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Beginning in the nineteenth century, mathematics' traditional domains of "number and figure" became vigorously displaced by altered settings in which former verities became discarded as no longer sacrosanct. These innovative recastings appeared everywhere, not merely within the familiar realm of the non-Euclidean geometries. How can mathematics retain its traditional status as a repository of necessary truth in the light of these revisions? The purpose of this Element is to provide a sketch of this developmental history.
PETSc for Partial Differential Equations : Numerical Solutions in C and Python by Ed BuelerThe Portable, Extensible Toolkit for Scientific Computation (PETSc) is an open-source library of advanced data structures and methods for solving linear and nonlinear equations and for managing discretizations. This book uses these modern numerical tools to demonstrate how to solve nonlinear partial differential equations (PDEs) in parallel. It starts from key mathematical concepts, such as Krylov space methods, preconditioning, multigrid, and Newton s method. In PETSc these components are composed at run time into fast solvers. Discretizations are introduced from the beginning, with an emphasis on finite difference and finite element methodologies. The example C programs of the first 12 chapters, listed on the inside front cover, solve (mostly) elliptic and parabolic PDE problems.
Journey from Natural Numbers to Complex Numbers by Nita H. Shah; Vishnuprasad D. ThakkarThis book is for those interested in number systems, abstract algebra, and analysis. It provides an understanding of negative and fractional numbers with theoretical background and explains rationale of irrational and complex numbers in an easy to understand format.This book covers the fundamentals, proof of theorems, examples, definitions, and concepts. It explains the theory in an easy and understandable manner and offers problems for understanding and extensions of concept are included. The book provides concepts in other fields and includes an understanding of handling of numbers by computers.Research scholars and students working in the fields of engineering, science, and different branches of mathematics will find this book of interest, as it provides the subject in a clear and concise way.
A Software Repository for Gaussian Quadratures and Christoffel Functions by Walter GautschiThis companion piece to the author's 2018 book, A Software Repository for Orthogonal Polynomials, focuses on Gaussian quadrature and the related Christoffel function. The book makes Gauss quadrature rules of any order easily accessible for a large variety of weight functions and for arbitrary precision. It also documents and illustrates known as well as original approximations for Gauss quadrature weights and Christoffel functions. The repository contains 60+ datasets, each dealing with a particular weight function. Included are classical, quasi-classical, and, most of all, nonclassical weight functions and associated orthogonal polynomials. Scientists, engineers, applied mathematicians, and statisticians will find the book of interest.
Mathematics of Casino Carnival Games by Mark BollmanThere are thousands of books relating to poker, blackjack, roulette and baccarat, including strategy guides, statistical analysis, psychological studies, and much more. However, there are no books on Pell, Rouleno, Street Dice, and many other games that have had a short life in casinos! While this is understandable -- most casino gamblers have not heard of these games, and no one is currently playing them -- their absence from published works means that some interesting mathematics and gaming history are at risk of being lost forever. Table games other than baccarat, blackjack, craps, and roulette are called carnival games, as a nod to their origin in actual traveling or seasonal carnivals. Mathematics of Casino Carnival Games is a focused look at these games and the mathematics at their foundation. Features * Exercises, with solutions, are included for readers who wish to practice the ideas presented * Suitable for a general audience with an interest in the mathematics of gambling and games * Goes beyond providing practical 'tips' for gamblers, and explores the mathematical principles that underpin gambling games
Tools and Problems in Partial Differential Equations by Thomas Alazard; Claude ZuilyThis textbook offers a unique learning-by-doing introduction to the modern theory of partial differential equations. Through 65 fully solved problems, the book offers readers a fast but in-depth introduction to the field, covering advanced topics in microlocal analysis, including pseudo- and para-differential calculus, and the key classical equations, such as the Laplace, Schrödinger or Navier-Stokes equations. Essentially self-contained, the book begins with problems on the necessary tools from functional analysis, distributions, and the theory of functional spaces, and in each chapter the problems are preceded by a summary of the relevant results of the theory. Informed by the authors' extensive research experience and years of teaching, this book is for graduate students and researchers who wish to gain real working knowledge of the subject.
Generators of Markov Chains : From a Walk in the Interior to a Dance on the Boundary by Adam BobrowskiElementary treatments of Markov chains, especially those devoted to discrete-time and finite state-space theory, leave the impression that everything is smooth and easy to understand. This exposition of the works of Kolmogorov, Feller, Chung, Kato, and other mathematical luminaries, which focuses on time-continuous chains but is not so far from being elementary itself, reminds us again that the impression is false: an infinite, but denumerable, state-space is where the fun begins. If you have not heard of Blackwell's example (in which all states are instantaneous), do not understand what the minimal process is, or do not know what happens after explosion, dive right in. But beware lest you are enchanted: 'There are more spells than your commonplace magicians ever dreamed of.'
Spatial Complexity : Theory, Mathematical Methods and Applications by Fivos PapadimitriouThis book delivers stimulating input for a broad range of researchers, from geographers and ecologists to psychologists interested in spatial perception and physicists researching in complex systems. How can one decide whether one surface or spatial object is more complex than another?What does it require to measure the spatial complexity of small maps, and why does this matter for nature, science and technology? Drawing from algorithmics, geometry, topology, probability and informatics, and with examples from everyday life, the reader is invited to cross the borders into the bewildering realm of spatial complexity, as it emerges from the study of geographic maps, landscapes, surfaces, knots, 3D and 4D objects. The mathematical and cartographic experiments described in this book lead to hypotheses and enigmas with ramifications in aesthetics and epistemology.
Linear Model Theory : With Examples and Exercises by Dale L. ZimmermanThis textbook presents a unified and rigorous approach to best linear unbiased estimation and prediction of parameters and random quantities in linear models, as well as other theory upon which much of the statistical methodology associated with linear models is based. The single most unique feature of the book is that each major concept or result is illustrated with one or more concrete examples or special cases. Commonly used methodologies based on the theory are presented in methodological interludes scattered throughout the book, along with a wealth of exercises that will benefit students and instructors alike. Generalized inverses are used throughout, so that the model matrix and various other matrices are not required to have full rank. Considerably more emphasis is given to estimability, partitioned analyses of variance, constrained least squares, effects of model misspecification, and most especially prediction than in many other textbooks on linear models. This book is intended for master and PhD students with a basic grasp of statistical theory, matrix algebra and applied regression analysis, and for instructors of linear models courses. Solutions to the book's exercises are available in the companion volume Linear Model Theory - Exercises and Solutions by the same author.
Linear Model Theory : Exercises and Solutions by Dale L. ZimmermanThis book contains 296 exercises and solutions covering a wide variety of topics in linear model theory, including generalized inverses, estimability, best linear unbiased estimation and prediction, ANOVA, confidence intervals, simultaneous confidence intervals, hypothesis testing, and variance component estimation. The models covered include the Gauss-Markov and Aitken models, mixed and random effects models, and the general mixed linear model. Given its content, the book will be useful for students and instructors alike. Readers can also consult the companion textbook Linear Model Theory - With Examples and Exercises by the same author for the theory behind the exercises.
Gödel, Tarski and the Lure of Natural Language : Logical Entanglement, Formalism Freeness by Juliette KennedyIs mathematics "entangled" with its various formalisations? Or are the central concepts of mathematics largely insensitive to formalisation, or "formalism free"? What is the semantic point of view and how is it implemented in foundational practice? Does a given semantic framework always have an implicit syntax? Inspired by what she calls the "natural language moves" of Gödel and Tarski, Juliette Kennedy considers what roles the concepts of '"entanglement" and "formalism freeness" play in a range of logical settings, from computability and set theory to model theory and second order logic, to logicality, developing an entirely original philosophy of mathematics along the way. The treatment is historically, logically and set-theoretically rich; topics such as naturalism and foundations receive their due, but now with a new twist.
A First Course in Random Matrix Theory for Physicists, Engineers and Data Scientists by Marc Potters; Jean-Philippe BouchaudThe real world is perceived and broken down as data, models and algorithms in the eyes of physicists and engineers. Data is noisy by nature and classical statistical tools have so far been successful in dealing with relatively smaller levels of randomness. The recent emergence of Big Data and the required computing power to analyse them have rendered classical tools outdated and insufficient. Tools such as random matrix theory and the study of large sample covariance matrices can efficiently process these big data sets and help make sense of modern, deep learning algorithms. Presenting an introductory calculus course for random matrices, the book focusses on modern concepts in matrix theory, generalising the standard concept of probabilistic independence to non-commuting random variables. Concretely worked out examples and applications to financial engineering and portfolio construction make this unique book an essential tool for physicists, engineers, data analysts, and economists.
Data Clustering : Theory, Algorithms, and Applications by Guojun Gan; Chaoqun Ma; Jianhong WuData clustering, also known as cluster analysis, is an unsupervised process that divides a set of objects into homogeneous groups. Since the publication of the first edition of this monograph in 2007, development in the area has exploded, especially in clustering algorithms for big data and open-source software for cluster analysis. This second edition reflects these new developments, covers the basics of data clustering, includes a list of popular clustering algorithms, and provides program code that helps users implement clustering algorithms. Data Clustering will be of interest to researchers, practitioners, and data scientists as well as undergraduate and graduate students.
Advanced Problem Solving Using Maple : Applied Mathematics, Operations Research, Business Analytics, and Decision Analysis by William P. Fox; William BauldryAdvanced Problem Solving Using Maple applies the mathematical modeling process by formulating, building, solving, analyzing, and criticizing mathematical models. Scenarios are developed within the scope of the problem-solving process. The text focuses on discrete dynamical systems, optimization techniques, single-variable unconstrained optimization and applied problems, and numerical search methods. Additional coverage includes multivariable unconstrained and constrained techniques. Linear algebra techniques to model and solve problems such as the Leontief model, and advanced regression techniques including nonlinear, logistics, and Poisson are covered. Game theory, the Nash equilibrium, and Nash arbitration are also included. Features: The text's case studies and student projects involve students with real-world problem solving Focuses on numerical solution techniques in dynamical systems, optimization, and numerical analysis The numerical procedures discussed in the text are algorithmic and iterative Maple is utilized throughout the text as a tool for computation and analysis All algorithms are provided with step-by-step formats About the Authors: William P. Fox is an emeritus professor in the Department of Defense Analysis at the Naval Postgraduate School. Currently, he is an adjunct professor, Department of Mathematics, the College of William and Mary. He received his PhD at Clemson University and has many publications and scholarly activities including twenty books and over one hundred and fifty journal articles. William C. Bauldry, Prof. Emeritus and Adjunct Research Prof. of Mathematics at Appalachian State University, received his PhD in Approximation Theory from Ohio State. He has published many papers on pedagogy and technology, often using Maple, and has been the PI of several NSF-funded projects incorporating technology and modeling into math courses. He currently serves as Associate Director of COMAP's Math Contest in Modeling (MCM).
Recent Advancements in Graph Theory by N. P. Shrimali; Nita H. Shah (Editors)Graph theory is a branch of discrete mathematics. It has many applications to many different areas of science and engineering. This book provides the most up-to-date research findings and applications in Graph Theory. This book focuses on the latest research in Graph theory. It provides recent findings that are occurring in the field, offers insights on an international and transnational levels, identifies the gaps in the results, and includes forthcoming international studies and research, along with its applications in networking, computer science, chemistry, biological sciences, etc. The book is written with researchers and post graduate students in mind.
Non-Diophantine Arithmetics in Mathematics, Physics and Psychology by Mark Burgin; Marek CzachorFor a long time, all thought there was only one geometry -- Euclidean geometry. Nevertheless, in the 19th century, many non-Euclidean geometries were discovered. It took almost two millennia to do this. This was the major mathematical discovery and advancement of the 19th century, which changed understanding of mathematics and the work of mathematicians providing innovative insights and tools for mathematical research and applications of mathematics.A similar event happened in arithmetic in the 20th century. Even longer than with geometry, all thought there was only one conventional arithmetic of natural numbers -- the Diophantine arithmetic, in which 2+2=4 and 1+1=2. It is natural to call the conventional arithmetic by the name Diophantine arithmetic due to the important contributions to arithmetic by Diophantus. Nevertheless, in the 20th century, many non-Diophantine arithmetics were discovered, in some of which 2+2=5 or 1+1=3. It took more than two millennia to do this. This discovery has even more implications than the discovery of new geometries because all people use arithmetic.This book provides a detailed exposition of the theory of non-Diophantine arithmetics and its various applications. Reading this book, the reader will see that on the one hand, non-Diophantine arithmetics continue the ancient tradition of operating with numbers while on the other hand, they introduce extremely original and innovative ideas.
Linear and Non-Linear System Theory by T. Thyagarajan; D. KalpanaLinear and Non-Linear System Theory focuses on the basics of linear and non-linear systems, optimal control and optimal estimation with an objective to understand the basics of state space approach linear and non-linear systems and its analysis thereof. Divided into eight chapters, materials cover an introduction to the advanced topics in the field of linear and non-linear systems, optimal control and estimation supported by mathematical tools, detailed case studies and numerical and exercise problems. This book is aimed at senior undergraduate and graduate students in electrical, instrumentation, electronics, chemical, control engineering and other allied branches of engineering. Topics covered include both linear and non-linear system theory; state feedback control and state estimator concepts; non-linear systems and phase plane analysis, and more.
Analyzing Spatial Models of Choice and Judgment by David A. Armstrong II; Ryan Bakker; Royce Carroll; Christopher Hare; Keith T. Poole; Howard RosenthalWith recent advances in computing power and the widespread availability of preference, perception and choice data, such as public opinion surveys and legislative voting, the empirical estimation of spatial models using scaling and ideal point estimation methods has never been more accessible.The second edition of Analyzing Spatial Models of Choice and Judgment demonstrates how to estimate and interpret spatial models with a variety of methods using the open-source programming language R. Requiring only basic knowledge of R, the book enables social science researchers to apply the methods to their own data. Also suitable for experienced methodologists, it presents the latest methods for modeling the distances between points. The authors explain the basic theory behind empirical spatial models, then illustrate the estimation technique behind implementing each method, exploring the advantages and limitations while providing visualizations to understand the results. This second edition updates and expands the methods and software discussed in the first edition, including new coverage of methods for ordinal data and anchoring vignettes in surveys, as well as an entire chapter dedicated to Bayesian methods. The second edition is made easier to use by the inclusion of an R package, which provides all data and functions used in the book. David A. Armstrong II is Canada Research Chair in Political Methodology and Associate Professor of Political Science at Western University. His research interests include measurement, Democracy and state repressive action. Ryan Bakker is Reader in Comparative Politics at the University of Essex. His research interests include applied Bayesian modeling, measurement, Western European politics, and EU politics. Royce Carroll is Professor in Comparative Politics at the University of Essex. His research focuses on measurement of ideology and the comparative politics of legislatures and political parties. Christopher Hare is Assistant Professor in Political Science at the University of California, Davis. His research focuses on ideology and voting behavior in US politics, political polarization, and measurement. Keith T. Poole is Philip H. Alston Jr. Distinguished Professor of Political Science at the University of Georgia. His research interests include methodology, US political-economic history, economic growth and entrepreneurship. Howard Rosenthal is Professor of Politics at NYU and Roger Williams Straus Professor of Social Sciences, Emeritus, at Princeton. Rosenthal's research focuses on political economy, American politics and methodology.
Numerical Homogenization by Localized Orthogonal Decomposition by Axel Målqvist; Daniel PeterseimThis book presents the first survey of the Localized Orthogonal Decomposition (LOD) method, a pioneering approach for the numerical homogenization of partial differential equations with multiscale data beyond periodicity and scale separation. The authors provide a careful error analysis, including previously unpublished results, and a complete implementation of the method in MATLAB. They also reveal how the LOD method relates to classical homogenization and domain decomposition. Illustrated with numerical experiments that demonstrate the significance of the method, the book is enhanced by a survey of applications including eigenvalue problems and evolution problems. Numerical Homogenization by Localized Orthogonal Decomposition is appropriate for graduate students in applied mathematics, numerical analysis, and scientific computing. Researchers in the field of computational partial differential equations will find this self-contained book of interest, as will applied scientists and engineers interested in multiscale simulation.
Memoirs of the American Mathematical Society
Sums of Reciprocals of Fractional Parts and Multiplicative Diophantine Approximation by Victor Beresnevich; Alan Haynes; Sanju VelaniThere are two main interrelated goals of this paper. Our theorems improve upon previous results of W. M. Schmidt and others, and are (up to constants) best possible. This first strand of the work is motivated by applications to multiplicative Diophantine approximation, which are also considered. In particular, we obtain complete Khintchine type results for multiplicative simultaneous Diophantine approximation on fibers in R2. The divergence result is the first of its kind and represents an attempt of developing the concept of ubiquity to the multiplicative setting.
Global Well-posedness of High Dimensional Maxwell-Dirac for Small Critical Data by Cristian Gavrus; Sung-Jin OhIn this paper, we prove global well-posedness of the massless Maxwell-Dirac equation in Coulomb gauge on R1+d (d≥4) for data with small scale-critical Sobolev norm, as well as modified scattering of the solutions. Main components of our proof are A) uncovering null structure of Maxwell-Dirac in the Coulomb gauge, and B) proving solvability of the underlying covariant Dirac equation. A key step for achieving both is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon.
The Riesz Transform of Codimension Smaller Than One and the Wolff Energy by Benjamin Jaye; Fedor Nazorov; Maria Carmen Reguera; Xavier Tolsa.We characterize the non-negative locally finite non-atomic Borel measures [mu] in Rd for which the associated s-Riesz transform is bounded in L²([mu]) in terms of the Wolff energy. This extends the range of s in which the Mateu-Prat-Verdera characterization of measures with bounded s-Riesz transform is known. As an application, we give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator (-[delta])[infinity]/2, [infinity] [epsilon] (1, 2), in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.
Affine Flag Varieties and Quantum Symmetric Pairs by Zhaobing Fan; Chun-Ju Lai; Yiqiang Li; Li Luo; Weiqiang WangThe quantum groups of finite and affine type A admit geometric realizations in terms of partial flag varieties of finite and affine type A. Recently, the quantum group associated to partial flag varieties of finite type B/C is shown to be a coideal subalgebra of the quantum group of finite type A. In this paper we study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type C. We show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine sl and gl types, respectively. In this way, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine sl type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotented quantum affine gl and its canonical basis.
Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi by David Joseph CarchediWe develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie (2004), but our approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived algebraic geometry, and the algebraic geometry of commutative ring spectra as in Lurie (2004), but also to differential topology, complex geometry, the theoryof supermanifolds, derived manifolds etc., where it produces a theory of higher generalized orbifolds appropriate for these settings.
An Elementary Recursive Bound for Effective Positivstellensatz and Hilbert's 17th Problem by Henri Lombardi; Daniel Perrucci Marie-Françoise RoyWe prove an elementary recursive bound on the degrees for Hilbert's 17th problem. More precisely we express a nonnegative polynomial as a sum of squares of rational functions, and we obtain as degree estimates for the numerators and denominators the following tower of five exponentials 222d4k where d is the degree and k is the number of variables of the input polynomial. Our method is based on the proof of an elementary recursive bound on the degrees for Stengle's Positivstellensatz. More precisely we give an algebraic certificate of the emptyness of the realization of a system of sign conditions and we obtain as degree bounds for this certificate a tower of five exponentials.
The Triangle-Free Process and the Ramsey Number R(3, k) by Gonzalo Fiz Pontiveros; Simon Griffiths; Robert MorrisThe areas of Ramsey theory and random graphs have been closely linked ever since Erdős' famous proof in 1947 that the "diagonal" Ramsey numbers R(k) grow exponentially in k. In the early 1990s, the triangle-free process was introduced as a model which might potentially provide good lower bounds for the "off-diagonal" Ramsey numbers R(3,k). In this model, edges of Kn are introduced one-by-one at random and added to the graph if they do not create a triangle; the resulting final (random) graph is denoted Gn,△. In 2009, Bohman succeeded in following this process for a positive fraction of its duration, and thus obtained a second proof of Kim's celebrated result that R(3,k)=Θ(k2/logk).
A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth by Jaroslav Nešetřil; Patrice Ossona de MendezIn this work, the authors introduce a general framework for the study of limits of relational structures and graphs in particular, which is based on a combination of model theory and (functional) analysis. The authors show how the various approaches to graph limits fit to this framework and naturally appear as “tractable cases” of a general theory. As an outcome of this, the authors provide extensions of known results, and believe that this puts these into a broader context.
Rigid Character Groups, Lubin-Tate Theory, and (φ,Γ)-Modules by Laurent Berger; Peter Schneider; Bingyong XieThe construction of the p-adic local Langlands correspondence for GL2(Qp) uses in an essential way Fontaine's theory of cyclotomic (φ,Γ)-modules. Here cyclotomic means that Γ=Gal(Qp(μp∞)/Qp) is the Galois group of the cyclotomic extension of Qp. In order to generalize the p-adic local Langlands correspondence to GL2(L), where L is a finite extension of Qp, it seems necessary to have at our disposal a theory of Lubin-Tate (φ,Γ)-modules. Such a generalization has been carried out, to some extent, by working over the p-adic open unit disk, endowed with the action of the endomorphisms of a Lubin-Tate group. The main idea of this article is to carry out a Lubin-Tate generalization of the theory of cyclotomic (φ,Γ)-modules in a different fashion. Instead of the p-adic open unit disk, the authors work over a character variety that parameterizes the locally L-analytic characters on oL. They study (φ,Γ)-modules in this setting and relate some of them to what was known previously.
Subgroup Decomposition in Out(Fn) by Michael Handel; Lee MosherThis is the introduction to a series of four papers that develop a decomposition theory for subgroups of Out(Fn) which generalizes the theory for elements of Out(Fn) found in the work of Bestvina, Feighn, and Handel on the Tits alternative, and which is analogous to the decomposition theory for subgroups of mapping class groups found in work of Ivanov. In this introduction we state the main theorems and we outline the contents of the whole series.
Degree Theory of Immersed Hypersurfaces by Harold Rosenberg; Graham Smith.The authors develop a degree theory for compact immersed hypersurfaces of prescribed K-curvature immersed in a compact, orientable Riemannian manifold, where K is any elliptic curvature function. They apply this theory to count the (algebraic) number of immersed hyperspheres in various cases: where K is mean curvature, extrinsic curvature and special Lagrangian curvature and show that in all these cases, this number is equal to −χ(M), where χ(M) is the Euler characteristic of the ambient manifold M.
Laminational Models for Some Spaces of Polynomials of Any Degree by Alexander Blokh; Lex Oversteegen; Ross Ptacek; Vladlen TimorinThe so-called "pinched disk" model of the Mandelbrot set is due to A. Douady, J. H. Hubbard and W. P. Thurston. It can be described in the language of geodesic laminations. The combinatorial model is the quotient space of the unit disk under an equivalence relation that, loosely speaking, "pinches" the disk in the plane (whence the name of the model). The significance of the model lies in particular in the fact that this quotient is planar and therefore can be easily visualized. The conjecture that the Mandelbrot set is actually homeomorphic to this model is equivalent to the celebrated MLC conjecture stating that the Mandelbrot set is locally connected. For parameter spaces of higher degree polynomials no combinatorial model is known.
New Complex Analytic Methods in the Study of Non-Orientable Minimal Surfaces in Rn by Antonio Alarcón; Franc Forstnerič; Francisco J. LópezThe aim of this work is to adapt the complex analytic methods originating in modern Oka theory to the study of non-orientable conformal minimal surfaces. These methods, which we develop essentially from the first principles, enable us to prove that the space of conformal minimal immersions of a given bordered non-orientable surface is a real analytic Banach manifold (see Theorem 1.1), obtain approximation results of Runge-Mergelyan type for conformal minimal immersions from non-orientable surfaces (see Theorem 1.2 and Corollary 1.3), and show general position theorems for non-orientable conformal minimal surfaces in (see Theorem 1.4).
Propagating Terraces and the Dynamics of Front-like Solutions of Reaction-Diffusion Equations on R by Peter PoláčikWe consider semilinear parabolic equations of the form ut = uxx + f(u), x ∈ R, t > 0, where f a C1 function. Assuming that 0 and γ > 0 are constant steady states, we investigate the large-time behavior of the front-like solutions, that is, solutions u whose initial values u(x, 0) are near γ for x ≈ −∞ and near 0 for x ≈ ∞. If the steady states 0 and γ are both stable, our main theorem shows that at large times, the graph of u(·, t) is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). We prove this result without requiring monotonicity of u(·, 0) or the nondegeneracy of zeros of f .
Conformal Graph Directed Markov Systems on Carnot Groups by Vasilionis Chousionis; Jeremy Tyson; Mariusz UrbanskiThe authors develop a comprehensive theory of conformal graph directed Markov systems in the non-Riemannian setting of Carnot groups equipped with a sub-Riemannian metric. In particular, they develop the thermodynamic formalism and show that, under natural hypotheses, the limit set of an Carnot conformal GDMS has Hausdorff dimension given by Bowen's parameter. They illustrate their results for a variety of examples of both linear and nonlinear iterated function systems and graph directed Markov systems in such sub-Riemannian spaces. These include the Heisenberg continued fractions introduced by Lukyanenko and Vandehey as well as Kleinian and Schottky groups associated to the non-real classical rank one hyperbolic spaces.
The Bounded and Precise Word Problems for Presentations of Groups by S.V. IvanovWe introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in NP, i.e., it can be solved in nondeterministic polynomial time, and the precise word problem is in PSPACE, i.e., it can be solved in polynomial space. The main technical result of the paper states that, for certain finite presentations of groups, which include the Baumslag-Solitar one-relator groups and free products of cyclic groups, the bounded word problem and the precise word problem can be solved in polylogarithmic space. As consequences of developed techniques that can be described as calculus of brackets, we obtain polylogarithmic space bounds for the computational complexity of the diagram problem for free groups, for the width problem for elements of free groups, and for computation of the area defined by polygonal singular closed curves in the plane. We also obtain polynomial time bounds for these problems.
Filtrations and Buildings by Christophe CornutWe construct and study a scheme theoretical version of the Tits vectorial building, relate it to filtrations on fiber functors, and use them to clarify various constructions pertaining to affine Bruhat-Tits buildings, for which we also provide a Tannakian description.
The Mother Body Phase Transition in the Normal Matrix Model by Pavel M. Bleher; Guilherme L. F. SilvaIn this present paper the authors consider the normal matrix model with cubic plus linear potential. In order to regularize the model, they follow Elbau & Felder and introduce a cut-off. In the large size limit, the eigenvalues of the model accumulate uniformly within a certain domain Ω that they determine explicitly by finding the rational parametrization of its boundary.
The authors also study in detail the mother body problem associated to Ω. It turns out that the mother body measure μ∗ displays a novel phase transition that we call the mother body phase transition: although ∂Ω evolves analytically, the mother body measure undergoes a “one-cut to three-cut” phase transition.
Localization for THH(ku) and the Topological Hochschild and Cyclic Homology of Waldhausen Categories by Andrew J. Blumberg; Michael A. MandellThe authors develop a theory of THH and TC of Waldhausen categories and prove the analogues of Waldhausen's theorems for K-theory. They resolve the longstanding confusion about localization sequences in THH and TC, and establish a specialized dévissage theorem. As applications, the authors prove conjectures of Hesselholt and Ausoni-Rognes about localization cofiber sequences surrounding THH(ku), and more generally establish a framework for advancing the Rognes program for studying Waldhausen's chromatic filtration on A(∗).