Slim exceptional set for sums of mixed powers of primes

2021 ◽  
Vol 131 (2) ◽  
Author(s):  
Jinjiang Li ◽  
Min Zhang ◽  
Yuetong Zhao
Keyword(s):  
Exceptional Set

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

scholarly journals A Korn-type inequality in SBD for functions with small jump sets

2017 ◽  
Vol 27 (13) ◽  
pp. 2461-2484 ◽  
Author(s):  
Manuel Friedrich
Keyword(s):  
Bounded Variation ◽  
Elastic Strain ◽  
Special Functions ◽  
Type Inequality ◽  
Rigid Motion ◽  
Exceptional Set ◽  
Finite Perimeter ◽  
Functions Of Bounded Deformation ◽  
Jump Set

We present a Korn-type inequality in a planar setting for special functions of bounded deformation. We prove that for each function in [Formula: see text] with a sufficiently small jump set the distance of the function and its derivative from an infinitesimal rigid motion can be controlled in terms of the linearized elastic strain outside of a small exceptional set of finite perimeter. Particularly, the result shows that each function in [Formula: see text] has bounded variation away from an arbitrarily small part of the domain.

scholarly journals SHARP LOGARITHMIC DERIVATIVE ESTIMATES WITH APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS IN THE UNIT DISC

2010 ◽  
Vol 88 (2) ◽  
pp. 145-167 ◽  
Author(s):  
I. CHYZHYKOV ◽  
J. HEITTOKANGAS ◽  
J. RÄTTYÄ
Keyword(s):  
Differential Equations ◽  
Unit Disc ◽  
Logarithmic Derivative ◽  
Maximum Modulus ◽  
Exceptional Set ◽  
Proximate Order ◽  
Upper Density ◽  
Linear Differential ◽  
Logarithmic Derivatives ◽  
Growth Of Solutions

AbstractNew estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f(k)/f(j), where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k>j≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.

ON EXCEPTIONAL SETS: THE SOLUTION OF A PROBLEM POSED BY K. MAHLER

2016 ◽  
Vol 94 (1) ◽  
pp. 15-19 ◽  
Author(s):  
DIEGO MARQUES ◽  
JOSIMAR RAMIREZ
Keyword(s):  
Entire Functions ◽  
Complex Conjugation ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Transcendental Numbers ◽  
Lecture Notes ◽  
Transcendental Entire Functions

In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

The exceptional set for the sum of a prime and a square

1989 ◽  
Vol 53 (3-4) ◽  
pp. 347-365 ◽  
Author(s):  
R. Brünner ◽  
A. Perelli ◽  
J. Pintz
Keyword(s):  
Exceptional Set

Exceptional sets related to the largest digits in Lüroth expansions

2021 ◽  
pp. 1-15
Author(s):  
Shuyi Lin ◽  
Jinjun Li ◽  
Manli Lou
Keyword(s):  
Number Theory ◽  
Hausdorff Dimension ◽  
Level Sets ◽  
Disjoint Union ◽  
Exceptional Set ◽  
Exceptional Sets ◽  
Lüroth Expansion

Let [Formula: see text] denote the largest digit of the first [Formula: see text] terms in the Lüroth expansion of [Formula: see text]. Shen, Yu and Zhou, A note on the largest digits in Luroth expansion, Int. J. Number Theory 10 (2014) 1015–1023 considered the level sets [Formula: see text] and proved that each [Formula: see text] has full Hausdorff dimension. In this paper, we investigate the Hausdorff dimension of the following refined exceptional set: [Formula: see text] and show that [Formula: see text] has full Hausdorff dimension for each pair [Formula: see text] with [Formula: see text]. Combining the two results, [Formula: see text] can be decomposed into the disjoint union of uncountably many sets with full Hausdorff dimension.

scholarly journals An exceptional set for inner functions

1971 ◽  
Vol 30 (3) ◽  
pp. 545-545
Author(s):  
Renate McLaughlin
Keyword(s):  
Inner Functions ◽  
Exceptional Set

scholarly journals Corrigendum of a note on the exceptional set for Goldbach's problem in short intervals

1995 ◽  
Vol 119 (3) ◽  
pp. 215-216
Author(s):  
J. Kaczorowski ◽  
A. Perelli ◽  
J. Pintz
Keyword(s):  
Exceptional Set ◽  
Short Intervals

A pair correlation hypothesis and the exceptional set in Goldbach's problem

Mathematika ◽  
1996 ◽  
Vol 43 (2) ◽  
pp. 349-361 ◽  
Author(s):  
A. Languasco ◽  
A. Perelli
Keyword(s):  
Pair Correlation ◽  
Exceptional Set

Penal Slavery in Early Modern Scandinavia

2021 ◽  
Vol 6 (3) ◽  
pp. 343-368
Author(s):  
Johan Heinsen
Keyword(s):  
Eighteenth Century ◽  
Early Modern ◽  
Distinct Form ◽  
Exceptional Set ◽  
Security Breaches ◽  
Multiple Voices ◽  
Penal Institution ◽  
The Status ◽  
The Creation ◽  
Atlantic Slavery

Abstract In Scandinavia, a penal institution known as “slavery” existed from the sixteenth to the nineteenth centuries. Penal slaves laboured in the creation and maintenance of military infrastructure. They were chained and often stigmatized, sometimes by branding. Their punishment was likened and, on a few occasions, linked to Atlantic slavery. Still, in reality, it was a wholly distinct form of enslavement that produced different experiences of coercion than those of the Atlantic. Such forms of penal slavery sit uneasily in historiographies of punishment but also offers a challenge for the dominant models of global labour history and its attempts to create comparative frameworks for coerced labour. This article argues for the need for contextual approaches to what such coercion meant to both coercers and coerced. Therefore, it offers an analysis of the meaning of early modern penal slavery based on an exceptional set of sources from 1723. In these sources, the status of the punished was negotiated and practiced by guards and slaves themselves. Court appearances by slaves were usually brief—typically revolving around escapes as authorities attempted to identify security breaches. The documents explored in this article are different: They present multiple voices speaking at length, negotiating their very status as voices. From that negotiation and its failures emerge a set of practiced meanings of penal “slavery” in eighteenth-century Copenhagen tied to competing yet intertwined notions of dishonour.

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