Singular Theory Research Articles

Equivariant Landau-Ginzburg mirror symmetry

We give a new proof of the computation of Hodge integrals we have previously obtained for the quantum singularity (FJRW) theory of chain polynomials. It uses the classical localization formula of Atiyah--Bott and we phrase our proof in a general framework that is suitable for future studies of gauged linear sigma models (GLSM). As a by-product, we obtain the first equivariant version of mirror symmetry without concavity, generalizing the work of Chiodo--Iritani--Ruan on the Landau--Ginzburg side.

Optimal Robust Filter of Uncertain Fractional Order Systems: A Penalized Deterministic Approach

Developing accurate and optimum state estimation methods for fractional order systems is highly relevant since it provides vital information related to memory effects. The optimum estimation of these systems can be guaranteed using the Kalman filter (KF) when all parameter matrices are not subject to uncertainties. Nevertheless, this fundamental principle of the filter is violated, and its performance can be degraded when the model is uncertain. In this light and to limit this deterioration, the present study introduces an optimal solution for filtering uncertain fractional order systems, which operates as follows. First, using the robust regularized least-squares (RLSs) problem combined with penalty functions, a robust penalty game approach is proposed. Then, in an independent framework of any auxiliary parameters, unified recursive Riccati equation and optimal robust filter are derived subject to norm-bounded uncertainties of parameter matrices. To accomplish this step, the stability and convergence analysis of the filter are illustrated based on the singular theory conception.

Investigation of ruled surfaces and their singularities according to Blaschke frame in Euclidean $ 3 $-space

<abstract><p>In this paper, we study the singularities on a non-developable ruled surface according to Blaschke's frame in the Euclidean 3-space. Additionally, we prove that singular points occur on this kind of ruled surface and use the singularity theory technique to examine these singularities. Finally, we construct an example to confirm and demonstrate our primary finding.</p></abstract>

Uniqueness criteria for initial value problem of conformable fractional differential equation

<abstract><p>This paper presents four uniqueness criteria for the initial value problem of a differential equation which depends on conformable fractional derivative. Among them is the generalization of Nagumo-type uniqueness theory and Lipschitz conditional theory, and advances its development in proving fractional differential equations. Finally, we verify the main conclusions of this paper by providing four concrete examples.</p></abstract>

The Main Parameters Affecting the Stability of Rotating Pretwisted Cylindrical Shells

In the paper, the main parameters affecting the stability of a rotating pretwisted cylindrical shell model with a presetting angle are found by the extended singularity theory. Based on the nonlinear second-order ordinary differential equations of rotating pretwisted cylindrical shell, the universal unfoldings of these equations are given, such that the derivative terms of these equations are treated as periodic perturbation parameters. Analysis results show that these equations have: one form of the universal unfolding codimension 4; one form of the universal unfolding codimension 2; one form of the universal unfolding codimension 2. Calculating the transition set of the universal unfolding codimension 2, the transition set covers all parameters which affect the stability of the rotating pretwisted cylindrical shell. The numerical results show that the amplitudes of periodic excitation and part of the functions of the geometric and physical parameters have strong effect on bifurcation and hysteresis of the rotating pretwisted cylindrical shell subjected to small perturbations near the singularity; however, small external disturbances have weak effect on bifurcation and hysteresis. Due to the periodicity of perturbation parameters, settled in different persistent regions by change in other parameters, the governing equations produce chaotic solutions. The present results are validated by comparison with those in the existing literature.

Pseudo‐spherical evolutes of lightlike loci on mixed type surfaces in Minkowski 3‐space

A surface with nonempty timelike, lightlike, and spacelike points in Minkowski 3‐space is a mixed type surface. The mixed type surface has a signature‐changing metric, and its lightlike points can be seen as singularities of such metric. In this paper, we study singular properties of pseudo‐spherical evolutes of lightlike loci on mixed type surfaces. We classify the generic singularities of pseudo‐spherical evolutes by the singularity theory. These singularities and the contact between lightlike loci and model submanifolds are closely associated. We also show the Legendrian duality among pseudo‐spherical evolutes, lightlike loci, and Darboux vectors. Moreover, the singularities of pseudo‐spherical evolutes are studied from the viewpoint of opening map‐germ. Finally, we give an example to demonstrate the theoretical results.

There Are No Real Black Holes!

In this present paper, I concluded that there are no real black holes by analyzing Einstein’s General Theory of Relativity, Quantum Physics and Naked Singularities.

Approach to Energy Objects’ Dynamics Modelling Based on Singular Systems’ Elements

Over the last decades there has been substantial progress on the development of theory and numerical methods for singular systems (known also as descriptor systems, semistate systems, differential alge-braic systems, generalized state-space systems, etc.). The need for such methodsarisen primarily from the increased practical interest for a more general system description which takes the intrinsic physical sys-tem model structure into account. Besides that, many physical process-es are most naturally and easily modelled as mixed systems of differen-tial and algebraic equations (DAE). As the title implies the paper de-scribes the singular systems theory application in power systems dy-namics simulation, particularly considered an alternative method for energy systems’ mathematical models formulation based on the singu-lar systems theory elements with some indicative examples illustrating feasibility and efficiency of this approach

Dissipation driven degeneracies

Dissipation alone can produce counterintuitive topological wave transport that is otherwise absent in a non-dissipative system. This work demonstrates the influence of dissipation on degeneracies that arise in the context of elastic wave transport. The conditions on the parameters necessary to observe non-Hermitian degeneracies in the Bloch spectrum are precisely derived. It will be shown—contrary to the expectation from singularity theory of a linear eigenvalue problem—that a proportionally damped system with commutative damping does not exhibit non-Hermitian degeneracies. The necessity of a non-commutative and non-proportional dissipation model to observe non-Hermitian degeneracies (or exceptional points) is emphasized. Non-proportional dissipation is used to induce a non-Hermitian degeneracy in a local resonance sub-Bragg bandgap of a linear chain, without using negative damping. While Bloch waves are chosen to illustrate the influence of dissipation, the results readily extend to waves in non-periodic media as well as other wave and vibration transport problems.

Assessment of Environmental, Social, Governance and Technogenic Components of Investment Risks

Introduction. To assess the investment attractiveness (IA) and development opportunities for investment objects (IO), non-financial factors characterizing the environmental, social, governance and technogenic (ESGT) features of objects of possible financing have been increasingly used recently. The purpose of this data analysis is to establish how the ESGT-parameters of IO may reflect their financial health and performance prospects in a rapidly changing world. Having built an ESGT risk profile with the help of mathematical models, the IA of the object of study and the strategy of practical measures to increase it are determined. When modeling these processes, one should consider the growing uncertainty of the modern world due to the emergence of new risks; a large number of systemic links between the structures of the modern technosphere; the power-law nature of the distribution of the probability density of catastrophe damage, which decreases more slowly than the Gaussian dependence. In addition, the efficiency of complex production systems is largely determined by the balance of their individual links. Therefore, to assess investment risks, new methods are required to formalize the dependence of IA on ESGT-factors for the integrated management of the level of credit, market, insurance and operational risks under conditions of uncertainty. The purpose of the article is to develop mathematical methods for quantifying IA and determining real costs to improve the management, social and technological structure of IO, and minimize environmental pollution. Results. A mathematical model has been developed for assessing the environmental, social, managerial and technogenic leaving risks of investment, which makes it possible to determine the optimal strategies for increasing the IA of a possible IO. For a comprehensive risk assessment, methods of the theory of singularities of smooth reflections (TOGO) and the method of analysis of hierarchies (MAH) are used. The following algorithm for estimating IA is proposed: 1) determining the indices of the ESGT-components of risk; 2) calculation of bifurcation index values; 3) determination of the weakest links, which are associated with a decrease in IA; 4) identification of priority measures to prevent the reduction of IA or restore it to a predetermined level and minimize the negative impacts of extreme events and ensure sustainable development. Conclusions. The obtained results show that mathematical models based on the use of TOGO and MAH methods are an important tool for estimating IA under conditions of uncertainty. They allow: 1) to calculate the degree of approximation of the parameters characterizing the functioning of the object to their critical values ??when the IA changes; 2) to determine effective controls to minimize the risk of losing IA or minimize the time and losses for returning IA; 3) to consider the uncertainty factor associated with the features of the decision-making process. The development of this work is aimed at creating an information system for assessing IA for the integrated management of the level of credit, market, insurance and operational risks in the face of uncertainty and determining effective scenarios for minimizing investment risks. Keywords: mathematical modeling, system analysis, investment risks.

Singularities

Singularity theory concerns local and global structure of singularities of (algebraic) varieties and maps. As such, it combines tools from algebraic geometry, complex analysis, topology, algebra and combinatorics.

Coupled Fixed Point Theorem for the Generalized Langevin Equation with Four-Point and Strip Conditions

By considering a metric space with partially ordered sets, we employ the coupled fixed point type to scrutinize the uniqueness theory for the Langevin equation that included two generalized orders. We analyze our problem with four-point and strip conditions. The description of the rigid plate bounded by a Newtonian fluid is provided as an application of our results. The exact solution of this problem and approximate solutions are compared.

Singularities for Focal Sets of Timelike Sabban Curves in de Sitter 3-Space

In the theory of cosmology, de Sitter space is the symmetrical model of accelerated expansions of the universe. It is derived from the solution of the Einstein field equation, which has a positive cosmological constant. In this paper, we define the evolutes and focal surfaces of timelike Sabban curves in de Sitter space. We find that de Sitter focal surfaces can be regarded as caustics and de Sitter evolutes corresponding to the locus of the polar vectors of osculating de Sitter subspaces. By using singularity theory, we classify the singularities of the de Sitter focal surfaces and de Sitter evolutes and show that there is a close relationship between a new geometric invariant and the types of singularities. Moreover, the Legendrian dual relationships between the hyperbolic tangent indicatrix of timelike Sabban curves and the focal surfaces are given. Finally, we provide an example to illustrate our main results.

Catastrophe conditions for vector fields in

Practical conditions are given here for finding and classifying high codimension intersection points of n hypersurfaces in n dimensions. By interpreting those hypersurfaces as the nullclines of a vector field in , we broaden the concept of Thom’s catastrophes to find bifurcation points of (non-gradient) vector fields of any dimension. We introduce a family of determinants , such that a codimension r bifurcation point is found by solving the system , subject to certain non-degeneracy conditions. The determinants generalize the derivatives that vanish at a catastrophe of a scalar function F(x). We do not extend catastrophe theory or singularity theory themselves, but provide a means to apply them more readily to the multi-dimensional dynamical models that appear, for example, in the study of various engineered or living systems. For illustration we apply our conditions to locate butterfly and star catastrophes in a second order partial differential equation.

Ulam Stability of Fractional Hybrid Sequential Integro-Differential Equations with Existence and Uniqueness Theory

In this paper, a variety of boundary value problems (BVPs) known as hybrid fractional sequential integro-differential equations (HFSIDs) with two point orders (p,q) are investigated. The uniqueness and existence of the solution are discussed via Banach fixed-point theorems. Certain particular theorems associated with Hyers–Ulam and Hyers–Ulam–Rassias stability to the solution, as well as the uniqueness and existence of the solution of the BVPs are studied. The results are illustrated with some particular examples, and the numerical data are analyzed for confirmation of the results. The results obtained in this work are simple and can easily be applicable to physical systems. Furthermore, symmetry analysis of fractional differential equations and HFSIDs are also presented. This is due to the fact that the aforementioned analysis plays a significant role in both the optimization and qualitative theory of fractional differential equations.

Qualitative analysis and numerical simulation of fractal-fractional COVID-19 epidemic model with real data from Pakistan

The dynamic of the fractal-fractional COVID-19 epidemic model is studied in this article. The future dynamics of the system are tested by taking the fractal-fractional derivative, which is very effective in formulating the real-life situation by their beneficial memory response. The qualitative analysis of solutions of the proposed COVID-19 model having fractal-fractional derivative has been derived. Also the approximate solution scheme for the considered Fractional-Fractal model has also established. The novel operator of fractal-fractional derivative can be applied to formulate several other dynamics in the real life of complex dimensions in the framework of scientific criteria, too. The threshold number is calculated by utilizing the next-generation matrix approach. The values of the parameters are estimated with the help of least square curve fitting tools. The considered fractal-fractional system was analyzed for theoretical investigation, comprising the uniqueness and existence theory along with Ulam Hyers stability. Sensitivity analysis has been done to examine the most reactive parameters with the basic reproduction number. The fractional Adams-Bashforth AB method has been taken into consideration for the numerical solution.

Application of Singularity Theory to the Distribution of Heavy Metals in Surface Sediments of the Zhongsha Islands

This research aimed to use nonlinear theory and technology to describe the spatial distribution of heavy metals in the surface sediments of the Zhongsha Islands Sea region. The goal of this study is to explore the spatial distribution characteristics of heavy metals in the surface sediments of the Zhongsha Islands. The singularity theory and method were used to delineate heavy metal geochemical anomalies and the generalized self-similarity analysis method was used to decompose heavy metal geochemical anomalies and background concentrations. The results showed that there were abnormally high concentrations of heavy metals in the deep-sea plain area and in the western central sand trough area. The results of this study can inform priority areas for environmental monitoring. The element anomalies extracted by the singularity analysis method in this paper can be a guide to the next step in the investigation and provide the basis for the regional environmental assessment.

Electromagnetic ultrasonic signal processing and imaging for debonding detection of bonded structures

This paper investigates an electromagnetic ultrasonic method for debonding detection of adhesive structures. Considering the non-stationary and nonlinear characteristics of the electromagnetic ultrasonic reflection signals, a joint adaptive noise reduction and reconstruction (JANRR) method, namely, complete ensemble empirical mode decomposition(CEEMDAN) and singular spectrum analysis(SSA) based on continuous mean square Error(CMSE) are proposed, in which, mirror extension is used to suppress the endpoint effect of decomposition, and the order of SSA noise reduction is determined based on singular entropy theory. For two side debonding detection of the adhesive layer, a back propagation (BP) neural network classification model is developed, and five characteristics such as signal skewness factor are taken as the network inputs based on Chi square test. Experiments were carried out for validation, which show that the proposed method can distinguish both signals of the upper and lower interfaces of the adhesive layer under different bonding states.

Dynamic Design of a Quad-Stable Piezoelectric Energy Harvester via Bifurcation Theory

The parameter tuning of a multi-stable energy harvester is crucial to enhancing harvesting efficiency. In this paper, the bifurcation theory is applied to quantitatively reveal the effects of structural parameters on the statics and dynamics of a quad-stable energy harvester (QEH). Firstly, a novel QEH system utilizing the geometric nonlinearity of springs is proposed. Static bifurcation analysis is carried out to design quad-stable working conditions. To investigate the cross-well and high-energy vibration, the complex dynamic frequency (CDF) method, suitable for both weakly and strongly nonlinear dynamic problems, is then applied to deduce the primary response solution. By using the unfolding analysis in singularity theory, four steady-state properties and dozens of primary resonance modes are demonstrated. Based on the transition set, the effective bandwidth for energy harvesting can be customized to adapt well to various vibration environments by parametric adjustment. Finally, the experimental tests verify that the output power can reach up to 1 mW. The proposed QEH and its mechanics optimization can guide energy supply for next-generation wireless systems and low-power sensors under magnetic forbidding environments.

Zariski’s dimensionality type of singularities. Case of dimensionality type 2

In the 1970s O. Zariski introduced a general theory of equisingularity for algebroid and algebraic hypersurfaces over an algebraically closed field of characteristic zero. His theory builds up on understanding the dimensionality type of hypersurface singularities, notion defined recursively by considering the discriminants loci of successive “generic” corank 1 1 projections. The theory of singularities of dimensionality type 1, that is the ones appearing generically in codimension 1, was developed by Zariski in his foundational papers on equisingular families of plane curve singularities. In this paper we completely settle the case of dimensionality type 2, by studying Zariski equisingular families of surfaces singularities, not necessarily isolated, in the three-dimensional space.