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"Processes, Infinite"
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The Cauchy-Schwarz Master Class
This lively, problem-oriented text, first published in 2004, is designed to coach readers toward mastery of the most fundamental mathematical inequalities. With the Cauchy-Schwarz inequality as the initial guide, the reader is led through a sequence of fascinating problems whose solutions are presented as they might have been discovered - either by one of history's famous mathematicians or by the reader. The problems emphasize beauty and surprise, but along the way readers will find systematic coverage of the geometry of squares, convexity, the ladder of power means, majorization, Schur convexity, exponential sums, and the inequalities of Hölder, Hilbert, and Hardy. The text is accessible to anyone who knows calculus and who cares about solving problems. It is well suited to self-study, directed study, or as a supplement to courses in analysis, probability, and combinatorics.
eBook
Strichartz Estimates for Wave Equations with Charge Transfer Hamiltonians
We prove Strichartz estimates (both regular and reversed) for a scattering state to the wave equation with a charge transfer
Hamiltonian in
eBook
Strichartz Estimates and the Cauchy Problem for the Gravity Water Waves Equations
This memoir is devoted to the proof of a well-posedness result for the gravity water waves equations, in arbitrary dimension and in
fluid domains with general bottoms, when the initial velocity field is not necessarily Lipschitz. Moreover, for two-dimensional waves,
we can consider solutions such that the curvature of the initial free surface does not belong to
The
proof is entirely based on the Eulerian formulation of the water waves equations, using microlocal analysis to obtain sharp Sobolev and
Hölder estimates. We first prove tame estimates in Sobolev spaces depending linearly on Hölder norms and then we use the dispersive
properties of the water-waves system, namely Strichartz estimates, to control these Hölder norms.
eBook
Sums of Reciprocals of Fractional Parts and Multiplicative Diophantine Approximation
There are two main interrelated goals of this paper. Firstly we investigate the sums
eBook
Positive definiteness of functions with applications to operator norm inequalities
Positive definiteness is determined for a wide class of functions relevant in the study of operator means and their norm comparisons.
Then, this information is used to obtain an abundance of new sharp (unitarily) norm inequalities comparing various operator means and
sometimes other related operators.
eBook
Some Historical Issues and Paradoxes regarding the Concept of Infinity: An Apos Analysis: Part 2
This is Part 2 of a two-part study of how APOS theory may be used to provide cognitive explanations of how students and mathematicians might think about the concept of infinity. We discuss infinite processes, describe how the mental mechanisms of interiorization and encapsulation can be used to conceive of an infinite process as a completed totality, explain the relationship between infinite processes and the objects that may result from them, and apply our analyses to certain mathematical issues related to infinity.
Journal Article
Positive Gaussian Kernels also Have Gaussian Minimizers
We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put
forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give
necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb’s results on maximizers of
multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends
several inverse inequalities.
eBook
Inequalities: A Journey into Linear Analysis
Contains a wealth of inequalities used in linear analysis, and explains in detail how they are used. The book begins with Cauchy's inequality and ends with Grothendieck's inequality, in between one finds the Loomis-Whitney inequality, maximal inequalities, inequalities of Hardy and of Hilbert, hypercontractive and logarithmic Sobolev inequalities, Beckner's inequality, and many, many more. The inequalities are used to obtain properties of function spaces, linear operators between them, and of special classes of operators such as absolutely summing operators. This textbook complements and fills out standard treatments, providing many diverse applications: for example, the Lebesgue decomposition theorem and the Lebesgue density theorem, the Hilbert transform and other singular integral operators, the martingale convergence theorem, eigenvalue distributions, Lidskii's trace formula, Mercer's theorem and Littlewood's 4/3 theorem. It will broaden the knowledge of postgraduate and research students, and should also appeal to their teachers, and all who work in linear analysis.
eBook
The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity
In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of
Brunn-Minkowski type for a nonlinear capacity,
In the first part of this article, we prove the Brunn-Minkowski inequality for this
capacity:
In the second part of this article we study a Minkowski problem for a certain measure associated with a compact
convex set
eBook