A global compactness result for an elliptic equation with double singular terms

In this paper, we establish a global compactness result for (P.S.) sequences of the variational functional of the elliptic problem $\left\{\begin{array}{cc}-\Delta u-\frac{\mu }{|x{|}^2}u=\frac{1}{|x{|}^s}|u{|}^{2_s^1-2}u+\lambda u,\hfill & x\in \Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$where $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, is a bounded smooth domain with $0\in \Omega$, $\mu \in [0,{(n-2)}^2∕4)$, $s\in [0,2)$ and $\lambda \in \mathbb{R}$ are constants. This extends the global compactness result of Cao and Peng (2003) to the case of elliptic problems with double singular critical terms. Our arguments adapt some refined Sobolev inequalities systematically developed quite recently by Palatucci and Pisante (2014) and blow-up analysis. In this way, our arguments turn out to be quite transparent and easy to be applied to many other problems.     

A global compactness result for singular elliptic problems involving critical Sobolev exponent

Let be a bounded domain such that . Let be a (P.S.) sequence of the functional . We study the limit behaviour of and obtain a global compactness result.

     

Continuity and compactness of singular integral operators

In the space of square integrable functions we establish effective sufficient continuity and compactness conditions for singular integral operators with Cauchy kernels on a segment of the real axis.     

New exact periodic wave solutions for the Dullin-Gottwald-Holm equation

In this paper, the Dullin-Gottwald-Holm equation is studied using semi-inverse method and integral bifurcation method. New periodic waves such as peakon-like periodic wave, compacton-like periodic wave and singular periodic wave are found and their dynamical behaviors and certain strange phenomena are explained using the proposed criterion. The exact parametric representations of these waves are also presented.     

An exact estimate result for a class of singular equations with critical exponents

We consider the singular boundary value problem

     

Perturbation of the SVD in the presence of small singular values

This paper gives SVD perturbation bounds and expansions that are of use when an m × n, m ? n matrix A has small singular values. The first part of the paper gives subspace bounds that are closely related to those of Wedin but are stated so as to isolate the effect of any small singular values to the left singular subspace. In the second part first and second order approximations are given for perturbed singular values. The subspace bounds are used to show that all approximations retain accuracy when applied to small singular values. The paper concludes by deriving a subspace bound for multiplicative perturbations and using that bound to give a simple approximation to a singular value perturbed by a multiplicative perturbation.     

Riesz-Nágy singular functions revisited

In 1952 F. Riesz and Sz.-Nágy published an example of a monotonic continuous function whose derivative is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-Nágy family of functions and we relate it to a system for real number representation which we call (τ,τ−1)-expansions. With the help of these real number expansions we generalize the family. The singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.     

A lower bound guaranteeing exact matrix completion via singular value thresholding algorithm

In this paper, we give a lower bound guaranteeing exact matrix completion via singular value thresholding (SVT) algorithm. The analysis shows that when the parameter in SVT algorithm is beyond some finite scalar, one can recover some unknown low-rank matrices exactly with high probability by solving a strictly convex optimization problem. Furthermore, we give an explicit expression for such a finite scalar. This result in the paper not only has theoretical interests, but also guides us to choose suitable parameters in the SVT algorithm.     

The exact solution of a linear integral equation with weakly singular kernel

A space , which is proved to be a reproducing kernel space with simple reproducing kernel, is defined. The expression of its reproducing kernel function is given. Subsequently, a class of linear Volterra integral equation (VIE) with weakly singular kernel is discussed in the new reproducing kernel space. The reproducing kernel method of linear operator equation Au=f, which request the image space of operator A is and operator A is bounded, is improved. Namely, the request for the image space is weakened to be L²[a,b], and the boundedness of operator A is also not required. As a result, the exact solution of the equation is obtained. The numerical experiments show the efficiency of our method.     

On local non‐compactness in recursive mathematics

A metric space is said to be locally non‐compact if every neighborhood contains a sequence that is eventually bounded away from every element of the space, hence contains no accumulation point. We show within recursive mathematics that a nonvoid complete metric space is locally non‐compact iff it is without isolated points. The result has an interesting consequence in computable analysis: If a complete metric space has a computable witness that it is without isolated points, then every neighborhood contains a computable sequence that is eventually computably bounded away from every computable element of the space. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Grothendieck compactness principle for the Mackey dual topology

Grothendieck [6] proved that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. In a recent paper [3], an analogous result for weak compactness in a Banach space is shown to be equivalent to the Schur property. In this article, we obtain a similar type result in the Mackey dual of a Banach space. A related result for weak^? compactness is also obtained.     

Variable step size initial value algorithm for singular perturbation problems using locally exact integration

We consider quasilinear singular perturbation problems of the form εy^″+p(x)y^′+q(x,y)=h(x),x[0,1];y(0)=,y(1)=β with a boundary layer at one end point. The original problem is reduced to an asymptotically equivalent linear first order initial-value problem (IVP). Then, a variable step size initial value algorithm is applied to solve this (IVP). The algorithm is based on the locally exact integration of quadratic linearized problem coefficients on a non-uniform mesh. Two term-recurrence relation with controlled step size is obtained. Several problems are solved to demonstrate the applicability and efficiency of the algorithm. It is observed that the present method approximates the exact solution very well.     

A theorem on sequentiality and compactness, with its fuzzy extension

An amended proof of a theorem of Franklin's on sequentiality and sequential compactness is presented, and a corresponding fuzzy version formulated using the theory of quasi-coincidence.     

The exact asymptotic behaviour of the unique solution to a singular nonlinear Dirichlet problem

By Karamata regular varying theory, a perturbed argument and constructing comparison functions, we show the exact asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem −Δu=b(x)g(u)+λf(u), u>0, x∈Ω, u_|∂Ω=0, which is independent on λf(u), and we also show the existence and uniqueness of solutions to the problem, where Ω is a bounded domain with smooth boundary in RN, λ>0, g∈C¹((0,∞),(0,∞)) and there exists γ>1 such that , ∀ξ>0, , the function is decreasing on (0,∞) for some s₀>0, and b is nonnegative nontrivial on Ω, which may be vanishing on the boundary.     

Countable choice and compactness

We work in set-theory without choice ZF. Denoting by the countable axiom of choice, we show in that the closed unit ball of a uniformly convex Banach space is compact in the convex topology (an alternative to the weak topology in ZF). We prove that this ball is (closely) convex-compact in the convex topology. Given a set I, a real number p1 (respectively p=0), and some closed subset F of [0,1]^I which is a bounded subset of ℓ^p(I), we show that (respectively DC, the axiom of Dependent Choices) implies the compactness of F.     

A new look at interpretability and saturation

We investigate the interpretability ordering ${?}^{?}$ using generalized Ehrenfeucht–Mostowski models. This gives a new approach to proving inequalities and investigating the structure of types.     

On exact controllability and complete stabilizability for linear systems

This work is concerned with the relations between exact controllability and complete stabilizability for linear systems in Hilbert spaces. We give an affirmative answer to the open problem posed by Rabah and Karrakchou [R. Rabah, J. Karrakchou, Exact controllability and complete stabilizability for linear systems in Hilbert spaces, Appl. Math. Lett. 10 (1997) 35–40]. More precisely, if the $C_0$-semigroup $S\left(t\right)$ generated by $A$ is surjective and the pair $(A,B)$ with a bounded operator $B$ is completely stabilizable, then $(A,B)$ is exactly controllable without any additional condition.     

Super weak compactness and uniform Eberlein compacta

We prove that a topological space is uniform Eberlein compact iff it is homeomorphic to a super weakly compact subset C of a Banach space such that the closed convex hull coC of C is super weakly compact. We also show that a Banach space X is super weakly compactly generated iff the dual unit ball B_X* of X^* in its weak star topology is affinely homeomorphic to a super weakly compactly convex subset of a Banach space.