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Functional Analysis, Sobolev Spaces and Partial Differential Equations
TLDR
In this article, the theory of conjugate convex functions is introduced, and the Hahn-Banach Theorem and the closed graph theorem are discussed, as well as the variations of boundary value problems in one dimension.Abstract:
Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.read more
Citations
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Fundamentals of Linear Algebra and Optimization
TL;DR: In this article, the affine hyperplane H0 parallel to H is given by the equation a1x1 + · · ·+ anxn = 0, with ai 6= 0 for some i, 1 ≤ i ≤ n.
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Passage times, exit times and Dirichlet problems for open quantum walks
TL;DR: In this article, the authors considered the probability that the passage time is finite, the expectation of that passage time and the number of visits, and discussed the notion of recurrence for open quantum walks.
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Well-posedness for a class of doubly nonlinear stochastic PDEs of divergence type
TL;DR: In this article, the authors prove well-posedness for doubly nonlinear parabolic stochastic partial differential equations of the form d X t − div γ ( ∇ X t ) d t + β ( X t ), d t ∋ B ( t, X t, d W t ), where W is a cylindrical Wiener process.
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Hamiltonian operator for spectral shape analysis
TL;DR: In this paper, the authors propose to adapt the classical Hamiltonian operator from quantum mechanics to the field of shape analysis, which is shown to produce better functional spaces to operate with, as demonstrated on different shape analysis tasks.
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Optimal control and pattern formation for a haptotaxis model of solid tumor invasion
TL;DR: The ringlike diffusion and aggregation patterns and the dynamics of tumor invasion as well as the optimal control strategies are presented numerically, which demonstrate that the optimal treatment strategies are capable of breaking the pattern formation, and preventing the tumor invading and metastasizing, even eliminating the tumor possibly.
References
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Book
Linear and Quasilinear Equations of Parabolic Type
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
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Non-homogeneous boundary value problems and applications
TL;DR: In this paper, the authors consider the problem of finding solutions to elliptic boundary value problems in Spaces of Analytic Functions and of Class Mk Generalizations in the case of distributions and Ultra-Distributions.
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Introduction to Fourier Analysis on Euclidean Spaces.
Elias M. Stein,Guido Weiss +1 more
TL;DR: In this paper, the authors present a unified treatment of basic topics that arise in Fourier analysis, and illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations.
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Theory of function spaces
TL;DR: In this article, the authors measure smoothness using Atoms and Pointwise Multipliers, Wavelets, Spaces on Lipschitz Domains, Wavelet and Sampling Numbers.