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In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space $ \mathbb{R}^{3} $. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings.

In this article, we define evolutoids and pedaloids of frontals on timelike surfaces in Minkowski 3-space. The evolutoids of frontals on timelike surfaces are not only the generalization of evolutoids of curves in the Minkowski plane but also the generalization of caustics in Minkowski 3-space. As an application of the singularity theory, we classify the singularities of evolutoids and reveal the relationships between the singularities and geometric invariants of frontals. Furthermore, we find that there exists a close connection between the pedaloids of frontals and the pedal surfaces of evolutoids. Finally, we give some examples to demonstrate the results.

This reflection article presents an approach to problematize the notions of ability/disability and the valorization of singularity to foster inclusive sensitivity among teachers in training. From a methodology based on autoethnography, it was proposed that students and educators reflect upon and write about significant experiences in their trajectories, exploring the possibilities of decentering both autoethnography and the context. The analysis of the narratives concludes that the invisibility of singularities has led to naturalized forms of exclusion within the teaching profession. Additionally, the autoethnographic approach contributes to teacher education, not only in understanding singularities but also in denaturalizing forms of exclusion in the realm of sensitivity.

‎In this paper‎, ‎we consider a Sturm-Liouville equation with non-separated boundary conditions on a finite interval‎. ‎We discuss some properties of solutions of the Sturm-Liouville equation‎, ‎where the potential function has a singularity in the finite interval‎. ‎We also calculate eigenvalues and prove the uniqueness of Borg's Theorem of this boundary value problem‎.

Singularities are configurations where the number of degrees of freedom of a robot changes instantaneously. In parallel manipulators, a singularity could reduce the mobility of the end-effector or produce uncontrolled motions of the mobile platform. Thus, a singularity is a critical problem for mechanical design and model-based control. This paper presents a general sensor-based method to identify singularities in the workspace of parallel manipulators with low computational cost. The proposed experimental method identifies a singularity by measuring sudden changes in the end-effector movements and huge increments in the forces applied by the actuators. This paper uses an inertial measurement unit and a 3D tracking system for measuring the end-effector movements, and current sensors for the forces exerted by the actuators. The proposed sensor-based identification of singularities is adjusted and implemented in three different robots to validate its effectiveness and feasibility for identifying singularities. The case studies are two prototypes for educational purposes—a five-bar mechanism and an L-CaPaMan parallel robot—and a four-degree-of-freedom robot for rehabilitation purposes. The tests showcase its potential as a practical solution for singularity identification in educational and industrial robots.

We are concerned with the study of existence and nonexistence of weak solutions for a class of hyperbolic inequalities with a Hardy potential singular on the boundary $ \partial B_1 $ of the annulus $ A = \left\{x\in \mathbb{R}^3: 1 < |x|\leq 2\right\} $, where $ \partial B_1 = \left\{x\in \mathbb{R}^3: |x| = 1\right\} $. A singular potential function of the form $ (|x|-1)^{-\rho} $, $ \rho\geq 0 $, is considered in front of the power nonlinearity. Two types of inhomogeneous boundary conditions on $ (0, \infty)\times \partial B_2 $, $ \partial B_2 = \left\{x\in \mathbb{R}^3: |x| = 2\right\} $, are studied: Dirichlet and Neumann. We use a unified approach to show the optimal criteria of Fujita-type for each case.

In this article, we propose to solve the three-dimensional time-dependent Maxwell equations in a singular axisymmetric domain with arbitrary data. Due to the axisymmetric assumption, the singular computational domain boils down to a subset of R2. However, the electromagnetic field and other vector quantities still belong to R3. Taking advantage that the domain is transformed into a two-dimensional one, by doing Fourier analysis in the third dimension, one arrives to a sequence of singular problems set in a 2D domain. The mathematical tools of such problems have been exposed in [4,19]. Here, we derive a variational method from which we propose an original finite element numerical approach to solve the problem. Numerical experiments are also shown to illustrate that the method is able to capture the singular part of the solution. This approach can also be viewed as a generalization of the Singular Complement Method to three-dimensional problem.

We consider Cauchy problem for ordinary differential equation with solution possessing a sequence of multiple poles. We propose the generalized reciprocal function method. It reduces calculation of a multiple pole to retrieval of a simple zero of accordingly chosen function. Advantages of this approach are illustrated by numerical examples. We propose two representative test problems which constitute interest for verification of other numerical methods for problems with poles.

Painlevé transcendents are usually considered as complex functions of a complex variable, but, in applications, it is often the real cases that are of interest. We propose a classification of the real solutions of Painlevé IV (or, more precisely, symmetric Painlevé IV) using a sequence of symbols describing their asymptotic behavior and singularities. We give rules for the sequences that are allowed (depending on the parameter values). We look in detail at the existence of globally nonsingular real solutions and determine the dimensions of the spaces of such solutions with different asymptotic behaviors. We show that for a generic choice of the parameters, there exists a unique finite sequence of singularities for which Painlevé IV has a two-parameter family of solutions with this singularity sequence. There also exist solutions with singly infinite and doubly infinite sequences of singularities, and we identify which such sequences are possible. We look at the singularity sequences (and asymptotics) of special solutions of Painlevé IV. We also show two other results concerning special solutions: rational solutions can be obtained as the solution of a set of polynomial equations, and other special solutions can be obtained as the solution of a first-order differential equation.

The behavior and bifurcations of solutions to three-dimensional (three-phase) quadratic polynomial dynamical systems (DSs) are considered. The integrability in elementary functions is proved for a class of autonomous polynomial DSs. The occurrence of bifurcations of the type-twisted fold is discovered on the basis and within the frames of the elements of the developed DS qualitative theory. The discriminant criterion applied originally to two-phase quadratic polynomial DSs is extended to three-phase DSs investigated in terms of their coefficient matrices. Specific classes of D- and S-vectors are introduced and a complete description of the symmetry relations inherent to the DS coefficient matrices is performed using the discriminant criterion.

A review of modern methods for effective calculations of Feynman integrals containing both massless propagators and propagators with masses is given. The effectiveness of these methods in various fields of their application is demonstrated by the examples under consideration.

In this paper, we obtain the explicit expression of the supersymmetric algebra associated with the recently proposed massive supermembrane including all surface terms. We formulate the theory as the limit of a supermembrane on a genus-two compact Riemann surface when one of the handles becomes a string attached to a torus. The formulation reduces to a supermembrane on a punctured torus with a “string spike” (in the sense of [1]), attached to it. In this limit, we identify all surface terms of the algebra and give the explicit expression of the Hamiltonian in agreement with the previous formulation of it. The symmetry under area preserving diffeomorphisms, connected and nonconnected to the identity, is also discussed. Only parabolic Sl(2,Z) discrete symmetries are preserved.

In this paper, we have parameterized a timelike (Tlike) circular surface (CIsurface) and have obtained its geometric properties, including striction curves, singularities, Gaussian and mean curvatures. Afterward, the situation for a Tlike roller coaster surface (RCOsurface) to be a flat or minimal surface is examined in detail. Further, we illustrate the approach’s outcomes with a number of pertinent examples.

In this paper, we study the surfaces of osculating circles, which are the sets of all osculating circles at all points of regular curves. Since the surfaces of osculating circles may be singular, it is necessary to investigate the singular points of these surfaces. However, traditional methods and tools for analyzing singular properties have certain limitations. To solve this problem, we define the framed surfaces of osculating circles in the Euclidean 3-space. Then, we discuss the types of singular points using the theory of framed surfaces and show that generic singular points of the surfaces consist of cuspidal edges and cuspidal cross-caps.

In this article, we mainly discuss the local differential geometrical properties of the lightlike Killing magnetic curve $ \mathit{\boldsymbol{\gamma }}(s) $ in $ \mathbb{S}^{3}_{1} $ with a magnetic field $ \boldsymbol{ V} $. Here, a new Frenet frame $ \{\mathit{\boldsymbol{\gamma }}, \boldsymbol{ T}, \boldsymbol{ N}, \boldsymbol{ B}\} $ is established, and we obtain the local structure of $ \mathit{\boldsymbol{\gamma }}(s) $. Moreover, the singular properties of the binormal lightlike surface of the $ \mathit{\boldsymbol{\gamma }}(s) $ are given. Finally, an example is used to understand the main results of the paper.

We show some examples of calculations of massless and massive Feynman integrals.

In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒp,qM⁢u:=-Δ⁢u+up-M⁢|∇⁡u|q=μ{{\mathcal{L}}^{{M}}_{p,q}u:=-\Delta u+u^{p}-M|\nabla u|^{q}=\mu} in a domain Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} where μ is a nonnegative Radon measure, when p>1{p>1}, q>1{q>1} and M≥0{M\geq 0}. We also give conditions under which nonnegative solutions of ℒp,qM⁢u=0{{\mathcal{L}}^{{M}}_{p,q}u=0} in Ω∖K{\Omega\setminus K}, where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.

The existence of positive periodic solutions of the Rayleigh equations $ x''+f(x')+g(x) = e(t) $ with singularities is investigated in this paper. Based on the continuation theorem of coincidence degree theory and the method of upper and lower solutions, the multiple periodic solutions of the singular Rayleigh equations can be determined under the weak conditions of the term $ g $. We discuss both the repulsive singular case and the attractive singular case. Some results in the literature are generalized and improved. Moreover, some examples and numerical simulations are given to illustrate our theoretical analysis.

In this paper we introduce a technique to deal with implicit differential equations exhibiting singularities. Our approach is a geometrical one, we use the concept of contact structure on a manifold associated with the differential equation. In this setting we prove an existence and uniqueness theorem. We also show how it relates to known geometric results for this kind of equation. We also indicate how the method can be implemented by using continuation methods techniques and the BDF (Backward Differentiation Formula).

We address the question of identifying non-smooth points in Vℝ(I){{\bf{V}}}_{{\mathbb{R}}}(I) the real part of an affine algebraic variety. Two simple algebraic criteria will be formulated and proven. As an application, we investigate the configuration spaces of the planar four-bar linkage and the delta robot and prove that all singularities are CS-singularities.

Abstract It is well known that quantum effects may lead to removal of the intrinsic singularity point of back holes. Also, the quintessence scalar field is a candidate model for describing late-time acceleration expansion. Accordingly, Kazakov and Solodukhin considered the existence of back-reaction of the spacetime due to the quantum fluctuations of the background metric to deform a Schwarzschild black hole, which led to a change of the intrinsic singularity of the black hole to a 2-sphere with a radius of the order of the Planck length. Also, Kiselev rewrote the Schwarzschild metric by taking into account the quintessence field in the background. In this study, we consider the quantum-corrected Schwarzschild black hole inspired by Kazakov–Solodukhin’s work, and the Schwarzschild black hole surrounded by quintessence deduced by Kiselev to study the mutual effects of quantum fluctuations and quintessence on the accretion onto the black hole. Consequently, the radial component of the 4-velocity and the proper energy density of the accreting fluid have a finite value on the surface of its central 2-sphere due to the presence of quantum corrections. Also, by comparing the accretion parameters in different kinds of black holes, we infer that the presence of a point-like electric charge in the spacetime is somewhat similar to some quantum fluctuations in the background metric.

Abstract Singularities in the Mohr–Coulomb (MC) yield criterion create numerical difficulties when implemented in the finite element model (FEM). In this research, a two-dimensional (2D) and a three-dimensional (3D) elasto-plastic FEM coupled with isotropic strain hardening–softening or simple strain-softening law incorporating non-associated flow rule including shear band, is used to investigate the effect of singularities, embedment, shape, soil frictional as well as dilation angle on vertically uploaded shallow anchor foundations buried in Toyoura sand. Mohr–Coulomb–Drucker Prager (MC–DP) model provides a more appropriate solution than Drucker Prager–Drucker Prager (DP–DP) model. Besides, the peak resistance factor and the settlement at the ultimate capacity have been found as the functions of embedment, shape, and sand density. Also, the numerical model gives satisfactory agreement with the previous studies. In particular, a simple strain-softening model cannot capture the settlement behavior of anchor foundations in sand efficiently.

The precise tracking 3D trajectory that changes dramatically within the boundary of flight performance is the essential capability of the UAV applied to air combat, which requires that flight control law design be adapted to nonlinear characteristics of flight dynamics and solve the singularities problem during the flight. Continuous trajectory control on the basis of nonlinear dynamic inversion (NDI) is studied. The singularities problem is caused by continuous trajectory expressed by flight velocity and flight-path angles. A sudden change of flight-path axis system happens when aircraft flies along a continuous trajectory which results in singularities. A method for amending flight-path axis system is presented to solve the sudden change problem of flight-path axis system. Finally, the combination of the maneuver generator based on quaternions and amending flight-path axis system realize the precise trajectory tracking control without singularities and obtain excellent tracking characteristics.

Scientific narratives about cosmology often present black holes as frightening objects of both creation and destruction, the centres of which are concealed behind event horizons. According to studies, black holes are capable of distorting time and tearing apart anything that plunges toward them. This article asks what the latest knowledge about the properties of black holes can contribute to philosophical understandings of being and time. Drawing on both scientific and narrative constructions of black holes in books written by physicists, the article analyzes black holes in relation to Martin Heidegger’s philosophy of ontology in Being and Time. The article argues that the topics of concealment, time, and mortality form interrelated points of intersection among scientific constructions of black holes and Heidegger’s philosophy. First, scientific commentaries on the concealment of singularities speak to Heidegger’s interpretation of the ways being is concealed in modern society. Second, with regard to time, the phenomenon of gravitational redshift invokes Heidegger’s criticisms of clock time and, by extension, notions of infinitude. On the other hand, theories about Hawking radiation and the decay of black holes also invoke Heidegger’s thesis that time subsists predominantly as a passing away toward the cessation of existence. Third and finally, Heidegger’s renowned and controversial explication of mortality helps frame black holes symbolically as forward-projections toward the impossibility of existence. Bringing these three subjects together, the article concludes that, as Heideggerian symbols of being and time, black holes represent projections toward the possibility of impossible modes and forms of existence. In this sense, black holes emblematize a relationship between being and time that marks the potential limits of existence, as humanity and science understand it.

Singular graphene metasurfaces, conductivity gratings realized by periodically suppressing the local doping level of a graphene sheet, were recently proposed to efficiently harvest THz light and couple it to surface plasmons over broad absorption bands, thereby achieving remarkably high field enhancement. However, the large momentum wavevectors thus attained are sensitive to the nonlocal behavior of the underlying electron liquid. Here, we extend the theory of singular graphene metasurfaces to account for the full nonlocal optical response of graphene and discuss the resulting impact on the plasmon resonance spectrum. Finally, we propose a simple local-analogue model that is able to reproduce the effect of nonlocality in local-response calculations by introducing a constant conductivity offset, which could prove a valuable tool in the modeling of more complex experimental graphene-based platforms.

This article presents the analysis of a parallel mechanism of type 3-SPS-U. The usual singularity approach is carried out with respect to the Euler angles of the universal joint. However, this approach is computationally expensive especially when stacked structures are analyzed. Thus, the positioning of the mobile platform for the mechanism is analyzed using the theory of Tilt and Torsion (T&T). The singularity-free workspace and the tilt limits of the mechanism are disclosed through this method. These workspaces can then be mapped to the Euler angles of the universal joint and the relation between the T&T space and the Euler space is demonstrated and validated in this study. Initially, simulations are performed using the results of singularity analysis to have a comparison between the T&T space and the Euler space. Experimental validation is then carried out on the prototype of the mechanism to perform a circular trajectory, in line with the simulations. The outcome of this study will be helpful for the integration of the mechanism for a piping inspection robot and also for the analysis of stacked architectures.

In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results.

The approach of the paper is on spacelike circular surfaces in the Minkowski 3-space. A spacelike circular surface is a one-parameter family of Lorentzian circles with a fixed radius regarding a non-null curve, which acts as the spine curve, and it has symmetrical properties. In the study, we have parametrized spacelike circular surfaces and have provided their geometric and singularity properties such as Gaussian and mean curvatures, comparing them with those of ruled surfaces and the classification of singularities. Furthermore, the conditions for spacelike roller coaster surfaces to be flat or minimal surfaces are obtained. Meanwhile, we support the results of the approach with some examples.

In this paper, we explore the gravitational collapse of matter (dust) under the effect of zero-point length l0. During the gravitational collapse, we neglect the backreaction effect of pre-Hawking radiation (in the sense that it is a small effect and cannot prevent the formation of an apparent horizon), then we recast the internal metric of a collapsing star as a closed FRW universe for any spherically symmetric case and, finally, we obtain the minimal value for the scale factor, meaning that the particles never hit the singularity. We argue that the object emerging at the end of the gravitational collapse can be interpreted as Planck stars (black hole core) hidden inside the event horizon of the black hole, with a radius proportional to (GMl02/c2)1/3. Quite interestingly, we found the same result for the radius of the Planck star using a free-falling observer point of view. In addition, we point out a correspondence between the modified Friedmann’s equations in loop quantum gravity and the modified Friedmann’s equation in string T-duality. In the end, we discuss two possibilities regarding the final stage of the black hole. The first possibility is that we end up with Planck-size black hole remnants. The second possibility is that the inner core can be unstable and, due to the quantum tunneling effect, the spacetime can undergo a black-hole-to-white-hole transition (a bouncing Planck star).

The formulations for three-dimensional (3D) scattering from uniaxial objects with a smooth boundary using a multiple infinitesimal dipole method (MIDM) are introduced. The proposed technique uses two sets of infinitesimal dipole triplets (IDTs), including three co-located orthogonally polarized electric infinitesimal dipoles, distributed inside and outside of a scatterer to construct simulated fields. The dyadic Green's functions of uniaxial materials are deployed in the MIDM so as to obtain the simulated fields. The singularity issues in using the uniaxial dyadic Green's functions, which cannot be solved analytically so far for a general uniaxial medium, can be easily eliminated by using the proposed MIDM. In comparison to the traditional single-layered distribution scheme of IDTs, the proposed multiple-layered distribution scheme can handle the scattering from uniaxial objects accurately and efficiently. Several numerical examples are presented to study bistatic radar cross section (RCS) responses under different scenarios. Excellent agreement is achieved by comparing numerical results with those obtained from commercial software packages, while the simulation performance including CPU time and required memory is drastically improved by using the MIDM when computing a general uniaxial material or a relatively larger object. The proposed technique has its merits on simplicity, conciseness and fast computation in comparison to existing numerical methods.

ABSTRAK Penelitian ini mengkaji singularitas semu pada metrik Reissner-Nordström, yang merupakan solusi persamaan medan Einstein untuk model partikel bermuatan. Kajian dilakukan dengan menganalisis titik-titik singular pada metrik, menghitung tensor kelengkungan Riemann, serta menghitung scalar Kretschmann pada titik-titik tersebut. Perhitungan dilakukan dengan bantuan program Maxima. Hasilnya, singularitas nyata hanya terjadi pada r = 0, sedangkan singularitas semu terjadi pada . Singularitas semu tersebut merupakan representasi dari horizon peristiwa. Terdapat tiga kemungkinan situasi pada horizon peristiwa. Hal menarik terdapat pada situasi r = M, dimana terjadi keseimbangan antara massa dan muatan yang memungkinkan tarikan gravitasi dan tolakan elektromagnetik saling meniadakan. Penelitian ini juga menghasilkan persamaan geodesik pada titik-titik yang tidak menghasilkan nilai infinite pada skalar Kretschmaan. Kata Kunci : Kelengkungan Riemann, Metrik Reisner-Nordström, Singularitas, Persamaan Geodesik ABSTRACT This study examines pseudo singularities on the Reissner-Nordström metric which is a solution to Einstein's field equations for charged particle models. The study was carried out by analyzing the singular points on the metric calculating the Riemann curvature tensor, and calculating Kretschmann's scalar at these points. The results show that real singularities only occur at r = 0, whereas pseudo singularity occurs at . There is a point of pseudo singularity that representing the event horizon. There are two possible situations on the event horizon. Interesting things are in the case r = M, where here is a balance between mass and charge which allows gravitational pull and electromagnetic repulsion to cancel each other. This study also yields the geodesic equation point that not yields infinite value of Kretschmaan scalar. Keywords: Riemann tensor, Reissner-Nordström Metrik, Singularities, Geodesik equation

Flying doughnuts (FDs) are exact propagating solutions of Maxwell equations in the form of single-cycle, space-time non-separable toroidal pulses. Here we review their properties and reveal the existence of a complex and robust fine topological structure. In particular, the electric and magnetic fields of the FD pulse vanish across a number of planes, spherical shells and rings, and display a number of point singularities including saddle points and vortices. Moreover, the instantaneous Poynting vector of the field exhibits a large number of singularities, which are often accompanied by extended areas energy backflow.

This paper proposes an approach to derive the Jacobian matrix of a hybrid mechanism by applying a velocity operator to the transformation matrix. This Jacobian matrix is capable of deducing hybrid singularities, which cannot be identified by using the screw-based Jacobian or velocity-based Jacobian. The transformation matrix was obtained based on the algebraic geometry approach, and it becomes the key point since it was used to not only formulate the Jacobian matrix, but also to define the motion type of hybrid mechanisms. In this paper, two hybrid mechanisms were investigated, which were composed of two distinct parallel mechanisms mounted in series. Hybrid Mechanisms 1 and 2 were composed of 3-PRP-3-RPS and 3-RPS-3-PRP (the underlined P is an actuated joint), respectively. The motion types of Hybrid Mechanisms 1 and 2 were determined from the product of the transformation matrices of the 3-PRP and 3-RPS parallel mechanisms, and vice versa. The developed method was employed to establish the Jacobian matrix to which the conditioning index was applied. Therefore, the kinematic performances of the two hybrid mechanisms can be compared for a given bone surgery trajectory within the workspace. It turns out that Hybrid Mechanism 1 has superior performance than that of Mechanism 2, which indicates that Mechanism 1 is better at transmitting power to the moving platform.

In this paper, we focus on a developable surface tangent to a timelike surface along a curve in Minkowski 3-space, which is called the osculating developable surface of the timelike surface along the curve. The ruling of the osculating developable surface is parallel to the osculating Darboux vector field. The main goal of this paper is to classify the singularities of the osculating developable surface. To this end, two new invariants of curves are defined to characterize these singularities. Meanwhile, we also research the singular properties of osculating developable surfaces near their lightlike points. Moreover, we give a relation between osculating Darboux vector fields and normal vector fields of timelike surfaces along curves from the viewpoint of Legendrian dualities. Finally, some examples with symmetrical structures are presented to illustrate the main results.

We present the results of a study of the Poynting vector field generic singularities at the resonant light scattering of a plane monochromatic linearly polarized electromagnetic wave by a subwavelength particle. We reveal the impact of the problem symmetry, the spatial dimension, and the energy conservation law on the properties of the singularities. We show that, in the cases when the problem symmetry results in the existence of an invariant plane for the Poynting vector field lines, a formation of a standing wave in the immediate vicinity of a singularity gives rise to a saddle-type singular point. All other types of singularities are associated with vanishing at the singular points, either (i) magnetic field, for the polarization plane parallel to the invariant plane, or (ii) electric field, at the perpendicular orientation of the polarization plane. We also show that in the case of two-dimensional problems (scattering by a cylinder), the energy conservation law restricts the types of possible singularities only to saddles and centers in the non-dissipative media and to saddles, foci, and nodes in dissipative. Finally, we show that dissipation affects the (i)-type singularities much stronger than the (ii)-type. The same conclusions are valid for the imaginary part of the Poynting vector in problems where the latter is regarded as a complex quantity. The singular points associated with the formation of standing waves are different for real and imaginary parts of this complex vector field, while all other singularities are common. We illustrate the general discussion by analyzing singularities at light scattering by a subwavelength Germanium cylinder with the actual dispersion of its refractive index.

By considering the Moore–Gibson–Thompson (MGT) equation, the current work introduces a modified fractional photothermal model. The construction model is based on the proportional Caputo fractional derivative, which is a new definition of the fractional derivative that is simple and works well. In addition, the theory of heat transfer in semiconductor materials was used in the context of optical excitation transfer and plasma processes. The proposed model was used to investigate the interaction of light and heat within a magnetized semiconductor sphere rotating at a constant angular speed. The Laplace transform was used to obtain solutions for optical excitation induced by physical field variables. Using a numerical method, Laplace transforms can be reversed. The figures show the effects of carrier lifetime, conformable fractional operator, and rotation on thermal and mechanical plasma waves, which are shown in the graphs. The theory’s predictions were compared and extensively tested against other existing models.

Hussain, Yau, and Zuo introduced the Lie algebra Lk(V) from the derivation of the local algebra Mk(V):=On/(g+J1(g)+⋯+Jk(g)). To find the dimension of a newly defined algebra is an important task in order to study its properties. In this regard, we compute the dimension of Lie algebra L5(V) and justify the sharp upper estimate conjecture for fewnomial isolated singularities. We also verify the inequality conjecture: δ5(V)<δ4(V) for a general class of singularities. Our findings are novel and an addition to the study of Lie algebra.

There are generally the mixed type surfaces with lightlike locus in the Lorentz-Minkowski 3-space. To investigate the geometry of lightlike locus, we define slant focal surfaces and slant evolutes associated to the oringinal mixed type surface by using a moving frame field along the lightlike locus defined by Honda etc. We obtain that singularities of slant focal surfaces and slant evolutes depend on the differential geometric properties of the lightlike locus. Furthermore, we investigate the relationship between slant focal surfaces and slant evolutes. We also consider the relationship between slant evolutes and the lightlike locus on the lightcone.

In this paper, we consider Frenet type framed base curves that may have singular points and define one-parameter developable surfaces associated with such curves. By using the singularity theory, we classify the generic singularities of the developable surfaces, which are cuspidal edges and swallowtails. In order to characterize these singularities, two geometric invariants are discovered. At last, an example is given to demonstrate the main results.

Singularities of the Poynting vector field subwavelength patterns in resonant light scattering by nanoparticles are discussed and classified. There are two generic types of the singularities, namely, (i) the singularities related to the vanishing of the magnetic (and/or electric) field at the singular points and (ii) the singularities related to the formation of standing waves in proximity to the singular points. The connection of these types of singularities to the topology of the singular points, space dimension (3D vs. 2D), and energy conservation law are revealed. In particular, it is shown that in 2D cases in non-dissipative media, the energy conservation reduces the possible types of generic singular points to saddles and centers only. In 3D cases, a universal expression connecting different components of the Poynting vector and valid for any generic singularities is derived and numerically checked for various types of singular points.

We investigate the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a positive parameter subject to nonlocal boundary conditions, which contain fractional derivatives and Riemann–Stieltjes integrals. The nonlinearity of the equation is nonnegative, and it may have singularities at its variables. In the proof of the main results, we use the fixed point index theory and the principal characteristic value of an associated linear operator. A related semipositone problem is also studied by using the Guo–Krasnosel’skii fixed point theorem.

Abstract Singularities in optics famously describe a broad range of intriguing phenomena, from vortices and caustics to field divergences near point charges. The diverging fields created by point charges are conventionally seen as a mathematical peculiarity that is neither needed nor related to the description of electromagnetic beams and pulses, and other effects in modern optics. This work disrupts this viewpoint by shifting point charges into the complex plane, and showing that their singularities then give rise to propagating, divergence‐free wavepackets. Specifically, point charges moving in complex space‐time trajectories are shown to map existing wavepackets to corresponding complex trajectories. Tailoring the complex trajectories in this “complex charge paradigm” leads to the discovery and design of new wavepacket families, as well as unprecedented electromagnetic phenomena, such as the combination of both nondiffracting behavior and abruptly‐varying behavior in a single wavepacket. As an example, the abruptly focusing X‐wave–a propagation‐invariant X‐wave‐like wavepacket with prechosen self‐disruptions that enhance its peak intensity by over 200 times–is presented. This work envisions a unified method that captures all existing wavepackets as corresponding complex trajectories, creating a new design tool in modern optics and paving the way to further discoveries of electromagnetic modes and waveshaping applications.

We discuss an approach to the problem of calculating the local topological index of vector fields given on complex spaces or varieties with singularities developed by the author over the past few years. Our method is based on the study of the homology of a contravariant version of the classical Poincaré–de Rham complex. This idea allows not only simplifying the calculations, but also clarifying the meaning of the basic constructions underlying many papers on the subject. In particular, in the graded case, the index can be expressed explicitly in terms of the elementary symmetric polynomials. We also considered some useful applications in physics, mechanics, control theory, the theory of bifurcations, etc.

Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of the first and second kind. In physics, however, commonly the only acceptable members of this set are Legendre polynomials. The quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone and in-depth argument for rejecting all solutions of Legendre’s equation in physics apart from the polynomial ones. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation are quintessential to the argument. We demonstrate that the evenness or oddness of Legendre polynomials is a consequence of the same premises.

The “ER = EPR” conjecture, conceived by Maldacena and Susskind, is grounded on the notion that a gravitational theory in the bulk is dual to the corresponding quantum field theory on the boundary in accordance to the AdS/CFT correspondence. The conjecture pertains to the idea that Einstein-Rosen (ER) spacetime bridges and Einstein-Podolsky-Rosen (EPR) quantum entanglement may be considered as dually equivalent. Since ER bridges refer to the connectivity between black holes, the “ER = EPR” conjecture implies that black holes connected by ER bridges are entangled, and conversely, that entangled black holes are connected by ER bridges. However, the instance of the maximally entangled tripartite (GHZ) quantum state points to the necessity of devising a model of non-classical Planck scale ER bridges going beyond the standard description of these bridges in spacetime. Based on the topological structure of the maximally entangled GHZ state, we propose that a universal topological link, called the Borromean rings, furnishes a particular linking structure that is able to unravel the equavalence between entanglement and wormholes, and thus, to address the validity of the “ER = EPR” conjecture beyond the initial context of the AdS/CFT correspondence. As a consequence, we propose the explicit construction of distinguishable extensions of the smooth classical spacetime manifold taking place in the transition to the quantum gravity regime according to a naturally induced physical criterion of gravitational generic properties following from this intrinsic topological qualification of the “ER = EPR” conjecture.

We study the intrinsic geometry of area minimizing hypersurfaces from a new point of view by relating this subject to quasiconformal geometry. Namely, for any such hypersurface H we define and construct a so-called S-structure. This new and natural concept reveals some unexpected geometric and analytic properties of H and its singularity set Ʃ. Moreover, it can be used to prove the existence of hyperbolic unfoldings of H\Ʃ. These are canonical conformal deformations of H\Ʃ into complete Gromov hyperbolic spaces of bounded geometry with Gromov boundary homeomorphic to Ʃ. These new concepts and results naturally extend to the larger class of almost minimizers.

Classical general relativity predicts the occurrence of spacetime singularities under very general conditions. Starting from the idea that the spacetime geometry must be described by suitable states in the complete quantum theory of matter and gravity, we shall argue that this scenario cannot be realised physically since no proper quantum state may contain the infinite momentum modes required to resolve the singularity.

The investigation of objects in Minkowski space is of great significance, especially for those objects with mathematical and physical backgrounds. In this paper, we study nullcone fronts, which are formed by the light rays emitted from points on a spacelike curve. However, if the spacelike curve is singular, then we cannot use the usual tools and methods to study related issues. To solve these problems, we show the definition of spacelike framed curves in Minkowski 3-space, whose original curves may contain singularities. Then, the singularities of the nullcone fronts are characterized by using framed curvatures of spacelike framed curves. Finally, we exhibit some examples to illustrate our results.

There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly nonlinear scalar field, which allow the appearance of singularities of a new type (“spherical singularities”) outside the center of curvature coordinates. As the example, we consider a scalar field potential ∼sinh(ϕ2n),n>2, which grows rapidly for large field values. The space-time is assumed to be asymptotically flat. We fulfill a numerical investigation of solutions with different n for different parameters, which define asymptotic properties at spatial infinity. Depending on the configuration parameters, we show that the distribution of the stable circular orbits of test bodies around the configuration is either similar to that in the case of the Schwarzschild solution (thus mimicking an ordinary black hole), or it contains additional rings of unstable orbits.

In this paper, we consider the following critical nonlocal problem: