Types Of Singularities Research Articles

Dressing the cusp: how paraxial sharp-edge diffraction theory solves a basic issue in catastrophe optics

The description of light diffraction using catastrophe optics is one of the most intriguing theoretical inventions in the field of classical optics of the last four decades. Its practical implementation has faced some resistance over the years, mainly due to the difficulty of decorating the different (topologically speaking) types of optical singularities (caustics) that concur to build the skeleton on which diffraction patterns stem. Such a fundamental dressing problem has been solved in the past only for the so-called fold, which lies at the bottom of the hierarchy of structurally stable caustics. Climbing this hierarchy implies considerably more challenging mathematical problems to be solved. An ancient mathematical theorem is employed here to find the complete solution of the dressing problem for the cusp, which is placed, in the stable caustic hierarchy, immediately after the fold. The other ingredient used for achieving such an important theoretical result is the paraxial version of the boundary diffraction wave theory, whose tight connection with catastrophe optics has recently been emphasized [Opt. Lett. 41, 3114 (2016)OPLEDP0146-959210.1364/OL.41.003114]. A significant example of the developed algorithm, aimed at demonstrating its effectiveness and ease of implementation, is also presented.

Optical solitons solution for the perturbed nonlinear Schrödinger’s equation

In this manuscript, the perturbed nonlinear Schrödinger’s equation (PNLSE) is considered, which has many implications in various fields such as ferromagnetic material, nonlinear optics, and optical fibers. The focus of this paper is to obtain soliton solutions of the perturbed nonlinear Schrödinger’s equation by implementing two analytical methods, namely, tanh–coth method and energy balance method. As an outcome, a various soliton solutions like, breather solitary wave, lump type soliton in periodic background, singular type soliton, and periodic soliton solution obtain via Mathematica. Additionally, the 2D, 3D, and contour plots are used to visualize the graphical propagation of the achieved soliton solutions by selecting appropriate parametric values.

Reductive quotients of klt singularities

We prove that the quotient of a klt type singularity by a reductive group is of klt type in characteristic 0. In particular, given a klt variety X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X$\\end{document} endowed with the action of a reductive group G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G$\\end{document} and admitting a quasi-projective good quotient X→X//G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X\\rightarrow X/\\!/G$\\end{document}, we can find a boundary B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$B$\\end{document} on X//G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$X/\\!/G$\\end{document} so that the pair (X//G,B)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(X/\\!/G,B)$\\end{document} is klt. This applies for example to GIT-quotients of klt varieties. Our main result has consequences for complex spaces obtained as quotients of Hamiltonian Kähler G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$G$\\end{document}-manifolds, for collapsings of homogeneous vector bundles as introduced by Kempf, and for good moduli spaces of smooth Artin stacks. In particular, it implies that the good moduli space parametrizing n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$n$\\end{document}-dimensional K-polystable smooth Fano varieties of volume v\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$v$\\end{document} has klt type singularities. As a corresponding result regarding global geometry, we show that quotients of Mori Dream Spaces with klt Cox rings are Mori Dream Spaces with klt Cox ring. This in turn applies to show that projective GIT-quotients of varieties of Fano type are of Fano type; in particular, projective moduli spaces of semistable quiver representations are of Fano type.

Explicit construction of a plane sextic model for genus-five Howe curves, II

A Howe curve is defined as the normalization of the fiber product over a projective line of two hyperelliptic curves. Howe curves are very useful to produce important classes of curves over fields of positive characteristic, e.g., maximal, superspecial, or supersingular ones. Determining their feasible equations explicitly is a basic problem, and it has been solved in the hyperelliptic case and in the non-hyperelliptic case with genus not greater than 4. In this paper, we construct an explicit plane sextic model for non-hyperelliptic Howe curves of genus 5. We also determine the number and type of singularities on our sextic model, and prove that the singularities are generically 4 double points. Our results together with Moriya-Kudo's recent ones imply that for each s∈{2,3,4,5}, there exists a non-hyperelliptic curve H of genus 5 with Aut(H)⊃V4 such that its associated plane sextic has s double points.

Approximate solution of multi-term fractional differential equations via a block-by-block method

In this paper we introduce a block-by-block method for the numerical solution of multi-term fractional differential equations (MFDEs). The main idea is to convert a MFDE to a Volterra integral equation of weakly singular type, to which a well known block-by-block method is applied. We also provide the error analysis and convergence of the method. Finally, numerical examples involving Bagley–Torvik and relaxation-oscillation equations are given to confirm applications and the theoretical results.

Dynamics of fractional optical solitary waves to the cubic–quintic coupled nonlinear Helmholtz equation

This work investigates the dynamics of optical waves to the generalized coupled nonlinear fractional Helmholtz equation with quintic and cubic nonlinear effects. The evolution of broad multicomponent self-trapping beams in Kerr-type nonlinear media is described by the coupled Helmholtz equation, which also take into account spatial dispersion due to non-paraxial effects. This phenomena is particularly relevant to the progressive miniaturization of optics, where the optical wavelength is similar to the beam width. It is essential to integrate non-Kerr terms, such as the self-steepening and the self frequency shift, into the coupled Helmholtz system in order to investigate the propagation of ultrashort optical pulses in the non-paraxial domain. For optical wave solutions, nonlinear ordinary equation of the governing model is achieved by applying the fractional transformation. The different types of the solutions like bright, dark, singular and mixed type solitons are extracted by applying advanced integration methods, namely modified Sardar subequation method and new Kudryashov method. These solutions provide very useful information on how the system operates. The applied approaches are highly efficient and have significant computational capability to efficiently tackle the nonlinear systems. Additionally, we include a diverse array of graphs to demonstrate the physical interpretation of the obtained solutions in relation to a number of significant parameters, thereby highlighting the impact of fractional derivatives. In the context of the proposed model, these visualizations assist with a comprehensive understanding of the solution’s behavior and characteristics. It is anticipated that these solutions may be significant in the study of wave propagation and related fields.

Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type, x″(t)+f(x(t))x′(t)+φ(t)xμ(t)−1xγ(t)=e(t),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ x''(t)+f(x(t))x'(t)+\\varphi (t)x^{\\mu}(t)-\\frac{1}{x^{\\gamma}(t)}=e(t), $$\\end{document} where f:(0,+∞)→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$f:(0,+\\infty )\\rightarrow R$\\end{document} is continuous, which may have a singularity at the origin, the sign of φ(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\varphi (t)$\\end{document}, e(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$e(t)$\\end{document} is allowed to change, and μ, γ are positive constants. By using a continuation theorem, as well as the techniques of a priori estimates, we show that this equation has a positive T-periodic solution when μ∈[0,+∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mu \\in [0,+\\infty )$\\end{document}.

From sight to insight: A multi-task approach with the visual language decoding model

Visual neural decoding aims to unlock the mysteries of how the human brain interprets the visual world through predicting perceived visual information from visual neural activity. While early studies made some progress in decoding visual activity for singular type of information, they failed to concurrently reveal the multi-level interweaving linguistic information in the brain. Here, we developed a novel Visual Language Decoding Model (VLDM) that can simultaneously decode the main categories, semantic labels, and textual descriptions of visual stimuli from visual activities. The large-scale NSD dataset was utilized to ensure the efficiency of the decoding model in joint training and evaluation across multiple tasks. For category decoding, we achieved the effective classification of 12 categories with an accuracy of nearly 70 %, significantly surpassing the chance level. For label decoding, we attained the precise prediction of 80 specific semantic labels with a 16-fold improvement over the chance level. For text decoding, the scores of the decoded text surpassed the corresponding baseline levels by remarkable margins on six evaluation metrics. These results highlight the complexity of how the brain processes visual information and the close connection between visual perception and language cognition. This study contributes significantly to extensive applications in multi-layered brain-computer interfaces, potentially leading to more natural and efficient human-computer interaction experiences.

Comparative study of control moment gyroscopes’ steering laws

Abstract The Control Moment Gyroscope (CMG) is superior to other attitude actuators in terms of torque amplification capabilities. Its operating margin is hampered by the singularity problem, which needs the proper selection of a steering law. This study analyses five steering laws for avoiding or escaping different types of singularity; pseudoinverse, pseudoinverse with the null space approach, singularity robust (SR) inverse, generalized SR inverse, and new generalized SR inverse. This analysis measures the effectiveness of these laws in terms of attitude performance and command generation. Simulation results are provided to show the performance of each steering law, with an emphasis on the benefits and drawbacks of each steering law.

Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties

Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X, we show that under certain conditions on the singularity type of the pair (f, X), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (f, X), i.e. to the existence of a coordinate system such that both f and X are quasihomogeneous with respect to the same positive rational weights.

Optical Solitons for the Dispersive Concatenation Model with Polarization Mode Dispersion by Sardar's Sub-Equation Approach

This paper recovers the spectrum of optical solitons for the dispersive concatenation model with the application of the Sardar sub-equation approach. The single soliton solutions that emerge from this scheme are bright, dark, and of singular type. The parameter constraints for the existence of such solitons are also included.

Reductive covers of klt varieties

In this article, we study G -covers of klt varieties, where G is a reductive group. First, we exhibit an example of a klt singularity admitting a \mathbb{P}{\mathrm{GL}}_{n}(\mathbb{K}) -cover that is not of klt type. Then, we restrict ourselves to G -quasi-torsors, a special class of G -covers that behave like G -torsors outside closed subsets of codimension two. Given a G -quasi-torsor X\rightarrow Y , where G is a finite extension of a torus \mathbb{T} , we show that X is of klt type if and only if Y is of klt type. We prove a structural theorem for \mathbb{T} -quasi-torsors over normal varieties in terms of Cox rings. As an application, we show that every sequence of \mathbb{T} -quasi-torsors over a variety with klt type singularities is eventually a sequence of \mathbb{T} -torsors. This is the torus version of a result due to Greb–Kebekus–Peternell regarding finite quasi-torsors of varieties with klt type singularities. On the contrary, we show that in any dimension there exists a sequence of finite quasi-torsors and \mathbb{T} -quasi-torsors over a klt type variety, such that infinitely many of them are not torsors. We show that every variety with klt type singularities is a quotient of a variety with canonical factorial singularities. We prove that a variety with Zariski locally toric singularities is indeed the quotient of a smooth variety by a solvable group. Finally, motivated by the work of Stibitz, we study the optimal class of singularities for which the previous results hold.

Mean field theory for strongly coupled systems: Holographic approach

In this paper, we develop the holographic mean field theory for strongly interacting fermion systems. We investigate various types of the symmetry-breakings and their effect on the spectral function. We found analytic expressions of fermion Green’s functions in the probe-limit for all types of tensor order parameter fields. We classified the spectral shapes and singularity types from the analytic Green’s function. We calculated the fermions spectral function in the full backreacted background and then compared it with the analytic results to show the reliability of analytic results in the probe limit. The fact that all the main features of the spectral features in the current condensed matter physics including gaps of s-,p- waves, nodal rings and nodal shells, the flat band of dimension 1,2,3, can be obtained in the absence of the lattice as consequences of the order and symmetry breaking pattern, is a pleaseant surprise.

Analysis of convolutional neural networks for fronto-temporal dementia biomarker discovery.

Frontotemporal lobe dementia (FTD) results from the degeneration of the frontal and temporal lobes. It can manifest in several different ways, leading to the definition of variants characterised by their distinctive symptomatologies. As these variants are detected based on their symptoms, it can be unclear if they represent different types of FTD or different symptomatological axes. The goal of this paper is to investigate this question with a constrained cohort of FTD patients in order to see if the heterogeneity within this cohort can be inferred from medical images rather than symptom severity measurements. An ensemble of convolutional neural networks (CNNs) is used to classify diffusion tensor images collected from two databases consisting of 72 patients with behavioural variant FTD and 120 healthy controls. FTD biomarkers were found using voxel-based analysis on the sensitivities of these CNNs. Sparse principal components analysis (sPCA) is then applied on the sensitivities arising from the patient cohort in order to identify the axes along which the patients express these biomarkers. Finally, this is correlated with their symptom severity measurements in order to interpret the clinical presentation of each axis. The CNNs result in sensitivities and specificities between 83 and 92%. As expected, our analysis determines that all the robust biomarkers arise from the frontal and temporal lobes. sPCA identified four axes in terms of biomarker expression which are correlated with symptom severity measurements. Our analysis confirms that behavioural variant FTD is not a singular type or spectrum of FTD, but rather that it has multiple symptomatological axes that relate to distinct regions of the frontal and temporal lobes. This analysis suggests that medical images can be used to understand the heterogeneity of FTD patients and the underlying anatomical changes that lead to their different clinical presentations.

Bifurcation, Traveling Wave Solutions and Dynamical Analysis in the (2+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2+1)$$\\end{document}-Dimensional Extended Vakhnenko–Parkes Equation

This paper is concerned with the bifurcation of the traveling wave solutions, as well as the dynamical behaviors and physical property of the soliton solutions of the (2+1)-dimensional extended Vakhnenko–Parkes (eVP) equation. Firstly, based on the traveling wave transformation, the planar dynamical system corresponding to the (2+1)-dimensional eVP equation is derived, and then the singularity type and trajectory map of this system are obtained and analyzed. Based on the bifurcation of this system, the analytical expression for the periodic wave solution is given and shown graphically. Secondly, the N-soliton solutions are obtained via the bilinear method, and some important physical quantities and asymptotic analysis of one-soliton and two-soliton solutions are discussed. The results obtained in this paper might be useful for understanding the propagation of high-frequency waves.

Exploring the Efficacy of Exposure and Response Prevention Therapy on Obsessive Compulsive Disorder

Exposure and response prevention therapy (EX/RP) is a well-established treatment method for combatting intrusive obsessions and repetitive, time-consuming compulsions that classify obsessive-compulsive disorder (OCD). Through therapist encouragement and self-monitoring, individuals confront their compulsive fears, in vivo or imaginal, without the ability to alleviate anxiety through obsessive behaviours. The following paper's purpose is to evaluate the efficacy of this therapy and its components for treating individuals diagnosed with OCD. Research showed that response prevention had a promising impact on those as young as 11. Other research emphasized the positive effects of EX/RP on all ages contingent on patients' ability to adhere to the therapy techniques, specifically response prevention. Moreover, studies have found that this singular type of therapy had more effectiveness and fewer disadvantages than other types of anxiety therapies, biological therapies, or integrative therapies. Cumulatively, the studies supported and affirmed EX/RP as a leading treatment method for OCD on the grounds of its efficacy.

Tight Closure, Coherence, and Localization at Single Elements

In this note, a condition (open persistence) is presented under which a (pre)closure operation on submodules (resp. ideals) over rings of global sections over a scheme X can be extended to a (pre)closure operation on sheaves of submodules of a coherent OX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_X$$\\end{document}-module (resp. sheaves of ideals in OX\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}_X$$\\end{document}). A second condition (glueability) is given for such an operation to behave nicely. It is shown that for an operation that satisfies both conditions, the question of whether the operation commutes with localization at single elements is equivalent to the question of whether the new operation preserves quasi-coherence. It is shown that both conditions hold for tight closure and some of its important variants, thus yielding a geometric reframing of the open question of whether tight closure localizes at single elements. A new singularity type (semi F-regularity) arises, which sits between F-regularity and weak F-regularity. The paper ends with (1) a case where semi F-regularity and weak F-regularity coincide, and (2) a case where they cannot coincide without implying a solution to a major conjecture.

Inverse problems for discrete Sturm–Liouville operator having Bessel-type potential

In this paper, inverse problems are studied for Sturm–Liouville difference operators having p 2 − 1 / 4 m 2 type singularity. First, we define the Green's function and the corresponding Weyl function, some of their properties are investigated, and we give a method to find the potential function. Then we give the solution of the inverse nodal problem and present a constructive procedure for the solution of the inverse problem by using nodes.

On Yau sequence over complete intersection surface singularities of Brieskorn type

On Yau sequence over complete intersection surface singularities of Brieskorn type

Fatigue Performance Evaluation of Fiber-Reinforced Asphalt Mixtures with Four-Point Bending Beam and Uniaxial Fatigue Tests

This study aimed to evaluate the fatigue cracking potential of aramid fiber–reinforced asphalt mixtures. Three fiber-reinforced asphalt mixtures (FRAMs) were prepared employing a singular aggregate type (diabase stone), one binder type (PG76-22), and two distinct combinations of aramid fibers at varying dosages. Two types of fiber were used at different dosages by weight of mix: polyolefin/aramid (PFA) fibers at a dosage 0.05% and sasobit-coated aramid (SCA) fibers at dosages of 0.01% and 0.02%. Additionally, an unreinforced (control) mix was also produced for comparison purposes. Each of the mixtures was produced at a batch plant, adhering to the fiber manufacturer’s recommended mixing methods, while maintaining a constant binder content of 5.5% based on the total mix weight. Four-point beam (4PB) fatigue, uniaxial fatigue (UF), and Texas overlay tests (OT) were performed to characterize the fatigue and reflective cracking behavior of FRAM. The 4PB test results suggest that FRAM improved the fatigue life at lower strain levels (400 µε and 600 µε); however, at higher microstrain (800 µε) level, FRAM exhibited lower fatigue life than the control mix. Damage characteristic curves obtained from the UF tests showed better fatigue tolerance for reinforced mixtures, regardless of microstrain level. For cycles to failure under tensile loading, UF and OT test results indicated lower fatigue life of reinforced mixtures compared with the control mix. Therefore, under flexural loading the fiber-reinforced asphalt mixtures tend to improve the fatigue life. However, for direct tension load, reinforced mixtures tend to deteriorate the fatigue life when comparing cycles to failure at constant strain amplitude.