Related papers: Variants of VC dimension and their applications to…
One unusual property of dynamic systems, whose state is characterized by a set of scalar dynamic variables satisfying a system of differential equations of a general form, is considered. This property is related to the behavior of equations…
In the present work we suggest a general covariant theory which can be used to study the stability of any physical system treated geometrically. Stability conditions are connected to the magnitude of the deviation vector. This theory is a…
Investigating properties of two-dimensional Dirac operators coupled to an electric and a magnetic field (perpendicular to the plane) requires in general unbounded (vector-) potentials. If the system has a certain symmetry, the fields can be…
The idea that quantum gravity can be realized at the TeV scale is extremely attractive to theorists and experimentalists alike. This proposal leads to extra spacial dimensions large compared to the electroweak scale. Here we give a very…
We propose a theory in electromagnetic dynamics, in which time and space are equivalent with each other and have totally twelve dimensions. Then, we solve that with realistic assumptions and find a steady state as a solution. The solution…
A compact version of the variation evolving method (VEM) is developed in the primal variable space for optimal control computation. Following the idea that originates from the Lyapunov continuous-time dynamics stability theory in the…
A new and extensive formalism is developed for monads and galaxies in non-standard enlargements. It is shown that monads and galaxies can be manipulated using order-preserving and order-reversing set-to-set maps, and that set properties…
It is well-known that the relativized variational principle established by Bogenschutz and Kifer connects the fiber topological entropy and fiber measure-theoretic entropy. In context of random dynamical systems, metric mean dimension was…
We review a growing theoretical motivation and evidence that the number of dimensions actually reduces at high energies. This reduction can happen near the Planck scale, or much before, the dimensions that are reduced can be effective,…
Conditional symmetries were introduced by Levi and Winternitz in their 1989 seminal paper to deal with nonlinear PDEs. Here we discuss their application in the framework of ODEs, and more specifically Dynamical Systems; it turns out they…
Variational analysis provides the theoretical foundations and practical tools for constructing optimization algorithms without being restricted to smooth or convex problems. We survey the central concepts in the context of a concrete but…
We describe a method, using periodic points and determinants, for giving alternative expressions for dynamical quantities (including Lyapunov exponents and Hausdorff dimension of invariant sets) associated to analytic hyperbolic systems.…
Multiple-view visualizations (MVs) have been widely used for visual analysis. Each view shows some part of the data in a usable way, and together multiple views enable a holistic understanding of the data under investigation. For example,…
Nowadays the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. It may lead to a confusion since there are at least two…
We study the theory of systems with constraints from the point of view of the formal theory of partial differential equations. For finite-dimensional systems we show that the Dirac algorithm completes the equations of motion to an…
Nonlocal and fractional-order models capture effects that classical partial differential equations cannot describe; for this reason, they are suitable for a broad class of engineering and scientific applications that feature multiscale or…
In the realm of machine learning theory, to prevent unnatural coding schemes between teacher and learner, No-Clash Teaching Dimension was introduced as provably optimal complexity measure for collusion-free teaching. However, whether…
The recent progress in artificial intelligence has led to an ever-increasing usage of images and videos by machine analysis algorithms, mainly neural networks. Nonetheless, compression, storage and transmission of media have traditionally…
We prove a variational principle for the upper and lower metric mean dimension of level sets \[ \left\{x\in X: \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))=\alpha\right\} \] associated to continuous potentials $\varphi:X\to…
This paper addresses a large class of vector optimization problems in infinite-dimensional spaces with respect to two important binary relations derived from domination structures. Motivated by theoretical challenges as well as by…