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Convolution-like structures, differential operators and diffusion processes / / Rúben Sousa, Manuel Guerra, Semyon Yakubovich
| Convolution-like structures, differential operators and diffusion processes / / Rúben Sousa, Manuel Guerra, Semyon Yakubovich |
| Autore | Sousa Rúben (Mathematician) |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (269 pages) |
| Disciplina | 512.86 |
| Collana | Lecture notes in mathematics |
| Soggetto topico |
Convolutions (Mathematics)
Differential operators Diffusion processes Convolucions (Matemàtica) Operadors diferencials Processos de difusió |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-031-05296-X |
| Formato | Materiale a stampa  |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Contents -- List of Symbols -- 1 Introduction -- 1.1 Motivation and Scope -- 1.2 Organization of the Book -- 2 Preliminaries -- 2.1 Continuous-Time Markov Processes -- 2.2 Sturm-Liouville Theory -- 2.2.1 Solutions of the Sturm-Liouville Equation -- 2.2.2 Eigenfunction Expansions -- 2.2.3 Diffusion Semigroups Generated by Sturm-Liouville Operators -- 2.2.4 Remarkable Particular Cases -- 2.3 Generalized Convolutions and Hypergroups -- 2.4 Harmonic Analysis with Respect to the Kingman Convolution -- 3 The Whittaker Convolution -- 3.1 A Special Case: The Kontorovich-Lebedev Convolution -- 3.2 The Product Formula for the Whittaker Function -- 3.3 Whittaker Translation -- 3.4 Index Whittaker Transforms -- 3.5 Whittaker Convolution of Measures -- 3.5.1 Infinitely Divisible Distributions -- 3.5.2 Lévy-Khintchine Type Representation -- 3.6 Lévy Processes with Respect to the Whittaker Convolution -- 3.6.1 Convolution Semigroups -- 3.6.2 Lévy and Gaussian Processes -- 3.6.3 Some Auxiliary Results on the Whittaker Translation -- 3.6.4 Moment Functions -- 3.6.5 Lévy-Type Characterization of the Shiryaev Process -- 3.7 Whittaker Convolution of Functions -- 3.7.1 Mapping Properties in the Spaces Lp(rα) -- 3.7.2 The Convolution Banach Algebra Lα,ν -- 3.8 Convolution-Type Integral Equations -- 4 Generalized Convolutions for Sturm-Liouville Operators -- 4.1 Known Results and Motivation -- 4.2 Laplace-Type Representation -- 4.3 The Existence Theorem for Sturm-Liouville Product Formulas -- 4.3.1 The Associated Hyperbolic Cauchy Problem -- 4.3.2 The Time-Shifted Product Formula -- 4.3.3 The Product Formula for wλ as the Limit Case -- 4.4 Sturm-Liouville Transform of Measures -- 4.5 Sturm-Liouville Convolution of Measures -- 4.5.1 Infinite Divisibility and Lévy-Khintchine Type Representation -- 4.5.2 Convolution Semigroups.
4.5.3 Additive and Lévy Processes -- 4.6 Sturm-Liouville Hypergroups -- 4.6.1 The Nondegenerate Case -- 4.6.2 The Degenerate Case: Degenerate Hypergroups of Full Support -- 4.7 Harmonic Analysis on Lp Spaces -- 4.7.1 A Family of L1 Spaces -- 4.7.2 Application to Convolution-Type Integral Equations -- 5 Convolution-Like Structures on Multidimensional Spaces -- 5.1 Convolutions Associated with Conservative Strong Feller Semigroups -- 5.2 Nonexistence of Convolutions: Diffusion Processes on Bounded Domains -- 5.2.1 Special Cases and Numerical Examples -- 5.2.2 Some Auxiliary Results -- 5.2.3 Eigenfunction Expansions, Critical Points and Nonexistence Theorems -- 5.3 Nonexistence of Convolutions: One-Dimensional Diffusions -- 5.4 Families of Convolutions on Riemannian Structures with Cone-Like Metrics -- 5.4.1 The Eigenfunction Expansion of the Laplace-Beltrami Operator -- 5.4.2 Product Formulas and Convolutions -- 5.4.3 Infinitely Divisible Measures and Convolution Semigroups -- 5.4.4 Special Cases -- 5.4.5 Product Formulas and Convolutions Associated with Elliptic Operators on Subsets of R2 -- A Some Open Problems -- References -- Index. |
| Record Nr. | UNISA-996483172603316 |
Sousa Rúben (Mathematician)
|
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| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. di Salerno | ||
| ||
Convolution-like structures, differential operators and diffusion processes / / Rúben Sousa, Manuel Guerra, Semyon Yakubovich
| Convolution-like structures, differential operators and diffusion processes / / Rúben Sousa, Manuel Guerra, Semyon Yakubovich |
| Autore | Sousa Rúben (Mathematician) |
| Pubbl/distr/stampa | Cham, Switzerland : , : Springer, , [2022] |
| Descrizione fisica | 1 online resource (269 pages) |
| Disciplina | 512.86 |
| Collana | Lecture notes in mathematics |
| Soggetto topico |
Convolutions (Mathematics)
Differential operators Diffusion processes Convolucions (Matemàtica) Operadors diferencials Processos de difusió |
| Soggetto genere / forma | Llibres electrònics |
| ISBN | 3-031-05296-X |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Preface -- Contents -- List of Symbols -- 1 Introduction -- 1.1 Motivation and Scope -- 1.2 Organization of the Book -- 2 Preliminaries -- 2.1 Continuous-Time Markov Processes -- 2.2 Sturm-Liouville Theory -- 2.2.1 Solutions of the Sturm-Liouville Equation -- 2.2.2 Eigenfunction Expansions -- 2.2.3 Diffusion Semigroups Generated by Sturm-Liouville Operators -- 2.2.4 Remarkable Particular Cases -- 2.3 Generalized Convolutions and Hypergroups -- 2.4 Harmonic Analysis with Respect to the Kingman Convolution -- 3 The Whittaker Convolution -- 3.1 A Special Case: The Kontorovich-Lebedev Convolution -- 3.2 The Product Formula for the Whittaker Function -- 3.3 Whittaker Translation -- 3.4 Index Whittaker Transforms -- 3.5 Whittaker Convolution of Measures -- 3.5.1 Infinitely Divisible Distributions -- 3.5.2 Lévy-Khintchine Type Representation -- 3.6 Lévy Processes with Respect to the Whittaker Convolution -- 3.6.1 Convolution Semigroups -- 3.6.2 Lévy and Gaussian Processes -- 3.6.3 Some Auxiliary Results on the Whittaker Translation -- 3.6.4 Moment Functions -- 3.6.5 Lévy-Type Characterization of the Shiryaev Process -- 3.7 Whittaker Convolution of Functions -- 3.7.1 Mapping Properties in the Spaces Lp(rα) -- 3.7.2 The Convolution Banach Algebra Lα,ν -- 3.8 Convolution-Type Integral Equations -- 4 Generalized Convolutions for Sturm-Liouville Operators -- 4.1 Known Results and Motivation -- 4.2 Laplace-Type Representation -- 4.3 The Existence Theorem for Sturm-Liouville Product Formulas -- 4.3.1 The Associated Hyperbolic Cauchy Problem -- 4.3.2 The Time-Shifted Product Formula -- 4.3.3 The Product Formula for wλ as the Limit Case -- 4.4 Sturm-Liouville Transform of Measures -- 4.5 Sturm-Liouville Convolution of Measures -- 4.5.1 Infinite Divisibility and Lévy-Khintchine Type Representation -- 4.5.2 Convolution Semigroups.
4.5.3 Additive and Lévy Processes -- 4.6 Sturm-Liouville Hypergroups -- 4.6.1 The Nondegenerate Case -- 4.6.2 The Degenerate Case: Degenerate Hypergroups of Full Support -- 4.7 Harmonic Analysis on Lp Spaces -- 4.7.1 A Family of L1 Spaces -- 4.7.2 Application to Convolution-Type Integral Equations -- 5 Convolution-Like Structures on Multidimensional Spaces -- 5.1 Convolutions Associated with Conservative Strong Feller Semigroups -- 5.2 Nonexistence of Convolutions: Diffusion Processes on Bounded Domains -- 5.2.1 Special Cases and Numerical Examples -- 5.2.2 Some Auxiliary Results -- 5.2.3 Eigenfunction Expansions, Critical Points and Nonexistence Theorems -- 5.3 Nonexistence of Convolutions: One-Dimensional Diffusions -- 5.4 Families of Convolutions on Riemannian Structures with Cone-Like Metrics -- 5.4.1 The Eigenfunction Expansion of the Laplace-Beltrami Operator -- 5.4.2 Product Formulas and Convolutions -- 5.4.3 Infinitely Divisible Measures and Convolution Semigroups -- 5.4.4 Special Cases -- 5.4.5 Product Formulas and Convolutions Associated with Elliptic Operators on Subsets of R2 -- A Some Open Problems -- References -- Index. |
| Record Nr. | UNINA-9910585774303321 |
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Sousa Rúben (Mathematician)
|
||
| Cham, Switzerland : , : Springer, , [2022] | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
| ||
Spectral Geometry of Graphs / / by Pavel Kurasov
| Spectral Geometry of Graphs / / by Pavel Kurasov |
| Autore | Kurasov Pavel |
| Edizione | [1st ed. 2024.] |
| Pubbl/distr/stampa | Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Birkhäuser, , 2024 |
| Descrizione fisica | 1 online resource (0 pages) |
| Disciplina |
006.3843
530.12 |
| Collana | Operator Theory: Advances and Applications |
| Soggetto topico |
Quantum computers
Mathematical analysis System theory Control theory Mathematical optimization Calculus of variations Quantum Computing Analysis Systems Theory, Control Calculus of Variations and Optimization Teoria espectral (Matemàtica) Operadors diferencials Mètodes gràfics |
| Soggetto genere / forma | Llibres electrònics |
| ISBN |
9783662678725
3662678721 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto |
Intro -- Notations -- Conventions -- Contents -- 1 Very Personal Introduction -- 2 How to Define Differential Operators on Metric Graphs -- 2.1 Schrödinger Operators on Metric Graphs -- 2.1.1 Metric Graphs -- 2.1.2 Differential Operators -- 2.1.3 Standard Vertex Conditions -- 2.1.4 Definition of the Operator -- 2.2 Elementary Examples -- 3 Vertex Conditions -- 3.1 Preliminary Discussion -- 3.2 Vertex Conditions for the Star Graph -- 3.3 Vertex Conditions Via the Vertex Scattering Matrix -- 3.3.1 The Vertex Scattering Matrix -- 3.3.2 Scattering Matrix as a Parameterin the Vertex Conditions -- 3.3.3 On Properly Connecting Vertex Conditions -- 3.4 Parametrisation Via Hermitian Matrices -- 3.5 Scaling-Invariant and Standard Conditions -- 3.5.1 Energy Dependence of the Vertex S-matrix -- 3.5.2 Scaling-Invariant, or Non-Robin Vertex Conditions -- 3.5.3 Standard Vertex Conditions -- 3.6 Signing Conditions for Degree Two Vertices -- 3.7 Generalised Delta Couplings -- 3.8 Vertex Conditions for Arbitrary Graphs and Definition of the Magnetic Schrödinger Operator -- 3.8.1 Scattering Matrix Parametrisationof Vertex Conditions -- 3.8.2 Quadratic Form Parametrisation of Vertex Conditions -- Appendix 1: Important Classes of Vertex Conditions -- δ and δ'-Couplings -- Circulant Conditions -- `Real' Conditions -- Indistinguishable Edges -- Equi-transmitting Vertices -- Appendix 2: Parametrisation of Vertex Conditions: Historical Remarks -- Parametrisation Via Linear Relations -- Parametrisation Using Hermitian Operators -- Unitary Matrix Parametrisation -- 4 Elementary Spectral Properties of Quantum Graphs -- 4.1 Quantum Graphs as Self-adjoint Operators -- 4.2 The Dirichlet Operator and the Weyl's Law -- 4.3 Spectra of Quantum Graphs -- 4.4 Laplacian Ground State -- 4.5 Bonus Section: Positivity of the Ground Statefor Quantum Graphs.
4.5.1 The Case of Standard Vertex Conditions -- 4.5.2 A Counterexample -- 4.5.3 Invariance of the Quadratic Form -- 4.5.4 Positivity of the Ground State for Generalised Delta-Couplings -- 4.6 First Spectral Estimates -- 5 The Characteristic Equation -- 5.1 Characteristic Equation I: Edge Transfer Matrices -- 5.1.1 Transfer Matrix for a Single Interval -- One-Dimensional Schrödinger Equation -- Magnetic Schrödinger Equation -- 5.1.2 The Characteristic Equation -- 5.1.3 The Characteristic Equation, Second Look -- 5.2 Characteristic Equation II: Scattering Approach -- 5.2.1 On the Scattering Matrix Associated with a Compact Interval -- 5.2.2 Positive Spectrum and Scattering Matrices for Finite Compact Graphs -- 5.3 Characteristic Equation III: M-Function Approach -- 5.3.1 M-Function for a Single Interval -- 5.3.2 The Edge M-Function -- 5.3.3 Characteristic Equation via the M-Function: General Vertex Conditions -- 5.3.4 Reduction of the M-Function for Standard Vertex Conditions -- 6 Standard Laplacians and Secular Polynomials -- 6.1 Secular Polynomials -- 6.2 Secular Polynomials for Small Graphs -- 6.3 Zero Sets for Small Graphs -- Appendix 1: Singular Sets on Secular Manifolds, Proof of Lemma 6.3 -- 7 Reducibility of Secular Polynomials -- 7.1 Contraction of Graphs -- 7.2 Extensions of Graphs -- 7.3 Secular Polynomials for the Watermelon Graphand Its Closest Relatives -- 7.4 Secular Polynomials for Flower Graphs and Their Extensions -- 7.5 Reducibility of Secular Polynomials for General Graphs -- 8 The Trace Formula -- 8.1 The Characteristic Equation: Multiplicityof Positive Eigenvalues -- 8.2 Algebraic and Spectral Multiplicities of the Eigenvalue Zero -- 8.3 The Trace Formula for Standard Laplacians -- 8.4 Trace Formula for Laplacians with Scaling-InvariantVertex Conditions -- 9 Trace Formula and Inverse Problems. 9.1 Euler Characteristic for Standard Laplacians -- 9.2 Euler Characteristic for Graphs with Dirichlet Vertices -- 9.3 Spectral Asymptotics and Schrödinger Operators -- 9.3.1 Euler Characteristic and Spectral Asymptotics -- 9.3.2 Schrödinger Operators and Euler Characteristic of Graphs -- 9.3.3 General Vertex Conditions: A Counterexample -- 9.4 Reconstruction of Graphs with RationallyIndependent Lengths -- 10 Arithmetic Structure of the Spectrumand Crystalline Measures -- 10.1 Arithmetic Structure of the Spectrum -- 10.2 Crystalline Measures -- 10.3 The Lasso Graph and Crystalline Measures -- 10.4 Graph's Spectrum as a Delone Set -- 11 Quadratic Forms and Spectral Estimates -- 11.1 Quadratic Forms (Integrable Potentials) -- 11.1.1 Explicit Expression -- 11.1.2 An Elementary Sobolev Estimate -- 11.1.3 The Perturbation Term Is Form-Bounded -- 11.1.4 The Reference Laplacian -- 11.1.5 Closure of the Perturbed Quadratic Form -- 11.2 Spectral Estimates (Standard Vertex Conditions) -- 11.3 Spectral Estimates for General Vertex Conditions -- 12 Spectral Gap and Dirichlet Ground State -- 12.1 Fundamental Estimates -- 12.1.1 Eulerian Path Technique -- 12.1.2 Symmetrisation Technique -- 12.2 Balanced and Doubly Connected Graphs -- 12.3 Graphs with Dirichlet Vertices -- 12.4 Cheeger's Approach -- 12.5 Topological Perturbations in the Case of Standard Conditions -- 12.5.1 Gluing Vertices Together -- 12.5.2 Adding an Edge -- 12.6 Bonus Section: Further Topological Perturbations -- 12.6.1 Cutting Edges -- 12.6.2 Deleting Edges -- 13 Higher Eigenvalues and Topological Perturbations -- 13.1 Fundamental Estimates for Higher Eigenvalues -- 13.1.1 Lower Estimates -- 13.1.2 Upper Bounds -- 13.1.3 Graphs Realising Extremal Eigenvalues -- 13.2 Gluing and Cutting Vertices with Standard Conditions -- 13.3 Gluing Vertices with Scaling-Invariant Conditions. 13.3.1 Scaling-Invariant Conditions Revisited -- 13.3.2 Gluing Vertices -- Gluing Vertices with One-Dimensional Vertex Conditions -- Gluing Vertices with Hyperplanar Vertex Conditions -- 13.3.3 Spectral Gap and Gluing Vertices with Scaling-Invariant Conditions -- 13.4 Gluing Vertices with General Vertex Conditions -- 14 Ambartsumian Type Theorems -- 14.1 Two Parameters Fixed, One Parameter Varies -- 14.1.1 Zero Potential Is Exceptional: Classical Ambartsumian Theorem -- 14.1.2 Interval-Graph Is Exceptional: Geometric Version of Ambartsumian Theorem for Standard Laplacians -- 14.1.3 Standard Vertex Conditions Are Not Exceptional -- 14.2 One Parameter Is Fixed, Two Parameters Vary -- 14.2.1 Standard Vertex Conditions Are Exceptional: Schrödinger Operators on Arbitrary Graphs -- 14.2.2 Zero Potential: Laplacians on Graphs that Are Isospectral to the Interval -- 14.2.3 Single Interval: Schrödinger Operators Isospectral to the Standard Laplacian -- Crum's Procedure -- Inverting Crum's Procedure -- 15 Further Theorems Inspired by Ambartsumian -- 15.1 Ambartsumian-Type Theorem by Davies -- 15.1.1 On a Sufficient Condition for the Potential to Be Zero -- 15.1.2 Laplacian Heat Kernel -- Heat Kernel for the Dirichlet Laplacian on an Interval -- Heat Kernel for the Standard Laplacian on the Graph -- 15.1.3 On Schrödinger Semigroups -- 15.1.4 A Theorem by Davies -- 15.2 On Asymptotically Isospectral Quantum Graphs -- 15.2.1 On the Zeroes of Generalised TrigonometricPolynomials -- 15.2.2 Asymptotically Isospectral Quantum Graphs -- 15.2.3 When a Schrödinger Operator Is Isospectral to a Laplacian -- 16 Magnetic Fluxes -- 16.1 Unitary Transformations via Multiplications and Magnetic Schrödinger Operators -- 16.2 Vertex Phases and Transition Probabilities -- 16.3 Topological Damping of Aharonov-Bohm Effect -- 16.3.1 Getting Started. 16.3.2 Explicit Calculation of the Spectrum -- 16.3.3 Topological Reasons for Damping -- 17 M-Functions: Definitions and Examples -- 17.1 The Graph M-Function -- 17.1.1 Motivation and Historical Hints -- 17.1.2 The Formal Definition -- 17.1.3 Examples -- 17.2 Explicit Formulas Using Eigenfunctions -- 17.3 Hierarchy of M-Functions for Standard Vertex Conditions -- 18 M-Functions: Properties and First Applications -- 18.1 M-Function as a Matrix-Valued Herglotz-Nevanlinna Function -- 18.2 Gluing Procedure and the Spectral Gap -- 18.2.1 Examples -- 18.3 Gluing Graphs and M-Functions -- 18.3.1 The M-Function for General Vertex Conditions at the Contact Set -- 18.3.2 Gluing Graphs with General Vertex Conditions -- Appendix 1: Scattering from Compact Graphs -- 19 Boundary Control: BC-Method -- 19.1 Inverse Problems: First Look -- 19.2 How to Use BC-Method for Graphs -- 19.3 The Response Operator and the M-Function -- 19.4 Inverse Problem for the One-DimensionalSchrödinger Equation -- 19.5 BC-Method for the Standard Laplacian on the Star Graph -- 19.6 BC-Method for the Star Graph with General Vertex Conditions -- 20 Inverse Problems for Trees -- 20.1 Obvious Ambiguities and Limitations -- 20.2 Subproblem I: Reconstruction of the Metric Tree -- 20.2.1 Global Reconstruction of the Metric Tree -- 20.2.2 Local Reconstruction of the Metric Tree -- 20.3 Subproblem II: Reconstruction of the Potential -- 20.4 Subproblem III: Reconstruction of the Vertex Conditions -- 20.4.1 Trimming a Bunch -- 20.4.2 Recovering the Vertex Conditions for an Equilateral Bunch -- 20.5 Cleaning and Pruning Using the M-functions -- 20.5.1 Cleaning the Edges -- 20.5.2 Pruning Branches and Bunches -- 20.6 Complete Solution of the Inverse Problem for Trees -- Appendix 1: Calculation of the M-function for the Cross Graph -- Appendix 2: Calderón Problem. 21 Boundary Control for Graphs with Cycles: Dismantling Graphs. |
| Record Nr. | UNINA-9910758501003321 |
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Kurasov Pavel
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| Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Birkhäuser, , 2024 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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Unified Theory for Fractional and Entire Differential Operators : An Approach via Differential Quadruplets and Boundary Restriction Operators / / by Arnaud Rougirel
| Unified Theory for Fractional and Entire Differential Operators : An Approach via Differential Quadruplets and Boundary Restriction Operators / / by Arnaud Rougirel |
| Autore | Rougirel Arnaud |
| Edizione | [1st ed. 2024.] |
| Pubbl/distr/stampa | Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2024 |
| Descrizione fisica | 1 online resource (502 pages) |
| Disciplina | 515.7 |
| Collana | Frontiers in Elliptic and Parabolic Problems |
| Soggetto topico |
Functional analysis
Operator theory Differential equations Functional Analysis Operator Theory Differential Equations Fractional calculus Operadors diferencials |
| Soggetto genere / forma | Llibres electrònics |
| ISBN |
9783031583568
9783031583551 |
| Formato | Materiale a stampa |
| Livello bibliografico | Monografia |
| Lingua di pubblicazione | eng |
| Nota di contenuto | Introduction -- Background on Functional Analysis -- Background on Fractional Calculus -- Differential Triplets on Hilbert Spaces -- Differential Quadruplets on Banach Spaces -- Fractional Differential Triplets and Quadruplets on Lebesgue Spaces -- Endogenous Boundary Value Problems -- Abstract and Fractional Laplace Operators. |
| Record Nr. | UNINA-9910869173003321 |
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Rougirel Arnaud
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| Cham : , : Springer International Publishing : , : Imprint : Birkhäuser, , 2024 | ||
| Materiale a stampa | ||
| Lo trovi qui: Univ. Federico II | ||
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