Institute of Mathematics AS CR, v. v. i.
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Conditional Fourier-Feynman transform given infinite dimensional conditioning function on abstract Wiener space
No full textsummary:We study a conditional Fourier-Feynman transform (CFFT) of functionals on an abstract Wiener space . An infinite dimensional conditioning function is used to define the CFFT. To do this, we first present a short survey of the conditional Wiener integral concerning the topic of this paper. We then establish evaluation formulas for the conditional Wiener integral on the abstract Wiener space . Using the evaluation formula, we next provide explicit formulas for CFFTs of functionals in the Kallianpur and Bromley Fresnel class and we finally investigate some Fubini theorems involving CFFT
A determinant formula from random walks
Get PDFsummary:One usually studies the random walk model of a cat moving from one room to another in an apartment. Imagine now that the cat also has the possibility to go from one apartment to another by crossing some corridors, or even from one building to another. That yields a new probabilistic model for which each corridor connects the entrance rooms of several apartments. This article computes the determinant of the stochastic matrix associated to such random walks. That new model naturally allows to compute the determinant of a large class of matrices. Two examples involving digraphs and hyperplane arrangements are provided
Fourier diffraction theorem for the tensor fields
Get PDFsummary:The paper is devoted to the electromagnetic inverse scattering problem for a dielectric anisotropic and magnetically isotropic media. The properties of an anisotropic medium with respect to electromagnetic waves are defined by the tensors, which give the relation between the inductions and the fields. The tensor Fourier diffraction theorem derived in the paper can be considered a useful tool for studying tensor fields in inverse problems of electromagnetic scattering. The method is based on the first Born approximation
On upper bounds for total -domination number via the probabilistic method
Get PDFsummary:For a fixed positive integer and a connected graph of order , whose minimum vertex degree is at least , a set is a total -dominating set, also known as a -tuple total dominating set, if every vertex has at least neighbors in . The minimum size of a total -dominating set for is called the total -domination number of , denoted by . The total -domination problem is to determine a minimum total -dominating set of . Since the exact problem is in general quite difficult to solve, it is also of interest to have good upper bounds on the total -domination number. In this paper, we present a probabilistic approach to computing an upper bound for the total -domination number that improves on some previous results
Bayesian Nash equilibrium seeking for multi-agent incomplete-information aggregative games
Get PDFsummary:In this paper, we consider a distributed Bayesian Nash equilibrium (BNE) seeking problem in incomplete-information aggregative games, which is a generalization of either Bayesian games or deterministic aggregative games. We handle the aggregation function to adapt to incomplete-information situations. Since the feasible strategies are infinite-dimensional functions and lie in a non-compact set, the continuity of types brings barriers to seeking equilibria. To this end, we discretize the continuous types and then prove that the equilibrium of the derived discretized model is an -BNE. On this basis, we propose a distributed algorithm for an -BNE and further prove its convergence
Equivalence of ill-posed dynamical systems
Get PDFsummary:The problem of topological classification is fundamental in the study of dynamical systems. However, when we consider systems without well-posedness, it is unclear how to generalize the notion of equivalence. For example, when a system has trajectories distinguished only by parametrization, we cannot apply the usual definition of equivalence based on the phase space, which presupposes the uniqueness of trajectories. In this study, we formulate a notion of “topological equivalence” using the axiomatic theory of topological dynamics proposed by Yorke [7], where dynamical systems are considered to be shift-invariant subsets of a space of partial maps. In particular, we study how the type of problems can be regarded as invariants under the morphisms between systems and how the usual definition of topological equivalence can be generalized. This article is intended to also serve as a brief introduction to the axiomatic theory of ordinary differential equations (or topological dynamics) based on the formalism presented in [6]
Deep learning for gradient flows using the Brezis–Ekeland principle
Get PDFsummary:We propose a deep learning method for the numerical solution of partial differential equations that arise as gradient flows. The method relies on the Brezis–Ekeland principle, which naturally defines an objective function to be minimized, and so is ideally suited for a machine learning approach using deep neural networks. We describe our approach in a general framework and illustrate the method with the help of an example implementation for the heat equation in space dimensions two to seven