scholarly journals ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC

2017 ◽  
Vol 5 ◽  
Author(s):  
KELLY MCKINNIE
Keyword(s):  
Lower Bound ◽  
Essential Dimension ◽  
Base Field ◽  
Witt Group ◽  
New Techniques ◽  
Characteristic 2 ◽  
Symbol Algebra ◽  
Bounded Below ◽  
Independent Parameters

In this article the $p$-essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$-essential dimension of the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by $n+\ell$ when the base field is algebraically closed of characteristic $p$. The proof uses new techniques for working with residues in Milne–Kato $p$-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on $p$-symbol algebras (i.e, degree 2 symbols) result from this work. The generic $p$-symbol algebra of length $\ell$ is shown to have $p$-essential dimension equal to $\ell +1$ as a $p$-torsion Brauer class. The second is a lower bound of $\ell +1$ on the $p$-essential dimension of the functor $\operatorname{Alg}_{p^{\ell },p}$. Roughly speaking this says that you will need at least $\ell +1$ independent parameters to be able to specify any given algebra of degree $p^{\ell }$ and exponent $p$ over a field of characteristic $p$ and improves on the previously established lower bound of 3.

scholarly journals A lower bound on the essential dimension of PGL4 in characteristic 2

2017 ◽  
Vol 16 (04) ◽  
pp. 1750063
Author(s):  
Sanghoon Baek
Keyword(s):  
Lower Bound ◽  
Cohomology Group ◽  
Positive Characteristic ◽  
Essential Dimension ◽  
Trace Form ◽  
Simple Algebras ◽  
Characteristic 2 ◽  
Field Of Positive Characteristic

In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato’s cohomology group, defined by cokernel of a general Artin–Schreier operator. Combining this with Tignol’s result on the second trace form of simple algebras of degree [Formula: see text], we show that [Formula: see text] over a field of characteristic [Formula: see text].

scholarly journals Badly approximable points on self-affine sponges and the lower Assouad dimension

2017 ◽  
Vol 39 (3) ◽  
pp. 638-657 ◽  
Author(s):  
TUSHAR DAS ◽  
LIOR FISHMAN ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Assouad Dimension ◽  
Badly Approximable Points ◽  
Bounded Below ◽  
Badly Approximable ◽  
Sierpinski Sponges

We highlight a connection between Diophantine approximation and the lower Assouad dimension by using information about the latter to show that the Hausdorff dimension of the set of badly approximable points that lie in certain non-conformal fractals, known as self-affine sponges, is bounded below by the dynamical dimension of these fractals. For self-affine sponges with equal Hausdorff and dynamical dimensions, the set of badly approximable points has full Hausdorff dimension in the sponge. Our results, which are the first to advance beyond the conformal setting, encompass both the case of Sierpiński sponges/carpets (also known as Bedford–McMullen sponges/carpets) and the case of Barański carpets. We use the fact that the lower Assouad dimension of a hyperplane diffuse set constitutes a lower bound for the Hausdorff dimension of the set of badly approximable points in that set.

scholarly journals A fiber dimension theorem for essential and canonical dimension

2012 ◽  
Vol 149 (1) ◽  
pp. 148-174 ◽  
Author(s):  
Roland Lötscher
Keyword(s):  
Algebraic Geometry ◽  
Lower Bound ◽  
Finite Type ◽  
Algebraic Groups ◽  
General Version ◽  
Essential Dimension ◽  
Algebraic Stacks ◽  
Algebraic Tori ◽  
Canonical Dimension

AbstractThe well-known fiber dimension theorem in algebraic geometry says that for every morphism f:X→Y of integral schemes of finite type the dimension of every fiber of f is at least dim X−dim Y. This has recently been generalized by Brosnan, Reichstein and Vistoli to certain morphisms of algebraic stacks f:𝒳→𝒴, where the usual dimension is replaced by essential dimension. We will prove a general version for morphisms of categories fibered in groupoids. Moreover, we will prove a variant of this theorem, where essential dimension and canonical dimension are linked. These results let us relate essential dimension to canonical dimension of algebraic groups. In particular, using the recent computation of the essential dimension of algebraic tori by MacDonald, Meyer, Reichstein and the author, we establish a lower bound on the canonical dimension of algebraic tori.

Variations on a theme of rationality of cycles

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Nikita Karpenko
Keyword(s):  
Function Field ◽  
Algebraic Cycles ◽  
Base Field ◽  
Arbitrary Characteristic ◽  
Steenrod Operations ◽  
Cobordism Theory ◽  
Characteristic 2 ◽  
Algebraic Cobordism ◽  
Variations On A Theme

AbstractWe prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.

scholarly journals Small eigenvalues of large Hankel matrices: The indeterminate case

2002 ◽  
Vol 91 (1) ◽  
pp. 67 ◽  
Author(s):  
Christian Berg ◽  
Yang Chen ◽  
Mourad E. H. Ismail
Keyword(s):  
Lower Bound ◽  
Positive Constant ◽  
Moment Problems ◽  
Hankel Matrices ◽  
Orthonormal Polynomials ◽  
Indeterminate Moment Problems ◽  
Explicit Lower Bound ◽  
Indeterminate Case ◽  
Bounded Below ◽  
Small Eigenvalues

In this paper we characterize the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant. An explicit lower bound is given in terms of the orthonormal polynomials and we find expressions for this lower bound in a number of indeterminate moment problems.

scholarly journals PRINCIPAL RADICAL SYSTEMS, LEFSCHETZ PROPERTIES, AND PERFECTION OF SPECHT IDEALS OF TWO-ROWED PARTITIONS

2021 ◽  
pp. 1-41
Author(s):  
CHRIS MCDANIEL ◽  
JUNZO WATANABE
Keyword(s):  
Lower Bound ◽  
Complex Numbers ◽  
Arbitrary Field ◽  
Cherednik Algebras ◽  
Weak Lefschetz Property ◽  
Lefschetz Property ◽  
Lefschetz Properties ◽  
Bounded Below ◽  
Rational Cherednik Algebras

Abstract We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

2021 ◽  
pp. 1-54 ◽  
Author(s):  
Michael Brannan ◽  
Li Gao ◽  
Marius Junge
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Von Neumann Algebras ◽  
Logarithmic Sobolev Inequality ◽  
Beltrami Operator ◽  
Free Products ◽  
Von Neumann ◽  
Amalgamated Free Products ◽  
Quantum Tori ◽  
Bounded Below

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

Introduction

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Quadratic Forms ◽  
General Structure ◽  
Root Systems ◽  
Simple Groups ◽  
Reductive Groups ◽  
Central Extensions ◽  
New Techniques ◽  
Characteristic 2 ◽  
Minimal Type

This book deals with the classification of pseudo-reductive groups. Using new techniques and constructions, it addresses a number of questions; for example, whether there are versions of the Isomorphism and Isogeny Theorems for pseudosplit pseudo-reductive groups and of the Existence Theorem for pseudosplit pseudo-simple groups; whether the automorphism functor of a pseudo-semisimple group is representable; or whether there is a Tits-style classification in the pseudo-semisimple case recovering the version due to Tits in the semisimple case. This introduction discusses the special challenges of characteristic 2 as well as root systems, exotic groups and degenerate quadratic forms, and tame central extensions. It also reviews generalized standard groups, minimal type and general structure theorem, and Galois-twisted forms and Tits classification.

scholarly journals Internally heated convection beneath a poor conductor

2015 ◽  
Vol 771 ◽  
pp. 36-56 ◽  
Author(s):  
David Goluskin
Keyword(s):  
Lower Bound ◽  
Rayleigh Number ◽  
Boundary Conditions ◽  
Static State ◽  
Long Wavelength ◽  
Various Boundary Conditions ◽  
Mean Temperature ◽  
Long Wavelength Asymptotics ◽  
The Mean ◽  
Bounded Below

We consider convection in an internally heated (IH) layer of fluid that is bounded below by a perfect insulator and above by a poor conductor. The poorly conducting boundary is modelled by a fixed heat flux. Using solely analytical methods, we find linear and energy stability thresholds for the static state, and we construct a lower bound on the mean temperature that applies to all flows. The linear stability analysis yields a Rayleigh number above which the static state is linearly unstable ($R_{L}$), and the energy analysis yields a Rayleigh number below which it is globally stable ($R_{E}$). For various boundary conditions on the velocity, exact expressions for $R_{L}$ and $R_{E}$ are found using long-wavelength asymptotics. Each $R_{E}$ is strictly smaller than the corresponding $R_{L}$ but is within 1 %. The lower bound on the mean temperature is proven for no-slip velocity boundary conditions using the background method. The bound guarantees that the mean temperature of the fluid, relative to that of the top boundary, grows with the heating rate ($H$) no slower than $H^{2/3}$.

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