Showing papers on "Geometry and topology published in 2018"
Book•
28 Mar 2018TL;DR: A survey of quasiisometric rigidity on linear groups can be found in this paper, where the authors present a mapping of groups to a metric space and a metric topology.
Abstract: Geometry and topology Metric spaces Differential geometry Hyperbolic space Groups and their actions Median spaces and spaces with measured walls Finitely generated and finitely presented groups Coarse geometry Coarse topology Ultralimits of metric spaces Gromov-hyperbolic spaces and groups Lattices in Lie groups Solvable groups Geometric aspects of solvable groups The Tits alternative Gromov's theorem The Banach-Tarski paradox Amenability and paradoxical decomposition Ultralimits, fixed point properties, proper actions Stallings's theorem and accessibility Proof of Stallings's theorem using harmonic functions Quasiconformal mappings Groups quasiisometric to $\mathbb{H}^n$ Quasiisometries of nonuniform lattices in $\mathbb{H}^n$ A survey of quasiisometric rigidity Appendix: Three theorems on linear groups Bibliography Index
149 citations
TL;DR: In this paper, the authors introduce a new class of problems called partial differential hemivariational inequalities (PDH inequalities) that combine evolution equations and hemivariations, and establish metric characterizations for these problems.
Abstract: Abstract The aim of this paper is to introduce and study a new class of problems called partial differential hemivariational inequalities that combines evolution equations and hemivariational inequalities. First, we introduce the concept of strong well-posedness for mixed variational quasi hemivariational inequalities and establish metric characterizations for it. Then we show the existence of solutions and meaningful properties such as measurability and upper semicontinuity for the solution set of the mixed variational quasi hemivariational inequality associated to the partial differential hemivariational inequality. Relying, on these properties we are able to prove the existence of mild solutions for partial differential hemivariational inequalities. Furthermore, we show the compactness of the set of the corresponding mild trajectories.
71 citations
TL;DR: In this article, a min-max theory for embedded hypersurfaces with a fixed boundary is proposed and applied to prove several theorems about the existence of embedded minimal surfaces with a given boundary.
Abstract: In this note we propose a min-max theory for embedded hypersurfaces with a fixed boundary and apply it to prove several theorems about the existence of embedded minimal hypersurfaces with a given boundary. A simpler variant of these theorems holds also for the case of the free boundary minimal surfaces.
41 citations
TL;DR: In this article, the authors propose a method to solve the problem of the EKG-based EKF model in the context of bioinformatics, which they call EPFL-ARTICLE-227878.
Abstract: Reference EPFL-ARTICLE-227878 URL: https://arxiv.org/abs/1703.09801 Record created on 2017-05-02, modified on 2017-10-27
39 citations
TL;DR: In this paper, the existence of positive solutions for systems of singular fractional differential equations with a parameter and integral boundary conditions is investigated by using fixed point theorem in cone, and sufficient conditions which guarantee that positive solutions are obtained are obtained.
Abstract: Abstract In this paper, the existence of positive solutions for systems of semipositone singular fractional differential equations with a parameter and integral boundary conditions is investigated. By using fixed point theorem in cone, sufficient conditions which guarantee the existence of positive solutions are obtained. An example is given to illustrate the results.
31 citations
TL;DR: In this paper, the authors considered the Timoshenko system of type III with frictional damping and delay terms and proved the global existence of solutions by using the Faedo-Galerkin approximations together with some energy estimates.
Abstract: Abstract In this paper, we consider the following Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms: { ρ 1 φ t t - K ( φ x + ψ ) x = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , ρ 2 ψ t t - b ψ x x + K ( φ x + ψ ) + β θ x = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , ρ 3 θ t t - δ θ x x + γ ψ t t x + ∫ 0 t g ( t - s ) θ x x ( s ) d s + μ 1 θ t ( x , t ) + μ 2 θ t ( x , t - τ ) = 0 , ( x , t ) ∈ ( 0 , 1 ) × ( 0 , ∞ ) , \\left\\{\\begin{aligned} &\\displaystyle\\rho_{1}\\varphi_{tt}-K(\\varphi_{x}+\\psi)_% {x}=0,&&\\displaystyle(x,t)\\in(0,1)\\times(0,\\infty),\\\\ &\\displaystyle\\rho_{2}\\psi_{tt}-b\\psi_{xx}+K(\\varphi_{x}+\\psi)+\\beta\\theta_{x}% =0,&&\\displaystyle(x,t)\\in(0,1)\\times(0,\\infty),\\\\ &\\displaystyle\\rho_{3}\\theta_{tt}-\\delta\\theta_{xx}+\\gamma\\psi_{ttx}+\\int_{0}^% {t}g(t-s)\\theta_{xx}(s)\\,\\mathrm{d}s+\\mu_{1}\\theta_{t}(x,t)+\\mu_{2}\\theta_{t}(% x,t-\\tau)=0,&&\\displaystyle(x,t)\\in(0,1)\\times(0,\\infty),\\end{aligned}\\right. together with initial datum and boundary conditions of Dirichlet type, where g is a positive non-increasing relaxation function and μ 1 , μ 2 {\\mu_{1},\\mu_{2}} are positive constants. Under a hypothesis between the weight of the delay term and the weight of the friction damping term, we prove the global existence of solutions by using the Faedo–Galerkin approximations together with some energy estimates. Then, by introducing appropriate Lyapunov functionals, under the imposed constrain on the above two weights, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.
30 citations
01 Jun 2018
TL;DR: The enumerative geometry of rational curves in equivariant symplectic resolutions was introduced in this paper, and its relation to the structures of geometric representation theory is discussed in Section 2.1.
Abstract: This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer Institute.
25 citations
TL;DR: In this article, the authors give a homological characterization of Stein spaces over any valued base field and obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces.
24 citations
TL;DR: In this paper, the geometry and topology of these spaces are described via the corresponding Higgs bundles and Hitchin fibrations, and the moduli space of flat G C -connections appearing from finite group actions on compact Riemann surfaces are studied.
23 citations
TL;DR: In this article, a variational method was used to prove the existence of ground state solutions for the Kirchhoff-type equations with positive constants, where a, b and V are positive constants and f has critical growth.
Abstract: Abstract In this paper, we concern ourselves with the following Kirchhoff-type equations: { - ( a + b ∫ ℝ 3 | ∇ u | 2 𝑑 x ) △ u + V u = f ( u ) in ℝ 3 , u ∈ H 1 ( ℝ 3 ) , \\left\\{\\begin{aligned} \\displaystyle-\\biggl{(}a+b\\int_{\\mathbb{R}^{3}}\\lvert% \
abla u\\rvert^{2}\\,dx\\biggr{)}\\triangle u+Vu&\\displaystyle=f(u)\\quad\\text{in % }\\mathbb{R}^{3},\\\\ \\displaystyle u&\\displaystyle\\in H^{1}(\\mathbb{R}^{3}),\\end{aligned}\\right. where a, b and V are positive constants and f has critical growth. We use variational methods to prove the existence of ground state solutions. In particular, we do not use the classical Ambrosetti–Rabinowitz condition. Some recent results are extended.
22 citations
01 Jun 2018
TL;DR: In this article, a generalization of the weighted Ostrowski type inequality and its companion inequalities on an arbitrary time scale for functions whose first delta derivatives are bounded has been presented.
Abstract: Abstract In this article, using two parameters, we obtain generalizations of a weighted Ostrowski type inequality and its companion inequalities on an arbitrary time scale for functions whose first delta derivatives are bounded. Our work unifies the continuous and discrete versions and can also be applied to the quantum calculus case.
TL;DR: In this article, the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk were compared and it was shown that all known models for (Formula presented.) -operads are equivalent.
Abstract: We compare two models for (Formula presented.) -operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for (Formula presented.) -operads are equivalent — for instance, it follows that the homotopy theory of Lurie's (Formula presented.) -operads is equivalent to that of dendroidal sets and that of simplicial operads.
Book•
28 Jun 2018
TL;DR: In this article, the authors introduce linear algebra, differential forms and tensors, Riemannian geometry, contact geometry, symplectic geometry, and Symplectic Geometry.
Abstract: Basic Objects and Notation.- Linear Algebra Essentials.- Advanced Calculus.- Differential Forms and Tensors.- Riemannian Geometry.- Contact Geometry.- Symplectic Geometry.- References.- Index.
TL;DR: In this article, the authors give easy new proofs to many known statements and also prove the following new statement: on a complete Riemannian manifold on nonconstant curvature, the index of affine transformations in the group of projective transformations is at most two.
Abstract: The paper is based on the lecture course “Metric projective geometry” which I conducted at the summer school “Finsler geometry with applications” at Karlovassi, Samos, in 2014, and at the workshop before the 8th seminar on Geometry and Topology of the Iranian Mathematical society at the Amirkabir University of Technology in 2015. The goal of this lecture course was to show how effective projectively invariant objects can be used to solve natural and named problems in differential geometry, and this paper also does it: I give easy new proofs to many known statements and also prove the following new statement: on a complete Riemannian manifold on nonconstant curvature, the index of the group of affine transformations in the group of projective transformations is at most two.
TL;DR: For linear perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is n\ge 7 and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.
Abstract: We build blowing-up solutions for linear perturbation of the Yamabe problem on manifolds with boundary, provided the dimension of the manifold is \(n\ge 7\) and the trace-free part of the second fundamental form is non-zero everywhere on the boundary.
Posted Content•
TL;DR: Recently, there has been a large expansion in diversity of Kendall-type Shape-and-Scale Theories as discussed by the authors, and the current article outlines this variety, and furthermore roots it in solving the poset of generalized Killing equations.
Abstract: Kendall's Shape Theory covers shapes formed by $N$ points in $\mathbb{R}^d$ upon quotienting out the similarity transformations. This theory is based on the geometry and topology of the corresponding configuration space: shape space. Kendall studied this to build a widely useful Shape Statistics thereupon. The corresponding Shape-and-Scale Theory -- quotienting out the Euclidean transformations -- is useful in Classical Dynamics and Molecular Physics, as well as for the relational `Leibnizian' side of the Absolute versus Relational Motion Debate. Kendall's shape spaces moreover recur withing this `Leibnizian' Shape-and-Scale Theory. There has recently been a large expansion in diversity of Kendall-type Shape(-and-Scale) Theories. The current article outlines this variety, and furthermore roots it in solving the poset of generalized Killing equations. This moreover also places a first great bound on how many more Shape(-and-Scale) Theories there can be. For it is nongeneric for geometrically-equipped manifolds -- replacements for Kendall's $\mathbb{R}^d$ carrier space (absolute space to physicists) - to possess any generalized Killing vectors. Article II places a second great bound, now at the topological level and in terms of which Shape(-and-Scale) Theories are technically tractable. Finally Article III explains how the diversity of Shape(-and-Scale) Theories - from varying which carrier space and quotiented-out geometrical automorphism group are in use - constitutes a theory of Comparative Background Independence: a topic of fundamental interest in Dynamics, Gravitation and Theoretical Physics more generally. Article I and II's great bounds moreover have significant consequences for Comparative Background Independence.
TL;DR: In this paper, the authors generalize Tian's result for toric varieties, and illustrate these concepts in the case of toric-Fano varieties, generalizing a result of Tian.
Abstract: Let -Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.
TL;DR: In this paper, the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature was studied and a Cauchy Domain of Outer Communications (CDOC) was defined.
Abstract: We study the relationship between initial data sets with horizons and the existence of metrics of positive scalar curvature. We define a Cauchy Domain of Outer Communications (CDOC) to be an asympt ...
12 Dec 2018
TL;DR: In this article, the singular moduli spaces of Higgs bundles of degree 0 and rank n on a compact Riemann surface of genus g are proved to be symplectic singularities, in the sense of Beauville [Bea00] and admit a projective projective resolution if and only if g = 1$ or n = 2,2.
Abstract: In this paper, we study the algebraic symplectic geometry of the singular moduli spaces of Higgs bundles of degree $0$ and rank $n$ on a compact Riemann surface $X$ of genus $g$. In particular, we prove that such moduli spaces are symplectic singularities, in the sense of Beauville [Bea00], and admit a projective symplectic resolution if and only if $g=1$ or $(g, n)=(2,2)$. These results are an application of a recent paper by Bellamy and Schedler [BS16] via the so-called Isosingularity Theorem.
TL;DR: In this article, the authors established a Harnack inequality for non-negative solutions of a non-divergence structure uniformly elliptic operator and a nondecreasing function that satisfies an appropriate growth condition at infinity.
Abstract: Abstract In this paper we establish a Harnack inequality for non-negative solutions of L u = f ( u ) {Lu=f(u)} where L is a non-divergence structure uniformly elliptic operator and f is a non-decreasing function that satisfies an appropriate growth conditions at infinity.
Posted Content•
TL;DR: In this paper, two gradient descent flows of quadratic disagreement functions on general Riemannian manifolds were studied and necessary conditions for the agents to synchronize from almost all initial conditions when the graph used to model the network is connected.
Abstract: Multi-agent systems on nonlinear spaces sometimes fail to synchronize. This is usually attributed to the initial configuration of the agents being too spread out, the graph topology having certain undesired symmetries, or both. Besides nonlinearity, the role played by the geometry and topology of the nonlinear space is often overlooked. This paper concerns two gradient descent flows of quadratic disagreement functions on general Riemannian manifolds. One system is intrinsic while the other is extrinsic. We derive necessary conditions for the agents to synchronize from almost all initial conditions when the graph used to model the network is connected. If a Riemannian manifold contains a closed curve of locally minimum length, then there is a connected graph and a dense set of initial conditions from which the intrinsic system fails to synchronize. The extrinsic system fails to synchronize if the manifold is multiply connected. The extrinsic system appears in the Kuramoto model on $\smash{\mathcal{S}^1}$, rigid-body attitude synchronization on $\mathsf{SO}(3)$, the Lohe model of quantum synchronization on the $n$-sphere, and the Lohe model on $\mathsf{U}(n)$. Except for the Lohe model on the $n$-sphere where $n\in\mathbb{N}\backslash\{1\}$, there are dense sets of initial conditions on which these systems fail to synchronize. The reason for this difference is that the $n$-sphere is simply connected for all $n\in\mathbb{N}\backslash\{1\}$ whereas the other manifolds are multiply connected.
Posted Content•
TL;DR: Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics as discussed by the authors.
Abstract: Geometric flows have proved to be a powerful geometric analysis tool, perhaps most notably in the study of 3-manifold topology, the differentiable sphere theorem, Hermitian-Yang-Mills connections and canonical Kaehler metrics. In the context of G_2 geometry, there are several geometric flows which arise. Each flow provides a potential means to study the geometry and topology associated with a given class of G_2 structures. We will introduce these flows, and describe some of the key known results and open problems in the field.
TL;DR: In this article, the authors give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between points in infinitesimally small neighborhoods of a point.
Abstract: We give a metric characterization of the scalar curvature of a smooth Riemannian manifold, analyzing the maximal distance between $(n+1)$ points in infinitesimally small neighborhoods of a point. Since this characterization is purely in terms of the distance function, it could be used to approach the problem of defining the scalar curvature on a non-smooth metric space. In the second part we will discuss this issue, focusing in particular on Alexandrov spaces and surfaces with bounded integral curvature.
TL;DR: In this paper, the initial coefficients of functions belonging to the pseudo bi-starlike functions with respect to symmetrical points in the open unit disk were obtained, and upper bounds for the coefficients of the functions in these new classes were obtained.
Abstract: In this paper, defining new interesting classes, $\lambda$-pseudo bi-starlike functions with respect to symmetrical points and $\lambda$-pseudo bi-convex functions with respect to symmetrical points in the open unit disk $\mathbb U$, we obtain upper bounds for the initial coefficients of functions belonging to these new classes.
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19 Nov 2018TL;DR: In this paper, the structure groups of almost even-Clifford Hermitian manifolds are computed and the authors determine when such groups lead to Spin structures. But they do not define spin structures.
Abstract: We compute the structure groups of almost even-Clifford Hermitian manifolds and determine when such groups lead to Spin structures.
TL;DR: In this article, the geometry and topology of extremal domains in a Hadamard manifold were investigated, i.e., domains that support a positive solution to an overdetermined elliptic problem (OEP).
TL;DR: The Toda lattice has been generalized in different directions, each with its own analytic, geometric, and topological characteristics as discussed by the authors, as well as a polytope structure of the solution spaces.
Abstract: In 1967, Japanese physicist Morikazu Toda published a pair of seminal papers in the Journal of the Physical Society of Japan that exhibited soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors In the fifty years that followed, Toda's system of particles has been generalized in different directions, each with its own analytic, geometric, and topological characteristics These are known collectively as the Toda lattice This survey recounts and compares the various versions of the finite nonperiodic Toda lattice from the perspective of their geometry and topology In particular, we highlight the polytope structure of the solution spaces as viewed through the moment map, and we explain the connection between the real indefinite Toda flows and the integral cohomology of real flag varieties
TL;DR: In this article, it was shown that the weak solution u is locally Hölder continuous besides a singular set by using the direct method and classical Morrey-type estimates, and this result not only holds in the interior, but also holds up to the boundary.
Abstract: Abstract In this paper, we are concerned with the nonlinear elliptic systems in divergence form under controllable growth condition. We prove that the weak solution u is locally Hölder continuous besides a singular set by using the direct method and classical Morrey-type estimates. Here the Hausdorff dimension of the singular set is less than n - p {n-p} . This result not only holds in the interior, but also holds up to the boundary.