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)
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].
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..
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.
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..
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 .
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 ."
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..
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" 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 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.
A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles (like the equator). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration.Geodesics preserve a direction on a surface (Tietze 1965, pp. 26-27) and have many other interesting properties. The normal vector to any point of a geodesic arc lies along the normal to a surface at that point (Weinstock 1974, p. 65).Furthermore, no matter how badly a sphere is distorted, there exist an infinite number of closed geodesics on it. This general result, demonstrated in the early 1990s, extended earlier work by Birkhoff, who proved in 1917 that there exists at least one closed geodesic on a distorted sphere, and Lyusternik and Schnirelmann, who proved in 1923 that..
There are at least two theorems known as Salmon's theorem. This first states that if and are two points, and are the perpendiculars from and to the polars of and , respectively, with respect to a circle with center , then (Durell 1928; Salmon 1954, §101, p. 93).The second Salmon's theorem states that, given a track bounded by two confocal ellipses, if a ball is rolled so that its trajectory is tangent to the inner ellipse, the ball's trajectory will be tangent to the inner ellipse following all subsequent caroms as well (Salmon 1954, §189, pp. 181-182).
The radical lines of three circles are concurrent in a point known as the radical center (also called the power center). This theorem was originally demonstrated by Monge (Dörrie 1965, p. 153). It is a special case of the three conics theorem (Evelyn et al. 1974, pp. 13 and 15).The point of concurrence of the three radical lines of three circles is the point(Kimberling 1998, p. 225).
Buffon's needle problem asks to find the probability that a needle of length will land on a line, given a floor with equally spaced parallel lines a distance apart. The problem was first posed by the French naturalist Buffon in 1733 (Buffon 1733, pp. 43-45), and reproduced with solution by Buffon in 1777 (Buffon 1777, pp. 100-104).Define the size parameter by(1)For a short needle (i.e., one shorter than the distance between two lines, so that ), the probability that the needle falls on a line is(2)(3)(4)(5)For , this therefore becomes(6)(OEIS A060294).For a long needle (i.e., one longer than the distance between two lines so that ), the probability that it intersects at least one line is the slightly more complicated expression(7)where (Uspensky 1937, pp. 252 and 258; Kunkel).Writing(8)then gives the plot illustrated above. The above can be derived by noting that(9)where(10)(11)are the probability functions for the..
Inspired by computer simulations of fossilized worms trails published by Raup and Seilacher (1969), computer scientist Mike Paterson at the University of Warwick and mathematician J. H. Conway created in early 1971 a simple set to rules to study idealized worms traveling along regular grids. Mike Beeler of the MIT Artificial Intelligence Laboratory subsequently published a study Paterson's worms in which he considered paths on a triangular grid (Beeler 1973).The following table summarizes the number of steps required for a number of long-running worms to terminate (Rokicki).patternsteps to terminate10420151042020104202212521211420221infinite14202241450221infinite14502241525115201414221451422454142Paterson's worms are featured in the 2003 Stephen Low IMAX film Volcanoesof the Deep Sea...
The game of life is the best-known two-dimensional cellular automaton, invented by John H. Conway and popularized in Martin Gardner's Scientific American column starting in October 1970. The game of life was originally played (i.e., successive generations were produced) by hand with counters, but implementation on a computer greatly increased the ease of exploring patterns.The life cellular automaton is run by placing a number of filled cells on a two-dimensional grid. Each generation then switches cells on or off depending on the state of the cells that surround it. The rules are defined as follows. All eight of the cells surrounding the current one are checked to see if they are on or not. Any cells that are on are counted, and this count is then used to determine what will happen to the current cell. 1. Death: if the count is less than 2 or greater than 3, the current cell is switched off. 2. Survival: if (a) the count is exactly 2, or (b) the count..
The terms "measure," "measurable," etc. have very precise technical definitions (usually involving sigma-algebras) that can make them appear difficult to understand. However, the technical nature of the definitions is extremely important, since it gives a firm footing to concepts that are the basis for much of analysis (including some of the slippery underpinnings of calculus).For example, every definition of an integral is based on a particular measure: the Riemann integral is based on Jordan measure, and the Lebesgue integral is based on Lebesgue measure. The study of measures and their application to integration is known as measure theory.A measure is defined as a nonnegative real function from a delta-ring such that(1)where is the empty set, and(2)for any finite or countable collection of pairwise disjoint sets in such that is also in .If is -finite and is bounded, then can be extended uniquely to a measure defined..
The descending chain condition, commonly abbreviated "D.C.C.," is the dual notion of the ascending chain condition. The descending chain condition for a partially ordered set requires that all decreasing sequences in become eventually constant.A module fulfilling the descending chain condition iscalled Artinian.
Given a short exact sequence of modules(1)let(2)(3)be projective resolutions of and , respectively. Then there is a projective resolution of (4)such that the above diagrams are commutative. Here, is the injection of the first summand, whereas is the projection onto the second factor for .The name of this lemma derives from the shape of the diagram formed by the shortexact sequence and the given projective resolutions.
A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequenceThis commutative diagram shows how the first commutative diagram (shown here in blue) can be modified to exhibit the long exact sequence (shown here in red) explicitly. The map is called a connecting homomorphism and describes a curve from the end of the upper row () to the beginning of the lower row (), which suggested the name given to this lemma.The snake lemma is explained in the first scene of Claudia Weill's film Itis My Turn (1980), starring Jill Clayburgh and Michael Douglas.
The ascending chain condition, commonly abbreviated "A.C.C.," for a partially ordered set requires that all increasing sequences in become eventually constant.A module fulfils the ascending chain condition if its set of submodules obeys the condition with respect to inclusion. In this case, is called Noetherian.
In the mice problem, also called the beetle problem, mice start at the corners of a regular -gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise direction at constant speed. The mice each trace out a logarithmic spiral, meet in the center of the polygon, and travel a distanceThe first few values for , 3, ..., aregiving the numerical values 0.5, 0.666667, 1, 1.44721, 2, 2.65597, 3.41421, 4.27432, 5.23607, .... The curve formed by connecting the mice at regular intervals of time is an attractive figure called a whirl.The problem is also variously known as the (three, four, etc.) (bug, dog, etc.) problem. It can be generalized to irregular polygons and mice traveling at differing speeds (Bernhart 1959). Miller (1871) considered three mice in general positions with speeds adjusted to keep paths similar and the triangle similar to the original...
An ideal is a subset of elements in a ring that forms an additive group and has the property that, whenever belongs to and belongs to , then and belong to . For example, the set of even integers is an ideal in the ring of integers . Given an ideal , it is possible to define a quotient ring . Ideals are commonly denoted using a Gothic typeface.A finitely generated ideal is generated by a finite list , , ..., and contains all elements of the form , where the coefficients are arbitrary elements of the ring. The list of generators is not unique, for instance in the integers.In a number ring, ideals can be represented as lattices, and can be given a finite basis of algebraic integers which generates the ideal additively. Any two bases for the same lattice are equivalent. Ideals have multiplication, and this is basically the Kronecker product of the two bases. The illustration above shows an ideal in the Gaussian integers generated by 2 and , where elements of the ideal..
Koh et al. (1980) and Gallian (2007) define a web graph as a stacked prism graph with the edges of the outer cycle removed.Web graphs are graceful.Precomputed properties of web graphs are available in the Wolfram Language as GraphData["Web", n].The term "web graph" is also used (e.g., Horvat and Pisanski 2010) to refer to the stacked prism graph itself, where is a cycle graph, is a path graph, and denotes a graph Cartesian product.The bipartite double graph of the web graph for odd is .
A tree to whose nodes and/or edges labels (usually number)are assigned.The word "weight" also has a more specific meaning when applied to trees, namely the weight of a tree at a point is the maximum number of edges in any branch at (Harary 1994, p. 35), as illustrated above. A point having minimal weight for the tree is called a centroid point, and the tree centroid is the set of all centroid points.
A rooted tree is a tree in which a special ("labeled") node is singled out. This node is called the "root" or (less commonly) "eve" of the tree. Rooted trees are equivalent to oriented trees (Knuth 1997, pp. 385-399). A tree which is not rooted is sometimes called a free tree, although the unqualified term "tree" generally refers to a free tree.A rooted tree in which the root vertex has vertexdegree 1 is known as a planted tree.The numbers of rooted trees on nodes for , 2, ... are 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, ... (OEIS A000081). Denote the number of rooted trees with nodes by , then the generating function is(1)(2)This power series satisfies(3)(4)where is the generating function for unrooted trees. A generating function for can be written using a product involving the sequence itself as(5)The number of rooted trees can also be calculated from the recurrencerelation(6)with and ,..
A weakly binary tree is a planted tree in which all nonroot graph vertices are adjacent to at most three graph vertices.Let(1)be the generating function for the number of weakly binary trees on nodes, where(2)(3)(4)(5)This gives the sequence 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, ... (OEIS A001190),sometimes also known as the Wedderburn-Etherington numbers.Otter (Otter 1948, Harary and Palmer 1973, Knuth 1997) showed that(6)where(7)(OEIS A086317; Knuth 1997, p. 583; Finch 2003, p. 297) is the unique positive root of(8)and(9)(OEIS A086318; Knuth 1997, p. 583). is also given by the rapidly converging limit(10)where is given by(11)(12)the first few terms of which are 6, 38, 1446, 2090918, 4371938082726, ... (OEIS A072191), giving(13)
A tree having four branches at each node. Quadtrees are used in the construction of some multidimensional databases (e.g., cartography, computer graphics, and image processing). For a -dimensional tree, the expected number of comparisons over all pairs of integers for successful and unsuccessful searches are known analytically for and numerically for .
A trivalent tree, also called a 3-valent tree or a 3-Cayley tree, is a tree for which each node has vertex degree . The numbers of trivalent trees on , 2, ... nodes are 1, 1, 1, 1, 2, 2, 4, 6, 11, 18, 37, 66, 135, 265, 552, ... (OEIS A000672).The number of trees with nodes of valency either 1 or 3 for , 2, ... are 1, 1, 0, 1, 0, 1, 0, 1, 0, 2, 0, 2, 0, 4, ... (OEIS A052120). These are sometimes called boron trees, since such a tree with nodes has nodes of valency 3 (corresponding to boron atoms) and nodes of valency (corresponding to hydrogen atoms) for , 4, ....
A planted tree is a rooted tree whose root vertex has vertex degree 1. The number of planted trees of nodes is , where is the number of rooted trees of vertices (Harary 1994, pp. 188-190), so there are 0, 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, ... (OEIS A000081) planted trees of , 2, 3, ... vertices.
The treewidth is a measure of the count of original graph vertices mapped onto any tree vertex in an optimal tree decomposition. Determining the treewidth of an arbitrary graph is an NP-hard problem. However, many NP-hard problems on graphs of bounded treewidth can be solved in polynomial time.Special cases include(1)(2)(3)(4)(5)(6)where denotes any tree, any Halin graph, is a cycle graph, is a complete graph, is a complete bipartite graph, and is an grid graph.
A planted plane tree is defined as a vertex set , edges set , root , and order relation on which satisfies 1. For if , then , where is the length of the path from to , 2. If , , and , then (Klarner 1969, Chorneyko and Mohanty 1975). The Catalannumbers give the number of planar trivalent planted trees.
The cotree of a spanning tree in a connected graph is the spacing subgraph of containing exactly those edges of which are not in (Harary 1994, p. 39).
A labeled ternary tree containing the labels 1 to with root 1, branches leading to nodes labeled 2, 3, 4, branches from these leading to 5, 6, 7 and 8, 9, 10 respectively, and so on (Knuth 1997, p. 401). The graph corresponding to the complete ternary tree on nodes is implemented in the Wolfram Language as KaryTree[n, 3].
A leaf of an unrooted tree is a node of vertex degree 1. Note that for a rooted or planted tree, the root vertex is generally not considered a leaf node, whereas all other nodes of degree 1 are.A function to return the leaves of a tree may be implemented in a future versionof the Wolfram Language as LeafVertex[g].The following tables gives the total numbers of leaves for various classes of graphs on , 2, ... nodes. Note that for rooted and planted trees, the root vertex is generally not counted as a leaf, even if it has vertex degree 1.graph typeOEISleaf count for , 2, ... nodesgraphA0555400, 2, 4, 14, 38, 153, 766, 6259, 88064, ...labeled graphA0953380, 2, 12, 96, 1280, 30720, ...labeled treeA0555410, 2, 6, 36, 320, 3750, ...planted treeA0032270, 1, 1, 3, 8, 22, 58, 160, 434, 1204, 3341, 9363, ...planted tree including root nodesA0953390, 2, 2, 5, 12, 31, 78, 208, 549, 1490, 4060, 11205, ...rooted treeA0032270, 1, 1, 3, 8, 22, 58, 160, 434, 1204, 3341, 9363,..
A labeled binary tree containing the labels 1 to with root 1, branches leading to nodes labeled 2 and 3, branches from these leading to 4, 5 and 6, 7, respectively, and so on (Knuth 1997, p. 401). The graph corresponding to the complete binary tree on nodes is implemented in the Wolfram Language as KaryTree[n, 2].
The complete bipartite graph is a tree known as the "claw." It is isomorphic to the star graph , and is sometimes known as the Y graph (Horton and Bouwer 1991; Biggs 1993, p. 147).More generally, the star graph is sometimes also known as a "claw" (Hoffmann 1960; Harary 1994, p. 17).The claw graph has chromatic number 2 and chromatic polynomialIts graph spectrum is .A graph that does not contain the claw as an inducedsubgraph is called a claw-free graph.
The score sequence of a tournament is a monotonic nondecreasing sequence of the outdegrees of the graph vertices of the corresponding tournament graph. Elements of a score sequence of length therefore lie between 0 and , inclusively. Score sequences are so named because they correspond to the set of possible scores obtainable by the members of a group of players in a tournament where each player plays all other players and each game results in a win for one player and a loss for the other. (The score sequence for a given tournament is obtained from the set of outdegrees sorted in nondecreasing order, and so must sum to , where is a binomial coefficient.)For example, the unique possible score sequences for is . For , the two possible sequences are and . And for , the four possible sequences are , , , and (OEIS A068029).Landau (1953) has shown that a sequence of integers () is a score sequence ifffor , ..., , where is a binomial coefficient, and equality for(Harary..
Throughout abstract algebra, the term "magma" is most often used as a synonym of the more antiquated term "groupoid," referring to a set equipped with a binary operator. The term is thought to have originated with Bourbaki.Unlike the term "groupoid" which has a number of different uses across algebra, the term "magma" has the benefit of being essentially unused in other contexts. On the other hand, the use of the term "magma" appears to be somewhat less common in literature.
The problem of determining how many nonattacking knights can be placed on an chessboard. For , the solution is 32 (illustrated above). In general, the solutions are(1)giving the sequence 1, 4, 5, 8, 13, 18, 25, ... (OEIS A030978,Dudeney 1970, p. 96; Madachy 1979).The minimal number of knights needed to occupy or attack every square on an chessboard (i.e., domination numbers for the knight graphs) are given for , 2, ... by 1, 4, 4, 4, 5, 8, 10, 12, 14, ... (OEIS A006075), with corresponding numbers of such solutions given by 1, 1, 2, 3, 8, 22, 3, ... (OEIS A006076).
A quantity which gives the inclination of a curve or line with respect to another curve or line. For a line in the -plane making an angle with the x-axis, the slope is a constant given by(1)where and are changes in the two coordinates over some distance.For a plane curve specified as , the slope is(2)for a curve specified parametrically as , the slope is(3)where and , for a curve specified as , the slope is(4)and for a curve given in polar coordinates as , the slope is(5)(Lawrence 1972, pp. 8-9).It is meaningless to talk about the slope of a curve in three-dimensional space unlessthe slope with respect to what is specified.J. Miller has undertaken a detailed study of the origin of the symbol to denote slope. The consensus seems to be that it is not known why the letter was chosen. One high school algebra textbook says the reason for is unknown, but remarks that it is interesting that the French word for "to climb" is "monter."..
The expansion of the two sides of a sum equality in terms of polynomials in and , followed by closed form summation in terms of and . For an example of the technique, see Bloom (1995).
A class of curve defined at integer values which hops from one value to another. Their name derives from the Greek word batrachion, which means "small frog." Many batrachions are fractal. Examples include the Blancmange function, Hofstadter-Conway $10,000 sequence, Hofstadter's Q-sequence, and Mallows' sequence.
A set of vectors in Euclidean -space is said to satisfy the Haar condition if every set of vectors is linearly independent (Cheney 1999). Expressed otherwise, each selection of vectors from such a set is a basis for -space. A system of functions satisfying the Haar condition is sometimes termed a Tchebycheff system (Cheney 1999).
A method of computing the determinant of a square matrix due to Charles Dodgson (1866) (who is more famous under his pseudonym Lewis Carroll). The method is useful for hand calculations because, for an integer matrix, all entries in submatrices computed along the way must also be integers. The method is also implemented efficiently in a parallel computation. Condensation is also known as the method of contractants (Macmillan 1955, Lotkin 1959).Given an matrix, condensation successively computes an matrix, an matrix, etc., until arriving at a matrix whose only entry ends up being the determinant of the original matrix. To compute the matrix (), take the connected subdeterminants of the matrix and divide them by the central entries of the matrix, with no divisions performed for . The matrices arrived at in this manner are the matrices of determinants of the connected submatrices of the original matrices.For example, the first condensation..
In the process of attaching a -handle to a manifold , the boundary of is modified by a process called -surgery. Surgery consists of the removal of a tubular neighborhood of a -sphere from the boundaries of and the standard sphere, and the gluing together of these two scarred-up objects along their common boundaries.
A topological space that contains a homeomorphicimage of every topological space of a certain class.A metric space is said to be universal for a family of metric spaces if any space from is isometrically embeddable in . Fréchet (1910) proved that , the space of all bounded sequences of real numbers endowed with a supremum norm, is a universal space for the family of all separable metric spaces. Holsztynski (1978) proved that there exists a metric on , inducing the usual topology, such that every finite metric space embeds in (Ovchinnikov 2000).
In continuum theory, a dendrite is a locally connected continuum that contains no simple closed curve. A semicircle is therefore a dendrite, while a triangle is not.The term dendrite is used by Steinhaus (1999, pp. 120-125) to refer to a system of line segments connecting a given set of points, where the total length of paths is as short as possible (therefore implying that no closed cycles are permitted) and the paths are not allowed to cross. This definition differs from the one in continuum theory since a semicircle is a dendritic continuum but is not a line segment.
Given a map from a space to a space and another map from a space to a space , a lift is a map from to such that . In other words, a lift of is a map such that the diagram (shown below) commutes. If is the identity from to , a manifold, and if is the bundle projection from the tangent bundle to , the lifts are precisely vector fields. If is a bundle projection from any fiber bundle to , then lifts are precisely sections. If is the identity from to , a manifold, and a projection from the orientation double cover of , then lifts exist iff is an orientable manifold.If is a map from a circle to , an -manifold, and the bundle projection from the fiber bundle of alternating n-forms on , then lifts always exist iff is orientable. If is a map from a region in the complex plane to the complex plane (complex analytic), and if is the exponential map, lifts of are precisely logarithms of ...
The cup product is a product on cohomology classes. In the case of de Rham cohomology, a cohomology class can be represented by a closed form. The cup product of and is represented by the closed form , where is the wedge product of differential forms. It is the dual operation to intersection in homology.In general, the cup product is a mapwhich satisfies , where is the th cohomology group.
In algebraic topology, a -skeleton is a simplicial subcomplex of that is the collection of all simplices of of dimension at most , denoted .The graph obtained by replacing the faces of a polyhedron with its edges and vertices is therefore the skeleton of the polyhedron. The polyhedral graphs corresponding to the skeletons of Platonic solids are illustrated above. The number of topologically distinct skeletons with graph vertices for , 5, 6, ... are 1, 2, 7, 18, 52, ... (OEIS A006869).
Home plate in the game of baseball is an irregular pentagon with two parallel sides, each perpendicular to a base. It seems reasonable to dub such a figure (i.e., a rectangle with a coincident isosceles triangle placed on one side) a "isosceles right pentagon."However, specification of the shape of home plate, illustrated above, as specified by both the Major League Baseball Official Rules and the Little League rulebook (Kreutzer and Kerley 1990) is not physically realizable, since it requires the existence of a (12, 12, 17) right triangle, whereas(Bradley 1996). More specifically, the standards require the existence of an isosceles right triangle with side lengths 8.5 inches and a hypotenuse of length 12 inches, which does not satisfy the Pythagorean theorem.
The goat problem (or bull-tethering problem) considers a fenced circular field of radius with a goat (or bull, or other animal) tied to a point on the interior or exterior of the fence by means of a tether of length , and asks for the solution to various problems concerning how much of the field can be grazed.Tieing a goat to a point on the interior of the fence with radius 1 using a chain of length , consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection(1)Taking gives(2)plotted above. Setting (i.e., half of ) leads to the equation(3)which cannot be solved exactly, but which has approximate solution(4)(OEIS A133731).Now instead consider tieing the goat to the exterior of the fence (or equivalently, to the exterior of a silo whose horizontal cross section is a circle) with radius . Assume that , so that the goat is not..
In a rectangular room (a cuboid) with dimensions , a spider is located in the middle of one wall one foot away from the ceiling. A fly is in the middle of the opposite wall one foot away from the floor. If the fly remains stationary, what is the shortest total distance (i.e., the geodesic) the spider must crawl along the walls, ceiling, and floor in order to capture the fly? The answer, , can be obtained by "flattening" the walls as illustrated above. Note that his distance is shorter than the the spider would have to travel if first crawling along the wall to the floor, then across the floor, then up one foot to get to the fly. The puzzle was originally posed in an English newspaper by Dudeney in 1903 (Gardner 1958).A twist to the problem can be obtained by a spider that suspends himself from strand of cobweb and thus takes a shortcut by not being forced to remain glued to a surface of the room. If the spider attaches a strand of cobweb to the wall at his starting..
666 is the occult "number of the beast," also called the "sign of the devil" (Wang 1994), associated in the Bible with the Antichrist. It has figured in many numerological studies. It is mentioned in Revelation 13:18: "Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is 666." The origin of this number is not entirely clear, although it may be as simple as the number containing the concatenation of one symbol of each type (excluding ) in Roman numerals: (Wells 1986).The first few numbers containing the beast number in their digits are 666, 1666,2666, 3666, 4666, 5666, 6660, ... (OEIS A051003)."666" is the combination of the mysterious suitcase retrieved by Vincent Vega (John Travolta) and Jules Winnfield (Samuel L. Jackson) in Quentin Tarantino's 1994 film Pulp Fiction. Various conspiracy theories, including the novel..
The White House switchboard constant is the name given by Munroe (2012) to the constant(1)(2)(OEIS A182064), the first few digits of which are 202-456-1414, which is the phone number of the switchboard of the White House (home of U.S. President).Other "simple" expressions that might vie for that moniker include(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)where is the Euler-Mascheroni constant, all of which are "better" than the canonical White House switchboard expression (E. Weisstein, Jul. 13, 2013).
Let denote the mean of a set of quantities , then the absolute deviation is defined by
Consider a game, first proposed by Nicolaus Bernoulli, in which a player bets on how many tosses of a coin will be needed before it first turns up heads. The player pays a fixed amount initially, and then receives dollars if the coin comes up heads on the th toss. The expectation value of the gain is then(1)dollars, so any finite amount of money can be wagered and the player will still come out ahead on average.Feller (1968) discusses a modified version of the game in which the player receives nothing if a trial takes more than a fixed number of tosses. The classical theory of this modified game concluded that is a fair entrance fee, but Feller notes that "the modern student will hardly understand the mysterious discussions of this 'paradox.' "In another modified version of the game, the player bets $2 that heads will turn up on the first throw, $4 that heads will turn up on the second throw (if it did not turn up on the first), $8 that heads will turn..
The EKG sequence is the integer sequence having 1 as its first term, 2 as its second, and with each succeeding term being the smallest number not already used that shares a factor with the preceding term. This results in the sequence 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, ... (OEIS A064413). When plotted as a connect-the-dots plot (left figure), the sequence looks somewhat like an electrocardiogram (abbreviated "EKG" in medical circles), so this sequence became known as the EKG sequence. Lagarias et al. have computed the first 10 million terms of the sequence (Lagarias et al. 2002, Peterson 2002).Every term appears exactly once in this sequence, and the primes occur in increasing order (Lagarias et al. 2002). The inverse permutation of the integers giving the sequence is 1, 2, 5, 3, 10, 4, 14, 8, 6, 9, 20, 7, 28, ... (OEIS A064664).Lagarias et al. (2002) established the boundsfor the term . For the first terms, whenever a prime occurs, it is immediately..
A real number that is -normal for every base 2, 3, 4, ... is said to be absolutely normal. As proved by Borel (1922, p. 198), almost all real numbers in are absolutely normal (Niven 1956, p. 103; Stoneham 1970; Kuipers and Niederreiter 1974, p. 71; Bailey and Crandall 2002).The first specific construction of an absolutely normal number was by Sierpiński (1917), with another method presented by Schmidt (1962). These results were both obtained by complex constructive devices (Stoneham 1970), and are by no means easy to construct (Stoneham 1970, Sierpiński and Schinzel 1988).
Let be an integer and letbe an integer polynomial that has at least one real root. Then has infinitely many prime divisors that are not congruent to 1 (mod ) (Nagell 1951, p. 168).
Let denote the set of the numbers less than and relatively prime to , where is the totient function. Define(1)Then a theorem of Lagrange states that(2)for an odd prime (Hardy and Wright 1979, p. 98). Actually, this relationship holds for some composite values as well. Value for which it holds are , 3, 4, 5, 6, 7, 10, 11, 13, 17, 19, 23, 29, ... (OEIS A158008).This can be generalized as follows. Let be an odd prime divisor of and the highest power which divides , then(3)and, in particular,(4)Now, if is even and is the highest power of 2 that divides , then(5)and, in particular,(6)
One of the numbers 1, 2, 3, ... (OEIS A000027), also called the counting numbers or natural numbers. 0 is sometimes included in the list of "whole" numbers (Bourbaki 1968, Halmos 1974), but there seems to be no general agreement. Some authors also interpret "whole number" to mean "a number having fractional part of zero," making the whole numbers equivalent to the integers.Due to lack of standard terminology, the following terms are recommended in preference to "counting number," "natural number," and "whole number."setnamesymbol..., , , 0, 1, 2, ...integersZ1, 2, 3, 4, ...positive integersZ-+0, 1, 2, 3, 4, ...nonnegative integersZ-*0, , , , , ...nonpositive integers, , , , ...negative integersZ--
One of the numbers ..., , , 0, 1, 2, .... The set of integers forms a ring that is denoted . A given integer may be negative (), nonnegative (), zero (), or positive (). The set of integers is, not surprisingly, called Integers in the Wolfram Language, and a number can be tested to see if it is a member of the integers using the command Element[x, Integers]. The command IntegerQ[x] returns True if has function head Integer in the Wolfram Language.Numbers that are integers are sometimes described as "integral" (instead of integer-valued), but this practice may lead to unnecessary confusions with the integrals of integral calculus.The ring of integers has cardinal number of aleph0. The generating function for the nonnegative integers isThere are several symbols used to perform operations having to do with conversion between real numbers and integers. The symbol ("floor ") means "the largest integer not greater than ,"..
A plot of the map winding number resulting from mode locking as a function of for the circle map(1)with . (Since the circle map becomes mode-locked, the map winding number is independent of the initial starting argument .) At each value of , the map winding number is some rational number. The result is a monotonic increasing "staircase" for which the simplest rational numbers have the largest steps. The Devil's staircase continuously maps the interval onto , but is constant almost everywhere (i.e., except on a Cantor set).For , the measure of quasiperiodic states ( irrational) on the -axis has become zero, and the measure of mode-locked state has become 1. The dimension of the Devil's staircase .Another type of devil's staircase occurs for the sum(2)for , where is the floor function (Böhmer 1926ab; Kuipers and Niederreiter 1974, p. 10; Danilov 1974; Adams 1977; Davison 1977; Bowman 1988; Borwein and Borwein 1993; Bowman..
Let be the volume of a ball of radius in a complete -dimensional Riemannian manifold with Ricci curvature tensor . Then , where is the volume of a ball in a space having constant sectional curvature. In addition, if equality holds for some ball, then this ball is isometric to the ball of radius in the space of constant sectional curvature .
The term "real line" has a number of different meanings in mathematics.Most commonly, "real line" is used to mean real axis, i.e., a line with a fixed scale so that every real number corresponds to a unique point on the line. The generalization of the real line to two dimensions is called the complex plane.The term "real line" is also used to distinguish an ordinary line from a so-called imaginary line which can arise in algebraic geometry.Renteln and Dundes (2005) give the following (bad) mathematical jokes about the real line:Q: What is green and homeomorphic to the open unit interval?A: The real lime.
The number (the cube root of 2) which is to be constructed in the cube duplication problem. This number is not a Euclidean number although it is an algebraic number of third degree.It has decimal digits 1.25992104989... (OEIS A002580).Its continued fraction is [1, 3, 1, 5, 1, 1,4, 1, 1, 8, 1, 14, 1, ...] (OEIS A002945).
The limaçon is a polar curve ofthe form(1)also called the limaçon of Pascal. It was first investigated by Dürer, who gave a method for drawing it in Underweysung der Messung (1525). It was rediscovered by Étienne Pascal, father of Blaise Pascal, and named by Gilles-Personne Roberval in 1650 (MacTutor Archive). The word "limaçon" comes from the Latin limax, meaning "snail."If , the limaçon is convex. If , the limaçon is dimpled. If , the limaçon degenerates to a cardioid. If , the limaçon has an inner loop. If , it is a trisectrix (but not the Maclaurin trisectrix).For , the inner loop has area(2)(3)(4)where . Similarly the area enclosed by the outer envelope is(5)(6)(7)Thus, the area between the loops is(8)In the special case of , these simplify to(9)(10)(11)Taking the parametrization(12)(13)gives the arc length as a function of as(14)where is an elliptic..
A curve with polar coordinates,(1)studied by the Greek mathematician Nicomedes in about 200 BC, also known as the cochloid. It is the locus of points a fixed distance away from a line as measured along a line from the focus point (MacTutor Archive). Nicomedes recognized the three distinct forms seen in this family for , , and . (For , it obviously degenerates to a circle.)The conchoid of Nicomedes was a favorite with 17th century mathematicians and could be used to solve the problems of cube duplication, angle trisection, heptagon construction, and other Neusis constructions (Johnson 1975).In Cartesian coordinates, the conchoid ofNicomedes may be written(2)or(3)The conchoid has as an asymptote, and the area between either branch and the asymptote is infinite.A conchoid with has a loop for , where , giving area(4)(5)(6)The curvature and tangentialangle are given by(7)(8)..
Bezdek and Kuperberg (1991) have constructed packings of identical ellipsoidsof densities arbitrarily close to(OEIS A093824), greater than the maximum density of (OEIS A093825) that is possible for sphere packing (Sloane 1998), as established by proof of the Kepler conjecture. Furthermore, J. Wills has modified the ellipsoid packing to yield an even higher density of (Bezdek and Kuperberg 1991).Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.
In dimensions for the arrangement of hyperspheres whose convex hull has minimal content is always a "sausage" (a set of hyperspheres arranged with centers along a line), independent of the number of -spheres. The conjecture was proposed by Fejes Tóth, and solved for dimensions by Betke et al. (1994) and Betke and Henk (1998).