Global smooth solutions of the compressible Navier-Stokes equations with density-dependent viscosity

In this paper we study a free boundary problem for the viscous, compressible, heat conducting, one-dimensional real fluids. More precisely, the viscosity is assumed to be a power function of density, i.e., μ(ρ)=ρα, where ρ denotes the density of fluids and α is a positive constant. In addition, the equations of state include and are more general than perfect flows which only depend linearly on temperature. The global existence (uniqueness) of smooth solutions is established with for general, large initial data, which improves the previous results. Moreover, it is also shown that the solutions will not develop vacuum, mass concentration or heat concentration in a finite time provided the initial data are bounded and smooth, and do not contain vacuum.     

Low rank perturbation of regular matrix polynomials

Let A(λ) be a complex regular matrix polynomial of degree ? with g elementary divisors corresponding to the finite eigenvalue λ₀. We show that for most complex matrix polynomials B(λ) with degree at most ? satisfying rank the perturbed polynomial (A+B)(λ) has exactly elementary divisors corresponding to λ₀, and we determine their degrees. If does not exceed the number of λ₀-elementary divisors of A(λ) with degree greater than 1, then the λ₀-elementary divisors of (A+B)(λ) are the elementary divisors of A(λ) corresponding to λ₀ with smallest degree, together with rank(B(λ)-B(λ₀)) linear λ₀-elementary divisors. Otherwise, the degree of all the λ₀-elementary divisors of (A+B)(λ) is one. This behavior happens for any matrix polynomial B(λ) except those in a proper algebraic submanifold in the set of matrix polynomials of degree at most ?. If A(λ) has an infinite eigenvalue, the corresponding result follows from considering the zero eigenvalue of the perturbed dual polynomial.     

A Robertson-type uncertainty principle and quantum Fisher information

Let A₁,…,A_N be complex self-adjoint matrices and let ρ be a density matrix. The Robertson uncertainty principle

     

Sequences of integers with missing quotients

Let S be any set of natural numbers, and A be a given set of rational numbers. We say that S is an A-quotient-free set if x,y∈S implies y/x∉A. Let and , where the supremum is taken over all A-quotient-free sets S, and are the upper and lower asymptotic densities of S respectively. Let ρ(A)=sup_Sδ(S), where the supremum is taken over all A-quotient-free sets S such that δ(S) exists. In this paper we study the properties of , and ρ(A).     

Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian

Let G be a simple algebraic group defined over C and T be a maximal torus of G. For a dominant coweight λ of G, the T-fixed point subscheme of the Schubert variety in the affine Grassmannian GrG is a finite scheme. We prove that for all such λ if G is of type A or D and for many of them if G is of type E, there is a natural isomorphism between the dual of the level one affine Demazure module corresponding to λ and the ring of functions (twisted by certain line bundle on GrG) of . We use this fact to give a geometrical proof of the Frenkel-Kac-Segal isomorphism between basic representations of affine algebras of A,D,E type and lattice vertex algebras.     

The case of equality in the Dobrushin-Deutsch-Zenger bound

Suppose that A=(a_i,j) is an n×n real matrix with constant row sums μ. Then the Dobrushin-Deutsch-Zenger (DDZ) bound on the eigenvalues of A other than μ is given by . When A a transition matrix of a finite homogeneous Markov chain so that μ=1,Z(A) is called the coefficient of ergodicity of the chain as it bounds the asymptotic rate of convergence, namely, , of the iteration , to the stationary distribution vector of the chain.In this paper we study the structure of real matrices for which the DDZ bound is sharp. We apply our results to the study of the class of graphs for which the transition matrix arising from a random walk on the graph attains the bound. We also characterize the eigenvalues λ of A for which |λ|=Z(A) for some stochastic matrix A.     

Extensions of strict polynomial functors

We compute Ext-groups between Frobenius twists of strict polynomial functors. The main result concerns the groups where Dd is the divided power functor, and F is an arbitrary functor. These groups are shown to be isomorphic to F(Ai) for certain explicitly described graded space Ai. We also calculate the groups where Wμ and Sλ are respectively the Weyl and Schur functors associated to diagrams μ, λ of the same weight.     

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d:=n_±(A)<∞. We show that if, for all λ in an open interval I⊂R, the dimension of defect subspaces Nλ(A) (=Ker(A^?−λ)) coincides with d, then every self-adjoint extension has no continuous spectrum in I and the point spectrum of is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator.     

The L resolvents of second-order elliptic operators of divergence form under the Dirichlet condition

Let A be the 2mth-order elliptic operator of divergence form with bounded measurable coefficients defined in a domain Ω of . For 1<p<∞ we regard A as a bounded linear operator from the L^p Sobolev space to H−m,p(Ω). It is known that when , we can construct the resolvent (A−λ)⁻¹ and estimate its operator norm for some λ if the leading coefficients are uniformly continuous. In this paper, we try to extend this result to a general domain. It is successful when m=1 if Ω is the half-space or a domain with C² bounded boundary. For m>1 it is shown that the problem is reduced to the case where Ω is the half-space and A is a homogeneous operator with constant coefficients. We also give a perturbation theorem.     

On a class of Köthe sequence spaces with normal structure

In this note, we investigate the generalized modulus of convexity δ ( λ ) and the generalized modulus smoothness ρ ( λ ) . We obtain some estimates of these moduli for X = lp . We obtain inequalities between WCS coefficient of a K¨othe sequence space X and δ ( λ ) X . We show that, for a wide class of K¨othe sequence spaces X, if for some ε∈ (0, 9 10 ] holds δ X (ε) > 1 3 1 √ 3 2 ε, then X has normal structure.     

How to regularize singular vectors and kill the dynamical Weyl group

Let be a simple Lie algebra, and let M_λ be the Verma module over with highest weight λ. For a finite-dimensional -module U we introduce a notion of a regularizing operator, acting in U, which makes the meromorphic family of intertwining operators holomorphic, and conjugates the dynamical Weyl group operators A_w(λ)∈End(U) to constant operators. We establish fundamental properties of regularizing operators, including uniqueness, and prove the existence of a regularizing operator in the case .     

Higher power moments of the Riesz mean error term of symmetric square L-function

Let be the error term of the Riesz mean of the symmetric square L-function. We give the higher power moments of and show that if there exists a real number A₀:=A₀(ρ)>3 such that , then we can derive asymptotic formulas for , 3?h<A₀, h∈N. Particularly, we get asymptotic formulas for , h=3,4,5 unconditionally.     

The symmetric Meixner-Pollaczek polynomials with real parameter

The Symmetric Meixner-Pollaczek polynomials for λ>0 are well-studied polynomials. These are polynomials orthogonal on the real line with respect to a continuous, positive real measure. For λ?0, are also polynomials, however they are not orthogonal on the real line with respect to any real measure. This paper defines a non-standard inner product with respect to which the polynomials for λ?0, become orthogonal polynomials. It examines the major properties of the polynomials, for λ>0 which are also shared by the polynomials, for λ?0.     

Inducing regulation of any digraphs

For a given structure D (digraph, multidigraph, or pseudodigraph) and an integer r large enough, a smallest inducing r-regularization of D is constructed. This regularization is an r-regular superstructure of the smallest possible order with bounded arc multiplicity, and containing D as an induced substructure. The sharp upper bound on the number, ρ, of necessary new vertices among such superstructures for n-vertex general digraphs D is determined, ρ being called the inducing regulation number of D. For being the maximum among semi-degrees in D, simple n-vertex digraphs D with largest possible ρ are characterized if either or (where the case is not a trivial subcase of ).     

Distributions and measures on the boundary of a tree

In this paper, we analyze the space of distributions on the boundary Ω of a tree and its subspace , which was introduced in [Amer. J. Math. 124 (2002) 999-1043] in the homogeneous case for the purpose of studying the boundary behavior of polyharmonic functions. We show that if , then μ is a measure which is absolutely continuous with respect to the natural probability measure λ on Ω, but on the other hand there are measures absolutely continuous with respect to λ which are not in . We then give the definition of an absolutely summable distribution and prove that a distribution can be extended to a complex measure on the Borel sets of Ω if and only if it is absolutely summable. This is also equivalent to the condition that the distribution have finite total variation. Finally, we show that for a distribution μ, Ω decomposes into two subspaces. On one of them, a union of intervals A_μ, μ restricted to any finite union of intervals extends to a complex measure and on A_μ we give a version of the Jordan, Hahn, and Lebesgue-Radon-Nikodym decomposition theorems. We also show that there is no interval in the complement of A_μ in which any type of decomposition theorem is possible. All the results in this article can be generalized to results on good (in particular, compact infinite) ultrametric spaces, for example, on the p-adic integers and the p-adic rationals.     

Global weak solutions and asymptotic behavior to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

This paper is concerned with existence of global weak solutions to a class of compressible Navier-Stokes equations with density-dependent viscosity and vacuum. When the viscosity coefficient μ is proportional to ρθ with , a global existence result is obtained which improves the previous results in Fang and Zhang (2004) [4], Vong et al. (2003) [27], Yang and Zhu (2002) [30]. Here ρ is the density. Moreover, we prove that the domain, where fluid is located on, expands outwards into vacuum at an algebraic rate as the time grows up due to the dispersion effect of total pressure. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., θ=1, γ=2).     

The crossed product by a partial endomorphism and the covariance algebra

Given a local homeomorphism where U⊆X is clopen and X is a compact and Hausdorff topological space, we obtain the possible transfer operators Lρ which may occur for given by α(f)=f○σ. We obtain examples of partial dynamical systems (XA,σA) such that the construction of the covariance algebra C^∗(XA,σA), proposed by B.K. Kwasniewski, and the crossed product by a partial endomorphism O(XA,α,L), recently introduced by the author and R. Exel, associated to this system are not equivalent, in the sense that there does not exist an invertible function ρ∈C(U) such that O(XA,α,Lρ)≅C^∗(XA,σA).