Related papers: Infinitesimal and tangential 16-th Hilbert problem…
We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a positive selfadjoint operator $H$, that is called the Hamiltonian of the…
We consider the problem of computing the partition function $\sum_x e^{f(x)}$, where $f: \{-1, 1\}^n \longrightarrow {\Bbb R}$ is a quadratic or cubic polynomial on the Boolean cube $\{-1, 1\}^n$. In the case of a quadratic polynomial $f$,…
In the framework of a real Hilbert space, we address the problem of finding the zeros of the sum of a maximally monotone operator $A$ and a cocoercive operator $B$. We study the asymptotic behaviour of the trajectories generated by a second…
We study the interior problem of tomography. The starting point is the Gelfand-Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function $f$ along a collection of lines. Pick one such…
We study the action of irreducible derivations X on some Hilbert's quasi-regular algebras QRH of germes at 0 of analytic functions on (U,0), where U is a semi-algebraic set: that is, we show that these algebras are X-finite or locally…
The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to…
In a Hilbert space $\mathcal H$, we study the asymptotic behaviour, as time variable $t$ goes to $+\infty$, of nonautonomous gradient-like dynamical systems involving inertia and multiscale features. Given $\mathcal H$ a general Hilbert…
We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K,…
For a graph G, we construct two algebras, whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of…
The finite Hilbert transform T is a singular integral operator which maps the Zygmund space $LlogL:=LlogL(-1,1)$ continuously into $L^1:=L^1(-1,1)$. By extending the Parseval and Poincar\'e-Bertrand formulae to this setting, it is possible…
Let $f$ be an entire transcendental function of finite order and $\Delta$ be a forward invariant bounded Siegel disk for $f$ with rotation number in Herman's class $\mathcal{H}$. We show that if $f$ has two singular values with bounded…
We consider one dimensional deformed Heisenberg algebra leading to existence of minimal length for coordinate operator and minimal and maximal uncertainty of momentum operator. For this algebra an exactly solvable Hamiltonian is…
The uniqueness problems on transcendental meromorphic or entire functions sharing at least two values with their derivatives or linear differential polynomials have been studied and many results have been obtained. In this paper, we study a…
We classify transcendental entire functions that are compositions of a polynomial and the exponential for which all singular values escape on disjoint rays. The construction involves an iteration procedure on an infinite-dimensional…
We study the displacement map associated to small one-parameter polynomial unfoldings of polynomial Hamiltonian vector fields on the plane. Its leading term, the generating function $M(t)$, has an analytic continuation in the complex plane…
Let $x'=S(t,x)$ be a differential equation in the cylinder, linear piecewise in $x$ and with trigonometric coefficients in $t$. In this paper, we provide an upper bound on the number of limit cycles in terms of the number of regions of the…
We investigate properties of differential and difference operators annihilating certain finite-dimensional subspaces of exponential functions in two variables that are connected to the representation of real-valued trigonometric and…
We introduce a systematic framework for counting and finding independent operators in effective field theories, taking into account the redundancies associated with use of the classical equations of motion and integration by parts. By…
By making use of some techniques based upon certain inverse new pairs of symbolic operators, the author investigate several decomposition formulas associated with Humbert hypergeometric functions $\Phi_1 $, $\Phi_2 $, $\Phi_3 $, $\Psi_1 $,…
Hilbert-Arnold (HA) problem, motivated by Hilbert 16-th problem, is to prove that for a generic k-parameter family of smooth vector fields {\dot x=v(x,\eps)}_{\eps\in B^k} on the 2-dimensional sphere S^2 has uniformly bounded number of…