scholarly journals A more intuitive proof of a sharp version of Halász’s theorem

2018 ◽  
Vol 146 (10) ◽  
pp. 4099-4104 ◽  
Author(s):  
Andrew Granville ◽  
Adam J. Harper ◽  
Kannan Soundararajan
Keyword(s):  
Intuitive Proof ◽  
Sharp Version

scholarly journals A PROOF OF ISBELL’S ZIGZAG THEOREM

2008 ◽  
Vol 84 (2) ◽  
pp. 229-232 ◽  
Author(s):  
PIOTR HOFFMAN
Keyword(s):  
Intuitive Proof

AbstractWe provide a short, intuitive proof of Isbell’s zigzag theorem.

scholarly journals A linear-optical proof that the permanent is # P -hard

2011 ◽  
Vol 467 (2136) ◽  
pp. 3393-3405 ◽  
Author(s):  
Scott Aaronson
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Universality Theorem ◽  
Linear Optical ◽  
Intuitive Proof

One of the crown jewels of complexity theory is Valiant's theorem that computing the permanent of an n × n matrix is # P -hard. Here we show that, by using the model of linear-optical quantum computing —and in particular, a universality theorem owing to Knill, Laflamme and Milburn—one can give a different and arguably more intuitive proof of this theorem.

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.

A sharp version of Ramanujan’s inequality for the factorial function

2014 ◽  
Vol 39 (1) ◽  
pp. 149-154 ◽  
Author(s):  
Chao-Ping Chen
Keyword(s):  
Factorial Function ◽  
Sharp Version

scholarly journals A new generalized and sharp version of Jordan’s inequality and its applications to the improvement of the Yang Le inequality, II

2007 ◽  
Vol 20 (5) ◽  
pp. 532-538 ◽  
Author(s):  
Shanhe Wu ◽  
Lokenath Debnath
Keyword(s):  
Jordan’S Inequality ◽  
Sharp Version

On the asymptotic distribution of the size of a stochastic epidemic

1983 ◽  
Vol 20 (2) ◽  
pp. 390-394 ◽  
Author(s):  
Thomas Sellke
Keyword(s):  
Population Size ◽  
Asymptotic Distribution ◽  
Epidemic Process ◽  
Stochastic Epidemic ◽  
Original Number ◽  
Intuitive Proof

For a stochastic epidemic of the type considered by Bailey [1] and Kendall [3], Daniels [2] showed that ‘when the threshold is large but the population size is much larger, the distribution of the number remaining uninfected in a large epidemic has approximately the Poisson form.' A simple, intuitive proof is given for this result without use of Daniels's assumption that the original number of infectives is ‘small'. The proof is based on a construction of the epidemic process which is more explicit than the usual description.

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 A Historical Note On The Proof Of The Area Of A Circle

2011 ◽  
Vol 8 (3) ◽  
Author(s):  
Yonah Wilamowsky ◽  
Sheldon Epstein ◽  
Bernard Dickman
Keyword(s):  
Twelfth Century ◽  
Historical Note ◽  
Mathematics Majors ◽  
Mathematical Literature ◽  
Intuitive Proof ◽  
Infinite Sequences

Proofs that the area of a circle is ?r2 can be found in mathematical literature dating as far back as the time of the Greeks. The early proofs, e.g. Archimedes, involved dividing the circle into wedges and then fitting the wedges together in a way to approximate a rectangle. Later more sophisticated proofs relied on arguments involving infinite sequences and calculus. Generally speaking, both of these approaches are difficult to explain to unsophisticated non-mathematics majors. This paper presents a less known but interesting and intuitive proof that was introduced in the twelfth century. It discusses challenges that were made to the proof and offers simple rebuttals to those challenges.

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.

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