L P Square Function Estimates On Spaces Of Homogeneous Type And On Uniformly Rectifiable Sets

Author : Steve Hofmann,Dorina Mitrea,Marius Mitrea
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 52,6 Mb
Release : 2017-01-18
Category : Function spaces
ISBN : 9781470422608

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$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets by Steve Hofmann,Dorina Mitrea,Marius Mitrea Pdf

The authors establish square function estimates for integral operators on uniformly rectifiable sets by proving a local theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, they consider integral operators associated with Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The local theorem is then used to establish an inductive scheme in which square function estimates on so-called big pieces of an Ahlfors-David regular set are proved to be sufficient for square function estimates to hold on the entire set. Extrapolation results for and Hardy space versions of these estimates are also established. Moreover, the authors prove square function estimates for integral operators associated with variable coefficient kernels, including the Schwartz kernels of pseudodifferential operators acting between vector bundles on subdomains with uniformly rectifiable boundaries on manifolds.

Author : Steve Hofmann,Dorina Mitrea,Marius Mitrea,Andrew Jordan Morris
Publisher : Unknown
Page : 108 pages
File Size : 51,7 Mb
Release : 2017
Category : Function spaces
ISBN : 1470436078

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Lp-square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets by Steve Hofmann,Dorina Mitrea,Marius Mitrea,Andrew Jordan Morris Pdf

Author : Denis R. Hirschfeldt,Karen Lange,Richard A. Shore
Publisher : American Mathematical Soc.
Page : 101 pages
File Size : 48,7 Mb
Release : 2017-09-25
Category : Computable functions
ISBN : 9781470426576

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Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem by Denis R. Hirschfeldt,Karen Lange,Richard A. Shore Pdf

Goncharov and Peretyat'kin independently gave necessary and sufficient conditions for when a set of types of a complete theory is the type spectrum of some homogeneous model of . Their result can be stated as a principle of second order arithmetic, which is called the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. The authors show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense and do the same for an analogous result of Peretyat'kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model.

Author : H. Hofer,K. Wysocki,E. Zehnder
Publisher : American Mathematical Soc.
Page : 218 pages
File Size : 41,5 Mb
Release : 2017-07-13
Category : Gromov-Witten invariants
ISBN : 9781470422035

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Applications of Polyfold Theory I: The Polyfolds of Gromov-Witten Theory by H. Hofer,K. Wysocki,E. Zehnder Pdf

In this paper the authors start with the construction of the symplectic field theory (SFT). As a general theory of symplectic invariants, SFT has been outlined in Introduction to symplectic field theory (2000), by Y. Eliashberg, A. Givental and H. Hofer who have predicted its formal properties. The actual construction of SFT is a hard analytical problem which will be overcome be means of the polyfold theory due to the present authors. The current paper addresses a significant amount of the arising issues and the general theory will be completed in part II of this paper. To illustrate the polyfold theory the authors use the results of the present paper to describe an alternative construction of the Gromov-Witten invariants for general compact symplectic manifolds.

Author : Akinari Hoshi,Aiichi Yamasaki
Publisher : American Mathematical Soc.
Page : 215 pages
File Size : 55,9 Mb
Release : 2017-07-13
Category : Affine algebraic groups
ISBN : 9781470424091

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Rationality Problem for Algebraic Tori by Akinari Hoshi,Aiichi Yamasaki Pdf

The authors give the complete stably rational classification of algebraic tori of dimensions and over a field . In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank and is given. The authors show that there exist exactly (resp. , resp. ) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension , and there exist exactly (resp. , resp. ) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension . The authors make a procedure to compute a flabby resolution of a -lattice effectively by using the computer algebra system GAP. Some algorithms may determine whether the flabby class of a -lattice is invertible (resp. zero) or not. Using the algorithms, the suthors determine all the flabby and coflabby -lattices of rank up to and verify that they are stably permutation. The authors also show that the Krull-Schmidt theorem for -lattices holds when the rank , and fails when the rank is ...

Author : Igor Burban,Yuriy Drozd
Publisher : American Mathematical Soc.
Page : 114 pages
File Size : 53,8 Mb
Release : 2017-07-13
Category : Cohen-Macaulay modules
ISBN : 9781470425371

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Maximal Cohen-Macaulay Modules Over Non-Isolated Surface Singularities and Matrix Problems by Igor Burban,Yuriy Drozd Pdf

In this article the authors develop a new method to deal with maximal Cohen–Macaulay modules over non–isolated surface singularities. In particular, they give a negative answer on an old question of Schreyer about surface singularities with only countably many indecomposable maximal Cohen–Macaulay modules. Next, the authors prove that the degenerate cusp singularities have tame Cohen–Macaulay representation type. The authors' approach is illustrated on the case of k as well as several other rings. This study of maximal Cohen–Macaulay modules over non–isolated singularities leads to a new class of problems of linear algebra, which the authors call representations of decorated bunches of chains. They prove that these matrix problems have tame representation type and describe the underlying canonical forms.

Author : Akihito Ebisu
Publisher : American Mathematical Soc.
Page : 96 pages
File Size : 40,5 Mb
Release : 2017-07-13
Category : Cohen-Macaulay modules
ISBN : 9781470425333

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Special Values of the Hypergeometric Series by Akihito Ebisu Pdf

In this paper, the author presents a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, the author gets identities for the hypergeometric series and shows that values of at some points can be expressed in terms of gamma functions, together with certain elementary functions. The author tabulates the values of that can be obtained with this method and finds that this set includes almost all previously known values and many previously unknown values.

Author : Dorina Mitrea,Irina Mitrea,Marius Mitrea
Publisher : Springer Nature
Page : 980 pages
File Size : 44,5 Mb
Release : 2023-05-12
Category : Mathematics
ISBN : 9783031227356

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Geometric Harmonic Analysis III by Dorina Mitrea,Irina Mitrea,Marius Mitrea Pdf

This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. Volume III is concerned with integral representation formulas for nullsolutions of elliptic PDEs, Calderón-Zygmund theory for singular integral operators, Fatou type theorems for systems of elliptic PDEs, and applications to acoustic and electromagnetic scattering. Overall, this amounts to a powerful and nuanced theory developed on uniformly rectifiable sets, which builds on the work of many predecessors.

Author : Irina Mitrea,Marius Mitrea
Publisher : Springer
Page : 430 pages
File Size : 45,5 Mb
Release : 2013-01-05
Category : Mathematics
ISBN : 9783642326660

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Multi-Layer Potentials and Boundary Problems by Irina Mitrea,Marius Mitrea Pdf

Many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator in a given domain. When the differential operator under discussion is of second order a variety of tools are available for dealing with such problems, including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. When the differential operator is of higher-order (as is the case, e.g., with anisotropic plate bending when one deals with a fourth order operator) only a few options could be successfully implemented. In the 1970s Alberto Calderón, one of the founders of the modern theory of Singular Integral Operators, advocated the use of layer potentials for the treatment of higher-order elliptic boundary value problems. The present monograph represents the first systematic treatment based on this approach. This research monograph lays, for the first time, the mathematical foundation aimed at solving boundary value problems for higher-order elliptic operators in non-smooth domains using the layer potential method and addresses a comprehensive range of topics, dealing with elliptic boundary value problems in non-smooth domains including layer potentials, jump relations, non-tangential maximal function estimates, multi-traces and extensions, boundary value problems with data in Whitney–Lebesque spaces, Whitney–Besov spaces, Whitney–Sobolev- based Lebesgue spaces, Whitney–Triebel–Lizorkin spaces,Whitney–Sobolev-based Hardy spaces, Whitney–BMO and Whitney–VMO spaces.

Author : Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor
Publisher : Walter de Gruyter GmbH & Co KG
Page : 528 pages
File Size : 49,7 Mb
Release : 2016-10-10
Category : Mathematics
ISBN : 9783110484380

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The Hodge-Laplacian by Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor Pdf

The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals. Contents: Preface Introduction and Statement of Main Results Geometric Concepts and Tools Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism Additional Results and Applications Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis Bibliography Index

Author : Ben Webster
Publisher : American Mathematical Soc.
Page : 141 pages
File Size : 40,9 Mb
Release : 2018-01-16
Category : Functor theory
ISBN : 9781470426507

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Knot Invariants and Higher Representation Theory by Ben Webster Pdf

The author constructs knot invariants categorifying the quantum knot variants for all representations of quantum groups. He shows that these invariants coincide with previous invariants defined by Khovanov for sl and sl and by Mazorchuk-Stroppel and Sussan for sl . The author's technique is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the KLR algebra. When the Lie algebra under consideration is sl , the author shows that these categories agree with certain subcategories of parabolic category for gl .

Author : Mikhail Ershov,Andrei Jaikin-Zapirain,Martin Kassabov
Publisher : American Mathematical Soc.
Page : 135 pages
File Size : 55,6 Mb
Release : 2017-09-25
Category : Root systems (Algebra)
ISBN : 9781470426040

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Property ($T$) for Groups Graded by Root Systems by Mikhail Ershov,Andrei Jaikin-Zapirain,Martin Kassabov Pdf

The authors introduce and study the class of groups graded by root systems. They prove that if is an irreducible classical root system of rank and is a group graded by , then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of . As the main application of this theorem the authors prove that for any reduced irreducible classical root system of rank and a finitely generated commutative ring with , the Steinberg group and the elementary Chevalley group have property . They also show that there exists a group with property which maps onto all finite simple groups of Lie type and rank , thereby providing a “unified” proof of expansion in these groups.

Author : A. M. Mason,N. C. Snaith
Publisher : American Mathematical Soc.
Page : 93 pages
File Size : 47,7 Mb
Release : 2018-02-23
Category : L-functions
ISBN : 9781470426859

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Orthogonal and Symplectic -level Densities by A. M. Mason,N. C. Snaith Pdf

In this paper the authors apply to the zeros of families of -functions with orthogonal or symplectic symmetry the method that Conrey and Snaith (Correlations of eigenvalues and Riemann zeros, 2008) used to calculate the -correlation of the zeros of the Riemann zeta function. This method uses the Ratios Conjectures (Conrey, Farmer, and Zimbauer, 2008) for averages of ratios of zeta or -functions. Katz and Sarnak (Zeroes of zeta functions and symmetry, 1999) conjecture that the zero statistics of families of -functions have an underlying symmetry relating to one of the classical compact groups , and . Here the authors complete the work already done with (Conrey and Snaith, Correlations of eigenvalues and Riemann zeros, 2008) to show how new methods for calculating the -level densities of eigenangles of random orthogonal or symplectic matrices can be used to create explicit conjectures for the -level densities of zeros of -functions with orthogonal or symplectic symmetry, including all the lower order terms. They show how the method used here results in formulae that are easily modified when the test function used has a restricted range of support, and this will facilitate comparison with rigorous number theoretic -level density results.

Author : Marco Bramanti,Luca Brandolini,Maria Manfredini,Marco Pedroni
Publisher : American Mathematical Soc.
Page : 79 pages
File Size : 45,8 Mb
Release : 2017-09-25
Category : Differential operators
ISBN : 9781470425593

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Fundamental Solutions and Local Solvability for Nonsmooth Hörmander’s Operators by Marco Bramanti,Luca Brandolini,Maria Manfredini,Marco Pedroni Pdf

The authors consider operators of the form in a bounded domain of where are nonsmooth Hörmander's vector fields of step such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution for and provide growth estimates for and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that also possesses second derivatives, and they deduce the local solvability of , constructing, by means of , a solution to with Hölder continuous . The authors also prove estimates on this solution.

Author : Nicola Gambino,André Joyal
Publisher : American Mathematical Soc.
Page : 110 pages
File Size : 50,8 Mb
Release : 2017-09-25
Category : Algebra, Homological
ISBN : 9781470425760

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On Operads, Bimodules and Analytic Functor by Nicola Gambino,André Joyal Pdf

The authors develop further the theory of operads and analytic functors. In particular, they introduce the bicategory of operad bimodules, that has operads as -cells, operad bimodules as -cells and operad bimodule maps as 2-cells, and prove that it is cartesian closed. In order to obtain this result, the authors extend the theory of distributors and the formal theory of monads.