52,345 research outputs found
Distributions in spaces with thick points
We present a theory of distributions in a space with a thick point in dimensions n≥2, generalizing the theory of thick distributions in one variable given in Estrada and Fulling (2007) [8]. The higher dimensional situation is quite different from the one dimensional case.We construct topological vector spaces of thick test functions and, by duality, spaces of thick distributions. We study several operations on these distributions, both algebraic and analytic, particularly partial differentiation. We introduce the notion of thick delta functions at the special point, not only of order 0 but of any integral order. We also consider the thick distributions constructed by the Hadamard finite part procedure. We give formulas for the derivatives of important thick distributions, including the finite part of power functions. We obtain the new formula ∂*2Pf(r-1)∂xi∂xj=(3xixj-δijr2)Pf(r-5)+4π(δij-4ninj)δ* for the second order thick derivatives of the finite part of r-1 in R3, where δ* is a thick delta of order 0. © 2013 Elsevier Ltd
Basic Properties of Periodic Functions
In this article we present definitions, basic properties and some examples of periodic functions according to [5].Li Bo - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaMen Yanhong - Qingdao University of Science and Technology, ChinaLiang Xiquan - Qingdao University of Science and Technology, ChinaGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Chuanzhang Chen. Mathematical Analysis. Higher Education Press, Beijing, 1978.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Peng Wang and Bo Li. Several differentiation formulas of special functions. Part V. Formalized Mathematics, 15(3):73-79, 2007, doi:10.2478/v10037-007-0009-4.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.Yuguang Yang and Yasunari Shidama. Trigonometric functions and existence of circle ratio. Formalized Mathematics, 7(2):255-263, 1998
Analytical two-center integrals over Slater geminal functions
We present analytical formulas for the calculation of the two-center
two-electron integrals in the basis of Slater geminals and products of Slater
orbitals. Our derivation starts with establishing a inhomogeneous fourth-order
ordinary differential equation that is obeyed by the master integral, the
simplest integral with inverse powers of all interparticle distances. To solve
this equation it was necessary to introduce a new family of special functions
which are defined through their series expansions around regular singular
points of the differential equation. To increase the power of the interparticle
distances under the sign of the integral we developed a family of open-ended
recursion relations. A handful of special cases of the integrals is also
analysed with some remarks on simplifications that occur. Additionally, we
present some numerical examples of the master integral that validate the
usefulness and correctness of the key equations derived in this paper. In
particular, we compare our results with the calculations based on the series
expansion of the exp(-\gamma r12) term in the master integral.Comment: 28 pages, 0 figures, 7 table
The Variable-Order Fractional Calculus of Variations
This book intends to deepen the study of the fractional calculus, giving
special emphasis to variable-order operators. It is organized in two parts, as
follows. In the first part, we review the basic concepts of fractional calculus
(Chapter 1) and of the fractional calculus of variations (Chapter 2). In
Chapter 1, we start with a brief overview about fractional calculus and an
introduction to the theory of some special functions in fractional calculus.
Then, we recall several fractional operators (integrals and derivatives)
definitions and some properties of the considered fractional derivatives and
integrals are introduced. In the end of this chapter, we review integration by
parts formulas for different operators. Chapter 2 presents a short introduction
to the classical calculus of variations and review different variational
problems, like the isoperimetric problems or problems with variable endpoints.
In the end of this chapter, we introduce the theory of the fractional calculus
of variations and some fractional variational problems with variable-order. In
the second part, we systematize some new recent results on variable-order
fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018).
In Chapter 3, considering three types of fractional Caputo derivatives of
variable-order, we present new approximation formulas for those fractional
derivatives and prove upper bound formulas for the errors. In Chapter 4, we
introduce the combined Caputo fractional derivative of variable-order and
corresponding higher-order operators. Some properties are also given. Then, we
prove fractional Euler-Lagrange equations for several types of fractional
problems of the calculus of variations, with or without constraints.Comment: The final authenticated version of this preprint is available online
as a SpringerBrief in Applied Sciences and Technology at
[https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos,
detected by the authors while reading the galley proofs, were corrected,
SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 201
Fredholm Determinants, Differential Equations and Matrix Models
Orthogonal polynomial random matrix models of NxN hermitian matrices lead to
Fredholm determinants of integral operators with kernel of the form (phi(x)
psi(y) - psi(x) phi(y))/x-y. This paper is concerned with the Fredholm
determinants of integral operators having kernel of this form and where the
underlying set is a union of open intervals. The emphasis is on the
determinants thought of as functions of the end-points of these intervals. We
show that these Fredholm determinants with kernels of the general form
described above are expressible in terms of solutions of systems of PDE's as
long as phi and psi satisfy a certain type of differentiation formula. There is
also an exponential variant of this analysis which includes the circular
ensembles of NxN unitary matrices.Comment: 34 pages, LaTeX using RevTeX 3.0 macros; last version changes only
the abstract and decreases length of typeset versio
Discontinuous collocation methods and gravitational self-force applications
Numerical simulations of extereme mass ratio inspirals, the mostimportant
sources for the LISA detector, face several computational challenges. We
present a new approach to evolving partial differential equations occurring in
black hole perturbation theory and calculations of the self-force acting on
point particles orbiting supermassive black holes. Such equations are
distributionally sourced, and standard numerical methods, such as
finite-difference or spectral methods, face difficulties associated with
approximating discontinuous functions. However, in the self-force problem we
typically have access to full a-priori information about the local structure of
the discontinuity at the particle. Using this information, we show that
high-order accuracy can be recovered by adding to the Lagrange interpolation
formula a linear combination of certain jump amplitudes. We construct
discontinuous spatial and temporal discretizations by operating on the
corrected Lagrange formula. In a method-of-lines framework, this provides a
simple and efficient method of solving time-dependent partial differential
equations, without loss of accuracy near moving singularities or
discontinuities. This method is well-suited for the problem of time-domain
reconstruction of the metric perturbation via the Teukolsky or
Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and
GPU architectures are discussed.Comment: 29 pages, 5 figure
Differentiation by integration using orthogonal polynomials, a survey
This survey paper discusses the history of approximation formulas for n-th
order derivatives by integrals involving orthogonal polynomials. There is a
large but rather disconnected corpus of literature on such formulas. We give
some results in greater generality than in the literature. Notably we unify the
continuous and discrete case. We make many side remarks, for instance on
wavelets, Mantica's Fourier-Bessel functions and Greville's minimum R_alpha
formulas in connection with discrete smoothing.Comment: 35 pages, 3 figures; minor corrections, subsection 3.11 added;
accepted by J. Approx. Theor
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