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

We 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.

Using a Blur Metric to Estimate Linear Motion Blur Parameters

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.

Refined forms of Oppenheim and Cusa-Huygens type inequalities

We rene Oppenheim's inequality as well as generalized Cusa-Huygens type inequalities established recently by some researchers. One of the results where the bounds of sin x / x are tractable will be used to obtain a sharp version of Yang's inequality.

On identities for derivative operators

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.

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

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 $\mathbb{R}^d$.

A Toy Model of the Information Paradox in Empty Space

A sharp version of the information paradox involves a seeming violation of the monogamy of entanglement during black hole evaporation. We construct an analogous paradox in empty anti-de Sitter space. In a local quantum field theory, Bell correlations between operators localized in mutually spacelike regions are monogamous. We show, through a controlled calculation, that this property can be violated by an order-1 factor in a theory of gravity. This example demonstrates that what appears to be a violation of the monogamy of entanglement may just be a subtle violation of locality in quantum gravity.

The Desiderata of Perfection

The first chapter opens with a challenge that any serious monist must face, one that Spinoza himself raised: how can a monist account for the world’s apparent diversity? It argues that Spinoza faces an especially sharp version of this long-standing question of the One and The Many, given his commitments to both maximal ontological parsimony and plenitude. After discussing the details of these ontological commitments, it is suggested that they ultimately stem not from the Principle of Sufficient Reason but from Spinoza’s account of metaphysical perfection, one that is similar to views held by the young Leibniz and, surprisingly enough, contemporary metaphysician Jonathan Schaffer.