Related papers: A Euclidean Skolem-Mahler-Lech-Chabauty method
We discuss the classes $\fC$, $\fM$, and $\fS$ of analytic functions that can be realized as the Liv\v{s}ic characteristic functions of a symmetric densely defined operator $\dot A$ with deficiency indices $(1,1)$, the Weyl-Titchmarsh…
The Chabauty--Kim method is a method for finding rational points on curves under certain technical conditions, generalising Chabauty's proof of the Mordell conjecture for curves with Mordell--Weil rank less than their genus. We show how the…
Consider an operator equation (*) $B(u)-f=0$ in a real Hilbert space. Let us call this equation ill-posed if the operator $B'(u)$ is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation…
We investigate some finitely-valued generalizations of propositional dynamic logic with tests. We start by introducing the (n+1)-valued Kripke models and a corresponding language based on a modal extension of {\L}ukasiewicz many-valued…
Let $K$ be a field of characteristic zero and suppose that $f:\mathbb{N}\to K$ satisfies a recurrence of the form $$f(n)\ =\ \sum_{i=1}^d P_i(n) f(n-i),$$ for $n$ sufficiently large, where $P_1(z),...,P_d(z)$ are polynomials in $K[z]$.…
It is known that a subharmonic function of finite order $\rho$ can be approximated by the logarithm of the modulus of an entire function at the point $z$ outside an exceptional set up to $C\log|z|$. In this article we prove that if such an…
We prove Veech's conjecture on the equivalence of Sarnak's conjecture on M\"obius orthogonality with a Kolmogorov type property of Furstenberg systems of the M\''obius function. This yields a combinatorial condition on the M\"obius function…
We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein-Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by such a noise. Our main contribution is on the…
A space of analytic functions in the unit disc with uniformly continuous derivatives is said to be quasianalytic if the boundary value of a non-zero function from the class can not have a zero of infinite multiplicity. Such classes were…
Multi-derivative one-step methods based upon Euler-Maclaurin integration formulae are considered for the solution of canonical Hamiltonian dynamical systems. Despite the negative result that simplecticity may not be attained by any…
In this article, we study the Euler's factorial series $F_p(t)=\sum_{n=0}^\infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $k\varphi(m)/(k+1)$ residue classes in the…
We consider dynamical systems $(X,T,\mu)$ which have exponential decay of correlations for either H\"older continuous functions or functions of bounded variation. Given a sequence of balls $(B_n)_{n=1}^\infty$, we give sufficient conditions…
It is well known that the non-spiraling leaves of real analytic foliations of codimension 1 all belong to the same o-minimal structure. Naturally, the question arises if the same statement is true for non-oscillating trajectories of real…
Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE's; for example, it allows to construct generalized solutions of differential equations and…
We regard Forcing Notions P adding real numbers and the algebras of P-measurable sets. As for Cohen- and Random-Forcing we can show that each analytic set is P-measurable using Solovay's Unfolding Trick for infinite games. To show this we…
We prove a monomialization theorem for mappings in general classes of infinitely differentiable functions that are called quasianalytic. Examples include Denjoy-Carleman classes, the class of $\cC^\infty$ functions definable in a…
High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of…
The Main Theorem for abelian fields (often called Main Conjecture despite proofs in most cases) has a long history which has found a solution by means of "elementary arithmetic", as detailed in Washington's book from Thaine's method having…
Let $T$ be the map defined on $\N=\{1,2,3, ...\}$ by $T(n) = \frac{n}{2} $ if $n$ is even and by $T(n) = \frac{3n+1}{2}$ if $n$ is odd. Consider the dynamical system $(\N, 2^{\N}, T,\mu)$ where $\mu$ is the counting measure. This dynamical…
We prove a stability theorem for finite-dimensional analytic inverse problems. Let \(U\subset\R^m\) be an open parameter set, let \(F(p)\) be a boundary measurement operator, and let \(R(p)\) be the finite-dimensional quantity to be…