Search results for "Mathematical analysis"
showing 10 items of 2409 documents
A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions
2010
Abstract In this paper, we establish the equivalence between the Hajlasz–Sobolev spaces or classical Triebel–Lizorkin spaces and a class of grand Triebel–Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when p ∈ ( n / ( n + 1 ) , ∞ ) , we give a new characterization of the Hajlasz–Sobolev spaces M ˙ 1 , p ( R n ) via a grand Littlewood–Paley function.
A continuous decomposition of the Menger curve into pseudo-arcs
2000
It is proved that the Menger universal curve M admits a continuous decomposition into pseudo-arcs with the quotient space homeomorphic to M. Wilson proved [8] Anderson's announcement [1] saying that for any Peano continuum X the Menger universal curve M admits a continuous decomposition into homeomorphic copies of M such that the quotient space is homeomorphic to X. Anderson also announced (unpublished) that the plane admits a continuous decomposition into pseudo-arcs. This result was proved by Lewis and Walsh [4]. In a previous paper [6] the author has proved that each locally planar Peano continuum with no local separating point admits a continuous decomposition into pseudo-arcs. Applying…
Diffusion processes with ultrametric jumps
2007
Abstract In the theory of spin glasses the relaxation processes are modelled by random jumps in ultrametric spaces. One may argue that at the border of glassy and nonglassy phases the processes combining diffusion and jumps may be relevant. Using the Dirichlet form technique we construct a model of diffusion on the real line with jumps on the Cantor set. The jumps preserve the ultrametric feature of a random process on unit ball of 2-adic numbers.
Indefinite integrals involving complete elliptic integrals of the third kind
2017
ABSTRACTA method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete …
A Lagrangian method for deriving new indefinite integrals of special functions
2015
A new method is presented for obtaining indefinite integrals of common special functions. The approach is based on a Lagrangian formulation of the general homogeneous linear ordinary differential equation of second order. A general integral is derived which involves an arbitrary function, and therefore yields an infinite number of indefinite integrals for any special function which obeys such a differential equation. Techniques are presented to obtain the more interesting integrals generated by such an approach, and many integrals, both previously known and completely new are derived using the method. Sample results are given for Bessel functions, Airy functions, Legendre functions and hype…
Fourier series for elliptic integrals and some generalizations via hypergeometric series
2008
Fourier series are derived for generalizations of the three canonical Legendre incomplete elliptic integrals using a hypergeometric series approach. The Fourier series for the incomplete Epstein–Hubbell integrals are obtained as special cases of the generalization of the Legendre integrals of the first and second kinds. The Fourier series for the integrals of the first and second kinds, and those for the Epstein–Hubbell integrals, were obtained recently using a different approach, but the series obtained for the generalization of the incomplete integral of the third kind is new. All cases of the integral of the third kind are given, with the modulus and the parameter being complex variables…
Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions
2017
Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the indep...
Indefinite integrals involving the Jacobi Zeta and Heuman Lambda functions
2017
ABSTRACTJacobian elliptic functions are used to obtain formulas for deriving indefinite integrals for the Jacobi Zeta function and Heuman's Lambda function. Only sample results are presented, mostly obtained from powers of the twelve Glaisher elliptic functions. However, this sample includes all such integrals in the literature, together with many new integrals. The method used is based on the differential equations obeyed by these functions when the independent variable is the argument u of elliptic function theory. The same method was used recently, in a companion paper, to derive similar integrals for the three canonical incomplete elliptic integrals.
INTEGRAL SOLUTIONS TO A CLASS OF NONLOCAL EVOLUTION EQUATIONS
2010
We study the existence of integral solutions to a class of nonlinear evolution equations of the form [Formula: see text] where A : D(A) ⊆ X → 2X is an m-accretive operator on a Banach space X, and f : [0, T] × X → X and [Formula: see text] are given functions. We obtain sufficient conditions for this problem to have a unique integral solution.
Existence and Regularity of Solutions of Cauchy Problems for Inhomogeneous Wave Equations with Interaction
1991
The main aim of this paper is a nonrecursive formula for the compatibility conditions ensuring the regularity of solutions of abstract inhomogeneous linear wave equations, which we derive using the theory of T. Kato [11]. We apply it to interaction problems for wave equations (cf. [3]), generalizing regularity results of Lions-Magenes [12].