Bulletin of L.N. Gumilyov Eurasian National University. Mathematics, computer science, mechanics series

Some remarks on linear closed subspaces in Bergman spaces

2026-06-13T19:49:25+00:00

Problems related to finite-dimensionality, compact embeddings, and quantitative estimates of dimensions of embedded subspaces in Bergman spaces play an important role in approximation theory, Banach space geometry, the theory of holomorphic function spaces, and the analysis of infinite-dimensional dynamical systems. Classical results of Grothendieck and Subramanian established fundamental principles for embeddings of subspaces in $L_p$ -spaces, while recent studies have extended these ideas to Bergman spaces of holomorphic functions on bounded complex domains. In this context, the investigation of quantitative dimension estimates for closed subspaces embedded into Bergman spaces with stronger integrability conditions becomes particularly relevant.

In this article the finite-dimensionality Grothendieck type problem for closed linear

subspaces of a Bergman space $A_{p}(\Omega ,d\lambda )\ $of holomorphic and $p$ - integrable with respect to the Lebesgue measure $d\lambda $ on a

bounded complex domain $\Omega \subset \mathbb{C}^{n},$ embedded into a

Bergman space $A_{q}(\Omega, d\lambda )$ \ for $q>p\geq 1,$ is analyzed. {It is shown that if a closed linear subspace }$S_{p}^{(q)}\subset A_{p}(\Omega, d\lambda )$ ${\hookrightarrow A}_{q}(\Omega, d\lambda ),q>p\geq 1,\ $ its dimension $\dim S_{p}^{(q)}=N\in $ $\mathbb{N}$ \ proves to satisfy the numerical inequality $\frac{\omega _{N}^{2(q-1)/q}}{N}\frac{|\Omega |^{\frac{

2-q}{q}}}{k_{N}(\xi _{0})^{2(q-1)/q}}\leq \tilde{K}_{p,q}^{2}$ \ for some bounded constant $\tilde{K}_{p,q}>0,$ where $\omega _{N}=|SU(N)|$ is the volume of the compact unitary group $SU(N),|\Omega |$ is the volume of the bounded domain $\Omega \subset \mathbb{C}^{n}$ and $k_{N}(\xi _{0})>0-$ is the corresponding homogeneity parameter at $\xi _{0}=(1,0,0,...,0)_{N}\in

\partial \mathbb{D}^{N}$ $\subset \mathbb{C}^{N}$.

TYPOLOGICAL DIFFERENCES BETWEEN SYNTHETIC AND ANALYTICAL CAUSAL STRUCTURES IN THE KAZAKH LANGUAGE

2026-07-15T21:03:37+00:00

This article presents a methodology for the manual annotation and structural-semantic systematization of causal constructions in the Kazakh language based on the KazCausal annotation scheme. The aim of the study is to formally describe the morphological, lexical, and syntactic means of expressing causality in Kazakh through an annotation model and to provide a scientific basis for the development of a corpus adapted to natural language processing tasks. The study is based on 1,250 causal sentences selected from scientific, journalistic, and normative texts. All sentences were considered as linguistic units expressing cause-and-effect relations and were classified into three main structural types: SYNTHETIC_CAUSE, ANALYTICO_SYNTHETIC_CAUSE, and ANALYTIC_CAUSE. The results show that synthetic constructions account for 431 sentences (34.5%), analytico-synthetic constructions for 425 sentences (34.0%), and analytic constructions for 394 sentences (31.5%). The proposed annotation scheme makes it possible to simultaneously annotate the cause fragment, the effect fragment, the causal marker, the structural type, the specific morphological form, and the semantic subtype. The quality of annotation was evaluated using an F1 score of 0.81 and Cohen’s kappa coefficient of k=0.72. The findings of the study provide a foundation for the development of typologically adapted NLP resources aimed at the automatic detection of cause-and-effect relations in the Kazakh language.

Mathematical Modeling and Effective Properties in Dispersed Composites

2026-07-15T21:03:23+00:00

Determining the effective properties of dispersed composites, such as conductivity

and permittivity, remains a central challenge in homogenization theory. This paper presents a

comprehensive revision of the Self-Consistent Method and its modern variants, demonstrating that

many so-called “new models” are, in fact, merely first-order approximations in concentration that

reproduce the classical results of Maxwell, Clausius, Mossotti, and others. We establish a rigorous

framework for assessing accuracy with respect to both inclusion concentration and contrast parameters.

A particular role of simulations addresses to standard packages, when the opacity of numerical

calculations often obscures the underlying probability distributions of inclusions. We conclude that

the concept of a mathematical model is frequently misapplied in the study of dispersed random

composites, especially when the fundamental principles of homogenization and asymptotic analysis

are overlooked. Finally, we demonstrate how symbolic computation can be effectively employed to

derive analytically correct formulas with precisely quantified accuracy.

Special attention is given to the relationship between analytical approaches and numerical simulations in the evaluation of effective material parameters. The paper emphasizes that reliable predictions require consistency with the fundamental principles of homogenization theory and the correct treatment of stochastic microstructures. By comparing classical and modern approaches, we identify the limits of applicability of commonly used approximations and clarify their asymptotic accuracy. The study also highlights the importance of transparent computational procedures, allowing the influence of inclusion geometry, concentration, and spatial distribution to be properly interpreted. The proposed methodology provides a unified perspective on dispersed composites and offers practical guidance for researchers working with effective medium theories, computational homogenization, and multiphase materials.