The space of integrable functions on a locally compact abelian group G is an algebra, where multiplication is convolution: if f, g are integrable functions then the convolution of f and g is defined as Theorem The Banach space L1(G) is an associative and commutative algebra under convolution. This algebra is referred to as the Group Algebra of G. By completeness of L1(G), it is a Banach algebra. The Ban ...

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Pontryagin duality, Pontryagin duality - Haar measure, Pontryagin duality - The dual group, Pontryagin duality - Fourier transform, Pontryagin duality - Examples, Pontryagin duality - The group algebra, Pontryagin duality - Plancherel and Fourier inversion theorems, Pontryagin duality - Bohr compactification and almost-periodicity, Pontryagin duality - Categorical considerations, Pontryagin duality - Non-commutative theory, Pontryagin duality - History

Read more here: » Pontryagin duality: Encyclopedia II - Pontryagin duality - The group algebra

One important application of Pontryagin duality is the following characterization of compact abelian topological groups: Theorem. A locally compact abelian group G is compact iff the dual group G^ is discrete. Conversely, G is discrete iff G^ is compact. The Bohr compactification is defined for any topological group G, regardless of whether G is locally compact or abelian. One use made of Pontryagin duality between compact abelian groups and discrete ...

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Pontryagin duality, Pontryagin duality - Haar measure, Pontryagin duality - The dual group, Pontryagin duality - Fourier transform, Pontryagin duality - Examples, Pontryagin duality - The group algebra, Pontryagin duality - Plancherel and Fourier inversion theorems, Pontryagin duality - Bohr compactification and almost-periodicity, Pontryagin duality - Categorical considerations, Pontryagin duality - Non-commutative theory, Pontryagin duality - History

Read more here: » Pontryagin duality: Encyclopedia II - Pontryagin duality - Bohr compactification and almost-periodicity

It is useful to regard the dual group functorially. In what follows, LCA is the category of locally compact abelian groups and continuous group homomorphisms. The dual group construction of G^ is a contravariant functor LCA → LCA. In particular, the iterated functor G → (G^)^ is covariant. Theorem. The dual group is a category isomorphism from LCA to LCAop. Theorem. The iterated dual funct ...

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Pontryagin duality, Pontryagin duality - Haar measure, Pontryagin duality - The dual group, Pontryagin duality - Fourier transform, Pontryagin duality - Examples, Pontryagin duality - The group algebra, Pontryagin duality - Plancherel and Fourier inversion theorems, Pontryagin duality - Bohr compactification and almost-periodicity, Pontryagin duality - Categorical considerations, Pontryagin duality - Non-commutative theory, Pontryagin duality - History

Read more here: » Pontryagin duality: Encyclopedia II - Pontryagin duality - Categorical considerations

The differential equation is said to be in Sturm-Liouville (or self-adjoint) form. All second-order linear ordinary differential equations can be recast in the form to the left of "=" above by multiplying both sides of the equation by an appropriate integrating factor (although the same is not true of second-order partial differential equations, or if y is a vector.) Sturm-Liouville theory - Exam ...

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Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville theory, Sturm-Liouville theory - Sturm-Liouville form, Sturm-Liouville theory - Examples, Sturm-Liouville theory - Sturm-Liouville differential operators, Sturm-Liouville theory - Some highly technical details, Sturm-Liouville theory - Useful consequences of the preceding technicalities, Sturm-Liouville theory - Example, Sturm-Liouville theory - Application to normal modes

Read more here: » Sturm-Liouville theory: Encyclopedia II - Sturm-Liouville theory - Sturm-Liouville form

The Riemann zeta function ζ(s) is defined for any complex number s with real part > 1 by the Dirichlet series: In the region {s in C: Re(s) > 1}, this infinite series converges and defines a function analytic in this region. Bernhard Riemann realized that the zeta function can be extended by analytic continuation in a unique way to a meromorphic function ζ(s) defined for all complex numbers s with s ≠ 1. It is this fu ...

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Riemann zeta function, Riemann zeta function - Definition, Riemann zeta function - Values at the integers, Riemann zeta function - Relationship to prime numbers, Riemann zeta function - Proving the Euler product formula, Riemann zeta function - An easier proof for the layperson, Riemann zeta function - The importance of the zeros of ζs, Riemann zeta function - Basic properties, Riemann zeta function - The Riemann zeta function as a Mellin transform, Riemann zeta function - Series expansions, Riemann zeta function - Globally convergent series, Riemann zeta function - Universality, Riemann zeta function - Applications, Riemann zeta function - Generalizations, Riemann zeta function - Zeta functions in fiction

Read more here: » Riemann zeta function: Encyclopedia II - Riemann zeta function - Definition

Given a finite group G, define the group algebra CG as the vector space over the complex numbers, with basis vectors {eg} corresponding to the elements . The algebra structure on this vector space is defined as . A representation of the algebra CG on a vector space V is the algebra homomorphism . That is, a representation is a left CG-module. Any group representation then extends linearly to an algeb ...

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Group algebra, Group algebra - Group algebra of a finite group, Group algebra - Group algebras of topological groups: CcG, Group algebra - The convolution algebra L1G, Group algebra - The group C*-algebra C*G, Group algebra - The maximal group C * -algebra, Group algebra - The reduced group C*-algebra C*rG, Group algebra - von Neumann algebras associated to groups

Read more here: » Group algebra: Encyclopedia II - Group algebra - Group algebra of a finite group

When one says "the Laplace transform" without qualification, the unilateral or one-sided transform is normally intended. The Laplace transform can be alternatively defined as the bilateral Laplace transform or two-sided Laplace transform by extending the limits of integration to be the entire real axis. If that is done the common unilateral transform simply becomes a special case of the bilateral transform where the definition of the function being transformed is multiplied by the Heaviside step function. The bilate ...

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Laplace transform, Laplace transform - Formal definition, Laplace transform - Region of convergence, Laplace transform - Inverse Laplace transform, Laplace transform - Bilateral Laplace transform, Laplace transform - Laplace transform of a function's derivative, Laplace transform - Applications, Laplace transform - Example #1: Solving a differential equation, Laplace transform - Example #2: Deriving the complex impedance for a capacitor, Laplace transform - Example #3: Finding the transfer function from the impulse response, Laplace transform - Relationship to other transforms, Laplace transform - Fourier transform, Laplace transform - Mellin transform, Laplace transform - Z-transform, Laplace transform - Fundamental relationships, Laplace transform - Properties and theorems, Laplace transform - Table of selected Laplace transforms, Laplace transform - Examples: How to apply the properties and theorems, Laplace transform - Example #1: Method of partial fraction expansion, Laplace transform - Example #2: Mixing sines cosines and exponentials, Laplace transform - Example #3, Laplace transform - Example #4: Phase delay

Read more here: » Laplace transform: Encyclopedia II - Laplace transform - Bilateral Laplace transform

It is often convenient to use the differentiation property of the Laplace transform to find the transform of a function's derivative. For the unilateral case, this approach becomes: And in the bilateral case, we have ...

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Laplace transform, Laplace transform - Formal definition, Laplace transform - Region of convergence, Laplace transform - Inverse Laplace transform, Laplace transform - Bilateral Laplace transform, Laplace transform - Laplace transform of a function's derivative, Laplace transform - Applications, Laplace transform - Example #1: Solving a differential equation, Laplace transform - Example #2: Deriving the complex impedance for a capacitor, Laplace transform - Example #3: Finding the transfer function from the impulse response, Laplace transform - Relationship to other transforms, Laplace transform - Fourier transform, Laplace transform - Mellin transform, Laplace transform - Z-transform, Laplace transform - Fundamental relationships, Laplace transform - Properties and theorems, Laplace transform - Table of selected Laplace transforms, Laplace transform - Examples: How to apply the properties and theorems, Laplace transform - Example #1: Method of partial fraction expansion, Laplace transform - Example #2: Mixing sines cosines and exponentials, Laplace transform - Example #3, Laplace transform - Example #4: Phase delay

Read more here: » Laplace transform: Encyclopedia II - Laplace transform - Laplace transform of a function's derivative

IFRS are used in many countries in the world, including Hong Kong and Russia, certain European countries, and recently Australia. In Africa, South Africa has adopted the IFRS standards. For a current overview see IAS PLUS's list of all countries that have adopted IFRS. International Financial Reporting Standards - Australia. The Australian Accounting standards, previous to 1 January 2005, were based around accounting standards developed by the Australian Accounting Standards Board (AASB). As ...

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International Financial Reporting Standards, International Financial Reporting Standards - Adaptation and convergence, International Financial Reporting Standards - Australia, International Financial Reporting Standards - European Union, International Financial Reporting Standards - Russia, International Financial Reporting Standards - Details

Read more here: » International Financial Reporting Standards: Encyclopedia II - International Financial Reporting Standards - Adaptation and convergence

Greek mathematicians such as Eudoxus and Archimedes made informal use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In the 12th century the Indian mathematician Bhaskara gave an example of what would now be called a "differential coefficient" and the basic idea behind what is now known as Rolle's theorem. The 14th century Indian mathematician Madhava of Sangamagrama expressed various trigonometric functions as infinite series, and estimated the magnitude of th ...

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Mathematical analysis, Mathematical analysis - History, Mathematical analysis - Subdivisions

Read more here: » Mathematical analysis: Encyclopedia II - Mathematical analysis - History

Every inner product <.,.> on a real or complex vector space H gives rise to a norm ||.|| as follows: We call H a Hilbert space if it is complete with respect to this norm. Completeness in this context means that every Cauchy sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero. Every Hilbert space is th ...

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Hilbert space, Hilbert space - Introduction, Hilbert space - Definition, Hilbert space - Examples, Hilbert space - Euclidean spaces, Hilbert space - Sequence spaces, Hilbert space - Lebesgue spaces, Hilbert space - Sobolev spaces, Hilbert space - Operations on Hilbert spaces, Hilbert space - Bases, Hilbert space - Orthogonal complements and projections, Hilbert space - Reflexivity, Hilbert space - Bounded operators, Hilbert space - Unbounded operators

Read more here: » Hilbert space: Encyclopedia II - Hilbert space - Definition

A topological group is locally compact if and only if the identity e of the group has a compact neighborhood. This means that there is some open set V containing e whose closure is relatively compact in the topology of G. One of the most remarkable facts about a locally compact group G is that it carries an essentially unique natural measure, the Haar measure, which allows one to consistently measure the "size" of sufficiently regular subsets of G. "Sufficiently regular subset" here means a Bo ...

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Pontryagin duality, Pontryagin duality - Haar measure, Pontryagin duality - The dual group, Pontryagin duality - Fourier transform, Pontryagin duality - Examples, Pontryagin duality - The group algebra, Pontryagin duality - Plancherel and Fourier inversion theorems, Pontryagin duality - Bohr compactification and almost-periodicity, Pontryagin duality - Categorical considerations, Pontryagin duality - Non-commutative theory, Pontryagin duality - History

Read more here: » Pontryagin duality: Encyclopedia II - Pontryagin duality - Haar measure

As we have stated, the dual group of a locally compact abelian group is a locally compact abelian group in its own right and thus has a Haar measure, or more precisely a whole family of scale-related Haar measures. Theorem. There is a scaling of Haar measure on the dual group so that the Fourier transform restricted to continuous functions of compact support on G, is an isometric linear map. It has a unique extension to a unitary operator ...

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Pontryagin duality, Pontryagin duality - Haar measure, Pontryagin duality - The dual group, Pontryagin duality - Fourier transform, Pontryagin duality - Examples, Pontryagin duality - The group algebra, Pontryagin duality - Plancherel and Fourier inversion theorems, Pontryagin duality - Bohr compactification and almost-periodicity, Pontryagin duality - Categorical considerations, Pontryagin duality - Non-commutative theory, Pontryagin duality - History

Read more here: » Pontryagin duality: Encyclopedia II - Pontryagin duality - Plancherel and Fourier inversion theorems

The standard notation for the general hypergeometric series is Here, the integers m and p refer to the degree of the polynomials P and Q, respectively, referring to the ratio If m>p+1, the radius of convergence is zero and so there is no analytic function. The series naturally terminates in case P(n) is ever 0 for n a natural number. If Q(n) we ...

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Hypergeometric series, Hypergeometric series - Introduction, Hypergeometric series - Notation, Hypergeometric series - Special cases and applications, Hypergeometric series - Identities, Hypergeometric series - History and generalizations

Read more here: » Hypergeometric series: Encyclopedia II - Hypergeometric series - Notation

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