331 research outputs found
Word maps in Kac-Moody setting
The paper is a short survey of recent developments in the area of word maps
evaluated on groups and algebras. It is aimed to pose questions relevant to
Kac--Moody theory.Comment: 16 pag
Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility
Word equations are a crucial element in the theoretical foundation of
constraint solving over strings, which have received a lot of attention in
recent years. A word equation relates two words over string variables and
constants. Its solution amounts to a function mapping variables to constant
strings that equate the left and right hand sides of the equation. While the
problem of solving word equations is decidable, the decidability of the problem
of solving a word equation with a length constraint (i.e., a constraint
relating the lengths of words in the word equation) has remained a
long-standing open problem. In this paper, we focus on the subclass of
quadratic word equations, i.e., in which each variable occurs at most twice. We
first show that the length abstractions of solutions to quadratic word
equations are in general not Presburger-definable. We then describe a class of
counter systems with Presburger transition relations which capture the length
abstraction of a quadratic word equation with regular constraints. We provide
an encoding of the effect of a simple loop of the counter systems in the theory
of existential Presburger Arithmetic with divisibility (PAD). Since PAD is
decidable, we get a decision procedure for quadratic words equations with
length constraints for which the associated counter system is \emph{flat}
(i.e., all nodes belong to at most one cycle). We show a decidability result
(in fact, also an NP algorithm with a PAD oracle) for a recently proposed
NP-complete fragment of word equations called regular-oriented word equations,
together with length constraints. Decidability holds when the constraints are
additionally extended with regular constraints with a 1-weak control structure.Comment: 18 page
Finding All Solutions of Equations in Free Groups and Monoids with Involution
The aim of this paper is to present a PSPACE algorithm which yields a finite
graph of exponential size and which describes the set of all solutions of
equations in free groups as well as the set of all solutions of equations in
free monoids with involution in the presence of rational constraints. This
became possible due to the recently invented emph{recompression} technique of
the second author.
He successfully applied the recompression technique for pure word equations
without involution or rational constraints. In particular, his method could not
be used as a black box for free groups (even without rational constraints).
Actually, the presence of an involution (inverse elements) and rational
constraints complicates the situation and some additional analysis is
necessary. Still, the recompression technique is general enough to accommodate
both extensions. In the end, it simplifies proofs that solving word equations
is in PSPACE (Plandowski 1999) and the corresponding result for equations in
free groups with rational constraints (Diekert, Hagenah and Gutierrez 2001). As
a byproduct we obtain a direct proof that it is decidable in PSPACE whether or
not the solution set is finite.Comment: A preliminary version of this paper was presented as an invited talk
at CSR 2014 in Moscow, June 7 - 11, 201
Computing the Noncomputable
We explore in the framework of Quantum Computation the notion of
computability, which holds a central position in Mathematics and Theoretical
Computer Science. A quantum algorithm that exploits the quantum adiabatic
processes is considered for the Hilbert's tenth problem, which is equivalent to
the Turing halting problem and known to be mathematically noncomputable.
Generalised quantum algorithms are also considered for some other mathematical
noncomputables in the same and of different noncomputability classes. The key
element of all these algorithms is the measurability of both the values of
physical observables and of the quantum-mechanical probability distributions
for these values. It is argued that computability, and thus the limits of
Mathematics, ought to be determined not solely by Mathematics itself but also
by physical principles.Comment: Extensively revised and enlarged with: 2 new subsections, 4 new
figures, 1 new reference, and a short biography as requested by the journal
edito
KAM for the quantum harmonic oscillator
In this paper we prove an abstract KAM theorem for infinite dimensional
Hamiltonians systems. This result extends previous works of S.B. Kuksin and J.
P\"oschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an
application we show that some 1D nonlinear Schr\"odinger equations with
harmonic potential admits many quasi-periodic solutions. In a second
application we prove the reducibility of the 1D Schr\"odinger equations with
the harmonic potential and a quasi periodic in time potential.Comment: 54 pages. To appear in Comm. Math. Phy
Discrete Quantum Mechanics
A comprehensive review of the discrete quantum mechanics with the pure
imaginary shifts and the real shifts is presented in parallel with the
corresponding results in the ordinary quantum mechanics. The main subjects to
be covered are the factorised Hamiltonians, the general structure of the
solution spaces of the Schroedinger equation (Crum's theorem and its
modification), the shape invariance, the exact solvability in the Schroedinger
picture as well as in the Heisenberg picture, the creation/annihilation
operators and the dynamical symmetry algebras, the unified theory of exact and
quasi-exact solvability based on the sinusoidal coordinates, the infinite
families of new orthogonal (the exceptional) polynomials. Two new infinite
families of orthogonal polynomials, the X_\ell Meixner-Pollaczek and the X_\ell
Meixner polynomials are reported.Comment: 61 pages, 1 figure. Comments and references adde
Quantitative behavior of unipotent flows and an effective avoidance principle
We give an effective bound on how much time orbits of a unipotent group
on an arithmetic quotient can stay near homogeneous subvarieties of
corresponding to -subgroups of . In particular, we
show that if such a -orbit is moderately near a proper homogeneous
subvariety of for a long time it is very near a different
homogeneous subvariety. Our work builds upon the linearization method of Dani
and Margulis.
Our motivation in developing these bounds is in order to prove quantitative
density statements about unipotent orbits, which we plan to pursue in a
subsequent paper. New qualitative implications of our effective bounds are also
given.Comment: 52 page
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