Search

The Kerr metric is a vacuum solution of the Einstein equations outside of a rotating black hole (BH), but what interior matter is actually rotating and sourcing the Kerr geometry? Here, a rotating exotic matter is described, which can source the Kerr geometry for the entire acceptable range of its spin parameter and be shown to saturate the radial null‐energy condition at every point in the interior, while being free of any obvious pathologies. The rotating frozen star is introduced, whose compactness is controlled by a perturbative parameter and whose outer surface can be arbitrarily close to the horizon of a Kerr BH. The interior geometry modifies Kerr's such that there is neither an inner ergosphere nor an inner horizon. The geometry of each radial slice of the interior is a nearly null surface with the same geometry, but different radial size, as that of the would‐be horizon on the outermost slice. The integral of the energy density leads to a rest mass that is equal to the irreducible mass of a Kerr BH, and the integral of the angular‐momentum density confirms that the ratio of the angular momentum to the mass is equal to the Kerr spin parameter. Including the rotational energy in the standard way, the total gravitational mass and angular momentum of a Kerr BH with the same mass and spin parameters are obtained. [ABSTRACT FROM AUTHOR]

We prove a criterion for the constancy of the Hilbert–Samuel function for locally Noetherian schemes such that the local rings are excellent at every point. More precisely, we show that the Hilbert–Samuel function is locally constant on such a scheme if and only if the scheme is normally flat along its reduction and the reduction itself is regular. Regularity of the underlying reduced scheme is a significant new property. [ABSTRACT FROM AUTHOR]

Singularities in robotics lead to a reduction in the number of degrees of freedom. When it comes to spacecraft employing control moment gyros (CMGs), singularities may occur due to the internal alignment of the gimbals, which inhibit the creation of torque in at least one direction. This translates into a loss of control authority that has a direct impact on the spacecraft's pointing performance. In this work, a convex optimization-based allocation framework for singularity avoidance is put forward. The presented solution aims to provide a singularity-robust allocation scheme that can be used as an add-on to a conventional attitude controller. To this end, the proposed algorithm employs a novel, computationally efficient, and numerically robust singularity metric to assess the proximity to the singularities of a cluster of CMGs. With this information, a model predictive controller determines control actions that guide the system towards singularity-free configurations. A particularity about this allocation algorithm is that its formulation boils down to a simple set of convex equalities and inequalities, given that the new singularity metric can be written in a linear form, unlike most of the literature solutions, such as the condition number, which are highly complex and nonlinear. Lastly, the proposed approach is applied to a two-dimensional CMG cluster in a realistic simulation environment. The results confirm that the system effectively avoids all the internal singularities of the cluster at a relatively low computational expense. • Detailed analysis of singularities across different systems of control moment gyros. • Novel singularity metric based on the distance to the closest singularity. • Derivation of a convex MPC-based allocation algorithm for singularity avoidance. • Proposed approach can circumvent internal singularities at a low computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we study the analytic classification of a class of nilpotent singularities of holomorphic foliations in (C 2 , 0) , those exhibiting a Poincaré-Dulac type singularity in their reduction process. This analytic classification is based in the holonomy of a certain component of the exceptional divisor. Finally, as a consequence, we show that these singularities exhibit a formal analytic rigidity. [ABSTRACT FROM AUTHOR]

Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided design, image detection, algebraic differential equations, etc.), there often appear unknown parameters. It is possible to adjoin these parameters to the coefficient field as transcendental elements. In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way. In this paper, we show when the singularities and thus the (geometric) genus of the curves might change. More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant. In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible. For the genus zero case, and for a given rational parametrization, a finer partition is possible such that the specialization of the parametrization parametrizes the specialized curve. Moreover, in this case, the set of parameters where Hilbert's irreducibility theorem does not hold can be identified. We conclude the paper by illustrating these results by some concrete applications. [ABSTRACT FROM AUTHOR]

We discuss the problem of singularity crossing in isotropic and anisotropic universes. We study at which conditions singularities can disappear in quantum cosmology and how quantum particles behave in the vicinity of singularities. Some attempts to develop general approach to the connection between the field reparametrization and the elimination of singularities is presented as well. [ABSTRACT FROM AUTHOR]

A non‐singular black hole solution is briefly presented which violates energy conditions only at its interior by postulating a consistent shift to negative energies and gravitationally repulsive negative masses at the event horizon. This shift is the unitary parity‐time PT$PT$ transformation of relativistic quantum mechanics and the proper antichronous transformation of the full Lorentz group. The transformation at the event horizon respects Einstein's equivalence principle, and the considered negative mass interaction does not result in the runaway motion paradox or vacuum instability. The correspondence of these regular black holes with observed ones is studied by proposing another mechanism of black hole growth independent of accretion and merging due to interior increasing entropy, which attempts to solve the unexplained size and formation of high‐redshift supermassive black holes, the intermediate mass gap, and the information paradox. [ABSTRACT FROM AUTHOR]

Classical curl forces are position-dependent Newtonian forces (accelerations) that are not the gradient of a scalar potential, and in general cannot be described by Hamiltonians. However, a special class of curl forces can be described by Hamiltonians, with the unusual feature that the kinetic energy is anisotropic in the momentum components. Therefore they can be quantised conventionally. We quantise the simplest such case: motion in the plane, with a curl force azimuthally directed and linear. As expected, the quantum propagator, and the way this drives Gaussian wavepackets, directly reflects the spiralling classical curl force dynamics. Two classes of stationary states—eigenfunctions of a continuous spectrum for the unbounded Hamiltonian—are described. They possess unusual singularities and an unfamiliar quantisation condition; their explanation requires asymptotics and unfamiliar singularities in the underlying families of classical trajectories. The analysis is supported and illustrated numerically. [ABSTRACT FROM AUTHOR]

In this paper, we define non-lightlike framed rectifying curves. They may have singularities. We give equivalent definitions and a construction method for non-lightlike framed rectifying curves. Moreover, we also study the relationship between the non-lightlike framed rectifying curves and the non-lightlike framed helices, as well as the properties of the centrode of non-lightlike framed rectifying curves. [ABSTRACT FROM AUTHOR]

We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains D ⊂ ℝd, d = 2, 3. We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in D, comprising the countably-normed spaces of I. M. Babuska and B. Q. Guo. As intermediate result, we prove that continuous, piecewise polynomial high order ("p-version") finite elements with elementwise polynomial degree p ∈ ℕ on arbitrary, regular, simplicial partitions of polyhedral domains D ⊂ ℝd, d ⩾ 2, can be exactly emulated by neural networks combining ReLU and ReLU2 activations. On shape-regular, simplicial partitions of polytopal domains D, both the number of neurons and the number of nonzero parameters are proportional to the number of degrees of freedom of the hp finite element space of I. M. Babuška and B. Q. Guo. [ABSTRACT FROM AUTHOR]

This article examines ‘the economics of singularities’ by Lucien Karpik, which is a sociological theory explaining the functioning of markets for a specific type of goods, labeled as ‘singularities’, defined according to three criteria: the plurality of qualities, quality uncertainty and the primacy of quality. According to Lucien Karpik, singularities complement the typology of goods traditionally used by economists, consisting of homogeneous and differentiated goods. He argues that singularities represent a ‘blind spot’ of what he calls ‘neoclassical economic theory.’ According to him, while neoclassical economic theory is adequate for studying homogeneous and differentiated goods, an autonomous framework is indispensable for studying singularities. In this article, we propose a comprehensive overview of the economics of singularities, and we formulate a critique of Karpik’s positioning in relation to neoclassical economic theory. [ABSTRACT FROM AUTHOR]

We study the algebraic sets of pairs of matrices defined by the vanishing of the anti-diagonal as well as the cross-diagonal of their commutator matrix. We prove that, over a field of prime characteristic, the coordinate ring of the latter is always F -regular and, with exactly one exception, so is that of the former, thus proving a conjecture of Kadyrsizova and Yerlanov. [ABSTRACT FROM AUTHOR]

Consider a Kleinian singularity ℂ2/Γ, where Γ is a finite subgroup of SL2(ℂ). In this paper, we introduce a natural stack compactifying the singularity by adding a smooth stacky divisor, and we show that sets of framed sheaves on this stack satisfying certain additional criteria are closely related to a class of Nakajima quiver varieties. This partially extends our previous work on punctual Hilbert schemes of Kleinian singularities. [ABSTRACT FROM AUTHOR]

We study the local structure of vector fields on that preserve the Martinet -form . We classify their singularities up to diffeomorphisms that preserve the form , as well as their transverse unfoldings. We are thus able to provide a fairly complete list of the bifurcations such vector fields undergo. [ABSTRACT FROM AUTHOR]

We examine the focusing of kinetic energy and the amplification of various quantities during the snapping motion of the free end of a flexible structure. This brief but violent event appears to be a regularized finite-time singularity, with remarkably large spikes in velocity, acceleration, and tension easily induced by generic initial and boundary conditions. A numerical scheme for the inextensible string equations is validated against available experimental data for a falling chain and further employed to explore the phenomenon. We determined that the discretization of the equations, equivalent to the physically discrete problem of a chain, does not provide the regularizing length scale, which in the absence of other physical effects must then arise from the geometry of the problem. An analytical solution for a geometrically singular limit, a falling perfectly folded string, accounts surprisingly well for the scalings of several quantities in the numerics, but can only indirectly suggest a behavior for the curvature, one which seems to explain prior experimental data but does not correspond to the evolution of the curvature peak in our system, which instead displays a newly observed anomalously slow scaling. A simple model, incorporating only knowledge of the initial conditions along with the anomalous and singular-limit scalings, provides reasonable estimates for the amplifications of relevant quantities. This is a first step to predict and harness arbitrarily large energy focusing in structures, with a practical limit set only by length scales present in the discrete mechanical system or the initial conditions. [ABSTRACT FROM AUTHOR]

In this paper, we define the evolute and focal surface of a spacelike framed curve with lightlike components in Minkowski 3-space. It is a generalization of the previous results of regular spacelike curves, since singularities are allowed in the original spacelike curves studied by spacelike framed curves with lightlike components. Meanwhile, we show a new geometric invariant to characterise singularities of the focal surface. Then, the classification theorem and recognition theorem for the singularities of the focal surface in generic are also given. [ABSTRACT FROM AUTHOR]

Given a perfect field $k$ with algebraic closure $\overline {k}$ and a variety $X$ over $\overline {k}$ , the field of moduli of $X$ is the subfield of $\overline {k}$ of elements fixed by field automorphisms $\gamma \in \operatorname {Gal}(\overline {k}/k)$ such that the Galois conjugate $X_{\gamma }$ is isomorphic to $X$. The field of moduli is contained in all subextensions $k\subset k'\subset \overline {k}$ such that $X$ descends to $k'$. In this paper, we extend the formalism and define the field of moduli when $k$ is not perfect. Furthermore, Dèbes and Emsalem identified a condition that ensures that a smooth curve is defined over its field of moduli, and prove that a smooth curve with a marked point is always defined over its field of moduli. Our main theorem is a generalization of these results that applies to higher-dimensional varieties, and to varieties with additional structures. In order to apply this, we study the problem of when a rational point of a variety with quotient singularities lifts to a resolution. As a consequence, we prove that a variety $X$ of dimension $d$ with a smooth marked point $p$ such that $\operatorname {Aut}(X,p)$ is finite, étale and of degree prime to $d!$ is defined over its field of moduli. [ABSTRACT FROM AUTHOR]

The Hilbert scheme Hilb n ℂ 3 of n points on ℂ 3 can be expressed as the critical locus of a regular function on a smooth variety . Recent development in birational geometry suggests a study of singularities of the pair (, Hilb n ℂ 3) using jet schemes. In this paper, we use a comparison between Hilb n ℂ 3 and the scheme C 3 , n of three commuting n × n matrices to estimate the log canonical threshold of (, Hilb n ℂ 3). As a consequence, we see that although both dim and dim Hilb n ℂ 3 have asymptotic growth O (n 2) , the largest multiplicity of any points on Hilb n ℂ 3 has at most linear growth O (n). [ABSTRACT FROM AUTHOR]

Cosmic censorship protects the outside world from black hole singularities and paves the way for assigning entropy to gravity at the event horizons. We point out a tension between cosmic censorship and the quantum backreacted geometry of Schwarzschild black holes, induced by vacuum polarization and driven by the conformal anomaly. A similar tension appears for the Weyl curvature hypothesis at the Big Bang singularity. We argue that the requirement of exact conformal symmetry resolves both conflicts and has major implications for constraining the set of fundamental constituents of the Standard Model. [ABSTRACT FROM AUTHOR]

Exploring infrastructure has much to offer for economic anthropology. Inspired by the convergence of literatures on value and infrastructure in studies of financialisation, we develop the analytics of 'infrastructures of value' on old terrain: agriculture. Infrastructure facilitates valuation practices and enables valorisation as fixed capital. Material networks emerging from practices of infrastructuring also mediate value by facilitating, channelling, or hindering the circulation – movement and metamorphoses – of objects, people and ideas. Shifting attention from the social life of things to the infrastructure undergirding their circulation fills a major gap in David Graeber's theory of value: it directs attention to how actions become incorporated into larger wholes. Various infrastructures and the frictions between them shape value by connecting producers and consumers, separating contents, and communicating evidence of qualities of food. Ethnographic attention to this material relationality of value invigorates dialogue between new and historical materialism and challenges binaries in economic thought. [ABSTRACT FROM AUTHOR]

We present nodal algebraic hypersurfaces in the complex projective space which are projectively rigid. Defects and Alexander polynomials associated with the hypersurfaces are obtained. There are families of nodal hypersurfaces with nontrivial Alexander polynomials and nodal threefolds with projective rigidity which are potentially infinite. [ABSTRACT FROM AUTHOR]

The phenomenological theory revealing the generic effects of the problem symmetry, its violation, and energy conservation law on the singularities of the Poynting vector field is presented. The bifurcation scenario of formation (annihilation) of the singularities under variations of the problem parameters is elucidated. The results describe the singularities in scattering a linearly polarized plane electromagnetic wave. However, they are valid for any configuration of the incident beam at its scattering by a subwavelength particle. The author shows that topological changes in the pattern of the Poynting vector field occur through a finite number of pitchfork bifurcations. It means that the patterns are topologically stable under variations of the problem parameter(s) that lie between the bifurcation values. The latter ensures that the discussed topological properties of the problem are robust to weak symmetry violation, which is inevitable in any actual experiment. The general consideration is illustrated by a detailed study of singularities in scattering by an infinite right circular germanium cylinder. The results open the possibility of fitting and controlling radiation patterns on subwavelength scales important for various nanotechnologies. [ABSTRACT FROM AUTHOR]

Founded on a novel analytical formulation that led to a rigorous yet compact path-integral representation of the time-domain elastodynamic half-space Green's function, a unified analysis of the possible occurrence of different singular wavefront behaviour in the response under arbitrary impulsive internal or surface point loads at arbitrary source-receiver locations is presented. With the decomposition of the general solution into distinct initiating and reflected wave group integrals that share a common factored format and simple contour definitions, the mathematical framework is shown to allow a straightforward identification of the specific conditions and the particular wave groups that are responsible for the singular wavefront phenomena without resorting to advanced analytic function theories or asymptotic methods. Analytic characterizations of the nature, strength and direction of all intrinsic singular wavefront behaviours of the three-dimensional Green's function in three canonical cases of source-receiver configurations are given in a dual integral-closed form format to facilitate their theoretical understanding as well as computational applications. Graphical illustrations of their variation with the source-receiver configuration and the medium's Poisson's ratio together with relevant comparison and clarifications of some classical treatments are included. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• Control moment gyroscope singularities can be complicated by the space environment. • Novel steering control law combines control moment gyroscope steering with gimbal compensation. • Efficient computation. • Logic-Based algorithm determines best option for singularity avoidance. Control Moment Gyroscopes (CMGs) are effective torque-producing actuators for satellite missions requiring agile maneuvering and accurate pointing. CMGs can develop relatively large torques compared to other momentum storage actuators like reaction wheels; however, using CMGs for three-axis attitude control is complicated by inherent mechanical singularities associated with CMGs. The singularity effect is worsened by common disturbances, such as aerodynamic or magnetic torques, which influence the CMG momentum state. A recent CMG design known as the Double-Gimbal Scissor-Pair CMG (DGSPCMG) has shown promise for use in small satellites because it has manageable singular conditions and has the potential to achieve three axis control using only two flywheels. This paper proposes a novel extension of the DGSPCMG control law and supplements it with a classical sliding-mode attitude controller to allow for the DGSPCMG momentum to be managed such that the gimbals may be restored to desirable positions by way of magnetic torques. The proposed solution achieves agile maneuvering by differentiating when singularity escape is possible through a steering control law and when magnetic torques are necessary for gimbal compensation in the DGSPCMG. The effectiveness of the control scheme is demonstrated by numerical simulations in MATLAB which subject the design to a detailed space environment. For the simulation conditions used in this research, the results indicate that the proposed control scheme would be an effective solution for attitude control of small satellites using a DGSPCMG in Low-Earth Orbit. [ABSTRACT FROM AUTHOR]

We study quotients of quadratic forms and associated polar lines in the projective plane. Our results, applied pointwise to quadratic differential forms, shed some light on classical binary differential equations (BDEs) associated to congruences of lines in Euclidean 3-space and lead us to construct a new one. The new BDE yields a new singular surface in the Euclidean 3-space associated to the congruence of lines. We determine the generic local configurations of the new and the classical BDEs. [ABSTRACT FROM AUTHOR]

Topology-optimisation strategies for structural design of discrete surface structures result in unstructured patterns that require post-rationalisation to limit the number and complexity of the various structural elements. Fabrication-related objectives still demand further processing. Designers need topology exploration, before topology optimisation, following a generative-design approach that is decoupled from density tuning and shape design, to produce high-quality patterns. Such an approach allows to flexibly embed multiple design constraints, benefit from state-of-the-art form-finding algorithms and explore design trade-offs of the multiple requirements related to architecture, engineering and construction. This research investigates a parameterisation strategy to encode topological exploration of structured patterns based on quad meshes. The focus is set on singular vertices, which connect an irregular number of blocks, cables, or beams. Singularities are independent from pattern density and geometry, and have a fundamental influence on the qualitative and quantitative performance of the structure. We introduce an L-system encoding a quad-mesh grammar into a string that describes topological transformations of these singularities. Design applications to a net and a gridshell demonstrate the influence of singularities on the design of surface structures, and highlight the flexibility and generality of the approach in terms of geometrical processing and performance metrics. Beyond exploration, this parameterisation strategy opens to novel applications of search and optimisation methods for generative design of singularities in structural patterns. [ABSTRACT FROM AUTHOR]

Chemical erosion, one of the two major erosion processes along with mechanical erosion, occurs when a soluble rock-like salt, gypsum, or limestone is dissolved in contact with a water flow. The coupling between the geometry of the rocks, the mass transfer, and the flow leads to the formation of remarkable patterns, like scallop patterns in caves. We emphasize the common presence of very sharp shapes and spikes, despite the diversity of hydrodynamic conditions and the nature of the soluble materials. We explain the generic emergence of such spikes in dissolution processes by a geometrical approach. Singularities at the interface emerge as a consequence of the erosion directed in the normal direction, when the surface displays curvature variations, like those associated with a dissolution pattern. First, we demonstrate the presence of singular structures in natural interfaces shaped by dissolution. Then, we propose simple surface evolution models of increasing complexity demonstrating the emergence of spikes and allowing us to explain at long term by coarsening the formation of cellular structures. Finally, we perform a dissolution pattern experiment driven by solutal convection, and we report the emergence of a cellular pattern following well the model predictions. Although the precise prediction of dissolution shapes necessitates performing a complete hydrodynamic study, we show that the characteristic spikes which are reported ultimately for dissolution shapes are explained generically by geometrical arguments due to the surface evolution. These findings can be applied to other ablation patterns, reported for example in melting ice. [ABSTRACT FROM AUTHOR]

In SGA 2, Grothendieck conjectures that the étale fundamental group of the punctured spectrum of a complete noetherian local domain of dimension at least two with algebraically closed residue field is topologically finitely generated. In this paper, we prove a weaker statement, namely that the maximal pro-nilpotent quotient of the fundamental group is topologically finitely generated. The proof uses p -adic nearby cycles and negative definiteness of intersection pairings over resolutions of singularities as well as some analysis of Lie algebras of certain algebraic group structures on deformation cohomology. [ABSTRACT FROM AUTHOR]

This research aims to define Al-imam Al-Daylami and Al-Mawrdi, stating the agreements and singularities in Irregular Qur'anic readings in their interpretations (Tafseer): (Al-Burhan, Al-Nukat, and Al-Uyoun) the importance of these two books and their sources, which both are considered as of the important sources in Tafseer. The researcher used the inductive historical approach when defining both two imams and their two books. The researcher used also the descriptive approach when presenting the agreements and singularities in the Irregular readings, and their style in presenting and orienting the readings. The researcher divided the research into two sections: First, defining Al-imam Al-Daylami and Al-Mawrdi and their two books. Second, stating the agreements and singularities in the Irregular Qur'anic readings. The most prominent findings of the research: both Imams mentioned the Irregular readings in their explanations, yet they didn't state their Irregular except in rarely cases. They also oriented the Irregular Qur'anic readings, and stating its impact on the meaning and on the linguistic and interpretive aspects. Moreover, they directed the Irregular readings using aphorisms and opinion, citing them sometimes in Arab words and poetry. As for the attribution of the Irregular readings to their readers, Imam Al-Mawardi mostly attributed them, while other Al-Daylami did not attribute it to its readers, and merely expressed it in the passive voice with the word (read). The researcher recommended singling out the agreements and singularities of Al-Daylami and Al-Mawardi in frequent readings, and directing them through their books Al-Burhan, Al-Nukat, and Al-Uyoun, with a scientific study. [ABSTRACT FROM AUTHOR]

In this paper, we have parameterized a timelike ( T l i k e ) circular surface ( CI surface ) and have obtained its geometric properties, including striction curves, singularities, Gaussian and mean curvatures. Afterward, the situation for a T l i k e roller coaster surface ( RCO surface ) to be a flat or minimal surface is examined in detail. Further, we illustrate the approach's outcomes with a number of pertinent examples. [ABSTRACT FROM AUTHOR]

This work deals with an exhaustive study of bouncing cosmology in the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker spacetime. The geometry of the bouncing point has been studied extensively and used as a tool to classify the models from the point of view of cosmology. Raychaudhuri equation (RE) has been furnished in these models to classify the bouncing point as regular point or singular point. Behavior of time-like geodesic congruence in the neighborhood of the bouncing point has been discussed using the Focusing Theorem which follows as a consequence of the RE. An analogy of the RE with the evolution equation for a linear harmonic oscillator has been made and an oscillatory bouncing model has been discussed in this context. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper is a practical guideline on how to analyze and evaluate the literature algorithms of singularity-robust inverse kinematics or to construct new ones. Additive, multiplicative, and based on the Singularity Value Decomposition (SVD) methods are examined to retrieve well-conditioning of a matrix to be inverted in the Newton algorithm of inverse kinematics. It is shown that singularity avoidance can be performed in two different, but equivalent, ways: either via properly modified manipulability matrix or not allowing the decrease of the minimal singular value below a given threshold. It is discussed which method can always be used and which can only be used when some pre-conditions are met. Selected methods are compared to with respect to the efficiency of coping with singularities based on a theoretical analysis as well as simulation results. Also, some questions important for mathematically and/or practically oriented roboticians are stated and answered. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive study of the forward and inverse kinematics of a six-degrees-of-freedom (DoF) spatial manipulator with a novel architecture. Developed by Systemantics India Pvt. Ltd., Bangalore, and designated as the H6A (i.e., Hybrid 6-Axis), this manipulator consists of two arm-like branches, which are attached to a rigid waist at the proximal end and are coupled together via a wrist assembly at the other. Kinematics of the manipulator is challenging due to the presence of two multi-DoF passive joints: a spherical joint in the right arm and a universal in the left. The forward kinematic problem has eight solutions, which are derived analytically in the closed form. The inverse kinematic problem leads to $160$ solutions and involves the derivation of a $40$ -degree polynomial equation, whose coefficients are obtained as closed-form symbolic expressions of the pose parameters of the end-effector, thus ensuring the generality of the results over all possible inputs. Furthermore, the analyses performed lead naturally to the conditions for various singularities involved, including certain non-trivial architecture singularities. The results are illustrated via numerical examples which are validated extensively. [ABSTRACT FROM AUTHOR]

Let ξ be an analytic vector field in R 3 with an isolated singularity at the origin and having only hyperbolic singular points after a reduction of singularities π : M → R 3. The union of the images by π of the local invariant manifolds at those hyperbolic points, denoted by Λ, is composed of trajectories of ξ accumulating to 0 ∈ R 3. Assuming that there are no cycles nor polycycles on the divisor of π , together with a Morse-Smale type property and a non-resonance condition on the eigenvalues at these points, in this paper we prove the existence of a fundamental system { V n } of neighborhoods well adapted for the description of the local dynamics of ξ : the frontier F r (V n) is everywhere tangent to ξ except around F r (V n) ∩ Λ , where transversality is mandatory. [ABSTRACT FROM AUTHOR]

This paper characterizes the possible blow-up of solutions for the 3D magneto-hydrodynamics (MHD for short) equations. We first establish some ϵ-regularity criteria in L q , ∞ spaces for suitable weak solutions, and then together with an embedding theorem from L p , ∞ space into a Morrey type space we characterize the local behaviors of solutions near a potential singular point. More precisely, we show that if z 0 = (t 0 , x 0) is a singular point, then for any r>0 it holds that lim sup t → t 0 − (‖ u (t , x) − u (t) x 0 , r ‖ L 3 , ∞ (B r (x 0)) + ‖ b (t , x) − b (t) x 0 , r ‖ L 3 , ∞ (B r (x 0)) ) > δ ∗ ; lim sup t → t 0 − (t 0 − t) 1 μ r 2 ν − 3 p ‖ (u , b) (t) ‖ L p , ∞ (B r (x 0)) > δ ∗ f o r 1 μ + 1 ν = 1 2 , 2 ≤ ν ≤ 2 p 3 , 3 < p ≤ ∞ ; lim sup t → t 0 − (t 0 − t) 1 μ r 2 ν − 3 p + 1 ‖ (∇ u , ∇ b) (t) ‖ L p (B r (x 0)) > δ ∗ f o r 1 μ + 1 ν = 1 2 , ν ∈ { [ 2 , ∞ ] , p ≥ 3 [ 2 , 2 p 3 − p ] , 3 2 ≤ p < 3 where δ ∗ is a positive constant independent on ν and p. [ABSTRACT FROM AUTHOR]

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

Machine learning algorithms especially deep neural networks have seen tremendous growth in their real-world deployment. While these algorithms have to yield high performances for the task at hand, it is of equal importance that they are robust against the different kinds of singularities or adversarial anomalies. Researchers have generally addressed singularity detection independently for each anomaly, however, for the algorithms to be effective in the real world, the singularity detection algorithm must be able to handle a wide variety of attacks whether trained individually on them or not seen at the time of training. With this objective, in this paper, we propose a unique end-to-end transformation domain network to detect a variety of well-known attacks. The proposed architecture utilizes a combination of two different wavelet transformations to simultaneously learn low-level and high-level image features in a deep stacked layer fashion. The proposed method is generalized to handle a broad set of singularities of several computer vision algorithms whether operating in the object recognition domain or biometrics recognition. We showcase the results on a wide range of singularity points including (i) adversarial perturbations which are learned using deep learning networks, (ii) physical presentation attacks on face recognition which aim to produce fake data for the sensor for acquisition, and identification, (iii) synthetic images, and (iv) digital retouching of the images. Even with a zero-day or open-world attack setting, the results of the proposed algorithm show improvement over state-of-the-art results. The proposed architecture is agnostic to parameters, computationally efficient to provide a sustainable detector to resource-constrained institutions, provides robustness to attacks, and supports green computing. • A novel parameter-agnostic architecture is proposed to defend anomalies. • The proposed algorithm can handle adversaries in seen, unseen, and open-set settings. • The proposed algorithm is domain agnostic i.e., can handle vision and face datasets. • Our unified algorithm surpasses the SOTA works developed to handle specific attacks. • Bring a computationally feasible algorithm, it can be deployed on mobile devices. [ABSTRACT FROM AUTHOR]

The time-optimal path following (OPF) problem is to find a time evolution along a prescribed path in task space with shortest time duration. Numerical solution algorithms rely on an algorithm-specific (usually equidistant) sampling of the path parameter. This does not account for the dynamics in joint space, that is, the actual motion of the robot, however. Moreover, a well-known problem is that large joint velocities are obtained when approaching singularities, even for slow task space motions. This can be avoided by a sampling in joint space, where the path parameter is replaced by the arc length. Such discretization in task space leads to an adaptive refinement according to the nonlinear forward kinematics and guarantees bounded joint velocities. The adaptive refinement is also beneficial for the numerical solution of the problem. It is shown that this yields trajectories with improved continuity compared to an equidistant sampling. The OPF is reformulated as a second-order cone programming and solved numerically. The approach is demonstrated for a 6-DOF industrial robot following various paths in task space. [ABSTRACT FROM AUTHOR]

In certain polytopal domains Ω , in space dimension d = 2 , 3 , we prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in H 1 (Ω) for weighted analytic function classes. These classes comprise in particular solution sets of source and eigenvalue problems for elliptic PDEs with analytic data. Functions in these classes are locally analytic on open subdomains D ⊂ Ω , but may exhibit isolated point singularities in the interior of Ω or corner and edge singularities at the boundary ∂ Ω . The exponential approximation rates are shown to hold in space dimension d = 2 on Lipschitz polygons with straight sides, and in space dimension d = 3 on Fichera-type polyhedral domains with plane faces. The constructive proofs indicate that NN depth and size increase poly-logarithmically with respect to the target NN approximation accuracy ε > 0 in H 1 (Ω) . The results cover solution sets of linear, second-order elliptic PDEs with analytic data and certain nonlinear elliptic eigenvalue problems with analytic nonlinearities and singular, weighted analytic potentials as arise in electron structure models. Here, the functions correspond to electron densities that exhibit isolated point singularities at the nuclei. [ABSTRACT FROM AUTHOR]

We show that the simplicial stratification associated to a triangulation of a PL pseudomanifold possesses a canonical system of trivializations of link bundles that satisfies a natural compatibility condition over nested singular strata. Consequently, Agustín Vicente and Fernández de Bobadilla's generalization of Banagl's intersection space construction is applicable to all PL pseudomanifolds (and in particular, to all complex algebraic varieties). [ABSTRACT FROM AUTHOR]

In this work, we present and prove results underlying a method which uses functionals derived from the interaction integral to approximate the stress intensity factors along a three-dimensional crack front. We first prove that the functionals possess a pair of important properties. The functionals are well-defined and continuous for square-integrable tensor fields, such as the gradient of a finite element solution. Furthermore, the stress intensity factors are representatives of such functionals in a space of functions over the crack front. Our second result is an error estimate for the numerical stress intensity factors computed via our method. The latter property of the functionals provides a recipe for numerical stress intensity factors; we apply the functionals to the gradient of a finite element approximation for a specific set of crack front variations, and we calculate the stress intensity factors by inverting the mass matrix for those variations. [ABSTRACT FROM AUTHOR]

We introduce consideration of dispersive aspects of standard perfect fluid Friedmann cosmology and study the new qualitative behaviours of cosmological solutions that emerge as the fluid parameter changes and zero eigenvalues appear in the linear part of the Friedmann equations. We find that due to their insufficient degeneracy, the Milne, flat, Einstein-static, and de Sitter solutions cannot properly bifurcate. However, the dispersive versions of Milne and flat universes contained in the versal unfolding of the standard Friedmann equations possess novel long-term properties not met in their standard counterparts. We apply these results to the horizon problem and show that unlike their hyperbolic versions, the dispersive Milne and flat solutions completely synchronize in the future, hence offering a solution to the homogeneity, isotropy, and causal connectedness puzzles. [ABSTRACT FROM AUTHOR]

A surface with nonempty timelike, lightlike, and spacelike points in Minkowski 3‐space is a mixed type surface. The mixed type surface has a signature‐changing metric, and its lightlike points can be seen as singularities of such metric. In this paper, we study singular properties of pseudo‐spherical evolutes of lightlike loci on mixed type surfaces. We classify the generic singularities of pseudo‐spherical evolutes by the singularity theory. These singularities and the contact between lightlike loci and model submanifolds are closely associated. We also show the Legendrian duality among pseudo‐spherical evolutes, lightlike loci, and Darboux vectors. Moreover, the singularities of pseudo‐spherical evolutes are studied from the viewpoint of opening map‐germ. Finally, we give an example to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

We characterize Hilbert polynomials that give rise to Hilbert schemes with two Borel-fixed points and determine when the associated Hilbert schemes or its irreducible components are smooth. In particular, we show that the Hilbert scheme is reduced and has at most two irreducible components. By describing the singularities in a neighbourhood of the Borel-fixed points, we prove that the irreducible components are Cohen-Macaulay and normal. We end by giving many examples of Hilbert schemes with three Borel-fixed points. [ABSTRACT FROM AUTHOR]

Assuming a certain form of resolution of singularities, we prove a general existential Ax-Kochen/Ershov principle for tamely ramified fields in all characteristics. This specializes to well-known results in residue characteristic 0 and unramified mixed characteristic. It also encompasses the conditional existential decidability results known for F p ((t)) and its finite extensions, due to Denef-Schoutens. On the other hand, it also applies to the setting of infinite ramification, thereby providing us with an abundance of existentially decidable infinitely ramified extensions of Q p and F p ((t)). [ABSTRACT FROM AUTHOR]

As is well known, self-similar solutions to the mean curvature flow, including self-shrinkers, translating solitons and self-expanders, arise naturally in the singularity analysis of the mean curvature flow. In this paper we give complete classifications of codimension one complete self-shrinkers with nonnegative constant scalar curvature. We also give alternative proofs of the classifications of codimension one translating solitons and self-expanders with nonnegative constant scalar curvature. [ABSTRACT FROM AUTHOR]