Separation of variables is a method of solving ordinary and partial differential equations.For an ordinary differential equation(1)where is nonzero in a neighborhood of the initial value, the solution is given implicitly by(2)If the integrals can be done in closed form and the resulting equation can be solved for (which are two pretty big "if"s), then a complete solution to the problem has been obtained. The most important equation for which this technique applies is , the equation for exponential growth and decay (Stewart 2001).For a partial differential equation in a function and variables , , ..., separation of variables can be applied by making a substitution of the form(3)breaking the resulting equation into a set of independent ordinary differential equations, solving these for , , ..., and then plugging them back into the original equation.This technique works because if the product of functions of independent variables..
An indirectly conformal mapping, sometimes called an anticonformal mapping, is a mapping that reverses all angles, whereas an isogonal mapping can reverse some angles and preserve others.For example, if is a conformal map, then is an indirectly conformal map, and is an isogonal mapping.
Multivariable calculus is the branch of calculus that studies functions of more than one variable. Partial derivatives and multiple integrals are the generalizations of derivative and integral that are used. An important theorem in multivariable calculus is Green's theorem, which is a generalization of the first fundamental theorem of calculus to two dimensions.
A vector basis of a vector space is defined as a subset of vectors in that are linearly independent and span . Consequently, if is a list of vectors in , then these vectors form a vector basis if and only if every can be uniquely written as(1)where , ..., are elements of the base field.When the base field is the reals so that for , the resulting basis vectors are -tuples of reals that span -dimensional Euclidean space . Other possible base fields include the complexes , as well as various fields of positive characteristic considered in algebra, number theory, and algebraic geometry.A vector space has many different vector bases, but there are always the same number of basis vectors in each of them. The number of basis vectors in is called the dimension of . Every spanning list in a vector space can be reduced to a basis of the vector space.The simplest example of a vector basis is the standard basis in Euclidean space , in which the basis vectors lie along each coordinate..
Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. For example, the real line is homotopic to a single point, as is any tree. However, the circle is not contractible, but is homotopic to a solid torus. The basic version of homotopy is between maps. Two maps and are homotopic if there is a continuous mapsuch that and .Whether or not two subsets are homotopic depends on the ambient space. For example, in the plane, the unit circle is homotopic to a point, but not in the punctured plane . The puncture can be thought of as an obstacle.However, there is a way to compare two spaces via homotopy without ambient spaces. Two spaces and are homotopy equivalent if there are maps and such that the composition is homotopic to the identity map of and is homotopic to the identity map of . For example, the circle is not homotopic to a point, for then the constant map would be homotopic to the identity map of a circle, which is impossible..
A topological space is semilocally simply connected (also called semilocally 1-connected) if every point has a neighborhood such that any loop with basepoint is homotopic to the trivial loop. The prefix semi- refers to the fact that the homotopy which takes to the trivial loop can leave and travel to other parts of .The property of semilocal simple connectedness is important because it is a necessary and sufficient condition for a connected, locally pathwise-connected space to have a universal cover.
Complex analysis is the study of complex numbers together with their derivatives, manipulation, and other properties. Complex analysis is an extremely powerful tool with an unexpectedly large number of practical applications to the solution of physical problems. Contour integration, for example, provides a method of computing difficult integrals by investigating the singularities of the function in regions of the complex plane near and between the limits of integration.The key result in complex analysis is the Cauchy integral theorem, which is the reason that single-variable complex analysis has so many nice results. A single example of the unexpected power of complex analysis is Picard's great theorem, which states that an analytic function assumes every complex number, with possibly one exception, infinitely often in any neighborhood of an essential singularity!A fundamental result of complex analysis is the Cauchy-Riemann..
An Abelian category is a category for which the constructions and techniques of homological algebra are available. The basic examples of such categories are the category of Abelian groups and, more generally, the category of modules over a ring. Abelian categories are widely used in algebra, algebraic geometry, and topology.Many of the same constructions that are found in categories of modules, such as kernels, exact sequences, and commutative diagrams are available in Abelian categories. A disadvantage that must be overcome is the fact that the objects in a category do not necessarily have elements that can be manipulated directly, so the traditional definitions do not work. As a result, methods must be developed that allow definition and manipulation of objects without the use of elements.As an example, consider the definition of the kernel of a morphism, which states that given , the kernel of is defined to be a morphism such that all morphisms..
A nonnegative measurable function is called Lebesgue integrable if its Lebesgue integral is finite. An arbitrary measurable function is integrable if and are each Lebesgue integrable, where and denote the positive and negative parts of , respectively.The following equivalent characterization of Lebesgue integrable follows as a consequence of monotone convergence theorem. A nonnegative measurable function is Lebesgue integrable iff there exists a sequence of nonnegative simple functions such that the following two conditions are satisfied: 1. . 2. almost everywhere.
A compact manifold is a manifold that is compact as a topological space. Examples are the circle (the only one-dimensional compact manifold) and the -dimensional sphere and torus. Compact manifolds in two dimensions are completely classified by their orientation and the number of holes (genus). It should be noted that the term "compact manifold" often implies "manifold without boundary," which is the sense in which it is used here. When there is need for a separate term, a compact boundaryless manifold is called a closed manifold.For many problems in topology and geometry, it is convenient to study compact manifolds because of their "nice" behavior. Among the properties making compact manifolds "nice" are the fact that they can be covered by finitely many coordinate charts, and that any continuous real-valued function is bounded on a compact manifold.For any positive integer , a distinct nonorientable..
Let , then the negative part of is the function defined byNote that the negative part is itself a nonnegative function. The negative part satisfies the identitywhere is the positive part of .
Dickson states "In a letter to Tanner [L'intermediaire des math., 2, 1895, 317] Lucas stated that Mersenne (1644, 1647) implied that a necessary and sufficient condition that be a prime is that be a prime of one of the forms , , ."Mersenne's implication has been refuted, but Bateman, Selfridge, and Wagstaff (1989) used the statement as an inspiration for what is now called the new Mersenne conjecture, which can be stated as follows.Consider an odd natural number . If two of the following conditions hold, then so does the third: 1. or , 2. is prime (a Mersenne prime), 3. is prime (a Wagstaff prime). This conjecture has been verified for all primes .Based on the distribution and heuristics of (cf. https://www.utm.edu/research/primes/mersenne/heuristic.html) the known Mersenne and Wagstaff prime exponents, it seems quite likely that there is only a finite number of exponents satisfying the criteria of the new Mersenne conjecture. In..
Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties.Mathematicians sometimes use the term "combinatorics" to refer to a larger subset of discrete mathematics that includes graph theory. In that case, what is commonly called combinatorics is then referred to as "enumeration."The Season 1 episode "Noisy Edge" (2005) of the television crime drama NUMB3RS mentions combinatorics.
If a complex function is analytic at all finite points of the complex plane , then it is said to be entire, sometimes also called "integral" (Knopp 1996, p. 112).Any polynomial is entire.Examples of specific entire functions are given in the following table.functionsymbolAiry functions, Airy function derivatives, Anger functionBarnes G-functionbeiberBessel function of the first kindBessel function of the second kindBeurling's functioncosinecoversineDawson's integralerferfcerfiexponential functionFresnel integrals, gamma function reciprocalgeneralized hypergeometric functionhaversinehyperbolic cosinehyperbolic sineJacobi elliptic functions, , , , , , , , , , , Jacobi theta functionsJacobi theta function derivativesMittag-Leffler functionmodified Struve functionNeville theta functions, , , Shisinesine integralspherical Bessel function of the first kindStruve functionversineWeber..
A group is nilpotent if the upper central sequenceof the group terminates with for some .Nilpotent groups have the property that each proper subgroup is properly contained in its normalizer. A finite nilpotent group is the direct product of its Sylow p-subgroups.
A Wagstaff prime is a prime number of the form for a prime number. The first few are given by , 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, and 4031399 (OEIS A000978), with and larger corresponding to probable primes. These values correspond to the primes with indices , 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, ... (OEIS A123176).The Wagstaff primes are featured in the newMersenne prime conjecture.There is no simple primality test analogous to the Lucas-Lehmer test for Wagstaff primes, so all recent primality proofs of Wagstaff primes have used elliptic curve primality proving.A Wagstaff prime can also be interpreted as a repunit prime of base , asif is odd, as it must be for the above number to be prime.Some of the largest known Wagstaff probable primes are summarized in the following..
A Fibonacci prime is a Fibonacci number that is also a prime number. Every that is prime must have a prime index , with the exception of . However, the converse is not true (i.e., not every prime index gives a prime ).The first few (possibly probable) prime Fibonacci numbers are 2, 3, 5, 13, 89, 233, 1597, 28657, 514229, ... (OEIS A005478), corresponding to indices , 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, ... (OEIS A001605). (Note that Gardner's statement that is prime (Gardner 1979, p. 161) is incorrect, especially since 531 is not even prime, which it must be for to be prime.) The following table summarizes Fibonacci (possibly probable) primes with index .termindexdigitsdiscovererstatus2453871126proven prime; https://primes.utm.edu/primes/page.php?id=511292593111946proven prime; https://primes.utm.edu/primes/page.php?id=374702696772023proven prime; https://primes.utm.edu/primes/page.php?id=3553727144313016proven..
A spectral sequence is a tool of homological algebra that has many applications in algebra, algebraic geometry, and algebraic topology. Roughly speaking, a spectral sequence is a system for keeping track of collections of exact sequences that have maps between them.There are many definitions of spectral sequences and many slight variations that are useful for certain purposes. The most common type is a "first quadrant cohomological spectral sequence," which is a collection of Abelian groups where , , and are integers, with and nonnegative and for some positive integer , usually 2. The groups come equipped with maps(1)such that(2)There is the further restriction that(3)The maps are called boundary maps.A spectral sequence may be visualized as a sequence of grids, one for each value of . The s and s denote positions on the grid, where is the -coordinate and is the -coordinate. The diagram above shows this for .The entire collection..
A pointed space is a topological space together with a choice of a basepoint . The notation for a pointed space is . Maps between two pointed spaces must take basepoints to basepoints. Pointed spaces are widely used in algebraic topology, homotopy theory, and topological K-theory.
A subspace of is called a deformation retract of if there is a homotopy (called a retract) such that for all and , 1. , 2. , and 3. . A tightening of the last condition gives a so-called strongdeformation retract (Bredon 1993, pp. 45-46).Note that a deformation retract is also a retract, because the homotopy defines a continuous map
A subspace of is called a strong deformation retract of if there is a homotopy (called a retract) such that for all , , and , 1. , 2. , and 3. . If the last equation is required only for , the retract is called simply a deformation retract.
There are several different kinds of magic numbers. The digital root and magic constant are sometimes known as magic numbers.In baseball, the magic number for a team in first place in a division is the number of games that team must win or the second place team must lose in order to clinch the division. The formula isFor example, the standings for the National League Central Division as of August 9, 2004 are summarized in the following table.teamwinslossesSt. Louis7238Chicago6150Houston5556Cincinnati5457Milwaukee5258Pittsburgh5158There were 162 games in the season so, for example, St. Louis's magic number on that date was
Just as many interesting integer sequences can be defined and their properties studied, it is often of interest to additionally determine which of their elements are prime. The following table summarizes the indices of the largest known prime (or probable prime) members of a number of named sequences.sequenceOEISdigitsdiscoverersearch limitcommentsalternating factorialA00127259961260448M. Rodenkirch (Sep. 18, 2017)100000 (M. Rodenkirch, Dec. 15, 2017)finite sequence; largest certified prime has index 661; the rest are probable primesApéry-constant primeA119334141141E. W. Weisstein (May 14, 2006)9089 (E. W. Weisstein, Mar. 22, 2008)status unknownApéry number A092825662410136E. W. Weisstein (Mar. 2004) (E. W. Weisstein, Mar. 2004)probable primeApéry number 87E. W. Weisstein..
Given a ring with identity, the special linear group is the group of matrices with elements in and determinant 1.The special linear group , where is a prime power, the set of matrices with determinant and entries in the finite field . is the corresponding set of complex matrices having determinant . is a subgroup of the general linear group and is a Lie-type group. Both and are genuine Lie groups.
A cyclic group is a group that can be generated by a single element (the group generator). Cyclic groups are Abelian.A cyclic group of finite group order is denoted , , , or ; Shanks 1993, p. 75), and its generator satisfies(1)where is the identity element.The ring of integers form an infinite cyclic group under addition, and the integers 0, 1, 2, ..., () form a cyclic group of order under addition (mod ). In both cases, 0 is the identity element.There exists a unique cyclic group of every order , so cyclic groups of the same order are always isomorphic (Scott 1987, p. 34; Shanks 1993, p. 74). Furthermore, subgroups of cyclic groups are cyclic, and all groups of prime group order are cyclic. In fact, the only simple Abelian groups are the cyclic groups of order or a prime (Scott 1987, p. 35).The th cyclic group is represented in the Wolfram Language as CyclicGroup[n].Examples of cyclic groups include , , , ..., and the modulo multiplication..
If is a map (a.k.a. function, transformation, etc.) over a domain , then the range of , also called the image of under , is defined as the set of all values that can take as its argument varies over , i.e.,Note that among mathematicians, the word "image"is used more commonly than "range."The range is a subset of and does not have to be all of .Unfortunately, term "range" is often used to mean domain--its precise opposite--in probability theory, with Feller (1968, p. 200) and Evans et al. (2000, p. 5) calling the set of values that a variate can assume (i.e., the set of values that a probability density function is defined over) the "range", denoted by (Evans et al. 2000, p. 5).Even worse, statistics most commonly uses "range" to refer to the completely different statistical quantity as the difference between the largest and smallest order statistics. In this work, this form..
A principal ideal domain is an integral domain in which every proper ideal can be generated by a single element. The term "principal ideal domain" is often abbreviated P.I.D. Examples of P.I.D.s include the integers, the Gaussian integers, and the set of polynomials in one variable with real coefficients.Every Euclidean ring is a principal ideal domain, but the converse is not true. Nevertheless, the notion of greatest common divisor arising from the Euclidean algorithm can be extended to the more general context of principal ideal domains as follows. Given two nonzero elements of a principal ideal domain , a greatest common divisor of and is defined as any element of such thatEvery principal ideal domain is a unique factorization domain, but not conversely. Every polynomial ring over a field is a unique factorization domain, but it is a principal ideal domain iff the number of indeterminates is one...
A conformal mapping, also called a conformal map, conformal transformation, angle-preserving transformation, or biholomorphic map, is a transformation that preserves local angles. An analytic function is conformal at any point where it has a nonzero derivative. Conversely, any conformal mapping of a complex variable which has continuous partial derivatives is analytic. Conformal mapping is extremely important in complex analysis, as well as in many areas of physics and engineering.A mapping that preserves the magnitude of angles, but not their orientation is called an isogonal mapping (Churchill and Brown 1990, p. 241).Several conformal transformations of regular grids are illustrated in the first figure above. In the second figure above, contours of constant are shown together with their corresponding contours after the transformation. Moon and Spencer (1988) and Krantz (1999, pp. 183-194) give tables of conformal..
An isogonal mapping is a transformation that preserves the magnitudes of local angles, but not their orientation. A few examples are illustrated above.A conformal mapping is an isogonal mapping that also preserves the orientations of local angles. If is a conformal mapping, then is isogonal but not conformal. This is due to the fact that complex conjugation is not an analytic function.
The word "base" in mathematics is used to refer to a particular mathematical object that is used as a building block. The most common uses are the related concepts of the number system whose digits are used to represent numbers and the number system in which logarithms are defined. It can also be used to refer to the bottom edge or surface of a geometric figure.A real number can be represented using any integer number as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base , the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ... are generally used as symbols representing the decimal numbers 10, 11, 12, ...).The digits of a number in base (for integer ) can be obtained in the Wolfram Language using IntegerDigits[x, b].Let the base representation of a number be written(1)(e.g., ). Then, for example, the number 10 is..
Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The term "discrete mathematics" is therefore used in contrast with "continuous mathematics," which is the branch of mathematics dealing with objects that can vary smoothly (and which includes, for example, calculus). Whereas discrete objects can often be characterized by integers, continuous objects require real numbers.The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatorics. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation. Topics in number theory such as congruences and recurrence relations are also considered part of discrete mathematics.The study of topics in discrete mathematics usually includes the study of algorithms, their..
The term analysis is used in two ways in mathematics. It describes both the discipline of which calculus is a part and one form of abstract logic theory.Analysis is the systematic study of real and complex-valued continuous functions. Important subfields of analysis include calculus, differential equations, and functional analysis. The term is generally reserved for advanced topics which are not encountered in an introductory calculus sequence, although many ideas from those courses, such as derivatives, integrals, and series are studied in more detail. Real analysis and complex analysis are two broad subdivisions of analysis which deal with real-values and complex-valued functions, respectively.Derbyshire (2004, p. 16) describes analysis as "the study of limits."Logicians often call second-order arithmetic "analysis." Unfortunately, this term conflicts with the more usual definition of analysis..
Let , then the positive part of is the function defined byThe positive part satisfies the identitywhere is the negative part of .
If is a map (a.k.a. function, transformation, etc.) over a domain , then the image of , also called the range of under , is defined as the set of all values that can take as its argument varies over , i.e.,"Image" is a synonym for "range," but"image" is the term preferred in formal mathematical writing.The notation denotes the image of the interval under the function . Formally,
The word "algebra" is a distortion of the Arabic title of a treatise by al-Khwārizmī about algebraic methods. In modern usage, algebra has several meanings.One use of the word "algebra" is the abstract study of number systems and operations within them, including such advanced topics as groups, rings, invariant theory, and cohomology. This is the meaning mathematicians associate with the word "algebra." When there is the possibility of confusion, this field of mathematics is often referred to as abstract algebra.The word "algebra" can also refer to the "school algebra" generally taught in American middle and high schools. This includes the solution of polynomial equations in one or more variables and basic properties of functions and graphs. Mathematicians call this subject "elementary algebra," "high school algebra," "junior high..
Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic structures rather than the usual number systems. The most important of these structures are groups, rings, and fields. Important branches of abstract algebra are commutative algebra, representation theory, and homological algebra.Linear algebra, elementary number theory, and discrete mathematics are sometimes considered branches of abstract algebra. Ash (1998) includes the following areas in his definition of abstract algebra: logic and foundations, counting, elementary number theory, informal set theory, linear algebra, and the theory of linear operators.
If is continuous on a closed interval , and is any number between and inclusive, then there is at least one number in the closed interval such that .The theorem is proven by observing that is connected because the image of a connected set under a continuous function is connected, where denotes the image of the interval under the function . Since is between and , it must be in this connected set.The intermediate value theorem (or rather, the space case with , corresponding to Bolzano's theorem) was first proved by Bolzano (1817). While Bolzano's used techniques which were considered especially rigorous for his time, they are regarded as nonrigorous in modern times (Grabiner 1983).
If a function is continuous on a closed interval , then has both a maximum and a minimum on . If has an extremum on an open interval , then the extremum occurs at a critical point. This theorem is sometimes also called the Weierstrass extreme value theorem.The standard proof of the first proceeds by noting that is the continuous image of a compact set on the interval , so it must itself be compact. Since is compact, it follows that the image must also be compact.
Let denote an -algebra, so that is a vector space over and(1)(2)Now define(3)where . An Associative -algebra is commutative if for all . Similarly, a ring is commutative if the multiplication operation is commutative, and a Lie algebra is commutative if the commutator is 0 for every and in the Lie algebra.The term "commutative algebra" also refers to the branch of abstract algebra that studies commutative rings. Commutative algebra is important in algebraic geometry.
The upper central series of a group is the sequence of groups (each term normal in the term following it)that is constructed in the following way: 1. is the center of . 2. For , is the unique subgroup of such that is the center of . If the upper central series of a group terminates with for some , then is called a nilpotent group.
Conjugation is the process of taking a complex conjugate of a complex number, complex matrix, etc., or of performing a conjugation move on a knot.Conjugation also has a meaning in group theory. Let be a group and let . Then, defines a homomorphism given byThis is a homomorphism becauseThe operation on given by is called conjugation by .Conjugation is an important construction in group theory. Conjugation defines a group action of a group on itself and this often yields useful information about the group. For example, this technique is how the Sylow Theorems are proven. More importantly, a normal subgroup of a group is a subgroup which is invariant under conjugation by any element. Normal groups are extremely important because they are the kernels of homomorphisms and it is possible to take the quotient of a group and one of its normal subgroups...
A linear algebraic group is a matrix group that is also an affine variety. In particular, its elements satisfy polynomial equations. The group operations are required to be given by regular rational functions. The linear algebraic groups are similar to the Lie groups, except that linear algebraic groups may be defined over any field, including those of positive field characteristic.The special linear group of matrices of determinant one is a linear algebraic group. This is because the equation for the determinant is a polynomial equation in the entries of the matrices. The general linear group of matrices with nonzero determinant is also a linear algebraic group. This can be seen by introducing an extra variable and writingThis is a polynomial equation in variables and is equivalent to saying that is nonzero. This equation describes as an affine variety...
A topological space, also called an abstract topological space, is a set together with a collection of open subsets that satisfies the four conditions: 1. The empty set is in . 2. is in . 3. The intersection of a finite number of sets in is also in . 4. The union of an arbitrary number of sets in is also in . Alternatively, may be defined to be the closed sets rather than the open sets, in which case conditions 3 and 4 become: 3. The intersection of an arbitrary number of sets in is also in . 4. The union of a finite number of sets in is also in . These axioms are designed so that the traditional definitions of open and closed intervals of the real line continue to be true. For example, the restriction in (3) can be seen to be necessary by considering , where an infinite intersection of open intervals is a closed set.In the chapter "Point Sets in General Spaces" Hausdorff (1914) defined his concept of a topological space based on the four Hausdorff axioms (which..
Let be the set of continuous mappings . Then the topological space supplied with the compact-open topology is called a mapping space. If is a pointed space, then the mapping space of pointed maps is called the path space of . In words, is the space of all paths which begin at . is a contractible space with the contraction given by .
A metric space is boundedly compact if all closed bounded subsets of are compact. Every boundedly compact metric space is complete. (This is a generalization of the Bolzano-Weierstrass theorem.)Every complete Riemannian manifold is boundedly compact. This is part of or a consequence of the Hopf-Rinow theorem.
Let be the set of continuous mappings . Then the topological space supplied with the compact-open topology is called a mapping space, and if is taken as the circle , then is called the "free loop space of " (or the space of closed paths).If is a pointed space, then a basepoint can be picked on the circle and the mapping space of pointed maps can be formed. This space is denoted and is called the "loop space of ."
An outlier is an observation that lies outside the overall pattern of a distribution (Moore and McCabe 1999). Usually, the presence of an outlier indicates some sort of problem. This can be a case which does not fit the model under study, or an error in measurement.Outliers are often easy to spot in histograms. Forexample, the point on the far left in the above figure is an outlier.A convenient definition of an outlier is a point which falls more than 1.5 times the interquartile range above the third quartile or below the first quartile.Outliers can also occur when comparing relationships between two sets of data. Outliers of this type can be easily identified on a scatter diagram.When performing least squares fitting to data, it is often best to discard outliers before computing the line of best fit. This is particularly true of outliers along the direction, since these points may greatly influence the result...
The trivial loop is the loop that takes every point to its basepoint. Formally, if is a topological space and , the trivial loop based at is the map given by for all .
An inner product is a generalization of the dot product. In a vector space, it is a way to multiply vectors together, with the result of this multiplication being a scalar.More precisely, for a real vector space, an inner product satisfies the following four properties. Let , , and be vectors and be a scalar, then: 1. . 2. . 3. . 4. and equal if and only if . The fourth condition in the list above is known as the positive-definite condition. Related thereto, note that some authors define an inner product to be a function satisfying only the first three of the above conditions with the added (weaker) condition of being (weakly) non-degenerate (i.e., if for all , then ). In such literature, functions satisfying all four such conditions are typically referred to as positive-definite inner products (Ratcliffe 2006), though inner products which fail to be positive-definite are sometimes called indefinite to avoid confusion. This difference, though subtle,..
A Hilbert space is a vector space with an inner product such that the norm defined byturns into a complete metric space. If the metric defined by the norm is not complete, then is instead known as an inner product space.Examples of finite-dimensional Hilbert spaces include1. The real numbers with the vector dot product of and . 2. The complex numbers with the vector dot product of and the complex conjugate of . An example of an infinite-dimensional Hilbert space is , the set of all functions such that the integral of over the whole real line is finite. In this case, the inner product isA Hilbert space is always a Banach space, but theconverse need not hold.A (small) joke told in the hallways of MIT ran, "Do you know Hilbert? No? Then what are you doing in his space?" (S. A. Vaughn, pers. comm., Jul. 31, 2005)...
A scatter diagram, also called a scatterplot or a scatter plot, is a visualization of the relationship between two variables measured on the same set of individuals. Scatter diagrams for lists of data , , ... can be generated with the Wolfram Language using ListPlot[x1, y1, x2, y2, ...].A scatter diagram makes it particularly easy to spot trends and correlations between the two variables. For example, the scatter diagram illustrated above plots wine consumption (in liters of alcohol from wine per person per year) against deaths from heart disease (in deaths per 100,000 people) for 19 developed nations (Moore and McCabe 1999, Ex. 2.5)There is clearly and inverse relationship between these two variables. Once such a relationship has been found, linear regression can be used to find curves of best fit. The graph above shows the same scatter diagram as above together with a line of best fit...
Given an ordinary differential equation , the slope field for that differential equation is the vector field that takes a point to a unit vector with slope . The vectors in a slope field are usually drawn without arrowheads, indicating that they can be followed in either direction. Using a visualization of a slope field, it is easy to graphically trace out solution curves to initial value problems. For example, the illustration above shows the slope field for the equation together with solution curves for various initial values of .
The term "product" refers to the result of one or more multiplications. For example, the mathematical statement would be read " times equals ," where is the product.More generally, it is possible to take the product of many different kinds of mathematical objects, including those that are not numbers. For example, the product of two sets is given by the Cartesian product. In topology, the product of spaces can be defined by using the product topology. The product of two groups, vector spaces, or modules is given by the direct product. In category theory, the product of objects is given using the category product.The product symbol is defined by(1)Useful product identities include(2)(3)