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Producción Científica, Let ξ be an analytic vector field in R3 with an isolated singularity at the origin and having only hyperbolic singular points after a reduction of singularities π : M → R3. 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 ∈ R3. 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 {Vn} of neighborhoods well adapted for the description of the local dynamics of ξ : the frontier F r(Vn) is everywhere tangent to ξ except around F r(Vn) ∩ Λ, where transversality is mandatory., Ministerio de Ciencia e Innovación (MTM2016-77642-C2-1-P and PID2019- 105621GB-I00), Junta de Castilla y León (VA083G19)

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.

We use birational geometry to show that the existence of rational points on proper rationally connected varieties over fields of characteristic 0 is a consequence of the existence of rational points on terminal Fano varieties. We discuss several consequences of this result, especially in relation to the C1-conjecture. We also provide evidence that supports the conjecture in dimension 3 for C1 fields of characteristic 0.

Using Monge-Amp\`ere geometry, we study the singular structure of a class of nonlinear Monge-Amp\`ere equations in three dimensions, arising in geophysical fluid dynamics. We extend seminal earlier work on Monge-Amp\`ere geometry by examining the role of an induced metric on Lagrangian submanifolds of the cotangent bundle. In particular, we show that the signature of the metric serves as a classification of the Monge-Amp\`ere equation, while singularities and elliptic-hyperbolic transitions are revealed by the degeneracies of the metric. The theory is illustrated by application to an example solution of the semigeostrophic equations., Comment: 22 pages, 5 figures

We study the existence and propagation of singularities of the solution to a one-dimensional linear stochastic wave equation driven by an additive Gaussian noise that is white in time and colored in space. Our approach is based on a simultaneous law of the iterated logarithm and general methods for Gaussian processes. (C) 2021 TheAuthor(s). Published by ElsevierB.V.

Resumo Cada vez mais, as pesquisas sobre a gestão de organizações culturais se destacam, à medida que reconhecemos as contribuições simbólicas e materiais da economia criativa para o desenvolvimento das sociedades contemporâneas. Contudo, carecemos de uma visão integrada e robusta dessa gestão, baseada numa caracterização clara do campo e de suas singularidades. O objetivo deste artigo é descrever a produção acadêmica sobre gestão de organizações culturais e demarcar um horizonte conceitual-teórico fecundo para estimular a pesquisa futura. A metodologia é baseada numa revisão sistemática de produções acadêmicas publicadas nas bases de dados nacionais e internacionais. Os resultados da pesquisa incluem a elaboração de três categorias integradoras da produção acadêmica: perspectivas sobre questões centrais na pesquisa sobre organizações culturais (técnico-operacional e política), as singularidades dessas organizações (hipersensibilidade, hipertensão e hiperincerteza) e o paradoxo como um eixo teórico-conceitual crucial para melhorar o conhecimento sobre as organizações culturais. Os resultados propõem uma agenda para pesquisas futuras com base no conceito de paradoxo. Resumen Cada vez más, la investigación sobre la gestión de las organizaciones culturales se destaca al reconocer las contribuciones simbólicas y materiales de la economía creativa al desarrollo de las sociedades contemporáneas. Sin embargo, necesitamos una visión integrada y robusta de esta gestión basada en una clara caracterización del campo y sus singularidades. El propósito de este artículo es caracterizar la producción académica sobre la gestión de las organizaciones culturales y trazar un horizonte conceptual-teórico fructífero para estimular futuras investigaciones. La metodología se basa en una revisión sistemática de publicaciones académicas en bases de datos nacionales e internacionales. Los resultados de la investigación incluyen el desarrollo de categorías que integran la producción académica, que se encuentran en forma de perspectivas de análisis (técnico-operativas y políticas) y singularidades (hipersensibilidad, hipertensión e hiperincertidumbre) de las organizaciones culturales. Los resultados proporcionan una agenda para futuras investigaciones basadas en el concepto de paradoja. Abstract Research on the management of cultural organizations is increasingly gaining interest as we recognize the symbolic and material contributions of the creative economy to the development of contemporary societies. However, we lack an integrated and robust vision of such management, a vision based on a clear characterization of the field and its singularities. This article aims to characterize the academic production on the management of cultural organizations and outline a fruitful conceptual-theoretical horizon to stimulate future research. The methodology is based on a systematic review of academic production published in national and international databases. The analysis of the literature revealed the emergence of three categories: the perspectives about the central issues of the research on cultural organizations (from the technical-operational and political viewpoints), these organizations’ singularities (hypersensitivity, hypertension, and hyper-uncertainty), and the paradox as a crucial theoretical-conceptual axis to increase knowledge of cultural organizations. The results provide an agenda for future research based on the concept of paradox.

This paper aims to analyse the U.S. writer and teacher Tom Spanbauer’s novel In the City of Shy Hunters (2001) by means of theoretical perspectives that embrace both queer and communitarian epistemologies. The novel, set mainly in New York City in the mid1980s, is narrated in first person by the protagonist and shows how HIV/AIDS, and the elevated social stigma surrounding it, affects different classes and ethnicities, as well as gender, sexual and corporal diversities. It experiments with a particular writing style and teaching method that Spanbauer calls dangerous writing; that is, how to expose our inner life and secrets, which are often related to social taboos. The main objective of this paper, which underlines its originality, is to demonstrate the connection between dangerous writing, queer studies and community theory by adopting a multidisciplinary approach to literary critical analysis. The novel is about an inwards as well as outwards journey taken by a young protagonist, who is both a queer singularity and a shy hunter, looking for the sore truth that lies within his own and other human hearts. Firstly, I will explore Spanbauer’s dangerous writing style, then I will investigate the HIV/AIDS crisis through a queer perspective (Judith Butler, Teresa de Lauretis, Michel Foucault and Annamarie Jagose). Furthermore, different community configurations (Jean-Luc Nancy’s in/operative community, Maurice Blanchot’s community of lovers and Roberto Esposito’s concept of immunitas) will be discussed. The main result demonstrates that the singularities represented in the novel, who are mostly queer and affected by HIV/AIDS, can create inoperative communities and communities of lovers, open to otherness and secret sharing, as well as being spontaneous, antisocial and momentary, with a recognition and acceptance of mortality.

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 have neglected the backreaction effect of the pre-Hawking radiation (in the sense that it is small effect and cannot prevent the formation of 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 radius proportional to $(GMl_0^{2}/c^2)^{1/3}$. Quite interestingly, we found the same result for radius of the Planck star using a free falling observer point of view. In addition, we pointed 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 a Planck-size black hole remnants. The second possibility is that the inner core can be unstable and, due to the quantum tunnelling effect, the spacetime can undergo a black hole-to-white hole transition (a bouncing Planck star)., Comment: 9 pages, no figures. Published in Universe https://www.mdpi.com/2218-1997/9/1/41

We show some examples of calculations of massless and massive Feynman integrals.

This is our second article devoted to the cardiac myoarchitecture considered as a nematic chiral liquid crystal (NCLC). While the first article focused on the myoarchitecture of the left ventricle (LV), this new article extends to the whole ventricular mass and introduces the concept of disclinations and topological singularities, which characterize the differences and relationships between the left and right ventricles (RV). At the level of the ventricular apices, we constantly observed a vortex shape at the LV apex, corresponding, in the terminology of liquid crystals, to a “+1 disclination”; we never observed this at the RV apex. At the level of the interventricular septum (IVS), we identified “−1/2 disclinations” at the anterior and posterior parts. During the perinatal period, there was a significant difference in their distribution, with more “−1/2 disclinations” in the posterior part of the IVS. After birth, concomitant to major physiological changes, the number of “−1/2 disclinations” significantly decreased, both in the anterior and posterior parts of the IVS. Finally, the description of the disclinations must be considered in any attempt to segment the whole ventricular mass, in biomechanical studies, and, more generally, for the characterization of myocardial remodeling.

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

We extend a result of Guan by showing that the second Betti number of a 4-dimensional primitively symplectic orbifold is at most 23 and there are at most 91 singular points. The maximal possibility 23 can only occur in the smooth case. In addition to the known smooth examples with second Betti numbers 7 and 23, we provide examples of such orbifolds with second Betti numbers 3, 5, 6, 8, 9, 10, 11, 14 and 16. In an appendix, we extend Salamon's relation among Betti/Hodge numbers of symplectic manifolds to symplectic orbifolds., Final version. Published online in Mathematische Zeitschrift

The existence of positive periodic solutions of the Rayleigh equations $ x''+f(x')+g(x) = e(t) $ with singularities is investigated in this paper. Based on the continuation theorem of coincidence degree theory and the method of upper and lower solutions, the multiple periodic solutions of the singular Rayleigh equations can be determined under the weak conditions of the term $ g $. We discuss both the repulsive singular case and the attractive singular case. Some results in the literature are generalized and improved. Moreover, some examples and numerical simulations are given to illustrate our theoretical analysis.

In this article, we mainly discuss the local differential geometrical properties of the lightlike Killing magnetic curve $ \mathit{\boldsymbol{\gamma }}(s) $ in $ \mathbb{S}^{3}_{1} $ with a magnetic field $ \boldsymbol{ V} $. Here, a new Frenet frame $ \{\mathit{\boldsymbol{\gamma }}, \boldsymbol{ T}, \boldsymbol{ N}, \boldsymbol{ B}\} $ is established, and we obtain the local structure of $ \mathit{\boldsymbol{\gamma }}(s) $. Moreover, the singular properties of the binormal lightlike surface of the $ \mathit{\boldsymbol{\gamma }}(s) $ are given. Finally, an example is used to understand the main results of the paper.

In this paper we continue the study of non-diagonalisable hyperbolic systems with variable multiplicity started by the authors in [1] . In the case of space dependent coefficients, we prove a representation formula for solutions that allows us to derive results of regularity and propagation of singularities.

Homogenization-based topology optimization has been shown to be effective but does not directly create mechanical structures. Instead, the method gives a multi-scale description of the optimized design, e.g., lamination thicknesses and directions. To obtain a realizable single-scale design, one can perform a subsequent de-homogenization step. This is done by converting the lamination directions to integrable vector fields from which it is possible to compute a parameterization of the domain. Unfortunately, however, singularities often make it impossible to find integrable vector fields that align with lamination directions. We present a short introduction to homogenization-based topology optimization followed by an overview of different types of singularities and how they impinge on the problem. Based on this, we propose a singularity aware de-homogenization pipeline, where we use a method for vector field combing which produces consistent labeling of the lamination directions but also introduces necessary seams in the domain. We demonstrate how methods from computer graphics can subsequently be used to compute the final parameterization from which the mechanical structure can easily be extracted. We demonstrate the method on several test cases.

The precise tracking 3D trajectory that changes dramatically within the boundary of flight performance is the essential capability of the UAV applied to air combat, which requires that flight control law design be adapted to nonlinear characteristics of flight dynamics and solve the singularities problem during the flight. Continuous trajectory control on the basis of nonlinear dynamic inversion (NDI) is studied. The singularities problem is caused by continuous trajectory expressed by flight velocity and flight-path angles. A sudden change of flight-path axis system happens when aircraft flies along a continuous trajectory which results in singularities. A method for amending flight-path axis system is presented to solve the sudden change problem of flight-path axis system. Finally, the combination of the maneuver generator based on quaternions and amending flight-path axis system realize the precise trajectory tracking control without singularities and obtain excellent tracking characteristics.

Let $k$ be a field of positive characteristic. Building on the work of the second named author, we define a new class of $k$-algebras, called diagonally $F$-regular algebras, for which the so-called Uniform Symbolic Topology Property (USTP) holds effectively. We show that this class contains all essentially smooth $k$-algebras. We also show that this class contains certain singular algebras, such as the affine cone over $\mathbb{P}^r_{k} \times \mathbb{P}^s_{k}$, when $k$ is perfect. By reduction to positive characteristic, it follows that USTP holds effectively for the affine cone over $\mathbb{P}^r_{\mathbb{C}} \times \mathbb{P}^s_{\mathbb{C}}$ and more generally for complex varieties of diagonal $F$-regular type., Comment: 22 pages. Comments welcome

The contemporary university privileges speed, precarity, competition, and performativity; it operates through modes of accelerationism, work intensification and productivity; and it is oriented to producing academic subjectivities rooted in self-commodification. Much of this is antithetical to feminist ethics and working practices which focus on care, relationality and working together. The article explores these tensions as a basis for moving forward with the question: What does a new material feminist approach offer as an ethical practice to work against these damaging conditions? In response, it draws on the work of Barad (2007) and Haraway (2016) to propose an embodied material feminist ethics of response-ability as an alternative approach to educational research, teaching and mentoring. Further, relating Stengers (2018) insights on the generativity of slow to Deleuze's (2004) concept of ‘singularities’ the article proposes slow singularities for collective mattering as a conceptual and practical means – as a material-discursive feminist praxis – to contest the un-liveable life of the neoliberal accelerated academy. In doing so, it makes the case for feminist work as an un/dutiful response-ability of nurturing decelerated forms of being which might help reimagine the aims and purpose of the university.

This article presents the analysis of a parallel mechanism of type 3-SPS-U. The usual singularity approach is carried out with respect to the Euler angles of the universal joint. However, this approach is computationally expensive especially when stacked structures are analyzed. Thus, the positioning of the mobile platform for the mechanism is analyzed using the theory of Tilt and Torsion (T&T). The singularity-free workspace and the tilt limits of the mechanism are disclosed through this method. These workspaces can then be mapped to the Euler angles of the universal joint and the relation between the T&T space and the Euler space is demonstrated and validated in this study. Initially, simulations are performed using the results of singularity analysis to have a comparison between the T&T space and the Euler space. Experimental validation is then carried out on the prototype of the mechanism to perform a circular trajectory, in line with the simulations. The outcome of this study will be helpful for the integration of the mechanism for a piping inspection robot and also for the analysis of stacked architectures.

Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of first and second kind. In physics however, commonly the only acceptable members of this set are the Legendre polynomials. Quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone, in-depth argument for rejecting all solutions of Legendre’s equation, but the polynomial ones, in physics. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation is quintessential to the argument. We demonstrate that the evenness or oddness of the Legendre polynomials is a consequence of the same ingredients.

We present the results of a study of the Poynting vector field generic singularities at the resonant light scattering of a plane monochromatic linearly polarized electromagnetic wave by a subwavelength particle. We reveal the impact of the problem symmetry, the spatial dimension, and the energy conservation law on the properties of the singularities. We show that, in the cases when the problem symmetry results in the existence of an invariant plane for the Poynting vector field lines, a formation of a standing wave in the immediate vicinity of a singularity gives rise to a saddle-type singular point. All other types of singularities are associated with vanishing at the singular points, either (i) magnetic field, for the polarization plane parallel to the invariant plane, or (ii) electric field, at the perpendicular orientation of the polarization plane. We also show that in the case of two-dimensional problems (scattering by a cylinder), the energy conservation law restricts the types of possible singularities only to saddles and centers in the non-dissipative media and to saddles, foci, and nodes in dissipative. Finally, we show that dissipation affects the (i)-type singularities much stronger than the (ii)-type. The same conclusions are valid for the imaginary part of the Poynting vector in problems where the latter is regarded as a complex quantity. The singular points associated with the formation of standing waves are different for real and imaginary parts of this complex vector field, while all other singularities are common. We illustrate the general discussion by analyzing singularities at light scattering by a subwavelength Germanium cylinder with the actual dispersion of its refractive index., Comment: 20 pages, 3 figures, 2 tables

Resumo A organização da música pop recorre ao termo world music para classificar produções não aderentes ao núcleo Europa/Estados Unidos. Este trabalho propõe três eixos de discussão sobre o uso do world music: a) revisar textos sobre o tema; b) identificar desigualdades narrativas históricas, de âmbito global, que o permeiam; c) análise empírica das obras de três artistas com base na Semiótica da Cultura e contribuições de linhas pós-estruturalista e decolonial para repensar o world music e sua função de forma estética. Abstract The organization of pop music uses the term world music to classify productions which do not adhere to the cores Europe/United States of America. This paper proposes three axes of discussion about the use of world music: a) to review texts on the theme; b) to identify historical narrative inequalities, from global scope, that permeate it; c) to propose an empirical analysis of the works of three artists based on Semiotics of Culture and with contributions of poststructuralist and decolonial thought to rethink world music and its function of aesthetic form.

Resumen: Dada una superficie S en R³ , una superficie paralela a S está formada por los puntos que se encuentran a la misma distancia euclidiana d de S. En este trabajo se estudiarán algunas propiedades geométricas de las superficies paralelas que dependen de las curvaturas de la superficie inicial S y de la distancia d. También se analizarán las superficies paralelas de las cuádricas y de otras familias de superficies. Además, se determinarán las singularidades y las autointersecciones de los distintos ejemplos de superficies considerados. En el ´último capítulo se estudiarán las superficies racionales que admiten superficies paralelas racionales. Abstract: Given a surface S in R³ , a surface parallel to S is formed by the points that are at the same Euclidean distance d from S. In this memory we will study some geometric properties of offset surfaces that depend on the curvatures of the initial surface S and on the distance d. Offsets to quadric surfaces and other families of surfaces will also be discussed. In addition, we will be considering different examples of surfaces and we will determine their singularities and self-intersections. In the last chapter of this memory we will study the rational surfaces that admit rational offsets. Grado en Matemáticas

[Resumen] Este artículo presenta un laboratorio remoto para realizar prácticas de control en el espacio de estados con dos robots paralelos reales, a través de Internet. Se presenta la arquitectura cliente-servidor de este laboratorio, en la que el cliente es una interfaz Java de control usada por el estudiante para comandar experimentos y visualizar resultados, mientras que el servidor es un PC industrial que controla ambos robots paralelos mediante una tarjeta de control dSPACE 1103 y una serie de componentes software que incluyen Matlab, Jimserver, y un gestor de reservas y de credenciales para garantizar un acceso seguro a los robots remotos. Se describen varias prácticas que pueden realizarse con este laboratorio remoto, como la identificación del modelo dinámico multivariable de los robots paralelos, el diseño de controladores integrales con estado realimentado, o el estudio de la pérdida de controlabilidad de dichos robots a medida que se aproximan a singularidades. [Abstract] This paper presents a remote laboratory for doing practices of state-space control with two real parallel robots over the Internet. The client-server architecture of this laboratory is presented, where the client is a Java control interface used by the student to command experiments and visualize results, whereas the server is an industrial PC that controls both parallel robots through a dSPACE 1103 control board and a series of software components including Matlab, Jimserver and a book manager that guarantees safe access to the remote robots. Different lab practices that can be done with this remote laboratory are described, such as the identification of the multivariate dynamic model of the parallel robots, the design of integral controllers with state feedback, or the study of loss of controllability of the robots when they approach singularities.

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.

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.

International audience; Given a general 3-RPR planar parallel robot with linear platforms, it is proven that there are no singularity-free paths between non-symmetrical direct kinematics solutions. We provide an alternative proof of this. We also provide a proof that shows that there is a singularity-free path between two symmetrical solutions if an appropriate condition is satisfied. This condition will be automatically satisfied for exactly one symmetrical pair of solutions if there are four real solutions to the direct kinematics. The topology of the kinematic singularities of these robots is described. We prove that the complement of the singularity space in the domain of the kinematic map for such robots consists of three connected components. An analysis of special robots is also provided, i.e. when the anchor points of the moving platform and the fixed platform have the same cross-ratios. For this case we show that, the singularity-free space consists of four connected components, no direct kinematics solutions can be connected without crossing the singularity surface and that the absolute value of the determinant of the Jacobian of the kinematic map evaluates to the same number for any of the direct kinematics solutions. The topological analysis of the singularities of 3-RPR planar parallel robots with linear platforms is based on the fibers of the natural map SE(2)->S^1 restricted to the singularity set of the kinematic map, and this is a conic fibration. In fact this is also the case when one or both platforms are triangular. We use similar methodology as in the linear platforms case and show that in the case that one or both platforms are triangles the singularity-free space has two connected components. This result was already proven when both platforms are triangles but here a different approach is followed.

Orientadores: Sueli Irene Rodrigues Costa, Maria del Carmen Romero-Fuster Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica Resumo: O projetivo de curvatura em um ponto de uma 3-variedade M de classe 'C POT. 2' imersa em 'IR POT. ?' , n >-4, é o lugar geométrico de todos os extremos dos vetores curvatura de secções normais ao longo de todas as direções tangentes a M em p. Mostramos que o projetivo de curvatura em p é isomorfo (difeomorfo) à superfície de Veronese clássica de ordem 2, composta com uma transformação linear. Conforme o posto desta transformação linear, o projetivo de curvatura será dado por projeções da superfície de Veronese em subespaços do espaço normal da variedade M. Quanto menor o posto, maior será a umbilicidade da variedade no ponto em questão. Também estudamos a natureza geométrica e singularidades para os diferentes casos de projetivos de curvatura em pontos de M, os quais incluem a superfície Romana de Steiner, a Cross-Cap, a superfície de Steiner de Tipo 5 e a Cross-Cup. Além disso, analisamos os pontos singulares de segunda ordem da imersão, no sentido de Feldman e estabelecemos condições relacionadas à natureza do projetivo de curvatura, para que uma 3-variedade imersa em 'IR POT. ?', n >_ 9, tenha contato de ordem _ 2 com k-planos e k-esferas de IRn, 3 _ k _ 8 Abstract: The curvature projective plane at each point p of three-manifolds M immersed in 'IR POT. ?', n _ 4, is the geometric locus of all end points of the curvature vectors of normal sections along of all tangent directions of M at p. In this study, we show that the curvature projective plane is isomorphic (diffeomorphic) to the classical Veronese surface of order two, composed with a linear transformation, and that according to the rank of this mapping, the curvature projective plane will be given by projections of the Veronese surface into subspaces of the normal space of M at p. Thus, the smaller the rank the greater the umbilicity of the manifold at this point. We also study the geometric nature and singularities of the curvature projective planes considering different possibilities, which include the Roman Steiner surface, the Cross-Cap, the Steiner surface of five-type, and the Cross-Cup. In addition, we analyze the order-two singularities of the immersion in the Feldman¿s sense and establish conditions related to the nature of the curvature projective plane for the existence of contacts of the three-manifolds in 'IR POT. ?', n _ 9, with k-planes and k-spheres, 3 _ k _ 8 Doutorado Geometria Doutor em Matemática

"This is the peer reviewed version of the following article: Abad C, Bravo A, Villamayor U. OV. Finite morphisms and simultaneous reduction of the multiplicity. Mathematische Nachrichten. 2020;293:8–38, which has been published in final form at https://doi.org/10.1002/mana.201800470. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited", Let X be a singular algebraic variety defined over a perfect field k, with quotient field K ( X ) . Let s ≥ 2 be the highest multiplicity of X and let F s ( X ) be the set of points of multiplicity s. If Y ⊂ F s ( X ) is a regular center and X ← X 1 is the blow up at Y, then the highest multiplicity of X1 is less than or equal to s. A sequence of blow ups at regular centers Y i ⊂ F s ( X i ) , say X ← X 1 ← ⋯ ← X n , is said to be a simplification of the multiplicity if the maximum multiplicity of X n is strictly lower than that of X, that is, if F s ( X n ) is empty. In characteristic zero there is an algorithm which assigns to each X a unique simplification of the multiplicity. However, the problem remains open when the characteristic is positive. In this paper we will study finite dominant morphisms between singular varieties β : X ′ → X of generic rank r ≥ 1 (i.e., [ K ( X ′ ) : K ( X ) ] = r ). We will see that, when imposing suitable conditions on β, there is a strong link between the strata of maximum multiplicity of X and X ′ , say F s ( X ) and F r s ( X ′ ) respectively. In such case, we will say that the morphism is strongly transversal. When β : X ′ → X is strongly transversal one can obtain information about the simplification of the multiplicity of X from that of X ′ and vice versa. Finally, we will see that given a singular variety X and a finite field extension L of K ( X ) of rank r ≥ 1 , one can construct (at least locally, in étale topology) a strongly transversal morphism β : X ′ → X , where X ′ has quotient field L., The authors were partially supported by the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV2015-0554), and through MTM2015-68524-P (MINECO/FEDER)

International audience; In this article, we discuss formal invariants of singularly-perturbed linear differential systems inneighborhood of turning points and give algorithms which allow their computation. The algo-rithms proposed are implemented in the computer algebra system Maple.

We provide a primer to numerical methods based on Taylor series expansions such as generalized finite difference methods and collocation methods. We provide a detailed benchmarking strategy for these methods as well as all data files including input files, boundary conditions, point distribution and solution fields, so as to facilitate future benchmarking of new methods. We review traditional methods and recent ones which appeared in the last decade. We aim to help newcomers to the field understand the main characteristics of these methods and to provide sufficient information to both simplify implementation and benchmarking of new methods. Some of the examples are chosen within a subset of problems where collocation is traditionally known to perform sub-par, namely when the solution sought is non-smooth, i.e. contains discontinuities, singularities or sharp gradients. For such problems and other simpler ones with smooth solutions, we study in depth the influence of the weight function, correction function, and the number of nodes in a given support. We also propose new stabilization approaches to improve the accuracy of the numerical methods. In particular, we experiment with the use of a Voronoi diagram for weight computation, collocation method stabilization approaches, and support node selection for problems with singular solutions. With an appropriate selection of the above-mentioned parameters, the resulting collocation methods are compared to the moving least-squares method (and variations thereof), the radial basis function finite difference method and the finite element method. Extensive tests involving two and three dimensional problems indicate that the methods perform well in terms of efficiency (accuracy versus computational time), even for non-smooth solutions., Comment: Arch Computat Methods Eng (2019)

We give an explicit formula for the motivic zeta function in terms of a log smooth model. It generalizes the classical formulas for snc-models, but it gives rise to much fewer candidate poles, in general. As an illustration, we explain how the formula for Newton non-degenerate polynomials can be viewed as a special case of our results., v3: Section 3 with background material on regular log schemes expanded, minor changes to the remainder of the paper

Flying doughnuts (FDs) are exact propagating solutions of Maxwell equations in the form of single-cycle, space-time non-separable toroidal pulses. Here we review their properties and reveal the existence of a complex and robust fine topological structure. In particular, the electric and magnetic fields of the FD pulse vanish across a number of planes, spherical shells and rings, and display a number of point singularities including saddle points and vortices. Moreover, the instantaneous Poynting vector of the field exhibits a large number of singularities, which are often accompanied by extended areas energy backflow.

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.

We consider plane divisorial valuations of Hirzebruch surfaces and introduce the concepts of non-positivity and negativity at infinity. We prove that the surfaces given by valuations of the last types have nice global and local geometric properties. Moreover, non-positive at infinity divisorial valuations are those divisorial valuations of Hirzebruch surfaces providing rational surfaces with minimal generated cone of curves. Non-positivity and negativity at infinity are also extended to the class of real valuations of the projective plane and the Hirzebruch surfaces. Finally, we compute the Seshadri-type constants for pairs formed by a big divisor and a divisorial valuation of a Hirzebruch surface and obtain the vertices of the Newton-Okounkov bodies of pairs as above under the non-positivity at infinity property. Introducimos los conceptos de no positividad y negatividad en el infinito para valoraciones planas divisoriales de una superficie de Hirzebruch. Probamos que las superficies dadas por valoraciones con las características anteriores poseen interesantes propiedades globales y locales. Además, las valoraciones divisoriales no positivas en el infinito son aquellas valoraciones divisoriales de superficies de Hirzebruch que dan lugar a superficies racionales tales que su cono de curvas está generado por un número mínimo de generadores. Los conceptos de no positividad y negatividad en el infinito también se extienden a valoraciones reales del plano proyectivo y de superficies de Hirzebruch. Por último, calculamos explícitamente las constantes de tipo Seshadri para pares formados por divisores big y valoraciones divisoriales de superficies de Hirzebruch y obtenemos los vértices de los cuerpos de Newton-Okounkov para pares como los anteriores bajo la condición de no positividad en el infinito. Programa de Doctorat en Ciències

We study spherically symmetric spacetimes in Einstein-aether theory in three different coordinate systems, the isotropic, Painlev\`e-Gullstrand, and Schwarzschild coordinates, in which the aether is always comoving, and present both time-dependent and time-independent exact vacuum solutions. In particular, in the isotropic coordinates we find a class of exact static solutions characterized by a single parameter $c_{14}$ in closed forms, which satisfies all the current observational constraints of the theory, and reduces to the Schwarzschild vacuum black hole solution in the decoupling limit ($c_{14} = 0$). However, as long as $c_{14} \not= 0$, a marginally trapped throat with a finite non-zero radius always exists, and in one side of it the spacetime is asymptotically flat, while in the other side the spacetime becomes singular within a finite proper distance from the throat, although the geometric area is infinitely large at the singularity. Moreover, the singularity is a strong and spacetime curvature singularity, at which both of the Ricci and Kretschmann scalars become infinitely large., Comment: revtex4-1, one figure and no tables. Some typos are corrected. Universe 7 (2021) 272

A unified approach is developed for the analysis of singularities of the scattered field and vanishing of scattering harmonics. Rigorous proofs are presented of the existence of complex resonance singularities of the solution to the problem of the plane wave scattering by a circular homogeneous dielectric cylinder. The method employs a mathematically correct approach of the spectral theory of open structures involving generalized conditions at infinity when complex resonance frequencies may be considered. The result is obtained by verifying the existence of complex singularities of the series solution coefficients considered as functions of the problem parameters. The recently developed theory of generalized cylindrical polynomials (GCPs) is used that enables one to reduce determination of singularities (resonances) to finding zeros (real or complex) of a particular subfamily of GCPs.

Singularities of the Poynting vector field at resonant light scattering by nanoparticles are discussed and classified. It is shown that there are two generic types of them, namely (i) the singularities related to the vanishing of the magnetic (and/or electric) field at the singular points and (ii) the singularities related to the formation of standing waves in the proximity of the singular points. The connection of these types of singularities to the topology of the singular points and the space dimension (3D vs. 2D) is revealed., Comment: 9 pages, 4 figures

There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly nonlinear scalar field, which allow the appearance of singularities of a new type (“spherical singularities”) outside the center of curvature coordinates. As the example, we consider a scalar field potential ∼sinh(ϕ2n),n&gt, 2, which grows rapidly for large field values. The space-time is assumed to be asymptotically flat. We fulfill a numerical investigation of solutions with different n for different parameters, which define asymptotic properties at spatial infinity. Depending on the configuration parameters, we show that the distribution of the stable circular orbits of test bodies around the configuration is either similar to that in the case of the Schwarzschild solution (thus mimicking an ordinary black hole), or it contains additional rings of unstable orbits.

Let $X$ be a strictly log canonical Fano variety, we show that every lc place of complements is dreamy, and there exists a correspondence between weakly special test configurations of $(X,-K_X)$ and lc places of complements., Comment: 18 pages. Final version, to appear in Algebraic & Number Theory

A general equation of state is considered for analysing the possible behaviors for (Early) dark energy that alleviates the Hubble parameter tension problem. By departing from the possible evolution for the (Early) dark energy density and the corresponding dynamical equations, the equation of state is obtained, which allow us to analyze qualitatively the cosmological evolution and the dominance of each term in the equation of state along the cosmic expansion, which show some interesting consequences as the occurrence of (past) future singularities. Then, by considering two general models, their free parameters are fit with different sources of data, showing the goodness of the fits in comparison to more standard models. Results might be considered as a promising starting point to get a better understanding of the cosmological evolution as a whole., 19 pages, 2 figures, analysis extended, version accepted for publication in Phys. of the Dark Universe

Agraïments: The third author is supported by NSERC. The fourth author is also supported by the grant 12.839.08.05F from SCSTD of ASM and partially by NSERC. In this work we classify, with respect to the geometric equivalence relation, the global configurations of singularities, finite and infinite, of quadratic differential systems possessing exactly three distinct finite simple singularities. This relation is finer than the topological equivalence relation which does not distinguish between a focus and a node or between a strong and a weak focus or between foci (or saddles) of different orders. Such distinctions are, however, important in the production of limit cycles close to the foci (or loops) in perturbations of the systems. The notion of geometric equivalence relation of configurations of singularities allows us to incorporate all these important geometric features which can be expressed in purely algebraic terms. The geomet ric classification of all configurations of singularities, finite and infinite, of quadratic systems was initiated in a work published in 2013 when the classification was done for systems with total multiplicity m f of finite singularities less than or equal to one. That work was continued in an article which is due to appear in 2014 where the geometric classification of configurations of singularities was done for the case m f = 2. In this article we go one step further and obtain the geometric classification of singularities, finite and infinite, for the subclass mentioned above. We obtain 147 geometrically distinct configurations of singularities for this family. We give here the global bifurcation diagram of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for this class of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants, a fact which gives us an algorithm for determining the geometric configuration of singularities for any quadratic system in this particular class.

Esta tesis cubre dos artículos conjuntos con Nuño-Ballesteros (The Image Milnor Number and Excellent Unfoldings, en 2021, y On whitney equisingular unfoldings of corank 1 germs, como prepublicación), un artículo que sigue en desarrollo conjunto con Nuño-Ballesteros y Lê Dũng Tráng (provisionalmente titulado Relative polar curves and monodromy, como prepublicación) y un trabajo en desarrollo con Mond. Estos tres trabajos delimitan las tres partes principales del texto. Como se ha mencionado, el texto está dividido en tres partes. La primera de ellas trata el estudio de singularidades de gérmenes de aplicaciones holomorfas en el contexto de la teoría de Thom-Mather, i.e., módulo cambio de coordenadas biholomorfo (llamado -equivalencia). En particular, nos centramos en gérmenes de corrango uno de en , pero también desarrollamos la teoría para gérmenes de en , con , y gérmenes con una intersección completa con singularidades aisladas (comúnmente conocidos como ICIS, de isolated complete intersection singularity en inglés) en el dominio. El principal objetivo de la primera parte del texto es encontrar una buena caracterización de la equisingularidad de Whitney para familias a un parámetro de gérmenes -finitos de corrango uno. Una caracterización de la equisingularidad de Whitney ya fue dada por Gaffney en 1993 en su artículo Polar multiplicities and equisingularity of map germs: una familia de gérmenes es Whitney equisingular si, y solo si, es excelente y todas las multiplicidades polares en el dominio y codominio son constantes a lo largo de la familia. A pesar de ello, esta caracterización tiene el inconveniente de necesitar una gran cantidad de invariantes para asegurar la equisingularidad, del orden de la dimensión del espacio de salida al cuadrado. Se han hecho algunos avances desde el resultado de Gaffney, por ejemplo, Jorge Pérez y Saia redujeron el número de invariantes en 2006 en su artículo Euler obstruction, polar multiplicities and equisingularity of map germs, necesitando todavía una gran cantidad de ellos (manteniendo el orden del número invariantes). Además, Houston tiene un artículo no publicado basado en una prepublicación no publicada de Gaffney (no disponible al público) en el que trata esta cuestión (véase su prepublicación de 2008, Equisingularity and the euler characteristic of a milnor fibre). Nuestra contribución ha sido, en primer lugar, encontrar una condición para deshacerse de la hipótesis de excelencia. Más concretamente, Houston conjeturó en su artículo de 2010 Stratification of unfoldings of corank 1 singularities que una familia de gérmenes de corrango uno era excelente si el número de Milnor en la imagen era constante a lo largo de la familia. Hemos resuelto esta conjetura para gérmenes de aplicaciones holomorfas que van de en , ergo usamos el número de Milnor en la imagen para asegurar la excelencia de la familia. Por otro lado, nos inspiramos en el trabajo de Teissier para hipersuperficies con singularidades aisladas en su artículo de 1982 Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney y en el de Gaffney para intersecciones completas con singularidades aisladas en su artículo de 1996 Multiplicities and equisingularity of ICIS germs: caracterizar la equisingularidad de Whitney en términos de una secuencia o, lo que es lo mismo, estudiar los números de Milnor de secciones hiperplanas genéricas con codimensión creciente. Así pues, probamos un resultado similar para gérmenes de aplicaciones holomorfas usando esta filosofía, utilizando el teorema de Gaffney y reduciendo el número de invariantes que necesitamos (y su orden, siendo ). Pese a ello, en el caso de gérmenes de aplicaciones holomorfas, necesitamos condiciones que controlen el dominio y el codominio por separado, y es por ello que usamos la secuencia para el codominio, usando el número de Milnor en la imagen usual, y la secuencia para el dominio, usando una definición análoga del número de Milnor en la imagen para gérmenes de aplicaciones holomorfas con una intersección completa con singularidad aislada en el dominio. Esta última secuencia fue la motivación para desarrollar la teoría de gérmenes de aplicaciones holomorfas definidos desde una intersección completa con singularidad aislada. Por último, cabe destacar que hemos probado algunos resultados interesantes a lo largo de nuestro trabajo en esta dirección. El primero de estos resultados es el principio de conservación del número de Milnor en la imagen, así como su semicontinuidad superior. Entre otros, también es de excepcional interés una versión más débil de la conjetura de Mond. Recordamos que la conjetura de Mond afirma que el número de Milnor en la imagen de un germen -finito es mayor o igual que su -codimensión (con igualdad en el caso homogéneo con pesos, llamado también casi homogéneo). Así pues, hemos probado que el número de Milnor en la imagen es cero si, y solo si, el germen es estable (o, equivalentemente, que su -codimensión es cero). La segunda parte de este texto trata la monodromía geométrica local de las fibraciones de Milnor-Lê: probamos que una monodromía geométrica local de un germen no fija ningún punto si . Esto es una generalización de un teorema de Lê Dũng Tràng en su artículo de 1975 La monodromie n'a pas de points fixes enunciado para gérmenes de funciones holomorfas con dominio suave, . Además, Tibar enunció este resultado en su tesis doctoral y en un artículo. Para probar esta generalización usamos una técnica desarrollada por Lê Dũng Tràng que permita hacer un razonamiento inductivo llamada el carrusel: un campo vectorial definido sobre un cilindro complejo que levanta el campo vectorial unitario de una circunferencia y es tangente a cierto conjunto de valores críticos. La idea principal de la prueba es levantar este campo vectorial especial a de una forma conveniente (esto es, que permita encajar el razonamiento inductivo) y tomar su flujo para tener una monodromía geométrica, por lo tanto, también usamos las técnicas mostradas en el libro de Gibson, Wirthmüller, du Plessis y Looijenga Topological stability of smooth mappings para probar los lemas de isotopía de Thom-Mather. Este teorema, así como su versión original dada por Lê Dũng Tràng en su artículo de 1975, tiene aplicaciones interesantes. Mediante un teorema clásico de Lefschetz sobre los puntos fijos de un endomorfismo continuo de un espacio topológico, el teorema que demostramos implica que el número de Lefschetz de una monodromía geométrica local es cero. Esto también es un resultado de A'Campo en su artículo de 1973 Le nombre de Lefschetz d'une monodromie, que da una versión más general usando maquinaria matemática muy técnica y cuya prueba, en la versión más general, atribuye a Deligne. Como corolario de esta aplicación, podemos probar que el hecho de ser suave es un invariante topológico de gérmenes de hipersuperficies usando también un teorema de Lê Dũng Tràng en su artículo de 1973 Calcul du nombre de cycles évanouissants d'une hypersurface complexe, esto también es un caso de la conjetura de Zariski sobre multiplicidades. Este corolario puede ser probado, también, con otro teorema de A'Campo en su artículo anteriormente mencionado, que es, a su vez, consecuencia de nuestro resultado principal. Finalmente, mostramos un teorema de no coalescencia en un contexto general, generalizando algunos conceptos ampliamente usados. Este resultado se traduce, a grandes rasgos, en que, bajo ciertas condiciones, una familia de singularidades, en algún sentido, no puede escindirse, o dividirse, a lo largo de una familia si se conservan ciertos invariantes, como el número de Milnor. Por ejemplo, Lê Dũng Tràng demostró en su artículo anteriormente mencionado que una familia de hipersuperficies con singularidades aisladas no tiene coalescencia si la suma de los números de Milnor es constante a lo largo de la familia (véase también el trabajo de Bey en su artículo de 1972 Sur l'irréductibilité de la monodromie locale y el de Lazzeri en 1973 A theorem on the monodromy of isolated singularities). Otro ejemplo de no coalescencia fue dado enpor Carvalho, Nuño-Ballesteros, Oréfice-Okamoto y Tomazella para familias de intersecciones completas con singularidad aislada con número de Milnor total constante en su prepublicación Families of ICIS with constant total Milnor number. Más precisamente, por contexto general nos referimos a una familia de hipersuperficies dadas por funciones con puntos críticos aislados en cierto espacio ambiente que además es un espacio de Milnor con ciertas hipótesis en la fibración y la familia. Finalmente, la tercera parte es una nueva forma de encarar el estudio de inestabilidades de gérmenes de aplicaciones holomorfas -finitas de en con . Como el lector verá, la principal herramienta que usamos para controlar el número de Milnor en la imagen en la primera parte del texto son los espacios de puntos múltiples de los gérmenes. Pese a ello, no los usamos como un todo, sino que solo nos preocupamos por la parte alternada de su homología porque usamos una secuencia espectral que calcula la imagen (ICSS, del inglés image-computing spectral sequences) para calcular los números de Milnor en la imagen. Esta es la razón para intentar usar toda la simetría de los espacios de puntos múltiples en lugar de solo su homología alternada. Esto lo hacemos utilizando la estructura de los espacios de puntos múltiples (de momento, en corrango uno) y teoría de representaciones. La filosofía de esta nueva forma de aproximarse a problemas de inestabilidades de gérmenes es intentar transformarlos en problemas de álgebra lineal. De hecho, tenemos éxito cuando tratamos de relacionar la constancia de y la de en familias, donde es el número de Milnor en los puntos dobles dado en la primera parte del texto como En particular, probamos que la constancia de implica la de en familias de monogérmenes de corrango uno. Además, probamos que cualquier espacio de puntos múltiples de un monogermen de corrango uno que tenga una singularidad dará homología alternada cuando tomemos su fibra de Milnor. Esto es también una generalización de un teorema dado en la primera parte del texto para familias de gérmenes que admiten un desdoblamiento estable a un parámetro. This thesis covers two joint papers with Nuño-Ballesteros ([GCNB21, GCNB20]), a joint paper with Nuño-Ballesteros and Lê Dung Tràng ([GCTNB21]) and a joint work in development with Mond. These three works delimit the three main parts of the text. As we were saying, the text is divided intro three parts. The ?rst of them is devoted to the study of singularities of holomorphic map germs in the context of the Thom-Mather theory, i.e., modulo A -equivalence. In particular, we focus on corank one germs from C n to C n+1 , but we also develop the theory for germs from C n to C p , with n < p, and germs with an isolated complete intersection singularity (icis) in the source. The main goal of the ?rst part of the text is ?nding a nice characterization of the Whitney equisingularity for one-parameter families of A -finite germs f t : (C n ,S) → (C n+1 ,0) of corank one. A characterization of the Whitney equisingularity was already given by Gaffney in [Gaf93]: a family of germs f t : (C n ,S) → (C n+1 ,0) is Whitney equisingular if, and only if, it is excellent and all the polar multiplicities in the source and target are constant along the family. However, this characterization has the inconvenient of needing a huge number of invariants to assure the equisingularity. Some developments have been made since Ga?ney's result, for example Jorge Pérez and Saia in [JPS06] reduced the number of invariants, still being a huge amount of inva- riants. Also, Houston has an unpublished paper based on a preprint of Ga?ney addressing this issue (see [Hou08]). Our contribution was, first of all, finding a condition to avoid the excellency hypot- hesis. To be more specific, Houston conjectured in [Hou10] that a family of corank one germs is excellent if the image Milnor number µ I is constant along the family. We sol- ve this conjecture for the dimensions (n,n + 1), so use the invariant µ I to assure the excellency of the family. On the other hand, we were inspired by the work of Teissier in [Tei82] for isolated singularities of hypersurfaces and by the work of Gaffney in [Gaf96] for icis: they cha- racterized Whitney equisingularity in terms of µ ∗ sequences. This means studying the Milnor numbers of generic sections with increasingly codimension. Hence, we proved a similar result for map germs using this approach, using Gaffney's theorem and reducing the number of invariants we need. However, in the case of map germs, one needs control conditions on the source and target separately, so we used the sequence µ ∗I (f t ) for the target, with the usual image Milnor number, and a sequence µ ∗ I ( D2 (f t ),π ) for the source, using an analogous definition of the image Milnor number defined for map germs with an icis in the source. This last sequence also motivated us to develop the theory of map germs on icis. Finally, we want to remark that we have proven some interesting results while going on this direction. The first one is the conservation principle of the image Milnor number, as well as its upper semi-continuity. Among other developments, it is of exceptional in- terest a weak version of Mond's conjecture. Recall that Mond's conjecture states that the image Milnor number of an A -?nite germ is greater than, or equal to, its A e -codimension (with equality in the weighted homogeneous case). Hence, we proved that the image Mil- nor number is zero if, and only if, the germ is stable (or, equivalently, the A e -codimension is zero). The second part of this text is about the local geometric monodromy of Milnor-Lê fibrations: we prove that a geometric local monodromy of a germ f : (X,x) → (C,0) does not have any fixed point if f ∈ m 2. This is a generalization of a theorem of Lê Dung Tràng in [Trá75] stated for germs on smooth source, f : (C n+1 ,x) → (C,0). Furthermore, Tibar stated this result in his PhD thesis and in one paper (see [Tib92, Tib93]). In order to prove this generalization, we use a technique developed by Lê Dung Tràng called the carousel, which is a vector field with convenient properties. The main idea of the proof is lifting this vector field to X and take its ?ow to have a geometric monodromy, hence, we also used the techniques shown in [GWdPL76] to prove Thom-Mather isotopy lemmas. This theorem, as its original version given by Lê Dung Tràng, has interesting ap- plications. By a classical result of Lefschetz, the theorem we prove implies that the Lefschetz number of the local geometric monodromy is equal to zero. This is also a result of A'Campo in [A'C73], which has a more general version using heavy mathematical machinery and whose proof, in the most general version, is attributed to Deligne. As corollary of this application, we can prove that being smooth is a topological invariant of germs of hypersurfaces (X,x), using also a theorem of Lê Dung Tràng in [Trá73a]. This corollary can also be proven using another theorem of A'Campo in [A'C73], which is, in turn, consequence of our main theorem. Finally, we show a theorem of no coalescence in a general context. This means that, in some conditions, a family of singularities, in some sense, cannot split along the family provided the conservation of some invariants, such as the Milnor number. For example, in [Trá73b], Lê Dung Tràng showed that a family of hypersurfaces with isolated singu- larities does not have coalescence provided the sum of the Milnor numbers is constant along the family (see also Bey's work in [Bey72] and Lazzeri's work in [Laz73a]). Anot- her example of no coalescence is given in [CNnBOOT21] by Carvalho, Nuño-Ballesteros, Oréfice-Okamoto, and Tomazella for families of icis with constant total Milnor number. To be more precise, by general context we mean a family of hypersurfaces f −1 t(0) given by functions f t : X t → C with isolated critical points inside an ambient space (X,x 0 ) that is a Milnor space with some hypothesis on the fibration and the family. Finally, the third part is a new approach to study instabilities of A -finite map germs f from C n to C p , with p > n. As the reader will see, the main tool we use to control the image Milnor number in the first part of the text is the multiple point spaces of the germs. However, we do not use them as a whole, but we only care about the alternating part of their homologies because we use an image-computing spectral sequence (icss) to compute the image Milnor numbers. This is a reason to try to use all the symmetric structure of the multiple point spaces instead of only their alternating homology. We do this using the structure of the multiple point spaces (at the moment, in corank one) and representation theory. The spirit of this new approach is trying to translate problems of instabilities of germs into problems of linear algebra. In fact, this attempt is successful when we try to relate the constancy of µ I and the constancy of µ D in families, where µ D is the double point Milnor number as given in the first part of the text by µ I ( D 2 (f t ),π ) . In particular, we prove that the constancy of µ I implies the constancy of µ D in families of corank one mono-germs. Furthermore, we prove that any multiple point space of a mono-germ of corank one that has a singularity will provide alternating homology when we take its Milnor fiber. This is also a generalization of a theorem given in the first part of the text for families of germs that admit a one-parameter stable unfolding.

We give a classification of absolutely dicritical foliations of cusp type, that is, the germ of singularities of complex foliations in the complex plane topologically equivalent to the singularity given by the level of the meromorphic function \frac{y^{2}+x^{3}}{xy}., 18 pages

The evolution of engineering applications is increasingly shifting towards the embedded nature, resulting in low-cost solutions, micro/nano dimensional and actuators being exploited as fundamental components to connect the physical nature of information with the abstract one, which is represented in the logical form in a machine. In this context, the scientific community has gained interest in modeling membrane Micro-Electro-Mechanical-Systems (MEMS), leading to a wide diffusion on an industrial level owing to their ease of modeling and realization. Physically, once the external voltage is applied, an electrostatic field, orthogonal to the tangent line of the membrane, is established inside the device, producing an electrostatic pressure that acts on the membrane, deforming it. Evidently, the greater the amplitude of the electrostatic field is, the greater the curvature of the membrane. Thus, it seems natural to consider the amplitude of the electrostatic field proportional to the curvature of the membrane. Starting with this principle, the authors are actively involved in developing a second-order semi-linear elliptic model in 1D and 2D geometries, obtaining important results regarding the existence, uniqueness and stability of solutions as well as evaluating the particular operating conditions of use of membrane MEMS devices. In this context, the idea of providing a survey matures to discussing the similarities and differences between the analytical and numerical results in detail, thereby supporting the choice of certain membrane MEMS devices according to the industrial application. Finally, some original results about the stability of the membrane in 2D geometry are presented and discussed.

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable., Final version

Singularities in the Mohr–Coulomb (MC) yield criterion create numerical difficulties when implemented in the finite element model (FEM). In this research, a two-dimensional (2D) and a three-dimensional (3D) elasto-plastic FEM coupled with isotropic strain hardening–softening or simple strain-softening law incorporating non-associated flow rule including shear band, is used to investigate the effect of singularities, embedment, shape, soil frictional as well as dilation angle on vertically uploaded shallow anchor foundations buried in Toyoura sand. Mohr–Coulomb–Drucker Prager (MC–DP) model provides a more appropriate solution than Drucker Prager–Drucker Prager (DP–DP) model. Besides, the peak resistance factor and the settlement at the ultimate capacity have been found as the functions of embedment, shape, and sand density. Also, the numerical model gives satisfactory agreement with the previous studies. In particular, a simple strain-softening model cannot capture the settlement behavior of anchor foundations in sand efficiently.

International audience; This article analyzes the singularities and workspace of two tensegrity mechanisms that employ a passive universal joint and either three or four tension springs. These two architectures are correlated to 3-SPS-U and 4-SPS-U parallel mechanisms for determining their geometric equations. By fixing the limits of prismatic joints, the workspace for the mechanisms is generated and the parallel singularities are analyzed. Based on the singularity boundaries obtained from the workspace, the joint limits are modified to generate the maximal singularity free workspaces for both the architectures. A comparison is done based on the tilt limits obtained from the workspace of the mechanisms. The mechanism with the maximum tilt limits is implemented for a piping inspection robot to pass through pipe bends and junctions.