Lifts of Non-Compact Convex Sets and Cone Factorizations

This paper generalizes the factorization theorem of Gouveia,Parrilo and Thomas to a broader class of convex sets.Given a general convex set,the authors define a slack operator associated to the set and its polar according to whether the convex set is full dimensional,whether it is a translated cone and whether it contains lines.The authors strengthen the condition of a cone lift by requiring not only the convex set is the image of an affine slice of a given closed convex cone,but also its recession cone is the image of the linear slice of the closed convex cone.The authors show that the generalized lift of a convex set can also be characterized by the cone factorization of a properly defined slack operator.

On Proximinality of Convex Sets in Superspaces

Abstract In this paper, we show that a closed convex subset C of a Banach space is strongly proximinal （proximinal, resp.） in every Banach space isometrically containing it if and only if C is locally （weakly, resp.） compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonempty set-valued and upper semi-continuous.

Structural eigenvalue analysis under the constraint of a fuzzy convex set model

In small-sample problems, determining and controlling the errors of ordinary rigid convex set models are difficult. Therefore, a new uncertainty model called the fuzzy convex set（FCS） model is built and investigated in detail. An approach was developed to analyze the fuzzy properties of the structural eigenvalues with FCS constraints. Through this method, the approximate possibility distribution of the structural eigenvalue can be obtained. Furthermore, based on the symmetric F-programming theory, the conditional maximum and minimum values for the structural eigenvalue are presented, which can serve as nonfuzzy quantitative indicators for fuzzy problems. A practical application is provided to demonstrate the practicability and effectiveness of the proposed methods.

CONVEX-MODEL ANALYSIS OF THE SEISMIC RESPONSE

An ellipsoidal Fourier-bound convex model (EFB model) is proposed in the present paper to express the uncertainty of seismic excitation, and several methods of selecting parameters of the model are explained. An analytical expression is obtained for the worst response of the single-degree-of-freedom (SDOF) system with the EFB model. A numerical simulation shows that the traditional prediction of maximum response can yield the value substantially lover than that predicted by the EFB model. This means that the traditional designing method based on standard seismic inputs may lead to unsafe design decisions.

SOME RESULTS ON BEST APPROXIMATION IN SPACES WITH CONVEX STRUCTURE

Some generalizations of the result proved by S.P. Singh [J. Approx. Theory 25（1979）, 89-90] are presented in convex metric spaces. The results proved contain several known results on the subject.

Convex Set Theory for Reliability Assessment of Steel Beam with Bounded Uncertainty

Probabilistic reliability model established by insufficient data is inaccessible. The convex model was applied to model the uncertainties of variables. A new non-probabilistic reliability model was proposed based on the robustness of system to uncertainty. The non-probabilistic reliability model,the infinite norm model,and the probabilistic model were used to assess the reliability of a steel beam,respectively. The results show that the resistance is allowed to couple with the action effect in the non-probabilistic reliability model. Additionally,the non-probabilistic reliability model becomes the same accurate as probabilistic model with the increase of the bounded uncertain information. The model is decided by the available data and information.

An algorithm for computed tomography image reconstruction from limited-view projections

With the development of the compressive sensing theory, the image reconstruction from the projections viewed in limited angles is one of the hot problems in the research of computed tomography technology. This paper develops an iterative algorithm for image reconstruction, which can fit the most cases. This method gives an image reconstruction flow with the difference image vector, which is based on the concept that the difference image vector between the reconstructed and the reference image is sparse enough. Then the l1-norm minimization method is used to reconstruct the difference vector to recover the image for flat subjects in limited angles. The algorithm has been tested with a thin planar phantom and a real object in limited-view projection data. Moreover, all the studies showed the satisfactory results in accuracy at a rather high reconstruction speed.

Iterative analytic extension in tomographic imaging

If a spatial-domain function has a finite support,its Fourier transform is an entire function.The Taylor series expansion of an entire function converges at every finite point in the complex plane.The analytic continuation theory suggests that a finite-sized object can be uniquely determined by its frequency components in a very small neighborhood.Trying to obtain such an exact Taylor expansion is difficult.This paper proposes an iterative algorithm to extend the measured frequency components to unmeasured regions.Computer simulations show that the proposed algorithm converges very slowly,indicating that the problem is too ill-posed to be practically solvable using available methods.

Optimization Processes of Tangible and Intangible Networks through the Laplace Problems for Regular Lattices with Multiple Obstacles along the Way

A systematic approach is proposed to the theme of safety,reliability and global quality of complex networks(material and immaterial)by means of special mathematical tools that allow an adequate geometric characterization and study of the operation,even in the presence of multiple obstacles along the path.To that end,applying the theory of graphs to the problem under study and using a special mathematical model based on stochastic geometry,in this article we consider some regular lattices in which it is possible to schematize the elements of the network,with the fundamental cell with six,eight or 2(n+2)obstacles,calculating the probability of Laplace.In this way it is possible to measure the“degree of impedance”exerted by the anomalies along the network by the obstacles examined.The method can be extended to other regular and/or irregular geometric figures,whose union together constitutes the examined network,allowing to optimize the functioning of the complex system considered.

NEW METHODS FOR CHARACTERIZING D-η-PROPERLY PREQUASI-INVEX FUNCTIONS

The D-η-proper prequasi invexity of vector-valued functions is characterized by means of （weak） nearly convexity and density of sets. Under weaker assumptions, some equivalent conditions for D-η-proper prequasi-invexity are derived.

A Comprehensive Model for Structural Non-Probabilistic Reliability and the Key Algorithms

It is very difficult to know the exact boundaries of the variable domain for problems with small sample size,and the traditional convex set model is no longer applicable.In view of this,a novel reliability model was proposed on the basis of the fuzzy convex set(FCS)model.This new reliability model can account for different relations between the structural failure region and variable domain.Key computational algorithms were studied in detail.First,the optimization strategy for robust reliability is improved.Second,Monte Carlo algorithms(i.e.,uniform sampling method)for hyper-ellipsoidal convex sets were studied in detail,and errors in previous reports were corrected.Finally,the Gauss-Legendre integral algorithm was used for calculation of the integral reliability index.Three numerical examples are presented here to illustrate the rationality and feasibility of the proposed model and its corresponding algorithms.

Lipschitz-continuity of Spherically Convex Functions

In this article,some basic and important properties of spherically convex functions,such as the Lipschitz-continuity,are investigated.It is shown that,under a weaker condition,every family of spherically convex functions is equi-Lipschitzian on each closed spherically convex subset contained in the relative interior of their common domain,and from which a powerful result is derived:the pointwise convergence of a sequence of spherically convex functions implies its uniform convergence on each closed spherically convex subset contained in the relative interior of their common domain.

Efficient Inverse Method for Structural Identification Considering Modeling and Response Uncertainties

The inverse problem analysis method provides an effective way for the structural parameter identification.However,uncertainties wildly exist in the practical engineering inverse problems.Due to the coupling of multi-source uncertainties in the measured responses and the modeling parameters,the traditional inverse method under the deterministic framework faces the challenges in solving mechanism and computing cost.In this paper,an uncertain inverse method based on convex model and dimension reduction decomposition is proposed to realize the interval identification of unknown structural parameters according to the uncertain measured responses and modeling parameters.Firstly,the polygonal convex set model is established to quantify the epistemic uncertainties of modeling parameters.Afterwards,a space collocation method based on dimension reduction decomposition is proposed to transform the inverse problem considering multi-source uncertainties into a few interval inverse problems considering response uncertainty.The transformed interval inverse problem involves the two-layer solving process including interval propagation and optimization updating.In order to solve the interval inverse problems considering response uncertainty,an efficient interval inverse method based on the high dimensional model representation and affine algorithm is further developed.Through the coupling of the above two strategies,the proposed uncertain inverse method avoids the time-consuming multi-layer nested calculation procedure,and then effectively realizes the uncertainty identification of unknown structural parameters.Finally,two engineering examples are provided to verify the effectiveness of the proposed uncertain inverse method.

ANALYTIC SOLUTIONS OF SYSTEMS OF FUNCTIONAL EQUATIONS

Let r be a given positive number. Denote by D=D r the closed disc in the complex plane C whose center is the origin and radius is r. For any subset K of C and any integer m≥1, write A(D m,K)={f|f∶D m→K is a continuous map, and f|(D m)° is analytic}. For H∈ A(D m,C)(m≥2), f∈A(D,D) and z∈D, write Ψ H(f)(z)=H(z,f(z),...,f m-1(z)). Suppose F,G∈A(D 2n+1,C), and H k,K k∈A(D k,C), k=2,...,n. In this paper, the system of functional equations F(z,f(z),f 2(Ψ H 2(f)(z)),...,f n(Ψ H n(f)(z)),g(z),g 2(Ψ K 2(g)(z)),..., g n(Ψ K n(g)(z)))=0 G(z,f(z),f 2(Ψ H 2(f)(z)),...,f n(Ψ H n(f)(z)),g(z),g 2(Ψ K 2(g)(z)),..., g n(Ψ K n(g)(z)))=0(z∈D) is studied and some conditions for the system of equations to have a solution or a unique solution in A(D,D)×A(D,D) are given.

Theorems of Helly's Type for Partially-closed Half-spaces

In this article, under quite weaker conditions, theorems of Helly’s type for(not necessarily finite) family of partially closed half-spaces are presented in both analytic and geometric forms. Examples are also provided to show that these conditions cannot be omitted in general.

On the Projections of Convex Lattice Sets

As a discrete analogue of Aleksandrov＇s projection theorem, it is natural to ask the following question： Can an o-symmetric convex lattice set of the integer lattice Z n be uniquely determined by its lattice projection counts？ In 2005, Gardner, Gronchi and Zong discovered a counterexample with cardinality 11 in the plane. In this paper, we will show that it is the only counterexample in Z2, up to unimodular transformations and with cardinality not larger than 17.

Image enhancement and post-processing for low resolution compressed video

This research paper recommends the point spread function(PSF)forecasting technique based on the projection onto convex set(POCS)and regularization to acquire low resolution images.As the environment for the production of user created contents(UCC)videos(one of the contents on the Internet)becomes widespread,resolution reduction and image distortion occurs,failing to satisfy users who desire high quality images.Accordingly,this research neutralizes the coding artifact through POCS and regularization processes by:1)factoring the local characteristics of the image when it comes to the noise that results during the discrete cosine transform(DCT)and quantization process;and 2)removing the blocking and ring phenomena which are problems with the existing video compression.Moreover,this research forecasts the point spread function to obtain low resolution images using the above-mentioned methods.Thus,a method is suggested for minimizing the errors found among the forecasting interpolation pixels.Low-resolution image quality obtained through the experiment demonstrates that significant enhancement was made on the visual level compared to the original image.

Existence and Uniqueness of Analytic Solutions of Systems of Iterative Functional Equations

Let r be a given positive number. Denote by D=D r the closed disc in the complex plane C whose center is the origin and radius is r. Write A(D,D)={f: f is a continuous map from D into itself, and f|D ° is analytic}. Suppose G,H: D 2n+1 →C are continuous maps (n≥2), and G|(D 2n+1 ) °, H|(D 2n+1 ) ° are analytic. In this paper, we study the system of iterative functional equationsG(z,f(z),…,f n(z), g(z),…,g n(z))=0, H(z,f(z),…,f n(z), g(z),…,g n(z))=0, for any z∈D,and give some conditions for the system of equations to have a solution or a unique solution in A(D,D) ×A(D,D).

SOME EQUIVALENT CONDITIONS OF PROXIMINALITY IN NONREFLEXIVE BANACH SPACES

In this paper,we discuss the relation between τ-strongly Chebyshev,approximatively τ-compact k-Chebyshev,approximatively τ-compact,τ-strongly proximinal and proximinal sets,where τ is the norm or the weak topology.We give some equivalent conditions regarding the above proximinality.Furthermore,we also propose the necessary and sufficient conditions that a half-space is τ-strongly proximinal,approximatively τ-compact and τ-strongly Chebyshev.

Distance Sets Relating to Orthogonal Exponentials

The aim of this paper is to investigate the size properties of a planar set whose distance set has some prescribed arithmetic combinatorics. Such research is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. By proving a shifted version of ErdSs-Solymosi＇s theorem on the distance sets, we give some grounds on the conjecture. The results obtained here extend the corresponding results of Iosevich and Jaming in a simple manner.