mathematical analysis

Early cultures identifed celestial objects with gods and spirits. They related these objects (and their movements) to phenomena such as rain, drought, seasons, and tides. It is generally believed that the first "professional" astronomers were priests (Magi), and that their understanding of the "heavens" was seen as "divine", hence astronomy's ancient connection to what is now called astrology. Ancient constructions with astronomical alineations (such as Stonehenge) proba ...

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History of astronomy, History of astronomy - Ancient history, History of astronomy - India, History of astronomy - Mesopotamia, History of astronomy - Sumer, History of astronomy - Chaldea Babylonia, History of astronomy - Mesoamerica, History of astronomy - Maya civilization, History of astronomy - East Asia, History of astronomy - China, History of astronomy - Ancient Greece, History of astronomy - Middle ages, History of astronomy - The Copernican revolution, History of astronomy - Physics marries astronomy, History of astronomy - Modern astronomy, History of astronomy - Cosmology and the expansion of the universe, History of astronomy - New windows into the Cosmos open

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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 point of M and L∈N. We say that "the limit of f at p is L" and write if and only if for every ε > 0 there exists a δ > 0 such that for all x∈M with 0 < dM(x, p) < δ, we have dN(f(x), L) < ε. See also:

Limit of a function, Limit of a function - History, Limit of a function - Formal definition, Limit of a function - Functions on metric spaces, Limit of a function - Real-valued functions, Limit of a function - Complex-valued functions, Limit of a function - Limit of a function of more than one variable, Limit of a function - Examples, Limit of a function - Real-valued functions, Limit of a function - Functions on metric spaces, Limit of a function - Properties

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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, ∞] such that the following properties are satisfied: The empty set has measure zero: Countable additivity or σ-additivity: if E1, E2, E3, ... is a countable sequence of pairwise disjoint set ...

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Measure mathematics, Measure mathematics - Formal definitions, Measure mathematics - Sigma-finite measures, Measure mathematics - Completeness, Measure mathematics - Examples, Measure mathematics - Counterexamples, Measure mathematics - Generalizations

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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), i.e. if g = O(f) and h = O(g), then h = O(f)See also:

Landau notation, Landau notation - Introduction, Landau notation - Definition, Landau notation - Generalizations, Landau notation - Remarks on notation, Landau notation - Properties, Landau notation - Applications

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Examples of harmonic functions of two variables are: the real and imaginary part of any holomorphic function the function f(x1, x2) = ln(x12 + x22) defined on R2 \ {0} (e.g. the electric potential due to a line charge, and the gravity potential due to a long cylindrical mass) the function f(x1, x2) = exp(x ...

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Harmonic function, Harmonic function - Examples, Harmonic function - Remarks, Harmonic function - Connections with complex function theory, Harmonic function - Properties of harmonic functions, Harmonic function - The maximum principle, Harmonic function - The mean value property, Harmonic function - Liouville's theorem, Harmonic function - General theory

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An alphabetical and subclassified list of mathematics articles is available. The following list of themes and links gives just one possible view. For a fuller treatment, see areas of mathematics or the list of mathematics lists. Mathematics - Quantity. This starts from explicit measurements of sizes of numbers or sets, or ways to find such measurements. See also:

Mathematics, Mathematics - History, Mathematics - Inspiration pure and applied mathematics and aesthetics, Mathematics - Notation language and rigor, Mathematics - Is mathematics a science?, Mathematics - Overview of fields of mathematics, Mathematics - Major themes in mathematics, Mathematics - Quantity, Mathematics - Structure, Mathematics - Space, Mathematics - Change, Mathematics - Foundations and methods, Mathematics - Discrete mathematics, Mathematics - Applied mathematics, Mathematics - Important theorems, Mathematics - Important conjectures, Mathematics - History and the world of mathematicians, Mathematics - Mathematics and other fields, Mathematics - Common misconceptions

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In the special case of a spherical surface with a central charge, the electric field is perpendicular to the surface, with the same magnitude at all points of it, giving the simpler expression: where E is the electric field strength at radius r, Q is the enclosed charge, and ε0 is the permittivity of free space. Thus the familiar inverse-square law dependence of the electric fi ...

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Gauss's law, Gauss's law - Integral Form, Gauss's law - Differential Form, Gauss's law - Coulomb's Law

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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. Spelled out, this means that the following hold: Closure of F under + and *  For all a, b belonging to F, both a + b and a * b belong to F (or more formally, + and * are binary operations on F). Both + and * are associative  For all a, b, c ...

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Field mathematics, Field mathematics - Introduction, Field mathematics - Definition, Field mathematics - Examples of fields, Field mathematics - Some first theorems

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Fractal - Contributions from classical analysis. Objects that are now called fractals were discovered and explored long before the word was coined. As Mandelbrot himself pointed out the idea of "recursive self similarity" was originally developed by the philosopher Leibniz and he even worked many details. In 1872, Karl Weierstrass found an example of a function with the non-intuitive property that it is everywhere continuous but nowhere differentiable — the graph of this function would now be called a fractal. ...

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Fractal, Fractal - History, Fractal - Contributions from classical analysis, Fractal - Aspects of set description, Fractal - Mandelbrot's contributions, Fractal - Definitions, Fractal - Categories of fractals, Fractal - Examples, Fractal - Fractals in nature, Fractal - Applications, Fractal - Fractal generation

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Factorial - Primorial. The primorial is similar to the factorial, but with the product taken only over the prime numbers. Factorial - Multifactorials. A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (n!!), three (n!!!), or more. n!! denotes the double factorial of nSee also:

Factorial, Factorial - Definition, Factorial - Non-integer factorials, Factorial - Applications, Factorial - Rate of growth, Factorial - Computation, Factorial - Factorial-like products, Factorial - Primorial, Factorial - Multifactorials, Factorial - Hyperfactorials, Factorial - Superfactorials, Factorial - Superfactorials alternative definition

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Since both gravity and electromagnetism propagate relative to the squared distance between two objects, we can relate the two using Gauss' Law by examining their respective fields, G(r) and E(r). In the same way that we evaluate the line integral for electromagnetism to get the result , we can choose a proper Gaussian Surface to quickly get an answer for gravitational flux. For a point mass, the most logical choice for our Gaussian Surface is a sphere of radius r centered at the point. ...

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Gauss's law, Gauss's law - Integral Form, Gauss's law - Differential Form, Gauss's law - Coulomb's Law, Gauss's law - Gravitational Analogue

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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 and a neighbourhood V of F that are also disjoint. In fancier terms, this condition says that E and F can be separated by neighbourhoods. X is a T4 s ...

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Normal space, Normal space - Definitions, Normal space - Examples of normal spaces, Normal space - Examples of non-normal spaces, Normal space - Properties, Normal space - Relationships to other separation axioms

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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 the roles of the two variables are uncoupled, present no problem in this context; and neither do their linear combinations. Quite generally, given compact spaces X and Y, we can use the Stone-Weierstrass theorem to show that such functions give a subalgebra of C(X×Y) that is dense in the uniform norm: or in other words any continuous function on X×Y can be uniformly approximated b ...

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Double integral, Double integral - Definitions, Double integral - Counterexample, Double integral - Explanation via Lebesgue theory, Double integral - In the positive sense

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After his schooling at Winchester, Hardy entered Trinity College, Cambridge in 1896 after standing fourth in the Tripos examination. Years later, Hardy sought to abolish the Tripos system as he felt that it was becoming more an end in itself than being a means to an end. While at university, Hardy joined the Cambridge Apostles, an elite, intellectual secret society. Hardy was Sadleirian Professor at Cambridge from 1931 to 1942; he had left Cambridge to take the Savilian Chair of Geometry at Oxford in the aftermath of the Bertran ...

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G. H. Hardy, G. H. Hardy - Life, G. H. Hardy - Work, G. H. Hardy - Attitudes, G. H. Hardy - Books

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Long before the earliest written records, there are drawings that indicate a knowledge of mathematics and of measurement of time based on the stars. For example, paleontologists have discovered ochre rocks in a cave in South Africa adorned with scratched geometric patterns dating back more than 70,000 years [1]. Also prehistoric artifacts discovered in Africa and France, dated between 35000 BC and 20000 BC, indicate early attempts to quantify time Evidence exists that early counting involved women who kept records of their monthly biological ...

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History of mathematics, History of mathematics - Mathematics in prehistory, History of mathematics - Early written mathematics 2000 BC - 600 BC, History of mathematics - India 800 BC - 300 BC, History of mathematics - Greece and Hellenistic mathematics 400 BC - 200 BC, History of mathematics - Arab and Persian mathematics 650 - 1200, History of mathematics - Developing the concept of number through equations, History of mathematics - Complex numbers, History of mathematics - Miscellaneous historical notes, History of mathematics - Notes

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Parameter - Mathematical. In mathematics, the difference in meaning between a parameter and an argument of a function is that the parameters are the symbols that are part of the function's definition, while arguments are the symbols that are supplied to the function when it is used. The value or objects assigned to the parameters by the corresponding arguments of a function or system are not reassigned during the function's evaluation. So, parameters are effectively constants during th ...

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Parameter, Parameter - Types of parameter, Parameter - Mathematical, Parameter - Computer science, Parameter - Logic, Parameter - Engineering, Parameter - Analytic geometry, Parameter - Mathematical analysis, Parameter - Probability theory, Parameter - Statistics

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Many people find this identity remarkable for its mathematical beauty. The identity links what are arguably the most fundamental mathematical constants: The number 0. The number 1. The number π is a fundamental constant of trigonometry, Euclidean geometry, and mathematical analysis. The number e is fundamental in the study of logarithms and occurs widely in mathematical analysis. The imaginary unit i (where i2 = −1) is a unit in the complex numbers. (Introducing this unit yields all non-constant polynomial equations soluble in the field ...

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Euler's identity, Euler's identity - Derivation, Euler's identity - Perceptions of the identity, Euler's identity - Notes

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As promised in the introduction, the limit of a pointwise convergent, equicontinuous sequence is continuous. Theorem 1: Let {fn} be an equicontinuous sequence of functions. If fn(x) → f(x) for every x ∈ X, then the function f is continuous. The condition in the above theorem can be slightly weakened. It suffices if the se ...

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Equicontinuity, Equicontinuity - Definitions, Equicontinuity - Properties, Equicontinuity - Generalizations

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Greek mathematics has origins that are presumed to go back to the 7th century BC, but are not easily documented. It is generally believed that it built on the computational methods of earlier Babylonian and Egyptian mathematics, and it may well have had Phoenician influences. Some of the most well-known figures in Greek mathematics are Pythagoras, a shadowy figure from the isle of Samos associated partly with number mysticism and numerology, but more commonly with his theorem, and Euclid, who is known for his Elements, a canon of geom ...

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Greek mathematics, Greek mathematics - Origins, Greek mathematics - Famous Greek mathematicians

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Jovan Karamata was born in Zagreb on February 1, 1902 to a Serbian-Aromanian father and a Serbian mother. His family descends from a merchant family from the city of Zemun in Serbia. His family's business affairs on the borders of the Austro-Hungarian and Ottoman empires were very well known. In 1914, he finished most of his primary school in Zemun but because of constant warfare on the borderlands, Karamata's father sent him, together with his brothers and his sister, to Switzerland for their own safety. In Lausanne, 1920, he finished prima ...

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Jovan Karamata, Jovan Karamata - Life, Jovan Karamata - Legacy, Jovan Karamata - Resources, Jovan Karamata - External links, Jovan Karamata - Further reading

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One thing that all of the examples listed above share in common is that in each case two or more known functions (sometimes multiplication by a constant, sometimes addition of two variables, sometimes the identity function) are substituted into the unknown function to be solved for. When it comes to asking for all solutions, it may be the case that conditions from mathematical analysis should be applied; for example, in the case of the Cauchy equation mentioned above, the solutions that are continuous functions are the ' ...

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Functional equation, Functional equation - Examples, Functional equation - Solving functional equations

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