Sharp bounds for the first non-zero Stekloff eigenvalues

Let (M,,) be an n(2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problems

where Δ is the Laplacian operator on M and ν denotes the outward unit normal on ∂M. The first non-zero eigenvalues of the above problems will be denoted by p₁ and q₁, respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by a positive constant c, then with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius , here λ₁ denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c then q₁nc with equality holding if and only if M is isometric to an n-dimensional Euclidean ball of radius . Finally, we show that q₁A/V and that if the equality holds and if there is a point x₀∂M such that the mean curvature of ∂M at x₀ is no less than A/{nV}, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively.     

The equation on weakly pseudo-convex CR manifolds of dimension 3

Let M be a smooth, compact, orientable, weakly pseudoconvex manifold of dimension 3, embedded in (N2), of codimension one or more, and endowed with the induced CR structure. Assuming that the tangential Cauchy-Riemann operator has closed range in L²(M) in order to rule out the Rossi example, we push regularity up to show has closed range in H^s(M) for all s>0. We then use the Szegö projection to show there is a smooth solution for the problem given smooth data. The results are obtained via microlocalization by piecing together estimates for functions and (0,1) forms that hold on different microlocal regions.     

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.     

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.     

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.     

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.     

Point spectra of partially power-bounded operators

Let T be an operator on a separable Banach space, and denote by σ_p(T) its point spectrum. We answer a question left open in (Israel J. Math. 146 (2005) 93–110) by showing that it is possible that be uncountable, yet Tⁿ∞. We further investigate the relationship between the growth of sequences (n_k) such that sup_kT^n_k<∞ and the possible size of .Analogous results are also derived for continuous operator semigroups (T_t)_t0.     

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.     

Probabilistic n-normed spaces, -n-compact sets and -n-bounded sets

In this paper, first we define and study the probabilistic n-normed spaces and -n-compactness, also we prove some theorems and inequalities. In the next section we define -n-boundedness and prove some results in relation between -n-compact and -n-bounded sets in these spaces.     

Construction of nonseparable dual -wavelet frames in

Suppose and are two pairs of dual M-wavelet frames and N-wavelet frames in and , respectively, where M and N are s₁×s₁ and s₂×s₂ dilation matrices with a₁(|det(M)|-1) and (2a₂+1)(|det(N)|-1). Moreover, their mask symbols both satisfy mixed extension principle (MEP). Based on the mask symbols, a family of nonseparable dual Ω-wavelet frames in are constructed, where s=s₁+s₂, and with Θ and M^-1Θ both being integer matrices. Then a convolution scheme for improving regularity of wavelet frames is given. From the nonseparable dual Ω-wavelet frames, nonseparable Ω-wavelet frames with high regularity can be constructed easily. We give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in . From the dual Ω-wavelet frames, nonseparable dual Ω-wavelet frames with symmetry can be obtained easily. In the end, two examples are given.     

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.     

Index theory for boundary value problems via continuous fields of -algebras

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semi-groupoid generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field of C^*-algebras over [0,1]. Its fiber in =0, , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ≠0 the fibers are isomorphic to the algebra of compact operators. We therefore obtain a natural map . Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.     

Hypercyclic sequences of PDE-preserving operators

A sequence of continuous linear operators is said to be hypercyclic if there exists a vector , called hypercyclic for , such that {T_nx:n0} is dense. A continuous linear operator, acting on some suitable function space, is PDE-preserving for a given set of convolution operators, when it map every kernel set for these operators invariantly. We establish hypercyclic sequences of PDE-preserving operators on , and study closed infinite-dimensional subspaces of, except for zero, hypercyclic vectors for these sequences.     

Some rigidity results for non-commutative Bernoulli shifts

We introduce the outer conjugacy invariants , for cocycle actions σ of discrete groups G on type II₁ factors N, as the set of real numbers t>0 for which the amplification σ^t of σ can be perturbed to an action, respectively, to a weakly mixing action. We calculate explicitly and the fundamental group of σ, , in the case G has infinite normal subgroups with the relative property (T) (e.g., when G itself has the property (T) of Kazhdan) and σ is an action of G on the hyperfinite II₁ factor by Connes–Størmer Bernoulli shifts of weights {t_i}_i. Thus, and coincide with the multiplicative subgroup S of generated by the ratios {t_i/t_j}_i,j, while if S={1} (i.e. when all weights are equal), and otherwise. In fact, we calculate all the “1-cohomology picture” of σ^t,t>0, and classify the actions (σ,G) in terms of their weights {t_i}_i. In particular, we show that any 1-cocycle for (σ,G) vanishes, modulo scalars, and that two such actions are cocycle conjugate iff they are conjugate. Also, any cocycle action obtained by reducing a Bernoulli action of a group G as above on to the algebra pNp, for p a projection in N, p≠0,1, cannot be perturbed to a genuine action.     

Projective change between two classes of -metrics

In this paper, we find equations to characterize projective change between (α,β)-metric and Randers metric on a manifold with dimension n3, where α and are two Riemannian metrics, β and are two nonzero one forms. Moreover, we consider this projective change when F has some special curvature properties.     

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.     

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.

Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains

We consider the problem in Ω_ε, u=0 on ∂Ω_ε, where Ω_ε:=ΩB(0,ε) and Ω is a bounded smooth domain in , which contains the origin and is symmetric with respect to the origin, N3 and ε is a positive parameter. As ε goes to zero, we construct sign changing solutions with multiple blow up at the origin.