Related papers: On the Complexity of some Geometrical Objects
Hilbert's epsilon-calculus is based on an extension of the language of predicate logic by a term-forming operator $\epsilon_{x}$. Two fundamental results about the epsilon-calculus, the first and second epsilon theorem, play a role similar…
Entropy notions for $\varepsilon$-incremental practical stability and incremental stability of deterministic nonlinear systems under disturbances are introduced. The entropy notions are constructed via a set of points in state space which…
For any complex number $c$, let $\sigma_c\colon\mathbb N\rightarrow\mathbb C$ denote the divisor function defined by $\sigma_c(n)=\displaystyle{\sum_{d|n}d^c}$ for all $n\in\mathbb N$, and define $R(c)=\{\sigma_c(n)\in\mathbb C\colon…
The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis,…
The interval approach to computation of dynamics of celestial bodies in the planetary problem has been considered. It is based on the refusal from idealization of infinitely high resolving capacity of measuring tools, and forms an…
We study hyperbolic mappings depending on a parameter $\varepsilon $. Each of them has an invariant Cantor set. As $\varepsilon $ tends to zero, the mapping approaches the boundary of hyperbolicity. We analyze the asymptotics of the gap…
Four point correlation functions for many electrons at finite temperature in periodic lattice are analyzed by the perturbation theory with respect to the coupling constant. The correlation functions are characterized as a limit of finite…
We introduce \emph{Term Coding}, a novel framework for analysing extremal problems in discrete mathematics by encoding them as finite systems of \emph{term equations} (and, optionally, \emph{non-equality constraints}). In its basic form,…
We formulate a general statement of the problem of defining invariant measures with certain properties and suggest an ergodic method of perturbations for describing such measures.
Infinite words, also known as streams, hold significant interest in computer science and mathematics, raising the natural question of how their complexity should be measured. We introduce cellular automaton reducibility as a measure of…
In this review article, we discuss connections between the physics of disordered systems, phase transitions in inference problems, and computational hardness. We introduce two models representing the behavior of glassy systems, the spiked…
The time evolution of complex systems usually can be described through stochastic processes. These processes are measured at finite resolution, what necessarily reduces them to finite sequences of real numbers. In order to relate these data…
We study entropy-bounded computational geometry, that is, geometric algorithms whose running times depend on a given measure of the input entropy. Specifically, we introduce a measure that we call range-partition entropy, which unifies and…
This note aims to bring attention to a simple class of discrete dynamical systems exhibiting some complex behaviour. Each of these systems is defined as a self-mapping of the unit square and is obtained by coupling two families of…
This article is concerned with the approximation of unbounded convex sets by polyhedra. While there is an abundance of literature investigating this task for compact sets, results on the unbounded case are scarce. We first point out the…
Reachable sets for a dynamical system describe collections of system states that can be reached in finite time, subject to system dynamics. They can be used to guarantee goal satisfaction in controller design or to verify that unsafe…
We give exact formulae for a wide family of complexity measures that capture the organization of hidden nonlinear processes. The spectral decomposition of operator-valued functions leads to closed-form expressions involving the full…
In this paper, we define a Mathematical model of program structure. Mathematical model of program structure defined here provides unified mathematical treatment of program structure, which reveals that a program is a large and finite set of…
Data complexity is an important concept in the natural sciences and related areas, but lacks a rigorous and computable definition. In this paper, we focus on a particular sense of complexity that is high if the data is structured in a way…
We propose to compute approximations to general invariant sets in dynamical systems by minimizing the distance between an appropriately selected finite set of points and its image under the dynamics. We demonstrate, through computational…