The Experts below are selected from a list of 36510 Experts worldwide ranked by ideXlab platform

Zhigang Wang - One of the best experts on this subject based on the ideXlab platform.

Kitazawa Naoki - One of the best experts on this subject based on the ideXlab platform.

  • Characterizing families of graph manifolds via suitable classes of simple fold maps into the plane and embeddability of Reeb spaces in some 3-dimensional manifolds
    2021
    Co-Authors: Kitazawa Naoki
    Abstract:

    Graph manifolds form important classes of 3-dimensional closed and orientable manifolds. For example, Seifert manifolds are graph manifolds where hyperbolic manifolds are not. In applying Singularity Theory of differentiable maps to understanding global topologies of manifolds, graph manifolds have been shown to be characterized as ones admitting so-called simple fold maps into the plane of explicit classes by Saeki and the author. The present paper presents several new results of this type. Fold maps are higher dimensional variants of Morse functions and simple ones form simple classes, generalizing the class of general Morse functions. Such maps into the plane on $3$-dimensional closed and orientable manifolds induce quotient maps to so-called simple polyhedra with no vertices, which are 2-dimensional. This is also closely related to the Theory of shadows of 3-dimensional manifolds. We also discuss invariants for graph manifolds via embeddability of these polyhedra in some 3-dimensional manifolds.Comment: 16 pages, title changed, a sketch of a proof of Theorem 1 (3) added (essentially a main theorem of arxiv:2105.00974), several expositions revised, this will be improved before submission to a refereed journa

  • Global topologies of Reeb spaces of stable fold maps with non-trivial top homology groups
    2021
    Co-Authors: Kitazawa Naoki
    Abstract:

    The Reeb space of a continuous map is the space of all (elements representing) connected components of preimages endowed with the quotient topology induced from the natural equivalence relation on the domain. These objects are strong tools in (differential) topological Theory of Morse functions, fold maps, which are their higher dimensional variants, and so on: they are in general polyhedra whose dimensions are same as those of the targets. In suitable cases Reeb spaces inherit topological information such as homology groups, cohomology rings, and so on, of the manifolds. This presents the following problem: what are global topologies of Reeb spaces of these smooth maps of suitable classes like? The present paper presents families of stable fold maps having Reeb spaces with non-trivial top homology groups with their (co)homology groups (and rings). Related studies on the global topologies from the viewpoints of the Singularity Theory of differentiable maps and differential topology have been presented by various researchers including the author. The author previously constructed families of fold maps with Reeb spaces with non-trivial top homology groups and with good topological properties. This paper presents new families, especially, generalized situations of some known situations.Comment: 18 pages, several phrases corrected

  • Realizing a homology class of a compact manifold by a homology class of an explicit closed submanifold--a new approach to Thom's works on homology classes of submanifolds-
    2020
    Co-Authors: Kitazawa Naoki
    Abstract:

    It is a classical important problem of differential topology by Thom; for a homology class of a compact manifold, can we realize this by a closed submanifold with no boundary? This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient ring is Z_2. If the degree of the class is smaller or equal to 6 or equal to k-2 or k-1 under the condition that the coefficient ring is the integer ring where k is the dimension of the manifold, then this is also true. As a specific study, for 4-dimensional closed manifolds, the topologies (genera) of closed and connected surfaces realizing given 2nd homology classes have been actively studied, for example. In the present paper, we consider the following similar problem; can we realize a homology class of a compact manifold by a homology class of an explicit closed manifold embedded in the (interior of the) given compact manifold? This problem is considered as a variant of previous problems. We present an affirmative answer via important Theory in the Singularity Theory of differentiable maps: lifting a given smooth map to an embedding or obtaining an embedding such that the composition of this with the canonical projection is the given map. Presenting this application of lifting smooth maps and related fundamental propositions is also a main purpose of the present paper.Comment: 12 pages, this is submitted to a refereed journal, incorrect stuffs are due to the carelessness of the autho

  • Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings
    2020
    Co-Authors: Kitazawa Naoki
    Abstract:

    Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions, fold maps and more general generic maps. It is one of important and interesting branches of the Singularity Theory of differentiable maps and applications to geometry of manifolds. In this paper we present fold maps with information of cohomology rings of their Reeb spaces. Reeb spaces are defined as the spaces of all connected components of all preimages, and in suitable situations inherit topological information such as homology groups and cohomology rings of the manifolds. Previously, the author demonstrated construction of fold maps in various cases : key methods are surgery operations to manifolds and maps and in this paper, we present more useful surgery operations and by them we construct new fold maps. More precisely, fold maps with singular value sets with crossings: the singular value set of a smooth map is the image of the set of all singular points and note that for fold maps, the set of all singular points are closed submanifolds without boundaries and the restrictions to them are immersions of codimension 1.Comment: 18 pages, 3 figures, the title changed and various errors including crucial errors in main theorems corrected, this is submitted to a (refereed academic) journa

  • Realizing a homology class of a compact manifold by a homology class of an explicit closed submanifold--a new approach to Thom's works on homology classes of submanifolds-
    2020
    Co-Authors: Kitazawa Naoki
    Abstract:

    It is a classical important problem of differential topology by Thom; for a homology class of a compact manifold, can we realize this by a closed submanifold with no boundary? This is true if the degree of the class is smaller or equal to the half of the dimension of the outer manifold under the condition that the coefficient ring is Z_2. If the degree of the class is smaller or equal to 6 or equal to k-2 or k-1 under the condition that the coefficient ring is the integer ring where k is the dimension of the manifold, then this is also true. As a specific study, for 4-dimensional closed manifolds, the topologies (genera) of closed and connected surfaces realizing given 2nd homology classes have been actively studied, for example. In the present paper, we consider the following similar problem; can we realize a homology class of a compact manifold by a homology class of an explicit closed manifold embedded in the (interior of the) given compact manifold? This problem is considered as a variant of previous problems. We present an affirmative answer via important Theory in the Singularity Theory of differentiable maps: lifting a given smooth map to an embedding or obtaining an embedding such that the composition of this with the canonical projection is the given map. Presenting this application of lifting smooth maps and related fundamental propositions is also a main purpose of the present paper.Comment: 14 pages, this is revised from the previous version and submitted to a refereed journa

Colombi Stéphane - One of the best experts on this subject based on the ideXlab platform.

  • Cold dark matter protohalo structure around collapse: Lagrangian cosmological perturbation Theory versus Vlasov simulations
    HAL CCSD, 2021
    Co-Authors: Saga Shohei, Taruya Atsushi, Colombi Stéphane
    Abstract:

    We explore the structure around shell-crossing time of cold dark matter protohaloes seeded by two or three crossed sine waves of various relative initial amplitudes, by comparing Lagrangian perturbation Theory (LPT) up to 10th order to high-resolution cosmological simulations performed with the public Vlasov code ColDICE. Accurate analyses of the density, the velocity, and related quantities such as the vorticity are performed by exploiting the fact that ColDICE can follow locally the phase-space sheet at the quadratic level. To test LPT predictions beyond shell-crossing, we employ a ballistic approximation, which assumes that the velocity field is frozen just after shell-crossing. In the generic case, where the amplitudes of the sine waves are all different, high-order LPT predictions match very well the exact solution, even beyond collapse. As expected, convergence slows down when going from quasi-1D dynamics where one wave dominates over the two others, to the axial-symmetric configuration, where all the amplitudes of the waves are equal. It is also noticed that LPT convergence is slower when considering velocity related quantities. Additionally, the structure of the system at and beyond collapse given by LPT and the simulations agrees very well with Singularity Theory predictions, in particular with respect to the caustic and vorticity patterns that develop beyond collapse. Again, this does not apply to axial-symmetric configurations, that are still correct from the qualitative point of view, but where multiple foldings of the phase-space sheet produce very high density contrasts, hence a strong backreaction of the gravitational force

  • Cold dark matter protohalo structure around collapse: Lagrangian cosmological perturbation Theory versus Vlasov simulations
    2021
    Co-Authors: Saga Shohei, Taruya Atsushi, Colombi Stéphane
    Abstract:

    We explore the structure around shell-crossing time of cold dark matter protohaloes seeded by two or three crossed sine waves of various relative initial amplitudes, by comparing Lagrangian perturbation Theory (LPT) up to 10th order to high-resolution cosmological simulations performed with the public Vlasov code ColDICE. Accurate analyses of the density, the velocity, and related quantities such as the vorticity are performed by exploiting the fact that ColDICE can follow locally the phase-space sheet at the quadratic level. To test LPT predictions beyond shell-crossing, we employ a ballistic approximation, which assumes that the velocity field is frozen just after shell-crossing. In the generic case, where the amplitudes of the sine waves are all different, high-order LPT predictions match very well the exact solution, even beyond collapse. As expected, convergence slows down when going from quasi-1D dynamics where one wave dominates over the two others, to the axial-symmetric configuration, where all the amplitudes of the waves are equal. It is also noticed that LPT convergence is slower when considering velocity related quantities. Additionally, the structure of the system at and beyond collapse given by LPT and the simulations agrees very well with Singularity Theory predictions, in particular with respect to the caustic and vorticity patterns that develop beyond collapse. Again, this does not apply to axial-symmetric configurations, that are still correct from the qualitative point of view, but where multiple foldings of the phase-space sheet produce very high density contrasts, hence a strong backreaction of the gravitational force.Comment: 30 pages, 18 figure

Todor Milanov - One of the best experts on this subject based on the ideXlab platform.

  • analyticity of the total ancestor potential in Singularity Theory
    Advances in Mathematics, 2014
    Co-Authors: Todor Milanov
    Abstract:

    Abstract K. Saito's Theory of primitive forms gives a natural semi-simple Frobenius manifold structure on the space of miniversal deformations of an isolated Singularity. On the other hand, Givental introduced the notion of a total ancestor potential for every semi-simple point of a Frobenius manifold and conjectured that in the settings of Singularity Theory his definition extends analytically to non-semisimple points as well. In this paper we prove Givental's conjecture by using the Eynard–Orantin recursion.

Rouot Jérémy - One of the best experts on this subject based on the ideXlab platform.

  • Accessibility, Abnormal Geodesics in Optimal Control. A Geometric Approach from Singularity Theory using Two Cases Studies'
    HAL CCSD, 2021
    Co-Authors: Bonnard Bernard, Rouot Jérémy, Wembe Boris
    Abstract:

    In this note, we use two case studies to analyze the relation between abnormal geodesiscs in time minimal control, accessibility and the regularity properties of the (time minimal) value function. The first problem concerns a Zermelo navigation problem in the plane. Using the historical problem of the calculus of variations set by Carathéodory and Zermelo aiming to compute the quickest nautical path of a ship on a river to be transferred from one shore to opposite one, we construct a semi-normal form to analyze the cusp Singularity of the abnormal geodesic when meeting the transition between the strong and weak current domains and to evaluate the time minimal value function. The problem is related to reaching the boundary of the two domains in minimum time. The second problem concerns the classification of the time minimal syntheses for chemical reactors near a terminal manifold of codimension one, in particular in relation with the McKeithan network and associated to maximize the production of one chemical species. Both cases studies have the common geometric frame and computations techniques illustrate the role of Singularity Theory in geometric optimal control, in relation with abnormal geodesics and regularity of the value function solution of Hamilton-Jacobi-Bellman equation

  • Toward Geometric Time Minimal Control without Legendre Condition and with Multiple Singular Extremals for Chemical Networks. An Extended Version
    HAL CCSD, 2021
    Co-Authors: Rouot Jérémy
    Abstract:

    This article deals with the problem of maximizing the production of a species for a chemical network by controlling the temperature. Under the socalled mass kinetics assumption the system can be modeled as a single-input control system using the Feinberg-Horn-Jackson graph associated to the reactions network. Thanks to Pontryagin's Maximum Principle, the candidates as minimizers can be found among extremal curves, solutions of a (non smooth) Hamiltonian dynamics and the problem can be stated as a time minimal control problem with a terminal target of codimension one. Using geometric control and Singularity Theory the time minimal syntheses (closed loop optimal control) can be classified near the terminal manifold under generic conditions. In this article, we focus to the case where the generalized Legendre-Clebsch condition is not satisfied, which paves the road to complicated syntheses with several singular arcs. In particular, it is related to the situation for a weakly reversible network like the McKeithan scheme

  • Towards Geometric Time Minimal Control without Legendre Condition and with Multiple Singular Extremals for Chemical Networks
    HAL CCSD, 2020
    Co-Authors: Rouot Jérémy
    Abstract:

    International audienceThis article deals with the problem of maximizing the production of a species for a chemical network by controlling the temperature. Under the so-called mass kinetics assumption the system can be modeled as a single-input control system using the Feinberg-Horn-Jackson graph associated to the reactions network. Thanks to Pontryagin's Maximum Principle, the candidates as minimizers can be found among extremal curves, solutions of a (non smooth) Hamiltonian dynamics and the problem can be stated as a time minimal control problem with a terminal target of codimension one. Using geometric control and Singularity Theory the time minimal syntheses (closed loop optimal control) can be classified near the terminal manifold under generic conditions. In this article we focus to the case where the generalized Legendre-Clebsch condition is not satisfied, which paves the road to complicated syntheses with several singular arcs. In particular it is related to the situation for a weakly reversible network like the McKeithan scheme of two reactions

  • Towards Geometric Time Minimal Control without Legendre Condition and with Multiple Singular Extremals for Chemical Networks
    HAL CCSD, 2020
    Co-Authors: Rouot Jérémy
    Abstract:

    This article deals with the problem of maximizing the production of a species for a chemical network by controlling the temperature. Under the so-called mass kinetics assumption the system can be modeled as a single-input control system using the Feinberg-Horn-Jackson graph associated to the reactions network. Thanks to Pontryagin's Maximum Principle, the candidates as minimizers can be found among extremal curves, solutions of a (non smooth) Hamiltonian dynamics and the problem can be stated as a time minimal control problem with a terminal target of codimension one. Using geometric control and Singularity Theory the time minimal syntheses (closed loop optimal control) can be classified near the terminal manifold under generic conditions. In this article we focus to the case where the generalized Legendre-Clebsch condition is not satisfied, which paves the road to complicated syntheses with several singular arcs. In particular it is related to the situation for a weakly reversible network like the McKeithan scheme of two reactions