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The asymptotic formula of the number of integral points in non-compact symmetric homogeneous spaces of semi-simple simply connected algebraic groups is given by the average of the product of the number of local solutions twisted by the…

Hayes equivalence is defined on monic polynomials over a finite field $\fq$ in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial $Q$. We study the distribution of the number of zeros in a…

Hyperasymptotics is an analytical method that incorporates exponentially small contributions into asymptotic approximations, thereby expanding their domain of validity, improving accuracy, and providing deeper insight into the underlying…

This is my dissertation. Its research object is a symmetric group of permutations acting on a finite set. The density of permutations with a given cycle structure pattern is explored when the group order tends to infinity. New and sharper…

The following class of sum-product statistics T_n(p)=\frac{1}{k}\sum_{h=1}^p \sum_{(s_1...s_h)\in P(p,h)} \sum_{i_1=l+1}^{i_0} ... \sum_{i_h=l+1}^{i_{h-1}} i_h \prod_{i=i_1}^{i_h} \frac{(Y_{n-i+1,n}-Y_{n-i,n})^{s_i}}{s_i!} (where $l,$…

Let $f \in \mathbb{Z}[y]$ be a polynomial such that $f(\mathbb{N}) \subseteq \mathbb{N}$, and let $p_{\mathcal{A}_{f}}(n)$ denote number of partitions of $n$ whose parts lie in the set $\mathcal{A}_f:=\{f(n):n \in \mathbb{N}\}$. Under…

Denote by $N(n)$ the number of integer solutions $(x_1,\,x_2,\ldots ,x_n)$ of the equation $x_1+x_2+\ldots+x_n=x_1x_2\cdot\ldots\cdot x_n$ such that $x_1\ge x_2\ge\ldots\ge x_n\ge 1$, $n \in \mathbb{Z}^+$. The aim of this paper are is…

Let U be a homogeneous variety over Q of a linear algebraic group. Choose an integral model and assume the existence of infinitely many integral points. Then one would like to give an asymptotic count of integral points of bounded height…

We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both…

We consider exponent equations in finitely generated groups. These are equations, where the variables appear as exponents of group elements and take values from the natural numbers. Solvability of such (systems of) equations has been…

We estimate the number of integer solutions to decomposable form inequalities (both asymptotic estimates and upper bounds are provided) when the degree of the form and the number of variables are relatively prime. These estimates display…

We employ an adapted version of H\"ormander's asymptotic systems method to show heuristically that the standard good-bad-ugly model admits formal polyhomogeneous asymptotic solutions near null infinity. In a related earlier approach, our…

The leading asymptotic behaviour of the Humbert functions $\Phi_2$, $\Phi_3$, $\Xi_2$ of two variables is found, when the absolute values of the two independent variables become simultaneosly large. New integral representations of these…

We study a family of partial differential equations in the complex domain, under the action of a complex perturbation parameter $\epsilon$. We construct inner and outer solutions of the problem and relate them to asymptotic representations…

We consider the error term of the asymptotic formula for the number of pairs of $k$-free integers up to $x$. Our error term improves results by Heath-Brown, Brandes and Dietmann/Marmon. We then extend our results to $r$-tuples of $k$-free…

The main physical result of this paper are exact analytical solutions of the heavenly equation, of importance in the general theory of relativity. These solutions are not invariant under any subgroup of the symmetry group of the equation.…

The aim of this article is to solve the system $XA=Y$ where $A=(a_{ij})\in M_{m\times n}(S)$, $Y\in S^{m}$ and $X$ is an unknown vector of size $n$, being $S$ an additively idempotent semiring. If the system has solutions then we completely…

Recently V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation $R$ of the set…

Let $H$ be an acylindrically hyperbolic group without nontrivial finite normal subgroups. We show that any finite system $S$ of equations with constants from $H$ is equivalent to a single equation. We also show that the algebraic set…

Let R be the polynomial ring in n variables, acted on by the symmetric group S_n. Soergel constructed a full monoidal subcategory of R-bimodules which categorifies the Hecke algebra, whose objects are now known as Soergel bimodules. Soergel…