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In this paper, we obtain the explicit expression of the supersymmetric algebra associated with the recently proposed massive supermembrane including all surface terms. We formulate the theory as the limit of a supermembrane on a genus-two compact Riemann surface when one of the handles becomes a string attached to a torus. The formulation reduces to a supermembrane on a punctured torus with a “string spike” (in the sense of [1]), attached to it. In this limit, we identify all surface terms of the algebra and give the explicit expression of the Hamiltonian in agreement with the previous formulation of it. The symmetry under area preserving diffeomorphisms, connected and nonconnected to the identity, is also discussed. Only parabolic Sl(2,Z) discrete symmetries are preserved.

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]

In this paper we have discussed the modified gravity and scalar field DE model specifically DBI essence model with the analysis of thermodynamics during cosmological evolution. We have used the modified gravity with the form f R , T = R + 2 λ T in our calculations. The basic aim behind this paper is to discuss a theory that unifies modified gravity with DE models including the solutions of some cosmological problems like thermodynamics energy conditions violation problems, finite time future singularity problems, initial singularity problem, cosmic inflation problem, decelerated expansion problem, graceful exit problem, reheating problem, bouncing nature problem, phase transformation-spontaneous symmetry breaking problem, negative heat capacity paradox problem and obviously the present-day universe problems (continuously decreasing temperature problem and present-day DE dominated expansion problem). We have established the viscous effects (both positive and negative viscosity) in our calculation and discussed the negative–positive viscosity by introducing two special type of energy cycles. Finally, we have discussed the stability conditions for universe evolution through cosmic perturbation and resolved the instability problems. We have also shown the state finder trajectories for both with and without viscous fluid on the basis of our calculations to compare our research with the results of other independent DE models. [ABSTRACT FROM AUTHOR]

An analytical and general form factor for any polyhedron is derived on the basis of a projection method, in terms of the vertex coordinates and topology of the polyhedron. An integral over the polyhedron equals the sum of the signed integrals over a set of dissected tetrahedra by defining a sign function, and a general tetrahedral form factor is established by defining a projection method. All possible singularities present in the formula are discussed in detail. Using a MATLAB implementation, illustrative examples are discussed to verify the accuracy and generality of the method. The use of the scalar product operation and the sign function in this work allows a general and neat formula to be obtained for any polyhedron, including convex and concave polyhedra. The formulas and discussions presented here will be useful for the characterization of nanoparticles using small‐angle scattering techniques. [ABSTRACT FROM AUTHOR]

In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results. [ABSTRACT FROM AUTHOR]

In this note we show finite time blow-up for a class of non-local active scalar equations on compact Riemannian manifolds. The strategy we follow was introduced by Silvestre and Vicol [Trans. Amer. Math. Soc. 368 (2016), pp. 6159–6188] to deal with the one dimensional Córdoba-Córdoba-Fontelos equation and might be regarded as an instance of De Giorgi's method. [ABSTRACT FROM AUTHOR]

Let R denote the generalized Radon transform, which integrates over a family of N-dimensional smooth submanifolds S y ~ ⊂ U , 1 ≤ N ≤ n - 1 , where an open set U ⊂ R n is the image domain. The submanifolds are parametrized by points y ~ ⊂ V ~ , where an open set V ~ ⊂ R n is the data domain. We assume that the canonical relation C ~ from T ∗ U to T ∗ V ~ of R is a local canonical graph (when R is viewed as a Fourier Integral Operator). The continuous data are denoted by g, and the reconstruction is f ˇ = R ∗ B g . Here R ∗ is a weighted adjoint of R , B is a pseudo-differential operator, and g is a conormal distribution. Discrete data consists of the values of g on a regular lattice with step size O (ϵ) . Let S denote the singular support of f ˇ , and f ˇ ϵ = R ∗ B g ϵ be the reconstruction from interpolated discrete data g ϵ (y ~) . Pick a point x 0 ∈ S , i.e. the singularity of f ˇ at x 0 is visible from the data. The main result of the paper is the computation of the limit DTB (x ˇ) : = lim ϵ → 0 ϵ κ f ˇ ϵ (x 0 + ϵ x ˇ). Here κ ≥ 0 is selected based on the strength of the reconstructed singularity, and x ˇ is confined to a bounded set. The limiting function DTB (x ˇ) , which we call the discrete transition behavior, contains full information about the resolution of reconstruction. [ABSTRACT FROM AUTHOR]

We provide a local description of the curves with minimal length based at singularities in a family of polyhedral surfaces. These singularities are accumulation points of vertices with conical angles equal to π and 4π (or 3π, in a variation). While a part of the minimizing curves behaves quite like the ones reaching conical vertices, the singularities present features such as being connected to points arbitrarily close to them by exactly two minimizing curves. The spaces containing such singularities are constructed as metric quotients of an euclidean half-disk by certain identification patterns along its edge. These patterns are examples of what is known as paper-folding schemes, and we provide the foundational aspects about them which are necessary for our analysis. The arguments are based on elementary metric geometry and calculus. [ABSTRACT FROM AUTHOR]

In this paper, we show that the gluing of germs of formal complex spaces is also a formal complex space. Moreover, we study the Betti numbers of the gluing of formal complex spaces and, for instance, we show that the Betti numbers satisfy βi(X,0)(0)≥di,$$\begin{equation*} \hspace*{11pc}\beta _i^{(\mathcal {X},0)}(0)\ge \binom{d}{i}, \end{equation*}$$for all 1≤i≤d$1\le i\le d$, where (X,0)$(\mathcal {X},0)$ is a d‐dimensional germ of formal complex space given by a regular and a singular germ of formal complex spaces with the same dimension. In particular, this proves the Buchsbaum–Eisenbud–Horrocks conjecture for certain gluing of germs of formal complex spaces. Some formulas for the Betti numbers of the gluing of germs of formal complex spaces are given in terms of the invariant multiplicities, Euler obstruction, Milnor number, and polar multiplicities. As an application, some upper bounds for the Betti numbers of the gluing of some germs of complete intersection are provided, using recent progress on the Watanabe's conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we explore the gravitational collapse of matter (dust) under the effect of zero-point length l 0 . During the gravitational collapse, we neglect the backreaction effect of pre-Hawking radiation (in the sense that it is a small effect and cannot prevent the formation of an apparent horizon), then we recast the internal metric of a collapsing star as a closed FRW universe for any spherically symmetric case and, finally, we obtain the minimal value for the scale factor, meaning that the particles never hit the singularity. We argue that the object emerging at the end of the gravitational collapse can be interpreted as Planck stars (black hole core) hidden inside the event horizon of the black hole, with a radius proportional to (G M l 0 2 / c 2) 1 / 3 . Quite interestingly, we found the same result for the radius of the Planck star using a free-falling observer point of view. In addition, we point out a correspondence between the modified Friedmann's equations in loop quantum gravity and the modified Friedmann's equation in string T-duality. In the end, we discuss two possibilities regarding the final stage of the black hole. The first possibility is that we end up with Planck-size black hole remnants. The second possibility is that the inner core can be unstable and, due to the quantum tunneling effect, the spacetime can undergo a black-hole-to-white-hole transition (a bouncing Planck star). [ABSTRACT FROM AUTHOR]

This article brings a reflective discussion about the singularities of these small cities, based on a comparative study between three cities of the Delta: Afuá and Ponta de Pedras, located in the state of Pará, and Mazagão, in the state of Amapá. Data collected through the application of questionnaires to urban residents and urban area mapping were used, in addition to census data. The results demonstrate that the three cities grew at similar intensities, as a result of unique processes, but their residents have, similarly, precarious access to infrastructure and services. The paper also discusses the invisible aspects of these cities, the solidary happening, the nature of the "ribeirurbano", a society that maintains relationships in networks, with activities in the informal circuit of the economy, alongside non-urban functions, in the production of a social space that, despite maintaining the riverine ethos, it is an urbanized riverside. [ABSTRACT FROM AUTHOR]

The approach of the paper is on spacelike circular surfaces in the Minkowski 3-space. A spacelike circular surface is a one-parameter family of Lorentzian circles with a fixed radius regarding a non-null curve, which acts as the spine curve, and it has symmetrical properties. In the study, we have parametrized spacelike circular surfaces and have provided their geometric and singularity properties such as Gaussian and mean curvatures, comparing them with those of ruled surfaces and the classification of singularities. Furthermore, the conditions for spacelike roller coaster surfaces to be flat or minimal surfaces are obtained. Meanwhile, we support the results of the approach with some examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system driven by a controlled Schrödinger equation with two external control inputs. Motivated by applications to the control of quantum systems having conical or semi-conical eigenvalue intersections, we propose to study the singularities and the parametric bifurcations of the associated non-mixing field, along whose integral curves in the space of controls the adiabatic approximation holds with higher precision. Our results can be applied to optimize the adiabatic control strategies of well known quantum systems such as Qubit systems, Stirap Processes and Eberly-Law models. [ABSTRACT FROM AUTHOR]

We provide a complete classification of the singularities of cluster algebras of finite type with trivial coefficients. Alongside, we develop a constructive desingularization of these singularities via blowups in regular centers over fields of arbitrary characteristic. Furthermore, from the same perspective, we study a family of cluster algebras which are not of finite type and which arise from a star shaped quiver. [ABSTRACT FROM AUTHOR]

In order to find useful information to complete the classification of Enriques– Fano threefolds, we will computationally study the singularities of some known Enriques–Fano threefolds of genus 6, 7, 8, 9, 10, 13, and 17. We will also deduce the projective normality of these threefolds. [ABSTRACT FROM AUTHOR]

We consider Cauchy problem for ordinary differential equation with solution possessing a sequence of multiple poles. We propose the generalized reciprocal function method. It reduces calculation of a multiple pole to retrieval of a simple zero of accordingly chosen function. Advantages of this approach are illustrated by numerical examples. We propose two representative test problems which constitute interest for verification of other numerical methods for problems with poles.

The present work deals with the Raychaudhuri equation (RE) to examine the space-time singularity both from classical and quantum point of view. The RE has been looked upon in terms of a classical linear Harmonic oscillator and Focusing theorem has been studied. Also, the Raychaudhuri scalar has been shown to affect the convergence and singularity formation in Black-holes and cosmology. Finally, resolution of initial big-bang singularity has been attempted quantum mechanically in two ways namely by canonical quantization and by formulation of Bohmian trajectories. [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.

A review of modern methods for effective calculations of Feynman integrals containing both massless propagators and propagators with masses is given. The effectiveness of these methods in various fields of their application is demonstrated by the examples under consideration.

In this paper, we have parameterized a timelike (Tlike) circular surface (CIsurface) and have obtained its geometric properties, including striction curves, singularities, Gaussian and mean curvatures. Afterward, the situation for a Tlike roller coaster surface (RCOsurface) to be a flat or minimal surface is examined in detail. Further, we illustrate the approach’s outcomes with a number of pertinent examples.

We define involutes and contrapedal curves of spherical curves and investigate some properties. We also give the classifications of the singular points of involutes. Moreover, we give the relationships between involute–evolute curve pairs and pedal–contrapedal curve pairs and give the formula of the rth involute. [ABSTRACT FROM AUTHOR]

In Nepal, there are many mathematics subjects taught at university level. Among them, complex analysis is the most powerful. In complex analysis, the Laurent series expansion is a well-known subject because it may be used to find the residues of complex functions around their singularities. It turns out that computing the Laurent series of a function around its singularities is an effective way to calculate the integral of the function along any closed contour around the singularities as well as the residue of the function. Learning the Laurent series concepts can be difficult, and many students struggle to develop adequate understanding, reasoning, and problem-solving skills. Therefore, this article presents multiple practical examples where the Laurent series of a function is found and then utilized to compute the integral of the function over any closed contour around the singularities of the function, based on the theory of the Laurent series. [ABSTRACT FROM AUTHOR]

The possibility of a theoretical analysis of the proximity of parallel mechanisms to singularities in the case when the group of all wrenches of the mechanism includes linearly dependent wrenches is considered. It is proposed to consider separate groups of wrenches as subspaces of a six-dimensional linear space. In this case, the angle between the subspaces can serve as a criterion of proximity to a singularity. The relation of this angle to the value of the generalized reaction associated with the corresponding wrench is noted. Using the example of a 4-RRR mechanism, it is shown that this measure is continuous, in contrast to the rank of the wrench matrix, which changes only in a singularity. [ABSTRACT FROM AUTHOR]

We define involutes and contrapedal curves of spherical curves and investigate some properties. We also give the classifications of the singular points of involutes. Moreover, we give the relationships between involute–evolute curve pairs and pedal–contrapedal curve pairs and give the formula of the rth involute. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a developable surface tangent to a timelike surface along a curve in Minkowski 3-space, which is called the osculating developable surface of the timelike surface along the curve. The ruling of the osculating developable surface is parallel to the osculating Darboux vector field. The main goal of this paper is to classify the singularities of the osculating developable surface. To this end, two new invariants of curves are defined to characterize these singularities. Meanwhile, we also research the singular properties of osculating developable surfaces near their lightlike points. Moreover, we give a relation between osculating Darboux vector fields and normal vector fields of timelike surfaces along curves from the viewpoint of Legendrian dualities. Finally, some examples with symmetrical structures are presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

We provide several applications of the minimal model program to the local and global study of co-rank 1 foliations on threefolds. Locally, we prove a singular variant of Malgrange's theorem, a classification of terminal foliation singularities and the existence of separatrices for log canonical singularities. Globally, we prove termination of flips, a connectedness theorem on log canonical centres, a non-vanishing theorem and some hyperbolicity properties of foliations. [ABSTRACT FROM AUTHOR]

This paper proposes an approach to derive the Jacobian matrix of a hybrid mechanism by applying a velocity operator to the transformation matrix. This Jacobian matrix is capable of deducing hybrid singularities, which cannot be identified by using the screw-based Jacobian or velocity-based Jacobian. The transformation matrix was obtained based on the algebraic geometry approach, and it becomes the key point since it was used to not only formulate the Jacobian matrix, but also to define the motion type of hybrid mechanisms. In this paper, two hybrid mechanisms were investigated, which were composed of two distinct parallel mechanisms mounted in series. Hybrid Mechanisms 1 and 2 were composed of 3-PRP-3-RPS and 3-RPS-3-PRP (the underlined P is an actuated joint), respectively. The motion types of Hybrid Mechanisms 1 and 2 were determined from the product of the transformation matrices of the 3-PRP and 3-RPS parallel mechanisms, and vice versa. The developed method was employed to establish the Jacobian matrix to which the conditioning index was applied. Therefore, the kinematic performances of the two hybrid mechanisms can be compared for a given bone surgery trajectory within the workspace. It turns out that Hybrid Mechanism 1 has superior performance than that of Mechanism 2, which indicates that Mechanism 1 is better at transmitting power to the moving platform. [ABSTRACT FROM AUTHOR]

Astrophysical objects and the surrounding geometry must be described by suitable states in the complete quantum theory of matter and gravity. We attempt at reconstructing such a state for a Schwarzschild black hole from the analysis of a collapsing ball of dust and describing the outer geometry by means of a coherent state. [ABSTRACT FROM AUTHOR]

In this work, we aim to discuss, qualitatively, possible associations between biological diversity and cultural diversity in the formation of biology teachers. These reflections can broaden the understanding of the relationship between society and nature, as well as the construction of curricula for the degree in biology. We approach some ecological perspectives of biodiversity and cultural diversity; in this one, we demarcate the perceptions of individual and collective singularities as a possible descriptor of a contemporary sociocultural profile. Seeking to approximate diversities, we present the biocultural and biocentric approaches, and the possibility of inserting integrating, disintegrating, biocentric and anthropocentric elements to analyze the perceptions of undergraduates. We conclude that the recognition of the relationships between particularities/collectivities and the themes related to the variability of living beings helps in the process of social emancipation of the teacher in training. Furthermore, we consider that a biocentric approach, based on equity, enables the recognition of the interspecific alterities of living beings, providing (re)connection, (re)integration and diplomatic coexistence. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of critical p(x)-Kirchhoff problems with a singular term. A nontrivial positive solution is obtained by combining variational methods with an appropriate truncation argument. [ABSTRACT FROM AUTHOR]

Effective sufficient conditions are established for the well-posedness of the Cauchy problem with weight for linear systems of generalized ordinary differential equations with singularities. The results are realized for impulsive differential systems. [ABSTRACT FROM AUTHOR]

One of the objectives of this paper is to establish the exact controllability for wave-type evolution equations on non-convex and/or cracked domains with non-concurrent support crack lines. Admittedly, we know that according to the work of Grisvard P., in domains with corners or cracks, the formulas of integrations by parts are subject to geometric conditions: the lines of cracks or their supports must be concurrent. In this paper, we have established the exact controllability for the wave equation in a domain with cracks without these additional geometric conditions. [ABSTRACT FROM AUTHOR]

We consider dispersing billiard tables whose boundary is piecewise smooth and the free flight function is unbounded. We also assume there are no cusps. Such billiard tables are called type D in the monograph of Chernov and Markarian [ 9 ]. For a class of non-degenerate type D dispersing billiards, we prove exponential decay of correlation and several other statistical properties. [ABSTRACT FROM AUTHOR]

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