The Convergence Rate of Fréchet Distribution under the Second-Order Regular Variation Condition

In this article we consider the asymptotic behavior of extreme distribution with the extreme value index γ>0 . The rates of uniform convergence for Fréchet distribution are constructed under the second-order regular variation condition.

Regularity Condition of Solutions to the Quasi-geostrophic Equations in Besov Spaces with Negative Indices

With a Hǒlder type inequality in Besov spaces, we show that every strong solution θ（t, x） on （0, T ） of the dissipative quasi-geostrophic equations can be continued beyond T provided that ⊥θ（t, x） ∈L 2γ/γ-2δ （（0, T ）; B^δ-γ/2 ∞∞（R^2）） for 0 〈 δ 〈 γ/2 .

Estimates for parametric Marcinkiewicz integrals in BMOand Campanato spaces

In this paper, the authors consider the behaviors of a class of parametricMarcinkiewicz integrals μ_Ω~ρ, μ_（Ω,）^（＊,）~ρ_λ and μ_Ω~ρ,S on BMO（Rn） and Campanato spaces with com-plex parameter ρ and the kernel Ω in Llog~＋ L（S^（n-1））. Here μ_（Ω,）^（＊,）~ρ_λand μ_Ω~ρ,S are parametricMarcinkiewicz functions corresponding to the Littlewood-Paley g_λ~＊-function and the Lusin areafunction S, respectively. Under certain weak regularity condition on Ω, the authors prove thatif f belongs to BMO（Rn） or to a certain Campanato space, then [μ_（Ω,）^（＊,）~ρ_λ（f）]~2, [μ_Ω~ρ,_S（f）]~2 and[μ_Ω~ρ（f）]~2 are either infinite everywhere or finite almost everywhere, and in the latter case, somekind of boundedness are also established.

A BMO ESTIMATE FOR THE MULTILINEAR SINGULAR INTEGRAL OPERATOR

The behavior on the space L^∞（R^n） for the multilinear singular integral operator defined by TAf（x）=∫RnΩ（x-y）/｜x-y｜^n＋1（A（x）-A（y）△A（y）（x-y））f（y）dy is considered, where 12 is homogeneous of degree zero, integrable on the unit sphere and has vanishing is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishingmoment of order one, A has derivatives of order one in BMO（R^n）. It is proved that if Ω satisfies some minimum size condition and an L1-Dini type regularity condition, then for f ∈ L^∞（R^n）, TAf is either infinite almost everywhere or finite almost everywhere, and in the latter case, TAf ∈ BMO（R^n）.

Conditional Regularity of Weak Solutions to the 3D Magnetic Bénard Fluid System

This paper concerns about the regularity conditions of weak solutions to the magnetic Benard fluid system in R^(3).We show that a weak solution(u,b,θ)(·,t)of the 3D magnetic Benard fluid system defined in[0,T),which satisfies some regularity requirement as(u,b,θ),is regular in R^(3)x(0,T)and can be extended as a C∞solution beyond T.

Regularity of Solutions to the Navier-Stokes Equations with a Nonstandard Boundary Condition

In this paper we are concerned with the regularity of solutions to the Navier-Stokes equations with the condition on the pressure on parts of the boundary where there is flow. For the steady Stokes problem a result similar to L q-theory for the one with Dirichlet boundary condition is obtained. Using the result, for the steady Navier-Stokes equations we obtain regularity as the case of Dirichlet boundary conditions. Furthermore,for the time-dependent 2-D Navier-Stokes equations we prove uniqueness and existence of regular solutions,which is similar to J.M.Bernard＇s results[6]for the time-dependent 2-D Stokes equations.

Existence theory for Rosseland equation and its homogenized equation

The global boundness and existence are presented for the kind of the Rosseland equation with a general growth condition. A linearized map in a closed convex set is defined. The image set is precompact, and thus a fixed point exists. A multi-scale expansion method is used to obtain the homogenized equation. This equation satisfies a similar growth condition.

Approximate Optimality Conditions for Composite Convex Optimization Problems

The purpose of this paper is to study the approximate optimality condition for composite convex optimization problems with a cone-convex system in locally convex spaces,where all functions involved are not necessarily lower semicontinuous.By using the properties of the epigraph of conjugate functions,we introduce a new regularity condition and give its equivalent characterizations.Under this new regularity condition,we derive necessary and sufficient optimality conditions ofε-optimal solutions for the composite convex optimization problem.As applications of our results,we derive approximate optimality conditions to cone-convex optimization problems.Our results extend or cover many known results in the literature.

A Regularity Criterion in Terms of Direction of Vorticity to the 3D Micropolar-fluid Equations

In this paper, we study the regularity of weak solutions to the 3D Micropolarfluid equations. We show that the weak solutions actually is strong solution if the corresponding vorticity field j = × u satisfies certain condition in the high vorticity region.

Some Results on Regular Conditional Probabilities

§ 1. IntroductionLet (Q, &~, P) be a separable probability space with &~=a (An> n>l), ^ be a sub o-field of &~. SetandF*-E\.I<t<t>m a.g..(2)By Follmer Lemma (o.f. [1], Th. 3.5 and Th. 3.7), we may assume that we have chosen a version of (Ft)t><> such that for all w, F.(w} is a right continuousincreasing function with F0(w~) =0 and Ft(w} =1 for all t^--. (Such a version of￡(Ft) can also be constructed by elementary method.) For a fixed w, denote by

MLE with datasets from populations having shared parameters

We consider maximum likelihood estimation with two or more datasets sampled from differ-ent populations with shared parameters.Although more datasets with shared parameters can increase statistical accuracy,this paper shows how to handle heterogeneity among different populations for correctness of estimation and inference.Asymptotic distributions of maximum likelihood estimators are derived under either regulan cases where regularityconditions are satis-fled or some non-regular situations.A bootstrap variance estimator for assessing performance of estimators and/or making large sample inferenceis also introduced and evaluated ina simulation study.

A ROBUST SQP METHOD BASED ON A SMOOTHING APPROXIMATE PENALTY FUNCTION FOR INEQUALITY CONSTRAINED OPTIMIZATION

A robust SQP method, which is analogous to Facchinei’s algorithm, is introduced. The algorithm is globally convergent. It uses automatic rules for choosing penalty parameter, and can efficiently cope with the possible inconsistency of the quadratic search subproblem. In addition, the algorithm employs a differentiable approximate exact penalty function as a merit function. Unlike the merit function in Facchinei’s algorithm, which is quite complicated and is not easy to be implemented in practice, this new merit function is very simple. As a result, we can use the Facchinei’s idea to construct an algorithm which is easy to be implemented in practice.