1,428 research outputs found

### Locus configurations and $\vee$-systems

We present a new family of the locus configurations which is not related to
$\vee$-systems thus giving the answer to one of the questions raised in
\cite{V1} about the relation between the generalised quantum Calogero-Moser
systems and special solutions of the generalised WDVV equations. As a
by-product we have new examples of the hyperbolic equations satisfying the
Huygens' principle in the narrow Hadamard's sense. Another result is new
multiparameter families of $\vee$-systems which gives new solutions of the
generalised WDVV equation.Comment: 12 page

### Deformations of the root systems and new solutions to generalised WDVV equations

A special class of solutions to the generalised WDVV equations related to a
finite set of covectors is investigated. Some geometric conditions on such a
set which guarantee that the corresponding function satisfies WDVV equations
are found (check-conditions). These conditions are satisfied for all root
systems and their special deformations discovered in the theory of the
Calogero-Moser systems by O.Chalykh, M.Feigin and the author. This leads to the
new solutions for the generalized WDVV equations.Comment: 8 page

### Goldfishing by gauge theory

A new solvable many-body problem of goldfish type is identified and used to
revisit the connection among two different approaches to solvable dynamical
systems. An isochronous variant of this model is identified and investigated.
Alternative versions of these models are presented. The behavior of the
alternative isochronous model near its equilibrium configurations is
investigated, and a remarkable Diophantine result, as well as related
Diophantine conjectures, are thereby obtained.Comment: 22 page

### Yang-Baxter maps: dynamical point of view

A review of some recent results on the dynamical theory of the Yang-Baxter
maps (also known as set-theoretical solutions to the quantum Yang-Baxter
equation) is given. The central question is the integrability of the transfer
dynamics. The relations with matrix factorisations, matrix KdV solitons,
Poisson Lie groups, geometric crystals and tropical combinatorics are discussed
and demonstrated on several concrete examples.Comment: 24 pages. Extended version of lectures given at the meeting
"Combinatorial Aspect of Integrable Systems" (RIMS, Kyoto, July 2004

### On integer programing with bounded determinants

Let $A$ be an $(m \times n)$ integral matrix, and let $P=\{ x : A x \leq b\}$
be an $n$-dimensional polytope. The width of $P$ is defined as $w(P)=min\{
x\in \mathbb{Z}^n\setminus\{0\} :\: max_{x \in P} x^\top u - min_{x \in P}
x^\top v \}$. Let $\Delta(A)$ and $\delta(A)$ denote the greatest and the
smallest absolute values of a determinant among all $r(A) \times r(A)$
sub-matrices of $A$, where $r(A)$ is the rank of a matrix $A$. We prove that if
every $r(A) \times r(A)$ sub-matrix of $A$ has a determinant equal to $\pm
\Delta(A)$ or $0$ and $w(P)\ge (\Delta(A)-1)(n+1)$, then $P$ contains $n$
affine independent integer points. Also we have similar results for the case of
\emph{$k$-modular} matrices. The matrix $A$ is called \emph{totally
$k$-modular} if every square sub-matrix of $A$ has a determinant in the set
$\{0,\, \pm k^r :\: r \in \mathbb{N} \}$. When $P$ is a simplex and $w(P)\ge
\delta(A)-1$, we describe a polynomial time algorithm for finding an integer
point in $P$. Finally we show that if $A$ is \emph{almost unimodular}, then
integer program $\max \{c^\top x :\: x \in P \cap \mathbb{Z}^n \}$ can be
solved in polynomial time. The matrix $A$ is called \emph{almost unimodular} if
$\Delta(A) \leq 2$ and any $(r(A)-1)\times(r(A)-1)$ sub-matrix has a
determinant from the set $\{0,\pm 1\}$.Comment: The proof of Lemma 4 has been fixed. Some minor corrections has been
don

### Tropical Markov dynamics and Cayley cubic

We study the tropical version of Markov dynamics on the Cayley cubic,
introduced by V.E. Adler and one of the authors. We show that this action is
semi-conjugated to the standard action of $SL_2(\mathbb Z)$ on a torus, and
thus is ergodic with the Lyapunov exponent and entropy given by the logarithm
of the spectral radius of the corresponding matrix.Comment: Extended version, accepted for publication in "Integrable Systems and
Algebraic Geometry" (Editors: R. Donagi, T. Shaska), Cambridge Univ. Press:
LMS Lecture Notes Series, 201

### Quasiinvariants of Coxeter groups and m-harmonic polynomials

The space of m-harmonic polynomials related to a Coxeter group G and a
multiplicity function m on its root system is defined as the joint kernel of
the properly gauged invariant integrals of the corresponding generalised
quantum Calogero-Moser problem. The relation between this space and the ring of
all quantum integrals of this system (which is isomorphic to the ring of
corresponding quasiinvariants) is investigated.Comment: 23 page

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