Computing convolutions of measures

30 Jul

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

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

An improved Type I estimate

30 Jul

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…

30 Jul

Spaun brain simulation

30 Jul

Spaun brain simulation

Spaun brain simulation

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

30 Jul

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/

The Riemann hypothesis in various settings

28 Jul

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 $latex {\{ s: \hbox{Re}(s)=\frac{1}{2} \}}&fg=000000$.