Differential geometry ≈ Differential geometry
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An Introduction to Optimization on Smooth Manifolds Open
Optimization on Riemannian manifolds-the result of smooth geometry and optimization merging into one elegant modern framework-spans many areas of science and engineering, including machine learning, computer vision, signal processing, dyna…
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A comprehensive introduction to sub-Riemannian geometry from the Hamiltonian viewpoint Open
Sub-Riemannian geometry is the geometry of a world with nonholonomic constraints. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other. In …
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An Elementary Introduction to Information Geometry Open
In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information scie…
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Hardy Inequalities on Homogeneous Groups Open
This open access book provides an extensive treatment of Hardy inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups. The place where Hardy inequalities and homogeneous groups meet i…
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How to compute invariant manifolds and their reduced dynamics in high-dimensional finite element models Open
Invariant manifolds are important constructs for the quantitative and qualitative understanding of nonlinear phenomena in dynamical systems. In nonlinear damped mechanical systems, for instance, spectral submanifolds have emerged as useful…
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Conformal Methods in General Relativity Open
This book offers a systematic exposition of conformal methods and how they can be used to study the global properties of solutions to the equations of Einstein's theory of gravity. It shows that combining these ideas with differential geom…
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Topological Hyperbolic Lattices Open
Non-Euclidean geometry, discovered by negating Euclid's parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general theo…
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Complexity geometry of a single qubit Open
The computational complexity of a quantum state quantifies how hard it is to make. `Complexity geometry', first proposed by Nielsen, is an approach to defining computational complexity using the tools of differential geometry. Here we demo…
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Fisher-Rao Metric, Geometry, and Complexity of Neural Networks Open
We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. We introduce a new notion of capacity --- the Fisher-Rao norm --- that possesses desirable invariance properties and is…
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Geometric Group Theory Open
The key idea in geometric group theory is to study infinite groups by endowing them with a metric and treating them as geometric spaces. This applies to many groups naturally appearing in topology, geometry, and algebra, such as fundamenta…
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Computational Conformal Geometry Open
This new volume presents thorough introductions to the theoretical foundations - as well as to the practical algorithms - of computational conformal geometry. These have direct applications to engineering and digital geometric processing, …
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Equivalence of critical and subcritical sharp Trudinger–Moser–Adams inequalities Open
Sharp Trudinger–Moser inequalities on the first order Sobolev spaces and their analogous Adams inequalities on high order Sobolev spaces play an important role in geometric analysis, partial differential equations and other branches of mod…
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K(π, 1) for Artin Groups of Finite Type Open
This paper is a continuation of a programme to construct new K(π, 1)’s for Artin groups of finite type which began in [4] with Artin groups on 2 and 3 generators and was extended to braid groups in [3]. These K(π, 1)’s differ from those in…
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Lorentzian Graph Convolutional Networks Open
Graph convolutional networks (GCNs) have received considerable research attention recently. Most GCNs learn the node representations in Euclidean geometry, but that could have a high distortion in the case of embedding graphs with scale-fr…
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Hyperbolic Topological Band Insulators Open
Recently, hyperbolic lattices that tile the negatively curved hyperbolic plane emerged as a new paradigm of synthetic matter, and their energy levels were characterized by a band structure in a four- (or higher-) dimensional momentum space…
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Chern insulator in a hyperbolic lattice Open
Motivated by the recent experimental realizations of hyperbolic lattices in\ncircuit quantum electrodynamics and the research interest in the non-Euclidean\ngeneralization of topological phenomena, we investigate the Chern insulator\nphase…
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Geometric formulation of the Cauchy invariants for incompressible Euler flow in flat and curved spaces Open
Cauchy invariants are now viewed as a powerful tool for investigating the Lagrangian structure of three-dimensional (3D) ideal flow (Frisch & Zheligovsky, Commun. Math. Phys. , vol. 326, 2014, pp. 499–505; Podvigina et al. , J. Comput. Phy…
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Short-time behavior of the heat kernel and Weyl’s law on $${{\mathrm{RCD}}}^*(K,N)$$ RCD ∗ ( K , N ) spaces Open
In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of RCD∗(K,N)-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in RCD∗(K,N)-spaces. We use then these results t…
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Common fixed points of G-nonexpansive mappings on Banach spaces with a graph Open
In this paper, we prove the weak and strong convergence of a sequence $\{x_{n}\}$ generated by the Ishikawa iteration to some common fixed points of two G -nonexpansive mappings defined on a Banach space endowed with a graph.
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Nonlinear definition of the shadowy mode in higher-order scalar-tensor theories Open
We study U-DHOST theories, i.e., higher-order scalar-tensor theories which are degenerate only in the unitary gauge and yield an apparently unstable extra mode in a generic coordinate system. We show that the extra mode satisfies a three-d…
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An Introduction to Geometric Topology Open
This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into …
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Evolutes and focal surfaces of framed immersions in the Euclidean space Open
We consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair …
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On solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces Open
In this paper, we introduce and study iterative schemes for solving split equilibrium problems and fixed point problems of nonspreading multi-valued mappings in Hilbert spaces and prove that the modified Mann iteration converges weakly to …
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Fixed point theorems of JS-quasi-contractions Open
In this paper, we introduce the concept of JS-quasi-contraction and prove some fixed point results for JS-quasi-contractions in complete metric spaces under the assumption that the involving function is nondecreasing and continuous. These …
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Log-concavity of volume and complex Monge-Amp\\`ere equations with\n prescribed singularity Open
Let $(X,\\omega)$ be a compact K\\"ahler manifold. We prove the existence and\nuniqueness of solutions to complex Monge-Amp\\`ere equations with prescribed\nsingularity type. Compared to previous work, the assumption of small unbounded\nlo…
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Browder and Göhde fixed point theorem for monotone nonexpansive mappings Open
Let X be a Banach space or a complete hyperbolic metric space. Let C be a nonempty, bounded, closed, and convex subset of X and $T: C \rightarrow C$ be a monotone nonexpansive mapping. In this paper, we show that if X is a Banach space whi…
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The Banach contraction principle in $C^{*}$-algebra-valued b-metric spaces with application Open
We introduce the notion of a $C^{*}$ -algebra-valued b -metric space. We generalize the Banach contraction principle in this new setting. As an application of our result, we establish an existence result for an integral equation in a $C^{*…
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Protein pocket detection via convex hull surface evolution and associated Reeb graph Open
Motivation Protein pocket information is invaluable for drug target identification, agonist design, virtual screening and receptor-ligand binding analysis. A recent study indicates that about half holoproteins can simultaneously bind multi…
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Wardowski type fixed point theorems in complete metric spaces Open
In this paper, we state and prove Wardowski type fixed point theorems in metric space by using a modified generalized F -contraction maps. These theorems extend other well-known fundamental metrical fixed point theorems in the literature (…
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Covariant-differential formulation of Lagrangian field theory Open
Building on the Utiyama principle we formulate an approach to Lagrangian field theory in which exterior covariant differentials of vector-valued forms replace partial derivatives, in the sense that they take up the role played by the latte…