The mean value of the zeta-function on <Emphasis Type="Italic">σ</Emphasis>=1

Let R(T): = ò₁^T|z(1+it)|² dt - z(2)T+plogTR(T):= \int_{1}^{T}|\zeta (1+it)|^{2}\,\mathrm{d}t - \zeta (2)T+\pi\log T. We derive a precise explicit expression for R(t) which is used to derive asymptotic formulas for ò₁^TR(t) dt\int_{1}^{T}R(t)\,\mathrm{d}t and ò₁^TR²(t) dt\int_{1}^{T}R^{2}(t)\,\mathrm{d}t. These results improve on earlier upper bounds of Balasubramanian, Ramachandra and the author for the integrals in question.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-harmonic forms and stable hypersurfaces in space forms

We prove that a complete noncompact orientable stable minimal hypersurface in \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. We also obtain that a complete noncompact strongly stable hypersurface with constant mean curvature in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} or \mathbbSⁿ⁺¹{\mathbb{S}^{n+1}} (n ≤ 4) admits no nontrivial L ²-harmonic forms. These results are generalized versions of Tanno’s result on stable minimal hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}}.     

Stability of unitary SK<Subscript>1</Subscript> of Azumaya algebras

For an Azumaya algebra A with center C of rank n ² and a unitary involution τ, we study the stability of the unitary SK₁ under reduction. We show that if R = C ^τ is a Henselian ring with maximal ideal \mathfrakm{\mathfrak{m}} and 2 and n are invertible in R then SK₁(A, t) @ SK₁(A/ \mathfrakm A, overline t){{{\rm SK}_1}(A, \tau) \cong {{\rm SK}_1}(A/ \mathfrak{m} A, overline \tau)}.     

Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime

We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))}We study the long-time asymptotics of the doubly nonlinear diffusion equation r_t=div(|?r^m |^p-2 ?(r^m)){\rho_t={\rm div}(|\nabla\rho^m |^{p-2} \nabla\left(\rho^m\right))} in \mathbbRⁿ{\mathbb{R}^n}, in the range \fracn-pn(p-1) < m < \fracn-p+1n(p-1){\frac{n-p}{n(p-1)} < m < \frac{n-p+1}{n(p-1)}} and 1 < p < ∞ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an L ¹-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval \frac2nn+1 < p < \frac2n+1n+1{\frac{2n}{n+1} < p < \frac{2n+1}{n+1}} where a rate of convergence towards self-similarity was still unknown for the p-Laplacian equation.     

Ring-Valued Assignments to the Points of a t-Design

We introduce the method of assigning values in a ring to the points of a t-design. We give the notion of an -design. We give a generalization of the concept of association scheme, and obtain a method to construct some of them from t-designs that generalizes the well-known construction for the case of ordinary association schemes.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

More large sets of resolvable MTS and DTS with even orders

In this paper, we first introduce a special structure that allows us to construct a large set of resolvable Mendelsohn triple systems of orders 2q ＋ 2, or LRMTS（2q ＋ 2）, where q = 6t ＋ 5 is a prime power. Using a computer, we find examples of such structure for t C T = {0, 1, 2, 3, 4, 6, 7, 8, 9, 14, 16, 18, 20, 22, 24}. Furthermore, by a method we introduced in [13], large set of resolvable directed triple systems with the same orders are obtained too. Finally, by the tripling construction and product construction for LRMTS and LRDTS introduced in [2, 20, 21], and by the new results for LR-design in [8], we obtain the existence for LRMTS（v）and LRDTS（v）, where v = 12（t ＋ 1） mi≥0（2.7mi＋1）mi≥0（2.13ni＋1）and t∈T,which provides more infinite family for LRMTS and LRDTS of even orders.     

Blow up of subcritical quantities at the first singular time of the mean curvature flow

Consider a family of smooth immersions F(·,t) : Mⁿ? \mathbbRⁿ⁺¹{F(\cdot,t)\,:\,{M^n\to \mathbb{R}^{n+1}}} of closed hypersurfaces in \mathbbRⁿ⁺¹{\mathbb{R}^{n+1}} moving by the mean curvature flow \frac?F(p,t)?t = -H(p,t)·n(p,t){\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)}, for t ? [0,T){t\in [0,T)}. We show that at the first singular time of the mean curvature flow, certain subcritical quantities concerning the second fundamental form, for example ò₀^tò_Ms\frac|A|^{n + 2} log (2 + |A|) dmds,{\int_{0}^{t}\int_{M_{s}}\frac{{\vert{\it A}\vert}^{n + 2}}{ log (2 + {\vert{\it A}\vert})}} d\mu ds, blow up. Our result is a log improvement of recent results of Le-Sesum, Xu-Ye-Zhao where the scaling invariant quantities were considered.     

Tight Gaussian 4-Designs

A Gaussian t-design is defined as a finite set X in the Euclidean space ℝⁿ satisfying the condition: for any polynomial f(x) in n variables of degree at most t, here α is a constant real number and ω is a positive weight function on X. It is easy to see that if X is a Gaussian 2e-design in ℝⁿ, then . We call X a tight Gaussian 2e-design in ℝⁿ if holds. In this paper we study tight Gaussian 2e-designs in ℝⁿ. In particular, we classify tight Gaussian 4-designs in ℝⁿ with constant weight or with weight . Moreover we classify tight Gaussian 4-designs in ℝⁿ on 2 concentric spheres (with arbitrary weight functions).     

Global classical solutions for reaction–diffusion systems with nonlinearities of exponential growth

The aim of this study is to prove global existence of classical solutions for systems of the form ${\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)}The aim of this study is to prove global existence of classical solutions for systems of the form \frac?u?t -a Du=-f(u,v){\frac{\partial u}{\partial t} -a \Delta u=-f(u,v)} , \frac?v?t -b Dv=g(u,v){\frac{\partial v}{\partial t} -b \Delta v=g(u,v)} in (0, +∞) × Ω where Ω is an open bounded domain of class C ¹ in \mathbbRⁿ{\mathbb{R}^n}, a > 0, b > 0 and f, g are nonnegative continuously differentiable functions on [0, +∞) × [0, +∞) satisfying f (0, η) = 0, g(x,h) ￡ C j(x)e^ahb{g(\xi,\eta) \leq C \varphi(\xi)e^{\alpha {\eta^\beta}}} and g(ξ, η) ≤ ψ(η)f(ξ, η) for some constants C > 0, α > 0 and β ≥ 1 where j{\varphi} and ψ are any nonnegative continuously differentiable functions on [0, +∞) such that j(0)=0{\varphi(0)=0} and lim_{h? +￥}h^b-1y(h) = l{ \lim_{\eta \rightarrow +\infty}\eta^{\beta -1}\psi(\eta)= \ell} where ℓ is a nonnegative constant. The asymptotic behavior of the global solutions as t goes to +∞ is also studied. For this purpose, we use the appropriate techniques which are based on semigroups, energy estimates and Lyapunov functional methods.     

Designs in Grassmannian Spaces and Lattices

We introduce the notion of a t-design on the Grassmann manifold of the m-subspaces of the Euclidean space ⁿ . It generalizes the notion of antipodal spherical design which was introduced by P. Delsarte, J.-M. Goethals and J.-J. Seidel. We characterize the finite subgroups of the orthogonal group which have the property that all its orbits are t-designs. Generalizing a result due to B. Venkov, we prove that, if the minimal m-sections of a lattice L form a 4-design, then L is a local maximum for the Rankin function _n,m .     

Semiconcavity estimates for viscous Hamilton–Jacobi equations

We present sharp Hessian estimates of the form D² S^e(t,x) ￡ g(t)I{D^2 S^\varepsilon(t,x)\leq g(t)I} for the solution of the viscous Hamilton–Jacobi equation

Regularity of weak solutions to the Navier–Stokes equations in exterior domains

Let u be a weak solution of the Navier–Stokes equations in an exterior domain ${\Omega \subset \mathbb{R}^3}Let u be a weak solution of the Navier–Stokes equations in an exterior domain W ì \mathbbR³{\Omega \subset \mathbb{R}^3} and a time interval [0, T[ , 0 < T ≤ ∞, with initial value u ₀, external force f = div F, and satisfying the strong energy inequality. It is well known that global regularity for u is an unsolved problem unless we state additional conditions on the data u ₀ and f or on the solution u itself such as Serrin’s condition || u ||_{L^s(0,T; L^q(W))} < ￥{\| u \|_{L^s(0,T; L^q(\Omega))} < \infty} with 2 < s < ￥, \frac2s + \frac3q = 1{2 < s < \infty, \frac{2}{s} + \frac{3}{q} =1}. In this paper, we generalize results on local in time regularity for bounded domains, see Farwig et al. (Indiana Univ Math J 56:2111–2131, 2007; J Math Fluid Mech 11:1–14, 2008; Banach Center Publ 81:175–184, 2008), to exterior domains. If e.g. u fulfills Serrin’s condition in a left-side neighborhood of t or if the norm || u ||_{L^s￠(t-d,t; L^q(W))}{\| u \|_{L^{s'}(t-\delta,t; L^q(\Omega))}} converges to 0 sufficiently fast as δ → 0 + , where ${\frac{2}{s'} + \frac{3}{q} > 1}${\frac{2}{s'} + \frac{3}{q} > 1}, then u is regular at t. The same conclusion holds when the kinetic energy \frac12|| u(t) ||₂²{\frac{1}{2}\| u(t) \|_2^2} is locally H?lder continuous with exponent ${\alpha > \frac{1}{2}}${\alpha > \frac{1}{2}}.     

Numerical integration with polynomial exactness over a spherical cap

This paper presents rules for numerical integration over spherical caps and discusses their properties. For a spherical cap on the unit sphere \mathbbS²\mathbb{S}^2, we discuss tensor product rules with n ²/2 + O(n) nodes in the cap, positive weights, which are exact for all spherical polynomials of degree ≤ n, and can be easily and inexpensively implemented. Numerical tests illustrate the performance of these rules. A similar derivation establishes the existence of equal weight rules with degree of polynomial exactness n and O(n ³) nodes for numerical integration over spherical caps on \mathbbS²\mathbb{S}^2. For arbitrary d ≥ 2, this strategy is extended to provide rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that have O(n ^d ) nodes in the cap, positive weights, and are exact for all spherical polynomials of degree ≤ n. We also show that positive weight rules for numerical integration over spherical caps on \mathbbS^d\mathbb{S}^d that are exact for all spherical polynomials of degree ≤ n have at least O(n ^d ) nodes and possess a certain regularity property.     

Solvability of boundary-value problems for nonlinear fractional differential equations

We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations

Self-similar solutions for a convection-diffusion equation with absorption inR N

We prove the existence of a positive and smooth solution for the following semilinear elliptic problem: % MathType!End!2!1! for anya∈R ^N , 1<p<1+2/N andq=(p+1)/2. This solution decays exponentially as |x|→+∞. Moreover, if |a| is sufficiently small, this positive and rapidly decaying solution is unique. The existence of a positive, self-similar solution % MathType!End!2!1! follows for the following convection-diffusion equation with absorption: % MathType!End!2!1!. It is also a very singular solution. This solution decays as |x|→+∞ for anyt>0 fixed. Because of the nonvariational nature of the elliptic problem, a fixed point method is used for proving the existence result. The uniqueness is proved applying the Implicit Function Theorem. The work of the first author has been partially supported by Grant 1273/00003/88 of the University of the Basque Country. The work of the second author has been supported by Grant PB 86-0112-C02-00 of the Dirección General de Investigación Científica y Técnica.     

Interpolating Thin-Shell and Sharp Large-Deviation Estimates for Lsotropic Log-Concave Measures

Given an isotropic random vector X with log-concave density in Euclidean space \mathbbRⁿ{\mathbb{R}^n} , we study the concentration properties of |X| on all scales, both above and below its expectation. We show in particular that

New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces

Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]^s, t_p, q(\mathbbRⁿ) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]^-s, t_p￠, q￠(\mathbbRⁿ){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .     

On Einstein Riemannian <Emphasis Type="Italic">g</Emphasis>-natural metrics on unit tangent sphere bundles

We prove that there is always a locally homogeneous Einstein g-natural metric on the unit tangent sphere bundle over any Riemannian space of constant positive sectional curvature. Furthermore, using the (1–1) correspondence between all SO(m + 1)-invariant homogeneous metrics on the Stiefel manifold V₂ \mathbbR^m+1 = SO(m+1)/SO(m-1){V_2 \mathbb{R}^{m+1} = {{SO}}(m+1)/{{SO}}(m-1)} and all g-natural metrics on T₁ S^m{T_1 S^m} (Abbassi and Kowalski, Diff. Geom. Appl., to appear [7]), we reconstruct, by purely local procedure, the same well-known unique SO(m + 1)-invariant homogeneous Einstein metric on V₂ \mathbbR^m+1, m 1 3{V_2 \mathbb{R}^{m+1}, m \neq 3}, initially constructed by Kobayashi.     

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities |u|t ≥ 2u + |v|q,|v|t ≥ 2v + |u|p in S = Rn × R+ with p,q > 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n4 ≤ max(ppq+11,pqq+11).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.