Topic
Periodic sequence
About: Periodic sequence is a research topic. Over the lifetime, 1309 publications have been published within this topic receiving 18854 citations. The topic is also known as: cycle.
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Papers
TL;DR: In this article, the relation between the solutions of the timeindependent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables.
Abstract: The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half‐width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well‐defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropicgermanium.
1,627 citations
Book•
1 Jan 1968
TL;DR: In this article, the Fourier series associated with an analytic almost periodic function with values in a Banach space is defined, and the uniqueness and approximation theorem for almost periodic functions on topological groups is given.
Abstract: Introduction. Periodic functions, Fourier series Almost Periodic Functions: 1 Definitions and fundamental properties 2 Two characteristic properties of almost periodic functions 3 The Fourier series associated with an almost periodic function 4 Convergence of Fourier series 5 Summability of Fourier series 6 Almost periodic numeric sequences and their connection with almost periodic functions Bibliographical notes Almost Periodic Functions Depending on Parameters: 1 Almost periodic functions depending uniformly on parameters 2 Almost periodic functions in the mean 3 Random functions almost periodic in probability Analytic Almost Periodic Functions: 1 Preliminary theorems 2 Elementary properties of analytic almost periodic functions 3 The Dirichlet series associated with an analytic almost periodic function 4 The behavior at infinity of analytic almost periodic functions in a half-plane Bibliographical notes Almost Periodic Solutions of Ordinary Differential Equations: 1 The primitive of an almost periodic function 2 Linear systems with constants coefficients 3 Quasilinear systems 4 An Amerio criterion for almost periodicity of bounded solutions 5 The equation of non-linear oscillations Bibliographical notes Almost Periodic Solutions of Partial Differential Equations: 1 Some properties of harmonic and polyharmonic functions 2 Harmonic and polyharmonic almost periodic functions 3 Almost periodic solutions of hyperbolic equations 4 Almost periodic solutions of certain parabolic equations Bibliographical notes Almost Periodic Functions with Values in Banach Spaces: 1 Definitions and general properties 2 The Banach space of almost periodic functions 3 The Fourier series attached to an almost periodic function with values in a Banach space. The approximation theorem 4 Almost periodic functions in the Muckenhoupt sense and in the Stepanov sense 5 Weakly almost periodic functions. The primitive of an almost periodic function Bibliographical notes Almost Periodic Functions on Groups: 1 Elementary properties of almost periodic functions on groups. The existence of the mean 2 Unitary representations of groups. The Fourier series associated with an almost periodic function 3 The uniqueness and approximation theorems 4 Almost periodic functions on topological groups Bibliographical notes Appendix References Index.
809 citations
TL;DR: A merit factor based on the sequence autocorrelation function, whose minimization leads to the reduction in the Cramer-Rao lower bound (CRLB) for the variance of "two-sided" intersymbol interference (ISI) channel estimation is introduced and it is shown that complementary sequences not only minimize the merit factor, but also the CRLB.
Abstract: A merit factor based on the sequence autocorrelation function, whose minimization leads to the reduction in the Cramer-Rao lower bound (CRLB) for the variance of "two-sided" intersymbol interference (ISI) channel estimation is introduced. Pairs of binary pilot symbol sequences (a preamble and a postamble) for channel estimation are jointly designed to minimize this merit factor. Given that the number of channel taps is L and the length of a pilot symbol sequence is (N+L-1), where N/spl ges/L, we distinguish between the case when N is even and the case when it is odd. For even N, we show that complementary sequences not only minimize the merit factor, but also the CRLB. For a subset of odd N we construct almost-complementary periodic sequence pairs that minimize the merit factor. The optimal pilot symbol block signaling requires alternating between two (in most cases) different binary sequences that form the merit-minimizing pair.
222 citations
TL;DR: In this article, Sarkovskii et al. provided complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT of an interval into itself (more generally, into the real line) has a periodic orbit of periodn, which other integers must occur as periods of the periodic orbits of T?
Abstract: Two theorems are proved—the first and the more important of them due to Sarkovskii—providing complete and surprisingly simple answers to the following two questions: (i) given that a continuous mapT of an interval into itself (more generally, into the real line) has a periodic orbit of periodn, which other integers must occur as periods of the periodic orbits ofT? (ii) given thatn is the least odd integer which occurs as a period of a periodic orbit ofT, what is the “shape” of that orbit relative to its natural ordering as a finite subset of the real line? As an application, we obtain improved lower bounds for the topological entropy ofT.
217 citations
TL;DR: A theory of almost periodic fuzzy functions, i.e. of the almost periodic functions of real variable and with values fuzzy real numbers, is developed and applications to fuzzy differential equations and to (fuzzy) dynamical systems are given.
213 citations
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| Year | Papers |
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| 2025 | 2 |
| 2024 | 6 |
| 2023 | 7 |
| 2022 | 14 |
| 2021 | 11 |
| 2020 | 19 |