If and are two absolutely integrable functions on a Euclidean space , then the convolution of the two functions is defined by the formula A simple application of the Fubini-Tonelli theorem shows that the convolution is well-defined almost everywhere, and yields another absolutely integrable function. In the case that…]]>

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…

View original post 2,091 more words

As in all previous posts in this series, we adopt the following asymptotic notation: is a parameter going off to infinity, and all quantities may depend on unless explicitly declared to be “fixed”. The asymptotic notation is then defined relative to this parameter. A quantity is said to be…]]>

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…

View original post 11,959 more words

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

]]>]]>

[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.]…]]>

[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…

View original post 12,730 more words

This salad is perfect for any summer gathering, party or lunch! The combination of fruit, nuts and dark leafy greens makes this salad not only delicious, but high up on a healthy scale. It also only took me 5 minutes to put it together. Easy and delicious, my favorite…]]>

This salad is perfect for any summer gathering, party or lunch!

The combination of fruit, nuts and dark leafy greens makes this salad not only delicious, but high up on a healthy scale.

It also only took me 5 minutes to put it together. Easy and delicious, my favorite combination

**Ingredients**:

- Apple Pecan Dressing
- Green Apple Slices
- Strawberries
- Cucumbers
- Sliced Almonds
- Dark Leafy Spring Lettuce
- Spinach Leaves

I found that this salad is better shaken. When you just pour the dressing over the lettuce, it tends to clump up on various ingredients and doesn’t reach to others. Trust me, this incredible dressing is so tasty, you’ll want it to reach every piece!

Make sure not to over-dress your salad. The delicate sweetness of the Apple Pecan dressing is delicious, but it’s easy to over do it. Start small, and add as needed.

Enjoy!

Ingredients: Fresh sliced red grapes Strawberries Cucumbers Feta Cheese Sliced Almonds Spinach Leaves Red Cabbage Black Pepper Apple Cider Vinaigrette Dressing I’ve got to admit this is one of my favorite salads yet! Perfect for a hot day, light and refreshing]]>

**Ingredients:**

- Fresh sliced red grapes
- Strawberries
- Cucumbers
- Feta Cheese
- Sliced Almonds
- Spinach Leaves
- Red Cabbage
- Black Pepper
- Apple Cider Vinaigrette Dressing

I’ve got to admit this is one of my favorite salads yet!

Perfect for a hot day, light and refreshing

John Huerta visited here for about a week earlier this month, and gave a couple of talks. The one I want to write about here was a guest lecture in the topics course Susama Agarwala and I were teaching this past semester. The course was about topics in category theory…]]>

John Huerta visited here for about a week earlier this month, and gave a couple of talks. The one I want to write about here was a guest lecture in the topics course Susama Agarwala and I were teaching this past semester. The course was about topics in category theory of interest to geometry, and in the case of this lecture, “geometry” means supergeometry. It follows the approach I mentioned in the previous post about looking at sheaves as a kind of generalized space. The talk was an introduction to a program of seeing supermanifolds as a kind of sheaf on the site of “super-points”. This approach was first proposed by Albert Schwartz, though see, for instance, this review by Christophe Sachse for more about this approach, and this paper (comparing the situation for real and complex (super)manifolds) for more recent work.

It’s amazing how many geometrical techniques can be…

View original post 3,029 more words