Scalar Versions Research Articles

On an equivalence of topological vector space valued cone metric spaces and metric spaces

Scalarization method is an important tool in the study of vector optimization as corresponding solutions of vector optimization problems can be found by solving scalar optimization problems. Recently this has been applied by Du (2010) [14] to investigate the equivalence of vectorial versions of fixed point theorems of contractive mappings in generalized cone metric spaces and scalar versions of fixed point theorems in general metric spaces in usual sense. In this paper, we find out that the topology induced by topological vector space valued cone metric coincides with the topology induced by the metric obtained via a nonlinear scalarization function, i.e any topological vector space valued cone metric space is metrizable, prove a completion theorem, and also obtain some more results in topological vector space valued cone normed spaces.

The Existence of Cone Critical Point and Common Fixed Point with Applications

We first establish some new critical point theorems for nonlinear dynamical systems in cone metric spaces or usual metric spaces, and then we present some applications to generalizations of Dancš‐Hegedüs‐Medvegyev′s principle and the existence theorem related with Ekeland′s variational principle, Caristi′s common fixed point theorem for multivalued maps, Takahashi′s nonconvex minimization theorem, and common fuzzy fixed point theorem. We also obtain some fixed point theorems for weakly contractive maps in the setting of cone metric spaces and focus our research on the equivalence between scalar versions and vectorial versions of some results of fixed point and others.

SCALARIZATION METHODS FOR VECTOR VARIATIONAL INEQUALITIES

1. Introduction and preliminariesIn studying vector problems including vector optimization prob-lems, vectorvariationalinequalityproblemsandvectorequilibriumprob-lems, there have been usually traditional concentrations on scalarizationapproaches which enable us to replace the vector problems under con-sideration with equivalent scalar problems [3, 4, 6, 8-10].In particular, in [10], Slavov discussed some scalarization techniquesand one application of the multi-objective optimization problem into amathematical economics.In [6], Jimenez et al. developed a scalarization method in order toobtain scalar versions of their results on the necessary and sufficientconditions for strict minimizers of a general vector optimization problem,through a variational approach.In [8], Konnov considered a scalarization approach to connect vectorvariational inequalities into an equivalent scalar variational inequalitieswith a set-valued cost mapping. He gave an equivalence between weakand strong solutions of set-valued vector variational inequalities andsuggested a new gap function for vector variational inequalities. Heapplied his results to vector optimization, vector network equilibriumand vector migration equilibrium problems.In [4], Guu et al. extended the scalarization approaches of Giannessiet. al [3] to set-valued vector optimization problems and set-valued weakvector optimization problems. The scalar variational inequalities givenby them is different from those given by Konnov [8].

Vibration based health monitoring for a lightweight truss structure: Experimental assessment of several statistical time series methods

An experimental assessment of several vibration based statistical time series methods for structural health monitoring (SHM) is presented via their application to a lightweight aluminum truss structure. A concise overview of the main non-parametric and parametric methods is provided, including response-only and excitation–response schemes. Damage detection and identification is based on univariate (scalar) versions of the methods, while results for three distinct vibration measurement positions on the structure are presented. The methods’ effectiveness is assessed via multiple experiments under various damage scenarios. The results of the study confirm the high potential and effectiveness of statistical time series methods for SHM.

SCALARIZATION METHODS FOR MINTY-TYPE VECTOR VARIATIONAL INEQUALITIES

Abstract. Many kinds of Minty’s lemmas show that Minty-type varia-tional inequality problems are very closely related to Stampacchia-typevariational inequality problems. Particularly, Minty-type vector varia-tional inequality problems are deeply connected with vector optimizationproblems. Liu et al. [10] considered vector variational inequalities for set-valued mappings by using scalarization approaches considered by Kon-nov [8]. Lee et al. [9] considered two kinds of Stampacchia-type vectorvariational inequalities by using four kinds of Stampacchia-type scalarvariational inequalities and obtain the relations of the solution sets be-tween the six variational inequalities, which are more generalized resultsthan those considered in [10]. In this paper, the author considers theMinty-type case corresponding to the Stampacchia-type case consideredin [9]. 1. Introduction and PreliminariesRecently, there have been usually traditional concentrations on scalarizationapproaches [3, 4, 6, 8-11] which enable us to replace the vector problems un-der consideration with equivalent scalar problems in studying vector problemsincluding vector optimization problems, vector variational inequality problemsand vector equilibrium problems.In particular, Slavov [11] discussed some scalarization techniques and oneapplication of multi-objective optimization problems into a mathematical eco-nomics.In 2009, Jimenez et al. [6] developed a scalarization method in order toobtain scalar versions of their results on the necessary and sucient condi-tions for strict minimizers of a general vector optimization problem, through avariational approach.Konnov [8] also considered a scalarization approach to connect vector vari-ational inequalities into an equivalent scalar variational inequalities with a

A note on cone metric fixed point theory and its equivalence

The main aim of this paper is to investigate the equivalence of vectorial versions of fixed point theorems in generalized cone metric spaces and scalar versions of fixed point theorems in (general) metric spaces (in usual sense). We show that the Banach contraction principles in general metric spaces and in T V S -cone metric spaces are equivalent. Our theorems also extend some results in Huang and Zhang (2007) [L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476], Rezapour and Hamlbarani (2008) [Sh. Rezapour, R. Hamlbarani, Some notes on the paper Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 345 (2008) 719–724] and others.

Scalarization and optimality conditions for strict minimizers in multiobjective optimization via contingent epiderivatives

In this paper necessary and sufficient conditions for strict minimizers of a general vector optimization problem are given by means of different notions of graphical or epigraphical derivatives, extending some existing results in the literature. In first place, these conditions are established by means of contingent derivatives and secondly, by applying a nonconvex separation result, in terms of contingent epiderivatives and hypoderivatives. Moreover, through a variational approach, a scalarization method is developed in order to obtain scalar versions of these results.

On some nonlinear problems induced by an abstract maximal element principle

In this paper, by applying an abstract maximal element principle on quasi-ordered sets established by Lin and Du, we obtain a generalized Brézis–Browder principle, system (vectorial) versions of Ekeland's variational principle and maximal element principle and a vectorial version of Takahashi's nonconvex minimization theorem. Moreover, we investigate the equivalence between scalar versions and vectorial versions of these results.

Comparative analysis of bitplane-based wavelet image coders

Wavelet image encoders based on bitplanes give excellent results in the compression of still images. The bitplane concept has recently been generalised to vectors, and wavelet image encoders based on vector bitplanes have been proposed, some achieving slightly better rate × distortion performances than scalar encoders. There remains the open question of whether the use of vector bitplanes has the potential of providing more significant rate × distortion improvements over scalar versions. The authors address this question by analysing in detail the performance of adaptations, for the use of vector bitplanes, of four popular wavelet-based bitplane encoders. From this analysis, they determine where the gains in performance of encoders based on vector bitplanes come from. It is concluded that performance improvements may come by increasing the vector dimension, provided that codebooks with good packing properties are used.

Coupled scalar and vector P N methods for solving multigroup transport problems in multislab geometry

A spherical-harmonics ( P N ) method that incorporates and couples existent scalar and vector versions of the method is developed. The scalar (group-by-group) version of method is applied to the slowing-down energy range and the vector version to the thermal energy range. The link of coupling between these methods is the slowing-down source to the thermal groups which becomes available once the problems for all of the slowing-down groups are solved with the scalar method. The advantage of using the coupled P N method over the vector method when solving transport problems defined in extended energy range is demonstarted in this work.

Application of vector parabolic equation method to urban radiowave propagation problems

The vector parabolic equation method is applied to the modelling of radiowave propagation in the urban environment. As a paraxial version of Maxwell's equations it allows full treatment of 3D electromagnetic scattering which was not possible with scalar versions of the algorithm. The method allows detailed specification of the building geometry and electrical parameters. This approach is particularly useful for accurate modelling of scattering by a single building or a group of buildings at microwave frequencies. The method is validated using an iterative integral equation method and known analytical results. Examples include scattering by a building with hemispherical roof and scattering by a group of buildings with sloping roofs.

On NP-Completeness for Linear Machines

We consider linear and scalar versions of the Blum–Shub–Smale model of computation over the reals. The permitted computing operations of linear machines are addition and multiplication by constants. The scalar machines can only multiply by constants. The size of an input is its dimension, and the cost of any instruction is one. For each of these structures we consider DNP and NP, the corresponding complexity classes with respect to digital nondeterminism and standard real nondeterminism, respectively. We give DNP- and NP-complete problems for linear and real scalar machines. On the other hand, we show that the NP-class restricted to scalar machines over the integers with equality-tests does not own a complete problem.

Clarkson and Random Clarkson Inequalities forLr(X)

We first show how (p,p′) Clarkson inequality for a Banach space X is inherited by Lebesgue-Bochner spaces Lr(X), which extends Clarkson's procedure deriving his inequalities for Lp from their scalar versions. Fairly many previous and new results on Clarkson's inequalities, and also those on Rademacher type and cotype at the same time (by a recent result of the authors), are obtained as immediate consequences. Secondly we show that if the (p, p') Clarkson inequality holds in X, then random Clarkson inequalities hold in Lr(X) for any 1 ≤ r ≤ ∞; the converse is true if r = p'. As corollaries the original Clarkson and random Clarkson inequalities for Lp are both directly derived from the parallelogram law for scalars.

Columnwise block LU factorization using blas kernels on VAX 6520/2VP

The LU factorization of a matrix A is a widely used algorithm, for instance in the solution of linear systems Ax = b. The increasing capacities of high performance computers allow us to use direct methods for systems of large and dense matrices. To build portable and efficient LU codes for vector and parallel computers, this method is rewritten in block versions and BLAS (Basic Linear Algebra Subprograms) kernels are used to mask the architectural details and allow good performance of codes such as the LAPACK (Linear Algebra PACKage) library. In the references it was proved that this strategy leads to portability and efficiency of codes using tuned BLAS kernels. After a short description of the block versions we will present some results obtained on the VAX 6520/2VP, comparing the block algorithm versus point algorithm, and vectorized versions versus scalar versions. The three columnwise versions of the block algorithm showed similar performance for this computer and large matrix dimensions. The block size used is a crucial parameter for these algorithms and the results show that the best performance is obtained with block size 64 (for large matrices) which is the vector registered size of the machine used.

On the Correlation Inequalities in Some Models Involving Fermions

AbstractUsing the Coleman (‐Fröhlich‐Seiler) theorem we have mapped chargeless sectors of some (generalized) Yukawa‐Thirring‐(Schwinger) two‐dimensional models onto pure Bose theories. Discussion of certain correlation inequalities in this framework is made. In particular, we proved that the first and second Griffiths‐Kelly‐Shermann correlation inequalities hold.Correlation inequalities suggest that there is on phase transition of the first order in scalar versions of these models. Some results on the D‐L‐R equations for these models are established, confirming previous suggestion.

Schwarz's Lemma for n-Ports

The scalar versions of Schwarz's Lemma have been extensively employed in the classical development of the synthesis of lumped RLC driving-point impedance function. A vector space generalization of Schwarz's Lemma, particularly suitable for application to linear passive n-port impedance functions, is derived in this paper. The concept of power dominant networks is introduced and a number of power inequalities derived.

Vacuum radiational processes in pair-generating fields

Diverse vacuum radiational processes in a constant homogeneous field are considered from the point of view of QED with an external pair-generating field. Computations are performed for the scalar and spinor versions of the theory.

Robustized recursive estimation with applications

In this paper two convergence theorems for the robustized scalar versions of the Robbins-Monro and the Kiefer-Wolfowitz procedures are presented. In addition, two information processing oriented examples are included which illustrate these recursive estimators and their robustness.

Radiation from a point source located on the boundary of a semi-infinite plane impedance system. Scalar versions of the problem

Radiation from a point source located on the boundary of a semi-infinite plane impedance system. Scalar versions of the problem