5-dimensional pseudo-Riemannian manifolds in the form of h-spaces H32 of the type {32} are considered. A classification of h-spaces H32,2 with respect to (nonhomothetic) Lie algebras of infinitesimal projective and affine transformations is given, all projectively mobile metrics are found, and the dimensions, basis elements and structural equations of the maximal nonhomothetic projective Lie algebras operating in them are indicated.

Quasi-discrete and π-projective modules are important and well-studied classes of modules, whose definitions are dual to the equivalent characterizations of the notion of a quasi-continuous module. In this paper, we introduce and study new classes of modules that also serve as dual concepts of quasi-continuous modules.

The article introduces a characteristic of a triangle, reflecting the measure of its nondegeneracy. The importance of studying this quantity is associated with the construction of high-quality computational grids. It is shown that if a Sobolev class mapping distorts this characteristic multiple times, then this mapping is a mapping with bounded distortion. In addition, it is proved that if the above condition and additionally the condition of bounded distortion of the area of the triangle are satisfied, then the mapping is bi-Lipschitz. The article establishes estimates for all constants characterizing the mappings under study.

Let $$({{x}_{n}})$$ be a sequence and $$\rho \geqslant 1$$ . For two fixed sequences $${{n}_{1}} < {{n}_{2}} < {{n}_{3}} < \ldots $$ , and $$M$$ define the oscillation operator $${{\mathcal{O}}_{\rho }}({{x}_{n}}) = {{\left( {\sum\limits_{k = 1}^\infty \,\mathop {\sup }\limits_{\substack{ {{n}_{k}} \leqslant m < {{n}_{{k + 1}}} \\ m \in M } } {{{\left| {{{x}_{m}} - {{x}_{{{{n}_{k}}}}}} \right|}}^{\rho }}} \right)}^{{1/\rho }}}.$$ Let $$(X,\mathcal{B},\mu ,\tau )$$ be a dynamical system with $$(X,\mathcal{B},\mu )$$ a probability space and $$\tau $$ a measurable, invertible, measure preserving point transformation from $$X$$ to itself. Suppose that the sequences $$({{n}_{k}})$$ is a lacunary, and $$M$$ is any sequence of positive real numbers such that there exists an $$\ell \in \mathbb{R}$$ satisfying $$\# \{ m \in M:{{n}_{k}} \leqslant m < {{n}_{{k + 1}}}\} \leqslant \ell $$ for all $$k \in \mathbb{N}$$ to obtain the above mentioned results, where $$\# $$ denotes cardinality. Then we prove the following results for $$\rho \geqslant 2$$ : (i) Define $${{\phi }_{n}}(x) = \frac{1}{n}{{\chi }_{{[0,n]}}}(x)$$ on $$\mathbb{R}$$ . Then there exists a constant $$C > 0$$ such that $${{\left\| {{{\mathcal{O}}_{\rho }}({{\phi }_{n}} * f)} \right\|}_{{{{L}^{1}}(\mathbb{R})}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(\mathbb{R})}}}$$ for all $$f \in {{H}^{1}}(\mathbb{R})$$ . (ii) Let $${{A}_{n}}f(x) = \frac{1}{n}\sum\limits_{k = 1}^n \,f({{\tau }^{k}}x)$$ be the usual ergodic averages in ergodic theory. Then $${{\left\| {{{\mathcal{O}}_{\rho }}({{A}_{n}}f)} \right\|}_{{{{L}^{1}}(X)}}} \leqslant C{{\left\| f \right\|}_{{{{H}^{1}}(X)}}}$$ for all $$f \in {{H}^{1}}(X)$$ . (iii) If $${{[f(x)\log (x)]}^{ + }}$$ is integrable, then $${{\mathcal{O}}_{\rho }}({{A}_{n}}f)$$ is integrable. In the author’s previously published article titled “Oscillation inequalities on real and ergodic $${{H}^{1}}$$ spaces” the above results have been obtained when both $$({{n}_{k}})$$ and $$M$$ are lacunary. Thus the results of this work extents those results to a nonlacunary sequence $$M$$ with a more general growth condition.

In this work, we study a family of multiparameter polynomial differential systems of degree eleven. We prove that the considered family has an invariant algebraic curve, which is given explicitly. Subsequently, we demonstrate the integrability of these systems and derive an explicit expression for a first integral. Moreover, we provide sufficient conditions for the systems to possess two limit cycles explicitly given. The applicability of our results is illustrated by a concrete example.

In this paper, a number of extreme problems related to the best polynomial approximation of functions analytical in the disk $$U: = \left\{ {z \in \mathbb{C}:\left| z \right| < 1} \right\}$$ and belonging to the Bergman space $${{B}_{2}}$$ are solved. The bilateral inequality is proved, which is a generalization of the result for periodic functions $$f \in {{L}_{2}}$$ obtained by Shabozov and Yusupov for the class $$L_{2}^{{(r)}}[0,2\pi ]$$ , in which the derivative $${{f}^{{(r - 1)}}}$$ is absolutely continuous and the derivative of the $$r$$ th order $${{f}^{{(r)}}} \in {{L}_{2}}$$ in the case of a polynomial approximation of $$f \in \mathcal{A}(U)$$ belongs to $$B_{2}^{{(r)}}(U)$$ . A number of cases are given when the bilateral inequality turns into equality. For some classes of functions belonging to $${{B}_{2}}$$ , the exact values of the known $$n$$ -widths are found, and the problem of joint approximation of functions and their intermediate derivatives is solved.

A problem with the Bitsadze–Samarskii and Frankl conditions for a mixed-type equation in a domain whose elliptic part is the first quadrant of the plane and the hyperbolic part is the characteristic triangle is studied. The correctness of the formulated problem is proved.

The aim of this paper is to give the Hyers–Ulam–Russias stability and the Hyers–Ulam stability results for Bernoulli’s differential equation. The argument makes use a fixed point approach. Some examples are given to illustrate the main results.

We provide a criterion for the equivalence of the second-kind involutions of upper triangular matrix algebras over commutative rings. For an algebra $${{T}_{n}}(F)$$ of upper triangular matrices over a field $$F$$ , it is proven that involutions are equivalent iff they coincide after restriction to $$F$$ .

In this paper, we study an inverse problem consisting of determining the unknown local changes of a population densities independent of the missing initial population age distribution from the knowledge of some measurement in the state. The main idea is to transform a study of the inverse problem into an optimal control problem with incomplete data. The proposed method to solve this kind of problem is the no-regret control approximated by a sequence of low-regret control, we show the existence and uniqueness of the sequence of low-regret control, which converges weakly to the unique no-regret control. The local changes will be characterized by a coupled optimality system.