Quadratic forms

We study unitary random matrix ensembles of the form −1 Zn,N | det M |2α e−N Tr V (M ) dM, where α −1/2 and V is such that the limiting mean eigenvalue density for n, N → ∞ and n/N → 1 vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight |x|2α e−N V (x) . ...

42p dontetvui 17-01-2013 73 7   Download

We settle an old question about the existence of certain ‘sums-of-squares’ formulas over a field F , related to the composition problem for quadratic forms. A classical theorem says that if such a formula exists over a field of characteristic 0, then certain binomial coefficients must vanish. We prove that this result also holds over fields of characteristic p

23p noel_noel 17-01-2013 49 7   Download

An exact sequence for KM /2 ∗ with applications to quadratic forms By D. Orlov,∗ A. Vishik,∗∗ and V. Voevodsky∗∗* Contents 1. Introduction 2. An exact sequence for KM /2 ∗ 3. Reduction to points of degree 2 4. Some applications 4.1. Milnor’s Conjecture on quadratic forms 4.2. The Kahn-Rost-Sujatha Conjecture 4.3. The J-filtration conjecture 1. Introduction Let k be a field of characteristics zero. For a sequence a = (a1 , . . . , an ) of invertible elements of k consider the homomorphism M M K∗ (k)/2 → K∗+n (k)/2 in Milnor’s K-theory modulo elements divisible by 2 defined by...

14p noel_noel 17-01-2013 59 5   Download

Introduction In 1903 Voronoi [42] postulated the existence of explicit formulas for sums of the form (1.1) n≥1 an f (n) , for any “arithmetically interesting” sequence of coefficients (an )n≥1 and every f in a large class of test functions, including characteristic functions of bounded intervals. He actually established such a formula when an = d(n) is the number of positive divisors of n [43]. He also asserted a formula for (1.2) an = #{(a, b) ∈ Z2 | Q(a, b) = n} , where Q denotes a positive definite integral quadratic form [44]; ...

67p noel_noel 17-01-2013 55 8   Download

The Oppenheim conjecture, proved by Margulis [Mar1] (see also [Mar2]), asserts that for a nondegenerate indefinite irrational quadratic form Q in n ≥ 3 variables, the set Q(Zn ) is dense. In [EMM] (where we used the lower bounds established in [DM]) a quantitative version of the conjecture was established. Namely: Let ρ be a continuous positive function on the sphere {v ∈ Rn | v = 1}, and let Ω = {v ∈ Rn | v

48p noel_noel 17-01-2013 40 5   Download

We show that any analytically integrable Hamiltonian system near an equilibrium point admits a convergent Birkhoff normalization. The proof is based on a new, geometric approach to the topic. 1. Introduction Among the fundamental problems concerning analytic (real or complex) Hamiltonian systems near an equilibrium point, one may mention the following two: 1) Convergent Birkhoff. In this paper, by “convergent Birkhoff” we mean a normalization, i.e.

17p noel_noel 17-01-2013 41 4   Download

Annals of Mathematics In our first article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields, while the spaces underlying those composition laws were closely related to certain exceptional Lie groups.

23p tuanloccuoi 04-01-2013 62 6   Download

Two centuries ago, in his celebrated work Disquisitiones Arithmeticae of 1801, Gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow. Even today, two centuries later, this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders. It is hence only natural to ask whether higher analogues of this composition law exist that could shed light on the structure of other algebraic number rings and fields. ...

35p tuanloccuoi 04-01-2013 52 8   Download

Under explicit diophantine conditions on (α, β) ∈ R2 , we prove that the local two-point correlations of the sequence given by the values (m − α)2 + (n−β)2 , with (m, n) ∈ Z2 , are those of a Poisson process. This partly confirms a conjecture of Berry and Tabor [2] on spectral statistics of quantized integrable systems, and also establishes a particular case of the quantitative version of the Oppenheim conjecture for inhomogeneous quadratic forms of signature (2,2). The proof uses theta sums and Ratner’s classification of measures invariant under unipotent flows. ...

54p tuanloccuoi 04-01-2013 56 8   Download

We prove that the Birkhoff normal form of hamiltonian flows at a nonresonant singular point with given quadratic part is always convergent or generically divergent. The same result is proved for the normalization mapping and any formal first integral. Introduction In this article we study analytic (R or C-analytic) hamiltonian flows xk ˙ yk ˙ ∂H , ∂yk ∂H = − , ∂xk = + where xk , yk ∈ C (resp. R), k = 1, 2, . . . n, and H is an analytic hamiltonian with power series expansion at 0 beginning with quadratic terms (so that 0...

19p tuanloccuoi 04-01-2013 50 5   Download

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Multidimensional Functional Equation Having Quadratic Forms as Solutions

8p sting10 29-02-2012 48 6   Download

Tham khảo luận văn - đề án 'báo cáo hóa học: "research article a multidimensional functional equation having quadratic forms as solutions"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả

8p sting10 29-02-2012 82 6   Download

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một số chỉ tiêu giới hạn và bất bình đẳng hình thức bậc hai cho các nhà khai thác Schroedinger...

20p matuot_266 27-08-2011 42 3   Download

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một số chỉ tiêu giới hạn và bất bình đẳng hình thức bậc hai cho các nhà khai thác Schroedinger...

16p matuot_266 27-08-2011 63 4   Download