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26001 results in Abstract analysis

Page 4 of 1301

  • 2 - The Abstract Setting

  • from Part I - Basics: Foundational Material, Elementary Aspects, and Examples
  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 14-49
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  • Summary

    We are interested in discrete structures like graphs, hypergraphs, and simplicial complexes, and we want to study them via spectral properties of Laplace type operators defined on them. These operators perform some kind of local averaging. The eigenvalues and eigenfunctions of such operators can be obtained as critical values and points of Rayleigh quotients. Symmetry groups of the underlying structures have representations on eigenspaces.

  • 5 - Eigenvalues and Eigenfunctions of the Laplace Operator via Floer Theory

  • from Part I - Basics: Foundational Material, Elementary Aspects, and Examples
  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 172-178
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  • Summary

    By considering Rayleigh quotients as Morse functions, we obtain eigenvalues as critical values and eigenfunctions as critical points. This allows us to apply Floer's theory of gradient flows to derive homological information from the relations between different eigenfunctions.

  • Preface

  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp ix-xiii
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  • Organization of the Book

  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp xv-xviii
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  • References

  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 389-402
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  • 13 - Applications

  • from Part III - Additional Topics: Interlacing, Tensors, Nonbacktracking Laplacians, and Applications
  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 373-387
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  • Summary

    Laplace operators and their spectral properties are powerful tools for the analysis of networks in the social and the biological sciences and in other domains. In computer science, the theory of families of expander graphs is particularly important. Eigenvalues are also a key for quantifying synchronization and other features of nonlinear dynamics.

  • Index

  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 403-406
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  • Part I - Basics: Foundational Material, Elementary Aspects, and Examples

  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 1-2
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  • 11 - Spectral Theory of Weighted Hypergraphs via Tensors

  • from Part III - Additional Topics: Interlacing, Tensors, Nonbacktracking Laplacians, and Applications
  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 327-350
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  • Summary

    This chapter generalizes spectral theory to hypergraphs via tensor notions.

  • 3 - The Classical Case: Eigenvalues on Euclidean Domains and Riemannian Manifold

  • from Part I - Basics: Foundational Material, Elementary Aspects, and Examples
  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 50-75
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  • Summary

    This chapter provides an overview of the spectral theory of the Laplacian on functions and differential forms. It also introduces the p-Laplacian for functions. We emphasize variational aspects of eigenvalues and Cheeger's inequality.

  • 4 - First Properties of the Spectrum

  • from Part I - Basics: Foundational Material, Elementary Aspects, and Examples
  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 76-171
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  • Summary

    This chapter discusses the elementary properties of Laplace operators on graphs and hypergraphs. Many interesting examples will illustrate how special eigenvalues emerge. We also introduce discrete Pólya–Cheeger constants and their dual versions and provide the initial steps relating spectral clustering, spectra of neighborhood graphs, signed Laplacians, and spectra of simplicial complexes and hypergraphs.

  • Remarks on Notation

  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 388-388
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  • Contents

  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp v-viii
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  • Acknowledgments

  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp xiv-xiv
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  • 7 - Discrete p-Laplacians

  • from Part II - Eigenvalues and Eigenfunctions on Simplicial Complexes and Hypergraphs
  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 205-231
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  • Summary

    This chapter provides an introduction to graph p-Laplacians and their analogues on hypergraphs. It describes them in variational terms and derives many spectral properties of these nonlinear operators.

  • 8 - Cheeger Inequalities

  • from Part II - Eigenvalues and Eigenfunctions on Simplicial Complexes and Hypergraphs
  • Jürgen Jost, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, Raffaella Mulas, Vrije Universiteit Amsterdam, Dong Zhang, Peking University
  • Book:
    Spectra of Discrete Structures
    Published online:
    21 May 2026
    Print publication:
    14 May 2026, pp 232-268
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  • Summary

    This chapter systematically treats Cheeger-type inequalities. Our theory combines higher-order Cheeger inequalities, Cheeger inequalities for signed graphs, and Cheeger inequalities for the 1-Laplacian. We introduce a new definition of a Cheeger constant for simplicial complexes. This leads us to a solution of the higher-order Cheeger problem on simplicial complexes.

  • The Rank-Ramsey problem and the Log-Rank conjecture

  • Part of
  • Gal Beniamini, Nati Linial, Adi Shraibman
  • Journal:
    Combinatorics, Probability and Computing , First View
    Published online by Cambridge University Press:
    13 May 2026, pp. 1-30
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  • A graph is called Rank-Ramsey if (i) Its clique number is small, and (ii) The adjacency matrix of its complement has small rank. We initiate a systematic study of such graphs. Our main motivation is that their constructions, as well as proofs of their non-existence, are intimately related to the famous log-rank conjecture from the field of communication complexity. These investigations also open interesting new avenues in Ramsey theory. We construct two families of Rank-Ramsey graphs exhibiting polynomial separation between order and complement rank. Graphs in the first family have bounded clique number (as low as $41$). These are subgraphs of certain strong products, whose building blocks are derived from triangle-free strongly-regular graphs. Graphs in the second family are obtained by applying Boolean functions to Erdős-Rényi graphs. Their clique number is logarithmic, but their complement rank is far smaller than in the first family, about $\mathcal{O}(n^{2/3})$. A key component of this construction is our matrix-theoretic view of lifts. We also consider lower bounds on the Rank-Ramsey numbers, and determine them in the range where the complement rank is $5$ or less. We consider connections between said numbers and other graph parameters, and find that the two best known explicit constructions of triangle-free Ramsey graphs turn out to be far from Rank-Ramsey.

Page 4 of 1301