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.     

Hilbert's Tenth Problem for function fields of varieties over number fields and p-adic fields

Let k be a subfield of a p-adic field of odd residue characteristic, and let be the function field of a variety of dimension n1 over k. Then Hilbert's Tenth Problem for is undecidable. In particular, Hilbert's Tenth Problem for function fields of varieties over number fields of dimension 1 is undecidable.     

Green kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs

We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel–Leader graph , where q,r2. The latter is the horocyclic product of two homogeneous trees with respective degrees q+1 and r+1. When q=r, it is the Cayley graph of the wreath product (lamplighter group) with respect to a natural set of generators. We describe the full Martin compactification of these random walks on -graphs and, in particular, lamplighter groups. This completes previous results of Woess, who has determined all minimal positive harmonic functions.     

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.     

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.     

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.     

An optimal symbolic calculus on Besov algebras

In this paper we consider Besov algebras on , that is Besov spaces for s>1/p. For s>1+(1/p), p>4/3, and qp we prove that the above algebras have a maximal symbolic calculus in the following sense: for any function f belonging locally to and such that f(0)=0, the associated superposition operator T_f(g):=f○g takes to itself.     

Orthogonal graphs of odd characteristic and their automorphisms

The (isotropic) orthogonal graph O(2ν+δ,q) over of odd characteristic, where ν1 and δ=0,1 or 2 is introduced. When ν=1, O(21+δ,q) is a complete graph. When ν2, O(2ν+δ,q) is strongly regular and its parameters are computed, as well as its chromatic number. The automorphism groups of orthogonal graphs are also determined.     

On a Liouville phenomenon for entire weak supersolutions of elliptic partial differential equations

We study a new Liouville-type phenomenon for entire weak supersolutions of elliptic partial differential equations of the form A(u)=0 on , n2. Typical examples of the operator A(u) are the p-Laplacian for p>1, the mean curvature operator, and their well-known modifications.     

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.     

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.     

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.

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.     

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.     

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.     

Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials

In this paper, we study oscillation of second-order functional differential equations with mixed nonlinearities

where τ0, p(t)C¹[0,∞), q(t),q_i(t),e(t)C[0,∞), p(t)>0, . Without assuming that q(t), q_i(t) and e(t) are nonnegative, the results given in [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 198 (2008) 375–381] have been extended to the aforementioned functional differential equation.     

On an extremal problem of Selberg II

Let m_p be the minimum of the product under the conditions that and . In our previous paper [J. Kaneko, On an extremal problem of Selberg, J. Approx. Theory 142 (2006) 129–137], we showed that the following estimates hold. provided p255. In this note, we prove that the limit of as p→∞ exists and is expressed by the (unique) solution of some simultaneous transcendental equations. By using this expression we obtain numerically.     

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.     

On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.