Topological Theory Research Articles

TOPOLOGICAL DEGREE METHOD FOR A CLASS OF Ψ-CAPUTO FRACTIONAL DIFFERENTIAL LANGEVIN EQUATION

This paper deals with the existence and uniqueness of solution for a new class of Ψ-Caputo fractional differential Langevin equation. The suggested study is based on some basic definitions of topological degree theory and fractional calculus. We established the existence result by using the topological degree method for condensing maps, and by means of Banach’s fixed point theorem we obtained the uniqueness result. As application, we give an illustrative example to demonstrate our theoretical result.

Extended Brauer analysis of some Dynkin and Euclidean diagrams

<p>The analysis of algebraic invariants of algebras induced by appropriated multiset systems called Brauer configurations is a Brauer analysis of the data defining the multisets. Giving a complete description of such algebraic invariants (e.g., giving a closed formula for the dimensions of algebras induced by significant classes of Brauer configurations) is generally a tricky problem. Ringel previously proposed an analysis of this type in the case of Dynkin algebras, for which so-called Dynkin functions were used to study the numerical behavior of invariants associated with such algebras. This paper introduces two additional tools (the entropy and the covering graph of a Brauer configuration) for Brauer analysis, which is applied to Dynkin and Euclidean diagrams to define Dynkin functions associated with Brauer configuration algebras. Properties of graph entropies defined by the corresponding covering graphs are given to establish relationships between the theory of Dynkin functions, the Brauer configuration algebras theory, and the topological content information theory.</p>

EXISTENCE OF SOLUTIONS FOR A SECOND ORDER THREE POINT BOUNDARY VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS ON A HALF LINE

In this paper, we consider the second order three point boundary value problem with integral bounday conditions on a half-line of the form u”(t) + q(t)f(t, u(t), u'(t)) = 0, t∈ (0,+∞), u(0) − au’ (0) = λη 0 u(s)ds, lim t→+∞ u (t) = u (+∞) = λ η0 u(s)ds, where λ > 0, 0 < λη < 1, a > 0, q : (0,+∞) → (0,+∞), f : [0,+∞) × R × R → R is continuous and satisfies Nagumo’s condition. We employ the Schauder’s fixed point theorem, the upper and lower solution method and topological degree theories to show the existence of at least one solution and then existence of three solutions of the above boundary value problem. Finally, we have shown the importance of our results through an examples.

Лигнаны и другие компоненты фитоэкстрактов для лечения фиброзно-кистозной мастопатии

Objective. Comparative analysis of data on various plant extracts and their components, characterized by antitumor and antiinflammatory effects. Materials and methods. Systematic data analysis using topological recognition theory methods, clinical study of patients with fibrocystic mastopathy (FCM). Results. The analysis showed that the anti-inflammatory, antioxidant and antitumor properties of extracts of oleander, hauttuynia cordata, mistletoe, bitter melon, ginkgo biloba, grape seeds, cohosh, chasteberry and common spruce may be useful in the treatment of mastopathy and the prevention of breast cancer. A new class of candidate molecules with special antitumor properties, resinol lignans, has been isolated from Norway spruce extracts. In particular, 7-hydroxymatairesinol can be used for FCM, accompanied by various manifestations of menopausal syndrome, incl. mastalgia. The presented results of the original clinical study showed that during therapy with FCM 7-hydroxymatairesinol, there was a decrease or complete disappearance of menopausal syndrome, pain in the mammary glands and other symptoms. Positive dynamics of estrogen metabolites, their ratios and blood lipid profile indicators were noted, indicating normalization of estrogen metabolism. According to ultrasound data, after therapy with 7-hydroxymatairesinol, a decrease in the number and size of cystic formations in FCM was noted. Conclusion. Despite the fact that lignans from spruce extracts are the least studied, this group of compounds is promising for use in FCM to normalize estrogen metabolism and reduce fibrocystic changes in the mammary glands. Key words: phytoextracts, fibrocystic mastopathy, mastalgia, menopausal syndrome, 7-hydroxymatairesinol

Evaluation and Enhancing Path of the Teaching Quality of English Courses in Colleges and Universities Based on Fuzzy Evaluation Model

Abstract As an important part of the college curriculum, the English course has certain practical significance in the exploration of its curriculum construction. In this paper, we first constructed a teaching quality index system for English courses in colleges and universities, solved the weights of the indexes based on triangular fuzzy hierarchical analysis, and then introduced the theory of topology and simple correlation to improve the comprehensive fuzzy evaluation method for the assessment of the teaching quality of English courses. Finally, an empirical study is conducted to evaluate indicators at every level and their comprehensive quality. According to the study, the first-level indicators have the highest weight of 0.6016 for courses. The teaching quality of English courses in all 20 universities is good, with 53.34% of universities evaluating it as good or above. The teaching quality index of course B among the three English courses is the highest, 81.69. 90% of colleges and universities have received an evaluation grade of medium or above when teaching English courses. Among the indicators, curriculum, teaching, and teaching scholarship were rated as excellent. The evaluation of the course’s pioneering innovation, equipment support, and evaluability was very poor. Optimizing course content, equipment support, and course assessment can improve the quality of teaching English in higher education, according to this study.

CHATGPT FOR COMPUTATIONAL TOPOLOGY.

ChatGPT represents a significant milestone in the field of artificial intelligence (AI), finding widespread applications across diverse domains. However, its effectiveness in mathematical contexts has been somewhat constrained by its susceptibility to conceptual errors. Concurrently, topological data analysis (TDA), a relatively new discipline, has garnered substantial interest in recent years. Nonetheless, the advancement of TDA is impeded by the limited understanding of computational algorithms and coding proficiency among theoreticians. This work endeavors to bridge the gap between theoretical topological concepts and their practical implementation in computational topology through the utilization of ChatGPT. We showcase how a pure theoretician, devoid of computational experience and coding skills, can effectively transform mathematical formulations and concepts into functional codes for computational topology with the assistance of ChatGPT. Our strategy outlines a productive process wherein a mathematician trains ChatGPT on pure mathematical concepts, steers ChatGPT towards generating computational topology codes, and subsequently validates the generated codes using established examples. Our specific case studies encompass the computation of Betti numbers, Laplacian matrices, and Dirac matrices for simplicial complexes, as well as the persistence of various homologies and Laplacians. Furthermore, we explore the application of ChatGPT in computing recently developed topological theories for hypergraphs and digraphs, as well as the persistent harmonic space, which has not been computed in the literature, to the best of our knowledge. This work serves as an initial step towards effectively transforming pure mathematical theories into practical computational tools, with the ultimate goal of enabling real applications across diverse fields.

Nonlinear elliptic equations by topological degree in Musielak-Orlicz-Sobolev spaces

"We prove by using the topological degree theory the existence of at least one weak solution for the nonlinear elliptic equation − div a1(x, ∇u) + a0(x, u) = f (x, u, ∇u) with homogeneous Dirichlet boundary condition in Musielak-Orlicz-Sobolev spaces. Keywords: Nonlinear elliptic equation, Musielak-Orlicz-Sobolev space, topological degree."

Theoretical aspects of designing the components of an aircraft onboard cable network

The article presents theoretical foundations of designing the components of an aircraft onboard cable network, proposed as a methodology for solving the problems of designing onboard equipment. The composition of the initial and developed technical documentation for the aircraft onboard cable network is determined. The article also shows the relationship between the stages of designing the onboard cable network and the stages of designing the aircraft. The formulation of the general task of creating the components of an onboard cable network is presented as a number of particular subtasks of a lower level of complexity. The method for solving the problem is based on a number of transformations performed with the original documentation in order to develop technical documentation for the introduction of the onboard system into the complex of onboard equipment. A number of successive transformations of the proposed bundle model using the apparatus of topology and multiset theory made it possible to form a model of the components of the onboard cable network of an aircraft that gives a detailed view of its future design. The proposed scientific approach to the design of aircraft onboard cable network components created the prerequisites for minimizing the heuristic component in the process of designing aircraft onboard cable network components.

Bifurcation from interval and multiple positive solutions for a nonlinear fractional differential equations with integral boundary conditions

This paper studies the bifurcation from interval and Multiple positive solutions for a boundary-value problem of nonlinear fractional differential equations with integral boundary conditions. Using the bifurcation technique and the topological degree theory, the existence of Multiple positive solutions is investigated and some sufficient conditions are obtained.

Overview and Perspectives of Chaos Theory and Its Applications in Economics

Starting from the contribution of such thinkers as the famous Giordano Bruno (1583) and the great mathematician and physicist Henri Poincaré (1889) and the surprising discovery of the meteorologist Edward Lorenz (1963), we consider the expansion of the mathematics of chaos in this article, paying attention to topology, qualitative geometry, and Catastrophe Theory, on the one hand, and addressing the possibilities derived from the new Computer Science as Quantum Algorithms and the advances in Artificial Intelligence, on the other. We especially highlight the section on computing chaos, which we consider to be new calculation and analysis instruments, such as machine learning and its algorithm called reservoir computing, through which we can know the dynamics of a chaotic system. With past data, with equations like Karamoto–Sivashinsky, one can improve predictions of the system eight times further ahead than in previous methods. Integrating the machine learning approach and traditional model-based prediction, one could obtain accurate predictions twelve Lyapunov times. As we know, in the framework of chaos theory, it is habitually accepted that the idea of long-term prediction seems impossible because we live under a veil of uncertainty. But with technological advances, the landscape begins to change, both in chaos theory and in its applications, especially in the field of economics, to which we devote particular attention, carrying out as an example the analysis of the evolution of the Madrid Stock Exchange in the 2006–2013 crisis. Above all this, a reflection of a general nature is necessary to enlighten us on the possibility of opening a new horizon.

A Deep Learning-Based Knowledge Graph Framework for Intelligent Management Scheduling Decision of Enterprises

Due to the huge magnitude expansion of data volume, the application of cloud computing and the Internet of Things is growing year by year. However, more and more industrial production requires real-time and efficient handling of resource scheduling. Therefore, this paper develops a deep learning-based knowledge graph framework for resource scheduling decision of enterprise. Single-objective and multi-objective problems for computing resources are studied, and the network nodes of computing resources are set with the help of network topology theory. For the single-objective problem, the mathematical model is constructed with the optimization objective of minimizing the time delay. For the multi-objective problem, the mathematical model is constructed with the optimization objectives of minimizing both time delay and energy consumption. Combining the historical scheduling scheme with the introduction of a genetic algorithm, an initial optimization method is proposed for the scheduling problem of mixed flow shop, and the optimization problem is solved to minimize the maximum completion time. The simulation experiments are conducted to evaluate the proposed method, and the obtained results show that the proposal can well realize intelligent management scheduling decision for enterprises.

Thermodynamic topology and photon spheres in the hyperscaling violating black holes

It was shown that a standard ring of light can be imagined outside the event horizon for stationary rotating four-dimensional black holes with axial symmetry using the topological method (Cunha, 2020; Wei and Shao-Wen, 2020). Based on this concept, in this paper, we investigate the topological charge and the conditions of existence of the photon sphere (PS) for a hyperscaling violation (HSV) black hole with various values of the parameters of this model. Then, after carrying out a detailed analysis, we show the conventional topological classes viz Q=−1 for the mentioned black hole and Q=0 for the naked singularities. Also, we propose a new topological class for naked singularities (Q=+1) with respect toz≥1. We also determined thatz≥2, it either shows a naked singularity form with total topological charge+1or has no solution. Therefore, we have the black hole solution only in1≤z<2. Then, we will use two different methods, namely the temperature (Duan’s topological current Φ-mapping theory) and the generalized Helmholtz free energy method, to study the topological classes of our black hole. By considering the black hole mentioned, we discuss the critical and zero points (topological charges and topological numbers) for different parameters of hyperscaling violating black holes, such as (z,θ¯) and other free parameters, and study their thermodynamic topology. We observe that for a given value of the parameters 1≤z<2, θ¯, and other free parameters, there exist two total topological charges (Qtotal=−1,0) with the same phase structure for the T method and total topological numbers (W=+1) for the generalized Helmholtz free energy method. Additionally, we summarize the results for each study as photon sphere, temperature, and generalized Helmholtz free energy in some figures and tables. Finally, we compare our findings with other related studies in the literature.

Twisted holography without conformal symmetry

We discuss the notion of translation-invariant vacua for 2d chiral algebras and relate it to the notion of the associated variety. The two-dimensional chiral algebra associated to four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 U(N) SYM has a conjectural holographic dual involving the B-model topological string theory. We study the effect of non-zero vacuum expectation values on the chiral algebra correlation functions and derive a holographic dual Calabi-Yau geometry. We test our proposal by a large N analysis of correlation functions of determinant operators, whose saddles can be matched with semi-classical configurations of “Giant Graviton” D-branes in the bulk.

Eulerian Spaces

We develop a unified theory of Eulerian spaces by combining the combinatorial theory of infinite, locally finite Eulerian graphs as introduced by Diestel and Kühn with the topological theory of Eulerian continua defined as irreducible images of the circle, as proposed by Bula, Nikiel and Tymchatyn. First, we clarify the notion of an Eulerian space and establish that all competing definitions in the literature are in fact equivalent. Next, responding to an unsolved problem of Treybig and Ward from 1981, we formulate a combinatorial conjecture for characterising the Eulerian spaces, in a manner that naturally extends the characterisation for finite Eulerian graphs. Finally, we present far-reaching results in support of our conjecture which together subsume and extend all known results about the Eulerianity of infinite graphs and continua to date. In particular, we characterise all one-dimensional Eulerian spaces.

The interconnection between local and global m-th powers and Grunwald-Wang theorem

The relation between local and global solution of an equation can be discussed with the method of class field theory and algebraic number theory. In this piece of writing, the author will introduce the behavior of both local and global m-th power in some specific number field. Of course, the result in this paper can be extended into the function field, but it will not be involved in this paper. This paper will prove that if k(_(2^t ))/k is cyclic then the Local m-th powers everywhere is equivalent to the global m-th power. In the Non-cyclic case this decomposition becomes P(m,S)=k^mk^m. This paper will also prove some useful propositions in topological group theory, which will be used in the proof of Grunwald-Wang theorem. Grunwald-Wang theorem states that we can find a cyclic extension with given local behaviors. To describe the extension, this paper combines character theory with a topological group, one can depict the cyclic extension. This theorem can be used in the further exploration of central simple algebra.

Algebraic and order properties of maps and structures related to dependence relations arising in topology, algebra and rough set theory

In this paper, given an arbitrary set [Formula: see text], we study the main order and algebraic properties of some maps and set structures that are strictly related to dependence set relations on [Formula: see text], which are binary relations between subsets of [Formula: see text] naturally arising when [Formula: see text] is a topological space or an attribute set in rough set theory and granular computing based on information systems. The previous maps, that we call granular maps, have the families of the set systems, set operators, binary set relations or also of information systems on the ground set [Formula: see text] as their domain and codomain. We make use of various algebraic methodologies on granular maps to determine the main order-theoretic and combinatorial properties of specific sub-collections of set systems, binary set relations and set operators naturally arising in the investigation of dependence set relations and of rough set theory. We introduce, in more detail, the notion of granular sub-bijection to formalize in all these situations the undefined notion of cryptomorphism, and through which we exhibit new equivalences between specific families of set systems, binary set relations and set operators strictly related to dependence set relations. By means of suitable granular maps we determine three granular sub-bijections between the family of all the closure operators, that of all the Moore set systems and that of all dependence set relations on the same ground set [Formula: see text]. Next, through a property of adjunctivity, we see that in order to generate a dependence set relation it suffices to consider pointed relations on [Formula: see text], namely collections of pairs in [Formula: see text] ℘([Formula: see text]). Because of that, we study order-theoretical properties of some relevant subclasses of pointed relations and analyze the granular maps on [Formula: see text] which determine two nontrivial granular sub-bijections between two subclasses of set operators and two corresponding subclasses of pointed relations. Next, we show that any dependence set relation has the form [Formula: see text] that is a dependence set relation induced by an information system [Formula: see text] on [Formula: see text] and generalizes the Pawlak dependence set relation frequently used in rough set theory. With regard to this representation result, we characterize some set systems of minimal subsets with respect to the Pawlak indiscernibility relation on information systems. Finally, given an arbitrary binary set relation [Formula: see text] on [Formula: see text], we consider the smallest dependence set relation [Formula: see text] on [Formula: see text] containing [Formula: see text] and call it dependence closure of [Formula: see text]. Then, when [Formula: see text] is a finite set, we show how to generate [Formula: see text] in four different and recursive ways by starting from [Formula: see text]. Moreover, again in the finite case, given an information system [Formula: see text] on [Formula: see text], we also determine a binary set relation [Formula: see text] on [Formula: see text] for which [Formula: see text] agrees with [Formula: see text] and whose cardinality is minimum with respect to that of all binary set relations whose dependence closure agrees with [Formula: see text].

P-form electrodynamics as edge modes of a topological field theory

p-form electrodynamics in d ≥ 2 dimensions is shown to emerge as the edge modes of a topological field theory with a precise set of boundary conditions, through the Hamiltonian reduction of its action. Electric and magnetic charges correspond to Noether ones in the topological field theory. For chiral p-forms, the topological action can be consistently truncated, so that the Henneaux-Teitelboim action is recovered from a pure Chern-Simons theory, with a manifestly covariant stress-energy tensor at the boundary. Topologically massive p-form electrodynamics as well as axion couplings are also shown to be described through this mechanism by considering suitable (self-)interaction terms in the topological theory.

Fractional Sobolev Spaces with Kernel Function on Compact Riemannian Manifolds

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability-type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional p(·,·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p(\\cdot , \\cdot )$$\\end{document}-Laplacian operator. As one of the main tools, topological degree theory is applied.

Various matching keys for asymmetric topology encryption

With the fast development of quantum computer technology, one has to focus on the security of information running in real networks. Topological coding is a technology of basic topological structure, number theory and algebra has potential application value in asymmetric encryption system. The public key and private key of topological graph encryption system designed by topological coding may be able to resist attacks equipped with artificial intelligence technology and quantum computer. In order to further investigate topology encoding, we propose new techniques for designing public and private keys using graph coloring and labeling.

Application of topological constraint theory to alkali borate and silicate glass systems

Principles of Topological Constraint Theory (TCT) were applied to alkali borate and silicate glass systems using intermediate range structural models over wide compositional ranges. The structural model for lithium borate was derived from the Feller, Dell, and Bray model [1] and extended to the terminal composition at R=3 where R is the molar ratio of lithium oxide to borate. The sodium borate structural model was built using both NMR [2] and Raman [3] data, and also included carbonate retention in the glass [4]. This model was extended to R=3 similarly to the lithium borate system. The silicate system models were created from 29Si NMR data [5] and also incorporated carbonate retention where necessary [6].Constraint models considered the effect of intermediate range structures on the system, and also incorporated the effect of “loose” alkali which is not directly associated with a non-bridging oxygen. Constraint models of the alkali borate, silicate, and borosilicate systems were then used to predict properties such as glass transition temperature and fragility.