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科学网-SCI投稿常用英语 - 0 views

www.sciencenet.cn/...user_content.aspx

featured mail-module

shared by arithwsun arithwsun on 11 May 08 - Cached
  • arithwsun arithwsun on 11 May 08
     
    1. Dear Dr. Defendi ML: I am sending a manuscript entitled "" by - which I should like to submit for possible publication in the journal of - . Yours sincerely 2. Dear Dr. A: Enclosed is a manuscript entitled "" by sb, which we are submitting for publication in the journal of - . We have chosen this journal because it deals with - . We believe that sth would be of interest to the journal's readers. 3. Dear Dr. A: Please find enclosed for your review an original research article, "" by sb. All authors have read and approve this version of the article, and due care has been taken to ensure the integrity of the work. No part of this paper has published or submitted elsewhere. No conflict of interest exits in the submission of this manuscript, and we have attached to this letter the signed letter granting us permission to use Figure 1 from another source. We appreciate your consideration of our manuscript, and we look forward to receiving comments from the reviewers. 二、询问有无收到稿件 Dear Editors, We dispatched our manuscript to your journal on 3 August 2006 but have not, as yet, receive acknowledgement of their safe arrival. We fear that may have been lost and should be grateful if you would let us know whether or not you have received them. If not, we will send our manuscript again. Thank you in advance for your help. 三、询问论文审查回音 Dear Editors， It is more than 12 weeks since I submitted our manuscript (No: ) for possible publication in your journal. I have not yet received a reply and am wondering whether you have reached a decision. I should appreciated your letting me know what you have decided as soon as possible. 四、关于论文的总体审查意见 1. This is a carefully done study and the findings are of considerable interest. A few minor revision are list below. 2. This is a well-written paper containing interesting results which merit publication. For the benefit of the reader, however, a numb
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Structure and randomness in the prime numbers « What's new - 0 views

terrytao.wordpress.com/...andomness-in-the-prime-numbers

featured rh zeta

shared by arithwsun arithwsun on 17 Jul 08 - Cached
  • 2 July, 2008 at 6:28 pm Terence Tao It unfortunately seems that the decomposition claimed in equation (6.9) on page 20 of that paper is, in fact, impossible; it would endow the function h (which is holding the arithmetical information about the primes) with an extremely strong dilation symmetry which it does not actually obey. It seems that the author was relying on this symmetry to make the adelic Fourier transform far more powerful than it really ought to be for this problem.
  • 3 July, 2008 at 3:41 am Gergely Harcos I also have some (perhaps milder) troubles with the proof. It seems to me as if Li had treated the Dirac delta on L^2(A) as a function. For example, the first 5 lines of page 28 make little sense to me. Am I missing something here?
  • 4 July, 2008 at 5:15 am Lior Silberman The function defined on page 20 does have a strong dilation symmetry: it is invariant by multiplication by ideles of norm one (since it is merely a function of the norm of ). In particular, it is invariant under multiplication by elements of . I’m probably missing something here. Probably the subtlety is in passing from integration over the nice space of idele classes to the singular space . The topologies on the spaces of adeles and ideles are quite different. There is a formal error in Theorem 3.1 which doesn’t affect the paper: the distribution discussed is not unique. A distribution supported at a point is a sum of derivatives of the delta distribution. Clearly there exist many such with a given special value of the Fourier transform. There is also something odd about this paper: nowhere is it pointed out what is the new contribution of the paper. Specifically, what is the new insight about number theory?
  • ...12 more annotations...
  • 4 July, 2008 at 6:09 am Emmanuel Kowalski A remark concerning Lior’s remark: the function h(u) in the current (v4) version of the paper is _not_ the same as the one that was defined when T. Tao pointed out a problem with it. This earlier one (still visible on arXiv, v1) was defined in different ways depending on whether the idele had at most one or more than one non-unit component, and was therefore not invariant under multiplication by . (It is another problem with looking at such a paper if corrections as drastic as that are made without any indication of when and why).
  • 4 July, 2008 at 8:15 am Terence Tao Dear Lior, Emmanuel is correct. The old definition of h was in fact problematic for a large number of reasons (the author was routinely integrating h on the idele class group C, which is only well-defined if h was -invariant). Changing the definition does indeed fix the problem I pointed out (and a number of other issues too). But Connes has pointed out a much more serious issue, in the proof of the trace formula in Theorem 7.3 (which is the heart of the matter, and is what should be focused on in any future revision): the author is trying to use adelic integration to control a function (namely, h) supported on the ideles, which cannot work as the ideles have measure zero in the adeles. (The first concrete error here arises in the equation after (7.13): the author has made a change of variables on the idele class group C that only makes sense when u is an idele, but u is being integrated over the adeles instead. All subsequent manipulations involving the adelic Fourier transform Hh of h are also highly suspect, since h is zero almost everywhere on the adeles.)
  • More generally, there is a philosophical objection as to why a purely multiplicative adelic approach such as this one cannot work. The argument only uses the multiplicative structure of , but not the additive structure of k. (For instance, the fact that k is a cocompact discrete additive subgroup of A is not used.) Because of this, the arguments would still hold if we simply deleted a finite number of finite places v from the adeles (and from ). If the arguments worked, this would mean that the Weil-Bombieri positivity criterion (Theorem 3.2 in the paper) would continue to hold even after deleting an arbitrary number of places. But I am pretty sure one can cook up a function g which (assuming RH) fails this massively stronger positivity property (basically, one needs to take g to be a well chosen slowly varying function with broad support, so that the Mellin transforms at Riemann zeroes, as well as the pole at 1 and the place at infinity, are negligible but which gives a bad contribution to a single large prime (and many good contributions to other primes which we delete).)
  • Emmanuel Kowalski That’s an interesting point indeed, if one considers that the RH doesn’t work over function fields once we take out a point of a (smooth projective) curve — there arise zeros of the zeta function which are not on the critical line.
  • 6 July, 2008 at 5:28 pm Chip Neville Terence, I have a question about your comment: “Because of this, the arguments would still hold if we simply deleted a finite number of finite places v from the adeles (and from k^*). … (basically, one needs to take g to be a well chosen slowly varying function with broad support, so that the Mellin transforms at Riemann zeroes, as well as the pole at 1 and the place at infinity, are negligible but which gives a bad contribution to a single large prime (and many good contributions to other primes which we delete).)” Does this mean that you would be considering the “reduced” (for lack of a better name) zeta function \prod 1/(1-1/p^{-s}), where the product is taken over the set of primes not in a finite subset S? If so, this “reduced” zeta function has the same zeroes as the standard Riemann zeta function, since the finite product \prod_S 1/(1-1/p^{-s}) is an entire function with no zeroes in the complex plane. Thus the classical situation in the complex plane seems to be very different in this regard from the situation with function fields over smooth projective curves alluded to by Emmanuel above. Does anyone have an example of an infinite set S and corresponding reduced zeta function with zeroes in the half plane Re z > 1/2? A set S of primes p so that \sum_S 1/p^{1/2} converges will not do, since \prod_S 1/(1-1/p^{-s}) is holomorphic in the half plane Re z > 1/2 with no zeroes there. Perhaps a set S of primes P thick enough so that \sum_S 1/p^{1/2} diverges, but thin enough so that \sum_S 1/p converges, might do. This seems to me to be a delicate and difficult matter. I hope these questions do not sound too foolish.
  • 6 July, 2008 at 7:44 pm Terence Tao Dear Chip, Actually, the product has a number of poles on the line , when s is a multiple of . Li’s approach to the RH was not to tackle it directly, but instead to establish the Weil-Bombieri positivity condition which is known to be equivalent to RH. However, the proof of that equivalence implicitly uses the functional equation for the zeta function (via the explicit formula). If one starts deleting places (i.e. primes) from the problem, the RH stays intact (at least on the half-plane ), but the positivity condition does not, because the functional equation has been distorted.
  • The functional equation, incidentally, is perhaps the one non-trivial way we do know how to exploit the additive structure of k inside the adeles, indeed I believe this equation can be obtained from the Poisson summation formula for the adeles relative to k. But it seems that the functional equation alone is not enough to yield the RH; some other way of exploiting additive structure is also needed, but I have no idea what it should be. [Revised, July 7:] Looking back at Li’s paper, I see now that Poisson summation was indeed used quite a few times, and in actually a rather essential way, so my previous philosophical objection does not actually apply here. My revised opinion is now that, beyond the issues with the trace formula that caused the paper to be withdrawn, there is another fundamental problem with the paper, which is that the author is in fact implicitly assuming the Riemann hypothesis in order to justify some facts about the operator E (which one can think of as a sort of Mellin transform multiplier with symbol equal to the zeta function, related to the operator on ). More precisely, on page 18, the author establishes that and asserts that this implies that , but this requires certain invertibility properties of E which fail if there is a zero off of the critical line. (A related problem is that the decomposition used immediately afterwards is not justified, because is merely dense in rather than equal to it.)
  • 7 July, 2008 at 9:59 am javier Dear Terence, I am not sure I understand your “philosophical” complain on using only the multiplicative structure and not the additive one. This is essentially the philosophy while working over the (so over-hyped lately) field with one element, which apparently comes into the game in the description of the Connes-Bost system on the latest Connes-Consani-Marcolli paper (Fun with F_un). From an algebraic point of view, you can often recover the additive structure of a ring from the multiplicative one provided that you fix the zero. There is an explanation of this fact (using the language of monads) in the (also famous lately) work by Nikolai Durov “A new approach to Arakelov geometry (Section 4.8, on additivity on algebraic monads). By the way, I wanted to tell you that I think you are doing an impressive work with this blog and that I really enjoy learning from it, even if this is the very first time I’ve got something sensible to say :-)
  • 7 July, 2008 at 11:01 am Terence Tao Dear Javier, I must confess I do not understand the field with one element much at all (beyond the formal device of setting q to 1 in any formula derived using and seeing what one gets), and don’t have anything intelligent to say on that topic. Regarding my philosophical objection, the point was that if one deleted some places from the adele ring A and the multiplicative group (e.g. if k was the rationals, one could delete the place 2 by replacing with the group of non-zero rationals with odd numerator and denominator) then one would still get a perfectly good “adele” ring in place of A, and a perfectly good multiplicative group in place of (which would be the invertible elements in the ring of rationals with odd denominator), but somehow the arithmetic aspects of the adeles have been distorted in the process (in particular, Poisson summation and the functional equation get affected). The Riemann hypothesis doesn’t seem to extend to this general setting, so that suggests that if one wants to use adeles to prove RH, one has to somehow exploit the fact that one has all places present, and not just a subset of such places. Now, Poisson summation does exploit this very fact, and so technically this means that my objection does not apply to Li’s paper, but I feel that Poisson summation is not sufficient by itself for this task (just as the functional equation is insufficient to resolve RH), and some further exploitation of additive (or field-theoretic) structure of k should be needed. I don’t have a precise formalisation of this feeling, though.
  • 7 July, 2008 at 1:22 pm Gergely Harcos Dear Terry, you are absolutely right that Poisson summation over k inside A is the (now) standard way to obtain the functional equation for Hecke L-functions. This proof is due to Tate (his thesis from 1950), you can also find it in Weil’s Basic Number Theory, Chapter 7, Section 5.
  • 15 July, 2008 at 7:57 am michele I think that the paper of Prof. Xian-Jin Li will be very useful for a future and definitive proof of the Riemann hypothesis. Furthermore, many mathematics contents of this paper can be applied for further progress in varios sectors of theoretical physics (p-adic and adelic strings, zeta strings).
  • Babak Hi Terrance, A few months ago I stumbled upon an interesting differential equation while using probability heuristics to explore the distribution of primes. It’s probably nothing, but on the off-chance that it might mean something to a better trained mind, I decided to blog about it: http://babaksjournal.blogspot.com/2008/07/differential-equation-estimating.html -Babak
arithwsun arithwsun

Szemeredi's theorem - 30 views

http://in-theory.blogspot.com/2006_05_28_archive.html in theory Saturday, June 03, 2006 Szemeredi's theorem Szemeredi's theorem on arithmetic progressions is one of the great triumphs of the "Hung...

szemeredi

started by arithwsun arithwsun on 03 Sep 07 no follow-up yet
Ke Gong

Papers of Paul Erdos - 0 views

www.renyi.hu/...Erdos4ht.html

collected erdos papers

shared by Ke Gong on 06 May 08 - Cached
  • Ke Gong on 06 May 08
     
    Papers of Paul Erdos
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A paper on the ArXiV « Gowers's Weblog - 0 views

gowers.wordpress.com/...a-paper-on-the-arxiv

ergodicpnt gowers szemeredi

shared by arithwsun arithwsun on 17 Oct 07 - Cached
  • 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.”
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[math/0606087] Quadratic Uniformity of the Mobius Function - 0 views

cn.arxiv.org/0606087

ergodicpnt mobius-function

shared by arithwsun arithwsun on 04 Mar 08 - Cached
  • Quadratic Uniformity of the Mobius Function Authors: Ben Green, Terence Tao (Submitted on 4 Jun 2006 (v1), last revised 22 Sep 2007 (this version, v2)) Abstract: This paper is a part of our programme to generalise the Hardy-Littlewood method to handle systems of linear questions in primes. This programme is laid out in our paper Linear Equations in Primes [LEP], which accompanies this submission. In particular, the results of this paper may be used, together with the machinery of [LEP], to establish an asymptotic for the number of four-term progressions p_1 < p_2 < p_3 < p_4 <= N of primes, and more generally any problem counting prime points inside a ``non-degenerate'' affine lattice of codimension at most 2. The main result of this paper is a proof of the Mobius and Nilsequences Conjecture for 1 and 2-step nilsequences. This conjecture is introduced in [LEP] and amounts to showing that if G/\Gamma is an s-step nilmanifold, s <= 2, if F : G/\Gamma -> [-1,1] is a Lipschitz function, and if T_g : G/\Gamma -> G/\Gamma is the action of g \in G on G/\Gamma, then the Mobius function \mu(n) is orthogonal to the sequence F(T_g^n x) in a fairly strong sense, uniformly in g and x in G/\Gamma. This can be viewed as a ``quadratic'' generalisation of an exponential sum estimate of Davenport, and is proven by the following the methods of Vinogradov and Vaughan.
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[0807.1736] The Mobius and Nilsequences Conjecture - 0 views

arxiv.org/0807.1736

featured green-tao mobius-function nilsequences

shared by arithwsun arithwsun on 15 Jul 08 - Cached
  • We show that the Mobius function mu(n) is strongly asymptotically orthogonal to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected nilpotent Lie group with a discrete and cocompact subgroup L (so G/L is a nilmanifold), g : Z -> G is a polynomial sequence and F: G/L -> R is a Lipschitz function. More precisely, we show that the inner product of mu(n) with F(g(n)L) over {1,...,N} is bounded by 1/log^A N, for all A > 0. In particular, this implies the Mobius and Nilsequence conjecture MN(s) from our earlier paper ``Linear equations in primes'' for every positive integer s. This is one of two major ingredients in our programme, outlined in that paper, to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collection \psi_1,...,\psi_t : Z^d -> Z of linear forms all take prime values. The proof is a relatively quick application of the results in our recent companion paper on the distribution of polynomial orbits on nilmanifolds. We give some applications of our main theorem. We show, for example, that the Mobius function is uncorrelated with any bracket polynomial. We also obtain a result about the distribution of nilsequences n -> a^nxL as n ranges only over the primes.
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Structure and randomness in combinatorics « What's new - 0 views

terrytao.wordpress.com/...nd-randomness-in-combinatorics

combinatorics pseudorandom tao

shared by arithwsun arithwsun on 01 Aug 07 - Cached
  • arithwsun arithwsun on 01 Aug 07
     
    I've just uploaded to the arXiv my lecture notes "Structure and randomness in combinatorics" for my tutorial at the upcoming FOCS 2007 conference in October. This tutorial covers similar ground as my ICM paper (or slides), or my first two Simons lectures, but focuses more on the "nuts-and-bolts" of how structure theorems actually work to separate objects into structured pieces and pseudorandom pieces, for various definitions of "structured" and "pseudorandom".  Given that the target audience consists of computer scientists, I have focused exclusively here on the combinatorial aspects of this dichotomy (applied for instance to functions on the Hamming cube) rather than, say, the ergodic theory aspects (which are covered in Bryna Kra's lecture notes from Montreal, or my notes from Montreal for that matter).  While most of the known applications of these decompositions are number-theoretic (e.g. my theorem with Ben Green), the number theory aspects are not covered in detail in these notes.  (For that, you can read Bernard Host's Bourbaki article, Ben Green's http
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Pseudorandom number generation by p-adic ergodic transformations. arXiv Mathematics - C... - 0 views

citeseerx.ist.psu.edu/...F61B8723F2551CBF2AD922A81A9AA6

ergodic p-adic pseudorandom

shared by arithwsun arithwsun on 12 May 08 - Cached
  • 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
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  • 22 The art of computer programming. Vol. 2: Seminumerical Algorithms – Knuth - 1981
  • 8 Uniformly distributed sequences of p-adic integers – Anashin - 1994
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SpringerLink - Journal Article - 0 views

www.springerlink.com/...e027w4211n7784h1

ergodic ergodicpnt paper primes

shared by arithwsun arithwsun on 01 May 08 - Cached
  • Abstract  The pointwise ergodic theorem is proved for prime powers for functions inL p,p>1. This extends a result of Bourgain where he proved a similar theorem forp>(1+√3)/2.
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[0812.2222] The Large Sieve and Galois Representations - 0 views

search.arxiv.org/paper.jsp

large-sieve galois-representations

shared by arithwsun arithwsun on 23 Dec 08 - Cached
  • We describe a generalization of the large sieve to situations where the underlying groups are nonabelian, and give several applications to the arithmetic of abelian varieties. In our applications, we sieve the set of primes via the system of representations arising from the Galois action on the torsion points of an abelian variety. The resulting upper bounds require explicit character sum calculations, with stronger results holding if one assumes the Generalized Riemann Hypothesis.
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Notes and unpublished papers of Emmanuel Kowalski - 0 views

www.math.ethz.ch/...notes-unpublished.html

kowalski large-sieve expositary

shared by arithwsun arithwsun on 24 Oct 08 - Cached
  • 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.
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Front: [arXiv:0711.3388] Inverse Conjecture for the Gowers norm is false - 0 views

front.math.ucdavis.edu/0711.3388

ergodicpnt gowers-norm inverse-conj

shared by arithwsun arithwsun on 25 Nov 07 - Cached
  • 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 $d = p^2$.
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Bryna Kra's web age. - 0 views

www.math.northwestern.edu/...papers.html

ergodicpnt homepage

shared by arithwsun arithwsun on 31 Jul 07 - Cached
  • arithwsun arithwsun on 31 Jul 07
     
    The Green-Tao Theorem on arithmetic progressions in the primes: an ergodic point of view.
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[0711.3388] Inverse Conjecture for the Gowers norm is false - 0 views

cn.arxiv.org/0711.3388

ergodicpnt gowers-norm

shared by arithwsun arithwsun on 07 Feb 08 - Cached
  • Inverse Conjecture for the Gowers norm is false Authors: Shachar Lovett, Roy Meshulam, Alex Samorodnitsky (Submitted on 21 Nov 2007) Abstract: 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 $d = p^2$. Comments: 20 pages
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[math/0603450] Pseudo Random test of prime numbers - 0 views

cn.arxiv.org/0603450

ergodicpnt primes pseudorandom

shared by arithwsun arithwsun on 28 Jan 08 - Cached
  • Pseudo Random test of prime numbers Authors: Wang Liang; Huang Yan (Submitted on 18 Mar 2006) Abstract: The prime numbers look like a randomly chosen sequence of natural numbers, but there is still no strict theory to determine 'Randomness'. In these years, cryptography has developed a battery of statistical tests for randomness. In this paper, we just apply these methods to study the distribution of primes. Here the binary sequence constructed by second difference of primes is used as samples. We find this sequence can't reach all the 'random standard' of FIPS 140-1/2, but still show obvious random feature. The interesting self-similarity is also observed in this sequence. These results add the evidence that prime numbers is a chaos system.
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[PAMQ] Obstructions to Uniformity and Arithmetic Patterns in the Primes - 0 views

qjpam.henu.edu.cn/onlineinfo.jsp

ergodicpnt published-version

shared by arithwsun arithwsun on 31 Jul 07 - Cached
  • arithwsun arithwsun on 31 Jul 07
     
    Published version, can be downloaded freely. PAMQ is a new journal with many beautiful papers.
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math.CO/0602037: A correspondence principle between (hyper)graph theory and probability... - 0 views

front.math.ucdavis.edu/...0602037

correspondence-principle ergodic ergodicpnt graph transference

shared by arithwsun arithwsun on 31 Jul 07 - Cached
  • arithwsun arithwsun on 31 Jul 07
     
    The setting of this paper was deliberately placed at a midpoint between graph theory and ergodic theory, and the author hopes that it illuminates the analogies and interconnections between these two subjects.
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