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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]

As commonly known, standard time integration of the kinematic equations of rigid bodies modeled with three rotation parameters is infeasible due to singular points. Common workarounds are reparameterization strategies or Euler parameters. Both approaches typically vary in accuracy depending on the choice of rotation parameters. To efficiently compute different kinds of multibody systems, one aims at simulation results and performance that are independent of the type of rotation parameters. As a clear advantage, Lie group integration methods are rotation parameter independent. However, few studies have addressed whether Lie group integration methods are more accurate and efficient compared to conventional formulations based on Euler parameters or Euler angles. In this paper, we close this gap using the R 3 × S O (3) Lie group formulation and several typical rigid multibody systems. It is shown that explicit Lie group integration methods outperform the conventional formulations in terms of accuracy. However, it turns out that the conventional Euler parameter-based formulation is the most accurate one in the case of implicit integration, while the Lie group integration method is computationally the more efficient one. It also turns out that Lie group integration methods can be implemented at almost no extra cost in an existing multibody simulation code if the Lie group method used to describe the configuration of a body is chosen accordingly. [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]

This article studies equilibrium singular configurations of gels and addresses open questions concerning gel energetics. We model a gel as an incompressible, immiscible and saturated mixture of a solid polymer and a solvent that sustain chemical interactions at the molecular level. We assume that the energy of the gel consists of the elastic energy of its polymer network plus the Flory-Huggins energy of mixing. The latter involves the entropic energies of the individual components plus that of interaction between polymer and solvent, with the temperature dependent Flory parameter, χ , encoding properties of the solvent. In particular, a good solvent promoting the mixing regime, is found below the threshold value χ = 0.5 , whereas the phase separating regime develops above that critical value. We show that cavities and singularities develop in the latter regime. We find two main classes of singularities: (i) drying out of the solvent, with water possibly exiting the gel domain through the boundary, leaving behind a core of exposed polymer at the centre of the gel; (ii) cavitation, in response to traction on the boundary or some form of negative pressure, with a cavity that can be either void or flooded by the solvent. The straightforward and unified mathematical approach to treat all such singularities is based on the construction of appropriate test functions, inspired by the particular states of uniform swelling or compression. The last topic of the article addresses a statistical mechanics rooted controversy in the research community, providing an experimental and analytic study in support of the phantom elastic energy versus the affine one. [ABSTRACT FROM AUTHOR]

We carry out complete membership to Kaplan classes of functions given by formula { ζ ∈ C : | ζ | < 1 } ∋ z ↦ ∏ k = 1 n (1 - z e - i t k ) p k , where n ∈ N , t k ∈ [ 0 ; 2 π) and p k ∈ R for k ∈ N ∩ [ 1 ; n ] . In this way we extend Sheil-Small’s, Jahangiri’s and our previous results. Moreover, physical and geometric applications of the obtained gap condition are given. The first one is an interpretation in terms of mass and density. The second one is a visualization in terms of angular inequalities between vectors in R 2 . [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]

The Gauss map of a generic immersion of a smooth, oriented surface into R 4 is an immersion. But this map takes values on the Grassmanian of oriented 2-planes in R 4 . Since this manifold has a structure of a product of two spheres, the Gauss map has two components that take values on the sphere. We study the singularities of the components of the Gauss map and relate them to the geometric properties of the generic immersion. Moreover, we prove that the singularities are generically stable, and we connect them to the contact type of the surface and J -holomorphic curves with respect to an orthogonal complex structure J on R 4 . Finally, we get some formulas of Gauss–Bonnet type involving the geometry of the singularities of the components with the geometry and topology of the surface. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the analysis and detailed study of a procedure to generate both the weighted arithmetic and harmonic means of n positive real numbers. Together with this interpretation, we prove some relevant properties that will allow us to define numerical approximation methods in several dimensions adapted to discontinuities. [ABSTRACT FROM AUTHOR]

We derive a variant of the Weierstrass–Enneper representation for generalized maximal immersions generated by a single harmonic function. We use this variant to construct examples of generalized maximal immersion from a single family. Subsequently, we proved a ‘superposition principle’ for generalized maximal immersions. It says that if we have given finitely many generalized maximal immersions, one can add their parametrizations to get another generalized maximal immersion under certain assumptions. We also study how the singularity of maxfaces behaves under superposition. [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]

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

The asymptotic Samuel function generalizes to arbitrary rings the usual order function of a regular local ring. In this paper, we use this function to introduce the notion of the Samuel slope of a Noetherian local ring, and we study some of its properties. In particular, we focus on the case of a local ring at singular point of a variety, and, among other results, we prove that the Samuel slope of these rings is related to some invariants used in algorithmic resolution of singularities. [ABSTRACT FROM AUTHOR]

We introduce a graph associated with any stable map defined from the connected sum # n (S 1 × S 2) of n copies of the product S 1 × S 2 to the Euclidean 3-space. This graph has a weight on each vertex and a pair of weights on each edge, and its properties provide a necessary and sufficient condition to be the graph of a stable map defined from # n (S 1 × S 2) to the Euclidean 3-space. [ABSTRACT FROM AUTHOR]

We focus on investigating the differential geometric properties of cuspidal edge in the three-sphere from a viewpoint of duality. Using Legendrian duality, we study a special kind of flat surface along cuspidal edge in three-dimensional sphere space. This kind of surface is dual to the singular set of the cuspidal edge surface. Thus, we call it the flat Δ -dual surface. Flatness of a surface can be defined by the degeneracy of the dual surface. It is similar to the case for the Gauss map of a flat surface in Euclidean space. Moreover, classifications of singularities of the flat Δ -dual surface are shown. We also investigate the dual relationships of singularities between flat Δ -dual surface and flat approximations of the original cuspidal edge surface. At last, we consider a global geometry of the singular set of a cuspidal edge surface using the flat Δ -dual surface. [ABSTRACT FROM AUTHOR]

In this research, we have constructed and studied special tubular surfaces in Euclidean 3-space R³. We examined the singularities and geometrical properties of these surfaces. We achieved some significant results for these surfaces via Darboux frame. Also, we have proposed a few geometric invariants that illustrate the geometric characteristics of these surfaces, such as tubular Weingarten surfaces, using the traditional methods of differential geometry. Additionally, taking advantage of the singularity theory, we have given the classification of generic singularities of these surfaces. At last, we have presented some computational examples as an instance of use to validate our theoretical findings. [ABSTRACT FROM AUTHOR]

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

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]

Branched coverings boast a rich history, ranging from the ramification of Riemann surfaces to the realization of 3-manifolds as coverings branched over knots and spanning both geometric topology and algebraic geometry. This work delves into branched coverings “à la Fox” of (G, X)-manifolds, encompassing three main avenues: Firstly, we introduce a comprehensive class of singular (G, X)-manifolds, elucidating elementary theory paired with illustrative examples to showcase its efficacy and universality. Secondly, building on Montesinos’ work, we revisit and augment the prevailing knowledge, formulating a Galois theory tailored for such branched coverings. This includes a detailed portrayal of the fiber above branching points. Lastly, we identify local attributes that guarantee the existence of developing maps for singular (G, X)-manifolds within the branched coverings framework. Notably, we pinpoint conditions that ensure the existence of developing maps for these singular manifolds. This research proves especially pertinent for non-metric singular (G, X)-manifolds like those of Lorentzian or projective nature, as discussed by Barbot, Bonsante, Suhyoung Choi, Danciger, Seppi, Schlenker, and the author, among others. While examples are sprinkled throughout, a standout application presented is a uniformization theorem “à la Mess” for singular locally Minkowski manifolds exhibiting BTZ-like singularities. [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]