Computing convolutions of measures

30 Jul

Originally posted on What's new:

If $latex {f: {\bf R}^n \rightarrow {\bf C}}&fg=000000$ and $latex {g: {\bf R}^n \rightarrow {\bf C}}&fg=000000$ are two absolutely integrable functions on a Euclidean space $latex {{\bf R}^n}&fg=000000$, then the convolution $latex {f*g: {\bf R}^n \rightarrow {\bf C}}&fg=000000$ of the two functions is defined by the formula

$latex \displaystyle f*g(x) := \int_{{\bf R}^n} f(y) g(x-y)\ dy = \int_{{\bf R}^n} f(x-z) g(z)\ dz.&fg=000000$

A simple application of the Fubini-Tonelli theorem shows that the convolution $latex {f*g}&fg=000000$ is well-defined almost everywhere, and yields another absolutely integrable function. In the case that $latex {f=1_F}&fg=000000$, $latex {g=1_G}&fg=000000$ are indicator functions, the convolution simplifies to

$latex \displaystyle 1_F*1_G(x) = m( F \cap (x-G) ) = m( (x-F) \cap G ) \ \ \ \ \ (1)&fg=000000$

where $latex {m}&fg=000000$ denotes Lebesgue measure. One can also define convolution on more general locally compact groups than $latex {{\bf R}^n}&fg=000000$, but we will restrict attention to the Euclidean…

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An improved Type I estimate

30 Jul

Originally posted on What's new:

As in all previous posts in this series, we adopt the following asymptotic notation: $latex {x}&fg=000000$ is a parameter going off to infinity, and all quantities may depend on $latex {x}&fg=000000$ unless explicitly declared to be “fixed”. The asymptotic notation $latex {O(), o(), \ll}&fg=000000$ is then defined relative to this parameter. A quantity $latex {q}&fg=000000$ is said to be of polynomial size if one has $latex {q = O(x^{O(1)})}&fg=000000$, and bounded if $latex {q=O(1)}&fg=000000$. We also write $latex {X \lessapprox Y}&fg=000000$ for $latex {X \ll x^{o(1)} Y}&fg=000000$, and $latex {X \sim Y}&fg=000000$ for $latex {X \ll Y \ll X}&fg=000000$.

The purpose of this (rather technical) post is both to roll over the polymath8 research thread from this previous post, and also to record the details of the latest improvement to the Type I estimates (based on exploiting additional averaging and using Deligne’s proof of the Weil conjectures) which lead…

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Spacetime, Relativity, and Quantum Physics

30 Jul

Spacetime, Relativity, and Quantum Physics

http://www.ws5.com/spacetime/

NASA Gravity Probe

Link

Spaun brain simulation

30 Jul

Spaun brain simulation

Spaun brain simulation

http://www.nature.com/news/simulated-brain-scores-top-test-marks-1.11914

Spaun brain simulation

Link

AI research at Google

30 Jul

AI research at Google

AI research at Google

http://www.artificialbrains.com/google

Google building 44

Link

Tetrapod Quantum Dots Light the Way to Stronger Polymers

30 Jul

Tetrapod Quantum Dots Light the Way to Stronger Polymers

http://newscenter.lbl.gov/feature-stories/2013/07/29/tetrapod-quantum-dots-light-the-way-to-stronger-polymers/

Tetrapod Quantum Dots Light the Way to Stronger Polymers

Fluorescent tetrapod quantum dots or tQDs (brown) serve as stress probes that allow precise measurement of polymer fiber tensile strength with minimal impact on mechanical properties. Inserts show relaxed tQDs (upper) and stressed tQDs (lower).

 

The Riemann hypothesis in various settings

28 Jul

Originally posted on What's new:

[Note: the content of this post is standard number theoretic material that can be found in many textbooks (I am relying principally here on Iwaniec and Kowalski); I am not claiming any new progress on any version of the Riemann hypothesis here, but am simply arranging existing facts together.]

The Riemann hypothesis is arguably the most important and famous unsolved problem in number theory. It is usually phrased in terms of the Riemann zeta function $latex {\zeta}&fg=000000$, defined by

$latex \displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}&fg=000000$

for $latex {\hbox{Re}(s)>1}&fg=000000$ and extended meromorphically to other values of $latex {s}&fg=000000$, and asserts that the only zeroes of $latex {\zeta}&fg=000000$ in the critical strip $latex {\{ s: 0 \leq \hbox{Re}(s) \leq 1 \}}&fg=000000$ lie on the critical line $latex {\{ s: \hbox{Re}(s)=\frac{1}{2} \}}&fg=000000$.

One of the main reasons that the Riemann hypothesis is so important to number theory is that the zeroes of…

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