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We introduce the notion of matrices graph, defining continued fraction algorithms where the past and the future are almost independent. We provide an algorithm to convert more general algorithms into matrices graphs. We present an algorithm…
Program behavior may depend on parameters, which are either configured before compilation time, or provided at run-time, e.g., by sensors or other input devices. Parametric program analysis explores how different parameter settings may…
We develop a method to construct algebraic invariants for hypermatrices. We then construct hyperdeterminants and exhibit a generalization of the Cayley-Hamilton theorem for hypermatrices.
Complexity bounds for many problems on matrices with univariate polynomial entries have been improved in the last few years. Still, for most related algorithms, efficient implementations are not available, which leaves open the question of…
This talk describes how a combination of symbolic computation techniques with first-order theorem proving can be used for solving some challenges of automating program analysis, in particular for generating and proving properties about the…
This article presents a strongly polynomial-time algorithm for the general linear programming problem. This algorithm is an implicit reduction procedure that works as follows. Primal and dual problems are combined into a special system of…
We present an algorithm for computing a holonomic system for a definite integral of a holonomic function over a domain defined by polynomial inequalities. If the integrand satisfies a holonomic difference-differential system including…
Contrary to the expected behavior, we show the existence of non-invertible deformations of Lie algebras which can generate invariants for the coadjoint representation, as well as delete cohomology with values in the trivial or adjoint…
The integrability condition called shape invariance is shown to have an underlying algebraic structure and the associated Lie algebras are identified. These shape-invariance algebras transform the parameters of the potentials such as…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…
Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized…
We present a new algorithm to decide isomorphism between finite graded algebras. For a broad class of nilpotent Lie algebras, we demonstrate that it runs in time polynomial in the order of the input algebras. We introduce heuristics that…
We present a deterministic algorithm which computes the multilinear factors of multivariate lacunary polynomials over number fields. Its complexity is polynomial in $\ell^n$ where $\ell$ is the lacunary size of the input polynomial and $n$…
We present a method to automatically approximate moment-based invariants of probabilistic programs with non-polynomial updates of continuous state variables to accommodate more complex dynamics. Our approach leverages polynomial chaos…
We consider the 2-generated free metabelian associative and Lie algebras over the complex field and the invariants of the dihedral groups of finite order acting on these algebras. In the associative case we find a finite set of generators…
We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains…
We consider tilings of Euclidean spaces by polygons or polyhedra, in particular, tilings made by a substitution process, such as the Penrose tilings of the plane. We define an isomorphism invariant related to a subgroup of rotations and…
Symbolic Mathematical tasks such as integration often require multiple well-defined steps and understanding of sub-tasks to reach a solution. To understand Transformers' abilities in such tasks in a fine-grained manner, we deviate from…
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three…
This paper shows that orbital equations generated by iteration of polynomial maps do not have necessarily a unique representation. Remarkably, they may be represented in an infinity of ways, all interconnected by certain nonlinear…