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Newton's iteration is an algorithm for computing the square root of a number via the recurrence equation(1)where . This recurrence converges quadratically as .Newton's iteration is simply an application of Newton'smethod for solving the equation(2)For example, when applied numerically, the first few iterations to Pythagoras's constant are 1, 1.5, 1.41667, 1.41422, 1.41421, ....The first few approximants , , ... to are given by(3)These can be given by the analytic formula(4)(5)These can be derived by noting that the recurrence can be written as(6)which has the clever closed-form solution(7)Solving for then gives the solution derived above.The following table summarizes the first few convergents for small positive integer OEIS, , ...11, 1, 1, 1, 1, 1, 1, 1, ...2A001601/A0510091, 3/2, 17/12, 577/408, 665857/470832, ...3A002812/A0715791, 2, 7/4, 97/56, 18817/10864, 708158977/408855776, .....

Min Max Min Max Re Im Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as(1)where and are the invariants of the Weierstrass elliptic function with modular discriminant(2)(Klein 1877). If , where is the upper half-plane, then(3)is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in (Apostol 1997, p. 15).Klein's absolute invariant is implemented in the WolframLanguage as KleinInvariantJ[tau].The function is the same as the j-function, modulo a constant multiplicative factor.Every rational function of is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40).Klein's invariant can be given explicitly by(4)(5)(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function(6) is a Jacobi theta function, the are..

The Reuleaux tetrahedron, sometimes also called the spherical tetrahedron, is the three-dimensional solid common to four spheres of equal radius placed so that the center of each sphere lies on the surface of the other three. The centers of the spheres are therefore located at the vertices of a regular tetrahedron, and the solid consists of an "inflated" tetrahedron with four curved edges.Note that the name, coined here for the first time, is based on the fact that the geometric shape is the three-dimensional analog of the Reuleaux triangle, not the fact that it has constant width. In fact, the Reuleaux tetrahedron is not a solid of constant width. However, Meißner (1911) showed how to modify the Reuleaux tetrahedron to form a surface of constant width by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. Depending on which three edge arcs are replaced (three that have a common..

Let(1)(2)(3)(OEIS A104457), where is the golden ratio, and(4)(5)(OEIS A002390).Define the Fibonacci hyperbolic sine by(6)(7)(8)The function satisfies(9)and for ,(10)where is a Fibonacci number. For , 2, ..., the values are therefore 1, 3, 8, 21, 55, ... (OEIS A001906).Define the Fibonacci hyperbolic cosine by(11)(12)(13)This function satisfies(14)and for ,(15)where is a Fibonacci number. For , 2, ..., the values are therefore 2, 5, 13, 34, 89, ... (OEIS A001519).Similarly, the Fibonacci hyperbolic tangent is defined by(16)and for ,(17)For , 2, ..., the values are therefore 1/2, 3/5, 8/13, 21/34, 55/89, ... (OEIS A001906 and A001519).

Schur's partition theorem lets denote the number of partitions of into parts congruent to (mod 6), denote the number of partitions of into distinct parts congruent to (mod 3), and the number of partitions of into parts that differ by at least 3, with the added constraint that the difference between multiples of three is at least 6. Then (Schur 1926; Bressoud 1980; Andrews 1986, p. 53).The values of for , 2, ... are 1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, ... (OEIS A003105). For example, for , there are nine partitions satisfying these conditions, as summarized in the following table (Andrews 1986, p. 54).15The identity can be established using the identity(1)(2)(3)(4)(5)(Andrews 1986, p. 54). The identity is significantly trickier.

The Kähler potential is a real-valued function on a Kähler manifold for which the Kähler form can be written as . Here, the operators(1)and(2)are called the del and del bar operator, respectively.For example, in , the function is a Kähler potential for the standard Kähler form, because(3)(4)(5)(6)

Picking two independent sets of points and from a unit uniform distribution and placing them at coordinates gives points uniformly distributed over the unit square.The distribution of distances from a randomly selected point in the unit square to its center is illustrated above.The expected distance to the square's center is(1)(2)(3)(4)(Finch 2003, p. 479; OEIS A103712), where is the universal parabolic constant. The expected distance to a fixed vertex is given by(5)(6)which is exactly twice .The expected distances from the closest and farthest vertices are given by(7)(8)Pick points at randomly in a unit square and take the convex hull . Let be the expected area of , the expected perimeter, and the expected number of vertices of . Then(9)(10)(11)(12)(13)(14)(OEIS A096428 and A096429), where is the multiplicative inverse of Gauss's constant, is the gamma function, and is the Euler-Mascheroni constant (Rényi and Sulanke..

Let two points and be picked randomly from a unit -dimensional hypercube. The expected distance between the points , i.e., the mean line segment length, is then(1)This multiple integral has been evaluated analytically only for small values of . The case corresponds to the line line picking between two random points in the interval .The first few values for are given in the following table.OEIS1--0.3333333333...2A0915050.5214054331...3A0730120.6617071822...4A1039830.7776656535...5A1039840.8785309152...6A1039850.9689420830...7A1039861.0515838734...8A1039871.1281653402...The function satisfies(2)(Anderssen et al. 1976), plotted above together with the actual values.M. Trott (pers. comm., Feb. 23, 2005) has devised an ingenious algorithm for reducing the -dimensional integral to an integral over a 1-dimensional integrand such that(3)The first few values are(4)(5)(6)(7)In the limit as , these..

The Wolstenholme numbers are defined as the numerators of the generalized harmonic number appearing in Wolstenholme's theorem. The first few are 1, 5, 49, 205, 5269, 5369, 266681, 1077749, ... (OEIS A007406).By Wolstenholme's theorem, for prime , where is the th Wolstenholme number. In addition, for prime .The first few prime Wolstenholme numbers are 5, 266681, 40799043101, 86364397717734821, ... (OEIS A123751), which occur at indices , 7, 13, 19, 121, 188, 252, 368, 605, 745, ... (OEIS A111354).

The term "Euler function" may be used to refer to any of several functions in number theory and the theory of special functions, including 1. the totient function , defined as the number of positive integers that are relatively prime to , where 1 is counted as being relatively prime to all numbers; 2. the function(1)(2)(3)where and are q-Pochhammer symbols; 3. the Euler L-function , which is a special case of the Artin L-function for the polynomial and is defined by(4)where(5)(6)with a Legendre symbol.

Based on a problem in particle physics, Dyson (1962abc) conjectured that the constantterm in the Laurent seriesis the multinomial coefficientThe theorem was proved by Wilson (1962) and independently by Gunson (1962). A definitive proof was subsequently published by Good (1970).

Any square matrix can be written as a sum(1)where(2)is a symmetric matrix known as the symmetric part of and(3)is an antisymmetric matrix known as the antisymmetric part of . Here, is the transpose.The symmetric part of a tensor is denoted using parenthesesas(4)(5)Symbols for the symmetric and antisymmetric partsof tensors can be combined, for example(6)(Wald 1984, p. 26).

Any square matrix can be written as a sum(1)where(2)is a symmetric matrix known as the symmetric part of and(3)is an antisymmetric matrix known as the antisymmetric part of . Here, is the transpose.Any rank-2 tensor can be written as a sum of symmetricand antisymmetric parts as(4)The antisymmetric part of a tensor is sometimes denoted using the special notation(5)For a general rank- tensor,(6)where is the permutation symbol. Symbols for the symmetric and antisymmetric parts of tensors can be combined, for example(7)(Wald 1984, p. 26).

The geodesic on an oblate spheroid can be computed analytically, although the resulting expression is much more unwieldy than for a simple sphere. A spheroid with equatorial radius and polar radius can be specified parametrically by(1)(2)(3)where . Using the second partial derivatives(4)(5)(6)(7)(8)(9)gives the geodesics functions as(10)(11)(12)(13)(14)(15)where(16)is the ellipticity.Since and and are explicit functions of only, we can use the special form of the geodesic equation(17)(18)(19)where is a constant depending on the starting and ending points. Integrating gives(20)where(21)(22) is an elliptic integral of the first kind with parameter , and is an elliptic integral of the third kind.Geodesics other than meridians of an oblate spheroid undulate between two parallels with latitudes equidistant from the equator. Using the Weierstrass sigma function and Weierstrass zeta function, the geodesic on the oblate spheroid..

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 developable surface is a ruled surface having Gaussian curvature everywhere. Developable surfaces therefore include the cone, cylinder, elliptic cone, hyperbolic cylinder, and plane.A developable surface has the property that it can be made out of sheet metal, since such a surface must be obtainable by transformation from a plane (which has Gaussian curvature 0) and every point on such a surface lies on at least one straight line.

A ruled surface is a surface that can be swept out by moving a line in space. It therefore has a parameterization of the form(1)where is called the ruled surface directrix (also called the base curve) and is the director curve. The straight lines themselves are called rulings. The rulings of a ruled surface are asymptotic curves. Furthermore, the Gaussian curvature on a ruled regular surface is everywhere nonpositive.Examples of ruled surfaces include the elliptic hyperboloidof one sheet (a doubly ruled surface)(2)the hyperbolic paraboloid (a doublyruled surface)(3)Plücker's conoid(4)and the Möbius strip(5)(Gray 1993).The only ruled minimal surfaces are the planeand helicoid (Catalan 1842, do Carmo 1986).

A surface in 3-space can be parameterized by two variables (or coordinates) and such that(1)(2)(3)If a surface is parameterized as above, then the tangentvectors(4)(5)are useful in computing the surface area and surfaceintegral.

The center of any sphere which has a contact of (at least) first-order with a curve at a point lies in the normal plane to at . The center of any sphere which has a contact of (at least) second-order with at point , where the curvature , lies on the polar axis of corresponding to . All these spheres intersect the osculating plane of at along a circle of curvature at . The osculating sphere has centerwhere is the unit normal vector, is the unit binormal vector, is the radius of curvature, and is the torsion, and radiusand has contact of (at least) third order with .

Given a system of two ordinary differential equations(1)(2)let and denote fixed points with , so(3)(4)Then expand about so(5)(6)To first-order, this gives(7)where the matrix, or its generalization to higher dimension, is called the stability matrix. Analysis of the eigenvalues (and eigenvectors) of the stability matrix characterizes the type of fixed point.

In a monoid or multiplicative group where the operation is a product , the multiplicative inverse of any element is the element such that , with 1 the identity element.The multiplicative inverse of a nonzero number is its reciprocal (zero is not invertible). For complex ,The inverse of a nonzero real quaternion (where are real numbers, and not all of them are zero) is its reciprocalwhere .The multiplicative inverse of a nonsingular matrixis its matrix inverse.To detect the multiplicative inverse of a given element in the multiplication table of finite multiplicative group, traverse the element's row until the identity element 1 is encountered, and then go up to the top row. In this way, it can be immediately determined that is the multiplicative inverse of in the multiplicative group formed by all complex fourth roots of unity...

Let and be nonnegative and continuous functions on the closed interval , then the solid of revolution obtained by rotating the curves and about the -axis from to and taking the region enclosed between them has volume given by

Let be a plane region bounded above by a continuous curve , below by the -axis, and on the left and right by and , then the volume of the solid of revolution obtained by rotating about the -axis is given by

Let be a nonnegative and continuous function on the closed interval , then the solid of revolution obtained by rotating the curve about the -axis from to has volume given by

Consider the average length of a line segment determined by two points picked at random in the interior of an arbitrary triangle. This problem is not affine, so a simple formula in terms of the area or linear properties of the original triangle apparently does not exist.However, if the original triangle is chosen to be an isosceles right triangle with unit legs, then the average length of a line with endpoints chosen at random inside it is given by(1)(2)(3)(OEIS A093063; M. Trott, pers. comm., Mar. 10, 2004), which is numerically surprisingly close to .Similarly, if the original triangle is chosen to be an equilateral triangle with unit side lengths, then the average length of a line with endpoints chosen at random inside it is given by(4)(5)The integrand can be split up into the four pieces(6)(7)(8)(9)As illustrated above, symmetry immediately gives and , so(10)With some effort, the integrals and can be done analytically to give..

The elastica formed by bent rods and considered in physics can be generalized to curves in a Riemannian manifold which are a critical point forwhere is the normal curvature of , is a real number, and is closed or satisfies some specified boundary condition. The curvature of an elastica must satisfywhere is the signed curvature of , is the Gaussian curvature of the oriented Riemannian surface along , is the second derivative of with respect to , and is a constant.

A number is called a barrier of a number-theoretic function if, for all , . Neither the totient function nor the divisor function has a barrier.Let be an open set and , then a function is called a barrier for at a point if 1. is continuous, 2. is subharmonic on , 3. , 4. (Krantz 1999, pp. 100-101).

The probability that a random integer between 1 and will have its greatest prime factor approaches a limiting value as , where for and is defined through the integral equation(1)for (Dickman 1930, Knuth 1998), which is almost (but not quite) a homogeneous Volterra integral equation of the second kind. The function can be given analytically for by(2)(3)(4)(Knuth 1998).Amazingly, the average value of such that is(5)(6)(7)(8)(9)which is precisely the Golomb-Dickman constant , which is defined in a completely different way!The Dickman function can be solved numerically by converting it to a delay differential equation. This can be done by noting that will become upon multiplicative inversion, so define to obtain(10)Now change variables under the integral sign by defining(11)(12)so(13)Plugging back in gives(14)To get rid of the s, define , so(15)But by the first fundamental theoremof calculus,(16)so differentiating both sides of equation..

Polynomials which form the Sheffer sequence for(1)(2)which have generating function(3)The first few are(4)(5)(6)

Polynomials which form the Sheffer sequence for(1)and have exponential generating function(2)The first few are(3)(4)(5)(6)(7)(8)

Polynomials which form a Sheffer sequence with(1)(2)and have generating function(3)The first few are(4)(5)(6)Jordan (1965) considers the related polynomials which form a Sheffer sequence with(7)(8)These polynomials have generating function(9)The first few are(10)(11)(12)(13)The Peters polynomials are a generalizationof the Boole polynomials.

The polynomials which form the Sheffer sequence for(1)(2)which have generating function(3)The first few are(4)(5)(6)

Polynomials which form the Sheffer sequence for(1)(2)where is the inverse function of , and have generating function(3)The first few polynomials are(4)(5)(6)(7)The Stirling polynomials are related to the Stirling numbers of the first kind by(8)where is a binomial coefficient and is an integer with (Roman 1984, p. 129).

Polynomials which form the Sheffer sequence for(1)(2)and have generating function(3)The are given in terms of the hypergeometricseries by(4)where is the Pochhammer symbol (Koepf 1998, p. 115). The first few are(5)(6)(7)Koekoek and Swarttouw (1998) defined the Meixner polynomials without the Pochhammersymbol as(8)The Krawtchouk polynomials are a specialcase of the Meixner polynomials of the first kind.

Polynomials which form the Sheffer sequence for(1)where is the inverse function of , and have generating function(2)The first few are(3)(4)(5)(6)(7)(8)

Polynomials which form a Sheffer sequence with(1)(2)giving generating function(3)Roman (1984) defines Bernoulli numbers of the second kind as . They are related to the Stirling numbers of the first kind by(4)(Roman 1984, p. 115), and obey the reflection formula(5)(Roman 1984, p. 119).The first few Bernoulli polynomials of the second kind are(6)(7)(8)(9)(10)

The polynomials given by the associated Sheffer sequence with(1)where . The inverse function (and therefore generating function) cannot be computed algebraically, but the generating function(2)can be given in terms of the sum(3)This results in(4)where is a falling factorial. The first few are(5)(6)(7)(8)(9)The binomial identity obtained from the Sheffer sequencegives the generalized Chu-Vandermonde identity(10)(Roman 1984, p. 69; typo corrected).In the special case , the function simplifies to(11)which gives the generating function(12)giving the polynomials(13)(14)(15)(16)(17)

The Lebesgue covering dimension is an important dimension and one of the first dimensions investigated. It is defined in terms of covering sets, and is therefore also called the covering dimension (as well as the topological dimension).A space has Lebesgue covering dimension if for every open cover of that space, there is an open cover that refines it such that the refinement has order at most . Consider how many elements of the cover contain a given point in a base space. If this has a maximum over all the points in the base space, then this maximum is called the order of the cover. If a space does not have Lebesgue covering dimension for any , it is said to be infinite dimensional.Results of this definition are: 1. Two homeomorphic spaces have the same dimension, 2. has dimension , 3. A topological space can be embedded as a closed subspace of a Euclidean space iff it is locally compact, T2, second countable, and is finite-dimensional (in the sense of the..

There are a number of point processes which are called Hawkes processes and while many of these notions are similar, some are rather different. There are also different formulations for univariate and multivariate point processes.In some literature, a univariate Hawkes process is defined to be a self-exciting temporal point process whose conditional intensity function is defined to be(1)where is the background rate of the process , where are the points in time occurring prior to time , and where is a function which governs the clustering density of . The function is sometimes called the exciting function or the excitation function of . Similarly, some authors (Merhdad and Zhu 2014) denote the conditional intensity function by and rewrite the summand in () as(2)The processes upon which Hawkes himself made the most progress were univariate self-exciting temporal point processes whose conditional intensity function is linear (Hawkes 1971)...

A smooth curve is a curve which is a smooth function, where the word "curve" is interpreted in the analytic geometry context. In particular, a smooth curve is a continuous map from a one-dimensional space to an -dimensional space which on its domain has continuous derivatives up to a desired order.

The Reynolds transport theorem, also called simply the Reynolds theorem, is an important result in fluid mechanics that's often considered a three-dimensional analog of the Leibniz integral rule. Given any scalar quantity associated with a moving fluid, the general form of Reynolds transport theorem saysHere, is the convective derivative, is the usual gradient, denotes the material volume at time , and denotes the velocity vector.Because of its relation to the Leibniz rule, the Reynolds transport theorem is sometimes called the Leibniz-Reynolds transport theorem.Worth noting is the large number of variants of Reynolds transport theorem present in the literature. Indeed, the formula is extremely general and can be applied to a variety of contexts in vastly many circumstances. As such, different literature will inevitably have equations which often look different than the above equation in both appearance and complexity...

The spherical harmonics can be generalized to vector spherical harmonics by looking for a scalar function and a constant vector such that(1)(2)(3)(4)so(5)Now interchange the order of differentiation and use the fact that multiplicative constants can be moving inside and outside the derivatives to obtain(6)(7)(8)and(9)(10)Putting these together gives(11)so satisfies the vector Helmholtz differential equation if satisfies the scalar Helmholtz differential equation(12)Construct another vector function(13)which also satisfies the vector Helmholtzdifferential equation since(14)(15)(16)(17)(18)which gives(19)We have the additional identity(20)(21)(22)(23)(24)In this formalism, is called the generating function and is called the pilot vector. The choice of generating function is determined by the symmetry of the scalar equation, i.e., it is chosen to solve the desired scalar differential equation. If is taken as(25)where..

The following vector integrals are related to the curltheorem. If(1)then(2)If(3)then(4)The following are related to the divergence theorem.If(5)then(6)Finally, if(7)then(8)

The curl of a vector field, denoted or (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of is the limiting value of circulation per unit area. Written explicitly,(1)where the right side is a line integral around an infinitesimal region of area that is allowed to shrink to zero via a limiting process and is the unit normal vector to this region. If , then the field is said to be an irrotational field. The symbol is variously known as "nabla" or "del."The physical significance of the curl of a vector field is the amount of "rotation" or angular momentum of the contents of given region of space. It arises in fluid mechanics and elasticity theory. It is also fundamental in the theory of electromagnetism, where it arises in..

The normal vector, often simply called the "normal," to a surface is a vector which is perpendicular to the surface at a given point. When normals are considered on closed surfaces, the inward-pointing normal (pointing towards the interior of the surface) and outward-pointing normal are usually distinguished.The unit vector obtained by normalizing the normal vector (i.e., dividing a nonzero normal vector by its vector norm) is the unit normal vector, often known simply as the "unit normal." Care should be taken to not confuse the terms "vector norm" (length of vector), "normal vector" (perpendicular vector) and "normalized vector" (unit-length vector).The normal vector is commonly denoted or , with a hat sometimes (but not always) added (i.e., and ) to explicitly indicate a unit normal vector.The normal vector at a point on a surface is given by(1)where and are partial derivatives.A..

A number defined by , where is a Bernoulli polynomial of the second kind (Roman 1984, p. 294), also called Cauchy numbers of the first kind. The first few for , 1, 2, ... are 1, 1/2, , 1/4, , 9/4, ... (OEIS A006232 and A006233). They are given bywhere is a falling factorial, and have exponential generating function

The reciprocal of the arithmetic-geometric mean of 1 and ,(1)(2)(3)(4)(5)(6)(7)(OEIS A014549), where is the complete elliptic integral of the first kind, is a Jacobi theta function, and is the gamma function. This correspondence was first noticed by Gauss, and was the basis for his exploration of the lemniscate function (Borwein and Bailey 2003, pp. 13-15).Two rapidly converging series for are given by(8)(9)(Finch 2003, p. 421).Gauss's constant has continued fraction [0,1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, ...] (OEIS A053002).The inverse of Gauss's constant is given by(10)(OEIS A053004; Finch 2003, p. 420; Borwein and Bailey 2003, p. 13), which has [1, 5, 21, 3, 4, 14, 1, 1, 1, 1, 1, 3, 1, 15, 1, ...] (OEIS A053003).The value(11)(OEIS A097057) is sometimes called the ubiquitous constant (Spanier and Oldham 1987; Schroeder 1994; Finch 2003, p. 421), and(12)(OEIS A076390) is sometimes called the secondlemniscate..

There are at least two results known as "the area principle."The geometric area principle states that(1)This can also be written in the form(2)where(3)is the ratio of the lengths and for with a plus or minus sign depending on if these segments have the same or opposite directions, and(4)is the ratio of signed areas of the triangles. Grünbaum and Shepard (1995) show that Ceva's theorem, Hoehn's theorem, and Menelaus' theorem are the consequences of this result.The area principle of complex analysis states that if is a schlicht function and if(5)then(6)(Krantz 1999, p. 150).

A technically defined group characterizing a system oflinear differential equationsfor , ..., , where are complex analytic functions of in a given complex domain.

Apéry's constant is defined by(1)(OEIS A002117) where is the Riemann zeta function. Apéry (1979) proved that is irrational, although it is not known if it is transcendental. Sorokin (1994) and Nesterenko (1996) subsequently constructed independent proofs for the irrationality of (Hata 2000). arises naturally in a number of physical problems, including in the second- and third-order terms of the electron's gyromagnetic ratio, computed using quantum electrodynamics.The following table summarizes progress in computing upper bounds on the irrationality measure for . Here, the exact values for is given by(2)(3)(Hata 2000).upper boundreference15.513891Rhin and Viola (2001)28.830284Hata (1990)312.74359Dvornicich and Viola (1987)413.41782Apéry (1979), Sorokin (1994), Nesterenko (1996), Prévost (1996)Beukers (1979) reproduced Apéry's rational approximation to using the triple..

A triple of positive integers satisfying is said to be harmonic ifIn particular, such a triple is harmonic if the reciprocals of its terms form an arithmetic sequence with common difference whereOne can show that there exists a one-to-one correspondence between the set of equivalence classes of harmonic triples and the set of equivalence classes of geometric triples where here, two triples and are said to be equivalent if , i.e., if there exists some positive real number such that .

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.

The conjecture made by Belgian mathematician Eugène Charles Catalan in 1844 that 8 and 9 ( and ) are the only consecutive powers (excluding 0 and 1). In other words,(1)is the only nontrivial solution to Catalan'sDiophantine problem(2)The special case and is the case of a Mordell curve.Interestingly, more than 500 years before Catalan formulated his conjecture, Levi ben Gerson (1288-1344) had already noted that the only powers of 2 and 3 that apparently differed by 1 were and (Peterson 2000).This conjecture had defied all attempts to prove it for more than 150 years, although Hyyrő and Makowski proved that no three consecutive powers exist (Ribenboim 1996), and it was also known that 8 and 9 are the only consecutive cubic and square numbers (in either order). Finally, on April 18, 2002, Mihăilescu sent a manuscript proving the entire conjecture to several mathematicians (van der Poorten 2002). The proof has now appeared in..

The function defined by Minkowski for the purpose of mapping the quadratic surds in the open interval into the rational numbers of in a continuous, order-preserving manner. takes a number having continued fraction to the number(1)The function satisfies the following properties (Salem 1943). 1. is strictly increasing. 2. If is rational, then is of the form , with and integers. 3. If is a quadratic surd, then the continued fraction is periodic, and hence is rational. 4. The function is purely singular (Denjoy 1938). can also be constructed as(2)where and are two consecutive irreducible fractions from the Farey sequence. At the th stage of this definition, is defined for values of , and the ordinates corresponding to these values are for , 1, ..., (Salem 1943).The function satisfies the identity(3)A few special values include(4)(5)(6)(7)(8)(9)(10)(11)where is the golden ratio...

Gauss's continued fraction is given by the continuedfractionwhere is a hypergeometric function. Many analytic expressions for continued fractions of functions can be derived from this formula.

Euler's continued fraction is the name given by Borwein et al. (2004, p. 30)to Euler's formula for the inverse tangent,An even more famous continued fraction related to Euler which is perhaps a more appropriate recipient of the appellation "Euler's continued fraction" is the simple continued fraction for e, namely

A triple of positive integers satisfying is said to be geometric if . In particular, such a triple is geometric if its terms form a geometric sequence with common ratio whereOne can show that there exists a one-to-one correspondence between the set of equivalence classes of geometric triples and the set of equivalence classes of harmonic triples where here, two triples and are said to be equivalent if , i.e., if there exists some positive real number such that .

An exponential sum of the form(1)where is a real polynomial (Weyl 1914, 1916; Montgomery 2001). Writing(2)a notation introduced by Vinogradov, Weyl observed that(3)(4)(5)(6)a process known as Weyl differencing (Montgomery 2001).Weyl was able to use this process to show that if(7)is a real polynomial and at least one of , ..., is irrational, then is uniformly distributed (mod 1).

Given a set of equations in variables , ..., , written explicitly as(1)or more explicitly as(2)the Jacobian matrix, sometimes simply called "the Jacobian" (Simon and Blume 1994) is defined by(3)The determinant of is the Jacobian determinant (confusingly, often called "the Jacobian" as well) and is denoted(4)The Jacobian matrix and determinant can be computed in the WolframLanguage using JacobianMatrix[f_List?VectorQ, x_List] := Outer[D, f, x] /; Equal @@ (Dimensions /@ {f, x}) JacobianDeterminant[f_List?VectorQ, x_List] := Det[JacobianMatrix[f, x]] /; Equal @@ (Dimensions /@ {f, x})Taking the differential(5)shows that is the determinant of the matrix , and therefore gives the ratios of -dimensional volumes (contents) in and ,(6)It therefore appears, for example, in the changeof variables theorem.The concept of the Jacobian can also be applied to functions in more than variables. For example, considering..

A unit circle is a circle of unit radius, i.e., of radius 1.The unit circle plays a significant role in a number of different areas of mathematics. For example, the functions of trigonometry are most simply defined using the unit circle. As shown in the figure above, a point on the terminal side of an angle in angle standard position measured along an arc of the unit circle has as its coordinates so that is the horizontal coordinate of and is its vertical component.As a result of this definition, the trigonometric functions are periodic with period .Another immediate result of this definition is the ability to explicitly write the coordinates of a number of points lying on the unit circle with very little computation. In the figure above, for example, points , , , and correspond to angles of , , , and radians, respectively, whereby it follows that , , , and . Similarly, this method can be used to find trigonometric values associated to integer multiples of..

The spherical distance between two points and on a sphere is the distance of the shortest path along the surface of the sphere (paths that cut through the interior of the sphere are not allowed) from to , which always lies along a great circle.For points and on the unit sphere, the spherical distance is given bywhere denotes a dot product.

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..

Baire's category theorem, also known as Baire's theorem and the category theorem, is a result in analysis and set theory which roughly states that in certain spaces, the intersection of any countable collection of "large" sets remains "large." The appearance of "category" in the name refers to the interplay of the theorem with the notions of sets of first and second category.Precisely stated, the theorem says that if a space is either a complete metric space or a locally compact T2-space, then the intersection of every countable collection of dense open subsets of is necessarily dense in .The above-mentioned interplay with first and second category sets can be summarized by a single corollary, namely that spaces that are either complete metric spaces or locally compact Hausdorff spaces are of second category in themselves. To see that this follows from the above-stated theorem, let be either a complete metric..

A quaternion Kähler manifold is a Riemannian manifold of dimension , , whose holonomy is, up to conjugacy, a subgroup ofbut is not a subgroup of . These manifolds are sometimes called quaternionic Kähler and are sometimes written hyphenated as quaternion-Kähler, quaternionic-Kähler, etc.Despite their name, quaternion-Kähler manifolds need not be Kähler due to the fact that all Kähler manifolds have holonomy groups which are subgroups of , whereas . Depending on the literature, such manifolds are sometimes assumed to be connected and/or orientable. In the above definition, the case for is usually excluded due to the fact that which, under Berger's classification of holonomy, implies merely that the manifold is Riemannian. The above classification can be extended to the case where by requiring that the manifold be both an Einstein manifold and self-dual.Some authors exclude this last criterion,..

Let be an involutive algebra over the field of complex numbers with involution . Then is a right Hilbert algebra if has an inner product satisfying: 1. For all , is bounded on . 2. . 3. The involution is closable. 4. The linear span of products , , is a dense subalgebra of .

Let be an involutive algebra over the field of complex numbers with involution . Then is a left Hilbert algebra if has an inner product satisfying: 1. For all , is bounded on . 2. . 3. The involution is closable. 4. The linear span of products , , is a dense subalgebra of . Left Hilbert algebras are historically known as generalized Hilbert algebras (Takesaki 1970).A basic result in functional analysis says that if the involution map on a left Hilbert algebra is an antilinear isometry with respect to the inner product , then is also a right Hilbert algebra with respect to the involution . The converse also holds.

A Kähler metric is a Riemannian metric on a complex manifold which gives a Kähler structure, i.e., it is a Kähler manifold with a Kähler form. However, the term "Kähler metric" can also refer to the corresponding Hermitian metric , where is the Kähler form, defined by . Here, the operator is the almost complex structure, a linear map on tangent vectors satisfying , induced by multiplication by . In coordinates , the operator satisfies and .The operator depends on the complex structure, and on a Kähler manifold, it must preserve the Kähler metric. For a metric to be Kähler, one additional condition must also be satisfied, namely that it can be expressed in terms of the metric and the complex structure. Near any point , there exists holomorphic coordinates such that the metric has the formwhere denotes the vector space tensor product; that is, it vanishes up to order two at . Hence, any geometric..

Regge calculus is a finite element method utilized in numerical relativity in attempts of describing spacetimes with few or no symmetries by way of producing numerical solutions to the Einstein field equations (Khavari 2009). It was developed initially by Italian mathematician Tullio Regge in the 1960s (Regge 1961).Modern forays into Regge's method center on the triangulation of manifolds, particularly on the discrete approximation of 4-dimensional Riemannian and Lorentzian manifolds by way of cellular complexes whose 4-dimensional triangular simplices share their boundary tetrahedra (i.e., 3-dimensional simplices) to enclose a flat piece of spacetime (Marinelli 2013). Worth noting is that Regge himself devised the framework in more generality, though noted that no such generality is lost by assuming a triangular approximation (Regge 1961).The benefit of this technique is that the structures involved are rigid and hence are..

A piecewise linear function is a function composed of some number of linear segments defined over an equal number of intervals, usually of equal size.For example, consider the function over the interval . If is approximated by a piecewise linear function over an increasing number of segments, e.g., 1, 2, 4, and 8, the accuracy of the approximation is seen to improve as the number of segments increases.In the first case, with a single segment, if we compute the Lagrangeinterpolating polynomial, the equation of the linear function results.The trapezoidal rule for numeric integrationis described in a similar manner.Piecewise linear functions are also key to some constructive derivations. The length of a "piece" is given by the(1)summing the length of a number of pieces gives(2)and taking the limit as , the sum becomes(3)which is simplify the usual arc length...

Let denote the group of all invertible maps and let be any group. A homomorphism is called an action of on . Therefore, satisfies 1. For each , is a map . 2. . 3. , where is the group identity in . 4. .

Green's theorem is a vector identity which is equivalent to the curl theorem in the plane. Over a region in the plane with boundary , Green's theorem states(1)where the left side is a line integral and the right side is a surface integral. This can also be written compactly in vector form as(2)If the region is on the left when traveling around , then area of can be computed using the elegant formula(3)giving a surprising connection between the area of a region and the line integral around its boundary. For a plane curve specified parametrically as for , equation (3) becomes(4)which gives the signed area enclosed by the curve.The symmetric for above corresponds to Green's theorem with and , leading to(5)(6)(7)(8)(9)However, we are also free to choose other values of and , including and , giving the "simpler" form(10)and and , giving(11)A similar procedure can be applied to compute the moment about the -axis using and as(12)and about the..

"The" Jacobi identity is a relationship(1)between three elements , , and , where is the commutator. The elements of a Lie algebra satisfy this identity.Relationships between the Q-functions are also known as Jacobi identities:(2)equivalent to the Jacobi triple product (Borweinand Borwein 1987, p. 65) and(3)where(4) is the complete elliptic integral of the first kind, and . Using Weber functions(5)(6)(7)(5) and (6) become(8)(9)(Borwein and Borwein 1987, p. 69).

For even ,(1)(Nagell 1951, p. 176). Writing out symbolically,(2)which gives(3)where is a q-Pochhammer symbol.For example, for ,(4)and for ,(5)

A particle is said to be undergoing uniform circular motion if its radius vector in appropriate coordinates has the form , where(1)(2)Geometrically, uniform circular motions means that moves in a circle in the -plane with some radius at constant speed. The quantity is called the angular velocity of . The speed of is(3)and the acceleration of P has constant magnitude(4)and is directed toward the center of the circle traced by . This is called centripetal acceleration.Ignoring the ellipticity of their orbits, planet show nearly uniform circular motion about the Sun. (Although due to orbital inclinations, the orbital planes of the different planets are not necessarily coplanar.)

A problem in the calculus of variations. Let a vessel traveling at constant speed navigate on a body of water having surface velocity(1)(2)The navigation problem asks for the course which travels between two points in minimal time.

A numberwhere is an integer or rational number, is the inverse tangent, and is the inverse cotangent. Gregory numbers arise in the determination of Machin-like formulas. Every Gregory number can be expressed uniquely as a linear combination of s where the s are Størmer numbers.

Lehmer (1938) showed that every positive irrational number has a unique infinite continued cotangent representation of the form(1)where the s are nonnegative and(2)Note that this growth condition on coefficients is essential for the uniqueness of Lehmer expansion.The following table summarizes the coefficients for various special constants.OEISeA0026682, 8, 75, 8949, 119646723, 15849841722437093, ...Euler-Mascheroni constant A0817820, 1, 3, 16, 389, 479403, 590817544217, ...golden ratio A0062671, 4, 76, 439204, 84722519070079276, ...Lehmer's constant A0020650, 1, 3, 13, 183, 33673, ...A0026673, 73, 8599, 400091364,371853741549033970, ...Pythagoras's constant A0026661, 5, 36, 3406, 14694817,727050997716715, ...The expansion for the golden ratio has the beautiful closed form(3)where is a Lucas number.An illustration of a different cotangent expansion for that is not a Lehmer expansion because its coefficients..

The Lehmer cotangent expansion for whichthe convergence is slowest occurs when the inequality in the recurrence equation(1)for(2)is replaced by equality, giving and(3)for .This recurrences gives values of corresponding to 0, 1, 3, 13, 183, 33673, ... (OEIS A002065), and defines the constant known as Lehmer's constant as(4)(5)(6)(OEIS A030125). is not an algebraic number of degree less than 4, but Lehmer's approach cannot show whether is transcendental.

(1)(2)(3)where is a gamma function and is a double factorial.

The logarithmic integral is defined as theCauchy principal value(1)(2)Soldner's constant, denoted (or sometimes ) is the root of the logarithmic integral,(3)so that(4)for (Soldner 1812; Nielsen 1965, p. 88). Ramanujan calculated (Hardy 1999, Le Lionnais 1983, Berndt 1994), while the correct value is 1.45136923488... (OEIS A070769; Derbyshire 2004, p. 114).

A plot in the complex plane of the points(1)where and are the Fresnel integrals (von Seggern 2007, p. 210; Gray 1997, p. 65). The Cornu spiral is also known as the clothoid or Euler's spiral. It was probably first studied by Johann Bernoulli around 1696 (Bernoulli 1967, pp. 1084-1086). A Cornu spiral describes diffraction from the edge of a half-plane.The quantities and are plotted above. The slope of the curve's tangentvector (above right figure) is(2)plotted below. The Cesàro equation for a Cornu spiral is , where is the radius of curvature and the arc length. The torsion is .Gray (1997) defines a generalization of the Cornu spiral given by parametricequations(3)(4)(5)(6)where is a generalized hypergeometric function.The arc length, curvature,and tangential angle of this curve are(7)(8)(9)The Cesàro equation is(10)Dillen (1990) describes a class of "polynomial spirals" for which the..

The decimal expansion of the natural logarithm of 10 is given by(1)(OEIS A002392).It is also given by the BBP-type formulas(2)(3)(4)(5)(6)(7)(E. W. Weisstein, Aug. 28, 2008).

The catacaustic of the natural logarithm specified parametrically as(1)(2)is a complicated expression for an arbitrary radiantpoint.However, for a point , the catacaustic becomes(3)(4)Making the substitution then gives the equivalent parametrization(5)(6)which is the equation of a catenary.

The osculating circle of a curve at a given point is the circle that has the same tangent as at point as well as the same curvature. Just as the tangent line is the line best approximating a curve at a point , the osculating circle is the best circle that approximates the curve at (Gray 1997, p. 111).Ignoring degenerate curves such as straight lines, the osculating circle of a given curve at a given point is unique.Given a plane curve with parametric equations and parameterized by a variable , the radius of the osculating circle is simply the radius of curvature(1)where is the curvature, and the center is just the point on the evolute corresponding to ,(2)(3)Here, derivatives are taken with respect to the parameter .In addition, let denote the circle passing through three points on a curve with . Then the osculating circle is given by(4)(Gray 1997)...

Special affine curvature, also called as the equi-affine or affine curvature, is a type of curvature for a plane curve that remains unchanged under a special affine transformation.For a plane curve parametrized by , the special affine curvature is given by(1)(2)(Blaschke 1923, Guggenheimer 1977), where the prime indicates differentiation with respect to t. This reduces for a curve to(3)(4)(Blaschke 1923, Shirokov 1988), where the prime here indicated differentiation with respect to .The following table summarizes the special affine curvatures for a number of curves.curveparametrizationcatenarycircleellipsehyperbolaparabola0

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."..

A positive number such that a lamina or solid body with moment of inertia about an axis and mass is given byPickover (1995) defines a generalization of as a function quantifying the spatial extent of the structure of a curve and given bywhere is the length distribution function. Small compact patterns have small .

Expanding the Riemann zeta function about gives(1)(Havil 2003, p. 118), where the constants(2)are known as Stieltjes constants.Another sum that can be used to define the constants is(3)These constants are returned by the WolframLanguage function StieltjesGamma[n].A generalization takes as the coefficient of is the Laurent series of the Hurwitz zeta function about . These generalized Stieltjes constants are implemented in the Wolfram Language as StieltjesGamma[n, a].The case gives the usual Euler-Mascheroni constant(4)A limit formula for is given by(5)where is the imaginary part and is the Riemann zeta function.An alternative definition is given by absorbing the coefficient of into the constant,(6)(e.g., Hardy 1912, Kluyver 1927).The Stieltjes constants are also given by(7)Plots of the values of the Stieltjes constants as a function of are illustrated above (Kreminski). The first few numerical values are given in the..

Multisection of a mathematical quantity or figure is division of it into a number of (usually) equal parts. Division of a quantity into two equal parts is known as bisection, and into three equal parts is known as trisection.The coordinates of the first -multisection of a line segment with endpoints given in trilinear coordinates by and is , where(1)(2)(3)

There are two sets of constants that are commonly known as Lebesgue constants. The first is related to approximation of function via Fourier series, which the other arises in the computation of Lagrange interpolating polynomials.Assume a function is integrable over the interval and is the th partial sum of the Fourier series of , so that(1)(2)and(3)If(4)for all , then(5)and is the smallest possible constant for which this holds for all continuous . The first few values of are(6)(7)(8)(9)(10)(11)(12)(13)Some sum formulas for include(14)(15)(Zygmund 1959) and integral formulas include(16)(17)(Hardy 1942). For large ,(18)This result can be generalized for an -differentiable function satisfying(19)for all . In this case,(20)where(21)(Kolmogorov 1935, Zygmund 1959).Watson (1930) showed that(22)where(23)(24)(25)(OEIS A086052), where is the gamma function, is the Dirichlet lambda function, and is the Euler-Mascheroni constant.Define..

A determinant used to determine in which coordinate systems the Helmholtz differential equation is separable (Morse and Feshbach 1953). A determinant(1)in which are functions of alone is called a Stäckel determinant. A coordinate system is separable if it obeys the Robertson condition, namely that the scale factors in the Laplacian(2)can be rewritten in terms of functions defined by(3)such that can be written(4)When this is true, the separated equations are of the form(5)The s obey the minor equations(6)(7)(8)which are equivalent to(9)(10)(11)(Morse and Feshbach 1953, p. 509). This gives a total of four equations in nine unknowns. Morse and Feshbach (1953, pp. 655-666) give not only the Stäckel determinants for common coordinate systems, but also the elements of the determinant (although it is not clear how these are derived)...

A determinant which arises in the solution of the second-order ordinary differential equation(1)Writing the solution as a power series(2)gives a recurrence relation(3)The value of can be computed using the Hill determinant(4)where(5)(6)(7)and is the variable to solve for. The determinant can be given explicitly by the amazing formula(8)where(9)leading to the implicit equation for ,(10)

How far can a stack of books protrude over the edge of a table without the stack falling over? It turns out that the maximum overhang possible for books (in terms of book lengths) is half the th partial sum of the harmonic series.This is given explicitly by(1)where is a harmonic number. The first few values are(2)(3)(4)(5)(OEIS A001008 and A002805).When considering the stacking of a deck of 52 cards so that maximum overhang occurs, the total amount of overhang achieved after sliding over 51 cards leaving the bottom one fixed is(6)(7)(8)(Derbyshire 2004, p. 6).In order to find the number of stacked books required to obtain book-lengths of overhang, solve the equation for , and take the ceiling function. For , 2, ... book-lengths of overhang, 4, 31, 227, 1674, 12367, 91380, 675214, 4989191, 36865412, 272400600, ... (OEIS A014537) books are needed.When more than one book or card can be used per level, the problem becomes much more complex. For..

There are (at least) two mathematical constants associated with Theodorus. The first Theodorus's constant is the elementary algebraic number , i.e., the square root of 3. It has decimal expansion(1)(OEIS A002194) and is named after Theodorus, who proved that the square roots of the integers from 3 to 17 (excluding squares 4, 9,and 16) are irrational (Wells 1986, p. 34). The space diagonal of a unit cube has length . has continued fraction [1, 1, 2, 1, 2, 1, 2, ...] (OEIS A040001). In binary, it is represented by(2)(OEIS A004547).Another constant sometimes known as the constant of Theodorus is the slope of a continuous analog of the discrete Theodorus spiral due to Davis (1993) at the point , given by(3)(4)(5)(6)(OEIS A226317; Finch 2009), where is the Riemann zeta function.

In this work, the name Pythagoras's constant will be given to the squareroot of 2,(1)(OEIS A002193), which the Pythagoreans provedto be irrational.In particular, is the length of the hypotenuse of an isosceles right triangle with legs of length one, and the statement that it is irrational means that it cannot be expressed as a ratio of integers and . Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. A slight generalization is sometimes known as Pythagoras's theorem.Theodorus subsequently proved that the square roots of the numbers from 3 to 17 (excluding 4, 9,and 16) are also irrational (Wells 1986, p. 34).It is not known if Pythagoras's constant is normalto any base (Stoneham 1970, Bailey and Crandall 2003).The continued fraction for..

If is a prime , then the numerator of the harmonic number(1)is divisible by and the numerator of the generalized harmonic number(2)is divisible by . The numerators of are sometimes known as Wolstenholme numbers.These imply that if is prime, then(3)

There are at least two statements which go by the name of Artin's conjecture.If is any complex finite-dimensional representation of the absolute Galois group of a number field, then Artin showed how to associate an -series with it. These -series directly generalize zeta functions and Dirichlet -series, and as a result of work by Richard Brauer, is known to extend to a meromorphic function on the complex plane. Artin's conjecture predicts that it is in fact holomorphic, i.e., has no poles, with the possible exception of a pole at (Artin 1923/1924). Compare with the generalized Riemann hypothesis, which deals with the locations of the zeros of certain -series.The second conjecture states that every integer not equal to or a square number is a primitive root modulo for infinitely many and proposes a density for the set of such which are always rational multiples of a constant known as Artin's constant. There is an analogous theorem for functions instead..

A closed two-form on a complex manifold which is also the negative imaginary part of a Hermitian metric is called a Kähler form. In this case, is called a Kähler manifold and , the real part of the Hermitian metric, is called a Kähler metric. The Kähler form combines the metric and the complex structure, indeed(1)where is the almost complex structure induced by multiplication by . Since the Kähler form comes from a Hermitian metric, it is preserved by , i.e., since . The equation implies that the metric and the complex structure are related. It gives a Kähler structure, and has many implications.On , the Kähler form can be written as(2)(3)where . In general, the Kähler form can be written in coordinates(4)where is a Hermitian metric, the real part of which is the Kähler metric. Locally, a Kähler form can be written as , where is a function called a Kähler potential. The Kähler form is..

The term domain has (at least) three different meanings in mathematics.The term domain is most commonly used to describe the set of values for which a function (map, transformation, etc.) is defined. For example, a function that is defined for real values has domain , and is sometimes said to be "a function over the reals." The set of values to which is sent by the function is then called the range.Unfortunately, the term range is sometimes used in probability theory to mean domain (Feller 1968, p. 200; Evans et al. 2000). To confuse matters even more, the term "range" is more commonly used in statistics to refer to a completely different quantity, known in this work as the statistical range. As if this wasn't confusing enough, Evans et al. (2000, p. 6) define a probability domain to be the range of the distribution function of a probability density function.The domain (in its usual established mathematical sense)..

The set of all Schwartz functions is called a Schwartz space and is denoted . If denotes the set of smooth functions of compact support on , then this is a subset of . Since is dense in , is dense in for any .

A linear function is a function which satisfiesandfor all and in the domain, and all scalars .

Given a subset and a real function which is Gâteaux differentiable at a point , is said to be pseudoconvex at ifHere, denotes the usual gradient of .The term pseudoconvex is used to describe the fact that such functions share many properties of convex functions, particularly with regards to derivative properties and finding local extrema. Note, however, that pseudoconvexity is strictly weaker than convexity as every convex function is pseudoconvex though one easily checks that is pseudoconvex and non-convex.Similarly, every pseudoconvex function is quasi-convex, though the function is quasi-convex and not pseudoconvex.A function for which is pseudoconvex is said to be pseudoconcave.

The map(1)where is computed modulo 1. A generalized Baker's map can be defined as(2)(3)where , , and and are computed mod 1. The q-dimension is(4)If , then the general q-dimension is(5)

If the coefficients of the polynomial(1)are specified to be integers, then rational roots must have a numerator which is a factor of and a denominator which is a factor of (with either sign possible). This follows since a polynomial of polynomial order with rational roots can be expressed as(2)where the roots are , , ..., and . Factoring out the s,(3)Now, multiplying through,(4)where we have not bothered with the other terms. Since the first and last coefficients are and , all the rational roots of equation (1) are of the form [factors of ]/[factors of ].

The sum-of-factorial powers function is defined by(1)For ,(2)(3)(4)where is the exponential integral, (OEIS A091725), is the En-function, is the real part of , and i is the imaginary number. The first few values are 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, ... (OEIS A007489). cannot be written as a hypergeometric term plus a constant (Petkovšek et al. 1996). The only prime of this form is , since(5)(6)(7)is always a multiple of 3 for .In fact, is divisible by 3 for and , 5, 7, ... (since the Cunningham number given by the sum of the first two terms is always divisible by 3--as are all factorial powers in subsequent terms ) and so contains no primes, meaning sequences with even are the only prime contenders.The sum(8)does not appear to have a simple closed form, but its values for , 2, ... are 1, 5, 41, 617, 15017, 533417, 25935017, ... (OEIS A104344). It is prime for indices 2, 3, 4, 5, 7, 8, 10, 18, 21, 42, 51, 91, 133, 177, 182, 310, 3175, 9566, 32841,..

The alternating factorial is defined as the sum of consecutive factorialswith alternating signs,(1)They can be given in closed form as(2)where is the exponential integral, is the En-function, and is the gamma function.The alternating factorial will is implemented in the WolframLanguage as AlternatingFactorial[n].A simple recurrence equation for is given by(3)where .For , 2, ..., the first few values are 1, 1, 5, 19, 101, 619, 4421, 35899, ... (OEIS A005165).The first few values for which are (probable) primes are , 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164, 43592, 59961, ... (OEIS A001272; extending Guy 1994, p. 100). Živković (1999) has shown that the number of such primes is finite. was verified to be prime in Jul. 2000 by team of G. La Barbera and others using the Certifix program developed by Marcel Martin.The following table summarizes the largest..

Borwein et al. (2004, pp. 4 and 44) term the expression of the integrals(1)(2)(3)(4)(OEIS A083648 and A073009)in terms of infinite sums "a sophomore's dream."For , write(5)(6)Integrating term by term then gives(7)(8)(9)(Borwein et al. 2004, p. 44).For , write(10)(11)Integrating term by term then gives(12)(13)(14)(Borwein et al. 2004, pp. 4 and 44).

Wynn's -method is a method for numerical evaluation of sums and products that samples a number of additional terms in the series and then tries to extrapolate them by fitting them to a polynomial multiplied by a decaying exponential.In particular, the method provides an efficient algorithm for implementing transformations of the form(1)where(2)is the th partial sum of a sequence , which are useful for yielding series convergence improvement (Hamming 1986, p. 205). In particular, letting , , and(3)for , 2, ... (correcting the typo of Hamming 1986, p. 206). The values of are there equivalent to the results of applying transformations to the sequence (Hamming 1986, p. 206).Wynn's epsilon method can be applied to the terms of a series using the Wolfram Language command SequenceLimit[l]. Wynn's method may also be invoked in numerical summation and multiplication using Method -> Fit in the Wolfram Language's NSum and NProduct..

The improvement of the convergence properties of a series, also called convergence acceleration or accelerated convergence, such that a series reaches its limit to within some accuracy with fewer terms than required before. Convergence improvement can be effected by forming a linear combination with a series whose sum is known. Useful sums include(1)(2)(3)(4)Kummer's transformation takes a convergent series(5)and another convergent series(6)with known such that(7)Then a series with more rapid convergence to the same value is given by(8)(Abramowitz and Stegun 1972).The Euler transform takes a convergent alternatingseries(9)into a series with more rapid convergence to the same value to(10)where(11)(Abramowitz and Stegun 1972; Beeler et al. 1972).A general technique that can be used to acceleration converge of series is to expand them in a Taylor series about infinity and interchange the order of summation. In cases where a symbolic..

A -variate multivariate normal distribution (also called a multinormal distribution) is a generalization of the bivariate normal distribution. The -multivariate distribution with mean vector and covariance matrix is denoted . The multivariate normal distribution is implemented as MultinormalDistribution[mu1, mu2, ..., sigma11, sigma12, ..., sigma12, sigma22, ..., ..., x1, x2, ...] in the Wolfram Language package MultivariateStatistics` (where the matrix must be symmetric since ).In the case of nonzero correlations, there is in general no closed-form solution for the distribution function of a multivariate normal distribution. As a result, such computations must be done numerically.

The bivariate normal distribution is the statistical distribution with probabilitydensity function(1)where(2)and(3)is the correlation of and (Kenney and Keeping 1951, pp. 92 and 202-205; Whittaker and Robinson 1967, p. 329) and is the covariance.The probability density function of the bivariate normal distribution is implemented as MultinormalDistribution[mu1, mu2, sigma11, sigma12, sigma12, sigma22] in the Wolfram Language package MultivariateStatistics` .The marginal probabilities are then(4)(5)and(6)(7)(Kenney and Keeping 1951, p. 202).Let and be two independent normal variates with means and for , 2. Then the variables and defined below are normal bivariates with unit variance and correlation coefficient :(8)(9)To derive the bivariate normal probability function, let and be normally and independently distributed variates with mean 0 and variance 1, then define(10)(11)(Kenney and Keeping..

The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being converted to decay constants. has numerical value(1)(OEIS A002162).The irrationality measure of is known to be less than 3.8913998 (Rukhadze 1987, Hata 1990).It is not known if is normal (Bailey and Crandall 2002).The alternating series and BBP-typeformula(2)converges to the natural logarithm of 2, where is the Dirichlet eta function. This identity follows immediately from setting in the Mercator series, yielding(3)It is also a special case of the identity(4)where is the Lerch transcendent.This is the simplest in an infinite class of such identities, the first few of which are(5)(6)(E. W. Weisstein, Oct. 7, 2007).There are many other classes of BBP-type formulas for , including(7)(8)(9)(10)(11)Plouffe (2006) found the beautiful sum(12)A rapidly converging Zeilberger-type sum..

Catalan's constant is a constant that commonly appears in estimates of combinatorial functions and in certain classes of sums and definite integrals. It is usually denoted (this work), (e.g., Borwein et al. 2004, p. 49), or (Wolfram Language).Catalan's constant may be defined by(1)(Glaisher 1877, who however did not explicitly identify the constant in this paper). It is not known if is irrational.Catalan's constant is implemented in the WolframLanguage as Catalan.The constant is named in honor of E. C. Catalan (1814-1894), who first gave an equivalent series and expressions in terms of integrals. Numerically,(2)(OEIS A006752). can be given analytically by the following expressions(3)(4)(5)where is the Dirichlet beta function, is Legendre's chi-function, is the Glaisher-Kinkelin constant, and is the partial derivative of the Hurwitz zeta function with respect to the first argument.Glaisher (1913) gave(6)(Vardi..

There are many formulas of of many types. Among others, these include series, products, geometric constructions, limits, special values, and pi iterations. is intimately related to the properties of circles and spheres. For a circle of radius , the circumference and area are given by(1)(2)Similarly, for a sphere of radius , the surface area and volume enclosed are(3)(4)An exact formula for in terms of the inverse tangents of unit fractions is Machin's formula(5)There are three other Machin-like formulas,as well as thousands of other similar formulas having more terms.Gregory and Leibniz found(6)(7)(Wells 1986, p. 50), which is known as the Gregory series and may be obtained by plugging into the Leibniz series for . The error after the th term of this series in the Gregory series is larger than so this sum converges so slowly that 300 terms are not sufficient to calculate correctly to two decimal places! However, it can be transformed..

The Gregory series is a pi formula found by Gregory and Leibniz and obtained by plugging into the Leibniz series,(1)(Wells 1986, p. 50). The formula converges very slowly, but its convergence can be accelerated using certain transformations, in particular(2)where is the Riemann zeta function (Vardi 1991).Taking the partial series gives the analytic result(3)Rather amazingly, expanding about infinity gives the series(4)(Borwein and Bailey 2003, p. 50), where is an Euler number. This means that truncating the Gregory series at half a large power of 10 can give a decimal expansion for whose decimal digits are largely correct, but where wrong digits occur with precise regularity. For example, taking gives where the sequence of differences is precisely twice the Euler (secant) numbers. In fact, just this pattern of digits was observed by J. R. North in 1988 before the closed form of the truncated series was known..

The BBP (named after Bailey-Borwein-Plouffe) is a formula for calculating pidiscovered by Simon Plouffe in 1995,Amazingly, this formula is a digit-extraction algorithm for in base 16.Following the discovery of this and related formulas, similar formulas in other bases were investigated. This class of formulas are now known as BBP-type formulas.

The figure eight knot, also known as the Flemish knot and savoy knot, is the unique prime knot of four crossings 04-001. It has braid word .The figure eight knot is implemented in the WolframLanguage as KnotData["FigureEight"].It is a 2-embeddable knot, and is amphichiral as well as invertible. It has Arf invariant 1. It is not a slice knot (Rolfsen 1976, p. 224).The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F of the figure eight knot are(1)(2)(3)(4)(5)(6)There are no other knots on 10 or fewer crossings sharing the same Alexander polynomial, BLM/Ho polynomial, bracket polynomial, HOMFLY polynomial, Jones polynomial, or Kauffman polynomial F.The figure eight knot has knot group(7)(Rolfsen 1976, p. 58).Helaman Ferguson's sculpture "Figure-Eight Complement II" illustrates the knot complement of the figure eight..

Newton's method, also called the Newton-Raphson method, is a root-finding algorithm that uses the first few terms of the Taylor series of a function in the vicinity of a suspected root. Newton's method is sometimes also known as Newton's iteration, although in this work the latter term is reserved to the application of Newton's method for computing square roots.For a polynomial, Newton's method is essentially the same as Horner's method.The Taylor series of about the point is given by(1)Keeping terms only to first order,(2)Equation (2) is the equation of the tangent line to the curve at , so is the place where that tangent line intersects the -axis. A graph can therefore give a good intuitive idea of why Newton's method works at a well-chosen starting point and why it might diverge with a poorly-chosen starting point.This expression above can be used to estimate the amount of offset needed to land closer to the root starting from an initial guess..

The transform inverting the sequence(1)into(2)where the sums are over all possible integers that divide and is the Möbius function.The logarithm of the cyclotomicpolynomial(3)is closely related to the Möbius inversion formula.

Let and be sequences of complex numbers such that for , and let the lower triangular matrices and be defined asandwhere the product over an empty set is 1. Then and are matrix inverses (Bhatnagar 1995, pp. 16-17).This result simplifies to the Gould and Hsu matrix inversion formula when , to Carlitz's -analog for (Carlitz 1972), and specialized to Bressoud's matrix theorem (Bressoud 1983) for and (Bhatnagar 1995, p. 17).The formula can also be extended to a summation theorem which generalizes Gosper's bibasic sum (Gasper and Rahman 1990, p. 240; Bhatnagar 1995, p. 19).

Given a subset and a point , the contingent cone at with respect to is defined to be the setwhere is the upper left Dini derivative of the distance functionA classical result in convex analysis characterizes as the collection of vectors in for which there are sequences in and in such that lies in for all (Borwein). Intuitively, then, the contingent cone consists of limits of directions to points near in .

A generalization of calculus of variations which draws the relationship between the stationary points of a smooth real-valued function on a manifold and the global topology of the manifold. For example, if a compact manifold admits a function whose only stationary points are a maximum and a minimum, then the manifold is a sphere. Technically speaking, Morse theory applied to a function on a manifold with and shows that every bordism can be realized as a finite sequence of surgeries. Conversely, a sequence of surgeries gives a bordism.There are a number of classical applications of Morse theory, including counting geodesics on a Riemann surface and determination of the topology of a Lie group (Bott 1960, Milnor 1963). Morse theory has received much attention in the last two decades as a result of the paper by Witten (1982) which relates Morse theory to quantum field theory and also directly connects the stationary points of a smooth function to..

Let be a locally Euclidean coordinate system. Then(1)Now plug in(2)(3)to obtain(4)Reading off the coefficients from(5)gives(6)(7)(8)Making a change of coordinates gives(9)(10)(11)(12)(13)(14)

A topology induced by the metric defined on a metric space . The open sets are all subsets that can be realized as the unions of open ballswhere , and .The metric topology makes a T2-space. Given two distinct points and of , their distance is certainly positive, so the open balls and are disjoint neighborhoods of and , respectively.

Replacing the logistic equation(1)with the quadratic recurrence equation(2)where (sometimes also denoted ) is a positive constant sometimes known as the "biotic potential" gives the so-called logistic map. This quadratic map is capable of very complicated behavior. While John von Neumann had suggested using the logistic map as a random number generator in the late 1940s, it was not until work by W. Ricker in 1954 and detailed analytic studies of logistic maps beginning in the 1950s with Paul Stein and Stanislaw Ulam that the complicated properties of this type of map beyond simple oscillatory behavior were widely noted (Wolfram 2002, pp. 918-919).The first few iterations of the logistic map (2) give(3)(4)(5)where is the initial value, plotted above through five iterations (with increasing iteration number indicated by colors; 1 is red, 2 is yellow, 3 is green, 4 is blue, and 5 is violet) for various values of .The..

There are at least two maps known as the Hénon map.The first is the two-dimensional dissipative quadraticmap given by the coupled equations(1)(2)(Hénon 1976).The strange attractor illustrated above is obtained for and .The illustration above shows two regions of space for the map with and colored according to the number of iterations required to escape (Michelitsch and Rössler 1989).The plots above show evolution of the point for parameters (left) and (right).The Hénon map has correlation exponent (Grassberger and Procaccia 1983) and capacity dimension (Russell et al. 1980). Hitzl and Zele (1985) give conditions for the existence of periods 1 to 6.A second Hénon map is the quadratic area-preserving map(3)(4)(Hénon 1969), which is one of the simplest two-dimensional invertible maps...

Let be a compact -dimensional oriented Riemannian manifold without boundary, let be a group representation of by orthogonal matrices, and let be the associated vector bundle. Suppose further that the Laplacian is strictly negative on where is the linear space of differential k-forms on with values in . In this context, the analytic torsion is defined as the positive real root ofwhere the -function is defined byfor the collection of eigenvalues of , the restriction of to the collection of bundle sections of the sheaf .Intrinsic to the above computation is that is a real manifold. However, there is a collection of literature on analytic torsion for complex manifolds, the construction of which is nearly identical to the construction given above. Analytic torsion on complex manifolds is sometimes called del bar torsion...

The fractal-like two-dimensional functionThe function is named for the appearance of a butterfly-like pattern centered around the origin (left figure). In the above illustration, the left plot runs from to 5 and the right plot runs from to 20.

There are two kinds of Bell polynomials.A Bell polynomial , also called an exponential polynomial and denoted (Bell 1934, Roman 1984, pp. 63-67) is a polynomial that generalizes the Bell number and complementary Bell number such that(1)(2)These Bell polynomial generalize the exponentialfunction.Bell polynomials should not be confused with Bernoulli polynomials, which are also commonly denoted .Bell polynomials are implemented in the Wolfram Language as BellB[n, x].The first few Bell polynomials are(3)(4)(5)(6)(7)(8)(9)(OEIS A106800). forms the associated Sheffer sequence for(10)so the polynomials have that exponentialgenerating function(11)Additional generating functions for are given by(12)or(13)with , where is a binomial coefficient.The Bell polynomials have the explicit formula(14)where is a Stirling number of the second kind.A beautiful binomial sum is given by(15)where is a binomial coefficient.The..

For , the Riemann zeta function is given by(1)(2)where is the th prime. This is Euler's product (Whittaker and Watson 1990), called by Havil (2003, p. 61) the "all-important formula" and by Derbyshire (2004, pp. 104-106) the "golden key."This can be proved by expanding the product, writing each term as a geometricseries, expanding, multiplying, and rearranging terms,(3)Here, the rearrangement leading to equation (1) follows from the fundamental theorem of arithmetic, since each product of prime powers appears in exactly one denominator and each positive integer equals exactly one product of prime powers.This product is related to the Möbius function via(4)which can be seen by expanding the product to obtain(5)(6)(7)(8)(9), but the finite product exists, giving(10)For upper limits , 1, 2, ..., the products are given by 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, ... (OEIS A060753 and..

Landau (1911) proved that for any fixed ,as , where the sum runs over the nontrivial Riemann zeta function zeros and is the Mangoldt function. Here, "fixed " means that the constant implicit in depends on and, in particular, as approaches a prime or a prime power, the constant becomes large.Landau's formula is roughly the derivative of the explicitformula.Landau's formula is quite extraordinary. If is not a prime or a prime power, then and the sum grows as a constant times . But if is a prime or a prime power, then and the sum grows much faster, like a constant times . This exhibits an amazing connection between the primes and the s; somehow the zeros "recognize" when is a prime and cause large contributions to the sum.

There are two types of singular values, one in the context of elliptic integrals, and the other in linear algebra.For a square matrix , the square roots of the eigenvalues of , where is the conjugate transpose, are called singular values (Marcus and Minc 1992, p. 69). The so-called singular value decomposition of a complex matrix is given by(1)where and are unitary matrices and is a diagonal matrix whose elements are the singular values of (Golub and Van Loan 1996, pp. 70 and 73). Singular values are returned by the command SingularValueList[m].If(2)where is a unitary matrix and is a Hermitian matrix, then the eigenvalues of are the singular values of .For elliptic integrals, a elliptic modulus such that(3)where is a complete elliptic integral of the first kind, and . The elliptic lambda function gives the value of . Abel (quoted in Whittaker and Watson 1990, p. 525) proved that if is an integer, or more generally whenever(4)where..

An Eisenstein series with half-period ratio and index is defined by(1)where the sum excludes , , and is an integer (Apostol 1997, p. 12).The Eisenstein series satisfies the remarkable property(2)if the matrix is in the special linear group (Serre 1973, pp. 79 and 83). Therefore, is a modular form of weight (Serre 1973, p. 83).Furthermore, each Eisenstein series is expressible as a polynomial of the elliptic invariants and of the Weierstrass elliptic function with positive rational coefficients (Apostol 1997).The Eisenstein series satisfy(3)where is the Riemann zeta function and is the divisor function (Apostol 1997, pp. 24 and 69). Writing the nome as(4)where is a complete elliptic integral of the first kind, , is the elliptic modulus, and defining(5)we have(6)(7)where(8)(9)(10)where is a Bernoulli number. For , 2, ..., the first few values of are , 240, , 480, -264, , ... (OEIS A006863 and A001067).The first..

Let be a complex number, then inequality(1)holds in the lens-shaped region illustrated above. Written explicitly in terms of real variables, this can be written as(2)where(3)The area enclosed is roughly(4)(OEIS A140133).This region can be parameterized in terms of a variable as(5)(6)Written parametrically in terms of the Cartesian coordinates,(7)(8)This region is intimately related to the study of Bessel functions and Kapteynseries (Plummer 1960, p. 47; Watson 1966, p. 270). reaches its maximum value at (OEIS A085984; Goursat 1959, p. 120; Le Lionnais 1983, p. 36), given by the root of(9)or equivalently by the root of(10)as noted by Stieltjes.The minimum value of corresponding to the maximum value is (OEIS A033259; Plummer 1960, p. 47; Watson 1966, p. 270), which is known as the Laplace limit constant. It is precisely the point at which Laplace's formula for solving Kepler's equation begins..

The conjugate gradient method is an algorithm for finding the nearest local minimum of a function of variables which presupposes that the gradient of the function can be computed. It uses conjugate directions instead of the local gradient for going downhill. If the vicinity of the minimum has the shape of a long, narrow valley, the minimum is reached in far fewer steps than would be the case using the method of steepest descent.For a discussion of the conjugate gradient method on vector and shared memory computers, see Dongarra et al. (1991). For discussions of the method for more general parallel architectures, see Demmel et al. (1993) and Ortega (1988) and the references therein.

The Cantor function is the continuous but not absolutely continuous function on which may be defined as follows. First, express in ternary. If the resulting ternary digit string contains the digit 1, replace every ternary digit following the 1 by a 0. Next, replace all 2's with 1's. Finally, interpret the result as a binary number which then gives .The Cantor function is a particular case of a devil's staircase (Devaney 1987, p. 110), and can be extended to a function for , with corresponding to the usual Cantor function (Gorin and Kukushkin 2004).Chalice (1991) showed that any real-valued function on which is monotone increasing and satisfies 1. , 2. , 3. is the Cantor function (Chalice 1991; Wagon 2000, p. 132).Gorin and Kukushkin (2004) give the remarkable identityfor integer . For and , 2, ..., this gives the first few values as 1/2, 3/10, 1/5, 33/230, 5/46, 75/874, ... (OEIS A095844 and A095845).M. Trott (pers. comm., June..

The transformation of a sequence into a sequence by the formula(1)where is a Stirling number of the second kind. The inverse transform is given by(2)where is a Stirling number of the first kind (Sloane and Plouffe 1995, p. 23).The following table summarized Stirling transforms for some common sequences, where denotes the Iverson bracket and denotes the primes.OEIS1A0001101, 1, 2, 5, 15, 52, 203, ...A0054930, 1, 3, 10, 37, 151, 674, ...A0001101, 2, 5, 15, 52, 203, 877, ...A0855070, 0, 1, 4, 13, 41, 136, 505, ...A0244301, 0, 1, 3, 8, 25, 97, 434, 2095, ...A0244290, 1, 1, 2, 7, 27, 106, 443, ...A0339991, , 1, , 1, , ...Here, gives the Bell numbers. has the exponential generating function(3)

The Borwein integrals are the class of definiteintegrals defined byfor odd . The integrals are curious because the terms , 3, ..., 13 all have unit numerators, but , 17, ... do not. The sequence of values of for , 3, ... is given by 1/2, 1/6, 1/30, 1/210, 1/1890, 1/20790, 1/270270, 467807924713440738696537864469/1896516717212415135141110350293750000, ... (OEIS A068214 and A068215; Borwein et al. 2004, p. 98; Bailey et al. 2006).

The th subfactorial (also called the derangement number; Goulden and Jackson 1983, p. 48; Graham et al. 2003, p. 1050) is the number of permutations of objects in which no object appears in its natural place (i.e., "derangements").The term "subfactorial "was introduced by Whitworth (1867 or 1878; Cajori 1993, p. 77). Euler (1809) calculated the first ten terms.The first few values of for , 2, ... are 0, 1, 2, 9, 44, 265, 1854, 14833, ... (OEIS A000166). For example, the only derangements of are and , so . Similarly, the derangements of are , , , , , , , , and , so .Sums and formulas for include(1)(2)(3)(4)where is a factorial, is a binomial coefficient, and is the incomplete gamma function.Subfactorials are implemented in the WolframLanguage as Subfactorial[n].A plot the real and imaginary parts of the subfactorial generalized to any real argument is illustrated above, with the usual integer-valued subfactorial..

A harmonic number is a number of the form(1)arising from truncation of the harmonic series.A harmonic number can be expressed analytically as(2)where is the Euler-Mascheroni constant and is the digamma function.The first few harmonic numbers are 1, , , , , ... (OEIS A001008 and A002805). The numbers of digits in the numerator of for , 1, ... are 1, 4, 41, 434, 4346, 43451, 434111, 4342303, 43428680, ... (OEIS A114467), with the corresponding number of digits in the denominator given by 1, 4, 40, 433, 4345, 43450, 434110, 4342302, 43428678, ... (OEIS A114468). These digits converge to what appears to be the decimal digits of (OEIS A002285).The first few indices such that the numerator of is prime are given by 2, 3, 5, 8, 9, 21, 26, 41, 56, 62, 69, ... (OEIS A056903). The search for prime numerators has been completed up to by E. W. Weisstein (May 13, 2009), and the following table summarizes the largest known values.decimal digitsdiscoverer6394227795E. W. Weisstein..

Exponentiation is the process of taking a quantity (the base) to the power of another quantity (the exponent). This operation most commonly denoted . In TeX, the Wolfram Language, and many other computer languages, exponentiation is denoted with a caret, i.e., as b^e. However, in FORTRAN, it is denoted b**e (Calderbank 1989, p. 29).

Polynomials which form the Sheffer sequence for(1)(2)and have generating function(3)The first few are(4)(5)(6)(7)The Pidduck polynomials are related to the Mittag-Leffler polynomials by(8)(Roman 1984, p. 127).

Polynomials which are a generalization of the Boole polynomials, form the Sheffer sequence for(1)(2)and have generating function(3)The first few are(4)(5)and(6)

The Cantor set , sometimes also called the Cantor comb or no middle third set (Cullen 1968, pp. 78-81), is given by taking the interval (set ), removing the open middle third (), removing the middle third of each of the two remaining pieces (), and continuing this procedure ad infinitum. It is therefore the set of points in the interval whose ternary expansions do not contain 1, illustrated above.The th iteration of the Cantor is implemented in the Wolfram Language as CantorMesh[n].Iterating the process 1 -> 101, 0 -> 000 starting with 1 gives the sequence 1, 101, 101000101, 101000101000000000101000101, .... The sequence of binary bits thus produced is therefore 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ... (OEIS A088917) whose th term is amazingly given by (mod 3), where is a (central) Delannoy number and is a Legendre polynomial (E. W. Weisstein, Apr. 9, 2006). The recurrence plot for this..

A subset of a topological space is said to be of first category in if can be written as the countable union of subsets which are nowhere dense in , i.e., if is expressible as a unionwhere each subset is nowhere dense in . Informally, one thinks of a first category subset as a "small" subset of the host space and indeed, sets of first category are sometimes referred to as meager. Sets which are not of first category are of second category.An important distinction should be made between the above-used notion of "category" and category theory. Indeed, the notions of first and second category sets are independent of category theory.The rational numbers are of first category and the irrational numbers are of second category in with the usual topology. In general, the host space and its topology play a fundamental role in determining category. For example, the set of integers with the subset topology inherited from is (vacuously) of..

Let be an involutive algebra over the field of complex numbers with involution . Then is a modular Hilbert algebra if has an inner product and a one-parameter group of automorphisms on , , satisfying: 1. . 2. For all , is bounded (hence, continuous) on . 3. The linear span of products , , is a dense subalgebra of . 4. for all , . 5. . 6. . 7. is an entire function of on . 8. For every real number , the set is dense in . The group is called the group of modular automorphisms.Note that the definition of modular Hilbert algebras is closely related to that of generalized Hilbert algebras in that every modular Hilbert algebra is a generalized Hilbert algebra provided that it satisfies one additional condition, namely that the involution is closable as a linear operator on the real pre-Hilbert space . This relationship is due, in part, to the fact that the properties of both structures were at the core of Tomita's original exposition of what is today the heart of Tomita-Takesaki..

A vector space with a T2-space topology such that the operations of vector addition and scalar multiplication are continuous. The interesting examples are infinite-dimensional spaces, such as a space of functions. For example, a Hilbert space and a Banach space are topological vector spaces.The choice of topology reflects what is meant by convergence of functions. For instance, for functions whose integrals converge, the Banach space , one of the L-p-spaces, is used. But if one is interested in pointwise convergence, then no norm will suffice. Instead, for each define the seminormon the vector space of functions on . The seminorms define a topology, the smallest one in which the seminorms are continuous. So is equivalent to for all , i.e., pointwise convergence. In a similar way, it is possible to define a topology for which "convergence" means uniform convergence on compact sets...

Let be a Hilbert space, the set of bounded linear operators from to itself, an operator on , and the operator spectrum of . Then if and is normal, there exists a unique resolution of the identity on the Borel subsets of which satisfiesFurthermore, every projection commutes with every that commutes with .

A projection matrix is an square matrix that gives a vector space projection from to a subspace . The columns of are the projections of the standard basis vectors, and is the image of . A square matrix is a projection matrix iff .A projection matrix is orthogonal iff(1)where denotes the adjoint matrix of . A projection matrix is a symmetric matrix iff the vector space projection is orthogonal. In an orthogonal projection, any vector can be written , so(2)An example of a nonsymmetric projection matrix is(3)which projects onto the line .The case of a complex vector space is analogous. A projection matrix is a Hermitian matrix iff the vector space projection satisfies(4)where the inner product is the Hermitian inner product. Projection operators play a role in quantum mechanics and quantum computing.Any vector in is fixed by the projection matrix for any in . Consequently, a projection matrix has norm equal to one, unless ,(5)Let be a -algebra. An element..

There are at least two distinct (though related) notions of the term Hilbert algebrain functional analysis.In some literature, a linear manifold of a (not necessarily separable) Hilbert space is a Hilbert algebra if the following conditions are satisfied: 1. is dense in . 2. is a ring so that, for any , there is defined an element such that , , , and for any complex number . 3. For any , there exists an adjoint element such that , and . 4. For any , there exists a positive number such that for all . 5. For every , there exists a unique bounded linear operator on such that for all . Moreover, if for an element and for all , then . At least one author defines a Hilbert algebra to be a quasi-Hilbertalgebrafor which for all (Dixmier 1981).

The group algebra , where is a field and a group with the operation , is the set of all linear combinations of finitely many elements of with coefficients in , hence of all elements of the form(1)where and for all . This element can be denoted in general by(2)where it is assumed that for all but finitely many elements of . is an algebra over with respect to the addition defined by the rule(3)the product by a scalar given by(4)and the multiplication(5)From this definition, it follows that the identity element of is the unit of , and that is commutative iff is an Abelian group.If the field is replaced by a unit ring , the addition and the multiplication defined above yield the group ring .If , and is the usual addition of integers, the group ring is isomorphic to the ring formed by all sums(6)where are integers, and for all indices .Let be a locally compact group and be a left invariant Haar measure on . Then the Banach space under the product given by the convolution..

Let be a sequence of complex numbers and let the lower triangular matrices and be defined asandwhere the product over an empty set is 1. Then and are matrix inverses (Bhatnagar 1995, pp. 15-16 and 50-51). The Krattenthaler matrix inversion formula is a generalization of this result.

Let be the incenter of a triangle and , , and be the intersections of the segments , , with the incircle. Also let the centroid lie inside the incircle and , , and be the intersections of the segments , , with the incircle. Then the perimeter of is less than or equal to that of , as proposed by Garfunkel (1981, 1982) and proved by Nyugen and Dergiades (2004).

If takes only nonnegative values, then(1)To prove the theorem, write(2)(3)Since is a probability density, it must be . We have stipulated that , so(4)(5)(6)(7)(8)Q.E.D.

The de Bruijn constant, also called the Copson-de Bruijn constant, is the minimal constant(OEIS A113276) such that the inequalityalways holds.

A quantified system of real algebraic equations and inequalities in variables is an expressionwhere is a quantifier ( or ) and is a system of real algebraic equations and inequalities in . By Tarski's theorem, the solution set of a quantified system of real algebraic equations and inequalities is a semialgebraic set.

The word polyhedron has slightly different meanings in geometry and algebraic geometry. In geometry, a polyhedron is simply a three-dimensional solid which consists of a collection of polygons, usually joined at their edges. The word derives from the Greek poly (many) plus the Indo-European hedron (seat). A polyhedron is the three-dimensional version of the more general polytope (in the geometric sense), which can be defined in arbitrary dimension. The plural of polyhedron is "polyhedra" (or sometimes "polyhedrons").The term "polyhedron" is used somewhat differently in algebraic topology, where it is defined as a space that can be built from such "building blocks" as line segments, triangles, tetrahedra, and their higher dimensional analogs by "gluing them together" along their faces (Munkres 1993, p. 2). More specifically, it can be defined as the underlying space..

Archimedes' axiom, also known as the continuity axiom or Archimedes' lemma, survives in the writings of Eudoxus (Boyer and Merzbach 1991), but the term was first coined by the Austrian mathematician Otto Stolz (1883). It states that, given two magnitudes having a ratio, one can find a multiple of either which will exceed the other. This principle was the basis for the method of exhaustion, which Archimedes invented to solve problems of area and volume.Symbolically, the axiom states thatiff the appropriate one of following conditions is satisfied for integers and : 1. If , then . 2. If , then . 3. If , then . Formally, Archimedes' axiom states that if and are two line segments, then there exist a finite number of points , , ..., on such thatand is between and (Itô 1986, p. 611). A geometry in which Archimedes' lemma does not hold is called a non-Archimedean Geometry...

If is a function on an open set , then the zero set of is the set . A subset of a topological space is called a zero set if it is equal to for some continuous function .

The Griewank function is a function widely used to test the convergence of optimization functions. The Griewank function of order is defined byfor (Griewank 1981), plotted above for . It has a global minimum of 0 at the point .The function has 191 minima, with global minimum at and local minima at for (OEIS A177889), 12.5601, 18.8401, 25.1202, .... Restricting the domain of the function to , the numbers of local minima for for , 2, ... are therefore given by 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7, ... (OEIS A178832).

Let be a rational number in the closed interval , and generate a sequence using the map(1)Then the number of periodic map orbits of period (for prime) is given by(2)(i.e., the number of period- repeating bit strings, modulo shifts). Since a typical map orbit visits each point with equal probability, the natural invariant is given by(3)

The level set of a differentiable function corresponding to a real value is the set of pointsFor example, the level set of the function corresponding to the value is the sphere with center and radius .If , the level set is a plane curve known as a level curve. If , the level set is a surface known as a level surface.

A web diagram, also called a cobweb plot, is a graph that can be used to visualize successive iterations of a function . In particular, the segments of the diagram connect the points , , , .... The diagram is so-named because its straight line segments "anchored" to the functions and can resemble a spider web. The animation above shows a web diagram for the logistic map with .

A Pólya plot is a plot of the vector field of of a complex function . Several examples are shown above.Pólya plots can be created in the WolframLanguage using the following code: PolyaFieldPlot[f_, {x_, xmin_, xmax_}, {y_, ymin_, ymax_}, opts : OptionsPattern[]] := VectorPlot[Evaluate @ {Re[f], -Im[f]}, {x, xmin, xmax}, {y, ymin, ymax}, VectorScale -> {Automatic, Automatic, Log[#5 + 1]&}, opts ]

There are (at least) two mathematical objects known as Weierstrass forms. The first is a general form into which an elliptic curve over any field can be transformed, given bywhere , , , , and are elements of .The second is the definition of the gamma functionaswhere is the Euler-Mascheroni constant (Krantz 1999, p. 157).

A symmetric polynomial on variables , ..., (also called a totally symmetric polynomial) is a function that is unchanged by any permutation of its variables. In other words, the symmetric polynomials satisfy(1)where and being an arbitrary permutation of the indices 1, 2, ..., .For fixed , the set of all symmetric polynomials in variables forms an algebra of dimension . The coefficients of a univariate polynomial of degree are algebraically independent symmetric polynomials in the roots of , and thus form a basis for the set of all such symmetric polynomials.There are four common homogeneous bases for the symmetric polynomials, each of which is indexed by a partition (Dumitriu et al. 2004). Letting be the length of , the elementary functions , complete homogeneous functions , and power-sum functions are defined for by(2)(3)(4)and for by(5)where is one of , or . In addition, the monomial functions are defined as(6)where is the set of permutations..

Orthogonal polynomials are classes of polynomials defined over a range that obey an orthogonality relation(1)where is a weighting function and is the Kronecker delta. If , then the polynomials are not only orthogonal, but orthonormal.Orthogonal polynomials have very useful properties in the solution of mathematical and physical problems. Just as Fourier series provide a convenient method of expanding a periodic function in a series of linearly independent terms, orthogonal polynomials provide a natural way to solve, expand, and interpret solutions to many types of important differential equations. Orthogonal polynomials are especially easy to generate using Gram-Schmidt orthonormalization.A table of common orthogonal polynomials is given below, where is the weighting function and(2)(Abramowitz and Stegun 1972, pp. 774-775).polynomialintervalChebyshev polynomial of the first kindChebyshev polynomial of the..

A real polynomial is said to be stable if all its roots lie in the left half-plane. The term "stable" is used to describe such a polynomial because, in the theory of linear servomechanisms, a system exhibits unforced time-dependent motion of the form , where is the root of a certain real polynomial . A system is therefore mechanically stable iff is a stable polynomial.The polynomial is stable iff , and the irreducible polynomial is stable iff both and are greater than zero. The Routh-Hurwitz theorem can be used to determine if a polynomial is stable.Given two real polynomials and , if and are stable, then so is their product , and vice versa (Séroul 2000, p. 280). It therefore follows that the coefficients of stable real polynomials are either all positive or all negative (although this is not a sufficient condition, as shown with the counterexample ). Furthermore, the values of a stable polynomial are never zero for and have..

Given a polynomial(1)of degree with roots , , ..., and a polynomial(2)of degree with roots , , ..., , the resultant , also denoted and also called the eliminant, is defined by(3)(Trott 2006, p. 26).Amazingly, the resultant is also given by the determinantof the corresponding Sylvester matrix.Kronecker gave a series of lectures on resultants during the summer of 1885 (O'Connor and Robertson 2005).An important application of the resultant is the elimination of one variable from a system of two polynomial equations (Trott 2006, p. 26).The resultant of two polynomials can be computed using the Wolfram Language function Resultant[poly1, poly2, var]. This command accepts the following methods: Automatic, SylvesterMatrix, BezoutMatrix, Subresultants, and Modular, where the optimal choice depends dramatically on the concrete polynomial pair under consideration and typically requires some experimentation. For high-order..

A differential ideal is an ideal in the ring of smooth forms on a manifold . That is, it is closed under addition, scalar multiplication, and wedge product with an arbitrary form. The ideal is called integrable if, whenever , then also , where is the exterior derivative.For example, in , the ideal(1)where the are arbitrary smooth functions, is an integrable differential ideal. However, if the second term were of the form , then the ideal would not be integrable because it would not contain .Given an integral differential ideal on , a smooth map is called integrable if the pullback of every form vanishes on , i.e., . In coordinates, an integral manifold solves a system of partial differential equations. For example, using above, a map from an open set in is integral if(2)(3)(4)(5)Conversely, any system of partial differential equations can be expressed as an integrable differential ideal on a jet bundle. For instance, on corresponds to on ...

Given two additive groups (or rings, or modules, or vector spaces) and , the map such that for all is called the zero map. It is a homomorphism in the category of groups (or rings or modules or vector spaces).

The term metric signature refers to the signature of a metric tensor on a smooth manifold , a tool which quantifies the numbers of positive, zero, and negative infinitesimal distances of tangent vectors in the tangent bundle of and which is most easily defined in terms of the signatures of a number of related structures.Most commonly, one identifies the signature of a metric tensor with the signature of the quadratic form induced by on any of the tangent spaces for points . Indeed, given an orthogonal vector basis for any tangent space , the action of on arbitrary vectors and in is given by(1)whereby the signature of is defined to be the signature of any of the forms , i.e., the ordered triple of positive, negatives, and zero values for the inner products . This value is well-defined due to the fact that the signature of remains the same for all points in . For non-degenerate quadratic forms, the value will always satisfy , whereby the signature of will be..

Lorentzian -space is the inner product space consisting of the vector space together with the -dimensional Lorentzian inner product.In the event that the metric signature is used, Lorentzian -space is denoted ; the notation is used analogously with the metric signature .The Lorentzian inner product induces a norm on Lorentzian space, whereby the squared norm of a vector has the form(1)Rewriting (where by definition), the norm in (0) can be written as(2)In particular, the norm induced by the Lorentzian inner product fails to be positive definite, whereby it makes sense to classify vectors in -dimensional Lorentzian space into types based on the sign of their squared norm, e.g., as spacelike, timelike, and lightlike. The collection of all lightlike vectors in Lorentzian -space is known as the light cone, which is further separated into lightlike vectors which are positive and negative lightlike. A similar distinction is made for positive..

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