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For the forst time, the general properties of degenerate related hypergeometric systems such as Horn, Whittaker, Bessel and Laguerre are investigated together. The joint research allowed to reveal their various common properties and to establish a number of new degenerate related systems. They are all private cases of the common system offered by the authors for consideration. For the full study, it is important to classify its regular and irregular special curves and to identify the types of corresponding solutions. In this paper, they are implemented using simple rules. Special attention is paid to the construction of normal and regular solutions, because the solutions of all related degenerate systems such as Horn, Whittaker, Bessel and Laguerre near the irregular singularity on infonity relate to this species. Peculiarities of building normal-regular solutions by the Frobenius-Latysheva method are shown. All constructed normal-regular solutions are expressed through the function of Humbert n variables, which is the solution of degenerate hypergeometric system of Horn type. As an example, the cases n = 2 where, along with the application of the Frobenius-Latysheva method, the possibility of outputting new degenerate related systems is demonstrated. [ABSTRACT FROM AUTHOR]

A section of a Hamiltonian system is a hypersurface in the phase space of the system, usually representing a set of one-sided constraints (e. g., a boundary, an obstacle or a set of admissible states). In this paper we give local classification results for all typical singularities of sections of regular (non-singular) Hamiltonian systems, a problem equivalent to the classification of typical singularities of Hamiltonian systems with one-sided constraints. In particular, we give a complete list of exact normal forms with functional invariants, and we show how these are related/obtained by the symplectic classification of mappings with prescribed (Whitney-type) singularities, naturally defined on the reduced phase space of the Hamiltonian system. [ABSTRACT FROM AUTHOR]

We obtain an explicit formula for the characteristic number of degree d curves in P 2 with prescribed singularities (of type A k) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. We are not aware of any explicit formula to enumerate plane curves of degree d with any number of A k singularities (beyond codimension 8); however, combined with the results of S. Basu and R. Mukherjee ([1] , [2] , and [3]), this gives us a complete formula for the characteristic number of curves with δ -nodes and one singularity of type A k , tangent to a given line, provided δ + k ≤ 8. We use a topological method to compute the degenerate contribution to the Euler class. We have made several low degree checks to verify special cases of our result. When the singularities are only nodes, we have verified that our numbers are logically consistent with those computed by L. Caporaso and J. Harris ([7]). We also verify that our answers for the characteristic number of cubics with a cusp tangent to a given line and the characteristic number of quartics with two nodes and a cusp, tangent to a given line is logically consistent with the characteristic number of rational cubics and quartics tangent to a given line that was computed by L. Ernström and G. Kennedy ([9]). [ABSTRACT FROM AUTHOR]

The Schrödinger equation admits smooth and finite solutions that spontaneously evolve into a singularity, even for a free particle. This blowup is generally ascribed to the intrinsic dispersive character of the associated time evolution. We resort to the notion of quantum Bohmian trajectories to relate this singular behavior to local phase variations, which generate an underlying velocity field responsible for driving the quantum flux toward the singular region. • The dispersive blowup of the Schrödinger equation is investigated. • Local phase variations undergone by the wave function generate an underlying velocity field responsible for driving the quantum flux toward singular regions. • Bohmian trajectories are used to relate the wave function singular behavior to local phase variations. [ABSTRACT FROM AUTHOR]

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.

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.

Black holes are unique among astrophysical sources: they are the simplest macroscopic objects in the Universe, and they are extraordinary in terms of their ability to convert energy into electromagnetic and gravitational radiation. Our capacity to probe their nature is limited by the sensitivity of our detectors. The LIGO/Virgo interferometers are the gravitational-wave equivalent of Galileo's telescope. The first few detections represent the beginning of a long journey of exploration. At the current pace of technological progress, it is reasonable to expect that the gravitational-wave detectors available in the 2035-2050s will be formidable tools to explore these fascinating objects in the cosmos, and space-based detectors with peak sensitivities in the mHz band represent one class of such tools. These detectors have a staggering discovery potential, and they will address fundamental open questions in physics and astronomy. Are astrophysical black holes adequately described by general relativity? Do we have empirical evidence for event horizons? Can black holes provide a glimpse into quantum gravity, or reveal a classical breakdown of Einstein's gravity? How and when did black holes form, and how do they grow? Are there new long-range interactions or fields in our Universe, potentially related to dark matter and dark energy or a more fundamental description of gravitation? Precision tests of black hole spacetimes with mHz-band gravitational-wave detectors will probe general relativity and fundamental physics in previously inaccessible regimes, and allow us to address some of these fundamental issues in our current understanding of nature. [ABSTRACT FROM AUTHOR]

The workspace of a plane lambda mechanism is studied. The analysis takes account of singularities that have a negative influence on the size of the zones and the performance of the mechanism. Parametric synthesis permits maximization of the workspace. [ABSTRACT FROM AUTHOR]

How can we understand the relations between economic things and different forms of exchange – commodities, assets, gifts and singularities – in the contemporary economy? The decline of industrial capitalism and the emergence of new types of intangible valuables challenge our understanding of what economic life is about. Analysing economies through one dominant form of exchange risks overlooking the interplay between different types of valuables, their materiality and interactions that form the basis of value creation and exchange. In contrast, this special issue highlights the mutability of things – their capacity to take on and abandon different forms – as a precondition for economic activity. Drawing on a variety of empirical case studies of markets for seeds, grains, fish, carbon emissions and cattle, the contributions set out to trace the social biographies of economic things in, between and beyond multiple economic forms. We argue that it is the very ability of economic things to shift in and out of particular forms of exchange that enables the complex globalised economies of our time. [ABSTRACT FROM AUTHOR]

A unified approach is developed for the analysis of singularities of the scattered field and vanishing of scattering harmonics. Rigorous proofs are presented of the existence of complex resonance singularities of the solution to the problem of the plane wave scattering by a circular homogeneous dielectric cylinder. The method employs a mathematically correct approach of the spectral theory of open structures involving generalized conditions at infinity when complex resonance frequencies may be considered. The result is obtained by verifying the existence of complex singularities of the series solution coefficients considered as functions of the problem parameters. The recently developed theory of generalized cylindrical polynomials (GCPs) is used that enables one to reduce determination of singularities (resonances) to finding zeros (real or complex) of a particular subfamily of GCPs. Plain Language Summary: Resonance and partial invisibility are inherent to dielectric scattereres. A hypothesis that they have similar mathematical nature and virtually the same background has been verified and confirmed for a dielectric cylinder of circular cross section: both these phenomena are governed by the properties of zeros of certain function families, namely, weighted cylindrical cross products. The resonance and partial invisibility frequencies are calculated as explicit numbers and may be treated as fundamental constants of the electromagnetic field theory. Key Points: Unified approach to the analysis of singularities of the scattered field and partial invisibility in the form of field‐harmonic suppressionRigorous proof of the existence of complex resonances and partial‐invisibility frequenciesContribution to the theory of cylindrical functions: novel proofs of the existence of zeros of cross products [ABSTRACT FROM AUTHOR]

We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. The 0-level of such a function defines a germ of analytic curve. Proceeding similarly to the homogeneous case Genzmer and Paul (2011) and the quasi homogeneous case Genzmer and Paul (2016) , we describe an algorithm which computes the dimension of the generic strata of the local moduli space of curves. [ABSTRACT FROM AUTHOR]

In this paper we study the well-posedness of the Cauchy problem for a wave equation with multiplicities and space-dependent irregular coefficients. As in Garetto and Ruzhansky [Arch. Ration. Mech. Anal. 217 (2015), pp. 113-154], in order to give a meaningful notion of solution, we employ the notion of very weak solution, which construction is based on a parameter dependent regularisation of the coefficients via mollifiers. We prove that, even with distributional coefficients, a very weak solution exists for our Cauchy problem and it converges to the classical one when the coefficients are smooth. The dependence on the mollifiers of very weak solutions is investigated at the end of the paper in some instructive examples. [ABSTRACT FROM AUTHOR]

Related to a Coxeter group are certain sets of tangents of the deltoid with evenly distributed orientations forming simplicial line configurations. These configurations are used to construct curves and surfaces with A D E singularities. Other surfaces associated with invariants of exceptional complex reflection groups are considered. A new lower bound for the maximal number of D 4 singularities in a sextic surface is obtained. Several Calabi–Yau threefolds defined as double coverings of the complex projective 3-space branched along nodal octic surfaces and Calabi–Yau quintic threefolds are analyzed. The Hodge numbers of a small resolution of all the nodes of the singular threefolds are obtained. [ABSTRACT FROM AUTHOR]

Hironaka's characteristic polyhedron is an important combinatorial object reflecting the local nature of a singularity. We prove that it can be determined without passing to the completion if the local ring is a G-ring and if additionally either it is Henselian, or a certain polynomiality condition (Pol) holds, or a mild condition (*) on the singularity holds. For example, the latter is fulfilled if the residue field is perfect. [ABSTRACT FROM AUTHOR]

This study aims to compute low-frequency energy with Hölder singularities in vibration signals of suspension system to predict the durability of coil spring. High frequencies in vibrations often had minimal contribution towards fatigue damage due to low amplitude range and thus induce errors in energy analysis of vibration signals. Since traditional low-pass method had not only been ineffective in reducing high frequencies, it also resulted in the loss of signal information. This study had therefore proposed characterising low-frequency energy for road excitations using Hölder singularities and power spectral analyses. Singularities and low-frequency energy of road vibration signals would first be identified through Hölder local regularity analysis. This was then followed by fatigue life prediction using the strain-life approaches (i.e. Coffin-Manson, Morrow and Smith–Watson–Topper models). The energy-based fatigue life prediction models had not only shown good fit with R2 values higher than 0.8, but had also demonstrated an accurate prediction of fatigue life with more than 95% of the data being within the acceptance boundary. The Morrow-based model provided the highest accuracy in fatigue life prediction because of its highest R2 value of 0.8625 and 100% data survival in the fatigue life correlation study. This showed that energy-based fatigue life prediction models provide an accurate and effective prediction of the durability performance. This study proposed a more precise energy characterisation method for energy-based durability prediction of suspension coil spring under random loading conditions. [ABSTRACT FROM AUTHOR]

We show that in any Q-Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed by Li, Liu, and Xu, we show that K-semistability is a very generic or empty condition in any Q-Gorenstein flat family of log Fano pairs. [ABSTRACT FROM AUTHOR]

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.

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.

Effective sufficient conditions are given for the unique solvability of the Cauchy problem for linear systems of generalized ordinary differential equations with singularities. [ABSTRACT FROM AUTHOR]

In this paper, a new differential model for electrostatic membrane micropumps for drug delivery systems is presented. In particular, a new two-dimensional nonlinear second-order differential model with singularity has been determined in steady-state conditions in which the electric field magnitude has been considered proportional to the mean curvature of the membrane. Then, a result of the existence of at least one solution for the model has been obtained, although, concerning the uniqueness of the solution, it is not guaranteed. Moreover, the stability of the solutions has been studied highlighting that when a solution exists, then it is unstable. Moreover, the problem was numerically solved by means of three-stage Lobatto IIIa formula achieving the ranges of the electromechanical parameters of the material constituting the membrane with or without ghost solutions. Finally, a criterion to select the intended use of the micropump starting from the electromechanical properties of the membrane and a criterion to choice the material constituting the membrane starting from the intended use of the micropump are presented. [ABSTRACT FROM AUTHOR]

The theory of intersection spaces assigns cell complexes to certain stratified topological pseudomanifolds depending on a perversity function in the sense of intersection homology. The main property of the intersection spaces is Poincaré duality over complementary perversities for the reduced singular (co)homology groups with rational coefficients. This (co)homology theory is not isomorphic to intersection homology, instead they are related by mirror symmetry. Using differential forms, Banagl extended the intersection space cohomology theory to 2-strata pseudomanifolds with a geometrically flat link bundle. In this paper, we use differential forms on manifolds with corners to generalize the intersection space cohomology theory to a class of 3-strata spaces with flatness assumptions for the link bundles. We prove Poincaré duality over complementary perversities for the cohomology groups. To do so, we investigate fiber bundles on manifolds with boundary. At the end, we give examples for the application of the theory. [ABSTRACT FROM AUTHOR]

We study Hironaka's idealistic exponents over Spec (Z). We give an idealistic interpretation of the tangent cone, the directrix, and the ridge. The main purpose is to introduce the notion of characteristic polyhedra of idealistic exponents and deduce from them intrinsic data on the idealistic exponent. [ABSTRACT FROM AUTHOR]

Let X be an elliptic surface over P 1 with κ (X) = 1 , and M = M (c 2) be the moduli scheme of rank-two stable sheaves E on X with (c 1 (E) , c 2 (E)) = (0 , c 2) in Pic (X) × ℤ. We look into defining equations of M at its singularity E , partly because if M admits only canonical singularities, then the Kodaira dimension κ (M) can be calculated. We show the following: (A) E is at worst canonical singularity of M if the restriction of E η to the generic fiber of X has no rank-one subsheaf, and if the number of multiple fibers of X is a few. (B) We obtain that κ (M) = { 1 + dim (M) } / 2 and the Iitaka program of M can be described in purely moduli-theoretic way for c 2 ≫ 0 , when χ (𝒪 X) = 1 , X has just two multiple fibers, and one of its multiplicities equals 2. (C) On the other hand, when E η has a rank-one subsheaf, it may be insufficient to look at only the degree-two part of defining equations to judge whether E is at worst canonical singularity or not. [ABSTRACT FROM AUTHOR]

For the spherical unit speed nonlightlike curve in pseudo-hyperbolic space and de Sitter space γ and a given point P, we can define naturally the pedal curve of γ relative to the pedal point P. When the pseudo-sphere dual curve germs are nonsingular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the pseudo-sphere dual curve germs are nonsingular. Furthermore, we obtain the extension results in dualities, which has wide influence on the open and closed string field theory and string dynamics in physics, and can be used to better solve the dynamics of trajectory particle condensation process. [ABSTRACT FROM AUTHOR]

In this paper, we consider a heat equation with mixed boundary conditions in a two-dimensional domain with a reentrant corner. This allows the solution to exhibit singularities near the corner as well as at the points where the mixed boundary conditions meet. The aim of this work is to establish a Carleman inequality by constructing a convenient weight function. As a consequence, we prove some control results. [ABSTRACT FROM AUTHOR]

Einstein's genius and penetrating physical intuition led to the general theory of relativity, which incorporates gravity into the geometry of spacetime. However, the theory of general relativity leads to perspectives which go far beyond the vision of its creator. Many of these insights came to light only after Einstein's death in 1955. These developments were due to a new breed of relativists, like Penrose, Hawking and Geroch, who approached the subject with a higher degree of mathematical sophistication than earlier workers. Some of these insights were made possible because of work by Amal Kumar Raychaudhuri (AKR) who derived an equation which turned out to be a key ingredient in the singularity theorems of general relativity. This article explains AKR's work in elementary terms. [ABSTRACT FROM AUTHOR]

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 FROM AUTHOR]

Copyright of Journal of Surra Man Raa is the property of Republic of Iraq Ministry of Higher Education & Scientific Research (MOHESR) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We show that there exists a natural Poisson–Lie algebra associated to a singular symplectic structure . We construct Poisson–Lie algebras for the Martinet and Roussarie types of singularities. In the special case when the singular symplectic structure is given by the pullback from the Darboux form, , this Poisson–Lie algebra is a basic symplectic invariant of the singularity of the smooth mapping into the symplectic space . The case of singularities of pullbacks is considered, and Poisson–Lie algebras for , and stable singularities of -forms are calculated. [ABSTRACT FROM AUTHOR]

Copyright of Revista Gama is the property of Revista Gama and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this note we introduce generalised pairs from the perspective of the evolution of the notion of space in birational algebraic geometry. We describe some applications of generalised pairs in recent years and then mention a few open problems. [ABSTRACT FROM AUTHOR]

We propose an approach to the problem of global reconstruction of an orientation field. The method is based on a geometric model called bisector line fields, which maps a pair of vector fields to an orientation field, effectively generalizing the notion of doubling phase vector fields. Endowed with a well-chosen energy minimization problem, we provide a polynomial interpolation of a target orientation field while bypassing the doubling phase step. The procedure is then illustrated with examples from fingerprint analysis. [ABSTRACT FROM AUTHOR]