L—Estimates for the Gradient of Weak Solutions to a Class of Degenerate Nonlinear Elliptic Systems

We call that λ(x)belongs to Muckenhoupt class A_p if λ(x)≥0 and satisfiesfor all balls in R,where|B_R| denotes the Lebesgue measure of a ball with centerx_0 and radius R,andLet be a bounded domain.Define normHere,denotes the Banach space of all measurable functions defined on     

On π-Subpairs of π-Blocks

Alperin and Broug have given the p-subpairs in a finite group, and proved that there is a Sylow theorem for p-subpairs. For a π-separable group with π-Hall subgroup nilpotent, we prove that there is a π-Sylow theorem for π-subpairs. Note that our π-subpairs are different from what Robinson and Staszewski gave.     

A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.     

Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

We establish a new 3G-Theorem for the Green’s function for the half space We exploit this result to introduce a new class of potentials that we characterize by means of the Gauss semigroup on . Next, we define a subclass of and we study it. In particular, we prove that properly contains the classical Kato class . Finally, we study the existence of positive continuous solutions in of the following nonlinear elliptic problem

On Extrapolation Properties of Schatten–von Neumann Classes

For a certain special class of symmetric sequence spaces, we give an explicit relation between the interpolation and extrapolation representations. This relation is carried over to symmetrically normed ideals of compact operators.     

Rate of Convergence of Nearest Neighbor Density Estimates

Let X_1,…,X_n be i.i.d.samples drawn from a one-dimensional population with unknowndensity function f(x),Loftsgarden and Quesenberry proposed the following estimate f_n(x)forf(x):Choose positive integer k(n)≤n,find the smallest a_n(x)such that N_n(x-a_n(x),x+a_n(x))≥k(n),and define.This is the so-called Nearest Neighbor Estimator.In this paper,we obtained the following theorem:     

Asymptotic Normality of Kernel Density Estimates

Let X_1,…,X_n be a random samples taking values in R~1 and having common density function f(x). A class of estimators of f(x) proposed by Rosenblatt has the form     

The derived π-length, nilpotent π-length, and simple π-length of finite π-soluble groups

The concept of the derived π-length for finite π-soluble groups is introduced and its elementary properties are described. The dependence between the π-length, nilpotent π-length, and derived π-length, and also between the derived and nilpotent lengths of a π-Hall subgroup, is determined.     

Slope Estimates of Artin–Schreier Curves

Let X/F_p be an Artin–Schreier curve defined by the affine equation y ^p –y=f(x) where f(x)F_p[x] is monic of degree d. In this paper we develop a method for estimating the first slope of the Newton polygon of X. Denote this first slope by NP₁(X/F_p). We use our method to prove that if p>d2 then NP₁(X/F_p)(p–1)/d/(p–1). If p>2d4, we give a sufficient condition for the equality to hold.     

Extrapolation of Nyström Solutions of Boundary Integral Equations on Non-Smooth Domains

The interior Dirichlet problem for Laplace's equation on a plane polygonal region $\Omega$ with boundary $\Gamma$ may be reformulated as a second kind integral equation on $\Gamma$. This equation may be solved by the Nyström method using the composite trapezoidal rule. It is known that if the mesh has $O(n)$ points and is graded appropriately, then $O(1/n^2)$ convergence is obtained for the solution of the integral equation and the associated solution to the Dirichlet problem at any $x\in \Omega$. We present a simple extrapolation scheme which increases these rates of convergence to $O(1/n^4)$ .     

Optimal L ∞(L 2)-Error Estimates for the DG Method Applied to Nonlinear Convection–Diffusion Problems with Nonlinear Diffusion

This article is concerned with the analysis of the discontinuous Galerkin finite element method (DGFEM) applied to the space semidiscretization of a nonstationary convection–diffusion problem with nonlinear convection and nonlinear diffusion. Optimal estimates in the L ^∞(L ²)-norm are derived for the symmetric interior penalty (SIPG) scheme in two dimensions. The error analysis is carried out for nonconforming triangular meshes under the assumption that the exact solution of the problem and the solution of a linearized elliptic dual problem are sufficiently regular.     

The logic of π1-conservativity

We show that the modal prepositional logicILM (interpretability logic with Montagna's principle), which has been shown sound and complete as the interpretability logic of Peano arithmetic PA (by Berarducci and Savrukov), is sound and complete as the logic of ₁-conservativity over eachbE ₁-sound axiomatized theory containingI ₁ (PA with induction restricted tobE ₁-formulas). Furthermore, we extend this result to a systemILMR obtained fromILM by adding witness comparisons in the style of Guaspari's and Solovay's logicR (this will be done in a separate continuation of the present paper).     

On π-regularity of General Rings

A general ring means an associative ring with or without identity.An idempotent e in a general ring I is called left （right） semicentral if for every x ∈ I,xe=exe （ex=exe）.And I is called semiabelian if every idempotent in I is left or right semicentral.It is proved that a semiabelian general ring I is π-regular if and only if the set N （I） of nilpotent elements in I is an ideal of I and I /N （I） is regular.It follows that if I is a semiabelian general ring and K is an ideal of I,then I is π-regular if and only if both K and I /K are π-regular.Based on this we prove that every semiabelian GVNL-ring is an SGVNL-ring.These generalize several known results on the relevant subject.Furthermore we give a characterization of a semiabelian GVNL-ring.     

Convergence and Stability Properties of a Vector Padé Extrapolation Method

In this paper we introduce a new convergence accelerator for vector sequences — vector Padé approximation method (VPA), discuss its convergence and stability properties, and show that it is a bona fide acceleration method for some vector sequences.     

Hybrid Dynamics of Stochastic π-Calculus

In this paper we describe a method to map stochastic π-calculus processes in chemical ground form into hybrid automata. Hybrid automata are tools widely employed to model systems characterized by both discrete and continuous evolution and their use in the context of Systems Biology allows us to address rather fundamental issues. Specifically, the key ingredient we use in this work is the possibility granted by hybrid automata to implement a separation of control and molecular terms in biochemical systems. The computational counterpart of our analysis turns out to be related to the determination of conservation properties of the system.