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# Topic: Picard theorem

 Springer Online Reference Works   (Site not responding. Last check: 2007-10-19) Theorems for functions regular in a disc, which establish some connections between geometrical properties of the conformal mapping that is induced by these functions and the initial coefficients of the power series that represent them. The Landau–Carathéodory theorem implies the Picard theorem on values that cannot be taken by entire functions. Landau has also established a number of covering theorems in the theory of conformal mapping that establish the existence of and bounds for the corresponding constants. eom.springer.de /l/l057430.htm   (422 words)

 Picard theorem - Wikipedia, the free encyclopedia In complex analysis, the Picard theorem, named after Charles Émile Picard, is either of two distinct yet related theorems, both of which pertain to the range of an analytic function. The first theorem, also referred to as "Little Picard", states that if a function f(z) is entire and non-constant, the range of f(z) is either the whole complex plane or the plane minus a single point. This is a substantial strengthening of the Weierstrass-Casorati theorem, which only guarantees that the range of f is dense in the complex plane. en.wikipedia.org /wiki/Picard_theorem   (308 words)

 [No title] "Little Picard Theorem": Every complex analytic function defined on the whole complex plane is either (a) onto C (i.e., for every w there is a z with f(z)=w) (b) onto C-{one point} (i.e., f misses at most one value) (c) or constant. Incidentally, the Great Picard Theorem assures that f attains all values (with, possibly, one exception) infinitely often in any neighborhood of an essential singularity, so functions which appear to be poor candidates for the Little Picard Theorem are still easy functions for which to demonstrate the existence of many zeros. By the Little Picard Theorem, the image of g must be the entire complex plane, with the possible exception of a single point. www.math.niu.edu /~rusin/known-math/95/picard   (1957 words)

 PlanetMath: Picard's theorem The above theorem can be generalized to a system of first order ordinary differential equations Theorem 2 (generalization of Picard's theorem [KF])   Let This is version 2 of Picard's theorem, born on 2005-02-03, modified 2005-02-04. planetmath.org /encyclopedia/PicardsTheorem2.html   (135 words)

 A Proof of the Fundamental Theorem of Algebra This note gives a proof, believed to be new, of the fundamental theorem of algebra; it is obtained by the use of the classical theorem of Picard: If there are two distinct values which a given entire function never assumes, the function is a constant. The proof is extremely simple and may be of interest as an application of Picard's theorem. By Picard's theorem, f(z), never assuming the distinct values 0 and 1/k, must be constant, contrary to the hypothesis that the degree of f(z) was at least 1. www.cut-the-knot.org /fta/picard.shtml   (293 words)

 Picard Iteration Revisited In differential equations, Picard iteration is a constructive procedure for establishing the existence of a solution to a differential equation The first type of Picard iteration uses computations to generate a "sequence of numbers" which converges to a solution. The goal of this article is to illustrate the second application of Picard iteration; i. math.fullerton.edu /mathews/n2003/PicardIterationMod.html   (925 words)

 PlanetMath: Picard's theorem as a consequence of the fundamental theorem of algebra. Cross-references: implies, fundamental theorem of algebra, polynomial, entire function, little Picard theorem, complex, words, contains, neighborhood, image, essential singularity, holomorphic function This is version 5 of Picard's theorem, born on 2002-12-11, modified 2006-04-09. planetmath.org /encyclopedia/PicardsTheorem.html   (144 words)

 Springer Online Reference Works   (Site not responding. Last check: 2007-10-19) on a two-dimensional complex projective manifold, the Kawamata–Viehweg vanishing theorem was well known as the Picard theorem on the regularity of the adjoint, [a13], Vol. Especially notable are results of C.P. Ramanujan [a14], which include the Kawamata–Viehweg vanishing theorem in the two-dimensional case; see also [a12]. The paper [a8] is particularly useful: it contains relative versions of the vanishing theorem with some singularities, for not necessarily Cartier divisors. eom.springer.de /k/k120060.htm   (587 words)

 Abstract for Jacob Sturm's UCVC talk   (Site not responding. Last check: 2007-10-19) Then the fundamental theorem of algebra says that for every complex number c, the equation f(z)=c has exactly n roots. Picard's big theorem says that for every complex number c (with possibly one exception), the equation f(z)=c has infinitely many roots. In my talk, I'll prove the fundamental theorem of algebra and then I will describe Picard's original proof of his big theorem. www.mtholyoke.edu /courses/quenell/club/proj/sturm.html   (95 words)

 Dept. of Mathematics: Academic Programs Limits of functions, continuity, uniform continuity, differentiation, the mean value theorem, Rolle's theorem, L'Hospital's rule, Taylor's theorem, Riemann Integral, properties of the Riemann Integral, the fundamental theorem of calculus, pointwise and uniform convergence, applications of uniform convergence. Montel's theorem, normal families, Riemann Mapping Theorem Picard's theorem, Mittag-Leffler's theorem, Weierstrass' theorem, simply connected domains, Riemann surfaces, meromorphic functions on compact Riemann surfaces. Banach spaces; the dual topology and weak topology; the Hahn-Banach, Krein- Milman and Alaoglu theorems; the Baire category theorem; the closed graph theorem; the open mapping theorem; the Banach-Steinhaus theorem; elementary spectral theory; and differential equations. www.coas.howard.edu /mathematics/programs_graduate_courses.html   (1023 words)

 mp_arc 95-146   (Site not responding. Last check: 2007-10-19) This theorem guarantees the existence of solutions which are elliptic of the second kind for second-order ordinary differential equations with elliptic coefficients associated with a common period lattice. The fundamental link between Picard's theorem and elliptic finite-gap solutions of completely integrable hierarchies of nonlinear evolution equations, as established in this paper, is without precedent in the literature. In addition, a detailed description of the singularity structure of the Green's function of the operator \$H=d^2/dx^2+q\$ in \$L^2(\bbR)\$ and its precise connection with the branch and singular points of the underlying hyperelliptic curve is given. www.ma.utexas.edu /mp_arc-bin/mpa?yn=95-146   (191 words)

 Initial Value Problems Whenever we talk of the Picard's theorem, we mean it in this local sense. Compute the interval of existence of the solution of the IVP by using Theorem 7.6.8. This again shows that the solution to an IVP may exist on a larger interval than what is being implied by Theorem 7.6.8. home.iitk.ac.in /~arlal/book/nptel/mth102/node66.html   (576 words)

 UCES Methods and Analysis Chap. 2.7: Roots and Newton's Algorithm This is in contrast to the Picard algorithm (recall our attempts to find the square root of 2). Quadratic convergence means that the error at the next step will be proportional to the square of the error at the current step. In general, the Picard algorithm only has first order convergence where the error at the next step is proportional to just the error at the present step (see the proof of the Picard convergence theorem). www.krellinst.org /UCES/archive/classes/CNA/dir2.7/uces2.7.html   (1074 words)

 Read This: Differential Equations: Theory, Technique, and Practice Simmons' original book began in a way that probably intimidated many students, with a discussion of Picard's existence theorem early on, an analysis of pendulum motion that included talk of elliptic integrals, and an extended example with the brachistochrone. What follows is a discussion of Picard's existence and uniqueness theorem that doesn't work very well. A well-chosen picture and a few comments about the heuristics of Picard's method would go a long way toward making the theorem and the method seem more plausible. www.maa.org /reviews/DiffEquationsSimmons.html   (1030 words)

 Final Examination Information Page   (Site not responding. Last check: 2007-10-19) have a good statement of the value to be gained from an existence and uniqueness theorem the wronskian, and the wronskian theorem on general solutions of linear equations, and its significance the second superposition theorem for solving non-homogeneous equations, and its significance www.math.uiuc.edu /~muncast/courses/math341f97/FinalExamInfo.html   (435 words)

 A Picard-Maclaurin theorem for initial value PDEs In this paper, we present an analogous theorem for partial differential equations (PDEs) with polynomial generators and analytic initial conditions. The initial conditions must be given in the designated component at zero and must be analytic in the nondesignated components. The power series solution of such a PDE, whose existence is guaranteed by the Cauchy theorem, can be generated to arbitrary degree by Picard iteration. www.hindawi.com /GetArticle.aspx?doi=10.1155/S1085337500000063   (204 words)

 Final remarks The Picard theorem follows from Bloch's theorem in one variable. However, here we can directly find a Picard theorem without invoking our Bloch theorem. In this connection, we have some interesting consequences of Theorem 11 which can be interpreted as an other kind of little Picard theorem for bicomplex numbers: www.3dfractals.com /bloch/node5.html   (144 words)

 Graduate Course Descriptions Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, nilpotent and solvable groups, generators and relations. These notions will be explained by examining simple concrete examples of dynamical systems such as translations and automorphisms of tori, expanding maps of the interval, topological Markov chains, etc. Fundamental theorems of ergodic theory such as the Poincare recurrence theorem, the Von Neumann mean ergodic theorem and the Birkhoff ergodic theorem will be presented. The orbit theorem and the theory of Lie determined systems (Chapters 2 and 3 of [J1]). www.math.toronto.edu /graduate/courses/descriptions.html   (4601 words)

 Nevanlinna theory and the Bott-Chern classes According to a classical theorem of Picard, a nonconstant holomorphic map The deficiency index is a normalized way of counting the number of points in the inverse image. A by-product of Bott and Chern's excursion in Nevanlinna theory is the notion of a refined Chern class, now called the Bott-Chern class, that has since been transformed into a powerful tool in Arakelov geometry and other aspects of modern number theory. www.math.harvard.edu /history/bott/bottbio/node16.html   (210 words)

 UIC Graduate College -- Courses: Mathematics   (Site not responding. Last check: 2007-10-19) Cohomology theory, universal coefficient theorems, cohomology products and their applications, orientation and duality for manifolds, homotopy groups and fibrations, the Hurewicz theorem, selected topics. Prerequisite: Math 411 and 417 and 481, or consent of the instructor. Prerequisites: Math 320 and 417 and 481, or consent of the instructor. www.uic.edu /depts/grad/courses/math.shtml   (2065 words)

 UNT Department of Mathematics: Dynamics and Analysis Seminar ABSTRACT: Recall that Picard's theorem says that an entire function with omits 0 and 1 must be constant. Analagous to Picard's theoorem is the fact that a holomorphic curve in n dimensional complex projective space omitting at least 2n+1 hyperplanes in general position must be constant. In 1944, Dufresnoy published a paper where he proves an analogous Landau theorem, but his estimate, using a normal families argument, is not effective and he comments that the undetermined constant depends in an "unknown way" on the set of omitted hyperplanes. www.math.unt.edu /seminars/dynamic.shtml   (4675 words)

 Write-ups, etc. The standard proof of the third Sylow theorem is perhaps shorter, but the beauty of the others more than makes up for it. We translate H.N. Shapiro's proof of Dirichlet's theorem on primes in progressions into the setting of the ring of polynomials over a finite field. Dirichlet's proof was analytic, but many special cases of this theorem have more elementary, purely algebraic proofs whose general strategy closely resembles Euclid's proof of the infinitude of the primes. www.math.dartmouth.edu /~ppollack/work.html   (1170 words)

 The Modified Picard Method for Solving Arbitrary Ordinary and Initial Value Partial Differential Equations   (Site not responding. Last check: 2007-10-19) The Modified Picard Method for Solving Arbitrary Ordinary and Initial Value Partial Differential Equations The Modified Picard Method of Edgar G. Parker and James Sochacki An Expository Document on Using the Modified Picard Method to Solve Ordinary Differential Equations and Initial Value Partial Differential Equations www.math.jmu.edu /~jim/picard.html   (83 words)

 PlanetMath 2004-01-12 Snapshot: Index of Contributors theorem for the direct sum of finite dimensional vector spaces \$n\$th root of \$2\$ is irrational for \$n\ge 3\$ (proof using Fermat's last theorem) proof of calculus theorem used in the Lagrange method simba.cs.uct.ac.za /~hussein/PlanetMath-snapshot_2004-01-12/people.html   (391 words)

 [No title] Prove the small Picard Theorem.) Basic facts ----------- Prove Cauchy's Thm. What theorem of multivariable calculus is this similar to? Maximum-modulus principle ------------------------- Hadamard 3-circles theorem, generalize to annuli with slits missing. www.princeton.edu /~missouri/Generals/generals/complex.txt   (1516 words)

 AUB - Faculty of Arts and Sciences - Department of Math - Courses/Programs A first course in measure theory, including general properties of measures, construction of Lebesgue measure in Rn, Lebesgue integration and convergence theorems, Lp-spaces, Hardy-Littlewood maximal function, Fubini's theorem, and convolutions. A second course in complex analysis, covering the homotopy version of Cauchy's theorem, the open mapping theorem, maximum principle, Schwarz's lemma, harmonic functions, normal families, Riemann mapping theorem, Riemannian metrics, method of negative curvature, Picard's theorem, analytic continuation, monodromy, and modular function. Vector spaces, Hamel basis, Hahn-Banach theorem, Banach spaces, continuous linear operators and functionals, Hilbert spaces, and weak topologies. wwwlb.aub.edu.lb /~webmath/course1.htm   (299 words)

 week 108 A fixed-point theorem is something that says there exists a solution, preferably unique, of this sort of equation. In this case, "Picard's theorem on the local existence and uniqueness of solutions of ordinary differential equations" is what comes to our rescue and asserts the existence of a unique fixed point. To prove Picard's theorem we need to assume the function g is reasonably nice (continuous isn't nice enough, we need something like "Lipschitz continuous"), and our initial guess should be reasonably nice (continuous will do here). math.ucr.edu /home/baez/week108.html   (2411 words)

 [No title]   (Site not responding. Last check: 2007-10-19) It's quite embarrassing when you do not remember the conjugacy classes of the symmetric group and have to work them out on the flboard, disturbed by questions like "do you know what the conjugacy class is ?". Know how to apply theorems to those simple examples. If you get a problem which you have not seen before it is more probable, that you choose the correct way to solve it, they usually do not ask difficult things. www.math.princeton.edu /generals/niepel_martin   (618 words)

 Picard–Lindelöf theorem - Wikipedia, the free encyclopedia In mathematics, the Picard–Lindelöf theorem or Picard's existence theorem on existence and uniqueness of solutions of differential equations (Picard 1890, Lindelöf 1894) states that an initial value problem It can then be shown, by using the Banach fixed point theorem, that the sequence of the (called the Picard iterates) is convergent and that the limit is a solution to the problem. en.wikipedia.org /wiki/Picard-Lindel%C3%B6f_theorem   (166 words)

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