Academician O.A. Ladyzhenskaya emphasized the importance of constructing a fundamental system in the space of solenoidal functions for simple domains such as squares, cubes, and similar regions. This article examines the problem of constructing such fundamental systems for a four-dimensional parallelepiped and cube. As is well known, applying the stream functions known from the two- and three-dimensional cases, the spectral problem for the Stokes operator reduces to the so-called clamped plate problem, which, in turn, has no solution in domains such as the square, cube, or parallelepiped. Thus, in higher-dimensional cases, the necessity of an analogous stream function becomes evident. In this work, the authors propose two curl operators that satisfy the above-mentioned requirements. Using the introduced curl operators, the spectral problem for the biharmonic operator in a four-dimensional parallelepiped and cube is formulated. Alternative approaches to constructing a fundamental system are presented, given the unsolvability of the spectral problem. Furthermore, the growth orders of the obtained eigenvalues are established.
Kelvin wakes are fluid motions generated by a moving disturbance at a free surface. We present a machine learning-based framework for inferring the properties of such moving disturbances from the Kelvin-wake patterns. We perform phase-resolved simulations to establish a dataset of nearly half a million Kelvin wakes generated by disturbances of varying propagating speed, length scale and geometry. Trained with the augmented data, the neural network achieves accuracies of 99.7% and 92.4% in predicting the velocity and the length scale of the disturbance, respectively, even if a random noise has been added to the training data. The explainability of the neural network is demonstrated by quantifying the contribution of the input data to the prediction, which shows a strong connection with the diverging and transverse waves. The accuracy of the neural network in predicting the disturbance length scale is sensitive to wave nonlinearity.
Telecommunications networks represent a significant fraction of global energy consumption which has resulted in a drive to develop environmentally sustainable lower cost networks. As part of this migration, the efficient depowering of large volumes of legacy equipment located in geographically dispersed exchange buildings is of interest in terms of reducing energy usage and cost. This requires taking into consideration the baseloads of different types of network device, the various activity times required to depower various device types, and the total level of field resource that is available to carry out the tasks within a given time period. A knapsack problem method is used to compare and contrast strategies based on selecting either single devices or aggregates of devices as items for inclusion in the knapsack. The device-based approach enables earlier energy reduction compared to the aggregated approach at the expense of a larger number of site visits. A hybrid of these two approaches is shown to have additional benefit compared to only adopting one of them as part of a depowering programme.
Lo´s’s theorem states that a first-order formula holds in an ultraproduct of structures if and only if it holds in “almost all” factors, where “almost all” is understood in terms of a given ultrafilter. This fundamental result plays a key role in understanding the behavior of first-order properties under ultraproduct constructions. Pseudofinite structures – those that are elementarily equivalent to ultraproducts of finite models–serve as an important bridge between the finite and the infinite, allowing the transfer of finite combinatorial intuition to the study of infinite models. In the context of unary algebras (unars), a classification of unar theories provides a foundation for analyzing pseudofiniteness within this framework. Based on this classification, a characterization of pseudofinite unar theories is obtained, along with several necessary and sufficient conditions for a unar theory to be pseudofinite. Furthermore, various forms of approximation to unar theories are investigated. These include approximations not only for arbitrary unar theories but also for the strongly minimal unar theory. Different types of approximating sequences of finite structures are examined, shedding light on the model-theoretic and algebraic properties of unars and enhancing our understanding of their finite counterparts.
This paper investigates the interaction between Formal Concept Analysis (FCA) and graph theory, with a focus on understanding the structure and representation of concept lattices derived from bipartite directed graphs. We establish connections between the complete formal contexts and their associated bipartite digraphs, providing a foundation for studying modular lattices. Particular attention is given to the structure of concept lattices arising from such contexts and their relationship to the combinatorial properties of the corresponding graphs. The results show that the concept lattice of a complete formal context is isomorphic to a modular lattice of height 2 if and only if its associated bipartite digraph is a disconnected union of bicliques. This establishes a precise correspondence between a specific class of formal contexts and well-studied objects in graph theory. Several examples are presented to illustrate these properties and demonstrate the application of the obtained results. The analysis opens the way for further exploration of lattices associated with more complex graph structures and contributes to a deeper understanding of the relationship between discrete mathematics and formal methods of knowledge representation.
Abstract Out-of-time-ordered-correlators (OTOCs) have been suggested as a means to diagnose chaotic behavior in quantum mechanical systems. Recently, it was found that OTOCs display exponential growth for the inverted quantum harmonic oscillator, mirroring the fact that this system is classically and quantum mechanically unstable. In this work, I study OTOCs for the inverted anharmonic (pure quartic) oscillator in quantum mechanics, finding only oscillatory behavior despite the classically unstable nature of the system. For higher temperature, OTOCs seem to exhibit saturation consistent with a value of –2⟨x 2⟩ T ⟨p 2⟩ T at late times. I provide analytic evidence from the spectral zeta-function and the WKB method as well as direct numerical solutions of the Schrödinger equation that the inverted quartic oscillator possesses a real and positive energy eigenspectrum, and normalizable wave-functions.
Nuclear and particle physics. Atomic energy. Radioactivity
In the present paper, the initial value problem for the hyperbolic type involutory in t second order linear partial differential equation is studied. The initial value problem for the fourth order partial differential equations equivalent to this problem is obtained. The stability estimates for the solution and its first and second order derivatives of this problem are established.
Strongly coupled sequences of shock waves, known as shock trains, are present in high-speed propulsion systems, where the presence of sidewalls substantially modifies the boundary layer thickness, skin friction and streamwise pressure distribution. In the present contribution, scale-resolved numerical simulations are performed on supersonic channel (infinite span) and square duct flows to evaluate the effect of sidewall confinement with and without shock trains. Comparable secondary flow vortices are observed in the duct case with and without the presence of the shock train. The absence of a separation region at the leading shock of the duct case results in lower flow deflection compared with the channel case, leading to a reduced shock strength. The principal effect of the sidewalls is to cause a shock train that is approximately twice as long and composed of a larger number of shocks. A modification of previous models, based on a momentum thickness-based blockage parameter, leads to an improved collapse of the channel and duct cases.
The article is committed to the study of model-theoretic properties of stable hereditary Jonsson theories, wherein we consider Jonsson theories that retain jonssonnes for any permissible enrichment. The paper proves a generalization of stability that relates stability and classical stability for Jonsson spectrum. This paper introduces new concepts such as “existentially finite cover property” and “semantic pair”. The basic properties of e.f.c.p and semantic pairs in the class of stable perfect Jonsson spectrum are studied.
As has been pointed out recently, a possible solution strategy to the wear–fatigue dilemma in fretting, operating on the level of contact mechanics and profile geometries, can be the introduction of “soft” sharp edges to the contact profiles, for example, by truncating an originally smooth profile. In that regard, analysis of possible mechanical failure of a structure, due to the contact interaction, requires the knowledge of the full subsurface stress state resulting from the contact loading. In the present manuscript, a closed-form exact solution for the subsurface stress state is given for the frictional contact of elastically similar truncated cylinders or wedges, within the framework of the half-plane approximation and a local-global Amontons–Coulomb friction law. Moreover, a fast and robust semi-analytical method, based on the appropriate superposition of solutions for parabolic contact, is proposed for the determination of the subsurface stress fields in frictional plane contacts with more complex profile geometries, and compared with the exact solution. Based on the analytical solution, periodic tangential loading of a truncated cylinder is considered in detail, and important scalar characteristics of the stress state, like the von-Mises equivalent stress, maximum shear stress, and the largest principal stress, are determined. Positive (i.e., tensile) principal stresses only exist in the vicinity of the contact edge, away from the pressure singularity at the edge of the profile, and away from the maxima of the von-Mises equivalent stress, or the maximum shear stress. Therefore, the fretting contact should not be prone to fatigue crack initiation.
In November 2021 it was the 110th anniversary of the birth of the outstanding Scientist, Doctor of Science in Physics and Mathematics, Professor E.I. Kim, who made a significant contribution to the development of mathematical science in Kazakhstan, created a school for the study of equations of mathematical physics and raised many students who continue his research.
We consider a boundary value problem for the fractional diffusion equation in an angle domain with a curvilinear boundary. Existence and uniqueness theorems for solutions are proved. It is shown that Holder continuity of the curvilinear boundary ensures the existence of solutions. The uniqueness is proved in the class of functions that vanish at infinity with a power weight. The solution to the problem is constructed explicitly in terms of the solution of the Volterra integral equation.
Abstract We study supersymmetric $$AdS_4$$ A d S 4 black holes in matter-coupled $$N=3$$ N = 3 and $$N=4$$ N = 4 gauged supergravities in four dimensions. In $$N=3$$ N = 3 theory, we consider $$N=3$$ N = 3 gauged supergravity coupled to three vector multiplets and $$SO(3)\times SO(3)$$ S O ( 3 ) × S O ( 3 ) gauge group. The resulting gauged supergravity admits two $$N=3$$ N = 3 supersymmetric $$AdS_4$$ A d S 4 vacua with $$SO(3)\times SO(3)$$ S O ( 3 ) × S O ( 3 ) and SO(3) symmetries. We find an $$AdS_2\times H^2$$ A d S 2 × H 2 solution with $$SO(2)\times SO(2)$$ S O ( 2 ) × S O ( 2 ) symmetry and an analytic solution interpolating between this geometry and the $$SO(3)\times SO(3)$$ S O ( 3 ) × S O ( 3 ) symmetric $$AdS_4$$ A d S 4 vacuum. For $$N=4$$ N = 4 gauged supergravity coupled to six vector multiplets with $$SO(4)\times SO(4)$$ S O ( 4 ) × S O ( 4 ) gauge group, there exist four supersymmetric $$AdS_4$$ A d S 4 vacua with $$SO(4)\times SO(4)$$ S O ( 4 ) × S O ( 4 ) , $$SO(4)\times SO(3)$$ S O ( 4 ) × S O ( 3 ) , $$SO(3)\times SO(4)$$ S O ( 3 ) × S O ( 4 ) and $$SO(3)\times SO(3)$$ S O ( 3 ) × S O ( 3 ) symmetries. We find a number of $$AdS_2\times S^2$$ A d S 2 × S 2 and $$AdS_2\times H^2$$ A d S 2 × H 2 geometries together with the solutions interpolating between these geometries and all, but the $$SO(3)\times SO(3)$$ S O ( 3 ) × S O ( 3 ) , $$AdS_4$$ A d S 4 vacua. These solutions provide a new class of $$AdS_4$$ A d S 4 black holes with spherical and hyperbolic horizons dual to holographic RG flows across dimensions from $$N=3,4$$ N = 3 , 4 SCFTs in three dimensions to superconformal quantum mechanics within the framework of four-dimensional gauged supergravity.
Astrophysics, Nuclear and particle physics. Atomic energy. Radioactivity
S. A. El-Tantawy, S. A. El-Tantawy, Alvaro H. Salas
et al.
In this work, two approaches are introduced to solve a linear damped nonlinear Schrödinger equation (NLSE) for modeling the dissipative rogue waves (DRWs) and dissipative breathers (DBs). The linear damped NLSE is considered a non-integrable differential equation. Thus, it does not support an explicit analytic solution until now, due to the presence of the linear damping term. Consequently, two accurate solutions will be derived and obtained in detail. The first solution is called a semi-analytical solution while the second is an approximate numerical solution. In the two solutions, the analytical solution of the standard NLSE (i.e., in the absence of the damping term) will be used as the initial solution to solve the linear damped NLSE. With respect to the approximate numerical solution, the moving boundary method (MBM) with the help of the finite differences method (FDM) will be devoted to achieve this purpose. The maximum residual (local and global) errors formula for the semi-analytical solution will be derived and obtained. The numerical values of both maximum residual local and global errors of the semi-analytical solution will be estimated using some physical data. Moreover, the error functions related to the local and global errors of the semi-analytical solution will be evaluated using the nonlinear polynomial based on the Chebyshev approximation technique. Furthermore, a comparison between the approximate analytical and numerical solutions will be carried out to check the accuracy of the two solutions. As a realistic application to some physical results; the obtained solutions will be used to investigate the characteristics of the dissipative rogue waves (DRWs) and dissipative breathers (DBs) in a collisional unmagnetized pair-ion plasma. Finally, this study helps us to interpret and understand the dynamic behavior of modulated structures in various plasma models, fluid mechanics, optical fiber, Bose-Einstein condensate, etc.
In this paper we study the interpolation properties of Nikol’skii-Besov spaces with a dominant mixed derivative and mixed metric with respect to anisotropic and complex interpolation methods. An interpolation theorem is proved for a weighted discrete space of vector-valued sequences lαq (A). It is shown that the Nikol’skii-Besov space under study is a retract of the space lαq (Lp). Based on the above results, interpolation theorems were obtained for Nikol’skii-Besov spaces with the dominant mixed derivative and mixed metric.
B. Shayakhmetova, N. Orumbayeva, Sh. Omarova
et al.
One of the most urgent tasks of our time is the problem of teaching in higher education. The task of teachers is to train young people in the field of the latest computer technologies. When teaching, the teacher must also change the methodology of training specialists in higher education in an adequately changing pace of development. Recently, there has been a definite bias towards the creation of software for complex systems, the tasks under consideration are becoming much more complicated, so there are not enough old techniques and methods used earlier to create simple or formalized programs. And in this regard, we will consider in detail the paradigm of object - oriented technology, which is developing at the present time.
We consider topological properties of hypergraphs of models of a theory. The separability of elements in these hypergraphs is characterized in terms of algebraic closures. Similarly we specify the separability of sets by the hypergraphs. The separability of finite sets is characterized for special hypergraphs defined by limit models as well as by countable models which are neither almost prime nor limit.