# Category: Mathematical analysis

Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name
Automorphic number
In mathematics, an automorphic number (sometimes referred to as a circular number) is a natural number in a given number base whose square "ends" in the same digits as the number itself.
Darboux's formula
In mathematical analysis, Darboux's formula is a formula introduced by Gaston Darboux for summing infinite series by using integrals or evaluating integrals using infinite series. It is a generalizati
Unconditional convergence
In mathematics, specifically functional analysis, a series is unconditionally convergent if all reorderings of the series converge to the same value. In contrast, a series is conditionally convergent
Series expansion
In mathematics, a series expansion is an expansion of a function into a series, or infinite sum. It is a method for calculating a function that cannot be expressed by just elementary operators (additi
Real coordinate space
In mathematics, the real coordinate space of dimension n, denoted Rn (/ɑːrˈɛn/ ar-EN) or , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. With componen
Continued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing t
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivati
Solving quadratic equations with continued fractions
In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is where a ≠ 0. The quadratic equation on a number can be solved using the well-known quadratic for
Carleman's condition
In mathematics, particularly, in analysis, Carleman's condition gives a sufficient condition for the determinacy of the moment problem. That is, if a measure satisfies Carleman's condition, there is n
Periodic continued fraction
In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form where the initial block of k + 1 partial denominators is followed by a block [ak+1, ak+2,
Stress majorization
Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of -dimensional data items, a configuration of points in -dimensional space is sought that minim
Lebesgue point
In mathematics, given a locally Lebesgue integrable function on , a point in the domain of is a Lebesgue point if Here, is a ball centered at with radius , and is its Lebesgue measure. The Lebesgue po
Mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions
Reduced derivative
In mathematics, the reduced derivative is a generalization of the notion of derivative that is well-suited to the study of functions of bounded variation. Although functions of bounded variation have
Infinite product
In mathematics, for a sequence of complex numbers a1, a2, a3, ... the infinite product is defined to be the limit of the partial products a1a2...an as n increases without bound. The product is said to
Regularization perspectives on support vector machines
Within mathematical analysis, Regularization perspectives on support-vector machines provide a way of interpreting support-vector machines (SVMs) in the context of other regularization-based machine-l
Arithmetization of analysis
The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century.
Quasisymmetric map
In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multi
Wirtinger derivatives
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of
Weighted geometric mean
In statistics, the weighted geometric mean is a generalization of the geometric mean using the weighted arithmetic mean. Given a sample and weights , it is calculated as: The second form above illustr
Timeline of calculus and mathematical analysis
A timeline of calculus and mathematical analysis.
Partition of an interval
In mathematics, a partition of an interval [a, b] on the real line is a finite sequence x0, x1, x2, …, xn of real numbers such that a = x0 < x1 < x2 < … < xn = b. In other terms, a partition of a comp
Applications of sensitivity analysis in epidemiology
Sensitivity analysis studies the relation between the uncertainty in a model-based the inference and the uncertainties in the model assumptions. Sensitivity analysis can play an important role in epid
Uniform continuity
In mathematics, a real function of real numbers is said to be uniformly continuous if there is a positive real number such that function values over any function domain interval of the size are as clo
Monogenic function
A monogenic function is a complex function with a single finite derivative. More precisely, a function defined on is called monogenic at , if exists and is finite, with: Alternatively, it can be defin
Treatise on Analysis
Treatise on Analysis is a translation by Ian G. Macdonald of the nine-volume work Éléments d'analyse on mathematical analysis by Jean Dieudonné, and is an expansion of his textbook Foundations of Mode
Hyperreal number
In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension o
Negligible set
In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose.As common examples, finite sets can be ignored when studying the limit of a sequence, and null se
Asymptote
In analytic geometry, an asymptote (/ˈæsɪmptoʊt/) of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity.
Ultrapolynomial
In mathematics, an ultrapolynomial is a power series in several variables whose coefficients are bounded in some specific sense.
Eikonal approximation
In theoretical physics, the eikonal approximation (Greek εἰκών for likeness, icon or image) is an approximative method useful in wave scattering equations which occur in optics, seismology, quantum me
Characteristic variety
In mathematical analysis, the characteristic variety of a P is an algebraic variety that is the zero set of the principal symbol of P in the cotangent bundle. It is invariant under a . The notion is a
Hyperbolic growth
When a quantity grows towards a singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth. More precisely, the reciprocal function has a hyperbola as a
Landau kernel
No description available.
Uniform boundedness
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than or equal to the absolute value of an
Ant on a rubber rope
The ant on a rubber rope is a mathematical puzzle with a solution that appears counterintuitive or paradoxical. It is sometimes given as a worm, or inchworm, on a rubber or elastic band, but the princ
Dirichlet kernel
In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of functions defined as where n is any nonnegative integer. The k
Caccioppoli set
In mathematics, a Caccioppoli set is a set whose boundary is measurable and has (at least locally) a finite measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli s
Total derivative
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total de
Metric space
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Met
Order unit
An order unit is an element of an ordered vector space which can be used to bound all elements from above. In this way (as seen in the first below) the order unit generalizes the unit element in the r
Hypercomplex analysis
In mathematics, hypercomplex analysis is the basic extension of real analysis and complex analysis to the study of functions where the argument is a hypercomplex number. The first instance is function
Second derivative
In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of
Bounding point
In functional analysis, a branch of mathematics, a bounding point of a subset of a vector space is a conceptual extension of the boundary of a set.
Chisini mean
In mathematics, a function f of n variables x1, ..., xn leads to a Chisini mean M if for every vector , there exists a unique M such that f(M,M, ..., M) = f(x1,x2, ..., xn). The arithmeti
Function of several real variables
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. Th
Summability kernel
In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are parti
Conical combination
Given a finite number of vectors in a real vector space, a conical combination, conical sum, or weighted sum of these vectors is a vector of the form where are non-negative real numbers. The name deri
Positively invariant set
In mathematical analysis, a positively (or positive) invariant set is a set with the following properties: Suppose is a dynamical system, is a trajectory, and is the initial point. Let where is a real
Quasiregular map
In the mathematical field of analysis, quasiregular maps are a class of continuous maps between Euclidean spaces Rn of the same dimension or, more generally, between Riemannian manifolds of the same d
Tannery's theorem
In mathematical analysis, Tannery's theorem gives sufficient conditions for the interchanging of the limit and infinite summation operations. It is named after Jules Tannery.
Hypostatic abstraction
Hypostatic abstraction in mathematical logic, also known as hypostasis or subjectal abstraction, is a formal operation that transforms a predicate into a relation; for example "Honey is sweet" is tran
Series multisection
In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series then its
Local boundedness
In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that p
Division by zero
In mathematics, division by zero is division where the divisor (denominator) is zero. Such a division can be formally expressed as , where a is the dividend (numerator). In ordinary arithmetic, the ex
Pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Regularization (mathematics)
In mathematics, statistics, finance, computer science, particularly in machine learning and inverse problems, regularization is a process that changes the result answer to be "simpler". It is often us
Epigraph (mathematics)
In mathematics, the epigraph or supergraph of a function valued in the extended real numbers is the set, denoted by of all points in the Cartesian product lying on or above its graph. The strict epigr
In mathematics, a function is superadditive if for all and in the domain of Similarly, a sequence is called superadditive if it satisfies the inequality for all and The term "superadditive" is also ap
Classification of discontinuities
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says
Fractal derivative
In applied mathematics and mathematical analysis, the fractal derivative or Hausdorff derivative is a non-Newtonian generalization of the derivative dealing with the measurement of fractals, defined i
List of continuity-related mathematical topics
In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways.
Fekete problem
In mathematics, the Fekete problem is, given a natural number N and a real s ≥ 0, to find the points x1,...,xN on the 2-sphere for which the s-energy, defined by for s > 0 and by for s = 0, is minimal
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded if it is, in a certain sense, of finite measure. Conversely, a set which is not bounded is called unbounded. The word
Restricted power series
In algebra, the ring of restricted power series is the subring of a formal power series ring that consists of power series whose coefficients approach zero as degree goes to infinity. Over a non-archi
Łojasiewicz inequality
In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Speci
Algebraic interior
In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior.
Exhaustion by compact sets
In mathematics, especially general topology and analysis, an exhaustion by compact sets of a topological space is a nested sequence of compact subsets of (i.e. ), such that is contained in the interio
Overcompleteness
Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics (usually in the form of overcomplete frames). It was introduced by R
Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the mul
Moment problem
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ to the sequences of moments More generally, one may consider for an arbitrary sequence of f
Convex combination
In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are no
Semi-continuity
In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function is upper (respectively, l
Equicontinuity
In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In
Extreme point
In mathematics, an extreme point of a convex set in a real or complex vector space is a point in which does not lie in any open line segment joining two points of In linear programming problems, an ex
Function of a real variable
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers , or a subs
Isoperimetric dimension
In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large-scale behavior of the manifold resembles that of a Euclidean space (unlike the to
In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to th
Barlow's formula
Barlow's formula (called "Kesselformel" in German) relates the internal pressure that a pipe can withstand to its dimensions and the strength of its material.This approximate formula is named after Pe
Logarithmically concave function
In convex analysis, a non-negative function f : Rn → R+ is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality for all x,y ∈ dom f and
P-variation
In mathematical analysis, p-variation is a collection of seminorms on functions from an ordered set to a metric space, indexed by a real number . p-variation is a measure of the regularity or smoothne
Fluxion
A fluxion is the instantaneous rate of change, or gradient, of a fluent (a time-varying quantity, or function) at a given point. Fluxions were introduced by Isaac Newton to describe his form of a time
Applications of sensitivity analysis to model calibration
Sensitivity analysis has important applications in model calibration. One application of sensitivity analysis addresses the question of "What's important to model or system development?" One can seek
Stationary phase approximation
In mathematics, the stationary phase approximation is a basic principle of asymptotic analysis, applying to the limit as . This method originates from the 19th century, and is due to George Gabriel St
Probability bounds analysis
Probability bounds analysis (PBA) is a collection of methods of uncertainty propagation for making qualitative and quantitative calculations in the face of uncertainties of various kinds. It is used t
Hemicontinuity
In mathematics, the notion of the continuity of functions is not immediately extensible to multivalued mappings or correspondences between two sets A and B. The dual concepts of upper hemicontinuity a
Superslow process
Superslow processes are processes in which values change so little that their capture is very difficult because of their smallness in comparison with the measurement error.
Constructive analysis
In mathematics, constructive analysis is mathematical analysis done according to some principles of constructive mathematics.This contrasts with classical analysis, which (in this context) simply mean
Intensity-duration-frequency curve
An intensity-duration-frequency curve (IDF curve) is a mathematical function that relates the rainfall intensity with its duration and frequency of occurrence. These curves are commonly used in hydrol
Hypertranscendental function
A hypertranscendental function or transcendentally transcendental function is a transcendental analytic function which is not the solution of an algebraic differential equation with coefficients in Z
Mean-periodic function
In mathematical analysis, the concept of a mean-periodic function is a generalization of the concept of a periodic function introduced in 1935 by Jean Delsarte. Further results were made by Laurent Sc
Cauchy index
In mathematical analysis, the Cauchy index is an integer associated to a real rational function over an interval. By the Routh–Hurwitz theorem, we have the following interpretation: the Cauchy index o
Finite difference
A finite difference is a mathematical expression of the form f (x + b) − f (x + a). If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by fini
International Workshop on Operator Theory and its Applications
International Workshop on Operator Theory and its Applications (IWOTA) was started in 1981 to bring together mathematicians and engineers working in operator theoretic side of functional analysis and
Oscillatory integral
In mathematical analysis an oscillatory integral is a type of distribution. Oscillatory integrals make rigorous many arguments that, on a naive level, appear to use divergent integrals. It is possible
Harmonic differential
In mathematics, a real differential one-form ω on a surface is called a harmonic differential if ω and its conjugate one-form, written as ω∗, are both closed.
Differential coefficient
In physics, the differential coefficient of a function f(x) is what is now called its derivative df(x)/dx, the (not necessarily constant) multiplicative factor or coefficient of the differential dx in
Engel expansion
The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that For instance, Euler's constant e has the Engel expansion 1, 1, 2, 3, 4, 5, 6, 7, 8,
Calculus on Manifolds (book)
Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) by Michael Spivak is a brief, rigorous, and modern textbook of multivariable calculus, differential forms, an
Polylogarithmic function
In mathematics, a polylogarithmic function in n is a polynomial in the logarithm of n, The notation logkn is often used as a shorthand for (log n)k, analogous to sin2θ for (sin θ)2. In computer scienc
Generality of algebra
In the history of mathematics, the generality of algebra was a phrase used by Augustin-Louis Cauchy to describe a method of argument that was used in the 18th century by mathematicians such as Leonhar
H square
In mathematics and control theory, H2, or H-square is a Hardy space with square norm. It is a subspace of L2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space
Mountain pass theorem
The mountain pass theorem is an existence theorem from the calculus of variations, originally due to Antonio Ambrosetti and Paul Rabinowitz. Given certain conditions on a function, the theorem demonst
Multiple rule-based problems
Multiple rule-based problems are problems containing various conflicting rules and restrictions. Such problems typically have an "optimal" solution, found by striking a balance between the various res
Christ–Kiselev maximal inequality
In mathematics, the Christ–Kiselev maximal inequality is a maximal inequality for filtrations, named for mathematicians Michael Christ and Alexander Kiselev.
Coincidence point
In mathematics, a coincidence point (or simply coincidence) of two functions is a point in their common domain having the same image. Formally, given two functions we say that a point x in X is a coin
Noetherian
In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descendin
Ditkin set
In mathematics, a Ditkin set, introduced by (Ditkin ), is a closed subset of the circle such that a function f vanishing on the set can be approximated by functions φnf with φ vanishing in a neighborh
Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Int
Least-squares spectral analysis
Least-squares spectral analysis (LSSA) is a method of estimating a frequency spectrum, based on a least squares fit of sinusoids to data samples, similar to Fourier analysis. Fourier analysis, the mos
Vanish at infinity
In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functio
A Course of Pure Mathematics
A Course of Pure Mathematics is a classic textbook in introductory mathematical analysis, written by G. H. Hardy. It is recommended for people studying calculus. First published in 1908, it went throu
Division by infinity
In mathematics, division by infinity is division where the divisor (denominator) is infinity. In ordinary arithmetic, this does not have a well-defined meaning, since infinity is a mathematical concep
Pompeiu problem
In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929, as follows. Suppose f is a nonzero continuous function defined on a
Exposed point
In mathematics, an exposed point of a convex set is a point at which some continuous linear functional attains its strict maximum over . Such a functional is then said to expose . There can be many ex
Identity channel
In quantum information theory, the identity channel is a noise-free quantum channel. That is, the channel outputs exactly what was put in. The identity channel is commonly denoted as , or .
Parker–Sochacki method
In mathematics, the Parker–Sochacki method is an algorithm for solving systems of ordinary differential equations (ODEs), developed by and , of the James Madison University Mathematics Department. The
Discrete calculus
Discrete calculus or the calculus of discrete functions, is the mathematical study of incremental change, in the same way that geometry is the study of shape and algebra is the study of generalization
Introductio in analysin infinitorum
Introductio in analysin infinitorum (Latin: Introduction to the Analysis of the Infinite) is a two-volume work by Leonhard Euler which lays the foundations of mathematical analysis. Written in Latin a
Wave front set
In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(f) characterizes the singularities of a generalized function f, not only in space, but also with respect to its
Weighted arithmetic mean
The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some
Damping
Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the ene
Workshop on Numerical Ranges and Numerical Radii
Workshop on Numerical Ranges and Numerical Radii (WONRA) is a biennial workshop series on numerical ranges and numerical radii which began in 1992.
Strip packing problem
The strip packing problem is a 2-dimensional geometric minimization problem. Given a set of axis-aligned rectangles and a strip of bounded width and infinite height, determine an overlapping-free pack
Paratingent cone
In mathematics, the paratingent cone and contingent cone were introduced by Bouligand, and are closely related to tangent cones.
Domain (mathematical analysis)
In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space Rn or the complex
Negligible function
In mathematics, a negligible function is a function such that for every positive integer c there exists an integer Nc such that for all x > Nc, Equivalently, we may also use the following definition.A
Zero to the power of zero
Zero to the power of zero, denoted by 00, is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines 00 = 1.
Dini continuity
In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is Dini continuous.
Elliptic boundary value problem
In mathematics, an elliptic boundary value problem is a special kind of boundary value problem which can be thought of as the stable state of an . For example, the Dirichlet problem for the Laplacian
Nowhere continuous function
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f is a function from real numbers t
Fluent (mathematics)
A fluent is a time-varying quantity or variable. The term was used by Isaac Newton in his early calculus to describe his form of a function. The concept was introduced by Newton in 1665 and detailed i
Cauchy–Schwarz inequality
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is considered one of the most important and widely used inequalities in mathematics. The inequality for sums was publi
Normal convergence
In mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed.
Regularization by spectral filtering
Spectral regularization is any of a class of regularization techniques used in machine learning to control the impact of noise and prevent overfitting. Spectral regularization can be used in a broad r
Macaulay brackets
Macaulay brackets are a notation used to describe the ramp function A popular alternative transcription uses angle brackets, viz. . Another commonly used notation is + or + for the positive part of ,
Sedrakyan's inequality
The following inequality is known as Sedrakyan's inequality, Bergström's inequality, Engel's form or Titu's lemma, respectively, referring to the article About the applications of one useful inequalit
Foias constant
In mathematical analysis, the Foias constant is a real number named after Ciprian Foias. It is defined in the following way: for every real number x1 > 0, there is a sequence defined by the recurrence
Aleksandrov–Rassias problem
The theory of isometries in the framework of Banach spaces has its beginning in a paper by Stanisław Mazur and Stanisław M. Ulam in 1932. They proved the Mazur–Ulam theorem stating that every isometry
Hyers–Ulam–Rassias stability
The stability problem of functional equations originated from a question of Stanisław Ulam, posed in 1940, concerning the stability of group homomorphisms. In the next year, Donald H. Hyers gave a par
Function series
In calculus, a function series is a series, where the summands are not just real or complex numbers but functions.
Vertical tangent
In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not different
Weight function
A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The resu
Regularized canonical correlation analysis
Regularized canonical correlation analysis is a way of using ridge regression to solve the singularity problem in the cross-covariance matrices of canonical correlation analysis. By converting and int
Asymptotic expansion
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite
List of convexity topics
This is a list of convexity topics, by Wikipedia page. * Alpha blending - the process of combining a translucent foreground color with a background color, thereby producing a new blended color. This
Principles of Mathematical Analysis
Principles of Mathematical Analysis, colloquially known as "PMA" or "Baby Rudin," is an undergraduate real analysis textbook written by Walter Rudin. Initially published by McGraw Hill in 1953, it is
Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
Quasi-exact solvability
A linear differential operator L is called quasi-exactly-solvable (QES) if it has a finite-dimensional invariant subspace of functions such that where n is a dimension of . There are two important cas
Maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest valu
Indefinite product
In mathematics, the indefinite product operator is the inverse operator of . It is a discrete version of the geometric integral of geometric calculus, one of the non-Newtonian calculi. Some authors us
Indefinite sum
In discrete calculus the indefinite sum operator (also known as the antidifference operator), denoted by or , is the linear operator, inverse of the forward difference operator . It relates to the for
Infinite expression
In mathematics, an infinite expression is an expression in which some operators take an infinite number of arguments, or in which the nesting of the operators continues to an infinite depth. A generic
Cauchy–Rassias stability
A classical problem of Stanislaw Ulam in the theory of functional equations is the following: When is it true that a function which approximately satisfies a functional equation E must be close to an
Interchange of limiting operations
In mathematics, the study of interchange of limiting operations is one of the major concerns of mathematical analysis, in that two given limiting operations, say L and M, cannot be assumed to give the
Journal of Mathematical Analysis and Applications
The Journal of Mathematical Analysis and Applications is an academic journal in mathematics, specializing in mathematical analysis and related topics in applied mathematics. It was founded in 1960, as
A Course of Modern Analysis
A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions (colloquially known as Whi
Uniform absolute-convergence
In mathematics, uniform absolute-convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is
Metric differential
In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. Wit
Zeta function regularization
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can b
Ehrenpreis's fundamental principle
In mathematical analysis, Ehrenpreis's fundamental principle, introduced by Leon Ehrenpreis, states: Every solution of a system (in general, overdetermined) of homogeneous partial differential equatio
Nonlocal operator
In mathematics, a nonlocal operator is a mapping which maps functions on a topological space to functions, in such a way that the value of the output function at a given point cannot be determined sol
Total variation
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous
Real-valued function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a
Coordinate singularity
A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame that can be removed by choosing a different frame. An example is the apparent (longitudinal
Singularity (mathematics)
In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lack
Stieltjes moment problem
In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (m0, m1, m2, ...) to be of the form for some measure μ. If
Characterizations of the exponential function
In mathematics, the exponential function can be characterized in many ways. The following characterizations (definitions) are most common. This article discusses why each characterization makes sense,
Hypograph (mathematics)
In mathematics, the hypograph or subgraph of a function is the set of points lying on or below its graph. A related definition is that of such a function's epigraph, which is the set of points on or a
Diffeomorphometry
Diffeomorphometry is the metric study of imagery, shape and form in the discipline of computational anatomy (CA) in medical imaging. The study of images in computational anatomy rely on high-dimension