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

Characterization of the Boundedness of the Discrete Fractional Maximal Operator Between Weighted Lorentz Sequence Spaces

Аmiran Gogatishvili — 2026-09-27

In this paper, we give the complete characterization of the boundedness of the discrete fractional maximal operator $M_\gamma $, defined on $\mathbb{Z}^n$, $n\in\mathbb{N}$, $\gamma\in [0,n)$, between the classical Lorentz sequence spaces $\lambda_p(v)$ and $\lambda_q(w)$, where $0<p,q<\infty$, $\{v(m)\}$ and $\{w(m)\}$ are non-negative sequences defined on $\mathbb{N}$. We first obtain a sharp upper estimate for the nonincreasing rearrangement of the discrete fractional maximal operator $M_\gamma x$, with a supremal operator involving a discrete Hardy operator and the nonincreasing rearrangement of the sequence $\{x(k)\}$, and we show that this estimate is sharp by showing a lower estimate for special sequences. Using this result, we can reduce the characterization of the boundedness of the discrete fractional maximal operator $ M_\gamma$ between the discrete weighted Lorentz sequence spaces to the boundedness of the discrete supremum operator between weighted Lebesgue sequence spaces, restricted to nonincreasing sequences. This problem was investigated in the recent paper by the authors, which enables us to obtain the final characterization.

On Bredon Cohomology with Mackey Functor in Morava $K$-Theory

Malkhaz Bakuradze — 2026-09-29

Utilizing the algebraic framework of Mackey functors, we analyze the cochain complex and explicitly compute the cellular differentials governed by restrictions and transfer maps within Morava $K$-theory $K(n)^*$. We focus on the geometric construction of the octahedral surface 2-complex, isolating purely equivariant phenomena driven by stabilizer jumps. By analyzing the complex directly over the Morava ring $K(n)_*[x,y]$, we determine the zero-th and first codifferentials. We show that while the underlying space is geometrically trivial in degree one, the equivariant chromatic classes persist due to the cluster's sector geometry. Finally, we provide a complete description of the first Bredon cohomology group $H_{Q_8}^1(X; \underline{K(n)}^*)$ expressed as a quotient of the coefficient ring. These results contribute to the structural classification of equivariant cohomology rings for finite $p$-groups with concentrated even degrees.

Master of Turbulent Forces: How Academician Yershin Brought Kazakhstan’s Hydromechanics to International Prominence

Uzak Kairbekovich Zhapbasbayev — 2026-10-31

Some scientists’ names remain only on the yellowed pages of dissertations. Others are builders capable of harnessing the intricate laws of physics to meet the needs of a vast country and making the world—from Moscow to Washington, the centres of the two most powerful opposing military superpowers—listen to the voice of Kazakhstan’s science. Academician of the National Academy of Sciences of the Republic of Kazakhstan, Doctor of Technical Sciences, Professor Shakhbaz Alimgireyevich Yershin (5 September 1926–2 August 2019) was one such person.

This year, a grateful scientific community commemorates the centenary of a man who established a distinctive national school of hydroaeromechanics, peered into the heart of raging flames, and put the Navier–Stokes equations to work for industry.

Drawing on the monograph Paradoxes of Aerohydrodynamics and publications by members of Yershin’s scientific school, this review discusses flows of viscous and viscoplastic media in channels, the effects of dissipative heating on viscosity and flow resistance, and the hydrodynamic conditions contributing to localised damage in oilfield pipelines. It examines transport through permeable walls, gas distribution in catalytic apparatuses, the development of free jets, and the evolution of vortical structures. Particular attention is paid to the effects of temperature on the structure of gas jets and the interplay between mixing, ignition, and fuel burnout during supersonic combustion. The review shows that apparent contradictions between conventional understanding and actual flow behaviour can be explained by accounting for interconnected physical processes and the limits of applicability of computational models. It emphasises the importance of comparing theory with experiment, distinguishing qualitative explanations from quantitative validation, and the role of a scientific school in cultivating a research culture. The review highlights the practical significance of Yershin’s legacy for the energy sector, the oil industry, and the improvement of technological apparatuses.