scholarly journals A sharp version of Bauer–Fike’s theorem

2012 ◽  
Vol 236 (13) ◽  
pp. 3218-3227 ◽  
Author(s):  
Xinghua Shi ◽  
Yimin Wei
Keyword(s):  
Sharp Version

A SHARP VERSION OF BONSALL’S INEQUALITY

2015 ◽  
Vol 92 (3) ◽  
pp. 397-404
Author(s):  
T. C. PEACHEY
Keyword(s):  
Reverse Inequality ◽  
Classical Inequality ◽  
The Best Possible Constant ◽  
Sharp Version

The best possible constant in a classical inequality due to Bonsall is established by relating that inequality to Young’s. Further, this extends the range of Bonsall’s inequality and yields a reverse inequality. It also provides a better constant in an inequality of Hardy, Littlewood and Pólya.

scholarly journals A Sharp Version of Price’s Law for Wave Decay on Asymptotically Flat Spacetimes

2021 ◽  
Author(s):  
Peter Hintz
Keyword(s):  
Order Term ◽  
Angular Frequency ◽  
Leading Order ◽  
Full Generality ◽  
Asymptotically Flat ◽  
Low Energies ◽  
Flat Spacetimes ◽  
Explicit Constant ◽  
Sharp Version ◽  
Linear Scalar

AbstractWe prove Price’s law with an explicit leading order term for solutions $$\phi (t,x)$$ ϕ ( t , x ) of the scalar wave equation on a class of stationary asymptotically flat $$(3+1)$$ ( 3 + 1 ) -dimensional spacetimes including subextremal Kerr black holes. Our precise asymptotics in the full forward causal cone imply in particular that $$\phi (t,x)=c t^{-3}+{\mathcal {O}}(t^{-4+})$$ ϕ ( t , x ) = c t - 3 + O ( t - 4 + ) for bounded |x|, where $$c\in {\mathbb {C}}$$ c ∈ C is an explicit constant. This decay also holds along the event horizon on Kerr spacetimes and thus renders a result by Luk–Sbierski on the linear scalar instability of the Cauchy horizon unconditional. We moreover prove inverse quadratic decay of the radiation field, with explicit leading order term. We establish analogous results for scattering by stationary potentials with inverse cubic spatial decay. On the Schwarzschild spacetime, we prove pointwise $$t^{-2 l-3}$$ t - 2 l - 3 decay for waves with angular frequency at least l, and $$t^{-2 l-4}$$ t - 2 l - 4 decay for waves which are in addition initially static. This definitively settles Price’s law for linear scalar waves in full generality. The heart of the proof is the analysis of the resolvent at low energies. Rather than constructing its Schwartz kernel explicitly, we proceed more directly using the geometric microlocal approach to the limiting absorption principle pioneered by Melrose and recently extended to the zero energy limit by Vasy.

scholarly journals On the Continuity of the Integrated Density of States in the Disorder

2019 ◽  
Author(s):  
Mira Shamis
Keyword(s):  
Density Of States ◽  
Schrödinger Operators ◽  
Random Schrödinger Operators ◽  
Integrated Density Of States ◽  
Schrodinger Operators ◽  
Quantitative Form ◽  
Alternative Approach ◽  
Ky Fan Inequalities ◽  
Ky Fan ◽  
Sharp Version

Abstract Recently, Hislop and Marx studied the dependence of the integrated density of states on the underlying probability distribution for a class of discrete random Schrödinger operators and established a quantitative form of continuity in weak* topology. We develop an alternative approach to the problem, based on Ky Fan inequalities, and establish a sharp version of the estimate of Hislop and Marx. We also consider a corresponding problem for continual random Schrödinger operators on $\mathbb{R}^d$.

scholarly journals Two hyperbolic Schwarz lemmas

2002 ◽  
Vol 66 (1) ◽  
pp. 17-24 ◽  
Author(s):  
L. Bernal-González ◽  
M. C. Calderón-Moreno
Keyword(s):  
Recent Result ◽  
Unit Disk ◽  
Sharp Version

In this paper, a sharp version of the Schwarz–Pick Lemma for hyperbolic derivatives is provided for holomorphic selfmappings on the unit disk with fixed multiplicity for the zero at the origin. This extends a recent result due to Beardon. A property of preserving hyperbolic distances also studied by Beardon is here completely characterised.

On the regularity of domains satisfying a uniform hour–glass condition and a sharp version of the Hopf–Oleinik boundary point principle

2011 ◽  
Vol 176 (3) ◽  
pp. 281-360 ◽  
Author(s):  
R. Alvarado ◽  
D. Brigham ◽  
V. Maz’ya ◽  
M. Mitrea ◽  
E. Ziadé
Keyword(s):  
Boundary Point ◽  
Sharp Version

scholarly journals Using a Blur Metric to Estimate Linear Motion Blur Parameters

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Taiebeh Askari Javaran ◽  
Hamid Hassanpour
Keyword(s):  
Image Processing ◽  
Health Services ◽  
Motion Blur ◽  
Linear Motion ◽  
Sensitive Stage ◽  
Blurred Image ◽  
The Given ◽  
Blur Direction ◽  
Sharp Version ◽  
Monotonic Relation

Motion blur is a common artifact in image processing, specifically in e-health services, which is caused by the motion of a camera or scene. In linear motion cases, the blur kernel, i.e., the function that simulates the linear motion blur process, depends on the length and direction of blur, called linear motion blur parameters. The estimation of blur parameters is a vital and sensitive stage in the process of reconstructing a sharp version of a motion blurred image, i.e., image deblurring. The estimation of blur parameters can also be used in e-health services. Since medical images may be blurry, this method can be used to estimate the blur parameters and then take an action to enhance the image. In this paper, some methods are proposed for estimating the linear motion blur parameters based on the extraction of features from the given single blurred image. The motion blur direction is estimated using the Radon transform of the spectrum of the blurred image. To estimate the motion blur length, the relation between a blur metric, called NIDCT (Noise-Immune Discrete Cosine Transform-based), and the motion blur length is applied. Experiments performed in this study showed that the NIDCT blur metric and the blur length have a monotonic relation. Indeed, an increase in blur length leads to increase in the blurriness value estimated via the NIDCT blur metric. This relation is applied to estimate the motion blur. The efficiency of the proposed method is demonstrated by performing some quantitative and qualitative experiments.

scholarly journals Freiman's Theorem in Finite Fields via Extremal Set Theory

2009 ◽  
Vol 18 (3) ◽  
pp. 335-355 ◽  
Author(s):  
BEN GREEN ◽  
TERENCE TAO
Keyword(s):  
Set Theory ◽  
Finite Fields ◽  
Common Theme ◽  
Additive Combinatorics ◽  
Extremal Set Theory ◽  
Extremal Set ◽  
Freiman’S Theorem ◽  
Sharp Version

Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asymptotically sharp version of Freiman's theorem in $\F_2^n$: if $A \subseteq \F_2^n$ is a set for which |A + A| ≤ K|A| then A is contained in a subspace of size $2^{2K + O(\sqrt{K}\log K)}|A|$; except for the $O(\sqrt{K} \log K)$ error, this is best possible. If in addition we assume that A is a downset, then we can also cover A by O(K46) translates of a coordinate subspace of size at most |A|, thereby verifying the so-called polynomial Freiman–Ruzsa conjecture in this case. A common theme in the arguments is the use of compression techniques. These have long been familiar in extremal set theory, but have been used only rarely in the additive combinatorics literature.

On identities for derivative operators

2019 ◽  
Vol 7 (4) ◽  
pp. 13-16
Author(s):  
Mirosław Baran ◽  
Paweł Ozorka
Keyword(s):  
Positive Integer ◽  
Commutative Algebra ◽  
Trigonometric Polynomials ◽  
Leibniz Rule ◽  
Bernstein’S Inequality ◽  
Bernstein's Inequality ◽  
Sharp Version

Let $X$ be a commutative algebra with unity $e$ and let $D$ be a derivative on $X$ that means the Leibniz rule is satisfied: $D(f\cdot g)=D(f)\cdot g+f\cdot D(g)$. If $D^{(k)}$ is $k$-th iteration of $D$ then we prove that the following identity holds for any positive integer $k$ $$\frac{1}{k!}\sum\limits_{j=0}^k(-1)^j\binom{k}{j}f^jD^{(m)}(gf^{k-j})=\Phi_{k,m}(g,f)=\begin{cases}0,\ 0\leq m <k,\\ gD(f)^k,\ k=m.\end{cases}$$ As an application we prove a sharp version of Bernstein's inequality for trigonometric polynomials.

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