Crooked binomials

A function f : GF(2 ^r ) → GF(2 ^r ) is called crooked if the sets {f(x) + f(x + a)|x ∈ GF(2 ^r )} is an affine hyperplane for any nonzero a ∈ GF(2 ^r ). We prove that a crooked binomial function f(x) = x ^d + ux ^e defined on GF(2 ^r ) satisfies that both exponents d, e have 2-weights at most 2.      

Arithmetical properties of the number of t-core partitions

Let Λ={λ ₁≥⋅⋅⋅≥λ _s ≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ _i nodes in the ith row. Let λ _j ′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ _i +λ _j ′−i−j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 ^t -core partitions of n are obtained. A simple convolution identity for t-cores is also given.      

Dependent percolation in two dimensions

For a natural number k, define an oriented site percolation on ℤ² as follows. Let x _i , y _j be independent random variables with values uniformly distributed in {1, …, k}. Declare a site (i, j) ∈ℤ² closed if x _i = y _j , and open otherwise. Peter Winkler conjectured some years ago that if k≥ 4 then with positive probability there is an infinite oriented path starting at the origin, all of whose sites are open. I.e., there is an infinite path P = (i ₀, j ₀)(i ₁, j ₁) · · · such that 0 = i ₀≤i ₁≤· · ·, 0 = j ₀≤j ₁≤· · ·, and each site (i _n , j _n ) is open. Rather surprisingly, this conjecture is still open: in fact, it is not known whether the conjecture holds for any value of k. In this note, we shall prove the weaker result that the corresponding assertion holds in the unoriented case: if k≤ 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive. Furthermore, we shall show that our method can be applied to a wide variety of distributions of (x _i ) and (y _j ). Independently, Peter Winkler [14] has recently proved a variety of similar assertions by different methods. Received: 4 March 1999 / Revised version: 27 September 1999 / Published online: 21 June 2000     

The second centralizer of a Bernoulli shift is just its powers

If ℐ is a collection of measure preserving transformations of a probability space, byC(ℐ), the centralizer of ℐ, we mean the group of all measure preserving transformationsS such thatTS=ST for allT ∈ ℐ. We show here that ifT is a Bernoulli shift, thenC(C(T))={T ⁱ |i ∈ Z}. The proof is carried out by constructing an action of Z², {T ₁ ⁱ °T ₂ ⁱ |i, j ∈ Z}, whereT ₁ is a Bernoulli shift of arbitrary entropy, but for anyj ≠ 0,C({T ₁,T ₂ ^i} ={T ₁ ⁱ °T ₂ ^k l, k ∈ Z}. The construction is a two-dimensional analogue of Ornstein’s “rank one mixing” transformation.     

Field induced aggregation in electrorheological suspension: kernel form and self similar solutions

Within the framework of the study of the fibrillation mechanism in an electrorheological (ER) suspension, this work presents a comparison between the self similar solutions when the kernel is K_i,j ~ (i⁻¹ + j⁻¹) and the behaviour of the chains growth. Till now, the field induced chains formation has only been studied by numerical or experimental methods. The work of Fournier and Lauren?ot (Communications in Mathematical Physics 256 2005) on the Smoluchowski’s equation allows us to present an analytical solution for the field induced pearl chains in a colloidal ER suspension.     

Regularity of the attractor for 3-D complex Ginzburg-Landau equation

In this paper,the existence of global attractor for 3-D complex Ginzburg Landau equation is considered.By a decomposition of solution operator,it is shown that the global attractor A_i in H~i(Ω) is actually equal to a global attractor Aj in H~j(Ω)(i≠j,i,j = 1,2,…m).     

Smooth convex bodies with proportional projection functions

For a convex body K ⊂ ℝⁿ and i ∈ {1, …, n − 1}, the function assigning to any i-dimensional subspace L of ℝⁿ, the i-dimensional volume of the orthogonal projection of K to L, is called the i-th projection function of K. Let K, K ₀ ⊂ ℝⁿ be smooth convex bodies with boundaries of class C ² and positive Gauss-Kronecker curvature and assume K ₀ is centrally symmetric. Excluding two exceptional cases, (i, j) = (1, n − 1) and (i, j) = (n − 2, n − 1), we prove that K and K ₀ are homothetic if their i-th and j-th projection functions are proportional. When K ₀ is a Euclidean ball this shows that a convex body with C ² boundary and positive Gauss-Kronecker with constant i-th and j-th projection functions is a Euclidean ball. The second author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.     

Some Physics Questions in Hyperbolic Complex Space

In hyperbolic complex space, the Clifford algebra is isomorphic to that of a corresponding Minkowski geometry. We define the hyperbolic imaginary unit j (j² = 1, j ≠   ±  1, j*  =   − j) to generate a class of Clifford algebras. We can introduce a class of non-Euclidean spaces and discuss the general form of 4-dimensional Lorentz transformation, and related special relativistic physics.     

The heat operator and mock Jacobi forms

We study a necessary and sufficient condition for Jacobi integrals of weight -r+\fracj2-r+\frac{j}{2}, r∈ℤ≥0, and index ℳ^(j) on ℋ×ℂ ^j to have a dual Jacobi form of weight r+\fracj2+2r+\frac{j}{2}+2 and index ℳ^(j). Such a meromorphic Jacobi integral with a dual Jacobi form is called a mock Jacobi form whose concept was first introduced by Zagier in Séminaire Bourbaki, 60éme année, 2006–2007, N° 986. In fact, we show the map L^r+1_M^(j)L^{r+1}_{\mathcal{M}^{(j)}} from the space of mock Jacobi forms to that of Jacobi forms is surjective by constructing the corresponding inverse image via Eichler integral of vector valued modular forms which are coming from the theta decomposition of Jacobi forms. We discuss Lerch sums as a typical example.     

Sums of almost equal prime cubes

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 5; 6; 7; 8, i.e., N = p ₁³ + ... + p _j ³ with |p _i − (N/j)^1/3| ≦ $ N^{1/3 - \delta _j + \varepsilon } $ N^{1/3 - \delta _j + \varepsilon } (1 ≦ i ≦ j), for δ _j = 1/45; 1/30; 1/25; 2/45, respectively.     

On the value distribution of positive definite quadratic forms

Denote by 0 = λ ₀ < λ ₁ ≤ λ ₂ ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l −1)-dimensional interval I = I ₂×. . .×I _l we consider the l-level correlation function K^(l)_I(R){K^{(l)}_I(R)} which counts the number of tuples (i ₁, . . . , i _l ) such that l_i1,?,l_il ￡ R²{\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2} and l_ij-l_i1 ? I_j{\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j} for 2 ≤ j ≤ l. We study the asymptotic behavior of K^(l)_I(R){K^{(l)}_I(R)} as R tends to infinity. If k ≥ 4 we prove K^(l)_I(R) ~ c_l(Q) vol(I)R^lk-2(l-1){K^{(l)}_I(R)\sim c_l(Q)\,{\rm vol}(I)R^{lk-2(l-1)}} for arbitrary l, where c _l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3.     

Field induced aggregation in electrorheological suspension: kernel form and self similar solutions

Within the framework of the study of the fibrillation mechanism in an electrorheological (ER) suspension, this work presents a comparison between the self similar solutions when the kernel is K_i,j ~ (i⁻¹ + j⁻¹) and the behaviour of the chains growth. Till now, the field induced chains formation has only been studied by numerical or experimental methods. The work of Fournier and Lauren?ot (Communications in Mathematical Physics 256 2005) on the Smoluchowski’s equation allows us to present an analytical solution for the field induced pearl chains in a colloidal ER suspension. René Limage: Chercheur indépendant, dipl?mé de l’Université de Liége.     

Nearly tight frames and space-frequency analysis on compact manifolds

Let M be a smooth compact oriented Riemannian manifold of dimension n without boundary, and let Δ be the Laplace–Beltrami operator on M. Say , and that f (0)  =  0. For t  >  0, let K _t (x, y) denote the kernel of f (t ² Δ). Suppose f satisfies Daubechies’ criterion, and b  >  0. For each j, write M as a disjoint union of measurable sets E _j,k with diameter at most ba ^j , and measure comparable to if ba ^j is sufficiently small. Take x _j,k ∈ E _j,k . We then show that the functions form a frame for (I  −  P)L ²(M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ (I  −  P)L ² is in space and in frequency, we can describe which terms in the summation are so small that they can be neglected. If n  =  2 and M is the torus or the sphere, and f (s)  =  se ^−s (the “Mexican hat” situation), we obtain two explicit approximate formulas for the φ _j,k , one to be used when t is large, and one to be used when t is small. A. Mayeli was partially supported by the Marie Curie Excellence Team Grant MEXT-CT-2004-013477, Acronym MAMEBIA.     

Riesz basis, Paley-Wiener class and tempered splines

The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ∮2m(T,R)∩L2(R) which is the space of polynomial splines with irregularly distributed nodes T={tj}j∈Z, where {tj}j∈Z is a real sequence such that {eitξ}j∈Z constitutes a Riesz basis for L2([-π,π]). From these results, the asymptotic relation E(f,Bπ,2)2=lim E(f,∮2m(T,R)∩L2(R))2 is proved, where Bπ,2 denotes the set of all functions from L2(R) which can be continued to entire functions of exponential type ≤π, i.e. the classical Paley-Wiener class.     

On the number of affine equivalence classes of spherical tube hypersurfaces

We consider Levi non-degenerate tube hypersurfaces in \mathbbCⁿ⁺¹{\mathbb{C}^{n+1}} that are (k, n − k)-spherical, i.e. locally CR-equivalent to the hyperquadric with Levi form of signature (k, n − k), with n ≤ 2k. We show that the number of affine equivalence classes of such hypersurfaces is infinite (in fact, uncountable) in the following cases: (i) k = n − 2, n ≥ 7; (ii) k = n − 3, n ≥ 7; (iii) k ≤ n − 4. For all other values of k and n, except for k = 3, n = 6, the number of affine classes was known to be finite. The exceptional case k = 3, n = 6 has been recently resolved by Fels and Kaup who gave an example of a family of (3, 3)-spherical tube hypersurfaces that contains uncountably many pairwise affinely non-equivalent elements. In this paper we deal with the Fels–Kaup example by different methods. We give a direct proof of the sphericity of the hypersurfaces in the Fels–Kaup family, and use the j-invariant to show that this family indeed contains an uncountable subfamily of pairwise affinely non-equivalent hypersurfaces.     

Sudakov-type minoration for gaussian chaos processes

Consider a setA of symmetricn×n matricesa=(a _i,j) _i,j≤n . Consider an independent sequence (g _i) _i≤n of standard normal random variables, and letM=Esup_a∈A|Σ_i,j⪯na_i,jg_ig_j|. Denote byN ₂(A, α) (resp.N _t(A, α)) the smallest number of balls of radiusα for thel ₂ norm ofR ^{n 2} (resp. the operator norm) needed to coverA. Then for a universal constantK we haveα(logN ₂(A, α))^1/4≤KM. This inequality is best possible. We also show that forδ≥0, there exists a constantK(δ) such thatα(logN _t≤K(δ)M. Work partially supported by an N.S.F. grant.     

Rook numbers and the normal ordering problem

For an element w in the Weyl algebra generated by D and U with relation DU=UD+1, the normally ordered form is w=∑c_i,jUⁱD^j. We demonstrate that the normal order coefficients c_i,j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients c_i,j. We calculate the Weyl binomial coefficients: normal order coefficients of the element (D+U)ⁿ in the Weyl algebra. We extend these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.     

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x ₁,…,x _n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x _i −x _j ‖≤‖L(x _i )−L(x _j )‖≤O(1)⋅‖x _i −x _j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 2^2O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E _n ⊆Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.     

Disk-like Tiles Derived from Complex Bases

For each positive integer k, the radix representation of the complex numbers in the base –k+i gives rise to a lattice self-affine tile T _k in the plane, which consists of all the complex numbers that can be expressed in the form ∑ _j≥1 d _j (–k+i)^–j , where d _j ∈{0, 1, 2, ...,k ²}. We prove that T _k is homeomorphic to the closed unit disk {z∈C:∣z∣ ≤ 1} if and only if k ≠ 2. The first author is supported by Youth Project of Tianyuan Foundation (10226031) and Zhongshan University Promotion Foundation for Young Teachers (34100-1131206); the second author is supported by National Science Foundation (10041005) and Guangdong Province Science Foundation (011221)     

Involutions on surfaces with pg?=?q?=?0 and K2?=?3

We study surfaces of general type S with p _g  = 0 and K ² = 3 having an involution i such that the bicanonical map of S is not composed with i. It is shown that, if S/i is not rational, then S/i is birational to an Enriques surface or it has Kodaira dimension 1 and the possibilities for the ramification divisor of the covering map S → S/i are described. We also show that these two cases do occur, providing an example. In this example S has a hyperelliptic fibration of genus 3 and the bicanonical map of S is of degree 2 onto a rational surface.