Related papers: $\mathcal C^m$ solutions of semialgebraic or defin…
Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family…
The main purpose of this article is concerned with the existence and the precise forms of the transcendental solutions of several refined versions of Fermat-type functional equations with polynomial coefficients in several complex variables…
The aim of this paper is to investigate weakly developable spaces. For a comparison with semi-metrizable spaces, we introduce and study a class of spaces among those of weakly developable spaces, semimetrizable spaces and first countable…
In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal…
A cosmologically viable hypergeometric model in the modified gravity theory $f(R)$ is found from the need for asintoticity towards $\Lambda$CDM, the existence of an inflection point in the $f(R)$ curve, and the conditions of viability given…
In this paper, we characterize meromorphic solutions $f(z_1,z_2),g(z_1,z_2)$ to the generalized Fermat Diophantine functional equations $h(z_1,z_2)f^m+k(z_1,z_2)g^n=1$ in $\mathbf{C}^2$ for integers $m,n\geq2$ and nonzero meromorphic…
We show that a wide range of overdetermined boundary problems for semilinear equations with position-dependent nonlinearities admits nontrivial solutions. The result holds true both on the Euclidean space and on compact Riemannian…
CFTs are naturally defined on Riemann surfaces. The rational ones can be solved using methods from algebraic geometry. One particular feature is the covariance of the partition function under the mapping class group. In genus $g=1$, this…
Let $G:\mathbb{R\rightarrow R}$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on $G$ such that \begin{equation*} \{G(f):f\in \dot{K}_{p,q}^{\alpha }F_{\beta }^{s}\}\subset…
Inspired by the Douglas lemma, we investigate the solvability of the operator equation $AX=C$ in the framework of Hilbert C*-modules. Utilizing partial isometries, we present its general solution when $A$ is a semi-regular operator. For…
We state conditions for which a definable local homomorphism between two locally definable groups $\mathcal{G}$, $\mathcal{G^{\prime}}$ can be uniquely extended when $\mathcal{G}$ is simply connected (Theorem 2.1). As an application of this…
Let $\mathcal F=(F, +. \cdot, <, 0, 1, \dots)$ be a definably complete locally o-minimal expansion of an ordered field. We demonstrate the existence of definable quotients of definable sets by definable equivalence relations when several…
We prove a variant of the standard Whitney extension theorem for $\mathcal C^m(\mathbb R^n)$, in which the norm of the extension operator has polynomial growth in $n$ for fixed $m$.
We consider two basic problems of algebraic topology, the extension problem and the computation of higher homotopy groups, from the point of view of computability and computational complexity. The extension problem is the following: Given…
We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators…
System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…
In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends…
In this paper, we show that there is a large class of fermionic systems for which it is possible to find, for any dimension, a finite closed set of eigenoperators and eigenvalues of the Hamiltonian. Then, the hierarchy of the equations of…
The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of…
We consider a definably complete locally o-minimal expansion of an ordered field. We treat two topics in this paper. The first topic is a definable $\mathcal C^r$ approximation of a definable $\mathcal C^{r-1}$ map between definable…