The Wiener-Hopf Method for the Transport Equation: a Finite Dimensional Version

1991 ◽  
pp. 20-33 ◽  
Author(s):  
I. Gohberg ◽  
M. A. Kaashoek
Keyword(s):  
Transport Equation ◽  
Finite Dimensional ◽  
Dimensional Version ◽  
Hopf Method

scholarly journals How low can vacuum energy go when your fields are finite-dimensional?

2019 ◽  
Vol 28 (14) ◽  
pp. 1944006
Author(s):  
ChunJun Cao ◽  
Aidan Chatwin-Davies ◽  
Ashmeet Singh
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Degrees Of Freedom ◽  
Vacuum Energy ◽  
Quantum Field ◽  
Locally Finite ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Klein Gordon ◽  
Dimensional Version

According to the holographic bound, there is only a finite density of degrees of freedom in space when gravity is taken into account. Conventional quantum field theory does not conform to this bound, since in this framework, infinitely many degrees of freedom may be localized to any given region of space. In this paper, we explore the viewpoint that quantum field theory may emerge from an underlying theory that is locally finite-dimensional, and we construct a locally finite-dimensional version of a Klein–Gordon scalar field using generalized Clifford algebras. Demanding that the finite-dimensional field operators obey a suitable version of the canonical commutation relations makes this construction essentially unique. We then find that enforcing local finite dimensionality in a holographically consistent way leads to a huge suppression of the quantum contribution to vacuum energy, to the point that the theoretical prediction becomes plausibly consistent with observations.

scholarly journals Morozov’s Discrepancy Principle For The Tikhonov Regularization Of Exponentially Ill-Posed Problems

2008 ◽  
Vol 8 (1) ◽  
pp. 86-98 ◽  
Author(s):  
S.G. SOLODKY ◽  
A. MOSENTSOVA
Keyword(s):  
Tikhonov Regularization ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
Operator Equations ◽  
Order Of Accuracy ◽  
Finite Dimensional ◽  
Dimensional Version ◽  
Ill Posed ◽  
Linear Operator Equations ◽  
Morozov’S Discrepancy Principle

Abstract The problem of approximate solution of severely ill-posed problems given in the form of linear operator equations of the first kind with approximately known right-hand sides was considered. We have studied a strategy for solving this type of problems, which consists in combinating of Morozov’s discrepancy principle and a finite-dimensional version of the Tikhonov regularization. It is shown that this combination provides an optimal order of accuracy on source sets

scholarly journals On Some Mutual Positions of Hyperplanes in a Finite-Dimensional Affine Space

2006 ◽  
Vol 13 (1) ◽  
pp. 101-108
Author(s):  
Alexander Kharazishvili
Keyword(s):  
Affine Space ◽  
Finite Dimensional ◽  
Dimensional Version

Abstract Several combinatorial questions and facts connected with certain types of mutual positions of finitely many hyperplanes in a finite-dimensional affine space are considered. An application of one of such facts to a multi-dimensional version of the well-known Sylvester theorem is presented.

scholarly journals UNCERTAINTY PRINCIPLES FOR GROUPS AND RECONSTRUCTION OF SIGNALS

2017 ◽  
Vol 21 (6) ◽  
pp. 102-109
Author(s):  
S.Y. Novikov ◽  
M.E. Fedina
Keyword(s):  
Abelian Groups ◽  
Summation Formula ◽  
Uncertainty Principles ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Finite Dimensional ◽  
Discrete Signals ◽  
Minimum Number ◽  
Dimensional Version ◽  
Indicator Functions

Uncertainty principles of harmonic analysis and their analogues for finite abelian groups are considered in the paper. Special attention is paid to the recent results of T. Tao and coauthors about cyclic groups of prime order. It is shown, that indicator functions of subgroups of finite Abelian groups are analogues of Gaussian functions. Finite-dimensional version of Poisson summation formula is proved. Opportunities of application of these results for reconstruction of discrete signals with incomplete number of coefficients are suggested. The principle of partial isometric whereby we can determine the minimum number of measurements for stable recovery of the signal are formulated.

Asymptotics of Gaussian integrals in infinite dimensions

2019 ◽  
Vol 22 (01) ◽  
pp. 1950004 ◽  
Author(s):  
Sergio Albeverio ◽  
Victoria Steblovskaya
Keyword(s):  
Hilbert Space ◽  
Phase Function ◽  
Original Form ◽  
Relevant Parameter ◽  
Laplace Method ◽  
Infinite Dimensions ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Dimensional Version ◽  
General Phase

We introduce an infinite-dimensional version of the classical Laplace method, in its original form, relative to a canonical Gaussian measure associated with a Hilbert space, and for a general phase function. Particular attention is given to the case of a phase function with finite-dimensional degeneracy. Explicit results on expansions in the form of power series in the relevant parameter, with estimates on remainders, are provided.

PECULARITIES OF STABILITY OF THE FINITE-DIMENSIONAL SYSTEM DESCRIBED BY THE NON-RESONANT EQUATION WITH THE LINEAR DIFFERENTIAL OPERATOR

2016 ◽  
Vol 9 (2) ◽  
pp. 42-51
Author(s):  
Котов ◽  
P. Kotov
Keyword(s):  
Differential Operator ◽  
Normal Form ◽  
Linear Equation ◽  
Linear Differential Operator ◽  
Dimensional System ◽  
Resonant System ◽  
Finite Dimensional ◽  
Positive Elements ◽  
Dimensional Version ◽  
Linear Differential

The linear system represented in a normal form with the material measurable elements of coefficient system is considered by the uniform linear equation and are offered aspects of stability of the finite-dimensional version of non-resonant system described by the linear equation resolved ralatively derivative with the constant positive elements of the coefficient system in a decimal numeral system at the example of the test of the known dynamic model resources of the portable personal computer are constructive.

scholarly journals The Finite Discrete KP Hierarchy and the Rational Functions

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Raúl Felipe ◽  
Nancy López
Keyword(s):  
Linear System ◽  
Moduli Space ◽  
Rational Functions ◽  
Kp Hierarchy ◽  
System A ◽  
Finite Dimensional ◽  
Infinite Point ◽  
Dimensional Version ◽  
Discrete Kp Hierarchy ◽  
Discrete Kp

The set of all rational functions with any fixed denominator that simultaneously nullify in the infinite point is parametrized by means of a well-known integrable system: a finite dimensional version of the discrete KP hierarchy. This type of study was originated in Y. Nakamura's works who used others integrable systems. Our work proves that the finite discrete KP hierarchy completely parametrizes the spaceRatΛ(n)of rational functions of the formf(x)=q(x)/zn, whereq(x)is a polynomial of ordern−1with nonzero independent coefficent. More exactly, it is proved that there exists a bijection fromRatΛ(n)to the moduli space of solutions of the finite discrete KP hierarchy and a compatible linear system.

scholarly journals On Resolution of an Extremum Norm Problem for the Terminal State of a Linear System

2020 ◽  
Vol 34 ◽  
pp. 3-17
Author(s):  
V. A. Srochko ◽  
◽  
E. V. Aksenyushkina ◽  
Keyword(s):  
Piecewise Linear ◽  
Terminal State ◽  
Linear Functions ◽  
Uniform Grid ◽  
Maximization Problem ◽  
Time Interval ◽  
Support Functions ◽  
Finite Dimensional ◽  
Minimum Norm Problem ◽  
Dimensional Version

We study extremum norm problems for the terminal state of a linear dynamical system using methods of parameterization of admissible controls. Piecewise continuous controls are approximated in the class of piecewise linear functions on a uniform grid of nodes of the time interval by linear combinations of special support functions. In this case, the restriction of a control of the original problem to the interval induces the same restrictions for the variables of the finite-dimensional problems. The finite-dimensional version of a minimum norm problem can effectively be resolved with the help of modern convex optimization programs. In the case of two variables, we propose an analytical method of resolution that uses a one-dimensional minimization problem for a parabola over a segment. For a non-convex norm maximization problem, the finite-dimensional version is resolved globally by exhaustive search over the vertices of a hypercube. The proposed approach provides further insights into global resolution of non-convex optimal control problems and is exemplified by some illustrative problems.

Sign in / Sign up

Export Citation Format

Share Document