Uniform Kazhdan constant for some families of linear groups

Let R be a ring generated by l elements with stable range r. Assume that the group EL_d(R) has Kazhdan constant ₀>0 for some dr+1. We prove that there exist (₀,l)>0 and , s.t. for every nd, EL_n(R) has a generating set of order k and a Kazhdan constant larger than . As a consequence, we obtain for where n3, a Kazhdan constant which is independent of n w.r.t. generating set of a fixed size.     

On hypersurfaces with two distinct principal curvatures in a unit sphere

We investigate the immersed hypersurfaces in a unit sphere . By using Otsuki's idea, we obtain the local and global classification results for immersed hypersurfaces in of constant m-th mean curvature and two distinct principal curvatures of multiplicities n−1,1 (in the local version, we assume that the principal curvatures are non-zero when m2). As the result, we prove that any local hypersurface in of constant mean curvature and two distinct principal curvatures is an open part of a complete hypersurface of the same curvature properties. The corresponding result does not hold for m-th mean curvature when m2.     

disjoint cycles containing specified independent vertices

Let k1 be an integer and G be a graph of order n3k satisfying the condition that σ₂(G)n+k-1. Let v₁,…,v_k be k independent vertices of G, and suppose that G has k vertex-disjoint triangles C₁,…,C_k with v_iV(C_i) for all 1ik.Then G has k vertex-disjoint cycles such that

(i) for all 1ik.

(ii) , and

(iii) At least k-1 of the k cycles are triangles.

The condition of degree sum σ₂(G)n+k-1 is sharp.

Eigenvalue estimates for non-normal matrices and the zeros of random orthogonal polynomials on the unit circle

We prove that for any n×n matrix, A, and z with |z|A, we have that . We apply this result to the study of random orthogonal polynomials on the unit circle.     

On the chromatic number of an oriented matroid

We prove that the chromatic number of an oriented matroid of rank r3 is at most r+1 with equality if and only if is the oriented matroid of an orientation of K_r+1, the complete graph on r+1 vertices.     

Arithmetic progressions consisting of unlike powers

In this paper we present some new results about unlike powers in arithmetic progression. We prove among other things that for given k 4 and L 3 there are only finitely many arithmetic progressions of the form with x_i , gcd(x₀, x_l) = 1 and 2 l_i L for i = 0, 1, …, k − 1. Furthermore, we show that, for L = 3, the progression (1, 1,…, 1) is the only such progression up to sign. Our proofs involve some well-known theorems of Faltings [9], Darmon and Granville [6] as well as Chabauty's method applied to superelliptic curves.     

The approximate and the Clarke subdifferentials can be different everywhere

We prove that, for a Lipschitz function on , n2, the approximate and the Clarke subdifferentials can differ everywhere. This completely answers a question by A.D. Ioffe, which was partially answered by G. Katriel.     

A supplement to the Davis–Gut law

Let {X,X_n;n1} be a sequence of i.i.d. real-valued random variables and set , n1. Let h() be a positive nondecreasing function such that . Define Lt=log_emax{e,t} for t0. In this note we prove that

if and only if E(X)=0 and E(X²)=1, where , t1. When h(t)≡1, this result yields what is called the Davis–Gut law. Specializing our result to h(t)=(Lt)^r, 0<r1, we obtain an analog of the Davis–Gut law.     

Discrepancies between metric dimension and partition dimension of a connected graph

Let and be graphs where the set of vertices is the set of points of the integer lattice and the set of edges consists of all pairs of vertices whose city block and chessboard distances, respectively, are 1.In this paper it is shown that the partition dimensions of these graphs are 3 and 4, respectively, while their metric dimensions are not finite. Also, for every n3 there exists an induced subgraph of of order 3n-1 with metric dimension n and partition dimension 3. These examples will answer a question raised by Chartrand, Salehi and Zhang. Furthermore, graphs of order n9 having partition dimension n-2 are characterized, thus completing the characterization of graphs of order n having partition dimension 2, n, or n-1 given by Chartrand, Salehi and Zhang. The list of these graphs includes 23 members.     

Existence of HSOLSSOMs of type

In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2ⁿu¹ (HSOLSSOM(2ⁿu¹)). For u2, necessary conditions for existence of such an HSOLSSOM are that u must be even and n3u/2+1. Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n9 and n3u/2+1 or (2) n263 and n2(u-2). In this paper we show that in (1) the condition n9 can be extended to n30 and that in (2), the condition n263 can be improved to n4, except possibly for 19 pairs (n,u), the largest of which is (53,28).     

Finite automata presentable abelian groups

We give new examples of FA presentable torsion-free abelian groups. Namely, for every n2, we construct a rank n indecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group in which every nontrivial cyclic subgroup is not FA recognizable.     

Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data

We consider the defocusing, -critical Hartree equation for the radial data in all dimensions (n5). We show the global well-posedness and scattering results in the energy space. The new ingredient in this paper is that we first take advantage of the term in the localized Morawetz identity to rule out the possibility of energy concentration, instead of the classical Morawetz estimate dependent of the nonlinearity.     

Cohomology and deformations of the infinite-dimensional filiform Lie algebra

Denote the infinite-dimensional -graded Lie algebra defined by basis e_i, i1, and relations [e₁,e_i]=e_i+1 for all i2. We compute in this article the bracket structure on , and in relation to this, we establish that there are only finitely many true deformations of in each non-positive weight by constructing them explicitly. It turns out that in weight 0 one gets exactly the other two filiform Lie algebras.     

Sur le groupe d'automorphismes des géométries paraboliques de rang 1

The aim of this article is to prove the following result, which generalizes the Ferrand–Obata theorem, concerning the conformal group of a Riemannian manifold, and the Schoen–Webster theorem about the automorphism group of a strictly pseudo-convex CR structure: let M be a connected manifold endowed with a regular Cartan geometry, modelled on the boundary of the hyperbolic space of dimension d2 over , being , , or the octonions . If the automorphism group of M does not act properly on M, then M is isomorphic, as a Cartan geometry, to X, or X minus a point.     

Refined blowup criteria and nonsymmetric blowup of an aggregation equation

We consider an aggregation equation in , d2, with fractional dissipation: u_t+(uK*u)=−νΛ^γu, where ν0, 0<γ<1, and K(x)=e^−|x|. We prove a refined blowup criteria by which the global existence of solutions is controlled by its norm, for any . We prove the finite time blowup of solutions for a general class of nonsymmetric initial data. The argument presented works for both the inviscid case ν=0 and the supercritical case ν>0 and 0<γ<1. Additionally, we present new proofs of blowup which does not use free energy arguments.     

On the mean value of

Let χ be the Dirichlet character modulo q3 and L(s,χ) denote the corresponding Dirichlet L-function. The mean value of is studied and a few asymptotic formulae are given. Hybrid mean value of , general Kloosterman sums and general quadratic Gauss sums are considered.     

On value sets of polynomials over a field

Let F be any field. Let p(F) be the characteristic of F if F is not of characteristic zero, and let p(F)=+∞ otherwise. Let A₁,…,A_n be finite nonempty subsets of F, and let

with k{1,2,3,…}, a₁,…,a_nF{0} and degg<k. We show that

When kn and |A_i|i for i=1,…,n, we also have

consequently, if nk then for any finite subset A of F we have

In the case n>k, we propose a further conjecture which extends the Erdős–Heilbronn conjecture in a new direction.     

Hamiltonicity and restricted block-intersection graphs of -designs

Given a combinatorial design with block set , its traditional block-intersection graph is the graph having vertex set such that two vertices b₁ and b₂ are adjacent if and only if b₁ and b₂ have non-empty intersection. In this paper, we consider the S-block-intersection graph, in which two vertices b₁ and b₂ are adjacent if and only if |b₁∩b₂|S. As our main result, we prove that {1,2,…,t−1}-block-intersection graphs of t-designs with parameters (v,t+1,λ) are Hamiltonian whenever t3 and vt+3, except possibly when (v,t){(8,5),(7,4),(7,3),(6,3)}.     

Tensor-product approximation to operators and functions in high dimensions

In recent papers tensor-product structured Nyström and Galerkin-type approximations of certain multi-dimensional integral operators have been introduced and analysed. In the present paper, we focus on the analysis of the collocation-type schemes with respect to the tensor-product basis in a high spatial dimension d. Approximations up to an accuracy are proven to have the storage complexity with q independent of d, where N is the discrete problem size. In particular, we apply the theory to a collocation discretisation of the Newton potential with the kernel , , d3. Numerical illustrations are given in the case of d=3.