Abstract:
We consider the problem of determining the asymptotics for the number of points moving along a metric graph. This problem is motivated by the problem of the evolution of wave packets, which at the initial moment of time are localized in a small neighborhood of one point. It turns out that the number of points, as a function of time, allows a polynomial approximation. This polynomial is expressed via Barnes’ multiple Bernoulli polynomials, which are related to the problem of counting the number of lattice points in expanding simplexes.
In this paper we give explicit formulas for the first two terms of the expansion for the counting function of the number of moving points. The leading term was found earlier and depends only on the number of vertices, the number of edges and the lengths of the edges. The second term in the expansion shows what happens to the graph when one or two edges are removed. In particular, whether it breaks up into several connected components or not. In this paper, examples of the calculation of the leading and second terms are given.
Citation:
Vsevolod L. Chernyshev, Anton A. Tolchennikov, “The Second Term in the Asymptotics for the Number of Points Moving Along a Metric Graph”, Regul. Chaotic Dyn., 22:8 (2017), 937–948
\Bibitem{CheTol17}
\by Vsevolod L. Chernyshev, Anton A. Tolchennikov
\paper The Second Term in the Asymptotics for the Number of Points Moving Along a Metric Graph
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 8
\pages 937--948
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\crossref{https://doi.org/10.1134/S1560354717080032}
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Linking options:
https://www.mathnet.ru/eng/rcd300
https://www.mathnet.ru/eng/rcd/v22/i8/p937
This publication is cited in the following 8 articles:
Andrew Eliseev, Vsevolod L. Chernyshev, “Upper bound on saturation time of metric graphs by intervals moving on them”, Journal of Mathematical Analysis and Applications, 531:2 (2024), 127873
D. V. Pyatko, V. L. Chernyshev, “Asymptotics of the Number of End Positions of a Random Walk on a Directed Hamiltonian Metric Graph”, Math. Notes, 113:4 (2023), 538–551
D. S. Minenkov, V. E. Nazaikinskii, T. W. Hilberdink, V. L. Chernyshev, “Restricted partions: the polynomial case”, Funct. Anal. Appl., 56:4 (2022), 299–309
V. L. Chernyshev, A. A. Tolchennikov, “A Metric Graph for Which the Number of Possible End Positions of a Random Walk Grows Minimally”, Russ. J. Math. Phys., 29:4 (2022), 426
V. L. Chernyshev, D. S. Minenkov, A. A. Tolchennikov, “The number of endpoints of a random walk on a semi-infinite metric path graph”, Theoret. and Math. Phys., 207:1 (2021), 487–493
V. L. Chernyshev, A. A. Tolchennikov, “Asymptotics of the number of endpoints of a random walk on a certain class of directed metric graphs”, Russ. J. Math. Phys., 28:4 (2021), 434–438
A. A. Izmaylov, L. W. Dworzanski, “Automated analysis of DP-systems using timed-arc Petri nets via TAPAAL tool”, Trudy ISP RAN, 32:6 (2020), 155–166
Vsevolod Chernyshev, Anton Tolchennikov, “Polynomial approximation for the number of all possible endpoints of a random walk on a metric graph”, Electronic Notes in Discrete Mathematics, 70 (2018), 31
Citation in format AMSBIB
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