An angle drawn on the coordinate plane is said to be in standard position if its initial side lies on the positive x-axis so that its vertex coincides with the origin and its rotation is in the counterclockwise direction.In the above image, the angle is in standard position due to the locations of its vertex and its initial side and because of the direction of its rotation.
The inverse of a square matrix , sometimes called a reciprocal matrix, is a matrix such that(1)where is the identity matrix. Courant and Hilbert (1989, p. 10) use the notation to denote the inverse matrix.A square matrix has an inverse iff the determinant (Lipschutz 1991, p. 45). The so-called invertible matrix theorem is major result in linear algebra which associates the existence of a matrix inverse with a number of other equivalent properties. A matrix possessing an inverse is called nonsingular, or invertible.The matrix inverse of a square matrix may be taken in the Wolfram Language using the function Inverse[m].For a matrix(2)the matrix inverse is(3)(4)For a matrix(5)the matrix inverse is(6)A general matrix can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination, or LU decomposition.The inverse of a product of matrices and can be expressed in terms of and . Let(7)Then(8)and(9)Therefore,(10)so(11)where..
A sphenic number is a positive integer which is the product of exactly three distinct primes. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, ... (OEIS A007304).In particular, if , , and are prime numbers, then every sphenic number has precisely eight positive divisors, namely , , , , , , , and itself.The Möbius function of a sphenic number is .
A prime number is called circular if it remains prime after any cyclic permutation of its digits. An example in base-10 is because , , and are all primes. The first few circular primes are 2, 3, 5, 7, 11, 13, 17, 31, 37, 71, 73, 79, 97, 113, 131, 197, ... (OEIS A068652).Base-10 circular primes not contain any digit 0, 2, 4, 5, 6, or 8, since having such a digit in the units place yields a number which is necessarily divisible by either or (and therefore not prime).Every prime repunit is a circular prime.Circular primes are rare. Including only the smallest number corresponding to each cycle gives the sequence 2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, ... (OEIS A016114; Darling 2004), together with repunits , , , , , , and (the last several of which are probable primes)...
An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive. The Riemann integral is the simplest integral definition and the only one usually encountered in physics and elementary calculus. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."The Riemann integral of the function over from to is written(1)Note that if , the integral is written simply(2)as opposed to .Every definition of an integral is based on a particular measure. For instance, the Riemann integral is based on Jordan measure, and the Lebesgue integral is based..
The decimal expansion of a number is its representation in base-10 (i.e., in the decimal system). In this system, each "decimal place" consists of a digit 0-9 arranged such that each digit is multiplied by a power of 10, decreasing from left to right, and with a decimal place indicating the s place. For example, the number with decimal expansion 1234.56 is defined as(1)(2)Expressions written in this form (where negative are allowed as exemplified above but usually not considered in elementary education contexts) are said to be in expanded notation.Other examples include the decimal expansion of given by 625, of given by 3.14159..., and of given by 0.1111.... The decimal expansion of a number can be found in the Wolfram Language using the command RealDigits[n], or equivalently, RealDigits[n, 10].The decimal expansion of a number may terminate (in which case the number is called a regular number or finite decimal, e.g., ), eventually..
The least common denominator of a collection of fractionsis the least common multiple of their denominators.
A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored (unlike a list or multiset). Members of a set are often referred to as elements and the notation is used to denote that is an element of a set . The study of sets and their properties is the object of set theory.Older words for set include aggregate and set class. Russell also uses the unfortunate term manifold to refer to a set.Historically, a single horizontal overbar was used to denote a set stripped of any structure besides order, and hence to represent the order type of the set. A double overbar indicated stripping the order from the set and hence represented the cardinal number of the set. This practice was begun by set theory founder Georg Cantor.Symbols used to operate on sets include (which means "and" or intersection), and (which means "or" or union). The symbol is used to denote the set containing..
In the IEEE 754-2008 standard (referred to as IEEE 754 henceforth), a signaling NaN or sNaN is a NaN which is signaling in the sense of being most commonly returned in conjunction with various exceptions and handling mechanisms defined therefor. This is in contrast to the quiet NaN (qNaN) which rarely signals a floating-point exception of any kind (IEEE Computer Society 2008).Within the framework documentation, it is suggested that sNaNs be implemented in such a way as to afford meaningful representations for uninitialized variables and arithmetic-like enhancements which may naturally fall beyond the scope of the standard. In particular, sNaNs are largely reserved for operands which signal exceptions for nearly every general-computational and signaling-computational operation though, in rare instances, qNaNs may also result from such contexts...
In the IEEE 754-2008 standard (referred to as IEEE 754 henceforth), a quiet NaN or qNaN is a NaN which is quiet in the sense of rarely signaling a floating-point exception. This is in contrast to the signaling NaN (sNaN) which often occurs in conjunction with any number of exceptions and handling mechanisms defined therefor (IEEE Computer Society 2008).Within the framework documentation, it is suggested that qNaNs be implemented in such a way as to afford useful diagnostic information regarding invalid or unavailable data and results. Under the default exception handling mechanisms provided within IEEE 754, any operation which signals an invalid operation exception and for which a floating-point result is expected shall return a quiet NaN; qNaNs may also result from operations not delivering a floating-point result, though in practice, these operations are far more likely to output sNaNs instead...
In the IEEE 754-2008 standard (referred to as IEEE 754 henceforth), NaN (or "not a number") is a symbolic floating-point representation which is neither a signed infinity nor a finite number. In general, NaNs occur as the output of computations which are somehow "indeterminate," e.g., when attempting to compute quantities such as , , or .The use of NaN representations is a relatively new development. Traditionally, the computation of indeterminate quantities such as or was treated as an unrecoverable error which caused a computation to halt. In practice, however, it sometimes makes sense for a computation to continue despite encountering such a scenario; in these situations, unnecessary halting can be avoided by specifying that the computation of expressions like and output NaN rather than halting the program (Goldberg 1991).Ostensibly, a NaN output should carry with it some degree of diagnostic information regarding..
Pascal's triangle is a number triangle with numbersarranged in staggered rows such that(1)where is a binomial coefficient. The triangle was studied by B. Pascal, although it had been described centuries earlier by Chinese mathematician Yanghui (about 500 years earlier, in fact) and the Persian astronomer-poet Omar Khayyám. It is therefore known as the Yanghui triangle in China. Starting with , the triangle is(2)(OEIS A007318). Pascal's formula shows that each subsequent row is obtained by adding the two entries diagonally above,(3)The plot above shows the binary representations for the first 255 (top figure) and 511 (bottom figure) terms of a flattened Pascal's triangle.The first number after the 1 in each row divides all other numbers in that row iff it is a prime.The sums of the number of odd entries in the first rows of Pascal's triangle for , 1, ... are 0, 1, 3, 5, 9, 11, 15, 19, 27, 29, 33, 37, 45, 49, ... (OEIS A006046). It is then..
The nesting of two or more functions to form a single new function is known as composition. The composition of two functions and is denoted , where is a function whose domain includes the range of . The notation(1)is sometimes used to explicitly indicate the variable.Composition is associative, so that(2)If the functions is continuous at and is continuous at , then is also continuous at .A function which is the composition of two other functions, say and , is sometimes said to be a composite function.Faà di Bruno's formula gives an explicit formula for the th derivative of the composition .A combinatorial composition is defined as an ordered arrangement of nonnegative integers which sum to (Skiena 1990, p. 60). It is therefore a partition in which order is significant. For example, there are eight compositions of 4,(3)(4)(5)(6)(7)(8)(9)(10)A positive integer has compositions.The number of compositions of into parts (where..
A quasigroup with an identity element such that and for any in the quasigroup. All groups are loops.In general, loops are considered to have very little in the way of algebraic structure and it is for that reason that many authors limit their investigation to loops which satisfy various other structural conditions. Common examples of such notions are the left- and right-Bol loop, the Moufang loop (which is both a left-Bol loop and a right-Bol loop simultaneously), and the generalized Bol loop.The above definition of loop is purely algebraic and shouldn't be confused with other notions of loop, such as a closed curves, a multi-component knot or hitch, a graph loop, etc.
The Wiener sausage of radius is the random process defined bywhere here, is the standard Brownian motion in for and denotes the open ball of radius centered at . Named after Norbert Wiener, the term is also intended to describe visually: Indeed, for a given Brownian motion , is essentially a sausage-like tube of radius having as its central line.
A real-valued stochastic process is a Brownian motion which starts at if the following properties are satisfied: 1. . 2. For all times , the increments , , ..., , are independent random variables. 3. For all , , the increments are normally distributed with expectation value zero and variance . 4. The function is continuous almost everywhere. The Brownian motion is said to be standard if . It is easily shown from the above criteria that a Brownian motion has a number of unique natural invariance properties including scaling invariance and invariance under time inversion. Moreover, any Brownian motion satisfies a law of large numbers so thatalmost everywhere. Moreover, despite looking ill-behaved at first glance, Brownian motions are Hölder continuous almost everywhere for all values . Contrarily, any Brownian motion is nowhere differentiable almost surely.The above definition is extended naturally to get higher-dimensional Brownian..
A complex matrix is said to be Hamiltonian if(1)where is the matrix of the form(2) is the identity matrix, and denotes the conjugate transpose of a matrix . An analogous definition holds in the case of real matrices by requiring that be symmetric, i.e., by replacing by in (1).Note that this criterion specifies precisely how a Hamiltonian matrix must look. Indeed, every Hamiltonian matrix (here assumed to have complex entries) must have the form(3)where satisfy and . This characterization holds for having strictly real entries as well by replacing all instances of the conjugate transpose operator in (1) by the transpose operator instead.
An matrix is an elementary matrix if it differs from the identity by a single elementary row or column operation.
The invertible matrix theorem is a theorem in linear algebra which gives a series of equivalent conditions for an square matrix to have an inverse. In particular, is invertible if and only if any (and hence, all) of the following hold: 1. is row-equivalent to the identity matrix . 2. has pivot positions. 3. The equation has only the trivial solution . 4. The columns of form a linearly independent set. 5. The linear transformation is one-to-one. 6. For each column vector , the equation has a unique solution. 7. The columns of span . 8. The linear transformation is a surjection. 9. There is an matrix such that . 10. There is an matrix such that . 11. The transpose matrix is invertible. 12. The columns of form a basis for . 13. The column space of is equal to . 14. The dimension of the column space of is . 15. The rank of is . 16. The null space of is . 17. The dimension of the null space of is 0. 18. fails to be an eigenvalue of . 19. The determinant of is not zero. 20. The orthogonal..
Two non-coincident plane angles and in angle standard position are said to be coterminal if the terminal side of is identically the same as the terminal side of .In general, given a plane angle measured in radians, is coterminal to if and only if for some positive integer . Similarly, if is a plane angle coterminal to a plane angle measured in degrees, then for some positive integer . In the event that , then and are coincident.In the figure above, the non-coincident angles and are coterminal angles.
The terminal side of an angle drawn in angle standard position is the side which isn't the initial side.When viewing an angle as the amount of rotation about the intersection point (the vertex) needed to bring one of two intersecting lines (or line segments) into correspondence with the other, the line (or line segment) towards which the initial side is being rotated the terminal side.
The initial side of an angle drawn in angle standard position is the side lying on the positive x-axis.In general, viewing an angle as the amount of rotation about the intersection point (the vertex) needed to bring one of two intersecting lines (or line segments) into correspondence with the other, the line (or line segment) being rotated about the vertex is thought of as the initial side.
Percolation, the fundamental notion at the heart of percolation theory, is a difficult idea to define precisely though it is quite easy to describe qualitatively.From the narrowest perspective, the term percolation can be defined as a model of a porous medium; indeed, it is from this perspective that the study of percolation theory blossomed, and is typically agreed to be the fundamental physical situation that modern percolation theory attempts to address.Likewise, it is not uncommon for the term percolation to describe the actual fluid flow within the random media or as the theoretical simulation of such a flow for a given simulated medium.One of the difficulties underlying the formulation of a precise definition is that some authors choose to define the term relative to the machinery used in its study. For instance, some authors choose to define percolation to be the result of independently removing vertices or edges from some sort of graph..
The disk model is the standard Boolean-Poisson model in two-dimensional continuum percolation theory. In particular, the disk model is characterized by the existence of a Poisson process in which distributes the centers of a collection of closed disks (i.e., two-dimensional closed balls) along with a random process which independently assigns random radii to each .The disks which make up the disk model are known as randomdisks.
An percolation is a discrete percolation model in which the underlying point lattice graph has the properties that each of its graph vertices is occupied by an atom either of type or of type , that there is a probability that any given vertex is occupied by an atom of type , and that different vertices are occupied independently of each other.In this model, a graph edge of is said to be open if its end vertices are occupied by atoms of different types and is said to be closed otherwise. The idea is based on the hypothesis that dissimilar atoms bond together whereas similar atoms repel one another.This model is sometimes studied under the title antipercolation.
Generally speaking, a Green's function is an integral kernel that can be used to solve differential equations from a large number of families including simpler examples such as ordinary differential equations with initial or boundary value conditions, as well as more difficult examples such as inhomogeneous partial differential equations (PDE) with boundary conditions. Important for a number of reasons, Green's functions allow for visual interpretations of the actions associated to a source of force or to a charge concentrated at a point (Qin 2014), thus making them particularly useful in areas of applied mathematics. In particular, Green's function methods are widely used in, e.g., physics, and engineering.More precisely, given a linear differential operator acting on the collection of distributions over a subset of some Euclidean space , a Green's function at the point corresponding to is any solution of(1)where denotes the delta..
Let and be fibered categories over a topological space . A morphism of fibered categories consists of: 1. a functor for each open subset and 2. a natural isomorphism for each inclusion . It is required that these structures satisfy a compatibility condition with respect to the 's, namely, that for the inclusions , , the above diagram should commute.
A fibered category over a topological space consists of 1. a category for each open subset , 2. a functor for each inclusion , and 3. a natural isomorphismfor each pair of inclusions , . In addition, for any three composable inclusions , , and , there exists a natural commuting as shown above.Sometimes, the pair is used to denote a fibered category with more precision while the shorthand is sometimes used for , , .
An axiomatic system is said to be categorical if there is only one essentially distinct representation for it. In particular, the names and types of objects within the system may vary while still being considered "the same," e.g., geometries and their plane duals.An example of an axiomatic system which isn't categorical is a geometrydescribed by the following four axioms (Smart): 1. There exist five points. 2. Each line is a subset of thosefive points. 3. There exist two lines. 4. Each line contains at least two points.One way to see that this is a non-categorical axiomatic system is to note that one can form a compatible system from two fundamentally different models, e.g., 1. Two disjoint lines each containing two points plus a separate point not on either line. 2. Two lines containing three pointseach which intersect in one of the points. The presence of an intersection in one model andnot the other implies that the models are fundamentally..
Let and be measure spaces, let be the collection of all measurable rectangles contained in , and let be the premeasure defined on byfor . By the product measure , one means the Carathéodory extension of defined on the sigma-algebra of -measurable subsets of where denotes the outer measure induced by the premeasure on .
Let be a collection of subsets of a set and let be a set function. The function is called a premeasure provided that is finitely additive, countably monotone, and that if , where is the empty set.
A sutured manifold is a tool in geometric topology which was first introduced by David Gabai in order to study taut foliations on 3-manifolds. Roughly, a sutured manifold is a pair with a compact, oriented 3-manifold with boundary and with a set of simple closed curves in which are oriented and which divide into pieces and (Juhász 2010).Defined precisely in a seminal work by Gabai (1983), a sutured manifold is a compact oriented 3-manifold together with a set of pairwise disjoint annuli and tori such that each component of contains a homologically nontrivial oriented simple closed curve (called a suture) and such that is oriented. Using this construction, the collection of a sutured manifold effectively splits into disjoint pieces and with , respectively , defined to be the components of whose normal vectors point into, respectively point out of, . Gabai's definition also requires that orientations on be coherent with respect to 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 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 function is a relation that uniquely associates members of one set with members of another set. More formally, a function from to is an object such that every is uniquely associated with an object . A function is therefore a many-to-one (or sometimes one-to-one) relation. The set of values at which a function is defined is called its domain, while the set of values that the function can produce is called its range. Here, the set is called the codomain of .In the context of univariate, real-valued functions , the fact that domain elements are mapped to unique range elements can be expressed graphically by way of the vertical line test.In some literature, the term "map" is synonymous with function. Some caution must be exhibited, however, as it is not uncommon for the term map to denote a function with some sort of unspoken regularity assumption, e.g., in point-set topology, where "map" sometimes refers to a function which is continuous..
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..
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.
In machine learning theory and artificial intelligence, a concept over a domain is a Boolean function . A collection of concepts is called a concept class.In context-specific applications, concepts are usually thought to assign either a "positive" or "negative" outcome (corresponding to range values of 1 or 0, respectively) to each element of the domain . In that way, concepts are the fundamental component of learning theory.
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 computation of points or values between ones that are known or tabulated using the surrounding points or values.In particular, given a univariate function , interpolation is the process of using known values to find values for at points , . In general, this technique involves the construction of a function called the interpolant which agrees with at the points and which is then used to compute the desired values.Unsurprisingly, one can talk about interpolation methods for multivariate functions as well, though these tend to be substantially more involved than their univariate counterparts.
In univariate interpolation, an interpolant is a function which agrees with a particular function at a set of known points and which is used to compute values for at points , .Modulo a change of notation, the above definition translates verbatim to multivariateinterpolation models as well.Generally speaking, the properties required of the interpolant are the most fundamental designations between various interpolation models. For example, the main difference between the linear and spline interpolation models is that the interpolant of the prior is required merely to be piecewise linear whereas spline interpolants are assumed to be piecewise polynomial and globally smooth.
A generalized eigenvector for an matrix is a vector for whichfor some positive integer . Here, denotes the identity matrix. The smallest such is known as the generalized eigenvector order of the generalized eigenvector. In this case, the value is the generalized eigenvalue to which is associated and the linear span of all generalized eigenvectors associated to some generalized eigenvalue is known as the generalized eigenspace for .As the name suggests, generalized eigenvectors are generalizations of eigenvectors of the usual kind; more precisely, an eigenvector is a generalized eigenvector corresponding to .Generalized eigenvectors are of particular importance for matrices which fail to be diagonalizable. Indeed, for such matrices, at least one eigenvalue has geometric multiplicity larger than its algebraic multiplicity, thereby implying that the collection of linearly independent eigenvectors of is "too small"..
A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions. A line is sometimes called a straight line or, more archaically, a right line (Casey 1893), to emphasize that it has no "wiggles" anywhere along its length. While lines are intrinsically one-dimensional objects, they may be embedded in higher dimensional spaces.Harary (1994) called an edge of a graph a "line."A line is uniquely determined by two points, and the line passing through points and is denoted . Similarly, the length of the finite line segment terminating at these points may be denoted . A line may also be denoted with a single lower-case letter (Jurgensen et al. 1963, p. 22).Euclid defined a line as a "breadthless length," and a straight line as a line that "lies evenly with the points on itself" (Kline 1956, Dunham 1990).Consider first lines in a two-dimensional plane. Two..
The standard form of a line in the Cartesian plane is givenbyfor real numbers .This form can be derived from any of the other forms (point-slope form, slope-intercept form, etc.), but can be seen most intuitively when starting from intercept form. Indeed, the intercept form of a line with x-intercept and y-intercept is given byand so by clearing denominators and setting , one gets precisely that .
A simple path is a path which is a simple curve. More precisely, a continuous mapping is a simple path if it has no self-intersections. Here, denotes the space of continuous functions.
There are no fewer than three distinct notions of curve throughout mathematics.In topology, a curve is a one-dimensional continuum(Charatonik and Prajs 2001).In algebraic geometry, an algebraic curve over a field is the zero locus of some polynomial of two variables which has its coefficients in .In analytic geometry, a curve is continuous map from a one-dimensional space to an -dimensional space. Loosely speaking, the word "curve" is often used to mean the function graph of a two- or three-dimensional curve. The simplest curves can be represented parametrically in -dimensional space as(1)(2)(3)(4)Other simple curves can be simply defined only implicitly, i.e., in the form(5)When discussing curves from the standpoint of analytic geometry, care must be exhibited to maintain the important distinction between the curve itself and its image within its codomain. For example, the curves , , defined, respectively, by(6)and(7)are..
The phrase dependent percolation is used in two-dimensional discrete percolation to describe any general model in which the states of the various graph edges (in the case of bond percolation models) or graph vertices (in site percolation models) are not independent.Many models of this type come about naturally in a number of fields. For example, a popular tool in statistical mechanics is the two-dimensional Ising model, a type of dependent site percolation model used to study the dipole moments of magnetic spins. Other examples include the Potts models-generalizations of the Ising model in which is allowed to take on different values rather than the usual two-and the random-cluster model.Ostensibly, it would also make sense to talk about dependent percolation models in contexts such as -dimensional discrete percolation theory for arbitrary , as well as in continuum percolation theory; even so, the literature on such generalizations..
A -dimensional discrete percolation model on a regular point lattice is said to be oriented if is an oriented lattice. One common such model takes place on the so-called north-east oriented lattice obtained by orienting each edge of an arbitrary (perhaps unoriented) point lattice in the direction of increasing coordinate-value.The above figure shows an example of a subset of a 2-dimensional oriented percolation model on the north-east lattice. Here, each edge has been deleted with probability for some , independently of all other edges.Oriented percolation models are especially common in several areas of physics including astrophysics, solid state physics, and particle physics. Worth noting is that, while obvious parallels exist between oriented and unoriented percolation models, the proofs of results in the presence of orientation offer differ greatly from those of their unoriented analogues; indeed, the existence of so-called..
Continuum percolation can be thought of as a continuous, uncountable version of percolation theory-a theory which, in its most studied form, takes place on a discrete, countable point lattice like . Unlike discrete percolation theory, continuum percolation theory involves notions of percolation for and for various non-discrete subsets thereof.There are a number of models used to study continuum percolation including but not limited to the disk model, the germ-grain model, and the random-connection model. Perhaps the most well-studied of these methods is the so-called Boolean-Poisson model which roughly consists of centering an independent copy of a random -dimensional shape at each point of a homogeneous Poisson process in -dimensional Euclidean space , the result of which is a collection of overlapping shapes spanning a subset of . Using this construction, one devises a percolation theory by considering whether a percolation occurs,..
In discrete percolation theory, site percolation is a percolation model on a regular point lattice in -dimensional Euclidean space which considers the lattice vertices as the relevant entities (left figure). The precise mathematical construction for the Bernoulli version of site percolation is as follows.First, designate each vertex of to be independently "open" with probability and closed otherwise. Next, define an open path to be any path in all of whose vertices are open, and define at the vertex the so-called open cluster to be the set of all vertices which may be attained following only open paths from . Write . The main objects of study in the site percolation model are then the percolation probability(1)and the critical probability(2)where here, is defined to be the product measure(3) is the Bernoulli measure which assigns whenever is closed and assigns when is open, and is the percolation threshold. Site models for which..
A 2-dimensional discrete percolation model is said to be mixed if both graph vertices and graph edges may be "blocked" from allowing fluid flow (i.e., closed in the sense of percolation theory). This is in contrast to the more-studied cases of bond percolation and site percolation, the standard models for which allow only edges and vertices, respectively, to be blocked.Considered a bridge between bond percolation and site percolation (Chayes and Schonmann 2000), mixed percolation models have become increasingly more studied since their inception in the earl 1980s. Indeed, many of the properties of and methods related to this type of percolation can be found in work done by Hammersley (1980).Some authors extend the above definition so as to allow for the faces of the underlying graph to also be viewed as random elements to which one can assign values of open and closed (Wierman 1984). Among such models, one assigns to each planar graph..
In most modern literature, a Boolean model is a probabilistic model of continuum percolation theory characterized by the existence of a stationary point process and a random variable which independently determine the centers and the random radii of a collection of closed balls in for some .In this case, the model is said to be driven by .Worth noting is that the most intuitive ideas about constructing a feasible model using and often lead to unexpected and undesirable results (Meester and Roy 1996). For that reason, some more sophisticated machinery and quite a bit of care is needed to translate from the language of and into a reasonable model of continuum percolation. The formal construction is as follows.Let be a stationary point process as discussed above and suppose that is defined on a probability space . Next, define the space to be the product space(1)and define associated to the usual product sigma-algebra and product measure where here,..
In statistical mechanics, the two-dimensional Ising model is a popular tool used to study the dipole moments of magnetic spins.The Ising model in two dimensions is a type of dependent site percolation model which is characterized by the existence of a random variable assigning to each point a value of and is driven by a distribution of the formwhere is a real constant, , and for site random variables , .Some authors differentiate between positive (or ferromagnetic) dependency and negative (or antiferromagnetic) dependency (Newman 1990) depending on the sign of the quantity , though little mention of this distinction appears overall.Other examples of dependent percolation models include the Potts models-generalizations of the Ising model in which is allowed to take on different values rather than the usual two-and the random-cluster model...
In discrete percolation theory, bond percolation is a percolation model on a regular point lattice in -dimensional Euclidean space which considers the lattice graph edges as the relevant entities (left figure). The precise mathematical construction for the Bernoulli percolation model version of bond percolation is given below.First, define the set of edges of to be the set(1)and designate each edge of to be independently "open" with probability and closed with probability . Next, define an open path to be any path in all of whose edges are open, and define the so-called open cluster to be the connected component of the random subgraph of consisting of only open edges and containing the vertex . Write . The main objects of study in the bond percolation model are then the percolation probability(2)and the critical probability(3)where is defined to be the product measure(4) is the Bernoulli measure which assigns whenever is closed..
A -dimensional discrete percolation model is said to be inhomogeneous if different graph edges (in the case of bond percolation models) or vertices (in the case of site percolation models) may have different probabilities of being open. This is in contrast to the typical bond and site percolation models which are homogeneous in the sense that openness of edges/vertices is determined by a random variable which is identically and independently distributed (i.i.d.).Unsurprisingly, the breadth of continuum percolation theory allows one to adapt the above definition to models thereof. Such an adaptation could consist either of distributing -dimensional shapes in to points determined by inhomogeneous point processes-point processes with time-dependent realizations-or of utilizing non-uniform probability distributions to determine the properties of the shapes themselves...
Intuitively, a model of -dimensional percolation theory is said to be a Bernoulli model if the open/closed status of an area is completely random. In particular, it makes sense to talk about a Bernoulli bond percolation, Bernoulli site percolation, as well as describing other models of both discrete and continuum percolation theory as being Bernoulli.Due to the vastness of the literature on percolation theory, however, there is a certain lack of uniformity in its terminology; as such, some authors choose to define -dimensional Bernoulli percolation strictly in terms of its behavior on the standard bond percolation model within the regular point lattice . According to this view, the term Bernoulli percolation refers to the independent assignment as either open (with probability ) or closed (with probability ) to each edge where here,This perspective, though framed relative to obvious graph theory terminology, is largely probabilistic..
In the field of percolation theory, the term percolation threshold is used to denote the probability which "marks the arrival" (Grimmett 1999) of an infinite connected component (i.e., of a percolation) within a particular model. The percolation threshold is commonly denoted and is sometimes called the critical phenomenon of the model.Special attention is paid to probabilities both below and above the percolation threshold; a percolation model for which is called a subcritical percolation while a model satisfying is called a supercritical percolation. Because of this distinction, the value is also sometimes called the phase transition of the model as it marks the exact point of transition between the subcritical phase and the supercritical phase . Note that by definition, subcritical percolation models are necessarily devoid of infinite connected components, whereas supercritical models always contain at least one..
The rascal triangle is a number triangle withnumbers arranged in staggered rows such that(1)The published study of this triangle seems to have originated relatively recently, having been added to Sloane's Online Encyclopedia of Integer Sequences (OEIS) as recently as 2002--where it was cataloged as [t]riangle with diagonal congruent to mod --and having been the subject of scholarly mathematical publication as recently as 2010 (Anggoro et al. 2010). The triangle is sometimes written without capitalization as the rascal triangle.One common point of exposition among literature regarding the rascal triangle is its similarity to Pascal's triangle. Indeed, the rascal triangle is topically similar to Pascal's triangle in that the configuration starting with begins(2)and that the rows afterwards have their first and last entries equal to(3)respectively.The similarities don't end there, however. One well-known fact about Pascal's..
A nonzero vector in -dimensional Lorentzian space is said to be positive timelike if it has imaginary (Lorentzian) norm and if its first component is positive. Symbolically, is positive timelike if bothandhold. Note that equation (6) above expresses the imaginary norm condition by saying, equivalently, that the vector has a negative squared norm.
A weak Riemannian metric on a smooth manifold is a tensor field which is both a weak pseudo-Riemannian metric and positive definite.In a very precise way, the condition of being a weak Riemannian metric is considerably less stringent than the condition of being a strong Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy but not vice versa. More precisely, any strong Riemannian metric provides an isomorphism between the tangent and cotangent spaces and , respectively, for all ; conversely, weak Riemannian metrics are merely injective linear maps from to (Marsden et al. 2002).
A nonzero vector in -dimensional Lorentzian space is said to be positive lightlike if it has zero (Lorentzian) norm and if its first component is positive. Symbolically, is positive lightlike if bothandhold. The collection of all positive lightlike vectors form the top half of the lightcone.
The total number of contravariant and covariant indices of a tensor. The rank of a tensor is independent of the number of dimensions of the underlying space.An intuitive way to think of the rank of a tensor is as follows: First, consider intuitively that a tensor represents a physical entity which may be characterized by magnitude and multiple directions simultaneously (Fleisch 2012). Therefore, the number of simultaneous directions is denoted and is called the rank of the tensor in question. In -dimensional space, it follows that a rank-0 tensor (i.e., a scalar) can be represented by number since scalars represent quantities with magnitude and no direction; similarly, a rank-1 tensor (i.e., a vector) in -dimensional space can be represented by numbers and a general tensor by numbers. From this perspective, a rank-2 tensor (one that requires numbers to describe) is equivalent, mathematically, to an matrix.rankobject0scalar1vector2 matrixtensorThe..
A nonzero vector in -dimensional Lorentzian space is said to be negative timelike if it has imaginary (Lorentzian) norm and if its first component is negative. Symbolically, is negative timelike if bothandhold. Note that equation (6) above expresses the imaginary norm condition by saying, equivalently, that the vector has a negative squared norm.
A nonzero vector in -dimensional Lorentzian space is said to be negative lightlike if it has zero (Lorentzian) norm and if its first component is negative. Symbolically, is negative lightlike if bothandhold. The collection of all negative lightlike vectors form the bottom half of the lightcone.
The so-called generalized Fourier integral is a pair of integrals--a "lower Fourier integral" and an "upper Fourier integral"--which allow certain complex-valued functions to be decomposed as the sum of integral-defined functions, each of which resembles the usual Fourier integral associated to and maintains several key properties thereof.Let be a real variable, let be a complex variable, and let be a function for which as , for which as , and for which has an analytic Fourier integral where here, are finite real constants. Next, define the upper and lower generalized Fourier integrals and associated to , respectively, by(1)and(2)on the complex regions and , respectively. Then, for and ,(3)where the first integral summand equals for and is zero for while the second integral summand is zero for and equals for . The decomposition () is called the generalized Fourier integral corresponding to .Note that some literature..
There are at least three definitions of "groupoid" currently in use.The first type of groupoid is an algebraic structure on a set with a binary operator. The only restriction on the operator is closure (i.e., applying the binary operator to two elements of a given set returns a value which is itself a member of ). Associativity, commutativity, etc., are not required (Rosenfeld 1968, pp. 88-103). A groupoid can be empty. The numbers of nonisomorphic groupoids of this type having , 2, ... elements are 1, 10, 3330, 178981952, ... (OEIS A001329), and the corresponding numbers of nonisomorphic and nonantiisomorphic groupoids are 1, 7, 1734, 89521056, ... (OEIS A001424). An associative groupoid is called a semigroup.The second type of groupoid is, roughly, a category which is "group-like" in the sense that every morphism (or arrow) is invertible. To make this notion more precise, one says that a groupoid is a category..
A algebraic loop is a generalized Bol loop if for all elements , , and of ,for some map . As the name suggests, these are generalizations of Bol loops; in particular, a Bol loop is a generalized Bol loop with respect to the identity map .One can show that there is an algebraic duality between generalized Bol loops and algebraic loops which satisfy the half-Bol identity (Adeniran and Solarin 1999).
There are at least two distinct notions known as the Whitehead group.Given an associative ring with unit, the Whitehead group associated to is the commutative quotient group(1)where is the union over all natural numbers of the general linear groups and where is the normal subgroup generated by all elementary matrices.Note that the commutativity of stems from the fact (proven by Whitehead) that is the commutator subgroup of .The second definition, though different, is related to the first. Given a multiplicative group with integral group ring , there exist natural homomorphisms(2)In this context, one can define the Whitehead group as the cokernel(3)
The quotient spaceof the Whitehead group is known as the reduced Whitehead group. Here, the element denotes the order-2 element corresponding to the unit where is the collection of invertible matrices with coefficients in an associative ring with unit, i.e., the collection of all units in an associative unit ring .
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 .
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 .
The Littlewood conjecture states that for any two real numbers ,where denotes the nearest integer function.In layman's terms, this conjecture concerns the simultaneous approximation of two real numbers by rationals, indeed saying that any two real numbers and can be simultaneously approximated at least moderately well by rationals having the same denominator (Venkatesh 2007).Though proof of the Littlewood conjecture still remains an open problem, many partial results exist. For example, Borel showed that the set of exceptional pairs of real numbers and for which the conjecture fails has Lebesgue measure zero. Much later, Einsiedler et al. (2006) proved that the set of pairs of exceptional points also has Hausdorff dimension zero.
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..
A lattice is said to be oriented if there exists a rule which assigns a specified direction to any edge connecting arbitrary lattice points . In that way, an oriented lattice is intimately connected to an oriented graph.Given a point lattice , one common such rule is the so-called north-east rule which orients each edge of in the direction of increasing coordinate-value. This type of ordered lattice is central in a number of fields, e.g.,in oriented percolation models within the field of discrete percolation theory.
A function is Fréchet differentiable at ifexists. This is equivalent to the statement that has a removable discontinuity at , whereIn literature, the Fréchet derivative is sometimes known as the strong derivative (Ostaszewski 2012) and can be seen as a generalization of the gradient to arbitrary vector spaces (Long 2009).Every function which is Fréchet differentiable is both Carathéodory differentiable and Gâteaux differentiable. The relationship between the Fréchet derivative and the Gâteaux derivative can be made even more explicit by noting that a function is Fréchet differentiable if and only if the limit used to describe the Gâteaux derivative exists uniformly with respect to vectors on the unit sphere of the domain space ; as such, this uniform limit (when it exists) is what's called the Fréchet Derivative (Andrews and Hopper 2011)..
Let and be measure spaces. A measurable rectangle is a set of the form for and .
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,..
Given a compact manifold and a transversely orientable codimension-one foliation on which is tangent to , the pair is called a generalized Reeb component if the holonomy groups of all leaves in the interior are trivial and if all leaves of are proper. Generalized Reeb components are obvious generalizations of Reeb components.The introduction of the generalized version of the Reeb component facilitates the proof of many significant results in the theory of 3-manifolds and of foliations. It is well-known that generalized Reeb components are transversely orientable and that a manifold admitting a generalized Reeb component also admits a nice vector field (Imanishi and Yagi 1976). Moreover, given a generalized Reeb component , is a fibration over .Like many notions in geometric topology, the generalized Reeb component can be presented in various contexts. One source describes a generalized Reeb component on a closed 3-manifold with foliation..
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.
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 tensor category is strict if the maps , , and are always identities.A related notion is that of a tensor R-category.
A short exact sequence of groups(1)is called split if it essentially presents as the direct sum of the groups and .More precisely, one can construct a commutative diagram as diagrammed above, where is the injection of the first summand and is the projection onto the second summand , and the vertical maps are isomorphisms.Not all short exact sequences of groups are split. For example the short exact sequence diagrammed above cannot be split, since and are non isomorphic finite groups. Note that this is also a short exact sequence of -modules: this shows that being split is a distinguished property of short exact sequences also in the category of modules. In fact, it is related to particular classes of modules.Given a module over a unit ring , all short exact sequences(2)are split iff is projective, and all short exact sequences(3)are split iff is injective.A short exact sequence of vectorspaces is always split...
A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequenceThis commutative diagram shows how the first commutative diagram (shown here in blue) can be modified to exhibit the long exact sequence (shown here in red) explicitly. The map is called a connecting homomorphism and describes a curve from the end of the upper row () to the beginning of the lower row (), which suggested the name given to this lemma.The snake lemma is explained in the first scene of Claudia Weill's film Itis My Turn (1980), starring Jill Clayburgh and Michael Douglas.
Let be a commutative ring, let be an R-module for , 1, 2, ..., and define a chain complex of the formA chain contraction is a collection of R-modules morphisms such that, for all ,Here, is the boundary map of .
There are no fewer than two closely related but somewhat different notions of gerbe in mathematics.For a fixed topological space , a gerbe on can refer to a stack of groupoids on satisfying the properties 1. for subsets open, and 2. given objects , any point has a neighborhood for which there is at least one morphism in . The second definition is due to Giraud (Brylinski 1993). Given a manifold and a Lie group , a gerbe with band is a sheaf of groupoids over satisfying the following three properties: 1. Given any object of , the sheaf of automorphisms of this object is a sheaf of groups on which is locally isomorphic to the sheaf of smooth -valued functions. Such a local isomorphism is unique up to inner automorphisms of . 2. Given two objects and of , there exists a surjective local homeomorphism such that and are isomorphic. In particular, and are locally isomorphic. 3. There exists a surjective local homeomorphism such that the category is non-empty. Clearly,..
A band over a fixed topological space is represented by a cover , , and for each , a sheaf of groups on along with outer automorphisms satisfying the cocycle conditions and . Here, restrictions of the cover to a finer cover should be viewed as defining the exact same band.The collection of all bands over the space with respect to a single cover has a natural category structure. Indeed, if and are two bands over with respect to , then an isomorphism consists of outer automorphisms compatible on overlaps so that . The collection of all such bands and isomorphisms thereof forms a category.The notion of band is essential to the study of gerbes (Moerdijk). In particular, for a gerbe over a topological space , one can choose an open cover of by open subsets , and for each , an object which together form a sheaf of groups on . One can then consider a collection of sheaf isomorphisms between any two groups and which forms a collection of well-defined outer automorphisms.In..
A diagram lemma also known as lemma. According to its most general statement, the commutative diagram illustrated above with exact rows and columns can be completed by two morphismswithout losing commutativity.Moreover, the short exact sequenceis exact.The lemma is also true if the roles of the first and the third row are interchanged.
Let be a commutative ring and let be an R-module for . A chain complex of the formis said to be acyclic if its th homology group is trivial for all values .A straightforward result in homological algebra states that a chain complex with each free is acyclic if and only if there exists a chain contraction .
There are several different definitions of the term "unital" used throughout various branches of mathematics.In geometric combinatorics, a block design of the form (, , 1) is said to be a unital. In particular, then, a unital is a collection consisting of points and arranged into subsets so that for all and every pair of distinct points is contained in exactly one .A completely separate notion of unital is used ubiquitously throughout abstract algebra as an adjective to refer to an algebraic structure which contains a unit, e.g., a unitary ring is a ring which contains at least one unit. Algebraic structures of this kind are sometimes called unitary, though caution must be exhibited due to numerous unrelated mathematical notions which are themselves called unitary, e.g., unitary matrices which collectively form the unitary group, unitary elements, unitary divisors, etc. One must also exhibit caution when consulting literature..
Throughout abstract algebra, the term "magma" is most often used as a synonym of the more antiquated term "groupoid," referring to a set equipped with a binary operator. The term is thought to have originated with Bourbaki.Unlike the term "groupoid" which has a number of different uses across algebra, the term "magma" has the benefit of being essentially unused in other contexts. On the other hand, the use of the term "magma" appears to be somewhat less common in literature.
Taylor's inequality is an estimate result for the value of the remainder term in any -term finite Taylor series approximation.Indeed, if is any function which satisfies the hypotheses of Taylor's theorem and for which there exists a real number satisfying on some interval , the remainder satisfieson the same interval .This result is an immediate consequence of the Lagrange remainder of and can also be deduced from the Cauchy remainder as well.
Given an matrix , the fundamental theorem of linear algebra is a collection of results relating various properties of the four fundamental matrix subspaces of . In particular: 1. and where here, denotes the range or column space of , denotes its transpose, and denotes its null space. 2. The null space is orthogonal to the row space . 1. There exist orthonormal bases for both the column space and the row space of . 4. With respect to the orthonormal bases of and , is diagonal. The third item on this list stems from Gram-Schmidt Orthonormalization; the fourth item stems from the singular value decomposition of . Also, while different, the first item is reminiscent of the rank-nullity theorem.The above figure summarizes some of the interactions between the four fundamental matrix subspaces for a real matrix including whether the spaces in question are subspaces of or , which subspaces are orthogonal to one another, and how the matrix maps various vectors..
The standard Lorentzian inner product on is given by(1)i.e., for vectors and ,(2) endowed with the metric tensor induced by the above Lorentzian inner product is known as Minkowski space and is denoted .The Lorentzian inner product on is nothing more than a specific case of the more general Lorentzian inner product on -dimensional Lorentzian space with metric signature : In this more general environment, the inner product of two vectors and has the form(3)The Lorentzian inner product of two such vectors is sometimes denoted to avoid the possible confusion of the angled brackets with the standard Euclidean inner product (Ratcliffe 2006). Analogous presentations can be made if the equivalent metric signature (i.e., for Minkowski space) is used.The four-dimensional Lorentzian inner product is used as a tool in special relativity, namely as a measurement which is independent of reference frame and which replaces the typical Euclidean notion..
Given a real matrix , there are four associated vector subspaces which are known colloquially as its fundamental subspaces, namely the column spaces and the null spaces of the matrices and its transpose . These four subspaces are important for a number of reasons, one of which is the crucial role they play in the so-called fundamental theorem of linear algebra.The above figure summarizes some of the interactions between the four fundamental matrix subspaces for a real matrix including whether the spaces in question are subspaces of or , which subspaces are orthogonal to one another, and how the matrix maps various vectors relative to the subspace in which lies.In the event that , all four of the fundamental matrix subspaces are lines in . In this case, one can write for some vectors , whereby the directions of the four lines correspond to , , , and . An elementary fact from linear algebra is that these directions are also represented by the eigenvectors..
An algebraic loop is a Moufang loop if all triples of elements , , and in satisfy the Moufang identities, i.e., if 1. , 2. , 3. , and 4. . One can show that an algebraic loop which satisfies both the left and right Bol identities is Moufang.
The term Bol loop refers to either of two classes of algebraic loops satisfying the so-called Bol identities. In particular, a left Bol loop is an algebraic loop which, for all , , and in , satisfies the left Bol relationSimilarly, is a right Bol loop provided it satisfies the right Bol relationAn algebraic loop which is both a left and rightBol loop is called a Moufang loop.Some sources use the term Bol loop to refer to a right Bol loop, whereas some reserve the term for algebraic loops that are Moufang.Although (left and right) Bol loops have relatively weak structural properties, one can show that such structures are power associative. Thus, given an algebraic loop , the element is well-defined for all elements and all integers independent of which order the multiplications are performed...
A temporal point process is a random process whose realizations consist of the times of isolated events.Note that in some literature, the values are assumed to be arbitrary real numbers while the index set is assumed to be the set of integers (Schoenberg 2002); on the other hand, some authors view temporal point processes as binary events so that takes values in a two-element set for each , and further assume that the index set is some finite set of points (Liam 2013). The prior perspective corresponds to viewing temporal point processes as how long events occur where the events themselves are spaced according to a discrete set of time parameters; the latter view corresponds to viewing temporal point processes as indications of whether or not a finite number of events has occurred.The behavior of a simple temporal point process is typically modeled by specifying its conditional intensity . Indeed, a number of specific examples of temporal point..
A point process is a probabilistic model for random scatterings of points on some space often assumed to be a subset of for some . Oftentimes, point processes describe the occurrence over time of random events in which the occurrences are revealed one-by-one as time evolves; in this case, any collectionof occurrences is said to be a realization of the point process.Poisson processes are regarded as archetypal examplesof point processes (Daley and Vere-Jones 2002).Point processes are sometimes known as counting processes or random scatters.
There are at least two distinct notions of when a pointprocess is stationary.The most commonly utilized terminology is as follows: Intuitively, a point process defined on a subset of is said to be stationary if the number of points lying in depends on the size of but not its location. On the real line, this is expressed in terms of intervals: A point process on is stationary if for all and for ,depends on the length of but not on the location .Stationary point processes of this kind were originally called simple stationary, though several authors call it crudely stationary instead. In light of the notion of crude stationarity, a different definition of stationary may be stated in which a point process is stationary whenever for every and for all bounded Borel subsets of , the joint distribution of does not depend on . This distinction also gives rise to a related notion known as interval stationarity.Some authors use the alternative definition of an intensity..
A multidimensional point process is a measurable function from a probability space into where is the set of all finite or countable subsets of not containing an accumulation point and where is the sigma-algebra generated over by the setsfor all bounded Borel subsets . Here, denotes the cardinality or order of the set .A multidimensional point process is sometimes abbreviated MPP, though care should be exhibited not to confuse the notion with that of a marked point process.Despite a number of apparent differences, one can show that multidimensional point processes are a special case of a random closed set on (Baudin 1984).
A simple point process (or SPP) is an almost surely increasing sequence of strictly positive, possibly infinite random variables which are strictly increasing as long as they are finite and whose almost sure limit is . Symbolically, then, an SPP is a sequence of -valued random variables defined on a probability space such that 1. , 2. , 3. . Here, and for each , can be interpreted as either the time point at which the th recording of an event takes place or as an indication that fewer than events occurred altogether if or if , respectively (Jacobsen 2006).
A marked point process with mark space is a double sequenceof -valued random variables and -valued random variables defined on a probability space such that is a simple point process (SPP) and: 1. for ; 2. for . Here, denotes probability, denotes the so-called irrelevant mark which is used to describe the mark of an event that never occurs, and .This definition is similar to the definition of an SPP in that it describes a sequence of time points marking the occurrence of events. The difference is that these events may be of different types where the type (i.e., the mark) of the th event is denoted by . Note that, because of the inclusion of the irrelevant mark , marking will assign values for all --even when , i.e., when the th event never occurs (Jacobsen 2006).
Given a marked point process of the formthe space is said to be the mark space of .
A random closed set (RACS) in is a measurable function from a probability space into where is the collection of all closed subsets of and where denotes the sigma-algebra generated over the by setsfor all compact subsets .Originally, RACS were defined not on but in the more general setting of locally compact and separable (LCS) topological spaces (Baudin 1984) which may or may not be T2. In this case, the above definition is modified so that is defined to be the collection of closed subsets of some ambient LCS space (Molchanov 2005).Despite a number of apparent differences, one can show that multidimensional point processes are a special case of RACS when talking about (Baudin 1984).
Given a random variable with continuous and strictly monotonic probability density function , a quantile function assigns to each probability attained by the value for which . Symbolically,Defining quantile functions for discrete rather than continuous distributions requires a bit more work since the discrete nature of such a distribution means that there may be gaps between values in the domain of the distribution function and/or "plateaus" in its range. Therefore, one often defines the associated quantile function to bewhere denotes the range of .
A point process on is said to be interval stationary if for every and for all integers , the joint distribution ofdoes not depend on , . Here, is an interval for all .As pointed out in a variety of literature (e.g., Daley and Vere-Jones 2002, pp 45-46), the notion of an interval stationary point process is intimately connected to (though fundamentally different from) the idea of a stationary point process in the Borel set sense of the term. Worth noting, too, is the difference between interval stationarity and other notions such as simple/crude stationarity.Though it has been done, it is more difficult to extend to the notion of interval stationarity; doing so requires a significant amount of additional machinery and reflects, overall, the significantly-increased structural complexity of higher-dimensional Euclidean spaces (Daley and Vere-Jones 2007)...
The intensity measure of a point process relative to a Borel set is defined to be the expected number of points of falling in . Symbolically,where here, denotes the expected value.The notion of an intensity measure is intimately connected to one oft-discussed notionof intensity function (Pawlas 2008).
There are at least two distinct notions of an intensity function related to the theoryof point processes.In some literature, the intensity of a point process is defined to be the quantity(1)provided it exists. Here, denotes probability. In particular, it makes sense to talk about point processes having infinite intensity, though when finite, allows to be rewritten so that(2)as where here, denotes little-O notation (Daley and Vere-Jones 2007).Other authors define the function to be an intensity function of a point process provided that is a density of the intensity measure associated to relative to Lebesgue measure, i.e.,if for all Borel sets in ,(3)where denotes Lebesgue measure (Pawlas 2008).
An integer is said to be highly cototient if the equationhas more solutions than the equations for all , where is the totient function.The first few highly cototient numbers are 2, 4, 8, 23, 35, 47, 59, 63, 83, 89, ...(OEIS A100827).The first few prime highly cototient numbers are 2, 23, 47, 59, 83, 89, 113, 167,269, 389, 419, 509, ... (OEIS A105440).
Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as "parameters." For example, while the equation of a circle in Cartesian coordinates can be given by , one set of parametric equations for the circle are given by(1)(2)illustrated above. Note that parametric representations are generally nonunique, so the same quantities may be expressed by a number of different parameterizations. A single parameter is usually represented with the parameter , while the symbols and are commonly used for parametric equations in two parameters.Parametric equations provide a convenient way to represent curves and surfaces, as implemented, for example, in the Wolfram Language commands ParametricPlot[x, y, t, t1, t2] and ParametricPlot3D[x, y, z, u, u1, u2, v, v1, v2]. Unsurprisingly, curves and surfaces obtained by way of parametric equation representations..
Cartesian coordinates are rectilinear two- or three-dimensional coordinates (and therefore a special case of curvilinear coordinates) which are also called rectangular coordinates. The two axes of two-dimensional Cartesian coordinates, conventionally denoted the x- and y-axes (a notation due to Descartes), are chosen to be linear and mutually perpendicular. Typically, the -axis is thought of as the "left and right" or horizontal axis while the -axis is thought of as the "up and down" or vertical axis. In two dimensions, the coordinates and may lie anywhere in the interval , and an ordered pair in two-dimensional Cartesian coordinates is often called a point or a 2-vector.The three-dimensional Cartesian coordinate system is a natural extension of the two-dimensional version formed by the addition of a third "in and out" axis mutually perpendicular to the - and -axes defined above. This new axis is conventionally..
A plot of a function expressed in polar coordinates, with radius as a function of angle . Polar plots can be drawn in the Wolfram Language using PolarPlot[r, t, tmin, tmax]. The plot above is a polar plot of the polar equation , giving a cardioid.Polar plots of give curves known as roses, while polar plots of produce what's known as Archimedes' spiral, a special case of the Archimedean spiral corresponding to . Other specially-named Archimedean spirals include the lituus when , the hyperbolic spiral when , and Fermat's spiral when . Note that lines and circles are easily-expressed in polar coordinates as(1)and(2)for the circle with center and radius , respectively. Note that equation () is merely a particular instance of the equation(3)defining a conic section of eccentricity and semilatus rectum . In particular, the circle is the conic of eccentricity , while yields a general ellipse, a parabola, and a hyperbola.The plotting of a complex number..
A point is a 0-dimensional mathematical object which can be specified in -dimensional space using an n-tuple (, , ..., ) consisting of coordinates. In dimensions greater than or equal to two, points are sometimes considered synonymous with vectors and so points in n-dimensional space are sometimes called n-vectors. Although the notion of a point is intuitively rather clear, the mathematical machinery used to deal with points and point-like objects can be surprisingly slippery. This difficulty was encountered by none other than Euclid himself who, in his Elements, gave the vague definition of a point as "that which has no part."The basic geometric structures of higher dimensional geometry--the line, plane, space, and hyperspace--are all built up of infinite numbers of points arranged in particular ways.These facts lead to the mathematical pun, "without geometry, life is pointless."The decimal point in a decimal..
The conditional intensity associated to a temporal point process is defined to be the expected infinitesimal rate at which events are expected to occur around time given the history of at times prior to time . Algebraically,provided the limit exists where here, is the history of over all times strictly prior to time .
A radial function is a function satisfying for points in some subset . Here, denotes the standard Euclidean norm in and is a discrete subset of whose elements are called centers.A collection of such functions which independently span a space is usually called a radial basis of . In this case, the functions are known as radial basis functions. Radial bases and radial basis functions play an important role in many areas of mathematics and approximation theory including statistics and partial differential equations.
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 method is an algorithm for factoring quadratic polynomials of the form with integer coefficients. As its name suggests, the crux of the algorithm is to consider the multiplicative factors of the product of the coefficients and . More precisely, the goal is to find an integer pair and satisfying and simultaneously, whereby one can rewrite in the form(1)and to factor the remaining four-term polynomial by grouping into a product of linear factors with integer coefficients.For example, consider the polynomial having coefficients , , and . To begin the factorization, consider the product . By observation, while ; in particular, this guarantees that can be rewritten so that(2)This four-term expression for can be factored by grouping:(3)and so(4)One can easily see that the above method generalizes to certain polynomials of the form for positive integers , though the result will be a factorization into pairs of polynomials of degree which aren't..
The zero product property asserts that, for elements and ,This property is especially relevant when considering algebraic structures because, e.g., integral domains are rings having the zero product property and are important objects of study because of that fact.
A nonzero element of a ring for which , where is some other nonzero element and the multiplication is the multiplication of the ring. A ring with no zero divisors is known as an integral domain. Let denote an -algebra, so that is a vector space over and(1)(2)Now define(3)where . is said to be -associative if there exists an -dimensional subspace of such that for all and . is said to be tame if is a finite union of subspaces of .The zero product property is intimately tethered to the notion of a zero divisor. For example, one may equivalently define an integral domain as a ring which satisfies the zero product property.
The so-called Malthusian equation is an antiquated term for the equationdescribing exponential growth. The constant is sometimes called the Malthusian parameter.The term itself refers to the influential work of Thomas Malthus in which the growth of the human population is speculated to be exponential.
Exponential growth is the increase in a quantity according to the law(1)for a parameter and constant (the analog of the decay constant), where is the exponential function and is the initial value. Exponential growth is common in physical processes such as population growth in the absence of predators or resource restrictions (where a slightly more general form is known as the law of growth). Exponential growth also occurs as the limit of discrete processes such as compound interest.Exponential growth is described by the first-order ordinary differential equation(2)which can be rearranged to(3)Integrating both sides then gives(4)and exponentiating both sides yields the functional form (1).A much more antiquated term for population growth modeled according to an exponential equation is the so-called Malthusian equation, a result of a 1798 philosophical text by Thomas Malthus which investigated population dynamics under the assumption..
The index associated to a metric tensor on a smooth manifold is a nonnegative integer for whichfor all . Here, the notation denotes the quadratic form index associated with .The index of a metric tensor provides an alternative tool by which to define a number of various notions typically associated to the signature of . For example, a Lorentzian manifold can be defined as a pair for which and for which , a definition equivalent to its more typical definition as a manifold of dimension no less than two equipped with a tensor of metric signature (or, equivalently, ).
A strong Riemannian metric on a smooth manifold is a tensor field which is both a strong pseudo-Riemannian metric and positive definite.In a very precise way, the condition of being a strong Riemannian metric is considerably more stringent than the condition of being a weak Riemannian metric due to the fact that strong non-degeneracy implies weak non-degeneracy but not vice versa. More precisely, strong Riemannian metrics provide an isomorphism between the tangent and cotangent spaces and , respectively, for all ; conversely, weak Riemannian metrics are merely injective linear maps from to .
A semi-Riemannian manifold is said to be Lorentzian if and if the index associated with the metric tensor satisfies .Alternatively, a smooth manifold of dimension is Lorentzian if it comes equipped with a tensor of metric signature (or, equivalently, ).
A Lorentz transformation is a four-dimensional transformation(1)satisfied by all four-vectors , where is a so-called Lorentz tensor. Lorentz tensors are restricted by the conditions(2)with the Minkowski metric (Weinberg 1972, p. 26; Misner et al. 1973, p. 68).Here, the tensor indices run over 0, 1, 2, 3, with being the time coordinate and being space coordinates, and Einstein summation is used to sum over repeated indices. There are a number of conventions, but a common one used by Weinberg (1972) is to take the speed of light to simplify computations and allow to be written simply as for . The group of Lorentz transformations in Minkowski space is known as the Lorentz group.An element in four-space which is invariant under a Lorentz transformation is said to be a Lorentz invariant; examples include scalars, elements of the form , and the interval between two events (Thorn 2012).Note that while some authors (e.g., Weinberg 1972,..
A four-vector is said to be lightlike if its four-vector norm satisfies .One should note that the four-vector norm is nothing more than a special case of the more general Lorentzian inner product on Lorentzian -space with metric signature : In this more general environment, the inner product of two vectors and has the formwhereby one defines a vector to be lightlike precisely when .Lightlike vectors are sometimes called null vectors. The collection of all lightlike vectors in a Lorentzian space (e.g., in the Minkowski space of special relativity) is known as the light cone. One often draws distinction between lightlike vectors which are positive and those which are negative.
The index associated to a symmetric, non-degenerate, and bilinear over a finite-dimensional vector space is a nonnegative integer defined bywhere the set is defined to beAs a concrete example, a pair consisting of a smooth manifold with a symmetric tensor field is said to be a Lorentzian manifold if and only if and the index associated to the quadratic form satisfies for all (Sachs and Wu 1977). This particular definition succinctly conveys the fact that Lorentzian manifolds have indefinite metric tensors of signature (or equivalently ) without having to make precise any definitions related to metric signatures, quadratic form signatures, etc.The above example also illustrates the deep connection between the index of a quadratic form and the notion of the index of a metric tensor defined on a smooth manifold . In particular, the index of a metric tensor is defined to be the quadratic form index associated to for any element . Because of this connection,..
A real-valued univariate function has a jump discontinuity at a point in its domain provided that(1)and(2)both exist and that .The notion of jump discontinuity shouldn't be confused with the rarely-utilized convention whereby the term jump is used to define any sort of functional discontinuity.The figure above shows an example of a function having a jump discontinuity at a point in its domain.Though less algebraically-trivial than removable discontinuities, jump discontinuities are far less ill-behaved than other types of singularities such as infinite discontinuities. This fact can be seen in a number of scenarios, e.g., in the fact that univariate monotone functions can have at most countably many discontinuities (Royden and Fitzpatrick 2010), the worst of which can be jump discontinuities (Zakon 2004).Unsurprisingly, the definition given above can be generalized to include jump discontinuities in multivariate real-valued..
A real-valued univariate function is said to have a removable discontinuity at a point in its domain provided that both and(1)exist while . Removable discontinuities are so named because one can "remove" this point of discontinuity by defining an almost everywhere identical function of the form(2)which necessarily is everywhere-continuous.The figure above shows the piecewise function(3)a function for which while . In particular, has a removable discontinuity at due to the fact that defining a function as discussed above and satisfying would yield an everywhere-continuous version of .Note that the given definition of removable discontinuity fails to apply to functions for which and for which fails to exist; in particular, the above definition allows one only to talk about a function being discontinuous at points for which it is defined. This definition isn't uniform, however, and as a result, some authors claim that, e.g.,..
A real-valued univariate function is said to have an infinite discontinuity at a point in its domain provided that either (or both) of the lower or upper limits of fails to exist as tends to .Infinite discontinuities are sometimes referred to as essential discontinuities, phraseology indicative of the fact that such points of discontinuity are considered to be "more severe" than either removable or jump discontinuities.The figure above shows the piecewise function(1)a function for which both and fail to exist. In particular, has an infinite discontinuity at .It is not uncommon for authors to say that univariate functions defined on a domain and admitting vertical asymptotes of the form have infinite discontinuities there though, strictly speaking, this terminology is incorrect unless such functions are defined piecewise so that . For example, the function has vertical asymptotes at , , though it has no discontinuities of any..
A discontinuity is point at which a mathematical object is discontinuous. The left figure above illustrates a discontinuity in a one-variable function while the right figure illustrates a discontinuity of a two-variable function plotted as a surface in . In the latter case, the discontinuity is a branch cut along the negative real axis of the natural logarithm for complex .Some authors refer to a discontinuity of a function as a jump,though this is rarely utilized in the literature.Though defined identically, discontinuities of univariate functions are considerably different than those of multivariate functions. One of the main differences between these cases exists with regards to classifying the discontinuities, a caveat discussed more at length below.In the case of a one-variable real-valued function , there are precisely three families of discontinuities that can occur. 1. The simplest type is the so-called removablediscontinuity...
The maximum number of pieces into which a cylinder can be divided by oblique cuts is given by(1)(2)(3)where is a binomial coefficient.This problem is sometimes also called cake cutting or pie cutting, and has the same solution as space division by planes. For , 2, ... cuts, the maximum number of pieces is 2, 4, 8, 15, 26, 42, ... (OEIS A000125). Unsurprisingly, the numbers of this sequence are called cake numbers.
The average number of regions into which randomly chosen planes divide a cube is(Finch 2003, p. 482).The maximum number of regions is presumably the same as for spacedivision by planes, namely(Yaglom and Yaglom 1987, pp. 102-106). For , 2, ... planes, this gives the values 2, 4, 8, 15, 26, 42, ... (OEIS A000125), a sequence whose values are sometimes called the "cake numbers" due to their relation to the cake cutting problem.
The maximal number of regions into which space can be divided by planes is(Yaglom and Yaglom 1987, pp. 102-106). For , 2, ..., these give the values 2, 4, 8, 15, 26, 42, ... (OEIS A000125), a sequence whose values are sometimes called the "cake numbers" due to their relation to the cake cutting problem. This is the same solution as for cylinder cutting.
Let and be linear operators from domains and , respectively, into a Hilbert space . It is said that extends if and if for any vector .
A vector on a Hilbert space is said to be cyclic if there exists some bounded linear operator on so that the set of orbitsis dense in . In this case, the operator is said to be a cyclic operator.
A bounded linear operator on a Hilbert space is said to be cyclic if there exists some vector for which the set of orbitsis dense in . In this case, the vector is said to be a cyclic vector.
A linear operator from its domain into a Hilbert space is closed if for any sequence of vectors such that and as , it follows that and .
A linear operator from its domain into a Hilbert space is closable if it has a closed extension where . Closable operators are sometimes called preclosed (Takesaki 1970), and the extension of is sometimes called the closure of .
Given a subalgebra of the algebra of bounded linear transformations from a Hilbert space onto itself, the vector is a separating vector for if the only operator satisfying is the zero operator .
Let be a real-valued function defined on an interval and let . The four one-sided limits(1)(2)(3)and(4)are called the Dini derivatives of at . Individually, they are referred to as the upper right, lower right, upper left, and lower left Dini derivatives of at , respectively, and any or all of the values may be infinite.It turns out that continuity at a point of a single Dini derivative of a continuous function implies continuity of the other three Dini derivatives of at , equality of the four Dini derivatives, and (usual) differentiability of the function . In addition, the Denjoy-Saks-Young theorem completely characterizes all possible Dini derivatives of finite real-valued functions defined on intervals and--as corollaries--the Dini derivatives of all monotone and continuous functions defined on intervals.Many other important properties of Dini derivatives have been studied and characterized. Banach showed that the Dini derivative..
A monotonic function is a function which is either entirely nonincreasing or nondecreasing. A function is monotonic if its first derivative (which need not be continuous) does not change sign.The term monotonic may also be used to describe set functions which map subsets of the domain to non-decreasing values of the codomain. In particular, if is a set function from a collection of sets to an ordered set , then is said to be monotone if whenever as elements of , . This particular definition comes up frequently in measure theory where many of the families of functions defined (including outer measure, premeasure, and measure) begin by considering monotonic set functions.
A set function is said to possess countable subadditivity if, given any countable disjoint collection of sets on which is defined,A function possessing countable subadditivity is said to be countably subadditive.Any countably subadditive function is also finitely subadditive presuming that where is the empty set.
Let and be Banach spaces and let be a function between them. is said to be Gâteaux differentiable if there exists an operator such that, for all ,(1)The operator is called the Gâteaux derivative of at . is sometimes assumed to be bounded, though much of the theory of Gâteaux differentiability remains unchanged without this assumption.If the Gâteaux derivative exists, it is unique.A basic result about Gâteaux derivatives is that is Gâteaux differentiable at a point if and only if all the directional operators(2)exist and form a bounded linear operator . In addition, the Gâteaux derivative satisfies analogues of many properties from basic calculus including a mean-value property of the form(3)One definition of the Fréchet derivative pertains to uniform existence of the Gâteaux derivative on the unit sphere of (Andrews and Hopper). In particular, then, Fréchet differentiability..
Let be a set and a collection of subsets of . A set function is said to possess countable monotonicity provided that, whenever a set is covered by a countable collection of sets in ,A function which possesses countable monotonicity issaid to be countably monotone.One can easily verify that any set function which is both monotone (in the sense of mapping subsets of the domain to subsets of the range) and countably additive is necessarily countably monotone. The converse is not true in general.
A set function is said to possess finite subadditivity if, given any finite disjoint collection of sets on which is defined,A set function possessing finite subadditivity is said to be finitely subadditive. In particular, every finitely additive set function is also finitely subadditive.
A set function possesses countable additivity if, given any countable disjoint collection of sets on which is defined,A function having countable additivity is said to becountably additive.Countably additive functions are countably subadditive by definition. Moreover, provided that where is the empty set, every countably additive function is necessarily finitely additive.
Let be a set and a collection of subsets of . A set function is said to possess finite monotonicity provided that, whenever a set is covered by a finite collection of sets in ,A set function possessing finite monotonicity is said to be finitely monotone. Note that a set function which is countably monotone is necessarily finitely monotone provided that and , where is the empty set.
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 set function is finitely additive if, given any finite disjoint collection of sets on which is defined,
Dini's theorem is a result in real analysis relating pointwise convergence of sequences of functions to uniform convergence on a closed interval.For an increasing sequence of continuous functions on an interval which converges pointwise on to a continuous function on , Dini's theorem states that converges to uniformly on .
A four-vector is said to be timelike if its four-vector norm satisfies .One should note that the four-vector norm is nothing more than a special case of the more general Lorentzian inner product on -dimensional Lorentzian space with metric signature . In this more general environment, the inner product of two vectors and has the formwhereby one defines a vector to be timelike precisely when .Geometrically, the collection of all timelike vectors lie in the open subset of formed by the interior of the light cone: In particular, the upper half of the interior consists of vectors which are positive timelike whereas the lower half consists of all negative timelike vectors.
In -dimensional Lorentzian space , the light cone is defined to be the subset consisting of all vectors(1)whose squared (Lorentzian) norm is identically zero:(2)Alternatively, is the collection of all lightlike vectors in . The decomposition of into Lorentzian space of signature leads to a natural decomposition of such a vector into its component and its -subvector . Using this notation, the squared norm of can be expressed as(3)whereby one can also define the light cone to be the collection of all vectors satisfying(4)This particular perspective makes natural the distinction between positiveand negative lightlike vectors.The open subset of formed by the interior of the light cone consists of all timelike vectors; the open subset formed by the exterior of consists of all vectors which are spacelike...
A four-vector is said to be spacelike if its four-vector norm satisfies .One should note that the four-vector norm is nothing more than a special case of the more general Lorentzian inner product on -dimensional Lorentzian space with metric signature . In this more general environment, the inner product of two vectors and has the formwhereby one defines a vector to be spacelike precisely when .Geometrically, the collection of all spacelike vectors lie in the open subset of formed by the exterior of the light cone.
Roughly speaking, the metric tensor is a function which tells how to compute the distance between any two points in a given space. Its components can be viewed as multiplication factors which must be placed in front of the differential displacements in a generalized Pythagorean theorem:(1)In Euclidean space, where is the Kronecker delta (which is 0 for and 1 for ), reproducing the usual form of the Pythagorean theorem(2)In this way, the metric tensor can be thought of as a tool by which geometrical characteristics of a space can be "arithmetized" by way of introducing a sort of generalized coordinate system (Borisenko and Tarapov 1979).In the above simplification, the space in question is most often a smooth manifold , whereby a metric tensor is essentially a geometrical object taking two vector inputs and calculating either the squared length of a single vector or a scalar product of two different vectors (Misner et al. 1978). In this..
The vertical line test is a graphical method of determining whether a curve in the plane represents the graph of a function by visually examining the number of intersections of the curve with vertical lines.The motivation for the vertical line test is as follows: A relation is a function precisely when each element is matched to at most one value and, as a result, any vertical line in the plane can intersect the graph of a function at most once. Therefore, the vertical line test concludes that a curve in the plane represents the graph of a function if and only if no vertical line intersects it more than once.A plane curve which doesn't represent the graph of a function is sometimes said to have failed the vertical line test.The figure above shows two curves in the plane. The leftmost curve fails the vertical line test due to the fact that the single vertical line drawn intersects the curve in two points. On the other hand, the vertical line test shows that the..
The Kubo-Martin-Schwinger (KMS) condition is a kind of boundary-value condition which naturally emerges in quantum statistical mechanics and related areas.Given a quantum system with finite dimensional Hilbert space , define the function as(1)where is the imaginary unit and where is the Hamiltonian, i.e., the sum of the kinetic energies of all the particles in plus the potential energy of the particles associated with . Next, for any real number , define the thermal equilibrium as(2)where denotes the matrix trace. From and , one can define the so-called equilibrium correlation function where(3)whereby the KMS boundary condition says that(4)In particular, this identity relates to the state the values of the analytic function on the boundary of the strip(5)where here, denotes the imaginary part of and denotes the signum function applied to .In various literature, the KMS boundary condition is stated in sometimes-different contexts...
Let be a pair consisting of finite, connected CW-complexes where is a subcomplex of . Define the associated chain complex group-wise for each by setting(1)where denotes singular homology with integer coefficients and where denotes the union of all cells of of dimension less than or equal to . Note that is free Abelian with one generator for each -cell of .Next, consider the universal covering complexes of and , respectively. The fundamental group of can be identified with the group of deck transformations of so that each determines a map(2)which then induces a chain map(3)The chain map turns each chain group into a module over the group ring which is -free with one generator for each -cell of and which is finitely generated over due to the finiteness of .Hence, there is a free chain complex(4)over , the homology groups of which are zero due to the fact that deformation retracts onto . A simple argument shows the existence of a so-called preferred basis..
A natural transformation is called unital if the leftmost diagram above commutes. Similarly, a natural transformation is called unital if the diagram on the right-hand side above commutes.Note that in these definitions, , , and are all objects in a tensor category , is the neutral (or identity) object in , and the juxtaposition is shorthand for the tensor product in . What's more, the subscripts attached to the transformations and denote the components of the functors (indexed with respect to the objects in ) in question.
The homomorphism which, according to the snake lemma, permits construction of an exact sequence(1)from the above commutative diagram with exact rows. The homomorphism is defined by(2)for all , denotes the image, and is obtained through the following construction, based on diagram chasing.1. Exploit the surjectivity of to find such that . 2. Since because of the commutativity of the right square, belongs to , which is equal to due to the exactness of the lower row at . This allows us to find such that . While the elements and are not uniquely determined, the coset is, as can be proven by using more diagram chasing. In particular, if and are other elements fulfilling the requirements of steps (1) and (2), then and , and(3)hence because of the exactness of the upper row at . Let be such that(4)Then(5)because the left square is commutative. Since is injective, it follows that(6)and so(7)..
A commutative diagram is a collection of maps in which all map compositions starting from the same set and ending with the same set give the same result. In symbols this means that, whenever one can form two sequences(1)and(2)the following equality holds:(3)Commutative diagrams are usually composed by commutative triangles and commutative squares.Commutative triangles and squares can also be combined to form plane figures or space arrangements.A commutative diagram can also contain multiple arrows that indicate different maps between the same two sets.A looped arrow indicates a map from a set to itself.The above commutative diagram expresses the fact that is the inverse map to , since it is a pictorial translation of the map equalities and .This can also be represented using two separate diagrams.Many other mathematical concepts and properties, especially in algebraic topology, homological algebra, and category theory, can be formulated..
In category theory, a tensor category consists of a category , an object of , a functor , and a natural isomorphism(1)(2)(3)where the data are subject to the following axioms: 1. Given four objects , , , and of , the top diagram above commutes. 2. Given two objects and of , the bottom diagram above commutes. In the above, is called the tensor product, is called the associator, is called the right unit, and is called the left unit of the tensor category. The object is referred to as the neutral element or the identity of the tensor product.If the maps , , and are always identities, the tensor category in question is said to be strict.A related notion is that of a tensor R-category.
In algebraic topology, the Reidemeister torsion is a notion originally introduced as a topological invariant of 3-manifolds which has now been widely adapted to a variety of contexts. At the time of its discovery, the Reidemeister torsion was the first 3-manifold invariant able to distinguish between manifolds which are homotopy equivalent but not homeomorphic. Since then, the notion has been adapted to higher-dimensional manifolds, knots and links, dynamical systems, Witten's equations, and so on. In particular, it has a number of different definitions for various contexts.For a commutative ring , let be a finite acyclic chain complex of based finitely generated free R-modules of the form(1)The Reidemeister torsion of is the value defined by(2)where is the set of units of , is a chain contraction, is the boundary map, and(3)is a map from to . In this context, Reidemeister torsion is sometimes referred to as the torsion of the complex (Nicolaescu..
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...
Let be functors between categories and . A natural transformation from to consists of a family of morphisms in which are indexed by the objects of so that, for each morphism between objects in , the equalityholds. The elements are called the components of the natural transformation.If all the components are isomorphisms in , then is called a natural isomorphism between and . In this case, one writes .
A linear equation is an algebraic equation of the forminvolving only a constant and a first-order (linear) term, where is the slope and is the -intercept. The above form is aptly known as slope-intercept form; alternatively, linear equations can be written in a number of other forms including standard form, intercept form, etc.Occasionally, the above is called a "linear equation of two variables," where and are the variables. Equations such as and are linear equations of a single variable, and is an example of a linear equation with three variables.
Euclidean -space, sometimes called Cartesian space or simply -space, is the space of all n-tuples of real numbers, (, , ..., ). Such -tuples are sometimes called points, although other nomenclature may be used (see below). The totality of -space is commonly denoted , although older literature uses the symbol (or actually, its non-doublestruck variant ; O'Neill 1966, p. 3). is a vector space and has Lebesgue covering dimension . For this reason, elements of are sometimes called -vectors. is the set of real numbers (i.e., the real line), and is called the Euclidean plane. In Euclidean space, covariant and contravariant quantities are equivalent so .
Minkowski space is a four-dimensional space possessing a Minkowskimetric, i.e., a metric tensor having the formAlternatively (though less desirably), Minkowski space can be considered to have a Euclidean metric with imaginary time coordinate where is the speed of light (by convention is normally used) and where i is the imaginary number . Minkowski space unifies Euclidean three-space plus time (the "fourth dimension") in Einstein's theory of special relativity.In equation (5) above, the metric signature is assumed; under this assumption, Minkowski space is typically written . One may also express equation (5) with respect to the metric signature by reversing the order of the positive and negative squared terms therein, in which case Minkowski space is denoted .The Minkowski metric induces an inner product, the four-dimensional Lorentzian inner product (sometimes referred to as the Minkowski inner product), which fails..
A diagram lemma which states that, given the above commutative diagram with exact rows, the following holds: 1. If is surjective, and and are injective, then is injective; 2. If is injective, and and are surjective, then is surjective. This lemma is closely related to the five lemma, whichis based on a similar diagram obtained by adding a single column.
A diagram lemma which states that, given the commutative diagram of additive Abelian groups with exact rows, the following holds: 1. If is surjective, and and are injective, then is injective; 2. If is injective, and and are surjective, then is surjective. If and are bijective, the hypotheses of (1) and (2) are satisfied simultaneously, and the conclusion is that is bijective. This statement is known as the Steenrod five lemma.If , , , and are the zero group, then and are zero maps, and thus are trivially injective and surjective. In this particular case the diagram reduces to that shown above. It follows from (1), respectively (2), that is injective (or surjective) if and are. This weaker statement is sometimes referred to as the "short five lemma."
A contingency table, sometimes called a two-way frequency table, is a tabular mechanism with at least two rows and two columns used in statistics to present categorical data in terms of frequency counts. More precisely, an contingency table shows the observed frequency of two variables, the observed frequencies of which are arranged into rows and columns. The intersection of a row and a column of a contingency table is called a cell.gendercupconesundaesandwichothermale5923002042480female4103351802055For example, the above contingency table has two rows and five columns (not counting header rows/columns) and shows the results of a random sample of adults classified by two variables, namely gender and favorite way to eat ice cream (Larson and Farber 2014). One benefit of having data presented in a contingency table is that it allows one to more easily perform basic probability calculations, a feat made easier still by augmenting a summary..
The score function is the partial derivativeof the log-likelihood function , where is the standard likelihood function.Defining the likelihood function(1)shows that(2)and thus that(3)(4)(5)Using the above formulation of , one can easily compute various statistical measurements associated with . For example, the mean can be shown to equal zero while the variance is precisely the Fisher information matrix. The score function has extensive uses in many areas of mathematics, both pure and applied, and is a key component of the field of likelihood theory.
The word quantile has no fewer than two distinct meanings in probability. Specific elements in the range of a variate are called quantiles, and denoted (Evans et al. 2000, p. 5). This particular meaning has close ties to the so-called quantile function, a function which assigns to each probability attained by a certain probability density function a value defined by(1)The th -tile is that value of , say , which corresponds to a cumulative frequency of (Kenney and Keeping 1962). If , the quantity is called a quartile, and if , it is called a percentile.A parametrized version of quantile is implemented as Quantile[list, q, a, b, c, d], which returns(2)where is the th order statistic, is the floor function, is the ceiling function, is the fractional part, and(3)There are a number of slightly different definitions of the quantile that are in common use, as summarized in the following table.#plotting positiondescriptionQ10010inverted empirical..
The term limit comes about relative to a number of topics from several different branches of mathematics.A sequence of elements in a topological space is said to have limit provided that for each neighborhood of , there exists a natural number so that for all . This very general definition can be specialized in the event that is a metric space, whence one says that a sequence in has limit if for all , there exists a natural number so that(1)for all . In many commonly-encountered scenarios, limits are unique, whereby one says that is the limit of and writes(2)On the other hand, a sequence of elements from an metric space may have several - even infinitely many - different limits provided that is equipped with a topology which fails to be T2. One reads the expression in (1) as "the limit as approaches infinity of is ."The topological notion of convergence can be rewritten to accommodate a wider array of topological spaces by utilizing the language..
The symbol used to indicate a root is called a radical, or sometimes a surd. The expression is therefore read " radical ," or "the nth root of ." In the radical symbol, the horizonal line is called the vinculum, the quantity under the vinculum is called the radicand, and the quantity written to the left is called the index.In general, the use of roots is equivalent to the use of fractional exponentsas indicated by the identity(1)a more generalized form of the standard(2)The special case is written and is called the square root of . is called the cube root.Some interesting radical identities are due to Ramanujan, and include the equivalent forms(3)and(4)Another such identity is(5)
Young's geometry is a finite geometry which satisfiesthe following five axioms: 1. There exists at least one line. 2. Every line of the geometryhas exactly three points on it. 3. Not all points of the geometryare on the same line. 4. For two distinct points, there exists exactly one line on both of them. 5. If a point does not lie on a given line, then there exists exactly one line on that point that does not intersect the given line. Cherowitzo (2006) notes that the last axiom bears a strong resemblance to the parallel postulate of Euclidean geometry.
Three point geometry is a finite geometry subjectto the following four axioms: 1. There exist exactly three points. 2. Two distinct points are on exactly one line.3. Not all the three points are collinear.4. Two distinct lines are on at least one point.Three point geometry is categorical.Like many finite geometries, the number of provable theorems in three point geometry is small. One can prove from this collection of axioms that two distinct lines are on exactly one point and that three point geometry contains exactly three lines. In this sense, three point geometry is extremely simple. On the other hand, note that the axioms say nothing about whether the lines are straight or curved, whereby it is possible that a number of different (but equivalent) visualizations of three point geometry may exist...
Four line geometry is a finite geometry subjectto the following three axioms: 1. there exist exactly four lines, 2. any two distinct lines have exactly one pointof on both of them, and 3. each point is on exactly two lines.Four line geometry is categorical.Like many finite geometries, the number of provable theorems in three point geometry is small. Of those, one can prove that there exist exactly six points and that each line has exactly three points on it. In that regard, four line geometry is among the simplest finite geometries.Note that by forming the plane dual of the four line geometry axioms (that is, by interchanging the terms "point" and "line" throughout the above discussion), one obtains axioms for a four point geometry. In this new (but equivalent) geometry, the plane duals of the above results still hold...
Five point geometry is a finite geometry subjectto the following three axioms: 1. there exist exactly five points, 2. each two distinct points have exactly one lineon both of them, and 3. each line has exactly two points.Five point geometry is categorical.Like many finite geometries, the number of provable theorems in five point geometry is small. One can show that in this scheme, there are exactly 10 lines and that each point has exactly four lines on it.
Fano's geometry is a finite geometry attributed to Fano from around the year 1892. This geometry comes with five axioms, namely: 1. There exists at least one line. 2. Every line has exactly three pointson it. 3. Not all the points are on the same line.4. For two distinct points, there exists exactly one line on both of them. 5. Each two lines have at least one pointon both of them. Fano's geometry is categorical. Even so, there are several different though equivalent visual representations of Fano's geometry. Perhaps the most common is the so-called Fano plane which shows, among other things, that lines in Fano's geometry need not be straight.Like many finite geometries, the number of provable theorems in Fano's geometry is small. One can show that in Fano's geometry, each two lines have exactly one point in common and that the geometry itself consists of exactly seven points and seven lines...
Let be a collection of subsets of a set , a set function, and the outer measure induced by . The measure that is the restriction of to the sigma-algebra of -measurable sets is called the Carathéodory measure induced by .Perhaps somewhat surprisingly, even though is a measure induced by the set function , it may not be the case that is an extension of . In the event that does extend , is called the Carathéodory extension of .
Given a set , a set function is said to be an outer measure provided that and that is countably monotone, where is the empty set.Given a collection of subsets of and an arbitrary set function , one can define a new set function by setting and defining, for each non-empty subset ,where the infimum is taken over all countable collections of sets in which cover . The resulting function is an outer measure and is called the outer measure induced by .
The term Borel hierarchy is used to describe a collection of subsets of defined inductively as follows: Level one consists of all open and closed subsets of , and upon having defined levels , level is obtained by taking countable unions and intersections of the previous level. In particular, level two of the hierarchy consists of the collections of all Fsigma and Gdelta sets while subsequent levels are described by way of the rather confusingly-named collection of sets of the form , , , , , etc.The collection of sets across all levels of the Borel hierarchy is the Borel sigma-algebra. As such, the Borel hierarchy is fundamental to the study of measure theory.More general notions of the Borel hierarchy (and thus of Borel sets, etc.) are introduced and studied as part of various areas of set theory, topology, and mathematical logic...
The norm topology on a normed space is the topology consisting of all sets which can be written as a (possibly empty) union of sets of the formfor some and for some number . Sets of the form are called the open balls in .
A subset of a topological space is said to be nonmeager if is of second category in , i.e., if cannot be written as the countable union of subsets which are nowhere dense in .
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..
A -algebra of operators on a Hilbert space is said to act nondegenerately if whenever for all , it necessarily implies that . Algebras which act nondegenerately are sometimes said to be nondegenerate.One can show that such an algebra is nondegenerate if and only if the subspaceis dense in .Any -algebra containing the identity operator necessarily acts nondegenerately.
Given a subset of a vector space , a nonempty subset is called an extreme set of if no point of is an internal point of any line interval whose endpoints are in except when both endpoints are in . Said another way, is an extreme set of if whenever andfor , it necessarily follows that .In the event that consists of a single point of , is called an extreme point of . Extreme points play an important role in a number of areas of math, e.g., in the Krein-Milman theorem in functional analysis.
An extreme point of a subset of a vector space is an extreme set of which consists of a single point in . The collection of all extreme points of is sometimes denoted .Extreme points play important roles in a number of areas of mathematics, e.g., in the Krein-Milman theorem which says that, despite their moniker implying a certain degree of rarity, the collection can be quite large relative the host space .
Let be a topological vector space whose continuous dual separates points (i.e., is T2). The weak topology on is defined to be the coarsest/weakest topology (that is, the topology with the fewest open sets) under which each element of remains continuous on . To differentiate the topologies and , is sometimes referred to as the strong topology on .Note that the weak topology is a special case of a more general concept. In particular, given a nonempty family of mappings from a set to a topological space , one can define a topology to be the collection of all unions and finite intersections of sets of the form with and an open set in . The topology -often called the -topology on and/or the weak topology on induced by -is the coarsest topology in which every element is continuous on and so it follows that the above-stated definition corresponds to the special case of for a topological vector space...
In real and functional analysis, equicontinuity is a concept which extends the notion of uniform continuity from a single function to collection of functions. Given topological vector spaces and , a collection of linear transformations from into is said to be equicontinuous if to every neighborhood of in there corresponds a neighborhood of in such that for all . In the special case that is a metric space and , this criterion can be restated as an epsilon-delta definition: A collection of real-valued continuous functions on is equicontinuous if, given , there is a such that whenever satisfy ,for all . It is often convenient to visualize an equicontinuous collection of functions as being "uniformly uniformly continuous," i.e., a collection for which a single can be chosen for any arbitrary so as to make all uniformly continuous simultaneously, independent of .In the latter case, equicontinuity is the ingredient needed to "upgrade"..
Given a Hilbert space , a -subalgebra of is said to be a von Neumann algebra in provided that is equal to its bicommutant (Dixmier 1981). Here, denotes the algebra of bounded operators from to itself.A non-trivial corollary of the so-called bicommutant theorem says that a nondegenerate -subalgebra of is a von Neumann algebra if and only if it is strongly closed. This is further equivalent to a number of other analytic properties of and of (Blackadar 2013), and due to its bijective equivalence is sometimes used as a definition for von Neumann algebras. In some literature, the assumption of being unital (i.e., containing the identity) is added to the hypotheses of this equivalence though, strictly speaking, the result holds in the somewhat more general case that is merely nondegenerate.One can easily show that every von Neumann algebra is a W-*-algebra and contrarily; as a result, some literature defines a von Neumann algebra as a C-*-algebra which..
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..
Given a topological vector space and a neighborhood of in , the polar of is defined to be the setand where denotes the magnitude of the scalar in the underlying scalar field of (i.e., the absolute value of if is a real vector space or its complex modulus if is a complex vector space) and where denotes the continuous dual space of (i.e., is the space of all continuous linear functionals from to the underlying scalar field of ).Worth noting is that the polar is essentially the norm unit ball in and is fundamental in functional analysis, e.g., in the Banach-Alaoglu theorem which says is weak-* compact for all neighborhoods of in .
Let be a locally convex topological vector space and let be a compact subset of . In functional analysis, Milman's theorem is a result which says that if the closed convex hull of is also compact, then contains all the extreme points of .The importance of Milman's theorem is subtle but enormous. One well-known fact from functional analysis is that where denotes the set of extreme points of . Ostensibly, however, one may have that has extreme points which are not in . This behavior is considered a pathology, and Milman's theorem states that this pathology cannot exist whenever is compact (e.g., when is a subset of a Fréchet space ).Milman's theorem should not be confused with the Krein-Milman theorem which says that every nonempty compact convex set in necessarily satisfies the identity ...
A normed vector space is said to be uniformly convex if for sequences , , the assumptions , , and together imply thatas tends to infinity.Such spaces are important in functional analysis. For example, the classical Banach-Saks theorem can be generalized so that the desired conclusion holds in the case that is a Banach space whose conjugate space (that is, the complex conjugate of the dual vector space ) is uniformly convex.
A subset of a topological space is said to be meager if is of first category in , i.e., if can be written as the countable union of subsets which are nowhere dense in . The terms "thin set," "meager set," and "first category" are equivalent.
A map from a metric space to a metric space is said to be uniformly continuous if for every , there exists a such that whenever satisfy .Note that the here depends on and on but that it is entirely independent of the points and . In this way, uniform continuity is stronger than continuity and so it follows immediately that every uniformly continuous function is continuous.Examples of uniformly continuous functions include Lipschitz functions and those satisfying the Hölder condition. Note however that not all continuous functions are uniformly continuous with two very basic counterexamples being (for ) and (for . On the other hand, every function which is continuous on a compact domain is necessarily uniformly continuous.