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The conjugacy problem for a finitely generated group $G$ is the two-variable problem of deciding for an arbitrary pair $(u,v)$ of elements of $G$, whether or not $u$ is conjugate to $v$ in $G$. We construct examples of finitely generated,…
Astronomy depends on ever increasing computing power. Processor clock-rates have plateaued, and increased performance is now appearing in the form of additional processor cores on a single chip. This poses significant challenges to the…
We introduce generalizations of powers and factor complexity via orbits of group actions. These generalizations include concepts like abelian powers and abelian complexity. It is shown that this notion of factor complexity cannot be used to…
We reconsider the classical problem of the continuation of degenerate periodic orbits in Hamiltonian systems. In particular we focus on periodic orbits that arise from the breaking of a completely resonant maximal torus. We here propose a…
The efficacy of using complexifications to understand the structure of real algebraic groups is demonstrated. In particular the following results are proved: a) If L is an algebraic subgroup of a connected real algebraic group G such that…
We describe an exact algorithm for finding the best 2-OPT move which, experimentally, was observed to be much faster than the standard quadratic approach. To analyze its average-case complexity, we introduce a family of heuristic procedures…
The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the…
We study a generalized motion planning problem involving multiple autonomous robots navigating in a $d$-dimensional Euclidean space in the presence of a set of obstacles whose positions are unknown a priori. Each robot is required to visit…
In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite…
We consider two natural Lagrangian intersection problems in the context of symplectic toric manifolds: displaceability of torus orbits and of a torus orbit with the real part of the toric manifold. Our remarks address the fact that one can…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
We study the computational complexity of (deterministic or randomized) algorithms based on point samples for approximating or integrating functions that can be well approximated by neural networks. Such algorithms (most prominently…
Graph routing problems have been investigated extensively in operations research, computer science and engineering due to their ubiquity and vast applications. In this paper, we study constant approximation algorithms for some variations of…
Models of computations over the integers are equivalent from a computability and complexity theory point of view by the Church-Turing thesis. It is not possible to unify discrete-time models over the reals. The situation is unclear but…
Periodic orbits are important objects of discrete dynamical systems, but finding them is not always easy. We present a self-contained introductory account, aimed at non-experts, to prove their existence and study their stability using the…
Many discrete optimization problems amount to selecting a feasible set of edges of least weight. We consider in this paper the context of spatial graphs where the positions of the vertices are uncertain and belong to known uncertainty sets.…
Non-compact symmetries cannot be fully broken by randomness since non-compact groups have no invariant probability distributions. In particular, this makes trickier the "Copernican" random choice of the place of the observer in infinite…
Periodic orbit action correlations are studied for the piecewise linear, area-preserving Baker map. Semiclassical periodic orbit formulae together with universal spectral statistics in the corresponding quantum Baker map suggest the…
This paper develops an efficient algorithm for computing the Euclidean projection onto the top-k-sum constraint, a key operation in financial risk management and matrix optimization problems. Existing projection methods rely on sorting and…
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates.…