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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.

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 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.

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

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).

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.

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.

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.

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.

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.

We study spherically symmetric spacetimes in Einstein-aether theory in three different coordinate systems, the isotropic, Painlevè-Gullstrand, and Schwarzschild coordinates, in which the aether is always comoving, and present both time-dependent and time-independent exact vacuum solutions. In particular, in the isotropic coordinates we find a class of exact static solutions characterized by a single parameter c14 in closed forms, which satisfies all the current observational constraints of the theory, and reduces to the Schwarzschild vacuum black hole solution in the decoupling limit (c14=0). However, as long as c14≠0, a marginally trapped throat with a finite non-zero radius always exists, and on one side of it the spacetime is asymptotically flat, while on the other side the spacetime becomes singular within a finite proper distance from the throat, although the geometric area is infinitely large at the singularity. Moreover, the singularity is a strong and spacetime curvature singularity, at which both of the Ricci and Kretschmann scalars become infinitely large.

In this paper, we focus on some geometrical properties of the partially null slant helices in semi-Euclidean 4-space. By structuring suitable height functions, we obtain the singularity types of the pseudonull hypersurfaces, which are generated by the partially null slant helices. An example is given to determine the main results.

We review the de Broglie–Bohm quantum theory. It is an alternative description of quantum phenomena in accordance with all the quantum experiments already performed. Essentially, it is a dynamical theory about objectively real trajectories in the configuration space of the physical system under investigation. Hence, it is not necessarily probabilistic, and it dispenses with the collapse postulate, making it suitable to be applied to cosmology. The emerging cosmological models are usually free of singularities, with a bounce connecting a contracting era with an expanding phase, which we are now observing. A theory of cosmological perturbations can also be constructed under this framework, which can be successfully confronted with current observations, and can complement inflation or even be an alternative to it.

In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered.

Parallel robots with configurable platforms are a class of robots in which the end-effector has an inner mobility, so that its overall shape can be reconfigured: in most cases, the end-effector is thus a closed-loop kinematic chain composed of rigid links. These robots have a greater flexibility in their motion and control with respect to rigid-platform parallel architectures, but their kinematics is more challenging to analyze. In our work, we consider n-RRR planar configurable robots, in which the end-effector is a chain composed of n links and revolute joints, and is controlled by n rotary actuators located on the base of the mechanism. In particular, we study the geometrical design of such robots and their direct and inverse kinematics for n=4, n=5 and n=6; we employ the bilateration method, which can simplify the kinematic analysis and allows us to generalize the approach and the results obtained for the 3-RRR mechanism to n-RRR robots (with n>3). Then, we study the singularity configurations of these robot architectures. Finally, we present the results from experimental tests that have been performed on a 5–RRR robot prototype.

I give a historical survey of the discussions about the existence of closed timelike curves in general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. Gödel in his rotating universe model of 1949. I emphasize that journeying into the past is intimately linked to spacetime models devoid of timelike singularities. Since such singularities arise as an inevitable consequence of the equations of general relativity given physically reasonable assumptions, time travel in the past becomes possible only when one or another of these assumptions is violated. It is the case with wormhole-type solutions. S. Hawking and other authors have tried to “save” the paradoxical consequences of time travel in the past by advocating physical mechanisms of chronological protection; however, such mechanisms remain presently unknown, even when quantum fluctuations near horizons are taken into account. I close the survey by a brief and pedestrian discussion of Causal Dynamical Triangulations, an approach to quantum gravity in which causality plays a seminal role.

The evolution of engineering applications is increasingly shifting towards the embedded nature, resulting in low-cost solutions, micro/nano dimensional and actuators being exploited as fundamental components to connect the physical nature of information with the abstract one, which is represented in the logical form in a machine. In this context, the scientific community has gained interest in modeling membrane Micro-Electro-Mechanical-Systems (MEMS), leading to a wide diffusion on an industrial level owing to their ease of modeling and realization. Physically, once the external voltage is applied, an electrostatic field, orthogonal to the tangent line of the membrane, is established inside the device, producing an electrostatic pressure that acts on the membrane, deforming it. Evidently, the greater the amplitude of the electrostatic field is, the greater the curvature of the membrane. Thus, it seems natural to consider the amplitude of the electrostatic field proportional to the curvature of the membrane. Starting with this principle, the authors are actively involved in developing a second-order semi-linear elliptic model in 1D and 2D geometries, obtaining important results regarding the existence, uniqueness and stability of solutions as well as evaluating the particular operating conditions of use of membrane MEMS devices. In this context, the idea of providing a survey matures to discussing the similarities and differences between the analytical and numerical results in detail, thereby supporting the choice of certain membrane MEMS devices according to the industrial application. Finally, some original results about the stability of the membrane in 2D geometry are presented and discussed.

This work builds upon previous studies of centered polygonal lacunary functions by presenting proofs of theorems showing how rotational and dihedral mirror symmetry manifest in these lacunary functions at the modulus level. These theorems then provide a general framework for constructing other lacunary functions that exhibit the same symmetries. These investigations enable one to better explore the effects of the gap behavior on the qualitative features of the associated lacunary functions. Further, two renormalized products of centered polygonal lacunary functions are defined and a connection to Ramanunjan’s triangular lacunary series is made via several theorems.

We study the singular Bertrand curves and Mannheim curves in the 3-dimensional space forms. We introduce the geometrical properties of such special curves. Moreover, we get the relationships between singularities of original curves and torsions of another mate curves.

This work investigates centered polygonal lacunary functions restricted from the unit disk onto symmetry angle space which is defined by the symmetry angles of a given centered polygonal lacunary function. This restriction allows for one to consider only the p-sequences of the centered polygonal lacunary functions which are bounded, but not convergent, at the natural boundary. The periodicity of the p-sequences naturally gives rise to a convergent subsequence, which can be used as a grounds for decomposition of the restricted centered polygonal lacunary functions. A mapping of the unit disk to the sphere allows for the study of the line integrals of restricted centered polygonal that includes analytic progress towards closed form representations. Obvious closures of the domain obtained from the spherical map lead to four distinct topological spaces of the “broom topology” type.

Positioning and tracking a moving target from limited positional information is a frequently-encountered problem. For given noisy observations of the target’s position, one wants to estimate the true trajectory and reconstruct the full phase space including velocity and acceleration. The shadowing filter offers a robust methodology to achieve such an estimation and reconstruction. Here, we highlight and validate important merits of this methodology for real-life applications. In particular, we explore the filter’s performance when dealing with correlated or uncorrelated noise, irregular sampling in time and how it can be optimised even when the true dynamics of the system are not known.

In this study, we derive, in the framework of mimetic theory, charged and non-charged black hole solutions for spherically symmetric as well as flat horizon spacetimes. The asymptotic behavior of those black holes behave as flat or (A)dS spacetimes and coincide with the solutions derived before in general relativity theory. Using the field equations of non-linear electrodynamics mimetic theory we derive new black hole solutions with monopole and quadrupole terms. The quadruple term of those black holes is related by a constant so that its vanishing makes the solutions coincide with the linear Maxwell black holes. We study the singularities of those solutions and show that they possess stronger singularity than the ones known in general relativity. Among many things, we study the horizons as well as the heat capacity to see if the black holes derived in this study have thermodynamical stability or not.

Reduced workspace is the main parallel robot disadvantage. It is generally due to the robot configuration, mainly the platform orientation constraint, the present work intends to find the maximum sphere within the orientation workspace, i.e., the singularity-free orientation regions. These regions are related to the platform orientation through Roll-Pitch-Yaw angles. Therefore, an optimization genetic algorithm is used to determine the initial platform orientation corresponding to the largest sphere volume. In this algorithm, the geometrical parameters and the direct and inverse singularities are the optimization constraints. The geometrical constraints may be studied using vectorial analysis. The reciprocity property from screw theory is implemented to analyze the direct and inverse kinematic. In this work it is used a methodology to verify the singularity closeness measure associated with direct kinematic. This measure is related to the rate of work done by each leg upon the platform twist. To determine how close is the parallel robot to a direct singularity a index value is proposed. It is considered that the passive joints reachable regions may be limited by a cone, whereby the cone symmetric axis is the same than the passive joint axis. In the optimization problem, the sphere volume, i.e., the maximal angular displacement of the moving platform around any axis is the objective function. Thus, the genetic algorithm individuals explore all feasible regions looking for an optimal solution.

In the last four decades, different programs have been carried out aiming at understanding the final fate of gravitational collapse of massive bodies once some prescriptions for the behaviour of gravity in the strong field regime are provided. The general picture arising from most of these scenarios is that the classical singularity at the end of collapse is replaced by a bounce. The most striking consequence of the bounce is that the black hole horizon may live for only a finite time. The possible implications for astrophysics are important since, if these models capture the essence of the collapse of a massive star, an observable signature of quantum gravity may be hiding in astrophysical phenomena. One intriguing idea that is implied by these models is the possible existence of exotic compact objects, of high density and finite size, that may not be covered by an horizon. The present article outlines the main features of these collapse models and some of the most relevant open problems. The aim is to provide a comprehensive (as much as possible) overview of the current status of the field from the point of view of astrophysics. As a little extra, a new toy model for collapse leading to the formation of a quasi static compact object is presented.

We review results about the development and asymptotic nature of singularities in “brane–bulk” systems. These arise for warped metrics obeying the five-dimensional Einstein equations with fluid-like sources, and including a brane four-metric that is either Minkowski, de Sitter, or Anti-de Sitter. We characterize all singular Minkowski brane solutions, and look for regular solutions with nonzero curvature. We briefly comment on matching solutions, energy conditions, and finite Planck mass criteria for admissibility, and we briefly discuss the connection of these results to ambient theory.

The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which the former may be embedded, satisfying the pertinent cohomological conditions required for the coordinatization of all of the tensorial physical quantities, such that the form of the field equations is preserved. We present in detail the construction of these distribution-like algebra sheaves in terms of residue classes of sequences of smooth functions modulo the information of singular loci encoded in suitable ideals. Finally, we consider the application of these distribution-like solution sheaves in geometrodynamics by modeling topologically-circular boundaries of singular loci in three-dimensional space in terms of topological links. It turns out that the Borromean link represents higher order wormhole solutions.

This is our second article devoted to the cardiac myoarchitecture considered as a nematic chiral liquid crystal (NCLC). While the first article focused on the myoarchitecture of the left ventricle (LV), this new article extends to the whole ventricular mass and introduces the concept of disclinations and topological singularities, which characterize the differences and relationships between the left and right ventricles (RV). At the level of the ventricular apices, we constantly observed a vortex shape at the LV apex, corresponding, in the terminology of liquid crystals, to a "+1 disclination"; we never observed this at the RV apex. At the level of the interventricular septum (IVS), we identified "-1/2 disclinations" at the anterior and posterior parts. During the perinatal period, there was a significant difference in their distribution, with more "-1/2 disclinations" in the posterior part of the IVS. After birth, concomitant to major physiological changes, the number of "-1/2 disclinations" significantly decreased, both in the anterior and posterior parts of the IVS. Finally, the description of the disclinations must be considered in any attempt to segment the whole ventricular mass, in biomechanical studies, and, more generally, for the characterization of myocardial remodeling.

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.

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.