ErgodicPNT

However, it turns out that if one of the sets, say A, is sufficiently “uniform” or “pseudorandom”, then one can always solve this Goldbach-type problem, regardless of what the other two sets are. This type of fact is often established by Fourier-analytic means (or by closely related techniques, such as spectral theory), but let me give a heuristic combinatorial argument to indicate why one would expect this type of phenomenon to occur.

quares Primes Lagrange’s four square theorem: For every positive integer N, there exists a pattern in of the form . Vinogradov’s theorem: For every sufficiently large integer N, there exists a pattern in of the form . Fermat’s two square theorem: For every prime number , there exists a pattern in of the form . Even Goldbach conjecture: For every even number , there exists a pattern in of the form . Fermat’s four square theorem: There does not exist any pattern in of the form with . Green-Tao theorem: For any , there exist infinitely many patterns in of the form with . Pell’s equation: There are infinitely many patterns in of the form . Sophie Germain conjecture: There are infinitely many patterns in of the form .

A new mathematical object, an elusive cousin of the Riemann zeta-function, was revealed to great acclaim recently at the American Institute of Mathematics. Ce Bian and Andrew Booker from the University of Bristol showed the first example of a third degree transcendental L-function.

Functional equations shed light on the properties of those functions that satisfy them, and for L-functions F(s) the functional equation is:   where q is an integer called the level, d is the degree, and the numbers  are Langland's parameters. is an analytic continuation of the factorial function  that is valid not only for integers but all complex numbers. There are two types of L-functions: algebraic and transcendental. These are classified according to their degree. If the Langland's parameters are rational or algebraic (that is, are complex numbers that are roots of non-zero polynomials with rational coefficients), then the L-function is algebraic. If these numbers are transcendental (that is, non-algebraic, such as  or  , then the L-function is transcendental. The Riemann zeta-function is the L-function where the level is 1, the degree is 1 and the Langland's parameters are 0 — that is, a first degree algebraic L-function. The Bristol researchers showed the first example of a third degree transcendental L-function.

The paper itself is called “Hypergraph regularity and the multidimensional Szemerédi theorem.” At the bottom level, the basic idea of the paper is due to Ruzsa, Szemerédi and Rödl. Ruzsa and Szemerédi started the ball rolling with a short and very clever argument that showed that Szemerédi’s famous theorem on arithmetic progressions, in the case of progressions of length 3, could be deduced from Szemerédi’s almost as famous regularity lemma, a remarkable result that allows any graph to be partitioned into a bounded number of pieces, almost all of which “behave randomly.”

Let $p$ be a fixed prime number, and $N$ be a large integer. The 'Inverse Conjecture for the Gowers norm' states that if the "$d$-th Gowers norm" of a function $f:\F_p^N \to \F_p$ is non-negligible, that is larger than a constant independent of $N$, then $f$ can be non-trivially approximated by a degree $d-1$ polynomial. The conjecture is known to hold for $d=2,3$ and for any prime $p$. In this paper we show the conjecture to be false for $p=2$ and for $d = 4$, by presenting an explicit function whose 4-th Gowers norm is non-negligible, but whose correlation any polynomial of degree 3 is exponentially small.Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao \cite{gt07}. Their analysis uses a modification of a Ramsey-type argument of Alon and Beigel \cite{ab} to show inapproximability of certain functions by low-degree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime $p$, for $d = p^2$.

Solving Mathematical Problems: A personal perspective. 2nd Edition, Terence Tao. Oxford University P ?The Journey of a thousand miles begins with one step? ? Lao Tzu Every so often, you come across a book that really stands out. I have recently been very fortunate to come across several such books, this being one of them. ?Solving mathematical problems? was written by Terence Tao when he was a 15 year old student and has now been slightly revised in this second edition. Like another of the reviewers at Amazon, I also came across this book after reading an article about Terence Tao winning the Fields medal (a bit like the Nobel prize for mathematics). Not only does it give a wonderful insight into the mind of a young Terence Tao, but also into the techniques used to elegantly solve some reasonably difficult problems, such as those posed as questions for the Maths Olympiad contests. [Terence competed in these challenges in his teens, winning bronze, silver and then gold.] Mathematical researchers are not always great e\ucators. Thankfully, Prof. Tao is.Mainly assuming only basic high-school pure mathematics, worked solutions to the problems are clearly and expertly described. Not only does he solve the problems but he also examines the steps, false starts and other solution possibilities that are part of the general approach to problem solving. I was only slightly disappointed that there were a handful of corrections in this second edition (available at Prof Tao?s blog here); one or two could perplex an unwary reader who might expect the work to be flawless. If you have an interest in mathematics, either as a high school student or a hobbyist, I would highly recommended reading this book. In the preface, Prof Tao remarks that if he wrote a book on the subject of competition problem-solving now, it would very different; now that is definitely a book I would like to read!….

Made to Stick: Why Some Ideas Survive and Others Die by Chip and Dan Heath helps us understand why our users (or our coworkers) can repeat the latest web hoax, but can’t remember anything about our projects. What we need to do is to create “sticky messages.” Sticky messages are not necessarily creative messages. In fact, there is formula that the brothers Heath have discovered that will help us to create sticky, memorable messages. That formula is:

An alternate argument for the arithmetic large sieve inequality September 2008 This short note describes a very natural and well-motivated derivation of the "arithmetic" large sieve inequality from the dual of the analytic inequality, which avoids the usual trick of submultiplicativity of Gallagher. This is also described in a blog post.

trick, that can be used in many mathematical situations. With such tricks, it is usually difficult, and in any case not desirable, to formalize them as lemmas: if you try to do so then almost certainly your formal lemma will not apply in all the situations where the trick does.

Of course, in many cases, the devil really is in the details, but nevertheless knowing the overall strategy of proof is extremely valuable when trying to read that proof.

Yong-Hui Says: November 3, 2008 at 5:57 pm | Reply I am in MSRI for the cofference discrete Rigity. Green will give the first lecture. I just happen to find a question for that tricki wiki: Whether is there a common-shared refference system for that tricki wiki? Similar to that of Mathscinet of ams math review It will be a basic instrument for a mathematical website.

\F_p^N

Behrend's construction

澳门赌场圈内人都说，“赌王”何鸿燊有识人之明，赌业由不赌的文人来运作，高!

We describe a very general abstract form of sieve based on a large sieve inequality which generalizes both the classical sieve inequality of Montgomery (and its higher-dimensional variants), and our recent sieve for Frobenius over function fields. The general framework suggests new applications. We get some first results on the number of prime divisors of ``most'' elements of an elliptic divisibility sequence, and we develop in some detail ``probabilistic'' sieves for random walks on arithmetic groups, e.g., estimating the probability of finding a reducible characteristic polynomial at some step of a random walk on SL(n,Z). In addition to the sieve principle, the applications depend on bounds for a large sieve constant. To prove such bounds involves a variety of deep results, including Property (T) or expanding properties of Cayley graphs, and the Riemann Hypothesis over finite fields. It seems likely that this sieve can have further applications.

Abstract. The paper study counter-dependent pseudorandom number generators based on m-variate (m> 1) ergodic mappings of the space of 2-adic integers Z2. The sequence of internal states of these generators is defined by the recurrence law xi+1 = H B i (xi) mod 2 n, whereas their output sequence is zi = F B i (xi) mod 2 n; here xj, zj are m-dimensional vectors over Z2. It is shown how the results obtained for a univariate case could be extended to a multivariate case. 1.

Foundations of Cryptography. Basic Tools. Cambridge Univ – Goldreich - 2001

129 Uniform distribution of sequences – Kuipers, Niederreiter - 1974