will further interest in these groups and facilitate research dealing with these subgroups.

The author would like to thank Junxian Li of the University of Illinois at Urbana-Champaign for many helpful discussions on improvements and corrections to initial implementations of the algorithms.

[1] | C. A. Kurth and L. Long, “Computations with Finite Index Subgroups of Using Farey Symbols.” arxiv.org/abs/0710.1835. |

[2] | T. Hsu, “Identifying Congruence Subgroups of the Modular Group,” Proceedings of the American Mathematical Society, 124(5), 1996 pp. 1351–1359. www.ams.org/journals/proc/1996-124-05/S0002-9939-96-03496-X/S0002-9939-96-03496-X.pdf. |

[3] | A. Lubotzky, “Counting Finite Index Subgroups,” London Mathematical Society Lecture Note Series 212: Groups ’93 Galway/St Andrews, Vol. 2 (C. M. Campbell, E. F. Robertson, T. C. Hurley, S. J. Tobin, and J. J. Ward, eds.), Cambridge: Cambridge University Press, 1995pp. 368–404. |

[4] | C. C. Caranica, “Algorithms Related to Subgroups of the Modular Group,” Ph.D. thesis, Louisiana State University, 2009. etd.lsu.edu/docs/available/etd-07092009-200839/unrestricted/Caranica_diss.pdf. |

[5] | S. A. Vidal, “Sur la classification et le dénombrement des sous-groupes du groupe modulaire et de leurs classes de conjugaison,” Publications IRMA, 66(II), 2006 pp. 1–35. Preprint: arxiv.org/abs/math.CO/0702223. |

[6] | N. W. M. Touikant, “A Fast Algorithm for Stallings’ Folding Process,” International Journal of Algebra and Computation, 16(6), 2006 pp. 1031–1045. doi:10.1142/S0218196706003396. |

D. Schultz, “Manipulating Subgroups of the Modular Group,” The Mathematica Journal, 2016. dx.doi.org/doi:10.3888/tmj.18-4. |

Daniel Schultz is a postdoctoral researcher at Pennsylvania State University who is interested in modular functions and automorphic forms in general.

**Daniel Schultz**

*Pennsylvania State University Department of Mathematics*

University Park

State College, PA 16802

*dps23@psu.edu*

We present an implementation of the Poisson-Influenced -Means Algorithm (PIKA), first developed to characterize the output of a superconducting transition edge sensor (TES) in the few-photon-counting regime. The algorithm seeks to group data into several clusters that minimize their distances from their means, as in classical -means clustering, but with the added knowledge that the cluster sizes should follow a Poisson distribution.

The algorithm proper is run when it is submitted using the button near the lower-right corner of the form. You also have the option to use a separate input file to manually override the form, which may be more useful for automated runs on multiple datasets. This function launches the program; evaluating it generates the form and then executes PIKA. The function is defined and documented in Section 7.9. After taking input through the form, `pika` calls `runPIKA`, the *de facto* main function for the program, defined in Section 3. You can also specify a separate options file in the form that overrides the variable assignments that the form makes. The first command ensures that the paclet for forms is current.

If you get a popup that asks, Do you want to automatically evaluate all the initialization cells…?, answer yes. When you get the form, it is necessary to replace [directory] with the pathname of the location of the data; you might also have to change the backslash to a forward slash.

The Poisson-Influenced -means Algorithm (PIKA) was first described in [1] as a way of calibrating a transition edge sensor (TES), a superconducting few-photon detector. A TES can discern the number of photons in a very weak pulse of light, but it must be calibrated in order to do so. Our implementation deals with photon counting, but many of its features are applicable to more general probability-assisted -means clustering situations.

A TES is a superconductor kept in its transition from its superconducting phase to the normal regime, where it loses its superconducting properties. Photons incident on the sensor heat it, causing its resistance to rise sharply and then slowly fall to superconducting levels as the heat dissipates. A current is run through the TES, and the change in resistance is captured by the voltage signal of a superconducting quantum interference device (SQUID) inductively coupled to the TES circuit.

Several groups of TES signal waveforms are shown in Figure 1 (each graph shows the set of signals elicited by an ensemble of laser pulses with an average number of photons per pulse given by ). For , one can clearly distinguish the different photon numbers and their relative frequencies; for higher numbers this is harder. Higher photon numbers create higher signal amplitudes, but at a certain point the TES saturates in the normal regime and additional photons change the signal maximum very little.

**Figure 1.** Several collections of TES waveforms resulting from pulses with particular mean photon numbers given by , from [2].

The goal of PIKA is to characterize individual TES waveforms by the integer photon numbers of the pulses that cause them. The photon numbers of individual pulses cannot be determined directly; we can only estimate the average photon number of all of the pulses, based on the nominal laser and attenuator parameters of the light source.

-means clustering, upon which PIKA is based, is a fundamental part of unsupervised machine learning. PIKA extends the -means algorithm to scenarios in which the ideal distribution that the clusters should follow is known, and though some of the implementation is specific to the context of TES calibration (e.g. the use of the Poisson distribution, the idea of ordering observations by photon number), much of it can be generalized without much difficulty to other situations with known probability distributions.

Traditional -means clustering consists of taking some amount of data and organizing it into clusters that minimize their members’ distance from the cluster mean. Essentially this is a minimization of an objective function, the sum over each piece of data of its deviation from its cluster mean (where deviation is measured by some relevant definition of distance). We can use a similar approach by considering each signal as a high-dimensional vector and its deviation from some mean as squared Euclidean distance. Then the -means component of the objective function becomes

(1) |

where is the signal vector for observation ( is the element of the vector ), is the mean of the cluster , and is the number of time points; and give the first cluster’s photon number and number of clusters, respectively, and are determined by which photon numbers we expect to be associated with at least one pulse based on the Poisson distribution. More physically, is an individual waveform and is the average of the waveforms with an assigned photon number .

To account for the Poisson-distributed cluster sizes, we introduce another term, , where is the likelihood, according to the Poisson distribution, of a group of clusters associated with a group of photon numbers having the particular sizes that a given clustering asserts that they do, given the mean photon number of the ensemble of pulses. The likelihood of a particular sequence of photon numbers occurring in an ensemble with mean is

(2) |

where is the number of waveforms in cluster . We also need a combinatorial component, since different photon-number sequences can yield the same eventual cluster sizes:

(3) |

where . Then , and the PIKA objective function is

(4) |

The constant relating the two terms can be estimated from the data, since the objective function is itself the negative log-likelihood of a normal distribution. (This also means that minimizing is equivalent to maximizing the product of two likelihoods.) PIKA minimizes the objective function by moving waveforms to neighboring clusters.

Once the clusters are optimized, each waveform is assigned an effective photon number by a linear interpolation between the two closest cluster means. First, we find the value that minimizes the root mean square deviation of from , where and are the closest and second-closest mean waveforms to . In practice, and . One can easily show that

(5) |

for each . The effective photon number is then given by

(6) |

PIKA needs an initial clustering upon which to improve. Random cluster assignment is an option, but a better alternative is to give the observations a rough order by photon number, so that our initial guess is actually a meaningful estimate. This is done via the dot product method: we assign each observation an initial effective photon number

(7) |

where is the entire ensemble’s mean, not a cluster mean . The initial clusters are sized to fit each observation and conform to the Poisson distribution, and the observations are placed in the clusters by order of effective photon number.

The geometric interpretation of PIKA and the dot product method is a curve and a line, respectively, evolving through hyperspace (shown in Figure 2). The dot product method projects each observation onto the mean waveform vector (a line) and then assumes that photon number scales linearly with distance along the mean vector (which is not actually true, but suffices for a first guess); that converts distance relative to the mean to photon number relative to the mean. PIKA, in contrast, finds a piecewise linear approximation of a curve that passes through the cluster means and projects each observation onto that. Both essentially measure photon number by progress along a one-dimensional path through high-dimensional space.

**Figure 2.** An illustration of the geometric differences between the dot product method (green arrow) and PIKA (blue curve). The 3D space here stands in for a high-dimensional space.

PIKA requires knowledge of the ensemble’s mean photon number in addition to an initial clustering. There are two ways of supplying that knowledge. The first is simply to give the exact mean photon number of the incoming pulses, if it is known; then PIKA clusters the data accordingly.

The second is to test several mean photon numbers on the data if the true mean is not known exactly. The test means should be close together in some range around a rough estimate of the true mean; PIKA clusters the data once for each value, returning a new (usually better) estimate of the mean based on each optimized clustering, as well as the value of the objective function associated with each new estimate. (In addition, since the test means are close together so that adjacent distributions should be very similar, the effective photon numbers for the waveforms from one round are used as the initial ordering for the next, instead of the dot product method.)

The optimized means are usually closer to the true mean than their initial seeds, but, depending on the structure of the probability distribution underlying the objective function, some optimized means may be moved farther away due to attraction to local minima. In the Poisson case, we have observed the objective function to have secondary minima at integer differences in mean photon number from the primary minimum and convex regions of width 1 centered around each minimum. Thus, test means that land in the same region as the true mean should be optimized toward the true mean, but those outside are diverted by secondary minima. Reference [3] estimates the range of test means to within half of a photon number from the nominal laser power and attenuation.

The following results and images (in Section 2.5) except Figure 6 come from [1].

Figure 3(a) shows the optimized cluster means from two PIKA runs (, solid red, and , dotted blue). The shapes of the mean waveforms appear independent of —reassuring, since the average photon number of an ensemble should not affect the shape of individual waveforms. This independence indicates that PIKA is properly identifying actual photon numbers in the data, and that the average photon number with which it is supplied is not unduly affecting the results.

Figure 3(b) is a histogram of the optimized effective photon numbers from (bin widths of 0.05), which follow a Poisson distribution but with some Gaussian spread around the integers (the red curves are Gaussians centered on the integers and fitted to the data), resulting in a comb-like shape.

**Figure 3.** (a) Optimized cluster mean waveforms for (solid red) and (dotted blue). (b) Optimized effective photon numbers for (black) and integer-centered Gaussians fitted to the data (red).

Figure 4 shows a similar comb structure for for both the dot product method and PIKA. As the photon number increases, the teeth of the comb grow less defined—that is, the peak visibility falls, and with it the photon-resolving capability. Figure 5 shows this drop in visibility. The power of PIKA is that it retains nonzero visibility (i.e. the uncertainty does not include 0) through , whereas the dot product method alone loses it after . PIKA has been used to explore the regime just beyond the loss of peak visibility [4].

**Figure 4.** Effective photon numbers from the dot product method and from PIKA for .

**Figure 5.** Peak visibility for the dot product method (blue) and PIKA (red) through (both lose visibility altogether after that point). Uncertainties are given at two standard deviations and are purely statistical.

Figure 6 illustrates the loss of visibility with high photon numbers. As increases, cluster means become closer to each other (saturation in the normal regime) and individual waveforms overlap with each other more and more, making it more difficult to differentiate between adjacent photon numbers.

**Figure 6.** Illustration of the loss of visibility as increases. Cluster means become closer together, so individual waveforms overlap more and more.

The code and documentation are written in the context of few-photon-counting with a TES, so variables and functions are often named for physical quantities and concepts specific to the original purpose of the algorithm instead of more abstract ones (e.g. `readTES` instead of `readData`). In addition, the terms “trace,” “observation,” and “waveform” are used interchangeably and synonymously to mean a single detector response in the set of signals—that is, a regular sampling of voltage over time that records the response to a single optical pulse fired at the TES. In the program a trace is a list of voltage values.

Variable and function names also follow certain conventions:

- A variable beginning with
`i`or`j`is an index (or list of indices) on the remainder of its name.`iObs`is single index or list of indices on a set of observations, for example.

- A variable beginning with
`n`is a number or size of the remainder of its name (or a list of numbers or sizes). - A variable beginning with
`m`is a maximum number of something.`mSample`gives the maximum allowed number of waveforms to sample for graphical output, but fewer are taken if the population of waveforms is smaller than`mSample`.

- A variable of the form
`xOfY`lists`x`for each`Y`.`iObsOfClust`gives a list of indices`"iObs"`corresponding to`"Of"`each cluster`"Clust"`.

`nClust`gives the number of clusters, but`nOfClust`lists the size of each cluster.

- A function beginning with
`get`returns something described by the rest of its name.`getIObsOfClust`returns`iObsOfClust`.

- A variable such as
`p1`is a plot.

This section describes the overarching form of the algorithm. Here, the process is broken into a series of more isolated procedures, whose implementation details are described in the next section. Note that the functions referenced in this section and defined in the next take no arguments and return no output—they merely break the whole algorithm into smaller pieces of code that operate on global variables.

Below is the skeleton of the process. We begin by defining constants and options (which can be overridden as necessary by the user), and then reading in the data with a noise filter. One can optionally filter out observations contaminated by background radiation; by default the algorithm does not do this. We then proceed to iterate (Loop 1) over the elements of `nPhotonAvgList`, a list of hypothetical mean photon numbers for the set of pulses incident on the sensor, running PIKA once on the dataset for each element. This lets us compare the optimized objective functions for each hypothesized mean and estimate the actual mean from the runs with the smallest objective functions. The dot-product method supplies the initial effective photon numbers (which create the initial clustering) for the first mean photon number; after that we use the optimized effective photon numbers from the previous mean on the list as our initial guess for the next mean (adjacent elements of `nPhotonAvgList` should be close together, so one round’s optimized ordering of the observations should be a reasonable seed for the next).

In Loop 1 (over `nPhotonAvgList`), we organize the traces by their clusters and then find the initial value of the objective function to be minimized. We then enter the actual optimization loop (Loop 2), which repeats some preset number of times, trying to lower the objective function by moving observations between clusters. For each iteration of Loop 2, the nested Loop 3 examines each observation and considers moving it to a neighboring cluster (one with a photon number one greater or less). The greedy algorithm [5] (the default option) accepts any move that reduces the objective function and rejects any that does not, while the simulated annealing algorithm [6] may accept disadvantageous moves so that it can find global minima instead of just local ones. If a move is accepted, the relevant pieces of the objective function are updated. When Loop 2 finishes, the observations have been organized into optimized clusters, and we create various graphical and textual outputs.

Here we define the functions referenced in the previous section. The structure of the following subsections mirrors that of the algorithm skeleton above: Subsection 1 defines the procedures called before Loop 1 begins, Subsections 2 through 4 define those used in Loop 1 (with Subsection 3 containing the functions for Loops 2 and 3), and Subsection 5 contains the (very minimal) end of the algorithm after exiting Loop 1. The functions are defined in the order in which they are called, and each is called exactly once. The functions called inside the bodies of these skeleton functions (i.e. those that take arguments, unlike the functions defined in this section) are defined in Sections 5

and 7.

The main execution function, `pika`, assigns values to the following variables based on the user’s response to the input form:

`iNPhoton`—the index of the dataset to read (assuming that multiple datasets are stored in the same location)`iDataSet`—a list of indices of parts of the dataset to read (assuming that datasets are split over multiple files)`nTime`—the number of time points to keep after filtration`mSample`—the maximum number of traces to randomly sample when outputting samples`nDatUse`—the maximum number of traces to read in (if this is equal to 0, we use all of the traces)`backgroundReject`—a Boolean, true if PIKA should reject response waveforms with background radiation`peakValCut`—the voltage cutoff for background rejection`peakPosCut`—the peak location cutoff for background rejection`peakNumCut`—the cutoff for number of peaks for background rejection`nCool`—the number of optimizing rounds to perform`nGreedy`—the number of rounds that should be run with the greedy algorithm`coolConst`—the cooling constant for simulated annealing`tAnneal`—the starting annealing temperature`probDistName`—the name of the probability distribution`nSigma`—the number of standard deviations from the mean of the probability distribution to generate`binFract`—the histogram bin size (fraction of a photon number)`outputImageSize`—the default graphics size for output`nPhotonAvgList`—a list of (close together) mean photon numbers with which to run PIKA on the dataset`partialFilePath`—the directory and the beginning of the name of the data files, without the`iNPhoton`or`iDataSet`markers`fileExt`—the file extension of the data files (without the`iDataSet`marker)`nSamplePerTrace`—the number of samples per trace`nTracePerFile`—the number of traces per file`useInputFile`—a Boolean, true if there is another options file to read in and override settings from the form`pikaInput`—the path to the options override file

`runPIKA` (defined in Section 3 with implementation in this section) begins with some initial setup, processing some of the input given in `pika` and setting up tables to store graphical output.

Now we read in the data and apply a noise filter. `readTESandFilter`, defined in Section 7.7, takes the data from the set of files specified in the options above and applies a Hanning filter to it. We can then take a random sample and reduce the dataset to a smaller size, if desired, but by default we keep all of the observations. `dat` is a list of lists—each sublist corresponds to a waveform in the data and lists its values at regular time points.

In addition to the noise filter, we can remove observations from the dataset that have been contaminated by background radiation. `getIObsKeep` returns the indices of the observations in `dat` that should not be thrown out, based on the parameters `peakPosCut`, `peakValCut`, and `peakNumCut` that characterize waveforms unduly influenced by background radiation. The graphical output is a random sample of rejected observations.

`meanTrace` is the average of all of the waveforms in `dat`. The output here is a graph of `meanTrace` with a sample of accepted traces underlaid.

For the first mean photon number in `nPhotonAvgList`, the effective photon number for observation `i` is found via the dot-product method. For subsequent means, the optimized effective photon number from the previous item on `nPhotonAvgList` serves as the initial estimate for the next mean, since the elements of `nPhotonAvgList` are expected to be in order and close together. In the beginning of Loop 1, we create the initial clustering based on these effective photon numbers.

Now we enter Loop 1, whose index, `iNPhotonAvgList`, is the position in `nPhotonAvgList` of the current mean photon number. Each iteration of Loop 1 runs PIKA on the given dataset with the assumption that the waveforms in the data come from pulses incident on the sensor with an average of `nPhotonAvgList` photons per pulse. The runs are independent of each other apart from the fact that one round’s optimized effective photon numbers are used as initial effective photon numbers for the next.

The initial clusters are formed from the effective photon numbers of the individual observations. The effective photon numbers order the observations, but the relative sizes of the clusters and the photon number to which each one corresponds are determined by the mean photon number, `nPhotonAvg`, and the Poisson distribution. `groupProbability`, defined and described in more detail in Section 7.4, returns:

`prob`—a list of ordered pairs, the first a photon number and the second the Poisson probability mass function at that photon number.`nPhotonOfClust`—a list of photon numbers for the clusters created.`iObsOfClust`—a list of lists. Each sublist corresponds to a cluster and contains the indices in`dat`of the waveforms in that cluster. The cluster has photon number .`clustMeanTrace`—a list of waveforms. is the average waveform of the cluster .

`nClust` is then the number of clusters, which `groupProbability` decides based on which photon numbers we expect to see in at least one observation.

The following code graphs samples of waveforms from the middle three clusters.

Here we graph the average waveform of each cluster.

The initial, unoptimized value of the objective function, from equation (4) above, is . Here, `sqDevOfClust` is a table containing the square deviation of each cluster from its own mean (each element is for cluster ), so the table’s sum is , the -means term of the objective function. `nOfClust` is a list of cluster sizes, and `logLikeProb` is , the Poisson-combinatorial term. `sigmaObjFtn` is , the regularization parameter for the objective function based on the deviations of the initial clusters (unlike , does not change as the objective function is minimized).

This graphs the root mean square (RMS) deviations of each cluster to its mean.

The optimization loop (Loop 2) is now ready to begin, apart from a few organizational tasks. First, since the optimization moves waveforms between clusters many times, we need a more convenient way to keep track of the clusters. `getIClustOfObs` converts `iObsOfClust`, a list of lists, into `iClustOfObs`, a one-dimensional list, each element of which corresponds to an observation in `dat` and gives the index of the cluster to which that observation belongs.

Next, for each cluster, `neighborOfClust` lists the acceptable clusters to transfer to during the optimization loop. By default, the neighbors for cluster are the clusters and , since the initial clustering (from the dot-product method or the previous round of PIKA) is expected to be at least a somewhat accurate arrangement, and so long-distance transfers should not be necessary.

Finally, a system for keeping track of transfer history lets us reduce unnecessary computation. `kMove` counts the number of transfers that have been made in the loop. `birthOfClust` is a table that records the move number when each cluster was last changed, and `birthOfObsClust` is a two-dimensional table that records, for each observation-cluster pair, the last time that the deviation from the observation to the cluster mean was calculated. `sqDevOfObsClust` is a table with the same dimensions that records the actual deviations so that they can be used later in the loop if they are still up to date.

Loop 2, nested inside Loop 1, begins here. Its index, `iCool`, specifies how many rounds of optimization have already passed. The optimization loop runs for a preset number of iterations (given by `nCool` in the initial setup), the first `nGreedy` of which employs the greedy algorithm. The remaining iterations are run with the simulated annealing algorithm. By default, `nGreedy` is equal to `nCool`, so all of the iterations use the greedy algorithm (i.e. simulated annealing is never used).

The bulk of Loop 2 is contained in the nested Loop 3, the waveform transfer loop that passes over all of the observations in `dat` and decides whether to move them to neighboring clusters. Before starting Loop 3, we randomly order the indices in `dat` of the waveforms upon which Loop 3 is to operate, so that the original ordering of the data does not bias the algorithm in any systematic way.

The index of Loop 3, `jObs`, is the current position of the loop in `iObsAll`, the random permutation of the observation indices. `iObs` is then set to the index of the observation itself, rather than the index of the index. `iClust` is the index of the associated cluster. If the cluster has only one observation then we skip to the next iteration of Loop 3, since no cluster is allowed to become empty. We then randomly pick a cluster `jClust` from the list of neighbors, and we consider moving the waveform from its current cluster to the new one.

To reduce the computation time, `birthOfObsClust` and `birthOfClust` record the “ages” of observation-to-mean deviation calculations and changes to cluster contents, respectively, so this step checks whether `iClust` and `jClust` have been modified since we last calculated the deviation of `iObs` from their means. If they are unchanged, then `sqDevOfObsClust` is up to date; this condition becomes increasingly common as the calculation approaches convergence. If they are changed, we need to recalculate one or both deviations and store them in `sqDevOfObsClust`.

We need to know how a proposed move would change the objective function before deciding whether to accept it. Section 5 gives more detail on the functions called here that efficiently compute the change in the objective function.

If the transfer would decrease the objective function, we automatically accept it, and if we use the greedy algorithm, we reject any transfer that does not decrease the objective function. If we use simulated annealing, however, we may accept a transfer that increases the objective function for the purpose of exploring a greater number of assignments and possibly avoiding local minima that would trap the greedy algorithm. In simulated annealing, it grows harder for a disadvantageous transfer to occur as time goes on and the “temperature” drops.

If the move is accepted, then we need to move the observation from one cluster to the other and update the clusters’ deviations and mean traces. Both clusters have been changed in this move, so we update `birthOfClust` accordingly.

After making a transfer or deciding not to, Loop 3 moves to the next waveform in the randomly ordered list or finishes if there is none.

Loop 3 is now over, after examining all of the observations and moving some to neighboring clusters. If we use the simulated annealing algorithm, the annealing temperature decreases after Loop 3, making it more difficult for a move that increases the objective function to be accepted.

Loop 2 repeats a preset number of times, randomly ordering the waveforms, running Loop 3 on each one, and then decreasing the annealing temperature in each iteration.

When Loop 2 is finished, PIKA has run for many rounds on the data and the observations should be arranged into optimal clusters. This subsection concerns itself primarily with creating graphical and numerical outputs to understand and visualize the clustering. Now that the optimization is finished, `iObsOfClust` (the list of lists) is a more useful format than `iClustOfObs` (the simple list). In order to compare the new clustering to the old, we generate `iObsOfClustNew` from `iClustOfObs`, which reflects the optimized clusters, and simply copy `iObsOfClustOld` from `iObsOfClust`, which has not been changed since before Loop 2. `freqNew` and `freqOld` are lists of the new and old cluster sizes, respectively, and `nPhotonAvgNew` is an estimate of the actual mean photon number based on the results of PIKA.

As before, we graph some sample waveforms from middle clusters.

Here we numerically output and then graph the optimized cluster mean waveforms, with a graph of both the optimized and initial means as well.

This graph compares the initial and final cluster counts to a Poisson distribution.

is the average mean square deviation of the observations in cluster `i` to the mean waveform in cluster `j`. This graph is of the RMS deviations of the traces in clusters to adjacent means.

is the mean square deviation of the mean of cluster `i` to that of cluster `j`. This graph is of the RMS deviations of cluster means to means one, two, and three clusters away.

`nPhotonEff` lists the effective photon number of each observation (`getNPhotonEff` is implemented in Section 7.2). The rest of this code creates a histogram of `nPhotonEff` with bin widths of `binFract`: `binEdge` specifies where the edges of the bins should be, `binCnt` counts the number of waveforms in each bin, and `binCtr` lists the centers of the bins.

Loop 1 runs PIKA on the data once with each mean photon number in `nPhotonAvgList` and then exits.

After Loop 1 finishes, the algorithm outputs the computation time used and displays the output from PIKA.

A naive implementation of the transfer of waveforms between clusters would recalculate the various components of the objective function from scratch with each change, resulting in a computationally expensive process whose running time would depend on the size of the clusters in question. We can avoid this excessive calculation by noticing that, given some basic summary information about the clusters and waveform to which a given transfer pertains, the changes to the objective function and the cluster means are very easy to compute without any knowledge of the other waveforms in the clusters. The running times of the updates in this section are independent of cluster size.

From equation (1), we can decompose the -means term of the objective function as

(8) |

where

(9) |

If we transfer a waveform into the cluster , the new cluster members form the set . It can be shown that

(10) |

If we transfer a waveform out of the cluster , the new cluster members form the set

, with

(11) |

In equations (10) and (11), and refer to the original clusters and , before the transfer of . See below for a proof that these equations indeed give the proper changes in cluster deviation. Equation (11) corrects equation (A4) in [1]. After the transfer, the cluster means for and become

(12) |

with plus signs for and minus signs for . No cluster is ever allowed to become empty (in such a case we would have not but means), so the denominator never becomes zero. Therefore, we can determine the appropriate change to the means and square deviations of the clusters between which the transfer takes place.

Suppose that cluster has nine waveforms and cluster has 14 (and each has some mean waveform in addition), and we want to move a waveform from to . When we make this transfer, changes by the following amount.

The mean waveforms of clusters and become the following.

Of course, the cluster sizes change as well: now has 10 waveforms, while has 13.

The derivation for equations (10) and (11) is slightly involved and requires some preliminaries. Voltage signals are considered as vectors for the derivation, denoted , where the discrete time index is not given explicitly in this section.

**Lemma:**

Let be the signal vector of the observation, and and be the cluster mean vector and sum of deviations, respectively, for cluster . Then,

**Proof:**

The first expression for is merely the definition; we now show that the second expression is equivalent.

which is equivalent to the first expression. (The final step is valid because .) □

**Lemma:**

Let , , , and be as before. Then,

**Proof:**

From the proof of the previous lemma, we have

and thus

Finally,

and the lemma follows. □

Now we can prove that equations (10) and (11) hold true.

**Theorem:**

Suppose we add or remove a waveform to or from a cluster , forming a cluster . Then,

with . (Throughout this theorem, the top symbol of a plus-or-minus sign indicates the addition of to , and the bottom symbol indicates removal.)

**Proof:**

By the first lemma,

(Note the summation over , not .) Since is the observation to add to or remove from , we can write

By the second lemma, we have

and the theorem follows. □

From equation (2), the Poisson log-likelihood is

(13) |

where . If some waveform is transferred from the to the cluster, then the change in the objective function consists of a term from the Poisson factor and a term from the combinatorial factor from equation (3). The term due to the Poisson factor is

(14) |

(The notation suggests , but the same formula applies for the case.)

The combinatorial factor’s treatment is similar. The logarithm of the factor is

(15) |

does not change with a transfer, so only the second term contributes to the change:

(16) |

The algorithm only considers moves to adjacent clusters, so we only need to calculate the change due to transfers to immediately preceding and succeeding clusters. These two functions give the change in the probabilistic/combinatorial component of the objective function for moves to higher and lower photon numbers.

Bear in mind that the term that contributes to the overall objective function is the negation of its logarithm, so a positive decreases .

Suppose we have a set of clusters, among them an cluster with five waveforms and an cluster with seven waveforms, with a mean of 4.4 photons. We are interested in moving a waveform from the cluster to the cluster. Then the change in the Poisson log-likelihood is given by `deltaPoissonUp`.

We would like to thank Boris L. Glebov and Alan L. Migdall for providing the TES data. Figure 1 courtesy Thomas Gerrits and the Optical Society of America.

The functions in this section either are used often, and so are given their own names, or perform very particular procedures (such as data reading) whose implementation details are tangential to the core operation of the algorithm, and so have been separated from the main body of the code.

`boundList` confines the elements of a list between two inclusive bounds.

`getDotToIdeal` takes a single trace or a set of traces and finds its or their dot product with an ideal trace, normalized by the square of the ideal trace’s magnitude. If the function is given a single trace as its first argument, the operation in the numerator is a simple dot product; if it is given a list of traces, the operation is matrix-vector multiplication, which returns a list of dot products.

`getMeanOfEachTime` averages all of the traces in `dat` at each time point, returning the overall mean waveform.

`getSquareDiff` returns a table that lists the squared magnitude of the difference between each observation and each cluster mean.

`groupCommon` is a small function that rearranges the result of `GatherBy`. It is best understood from its definition.

`meanSquare` takes a vector and returns its mean square (magnitude squared divided by length).

`pair` takes two lists (of equal length) and pairs elements at the same position, returning a list of pairs.

`randomSample` is a wrapper for `RandomSample` that returns a random sample of some size from a list, reducing the sample size if it is greater than the length of the list.

`twoMin` takes a list and returns the positions of the smallest and second-smallest elements, in that order.

The three-argument version of `getNPhotonEff` takes a list of observations, a list of photon numbers associated with clusters, and a list of cluster means. It returns a list specifying the effective photon number of each observation, found via a linear interpolation between the two cluster means closest to a given observation.

`alphaByClustPair` is a list that gives the value of the interpolation parameter for each observation, organized by the two cluster means to which each observation is closest. `getNPhotonEff` (the two-argument version) takes that information and finds the effective photon number for each observation.

`getAlphaByClustPair` returns a list of ordered triples that give the cluster means closest to a group of observations and the alpha values for those observations. is the mean square deviation of observation `i` to cluster mean `j`, and is a pair of indices giving the two cluster means with the lowest mean square deviations to observation `i` (i.e. a pair of indices giving the two columns with the smallest elements in row `i` of `msDev`). `iObsTwoMinIndex` pairs the relevant observation index with each pair in `twoMinIndex`. In the next line, the `GatherBy` groups the elements of `iObsTwoMinIndex` by their lowest-deviation index pairs, so that each element of `iObsTwoMinIndexGroup` is a list of elements of `iObsTwoMinIndex` that all have the same pair of nearest cluster means. Mapping `groupCommon` onto each of these lists in `iObsTwoMinIndexGroup` converts them into a more useful form; each element of `twoMinIndexIObsList` is now a list of length two, with the first entry being a pair of nearest-mean indices and the second being a list of observation indices whose nearest two means correspond to the pair in the first entry. The output is a list with an entry for each distinct ordered pair of nearest means. Each entry contains a pair of nearest-mean indices, a list of observations whose nearest means are those two, and a list giving the alpha value for each of those observations.

`getAlpha` takes a list of observations and a pair of ideal waveforms and finds the value of alpha for each observation. Alpha for a waveform is the value that minimizes the RMS deviations of from , where is the closest mean waveform (the first entry in the second argument of `getAlpha`) and is the second closest. `getAlpha` finds the minimizing value from equation (5) for each observation in `traces`. A value of 0 indicates perfect agreement with the first ideal trace, and a value of 1 indicates perfect agreement with the second.

The two-argument `getNPhotonEff` takes the formatted list of triples `alphaByClustPair` and a list of cluster photon numbers and determines the effective photon number of each observation. `alpha` is a list of alpha values for the observations, `iObs` is a list of observation indices corresponding to those alpha values, and `clustPair` is a list of nearest-mean-index pairs for those observations. `alpha` and `clustPair` are in the same order as `iObs`, but we want them in the same order as the waveforms from the original dataset, so we reorder them with `iiObs`. `nPhotonPair` takes the cluster indices in `clustPair` and converts them to the actual photon numbers associated with those clusters. The output is a list specifying the effective photon number for each observation, derived from a linear interpolation between the closest () and second-closest () photon numbers.

If background rejection is enabled, `getIObsKeep` returns the indices of the observations that should stay in the dataset, based on the preset parameters for rejection. `peakPosList` and `peakValList` are lists specifying the position and value of the maximum in each waveform in `dat`. `getIObsDrop` returns the indices of the waveforms that should be removed because their peak positions and values exceed their respective cutoff values, and `getIObsDropA` lists those whose endpoints are greater than the value cutoff, indicating that the sensor was registering a background photon at the beginning or end of the pulse. These two lists to remove are combined into `iObsDrop`. Each element of `datPeakLoc` lists the positions of the local maxima of a waveform in `dat`, `datPeakLocVal` pairs each of those positions with the waveform’s value there, and `datPeakLocValBig` reduces `datPeakLocVal` to the maxima that exceed the peak value cutoff. `datPeakNum` gives the number of maxima in each waveform, and `getIObsKeepA` returns the indices that should stay in the dataset (i.e. those that are not in `iObsDrop` and that have fewer maxima than `peakNumCut`).

`posMax` returns the position of the first occurrence of a list’s maximum.

`getMaxAndPosList` returns two lists, the first giving the position of each waveform’s maximum and the second giving the maxima themselves.

`getIObsDrop` returns the indices of the waveforms whose peaks have values and positions greater than the cutoffs.

`getIObsDropA` returns the indices of the waveforms whose endpoints have values greater than the cutoff, indicating that a background photon may have registered with the sensor at the beginning or end of the pulse.

`peakList` finds the local maxima of a list by comparing each element to the ones before and after it.

`getDatPeakLoc` takes a list of waveforms and returns the locations of the maxima for each.

`getDatPeakLocVal` pairs each of the locations returned by `getDatPeakLoc` with the value of the corresponding waveform at that point.

`getDatPeakLocValBig` finds the maxima that exceed the cutoff among those returned by `getDatPeakLocVal`.

`getIObsKeepA` takes the list of waveforms rejected because of misplaced peaks and too-large endpoints and combines that with the information on each waveform’s number of maxima to return a list of the indices of all of the waveforms that should be rejected due to background radiation.

These two functions convert between `iObsOfClust` and `iClustOfObs`, two different ways of organizing clusters of waveforms. `iObsOfClust` is a list of lists. Each sublist is associated with a cluster and has the indices (in `dat`) of the traces in that cluster. `iClustOfObs` is a simple list, each of whose entries corresponds to a trace and states what cluster that trace is in (0 if the trace is not in any cluster). `getIClustOfObs` takes `iObsOfClust` and returns `iClustOfObs`, and `getIObsOfClust` is its inverse function.

`getIClustOfObs` creates a table with an entry for each observation and then iterates through the clusters. In the table, it assigns the contents of each cluster to the appropriate cluster number.

`getIObsOfClust` creates a list of observation number/cluster number pairs and then applies a `GatherBy` to group waveforms in the same clusters together. Mapping `groupCommon` onto the list creates a list of lists, each of whose first element is a cluster number and each of whose second element is a list of the indices of the waveforms in that cluster. We then sort the overall list by cluster number and extract the second entries in the sublists to create a list of lists of observation indices.

`groupProbability` takes the data, the hypothesized mean photon number for the current round, and the effective-photon-number ordering of the data, and returns a list of four things: the Poisson probability distribution (ordered pairs of photon numbers and probabilities), a list of photon numbers for the clustering it creates, a list of lists of constituent waveforms’ indices for the clusters (`iObsOfClust`), and a list containing the average waveform in each cluster. `groupProbability` finds the photon number/value pairs for the Poisson distribution with the given mean photon number and then calls `groupProb` to organize the clusters.

`groupProb` processes the data into clusters based on the given effective photon number list and the Poisson distribution. First we find the proper ordering of the effective photon numbers by size (`nPhotonEffSortIndex`); this will be needed once we have determined the sizes of the clusters to be created. The probability distribution `prob` is broken up into its component parts: `nPhoton` lists photon numbers, and `probList` lists their corresponding probabilities. `probCum` is the cumulative distribution function of `probList`, which we normalize to ensure that the final element is one. gives the index (in the sorted list of effective photon numbers, `nPhotonEffSortIndex`) where cluster `i` should start; gives the index where it should stop. Both indices are inclusive, which is why `probIndexStart` is offset by one from `probIndexStop`. (`boundList` ensures that the starting and stopping indices stay within the confines of the number of observations available to group.) `keep` weeds out the zero-length bins, and we use it to reduce `nPhoton`, `probIndexStart`, and `probIndexStop` down to only the nontrivial clusters. We then form `iObsOfClust` by finding the range of *sorted* effective photon numbers that each cluster should encompass and then using `nPhotonEffSortIndex` to find the observation numbers to which that range corresponds. `clustMeanTrace` is formed by taking the waveforms in `dat` of each cluster in `iObsOfClust` and averaging them. We then return `nPhotonUse` (the photon numbers of the nonempty clusters), `iObsOfClust`, and `clustMeanTrace`.

This is the combinatorial term of the log-likelihood objective function. From equation (15) above, , where the are the cluster sizes and .

The Poisson log-likelihood term is .

`probComboLogLikelihood` finds the Poisson/combinatorial log-likelihood term of the objective function.

`poisson` generates about `nSigma` standard deviations of a Poisson distribution on either side of the given mean. The output is a list of ordered pairs of photon numbers and associated values of the Poisson probability mass function. The output is normalized so that it sums to one.

`getSqDevClustObs` takes a list of observation traces and a list of cluster mean traces and returns a table whose element in position is the mean square deviation of observation `j` from the mean trace of cluster `i`.

`getSqDevClustClust` returns a table whose element in position is the average of the mean square deviations of the traces in cluster `i` from the mean trace of cluster `j`.

`getObjFtn` takes a list of each cluster’s square deviation from its mean, the constant relating the -means and probabilistic components of the objective function (which can be a scalar or a list that takes on a different value for each cluster), and the log-likelihood of the Poisson distribution (with the combinatorial term included). It returns the value of the objective function.

`getSqDevInClust` takes a list of observations and a mean waveform and totals up the mean square deviations of the observations to the mean. (It returns , where is the list of observations, is the number of time points, the are the observations, and is the mean.)

`getSqDevInClust` returns a list that gives , the total mean square deviation of a cluster to its mean, for each cluster . The list returned becomes the first argument of `getObjFtn`.

`readTES` reads in the data from a set of files specified in the options, returning `dat`, a table whose rows give the values of particular waveforms at regular time intervals. `readTES` assumes that several different datasets may share a directory and that each dataset is split over some number of files, each of which consists of an equal number of unsigned 16-bit integers concatenated together in string form. `iNPhoton` gives the numeric label of the dataset to read (it is not equal to the mean photon number of the dataset). `iDataSet` lists the indices of the particular files in the dataset that should be read in. `fileInfo` is a list of three things: `fileNamePart`, a list of two strings that combine with the dataset label and the file index to create the whole filename; `nSamplePerTrace`, the number of time points in each waveform; and `nTracePerFile`, the number of waveforms in each file of the dataset.

For example, if `iNPhoton` were `7`, `iDataSet` were , and `fileInfo` were , then we would be looking for the files , , and in , with 200 time points in each waveform and 512 waveforms in each of the three files.

For each file, we import the data as a list of samples, organize the samples into sublists (waveforms) of length `nSamplePerTrace`, and assign the list of waveforms to the appropriate section of `dat`. When all of the files have been read, `dat` is full and properly formatted.

We can also filter the data as we read it in. `readTES` applies a “short” filter to a list, one that omits half of the frequency domain.

This generates a Hanning filter of a given length.

`readTESandFilter` reads in the dataset, applying a filter to each file’s data before it is stored. This halves the memory consumption of the reading process, since filtration reduces the amount of information to store by half.

`outputCreateTabs` creates several list structures (global variables) of the appropriate dimensions to accommodate output from the specified number of runs of PIKA. The output is organized into a `TabView`, with each run receiving a tab that contains a nested view with graphical and textual output. `outputNameList` stores the names of the labels in the subview (which are the same for all runs), and `outputContentTable` stores the output content for each run and sublabel. (Content is stored as a list that later becomes a `Column`.) Each run’s tab also has a space above its subview for other information, which is stored in `outputTopList`. Finally, there is another tab on the level of the runs with general textual (`outputGeneralLog`) and graphical (`outputGeneralGraphics`) information about the data read in.

`outputAdd` adds some expression to the content list for a particular name and run, creating the name’s content list if it has no content already. A run number of 0 indicates output to the general information tab, and a blank name indicates output to the top space of a run’s tab.

`outputShowTabView` is called at the end of PIKA, taking the name and content lists and formatting them into a nested `TabView`.

`print` takes any number of arguments (`args`), turns them into strings, concatenates them, and prints the result to the log. `printSp` prints to an arbitrary section of output, not just the log. `iRun` and `name` indicate the run number and tab name, respectively, to which to print.

`pika` creates a form with a field for each constant and option that needs to be set for PIKA to run. It then uses `runFromAssociation` to assign the proper values to the proper variables and run PIKA.

`makeField` creates a rule that, when used in a `FormObject`, generates a field to set the variable with name `name` and type `type` to the entered value, with descriptive text `labelPart` and default value `defaultContents`. `args` takes optional extra arguments that specify additional options for the field.

`runFromAssociation` takes an association between variable names (strings) and the values that should be assigned to them and assigns the proper value to the symbolic variable associated with each name. It then runs PIKA (using the main function from Section 3) with those global variables set.

[1] | Z. H. Levine, T. Gerrits, A. L. Migdall, D. V. Samarov, B. Calkins, A. E. Lita, and S. W. Nam, “Algorithm for Finding Clusters with a Known Distribution and Its Application to Photon-Number Resolution Using a Superconducting Transition-Edge Sensor,” Journal of the Optical Society of America B, 29(8), 2012 pp. 2066–2073. doi:10.1364/JOSAB.29.002066. |

[2] | T. Gerrits, B. Calkins, N. Tomlin, A. E. Lita, A. Migdall, R. Mirin, and S. W. Nam, “Extending Single-Photon Optimized Superconducting Transition Edge Sensors beyond the Single-Photon Counting Regime,” Optics Express, 20(21), 2021 pp. 23798–23810. doi:10.1364/OE.20.023798. |

[3] | Z. H. Levine, B. L. Glebov, A. L. Pintar, and A. L. Migdall, “Absolute Calibration of a Variable Attenuator Using Few-Photon Pulses,” Optics Express, 23(12), 2015 pp. 16372–16382. doi:10.1364/OE.23.016372. |

[4] | Z. H. Levine, B. L. Glebov, A. L. Migdall, T. Gerrits, B. Calkins, A. E. Lita, and S. W. Nam, “Photon-Number Uncertainty in a Superconducting Transition Edge Sensor beyond Resolved-Photon-Number Determination,” Journal of the Optical Society of America B, 31(10), 2014 pp. B20–B24. doi:10.1364/JOSAB.31.000B20. |

[5] | P. E. Black, “Greedy Algorithm,” in Dictionary of Algorithms and Data Structures (V. Pieterse and P. E. Black, eds.), Feb 2, 2005. www.nist.gov/dads/HTML/greedyalgo.html. |

[6] | S. Kirkpatrick, C. D. Gelatt Jr., and M. P. Vecchi, “Optimization by Simulated Annealing,” Science, 220, 1983 pp. 671–680. doi:10.1126/science.220.4598.671. |

B. P. M. Morris and Z. H. Levine, “The Poisson-Influenced -Means Algorithm,” The Mathematica Journal, 2016. dx.doi.org/doi:10.3888/tmj.18-3. |

www.mathematica-journal.com/data/uploads/2016/03/TES_E0019-16.daq00.zip

www.mathematica-journal.com/data/uploads/2016/03/TES_E0019-16.daq01.zip

www.mathematica-journal.com/data/uploads/2016/03/TES_E0019-16.daq02.zip

www.mathematica-journal.com/data/uploads/2016/03/TES_E0019-16.daq03.zip

www.mathematica-journal.com/data/uploads/2016/03/TES_E0019-19.daq00.zip

www.mathematica-journal.com/data/uploads/2016/03/TES_E0019-19.daq01.zip

www.mathematica-journal.com/data/uploads/2016/03/TES_E0019-19.daq02.zip

www.mathematica-journal.com/data/uploads/2016/03/TES_E0019-19.daq03.zip

Brian P. M. Morris is a senior in the Science, Mathematics, and Computer Science Magnet Program at Montgomery Blair High School in Silver Spring, Maryland, was in the Summer High School Intern Program at the National Institute of Standards and Technology, and is a 2016 Intel Science Talent Search Semifinalist.

Zachary H. Levine is a physicist at the National Institute of Standards and Technology in the Infrared Technology Group and is a Fellow of the American Physical Society. He is a graduate of MIT and the University of Pennsylvania.

**Brian P. M. Morris**

*Montgomery Blair High School
Silver Spring, Maryland 20901*

**Zachary H. Levine**

*National Institute of Standards and Technology
Gaithersburg, Maryland 20899-8441
*

Picard’s iteration is used to find the analytical solutions of some Abel–Volterra equations. For many equations, the integrals involved in Picard’s iteration cannot be evaluated. The author approximates the solutions of those equations employing a semi-implicit product midpoint rule.The Aitken extrapolation is used to accelerate the convergence of both methods. A blow-up phenomena model, a wave propagation, and a superfluity equation are solved to show the practicality of the methods. Programs offered solve the general equations. A user needs only to enter the particular forcing and kernel functions, constants, and a step size for a particular problem.

Abel–Volterra equations are normally represented by

(1) |

Equation (1) is called *regular* if and *weakly singular* (or of *Abel type*) if . Equation (1) is *linear* if ; otherwise, it is *nonlinear*. In most practical applications, is either 0 or 1/2. See [1–4] for conditions on existence, uniqueness, and continuity of a solution for equation (1). To solve equation (1) analytically, one normally employs the Picard method, a method of successive iterations, given by

(2) |

(3) |

From a theoretical point of view, the successive iterations given by equations (2) and (3) always converge for linear equations; see theorem 10.15, page 152 of [5] and pages 92–95 of [4]; see also solutions of some integral equations by the Picard method in [2]. In practice the convergence of successive iterations depends on the computability of the corresponding integrals in equation (3). One might be able to evaluate the integrals in some cases. Successive iterations may also be effective for some nonlinear equations. In what follows, we first introduce a simple program that implements the successive iterations and solve two examples using this program. For many integral equations whose exact solutions cannot be found by the Picard method, we approximate their analytical solution using a semi-implicit product midpoint rule. Two practical examples are solved to test the validity of this numerical approach.

The following program implements equations (2) and (3).

One needs only to introduce the kernel `Ker`, the forcing function `g`, and the real value `α` to solve the corresponding equation.

**Example 1**

To solve the equation

(4) |

note that

(5) |

where is the well-known beta function defined by

(6) |

and is the gamma function.

Then it is easy to verify that the outputs produced by the Picard method for this example can be written as

(7) |

where , , , , , and .

Now substitute the corresponding kernel, forcing function, and the value of . The evaluation takes a minute or two.

Aitken’s method accelerates the convergence [6], but accelerators like this one are highly unstable numerically. One has to pay careful attention (in particular for division by zero) when working with such accelerators; it is important to use high precision. The following is a program to accelerate Picard’s iteration using the already developed `PicardIteration` program.

The analytic solution of is (see [3]). Use that to compare with the result of the Aitken accelerator.

Define `res1` and plot the error.

This graph shows the error in applying the accelerated Picard’s method on equation (4). We note that uses only the values of to and provides an excellent approximation for , the exact solution of equation (4).

Equation (7) can be used to find , for very large values of . This equation was obtained as a result of analyzing the corresponding Picard iteration. The same type of analysis will be used for our next nonlinear example.

**Example 2**

We solve a nonlinear *blow-up phenomena *model,

(8) |

Equation (8) is mentioned as a a blow-up phenomena model on p. 417 of [7]. The analytical solution exhibits blow-up at finite time. A blow-up means there is a finite time such that

(9) |

Our goal is to find the value of as accurately as possible.

Implementing a few steps of Picard’s iteration to equation (8) shows that the solution is of the form

(10) |

which implies that

(11) |

Make a change of variable from to in equation (8) and replace and using equations (10) and (11), respectively, on both sides of equation (8). We obtain

(12) |

where

(13) |

Equating coefficients corresponding to equal powers of from both sides, we get

(14) |

Note that

(15) |

and that is a power series in . Therefore, the radius of convergence for given by equation (10) is , which is the blow-up number in equation (9).

All terms in equation (10) are positive, and the series is wildly divergent beyond 0.897677. For a series with all positive terms, the nonlinear accelerators are not useful in evaluating beyond the radius of convergence. For an alternating series, the situation is different. Nonlinear extrapolations are normally quite effective in evaluating the sums of alternating divergent series for variable values far beyond the radius of convergence.

We evaluate

(16) |

for large values of , where are given by equation (14).

For the following tables, the execution time is provided for cases and .

The following program is for the accelerated Picard method for example 2. All terms in equation (16) are positive, and the acceleration is not as helpful as in example 1, where the series was alternating.

For most integral equations, Picard iteration is not practical, and we approximate the solution of equation (1) using a semi-implicit product midpoint rule. To understand the method, we start by subdividing the interval of integration into equal subintervals using a step size . Equation (1) becomes

(17) |

note that . For , we only have the subinterval . Let (the middle variable of the kernel function ) be the midpoint of the interval ; that is, . Let (the third variable of ) be the midpoint of and ; that is, , and recall that . The denominator of the integral that contains the singularity stays as is, so

(18) |

We solve equation (17) for using Mathematica’s built-in function `FindRoot` with an initial guess of . The integral that contains the singularity is solved exactly and therefore does not introduce any inaccuracy in the method, which is why the word “product rule” is added to the name of this technique; see [8] for more details on product integration. Also, at each step only the very last variable (in this case ) needs to be found, which is why the name “semi-implicit” is used. We also use the midpoints of the intervals; hence the term “a semi-implicit product midpoint rule.” Now let to get

(19) |

Or

(20) |

Then equation (19) is solved to find an approximation for using `FindRoot` with an initial guess of from the previous step. Continuing the same procedure, we arrive at the following program.

**Example 3**

The nonlinear integral equation

(21) |

arises in the theory of superfluity [9].

We use the semi-implicit product midpoint rule program to approximate the solution of equation (1).

Graph of superfluity equation with no acceleration and a step size of 0.01 and .

Graph of superfluity equation with acceleration and a step size of 0.01.

This graph shows the difference between the accelerated and non-accelerated methods for the superfluity equation. The difference is of order , which shows that iterating the extrapolation once more is not sensible. The execution time goes up exponentially, and not much is gained in accuracy.

For our next example, we were able to find the corresponding analytical solution. Therefore, we can demonstrate the efficiency of our method by comparing the numerical solution with the exact solution.

Before solving example 4, we prove the following theorem for a class of inhomogeneous equations to find its analytic solution.

**Theorem 1**

(22) |

(23) |

**Proof**

From the general theory of the Abel equations ([2], p. 224), if

(24) |

(25) |

Equation (22) can be written as

(26) |

(27) |

Now on the right side of equation (27), replace the integral by from equation (26) to get the desired result, equation (23).

□

We now use Theorem 1 to find the exact solution of wave propagation for example 4 and test the efficiency of our midpoint rule and its extrapolated version.

**Example 4**

The equation

(28) |

represents wave propagation over a flat surface (see p. 229 and p. 235, exercise 3 of [2]). Using Theorem 1, the exact solution of equation (28) can be obtained as

(29) |

Solving equation (28) with our midpoint rule and an Aitken extrapolation with an step size , we obtain an approximation with a maximum absolute error of order .

Example 4 is the three-dimensional case. Therefore, the program for the general midpoint rule given for equation (1) must be adjusted a little.

This plots the error in applying the midpoint rule with a step size ; for and , with , the maximum absolute error is of order 0.001.

This plots the error in applying the Aitken extrapolation with a step size ; for and , , the maximum absolute error is of order .

We studied two methods: Picard’s iteration and a semi-implicit product midpoint rule. The Aitken extrapolation was used to accelerate the convergence of these methods. In some cases, the extrapolation demonstrated a significant improvement. A blow-up phenomena model, a wave propagation, and a superfluity equation were solved, and the efficiency and the practicality of the methods were established. The user-friendly programs created here solve the general equations. One only needs to enter a forcing function, a kernel function, the value, and a step size for a particular problem. We used Mathematica 10.2 on the Mac OS X operating system with 16 GB RAM and a 2.8 GHz processor. The execution time was always under five minutes, and the vast majority of problems were executed in less than five seconds.

I am grateful for constructive suggestions by a reviewer, resulting in more transparent coding.

The author dedicates his work to Mahshid, Arman, and Ida; wife, son, and daughter.

[1] | W. Hackbusch, Integral Equations: Theory and Numerical Treatment, Boston: Birkhäuser Verlag, 1995. |

[2] | R. P. Kanwal, Linear Integral Equations, 2nd ed., Boston: Birkhäuser, 1997. |

[3] | R. K. Miller, Nonlinear Volterra Integral Equations, Menlo Park, CA: W. A. Benjamin, Inc., 1971. |

[4] | A. Pipkin, A Course on Integral Equations, New York: Springer-Verlag, 1991. |

[5] | R. Kress, Linear Integral Equations, Berlin: Springer-Verlag, 1989. |

[6] | J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 2nd ed., New York: Springer, 1996. |

[7] | H. Brunner, Collocation Methods for Volterra and Related Functional Equations, Cambridge: Cambridge University Press, 2004. |

[8] | P. Linz, Analytical and Numerical Methods for Volterra Equations, Philadelphia: SIAM, 1985. |

[9] | N. Levinson, “A Nonlinear Volterra Equation Arising in the Theory of Superfluidity,” Journal of Mathematical Analysis and Applications, 1(1), 1960 pp. 1–11.doi:10.1016/0022-247X(60)90028-7. |

J. Abdalkhani, “Exact and Approximate Solutions of the Abel–Volterra Equations,” The Mathematica Journal, 2016. dx.doi.org/doi:10.3888/tmj.18-2. |

Javad Abdalkhani is an associate professor of mathematics at the Ohio State University, Lima campus and a Distinguished Alumni teacher at the Ohio State University. His area of research is numerical analysis. His hobbies are reading and cycling.

**Javad Abdalkhani**

*Department of Mathematics
The Ohio State University, Lima
4240 Campus Drive
Lima, Ohio 45804
*

Biological systems possess traits that are a product of coordinated gene expression, heavily influenced by environmental pressures. This is true of both desirable and undesirable traits (i.e. disease). In cases of unwanted traits, it would be tremendously useful to have a means of systematically lowering the expression of genes implicated in producing the undesirable traits. This article describes the implementation of a computational biology algorithm designed to compute small interfering RNA sequences, capable of systematically silencing the expression of specific genes.

Small interfering RNAs (siRNAs) are short, single-stranded fragments of RNA, typically 20–30 nucleotides in length, that have created tremendous excitement in the biological sciences because of their ability to down-regulate the expression of genes in a targeted, systematic manner. siRNAs mediate their knockdown effects by activating a gene expression surveillance mechanism widely conserved in eukaryotic cells, known as RNA interference (RNAi). Once activated by siRNA, the RNAi machinery of the cell suppresses the expression of a specific gene by blocking translation (the synthesis of protein by ribosomes) or by promoting the degradation of mRNA needed to synthesize a protein. Figure 1 highlights the basic interplay between siRNA and RNAi. Further excellent reviews of siRNA and RNAi can be found in [1–3]. Having the ability to purposefully engineer siRNAs to “turn off” (or “turn down”) the expression of genes promoting disease phenotypes like cancer, multiple sclerosis, diabetes, inflammation, etc. is of immediate and widespread interest to scientists, clinicians, and patients living with the burden of disease. Indeed, several clinical trials are currently evaluating the effectiveness of siRNA therapy for conditions such as transthretin amyloidosis [4], elevated intraocular pressure [5], and non-small cell lung cancer [6].

**Figure 1.** (A) A genetically modified retroviral vector is used to insert DNA instructions into the nucleus of a eukaryotic cell. (B) The cell uses the inserted DNA instructions to manufacture short hairpin RNA (shRNA). (C) Following enzymatic trimming in the nucleus, the shRNA is exported from the nucleus. (D) An enzyme named Dicer removes the loop from the shRNA hairpin, releasing a double-stranded version of siRNA. (E) A complex of proteins named RISC (short for RNA-induced silencing complex) enzymatically removes the passenger strand of the siRNA duplex. (F) The mature single-stranded siRNA binds to its target—a specific mRNA from a gene being expressed by the cell. (G) The presence of siRNA attached to the mRNA interferes with translation (protein production), effectively short-circuiting the cell’s intention to fully express the information contained in its DNA.

This article describes a Mathematica implementation (named siRNA Silencer) of the seminal algorithm created by Naito et al. [7] to design siRNA sequences for the express purpose of silencing a specific gene. Like Naito’s algorithm, siRNA Silencer (SRS for short) uses “rules” of effective siRNA design that have been elucidated through meticulous experimentation [8]. To generate viable siRNA candidates, SRS progressively works through the four steps described in Table 1.

These 23-mer sequences correspond to the complementary sequence of 21-nucleotide guide strand and 2-nucleotide 3′ overhang of the passenger strand within the target sequence (Figures 1 and 2).Step 2 (select functional siRNAs)

Rule C: Choose 23-mers that have at least 4 A or U residues in the 5′ terminal 7 base pairs of the guide strand.Step 3 (reduce seed-dependent off-target effects)

Calculate the melting temperature (Tm) of the seed-target duplex (i.e. nucleotide pairs 2–8 of the siRNA guide and passenger strands as bound to their target sequence) for the 23-mers that survive Step 2. Accept only those sequences with a Tm less than 21.5 °C.

As an example of how the steps in Table 1 are applied, Figure 2 presents a detailed illustration of some of the output generated by SRS when it is tasked with silencing the human gene IFNβ1. (Interferon beta 1 is a powerful antiviral protein, manufactured from instructions in the IFNβ1

gene [9].)

**Figure 2.** One of 23 siRNA candidates generated by siRNA Silencer to silence IFNβ1 expression is presented in detail here. The middle strip of letters contains a portion of the mRNA sequence manufactured from instructions in the human IFNβ1 gene. The black box surrounds nucleotides 591 to 613 of the mRNA target, which is precisely 23 nucleotides in length, corresponding to Step 1 above. The sequence outlined in red is the guide strand of RNA that will eventually bind to the target mRNA and block protein synthesis (Figure 1). Notice how the guide sequence satisfies rules A, C, and D in Step 2 above. The passenger sequence (blue outline) satisfies rules B and D in Step 2. The predicted melting temperatures (details described below) of the guide-target seed sequence (green bar) and the passenger-target seed sequence (purple bar) are 8.98 °C and 13.3 °C, respectively, satisfying Step 3 above. Step 4 output will be discussed below.

As the end user interprets the output of siRNA Silencer and ultimately makes a decision about which presented siRNA candidate to use, it becomes a (relatively) simple cloning project to engineer the guide and passenger sequences into an expression vector (Figure 1) to silence the gene in a living cell.

SRS is template driven, meaning the algorithm expects several pieces of user-defined information to be provided in a notebook cell that is used as a template for entering information. The features of SRS will be illustrated using the pre-mRNA sequence of the human gene IFNβ1 (interferon beta 1) to design siRNA candidates capable of silencing IFNβ1.

The example above contains several variables that must be completed by the user to let SRS do its job. The reader may modify them to run the code.

The variables requiring user input are:

**1.** `refseqlocation`: This variable holds the directory location for finding pre-parsed RNA data files from specific organisms of interest. The pre-parsed RNA data files were generated by downloading transcript source files from NCBI’s Map Viewer FTP (ftp://ftp.ncbi.nih.gov/genomes/MapView). The data contained in these files is part of NCBI’s RefSeq database, which maintains a nonredundant catalog of all known biological molecules in a species-specific manner [10]. Original source files include:

Once downloaded, the source files were parsed by custom Mathematica code (not shown here) to generate expressions containing transcript accession values, annotation information, and sequence information for the genes contained in the source files. These expressions were saved and serve as the primary data source for the SRS program.

**2.** `refdatabaseversion`: This variable contains the name of the specific organism from which a gene is to be silenced. Options available include: `"humantranscripts"`, `"mousetranscripts"`, `"rattranscripts"`, `"dogtranscripts"`, and `"cattranscripts"`. The option chosen must correspond to the name of a file present in defined above.

**3.** `savelocationroot`: This variable holds the location where the user would like the final results of the analysis to be saved.

**4.** `studyname`: This variable lets the user name the output generated by SRS. The output of SRS is saved using this name to the location provided in `"saverootlocation"` above.

**5.** `query`: This variable contains the transcript sequence, in string format, of the gene to be silenced.

SRS starts by loading the pre-parsed RNA data file from which the query gene sequence is to be silenced and then proceeds to create a preliminary list of siRNA candidates that are filtered through the four steps referenced in Table 1.

For the specific gene highlighted here (IFNβ1), successive pruning of the initial list of 818 23-nucleotide-long sequences is highlighted by a decline in the length of variables containing the pruning results. The variables highlighted below contain the results of completing Step 1, Step 2 (Rules A and B), and Step 2 (Rules C and D).

A small portion of initial siRNA candidates is shown here, where each row contains a guide and passenger sequence (Figure 2) generated from IFNβ1 that may, if they pass additional requirements, prove useful in silencing the expression of IFNβ1.

The algorithm moves next to complete Step 3, in which the predicted melting temperatures of the guide and passenger seed sequences (Figure 2) are calculated and screened to accept only those sequences with predicted melting temperatures below 21.5 °C. Melting temperature is a reflection of thermodynamic stability, and for the purposes of gene silencing, experiments have suggested melting temperatures below 21.5 °C are optimal [11]. If no candidate siRNAs can meet that demand, the requirement is waived, but the algorithm will print a warning message that the requirement could not be met.

Inspection of the first five candidates that survived Step 3 screening reveals that none of them match the first five candidates that survived Step 2 screening. This means that none of the first five candidates from Step 2 screening had melting temperatures below 21.5 °C, causing SRS to remove them from further consideration.

Indeed, there is considerable reduction in the size of the candidate list from Step 2 to Step 3.

After the candidate list of potential siRNAs is pruned through Step 3, the algorithm presents a visual representation of the current list of siRNA candidates as they are mapped to their respective positions within the query sequence.

For publication purposes, the next output is reduced in size to allow the data to fit within printable margins.

**Figure 3.** A map of the siRNA candidates (shown in red) that have past the first three steps of Table 1, aligned to their positions within the target gene sequence (shown in black) to be silenced. The example here shows siRNA candidates suggested for silencing IFNβ1.

From here, SRS begins an exhaustive search for genes within `<refdatabaseversion>` that contain subsequences that match the siRNA candidates graphically presented above. In our current example, SRS searches the siRNA candidates generated from IFNβ1 that may also bind to alternative human genes with similar subsequences. The results of this search are presented for inspection so the user can decide which of the potential siRNA candidates is least likely to create undesirable effects if used to silence the gene of interest. Making this decision is largely based on human experience, and is an area that highlights the “art of doing science.”

**Caution:** Due to the sheer volume of computation that needs to be completed using the data described in this article, the next segment of code will likely take 20–40 minutes to complete (depending on the speed of your computer) and consume roughly 24 GB of RAM. Computations on machines with less RAM will finish, but will require significant use of the hard drive, slowing computation considerably.

For publication purposes, the next output is reduced in size to allow the data to fit within printable margins.

SRS then finishes by saving its results to the location specified by the user in the template described above. SRS will output a JPEG version of Figure 3, as well as Mathematica and Microsoft Excel versions of the data in Table 2. The Mathematica version of Table 2 is complete, including all the potential off-targets that were found. The Excel version of Table 2 lacks the data of the potential off-targets because there is no reasonable way to organize such a large volume of information within Excel.

Table 2 provides summary information about the siRNA candidates the algorithm is proposing as effector mediators of gene silencing. In the example of this article, the IFNβ1 gene generates 23 siRNA candidates that should be capable of effectively reducing the expression of the IFNβ1 gene; all 23 candidates have passed the four steps of Table 1. It is left to the end user to make the final decision about which siRNA sequence to use. While much is known about RNA interference and the use of siRNAs to down-regulate the expression of a gene, there are still aspects that benefit from human experience and intuition, which is why Table 2 presents information that may be contextually important to the researcher.

The first column of Table 2 conveys the positions of the siRNA candidates in relation to the target gene sequence. This also corresponds to the same graphical information presented in Figure 3. These 23-mer candidates represent subregions of the gene’s transcript that are likely to be suitable targets for the guide RNA (in combination with RISC, Figure 1) to actually silence a gene. While the position of the siRNA candidate may be important to effect gene silencing, there are no universally accepted rules to develop algorithmic prediction based on position alone, which is one reason the final decision about which siRNA candidate to use is best left to the end user. Maximal suppression of gene expression by the siRNA candidate list will likely require empirical comparison of several siRNA candidates.

The second column of Table 2 provides 23-mer sequences (written 5′ to 3′) specific to the gene of interest to be silenced. The black box of Figure 2 highlights one example of a 23-mer from within IFNβ1. Each 23-mer represents a region of the mRNA transcript to which a guide sequence of RNA (i.e. the guide strand, Figure 1) can bind and mediate repression of protein synthesis.

The third column of Table 3 reports the 21-nucleotide guide and passenger sequences (Figures 1 and 2) that are found originally in the small hairpin RNA (shRNA), itself manufactured from the vector referenced in Figure 1. DNA versions of these RNA sequences can be subsequently used to construct vectors for permanent or transient repression of a specific gene. As this sounds more complicated than it actually is, Figure 4 provides a graphical description of how to take the information from Table 2 and create a vector capable of using the information provided by SRS to silence a gene.

**Figure 4.** Basic workflow involved in using the output of SRS to silence expression of IFNβ1. Here, the siRNA candidate at position 591–613 of the target gene (referenced in Figure 2 and Table 2) is used to compute the passenger and guide strand RNA sequences that a cell would need to manufacture from instructions contained in a suitable expression vector (Figure 1). The sequences outlined in red refer to the passenger and guide strand RNAs, respectively, while the sequence outlined in green represents one of many possible loop structures that could be designed to ensure the expression vector contains the necessary information for a cell to manufacture short hairpin RNA (shRNA, Figure 1). Where indicated, “Column 2” and “Column 3” refer to the columns of Table 2 that contain information displayed here.

The fourth and fifth columns of Table 2 report the melting temperatures (Tm) for the seed-target duplexes (Figure 2) of both the guide and passenger RNA sequences. The lower the Tm value, the more stable the seed-target duplex is. A good rule of thumb for selecting siRNA candidates, which is computationally enforced by SRS, is to use candidates that have seed-target melting temperatures below 21.5 °C. All things considered, the lower the Tm, the better.

The final two columns of Table 2 (columns 6 and 7) contain checkboxes that will, when selected, reveal the results of SRS’s attempt to find similar genes that may be inadvertently silenced by the siRNA candidates under consideration. Checkboxes are used because the amount of information to be displayed can be considerable. When a checkbox is selected, a pop-up window reveals the data that was discovered. Figure 5 shows a typical result displayed when selecting a checkbox in Table 2. SRS will discard any similar gene sequences with five or more differences, as the possibility of those sequences being repressed by the siRNA sequence is virtually zero.

**Figure 5.** Example output produced when the first checkbox contained in the third row of Table 2 is selected. The first column of output displays sequence alignments between the siRNA candidate chosen (in this case the 23-mer sequence positioned between nucleotides 68–90 of IFNβ1) and other gene sequences contained within the RefSeq database used by SRS. Mismatch positions are highlighted in green. The second column contains annotation information about the genes that are aligned to the siRNA candidate. As it is possible that siRNA candidates with a limited number of mismatches to other gene sequences could inappropriately silence these other genes, SRS provides critical information to allow the end user to choose the siRNA candidate that best fits their needs.

To gauge the performance of SRS, two genes were chosen randomly from each of the five species the program can accommodate, and each of the genes was used as an input query to generate a list of siRNA candidates (Table 3). Timings came from running Mathematica 10 under Windows 7 (64 bit) using an Intel 2500K processor overclocked to 4.48 GHz. Total system memory is 32 GB. All reported timings use a fresh kernel.

As biologists continue to develop better methods to identify causal genes driving changes in phenotype, facile methods of altering the expression of those genes are necessary to interrupt undesirable phenotypes, most notably those of disease. SRS brings to the Mathematica community a contemporary algorithm enabling the end user to quickly design small interfering RNAs capable of suppressing a gene’s output, while simultaneously minimizing off-targeting effects that have challenged earlier attempts at using siRNA as a gene suppression technology.

[1] | E. M. Youngman and J. M. Claycomb, “From Early Lessons to New Frontiers: The Worm as a Treasure Trove of Small RNA Biology,” Frontiers in Genetics, Nov 27, 2014. doi:10.3389/fgene.2014.00416. |

[2] | D. Barbosa Dogini, V. D’Avila Bittencourt Pascoal, S. H. Avansini, A. Schwambach Vieira, T. Campos Pereira, and I. Lopes-Cendes, “The New World of RNAs,” Genetics and Molecular Biology, 37(1 Suppl), 2014 pp. 285–293. www.ncbi.nlm.nih.gov/pmc/articles/PMC3983583. |

[3] | K. Gavrilov and W. M. Saltzman, “Therapeutic siRNA: Principles, Challenges, and Strategies,” Yale Journal of Biology and Medicine, 85(2), 2012 pp. 187–200. www.ncbi.nlm.nih.gov/pmc/articles/PMC3375670. |

[4] | T. Coelho, D. Adams, A. Silva, P. Lozeron, P. N. Hawkins, T. Mant, J. Perez, J. Chiesa, S. Warrington, E. Tranter, M. Munisamy, R. Falzone, J. Harrop, J. Cehelsky, B. R. Bettencourt, M. Geissler, J. S. Butler, A. Sehgal, R. E. Meyers, Q. Chen, T. Borland, R. M. Hutabarat, V. A. Clausen, R. Alvarez, K. Fitzgerald, C. Gamba-Vitalo, S. V. Nochur, A. K. Vaishnaw, D. W. Y. Sah, J. A. Gollob, and O. B. Suhr, “Safety and Efficacy of RNAi Therapy for Transthyretin Amyloidosis,” New England Journal of Medicine, 369, 2013 pp. 819–829. doi:10.1056/NEJMoa1208760. |

[5] | J. Moreno-Montañés, B. Sádaba, V. Ruz, A. Gómez-Guiu, J. Zarranz, M. V. González, C. Pañeda, and A. I. Jimenez, “Phase I Clinical Trial of SYL040012, a Small Interfering RNA Targeting -Adrenergic Receptor 2, for Lowering Intraocular Pressure,” Molecular Therapy, 22(1), 2014 pp. 226–232. www.nature.com/mt/journal/v22/n1/full/mt2013217a.html. |

[6] | J. A. McCarroll, T. Dwarte, H. Baigude, J. Dang, L. Yang, R. B. Erlich, K. Kimpton, J. Teo, S. M. Sagnella, M. C. Akerfeldt, J. Liu, P. A. Phillips, T. M. Rana, and M. Kavallaris, “Therapeutic Targeting of Polo-Like Kinase 1 Using RNA-Interfering Nanoparticles (iNOPs) for the Treatment of Non-Small Cell Lung Cancer,” Oncotarget, 6(14), 2014 pp. 12020–12034. www.ncbi.nlm.nih.gov/pmc/articles/PMC4494920. |

[7] | Y. Naito, J. Yoshimura, S. Morishita, and K. Ui-Tei, “siDirect 2.0: Updated Software for Designing Functional siRNA with Reduced Seed-Dependent Off-Target Effect,” BMC Bioinformatics, 10(392), 2009. doi:10.1186/1471-2105-10-392. |

[8] | K. Ui-Tei, Y. Naito, F. Takahashi, T. Haraguchi, H. Ohki-Hamazaki, A. Juni, R. Ueda, and K. Saigo, “Guidelines for the Selection of Highly Effective siRNA Sequences for Mammalian and Chick RNA Interference,” Nucleic Acids Research, 32(3), 2004 pp. 936–948. doi:10.1093/nar/gkh247. |

[9] | J. Bekisz, H. Schmeisser, J. Hernandez, N. D Goldman, and K. C Zoon, “Human Interferons Alpha, Beta and Omega,” Growth Factors, 22(4), 2004 pp. 243–251. doi:10.1080/08977190400000833. |

[10] | K. D. Pruitt, T. Tatusova, G. R. Brown, and D. R. Maglott, “NCBI Reference Sequences (RefSeq): Current Status, New Features and Genome Annotation Policy,” Nucleic Acids Research, 40(D1), 2012 pp. D130–D135. doi:10.1093/nar/gkr1079. |

[11] | K. Ui-Tei, Y. Naito, K. Nishi, A. Juni, and K. Saigo, “Thermodynamic Stability and Watson–Crick Base Pairing in the Seed Duplex Are Major Determinants of the Efficiency of the siRNA-Based Off-Target Effect,” Nucleic Acids Research, 36(22), 2008 pp. 7100–7109. doi:10.1093/nar/gkn902. |

T. D. Allen, “A Computational Strategy for Effective Gene Silencing through siRNAs,” The Mathematica Journal, 2016. dx.doi.org/doi:10.3888/tmj.18-1. |

Todd Allen is an associate professor of biology at HACC, Lancaster. His interest in computational biology using Mathematica took shape during his postdoctoral research years at the University of Maryland, where he developed a custom cDNA microarray chip to study gene expression changes in the chestnut blight pathogen, *Cryphonectria parasitica*.

**Todd D. Allen, Ph.D.**

*Harrisburg Area Community College (Lancaster Campus)
East 206R
1641 Old Philadelphia Pike
Lancaster, PA 17602*

Domain coloring is a technique for constructing a tractable visual object of the graph of a complex function. The package *complexVisualize.m* improves on existing domain coloring techniques by rendering a global picture on the Riemann sphere (the compactification of the complex plane). Additionally, the package allows dynamic visualization of families of Möbius transformations. In this article we discuss the implementation of the package and illustrate its usage with some examples.

Domain coloring is a technique that uses a color spectrum to compensate for the missing dimension in the graph of a complex function. It was first introduced in [1], with a detailed description of the complex‐plane case in [2]. A general definition is found in [3]. More precisely, consider a function between two sets (for example, two complex manifolds). Choose a “color” function, , where denotes the Hue‐Saturation‐Brightness standard color space. Next, for any , compute and evaluate the resulting color , assigning this color to the preimage . The final colored domain has all the information (through the color) needed to say where the point gets mapped. Of course, the effectiveness of domain coloring depends on the choice of an adequate color scheme .

Geometrically, the HSB color space is identified with an inverted solid cone. In cylindrical coordinates, HSB is parametrized by , where , , and . Then corresponds to the hue value of the visible spectrum, with , , and corresponding to red, cyan, and violet, respectively. The axis of the cone corresponds to zero saturation (purely a gradient of grays), and the external surface of the cone corresponds to full saturation. The brightness corresponds to the height of the cone, with (the vertex) being black and (the top) corresponding to full brightness.

This shows the HSB color space.

In other words, , with the equivalence given by for all , and identified as a single point, the vertex.

This article considers , where is the Riemann sphere: the complex numbers plus a single point at infinity (the north pole), embedded in , and , the *color scheme*, a usually injective map from the sphere to the solid inverted cone. Given , we call and the *domain* and *image* Riemann spheres, respectively. (The terminology *source* and *target* spheres is also used.)

The package *complexVisualize.m* renders the domain coloring of a complex function on the Riemann sphere. The two main commands defined in the package are `complexVisualize` and `mobiusVisualize`. The former renders the domain coloring of a generic ; the latter is optimized for Möbius transformations for the purpose of dynamic rendering. This article describes the package’s implementation and illustrates its use with some examples.

In the color schemes that we explicitly consider here, the value of the hue is directly related to the argument of the image of the function, while the saturation and brightness are related to the modulus of the image values. Since a complex-valued function is determined by the argument and modulus of its image, the color scheme gives a complete global picture of the function.

The program also draws the preimage of the grid on the image sphere determined by its longitudes and latitudes. Furthermore, the preimages of three special geodesics are emphasized: the preimages of the equator are red, and the meridians corresponding to the intersections of the ‐ and ‐ planes with the image sphere are blue and green, respectively. These correspond to the unit circle and the real and imaginary axes on the image sphere.

To use the functions in this article, make sure the `$Path` variable points to the file complexVisualize.m (*Mathematica*’s front end menu command File▶Install does this automatically) and load it.

In this section we continue to elaborate on the theory of domain coloring and discuss the main ideas behind the implementation of the package *complexVisualize.m*. We also take a close look at the implementation of some color schemes.

Let us now formalize the concepts behind domain coloring and develop the code that provides its basic implementation.

The code in the next two subsections is self-contained and should be evaluated sequentially. The code in the third subsection requires loading the package *complexVisualize.m*.

Given a complex function , let

satisfy the condition , where is the projection . Recall that is interpreted as the Hue‐Saturation‐Brightness space. Such a function is referred to as the *color scheme*. Domain coloring can be understood as the implementation of the composition to assign color to the domain of .

This technique can be extended to the Riemann sphere . Identifying with the ‐ plane in , the *stereographic projection*

is defined by the requirement that , , and are collinear, while . The function induces

defined by , with , provided that the limit exists in . (In the package’s documentation, the domain and image of are referred to as *source* and *target*, respectively.)

**Figure 1**. Stereographic projection.

The coordinate realization of the stereographic projection is deduced from Figure 1 as follows. Let and . Let , , and .

By similar triangles,

so

Note that for . Thus,

and hence

So define a function from the plane to the sphere and its inverse.

A color scheme on the Riemann sphere is a function such that , where are spherical coordinates on , and again denotes projection onto . Domain coloring is implemented, essentially, by a parametric plot.

Here *colorfunction* is, conceptually, given by .

For example, consider the function and color scheme :

(1) |

where the function corresponds to the command `Hue[x]`. To start, transform to its vector form.

For example, here is the vector form of .

Choose the color scheme to be .

Therefore this is .

This is the domain coloring of .

**Figure 2**. Domain coloring associated with `theFunction` using the chosen `colorScheme1`; see equations (1).

The information given by the domain coloring can be complemented by displaying the preimages of the standard *reference lines*.

These preimages are given, respectively, by , , and , where , and . This defines the projections , , and .

This defines the reference lines.

This shows the domain coloring in Figure 2 complemented with the preimages of the reference lines.

**Figure 3**. Domain coloring of function and color scheme (1), showing reference lines.

The command `complexVisualize`, provided by the package *complexVisualize.m*, automates the domain coloring so far described and adds the following features:

- The inclusion of the Cartesian , , and axes (colored blue, green, and black).
- The rendering of the pullback of the standard mesh determined by the spherical coordinate parametrization of the target .
- Most importantly, a simple and flexible syntax, with the ability to call various options defined by the package.

These Cartesian axes serve to orient the reader with respect to the location of 1, , and infinity on the domain Riemann sphere. (We also refer to the Cartesian and axes as the *real* and *imaginary* axes, since the inverse stereographic projection sends those axes to the meridians representing the real and imaginary lines on the domain .)

This pulled-back mesh complements the visual cues given by the pullback of the three reference lines. Its implementation follows the same idea used previously to draw `theReferenceLines`, that is to say, using the options `Mesh` and `MeshFunctions` within the sphere rendering by `ParametricPlot3D`.

These options (specifying the number or color of mesh lines, for example) let you fine-tune the domain coloring of a given complex function. Standard *Mathematica* options can also be called.

The command `complexVisualize` has the following simple syntax.

To illustrate it, consider the working example, the domain coloring of with the same color scheme .

Here is an alternate rendering with a different color scheme.

We omit here the discussion of how the various options are implemented. The interested reader may consult the help documentation for individual options or review their implementation in the package.

An exception must however be granted to the option `colorScheme`, which deserves a careful discussion; it is clear that different color schemes will emphasize different features of a given complex function.

The “color scheme” is specified with the option `colorScheme`.

With the setting , the color scheme function is implemented with `azimuthColorScheme`.

This coloring only keeps track of and not the modulus. Although this seems rather restrictive, it is satisfactory in many examples. Indeed, the *argument principle* (see [4]) is usually enough to distinguish between zeros and poles without a visual cue coming from .

To keep track of both the argument and the modulus of the function, use the option with , .

This implements the corresponding color scheme function .

The *saturation* and *brightness* are given by bump functions.

**Figure 4**. Saturation and brightness for color scheme `azimuthLatitude[a,r]`.

The net effect is that close to zeros of the function, the coloring tends to *darken*, while close to poles the coloring tends to *whiten*. This is because on the target , zero corresponds to (the south pole) and infinity corresponds to (the north pole). The parameters and fine-tune how this effect is achieved. The rational function shown in Section 3.2 gives a nice illustration of the `azimuthLatitude` color scheme.

A somewhat less sophisticated but attractive method of getting the “dark zeros and bright poles” effect is to use the color scheme function .

This is invoked with the setting . The saturation and brightness curves associated with this color scheme are shown in Figure 5, and its usage is illustrated in Section 3.2.

**Figure 5.** Saturation and brightness for color scheme “metallic”.

This section lists domain coloring examples, which illustrate the concepts discussed so far.

Here is the domain coloring of the identity function .

This seemingly trivial example familiarizes us with the essence of domain coloring: is represented by the hue, and as () the color darkens (brightens), assuming you use . The real and imaginary axes are clearly shown in blue and green, respectively. The unit circle is the equator, shown in red.

Additionally, the , , and axes are always drawn by `complexVisualize` in blue, green, and black, respectively. Their purpose is to keep track of the Riemann sphere in its role as the domain of the function, with , , and located, respectively, at the intersection of the , , and axes with the sphere.

The domain coloring of the rational function

serves to illustrate the global information that we get working on the Riemann sphere, as well as the nature of the zeros and poles.

Here is a rational function.

This example shows an important result in complex analysis, the *argument principle* [4]: the number of total hue variations and orientation show the nature (type and order) of isolated singularities. The brightness and saturation complement this information, with total brightness with zero saturation corresponding to a pole, and zero brightness corresponding to a zero. See Section 2.2 for a description of the color scheme `azimuthLatitude` used in this example.

Next we illustrate the domain coloring of the rational function

using an alternate color scheme that we have called `metallic`, whose definition is also discussed in Section 2.2. The subtler variations in saturation and brightness still give a visual cue to the location of the zeros and poles.

Next, let us look at a polynomial like .

This example illustrates the relevance of having a compact domain, the Riemann sphere, to deal with isolated singularities. Indeed, whenever a function has no essential singularities, the number of zeros and poles, counted with multiplicity, coincide. This is of course a consequence of the fact that the Euler characteristic of the sphere is , so its genus is zero.

Essential singularities have a visual interpretation closely related with the great Picard theorem [4]. Indeed, at an essential singularity we expect to observe an infinite variation of color.

Here are the circular and hyperbolic cosines.

Apart from observing the essential singularities at infinity, the relation becomes evident as a rotation of the coloring around the axis. Other identities between circular and hyperbolic trigonometric functions can be visualized in a similar fashion.

To give one more example of a transcendental function, let us look at the “baroque” coloring that we get with the function

Given a family of complex functions, an attractive possibility is the use of *Mathematica*’s dynamic graphical capabilities provided by the `Manipulate` command to observe how the domain coloring changes as we move around the given function family.

One class of complex functions that lends itself to dynamic visualization is the class of Möbius transformations. There have been important efforts to construct optimal computational models that characterize the geometry of the Möbius group, one of the most popular being [5]. Our approach here is to construct a `Manipulate` in which the *reference curves* (i.e., the preimages of the real and imaginary axes, as well as the preimage of the unit circle) are drawn while parameters are dynamically varied. Once variation of the parameters stops, the rest of the elements of domain coloring are rendered.

The Möbius transformations constitute the set of all linear rational functions on the extended complex plane, which is identified with the set of all conformal automorphisms on the Riemann sphere [6], also called the *Möbius group*:

It is well known [7] that

where is the general (special) linear group with complex entries and denotes its center.

A Möbius transformation is completely characterized by the preimages of , , and , which we will refer to as , , and , respectively. Indeed, it is easy to see that in terms of these parameters

Thus, dynamic visualization of Möbius transformations can be implemented by assigning a 2D controller to each of the parameters , , . Note that a point on the rectangle gets mapped to through the use of cylindrical coordinates:

Since a Möbius transformation, viewed as conformal automorphisms in , sends circles to circles, the preimages of the reference curves (the real axis, the imaginary axis, and the unit circle) are circles on . We call them *reference circles*. Using the `Manipulate` option `ControlActive`, a fast-rendering version of `complexVisualize` can be implemented, in which only the reference circles are drawn. To this effect, it suffices to characterize each reference circle by three points.

Here . Each reference circle is drawn by the elementary geometric construction described in the following. The simplicity of the construction translates in an instantaneous rendering, which allows a dynamic visualization with the command `Manipulate`.

We now describe the implementation of the geometric construction that quickly renders the three reference circles associated with a given Möbius transformation.

Given three points , , on the unit sphere, the circle passing through them is given by

where is the position vector of the center of the circle

and

with , and where denotes the normalization operator .

Here is the corresponding code.

The function `circleThroughThreePoints` fails when , , and are collinear. This, however, is guaranteed not to occur, because the command is only applied to three distinct points on the sphere, which are thus never collinear.

Given a Möbius transformation , this recovers its coefficients.

This gives its five preimages , , , , .

Since `ComplexInfinity` is an acceptable point in , we switch off the warning in the declaration of `fivePreImages`.

These five preimages need to be located on the Riemann sphere , and this is achieved by `stereoInv`, the inverse of the stereographic projection.

According to Table 2, given , each reference circle is in correspondence with each triple in the list.

This finishes all the necessary ingredients needed to draw the reference circles associated with a given Möbius transformation. Let us implement the procedure with an example.

Let us write the code that renders the reference circles for the Möbius transformation

(2) |

This defines the function.

Here are the preimages under of , , , , and , regarded as points on the Riemann sphere .

Hence, using the color code in Table 1, here is how the three reference circles are rendered.

Finally, here are the three reference circles rendered on the Riemann sphere .

The procedure discussed is implemented with the command `mobiusVisualize`, which is also part of the package *complexVisualize.m*.

The main option for this command is `fastRender`. With , `mobiusVisualize` simply invokes `complexVisualize` to do the rendering. With the (default) setting , the procedure described in the previous two subsections is invoked in order to draw the reference circles.

For example, here is the Möbius transformation defined in (2).

The command `mobiusVisualize` can be customized like `complexVisualize` through options, both the ones defined in the package and most of the standard `ParametricPlot3D` options. The interested reader can consult all the details in the package.

The point of having the ability to switch off the coloring (through the option `fastRender`) is that we can now implement a fast dynamic `Manipulate` to visualize families of Möbius transformations.

The command `cylC` handles the 2D controllers representing complex parameters on .

Here is a `Manipulate` allowing dynamic visualization of the group .

The wrapper `ControlActive` in the option assignment to `fastRender` enables an instantaneous redrawing of the reference circles as the parameters are varied.

This example can be regarded as a first step in an exploration of how `Manipulate` and `mobiusVisualize` can be used together to dynamically visualize the generators and subgroups of .

To further illustrate the dynamic visualization of Möbius transformations using a `Manipulate`, consider the generators of the group of : translations, complex multiplication, and complex inversion.

This shows all pure translations , with . Experiment by varying the value of the parameter .

This shows complex multiplication , with .

On the complex plane, complex multiplication is a homothety followed by a rotation, both with respect to the origin. The preceding `Manipulate` shows how rotations with respect to the origin become rotations around the axis. The effect of a homothety also becomes visually evident as a transformation in .

The complex inverse function is . A homotopy continuously relating the identity with inversion is given by

so that and .

The following animation shows this homotopy.

It becomes evident that on , the geometry of a complex inversion is simply a rotation around the real axis. Note that the simplicity of this homotopy is lost when we try to visualize it purely in the complex plane.

The package *complexVisualize.m* is a robust and flexible tool for generating global domain coloring for holomorphic, meromorphic, and conformal functions of one complex variable, which makes it a valuable exploratory and didactic tool. We have described the main ideas behind its implementation and examples of its use.

You can experiment with further applications. Some ideas for classroom demonstrations are: (a) a sequential rendering of the Taylor expansion of a transcendental function in increasing degree; (b) exploration of the subgroups of Möbius transformations; and (c) exploration of different color schemes.

With the `colorScheme` option it is possible to implement any color rule with an appropriate function between the HSB space and the Riemann sphere. This allows the program to generalize color schemes that have appeared in the literature in the planar case.

Being on the Riemann sphere, the global character of domain coloring gives a visual interpretation of various fundamental complex analysis results. Here we have shown only a few.

When working with automorphisms on the Riemann sphere, it is possible to construct `Manipulate` animations of families of domain colorings. This dynamic visualization allows geometrical interpretations of the fundamental properties and results of these transformations to be given in an accessible and comprehensive manner.

There are many possibilities for extending the reach of the visualizations considered here. Probably the most obvious line of exploration is a deeper analysis of multifunctions on general Riemann surfaces, for which [3] is a good starting point.

The authors wish to thank Prof. Ricardo Berlanga Z. for encouragement and useful insight on the subject matter of this paper. The first author acknowledges the support of Ixtli‐UNAM under project IX100310.

[1] | F. A. Farris. “Visualizing Complex‐Valued Functions in the Plane.” (Jun 15, 2015) www.maa.org/visualizing-complex-valued-functions-in-the-plane. |

[2] | H. Lundmark. “Visualizing Complex Analytic Functions Using Domain Coloring.” (Jun 15, 2015) users.mai.liu.se/hanlu09/complex/domain_coloring.html. |

[3] | K. Poelke and K. Polthier, “Lifted Domain Coloring,” EuroVis’09, Proceedings of the 11th Eurographics / IEEE-VGTC Conference on Visualization, Berlin, Germany, June 10–12, 2009, Computer Graphics Forum, 28(3), 2009 pp. 735–742.doi:10.1111/j.1467-8659.2009.01479.x. |

[4] | L. V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable (International Series in Pure and Applied Mathematics), New York: McGraw‐Hill, 1953. |

[5] | D. Arnold and J. Rogness. “Möbius Transformations Revealed.” (Jun 15, 2015) www.ima.umn.edu/~arnold/moebius. |

[6] | J. Milnor, Dynamics in One Complex Variable, 3rd ed., Princeton: Princeton University Press, 2006. |

[7] | G. Toth, Finite Möbius Groups, Minimal Immersions of Spheres, and Moduli, New York: Springer‐Verlag, 2002. |

A. Sandoval-Romero and A. Hernández-Garduño, “Domain Coloring on the Riemann Sphere,” The Mathematica Journal, 2015. dx.doi.org/doi:10.3888/tmj.17-9. |

Additional electronic files:

Available at: www.mathematica-journal.com/data/uploads/2015/11/complexVisualize.m

Ángeles Sandoval-Romero is a full professor at Universidad Nacional Autónoma de Mexico, Facultad de Ciencias, Department of Mathematics. She obtained her Ph.D. in mathematics from UNAM in 2006 with a dissertation on scattering theory applied to a system of nonlinear partial differential equations. Her research interests include geometric complex analysis, differential equations, and their applications.

Antonio Hernández-Garduño is a professor in the Mathematics Department of Universidad Autónoma Metropolitana (campus Iztapalapa) in Mexico City. He obtained his Ph.D. from Caltech in 2002 with a dissertation on the bifurcation of relative equilibria in mechanical systems. His research interests include geometric mechanics, vortex dynamics, celestial mechanics, and computer-based math education.

**Ángeles Sandoval-Romero**

*Departamento de Matemáticas
Facultad de Ciencias – UNAM*

Circuito Exterior, Ciudad Universitaria

México D.F. 04510

**Antonio Hernández-Garduño**

*Departamento de Matemáticas
Universidad Autónoma Metropolitana – Iztapalapa*

México D.F. 09340

This article describes how *Mathematica* can be used to develop an understanding of the basic steps involved in solving Navier–Stokes equations using a finite-volume approach for incompressible steady-state flow. The main aim is to let students follow from a mathematical description of a given problem through to the method of solution in a transparent way. The well-known “driven cavity” problem is used as the problem for testing the coding, and the Navier–Stokes equations are solved in vorticity-streamfunction form. Building on what the students were familiar with from a previous course, the solution algorithm for the vorticity-streamfunction equations chosen was a relaxation procedure. However, this approach converges very slowly, so another method using matrix and linear algebra concepts was also introduced to emphasize the need for efficient and optimized code.

*Mathematica* is used to help with an initial understanding of the process of solving Navier–Stokes equations. For 2D incompressible flows, it is possible to recast the Navier–Stokes equations in an alternative form in terms of the streamfunction and the vorticity. In many applications, the vorticity-streamfunction form of the Navier–Stokes equations provides better insight into the physical mechanisms driving the flow than the “primitive variable” formulation in terms of the mean velocities , , and pressure . The streamfunction and vorticity formulation is also useful for numerical work, since it avoids some problems resulting from the discretization of the continuity equation.

The streamfunction is defined as

(1) |

where the integral has to be evaluated along a curve from the arbitrary but fixed point to point , ** **is the velocity vector, and is the unit normal on the curve from to ; see Figure 1. We regard as a function of the location of point .

**Figure 1.** Sketch illustrating the definitions of a streamfunction.

Figure 1 shows that is equal to the component of the velocity that crosses . Therefore represents the volume flux (per unit depth in the direction) through . Evaluating along two different paths and invoking the integral form of the incompressibility constraint shows that is path independent; that is, its value only depends on the locations of the points and . Changing the position of point only changes by a constant. It turns out that for all applications such changes are irrelevant. It is therefore common to suppress the explicit reference to . Hence, we regard as a function of the spatial coordinates only; that is, . Streamlines are lines that are everywhere tangential to the velocity field, that is, , where is the unit normal to the streamline. Hence the streamfunction is constant along streamlines. Note that stationary impermeable boundaries are also characterized by , where is the unit normal on the boundary. Therefore, is also constant along such boundaries. Invoking the integral incompressibility constraint for an infinitesimally small triangle shows that is related to the two Cartesian velocity components and via

(2) |

Flows that are specified by a streamfunction automatically satisfy the continuity equation, since

(3) |

For 2D flows, the vorticity vector only has one nonzero component (in the direction); that is, , where

(4) |

Using the definition of the velocities in terms of the streamfunction shows that

(5) |

where is the 2D Laplace operator.

The understanding of the steps involved in solving the Navier–Stokes equations using the vorticity-streamfunction form is one of the topics used in a third-year undergraduate course on computational fluid mechanics, solely for students majoring in mechanical engineering. The use of *Mathematica* makes the assumption that the students are familiar with the package, as it generally takes a good deal of exposure to *Mathematica* to become comfortable using it at the level required here [1]. The students taking the computational fluid mechanics course are indeed very familiar with *Mathematica*, as the computer algebra system is used during year one in the modules Calculus and Applications and Vector Calculus, and during year two in the module Numerical Methods for Engineering. In addition, there are many notes, explanations, and examples on the in-house Moodle open-source learning platform. Similar work to this can be found in Fearn [2], while background reading on computational fluid dynamics and vorticity may be found in Ferziger and Perić [3] and Chorin [4], respectively.

The aim here is to give a continuous and comprehensive process involving the mathematics of one formulation of the Navier–Stokes equations and a solution using *Mathematica*. In this way, the students can actually see the development of the mathematical description linked to a programming environment solution process so often hidden in commercial code used for training CFD students.

The Navier–Stokes equations for incompressible steady-state flow in vorticity-streamfunction form are

*Advection-Diffusion Equation*

(6) |

*Elliptic Equation*

(7) |

where is the Reynolds number. For a given problem, boundary conditions need to be specified to solve equations (6) and (7). The problem chosen here introduces the student to a classical computational fluid dynamics (CFD) case, namely the lid-driven cavity flow [5]. This flow is commonly used to test, for example, a novel method of discretization of the equations or new computer programming, as the resulting flow is well known from experiments. Consider a rectangular box as shown in Figure 2, where a lid is allowed to move in the horizontal plane from left to right. When the lid is not moving, the fluid in the box is stationary, whereas when the lid is moving, the fluid circulates inside the box. Here the boundary conditions needed for solution are summarized in terms of vorticity and streamfunction. As all four walls of the cavity touch, the streamfuction must be equal for all four walls, as indicated by the in Figure 2. The streamfuction is constant at the walls, as its gradient is velocity, which is zero relative to a given wall (no-slip condition), as indicated in the outer boxes also shown in Figure 2. The vorticity boundary conditions for each wall are also shown in each of these outer boxes and were derived from the streamfunction.

**Figure 2.** Summary of boundary conditions in terms of vorticity and streamfunction for cavity with moving lid.

It is convenient from a numerical point of view to make equations (6) and (7) nondimensional. If and are the height and width of the cavity, respectively, and is a reference velocity, then

.

Equations (6) and (7) become

(8) |

(9) |

defined on the domain . The boundary equations are defined by

(10) |

Here is an iterative solution for solving the incompressible steady 2D Navier–Stokes equations. The method is based on Burggraf’s proposal [6]. So as to include the boundary nodes of the cavity, central differencing is used to discretize equations (8) and (9). The number of nodes in the and directions is set at and , respectively. The mesh is a regular Cartesian grid with the nodes equally spaced at a distance , as illustrated in Figure 3.

**Figure 3.** Typical Cartesian mesh used for the lid-driven cavity flow.

To discretize equations (8) and (9), the following general finite central-difference approximations were introduced for spatial dimensions, obtained by adding or subtracting one Taylor series from another and invoking the intermediate value theorem. As can be seen from equation (11), the order of accuracy for both the first and second derivatives is .

(11) |

Using the approximations from equation (11), the discretized advection-diffusion equation (equation (8)) at the node is

(12) |

The elliptic equation (equation (9)) at the node is

(13) |

The equations are applied to the internal nodes of the Cartesian mesh; that is, and .

The solution method uses residual functions; that is, if the values of and are exact on the nodes spanned by the residual functions and , then

(14) |

where

(15) |

(16) |

The following fixed-point iterative procedure, based on the Gauss–Seidel scheme, is then constructed:

(17) |

where is a relaxation parameter lying in the range , and and refer to the respective iterations. In this article, the actual value of depends on and can be obtained by numerical experimentation. The use of this relaxation parameter is to improve the stability of a computation, particularly in solving steady-state problems. It works by limiting, when necessary, the amount that a variable changes from one iteration to the next. The optimum choice of is one that is small enough to ensure stable computation but large enough to move the iterative process forward quickly.

The boundary conditions now need to be determined. For the streamfunction from equation (11),

(18) |

For vorticity, the following needs to be considered for, say, the left wall of the cavity with one node outside the computational domain:

(19) |

The value of can be accounted for using the no-slip boundary condition on the cavity wall; that is,

(20) |

Using central differences, this condition can be written as

(21) |

where again the order of accuracy is .

This finding, together with , gives

(22) |

Similarly, for the other walls of the cavity the boundary conditions for vorticity can be deduced from

(23) |

to give

(24) |

where and are defined in Figure 3. Note that the final boundary condition shown in equation (24) is for the moving lid, and the extra term is due to the tangential velocity being nonzero.

This section starts with the definition of mesh size and grid spacing; for this problem the mesh spacing is set equal in the and directions. The parameters `Nx` and `Ny` denote the mesh size. Initially, `ω` and `ψ` are set to zero for all nodes except at the top wall, where vorticity is not zero. The Reynolds number is set at 100, the relaxation parameter `p` is set to 1, and the aspect ratio `γ` is set at 1. A maximum allowable residual value `e` is set, and nested loops are used to execute the iterative algorithm. At every iteration step `k`, if the maximum value of the absolute value of is less than `e`, the calculations are halted. The iterative loops are wrapped with the function `Timing` to give an estimate of the time taken to do the calculations.

Extraction of the centerline velocities is also instructive. First of all, students are given more experience viewing velocity profiles, and second, there are ample calculations and measurements in the literature [7, 8, 9] for comparison with the results obtained here. The centerline velocities (in the direction) and (in the direction) were derived from the streamfunction values using the equations

(25) |

(26) |

The velocity distributions for and in the following figure are drawn along the vertical and horizontal centerlines, respectively.

Importantly, the students were made aware that the profiles they calculated must be compared, preferably with experimental measurements, to test the validity of the calculation technique. As can be seen from the preceding figure, the calculations for the horizontal centerline velocities compare very favorably with those reported in the literature [7, 9] for .

It can be clearly seen that the relaxation procedure, though reasonably easy for students to follow and therefore educationally productive, is indeed very slow to converge. However, students must be aware that code should be written efficiently and optimized, making use of all available concepts and procedures. Therefore, as part of their understanding of solving Navier–Stokes equations in the best way, they were also introduced to the following *Mathematica* code. The preceding exercise was a small-scale problem using a fairly primitive method of solution, and even for this simple problem, the convergence time is prohibitive. Therefore, for large numerical computations and those with more complex geometry, it is important to use a solution method that will run more efficiently. This can be achieved using linear algebra and matrices and highly optimized functionalities built into *Mathematica*, namely `NDSolve` and `LinearSolve`. The following program solves the same problem as that described in Figure 2, except the Reynolds number is set at 400 and the computational grid is slightly more dense. The time to solve this same problem was reduced by over an order of magnitude. It should be understood that what the code is solving is the unsteady 2D Navier–Stokes equations. Each iteration is a time step from impulse-starting conditions to steady state, and for the lid-driven cavity flow, the unsteady solution converges to the steady state. The animation shown is for the streamfunction with a Reynolds number of 400.

Having established the method of solution, the student would then experiment by varying the grid size and seeking a solution independent of the grid, which is very important in CFD calculations. Results would also be obtained for different aspect ratios of the container. Again, where possible, calculations must be compared with experimental measurements to establish validity. As can be seen, the calculated horizontal velocity along the vertical centerline for was in good agreement with experimental results [7, 9].

This article outlines a well-defined sequence of steps needed to solve the Navier–Stokes equations cast in the vorticity-streamfunction form. The sequence is integrated with the use of *Mathematica* at appropriate stages to take care of the tedious computations, and hence to allow the students to concentrate on the overall details of the solution process. In addition, it is now important to integrate computer technology so as to complete lectures and theory. This has the advantages of helping with the computations, aiding presentation for reports and analysis, and motivating students. Incorporating *Mathematica* also takes away the “black-box” approach so often being used by students with full CFD commercial codes, which give no real understanding of the numerics involved. The idea of efficient and optimized coding was also introduced.

[1] | S. Pomeranz, “Using a Computer Algebra System to Teach the Finite Element Method,” International Journal of Engineering Education, 16(4), 2000 pp. 362–368. www.ijee.ie/articles/Vol16-4/IJEE1162.pdf. |

[2] | R. L. Fearn, “Airfoil Aerodynamics Using Panel Methods,” The Mathematica Journal, 10(4), 2008 pp. 725–739.www.mathematica-journal.com/2008/11/airfoil-aerodynamics-using-panel-methods. |

[3] | J. H. Ferziger and M. Perić, Computational Methods for Fluid Dynamics, 3rd ed., Berlin: Springer, 2002. |

[4] | A. J. Chorin, Vorticity and Turbulence (Applied Mathematical Sciences, Vol. 103), New York: Springer-Verlag, 1994. |

[5] | J. D. Bozeman and C. Dalton, “Numerical Study of Viscous Flow in a Cavity,” Journal of Computational Physics, 12(3), 1973 pp. 348–363. doi:10.1016/0021-9991(73)90157-5. |

[6] | O. R. Burggraf, “Analytical and Numerical Studies of the Structure of Steady Separated Flows,” Journal of Fluid Mechanics, 24(1), 1966 pp. 113–151. doi:10.1017/S0022112066000545. |

[7] | U. Gia, K. N. Ghia, and C. T. Shin, “High-Re Solutions for Incompressible Flow Using the Navier–Stokes Equations and a Multigrid Method,” Journal of Computational Physics, 48(3), 1982 pp. 387–411. doi:10.1016/0021-9991(82)90058-4. |

[8] | W. F. Spotz, “Accuracy and Performance of Numerical Wall Boundary Conditions for Steady, 2D, Incompressible Streamfunction Vorticity,” International Journal for Numerical Methods in Fluids, 28(4), 1998 pp. 737–757.onlinelibrary.wiley.com/doi/10.1002/%28SICI %291097-0363%2819980930 %2928:4%3 C737::AID-FLD744 %3 E3 .0.CO;2-L/abstract. |

[9] | D. C. Wan, Y. C. Zhou, and G. W. Wei, “Numerical Solution of Incompressible Flows by Discrete Singular Convolution,” International Journal for Numerical Methods in Fluids, 38(8), 2002 pp. 789–810. doi:10.1002/fld.253. |

D. Adair and M. Jaeger, “Developing an Understanding of the Steps Involved in Solving Navier–Stokes Equations,” The Mathematica Journal, 2015.dx.doi.org/doi:10.3888/tmj.17-8. |

Desmond Adair is a professor of mechanical engineering in the School of Engineering, Nazarbayev University, Astana, Republic of Kazakhstan. His recent research interests include investigations of airborne pollution for both passive and reacting flows, and developing engineering mathematics by the incorporation of computer algebra systems.

Martin Jaeger is an associate professor of civil engineering and head of department in the School of Engineering, Australian College of Kuwait, Misref, Kuwait. His recent research interests include construction management and total quality, as well as developing strategies for engineering education.

**Desmond Adair**

*School of Engineering
Nazarbayev University
53 Kabanbay batyr Ave.
Astana, 010000, Republic of Kazakhstan*

**Martin Jaeger**

*School of Engineering*

*Australian College of Kuwait
Al Aqsa Mosque Street
Misref, Kuwait City, Kuwait*

We describe the use of classical and metric multidimensional scaling methods for graphical representation of the proximity between collections of data consisting of cases characterized by multidimensional attributes. These methods can preserve metric differences between cases, while allowing for dimensional reduction and projection to two or three dimensions ideal for data exploration. We demonstrate these methods with three datasets for: (i) the immunological similarity of influenza proteins measured by a multidimensional assay; (ii) influenza protein sequence similarity; and (iii) reconstruction of airport-relative locations from paired proximity measurements. These examples highlight the use of proximity matrices, eigenvalues, eigenvectors, and linear and nonlinear mappings using numerical minimization methods. Some considerations and caveats for each method are also discussed, and compact *Mathematica* programs are provided.

One “curse” of high- and multidimensional data is the difficulty in graphically displaying how the data points are related to each other. Each data point (in some fields referred to as a case) is characterized by a vector of attributes (e.g. shape, color, temperature, etc.) with numerical values that form a set of coordinates. These coordinates specify a location in an -dimensional space, with being the number of variables used to describe each case. Visualization of such -dimensional data is problematic as, for example, we do not think in 17 dimensions! For human comprehension, we are limited to two- or three-dimensional graphics, sometimes referred to as visualizable dimensions, in which each case is represented by a data point. The distances between points and their relative locations reflect their proximity (similarity or dissimilarity), as measured by a metric function of their attributes, such as Euclidean distance. Larger distances between cases indicate that they are less related. Fortunately, multidimensional scaling (MDS) methods can be used to visualize the degree of proximity for high-dimensional datasets [1, 2, 3]. MDS methods can be used to project the -dimensional coordinates of each case to two or three dimensions, while preserving the relative distances between cases [4, 5]. These methods are increasingly employed in such diverse fields as immunology, molecular biology, marketing, perceptual mapping, and ecology to visualize relative “distances” between viruses, molecules, species, and customers’ behavior. In this article, we discuss some mathematical details of classical and metric MDS methods and build generalizable *Mathematica* programs that can be used for each. We incorporate use cases from geography, molecular virology, and immunology.

Before covering the analytical details, it is worth briefly discussing the variety of MDS methods. The simplest method is referred to as classical multidimensional scaling, also known as principal *coordinate* analysis (PCoA) [1]. PCoA is not to be confused with principal *component* analysis (PCA). The difference between PCA and classical MDS/PCoA is primarily based on the input data. PCA starts with a set of cases or data points , each described by a set of attributes , and uses a Euclidean distance to project the data to a lower-dimensional space oriented to maximize the variance observed between data points. In contrast, classical MDS begins with an input matrix that specifies proximities between pairs of items, often referred to as dissimilarities. It outputs a coordinate matrix in a lower dimension than the starting data (e.g. 120 dimensions projected to two) that preserves the relative distances between all data points. Classical MDS was originally developed by Gower [1] and others and is often demonstrated with a simple problem: given only the distances between entities (e.g. cities), is it possible to calculate the map showing their relative distances? We use this example in Section 4. In the specific case where the dissimilarity is measured by a Euclidean distance, PCA and classical MDS/PCoA are mathematically identical to another dimensional reduction method, singular value decomposition (SVD). In that case, there may be computational advantages to using SVD, which are discussed in this article.

While classical MDS relies on linear transformation of data using matrix operations, metric MDS methods depend on computational optimization. Metric MDS takes a proximity matrix and calculates a two- or three-dimensional coordinate matrix whose configuration minimizes the difference between the -dimensional and the calculated distances in the reduced dimensions for each case or data point [2, 3]. Thus, similar cases are located next to each other in a reduced-dimensional plot, while disparate cases are farther away, and the final coordinates can be projected to a planar or three-dimensional space. Other varieties of MDS, not discussed here, include nonmetric MDS, which uses isotonic regression methods, and generalized MDS, which projects the coordinates to an arbitrary smooth target space [3–7]. Metric MDS is more flexible than classical MDS for certain problems, including when the proximity matrix contains missing data or reflects a distance function that is not Euclidean. In addition, it can accommodate nonlinear mappings from the -dimensional data space to the visualization. However, because metric MDS uses numerical optimization, it is computationally expensive. These tradeoffs are explored in Section 7.

A byproduct of both MDS methods is the proximity matrix, which can also be used for other high-dimensional visualizations, such as heat mapping and multidimensional clustering methods. Combining MDS with these methods often reveals different facets of the data and aids investigators in gaining understanding of the underlying data structure and exploratory data analysis.

In the following sections, we discuss construction of proximity matrices, classical MDS, and metric MDS. Several examples illustrate the strengths and pitfalls of these methods; the full datasets are included in the article. The tradeoff between computational efficiency and accuracy is touched on as well. Each section briefly covers the mathematical underpinning of the method, a step-by-step example, a self-contained function using compact code, considerations for the method and alternate approaches, and a synopsis of the most salient points. A separate section discusses accuracy and computational efficiency and compares these methods.

This article was created using Wolfram *Mathematica* Version 10.0.2, running Macintosh OS X 10.9.5 on a MacBook Pro with a 2.8 GHz Intel Core i7 processor with 16 GB RAM. It contains the entire code and all the data needed for execution of all examples. Internet connectivity is required to retrieve the airport distances and perform the mapping of the airport locations within *Mathematica*.

For the sake of aesthetics and clarity, we have collapsed cells containing code related to the display of tables and figures, which appear in their final form.

Execution of the last section of the article may take 4–8 minutes, given the number of minimization problems that are calculated. Also, earlier versions of *Mathematica* may not execute the `GeoListPlot` commands properly (or at all for Version 9.0 or lower), given that this functionality was added in Version 10.0 and several options were added between Versions 10.0 and 10.0.2.

For the sake of brevity, the datasets are placed here in collapsed cells, along with several house-keeping functions for displaying tables and figures. Some of the data for intra-airport distances is retrieved by using the `AirportData` function and requires internet access if executing the

notebook again.

[collapsed cells]

Classical and metric MDS methods rely on a symmetric input matrix (referred to as a proximity matrix) that specifies the -dimensional distance between pairs of objects and , denoted . Proximity may be defined as a distance or dissimilarity.

In some applications, is composed of distances that are specified directly, for example, the flying distances between airport pairs in the United States, as shown in Table 1. The proximity matrix is a symmetric distance matrix with the airport names in the first column and the corresponding airport abbreviations for the column labels.

Often, is not the primary data but must be calculated from a set of cases, each of which is described by a vector of attributes, with the primary data specified in an cases by attributes data matrix. The attributes can be binary membership in a category or group, physical properties, or string sequences. The methods used to calculate vary by the data and the application. The most commonly used metric is the Euclidean distance for continuous variables, but other distance functions may also be used, such as the Hamming and Manhattan distances, Boolean distances for Boolean data, Smith–Waterman string sequence similarity, or color distance functions for images.

Table 2 is an example of such a cases by attributes matrix. The cases (rows) represent the influenza virus surface hemagglutinin proteins from various influenza strains (H1, H3, H5) and substrains (e.g. A/California/04/2009). The data is taken from an antibody binding assay, with the values reflecting the amount of antibody binding to each influenza hemagglutinin protein. The antibodies were isolated from the serum of ferrets infected with an influenza virus strain, shown in the columns. Each case (row) is characterized by a 15-dimensional attribute vector, with each attribute corresponding to the serum antibody binding values against that particular influenza strain hemagglutinin from a biological assay. Higher values indicate more antibody binding. Such assays are used to select vaccines each year against strains predicted to circulate in the fall. Note that there are 15 different cases characterized by 14 sera.

Before performing the MDS calculations, it is important to rescale each attribute to span the same range across all the cases, especially if the attributes have different units or variable ranges. This rescaling does not assume any specific statistical distribution. It is designed so that each attribute has equal weight in the proximity calculation, although weighted proximity matrices can also be used if the application requires this. Here we rescale the results for the above attributes so that the range of each attribute column (antibody binding test result for a single hemagglutinin) has a minimum of 0 and a maximum of 1000.

For multidimensional scaling, this data is transformed into a proximity matrix using a `EuclideanDistance` calculation. The larger the proximity measurement, the greater the difference between the hemagglutinin proteins from the two viral infection strains, as specified by the reactivity of immune sera against the probe hemagglutinin proteins. Table 4 uses the `DistanceMatrix` function to create a proximity matrix for the influenza. There are many ways to calculate distance matrices, and custom functions are often used. One convenient option is to use the `DistanceMatrix` function, which allows access to a number of standardized distance metrics (e.g. `EuclideanDistance`, `ManhattanDistance`, `CorrelationDistance`, etc.), which can be directly set by the option `DistanceFunction`. The `DistanceMatrix` function is contained in the `HierarchicalClustering` package.

It is often helpful to visualize a proximity matrix as a heat map using the `ArrayPlot` function. In Figure 1 the most similar hemagglutinin protein pairs are blue and the most different are red. As can be seen, this representation of the data suggests a similarity of influenza virus hemagglutinin proteins from similar strains (H1, H3, H5) and the occurrence of year-to-year differences, especially for the

H1 strains.

**Figure 1.** Heat map of proximity matrix data.

Another common application of multidimensional scaling is to compare protein or DNA sequences. Such sequences are essentially linear lists of strings that specify the protein or nucleic acid composition. Sequence differences are used to estimate molecular evolution, how the immune system may respond to a new virus after immunization, and cross-species comparison of molecules.

To construct a protein sequence proximity matrix, we begin with the amino acid sequences of the hemagglutinin proteins from influenza strains specified in the proximity matrix of the previous section. These were retrieved from the UniProt database (www.uniprot.org). Each letter in the sequence represents an amino acid, and each sequence is approximately 566 amino acids long.

To calculate the proximity matrix, we use a rescaled inverse of the Smith–Waterman similarity calculation, with the PAM70 similarity rules for amino acid comparison. Sequence similarity calculations have larger values with greater similarity, and the minimum score is generally the length of the sequence. Because MDS methods use proximity measures that increase monotonically with dissimilarity, the inverse of the sequence similarity function is used. Thus, for clarity in this example, we have transformed the result by subtracting a constant (528), which is one less than the minimum sequence length, and multiplying the inverse by a scale factor of . Note that nonzero values do

not occur.

The first step in multidimensional scaling is to create a proximity matrix. The value of each matrix element is a distance measure; the greater the difference between the two elements being compared, the greater the value of the metric. We have demonstrated three different cases of proximity matrix construction, including: (i) the airport distance case, where the proximities are known *a priori* and do not need to be calculated; (ii) the influenza protein antibody reactivity case, with proximity calculation from a cases by attributes matrix; and (iii) the influenza protein sequence example, with calculation from a sequence similarity measure. Heat maps are useful for displaying proximity matrices, giving a visual view of similarities. All the resulting proximity matrices can be used for MDS calculations, as will be discussed next.

Here we briefly describe the mathematics of classical multidimensional scaling, which uses a linear coordinate transformation calculation to reduce case coordinates from dimensions down to two or three dimensions. The approach begins with calculating a proximity matrix. The orientation of the new coordinate system is such that it accounts for the greatest amount of variance between cases, as defined by attributes, in the reduced number of dimensions. The new coordinates can be used to plot the relative locations of the data points and help visualize differences between data points and clusters of similar cases. The proximity matrix can also be used for data clustering and heat mapping, which provide alternate visualizations of case relationships.

Let be a matrix, where is the number of individual cases (e.g. data points) and is the number of measured variables or attributes (e.g. financial variables, consumer product attributes, antibody binding measurements, etc.), which are used to discriminate between each case , .

If the data attributes are of different scales and units, they are generally rescaled with a range of . This assures that each attribute contributes equally to the proximity measure.

Alternatively, may remain in the original units if each attribute has the same range and units, or if a weighted proximity matrix is desired. Next, let the proximity matrix of or rescaled be a matrix of dimensions with elements

(1) |

One can use a variety of applicable proximity measures (e.g. Euclidean, Mahalanobis, etc.) to create the proximity matrix. If the proximity measure is a Euclidean distance, then classical MDS is equivalent to principal component analysis (PCA). If any other distance function is used, then classical MDS will result in a different transformation that is not equivalent to PCA.

Let be the identity matrix of dimensions and be the constant matrix of dimensions , where . Let be the matrix of the squared proximities:

(2) |

The kernel matrix is derived by first calculating the matrix from the identity matrix and the number of variables :

(3) |

is the matrix that double-centers the input matrix by subtracting the row and column means from every element of the matrix. Introducing the constant creates an equivalence to PCA if is a Euclidean distance function:

(4) |

The eigenvalues and eigenvectors of are calculated. To reduce to dimension with , only the first of a ranked list of the and are selected (e.g. for a two-dimensional mapping).

The diagonal matrix of the square roots of the eigenvalues is

(5) |

The matrix of relative coordinates in dimensions is then calculated by the product of the matrix of eigenvectors and the diagonal matrix of the square roots of the eigenvalues :

(6) |

We begin with one of the original motivating problems related to MDS, reconstructing the relative locations between cities in two dimensions, knowing only the relative distances between each one. We use the United States airport distance data from Section 3 and reconstruct the cities’ relative geographic positions. We begin by squaring the distance functions and calculating the scale of the dataset.

Next, we create an identity matrix, with all elements being 0 except the diagonal elements, which are 1, and a constant matrix with all elements equal to 1. Both matrices are of the same dimensions and are used in subsequent calculations to transform data from 28 down to two dimensions.

We then calculate the kernel matrix and double-center the new -dimensional coordinates.

With the number of dimensions set to 2, the matrix eigenvectors and eigenvalues are calculated. The sign of the first eigenvector is adjusted to lay out the cities from west to east. (*Mathematica* does not guarantee the direction of an eigenvector.)

The transformation matrix `dMTX` is calculated from a diagonal matrix formed by the eigenvalues `Ev`, and the coordinate matrix `EmM` is calculated from the eigenvectors `Em`.

The graph coordinates `cMDScoord` are calculated and projected to two dimensions.

Finally, we plot the two-dimensional airport locations in `cMDScoord`, which shows the airports in the new coordinate system.

**Figure 2.** Classical MDS display of relative airport locations.

Classical multidimensional scaling gives a solution that has an excellent projection of the cases in a lower dimension, preserving the between-cases distances from dimensions. It is important to note, however, that the solution is not unique with respect to the rotation and reflection of the visualization. It is often the case when one knows the “true” orientation of the case locations (e.g. the previous map example), that classical (and metric) MDS methods yield a visualization that is reflected or rotated. In these cases, use of other boundary conditions (fixing known coordinates) helps constrain the orientation of the solution. Often, however, such boundary conditions are not available.

Given the above solution, how well does classical MDS reproduce the airport coordinates? Because we have a gold standard, the actual airport locations, comparison of the coordinate estimates is possible. However, the airport locations are represented in a geographic coordinate system, making a comparison of calculated versus actual airport locations a bit more complicated. One approximate way to approach this issue is by first examining the correlation between the actual airport locations, in longitude and latitude coordinates, and the derived coordinates from the classical MDS calculation, using the `PearsonCorrelationTest`. The units are different, which we will address in a moment, but this does not affect the Pearson correlation test.

**Figure 3.** Comparison of actual and estimated coordinates.

While the correlations are high, they are not perfect, as we can see by the off-diagonal locations of some of the points. To better visualize these differences, we transform the classical MDS coordinates into longitude and latitude units. This is a transformation that uses the pseudoinverse matrix to translate and rotate to find a good alignment between the two coordinate sets.

This type of transformation preserves the relative distances between the classical MDS derived city locations and thus provides a relatively accurate estimate of how close the estimates are to the actual city locations. A more accurate solution would involve mapping of the coordinates to an appropriate spherical projection system; however, this is outside the focus of this article.

The transformation coefficients can now be used to transform the classical MDS derived coordinates into `GeoPosition` values. To compare the actual (●) with the cMDS calculated (◆) airport locations, we use the `GeoListPlot` function, which plots both sets of points on a map of North America.

**Figure 4.** Comparison of airport geo locations.

The accuracy of the classical MDS estimates is fairly good, but not perfect. Part of this error is due to the approximate transformation from arbitrary MDS coordinates to geographic coordinates, and the complex projection to a flat map.

The preceding code can be summarized in a succinct *Mathematica* function for classical MDS calculation. This function takes a proximity matrix as input and returns coordinates mapped to a space of the specified number of dimensions. The default number of dimensions is two, but the function also works with three or more dimensions.

It is important to note that if the proximity matrix used for classical MDS is formed from a cases by attributes matrix using a Euclidean distance, then classical MDS is equivalent to principal component analysis. Principal component analysis is a specific case of the more general singular value decomposition method (SVD). Details of the SVD mathematics and computational method is not discussed in detail here, but rather we will outline the computational steps necessary to use the method with the function `SingularValueDecomposition`.

As with classical MDS, a transformation matrix is calculated, and the data is centered to a mean of zero.

The SVD function is then applied to the normalized data and yields three matrices. Here `ww` is the diagonal matrix of eigenvalues and `vv` is the coordinate matrix.

Next, the two-dimensional projection of the coordinates is calculated.

When the dataset to be analyzed is very large, then the computational efficiency degrades quickly if the proximity matrix must be directly calculated. In this case, a matrix transformation can be applied to avoid a cell-by-cell proximity matrix calculation; this greatly increases computational efficiency. The computational efficiency for both classical and metric MDS is discussed, with a direct comparison, in Section 7.

Classical MDS is a data mining and data exploration method allowing dimensional reduction to be used to highlight possible clusters, similarities, or differences between cases described by high-dimensional attribute vectors. It is a starting point for more rigorous statistical analysis and hypothesis testing. When the input proximity matrix is composed of Euclidean distances, classical MDS is equivalent to both principal coordinate analysis and singular value decomposition. Computational efficiency and possible limitations of classical MDS methods are discussed in Section 7.

Metric multidimensional scaling is a second type of MDS. While classical MDS relies on matrix algebra, metric MDS involves computational minimization of the difference between -dimensional and two-dimensional proximity measures (e.g. Euclidean distance) of the case coordinates in each dimensional system. In essence, the method attempts to minimize the total error between actual and reduced intercase distances for the group of cases as a whole. Metric MDS is also flexible, accommodating different types of proximity metrics (e.g. not Euclidean), as well as different stress functions (nonlinear transformation and distance metrics other than Euclidean) for minimization. This permits nonlinear dimensional reduction, where the projection of data from an -dimensional space to lower dimensions can be done using a nonlinear transformation. The end result is a dimensional reduction that allows for data visualization.

As in the previous section, we first describe the mathematics with some references to implementation, then demonstrate a practical example.

Metric MDS also begins with a proximity matrix , which represents the original -dimensional proximity measure for the cases , where . This proximity matrix can be given *de novo*, as in distances between airports, or can be calculated from a cases by attributes matrix as described previously,

(7) |

Initial values for the case coordinates, used to seed the minimization algorithm, can be generated at random or calculated using values from the classical MDS algorithm. So we next set up the initial coordinate matrix for a two-dimensional projection, for the minimization algorithm,

(8) |

where . Next, we create a matrix of variables representing the coordinates that the minimization algorithm estimates:

(9) |

We then calculate a matrix of distance functions for using Euclidean distance between any two elements and ,

(10) |

The distance matrix is now given by

(11) |

For the minimization, several different types of stress functions may be used. Here we select the stress function to be the sum of the squares of the errors (SSE):

(12) |

We use the efficient and robust `"DifferentialEvolution"` method, with sufficient search points to avoid local minima. Some aspects of the choice of minimization methods are discussed later.

This example is motivated by the need to determine if an influenza strain is sufficiently different, as determined by the immune system, so that even if a person has prior immunity from vaccination against other similar strains, it will be insufficient to protect against the new strain. Influenza vaccination relies on the immune system’s ability to generate protective antibodies, which are proteins that bind to the virus and block its ability to infect cells in the respiratory system. The major protection comes from antibodies that bind the viral surface protein hemagglutinin, the protein responsible for attaching the virus to the target cells. Variations in the hemagglutinin protein structure between influenza strains allow the viruses to evade the immune system and cause an infection. This is why the influenza vaccine composition is changed every year. Figuring out which strains to include in the influenza vaccine is an annual problem for the international organizations that recommend changes in the seasonal influenza vaccine.

Metric MDS provides a graphical way of visualizing influenza strain similarity, derived from experiments measuring the ability of serum from animals infected with influenza to bind to the target virus hemagglutinin. There has been extensive literature over the last decade on the use of metric MDS for this purpose, referred to as antigenic cartography [8–11]. Metric MDS was chosen for dimensional reduction and visualization by several groups, as classical MDS methods could not be easily adapted to solve several issues [8, 9]. These included the need for complete datasets [10, 11], where in some cases data was missing due to experimental considerations. Metric MDS methods could be adapted to impute relationships [10]. In addition, metric MDS was viewed as a more accurate estimator of influenza strain antigenic distance due to correlations with the binary logarithm of the hemagglutinin inhibition assay serum titers [8, 9]. Use of metric MDS continues in the influenza literature [10, 11], although newer methods of measuring antibody reactivity do not have the same issues as older assays, and classical MDS could be used to the same end. To illustrate the method of metric MDS, however, we will use it in this example. We discuss method selection in Section 7, as well as considerations of computational efficiency.

We begin with the proximity matrix derived in the previous section from the antibody reactivity to influenza hemagglutinin data. We next calculate the relative positions of each influenza strain with respect to the entire set by minimizing a stress function. The `Array` function is used to create an array of variables that specify the coordinates for each case as a 2D set of points.

Each `sr` variable pair will hold a pair of coordinates in two-dimensional space. Finally, we flatten the list of variable pairs to input into the function `NMinimize`.

The next section of code creates a series of Euclidean distance calculations for the distances between all combinations of case locations, using the estimated coordinates for each point. For succinctness, only one function within the full matrix is displayed.

The stress function is a very large least-squares expression.

The first nonzero element of the stress function is shown here. The reader may expand the formula to see the entire function, if desired.

Finally, we must choose the method for minimizing the stress function. We discuss two options, `FindMinimum` and `NMinimize`. A known issue with metric MDS is the existence of local minima, and thus identifying a global minimum cannot be guaranteed [7]. While `FindMinimum` may be computationally more efficient, it also lacks the ability to specify the number of search points and is prone to finding local minima instead of global minima. These issues are discussed in some detail in Section 7. In contrast, the `NMinimize` function allows specification of `"SearchPoints"` to address this issue and may be substantially more robust. For these reasons, we chose `NMinimize` to optimize the stress function.

The viral strains are then plotted in the two-dimensional space and are color-coded for clarity (H1, H3, and H5). Note that the coordinate system is arbitrary, in the sense that what is preserved and important are the relative distances between the data points. Thus, we have omitted the axes, which have the same scale in each dimension.

**Figure 5.** Relative antigenic distances for influenza strains.

Examining the plot, it is immediately apparent that the influenza strains fall into four distinct clusters. Two of these correspond to major hemagglutinin protein strains (H3, H5). In addition, we can see differences between temporally distinct strains in the H1 influenza strains, giving perhaps two clusters. For example, note the antigenic distance between the pandemic A/California/07/2009 (CAL709), A/Utah/20/2009 (UT2009), and A/Mexico/41/2009 (MEX4109) strains and the other H1 influenza strains. This likely reflects molecular mutations in the hemagglutinin proteins for the 2009 strains. These mutations resulted in decreased binding of antibodies from ferrets infected with earlier influenza strains. This finding was consistent with the decreased population immunity observed in humans to A/California/07/2009 and demonstrates the pandemic nature of that particular influenza strain.

Given that the influenza strains seem to cluster together, what is the relationship between MDS and methods of unsupervised data clustering used in data mining? The answer lies in the use of the proximity matrix, which is used by both hierarchical and agglomerative clustering methods to determine relatedness. This relationship was noted by the developers of the MDS methods [1, 2, 8]. To briefly demonstrate, we apply hierarchical clustering to the same proximity matrix `Dm`, using the `DirectAgglomerate` function. Ward’s minimum variance method is used for determining cluster linkage.

The resulting dendrogram is displayed in Figure 6. Note the grouping of the different virus types (H1, H3, and H5) based on the reactivity of ferret serum after infection with a single virus and the resulting antibodies against the hemagglutinin proteins.

**Figure 6.** Dendrogram of influenza virus relationships.

To explore which viruses are antigenically closer to each other, a heat map of the relative distances from the calculated coordinates (or the original proximity matrix) can then be created with `DistanceMatrix` and `ArrayPlot`. Note that more antigenically similar influenza strains have smaller distances between each other.

**Figure 7.** Heat map of proximity matrix data.

As in the previous section, we end with a general function for the metric MDS calculation. The function requires several input variables, including precision, accuracy, the minimization method, and the maximum number of iterations. This function uses the classical MDS routine for the initial coordinate estimates. Some suggested defaults are given in the function definition to be passed to `NMinimize`. The metric MDS function is written for two-dimensional mapping that could easily be generalized for three dimensions.

One of the advantages of metric MDS methods is the ability to use nonlinear mappings from -dimensional to visualizable space. We now use this function to create an MDS map comparing the influenza hemagglutinin proteins by sequence similarity. Recall that the proximity measure for sequence comparison is the Smith–Waterman sequence distance, which is not a Euclidean distance function. Thus, this violates the assumptions of classical MDS and makes metric MDS the appropriate method to use, albeit at computational cost. Taking the hemagglutinin protein proximity matrix, `fluPm`, we apply the `metricMDS` function and obtain the coordinates. We also apply hierarchical clustering to the proximity matrix `fluPm`.

One of the advantages of metric MDS methods is that one can use nonlinear stress functions to emphasize particular regions of data relationships. One of the first nonlinear metric MDS methods was that of Sammon’s mapping [12, 13]. Sammon’s mapping can be useful in revealing underlying data structure or differences with nonlinear relationships. This mapping minimizes the following general nonlinear form of stress functions:

(13) |

where is a weighting function and is a constant. In the case of Sammon’s mapping, the stress function is defined by , , with specified as [12]:

(14) |

Many related nonlinear mappings exist of the same form [12, 14]. For the purposes of this example, we explore applying a nonlinear exponential MDS mapping with a stress function defined by , , , , and :

(15) |

Note that is an empirically specified tuning factor. This nonlinear mapping function decreases the contribution to the overall stress function of larger and has the effect of expanding the mapped distances between data points with smaller . The advantage of this mapping is that the weight of any point in the minimization is inversely proportional to its magnitude. Thus, smaller differences between data elements are spread out. The coding of the exponential MDS function lets you specify .

To demonstrate, we apply the metric nonlinear MDS method to the above `fluPm` dataset of protein sequence comparisons. First, we apply the standard metric MDS function defined in the previous section, `metricMDS`, and generate a dendrogram to highlight the sequence differences.

**Figure 8.** Influenza virus relationships from protein sequences.

While the major influenza viral subtypes (H1, H3, and H5) all cluster together, minimal differences can be observed with respect to sequences differences within each subtype. Using the nonlinear mapping, as shown in Figure 9, accentuates the small differences between the hemagglutinin protein sequences. Note the negative exponential weighting function, with the large distances between hemagglutinins being given less weight in the minimization. We are now able to visualize the division of the H1 influenza hemagglutinins into two major clusters and the split between clusters of the H3 substrains within the visualization. It is worth noting that the hierarchical clustering may capture these differences.

**Figure 9.** Weighting function and nonlinear mapping of protein sequences.

Metric multidimensional scaling is a more flexible method compared to classical MDS. While computationally less efficient, it allows nonlinear dimensional reduction that is not possible with the SVD or PCA method of classical MDS. This functionality can be used to highlight possible clusters, similarities, or differences between cases described by high-dimensional attribute vectors and to weight or penalize the stress function based on data attributes. As with classical MDS, complementary visualization methods allow for a fuller picture of case differences but must be carefully interpreted when nonlinear stress functions are used.

From a computational perspective, several features of classical and metric MDS methods should be considered when selecting which method to use for a specific analysis.

Classical MDS is computationally straightforward but has some subtle mathematical constraints. For a solution to exist in Euclidean space, the distance matrix must follow a set of necessary and sufficient conditions [1, 15]. In practice, this is rarely an issue. When it is an issue, it often manifests by a computational result expressed in imaginary numbers. If, however, the proximity matrix is composed of Euclidean distances, classical MDS becomes principal component analysis, and the support vector machine (SVM) function can be used. This is much more computationally efficient, as we demonstrate in Section 7.3, and is the preferred method of MDS for visualization of most data.

Metric MDS has somewhat more relaxed constraints than classical MDS. For very high numbers of initial dimensions , metric MDS is computationally and memory intensive. This is primarily due to the need to use iterative optimization to minimize the stress function, whereas classical MDS uses more efficient matrix algebra operations. In addition, one needs to pay attention to the algorithms used to perform metric MDS to ensure that minimization finds global rather than local minima.

We next examine the selection of minimization algorithms for metric MDS, compare the computational efficiency of classical and metric MDS methods, and discuss the particular circumstances where metric MDS may be the method of choice.

With metric MDS, selecting a minimization algorithm with an appropriate `Method` setting for *Mathematica*’s built-in function `NMinimize` may require some investigation. In addition, a known issue with metric MDS is the existence of local minima, so that a global minimum cannot be guaranteed [7]. Using a large number of starting points or selecting the appropriate minimization function and method may help. In this article we have used `NMinimize` with `Method` set to the `"RandomSearch"` option. We also specified sufficient search points to avoid local minima. This can be an important issue when no other external data is available to constrain the solution and when it is not known if other minimization methods will routinely converge to a solution.

It is worth exploring this tradeoff between convergence, accuracy, and computational efficiency. For example, the execution time of `NMinimize` increases with the number of initial search points. The next example uses `NMinimize` on the influenza antibody dataset distance matrix `Dm` and the method `"RandomSearch"`. In this case, however, the number of search points appears to make little difference in the residual after optimizing the stress function.

**Figure 10.** Efficiency of `NMinimize` (`RandomSearch`).

In the case of `RandomSearch`, there appears to be no tradeoff with this particular dataset. In contrast, the theoretically more robust setting finds a small number of varying minima, which improves after specifying a larger number of search points. Also note the improved execution times, which are an order of magnitude less than with `RandomSearch`.

**Figure 11.** Efficiency of `NMinimize` (`DifferentialEvolution`).

Another consideration is the speed of computation. The `FindMinimum` function is often much faster than `NMinimize`, but the tradeoff is the risk that the algorithm will not converge or that it will find local instead of global minima. For the influenza data, `FindMinimum` can suffer from both nonconvergence and local minima, although the former issue may be sporadic. The sporadic nature of convergence can be frustrating and problematic. The issue of finding local minima is potentially more serious. To explore this further, we minimized 300 times with `FindMinimum`.

The results are displayed as a histogram in Figure 13. We found that a number did not converge with the influenza antibody-reactivity dataset. Of even more concern is that `FindMinimum` was not consistent, and often appeared to provide local rather than global minima. This may in some cases be dataset specific. The starting points selected, here a `RandomReal` number between and for each coordinate, are also critical for `FindMinimum`.

**Figure 12.** Residuals and convergence using `FindMinimum`.

The convergence can be improved by minimizing the sum of the squares rather than the Euclidean distance. Note that the minimum residual is higher than when minimizing the sum of the differences between distances rather than the square of the difference between distances. This may be due to computational considerations, in that there is differentiability at the optima when minimizing the sum of the squares of the distances, while this is lacking if minimizing the difference between the distances. Still, a moderate number (2% to 15% from 300 attempts) of the solution attempts did not converge or find global minima. Whether this is an issue for your particular application depends on both the level of precision required and the computational efficiency required.

**Figure 13.** `FindMinimum` and minimizing the sum of the squares.

Given the low percentage of nonconvergence and the presence of local minima, one could execute `FindMinimum` several times and take the results that converge with the lowest minimum. This does increase the time for execution. An alternative is to use the function `NMinimize`. The tradeoff is an increased computational time for consistent convergence on global minima, as shown here. This demonstration takes several minutes to perform on this dataset, even with parallelization.

**Figure 14.** Residuals and convergence using `NMinimize`.

A single minimum residual is obtained for all 300 separate runs of `NMinimize`. Thus, unless local minima are known not to be an issue, `NMinimize` may be a more consistent minimization method for metric MDS routines.

The issues of optimization and computational efficiency make it useful to benchmark classical and metric MDS methods with respect to execution time. Therefore, we compared SVD with classical and metric MDS across the three example datasets. For metric MDS, we include both `FindMinimum` and `NMinimize`. The measured execution times encompass all the steps from the input of the proximity matrix through final coordinate output, but not plotting. The previously created `classicalMDS` and `metricMDS` functions defined previously were used, along with new functions created for singular value decomposition (`mdsSVD`) and metric MDS using `FindMinimum` (`findMinMDS`).

Note that `FindMinimum` failed to converge for the airport dataset but did converge for both the flu antibody and sequence datasets, a behavior that may vary between executions. Timings are shown here.

Overall, there is a speed advantage to using classical MDS and SVD, which are several orders of magnitude faster than metric MDS using either `FindMinimum` or `NMinimize`. This advantage assumes that the proximity matrix can be formulated using a Euclidean distance, and thus other measurements (e.g. Manhattan distance, etc.) may require other methods. It also assumes that you require a linear mapping from the -dimensional space to the visualizable space. SVD has the advantage that, for many applications, the proximity matrix may not have to be directly calculated, saving additional memory and computational time [8]. With respect to metric MDS, there are also tradeoffs. For moderately large datasets, if one can optimize the conditions to avoid nonconvergence and local minima, `FindMinimum` is a good choice. Note that the execution time for `FindMinimum` in the airport and influenza antibody dataset examples occurred in the setting of failure to converge. Although slower, metric MDS using `NMinimize` had no issues with nonconvergence or finding local instead of global minima.

Given the computational advantages of classical MDS in its various forms (principal component analysis, singular value decomposition, etc.), why would you choose metric MDS, which requires numerical optimization? For those wishing an in-depth and lucid explanation, we refer you to the initial work of Gower [15] and the outstanding article by Shiens [16]. In short, classical MDS methods are usually preferable to and always computationally more efficient than metric MDS methods. The method is computationally efficient and robust; there are very few cases where classical MDS methods fail to provide a visualization. For some datasets, classical MDS methods may not provide optimal visualization, and metric MDS should be considered. Some examples include datasets whose variables have a non-Gaussian distribution, those where the distance metric is not Euclidean, those requiring imputation of missing data [10, 11], or cases with a parametrization in a nonorthogonal coordinate system [16]. For nonlinear dimensional reduction, which is often used in graph layout algorithms, metric MDS with stress function minimization is the method of choice [14].

Thus, selection of an MDS method for visualization should consider several factors. If the distance function is Euclidean, and especially if the dataset is large, SVD or classical MDS are the most appropriate and computationally efficient methods. In the less common case where metric MDS is used, careful consideration should be given to the choice of stress function minimization method (in this example `NMinimize` versus `FindMinimum`) to avoid local minima.

We have demonstrated the application of two methods of multidimensional scaling, classical and metric, for the visualization of similarity/proximity of high-dimensional data, with reduction to two or three dimensions. We have shown how these methods can be used to visualize the relatedness of influenza virus strains with respect to antibody-mediated immunity, as well as their utility in reconstructing relative spatial-geographic locations using only interlocation distances. These MDS methods are, however, quite generalizable. Both classical and metric MDS rely on a proximity matrix. While the examples in this article use continuous variables and sequences or case attributes, a variety of data types with appropriate proximity metrics can be visualized with MDS methods, such as molecular DNA or protein sequence data (Smith–Waterman similarity, Needleman–Wunsch similarity, or Damerau–Levenshtein distance), Boolean data (Hamming distance), and images (image distance, color distance). For cases with a single data attribute (e.g. sequence similarity, distance), no data scaling is necessary. For cases with multiple attributes having disparate units, standardization (e.g. -score) and rescaling are needed to equally weight each attribute.

In our examples, we used reduction down to two dimensions for multidimensional data visualization. Reduction to three dimensions is easily accomplished with both classical and metric MDS, although the computational cost may increase with the added dimension, depending on the method used. While we have emphasized data visualization, the classical MDS method can also be used for reduction to four or more dimensions as a method of identifying the weighted combinations of components that contribute the most to the data variance. In these cases, the goal is to select a small set of variables that explain the most variance in the dataset, such that this minimum set of variables can be further used for statistical or other modeling.

We have also shown how MDS methods are related to clustering algorithms, which also use proximity matrices to compare and classify cases by their attributes. This relationship also allows creative graphical display of multidimensional data from several vantage points. For example, one can use the MDS plot to display the relative proximity of cases to each other and plot marker coloring or other methods to add information regarding other case attributes. Some caution is in order, however, as different proximity measures and data transformations may give different clustering and classification. Parallel display of dendrograms and heat maps may also enhance understanding of the relationship of data clusters to each other. Similarly, heat maps, combined with MDS displays, are particularly helpful for data exploration, in that they enhance visual identification of those data attributes (dimensions) that contribute the most to differentiating between case clusters.

Care should be taken when selecting the MDS method. In most cases, classical MDS will be the most computationally efficient method, especially for very large datasets. In the cases where metric MDS is optimal, such as the use of nonlinear mapping, care should be taken to choose a minimization method that is robust and avoids local minima. Performing some testing on a subset of the data can be very informative regarding convergence, accuracy, and computational efficiency. While we did not discuss in detail how constraining optimization problems can improve computational efficiency and accuracy, this should also be considered whenever boundary conditions or other information is available.

Finally, one must remain aware that these methods reveal only associative patterns. Further analysis with rigorous statistical inference methods is needed to test the validity and specify the error boundaries of these associations. Mechanistic studies should be performed, if possible, to confirm suspected causal relationships. Overall, however, MDS methods are excellent for dimensional reduction and data exploration with the goal of creating comprehensible and informative quantitative graphical representations of multidimensional data.

We would like to thank Ollivier Hyrien, Alex Rosenberg, Chris Fucile, and Melissa Trayhan for their suggestions and critical reading of the manuscript. We would also like to acknowledge the reviewer at the *Mathematica Journal*, who suggested several improvements and additions, including the evaluation of execution time and accuracy for the MDS methods, and the discussion regarding the relative merits of `SingularValueDecomposition`, `NMinimize`, and `FindMinimum`. This work was supported by grants to the authors from the Philip Templeton Foundation, as well as the National Institute of Allergy and Infectious Diseases and the National Institute of Immunology, grant and contract numbers HHSN272201000055C, AI087135, AI098112, AI069351.

[1] | J. C. Gower, “Some Distance Properties of Latent Root and Vector Methods Used in Multivariate Analysis,” Biometrika 53(3–4), 1966 pp. 325–338.doi:10.1093/biomet/53.3-4.325. |

[2] | W. S. Torgerson, “Multidimensional Scaling: I. Theory and Method,” Psychometrika, 17(4), 1952 pp. 401–419. doi:10.1007/BF02288916. |

[3] | W. S. Torgerson, Theory and Methods of Scaling, New York: Wiley, 1958. |

[4] | T. F. Cox and M. A. A. Cox, Multidimensional Scaling, Boca Raton: Chapman and Hall, 2001. |

[5] | I. Borg, P. Groenen, Modern Multidimensional Scaling: Theory and Applications, 2nd ed., New York: Springer, 2005 pp. 207–212. |

[6] | M. Wish and J. D. Carroll, “Multidimensional Scaling and Its Applications,” Handbook of Statistics, Vol. 2: Classification, Pattern Recognition and Reduction of Dimensionality (P. R. Krishnaiah and L. N. Kanal, eds.), Amsterdam: North-Holland, 1982 pp. 317–345. |

[7] | V. Saltenis, “Constrained Optimization of the Stress Function for Multidimensional Scaling,” in Computational Science–ICCS 2006 (V. Alexandrov, G. van Albada, P. A. Sloot, and J. Dongarra, eds.), Berlin: Springer, 2006 pp. 704–711.link.springer.com/chapter/10.1007/11758501_ 94. |

[8] | D. J. Smith, A. S. Lapedes, J. C. de Jong, T. M. Bestebroer, G. F. Rimmelzwaan, A. D. M. E. Osterhaus, and R. A. M. Fouchier, “Mapping the Antigenic and Genetic Evolution of Influenza Virus,” Science, 305(5682), 2004 pp. 371–376. doi:10.1126/science.1097211. |

[9] | T. Bedford, M. A. Suchard, P. Lemey, G. Dudas, V. Gregory, A. J. Hay, J. W. McCauley, C. A. Russell, D. J. Smith, and A. Rambaut, “Integrating Influenza Antigenic Dynamics with Molecular Evolution,” eLife, 3:e01914, 2014. doi:10.7554/eLife.01914. |

[10] | Z. Cai, T. Zhang, and X.-F. Wan, “Concepts and Applications for Influenza Antigenic Cartography,” Influenza and Other Respiratory Viruses, 5(Suppl. s1), 2011 pp. 204–207. www.ncbi.nlm.nih.gov/pmc/articles/PMC3208348. |

[11] | D. W. Fanning, J. A. Smith, and G. D. Rose, “Molecular Cartography of Globular Proteins with Application to Antigenic Sites,” Biopolymers, 25(5), 1986 pp. 863–883. doi:10.1002/bip.360250509. |

[12] | S. Lespinats and M. Aupetit, “False Neighbourhoods and Tears are the Main Mapping Defaults. How to Avoid It? How to Exhibit Remaining Ones?,” QIMIE/PAKDD 2009 (PAKDD Workshops, Thailand), Bangkok, 2009 pp. 55–64.conferences.telecom-bretagne.eu/data/qimie09/lespinats_aupetit_QIMIE _ 2009.pdf. |

[13] | J. W. Sammon, “A Nonlinear Mapping for Data Structure Analysis,” IEEE Transactions on Computers, C-18(5), 1969 pp. 401–409. doi:10.1109/T-C .1969.222678. |

[14] | L. Chen and A. Buja, “Stress Functions for Nonlinear Dimension Reduction, Proximity Analysis, and Graph Drawing,” Journal of Machine Learning Research, 14(1), 2013pp. 1145–1173. dl.acm.org/citation.cfm?id=2502616. |

[15] | J. C. Gower, “A Comparison of Some Methods of Cluster Analysis,” Biometrics, 23(4), 1967 pp. 623–637. doi:10.2307/2528417. |

[16] | J. Shiens, “A Tutorial on Principal Component Analysis.” arxiv.org/pdf/1404.1100.pdf. |

M. S. Zand, J. Wang, and S. Hilchey, “Graphical Representation of Proximity Measures for Multidimensional Data,” The Mathematica Journal, 2015. dx.doi.org/doi:10.3888/tmj.17-7. |

Martin S. Zand is a professor of medicine and director of the Rochester Center for Health Informatics at the University of Rochester. His research includes the application of informatics, graph theory, and computational modeling to vaccine immunology, gene regulatory networks, and health care delivery.

Jiong Wang is a molecular virologist and research assistant professor at the University of Rochester. She works on influenza vaccination responses and developing high-dimensional measurements of vaccine immune responses.

Shannon Hilchey is a cellular immunologist and research associate professor at the University of Rochester. He studies human immune responses to vaccination, hematologic cancers, and organ transplants in human subjects.

**Martin S. Zand, MD, PhD**

University of Rochester Medical Center

601 Elmwood Avenue – Box 675

Rochester, NY 14618

*martin_zand@urmc.rochester.edu*

**Jiong Wang, PhD**

University of Rochester Medical Center

601 Elmwood Avenue – Box 675

Rochester, NY 14618

*jiong_wang@urmc.rochester.edu*

**Shannon Hilchey, PhD**

University of Rochester Medical Center

601 Elmwood Avenue – Box 675

Rochester, NY 14618

*shannon_hilchey@urmc.rochester.edu*

We propose and implement an algorithm to compute the exact cumulative density function (CDF) of the distance from an arbitrary reference point to a randomly located node within an arbitrarily shaped (convex or concave) simple polygon. Using this result, we also obtain the closed-form probability density function (PDF) of the Euclidean distance between an arbitrary reference point and its neighbor node when nodes are uniformly and independently distributed inside the arbitrarily shaped polygon. The implementation is based on the recursive approach proposed by Ahmadi and Pan [1] in order to obtain the distance distributions associated with arbitrary triangles. The algorithm in [1] is extended for arbitrarily shaped polygons by using a modified form of the shoelace formula. This modification allows tractable computation of the overlap area between a disk of radius centered at the arbitrary reference point and the arbitrarily shaped polygon, which is a key part of the implementation. The obtained distance distributions can be used in the modeling of wireless networks, especially in the context of emerging ultra-dense small cell deployment scenarios, where network regions can be arbitrarily shaped. They can also be applied in other branches of science, such as forestry, mathematics, operations research, and material sciences.

Wireless networks are generally modeled as a finite region in Euclidean space (this article considers those regions that are simple polygons in two-dimensional Euclidean space ) with nodes independently and uniformly distributed throughout the region. The random distances between nodes or users, therefore, significantly affect the modeling of the wireless network, since the received signal power and interference depend critically on the distances between the transmitter and the receiver nodes [2]. As shown recently in [3], when nodes are independently and uniformly distributed within the network, the important distance distribution is the cumulative density function (CDF) of the distance from an arbitrary reference point to a randomly located node within the polygon (in this article, we use the phrase *distance distribution* to denote this CDF distribution). It can be obtained by finding the ratio of the overlap area between a disk of radius centered at the arbitrary reference point and the area of the polygon. This CDF can then be used to obtain the probability density function (PDF) of the Euclidean distance between an arbitrary reference point and its neighbor node when nodes are uniformly and independently distributed inside the polygon.

Recently, there has been increasing interest in wireless communications to model the distance distributions in polygonal regions. In traditional cellular networks, the network region is often modeled as an equilateral triangle, a square, a hexagon, or a disk. For regular -sided polygons, the distance distributions were obtained for the special case when the arbitrary reference point is located at the center of the polygon in [4] and for the general case when the arbitrary reference point is located anywhere inside the polygon in [3]. Note that a *Mathematica* implementation of the algorithm in [4] is available [5]. In emerging wireless network paradigms, such as ultra-dense small cell deployments, the network regions can be arbitrarily shaped. For arbitrarily shaped convex polygons, when the arbitrary reference point is located anywhere inside the polygon, an algorithm to obtain the distance distributions was proposed in [6]. For triangular regions, when the arbitrary reference point is located anywhere, an algorithm to obtain the distance distributions was proposed in [1]. The authors in [1] argued that since any polygon can be triangulated (i.e., broken up into non-overlapping triangles), their approach in principle could be applied to determine the distance distributions for the general case of arbitrary reference point location and arbitrarily shaped (convex or concave) polygons.

In this article, we extend the algorithm in [1] for arbitrarily shaped polygons by using a modified form of the shoelace formula. The shoelace formula, also known as Gauss’s area formula, is a well-known mathematical method to determine the area of a polygon whose vertices are described by ordered pairs in [7]. Our modification of the shoelace formula allows tractable computation of the overlap area between a disk of radius centered at the arbitrary reference point and the arbitrarily shaped polygon, allowing the algorithm in [1] to be generalized and implemented.

This article is organized as follows. In the remainder of Section 1, we briefly summarize the algorithm in [1] and define the commonly used notation. In Section 2, we discuss the shoelace formula and its proposed modification. In Section 3, we present the proposed algorithm and its *Mathematica* implementation. The simulation results, which are used to verify the analytical results, are discussed in Section 4. An example illustrating the proposed *Mathematica* implementation is discussed in Section 5. Finally, conclusions are presented in Section 6.

Calculating the distance distribution (i.e. the CDF) evaluated at is equivalent to finding the ratio of the area within the polygon that is less than a distance from the reference point to the area of the polygon. The latter area is readily calculated if the polygon vertices are known. (Generally the polygon defining the network area has known coordinates, so the area may be calculated.) Hence, the challenge is calculating the former area.

It is clear that the CDF has an obvious geometric interpretation; if we let be a polygon and be the disc of radius centered at some reference point , then the CDF is the area of divided by the area of the polygon. Ahmadi and Pan [1] perform this calculation for an arbitrary triangle and arbitrary reference point by first establishing the case for which the reference point is one of the triangle vertices, as illustrated in Figure 1.

**Figure 1.** Depiction of the two characteristic cases considered by Ahmadi and Pan [1]. (a) shows the case that the altitude from to the side is inside the triangle. (b) shows the case that the altitude from to the side is outside the triangle.

They assume, without loss of generality, that the side length of is less than or equal to the side length of . The possible cases are then characterized by whether the altitude from to the side is inside or outside the triangle, and considered separately (see Figure 1). For a disc centered at , the area of intersection of the triangle and the disc as a function of the radius is derived for each case. The result for the former case is

(1) |

and for the latter case the result is

(2) |

All the symbols in (1) and (2) are defined in Figure 1. The equations (1) and (2) are extended to an arbitrary triangle with an arbitrary reference point using triangle decomposition and adding and subtracting the areas appropriately [1]. The three possible cases are that the reference point is inside the triangle, the reference point is outside the triangle and in the area formed by the extension of two edges from a vertex, or otherwise (see Figure 2). For these three cases, the area is given by

respectively, where the individual intersection areas can be found using (1) and (2) appropriately.

**Figure 2.** Possible cases for triangle decompositions given an arbitrary triangle and reference point. (a) is the case of an interior reference point. (b) and (c) show the case for an exterior reference point. (c) is the case that the reference point is in the area formed by the extension of the edges from one vertex.

Using this result, an algorithm to compute the CDF for the general case of an arbitrary polygon and arbitrary reference point is proposed and implemented. This is achieved by first establishing a modification of the shoelace formula, which is described and proved in Section 2.

In this subsection, we define some functions and notation that are used throughout this article.

**Definition 1**

The area of a region is denoted by . In the case of a triangle with vertices , the area can be calculated as in [8].

(3) |

**Definition 2**

(4) |

(The subscript stands for “signed”.) We note that from the above definition, has the same magnitude as the area of the triangle, , but is positive if the vertices are given in counterclockwise order, and negative if the vertices are given in clockwise order.

**Definition 3**

The signed area of the region defined by the intersection of a triangle and a region , that is, , is denoted and is given by

(5) |

Essentially, this says that , where is positive if the signed area of is positive and negative if the signed area of is negative.

The shoelace formula is a useful formula for calculating the area of a polygon (see the illustration in [6] for an explanation of the name). It is stated in the theorem below [9].

**Theorem 1**

The shoelace formula: Let be an ordered set of vertices in the plane that define an -sided polygon. Define the triangles as the triangles formed from two adjacent points from and the origin. That is,

(6) |

The shoelace formula holds if instead of using the origin to define each , we used an arbitrary point in .

Notice that

Hence, the shoelace formula can alternatively be stated as

(7) |

A visual illustration of (7) is shown in Figure 3. The triangles with positively signed area are shaded in green and shown in (a), and the triangles with negatively signed area are shaded in orange and shown in (b). In both cases, the darker regions indicate where triangles overlap. We can thus think of the shoelace formula as stating that if we add the green regions and subtract the orange regions, we obtain the region defined by the polygon (shown in (c)). In the given example in Figure 3 we can see this visually because the green regions outside the polygon “cancel” with the orange regions outside the polygon, and the dark green regions inside the polygon are “canceled” by orange regions inside the polygon.

**Figure 3. **Visual illustration of the shoelace formula for calculating the area of a polygon.

We now build upon the shoelace formula in (7) to obtain a useful modification.

**Theorem 2**

The modified shoelace formula: Let be an ordered set of vertices in the plane that define an -sided polygon. Let be a disc of radius centered at the point . Define the triangles as the triangles formed from two adjacent points from and the reference point . That is,

(8) |

Thus, just as the area of the polygon was equal to the sum of the signed areas of the triangles in the original shoelace formula, we have that the area of intersection of the polygon and a disc is given by the sum of the signed areas of intersection of each and the disc in the modified shoelace formula. A visual illustration of this modified formula is shown in Figure 4. If we consider the same example as in Figure 3 but with the addition of a disc, as depicted in Figure 4, then the orange regions (areas with negative signed area, shown in (b)) “cancel” the green regions (areas with positive signed area, shown in (a)) that are outside the polygon and “cancel” the darker regions inside the polygon, giving the desired area of intersection, shown in (c). We prove theorem 2 using induction.

**Figure 4. **Visual illustration of the modified shoelace formula for calculating overlap areas.

**Proof of Theorem 2**

We use strong induction on (the number of sides of the polygon). The base case is immediate, as Ahmadi and Pan [1] show exhaustively in their paper. We now assume that the result is true for all for some , and let be an arbitrary -sided polygon. We may choose two vertices , of the polygon such that the straight line joining and is contained within the polygon (for a proof of this fact, see [10]).

To complete the induction step, let and pick any diagonal contained within , where appears before in the list. Then let , where and are the two polygons formed by adding the diagonal. We can thus write and . These two polygons correspond to (a) and (b), respectively, in the example given in Figure 5.

**Figure 5.** Example polygon decomposition given the diagonal . The arrows denote the order of the vertices.

Since both and have at most sides (say and sides, respectively), then for any and any , we know that for the disc of radius centered on (denoted ), we have

where and are the triangles defined in the same way as in the statement of the theorem for the polygons and , respectively. From their construction, the vertices in and are either both in clockwise order, or both in counterclockwise order. So and are either both positive or both negative. Hence

(9) |

Using (7) we deduce that

But all of the triangles are precisely the analogous triangles for the polygon , and also include the triangles added by the diagonal, namely and . Thus

In this section, we extend the algorithm in [1] for arbitrarily shaped polygons by using the modified form of the shoelace formula in theorem 2. Since any polygon can be triangulated, the area of intersection of a disc of radius centered at a reference point and the polygon can be found by summing the areas of the intersections of the disc and each triangle of the triangulation. Since the latter areas can be found using theorem 2, the generalization follows.

We know that given a polygon and a reference point , the CDF is

(10) |

Using the modified shoelace formula, we may write (10) as

(11) |

where and

In (11), we can calculate using the shoelace formula in theorem 1. The method to calculate each was given in [1] and is summarized in Section 1.1. Once the CDF is found, the corresponding PDF can be found by differentiation.

In this section, we describe the implementation of the algorithm. The functions in (1) and (2) are implemented as the functions `InsideAltitudeArea` and `OutsideAltitudeArea`, respectively. These functions return the piecewise functions (1) and (2) as a function of the argument . Each function returns a list of sublists; here each sublist is of the form , where is the piece of the function that corresponds to the range . The function (1) has four pieces and so `InsideAltitudeArea` returns a array. Similarly, `OutsideAltitudeArea` returns a array, as it corresponds to (2), which has three pieces.

The function `PolygonArea`, given the vertices, calculates the area of a polygon with the shoelace formula.

`CombinePieces` is responsible for simplifying a “pseudo-piecewise” function by determining the distinct ranges of the equivalent piecewise function, sorting these ranges, and finding the corresponding function for each range. This piecewise function is then converted to either the CDF or the PDF, depending on the argument `case`. The function takes four arguments: is a “pseudo-piecewise” function of the form that is returned by `InsideAltitudeArea` and `OutsideAltitudeArea`, namely, a list of sublists, where each sublist is of the form where is the piece of the function that corresponds to the range . The argument `area` is set equal to the area of the polygon. The returned CDF is a function of the argument . Finally, `case` is a Boolean argument that determines whether the CDF or PDF is returned; when it is true the function returns the CDF, and when it is false the function returns the PDF.

This gives the sign of a triangle’s area.

`AltitudeInsideQ` returns `True` if the altitude of the triangle intersects the opposite edge inside the triangle.

`PolygonCDF` is the main function used to compute the CDF; it uses the method outlined in Subsection 3.1.

This section discusses how to generate the simulation results, which are used to verify the analytical results. In general, we need uniformly distributed points inside arbitrarily shaped polygons.

Generating uniformly distributed points inside a triangle is straightforward and can be accomplished in a number of ways [11, 12]. The method used here selects two numbers at random from to measure off lengths on two sides of the triangle to use as weights on the vertices. `RandomPointsTriangle` has two arguments: `T` is a list of three vertices describing a triangle and `n` is the number of points to generate.

The method for triangles is extended to arbitrary polygons by triangulating the polygon , uniformly picking a triangle, and then generating a point in that triangle. Uniformly picking a triangle means choosing each triangle with probability such that the points generated are uniform for the whole polygon; this means that the probability of picking a triangle must be proportional to its area. Precisely, the probability is . This is done efficiently using `TriangulateMesh`.

For convenience, here is a list of the different special examples that have been considered in the literature that can be verified using our implementation. The polygon vertices must be specified in clockwise order.

**1.** Example 1 in [1]: equilateral triangle with interior reference point

**2.** Example 2 in [1]: equilateral triangle with exterior reference point.

**3.** Section 2.5.1, p. 263 in [13]: triangle with reference point at a vertex.

**4.** Example in [3]: square with reference point on the boundary.

**5.** Example in [4]: hexagon with reference point at the center.

**6.** Example in [6], [14], and [15]: arbitrarily shaped convex polygon.

**7.** Example in [18]: arbitrarily shaped convex polygon.

**8.** Example in this report: arbitrarily shaped polygon shaped like the letter N.

**9.** Example in this report: star-shaped (concave) polygon region with reference point at the center.

**10.** Example in this report: arbitrarily shaped concave polygon with exterior reference point.

We illustrate using Example 10.

The simulation data is calculated. The code checks to see if the polygon is convex. If it is, the points are simulated using `RandomPointsConcave`, which is faster than `RandomPointsPolygon`. Otherwise, the code uses `RandomPointsPolygon`.

This shows the simulated uniformly distributed points for the given polygon.

A function that converts the simulated points to the CDF is defined. The number of simulation trials

is .

The closed form of the CDF is displayed as calculated by the algorithm.

Also find the corresponding closed form of the PDF, which is needed for the neighbor PDF.

The analytical result for the CDF (red) is compared with the result obtained from simulations (blue).

Using the CDF, the PDF of the Euclidean distance between an arbitrary reference point and its neighbor node can be found.

Define the neighbor PDF: equation (12) in [4].

The result can be plotted as follows.

In this article, we have reported on the use of *Mathematica* for distance distribution modeling in wireless networks. We have proposed and implemented an algorithm to compute the exact cumulative density function (CDF) of the distance from an arbitrary reference point to a randomly located node within an arbitrarily shaped (convex or concave) polygon. Examples of how the obtained distance distributions can be used in the modeling of finite-area wireless networks are provided in [6], [15–19]. The distance distribution results can also be applied in other branches of science, such as forestry, mathematics, operations research, and material sciences [13], [20].

We would like to thank the anonymous reviewer for his comments, especially for suggesting an efficient method of uniformly distributing points in a triangle. We would also like to thank Ms. Maryam Ahmadi (University of Victoria, Canada) and Prof. Jianping Pan (University of Victoria, Canada) for their constructive feedback, which helped us to further improve the implementation.

[1] | M. Ahmadi and J. Pan, “Random Distances Associated with Arbitrary Triangles: A Recursive Approach with an Arbitrary Reference Point,” UVic-PanLab-TR-14-01, 2014. dspace.library.uvic.ca//handle/1828/5134. |

[2] | J. G. Andrews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Weber, “A Primer on Spatial Modeling and Analysis in Wireless Networks,” IEEE Communications Magazine, 48(11), 2010 pp. 156–163. doi:10.1109/MCOM.2010.5621983. |

[3] | Z. Khalid and S. Durrani, “Distance Distributions in Regular Polygons,” IEEE Transactions on Vehicular Technology, 62(5), 2013 pp. 2363–2368. doi:10.1109/TVT.2013.2241092. |

[4] | S. Srinivasa and M. Haenggi, “Distance Distributions in Finite Uniformly Random Networks: Theory and Applications,” IEEE Transactions on Vehicular Technology, 59(2), 2010pp. 940–949. doi:10.1109/TVT.2009.2035044. |

[5] | S. Durrani, J. Guo, and Z. Khalid, “Mathematica and Matlab Software for Computing Distance Distributions,” (May 12, 2015). users.cecs.anu.edu.au/~Salman.Durrani/software.html. |

[6] | J. Guo, S. Durrani, and X. Zhou, “Outage Probability in Arbitrarily-Shaped Finite Wireless Networks,” IEEE Transactions on Communications, 62(2), 2014 pp. 699–712. doi:10.1109/TCOMM.2013.122913.130298. |

[7] | Wikipedia. “Shoelace Formula.” (Jun 18, 2015) en.wikipedia.org/wiki/Shoelace_formula. |

[8] | E. W. Weisstein. “Triangle Area” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/TriangleArea.html. |

[9] | E. W. Weisstein. “Polygon Area” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/PolygonArea.html. |

[10] | J. O’Rourke, Art Gallery Theorems and Algorithms, New York: Oxford University Press, 1987. |

[11] | E. W. Weisstein. “Triangle Point Picking” from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/TrianglePointPicking.html. |

[12] | G. Turk, “Generating Random Points in Triangles,” in Graphics Gems (A. S. Glassner, ed.), Boston: Academic Press, 1990 pp. 24–28. inis.jinr.ru/sl/vol1/CMC/Graphics_Gems_1,ed_A.Glassner.pdf. |

[13] | A. M. Mathai, An Introduction to Geometrical Probability: Distributional Aspects with Applications, Amsterdam: Gordon and Breach Science Publishers, 1999. |

[14] | J. Guo, S. Durrani, and X. Zhou, “Performance Analysis of Arbitrarily-Shaped Underlay Cognitive Networks: Effect of Secondary User Activity Protocols,” IEEE Transactions on Communications, 63(2), 2015 pp. 376–389. doi:10.1109/TCOMM.2014.2385065. |

[15] | J. Guo, S. Durrani, and X. Zhou, “Characterization of Aggregate Interference in Arbitrarily-Shaped Underlay Cognitive Networks,” Proceedings of IEEE Global Communications Conference (GLOBECOM), Austin, TX: IEEE, 2014. pp. 961–966. doi:10.1109/GLOCOM.2014.7036933. |

[16] | M. C. Valenti, D. Torrieri, and S. Talarico, “A Direct Approach to Computing Spatially Averaged Outage Probability,” IEEE Communications Letters, 18(7), 2014 pp. 1103–1106. doi:10.1109/LCOMM.2014.2317740. |

[17] | Z. Khalid, S. Durrani, and J. Guo, “A Tractable Framework for Exact Probability of Node Isolation and Minimum Node Degree Distribution in Finite Multi-hop Networks,” IEEE Transactions on Vehicular Technology, 63(6), 2014 pp. 2836–2847. doi:10.1109/TVT.2013.2293580. |

[18] | K. B. Baltzis, “Distance Distribution in Convex -Gons: Mathematical Framework and Wireless Networking Applications,” Wireless Personal Communications, 71(2), 2012pp. 1487–1503. doi:10.1007/s11277-012-0887-9. |

[19] | D. Moltchanov, “Distance Distribution in Random Networks,” Ad Hoc Networks, 10(6), 2012 pp. 1146–1166. doi:10.1016/j.adhoc.2012.02.005. |

[20] | U. Basel, “Random Chords and Point Distances in Regular Polygons,” Acta Mathematica Universitatis Comenianae, LXXXIII(1), 2014 pp. 1–18.www.emis.de/journals/AMUC/_vol-83/_no_1/_baesel/baesel.html. |

R. Pure and S. Durrani, “Computing Exact Closed-Form Distance Distributions in Arbitrarily Shaped Polygons with Arbitrary Reference Point,” The Mathematica Journal, 2015. dx.doi.org/doi:10.3888/tmj.17-6. |

Ross Pure is a second-year Bachelor of Engineering (Research & Development)/Bachelor of Science student in the Research School of Engineering, the Australian National University (ANU), Canberra, Australia.

Salman Durrani (SMIEEE, SFHEA, MIEAust) is a Senior Lecturer in the Research School of Engineering, ANU. He has coauthored more than 80 publications to date in refereed international journals and conferences. He currently serves as an editor of *IEEE Transactions on Communications*. His research interests include wireless and mobile communications, wireless energy harvesting, and synchronization and signal processing on the unit sphere.

**Ross Pure**

*Research School of Engineering
The Australian National University
Canberra, Australia*

**Salman Durrani**

*Research School of Engineering
The Australian National University
Canberra, Australia*

This article presents new symbolic solutions for the problem of pore elasticity and pore pressure. These techniques are based on the classic theoretical approach proposed by M. A. Biot [1]. The new symbolic solutions differ from the well-known approximations of the functions proposed for the 2D pore elasticity problem. Both new symbolic and numerical solutions are then applied to solve problems arising in offshore design technology, specifically dealing with the penetration of a gravity-based rig installed in the Arctic region of the North Sea of Russia. All symbolic approaches are based on solutions of the linear problem of the pore elasticity for homogeneous soil. The new symbolic solutions are compared with Biot’s solutions.

The main purpose of this article is to derive new symbolic solutions for the classic problem of pore elasticity set up in [1]. Approximate solutions proposed by Biot have been widely used to solve various linear initial-boundary problems involving pore elasticity. But more accurate solutions of the problem are still of interest.

In offshore technology, a practical example deals with the penetration of huge oil rigs into the soil of the sea bed. Such cases are usually based on solutions given by the linear theory of pore elasticity.

First, consider Biot’s problem for a rectangular load on homogeneous soil [1].

In the three-dimensional case, the depth of penetration of a rigid body into soil may be found by solving the equations

(1) |

where , , are the components of the displacements of the saturated soil in the , , directions and

(2) |

is the divergence of the displacements of the soil. The initial and boundary conditions will be given later.

The other variables are:

- , the pore pressure
- , the shear modulus of the rigid skeleton of the saturated soil
- , the Poisson coefficient of the rigid skeleton of the soil
- , the coefficient of consolidation
- , the coefficient of permeability
- , the final compressibility

As usual, define the Laplace operator:

(3) |

Consider an infinite half-space (e.g. clay) bounded by the horizontal - plane, and let the axis be directed vertically downward.

The vertical deflection of the horizontal plane is to be found when a vertical load acts on the rectangular plane at time ; the rectangular load is distributed in the strip on the surface.

Assume that water saturates the clay and may be freely filtered by the neighboring half-area, and that the water pressure on the - surface equals atmospheric pressure.

Then the original problem becomes two dimensional, with . The equations in (1) become

(4) |

Taking the Laplace transform and noting that changes the third equation, giving the system

(5) |

where is a Laplace-transformed parameter.

The boundary conditions for the 2D case (reduced for the full symmetric geometry of the body) of the pore elasticity are:

1. The displacements and pore pressure are 0 as :

(6) |

2. The pore pressure at the surface is

(7) |

3. The skeleton stress at the surface is equal to the external load given by

(8) |

4. The shear stress at the surface is 0:

(9) |

First, clear any old values of and .

Define three operators and three equations.

Consider general solutions of the system of equations (5) in the form of functions with separated variables.

Then (5) is transformed into the following system of ODEs.

(10) |

This determines the general solutions of (10).

This allows us to define the three previously unknown functions.

New forms of the symbolic solutions are presented in section 3.3.

The symbolic solutions derived in section 3.2 may be combined in the function .

(11) |

To satisfy the boundary condition (6), the coefficients and in (11) must be zero.

Here is a linear system for the unknown coefficients in .

This finds solutions for three of the coefficients.

We can express the functions in a different form using .

The other three unknown constants are found from the boundary conditions (7), (8), and (9). These conditions lead to three linear equations.

Finally, here is the system of linear equations.

It leads to the general solutions for all the coefficients of the problem of pore elasticity.

Following [1], we introduce a transformed solution for the vertical deflection of the plane.

This gives the time dependence of the vertical displacement of the saturated soil.

This is the formula for the initial distribution at the surface.

Based on [1], here is the general vertical deflection for the horizontal level of the soil.

Finally, a new symbolic solution for a rectangular load acting on the plane (with zero Poisson coefficient) is derived by integrating.

The result is a new symbolic form for problem [1].

(12) |

Originally in [1] the approximate symbolic solution was given by:

(13) |

This is the new function for practical solutions based on the exact solution derived above.

Here is the displacement of the horizontal level of the porous space under a rectangular load.

Let us consider an application of the new symbolic solution of the pore elasticity problem obtained above to the study of the penetration of the gravity rig into the sea bed. The platform Prirazlomnaya has a mass of 250,000 tons and its bottom measures (see [2]). We also define the parameters of the clay soil and sediment.

This is the average pressure of the rig on the surface of the sediment.

(14) |

Here is a plot of the rectangular load.

This plots the family of vertical displacements of the rig penetrated into clay and sediment.

For a complex load acting on the rig, the general penetration is the sum of each of the component penetrations.

(15) |

where and are the sizes of the mechanical components and living quarters of the platform.

Here is a plot of the loads on the platform from the housing and mechanical blocks.

Let the production unit have length 30 meters and suppose the distributed load is .

This shows the penetration of the platform on the sediment due to the production unit over a two-week period.

This shows the penetration of the platform, taking into account the nonuniform load distribution on the sediment, where is the side length of the square of the bottom of the rig in (15).

**Figure 1.** Here is a photo of the rig Prirazlomnaya.

Since [1] does not take account of porous pressure, here is a new symbolic solution for this case, substituting 0 for .

Following [1], here is an approximate symbolic solution for the pressure.

The explicit form of the pore pressure function is computed by taking the inverse Laplace transform.

So this is a new symbolic solution for the porous pressure for a rectangular load at the surface.

This section shows how the pore pressure changes with time and depth of the soil layer near the horizon.

This determines the distribution function of pore pressure 1.5 meters below the bottom of the gravity platform.

This shows the family of porous pressure distributions at the bottom of the gravity platform.

Obviously, the pore pressure is the load-bearing frame factor, which restrains the platform to the surface of the ground.

In this article, new symbolic solutions for the penetration of a gravity platform into soil and a determination of the porous pressure in the saturated soil are found by computer algebra techniques. These solutions improve upon earlier solutions obtained by M. Biot and give us new possibilities to apply symbolic computer applications to diverse problems in pore elasticity theory.

Both symbolic solutions are applicable to the design of offshore gravity structures installed in shallow water.

[1] | M. A. Biot, “General Theory of Three-Dimensional Consolidation,” Journal of Applied Physics, 12(2), 1941 pp. 155–164. doi:10.1063/1.1712886. |

[2] | Wikipedia. “Prirazlomnoye Field.” (Jan 12, 2015) en.wikipedia.org/wiki/Prirazlomnoye_field. |

A. N. Papusha and D. P. Gontarev, “New Symbolic Solutions of Biot’s 2D Pore Elasticity Problem,” The Mathematica Journal, 2015. dx.doi.org/doi:10.3888/tmj.17-5. |

Professor Alexander N. Papusha is the head of the Continuum Mechanics and Offshore Exploration Department at Murmansk State Technical University (MSTU). He received an M.S. in mechanics from Kiev State University in 1972 and a Ph.D. in theoretical mechanics in 1979 from the Institute of Mechanics, Ukrainian Academy of Sciences, as well as a Doctor of Sciences from the Institute of Machinery (St. Petersburg), Russian Academy of Sciences in 1999. Since 1983, he has been a full professor at MSTU.

Denis P. Gontarev is a postgraduate student in the Continuum Mechanics and Offshore Exploration Department at MSTU.

**Alexander N. Papusha**

*Continuum Mechanics and Offshore Exploration Department
Murmansk State Technical University
13, Sportyvnay
183010 Murmansk, Russia
*

**Denis P. Gontarev**

*Continuum Mechanics and Offshore Exploration Department
Murmansk State Technical University
13, Sportyvnay
183010 Murmansk, Russia
*