mathematical analysis - Article Index

Encyclopedia - Cauchy Sequence: Encyclopedia - Cauchy Sequence
In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progr...   » Read the article

Encyclopedia - Noam Chomsky: Encyclopedia - Noam Chomsky
Avram Noam Chomsky (born December 7, 1928) is the Institute Professor Emeritus of linguistics at the Massachusetts Institute of Technolog...   » Read the article

Encyclopedia - William Thomson 1st Baron Kelvin: Encyclopedia - William Thomson 1st Baron Kelvin
The Right Honourable William Thomson, 1st Baron Kelvin, GCVO, OM, PC, PRS (26 June 1824–17 December 1907) was a Scottish-Irish mathemat...   » Read the article

Encyclopedia - Nicolas Bourbaki: Encyclopedia - Nicolas Bourbaki
Nicolas Bourbaki is the collective allonym under which a group of mainly French 20th-century mathematicians wrote a series of books prese...   » Read the article

Encyclopedia - Complete Space: Encyclopedia - Complete Space
In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is...   » Read the article

Encyclopedia - Almost Everywhere: Encyclopedia - Almost Everywhere
In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which ...   » Read the article

Encyclopedia - 2nd Millennium: Encyclopedia - 2nd Millennium
(1st millennium – 2nd millennium – 3rd millennium – other millennia) 2nd millennium - Events. European crusades in Middle East Mo...   » Read the article

Encyclopedia - Mathematics: Encyclopedia - Mathematics
Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematic...   » Read the article

Encyclopedia - Adrien-marie Legendre: Encyclopedia - Adrien-marie Legendre
Adrien-Marie Legendre (September 18, 1752 - January 10, 1833) was a French mathematician. He made important contributions to statistics, ...   » Read the article

Encyclopedia - Algebra: Encyclopedia - Algebra
Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstract...   » Read the article

Encyclopedia - Vanish At Infinity: Encyclopedia - Vanish At Infinity
In mathematics, a function on a normed vector space is said to vanish at infinity if as For example, the function defined on the rea...   » Read the article

Encyclopedia - Analysis: Encyclopedia - Analysis
An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent part...   » Read the article

Encyclopedia - Almost Periodic Function: Encyclopedia - Almost Periodic Function
In mathematics, almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exa...   » Read the article

Encyclopedia - Divergent Series: Encyclopedia - Divergent Series
In mathematics, a divergent series is an infinite series that does not converge. That is, divergent series and convergent series are anto...   » Read the article

Encyclopedia - Feedback: Encyclopedia - Feedback
In cybernetics and control theory, feedback is a process whereby some proportion or in general, function, of the output signal of a syste...   » Read the article

Encyclopedia - Henri Lebesgue: Encyclopedia Ii - Henri Lebesgue - Lebesgue's Theory Of Integration
This is a non-technical treatment from a historical point of view; see the article Lebesgue integration for a technical treatment from a ...   » Read the article

Encyclopedia - Heine–borel Theorem: Encyclopedia Ii - Heine–borel Theorem - Discussion Of The Theorem
If a set is not closed, then it cannot be compact. If a set is not closed, then it is either an open set, or it is partially open: part o...   » Read the article

Encyclopedia - Extended Real Number Line: Encyclopedia Ii - Extended Real Number Line - Motivation
Extended real number line - Limits. We often wish to describe the behavior of a function f(x), as either the argument x or the function...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - History
The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matt...   » Read the article

Encyclopedia - Logarithm: Encyclopedia Ii - Logarithm - History
Jaina mathematicians in ancient India first conceived of logarithms between 200 BC and 400 CE. They performed a number of operations usin...   » Read the article

Encyclopedia - History Of Astronomy: Encyclopedia Ii - History Of Astronomy - Ancient History
Early cultures identifed celestial objects with gods and spirits. They related these objects (and their movements) to phenomena such as r...   » Read the article

Encyclopedia - Limit Of A Function: Encyclopedia Ii - Limit Of A Function - Formal Definition
Limit of a function - Functions on metric spaces. Suppose f : (M,dM) -> (N,dN) is a map between two metric spaces, p is a limit...   » Read the article

Encyclopedia - Measure Mathematics: Encyclopedia Ii - Measure Mathematics - Formal Definitions
Formally, a countably additive measure μ is a function defined on a σ-algebra Σ over a set X with values in the extended interval [0, ...   » Read the article

Encyclopedia - Landau Notation: Encyclopedia Ii - Landau Notation - Properties
Basic properties concerning this notation are the following: , i.e. if g = o(f), then g = O(f). Transitivity of O and o: O(O(f)) = O(f...   » Read the article

Encyclopedia - Harmonic Function: Encyclopedia Ii - Harmonic Function - Examples
Examples of harmonic functions of two variables are: the real and imaginary part of any holomorphic function the function f(x1, x2) ...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Major Themes In Mathematics
An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possib...   » Read the article

Encyclopedia - Gauss's Law: Encyclopedia Ii - Gauss's Law - Coulomb's Law
In the special case of a spherical surface with a central charge, the electric field is perpendicular to the surface, with the same magni...   » Read the article

Encyclopedia - Field Mathematics: Encyclopedia Ii - Field Mathematics - Definition
A field is a commutative ring (F, +, *) such that 0 does not equal 1 and all elements of F except 0 have a multiplicative inverse. Spelle...   » Read the article

Encyclopedia - Fractal: Encyclopedia Ii - Fractal - History
Fractal - Contributions from classical analysis. Objects that are now called fractals were discovered and explored long before the word...   » Read the article

Encyclopedia - Factorial: Encyclopedia Ii - Factorial - Factorial-like Products
Factorial - Primorial. The primorial is similar to the factorial, but with the product taken only over the prime numbers. Factorial - ...   » Read the article

Encyclopedia - Gauss's Law: Encyclopedia Ii - Gauss's Law - Gravitational Analogue
Since both gravity and electromagnetism propagate relative to the squared distance between two objects, we can relate the two using Gauss...   » Read the article

Encyclopedia - Normal Space: Encyclopedia Ii - Normal Space - Definitions
Suppose that X is a topological space. X is a normal space if, given any disjoint closed sets E and F, there are a neighbourhood U of E a...   » Read the article

Encyclopedia - Double Integral: Encyclopedia Ii - Double Integral - In The Positive Sense
One can give a further explanation, however from the other direction, based on the special role of functions f(x)g(y). These, in which th...   » Read the article

Encyclopedia - G. H. Hardy: Encyclopedia Ii - G. H. Hardy - Life
After his schooling at Winchester, Hardy entered Trinity College, Cambridge in 1896 after standing fourth in the Tripos examination. Year...   » Read the article

Encyclopedia - History Of Mathematics: Encyclopedia Ii - History Of Mathematics - Mathematics In Prehistory
Long before the earliest written records, there are drawings that indicate a knowledge of mathematics and of measurement of time based on...   » Read the article

Encyclopedia - Parameter: Encyclopedia Ii - Parameter - Types Of Parameter
Parameter - Mathematical. In mathematics, the difference in meaning between a parameter and an argument of a function is that the param...   » Read the article

Encyclopedia - Euler's Identity: Encyclopedia Ii - Euler's Identity - Perceptions Of The Identity
Many people find this identity remarkable for its mathematical beauty. The identity links what are arguably the most fundamental mathemat...   » Read the article

Encyclopedia - Equicontinuity: Encyclopedia Ii - Equicontinuity - Properties
As promised in the introduction, the limit of a pointwise convergent, equicontinuous sequence is continuous. Theorem 1: Let {fn} be ...   » Read the article

Encyclopedia - Greek Mathematics: Encyclopedia Ii - Greek Mathematics - Origins
Greek mathematics has origins that are presumed to go back to the 7th century BC, but are not easily documented. It is generally believed...   » Read the article

Encyclopedia - Jovan Karamata: Encyclopedia Ii - Jovan Karamata - Life
Jovan Karamata was born in Zagreb on February 1, 1902 to a Serbian-Aromanian father and a Serbian mother. His family descends from a merc...   » Read the article

Encyclopedia - Functional Equation: Encyclopedia Ii - Functional Equation - Examples
One thing that all of the examples listed above share in common is that in each case two or more known functions (sometimes multiplicatio...   » Read the article

Encyclopedia - Limit Mathematics: Encyclopedia Ii - Limit Mathematics - Limit Of A Function
Main article: limit of a function Suppose f(x) is a real function and c is a real number. The expression: means that f(x) can be made...   » Read the article

Encyclopedia - Lie Group: Encyclopedia Ii - Lie Group - The Lie Algebra Associated To A Lie Group
To every Lie group, we can associate a Lie algebra which completely captures the local structure of the group, at least if the Lie group ...   » Read the article

Encyclopedia - Fubini's Theorem: Encyclopedia Ii - Fubini's Theorem - Tonelli's Theorem
Tonelli's theorem (named after Leonida Tonelli) is a predecessor of Fubini's theorem. The conclusion of Tonelli's theorem is identical to...   » Read the article

Encyclopedia - Lebesgue Integration: Encyclopedia Ii - Lebesgue Integration - Construction Of The Lebesgue Integral
The discussion that follows parallels the most common expository approach to the Lebesgue integral. In this approach the theory of integr...   » Read the article

Encyclopedia - History Of Mathematics: Encyclopedia Ii - History Of Mathematics - Mathematics In Prehistory
Long before the earliest written records, there are drawings that indicate a knowledge of mathematics and of measurement of time based on...   » Read the article

Encyclopedia - Hardy Space: Encyclopedia Ii - Hardy Space - Factorization
For , every function can be written as the product f = Gh where G is an outer function and h is an inner function, as defined below. One...   » Read the article

Encyclopedia - Haar Measure: Encyclopedia Ii - Haar Measure - Preliminaries
Let G be a locally compact topological group. In this article, the σ-algebra generated by all compact subsets of G is called the Borel a...   » Read the article

Encyclopedia - Generalized Fourier Series: Encyclopedia Ii - Generalized Fourier Series - Example Fourier-legendre Series
The Legendre polynomials are solutions to the Sturm-Liouville problem and because of the theory, these polynomials are eigenfunctions ...   » Read the article

Encyclopedia - G. H. Hardy: Encyclopedia Ii - G. H. Hardy - Life
After his schooling at Winchester, Hardy entered Trinity College, Cambridge in 1896 after standing fourth in the Tripos examination. Year...   » Read the article

Encyclopedia - List Of Publications In Mathematics: Encyclopedia Ii - List Of Publications In Mathematics - Geometry
List of publications in mathematics - Euclid's Elements. Euclid Publication data: c. 300 BC Online version: Interactive Java version ...   » Read the article

Encyclopedia - Linear Algebra: Encyclopedia Ii - Linear Algebra - Elementary Introduction
Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space. A vector, here, is a directed line segment, c...   » Read the article

Encyclopedia - Logarithm: Encyclopedia Ii - Logarithm - History
Joost Bürgi, a Swiss clockmaker in the employ of the Duke of Hesse-Kassel, first conceived of logarithms. The method of natural logarith...   » Read the article

Encyclopedia - Kerala School: Encyclopedia Ii - Kerala School - Keralese Mathematicians
Kerala School - Narayana Pandit c. 1340-1400. Narayana Pandit, the earliest of the notable Keralese mathematicians, is known to have de...   » Read the article

Encyclopedia - Indeterminate Form: Encyclopedia Ii - Indeterminate Form - Discussion
To say that "0/0" is an indeterminate form does not just mean that "0/0" by itself can represent any number, or may represent no number. ...   » Read the article

Encyclopedia - Integration By Parts: Encyclopedia Ii - Integration By Parts - Examples
In order to calculate: Let: u = x, so that du = dx, dv = cos(x) dx, so that v = sin(x). Then: where C is an arbitr...   » Read the article

Encyclopedia - Formal Power Series: Encyclopedia Ii - Formal Power Series - Formal Development
Formal power series - Two definitions of the formal power series ring. We start with a commutative ring R. We want to define the ring o...   » Read the article

Encyclopedia - Jet Mathematics: Encyclopedia Ii - Jet Mathematics - Jets Of Functions Between Two Manifolds
If M and N are two smooth manifolds, how do we define the jet of a function ? We could perhaps attempt to define such a jet by using loca...   » Read the article

Encyclopedia - Feedback: Encyclopedia Ii - Feedback - Feedback In Nature
In biological systems such as organisms, ecosystems, or the biosphere, most parameters must stay under control within a narrow range arou...   » Read the article

Encyclopedia - Madhava Of Sangamagrama: Encyclopedia Ii - Madhava Of Sangamagrama - Contributions
He discovered the infinite series for arctan and sin and many methods for calculating the circumference of the circle. One of Madhava's s...   » Read the article

Encyclopedia - Indian Mathematics: Encyclopedia Ii - Indian Mathematics - Ancient Period 500 Bc - 400 Ce
Sathanang Sutra, Bhagvati Sutra and Anoyogdwar Sutra are famous books of this time. Apart from these the book titled Tatvarthaadigyam Sut...   » Read the article

Encyclopedia - Fractional Calculus: Encyclopedia Ii - Fractional Calculus - Heuristics
A fairly natural question to ask is, does there exist an operator H, or half-derivative, such that ? It turns out that there is suc...   » Read the article

Encyclopedia - William Thomson 1st Baron Kelvin: Encyclopedia Ii - William Thomson 1st Baron Kelvin - Early Life And Work
William Thomson 1st Baron Kelvin - Family. William's father was Dr. James Thomson, the son of a Belfast farmer. James received little y...   » Read the article

Encyclopedia - Series Mathematics: Encyclopedia Ii - Series Mathematics - History Of The Theory Of Infinite Series
Series mathematics - Convergence criteria. The investigation of the validity of infinite series is considered to begin with Gauss. Eule...   » Read the article

Encyclopedia - Improper Integral: Encyclopedia Ii - Improper Integral - Infinite Bounds Of Integration
The most basic of improper integrals are integrals such as: As stated above, this need not be defined as an improper integral, since i...   » Read the article

Encyclopedia - Extended Real Number Line: Encyclopedia Ii - Extended Real Number Line - Motivation
Extended real number line - Limits. We often wish to describe the behavior of a function f(x), as either the argument x or the function...   » Read the article

Encyclopedia - Fractal: Encyclopedia Ii - Fractal - History
Fractal - Contributions from classical analysis. Objects that are now called fractals were discovered and explored long before the word...   » Read the article

Encyclopedia - Boulevard Saint-michel: Encyclopedia Ii - Boulevard Saint-michel - Literature
1. Mentioned in Of Human Bondage, Chapter 44 by W. Somerset Maugham, 1915. 2. Extract from Noctambule, Ballads of a Bohemian by Robert S...   » Read the article

Encyclopedia - Bounded Set: Encyclopedia Ii - Bounded Set - Definition
A set S of real numbers is called bounded above if there is a real number k such that k ≥ s for all s in S. The number k is called an u...   » Read the article

Encyclopedia - Support Mathematics: Encyclopedia Ii - Support Mathematics - Compact Support
Functions with compact support in X are those with support that is a compact subset of X. For example, if X is the real line, they are ex...   » Read the article

Encyclopedia - Uniform Convergence: Encyclopedia Ii - Uniform Convergence - Theorems
If S is a real interval (or indeed any topological space), we can talk about the continuity of the functions fn and f. The following is t...   » Read the article

Encyclopedia - Triangle Inequality: Encyclopedia Ii - Triangle Inequality - Normed Vector Space
In a normed vector space V, the triangle inequality is ||x + y|| ≤ ||x|| + ||y||     for all x, y in V that is, the no...   » Read the article

Encyclopedia - Squeeze Theorem: Encyclopedia Ii - Squeeze Theorem - Examples And Applications
The following examples illustrate how the squeeze theorem is applied in practice. Squeeze theorem - Example 1. Consider f(x) = x2 sin(1...   » Read the article

Encyclopedia - Asymptotic Notation: Encyclopedia Ii - Asymptotic Notation - Definitions
The O-notation means that the function is bounded from above (also known as dominated) by a function which varies like f, i.e., Thus, ...   » Read the article

Encyclopedia - Semi-continuity: Encyclopedia Ii - Semi-continuity - Formal Definition
Suppose X is a topological space, x0 is a point in X and f : X → R is a real-valued function. We say that f is upper sem...   » Read the article

Encyclopedia - Sigma-algebra: Encyclopedia Ii - Sigma-algebra - Examples
If X is any set, then the family consisting only of the empty set and X is a σ-algebra over X, the so-called trivial σ-algebra. Another...   » Read the article

Encyclopedia - Noam Chomsky: Encyclopedia Ii - Noam Chomsky - Political Views
Related article: Criticism of Noam Chomsky. Chomsky is one of the best known figures of radical American politics. He defines himself as ...   » Read the article

Encyclopedia - Unifying Theories In Mathematics: Encyclopedia Ii - Unifying Theories In Mathematics - Mathematical Theories
The term theory is used informally within mathematics to mean a self-consistent body of definitions, axioms, theorems, examples, and so o...   » Read the article

Encyclopedia - Nicolas Bourbaki: Encyclopedia Ii - Nicolas Bourbaki - Books By Bourbaki
Aiming at a completely self-contained treatment of most of modern mathematics based on set theory, the group produced the following volum...   » Read the article

Encyclopedia - William Feller: Encyclopedia Ii - William Feller - Work
Feller held a docent position at the University of Kiel beginning in 1928. He fled the Nazis and went to Denmark, (Copenhagen) in 1933. H...   » Read the article

Encyclopedia - Measure Mathematics: Encyclopedia Ii - Measure Mathematics - Generalizations
For certain purposes, it is useful to have a "measure" whose values are not restricted to the non-negative reals or infinity. For instanc...   » Read the article

Encyclopedia - Complete Space: Encyclopedia Ii - Complete Space - Examples
The space Q of rational numbers, with the standard metric given by the absolute value, is not complete. Consider for instance the sequenc...   » Read the article

Encyclopedia - Cauchy Sequence: Encyclopedia Ii - Cauchy Sequence - Cauchy Sequence In A Metric Space
Cauchy sequence - Formal definition. Formally, a Cauchy sequence is a sequence in a metric space (M, d) such that for every positive...   » Read the article

Encyclopedia - Courant Institute Of Mathematical Sciences: Encyclopedia Ii - Courant Institute Of Mathematical Sciences - Academics
The institute is highly regarded throughout the world as a leader in applied mathematics, mathematical analysis, and scientific computati...   » Read the article

Encyclopedia - Dirichlet Kernel: Encyclopedia Ii - Dirichlet Kernel - Relation To The Delta Function
Take the periodic Dirac delta function, which is not really a function, in the sense of mapping one set into another, but is rather a "ge...   » Read the article

Encyclopedia - Tychonoff Space: Encyclopedia Ii - Tychonoff Space - Definitions
Suppose that X is a topological space. X is a completely regular space iff, given any closed set F and any point x that does not belong t...   » Read the article

Encyclopedia - Limit Of A Sequence: Encyclopedia Ii - Limit Of A Sequence - Formal Definition
Suppose x1, x2, ... is a sequence of elements in a topological space T. We say that L∈T is a limit of this sequence and write if and...   » Read the article

Encyclopedia - Distribution Mathematics: Encyclopedia Ii - Distribution Mathematics - Formal Definition
In the sequel, real-valued distributions on an open subset U of Rn will be formally defined. (With minor modifications, one can also defi...   » Read the article

Encyclopedia - Euler's Identity: Encyclopedia Ii - Euler's Identity - Perceptions Of The Identity
Benjamin Peirce, the noted nineteenth century mathematician and Harvard professor, after proving the identity in a lecture, said, "Gentle...   » Read the article

Encyclopedia - Uniform Continuity: Encyclopedia Ii - Uniform Continuity - Properties
Every uniformly continuous function is continuous, but the converse is not true. Consider for instance the function f(x) = 1/x with domai...   » Read the article

Encyclopedia - Exponentiation: Encyclopedia Ii - Exponentiation - Exponentiation In Abstract Algebra
Exponentiation can also be understood purely in terms of abstract algebra, if we limit the exponents to integers. Specifically, suppose t...   » Read the article

Encyclopedia - Topology: Encyclopedia Ii - Topology - History
The root of topology was in the study of geometry in ancient cultures. Leibniz was the first to employ the term analysus situs, later emp...   » Read the article

Encyclopedia - Exponentiation: Encyclopedia Ii - Exponentiation - Advanced Topics
Exponentiation - Computation. an can be computed by means of n − 1 multiplications, but, if n is a large number, the work r...   » Read the article

Encyclopedia - Sigma-algebra: Encyclopedia Ii - Sigma-algebra - Examples
If X is any set, then the family consisting only of the empty set and X is a σ-algebra over X, the so-called trivial σ-algebra. Another...   » Read the article

Encyclopedia - Regular Space: Encyclopedia Ii - Regular Space - Relationships To Other Separation Axioms
A regular space is necessarily also preregular. Since a Hausdorff space is the same as a preregular T0 space, a regular space that is als...   » Read the article

Encyclopedia - Algebra: Encyclopedia Ii - Algebra - Classification
Algebra may be roughly divided into the following categories: elementary algebra, in which the properties of operations on the real numb...   » Read the article

Encyclopedia - Pure Mathematics: Encyclopedia Ii - Pure Mathematics - The Twentieth Century
At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example. The lo...   » Read the article

Encyclopedia - Sas System: Encyclopedia Ii - Sas System - Description Of Sas
Like other database-oriented fourth-generation programming languages such as SQL or Focus, SAS assumes a default file structure, and auto...   » Read the article

Encyclopedia - Even And Odd Functions: Encyclopedia Ii - Even And Odd Functions - Some Facts
Even and odd functions - Basic properties. The only function which is both even and odd is the constant function which is identically ...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Inspiration, Pure And Applied Mathematics, And Aesthetics
Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in...   » Read the article

Encyclopedia - Mathematics: Encyclopedia Ii - Mathematics - Notation, Language, And Rigor
Most of the mathematical notation we use today was not invented until the 16th Century. Before that, mathematics was written out in words...   » Read the article