Topological Theory Research Articles

Periodic Solutions of Certain Higher Order Autonomous Differential Equations via Topological Degree Theory

Periodic Solutions of Certain Higher Order Autonomous Differential Equations via Topological Degree Theory

Resurgence, Stokes constants, and arithmetic functions in topological string theory

Resurgence, Stokes constants, and arithmetic functions in topological string theory

On periodic Ambrosetti-Prodi-type problems

<abstract><p>This work presents a discussion of Ambrosetti-Prodi-type second-order periodic problems. In short, the existence, non-existence, and multiplicity of solutions will be discussed on the parameter $ \lambda $. The arguments rely on a Nagumo condition, to guarantee an apriori bound on the first derivative, lower and upper-solutions method, and the Leray-Schauder's topological degree theory. There are two types of new results based on the parameter's variation: An existence and non-existence theorem and a multiplicity theorem, proving the existence of a bifurcation point. An application to a damped and forced pendulum is studied, suggesting a method to estimate the critical values of the parameter.</p></abstract>

On the boundedness of the solution set for the $ \psi $-Caputo fractional pantograph equation with a measure of non-compactness via simulation analysis

<abstract><p>A large number of physical phenomena can be described and modeled by differential equations. One of these famous models is related to the pantograph, which has been investigated in the history of mathematics and physics with different approaches. Optimizing the parameters involved in the pantograph is very important due to the task of converting the type of electric current in the relevant circuit. For this reason, it is very important to use fractional operators in its modeling. In this work, we will investigate the existence of the solution for the fractional pantograph equation by using a new $ \psi $-Caputo operator. The novelty of this work, in addition to the $ \psi $-Caputo fractional operator, is the use of topological degree theory and numerical results from simulations. Techniques in fixed point theory and the use of inequalities will also help to prove the main results. Finally, we provide two examples with some graphical and numerical simulations to make our results more objective. Our data indicate that the boundedness of the solution set for the desired problem depends on the choice of the $ \psi(\kappa) $ function.</p></abstract>

Existence of solutions for Caputo fractional iterative equations under several boundary value conditions

<abstract><p>In this paper, we investigate the existence and uniqueness of solutions for nonlinear quadratic iterative equations in the sense of the Caputo fractional derivative with different boundary conditions. Under a one-sided-Lipschitz condition on the nonlinear term, the existence and uniqueness of a solution for the boundary value problems of Caputo fractional iterative equations with arbitrary order is demonstrated by applying the Leray-Schauder fixed point theorem and topological degree theory, where the solution for the case of fractional order greater than 1 is monotonic. Then, the existence and uniqueness of a solution for the period and integral boundary value problems of Caputo fractional quadratic iterative equations in $ R^N $ are also demonstrated. Furthermore, the well posedness of the control problem of a nonlinear iteration system with a disturbance is established by applying set-valued theory, and the existence of solutions for a neural network iterative system is guaranteed. As an application, an example is provided at the end.</p></abstract>

On a general class of $ n $th order sequential hybrid fractional differential equations with boundary conditions

<abstract><p>This manuscript is related to consider a general class of $ n $th order sequential hybrid fractional differential equations (S-HFDEs) with boundary conditions. With the help of the coincidence degree theory of topology, some appropriate results for the existence theory of the aforementioned class are developed. The mentioned degree theory is a powerful tool to investigate nonlinear problems for qualitative theory. A result related to Ulam-Hyers (U-H) stability is also developed for the considered problem. It should be kept in mind that the considered degree theory relaxes the strong compact condition by some weaker one. Hence, it is used as a sophisticated tool in the investigation of the existence theory of solutions to nonlinear problems. Also, an example is given.</p></abstract>

Retracted: Wireless Sensor Network Topology Theory for Data Collection and Analysis of Sports Training Human Body

Retracted: Wireless Sensor Network Topology Theory for Data Collection and Analysis of Sports Training Human Body

매듭사영의 BLL 그래프의 정의와 그 응용에 대한 연구

This paper is the research on the Knot theory in Topology. A knot is a simple closed curve in ℝ and its projection onto a plane in ℝ is called a knot projection. As the results of this paper we define a BLL(Bidirectional Linear Link) graph for a knot projection which is a bidirectional linear link representing the relations between arcs of a knot projection and obtain some properties of the BLL graphs. We also define an Eulerian cycle of the BLL graph and an Eulerian cycle of a knot projection. As the main results of this paper, we obtain the equivalent conditions of being an alternation knot projection as follows: (1) an out-degree of every vertex of the corresponding BLL graph is 2; (2) the corresponding BLL graph has an Eulerian cycle; (3) the knot projection has an Eulerian cycle. As the subsequent study, using these results of the BLL graphs, we propose the analysis on the BLL graphs for deformation operation obtaining a new alternating knot projection, decision on the tricolorability of a knot projection, and a polynomial of a knot projection.

High-throughput first-principle prediction of collinear magnetic topological materials

The success of topological band theory and symmetry-based topological classification significantly advances our understanding of the Berry phase. Based on the critical concept of topological obstruction, efficient theoretical frameworks, including topological quantum chemistry and symmetry indicator theory, were developed, making a massive characterization of real materials possible. However, the classification of magnetic materials often involves the complexity of their unknown magnetic structures, which are often hard to know from experiments, thus, hindering the topological classification. In this paper, we design a high-throughput workflow to classify magnetic topological materials by automating the search for collinear magnetic structures and the characterization of their topological natures. We computed 1049 chosen transition-metal compounds (TMCs) without oxygen and identified 64 topological insulators and 53 semimetals, which become 73 and 26 when U correction is further considered. Due to the lack of magnetic structure information from experiments, our high-throughput predictions provide insightful reference results and make the step toward a complete diagnosis of magnetic topological materials.

Spectroscopic ellipsometry-based investigations into the scattering characteristics of topologically distinct photonic stopbands

Topological band theory provides a framework to establish the equivalence/inequivalence of bandgaps in photonic topological insulators. However, experimental discernment of bandgap topological characteristics encounters inherent measurement complexities, particularly beyond the terahertz frequencies. To surmount this difficulty, we resort to the prolific optical technique of spectroscopic ellipsometry and carry out detailed experimental examination of attributes of one-dimensional photonic crystal stopbands and, in consequence, identify an appropriate classifier of the implicit topological characteristics. It is found that governed by the bulk topology, the band edge locations in the dispersion diagram provide a conditional site for the appearance of zeros of a complex reflection ratio. This leads to a selective appearance of topologically robust phase singularities with integer (unity positive) topological charge. We demonstrate that the presence of these phase singularities on either the blue or the red band edges of the stopbands provides us with an experimental marker of their distinctive topological characteristics.

Hausdorff of a cycle in topological graph

The main purpose of the paper is to bring together two areas in which strong relation, graph theory and topological spaces. And derive interesting formula for the set that contains all minimal dominating sets(MDS) and ( -set). Some separation axioms are discussed in topological graph theory, especially in a cycle graph Cn.

Nonexistence and Existence of Solutions with Prescribed Norms for Nonlocal Elliptic Equations with Combined Nonlinearities

In this paper, we study the nonlocal equation ∫RN|u(x)|2dxγΔu=λu+μ|u|q−2u+|u|p−2u, xinRN having a prescribed mass ∫RN|u(x)|2dx=c2, where N≥3, μ, γ∈(0,+∞), q∈(2,2∗), c is a positive constant, p, q∈(2,2∗) with p≠q and 2∗=2NN−2. This research is meaningful from a physical point of view. Using variational methods, we present some results on the nonexistence and existence of solutions under different cases p and q which improve upon the previous ones via topological theory.

Synthesis and Crystal Structure of Adamantylated 4,5,6,7-Tetrahalogeno-1H-benzimidazoles Novel Multi-Target Ligands (Potential CK2, M2 and SARS-CoV-2 Inhibitors); X-ray/DFT/QTAIM/Hirshfeld Surfaces/Molecular Docking Study.

A series of new congeners, 1-[2-(1-adamantyl)ethyl]-1H-benzimidazole (AB) and 1-[2-(1-adamantyl)ethyl]-4,5,6,7-tetrahalogeno-1H-benzimidazole (Hal=Cl, Br, I; tClAB, tBrAB, tIAB), have been synthesized and studied. These novel multi-target ligands combine a benzimidazole ring known to show antitumor activity and an adamantyl moiety showing anti-influenza activity. Their crystal structures were determined by X-ray, while intermolecular interactions were studied using topological Bader's Quantum Theory of Atoms in Molecules, Hirshfeld Surfaces, CLP and PIXEL approaches. The newly synthesized compounds crystallize within two different space groups, P-1 (AB and tIAB) and P21/c (tClAB and tBrAB). A number of intramolecular hydrogen bonds, C-H⋯Hal (Hal=Cl, Br, I), were found in all halogen-containing congeners studied, but the intermolecular C-H⋯N hydrogen bond was detected only in AB and tIAB, while C-Hal⋯π only in tClAB and tBrAB. The interplay between C-H⋯N and C-H⋯Hal hydrogen bonds and a shift from the strong (C-H⋯Cl) to the very weak (C-H⋯I) attractive interactions upon Hal exchange, supplemented with Hal⋯Hal overlapping, determines the differences in the symmetry of crystalline packing and is crucial from the biological point of view. The hypothesis about the potential dual inhibitor role of the newly synthesized congeners was verified using molecular docking and the congeners were found to be pharmaceutically attractive as Human Casein Kinase 2, CK2, inhibitors, Membrane Matrix 2 Protein, M2, blockers and Severe Acute Respiratory Syndrome Coronavirus 2, SARS-CoV-2, inhibitors. The addition of adamantyl moiety seems to broaden and modify the therapeutic indices of the 4,5,6,7-tetrahalogeno-1H-benzimidazoles.

Radial positive solutions for (p(x),q(x))-Laplacian systems

In this paper, we study the existence of radial positive solutions for nonvariational elliptic systems involving the p(x)-Laplacian operator, we show the existence of solutions using Leray-Schauder topological degree theory, sustained by Gidas-Spruck Blow-up technique.

Topological gauge theory for mixed Dirac stationary states in all dimensions

We derive the universal real-time $\text{U}(1)$ topological gauge field action for mixed Dirac states of fermions, described by a density matrix weighted by a gapped Dirac operator, in all dimensions. The key prerequisites are charge quantization and charge conservation; in fact, charge quantization is the root of the action's topological properties. The gauge action encodes nonquantized linear responses as expected for mixed states, but also quantized nonlinear responses, which are accessible in experiment. Our construction furthermore demonstrates how the physical pictures of anomaly inflow and bulk-boundary correspondence extend to nonequilibrium systems. Finally, we demonstrate the independence of the gauge action of the underlying equilibrium or nonequilibrium nature of dynamics stabilizing the state.

Topology design for leader-follower coherence in noisy asymmetric networks

In this paper, we aim to study the effect of the leader’s positions in leader-follower coherence quantified by the spectrum in noisy asymmetric networks with a set of hub nodes. In order to compare the impact of leader selection in different ways on the studied coherence, we choose a family of ring-trees networks to conveniently assign the leaders and hubs. Based on the regular network topology and matrix theories, we obtain analytical solutions for the leader-follower coherence regarding network parameters and the number of leaders. Using these expressions, we then obtain exact relations among the coherences and show that the leader’s positions and network parameters have a profound impact on the coherence. More specifically, the network with one hub displays better coherence than the networks with two hubs. In addition, two adjacent and nonadjacent hubs lead to distinct performance of leader-follower consensus dynamics that depends on network parameters and assigned leaders in the ring or the tree network.

Pseudo links and singular links in the Solid Torus

In this paper we introduce and study the theories of pseudo links and singular links in the Solid Torus, ST. Pseudo links are links with some missing crossing information that naturally generalize the notion of knot diagrams, and that have potential use in molecular biology, while singular links are links that contain a finite number of self-intersections. We consider pseudo links and singular links in ST and we set up the appropriate topological theory in order to construct invariants for these types of links in ST. In particular, we formulate and prove the analogue of the Alexander theorem for pseudo links and for singular links in ST. We then introduce the mixed pseudo braid monoid and the mixed singular braid monoid, with the use of which, we formulate and prove the analogue of the Markov theorem for pseudo links and for singular links in ST. \smallbreak Moreover, we introduce the pseudo Hecke algebra of type A, $P\mathcal{H}_n$, the cyclotomic and generalized pseudo Hecke algebras of type B, $P\mathcal{H}_{1, n}$, and discuss how the pseudo braid monoid (cor. the mixed pseudo braid monoid) can be represented by $P\mathcal{H}_{n}$ (cor. by $P\mathcal{H}_{1, n}$). This is the first step toward the construction of HOMFLYPT-type invariants for pseudo links in $S^3$ and in ST. We also introduce the cyclotomic and generalized singular Hecke algebras of type B, $S\mathcal{H}_{1, n}$, and we present two sets that we conjecture that they form linear bases for $S\mathcal{H}_{1, n}$. Finally, we generalize the bracket polynomial for pseudo links in ST.

ИННОВАЦИОННАЯ КОНСОЛИДАЦИЯ КАК ИНДИКАТОР ИННОВАЦИОННОГО РАЗВИТИЯ РЕГИОНАЛЬНОГО ПРОСТРАНСТВА (ПО ИТОГАМ ЭКСПЕРТНОГО ОПРОСА В РЕСПУБЛИКЕ КРЫМ)

Objective of the study is to identify the ideolog­ical foundations of social consolidation and in­novative development of the Crimean region to determine the dominant idea of innovative con­solidation. Innovative development of Russian regions is in the zone of influence of the inertial factor that determines promising scenarios of regional development in the format of an iner­tial model. This model is not capable of ensur­ing a qualitative breakthrough of Russia in the global space and the effective development of local regional communities. In this regard, the question of the causes of the current situation and the prospects for overcoming it is acute. This paper proposes a model of innovative con­solidation, tested as an experimental one during an expert survey in the Republic of Crimea.   The methodological basis of the research con­sists of the provisions of the spatial approach to the study of the region and regional processes and the theory of social topology of the region. Research results. They allowed us to formulate the main idea of innovative consolidation of the region as “Russia is an economic power” in the ratio of the meanings of economic breakthrough and the greatness of the Russian power. This type of innovative consolidation, which is be­ing implemented at the present stage, is of an inertial nature, since it basically refers to the schemes of innovative development that existed in the history of Russia. This is the reason for the low efficiency of the innovative develop­ment of the Crimea.   Prospects of the study. They are connected with the implementation of a series of empirical measurements within the presented conceptual boundaries in other regions of Southern Russia in order to determine the prospects for the for­mation of an innovation consolidation model adequate to the interests and needs of the re­gional society.  

Introduction to Mirror Symmetry in Aspects of Topological String Theory

Under the compactification by the Calabi-Yau threefold, the string theory shows there is duality called mirror symmetry, which implies there is an isomorphism between two string theories under the compactifications of two topologically different internal manifolds. By twisting the topological string theory in two methods, the twisted theories named A-model and B-model have an isomorphism to each other under the mirror symmetry.

Separated and complete adelic models for one-dimensional Noetherian tensor-triangulated categories

We prove the existence of various adelic-style models for rigidly small-generated tensor-triangulated categories whose Balmer spectrum is a one-dimensional Noetherian topological space. This special case of our general programme of giving adelic models is particularly concrete and accessible, and we illustrate it with examples from algebra, geometry, topology and representation theory.