The Bounded And Precise Word Problems For Presentations Of Groups

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The Bounded and Precise Word Problems for Presentations of Groups

S. V. Ivanov
The Bounded and Precise Word Problems for Presentations of Groups Book PDF

Author : S. V. Ivanov
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 55,6 Mb
Release : 2020-05-13
Category : Education
ISBN : 9781470441432

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The Bounded and Precise Word Problems for Presentations of Groups by S. V. Ivanov Pdf

The author introduces and studies 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, the author obtains 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. The author also obtains polynomial time bounds for these problems.

Conformal Graph Directed Markov Systems on Carnot Groups

Vasileios Chousionis,Jeremy T. Tyson,Mariusz Urbanski
Conformal Graph Directed Markov Systems on Carnot Groups Book PDF

Author : Vasileios Chousionis,Jeremy T. Tyson,Mariusz Urbanski
Publisher : American Mathematical Soc.
Page : 153 pages
File Size : 49,6 Mb
Release : 2020-09-28
Category : Mathematics
ISBN : 9781470442156

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Conformal Graph Directed Markov Systems on Carnot Groups by Vasileios Chousionis,Jeremy T. Tyson,Mariusz Urbanski Pdf

The 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 Irreducible Subgroups of Exceptional Algebraic Groups

Adam R. Thomas
The Irreducible Subgroups of Exceptional Algebraic Groups Book PDF

Author : Adam R. Thomas
Publisher : American Mathematical Soc.
Page : 191 pages
File Size : 51,7 Mb
Release : 2021-06-18
Category : Education
ISBN : 9781470443375

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The Irreducible Subgroups of Exceptional Algebraic Groups by Adam R. Thomas Pdf

This paper is a contribution to the study of the subgroup structure of excep-tional algebraic groups over algebraically closed fields of arbitrary characteristic. Following Serre, a closed subgroup of a semisimple algebraic group G is called irreducible if it lies in no proper parabolic subgroup of G. In this paper we com-plete the classification of irreducible connected subgroups of exceptional algebraic groups, providing an explicit set of representatives for the conjugacy classes of such subgroups. Many consequences of this classification are also given. These include results concerning the representations of such subgroups on various G-modules: for example, the conjugacy classes of irreducible connected subgroups are determined by their composition factors on the adjoint module of G, with one exception. A result of Liebeck and Testerman shows that each irreducible connected sub-group X of G has only finitely many overgroups and hence the overgroups of X form a lattice. We provide tables that give representatives of each conjugacy class of connected overgroups within this lattice structure. We use this to prove results concerning the subgroup structure of G: for example, when the characteristic is 2, there exists a maximal connected subgroup of G containing a conjugate of every irreducible subgroup A1 of G.

Double Affine Hecke Algebras and Congruence Groups

Bogdan Ion,Siddhartha Sahi
Double Affine Hecke Algebras and Congruence Groups Book PDF

Author : Bogdan Ion,Siddhartha Sahi
Publisher : American Mathematical Soc.
Page : 90 pages
File Size : 47,6 Mb
Release : 2021-06-18
Category : Education
ISBN : 9781470443269

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Double Affine Hecke Algebras and Congruence Groups by Bogdan Ion,Siddhartha Sahi Pdf

The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by auto-morphisms of a finite index subgroup of the Artin group of type A2, which descends to a faithful outer action of a congruence subgroup of SL(2, Z)or PSL(2, Z). This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. It turns out that the structural intricacies of DAAG/DAHA are captured by the underlying semisimple data and, to a large extent, even by adjoint data; we prove our main result by reduction to the adjoint case. Adjoint DAAG/DAHA correspond in a natural way to affine Lie algebras, or more precisely to their affinized Weyl groups, which are the semi-direct products W 􀀁 Q∨ of the Weyl group W with the coroot lattice Q∨. They were defined topologically by van der Lek, and independently, algebraically, by Cherednik. We now describe our results for the adjoint case in greater detail. We first give a new Coxeter-type presentation for adjoint DAAG as quotients of the Coxeter braid groups associated to certain crystallographic diagrams that we call double affine Coxeter diagrams. As a consequence we show that the rank two Artin groups of type A2,B2,G2 act by automorphisms on the adjoint DAAG/DAHA associated to affine Lie algebras of twist number r =1, 2, 3, respec-tively. This extends a fundamental result of Cherednik for r =1. We show further that the above rank two Artin group action descends to an outer action of the congruence subgroup Γ1(r). In particular, Γ1(r) acts naturally on the set of isomorphism classes of representations of an adjoint DAAG/DAHA of twist number r, giving rise to a projective representation of Γ1(r)on the spaceof aΓ1(r)-stable representation. We also provide a classification of the involutions of Kazhdan-Lusztig type that appear in the context of these actions.

Bounded Littlewood Identities

Eric M. Rains,S. Ole Warnaar
Bounded Littlewood Identities Book PDF

Author : Eric M. Rains,S. Ole Warnaar
Publisher : American Mathematical Soc.
Page : 115 pages
File Size : 45,5 Mb
Release : 2021-07-21
Category : Education
ISBN : 9781470446901

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Bounded Littlewood Identities by Eric M. Rains,S. Ole Warnaar Pdf

We describe a method, based on the theory of Macdonald–Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald’s partial fraction technique and results in the first examples of bounded Littlewood identities for Macdonald polynomials. These identities, which take the form of decomposition formulas for Macdonald polynomials of type (R, S) in terms of ordinary Macdonald polynomials, are q, t-analogues of known branching formulas for characters of the symplectic, orthogonal and special orthogonal groups. In the classical limit, our method implies that MacMahon’s famous ex-conjecture for the generating function of symmetric plane partitions in a box follows from the identification of GL(n, R), O(n) as a Gelfand pair. As further applications, we obtain combinatorial formulas for characters of affine Lie algebras; Rogers–Ramanujan identities for affine Lie algebras, complementing recent results of Griffin et al.; and quadratic transformation formulas for Kaneko–Macdonald-type basic hypergeometric series.

The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners

Paul Godin
The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners Book PDF

Author : Paul Godin
Publisher : American Mathematical Soc.
Page : 72 pages
File Size : 55,8 Mb
Release : 2021-06-21
Category : Education
ISBN : 9781470444211

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The 2D Compressible Euler Equations in Bounded Impermeable Domains with Corners by Paul Godin Pdf

We study 2D compressible Euler flows in bounded impermeable domains whose boundary is smooth except for corners. We assume that the angles of the corners are small enough. Then we obtain local (in time) existence of solutions which keep the L2 Sobolev regularity of their Cauchy data, provided the external forces are sufficiently regular and suitable compatibility conditions are satisfied. Such a result is well known when there is no corner. Our proof relies on the study of associated linear problems. We also show that our results are rather sharp: we construct counterexamples in which the smallness condition on the angles is not fulfilled and which display a loss of L2 Sobolev regularity with respect to the Cauchy data and the external forces.

Global Smooth Solutions for the Inviscid SQG Equation

Angel Castro,Diego Cordoba,Javier Gomez-Serrano
Global Smooth Solutions for the Inviscid SQG Equation Book PDF

Author : Angel Castro,Diego Cordoba,Javier Gomez-Serrano
Publisher : American Mathematical Soc.
Page : 89 pages
File Size : 52,7 Mb
Release : 2020-09-28
Category : Mathematics
ISBN : 9781470442149

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Global Smooth Solutions for the Inviscid SQG Equation by Angel Castro,Diego Cordoba,Javier Gomez-Serrano Pdf

In this paper, the authors show the existence of the first non trivial family of classical global solutions of the inviscid surface quasi-geostrophic equation.

Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case

Jacob Bedrossian,Pierre Germain,Nader Masmoudi
Dynamics Near the Subcritical Transition of the 3D Couette Flow I Below Threshold Case Book PDF

Author : Jacob Bedrossian,Pierre Germain,Nader Masmoudi
Publisher : American Mathematical Soc.
Page : 154 pages
File Size : 40,6 Mb
Release : 2020-09-28
Category : Mathematics
ISBN : 9781470442170

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Dynamics Near the Subcritical Transition of the 3D Couette Flow I: Below Threshold Case by Jacob Bedrossian,Pierre Germain,Nader Masmoudi Pdf

The authors study small disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number Re. They prove that for sufficiently regular initial data of size $epsilon leq c_0mathbf {Re}^-1$ for some universal $c_0 > 0$, the solution is global, remains within $O(c_0)$ of the Couette flow in $L^2$, and returns to the Couette flow as $t rightarrow infty $. For times $t gtrsim mathbf {Re}^1/3$, the streamwise dependence is damped by a mixing-enhanced dissipation effect and the solution is rapidly attracted to the class of ``2.5 dimensional'' streamwise-independent solutions referred to as streaks.

The Riesz Transform of Codimension Smaller Than One and the Wolff Energy

Benjamin Jaye,Fedor Nazarov,Maria Carmen Reguera,Xavier Tolsa
The Riesz Transform of Codimension Smaller Than One and the Wolff Energy Book PDF

Author : Benjamin Jaye,Fedor Nazarov,Maria Carmen Reguera,Xavier Tolsa
Publisher : American Mathematical Soc.
Page : 97 pages
File Size : 52,6 Mb
Release : 2020-09-28
Category : Mathematics
ISBN : 9781470442132

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The Riesz Transform of Codimension Smaller Than One and the Wolff Energy by Benjamin Jaye,Fedor Nazarov,Maria Carmen Reguera,Xavier Tolsa Pdf

Fix $dgeq 2$, and $sin (d-1,d)$. The authors characterize the non-negative locally finite non-atomic Borel measures $mu $ in $mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(mu )$ 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, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-Delta )^alpha /2$, $alpha in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.

Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields

Lisa Berger,Chris Hall,Rene Pannekoek,Rachel Pries,Shahed Sharif
Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields Book PDF

Author : Lisa Berger,Chris Hall,Rene Pannekoek,Rachel Pries,Shahed Sharif
Publisher : American Mathematical Soc.
Page : 131 pages
File Size : 40,6 Mb
Release : 2020-09-28
Category : Mathematics
ISBN : 9781470442194

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Explicit Arithmetic of Jacobians of Generalized Legendre Curves Over Global Function Fields by Lisa Berger,Chris Hall,Rene Pannekoek,Rachel Pries,Shahed Sharif Pdf

The authors study the Jacobian $J$ of the smooth projective curve $C$ of genus $r-1$ with affine model $y^r = x^r-1(x + 1)(x + t)$ over the function field $mathbb F_p(t)$, when $p$ is prime and $rge 2$ is an integer prime to $p$. When $q$ is a power of $p$ and $d$ is a positive integer, the authors compute the $L$-function of $J$ over $mathbb F_q(t^1/d)$ and show that the Birch and Swinnerton-Dyer conjecture holds for $J$ over $mathbb F_q(t^1/d)$.

Filtrations and Buildings

Christophe Cornut
Filtrations and Buildings Book PDF

Author : Christophe Cornut
Publisher : American Mathematical Soc.
Page : 150 pages
File Size : 44,7 Mb
Release : 2020-09-28
Category : Mathematics
ISBN : 9781470442217

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Filtrations and Buildings by Christophe Cornut Pdf

The author constructs and studies a scheme theoretical version of the Tits vectorial building, relates it to filtrations on fiber functors, and uses them to clarify various constructions pertaining to affine Bruhat-Tits buildings, for which he also provides a Tannakian description.

Operator Theory on One-Sided Quaternion Linear Spaces: Intrinsic $S$-Functional Calculus and Spectral Operators

Jonathan Gantner
Operator Theory on One Sided Quaternion Linear Spaces Intrinsic S Functional Calculus and Spectral Operators Book PDF

Author : Jonathan Gantner
Publisher : American Mathematical Society
Page : 114 pages
File Size : 48,8 Mb
Release : 2021-02-10
Category : Mathematics
ISBN : 9781470442385

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Operator Theory on One-Sided Quaternion Linear Spaces: Intrinsic $S$-Functional Calculus and Spectral Operators by Jonathan Gantner Pdf

Two major themes drive this article: identifying the minimal structure necessary to formulate quaternionic operator theory and revealing a deep relation between complex and quaternionic operator theory. The theory for quaternionic right linear operators is usually formulated under the assumption that there exists not only a right- but also a left-multiplication on the considered Banach space $V$. This has technical reasons, as the space of bounded operators on $V$ is otherwise not a quaternionic linear space. A right linear operator is however only associated with the right multiplication on the space and in certain settings, for instance on quaternionic Hilbert spaces, the left multiplication is not defined a priori, but must be chosen randomly. Spectral properties of an operator should hence be independent of the left multiplication on the space.

Łojasiewicz-Simon Gradient Inequalities for Coupled Yang-Mills Energy Functionals

Paul M Feehan,Manousos Maridakis
ojasiewicz Simon Gradient Inequalities for Coupled Yang Mills Energy Functionals Book PDF

Author : Paul M Feehan,Manousos Maridakis
Publisher : American Mathematical Society
Page : 138 pages
File Size : 52,5 Mb
Release : 2021-02-10
Category : Mathematics
ISBN : 9781470443023

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Łojasiewicz-Simon Gradient Inequalities for Coupled Yang-Mills Energy Functionals by Paul M Feehan,Manousos Maridakis Pdf

The authors' primary goal in this monograph is to prove Łojasiewicz-Simon gradient inequalities for coupled Yang-Mills energy functions using Sobolev spaces that impose minimal regularity requirements on pairs of connections and sections.

Theory of Fundamental Bessel Functions of High Rank

Zhi Qi
Theory of Fundamental Bessel Functions of High Rank Book PDF

Author : Zhi Qi
Publisher : American Mathematical Society
Page : 123 pages
File Size : 54,5 Mb
Release : 2021-02-10
Category : Mathematics
ISBN : 9781470443252

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Theory of Fundamental Bessel Functions of High Rank by Zhi Qi Pdf

In this article, the author studies fundamental Bessel functions for $mathrm{GL}_n(mathbb F)$ arising from the Voronoí summation formula for any rank $n$ and field $mathbb F = mathbb R$ or $mathbb C$, with focus on developing their analytic and asymptotic theory. The main implements and subjects of this study of fundamental Bessel functions are their formal integral representations and Bessel differential equations. The author proves the asymptotic formulae for fundamental Bessel functions and explicit connection formulae for the Bessel differential equations.