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The Ritt problem asks if there is an algorithm that tells whether one prime differential ideal is contained in another one if both are given by their characteristic sets. We give several equivalent formulations of this problem. In…
We show that only a rather small proportion of linear equations are solvable in elements of a fixed finitely generated subgroup of a multiplicative group of a number field. The argument is based on modular techniques combined with a…
The regular separability problem asks, for two given languages, if there exists a regular language including one of them but disjoint from the other. Our main result is decidability, and PSpace-completeness, of the regular separability…
Let $S \subset R$ be an arbitrary subset of a unique factorization domain $R$ and $\K$ be the field of fractions of $R$. The ring of integer-valued polynomials over $S$ is the set $\mathrm{Int}(S,R)= \{ f \in \mathbb{K}[x]: f(a) \in R\…
We consider some variations on the classical method of Runge for effectively determining integral points on certain curves. We first prove a version of Runge's theorem valid for higher-dimensional varieties, generalizing a uniform version…
Boundary differentiability is shown for solutions of nondivergence elliptic equations with unbounded drift
The differential equations with piecewise constant argument (DEPCAs, for short) is a class of hybrid dynamical systems (combining continuous and discrete). In this paper, under the assumption that the nonlinear term is partially unbounded,…
The dynamics of physical theories is usually described by differential equations. Difference equations then appear mainly as an approximation which can be used for a numerical analysis. As such, they have to fulfill certain conditions to…
Over increasing domain interpretations, \exists\Box and \forall\Box bundled fragments are decidable and over constant domain interpretations, \exists\Box bundled fragment is decidable while \forall\Box bundled fragment is undecidable. Based…
Larrauri and \v{Z}ivn\'y [ICALP'25/ACM ToCL'24] recently established a complete complexity classification of the problem of solving a system of equations over a monoid $N$ assuming that a solution exists over a monoid $M$, where both…
We study global scale invariance along with the unimodular gravity in the vacuum. The global scale invariant gravitational action which follows the unimodular general coordinate transformations is considered without invoking any scalar…
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
The predictability of a sequence is defined as the asymptotic performance of the best performing predictor in a given class. The value of the predictability of a sequence will in general depend on the choice of this predictor class. The…
We consider the uniqueness of solutions of ordinary differential equations where the coefficients may have singularities. We derive upper bounds on the the order of singularities of the coefficients and provide examples to illustrate the…
We give upper bounds for the differential Nullstellensatz in the case of ordinary systems of differential algebraic equations over any field of constants $K$ of characteristic $0$. Let $\vec{x}$ be a set of $n$ differential variables,…
For fragments L of first-order logic (FO) with counting quantifiers, we consider the definability problem, which asks whether a given L-formula can be equivalently expressed by a formula in some fragment of L without counting, and the more…
An infinite dimensional algebra, which is useful for deriving exact solutions of the generalized pairing problem, is introduced. A formalism for diagonalizing the corresponding Hamiltonian is also proposed. The theory is illustrated with…
We determine a considerable class of nonlinear partial differential equation systems which have global regular solutions. Uniqueness is not a direct general consequence of this method. The scheme can be applied to the incompressible Navier…
Using continuation methods and bifurcation theory, we study the exact multiplicity of periodic solutions, and the global solution structure, for three classes of periodically forced equations with singularities, including the equations…
We prove Holder regularity for solutions of non divergence integro-differential equations with non necessarily even kernels. The even/odd decomposition of the kernel can be understood as a sum of a diffusion and a drift term. In our case we…