A new partial regularity result for non-autonomous convex integrals with non-standard growth conditions

We establish C1,γ-partial regularity of minimizers of non-autono-mous convex integral functionals of the type: , with non-standard growth conditions into the gradient variable

     

Scattering above energy norm of solutions of a loglog energy-supercritical Schrödinger equation with radial data

We prove global well-posedness and scattering of -solutions of the loglog energy-supercritical Schrödinger equation , 0<c<cn, n={3,4}, with radial data , . This is achieved, roughly speaking, by extending Bourgain's argument in Bourgain (1999) [1] (see also Grillakis, 2000 [5]) and Tao's argument in Tao (2005) [10] in high dimensions.     

The local variational principle of topological pressure for sub-additive potentials

Let (X,T) be a topological dynamical system and be a sub-additive potential on C(X,R). Let U be an open cover of X. Then for any T-invariant measure μ, let . The topological pressure for open covers U is defined for sub-additive potentials. Then we have a variational principle:

     

Extinction, decay and blow-up for Keller-Segel systems of fast diffusion type

We consider the quasi-linear Keller-Segel system of singular type, where the principal part Δum represents a fast diffusion like 0<m<1. We first construct a global weak solution with small initial data in the scaling invariant norm for all dimensions N?2 and all exponents q?2. As for the large initial data, we show that there exists a blow-up solution in the case of N=2. In the second part, the decay property in Lr with 1<r<∞ for with the mass conservation is shown. On the other hand, in the case of , the extinction phenomenon of solution is proved. It is clarified that the case of exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across . For the borderline case of , our solution decays in Lr exponentially as t→∞.     

The coarse shape groups

The (pointed) coarse shape category Sh^* (), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory, was recently constructed. Its isomorphisms classify (pointed) topological spaces strictly coarser than the (pointed) shape type classification. In this paper we introduce a new algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space (X,?) and for every k∈N₀, the coarse shape group , having the standard shape group for its subgroup, is defined. Furthermore, a functor is constructed. The coarse shape and shape groups already differ on the class of polyhedra. An explicit formula for computing coarse shape groups of polyhedra is given. The coarse shape groups give us more information than the shape groups. Generally, does not imply (e.g. for solenoids), but from pro-πk(X,?)=0 follows . Moreover, for pointed metric compacta (X,?), the n-shape connectedness is characterized by , for every k?n.     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

Multiple solutions for an elliptic problem related to vortex pairs

Let Ω be a bounded domain in RN (N?2), φ a harmonic function in . In this paper we study the existence of solutions to the following problem arising in the study of vortex pairs

     

Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices

Let be the set of entrywise nonnegative n×n matrices. Denote by r(A) the spectral radius (Perron root) of . Characterization is obtained for maps such that r(f(A)+f(B))=r(A+B) for all . In particular, it is shown that such a map has the form

     

Critical points between varieties generated by subspace lattices of vector spaces

We denote by the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by the class of all semilattices isomorphic to for some A∈V. Given varieties V and W of algebras, the critical point of V under W is defined as . Given a finitely generated variety V of modular lattices, we obtain an integer ?, depending on V, such that for any n≥? and any field F.In a second part, using tools introduced in Gillibert (2009) [5], we prove that:

     

The Cauchy problem for the Schrödinger-KdV system

In this paper we prove that in the general case (i.e. β not necessarily vanishing) the Cauchy problem for the Schrödinger-Korteweg-de Vries system is locally well-posed in , and if β=0 then it is locally well-posed in with . These results improve the corresponding results of Corcho and Linares (2007) [5]. Idea of the proof is to establish some bilinear and trilinear estimates in the space Gs×Fs, where Gs and Fs are dyadic Bourgain-type spaces related to the Schrödinger operator and the Airy operator , respectively, but with a modification on Fs in low frequency part of functions with a weaker structure related to the maximal function estimate of the Airy operator.     

Large energy entire solutions for the Yamabe equation

We consider the Yamabe equation in Rn, n?3. Let k?1 and . For all large k we find a solution of the form , where , for n?4, for n=3 and o(1)→0 uniformly as k→+∞.     

Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials

Denote by , k=1,…,n, the zeros of the Laguerre-Sobolev-type polynomials orthogonal with respect to the inner product

     

More on almost self-complementary graphs

A graph X is called almost self-complementary if it is isomorphic to one of its almost complements , where denotes the complement of X and I a perfect matching (1-factor) in . If I is a perfect matching in and is an isomorphism, then the graph X is said to be fairly almost self-complementary if φ preserves I setwise, and unfairly almost self-complementary if it does not.In this paper we construct connected graphs of all possible orders that are fairly and unfairly almost self-complementary, fairly but not unfairly almost self-complementary, and unfairly but not fairly almost self-complementary, respectively, as well as regular graphs of all possible orders that are fairly and unfairly almost self-complementary.Two perfect matchings I and J in are said to be X-non-isomorphic if no isomorphism from X+I to X+J induces an automorphism of X. We give a constructive proof to show that there exists a graph X that is almost self-complementary with respect to two X-non-isomorphic perfect matchings for every even order greater than or equal to four.     

Indecomposable and noncrossed product division algebras over function fields of smooth p-adic curves

We construct indecomposable and noncrossed product division algebras over function fields of connected smooth curves X over Zp. This is done by defining an index preserving morphism which splits , where is the completion of K(X) at the special fiber, and using it to lift indecomposable and noncrossed product division algebras over .     

Exact rates in log law for positively associated random variables

Let be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set , Mn=maxk?n|Sk|, n?1. Suppose . In this paper, we study the exact convergence rates of a kind of weighted infinite series of , and as ε↘0, respectively.     

A new hybrid iterative algorithm for variational inequalities

Let H be a real Hilbert space. Let F:H→H be a strongly monotone and Lipschitzian mapping. Let be an infinite family of non-expansive mappings with common fixed points set . We devise an iterative algorithm

     

Cap pairings and spectral sequences

Let be a small category. For an -diagram X and -diagrams A and B of pointed spaces, each pairing X∧A→B satisfying the projection formula induces a pairing . In this note we show that there is an induced pairing of homotopy spectral sequences compatible with abutments in the sense that

     

On the relation between cluster and classical tilting

Let be a triangulated category with a cluster tilting subcategory U. The quotient category is abelian; suppose that it has finite global dimension.We show that projection from to sends cluster tilting subcategories of to support tilting subcategories of , and that, in turn, support tilting subcategories of can be lifted uniquely to weak cluster tilting subcategories of .