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### Lyapunov condition

The Lyapunov condition, sometimes known as Lyapunov's central limit theorem, states that if the th moment (with ) exists for a statistical distribution of independent random variates (which need not necessarily be from same distribution), the means and variances are finite, and(1)then if(2)where(3)the central limit theorem holds.

### Teardrop curve

A plane curve given by the parametric equations(1)(2)The plots above show curves for values of from 0 to 7.The teardrop curve has area(3)

### Lindeberg condition

A sufficient condition on the Lindeberg-Feller central limit theorem. Given random variates , , ..., let , the variance of be finite, and variance of the distribution consisting of a sum of s(1)be(2)In the terminology of Zabell (1995), let(3)where denotes the expectation value of restricted to outcomes , then the Lindeberg condition is(4)for all (Zabell 1995).In the terminology of Feller (1971), the Lindeberg condition assumed that for each ,(5)or equivalently(6)Then the distribution(7)tends to the normal distribution with zero expectation and unit variance (Feller 1971, p. 256). The Lindeberg condition (5) guarantees that the individual variances are small compared to their sum in the sense that for given for all sufficiently large , for , ..., (Feller 1971, p. 256).

### Right angle

A right angle is an angle equal to half the angle from one end of a line segment to the other. A right angle is radians or . A triangle containing a right angle is called a right triangle. However, a triangle cannot contain more than one right angle, since the sum of the two right angles plus the third angle would exceed the total possessed by a triangle.The patterns of cracks observed in mud that has been dried by the sun form curves that often intersect in right angles (Williams 1979, p. 45; Steinhaus 1999, p. 88; Pearce 1990, p. 12).

### Angle

Given two intersecting lines or line segments, the amount of rotation about the point of intersection (the vertex) required to bring one into correspondence with the other is called the angle between them. The term "plane angle" is sometimes used to distinguish angles in a plane from solid angles measured in space (International Standards Organization 1982, p. 5).The term "angle" can also be applied to the rotational offset between intersecting planes about their common line of intersection, in which case the angle is called the dihedral angle of the planes.Angles are usually measured in degrees (denoted ), radians (denoted rad, or without a unit), or sometimes gradians (denoted grad).The concept of an angle can be generalized from the circle to the sphere, in which case it is known as solid angle. The fraction of a sphere subtended by an object (its solid angle) is measured in steradians, with the entire sphere..

### Crookedness

Let a knot be parameterized by a vector function with , and let be a fixed unit vector in . Count the number of local minima of the projection function . Then the minimum such number over all directions and all of the given type is called the crookedness . Milnor (1950) showed that is the infimum of the total curvature of . For any tame knot in , where is the bridge index.

### Queens problem

What is the maximum number of queens that can be placed on an chessboard such that no two attack one another? The answer is queens for or and queens otherwise, which gives eight queens for the usual board (Madachy 1979; Steinhaus 1999, p. 29). The number of different ways the queens can be placed on an chessboard so that no two queens may attack each other for the first few are 1, 0, 0, 2, 10, 4, 40, 92, ... (OEIS A000170; Madachy 1979; Steinhaus 1999, p. 29). The number of rotationally and reflectively distinct solutions of these are 1, 0, 0, 1, 2, 1, 6, 12, 46, 92, ... (OEIS A002562; Dudeney 1970; p. 96). The 12 distinct solutions for are illustrated above, and the remaining 80 are generated by rotation and reflection (Madachy 1979, Steinhaus 1999).The minimum number of queens needed to occupy or attack all squares of an chessboard (i.e., domination numbers for the queen graphs) are given for , 2, ... by 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 8, 9,..

### Prince rupert's cube

Prince Rupert's cube is the largest cube that can be made to pass through a given cube. In other words, the cube having a side length equal to the side length of the largest hole of a square cross section that can be cut through a unit cube without splitting it into two pieces.Prince Rupert's cube cuts a hole of the shape indicated in the above illustration (Wells 1991). Curiously, it is slightly larger than the original cube, with side length (OEIS A093577). Any cube this size or smaller can be made to pass through the original cube.

### Salinon

The salinon is the figure illustrated above formed from four connected semicircles. The word salinon is Greek for "salt cellar," which the figure resembles. If the radius of the large enclosing circle is and the radius of the small central circle is , then the radii of the two small side circles are .In his Book of Lemmas, Archimedes proved that the salinon has an area equal to the circle having the line segment joining the top and bottom points as its diameter (Wells 1991), namely

### Vesica piscis

The term "vesica piscis," meaning "fish bladder" in Latin, is used for the particular symmetric lens formed by the intersection of two equal circles whose centers are offset by a distance equal to the circle radii (Pedoe 1995, p. xii). The height of the lens is given by letting in the equation for a circle-circle intersection(1)giving(2)The vesica piscis therefore has two equilateral triangles inscribed in it as illustrated above.The area of the vesica piscis is given by plugging into the circle-circle intersection area equation with ,(3)giving(4)(5)(OEIS A093731). Since each arcof the lens is precisely 1/3 of a circle, perimeter is given by(6)Renaissance artists frequently surrounded images of Jesus with the vesica piscis (Pedoe 1995, p. xii; Rawles 1997).

### Lens

A (general, asymmetric) lens is a lamina formed by the intersection of two offset disks of unequal radii such that the intersection is not empty, one disk does not completely enclose the other, and the centers of curvatures are on opposite sides of the lens. If the centers of curvature are on the same side, a lune results.The area of a general asymmetric lens obtained from circles of radii and and offset can be found from the formula for circle-circle intersection, namely(1)(2)Similarly, the height of such a lens is(3)(4)A symmetric lens is lens formed by the intersection of two equal disk. The area of a symmetric lens obtained from circles with radii and offset is given by(5)and the height by(6)A special type of symmetric lens is the vesica piscis (Latin for "fish bladder"), corresponding to a disk offset which is equal to the disk radii.A lens-shaped region also arises in the study of Bessel functions, is very important in the theory of..

### Honeycomb

The regular tessellation consisting of regular hexagons (i.e., a hexagonal grid).In general, the term honeycomb is used to refer to a tessellation in dimensions for . The only regular honeycomb in three dimensions is , which consists of eight cubes meeting at each polyhedron vertex. The only quasiregular honeycomb (with regular cells and semiregular vertex figures) has each polyhedron vertex surrounded by eight tetrahedra and six octahedra and is denoted .Ball and Coxeter (1987) use the term "sponge" for a solid that can be parameterized by integers , , and that satisfy the equationThe possible sponges are , , , , and .There are many semiregular honeycombs, such as , in which each polyhedron vertex consists of two octahedra and four cuboctahedra .

### Cup

The symbol , used for the union of sets, and, sometimes, also for the logical connective OR instead of the symbol (vee). In fact, for any two sets and and this equivalence demonstrates the connection between the set-theoretical and the logical meaning.

### Snake

A snake is an Eulerian path in the -hypercube that has no chords (i.e., any hypercube edge joining snake vertices is a snake edge). Klee (1970) asked for the maximum length of a -snake. Klee (1970) gave the bounds(1)for (Danzer and Klee 1967, Douglas 1969), as well as numerous references. Abbott and Katchalski (1988) show(2)and Snevily (1994) showed that(3)for , and conjectured(4)for . The first few values for for , 2, ..., are 2, 4, 6, 8, 14, 26, ... (OEIS A000937).

### Pseudotree

A pseudotree is a connected pseudoforest, i.e., an undirected connected graph that contains at most one graph cycle. Connected acyclic graphs (i.e., trees), are therefore pseudotrees.Some care is needed when encountering pseudotrees as some authors use the term to mean "a pseudotree that is not a tree." Such graphs are perhaps better known as connected unicyclic graphs for clarity.The numbers of pseudotrees on 1, 2, 3, ... vertices are 1, 1, 2, 4, 8, 19, 44, 112, ... (OEIS A005703), the first few of which are illustrated above.

### Pseudoforest

A pseudoforest is an undirected graph in which every connected component contains at most one graph cycle. A pseudotree is therefore a connected pseudoforest and a forest (i.e., not-necessarily-connected acyclic graph) is a trivial pseudoforest.Some care is needed when encountering pseudoforests as some authors use the term to mean "a pseudoforest that is not a forest."The numbers of pseudoforests on 1, 2, 3, ... vertices are 1, 2, 4, 9, 19, 46, 108, 273 ... (OEIS A134964), the first few of which are illustrated above.

### Antelope graph

An antelope graph is a graph formed by all possible moves of a hypothetical chess piece called an "antelope" which moves analogously to a knight except that it is restricted to moves that change by three squares along one axis of the board and four squares along the other. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable antelope moves are considered edges. The plots above show the graphs corresponding to antelope graph on chessboards for to 7.The antelope graph is connected for , Hamiltonian for (trivially) and 14 (but for no odd or other even values ), and traceable for and 21 (with the status for unknown and unknown).Precomputed properties of antelope graphs are implemented in the Wolfram Language as GraphData["Antelope", m, n]...

### Giraffe graph

A giraffe graph is a graph formed by all possible moves of a hypothetical chess piece called a "giraffe" (a.k.a. -leaper) which moves analogously to a knight except that it is restricted to moves that change by one square along one axis of the board and four squares along the other. To form the graph, each chessboard square is considered a vertex, and vertices connected by allowable giraffe moves are considered edges.The smallest board allowing a closed tour for the giraffe (i.e., the giraffe graph is Hamiltonian) is the , first solved by A. H. Frost in 1886.

### Forest

A forest is an acyclic graph (i.e., a graph without any graph cycles). Forests therefore consist only of (possibly disconnected) trees, hence the name "forest."Examples of forests include the singleton graph,empty graphs, and all trees.A forest with components and nodes has graph edges. The numbers of forests on , 2, ... nodes are 1, 2, 3, 6, 10, 20, 37, ... (OEIS A005195).A graph can be tested to determine if it is acyclic (i.e., a forest) in the Wolfram Language using AcylicGraphQ[g]. A collection of acyclic graphs is available as GraphData["Acyclic"] or GraphData["Forest"].The total numbers of trees in all the forests of orders , 2, ... are 1, 3, 6, 13, 24, 49, 93, 190, 381, ... (OEIS A005196). The average numbers of trees are therefore 1, 3/2, 2, 13/6, 12/5, 49/20, 93/37, 5/2, ... (OEIS A095131 and A095132).The triangle of numbers of -node forests containing trees is 1; 1, 1; 1, 1, 1; 2, 2, 1, 1; 3, 3, 2, 1, 1; ... (OEIS..

### Universality

Universality is the property of being able to perform different tasks with the same underlying construction just by being programmed in a different way. Universal systems are effectively capable of emulating any other system. Digital computers are universal, but proving that idealized computational systems are universal can be extremely difficult and technical. Nonetheless, examples have been found in many systems, and any system that can be translated into another system known to be universal must itself be universal. Specific universal Turing machines, universal cellular automata (in both one and two dimensions), and universal cyclic tag systems are known, although the smallest universal example is known only in the case of elementary cellular automata (Wolfram 2002, Cook 2004).

### Rice's theorem

If is a class of recursively enumerable sets, then the set of Gödel numbers of functions whose domains belong to is called its index set. If the index set of is a recursive set, then either is empty or contains all recursively enumerable sets.Rice's theorem is an important result for computer science because it sets up boundaries for research in that area. It basically states that only trivial properties of programs are algorithmically decidable.

### Winkler conditions

Conditions arising in the study of the Robbins axiom and its connection with Boolean algebra. Winkler studied Boolean conditions (such as idempotence or existence of a zero) which would make a Robbins algebra become a Boolean algebra. Winkler showed that each of the conditionswhere denotes OR and denotes NOT, known as the first and second Winkler conditions, suffices. A computer proof demonstrated that every Robbins algebra satisfies the second Winkler condition, from which it follows immediately that all Robbins algebras are Boolean.

### Bishops problem

Find the maximum number of bishops that can be placed on an chessboard such that no two attack each other. The answer is (Dudeney 1970, Madachy 1979), giving the sequence 2, 4, 6, 8, ... (the even numbers) for , 3, .... One maximal solution for is illustrated above. The numbers of distinct maximal arrangements for , 2, ... bishops are 1, 4, 26, 260, 3368, ... (OEIS A002465). The numbers of rotationally and reflectively distinct solutions on an board for is(1)for (Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 1995). An equivalent formula also valid for is(2)where is the floor function, giving the sequence for , 2, ... as 1, 1, 2, 3, 6, 10, 20, 36, ... (OEIS A005418).The minimum number of bishops needed to occupy or attack all squares on an chessboard is , arranged as illustrated above...

### Solomon's seal knot

Solomon's seal knot is the prime (5,2)-torus knot with braid word . It is also known as the cinquefoil knot (a name derived from certain herbs and shrubs of the rose family which have five-lobed leaves and five-petaled flowers) or the double overhand knot. It has Arf invariant 1 and is not amphichiral, although it is invertible.The knot group of Solomon's seal knot is(1)(Livingston 1993, p. 127).The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F of the Solomon's seal knot are(2)(3)(4)(5)(6)(7)Surprisingly, the knot 10-132 shares the same Alexander polynomial and Jones polynomial with the Solomon's seal knot. However, no knots on 10 or fewer crossings share the same BLM/Ho polynomial with it.

### Singleton graph

The singleton graph is the graph consisting of a single isolated node with no edges. It is therefore the empty graph on one node. It is commonly denoted (i.e., the complete graph on one node).By convention, the singleton graph is considered to be Hamiltonian(B. McKay, pers. comm., Mar. 22, 2007).

### Transformation

A transformation (a.k.a., map, function) over a domain takes the elements to elements , where the range (a.k.a., image) of is defined asNote that when transformations are specified with respect to a coordinate system, it is important to specify whether the rotation takes place on the coordinate system, with space and objects embedded in it being viewed as fixed (a so-called alias transformation), or on the space itself relative to a fixed coordinate system (a so-called alibi transformation).Examples of transformations are summarized in the following table.TransformationCharacterizationdilationcenter of dilation, scale decrease factorexpansioncenter of expansion, scale increase factorreflectionmirror line or planerotationcenter of rotation, rotation angleshearinvariant line and shear factorstretch (1-way)invariant line and scale factorstretch (2-way)invariant lines and scale factorstranslationdisplacement..

### Gabriel's horn

Gabriel's horn, also called Torricelli's trumpet, is the surface of revolution of the function about the x-axis for . It is therefore given by parametric equations(1)(2)(3)The surprising thing about this surface is that it (taking for convenience here) has finite volume(4)(5)(6)but infinite surface area,since(7)(8)(9)(10)(11)(12)This leads to the paradoxical consequence that while Gabriel's horn can be filled up with cubic units of paint, an infinite number of square units of paint are needed to cover its surface!The coefficients of the first fundamental formare,(13)(14)(15)and of the second fundamental form are(16)(17)(18)The Gaussian and meancurvatures are(19)(20)The Gaussian curvature can be expressed implicitly as(21)

### Tube

A tube of radius of a set is the set of points at a distance from . In particular, if is a regular space curve whose curvature does not vanish, then the normal vector and binormal vector are always perpendicular to , and the circle is perpendicular to at . So as the circle moves around , it traces out a tube, provided the tube radius is small enough so that the tube is not self-intersecting. A formula for the tube around a curve is therefore given byfor over the range of the curve and . The illustrations above show tubes corresponding to a circle, helix, and two torus knots.The surface generated by constructing a tube around a circleis known as a torus.

### Star of lakshmi

The Star of Lakshmi is the star figure , that is used in Hinduism to symbolize Ashtalakshmi, the eight forms of wealth. This symbol appears prominently in the Lugash national museum portrayed in the fictional film The Return of the Pink Panther.The interior of a Star of Lakshmi with edges of length is a regular octagon with side lengths(1)The areas of the intersection and union of the two constituent squares are(2)(3)

### Ear

A principal vertex of a simple polygon is called an ear if the diagonal that bridges lies entirely in . Two ears and are said to overlap ifThe two-ears theorem states that, except for triangles, every simple polygon has at least two nonoverlapping ears.

### Mouth

A principal vertex of a simple polygon is called a mouth if the diagonal is an extremal diagonal (i.e., the interior of lies in the exterior of ).

### Maltese cross

The Maltese cross is a symbol identified with the Christian warrior whose outward points form an octagon (left figure). Another class of cross sometimes (incorrectly) known as the Maltese cross is the cross pattée (from the French word meaning "paw," which each arm of the cross resembles). The TeX macro gives the form of the cross pattée illustrated in the middle figure. Around 1901, Dudeney published a seven-piece dissection of what he termed a "Maltese cross" (but which is actually a variant of the cross pattée) to a square (right figure) due to A. E. Hill (Gardner 1991, p. 46).

### Latin cross

An irregular dodecagonal cross in the shape of a dagger . The six faces of a cube can be cut along seven edges and unfolded into a Latin cross (i.e., the Latin cross is the net of the cube). Similarly, eight hypersurfaces of a hypercube can be cut along 17 squares and unfolded to form a three-dimensional Latin cross.Another cross also called the Latin cross is illustrated above. It is a Greekcross with flared ends, and is also known as the crux immissa or cross patée.

### Set

A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). Members of a set are often referred to as elements and the notation is used to denote that is an element of a set . The study of sets and their properties is the object of set theory.Older words for set include aggregate and set class. Russell also uses the unfortunate term manifold to refer to a set.Historically, a single horizontal overbar was used to denote a set stripped of any structure besides order, and hence to represent the order type of the set. A double overbar indicated stripping the order from the set and hence represented the cardinal number of the set. This practice was begun by set theory founder Georg Cantor.Symbols used to operate on sets include (which means "and" or intersection), and (which means "or" or union). The symbol is used to denote the set containing..

### Cayley tree

A tree in which each non-leaf graph vertex has a constant number of branches is called an -Cayley tree. 2-Cayley trees are path graphs. The unique -Cayley tree on nodes is the star graph. The illustration above shows the first few 3-Cayley trees (also called trivalent trees, binary trees, or boron trees). The numbers of binary trees on , 2, ... nodes (i.e., -node trees having vertex degree either 1 or 3; also called 3-Cayley trees, 3-valent trees, or boron trees) are 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0 ,4, 0, 6, 0, 11, ... (OEIS A052120).The illustrations above show the first few 4-Cayley and 5-Cayley trees.The percolation threshold for a Cayley tree having branches is

### Taylor's condition

For a given positive integer , does there exist a weighted tree with graph vertices whose paths have weights 1, 2, ..., , where is a binomial coefficient? Taylor showed that no such tree can exist unless it is a perfect square or a perfect square plus 2. No such trees are known except , 3, 4, and 6.Székely et al. showed computationally that there are no such trees with and 11. They also showed that if there is such a tree on vertices then the maximum vertex degree is at most and that there is no path of length larger than . They conjecture that there are only finitely many such trees.

### Caterpillar graph

A caterpillar graph, caterpillar tree, or simply "caterpillar," is a tree in which every graph vertex is on a central stalk or only one graph edge away from the stalk (in other words, removal of its endpoints leaves a path graph; Gallian 2007). A tree is a caterpillar iff all nodes of degree are surrounded by at most two nodes of degree two or greater.Caterpillar graphs are graceful.The number of caterpillar trees on nodes iswhere is the floor function (Harary and Schwenk 1973). For , 2, ... nodes, this gives 1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 136, ... (OEIS A005418). The first few caterpillars are illustrated above.The number of noncaterpillar trees on , 8, ... as 1, 3, 11, 34, 99, ... (OEIS A052471). The noncaterpillar trees on nodes are illustrated above.

### Strongly binary tree

A strongly binary tree is a rooted tree for which the root is adjacent to either zero or two vertices, and all non-root vertices are adjacent to either one or three vertices (Finch 2003, p. 298). The numbers of strongly binary trees on , 2, ... nodes are 1, 0, 1, 0, 1, 0, 2, 0, 3, 0, 6, 0, ... (OEIS A001190). The counts are 0 for even, and for odd , where is the number of weakly binary trees on nodes (Finch 2003, p. 298).

### Labeled tree

A tree with its nodes labeled. The number of labeled trees on nodes is , the first few values of which are 1, 1, 3, 16, 125, 1296, ... (OEIS A000272). Cayley (1889) provided the first proof of the number of labeled trees (Skiena 1990, p. 151), and a constructive proof was subsequently provided by Prüfer (1918). Prüfer's result gives an encoding for labeled trees known as Prüfer code (indicated underneath the trees above, where the trees are depicted using an embedding with root at the node labeled 1).The probability that a random labeled tree is centeredis asymptotically equal to 1/2 (Szekeres 1983; Skiena 1990, p. 167).

### Steiner tree

The Steiner tree of some subset of the vertices of a graph is a minimum-weight connected subgraph of that includes all the vertices. It is always a tree. Steiner trees have practical applications, for example, in the determination of the shortest total length of wires needed to join some number of points (Hoffman 1998, pp. 164-165).The determination of a Steiner tree is NP-complete and hard even to approximate. There is 1.55-approximate algorithm due to Robins and Zelikovski (2000), but approximation within 95/94 is known to be NP-hard (Chlebik and Chlebikova 2002).

### Binary tree

A binary tree is a tree-like structure that is rooted and in which each vertex has at most two children and each child of a vertex is designated as its left or right child (West 2000, p. 101). In other words, unlike a proper tree, the relative positions of the children is significant.Dropping the requirement that left and right children are considered unique gives a true tree known as a weakly binary tree (in which, by convention, the root node is also required to be adjacent to at most one graph vertex).The height of a binary tree is the number of levels within the tree. The numbers of binary trees of height , 2, ... nodes are 1, 3, 21, 651, 457653, ... (OEIS A001699). A recurrence equation giving these counts is(1)with .The number of binary trees with nodes are 1, 2, 5, 14, 42, ... (OEIS A000108), which are the Catalan number .For a binary tree of height with nodes,(2)These extremes correspond to a balanced tree (each node except the tree leaves has a left..

### Rsa number

RSA numbers are difficult to-factor composite numbers having exactly two prime factors (i.e., so-called semiprimes) that were listed in the Factoring Challenge of RSA Security®--a challenge that is now withdrawn and no longer active.While RSA numbers are much smaller than the largest known primes, their factorization is significant because of the curious property of numbers that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization"). Thus, while it is trivial to multiply two large numbers and together, it can be extremely difficult to determine the factors if only their product is given. With some ingenuity, this property can be used to create practical and efficient encryption systems for electronic data.RSA Laboratories sponsored the RSA Factoring Challenge to encourage research into computational..

### Wythoff symbol

A symbol consisting of three rational numbers that can be used to describe uniform polyhedra based on how a point in a spherical triangle can be selected so as to trace the vertices of regular polygonal faces. For example, the Wythoff symbol for the tetrahedron is . There are four types of Wythoff symbols, , , and , and one exceptional symbol, (which is used for the great dirhombicosidodecahedron).The meaning of the bars may be summarized as follows (Wenninger 1989, p. 10; Messer 2002). Consider a spherical triangle whose angles are , , and . 1. : is a special point within that traces snub polyhedra by even reflections. 2. (or ): is the vertex . 3. (or ): lies on the arc and the bisector of the opposite angle . 4. (or any permutation of the three letters): is the incenter of the triangle . Some special cases in terms of Schläfli symbolsare(1)(2)(3)(4)(5)(6)..

### Parallel

Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect. In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. Lines in three-space that are not parallel but do not intersect are called skew lines.If lines and are parallel, the notation is used.In a non-Euclidean geometry, the concept of parallelism must be modified from its intuitive meaning. This is accomplished by changing the so-called parallel postulate. While this has counterintuitive results, the geometries so defined are still completely self-consistent.In a triangle , a triangle median bisects all segments parallel to a given side (Honsberger 1995, p. 87).

### Variance

For a single variate having a distribution with known population mean , the population variance , commonly also written , is defined as(1)where is the population mean and denotes the expectation value of . For a discrete distribution with possible values of , the population variance is therefore(2)whereas for a continuous distribution,it is given by(3)The variance is therefore equal to the second central moment .Note that some care is needed in interpreting as a variance, since the symbol is also commonly used as a parameter related to but not equivalent to the square root of the variance, for example in the log normal distribution, Maxwell distribution, and Rayleigh distribution.If the underlying distribution is not known, then the samplevariance may be computed as(4)where is the sample mean.Note that the sample variance defined above is not an unbiased estimator for the population variance . In order to obtain an unbiased estimator for..

### Tau function

A function related to the divisor function , also sometimes called Ramanujan's tau function. It is defined via the Fourier series of the modular discriminant for , where is the upper half-plane, by(1)(Apostol 1997, p. 20). The tau function is also given by the Cauchyproduct(2)(3)where is the divisor function (Apostol 1997, pp. 24 and 140), , and .The tau function has generating function(4)(5)(6)(7)(8)where is a q-Pochhammer symbol. The first few values are 1, , 252, , 4830, ... (OEIS A000594). The tau function is given by the Wolfram Language function RamanujanTau[n].The series(9)is known as the tau Dirichlet series.Lehmer (1947) conjectured that for all , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of for which this condition holds.reference3316799Lehmer (1947)214928639999Lehmer..

### Mulliken symbols

Symbols used to identify irreducible representations of groups: singly degenerate state which is symmetric with respect to rotation about the principal axis, singly degenerate state which is antisymmetric with respect to rotation about the principal axis, doubly degenerate, triply degenerate, (gerade, symmetric) the sign of the wavefunction does not change on inversion through the center of the atom, (ungerade, antisymmetric) the sign of the wavefunction changes on inversion through the center of the atom, (on or ) the sign of the wavefunction does not change upon rotation about the center of the atom, (on or ) the sign of the wavefunction changes upon rotation about the center of the atom, ' = symmetric with respect to a horizontal symmetry plane , " = antisymmetric with respect to a horizontal symmetry plane . ..

### Minus

The operation of subtraction, i.e., minus . The operation is denoted . The minus sign "" is also used to denote a negative number, i.e., .

### Minimum

The smallest value of a set, function, etc. The minimum value of a set of elements is denoted or , and is equal to the first element of a sorted (i.e., ordered) version of . For example, given the set , the sorted version is , so the minimum is 1. The maximum and minimum are the simplest order statistics.The minimum value of a variable is commonly denoted (cf. Strang 1988, pp. 286-287 and 301-303) or (Golub and Van Loan 1996, p. 84). In this work, the convention is used.The minimum of a set of elements is implemented in the Wolfram Language as Min[list] and satisfies the identities(1)(2)A continuous function may assume a minimum at a single point or may have minima at a number of points. A global minimum of a function is the smallest value in the entire range of the function, while a local minimum is the smallest value in some local neighborhood.For a function which is continuous at a point , a necessary but not sufficient condition for to have a local..

### Mega

A large number defined as where the circle notation denotes " in squares," and triangles and squares are expanded in terms of Steinhaus-Moser notation (Steinhaus 1999, pp. 28-29). Here, the typographical error of Steinhaus has been corrected.

### Sign

Min Max Re Im The sign of a real number, also called sgn or signum, is for a negative number (i.e., one with a minus sign ""), 0 for the number zero, or for a positive number (i.e., one with a plus sign ""). In other words, for real ,(1)For real , this can be written(2)and satisfies(3) for real can also be defined as(4)where is the Heaviside step function.The sign function is implemented in the Wolfram Language for real as Sign[x]. For nonzero complex numbers, Sign[z] returns , where is the complex modulus of . can also be interpreted as an unspecified point on the unit circle in the complex plane (Rich and Jeffrey 1996).

### Stomachion

The stomachion is a 14-piece dissection puzzle similar to tangrams. It is described in fragmentary manuscripts attributed to Archimedes as noted by Magnus Ausonius (310-395 A.D.). The puzzle is also referred to as the "loculus of Archimedes" (Archimedes' box) or "syntemachion" in Latin texts. The word stomachion has as its root the Greek word , meaning "stomach." Note that Ausonius refers to the figure as the "ostomachion," an apparent corruption of the original Greek.The puzzle consists of 14 flat pieces of various shapes arranged in the shape of a square, with the vertices of pieces occurring on a grid. Two pairs of pieces are duplicated. Like tangrams, the object is to rearrange the pieces to form interesting shapes such as the elephant illustrated above (Andrea).Taking the square as having edge lengths 12, the pieces have areas 3, 3, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 21, and 24, giving them relative..

### Foliation leaf

Let be an -manifold and let denote a partition of into disjoint pathwise-connected subsets. Then if is a foliation of , each is called a leaf and is not necessarily closed or compact.

### Foliation

Let be an -manifold and let denote a partition of into disjoint pathwise-connected subsets. Then is called a foliation of of codimension (with ) if there exists a cover of by open sets , each equipped with a homeomorphism or which throws each nonempty component of onto a parallel translation of the standard hyperplane in . Each is then called a foliation leaf and is not necessarily closed or compact (Rolfsen 1976, p. 284).

### Rats sequence

A sequence produced by the instructions "reverse, add to the original, then sort the digits." For example, after 668, the next iteration is given byso the next term is 1345.Applied to 1, the sequence gives 1, 2, 4, 8, 16, 77, 145, 668, 1345, 6677, 13444, 55778, 133345, 666677, 1333444, 5567777, 12333445, 66666677, 133333444, 556667777, 1233334444, 5566667777, 12333334444, 55666667777, 123333334444, 556666667777, 1233333334444, ... (OEIS A004000).Conway conjectured that an initial number leads to a divergent period-two pattern (such as the above in which the numbers of threes and sixes in the middles of alternate terms steadily increase) or to a cycle (Guy 2004, p. 404).The lengths of the cycles obtained by starting with , 2, ... are 0, 0, 8, 0, 0, 8, 0, 0, 2, 0, ... (OEIS A114611), where a 0 indicates that the sequence diverges.The following table summarizes the first few values of leading to a period of length . There are no..

### Rabbit constant

The limiting rabbit sequence written as a binary fraction (OEIS A005614), where denotes a binary number (a number in base-2). The decimal value is(1)(OEIS A014565).Amazingly, the rabbit constant is also given by the continued fraction [0; , , , , ...] = [2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, ...] (OEIS A000301), where are Fibonacci numbers with taken as 0 (Gardner 1989, Schroeder 1991). Another amazing connection was discovered by S. Plouffe. Define the Beatty sequence by(2)where is the floor function and is the golden ratio. The first few terms are 1, 3, 4, 6, 8, 9, 11, ... (OEIS A000201). Then(3)This is a special case of the Devil's staircase function with .The irrationality measure of is (D. Terr, pers. comm., May 21, 2004).

### Fish curve

The fish curve is a term coined in this work for the ellipse negative pedal curve with pedal point at the focus for the special case of the eccentricity . For an ellipse with parametric equations(1)(2)the corresponding fish curve has parametric equations(3)(4)The Cartesian equation is(5)which, when the origin is translated to the node, canbe written(6)(Lockwood 1957).The interior of the curve is not consistently oriented in the above parametrization, with the fish's head being on the left of the curve and the tail on the right as the curve is traversed. Treating the two pieces separately then gives the areas of the tail and head as(7)(8)giving an overall area for the fish as(9)(Lockwood 1957).The arc length of the curve is given by(10)(11)(12)(Lockwood 1957).The curvature and tangentialangle are given by(13)(14)where is the complex argument.The Tschirnhausen cubic, illustrated above,also resembles a fish, as does the trefoil curve...

### D&uuml;rer's conchoid

The class of curve known as Dürer's conchoid appears in Dürer's work Instruction in Measurement with Compasses and Straight Edge (1525) and arose in investigations of perspective. Dürer constructed the curve by drawing lines and of length 16 units through and , where . The locus of and is the curve, although Dürer found only one of the two branches of the curve.The envelope of the lines and is a parabola, and the curve is therefore a glissette of a point on a line segment sliding between a parabola and one of its tangents.Dürer called the curve "muschellini," which means conchoid. However, it is not a true conchoid and so is sometimes called Dürer's shell curve. The Cartesian equation isThere are a number of interesting special cases. For , the curve becomes the line pair together with the circle . If , the curve becomes two coincident straight lines . If , the curve has a cusp at ...

### Altitude

The altitudes of a triangle are the Cevians that are perpendicular to the legs opposite . The three altitudes of any triangle are concurrent at the orthocenter (Durell 1928). This fundamental fact did not appear anywhere in Euclid's Elements.The triangle connecting the feet of the altitudes is known as the orthic triangle.The altitudes of a triangle with side length , , and and vertex angles , , have lengths given by(1)(2)(3)where is the circumradius of . This leads to the beautiful formula(4)Other formulas satisfied by the altitude include(5)where is the inradius, and(6)(7)(8)where are the exradii (Johnson 1929, p. 189). In addition,(9)(10)(11)where is again the circumradius.The points , , , and (and their permutations with respect to indices; left figure) all lie on a circle, as do the points , , , and (and their permutations with respect to indices; right figure).Triangles and are inversely similar.Additional properties involving..

### Barrel

A barrel solid of revolution composed of parallel circular top and bottom with a common axis and a side formed by a smooth curve symmetrical about the midplane.The term also has a technical meaning in functional analysis. In particular, a subset of a topological linear space is a barrel if it is absorbing, closed, and absolutely convex (Taylor and Lay 1980, p. 111). (A subset of a topological linear space is absorbing if for each there is an such that is in if for each such that . A subset of a topological linear space is absolutely convex if for each and in , is in if .)When buying supplies for his second wedding, the great astronomer Johannes Kepler became unhappy about the inexact methods used by the merchants to estimate the liquid contents of a wine barrel. Kepler therefore investigated the properties of nearly 100 solids of revolution generated by rotation of conic sections about non-principal axes (Kepler, MacDonnell, Shechter, Tikhomirov..

### Triangulation

Triangulation is the division of a surface or plane polygon into a set of triangles, usually with the restriction that each triangle side is entirely shared by two adjacent triangles. It was proved in 1925 that every surface has a triangulation, but it might require an infinite number of triangles and the proof is difficult (Francis and Weeks 1999). A surface with a finite number of triangles in its triangulation is called compact.Wickham-Jones (1994) gives an algorithm for triangulation ("otectomy"), and O'Rourke (1998, p. 47) sketches a method for improving this to , as first done by Lennes (1911). Garey et al. (1978) gave an algorithmically straightforward method for triangulation, which was for many years believed optimal. However, Tarjan and van Wyk (1988) produced an algorithm. This was followed by an unexpected result due to Chazelle (1991), who showed that an arbitrary simple polygon can be triangulated in . However,..

### Greek cross

A Greek cross, also called a square cross, is a cross inthe shape of a plus sign. It is a non-regular dodecagon.A square cross appears on the flag of Switzerland, and also on the key to the Swiss Bank deposit box in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 146 and 171-172).Greek crosses can tile the plane, as noted by the protagonist Christopher in The Curious Incident of the Dog in the Night-Time (Haddon 2003, pp. 203-204).

### Cardinal exponentiation

Let and be any sets, and let be the cardinal number of a set . Then cardinal exponentiation is defined by(Ciesielski 1997, p. 68; Dauben 1990, p. 174; Moore 1982, p. 37; Rubin 1967, p. 275, Suppes 1972, p. 116).It is easy to show that the cardinal number of the power set of is , since and there is a natural bijection between the subsets of and the functions from into .

### Cardinal comparison

For any sets and , their cardinal numbers satisfy iff there is a one-to-one function from into (Rubin 1967, p. 266; Suppes 1972, pp. 94 and 116). It is easy to show this satisfies the reflexive and transitive axioms of a partial order. However, it is difficult to show the antisymmetry property, whose proof is known as the Schröder-Bernstein theorem. To show the trichotomy property, one must use the axiom of choice.Although an order type can be defined similarly, it does not seem usual to do so.

Let and be any sets with empty intersection, and let denote the cardinal number of a set . Then(Ciesielski 1997, p. 68; Dauben 1990, p. 173; Rubin 1967, p. 274; Suppes 1972, pp. 112-113).It is an interesting exercise to show that cardinal addition is well-defined. The main steps are to show that for any cardinal numbers and , there exist disjoint sets and with cardinal numbers and , and to show that if and are disjoint and and disjoint with and then . The second of these is easy. The first is a little tricky and requires an appeal to the axioms of set theory. Also, one needs to restrict the definition of cardinal to guarantee if is a cardinal, then there is a set satisfying .

### Piriform surface

A generalization to a quartic three-dimensional surface is the quartic surface of revolution(1)illustrated above. With , this surface is termed the "zeck" surface by Hauser. It has volume(2)geometric centroid(3)(4)(5)and inertia tensor(6)for constant density and mass .

### Piriform curve

A quartic algebraic curve also called the peg-top curve and given by the Cartesian equation(1)and the parametric curves(2)(3)for . It was studied by G. de Longchamps in 1886.The area of the piriform is(4)which is exactly the same as the ellipse with semiaxes and .The curvature of the piriform is given by(5)

### Bifolium

The bifolium is a folium with . The bifolium is a quartic curve and is given by the implicit equation is(1)and the polar equation(2)The bifolium has area(3)(4)(5)Its arc length is(6)(7)(OEIS A118307), where , , , and are elliptic integrals with(8)(9)The curvature is given by(10)(11)The bifolium is the pedal curve of the deltoid where the pedal point is the midpoint of one of the three curved sides.

### Rabbit sequence

A sequence which arises in the hypothetical reproduction of a population of rabbits. Let the substitution system map correspond to young rabbits growing old, and correspond to old rabbits producing young rabbits. Starting with 0 and iterating using string rewriting gives the terms 1, 10, 101, 10110, 10110101, 1011010110110, .... A recurrence plot of the limiting value of this sequence is illustrated above.Converted to decimal, this sequence gives 1, 2, 5, 22, 181, ... (OEIS A005203), with the th term given by the recurrence relationwith , , and the th Fibonacci number.The limiting sequence written as a binary fraction (OEIS A005614), where denotes a binary number (i.e., a number written in base 2, so or 1), is called the rabbit constant.

### Mephisto waltz sequence

The Mephisto waltz sequence is defined by beginning with 0 and then iterating the maps and . This gives 0, 001, 001001110, 001001110001001110110110001, ... (OEIS A064990). These words are fourth power-free (Allouche and Shallit 2003, p. 25).The numbers of 0s and 1s in step , 1, ... are given by 1, 2, 5, 14, 41, 122, ... (OEIS A007051) and 0, 1, 4, 13, 40, 121, ... (OEIS A003462), respectively, which are given in closed form by and , respectively.A recurrence plot of the Mephisto waltz sequenceis illustrated above.

### Laplace transform

The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.The (unilateral) Laplace transform (not to be confused with the Lie derivative, also commonly denoted ) is defined by(1)where is defined for (Abramowitz and Stegun 1972). The unilateral Laplace transform is almost always what is meant by "the" Laplace transform, although a bilateral Laplace transform is sometimes also defined as(2)(Oppenheim et al. 1997). The unilateral Laplace transform is implemented in the Wolfram Language as LaplaceTransform[f[t], t, s] and the inverse Laplace transform as InverseRadonTransform.The inverse Laplace transform is known as the Bromwich integral, sometimes known as the Fourier-Mellin integral..

### Divisible

A number is said to be divisible by if is a divisor of .The function Divisible[n, d] returns True if an integer is divisible by an integer .The product of any consecutive integers is divisible by . The sum of any consecutive integers is divisible by if is odd, and by if is even.

### Laplace distribution

The Laplace distribution, also called the double exponential distribution, is the distribution of differences between two independent variates with identical exponential distributions (Abramowitz and Stegun 1972, p. 930). It had probability density function and cumulative distribution functions given by(1)(2)It is implemented in the Wolfram Language as LaplaceDistribution[mu, beta].The moments about the mean are related to the moments about 0 by(3)where is a binomial coefficient, so(4)(5)where is the floor function and is the gamma function.The moments can also be computed using the characteristicfunction,(6)Using the Fourier transform ofthe exponential function(7)gives(8)(Abramowitz and Stegun 1972, p. 930). The momentsare therefore(9)The mean, variance, skewness,and kurtosis excess are(10)(11)(12)(13)..

### Disjunction

The term in logic used to describe the operation commonly known as OR. A literal is considered a (degenerate) disjunction (Mendelson 1997, p. 30).The Wolfram Language command Disjunction[expr, a1, a2, ...] gives the disjunction of expr over all choices of the Boolean variables .

### Kurtosis

Kurtosis is defined as a normalized form of the fourth central moment of a distribution. There are several flavors of kurtosis, the most commonly encountered variety of which is normally termed simply "the" kurtosis and is denoted (Pearson's notation; Abramowitz and Stegun 1972, p. 928) or (Kenney and Keeping 1951, p. 27; Kenney and Keeping 1961, pp. 99-102). The kurtosis of a theoretical distribution is defined by(1)where denotes the th central moment (and in particular, is the variance). This form is implemented in the Wolfram Language as Kurtosis[dist].The "kurtosis excess" (Kenney and Keeping1951, p. 27) is defined by(2)(3)and is commonly denoted (Abramowitz and Stegun 1972, p. 928) or . Kurtosis excess is commonly used because of a normal distribution is equal to 0, while the kurtosis proper is equal to 3. Unfortunately, Abramowitz and Stegun (1972) confusingly refer to as..

### Kronecker symbol

The Kronecker symbol is an extension of the Jacobi symbol to all integers. It is variously written as or (Cohn 1980; Weiss 1998, p. 236) or (Dickson 2005). The Kronecker symbol can be computed using the normal rules for the Jacobi symbol(1)(2)(3)plus additional rules for ,(4)and . The definition for is variously written as(5)or(6)(Cohn 1980). Cohn's form "undefines" for singly even numbers and , probably because no other values are needed in applications of the symbol involving the binary quadratic form discriminants of quadratic fields, where and always satisfies .The Kronecker symbol is implemented in the Wolfram Language as KroneckerSymbol[n, m].The Kronecker symbol is a real number theoretic character modulo , and is, in fact, essentially the only type of real primitive character (Ayoub 1963).The illustration above and table below summarize for , 2, ... and small .OEISperiodA109017241, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1,..

### Discrete uniform distribution

The discrete uniform distribution is also known as the "equally likely outcomes" distribution. Letting a set have elements, each of them having the same probability, then(1)(2)(3)(4)so using gives(5)Restricting the set to the set of positive integers 1, 2, ..., , the probability distribution function and cumulative distributions function for this discrete uniform distribution are therefore(6)(7)for , ..., .The discrete uniform distribution is implemented in the WolframLanguage as DiscreteUniformDistribution[n].Its moment-generating function is(8)(9)(10)(11)The moments about 0 are(12)so(13)(14)(15)(16)and the moments about the meanare(17)(18)(19)The mean, variance, skewness,and kurtosis excess are(20)(21)(22)(23)The mean deviation for a uniform distribution on elements is given by(24)To do the sum, consider separately the cases of odd and even. For odd,(25)(26)(27)(28)Similarly, for even,(29)(30)(31)(32)The..

### Power

A power is an exponent to which a given quantity is raised. The expression is therefore known as " to the th power." A number of powers of are plotted above (cf. Derbyshire 2004, pp. 68 and 73).The power may be an integer, real number, or complex number. However, the power of a real number to a non-integer power is not necessarily itself a real number. For example, is real only for .A number other than 0 taken to the power 0 is defined to be 1, which followsfrom the limit(1)This fact is illustrated by the convergence of curves at in the plot above, which shows for , 0.4, ..., 2.0. It can also be seen more intuitively by noting that repeatedly taking the square root of a number gives smaller and smaller numbers that approach one from above, while doing the same with a number between 0 and 1 gives larger and larger numbers that approach one from below. For square roots, the total power taken is , which approaches 0 as is large, giving in the limit that..

### Hypergeometric distribution

Let there be ways for a "good" selection and ways for a "bad" selection out of a total of possibilities. Take samples and let equal 1 if selection is successful and 0 if it is not. Let be the total number of successful selections,(1)The probability of successful selections is then(2)(3)(4)The hypergeometric distribution is implemented in the Wolfram Language as HypergeometricDistribution[N, n, m+n].The problem of finding the probability of such a picking problem is sometimes called the "urn problem," since it asks for the probability that out of balls drawn are "good" from an urn that contains "good" balls and "bad" balls. It therefore also describes the probability of obtaining exactly correct balls in a pick- lottery from a reservoir of balls (of which are "good" and are "bad"). For example, for and , the probabilities of obtaining correct balls..

### Hyperfactorial

The hyperfactorial (Sloane and Plouffe 1995) is the function defined by(1)(2)where is the K-function.The hyperfactorial is implemented in the WolframLanguage as Hyperfactorial[n].For integer values , 2, ... are 1, 4, 108, 27648, 86400000, 4031078400000, 3319766398771200000, ... (OEIS A002109).The hyperfactorial can also be generalized to complex numbers, as illustrated above.The Barnes G-function and hyperfactorial satisfy the relation(3)for all complex .The hyperfactorial is given by the integral(4)and the closed-form expression(5)for , where is the Riemann zeta function, its derivative, is the Hurwitz zeta function, and(6) also has a Stirling-like series(7)(OEIS A143475 and A143476). has the special value(8)(9)(10)where is the Euler-Mascheroni constant and is the Glaisher-Kinkelin constant.The derivative is given by(11)..

### Block matrix

A block matrix is a matrix that is defined using smallermatrices, called blocks. For example,(1)where , , , and are themselves matrices, is a block matrix. In the specific example(2)(3)(4)(5)therefore, it is the matrix(6)Block matrices can be created using ArrayFlatten.When two block matrices have the same shape and their diagonal blocks are square matrices, then they multiply similarly to matrix multiplication. For example,(7)Note that the usual rules of matrix multiplication hold even when the block matrices are not square (assuming that the block sizes correspond). Of course, matrix multiplication is in general not commutative, so in these block matrix multiplications, it is important to keep the correct order of the multiplications.When the blocks are square matrices, the set of invertible block matrices is a group isomorphic to the general linear group , where is the ring of square matrices...

### Histogram

The grouping of data into bins (spaced apart by the so-called class interval) plotting the number of members in each bin versus the bin number. The above histogram shows the number of variates in bins with class interval 1 for a sample of 100 real variates with a uniform distribution from 0 and 10. Therefore, bin 1 gives the number of variates in the range 0-1, bin 2 gives the number of variates in the range 1-2, etc. Histograms are implemented in the Wolfram Language as Histogram[data].

### Hilbert curve

The Hilbert curve is a Lindenmayer system invented by Hilbert (1891) whose limit is a plane-filling function which fills a square. Traversing the polyhedron vertices of an -dimensional hypercube in Gray code order produces a generator for the -dimensional Hilbert curve. The Hilbert curve can be simply encoded with initial string "L", string rewriting rules "L" -> "+RF-LFL-FR+", "R" -> "-LF+RFR+FL-", and angle (Peitgen and Saupe 1988, p. 278). The th iteration of this Hilbert curve is implemented in the Wolfram Language as HilbertCurve[n].A related curve is the Hilbert II curve, shown above (Peitgen and Saupe 1988, p. 284). It is also a Lindenmayer system and the curve can be encoded with initial string "X", string rewriting rules "X" -> "XFYFX+F+YFXFY-F-XFYFX", "Y" -> "YFXFY-F-XFYFX+F+YFXFY",..

### Bit length

The number of binary bits necessary to represent a number, given explicitly by(1)(2)where is the ceiling function, is the floor function, and is lg, the logarithm to base 2. For , 2, ..., the sequence of bit lengths is given by 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, ... (OEIS A036377). The function is given by the Wolfram Language function BitLength[n].

### Polygonal number

A polygonal number is a type of figurate number that is a generalization of triangular, square, etc., to an -gon for an arbitrary positive integer. The above diagrams graphically illustrate the process by which the polygonal numbers are built up. Starting with the th triangular number , then(1)Now note that(2)gives the th square number,(3)gives the th pentagonal number, and so on. The general polygonal number can be written in the form(4)(5)where is the th -gonal number (Savin 2000). For example, taking in (5) gives a triangular number, gives a square number, etc.Polygonal numbers are implemented in the WolframLanguage as PolygonalNumber.Call a number -highly polygonal if it is -polygonal in or more ways out of , 4, ... up to some limit. Then the first few 2-highly polygonal numbers up to are 1, 6, 9, 10, 12, 15, 16, 21, 28, (OEIS A090428). Similarly, the first few 3-highly polygonal numbers up to are 1, 15, 36, 45, 325, 561, 1225, 1540, 3025, ... (OEIS..

### Binomial distribution

The binomial distribution gives the discrete probability distribution of obtaining exactly successes out of Bernoulli trials (where the result of each Bernoulli trial is true with probability and false with probability ). The binomial distribution is therefore given by(1)(2)where is a binomial coefficient. The above plot shows the distribution of successes out of trials with .The binomial distribution is implemented in the Wolfram Language as BinomialDistribution[n, p].The probability of obtaining more successes than the observed in a binomial distribution is(3)where(4) is the beta function, and is the incomplete beta function.The characteristic function for the binomialdistribution is(5)(Papoulis 1984, p. 154). The moment-generating function for the distribution is(6)(7)(8)(9)(10)(11)The mean is(12)(13)(14)The moments about 0 are(15)(16)(17)(18)so the moments about the meanare(19)(20)(21)The skewness..

### B&eacute;zier curve

Given a set of control points , , ..., , the corresponding Bézier curve (or Bernstein-Bézier curve) is given bywhere is a Bernstein polynomial and . Bézier splines are implemented in the Wolfram Language as BezierCurve[pts].A "rational" Bézier curve is defined bywhere is the order, are the Bernstein polynomials, are control points, and the weight of is the last ordinate of the homogeneous point . These curves are closed under perspective transformations, and can represent conic sections exactly.The Bézier curve always passes through the first and last control points and lies within the convex hull of the control points. The curve is tangent to and at the endpoints. The "variation diminishing property" of these curves is that no line can have more intersections with a Bézier curve than with the curve obtained by joining consecutive points with straight line segments...

### Poisson distribution

Given a Poisson process, the probability of obtaining exactly successes in trials is given by the limit of a binomial distribution(1)Viewing the distribution as a function of the expected number of successes(2)instead of the sample size for fixed , equation (2) then becomes(3)Letting the sample size become large, the distribution then approaches(4)(5)(6)(7)(8)which is known as the Poisson distribution (Papoulis 1984, pp. 101 and 554; Pfeiffer and Schum 1973, p. 200). Note that the sample size has completely dropped out of the probability function, which has the same functional form for all values of .The Poisson distribution is implemented in the WolframLanguage as PoissonDistribution[mu].As expected, the Poisson distribution is normalized so that the sum of probabilities equals 1, since(9)The ratio of probabilities is given by(10)The Poisson distribution reaches a maximum when(11)where is the Euler-Mascheroni..

### Beta distribution

A general type of statistical distribution which is related to the gamma distribution. Beta distributions have two free parameters, which are labeled according to one of two notational conventions. The usual definition calls these and , and the other uses and (Beyer 1987, p. 534). The beta distribution is used as a prior distribution for binomial proportions in Bayesian analysis (Evans et al. 2000, p. 34). The above plots are for various values of with and ranging from 0.25 to 3.00.The domain is , and the probability function and distribution function are given by(1)(2)(3)where is the beta function, is the regularized beta function, and . The beta distribution is implemented in the Wolfram Language as BetaDistribution[alpha, beta].The distribution is normalized since(4)The characteristic function is(5)(6)where is a confluent hypergeometric function of the first kind.The raw moments are given by(7)(8)(Papoulis 1984,..

### Beta binomial distribution

A variable with a beta binomial distribution is distributed as a binomial distribution with parameter , where is distribution with a beta distribution with parameters and . For trials, it has probability density function(1)where is a beta function and is a binomial coefficient, and distribution function(2)where is a gamma function and(3)is a generalized hypergeometricfunction.It is implemented as BetaBinomialDistribution[alpha,beta, n].The first few raw moments are(4)(5)(6)giving the mean and varianceas(7)(8)

### Planar graph

A graph is planar if it can be drawn in a plane without graph edges crossing (i.e., it has graph crossing number 0). The number of planar graphs with , 2, ... nodes are 1, 2, 4, 11, 33, 142, 822, 6966, 79853, ... (OEIS A005470; Wilson 1975, p. 162), the first few of which are illustrated above.The corresponding numbers of planar connected graphs are 1, 1, 1, 2, 6, 20, 99, 646, 5974, 71885, ... (OEIS A003094; Steinbach 1990, p. 131)There appears to be no term in standard use for a graph with graph crossing number 1 (the terms "almost planar" and "1-planar" are used in the literature for other concepts).Note that while graph planarity is an inherent property of a graph, it is still sometimes possible to draw nonplanar embeddings of planar graphs. For example, the two embeddings above both correspond to the planar tetrahedral graph, but while the left embedding is planar, the right embedding is not.A planar embedding of a..

### Hamburger moment problem

A necessary and sufficient condition that there should exist at least one nondecreasing function such thatfor , 1, 2, ..., with all the integrals converging, is that sequence is positive definite (Widder 1941, p. 129).

### Operation

Let be a set. An operation on is a function from a power of into . More precisely, given an ordinal number , a function from into is an -ary operation on . If is a finite ordinal, then the -ary operation is a finitary operation on .

### Vector space span

The span of subspace generated by vectors and isA set of vectors can be tested to see if they span -dimensional space using the following Wolfram Language function: SpanningVectorsQ[m_List?MatrixQ] := (NullSpace[m] == {})

### Cross product

For vectors and in , the cross product in is defined by(1)(2)where is a right-handed, i.e., positively oriented, orthonormal basis. This can be written in a shorthand notation that takes the form of a determinant(3)where , , and are unit vectors. Here, is always perpendicular to both and , with the orientation determined by the right-hand rule.Special cases involving the unit vectors in three-dimensionalCartesian coordinates are given by(4)(5)(6)The cross product satisfies the general identity(7)Note that is not a usual polar vector, but has slightly different transformation properties and is therefore a so-called pseudovector (Arfken 1985, pp. 22-23). Jeffreys and Jeffreys (1988) use the notation to denote the cross product.The cross product is implemented in the Wolfram Language as Cross[a, b].A mathematical joke asks, "What do you get when you cross a mountain-climber with a mosquito?" The answer is, "Nothing:..

### Cross

In general, a cross is a figure formed by two intersecting line segments. In linear algebra, a cross is defined as a set of mutually perpendicular pairs of vectors of equal magnitude from a fixed origin in Euclidean -space.The word "cross" is also used to denote the operation of the cross product, so would be pronounced " cross ."

### Null space

If is a linear transformation of , then the null space Null(), also called the kernel , is the set of all vectors such thati.e.,The term "null space" is most commonly written as two separate words (e.g., Golub and Van Loan 1989, pp. 49 and 602; Zwillinger 1995, p. 128), although other authors write it as a single word "nullspace" (e.g., Anton 1994, p. 259; Robbin 1995, pp. 123 and 180).The Wolfram Language command NullSpace[v1, v2, ...] returns a list of vectors forming a vector basis for the nullspace of a set of vectors over the rationals (or more generally, over whatever base field contains the input vectors).

### Vector

A vector is formally defined as an element of a vector space. In the commonly encountered vector space (i.e., Euclidean n-space), a vector is given by coordinates and can be specified as . Vectors are sometimes referred to by the number of coordinates they have, so a 2-dimensional vector is often called a two-vector, an -dimensional vector is often called an n-vector, and so on.Vectors can be added together (vector addition), subtracted (vector subtraction) and multiplied by scalars (scalar multiplication). Vector multiplication is not uniquely defined, but a number of different types of products, such as the dot product, cross product, and tensor direct product can be defined for pairs of vectors.A vector from a point to a point is denoted , and a vector may be denoted , or more commonly, . The point is often called the "tail" of the vector, and is called the vector's "head." A vector with unit length is called a unit vector..

### Huffman coding

A lossless data compression algorithm which uses a small number of bits to encode common characters. Huffman coding approximates the probability for each character as a power of 1/2 to avoid complications associated with using a nonintegral number of bits to encode characters using their actual probabilities.Huffman coding works on a list of weights by building an extended binary tree with minimum weighted external path length and proceeds by finding the two smallest s, and , viewed as external nodes, and replacing them with an internal node of weight . The procedure is them repeated stepwise until the root node is reached. An individual external node can then be encoded by a binary string of 0s (for left branches) and 1s (for right branches).The procedure is summarized below for the weights 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, and 41 given by the first 13 primes, and the resulting tree is shown above (Knuth 1997, pp. 402-403). As is clear..

### Multiplicative order

Let be a positive number having primitive roots. If is a primitive root of , then the numbers 1, , , ..., form a reduced residue system modulo , where is the totient function. In this set, there are primitive roots, and these are the numbers , where is relatively prime to .The smallest exponent for which , where and are given numbers, is called the multiplicative order (or sometimes haupt-exponent or modulo order) of (mod ).The multiplicative order is implemented in the Wolfram Language as MultiplicativeOrder[g, n].The number of bases having multiplicative order is , where is the totient function. Cunningham (1922) published the multiplicative order for primes to 25409 and bases 2, 3, 5, 6, 7, 10, 11, and 12.Multiplicative orders exist for that are relatively prime to . For example, the multiplicative order of 10 (mod 7) is 6, since(1)The multiplicative order of 10 mod an integer relatively prime to 10 gives the period of the decimal expansion of the..

### Flow polynomial

Let denote the number of nowhere-zero -flows on a connected graph with vertex count , edge count , and connected component count . This quantity is called the flow polynomial of the graph , and is given by(1)(2)where is the rank polynomial and is the Tutte polynomial (extending Biggs 1993, p. 110).The flow polynomial of a graph can be computed in the Wolfram Language using FlowPolynomial[g, u].The flow polynomial of a planar graph is related to the chromatic polynomial of its dual graph by(3)The flow polynomial of a bridged graph, and therefore also of a tree on nodes, is 0.The flow polynomials for some special classes of graphs are summarized in the table below.graphflow polynomialbook graph cycle graph ladder graph prism graph web graph0wheel graph Linear recurrences for some special classes of graphs are summarized below.graphorderrecurrenceantiprism graph4book graph 2ladder graph 1prism graph 3wheel graph 2..

### Moving average

Given a sequence , an -moving average is a new sequence defined from the by taking the arithmetic mean of subsequences of terms,(1)So the sequences giving -moving averages are(2)(3)and so on. The plot above shows the 2- (red), 4- (yellow), 6- (green), and 8- (blue) moving averages for a set of 100 data points.Moving averages are implemented in the Wolfram Language as MovingAverage[data, n].

### Weibull distribution

The Weibull distribution is given by(1)(2)for , and is implemented in the Wolfram Language as WeibullDistribution[alpha, beta]. The raw moments of the distribution are(3)(4)(5)(6)and the mean, variance, skewness, and kurtosis excess of are(7)(8)(9)(10)where is the gamma function and(11)A slightly different form of the distribution is defined by(12)(13)(Mendenhall and Sincich 1995). This has raw moments(14)(15)(16)(17)so the mean and variance forthis form are(18)(19)The Weibull distribution gives the distribution of lifetimes of objects. It was originally proposed to quantify fatigue data, but it is also used in analysis of systems involving a "weakest link."

### Volume

The volume of a solid body is the amount of "space" it occupies. Volume has units of length cubed (i.e., , , , etc.) For example, the volume of a box (cuboid) of length , width , and height is given byThe volume can also be computed for irregularly-shaped and curved solids such as the cylinder and cone. The volume of a surface of revolution is particularly simple to compute due to its symmetry.The volume of a region can be computed in the WolframLanguage using Volume[reg].The following table gives volumes for some common surfaces. Here denotes the radius, the height, and the base area, and, in the case of the torus, the distance from the torus center to the center of the tube (Beyer 1987).surfacevolumeconeconical frustumcubecylinderellipsoidoblate spheroidprolate spheroidpyramidpyramidal frustumspherespherical capspherical sectorspherical segmenttorusEven simple surfaces can display surprisingly counterintuitive properties...

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