Surfaces on the pseudo-Helmholtz Lie group are studied. The Dirac, Weingarten, and Codazzi equations are derived. The minimal surfaces of this group are investigated, and the energy functional is derived.

Let $$G$$ be a finite simple graph on the vertex set $$V$$ and $$I_G$$ its edge ideal in the polynomial ring $$S=\mathbb{K}[x_V]$$ . In this paper, we compute the Castelnuovo–Mumford regularity and depth of $$S/I_G$$ when $$G=F_{k}^{W}(K_n)$$ is a $$k$$ -fan graph, or $$G=G_1\circ G_2$$ or $$G=G_1* G_2$$ is the graph obtained from fan graphs $$G_1$$ , $$G_2$$ by the $$\circ$$ operation or the $$*$$ operation, respectively.

The presence of a discrete rational component in the spectrum of an ergodic automorphism $$S$$ is inconsistent with the existence of certain roots of $$S$$ . If $$T$$ is an ergodic automorphism of a space with $$\sigma$$ -finite measure, then the discrete spectrum disappears from the product $$S\otimes T$$ , but the memory of it may remain in the form of the absence of roots, like Cheshire Cat’s grin. Under certain additional assumptions, this effect is inherited by the Poisson suspension over such a product. Based on this idea, we propose a simple rank-one construction for which the Poisson suspension has no roots and is rigid.

We consider the problem of calculating the parameters of the Schwarz–Christoffel integral implementing a conformal mapping of the upper half-plane onto a polygon containing an interior point at infinity. The paper proposes a solution to this problem based on new formulas for the analytic continuation of the Lauricella function $$F_D^{(N)}$$ —the hypergeometric function of $$N$$ complex variables. A set of new identities and continuation formulas for this function is obtained, aimed at calculating the parameters of the Schwarz–Christoffel integral in the “crowding” situation. Representations via the Lauricella function are found for the Schwarz–Christoffel integral that are convenient for calculating such a conformal mapping.

The paper considers the development of an analytical-numerical method for solving the Dirichlet–Neumann problem for the equation $$\operatorname{div} (\varkappa \nabla u) = 0$$ in a domain $$\Omega$$ containing a cone with base of general form, including a polyhedral corner. The coefficient $$\varkappa$$ of the equation is piecewise constant with discontinuity along a conical surface lying in $$\Omega$$ and having the vertex in common with the original cone. The solution of the problem is represented as the limit of a sequence of linear combinations of functions $$\Psi_k$$ that make up an approximation system and are constructed explicitly. The method permits one to obtain not only the solution in the domain $$\Omega$$ but also its expansion near the vertex of the cone and to calculate the corresponding singularity exponents and intensity factors. Some numerical results are presented.

In this paper, new formulas are obtained for estimating the remainder term arising in the summation of the Lauricella hypergeometric series $$F_D^{(N)}$$ . Such formulas allow one to effectively estimate the remainder of the summation when calculating the value of the function $$F_D^{(N)}$$ in a unit polydisk and have applications to calculating Euler-type integrals and certain solutions of systems of differential equations that the Lauricella function satisfies. The results can be applied to problems in mathematical physics and function theory where the Lauricella function arises, including those which are needed for calculations of conformal mappings of polygons.

We prove that a real hypersurface in a nonflat complex space form is of type $$(A)$$ if and only if the structure Lie operator is cyclic parallel.

The Rock–Paper–Scissors game serves as a crucial toy model in understanding evolutionary dynamics, leading to the formulation of a replicator equation. This paper introduces a comprehensive and enduring category of replicator equations derived from expanding functions. It is established that such equations are bijections of the 2D simplex. The current investigation allows further exploration of inverse dynamical behavior of the replicator equations.

An integer random walk $$\{S_{i},\,i\geq 0\}$$ with zero drift and finite variance $$\sigma^{2}$$ is considered. For a random process that assigns, to a variable $$u\in \mathbb{R}$$ , the number of hits of the specified walk into the state $$\lfloor u\sigma\sqrt n\rfloor $$ up to time $$n$$ and is considered under the condition that $$S_{n}=0$$ , a functional limit theorem concerning convergence of the process to the local time of the Brownian bridge is proved.

The paper proposes an approach to lower bounds for the expectation of the maximum of a Gaussian process based on classical results from the geometry of convex bodies.