The Stieltjes spectral matrix measure of the doubly infinite Jacobi matrix associated with a Toda $g$-soliton is computed, using Sato theory. The result is used to give an explicit expansion of the fundamental solution of some discrete heat equations, in a series of Jackson’s $q$-Bessel functions. For Askey-Wilson type solitons, this expansion reduces to a finite sum.

The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of...

Let $H$ be a real Hilbert space, ${\Phi }_1:H\to ℝ$ a convex function of class ${𝒞}^1$ that we wish to minimize under the convex constraint $S$. A classical approach consists in following the trajectories of the generalized steepest descent system (cf. Brézis [5]) applied to the non-smooth function ${\Phi }_1+{\delta }_S$. Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function ${\Phi }_0:H\to ℝ$ whose critical points coincide with $S$ and a control...

Let H be a real Hilbert space, ${\Phi }_1:H\to$ a convex function of class ${𝒞}^1$ that we wish to minimize under the convex constraint S. A classical approach consists in following the trajectories of the generalized steepest descent system (cf.   Brézis [CITE]) applied to the non-smooth function ${\Phi }_1+{\delta }_S$. Following Antipin [1], it is also possible to use a continuous gradient-projection system. We propose here an alternative method as follows: given a smooth convex function ${\Phi }_0:H\to$ whose critical points coincide with S and...

The aim of the paper is to investigate the structure of disjoint iteration groups on the unit circle ${𝕊}^1$, that is, families $ℱ=\{F^v{𝕊}^1⟶{𝕊}^{1}v\in V\}$ of homeomorphisms such that $$F^{v_1}\circ F^{v_2}=F^{v_1+v_2},v_1,v_2\in V,$$ and each $F^v$ either is the identity mapping or has no fixed point ($(V,+)$ is an arbitrary $2$-divisible nontrivial (i.e., $\mathrm{c}ardV>1$) abelian group).

We prove that every infinite nowhere dense compact subset of the interval $I$ is an $\omega$-limit set of homoclinic type for a continuous function from $I$ to $I$.

Let ${\left(f_{\lambda }\right)}_{\lambda \in \Lambda }$ be a holomorphic family of rational mappings of degree $d$ on ${\mathbb{P}}^1\left(ℂ\right)$, with $k$ marked critical points $c_1,...,c_k$. To this data is associated a closed positive current $T_1\wedge \cdots \wedge T_k$ of bidegree $(k,k)$ on $\Lambda$, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which $c_1,...,c_k$ eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of $\mathrm{Supp}(T_1\wedge \cdots \wedge T_k)$.

Equivalence is established between a special class of Painlevé VI equations parametrized by a conformal dimension $\mu$, time dependent Euler top equations, isomonodromic deformations and three-dimensional Frobenius manifolds. The isomonodromic tau function and solutions of the Euler top equations are explicitly constructed in terms of Wronskian solutions of the 2-vector 1-constrained symplectic Kadomtsev-Petviashvili (CKP) hierarchy by means of Grassmannian formulation. These Wronskian solutions give...