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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.

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)

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).

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).

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..

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.

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 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.

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

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).

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..

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 .

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.

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).

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.

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.

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]...

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.

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 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).

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.

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.

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 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.

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).

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, 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)

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.

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)

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.

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 ).

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).

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.

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..

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

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.

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.

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).

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).

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).

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..

A spanning tree of a graph on vertices is a subset of edges that form a tree (Skiena 1990, p. 227). For example, the spanning trees of the cycle graph , diamond graph, and complete graph are illustrated above.The number of nonidentical spanning trees of a graph is equal to any cofactor of the degree matrix of minus the adjacency matrix of (Skiena 1990, p. 235). This result is known as the matrix tree theorem. A tree contains a unique spanning tree, a cycle graph contains spanning trees, and a complete graph contains spanning trees (Trent 1954; Skiena 1990, p. 236). A count of the spanning trees of a graph can be found using the command NumberOfSpanningTrees[g] in the Wolfram Language package Combinatorica` . For a connected graph, it is also given bywhere is the Tutte polynomial.Kruskal's algorithm can be used to find a maximum or minimum spanning tree of graph.The following table summarizes the numbers of spanning trees for various..

Given a set , let be the set of neighbors of . Then the bipartite graph with bipartitions and has a perfect matching iff for all subsets of .

A block is a maximal biconnected subgraph of a given graph . In the illustration above, the blocks are , , and .If a graph is biconnected, then itself is called a block (Harary 1994, p. 26) or a biconnected graph (Skiena 1990, p. 175).

A graph is a minor of a graph if a copy of can be obtained from via repeated edge deletion and/or edge contraction.The Kuratowski reduction theorem states that any nonplanar graph has the complete graph or the complete bipartite graph as a minor. In addition, any snark has the Petersen graph as a minor, as conjectured by Tutte (1967; West 2000, p. 304) and proved by Robertson et al. The determination of graph minors is an NP-hard problem for which no good algorithms are known, although brute-force methods such as those due to Robertson, Sanders, and Thomas exist.

Assume that numbered pancakes are stacked, and that a spatula can be used to reverse the order of the top pancakes for . Then the pancake sorting problem asks how many such "prefix reversals" are sufficient to sort an arbitrary stack (Skiena 1990, p. 48).The maximum numbers of flips needed to sort a random stack of , 2, 3, ... pancakes are 0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (OEIS A058986), with the number of maximal stacks for , 3, ... being 1, 1, 3, 20, 2, 35, 455, ... (OEIS A067607).The following table (OEIS A092113) gives the numbers of stacks of pancakes requiring flips. A flattened version is shown above as a binary plot.0123456781121131221413611351412354820615207919928113327163014954313571903101635For example, the three stacks of four pancakes requiring the maximum of four flips are , , and , which can be ordered using the flip sequences , , and , respectively (illustrated above). Similarly, the two stacks of six pancakes..

In database structures, two quantities are generally of interest: the average number of comparisons required to 1. Find an existing random record, and 2. Insert a new random record into a data structure. Some constants which arise in the theory of digital tree searching are(1)(2)(3)(4)(5)(6)(OEIS A065442 and A065443), where is a q-polygamma function. Erdős (1948) proved that is irrational, and is sometimes known as the Erdős-Borwein constant.The expected number of comparisons for a successful search is(7)(8)and for an unsuccessful search is(9)(10)(OEIS A086309 and A086310). Here is the Euler-Mascheroni constant, , , and are small-amplitude periodic functions, and lg is the base 2 logarithm.The variance for searching is(11)(12)(OEIS A086311) and for inserting is(13)(14)(OEIS A086312).The expected number of pairs of twin vacancies in a digital search tree is(15)where(16)(17)(OEIS A086313), which can also be written(18)(Flajolet..

A tree is a mathematical structure that can be viewed as either a graph or as a data structure. The two views are equivalent, since a tree data structure contains not only a set of elements, but also connections between elements, giving a tree graph.Trees were first studied by Cayley (1857). McKay maintains a database of trees up to 18 vertices, and Royle maintains one up to 20 vertices.A tree is a set of straight line segments connected at their ends containing no closed loops (cycles). In other words, it is a simple, undirected, connected, acyclic graph (or, equivalently, a connected forest). A tree with nodes has graph edges. Conversely, a connected graph with nodes and edges is a tree. All trees are bipartite graphs (Skiena 1990, p. 213). Trees with no particular node singled out are sometimes called free trees (or unrooted tree), by way of distinguishing them from rooted trees (Skiena 1990, Knuth 1997).The points of connection are known..

For a set partition of elements, the -character string in which each character gives the set block (, , ...) in which the corresponding element belongs is called the restricted growth string (or sometimes the restricted growth function). For example, for the set partition , the restricted growth string would be 0122. If the set blocks are "sorted" so that , then the restricted growth string satisfies the inequality for , 2, ..., .

The problem of determining how many nonattacking kings can be placed on an chessboard. For , the solution is 16, as illustrated above (Madachy 1979). In general, the solutions are(1)(Madachy 1979), giving the sequence of doubled squares 1, 1, 4, 4, 9, 9, 16, 16, ... (OEIS A008794). This sequence has generating function(2)The minimal number of kings needed to occupy or attack every square on an chessboard (i.e., domination numbers for the king graphs) are given for , 2, ... by 1, 1, 1, 4, 4, 4, 9, 9, 9, 16, ... (OEIS A075561), with the case illustrated above and noted by (Madachy 1979, p. 39). In general, for an chessboard,(3)

The Christmas stocking theorem, also known as the hockey stick theorem, states that the sum of a diagonal string of numbers in Pascal's triangle starting at the th entry from the top (where the apex has ) on left edge and continuing down rows is equal to the number to the left and below (the "toe") bottom of the diagonal (the "heel"; Butterworth 2002). This follows from the identitywhere is a binomial coefficient.

As originally stated by Gould (1972),(1)where GCD is the greatest common divisor and is a binomial coefficient. This was subsequently extended by D. Singmaster to(2)(Sato 1975), and generalized by Sato (1975) to(3)An even larger generalization was obtained by Hitotumatu and Sato (1975), who defined(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)with(16)and showed that each of the twelve binomial coefficients , , , , , , , , , , , and has equal greatest common divisor.A second star of David theorem states that if two triangles are drawn centered on a given element of Pascal's triangle as illustrated above, then the products of the three numbers in the associated points of each of the two stars are the same (Butterworth 2002). This follows from the fact that(17)(18)(19)The second star of David theorem holds true not only for the usual binomial coefficients, but also for q-binomial coefficients, where the common product is given by(20)In..

A great circle is a section of a sphere that contains a diameter of the sphere (Kern and Bland 1948, p. 87). Sections of the sphere that do not contain a diameter are called small circles. A great circle becomes a straight line in a gnomonic projection (Steinhaus 1999, pp. 220-221).The shortest path between two points on a sphere, also known as an orthodrome, is a segment of a great circle. To find the great circle (geodesic) distance between two points located at latitude and longitude of and on a sphere of radius , convert spherical coordinates to Cartesian coordinates using(1)(Note that the latitude is related to the colatitude of spherical coordinates by , so the conversion to Cartesian coordinates replaces and by and , respectively.) Now find the angle between and using the dot product,(2)(3)(4)The great circle distance is then(5)For the Earth, the equatorial radius is km, or 3963 (statute) miles. Unfortunately, the flattening..

A linear transformation(1)(2)(3)is said to be an orthogonal transformationif it satisfies the orthogonality condition(4)where Einstein summation has been used and is the Kronecker delta.

Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. For example, the real line is homotopic to a single point, as is any tree. However, the circle is not contractible, but is homotopic to a solid torus. The basic version of homotopy is between maps. Two maps and are homotopic if there is a continuous mapsuch that and .Whether or not two subsets are homotopic depends on the ambient space. For example, in the plane, the unit circle is homotopic to a point, but not in the punctured plane . The puncture can be thought of as an obstacle.However, there is a way to compare two spaces via homotopy without ambient spaces. Two spaces and are homotopy equivalent if there are maps and such that the composition is homotopic to the identity map of and is homotopic to the identity map of . For example, the circle is not homotopic to a point, for then the constant map would be homotopic to the identity map of a circle, which is impossible..

Let and be the number of successes in variates taken from two populations. Define(1)(2)The estimator of the difference is then . Doing a so-called -transform,(3)where(4)The standard error is(5)(6)(7)

Given an matrix and a matrix , their Kronecker product , also called their matrix direct product, is an matrix with elements defined by(1)where(2)(3)For example, the matrix direct product of the matrix and the matrix is given by the following matrix,(4)(5)The matrix direct product is implemented in the Wolfram Language as KroneckerProduct[a, b].The matrix direct product gives the matrix of the linear transformation induced by the vector space tensor product of the original vector spaces. More precisely, suppose that(6)and(7)are given by and . Then(8)is determined by(9)

A spherical shell is a generalization of an annulus to three dimensions. A spherical shell is therefore the region between two concentric spheres of differing radii.The spherical shell is implemented in the Wolfram Language as SphericalShell[x, y, z, b, a].

A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius") from a given point (the "center"). Twice the radius is called the diameter, and pairs of points on the sphere on opposite sides of a diameter are called antipodes.Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). As a result, geometers call the surface of the usual sphere the 3-sphere, while topologists refer to it as the..

The word "degree" has many meanings in mathematics.The most common meaning is the unit of angle measure defined such that an entire rotation is . This unit harks back to the Babylonians, who used a base 60 number system. likely arises from the Babylonian year, which was composed of 360 days (12 months of 30 days each). The degree is subdivided into 60 arc minutes per degree, and 60 arc seconds per arc minute. In the Wolfram Language, the symbol giving the number of radians in one degree is Degree.The word "degree" is also used in many contexts where it is synonymous with "order," as applied for example to polynomials.

The term "cylinder" has a number of related meanings. In its most general usage, the word "cylinder" refers to a solid bounded by a closed generalized cylinder (a.k.a. cylindrical surface) and two parallel planes (Kern and Bland 1948, p. 32; Harris and Stocker 1998, p. 102). A cylinder of this sort having a polygonal base is therefore a prism (Zwillinger 1995, p. 308). Harris and Stocker (1998, p. 103) use the term "general cylinder" to refer to the solid bounded a closed generalized cylinder.Unfortunately, the term "cylinder" is commonly used not only to refer to the solid bounded by a cylindrical surface, but to the cylindrical surface itself (Zwillinger 1995, p. 311). To make matters worse, according to topologists, a cylindrical surface is not even a true surface, but rather a so-called surface with boundary (Henle 1994, pp. 110 and 129).As if this were..

A regular polygon is an -sided polygon in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both equiangular and equilateral). Only certain regular polygons are "constructible" using the classical Greek tools of the compass and straightedge.The terms equilateral triangle and square refer to the regular 3- and 4-polygons, respectively. The words for polygons with sides (e.g., pentagon, hexagon, heptagon, etc.) can refer to either regular or non-regular polygons, although the terms generally refer to regular polygons in the absence of specific wording.A regular -gon is implemented in the Wolfram Language as RegularPolygon[n], or more generally as RegularPolygon[r, n], RegularPolygon[x, y, rspec, n], etc.The sum of perpendiculars from any point to the sides of a regular polygon of sides is times the apothem.Let be the side length, be the inradius, and the circumradius..

A (finite, circular) conical surface is a ruled surface created by fixing one end of a line segment at a point (known as the vertex or apex of the cone) and sweeping the other around the circumference of a fixed circle (known as the base). When the vertex lies above the center of the base (i.e., the angle formed by the vertex, base center, and any base radius is a right angle), the cone is known as a right cone; otherwise, the cone is termed "oblique." When the base is taken as an ellipse instead of a circle, the cone is called an elliptic cone.In discussions of conic sections, the word "cone" is commonly taken to mean "double cone," i.e., two (possibly infinitely extending) cones placed apex to apex. The infinite double cone is a quadratic surface, and each single cone is called a "nappe." The hyperbola can then be defined as the intersection of a plane with both nappes of the double cone.As can be seen from the above,..

The previous prime function gives the largest prime less than . The function can be given explicitly aswhere is the th prime and is the prime counting function. For , 4, ... the values are 2, 3, 3, 5, 5, 7, 7, 7, 7, 11, 11, 13, 13, 13, 13, 17, 17, 19, ... (OEIS A007917).The previous prime function was implemented in versions of the Wolfram Language prior to 6 as PreviousPrime[n] (after loading the package NumberTheory`NumberTheoryFunctions).Finding the previous prime before gives the largest -digit prime. For , 2, ..., the first few of these are 7, 97, 997, 9973, 99991, 999983, 9999991, 99999989, ... (OEIS A003618). The amounts by which these are less than are 3, 3, 3, 3, 27, 9, 17, 9, 11, 63, ... (OEIS A033873).

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..

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)..

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).

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..

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..

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 . ..

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

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..

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.

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).

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..

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.

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).

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..

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).

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...

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 ...

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..

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 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,..

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).

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 .

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 .

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 .

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)

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.

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.

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.

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..

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.

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)..

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 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..

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,..

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..

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..

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..

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)..

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...

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].

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",..

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].

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..

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..

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...

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..

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,..

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)

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..

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).

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 .

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] == {})

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:..

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 ."

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).

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..

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..

The ratio of the largest to smallest singular value in the singular value decomposition of a matrix. The base- logarithm of is an estimate of how many base- digits are lost in solving a linear system with that matrix. In other words, it estimates worst-case loss of precision. A system is said to be singular if the condition number is infinite, and ill-conditioned if it is too large, where "too large" means roughly the precision of matrix entries.An estimate of the -norm condition number of a matrix can be computed in the Wolfram Language prior to Version 11.2 using LinearAlgebra`MatrixConditionNumber[m, p] for , 2, or , where omitting the is equivalent to specifying Infinity. A similar approximation for the condition number can be computed using LUDecomposition[mat][[-1]].

The curve given by the polar equation(1)sometimes also written(2)where .The cardioid has Cartesian equation(3)and the parametric equations(4)(5)The cardioid is a degenerate case of the limaçon. It is also a 1-cusped epicycloid (with ) and is the catacaustic formed by rays originating at a point on the circumference of a circle and reflected by the circle.The cardioid has a cusp at the origin.The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741. Its arc length was found by la Hire in 1708. There are exactly three parallel tangents to the cardioid with any given gradient. Also, the tangents at the ends of any chord through the cusp point are at right angles. The length of any chord through the cusp point is .The cardioid may also be generated as follows. Draw a circle and fix a point on it. Now draw a set of circles centered on the circumference of and passing through . The envelope of these circles..

In general, "a" calculus is an abstract theory developed in a purely formal way."The" calculus, more properly called analysis (or real analysis or, in older literature, infinitesimal analysis), is the branch of mathematics studying the rate of change of quantities (which can be interpreted as slopes of curves) and the length, area, and volume of objects. The calculus is sometimes divided into differential and integral calculus, concerned with derivativesand integralsrespectively.While ideas related to calculus had been known for some time (Archimedes' method of exhaustion was a form of calculus), it was not until the independent work of Newton and Leibniz that the modern elegant tools and ideas of calculus were developed. Even so, many years elapsed until the subject was put on a mathematically rigorous footing by mathematicians such as Weierstrass...

A strophoid of a circle with the pole at the center of the circle and the fixed point on the circumference of the circle. Freeth (1878, pp. 130 and 228) described this and various other strophoids (MacTutor Archive).It has polar equation(1)The area enclosed by the outer boundary of the curve is(2)and the total arc length is(3)(4)(OEIS A138498), where , is a complete elliptic integral of the first kind, is a complete elliptic integral of the second kind, and is a complete elliptic integral of the third kind.If the line through parallel to the y-axis cuts the nephroid at , then angle is , so this curve can be used to construct a regular heptagon.The curvature and tangentialangle are given by(5)(6)where is the floor function.

The 2-cusped epicycloid is called a nephroid. The name nephroid means "kidney shaped" and was first used for the two-cusped epicycloid by Proctor in 1878 (MacTutor Archive).The nephroid is the catacaustic for rays originating at the cusp of a cardioid and reflected by it. In addition, Huygens showed in 1678 that the nephroid is the catacaustic of a circle when the light source is at infinity, an observation which he published in his Traité de la luminère in 1690 (MacTutor Archive). (Trott 2004, p. 17, mistakenly states that the catacaustic for parallel light falling on any concave mirror is a nephroid.) The shape of the "flat visor curve" produced by a pop-up card dubbed the "knight's visor" is half a nephroid (Jakus and O'Rourke 2012).Since the nephroid has cusps, , and the equation for in terms of the parameter is given by epicycloid equation(1)with ,(2)where(3)This can be written(4)The..

Napier's bones, also called Napier's rods, are numbered rods which can be used to perform multiplication of any number by a number 2-9. By placing "bones" corresponding to the multiplier on the left side and the bones corresponding to the digits of the multiplicand next to it to the right, and product can be read off simply by adding pairs of numbers (with appropriate carries as needed) in the row determined by the multiplier. This process was published by Napier in 1617 an a book titled Rabdologia, so the process is also called rabdology.There are ten bones corresponding to the digits 0-9, and a special eleventh bone that is used the represent the multiplier. The multiplier bone is simply a list of the digits 1-9 arranged vertically downward. The remainder of the bones each have a digit written in the top square, with the multiplication table for that digits written downward, with the digits split by a diagonal line going from the lower left..

For some range of , the Mandelbrot set lemniscate in the iteration towards the Mandelbrot set is a pear-shaped curve. In Cartesian coordinates with a constant , the equation is given byThe plots above show the resulting curve for (left figure) and for a range of between 0 and 2 (right figure).

The thin plate spline is the two-dimensional analog of the cubic spline in one dimension. It is the fundamental solution to the biharmonic equation, and has the formGiven a set of data points, a weighted combination of thin plate splines centered about each data point gives the interpolation function that passes through the points exactly while minimizing the so-called "bending energy." Bending energy is defined here as the integral over of the squares of the second derivatives,Regularization may be used to relax the requirement that the interpolant pass through the data points exactly.The name "thin plate spline" refers to a physical analogy involving the bending of a thin sheet of metal. In the physical setting, the deflection is in the direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the or coordinates..

An algorithm for making tables of primes. Sequentially write down the integers from 2 to the highest number you wish to include in the table. Cross out all numbers which are divisible by 2 (every second number). Find the smallest remaining number . It is 3. So cross out all numbers which are divisible by 3 (every third number). Find the smallest remaining number . It is 5. So cross out all numbers which are divisible by 5 (every fifth number).Continue until you have crossed out all numbers divisible by , where is the floor function. The numbers remaining are prime. This procedure is illustrated in the above diagram which sieves up to 50, and therefore crosses out composite numbers up to . If the procedure is then continued up to , then the number of cross-outs gives the number of distinct prime factors of each number.The sieve of Eratosthenes can be used to compute the primecounting function aswhich is essentially an application of the inclusion-exclusionprinciple..

A McNugget number is a positive integer that can be obtained by adding together orders of McDonald's® Chicken McNuggetsTM (prior to consuming any), which originally came in boxes of 6, 9, and 20 (Vardi 1991, pp. 19-20 and 233-234; Wah and Picciotto 1994, p. 186). All integers are McNugget numbers except 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43. The value 43 therefore corresponds to the Frobenius number of .Since the Happy MealTM-sized nugget box (4 to a box) can now be purchased separately, the modern McNugget numbers are linear combinations of 4, 6, 9, and 20. These new-fangled numbers are much less interesting than before, with only 1, 2, 3, 5, 7, and 11 remaining as non-McNugget numbers. The value 11 therefore corresponds to the Frobenius number of .The greedy algorithm can be used to find a McNugget expansion of a given integer . This can also be done in the Wolfram Language using FrobeniusSolve[6,..

where is a modified Bessel function of the first kind and . For a derivation, see Papoulis (1962). For = 0, this reduces to the Rayleigh distribution.

A sieving procedure that can be used in conjunction with Dixon's factorization method to factor large numbers . Pick values of given by(1)where , 2, ... and is the floor function. We are then looking for factors such that(2)which means that only numbers with Legendre symbol (less than for trial divisor , where is the prime counting function) need be considered. The set of primes for which this is true is known as the factor base. Next, the congruences(3)must be solved for each in the factor base. Finally, a sieve is applied to find values of which can be factored completely using only the factor base. Gaussian elimination is then used as in Dixon's factorization method in order to find a product of the s, yielding a perfect square.The method requires about steps, improving on the continued fraction factorization algorithm by removing the 2 under the square root (Pomerance 1996). The use of multiple polynomials gives a better chance of factorization,..

The flower snarks are a family of snarks discovered by Isaacs (1975) and denoted . is Tietze's graph, which is a "reducible snark" since it contains a cycle of length less than 5. is illustrated above in two embeddings, the second of which appears in Scheinerman and Ullman (2011, p. 96) as an example of a graph with edge chromatic number and fractional edge chromatic number (4 and 3, respectively) both integers but not equal. is maximally nonhamiltonian for odd (Clark and Entringer 1983).

Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an chessboard? Since this is a geometric series, the answer for squares isa Mersenne number. Plugging in then gives .

The trefoil knot , also called the threefoil knot or overhand knot, is the unique prime knot with three crossings. It is a (3, 2)-torus knot and has braid word . The trefoil and its mirror image are not equivalent, as first proved by Dehn (1914). In other words, the trefoil knot is not amphichiral. It is, however, invertible, and has Arf invariant 1.Its laevo form is implemented in the WolframLanguage, as illustrated above, as KnotData["Trefoil"].M. C. Escher's woodcut "Knots" (Bool et al. 1982, pp. 128 and 325; Forty 2003, Plate 71) depicts three trefoil knots composed of differing types of strands. A preliminary study (Bool et al. 1982, p. 123) depicts another trefoil.The animation above shows a series of gears arranged along a Möbiusstrip trefoil knot (M. Trott).The bracket polynomial can be computed as follows.(1)(2)Plugging in(3)(4)gives(5)The corresponding Kauffman polynomial..

The "trefoil" curve is the name given by Cundy and Rollett (1989, p. 72) to the quartic plane curve given by the equation(1)As such, it is a simply a modification of the folium with and (2)obtained by dropping the coefficients 2.The area enclosed by the trefoil curve is(3)the geometric centroid of the enclosed region is(4)(5)and the area moment of inertia elements by(6)(7)(8)(E. Weisstein, Feb 3, 2018).

Given two curves and and a fixed point , let a line from cut at and at . Then the locus of a point such that is the cissoid. The word cissoid means "ivy shaped."curve 1curve 2polecissoidlineparallel lineany pointlinelinecirclecenter of circleconchoid of Nicomedescirclecircle tangent lineon circumferenceoblique cissoidcirclecircle tangent lineon circumference opposite tangentcissoid of Dioclescircleradial lineon circumferencestrophoidcircleconcentric circlecenter of circlescirclecirclesame circlelemniscate

A cactus graph, sometimes also called a cactus tree, a mixed Husimi tree, or a polygonal cactus with bridges, is a connected graph in which any two graph cycles have no edge in common. Equivalently, it is a connected graph in which any two (simple) cycles have at most one vertex in common.The inequalitywhere is the circuit rank and is the total number of undirected graph cycles holds for a connected graph iff it is a cactus graph (Volkmann 1996).Every cycle of a cactus graph is therefore chordless. However, there exist graphs (e.g., the -graph and Pasch graph) whose cycles are all chordless but which are not cactus graphs.Every cactus graph is a unit-distance graph(Erdős et al. 1965).Every pseudotree is a cactus graph.The numbers of cactus graphs on 1, 2, ... nodes are 1, 1, 2, 4, 9, 23, 63, 188, ...(OEIS A000083)...

A plane curve proposed by Descartes to challenge Fermat's extremum-finding techniques. In parametric form,(1)(2)The curve has a discontinuity at . The left wing is generated as runs from to 0, the loop as runs from 0 to , and the right wing as runs from to .In Cartesian coordinates,(3)(MacTutor Archive). The equation of the asymptote is(4)The curvature and tangentialangle of the folium of Descartes are(5)(6)where is the Heaviside step function.Converting the parametric equations to polar coordinates gives(7)(8)so the polar equation is(9)The area enclosed by the curve is(10)(11)(12)The arc length of the loop is given by(13)(14)

The folium (meaning leaf-shaped, referring to the lobes present in this curve), also known as Kepler's folium, is the curve with polar equation(1)Its Cartesian equation is(2)If , it is a single folium. If , it is a bifolium. If , it is a three-lobed curve sometimes called a trifolium. A modification of the case , is sometimes called the trefoil curve (Cundy and Rollett 1989, p. 72).The area of the folium is(3)The plots above show families of the folium for between 0 and 4.The simple folium is the pedal curve of the deltoidwhere the pedal point is one of the cusps.

A curve which has the shape of a petalled flower. This curve was named rhodonea by the Italian mathematician Guido Grandi between 1723 and 1728 because it resembles a rose (MacTutor Archive). The polar equation of the rose isorIf is odd, the rose is -petalled. If is even, the rose is -petalled.If is a rational number, then the curve closes at a polar angle of , where if is odd and if is even.If is irrational, then there are an infinite number of petals.The following table summarizes special names gives to roses with various values of .curve2quadrifolium3trifolium, paquerette de mélibée

A curve whose name means skull-like. It is given by the polar equationwhere , , , , and . The top of the curve corresponds to , while the bottom corresponds to .It has area given bywhere is an Appell hypergeometric function.

A curve whose name means "shell form." Let be a curve and a fixed point. Let and be points on a line from to meeting it at , where , with a given constant. For example, if is a circle and is on , then the conchoid is a limaçon, while in the special case that is the diameter of , then the conchoid is a cardioid. The equation for a parametrically represented curve with is(1)(2)

The inverse curve of the cochleoid(1)with inversion center at the origin and inversion radius is the quadratrix of Hippias.(2)(3)

The cochleoid, whose name means "snail-form" in Latin, was first considered by John Perks as referenced in Wallis et al. (1699). The cochleoid has also been called the oui-ja board curve (Beyer 1987, p. 215). The points of contact of parallel tangents to the cochleoid lie on a strophoid.Smith (1958, p. 327) gives historical references for the cochleoid, but corrections to the name and date mentioned as "discussed by J. Perk Phil. Trans. 1700" (actually John Perks, as mentioned in Wallis et al. 1699 and Pedersen 1963), the separateness of papers by Falkenburg (1844) and Benthem (1844), and the spelling of the latter's name should all be noted.In polar coordinates, the curve is given by(1)For the parametric form(2)(3)the curvature is(4)

Let be a function and let , and define the cardinal series of with respect to the interval as the formal serieswhere is the sinc function. If this series converges, it is known as the cardinal function (or Whittaker cardinal function) of , denoted (McNamee et al. 1971).

A surface which a monkey can straddle with both legsand his tail. A simple Cartesian equation for such a surface is(1)which can also be given by the parametric equations(2)(3)(4)The monkey saddle has a single stationary point as summarized in the table below. While the second derivative test is not sufficient to classify this stationary point, it turns out to be a saddle point.point20saddle pointThe coefficients of the first fundamental formof the monkey saddle are(5)(6)(7)and the second fundamental form coefficientsare(8)(9)(10)giving Riemannian metric(11)area element(12)and Gaussian and meancurvatures(13)(14)(Gray 1997). The Gaussian curvature can be written implicitly as(15)so every point of the monkey saddle except the origin has negative Gaussian curvature.

A surface of revolution defined by Kepler. It consists of less than half of a circular arc rotated about an axis passing through the endpoints of the arc. The equations of the upper and lower boundaries in the plane are(1)for and . The cross section of a lemon is a lens. The lemon is the inside surface of a spindle torus. The American football is shaped like a lemon.Two other lemon-shaped surfaces are given by the sexticsurface(2)called the "citrus" (or zitrus) surface by Hauser (left figure), and thesextic surface(3)whose upper and lower portions resemble two halves of a lemon, called the limão surface by Hauser (right figure).The citrus surface had bounding box , centroid at , volume(4)and a moment of inertia tensor(5)for a uniform density solid citrus with mass .

The kiss surface is the quintic surfaceof revolution given by the equation(1)that is closely related to the ding-dong surface. It is so named because the shape of the lower portion resembles that of a Hershey's Chocolate Kiss.It can be represented parametrically as(2)(3)(4)The coefficients of the first fundamental formare(5)(6)(7)and of the second fundamental form are(8)(9)(10)The Gaussian and meancurvatures are given by(11)(12)The Gaussian curvature can be given implicitlyby(13)The surface area and volumeenclosed of the top teardrop are given by(14)(15)Its centroid is at and the moment of inertia tensor is(16)for a solid kiss with uniform density and mass .

The surface given by the parametricequations(1)(2)(3)For , the coefficients of the first fundamental form are(4)(5)(6)and of the second fundamental form are(7)(8)(9)The Gaussian and meancurvatures are given by(10)(11)and the principal curvatures are(12)(13)

A heart-shaped surface given by the sextic equation(Taubin 1993, 1994). The figures above show a ray-traced rendering (left) and the cross section (right) of the surface.A slight variation of the same surface is given by(Nordstrand, Kuska 2004).

A surface of revolution defined by Kepler. It consists of more than half of a circular arc rotated about an axis passing through the endpoints of the arc. The equations of the upper and lower boundaries in the - plane are(1)for and . It is the outside surface of a spindle torus.It is also a quartic surface given by Cartesianequation(2)or(3)

A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions. A line is sometimes called a straight line or, more archaically, a right line (Casey 1893), to emphasize that it has no "wiggles" anywhere along its length. While lines are intrinsically one-dimensional objects, they may be embedded in higher dimensional spaces.Harary (1994) called an edge of a graph a "line."A line is uniquely determined by two points, and the line passing through points and is denoted . Similarly, the length of the finite line segment terminating at these points may be denoted . A line may also be denoted with a single lower-case letter (Jurgensen et al. 1963, p. 22).Euclid defined a line as a "breadthless length," and a straight line as a line that "lies evenly with the points on itself" (Kline 1956, Dunham 1990).Consider first lines in a two-dimensional plane. Two..

The 27 real or imaginary lines which lie on the general cubic surface and the 45 triple tangent planes to the surface. All are related to the 28 bitangents of the general quartic curve.Schoute (1910) showed that the 27 lines can be put into a one-to-one correspondence with the vertices of a particular polytope in six-dimensional space in such a manner that all incidence relations between the lines are mirrored in the connectivity of the polytope and conversely (Du Val 1933). A similar correspondence can be made between the 28 bitangents and a seven-dimensional polytope (Coxeter 1928) and between the tritangent planes of the canonical curve of genus four and an eight-dimensional polytope (Du Val 1933).

The lituus is an Archimedean spiral with , having polar equation(1)Lituus means a "crook," in the sense of a bishop's crosier. The lituus curve originated with Cotes in 1722. Maclaurin used the term lituus in his book Harmonia Mensurarum in 1722 (MacTutor Archive). The lituus is the locus of the point moving such that the area of a circular sector remains constant.The arc length, curvature,and tangential angle are given by(2)(3)(4)where the arc length is measured from .

The beautiful arrangement of leaves in some plants, called phyllotaxis, obeys a number of subtle mathematical relationships. For instance, the florets in the head of a sunflower form two oppositely directed spirals: 55 of them clockwise and 34 counterclockwise. Surprisingly, these numbers are consecutive Fibonacci numbers. The ratios of alternate Fibonacci numbers are given by the convergents to , where is the golden ratio, and are said to measure the fraction of a turn between successive leaves on the stalk of a plant: 1/2 for elm and linden, 1/3 for beech and hazel, 2/5 for oak and apple, 3/8 for poplar and rose, 5/13 for willow and almond, etc. (Coxeter 1969, Ball and Coxeter 1987). A similar phenomenon occurs for daisies, pineapples, pinecones, cauliflowers, and so on.Lilies, irises, and the trillium have three petals; columbines, buttercups, larkspur, and wild rose have five petals; delphiniums, bloodroot, and cosmos have eight petals;..

The Kakeya needle problems asks for the plane figure of least area in which a line segment of width 1 can be freely rotated (where translation of the segment is also allowed). Surprisingly, there is no minimum area (Besicovitch 1928). Another iterative construction which tends to as small an area as desired is called a Perron tree (Falconer 1990, Wells 1991).When the figure is restricted to be convex, the smallest region is an equilateral triangle of unit height. Wells (1991) states that Kakeya discovered this, while Falconer (1990) attributes it to Pál.If convexity is replaced by the weaker assumption of simply-connectedness, then the area can still be arbitrarily small, but if the set is required to be star-shaped, then is a known lower bound (Cunningham 1965).The smallest simple convex domain in which one can put a segment of length 1 which will coincide with itself when rotated by has area(OEIS A093823; Le Lionnais 1983). ..

Given a chord of a circle, draw any other two chords and passing through its midpoint. Call the points where and meet and . Then is also the midpoint of . There are a number of proofs of this theorem, including those by W. G. Horner, Johnson (1929, p. 78), and Coxeter (1987, pp. 78 and 144). The latter concise proof employs projective geometry.The following proof is given by Coxeter and Greitzer (1967, p. 46). In the figure at right, drop perpendiculars and from and to , and and from and to . Write , , and , and then note that by similar triangles(1)(2)(3)so(4)(5)so . Q.E.D.

An equilateral parallelogram whose acute angles are . Sometimes, the restriction to is dropped, and it is required only that two opposite angles are acute and the other two obtuse. The term rhombus is commonly used for an arbitrary equilateral parallelogram.The area of a lozenge of side length is(1)its diagonals have lengths(2)(3)and it has inradius(4)

In common usage, a cardinal number is a number used in counting (a countingnumber), such as 1, 2, 3, ....In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set. A set has (aleph-0) members if it can be put into a one-to-one correspondence with the finite ordinal numbers. The cardinality of a set is also frequently referred to as the "power" of a set (Moore 1982, Dauben 1990, Suppes 1972).In Georg Cantor's original notation, the symbol for a set annotated with a single overbar indicated stripped of any structure besides order, hence it represented the order type of the set. A double overbar then indicated stripping the order from the set and thus indicated..

Let and be any sets. Then the product of and is defined as the Cartesian product(Ciesielski 1997, p. 68; Dauben 1990, p. 173; Moore 1982, p. 37; Rubin 1967, p. 274; Suppes 1972, pp. 114-115).

Given a weighted, undirected graph and a graphical partition of into two sets and , the cut of with respect to and is defined aswhere denotes the weight for the edge connecting vertices and . The weight of the cut is the sum of weights of edges crossing the cut.

If is a weighted tree with weights assigned to each vertex , then is perfectly weighted if the matrixwhere is the adjacency matrix of (Butske et al. 1999).

The Lovász number of a graph satisfieswhere is the clique number, is the chromatic number of , and is the graph complement of . Furthermore, can be computed efficiently despite the fact that the computation of the two numbers it lies between is an NP-hard problem.The squeezing theorem is also sometimes knownas the sandwich theorem.

Let there be doctors and patients, and let all possible combinations of examinations of patients by doctors take place. Then what is the minimum number of surgical gloves needed so that no doctor must wear a glove contaminated by a patient and no patient is exposed to a glove worn by another doctor (where it is assumed that each doctor wears a glove on a single hand only)? In this problem, the gloves can be turned inside out and even placed on top of one another if necessary, but no "decontamination" of gloves is permitted. The optimal solution is(1)where is the ceiling function (Vardi 1991).The case is straightforward since two gloves have a total of four surfaces, which is the number needed for examinations. With doctors AB, patients ab, and gloves 12, a solution is A12a, A1b, B2a, B21b...

A branch cut is a curve (with ends possibly open, closed, or half-open) in the complex plane across which an analytic multivalued function is discontinuous. For convenience, branch cuts are often taken as lines or line segments. Branch cuts (even those consisting of curves) are also known as cut lines (Arfken 1985, p. 397), slits (Kahan 1987), or branch lines.For example, consider the function which maps each complex number to a well-defined number . Its inverse function , on the other hand, maps, for example, the value to . While a unique principal value can be chosen for such functions (in this case, the principal square root is the positive one), the choices cannot be made continuous over the whole complex plane. Instead, lines of discontinuity must occur. The most common approach for dealing with these discontinuities is the adoption of so-called branch cuts. In general, branch cuts are not unique, but are instead chosen by convention..

In complex analysis, a branch (also called a sheet) is a portion of the range of a multivalued function over which the function is single-valued. Combining all the sheets gives the full structure of the function. It is often convenient to choose a particular branch of a function to work with, and this choice is often designated the "principal branch" (or "principal sheet").In graph theory, a branch at a point in a tree is a maximal subtree containing as an endpoint (Harary 1994, p. 35).

If (with possible repetitions) satisfieswhere is the unit open disk, and no , then there is a bounded analytic function on which has zero set consisting precisely of the s, counted according to their multiplicities. More specifically, the infinite productwhere is a Blaschke factor and is the complex conjugate, converges uniformly on compact subsets of to a bounded analytic function .

Triskaidekaphobia is the fear of 13, a number commonly associated with bad luck in Western culture. While fear of the number 13 can be traced back to medieval times, the word triskaidekaphobia itself is of recent vintage, having been first coined by Coriat (1911; Simpson and Weiner 1992). It seems to have first appeared in the general media in a Nov. 8, 1953 New York Times article covering discussions of a United Nations committee.This superstition leads some people to fear or avoid anything involving the number 13. In particular, this leads to interesting practices such as the numbering of floors as 1, 2, ..., 11, 12, 14, 15, ... (OEIS A011760; the "elevator sequence"), omitting the number 13, in many high-rise American hotels, the numbering of streets avoiding 13th Avenue, and so on.Apparently, 13 hasn't always been considered unlucky. In fact, it appears that in ancient times, 13 was either considered in a positive light or..

A number in which the first decimal digits of the fractional part sum to 666 is known as an evil number (Pegg and Lomont 2004).However, the term "evil" is also used to denote nonnegative integers that have an even number of 1s in their binary expansions, the first few of which are 0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, ... (OEIS A001969), illustrated above as a binary plot. Numbers that are not evil are then known as odious numbers.Returning to Pegg's definition of evil, the fact that is evil was noted by Keith, while I. Honig (pers. comm., May 9, 2004) noted that the golden ratio is also evil. The following table gives a list of some common evil numbers (Pegg and Lomont 2004).Ramanujan constant 132hard hexagon entropy constant 137139140Stieltjes constant 142pi 144golden ratio 146146151Glaisher-Kinkelin constant 153cube line picking average length155Delian constant 156The probability of the digits of a given real number summing..

An extremely fast factorization method developed by Pollard which was used to factor the RSA-130 number. This method is the most powerful known for factoring general numbers, and has complexity(1)reducing the exponent over the continued fraction factorization algorithm and quadratic sieve. There are three values of relevant to different flavors of the method (Pomerance 1996). For the "special" case of the algorithm applied to numbers near a large power,(2)for the "general" case applicable to any odd positive number which is not a power,(3)and for a version using many polynomials (Coppersmith1993),(4)

Specifying two adjacent side lengths and of a triangle (with ) and one acute angle opposite does not, in general, uniquely determine a triangle.If , there are two possible triangles satisfying the given conditions (left figure). If , there is one possible triangle (middle figure). If , there are no possible triangles (right figure).Remember: Don't try to prove congruence with the ASS theorem or you will make an ASS out of yourself.An ASS triangle with sides and and excluded angle with has two possible side lengths ,The SSS or SAS theorems can then be used with either choice of to determine the angles and and triangle area .

Fermat's sandwich theorem states that 26 is the only number sandwiched between a perfect square number ( and a perfect cubic number (). According to Singh (1997), after challenging other mathematicians to establish this result while not revealing his own proof, Fermat took particular delight in taunting the English mathematicians Wallis and Digby with their inability to prove the result.

Archimedes' cattle problem, also called the bovinum problema, or Archimedes' reverse, is stated as follows: "The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown; the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the number of the spotted, one sixth and one seventh the number of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number of the black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number of the brown, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?"Solution consists..

A Diophantine problem (i.e., one whose solution must be given in terms of integers) which seeks a solution to the following problem. Given men and a pile of coconuts, each man in sequence takes th of the coconuts left after the previous man removed his (i.e., for the first man, , for the second, ..., for the last) and gives coconuts (specified in the problem to be the same number for each man) which do not divide equally to a monkey. When all men have so divided, they divide the remaining coconuts ways (i.e., taking an additional coconuts each), and give the coconuts which are left over to the monkey. If is the same at each division, then how many coconuts were there originally? The solution is equivalent to solving the Diophantine equations(1)(2)(3)(4)(5)which can be rewritten as(6)(7)(8)(9)(10)(11)Since there are equations in the unknowns , , ..., , , and , the solutions span a one-dimensional space (i.e., there is an infinite family of solution parameterized..

The Zipf distribution, sometimes referred to as the zeta distribution, is a discrete distribution commonly used in linguistics, insurance, and the modelling of rare events. It has probability density function(1)where is a positive parameter and is the Riemann zeta function, and distribution function(2)where is a generalized harmonic number.The Zipf distribution is implemented in the WolframLanguage as ZipfDistribution[rho].The th raw moment is(3)giving the mean and varianceas(4)(5)The distribution has mean deviation(6)where is a Hurwitz zeta function and is the mean as given above in equation (4).

The neighborhood graph of a given graph from a vertex is the subgraph induced by the neighborhood of a graph from vertex , most commonly including itself. Such graphs are sometimes also known in more recent literature as ego graphs or ego-centered networks (Newman 2010, pp. 44-46).A graph for which the neighborhood graph at each point excluding the point itself is isomorphic to a graph is said to be a local H graph, or simply "locally ."Neighborhood graphs are implemented in the Wolfram Language as NeighborhoodGraph[g, v].

The Frobenius number is the largest value for which the Frobenius equation(1)has no solution, where the are positive integers, is an integer, and the solutions are nonnegative integer. As an example, if the values are 4 and 9, then 23 is the largest unsolvable number. Similarly, the largest number that is not a McNugget number (a number obtainable by adding multiples of 6, 9, and 20) is 43.Finding the Frobenius number of a given problem is known as the coinproblem.Computation of the Frobenius number is implemented in the Wolfram Language as FrobeniusNumber[a1, ..., an].Sylvester (1884) showed(2)(3)

The negative binomial distribution, also known as the Pascal distribution or Pólya distribution, gives the probability of successes and failures in trials, and success on the th trial. The probability density function is therefore given by(1)(2)(3)where is a binomial coefficient. The distribution function is then given by(4)(5)(6)where is the gamma function, is a regularized hypergeometric function, and is a regularized beta function.The negative binomial distribution is implemented in the Wolfram Language as NegativeBinomialDistribution[r, p].Defining(7)(8)the characteristic function is given by(9)and the moment-generating functionby(10)Since ,(11)(12)(13)(14)The raw moments are therefore(15)(16)(17)(18)where(19)and is the Pochhammer symbol. (Note that Beyer 1987, p. 487, apparently gives the mean incorrectly.)This gives the central moments as(20)(21)(22)The mean, variance, skewnessand..

The function giving the fractional (noninteger) part of a real number . The symbol is sometimes used instead of (Graham et al. 1994, p. 70; Havil 2003, p. 109), but this notation is not used in this work due to possible confusion with the set containing the element .Unfortunately, there is no universal agreement on the meaning of for and there are two common definitions. Let be the floor function, then the Wolfram Language command FractionalPart[x] is defined as(1)(left figure). This definition has the benefit that , where is the integer part of . Although Spanier and Oldham (1987) use the same definition as the Wolfram Language, they mention the formula only very briefly and then say it will not be used further. Graham et al. (1994, p. 70), and perhaps most other mathematicians, use the different definition(2)(right figure). Min Max Re Im The fractional part function can also be extended to the complexplane as(3)as illustrated..

If for , ..., has a multivariate normal distribution with mean vector and covariance matrix , and denotes the matrix composed of the row vectors , then the matrix has a Wishart distribution with scale matrix and degrees of freedom parameter . The Wishart distribution is most typically used when describing the covariance matrix of multinormal samples. The Wishart distribution is implemented as WishartDistribution[sigma, m] in the Wolfram Language package MultivariateStatistics` .

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..

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..

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].

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."

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...

A vertex coloring is an assignment of labels or colors to each vertex of a graph such that no edge connects two identically colored vertices. A vertex coloring that minimize the number of colors needed for a given graph is known as a minimum vertex coloring of . The minimum number of colors itself is called the chromatic number, denoted , and a graph with chromatic number is said to be a k-chromatic graph.Brelaz's heuristic algorithm can be used to find a good, but not necessarily minimum vertex coloring. Finding a minimal coloring can be done using brute-force search (Christofides 1971; Wilf 1984; Skiena 1990, p. 214), though more sophisticated methods can be substantially faster. A minimal vertex coloring can be found for small graphs using backtracking with MinimumVertexColoring[g] in the Wolfram Language package Combinatorica` and Brelaz's algorithm can be applied using BrelazColoring[g].Mehrotra and Trick (1996) devised a column..

The minimum spanning tree of a weighted graph is a set of edges of minimum total weight which form a spanning tree of the graph. When a graph is unweighted, any spanning tree is a minimum spanning tree.The minimum spanning tree can be found in polynomial time. Common algorithms include those due to Prim (1957) and Kruskal's algorithm (Kruskal 1956). The problem can also be formulated using matroids (Papadimitriou and Steiglitz 1982). A minimum spanning tree can be found in the Wolfram Language using the command FindSpanningTree[g].The Season 1 episodes "Vector" and "Man Hunt" (2005) and Season 2 episode "Rampage" (2006) of the television crime drama NUMB3RS feature minimal spanning trees.

The vertex height of a vertex in a rooted tree is the number of edges on the longest downward path between and a tree leaf.The height of the root vertex of a rootedtree is known as the tree height.A function to return the height of a vertex in a tree may be implemented in a future version of the Wolfram Language as TreeHeight[g, v].

The depth of a vertex in a rooted tree as the number of edges from to the root vertex.A function to return the depth of a vertex in a tree may be implemented in a future version of the Wolfram Language as VertexDepth[g, v].

The vertex count of a graph , commonly denoted or , is the number of vertices in . In other words, it is the cardinality of the vertex set.The vertex count of a graph is implemented in the Wolfram Language as VertexCount[g]. The numbers of vertices for many named graphs are given by the command GraphData[graph, "VertexCount"].

The vertex connectivity of a graph is the minimum number of nodes whose deletion disconnects it. Vertex connectivity is sometimes called "point connectivity" or simply "connectivity."A graph with is said to be connected, a graph with is said to be biconnected (Skiena 1990, p. 177), and in general, a graph with vertex connectivity is said to be -connected.Let be the edge connectivity of a graph and its minimum degree, then for any graph,(Whitney 1932, Harary 1994, p. 43).For a connected strongly regular graph or distance-regular graph with vertex degree , (A. E. Brouwer, pers. comm., Dec. 17, 2012).The vertex connectivity of a graph can be determined in the Wolfram Language using VertexConnectivity[g]. Precomputed vertex connectivities are available for many named graphs via GraphData[graph, "VertexConnecitivity"]...

The Mellin transform is the integral transformdefined by(1)(2)It is implemented in the Wolfram Language as MellinTransform[expr, x, s]. The transform exists if the integral(3)is bounded for some , in which case the inverse exists with . The functions and are called a Mellin transform pair, and either can be computed if the other is known.The following table gives Mellin transforms of common functions (Bracewell 1999, p. 255). Here, is the delta function, is the Heaviside step function, is the gamma function, is the incomplete beta function, is the complementary error function erfc, and is the sine integral.convergenceAnother example of a Mellin transform is the relationship between the Riemann function and the Riemann zeta function ,(4)(5)A related pair is used in one proof of the primenumber theorem (Titchmarsh 1987, pp. 51-54 and equation 3.7.2)...

The uniform polyhedra are polyhedra with identical polyhedron vertices. Badoureau discovered 37 nonconvex uniform polyhedra in the late nineteenth century, many previously unknown (Wenninger 1983, p. 55). The uniform polyhedra include the Platonic solids and Kepler-Poinsot solids.Coxeter et al. (1954) conjectured that there are 75 such polyhedra in which only two faces are allowed to meet at an polyhedron edge, and this was subsequently proven. The five pentagonal prisms can also be considered uniform polyhedra, bringing the total to 80. In addition, there are two other polyhedra in which four faces meet at an edge, the great complex icosidodecahedron and small complex icosidodecahedron (both of which are compounds of two uniform polyhedra).The polyhedron vertices of a uniform polyhedron all lie on a sphere whose center is their geometric centroid (Coxeter et al. 1954, Coxeter 1973, p. 44. The polyhedron vertices joined..

Two geometric figures are said to exhibit geometric congruence (or "be geometrically congruent") iff one can be transformed into the other by an isometry (Coxeter and Greitzer 1967, p. 80). This relationship is written . (Unfortunately, the symbol is also used to denote an isomorphism.)

The study of angles and of the angular relationships of planar and three-dimensional figures is known as trigonometry. The trigonometric functions (also called the circular functions) comprising trigonometry are the cosecant , cosine , cotangent , secant , sine , and tangent . The inverses of these functions are denoted , , , , , and . Note that the notation here means inverse function, not to the power.The trigonometric functions are most simply defined using the unit circle. Let be an angle measured counterclockwise from the x-axis along an arc of the circle. Then is the horizontal coordinate of the arc endpoint, and is the vertical component. The ratio is defined as . As a result of this definition, the trigonometric functions are periodic with period , so(1)where is an integer and func is a trigonometric function.A right triangle has three sides, which can be uniquely identified as the hypotenuse, adjacent to a given angle , or opposite . A helpful..

"SOHCAHTOA" is a helpful mnemonic for remembering the definitions of the trigonometric functions sine, cosine, and tangent i.e., sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, and tangent equals opposite over adjacent,(1)(2)(3)Other mnemonics include 1. "Tommy On A Ship Of His Caught A Herring" (probably more common in Great Britain than the United States). 2. "Oscar Has A Hold On Angie." 3. "Oscar Had A Heap of Apples." 4. "The Old Army Colonel And His Son Often Hiccup" (which gives the functions in the order tangent, cosine, sine). 5. "Studying Our Homework Can Always Help To Obtain Achievement." 6. "Some Old Hippy Caught Another Hippy Tripping On Acid."

Given a group of men arranged in a circle under the edict that every th man will be executed going around the circle until only one remains, find the position in which you should stand in order to be the last survivor (Ball and Coxeter 1987). The list giving the place in the execution sequence of the first, second, etc. man can be given by Josephus[n, m] in the Wolfram Language package Combinatorica` . For example, consider men numbered 1 to 4 such that each second () man is iteratively slaughtered, as illustrated above. As can be seen, the first man is slaughtered 4th, the second man 1st, the third man 3rd, and the fourth man 2nd, so Josephus[4, 2] returns 4, 1, 3, 2.To obtain the ordered list of men who are consecutively slaughtered, InversePermutation can be applied to the output of Josephus. So, in the above example, InversePermutation[Josephus[4, 2]] returns 2, 4, 3, 1 since the 2nd man is slaughtered first, the 4th man is slaughtered second, the 3rd man..

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