|
List Of Topics Named After Augustin-Louis Cauchy
Things named after the 19th-century French mathematician Augustin-Louis Cauchy include: * Binet–Cauchy identity * Intermediate value theorem, Bolzano–Cauchy theorem * Cauchy's argument principle * Cauchy–Binet formula * Cauchy–Born rule * Cauchy boundary condition * Geometrical_properties_of_polynomial_roots#Lagrange's_and_Cauchy's_bounds, Cauchy bounds * Completeness_of_the_real_numbers#Cauchy_completeness, Cauchy completeness * Karoubi envelope, Cauchy completion * Cauchy condensation test * Cauchy-continuous function * Cauchy's convergence test * Cauchy (crater) * Restricted_sumset#Cauchy–Davenport_theorem, Cauchy–Davenport theorem * Cauchy determinant * Cauchy distribution **Log-Cauchy distribution **Wrapped Cauchy distribution * Cauchy elastic material * Cauchy's equation * Cauchy–Euler equation * Cauchy's functional equation * Uniform_space#Cauchy_filter, Cauchy filter * Cauchy formula for repeated integration * Cauchy–Frobenius lemma * Finite_strain_theory#Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Log-Cauchy Distribution
In probability theory, a log-Cauchy distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Cauchy distribution. If ''X'' is a random variable with a Cauchy distribution, then ''Y'' = exponential function, exp(''X'') has a log-Cauchy distribution; likewise, if ''Y'' has a log-Cauchy distribution, then ''X'' = log(''Y'') has a Cauchy distribution. Characterization The log-Cauchy distribution is a special case of the log-t distribution where the degrees of freedom parameter is equal to 1. Probability density function The log-Cauchy distribution has the probability density function: :\begin f(x; \mu,\sigma) & = \frac, \ \ x>0 \\ & = \left[ \right], \ \ x>0 \end where \mu is a real number and \sigma >0. If \sigma is known, the scale parameter is e^. \mu and \sigma correspond to the location parameter and scale parameter of the associated Cauchy distribution. Some authors define \mu and \sigma as the locat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 of :''r''(''x'') = ''p''(''x'')/''q''(''x'') over the real line is the difference between the number of roots of ''f''(''z'') located in the right half-plane and those located in the left half-plane. The complex polynomial ''f''(''z'') is such that :''f''(''iy'') = ''q''(''y'') + ''ip''(''y''). We must also assume that ''p'' has degree less than the degree of ''q''. Definition * The Cauchy index was first defined for a pole ''s'' of the rational function ''r'' by Augustin-Louis Cauchy in 1837 using one-sided limits as: : I_sr = \begin +1, & \text \displaystyle\lim_r(x)=-\infty \;\land\; \lim_r(x)=+\infty, \\ -1, & \text \displaystyle\lim_r(x)=+\infty \;\land\; \lim_r(x)=-\infty, \\ 0, & \text \end * A generalization over the compact interval 'a'',''b''is direct (when neither ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schur Polynomial
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in ''n'' variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials. Definition (Jacobi's bialternant formula) Schur polynomials are indexed by integer partitions. Given a partition , where , and each is a non-negative integer, the functions a_ (x_1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Horizon
In physics, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem (a particular boundary value problem of the theory of partial differential equations). One side of the horizon contains closed space-like geodesics and the other side contains closed time-like geodesics. The concept is named after Augustin-Louis Cauchy. Under the averaged weak energy condition (AWEC), Cauchy horizons are inherently unstable. However, cases of AWEC violation, such as the Casimir effect caused by periodic boundary conditions, do exist, and since the region of spacetime inside the Cauchy horizon has closed timelike curves it is subject to periodic boundary conditions. If the spacetime inside the Cauchy horizon violates AWEC, then the horizon becomes stable and frequency boosting effects would be canceled out by the tendency of the spacetime to act as a divergent lens. Were this conjecture to be shown empirically true, it would provide a counter-example to the strong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Hadamard Theorem
In mathematics, the Cauchy–Hadamard theorem is a result in complex analysis named after the French mathematicians Augustin Louis Cauchy and Jacques Hadamard, describing the radius of convergence of a power series. It was published in 1821 by Cauchy, but remained relatively unknown until Hadamard rediscovered it. Hadamard's first publication of this result was in 1888; he also included it as part of his 1892 Ph.D. thesis. Theorem for one complex variable Consider the formal power series in one complex variable ''z'' of the form f(z) = \sum_^ c_ (z-a)^ where a, c_n \in \Complex. Then the radius of convergence R of ''f'' at the point ''a'' is given by \frac = \limsup_ \left( , c_ , ^ \right) where denotes the limit superior, the limit as approaches infinity of the supremum of the sequence values after the ''n''th position. If the sequence values are unbounded so that the is ∞, then the power series does not converge near , while if the is 0 then the radius of convergence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Strain Theory
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. Displacement The displacement of a body has two components: a rigid-body displacement and a deformation. * A rigid-body displacement consists of a simultaneous translation (physics) and rotation of the body without changing its shape or size. * Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration \kappa_0(\mathcal B) to a current or deformed configuration \kappa_t(\mathcal B) (Figure 1). A change in the conf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Frobenius Lemma
Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, the orbit-counting theorem, or the Lemma that is not Burnside's, is a result in group theory that is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms are based on William Burnside, George Pólya, Augustin Louis Cauchy, and Ferdinand Georg Frobenius. The result is not due to Burnside himself, who merely quotes it in his book 'On the Theory of Groups of Finite Order', attributing it instead to . Burnside's Lemma counts "orbits", which is the same thing as counting distinct objects taking account of a symmetry. Other ways of saying it are counting distinct objects up to an equivalence relation ''R'', or counting objects that are in canonical form. In the following, let ''G'' be a finite group that acts on a set ''X''. For each ''g'' in ''G'', let ''Xg'' denote the set of elements in ''X'' that are fixed by ''g'' (also said to be left in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Formula For Repeated Integration
The Cauchy formula for repeated integration, named after Augustin-Louis Cauchy, allows one to compress ''n'' antidifferentiations of a function into a single integral (cf. Cauchy's formula). Scalar case Let ''f'' be a continuous function on the real line. Then the ''n''th repeated integral of ''f'' with basepoint ''a'', f^(x) = \int_a^x \int_a^ \cdots \int_a^ f(\sigma_) \, \mathrm\sigma_ \cdots \, \mathrm\sigma_2 \, \mathrm\sigma_1, is given by single integration f^(x) = \frac \int_a^x\left(x-t\right)^ f(t)\,\mathrmt. Proof A proof is given by induction. The base case with ''n=1'' is trivial, since it is equivalent to: f^(x) = \frac1\int_a^x f(t)\,\mathrmt = \int_a^x f(t)\,\mathrmtNow, suppose this is true for ''n'', and let us prove it for ''n''+1. Firstly, using the Leibniz integral rule, note that \frac \left \frac \int_a^x \left(x-t\right)^n f(t)\,\mathrmt \right= \frac \int_a^x\left(x-t\right)^ f(t)\,\mathrmt . Then, applying the induction hypothesis, \begin f^(x) &= \int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Space
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and uniform convergence. Uniform spaces generalize metric spaces and topological groups, but the concept is designed to formulate the weakest axioms needed for most proofs in analysis. In addition to the usual properties of a topological structure, in a uniform space one formalizes the notions of relative closeness and closeness of points. In other words, ideas like "''x'' is closer to ''a'' than ''y'' is to ''b''" make sense in uniform spaces. By comparison, in a general topological space, given sets ''A,B'' it is meaningful to say that a point ''x'' is ''arbitrarily close'' to ''A'' (i.e., in the closure of ''A''), or perhaps that ''A'' is a ''smaller neighborhood'' of ''x'' than ''B'', but notions of closeness of points and relative closeness ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy's Functional Equation
Cauchy's functional equation is the functional equation: f(x+y) = f(x) + f(y).\ A function f that solves this equation is called an additive function. Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely f:x\mapsto cx for any rational constant c. Over the real numbers, the family of linear maps f : x \mapsto cx, now with c an arbitrary real constant, is likewise a family of solutions; however there can exist other solutions not of this form that are extremely complicated. However, any of a number of regularity conditions, some of them quite weak, will preclude the existence of these pathological solutions. For example, an additive function f:\R\to\R is linear if: * f is continuous (proven by Cauchy in 1821). This condition was weakened in 1875 by Darboux who showed that it is only necessary for the function to be continuous at one point. * f is monotonic on any interval. * f is bounded on any interval. * f is L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy–Euler Equation
In mathematics, an Euler–Cauchy equation, or Cauchy–Euler equation, or simply Euler's equation is a linear homogeneous ordinary differential equation with variable coefficients. It is sometimes referred to as an '' equidimensional'' equation. Because of its particularly simple equidimensional structure, the differential equation can be solved explicitly. The equation Let be the ''n''th derivative of the unknown function . Then a Cauchy–Euler equation of order ''n'' has the form a_ x^n y^(x) + a_ x^ y^(x) + \dots + a_0 y(x) = 0. The substitution x = e^u (that is, u = \ln(x); for x < 0, one might replace all instances of by , which extends the solution's domain to ) may be used to reduce this equation to a linear differential equation with constant coefficients. Alternatively, the trial solution may be used to directly solve for the basic solutions. Second ...
|
