Representation Of Operators Research Articles

Nonlinear localized modes in a higher-order anisotropic ferromagnetic nanowire with octupole–dipole interaction

Increasing demand for advanced technologies that depends on magnetic phenomena, understanding and controlling the behavior of discrete breather in ferromagnetic nanowires are crucial for enhancing the efficiency and performance of such devices. The presence of octupole–dipole interactions signifies a unique aspect that could potentially influence the stability and localization of breather excitations. Hence, we adopted a multifaceted approach to investigate the Heisenberg anisotropic ferromagnetic nanowire discrete model with the following interactions: bilinear, octupole–dipole, anisotropy and its higher-order terms. The dynamics is governed by a discrete nonlinear Schrödinger equation (DNLS) arrived with the aid of Holstein–Primakoff transformation. This transformation was facilitated by utilizing the Glauber coherent representation of the boson operators. Subsequently, the dynamical equation is incorporated to the Modulational Instability (MI) analysis which is a systematical gateway to explore the breather excitation in the region of instability influenced by the octupole–dipole interaction coupling parameter. Then, we pictorially demonstrated that the octupole–dipole interaction plays a pivotal role in promoting the localization of discrete breather on the surface of the spin lattice sites in the discrete ferromagnetic nanowire. The energy density distribution also implies that the increase in octupole–dipole interaction results in the highly dense breather localization. The result shows that the increment in the octupole–dipole interaction parameter increases the amplitude of the localized breathers. These discrete breathers could hold immense promise for applications in magnetic storage and Spintronic devices, where maintaining stable localized modes is crucial for the device functionality. Our novelty lies in being pioneers in the exploration of a fully discrete model that encompasses higher-order interactions, such as the octupole–dipole interaction. We already have confirmed the existence of instability region on the discrete spin lattice by incorporating the octupole–dipole interaction [T. Pavithra, L. Kavitha, Prabhu and A. Mani, Modulational instability analysis in an isotropic ferromagnetic nanowire with higher order octopole-dipole interaction, in Nonlinear Dynamics and Applications: Proceedings of the ICNDA 2022 (Springer, 2022), p. 1209], we attempting to explore the generation of discrete breathers in a discrete anisotropic ferromagnetic nanowire. This effectively bridges the gap between theoretical understanding and practical implications, paving the way for innovative advancements in magnetic technology.

Matrix Representation of Magnetic Pseudo-Differential Operators via Tight Gabor Frames

In this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^d$$\\end{document} (d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\ge 1$$\\end{document}), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.

Enhancing predictive capabilities in data-driven dynamical modeling with automatic differentiation: Koopman and neural ODE approaches.

Data-driven approximations of the Koopman operator are promising for predicting the time evolution of systems characterized by complex dynamics. Among these methods, the approach known as extended dynamic mode decomposition with dictionary learning (EDMD-DL) has garnered significant attention. Here, we present a modification of EDMD-DL that concurrently determines both the dictionary of observables and the corresponding approximation of the Koopman operator. This innovation leverages automatic differentiation to facilitate gradient descent computations through the pseudoinverse. We also address the performance of several alternative methodologies. We assess a "pure" Koopman approach, which involves the direct time-integration of a linear, high-dimensional system governing the dynamics within the space of observables. Additionally, we explore a modified approach where the system alternates between spaces of states and observables at each time step-this approach no longer satisfies the linearity of the true Koopman operator representation. For further comparisons, we also apply a state-space approach (neural ordinary differential equations). We consider systems encompassing two- and three-dimensional ordinary differential equation systems featuring steady, oscillatory, and chaotic attractors, as well as partial differential equations exhibiting increasingly complex and intricate behaviors. Our framework significantly outperforms EDMD-DL. Furthermore, the state-space approach offers superior performance compared to the "pure" Koopman approach where the entire time evolution occurs in the space of observables. When the temporal evolution of the Koopman approach alternates between states and observables at each time step, however, its predictions become comparable to those of the state-space approach.

Matrix representation of Toeplitz operators on Newton spaces

In this paper, we study several properties of an orthonormal basis {Nn(z)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\{N_{n}(z)\\}$\\end{document} for the Newton space N2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N^{2}({\\mathbb{P}})$\\end{document}. In particular, we investigate the product of Nm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N_{m}$\\end{document} and Nm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N_{m}$\\end{document} and the orthogonal projection P of Nn‾Nm\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\overline{N_{n}}N_{m}$\\end{document} that maps from L2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L^{2}(\\mathbb{P})$\\end{document} onto N2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N^{2}(\\mathbb{P})$\\end{document}. Moreover, we find the matrix representation of Toeplitz operators with respect to such an orthonormal basis on the Newton space N2(P)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$N^{2}({\\mathbb{P}})$\\end{document}.

Tensor networks for solving the time-independent Boltzmann neutron transport equation

Tensor network techniques, known for their low-rank approximation ability that breaks the curse of dimensionality, are emerging as a foundation of new mathematical methods for ultra-fast numerical solutions of high-dimensional Partial Differential Equations (PDEs). Here, we present a mixed Tensor Train (TT)/Quantized Tensor Train (QTT) approach for the numerical solution of time-independent Boltzmann Neutron Transport equations (BNTEs) in Cartesian geometry. Discretizing a realistic three-dimensional (3D) BNTE by (i) diamond differencing in space, (ii) multigroup-in-energy, and (iii) discrete ordinates collocation in angle leads to large generalized eigenvalue problems that generally require a matrix-free approach and large computer clusters. Starting from this discretization, we construct a TT representation of the PDE fields and discrete operators, followed by a QTT representation of the TT cores. We then solve the tensorized generalized eigenvalue problem using a fixed-point scheme with tensor network optimization techniques. We validate our approach by applying the method to two examples of 3D neutron transport problems, currently solved by the Los Alamos National Laboratory PARallel TIme-dependent SN (PARTISN) solver.1 We demonstrate that our TT/QTT method, executed on a standard desktop computer, leads to large compression. This allows for the storage of terrabyte-sized neutron angular flux eigenvectors in megabytes. Additionally, we create megabyte-sized full access TT representations of yottabyte-sized transport matrix operators. By leveraging the TT operators and solution methods, we obtain a 7500 times speedup when compared to the PARTISN solution time with an error of less than 10−5.

Features of calculating the strength of gantry and ship cranes gear units in the probabilistic aspect

The performed study is the result of collecting statistical data on failures of port transshipment and floating cranes. It is found that many accidents of components and parts of crane mechanisms can be associated with an erroneous determination of design loads. In order to increase the strength reliability, numerous field tests were carried out in river and sea ports under various modes of transshipment for different types of cargo. Tests have shown that operational loads are random in nature, and their laws of distribution over time of individual crane mechanisms can be approximated by truncated normal Gauss laws, accounting for which at the stage of new design developments, as shown by the practice of performing strength fatigue calculations, is not possible. In the article, in order to use the obtained laws at the design stage, it is proposed to replace the representation of operational loads with initial moments that are common to the entire crane mechanism, and transfer individual characteristics consideration such as stress concentration coefficients, fatigue curve parameters, operating time or test base, to subsequent stages of machine components development. For practical testing of the operational loads representation by the normal Gauss laws, a critical analysis of their initial moments, which makes it possible to identify inconsistencies in the representations of variable loads proposed for strength calculation, is given. For the corrected truncated normal load distribution law, the initial moments, the value of the order of which corresponds to the formulas structure used in calculating fatigue resistance at variable load amplitudes (in particular, under alternating loading modes), are determined. The presentation of corrected normal distribution laws with initial moments complements the normative documents used in practice when performing strength calculations under variable loading conditions, significantly simplifies calculations and makes it possible to take it into account at the stage of new design developments in accordance with the formulas established in the current normative documents.

Quantum α-fidelity of diagonalizable second-order Hamiltonians in multi-dimensional systems

How to characterize the similarity between two quantum states has received much attention recently. Motivated by the continued push in this direction, we propose a computable analytical formula for the quantum α-fidelity between diagonalizable second-order Hamiltonians, which is simply expressed in terms of the matrix representation of operators. Further, using this analytical formula, we calculate the α-fidelity of the one-dimensional and two-dimensional displaced-squeezed thermal states, the Liu-Tombesi oscillators, and twin-beam amplitude-squeezed states. Since α-fidelity can precisely characterize the approximation accuracy between two quantum states by adjusting the value of the parameter α, our results offer a richer fidelity family to quantify the similarity between diagonalizable second-order Hamiltonians in multi-dimensional systems, and hence may have potential applications in quantum information and quantum computing.

Military logistics planning models for enemy targets attack by a swarm of combat drones

This article describes and investigates the planning aspect of military actions aimed at destroying enemy targets with the help of an attack drone swarm. This study attempts to solve the task of operational-tactical planning of a massive attack on enemy targets with the help of combat drones, which have different combat potential characteristics. It analyzes the problems of unmanned aerial vehicles (UAVs) swarms’ usage, which ensures maximum efficiency during combat operations. The article shows that in order to plan effective military operations, it is necessary to form the following logistical sequence: identification of relevant targets set, formation of drones into a swarm to attack targets, distribution of drones by targets, and planning flight routes of a drone swarm in conditions of military threats. It concludes that for the effective use of a combat drone swarm, it is necessary to plan logistical actions in advance to inflict maximum damage on the enemy and successfully fulfill the operational and tactical goals of the military leadership. The purpose of this study is to create information technology models that will allow planning logistical military actions for the effective use of combat drone swarms to defeat enemy targets. This article describes a systematic representation of logistical military operations for combat drone swarms. It also analyzes enemy targets, which are represented in the form of a priority list with the characteristics of relevance, the necessary combat potential to hit the targets, the risks of approaching the targets, and the flight time of the drones. From the list of targets, a sublist is formed, considering the combat potential of the drone swarm and the necessary potential to defeat the selected enemy targets. The optimization model helps to distribute the swarm of drones into groups to achieve the enemy targets and destroy them. The movement of drones is planned considering flight zones, possible anti-drone actions of the enemy, and the risks of military threats. Any Logic agent simulation platform can be used to create a simulated flight model of a drone swarm to selected enemy targets. Modeling makes it possible to form rational flight routes of a drone swarm under conditions of military threats from the enemy. An example is given to illustrate the formation of logistical actions for planning a massive attack on enemy targets with the help of a drone swarm. The scientific novelty of this study is related to the solution of the urgent problem of planning logistical military operations for the effective use of a combat drone swarm to destroy enemy targets. The results of this study should be used for the operational and tactical planning of logistical military operations to defeat enemy targets with the help of a combat drone swarm.

Hidden Tensor Structures.

Any single system whose space of states is given by a separable Hilbert space is automatically equipped with infinitely many hidden tensor-like structures. This includes all quantum mechanical systems as well as classical field theories and classical signal analysis. Accordingly, systems as simple as a single one-dimensional harmonic oscillator, an infinite potential well, or a classical finite-amplitude signal of finite duration can be decomposed into an arbitrary number of subsystems. The resulting structure is rich enough to enable quantum computation, violation of Bell's inequalities, and formulation of universal quantum gates. Less standard quantum applications involve a distinction between position and hidden position. The hidden position can be accompanied by a hidden spin, even if the particle is spinless. Hidden degrees of freedom are, in many respects, analogous to modular variables. Moreover, it is shown that these hidden structures are at the roots of some well-known theoretical constructions, such as the Brandt-Greenberg multi-boson representation of creation-annihilation operators, intensively investigated in the context of higher-order or fractional-order squeezing. In the context of classical signal analysis, the discussed structures explain why it is possible to emulate a quantum computer by classical analog circuit devices.

Linear to multi-linear algebra and systems using tensors

In the past few decades, multi-linear algebra also known as tensor algebra has been adapted and employed as a tool for various engineering applications. Recent developments in tensor algebra have indicated that several well-known concepts from linear algebra can be extended to a multi-linear setting with the help of a special form of tensor contracted product, known as the Einstein product. Thus, the tensor contracted product and its properties can be harnessed to define the notions of multi-linear system theory where the input, output signals, and the system are inherently multi-domain or multi-modal. This study provides an overview of tensor algebra tools which can be seen as an extension of linear algebra, at the same time highlighting the differences and advantages that the multi-linear setting brings forth. In particular, the notions of tensor inversion, tensor singular value, and tensor eigenvalue decomposition using the Einstein product are explained. In addition, this study also introduces the notion of contracted convolution for both discrete and continuous multi-linear system tensors. Tensor network representation of various tensor operations is also presented. In addition, application of tensor tools in developing transceiver schemes for multi-domain communication systems, with an example of MIMO CDMA system, is presented. This study provides a foundation for professionals whose research involves multi-domain or multi-modal signals and systems.

Finite dimensional irreducible representations of Lie superalgebra D (2, 1; α)

This paper focuses on the finite dimensional irreducible representations of Lie superalgebra D(2, 1; α). We explicitly construct the finite dimensional representations of the superalgebra D(2, 1; α) by using the shift operator and differential operator representations. Unlike ordinary Lie algebra representation, there are typical and atypical representations for most superalgebras. Therefore, its typical and atypical representation conditions are also given. Our results are expected to be useful for the construction of primary fields of the corresponding current superalgebra of D(2, 1; α).

An efficient algorithm for areal morphological filter

This article discusses morphological filtering for assessing a areal parameters of a texture of engineering products. A data organization structure has been proposed that makes it possible to increase the efficiency of an iterative algorithm for enumerating a surface coordinates. Morphological filters complement the standard Gaussian filter to evaluate the functional properties of a surface. The widespread use of the areal morphological filtering in metrological practice is currently hampered by the lack of an effective algorithm. An algorithm based on a matrix representation of morphological operations on the surface coordinates is presented. For this purpose, different indexing of points on a primary surface and a structuring element has been introduced. A sphere or flat segment is used as the structuring element. The high performance of the developed algorithm is ensured due to the fact that all calculations are carried out by enumerating the surface coordinates in one pass, without nested loops, as in some other algorithms. Algorithms for morphological operations of dilation and erosion are presented, on the basis of which a closing and opening filters are constructed. Simulations carried out for various data sets and comparison with known morphological filtering algorithms confirmed the high efficiency of the designed algorithm. The results obtained can be used to analyze the functional properties of product surfaces.

Technical note: Coordinate representations of important differential operations in continuum mechanics

This note is the last part in the trilogy Tensors in Continuum Mechanics. The first part was focused on the representations of m-linear functions on tensor spaces, especially on their duals and transpositions. The second part was dedicated to the memory of Prof. T. Salmi, and it was focused on the curvilinear coordinate systems in continuum mechanics (in Finnish). Because the relationship between stress and their strain conjugates has been discussed in several papers, the focus in the present third part is on a more specific area of continuum mechanics: representations of important differential operations (the gradient, divergence, rotor, and Laplace) in curvilinear coordinate systems.

Information flow in non-unitary quantum cellular automata

The information flow in a quantum system is a fundamental feature of its dynamics. An important class of dynamics are quantum cellular automata (QCA), systems with discrete updates invariant in time and space, for which an index theory has been proposed for the quantification of the net flow of quantum information across a boundary. While the index is rigid in the sense of begin invariant under finite-depth local circuits, it is not defined when the system is coupled to an environment, i.e.for non-unitary time evolution of open quantum systems. We propose a new measure of information flow for non-unitary QCA denoted the information current which is not rigid, but can be computed locally based on the matrix-product operator representation of the map.

Fraunhofer approximation of Fresnel integrals

A method to derive the Fraunhofer diffraction integral by making use of the theory of distributions is presented. With respect to standard textbooks, the pedagogical benefit of this method is that it uniquely uses the far-field approximation of the Fresnel diffraction integral. An operator representation of Fresnel diffraction is also derived.

Two-variable q-Laguerre polynomials from the context of quasi-monomiality

We introduce the theory of two-variable q-Laguerre polynomials by means of a generating function involving zeroth-order q-Bessel Tricomi functions and we establish two-variable q-Laguerre polynomials from the context of quasi-monomiality. Then we obtain the operational representations and q-integrodifferential equations for two-variable q-Laguerre polynomials. Moreover, we introduce mth-order two-variable q-Laguerre polynomials, we deduce the q-partial differential equations and q-integrodifferential equations for mth-order two-variable q-Laguerre polynomials and we also examine the quasi-monomiality characteristics of these new q-special polynomials. Finally, we give some graphical representations of q-Laguerre polynomials.

Methodological Framework for Sustainable Transport Corridor Modeling Using Petri Nets

Current models for evaluating sustainable transport corridors often lack a comprehensive framework capturing the multifaceted performance measures vital for holistic assessment especially at the early stages of operation, when detailed information on the performance indicators of these transport corridors is not available. This gap motivates a Petri net-based modeling approach that integrates key sustainability indicators into a flexible simulation tool. This paper details a versatile methodology harnessing Petri nets, specifically Evaluation Petri nets (E-Nets), to assess corridor sustainability across environmental, social, and economic dimensions. The proposed framework equips planners and policymakers to explore diverse green corridor configurations under varying conditions. Case studies showcase the model’s capabilities in analyzing real-world corridors, identifying performance bottlenecks, and comparing alternative solutions. The model provides a practical decision-support tool to strengthen strategies for efficient, socially responsible, and environmentally sound transport infrastructure. This research advances the theoretical foundations and demonstrates the practical value of Petri nets as an enabling methodology for modeling the intricate dynamics of sustainable transport corridors. This paper demonstrates how E-Nets provide a visual and quantitative representation of transport operations, enabling stakeholders to identify inefficiencies and potential improvements. This paper discusses the theoretical underpinnings of the E-Net modeling of transport corridors, the advantages and limitations of its application, and suggests avenues for future research to enhance the model’s predictive power and real-world application. The paper concludes that the E-Net approach is a scalable, adaptable tool that can significantly con-tribute to the sustainable development of international transport corridors, providing a framework for continuous improvement in alignment with global sustainability objectives.

Representation of a Special Military Operation in the Media of Southern Russia

The following paper based on the analysis of the publications of six leading media outlets in the mega-region of Southern Russia, reveals the features of journalistic work related to the Special Military Operation. The total number of publications that examined by the author is within a three-month period September 2023. The media outlets were selected based on the citation indicators measured by the rating agency. Publications for the3 content analysis were selected chosen according to thematic criteria and keywords “special military operation”, “special operation in Ukraine”. The theoretical basis of the study was formed by the scientific works on agenda setting in the media and including the specifics of regional agenda. The author considered the understanding of an emergency agenda as a list of issues formulated in the media that are subject to the primary public discussion. The scientific significance of the topic mentioned in the study is due to the priority attention that the state and society pays to the armed conflict in Ukraine and medialization of the Special Military Operation as a phenomenon that determines not only the mind-set of the mass audience today, but also the future of our country.

Integral Transforms and Extended Hermite-Apostol Type Frobenius-Genocchi Polynomials

The schemata for applications of the integral transforms of mathematical physics to recurrence relations, differential, integral, integro-differential equations and in the theory of special functions has been developed. The article aims to introduce and present operational representations for a new class of extended Hermite-Apostol type Frobenius-Genocchi polynomials via integral transforms. The recurrence relations and some identities involving these polynomials are established. The article concludes by establishing a determinant form and quasi-monomial properties for the Hermite-Apostol type Frobenius-Genocchi polynomials and for their extended forms.

Simulation of discrete control systems with parallelism of behavior

Objectives. The problem of functional verification of control devices with respect to their design specification is considered. When solving the problems of implementing and testing of discrete systems, one has to deal with the presence of parallelism in the behavior of interacting control objects, which is also displayed in the assignment for designing control systems. The aim of the work is to develop a method for simulating descriptions of such systems, which allows their behavior testing dynamically on the area limited by their possible functioning.Methods. The paper considers a class of control systems with parallelism of the processes occurring in them, which permits linearization of their execution. To specify the behavior of such control systems, it is proposed to use the PRALU language of parallel control algorithms, which is based on Petri nets and which allows to order events occurring during the device operation. An object-oriented approach to simulation of the description of the control algorithm at the transaction level is proposed. For this purpose, a TLM (Transaction-Level Modeling) model has been developed for describing the devices with behavior parallelism in PRALU language. The transaction level model describes a system as a set of interacting processes that run in parallel and determine the behavior of the system over time.Results. The key concepts of the TLM model for simulating the descriptions of control algorithms in the PRALU language are defined: data structure, transactions, processes, and a barrier mechanism for synchronization of parallel processes. A method is proposed for transforming the description of an algorithm in the language into a TLM model, which is based on the representation of language operations as compositions of elementary operations that are performed sequentially. The set of these operations forms the basis for the algorithmic decomposition of a parallel algorithm in PRALU language into intermediate language program that is executed strictly sequentially. Translators of this program into the Verilog and C languages have been developed, the results of their compilation are simulators of the behavior of control system.Conclusion. The proposed simulation method can be used to create a test bench for functional verification of the circuit implementation of control devices with behavior parallelism. In this case, test sequences for verifying the circuit implementation can be generated dynamically – in the process of simulating the description of the algorithm in the PRALU language directly the control device or system, which include the control algorithm and the algorithms of controlled objects behavior.