Singular Line Research Articles

Closed timelike curves and ‘effective’ superluminal travel with naked line singularities

We examine closed timelike curves (CTCs) and ‘effective’ superluminal travel in a spacetime containing naked line singularities, which we call ‘wires’. Each wire may be straight-line singularity or a ring singularity. The weak energy condition (WEC) is preserved in all well-defined regions of the spacetime. (The singularities themselves are not well-defined, so the WEC is undefined there, but it is never explicitly violated.) Parallel to the wire, ‘effective’ superluminal travel is possible, in that the wire may be used as a shortcut between distant regions of spacetime. Our purpose in presenting the superluminal aspects of the wire is to dispel the commonly held view that explicit WEC violation is necessary for effective superluminal travel, whereas in truth the strictures against superluminal travel are more complicated. We also demonstrate how the existence of such ‘wires’ could create CTCs. We present a model spacetime which contains two wires which are free to move relative to each other. This spacetime is asymptotically flat: it becomes a Minkowski spacetime a finite distance away from each of the wires. The CTCs under investigation do not need to enter the wires’ singularities, and can be confined to regions that are weak-field: this means that if these wires were physically possible, they would present causality problems even in nonsingular, energetically realistic regions of the spacetime. We conclude that the weak energy condition alone is not sufficient to prevent superluminal travel in asymptotically flat spacetimes.

On the Usage of Different Coordinate Systems for 3D Plots of Functions of Two Real Variables

Students learning towards a degree in a STEM related domain learn quite early a course in Advanced Calculus, i.e. a course where the main object of study are multivariate functions. It happens that students do not see the connection between the properties of the function such as continuity or differentiability and the 3-dimensional graphical representation. In particular, difficulties appear in classroom for functions having different limits at a point, according to the path approaching the point. One of the ways for this analysis is to use different coordinate systems to study the behavior of the function. Constraints exist on the hardware, the software and on the numerical approximation methods, which may lead to completely different graphs for the same function, when using two different coordinate systems. In this contribution we analyze the different types of graphs created with a CAS such as MATLAB 14 and Maple 2017 for different coordinate systems, in terms of properties of the 2-variable function under study. We prove that when the function is continuous, the 3D graphical representation does not depend on the coordinate system in use. However, when the function is discontinuous,the graphical representation depends strongly on the coordinate system and on the type of discontinuity (point singularity, line singularity, multiple singularity, etc.). Moreover, without a suitable analysis prior to plotting, the 3D graph may be uncorrect. The software shows a strange behavior around the singular points. It happens also that a line singularity is replaced either by a point singularity or by no singularity. We analyse the software behavior using numerical analysis of the meshing algorithm.

Reconstructing the topology of optical polarization knots

Knots are topological structures describing how a looped thread can be arranged in space. Although most familiar as knotted material filaments, it is also possible to create knots in singular structures within three-dimensional physical fields such as fluid vortices1 and the nulls of optical fields2–4. Here we produce, in the transverse polarization profile of optical beams, knotted lines of circular transverse polarization. We generate and observe both simple torus knots and links as well as the topologically more complicated figure-eight knot. The presence of these knotted polarization singularities endows a nontrivial topological structure on the entire three-dimensional propagating wavefield. In particular, the contours of constant polarization azimuth form Seifert surfaces of high genus5, which we are able to resolve experimentally in a process we call seifertometry. This analysis reveals a level of topological complexity, present in all experimentally generated polarization fields, that goes beyond the conventional reconstruction of polarization singularity lines. Knotted lines representing torus knot and figure-eight knot are produced in the polarization profile of optical beams, leading to a topological characterization of the structure of the polarization field.

On the approximation of functions with line singularities by ridgelets

In Grohs and Obermeier (2015), the authors discussed the existence of numerically feasible solvers for advection equations that run in optimal computational complexity. In this paper, we complete the last remaining requirement to achieve this goal — by showing that ridgelets, on which the solver is based, approximate functions with line singularities (which may appear as solutions to the advection equation) with the best possible approximation rate (up to arbitrarily small δ>0) if the functions decay sufficiently fast, and with a slight penalty if not.Structurally, the proof resembles Candès (2001), where a similar result was proved for a different ridgelet construction, which is however not well-suited for use in a PDE solver (and in particular, not suitable for the CDD-schemes (Cohen et al., 2001) we are interested in). Due to the differences between the two ridgelet constructions, we have to deal with quite a different set of issues, but are also able to relax the (support) conditions on the function being approximated. Finally, the proof employs a new convolution-type estimate that could be of independent interest due to its sharpness.

Laplace’s equation for a point source near a sphere: improved internal solution using spheroidal harmonics

As shown recently [Phys. Rev. E 95, 033307 (2017)], spheroidal harmonics expansions are well suited for the external solution of Laplace's equation for a point source outside a spherical object. Their intrinsic singularity matches the line singularity of the analytic continuation of the solution and the series solution converges much faster than the standard spherical harmonic solution. Here we extend this approach to internal potentials using the Kelvin transformation, i.e. radial inversion, of the spheroidal coordinate system. This transform converts the standard series solution involving regular solid spherical harmonics into a series of irregular spherical harmonics. We then substitute the expansion of irregular spherical harmonics in terms of transformed irregular spheroidal harmonics into the potential. The spheroidal harmonic solution fits the image line singularity of the solution exactly and converges much faster. We also discuss why a solution in terms of regular solid spheroidal harmonics cannot work, even though these functions are finite everywhere in the sphere. We also present the analogous solution for an internal point source, and two new relationships between the solid spherical and spheroidal harmonics.

Special Classes of Monogenic Functions in $$\mathbb {H}$$ H

Recently, the classical orthogonal function systems of inner and outer monogenic Appell functions were used to find a new class of monogenic functions with (logarithmic) line singularities, which naturally extends the class of outer monogenic Appell functions. Based on these results, this article constructs another novel class of monogenic functions with line singularities, which now also relates the class of inner monogenic Appell functions and thus complements the previously known function classes. It is shown that a subset of the constructed functions are rational monogenic functions of the form $$\varvec{p}_{k}^{l}(\varvec{x})\,\varvec{q}^{l}(\overline{\varvec{\zeta }})^{-1}$$ , where $$\varvec{p}_{k}^{l}(\varvec{x})$$ and $$\varvec{q}^{l}(\overline{\varvec{\zeta }})$$ are homogeneous polynomials in the respective variables. Finally, the classical and special classes of monogenic functions are brought together in a unified schematic representation and essential features of the individual function classes and their relationships to each other are shown.

Connecting localization and wall-crossing via D-branes

We demonstrate explicitly that the vacuum expectation values (vevs) of BPS line operators in 4d N=2 super Yang–Mills theory compactified on a circle, computed by localization techniques, can be expanded in terms of Darboux coordinates as proposed by Gaiotto, Moore, and Neitzke [1]. However, we need to augment the expressions for Darboux coordinates with additional monopole bubbling contributions to obtain a precise match. Using D-brane realization of these singular BPS line operators, we derive and incorporate the monopole bubbling contributions as well as predict the degeneracies of framed BPS states contributing to the line operator vevs in the limit of vanishing simultaneous spatial and R-symmetry rotation fugacity parameter.

On the Five-Moment Hamburger Maximum Entropy Reconstruction

We consider the Maximum Entropy Reconstruction (MER) as a solution to the five-moment truncated Hamburger moment problem in one dimension. In the case of five monomial moment constraints, the probability density function (PDF) of the MER takes the form of the exponential of a quartic polynomial. This implies a possible bimodal structure in regions of moment space. An analytical model is developed for the MER PDF applicable near a known singular line in a centered, two-component, third- and fourth-order moment (μ_3, μ_4) space, consistent with the general problem of five moments. The model consists of the superposition of a perturbed, centered Gaussian PDF and a small-amplitude packet of PDF-density, called the outlying moment packet (OMP), sitting far from the mean. Asymptotic solutions are obtained which predict the shape of the perturbed Gaussian and both the amplitude and position on the real line of the OMP. The asymptotic solutions show that the presence of the OMP gives rise to an MER solution that is singular along a line in ( μ_3, μ_4) space emanating from, but not including, the point representing a standard normal distribution, or thermodynamic equilibrium. We use this analysis of the OMP to develop a numerical regularization of the MER, creating a procedure we call the Hybrid MER (HMER). Compared with the MER, the HMER is a significant improvement in terms of robustness and efficiency while preserving accuracy in its prediction of other important distribution features, such as higher order moments.

Конструирование одномерных обводов, принадлежащих поверхностям, путем их отображения на плоскость

As is known, differential geometry studies the properties of curve lines (tangent, curvature, torsion), surfaces (bending, first and second basic quadratic forms) and their families in small, that is, in the neighborhood of the point by means of differential calculus. Algebraic geometry studies properties of algebraic curves, surfaces, and algebraic varieties in general [1; 17]: order, class, genre, existence of singular points and lines, curves and surfaces family intersections (sheaves, bundles, congruences, complexes and their characteristics). Rational curves and surfaces occupy a special place among them: • their design by bi-rational (Cremona) transformations [10; 21]; • investigation of their properties by mapping to lines and planes [9; 21; 22]; • construction of smooth contours from arcs of rational curves belonging to surfaces [10]. It seems that the main results obtained in this direction by mathematicians in the second half of the 19th century by structural and geometric methods should be the theoretical support for the design of technical forms that meet a number of pre-set requirements using modern computational tools and information technologies. It is obvious that application of Cremona transformations’ powerful apparatus is useful when designing, for example, pipes of complex geometry according to set of streamlines, thin-walled shells for a given mesh manifold of curvature lines etc. Apparently, this stage should precede computer graphics’ calculation procedures. However, in Russian publications on applied (engineering) geometry, only a little attention is paid to the study of surfaces in general. The author knows nothing about the use of this approach for solving of these applied problems. In this regard, the aims of this paper are: • illustration of method for mapping a surface to a plane to study its properties in general by the example of construction a flat model for a hyperboloid of one sheet; • constructive approach to the construction of smooth one-dimensional contours on rational surfaces.

Multi-channel multi-scale fully convolutional network for 3D perivascular spaces segmentation in 7T MR images.

Accurate segmentation of perivascular spaces (PVSs) is an important step for quantitative study of PVS morphology. However, since PVSs are the thin tubular structures with relatively low contrast and also the number of PVSs is often large, it is challenging and time-consuming for manual delineation of PVSs. Although several automatic/semi-automatic methods, especially the traditional learning-based approaches, have been proposed for segmentation of 3D PVSs, their performance often depends on the hand-crafted image features, as well as sophisticated preprocessing operations prior to segmentation (e.g., specially defined regions-of-interest (ROIs)). In this paper, a novel fully convolutional neural network (FCN) with no requirement of any specified hand-crafted features and ROIs is proposed for efficient segmentation of PVSs. Particularly, the original T2-weighted 7T magnetic resonance (MR) images are first filtered via a non-local Haar-transform-based line singularity representation method to enhance the thin tubular structures. Both the original and enhanced MR images are used as multi-channel inputs to complementarily provide detailed image information and enhanced tubular structural information for the localization of PVSs. Multi-scale features are then automatically learned to characterize the spatial associations between PVSs and adjacent brain tissues. Finally, the produced PVS probability maps are recursively loaded into the network as an additional channel of inputs to provide the auxiliary contextual information for further refining the segmentation results. The proposed multi-channel multi-scale FCN has been evaluated on the 7T brain MR images scanned from 20 subjects. The experimental results show its superior performance compared with several state-of-the-art methods.

On a Class of Monogenic Functions with (Logarithmic) Line Singularities

In the article a new class of monogenic functions with (logarithmic) line singularities is studied, which naturally extend the classical system of outer solid spherical monogenics. These functions have special properties with respect to the hypercomplex derivative (generalized Appell property) and can be generated by a three-term recurrence relation. Furthermore, an explicit decomposition of the harmonic Newton kernel is given by means of these functions and a connection to the Cauchy kernel is shown.

Topological transformations of Hopf solitons in chiral ferromagnets and liquid crystals

Liquid crystals are widely known for their facile responses to external fields, which forms a basis of the modern information display technology. However, switching of molecular alignment field configurations typically involves topologically trivial structures, although singular line and point defects often appear as short-lived transient states. Here, we demonstrate electric and magnetic switching of nonsingular solitonic structures in chiral nematic and ferromagnetic liquid crystals. These topological soliton structures are characterized by Hopf indices, integers corresponding to the numbers of times that closed-loop-like spatial regions (dubbed "preimages") of two different single orientations of rod-like molecules or magnetization are linked with each other. We show that both dielectric and ferromagnetic response of the studied material systems allow for stabilizing a host of topological solitons with different Hopf indices. The field transformations during such switching are continuous when Hopf indices remain unchanged, even when involving transformations of preimages, but discontinuous otherwise.

Configurational Forces on Elastic Line Singularities

Configurational forces acting on two-dimensional (2D) elastic line singularities are evaluated by path-independent J-, M-, and L-integrals in the framework of plane strain linear elasticity. The elastic line singularities considered in this study are the edge dislocation, the line force, the nuclei of strain, and the concentrated couple moment that are subjected to far-field loads. The interaction forces between two similar parallel elastic singularities are also calculated. Self-similar expansion force, M, evaluated for the line force shows that it is exactly the negative of the strain energy prelogarithmic factor as in the case for the well-known edge dislocation result. It is also shown that the M-integral result for the nuclei of strain and the L-integral result for the line force yield interesting nonzero expressions under certain circumstances.

Chemotactic Aggregation versus Logistic Damping on Boundedness in the 3D Minimal Keller--Segel Model

We study chemotactic aggregation versus logistic damping on boundedness for the 3D minimal Keller--Segel (KS) model with logistic source, $\{ u_t = \nabla \cdot (\nabla u-\chi u\nabla v)+ u-\mu u^2, \, x\in \Omega, t>0; v_t =\Delta v -v+u, \, x\in \Omega, t>0\}, $ in a smooth, bounded, but not necessarily convex domain $\Omega\subset \mathbb{R}^3$ with $\chi, \mu>0$, nonnegative initial data $u_0, v_0$, and homogeneous Neumann boundary data. In a previous work [T. Xiang, J. Math. Anal. Appl., 459 (2018), pp. 1172--1200], the global boundedess of the above KS system in nonconvex domains is guaranteed under the explicit condition $ (*) \;\mu>\vartheta_0\chi=\frac{9}{\sqrt{10}-2}\chi. %\eqno(*) $ As a continuation, under the critical condition $(*)$, up to a scaling constant depending only on $u_0$, $v_0$, and $\Omega$, we here establish explicit uniform-in-time upper bounds for the quantities $\|u(\cdot,t)\|_{L^\infty(\Omega)}$ and $\|v(\cdot,t)\|_{W^{1,\infty}(\Omega)}$ in terms of $\chi$ and $\mu$; these bounds are defined for all $\chi\geq 0$ and $\mu>\vartheta_0\chi$, increasing in $\chi$ and decreasing in $\mu$, and have only one singular line in $\mu$ at $\mu=\vartheta_0\chi$. The corresponding 2D qualitative boundedness has been investigated [H. Jin and T. Xiang, C. R. Math. Acad. Sci. Paris, 356 (2018), pp. 875--885], wherein the only singular line of the upper bounds in $\mu$ is shown to be $\mu=0$. By comparison, an interesting new feature is that the singular line $\mu=0$ of the corresponding 2D upper bounds has moved up to $\mu=\vartheta_0\chi$ in the 3D setting. It is worth mentioning that, in the absence of a logistic source, the corresponding classical KS model (by setting $\mu=0$ and removing the proliferation term $+u$ in the first equation) is now well known to possess blow-up solutions for even small initial data [M. Winkler, J. Math. Pures Appl., 100 (2013), pp. 748--767].

PEAKON AND CUSPON SOLUTIONS OF A GENERALIZED CAMASSA-HOLM-NOVIKOV EQUATION

The bounded traveling wave solutions of a generalized CamassaHolm-Novikov equation with <i>p</i>=2 and <i>p</i>=3 are derived via the dynamical system approach. The singular wave solutions including peakons and cuspons are obtained by the bifurcation analysis of the corresponding singular dynamical system and the orbits intersecting with or approaching the singular lines. The results show that the generalized Camassa-Holm-Novikov equation with <i>p</i>=2 and <i>p</i>=3 both admit smooth solitary wave, smooth periodic wave solutions, solitary peakons, periodic peakons, solitary cuspons and periodic cuspons as well. It is worth pointing out that the Novikov equation has no bounded traveling wave solutions with negative wave speed, but has a family of new periodic cuspons which are distinguished with the normal periodic cuspons for their discontinuous first-order derivatives at both maximum and minimum.

Exact Solutions and Dynamics of the Raman Soliton Model in Nanoscale Optical Waveguides, with Metamaterials, Having Polynomial Law Nonlinearity

Raman soliton model in nanoscale optical waveguides, with metamaterials, having polynomial law nonlinearity is investigated by the method of dynamical systems. The functions [Formula: see text] in the solutions [Formula: see text] [Formula: see text] satisfy a singular planar dynamical system having two singular straight lines. By using the bifurcation theory method of dynamical systems to the equations of [Formula: see text], under 23 different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons and peakons as well as compacton solutions for this planar dynamical system are obtained. 92 exact explicit solutions of system (6) are derived.

Reverse orbiting and spinning of a Rayleigh dielectric spheroid in a J_0 Bessel optical beam

Based on the electric dipole approximation, numerical computations for the optical orbital and spin radiation torques induced by a zeroth-order Bessel beam on a lossless dielectric subwavelength spheroid with arbitrary orientation in space are performed. The transverse optical force components are determined first, and then used to compute the longitudinal orbital torque component. Moreover, the Cartesian components of the spin radiation torque vector are evaluated. Numerical calculations illustrate the analysis with particular emphasis on the beam parameters, polarization of the magnetic vector potential forming the beam, and aspect ratio of the spheroid. The results demonstrate that the lossless dielectric subwavelength spheroid becomes irresponsive to the transfer of angular momentum, where the longitudinal orbital and spin radiation torque components vanish along singularity lines. Moreover, depending on the beam parameters and the spheroid location in space, the longitudinal orbital and spin torque components reverse sign, indicating a direction of rotation around the central axis of the beam and center of mass of the spheroid, respectively, in either the counterclockwise or clockwise directions. The results are important in particle dynamics and trapping applications, optical tweezers and spanners, and other related fields.

Approximation and equidistribution results for pseudo-effective line bundles

We study the distribution of the common zero sets of m-tuples of holomorphic sections of powers of m singular Hermitian pseudo-effective line bundles on a compact Kähler manifold. As an application, we obtain sufficient conditions which ensure that the wedge product of the curvature currents of these line bundles can be approximated by analytic cycles.

Uniform stable radius, Lê numbers and topological triviality for line singularities

Let $\{f_t\}$ be a family of complex polynomial functions with line singularities. We show that if $\{f_t\}$ has a uniform stable radius (for the corresponding Milnor fibrations), then the L\^e numbers of the functions $f_t$ are independent of $t$ for all small $t$. In the case of isolated singularities --- a case for which the only non-zero L\^e number coincides with the Milnor number --- a similar assertion was proved by M. Oka and D. O'Shea. By combining our result with a theorem of J. Fern\'andez de Bobadilla --- which says that families of line singularities in $\mathbb{C}^n$, $n\geq 5$, with constant L\^e numbers are topologically trivial --- it follows that a family of line singularities in $\mathbb{C}^n$, $n\geq 5$, is topologically trivial if it has a uniform stable radius. As an important example, we show that families of weighted homogeneous line singularities have a uniform stable radius if the nearby fibres $f_t^{-1}(\eta)$, $\eta\not=0$, are "uniformly" non-singular with respect to the deformation parameter $t$.

Next-nearest-neighbor resonance coupling and exceptional singularities in degenerate optical microcavities

We report a specially configured non-Hermitian optical microcavity, imposing spatially imbalanced gain-loss profile, to host an exclusively proposed next nearest neighbor resonances coupling scheme. Adopting scattering matrix (S-matrix) formalism, the effect of interplay between such proposed resonance interactions and the incorporated non-Hermiticity in the microcavity is analyzed drawing a special attention to the existence of hidden singularities, namely exceptional points (EPs); where at least two coupled resonances coalesce. We establish adiabatic flip-of-states phenomena of the coupled resonances in the complex frequency plane (k-plane) which is essentially an outcome of the fact that the respective EP is being encircled in system parameter plane. Encountering such multiple EPs, the robustness of flip-of-states phenomena have been analyzed via continuous tuning of coupling parameters along a special hidden singular line which connects all the EPs in the cavity. Such a numerically devised cavity, incorporating the exclusive next neighbor coupling scheme, have been designed for the first time to study the unconventional optical phenomena in the vicinity of EPs.