A simplex, sometimes called a hypertetrahedron (Buekenhout and Parker 1998), is the generalization of a tetrahedral region of space to dimensions. The boundary of a -simplex has 0-faces (polytope vertices), 1-faces (polytope edges), and -faces, where is a binomial coefficient. An -dimensional simplex can be denoted using the Schläfli symbol . The simplex is so-named because it represents the simplest possible polytope in any given space.The content (i.e., hypervolume) of a simplex can be computedusing the Cayley-Menger determinant.In one dimension, the simplex is the line segment . In two dimensions, the simplex is the convex hull of the equilateral triangle. In three dimensions, the simplex is the convex hull of the tetrahedron. The simplex in four dimensions (the pentatope) is a regular tetrahedron in which a point along the fourth dimension through the center of is chosen so that . The regular simplex in dimensions with is denoted..
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.
Algebraic topology is the study of intrinsic qualitative aspects of spatial objects (e.g., surfaces, spheres, tori, circles, knots, links, configuration spaces, etc.) that remain invariant under both-directions continuous one-to-one (homeomorphic) transformations. The discipline of algebraic topology is popularly known as "rubber-sheet geometry" and can also be viewed as the study of disconnectivities. Algebraic topology has a great deal of mathematical machinery for studying different kinds of hole structures, and it gets the prefix "algebraic" since many hole structures are represented best by algebraic objects like groups and rings.Algebraic topology originated with combinatorial topology, but went beyond it probably for the first time in the 1930s when Čech cohomology was developed.A technical way of saying this is that algebraic topology is concerned with functors from the topological..
A barrel solid of revolution composed of parallel circular top and bottom with a common axis and a side formed by a smooth curve symmetrical about the midplane.The term also has a technical meaning in functional analysis. In particular, a subset of a topological linear space is a barrel if it is absorbing, closed, and absolutely convex (Taylor and Lay 1980, p. 111). (A subset of a topological linear space is absorbing if for each there is an such that is in if for each such that . A subset of a topological linear space is absolutely convex if for each and in , is in if .)When buying supplies for his second wedding, the great astronomer Johannes Kepler became unhappy about the inexact methods used by the merchants to estimate the liquid contents of a wine barrel. Kepler therefore investigated the properties of nearly 100 solids of revolution generated by rotation of conic sections about non-principal axes (Kepler, MacDonnell, Shechter, Tikhomirov..
Let be a bounded set in the plane, i.e., is contained entirely within a rectangle. The outer Jordan measure of is the greatest lower bound of the areas of the coverings of , consisting of finite unions of rectangles. The inner Jordan measure of is the difference between the area of an enclosing rectangle , and the outer measure of the complement of in . The Jordan measure, when it exists, is the common value of the outer and inner Jordan measures of .If is a bounded nonnegative function on the interval , the ordinate set of f is the setThen is Riemann integrable on iff is Jordan measurable, in which case the Jordan measure of is equal to .There are analogous versions of Jordan measure in all other dimensions.
Measure theory is the study of measures. It generalizes the intuitive notions of length, area, and volume. The earliest and most important examples are Jordan measure and Lebesgue measure, but other examples are Borel measure, probability measure, complex measure, and Haar measure.
A univariate function is said to be odd provided that . Geometrically, such functions are symmetric about the origin. Examples of odd functions include , , the sine , hyperbolic sine , tangent , hyperbolic tangent , error function erf , inverse erf , and the Fresnel integrals , and .An even function times an odd function is odd, and the product of two odd functions is even while the sum or difference of two nonzero functions is odd if and only if each summand function is odd. The product and quotient of two odd functions is an even function.If an even function is differentiable, then its derivative is an odd function; what's more, if an odd function is integrable, then its integral over a symmetric interval , , is identically zero. Similarly, if an even function is differentiable, then its derivative is an odd function while the integral of such a function over a symmetric interval is twice the value of its integral over the interval .Ostensibly, one can define..
A univariate function is said to be even provided that . Geometrically, such functions are symmetric about the -axis. Examples of even functions include 1 (or, in general, any constant function), , , , and .An even function times an odd function is odd, while the sum or difference of two nonzero functions is even if and only if each summand function is even. The product or quotient of two even functions is again even.If a univariate even function is differentiable, then its derivative is an odd function; what's more, if an even function is integrable, then its integral over a symmetric interval , , is precisely the same as twice the integral over the interval . Similarly, if an odd function is differentiable, then its derivative is an even function while the integral of such a function over a symmetric interval is identically zero.Ostensibly, one can define a similar notion for multivariate functions by saying that such a function is even if and only ifEven..
For vectors and in , the cross product in is defined by(1)(2)where is a right-handed, i.e., positively oriented, orthonormal basis. This can be written in a shorthand notation that takes the form of a determinant(3)where , , and are unit vectors. Here, is always perpendicular to both and , with the orientation determined by the right-hand rule.Special cases involving the unit vectors in three-dimensionalCartesian coordinates are given by(4)(5)(6)The cross product satisfies the general identity(7)Note that is not a usual polar vector, but has slightly different transformation properties and is therefore a so-called pseudovector (Arfken 1985, pp. 22-23). Jeffreys and Jeffreys (1988) use the notation to denote the cross product.The cross product is implemented in the Wolfram Language as Cross[a, b].A mathematical joke asks, "What do you get when you cross a mountain-climber with a mosquito?" The answer is, "Nothing:..
If , , , and are points in the extended complex plane , their cross ratio, also called the cross-ratio (Courant and Robbins 1996, p. 172; Durell 1928, p. 73), anharmonic ratio, and anharmonic section (Casey 1888), is defined by(1)Here if , the result is infinity, and if one of , , , or is infinity, then the two terms on the right containing it are cancelled.For a linear fractional transformation ,(2)The function is the unique linear fractional transformation which takes to 0, to 1, and to infinity. Moreover, is real if and only if the four points lie on a straight line or a generalized circle.There are six different values which the cross ratio may take, depending on the order in which the points are chosen. Let . Possible values of the cross-ratio are then , , , , , and .Given four collinear points , , , and , let the distance between points and be denoted , etc. Then the cross ratio can be defined by(3)The notation is sometimes also used (Coxeter and..
Min Max Min Max Re Im The exponential function is the entire functiondefined by(1)where e is the solution of the equation so that . is also the unique solution of the equation with .The exponential function is implemented in the WolframLanguage as Exp[z].It satisfies the identity(2)If ,(3)The exponential function satisfies the identities(4)(5)(6)(7)where is the Gudermannian (Beyer 1987, p. 164; Zwillinger 1995, p. 485).The exponential function has Maclaurin series(8)and satisfies the limit(9)If(10)then(11)(12)(13)The exponential function has continued fraction(14)(Wall 1948, p. 348). Min Max Re Im The above plot shows the function (Trott 2004, pp. 165-166).Integrals involving the exponential function include(15)(16)(Borwein et al. 2004, p. 55)...
A solvable group is a group having a normal series such that each normal factor is Abelian. The special case of a solvable finite group is a group whose composition indices are all prime numbers. Solvable groups are sometimes called "soluble groups," a turn of phrase that is a source of possible amusement to chemists.The term "solvable" derives from this type of group's relationship to Galois's theorem, namely that the symmetric group is unsolvable for while it is solvable for , 2, 3, and 4. As a result, the polynomial equations of degree are (in general) not solvable using finite additions, multiplications, divisions, and root extractions.A major building block for the classification of finite simple groups was the Feit-Thompson theorem, which proved that every group of odd order is solvable. This proof took up an entire journal issue.Every finite group of order , every Abelian group, and every subgroup of a solvable group..
Let be a prime with digits and let be a constant. Call an "-prime" if the concatenation of the first digits of (ignoring the decimal point if one is present) give . Constant primes are therefore a special type of integer sequence primes, with e-primes, pi-primes, and phi-primes being perhaps the most prominent examples.The following table summarizes the indices of known constant primes for some named mathematical constants.constantname of primesOEIS giving primeApéry's constantA11933410, 55, 109, 141Catalan's constantA11832852, 276, 25477Champernowne constantA07162010, 14, 24, 235, 2804, 4347, 37735, 68433Copeland-Erdős constantA2275301, 2, 4, 11, 353, 355, 499, 1171, 1543, 5719, 11048ee-primeA0641181, 3, 7, 85, 1781, 2780, 112280, 155025Euler-Mascheroni constantA0658151, 3, 40, 185, 1038, 22610, 179849Glaisher-Kinkelin constantA1184207, 10, 18, 64, 71, 527, 1992, 5644, 8813, 19692Golomb-Dickman..
A hypersphere is parallelizable if there are vector fields that are linearly independent at each point. There exist only three parallelizable spheres: , , and (Adams 1958, 1960, Le Lionnais 1983).More generally, an -dimensional manifold is parallelizable if its tangent bundle is a trivial bundle (i.e., if is globally of the form ).
The greatest common divisor, sometimes also called the highest common divisor (Hardy and Wright 1979, p. 20), of two positive integers and is the largest divisor common to and . For example, , , and . The greatest common divisor can also be defined for three or more positive integers as the largest divisor shared by all of them. Two or more positive integers that have greatest common divisor 1 are said to be relatively prime to one another, often simply just referred to as being "relatively prime."Various notational conventions are summarized in the following table.notationsourcethis work, Zwillinger (1996, p. 91), Råde and Westergren (2004, p. 54)Gellert et al. (1989, p. 25), D'Angelo and West (1990, p. 13), Graham et al. (1990, p. 103), Bressoud and Wagon (2000, p. 7), Yan (2002, p. 30), Bronshtein et al. (2007, pp. 323-324), Wolfram Languageg.c.d.Andrews 1994,..
The Gauss map is a function from an oriented surface in Euclidean space to the unit sphere in . It associates to every point on the surface its oriented unit normal vector. Since the tangent space at a point on is parallel to the tangent space at its image point on the sphere, the differential can be considered as a map of the tangent space at into itself. The determinant of this map is the Gaussian curvature, and negative one-half of the trace is the mean curvature.Another meaning of the Gauss map is the function(Trott 2004, p. 44), where is the floor function, plotted above on the real line and in the complex plane.The related function is plotted above, where is the fractional part.The plots above show blowups of the absolute values of these functions (a version of the left figure appears in Trott 2004, p. 44)...
The Euclidean algorithm, also called Euclid's algorithm, is an algorithm for finding the greatest common divisor of two numbers and . The algorithm can also be defined for more general rings than just the integers . There are even principal rings which are not Euclidean but where the equivalent of the Euclidean algorithm can be defined. The algorithm for rational numbers was given in Book VII of Euclid's Elements. The algorithm for reals appeared in Book X, making it the earliest example of an integer relation algorithm (Ferguson et al. 1999).The Euclidean algorithm is an example of a P-problem whose time complexity is bounded by a quadratic function of the length of the input values (Bach and Shallit 1996).Let , then find a number which divides both and (so that and ), then also divides since(1)Similarly, find a number which divides and (so that and ), then divides since(2)Therefore, every common divisor of and is a common divisor of and , so the procedure..
The derivative of a function represents an infinitesimalchange in the function with respect to one of its variables.The "simple" derivative of a function with respect to a variable is denoted either or(1)often written in-line as . When derivatives are taken with respect to time, they are often denoted using Newton's overdot notation for fluxions,(2)The "d-ism" of Leibniz's eventually won the notation battle against the "dotage" of Newton's fluxion notation (P. Ion, pers. comm., Aug. 18, 2006).When a derivative is taken times, the notation or(3)is used, with(4)etc., the corresponding fluxion notation.When a function depends on more than one variable, a partial derivative(5)can be used to specify the derivative with respect to one or more variables.The derivative of a function with respect to the variable is defined as(6)but may also be calculated more symmetrically as(7)provided the..
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..
Let be a T2-topological space and let be the space of all bounded complex-valued continuous functions defined on . The supremum norm is the norm defined on F byThen is a commutative Banach algebra with identity.
An algebraic function is a function which satisfies , where is a polynomial in and with integer coefficients. Functions that can be constructed using only a finite number of elementary operations together with the inverses of functions capable of being so constructed are examples of algebraic functions. Nonalgebraic functions are called transcendental functions.
A series is said to be conditionally convergent iff it is convergent, the series of its positive terms diverges to positive infinity, and the series of its negative terms diverges to negative infinity.Examples of conditionally convergent series include the alternating harmonic seriesand the logarithmic serieswhere is the Euler-Mascheroni constant.The Riemann series theorem states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge. The Riemann series theorem can be proved by first taking just enough positive terms to exceed the desired limit, then taking just enough negative terms to go below the desired limit, and iterating this procedure. Since the terms of the original series tend to zero, the rearranged series converges to the desired limit. A slight variation works to make the new series diverge to positive infinity or to negative infinity...
A sequence of functions , , 2, 3, ... is said to be uniformly convergent to for a set of values of if, for each , an integer can be found such that(1)for and all .A series converges uniformly on if the sequence of partial sums defined by(2)converges uniformly on .To test for uniform convergence, use Abel's uniform convergence test or the Weierstrass M-test. If individual terms of a uniformly converging series are continuous, then the following conditions are satisfied. 1. The series sum(3)is continuous. 2. The series may be integrated term by term(4)For example, a power series is uniformly convergent on any closed and bounded subset inside its circle of convergence. 3. The situation is more complicated for differentiation since uniform convergence of does not tell anything about convergence of . Suppose that converges for some , that each is differentiable on , and that converges uniformly on . Then converges uniformly on to a function , and for each..
An extension of a group by a group is a group with a normal subgroup such that and . This information can be encoded into a short exact sequence of groupswhere is injective and is surjective.It should be noted that some authors reverse the roles and say that is an extension of (Spanier 1994, Mac Lane and Birkhoff 1993).Given groups and there are (often) many extensions of by . Examples include the direct product of and and a semidirect product of and . A function such that is the identity function on is called a transversal function. A group extension is said to be split if there is a transversal function which is a homomorphism. A group extension is split iff it is a semidirect product.The study of group extensions has connections with group cohomology.
A metric space is boundedly compact if all closed bounded subsets of are compact. Every boundedly compact metric space is complete. (This is a generalization of the Bolzano-Weierstrass theorem.)Every complete Riemannian manifold is boundedly compact. This is part of or a consequence of the Hopf-Rinow theorem.
If a matrix has a matrix of eigenvectors that is not invertible (for example, the matrix has the noninvertible system of eigenvectors ), then does not have an eigen decomposition. However, if is an real matrix with , then can be written using a so-called singular value decomposition of the form(1)Note that there are several conflicting notational conventions in use in the literature. Press et al. (1992) define to be an matrix, as , and as . However, the Wolfram Language defines as an , as , and as . In both systems, and have orthogonal columns so that(2)and(3)(where the two identity matrices may have different dimensions), and has entries only along the diagonal.For a complex matrix , the singular value decomposition is a decomposition into the form(4)where and are unitary matrices, is the conjugate transpose of , and is a diagonal matrix whose elements are the singular values of the original matrix. If is a complex matrix, then there always exists..
Min Max Min Max Re Im Given a number , the cube root of , denoted or ( to the 1/3 power), is a number such that . The cube root is therefore an nth root with . Every real number has a unique real cube root, and every nonzero complex number has three distinct cube roots.The schoolbook definition of the cube root of a negative number is . However, extension of the cube root into the complex plane gives a branch cut along the negative real axis for the principal value of the cube root as illustrated above. By convention, "the" (principal) cube root is therefore a complex number with positive imaginary part. As a result, the Wolfram Language and other symbolic algebra languages and programs that return results valid over the entire complex plane therefore return complex results for . For example, in the Wolfram Language, ComplexExpand[(-1)^(1/3)] gives the result .When considering a positive real number , the Wolfram Language function CubeRoot[x],..
A ring in the mathematical sense is a set together with two binary operators and (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions: 1. Additive associativity: For all , , 2. Additive commutativity: For all , , 3. Additive identity: There exists an element such that for all , , 4. Additive inverse: For every there exists such that , 5. Left and right distributivity: For all , and , 6. Multiplicative associativity: For all , (a ring satisfying this property is sometimes explicitly termed an associative ring). Conditions 1-5 are always required. Though non-associative rings exist, virtually all texts also require condition 6 (Itô 1986, pp. 1369-1372; p. 418; Zwillinger 1995, pp. 141-143; Harris and Stocker 1998; Knuth 1998; Korn and Korn 2000; Bronshtein and Semendyayev 2004).Rings may also satisfy various optional conditions: 7. Multiplicative commutativity:..