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Mathematical psychology has a long tradition of modeling probabilistic choice via distribution-free random utility models and associated random preference models. For such models, the predicted choice probabilities often form a bounded and…
First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general…
We study central configurations when the set of positions is symmetric. We use a theorem from representation theory of finite groups to explore the symmetry properties of equations for central configurations. This approach simplifies…
The aim of this study is to provide a perspective to help understand the singular average operator over polynomial hypersurfaces. In particular, this perspective will provide brevity and the possibility of generalizing previous results…
We prove a family of identities, expressing generating functions of powers of characteristic polynomials of permutations, as finite or infinite products. These generalize formulae first obtained in a study of the geometry/topology of…
The mixed volume counts the roots of generic sparse polynomial systems. Mixed cells are used to provide starting systems for homotopy algorithms that can find all those roots, and track no unnecessary path. Up to now, algorithms for that…
We characterize the algebraic structure of semi-direct product of cyclic groups, $\Z_{N}\rtimes\Z_{p}$, where $p$ is an odd prime number which does not divide $q-1$ for any prime factor $q$ of $N$, and provide a polynomial-time quantum…
Solution and analysis of mathematical programming problems may be simplified when these problems are symmetric under appropriate linear transformations. In particular, a knowledge of the symmetries may help reduce the problem dimension, cut…
By a numerical continuation method called a diagonal homotopy we can compute the intersection of two positive dimensional solution sets of polynomial systems. This paper proposes to use this diagonal homotopy as the key step in a procedure…
Quantum algorithm is an algorithm for solving mathematical problems using quantum systems encoded as information, which is found to outperform classical algorithms in some specific cases. The objective of this study is to develop a quantum…
We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases.…
The volume of a cyclic polytope can be obtained by forming an iterated integral along a suitable piecewise linear path running through its edges. Different choices of such a path are related by the action of a subgroup of the combinatorial…
A large number of NP-hard graph problems can be solved in $f(w)n^{O(1)}$ time and space when the input graph is provided together with a tree decomposition of width $w$, in many cases with a modest exponential dependence $f(w)$ on $w$.…
In this work we discuss the possibility to reduce the computational complexity of modal methods, i.e. methods based on eigenmodes expansion, from the third power to the second power of the number of eigenmodes. The proposed approach is…
We give a geometric method for determining the cohomology groups of a polyhedral product under suitable freeness conditions or with coefficients taken in a field. This is done by considering first the special case for which the pairs of…
Iteration of the quadratic map produces sequences of polynomials whose degrees {\sl explode} as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree $4020$, while for the $52,377$ period-20…
We consider positive solutions to parametrized systems of generalized polynomial equations (with real exponents and positive parameters). By a fundamental result obtained in parallel work, polynomial systems are determined by geometric…
With every family of finitely many subsets of a finite-dimensional vector space over the Galois-field with two elements we associate a cyclic transversal polytope. It turns out that those polytopes generalize several well-known polytopes…
We present an exact general remeshing scheme to compute analytic integrals of polynomial functions over the intersections between convex polyhedral cells of old and new meshes. In physics applications this allows one to ensure global mass,…
We introduce a new iterative root-finding method for complex polynomials, dubbed {\it Newton-Ellipsoid} method. It is inspired by the Ellipsoid method, a classical method in optimization, and a property of Newton's Method derived in…