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Differential equations workbook for dummies
\"Need to know how to solve differential equations? This easy-to-follow, hands-on workbook helps you master the basic concepts and work through the types of problems you'll encounter in your coursework. You get valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step solutions to every equation. You'll also memorize the most-common types of differential equations, see how to avoid common mistakes, get tips and tricks for advanced problems, improve your exam scores, and much more\"--Resource description page.
Book
Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume
The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper
The first aim of this paper is to lay the foundation of the theory of
multicomplexes. After setting the main definitions, we construct the singular multicomplex
In the second part of this work we apply the theory of multicomplexes to the study of the bounded
cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which
implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem,
which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity.
The
third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume
of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the
finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the
eBook
Asymptotic Spreading for General Heterogeneous Fisher-KPP Type Equations
In this monograph, we review the theory and establish new and general results regarding spreading properties for heterogeneous
reaction-diffusion equations:
The characterizations of these sets involve two new notions of generalized principal eigenvalues
for linear parabolic operators in unbounded domains. In particular, it allows us to show that
eBook
Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation
The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are
given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the
Airy
In this paper, we employ the Brownian Gibbs property to make a close
comparison between the Airy line ensemble’s curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially
growing moment bound on Radon-Nikodym derivatives.
We also determine the value of a natural exponent describing in Brownian last
passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common
endpoints, where the notion of ‘near’ refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness
tending to zero.
To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on
probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property.
Several results in this article
play a fundamental role in a further study of Brownian last passage percolation in three companion papers (Hammond 2017a,b,c), in which
geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.
eBook
An introduction to partial differential equations
\"This book is an introduction to methods for solving partial differential equations (PDEs). After the introduction of the main four PDEs that could be considered the cornerstone of Applied Mathematics, the reader is introduced to a variety of PDEs that come from a variety of fields in the Natural Sciences and Engineering and is a springboard into this wonderful subject. The chapters include the following topics: First-order PDEs, Second-order PDEs, Fourier Series, Separation of Variables, and the Fourier Transform. The reader is guided through these chapters where techniques for solving first- and second-order PDEs are introduced. Each chapter ends with a series of exercises illustrating the material presented in each chapter. The book can be used as a textbook for any introductory course in PDEs typically found in both science and engineering programs and has been used at the University of Central Arkansas for over ten years.\"--Abstract (page vi).
Book
Recent trends in formal and analytic solutions of diff. equations : Virtual Conference Formal and Analytic Solutions of Diff. Equations, June 28-July 2, 2021, University of Alcalá, Alcalá de Henares, Spain
This volume contains the proceedings of the conference on Formal and Analytic Solutions of Diff. Equations, held from June 28-July 2, 2021, and hosted by University of Alcala, Alcala de Henares, Spain. The manuscripts cover recent advances in the study of formal and analytic solutions of different kinds of equations such as ordinary differential equations, difference equations, $q$-difference equations, partial differential equations, moment differential equations, etc. Also discussed are related topics such as summability of formal solutions and the asymptotic study of their solutions. The volume is intended not only for researchers in this field of knowledge but also for students who aim to acquire new techniques and learn recent results.
eBook
Tunneling estimates and approximate controllability for hypoelliptic equations
This memoir is concerned with quantitative unique continuation estimates for equations involving a “sum of squares” operator
The first result is the tunneling estimate
The main
result is a stability estimate for solutions to the hypoelliptic wave equation
We then prove the approximate controllability of the
hypoelliptic heat equation
We also explain how the analyticity
assumption can be relaxed, and a boundary
Most results turn out to be optimal on a family of Grushin-type operators.
The main proof relies on the
general strategy to produce quantitative unique continuation estimates, developed by the authors in Laurent-Léautaud (2019).
eBook