Related papers: Numerically stable conditions on rational and esse…
We use weighted polynomial approximation to prove the existence of a compact set K with non-empty interior and a function f is dense in the space A(K) of all continuous functions on K that are holomorphic in the interior of K, endowed with…
Let A be the category of modules over a complex, finite-dimensional algebra. We show that the space of stability conditions on A parametrises an isomonodromic family of irregular connections on P^1 with values in the Hall algebra of A. The…
Starting from a representation of the early time evolution of a dynamical system in terms of the polynomial expression of some observable f (t) as a function of the time variable in some interval 0 < t < T, we investigate how to…
A general solution is found for a large class of time continuous autonomous nonlinear dynamical systems, the so-called quasi-polynomial systems. This solution is expressed in terms of a new type of special functions defined via their Taylor…
A generalization of Hurwitz stable polynomials to real rational functions is considered. We establishe an analogue of the Hurwitz stability criterion for rational functions and introduce a new type of determinants that can be treated as a…
During 2022--2023 Z.-W. Sun posed many conjectures on infinite series with summands involving generalized harmonic numbers. Motivated by this, we deduce $58$ series identities involving harmonic numbers, eight of which were previously…
We consider analytic functions of the Riemann zeta type, for which, if $s$ is a zero, so is $1-s$. We use infinite product representations of these functions, assuming their zeros to be of first order. We use exponential factors to…
We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form \varepsilon zf^{\prime} = F(\varepsilon,z,f) with F…
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 $K[z]$.…
We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex…
For functions $f(z)=z^p+a_{n+1}z^{p+1}+...$ defined on the open unit disk, the condition $\Re (f'(z)/z^{p-1})>0$ is sufficient for close-to-convexity of $f$. By making use of this result, several sufficient conditions for close-to-convexity…
We prove a factorization formula for the Taylor series coefficients of a zero of a polynomial as a function of the polynomial's coefficients. This result extends to more general functions which we call "complex-exponent polynomials". To…
We prove that the negative resonances of the Chazy equation (in thesense of Painlev\'e analysis) can be related directly to it sgroup-invariance properties. These resonances indicate in this case the instability of pole singularities.…
We introduce a setting based on the one-dimensional (1D) nonlinear Schroedinger equation (NLSE) with the self-focusing (SF) cubic term modulated by a singular function of the coordinate, |x|^{-a}. It may be additionally combined with the…
Consistency conditions for nonminimally coupled $f(R)$ theories have been derived by requiring the absence of tachyons and instabilities in the scalar fluctuations. This note confirms these results and clarifies a subtlety regarding…
We prove the existence of holomorphic functions $f$ defined on any open convex subset ${\rm \Omega}\subset {{\mathbb C}}^n$, whose partial sums of the Taylor developments approximate uniformly any complex polynomial on any convex compact…
We generalize a theorem of Delzant classifying compact connected symplectic manifolds with completely integrable torus actions to certain singular symplectic spaces. The assumption on singularities is that if they are not finite quotient…
We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming…
We give an overview of the fundamental definitions and results concerning hypersurface singularities, defined by convergent power series over an arbitrary real valued field. This approach combines, on the one hand, the classical case of…
From a sequence $\left\{ a_{n}\right\} _{n=0}^{\infty}$ of real numbers satisfying a three-term recurrence, we form a sequence of polynomials $\left\{ P_{m}(z)\right\} _{m=0}^{\infty}$ whose coefficients are numbers in this sequence. We…