Minimum and maximum principle sufficiency properties for nonsmooth variational inequalities

This paper deals with the study of minimum and maximum principle sufficiency properties for nonsmooth variational inequalities (in short, NVI) by using gap functions. Several characterizations of these two sufficiency properties are provided. We also discuss the error bound for nonsmooth variational inequalities. Two open questions are given at the end.     

Optimality Conditions for Isolated Local Minimum of Nonsmooth Multiobjective Problems

This article is devoted to the study of Fritz John and strong Kuhn-Tucker necessary conditions for properly efficient solutions, efficient solutions and isolated efficient solutions of a nonsmooth multiobjective optimization problem involving inequality and equality constraints and a set constraints in terms of the lower Hadamard directional derivative. Sufficient conditions for the existence of such solutions are also provided where the involved functions have pseudoconvex sublevel sets. Our results are based on the concept of pseudoconvex sublevel sets. The functions with pseudoconvex sublevel sets are a class of generalized convex functions that include quasiconvex functions.     

Promises and potential pitfalls of a ‘cognitive neuroscience of mathematics learning’

The present commentary discusses the papers of the special issue on ‘cognitive neuroscience and mathematics learning’ with respect to methodological and theoretical constraints of using neuroscientific methods to study educationally relevant processes associated with mathematics learning. A special focus is laid on the relevance of subject populations, methodological limitations of current neuroimaging methods and theoretical questions concerning the relationship between the well-studied neural correlates of numerical magnitude processing and the less-investigated neural processes underlying higher level mathematical skills, such as algebraic reasoning.     

A search for mechanoluminophors capable of pressure-induced thermal population of excited states

During the process of deforming a crystal, a high pressure is developed near the tip of mobile cracks, which may in turn produce a new ground state by thermal electron transfer. Upon sudden release of pressure, the electron can either relax to one atmosphere ground state or remain in the excited state potential well long enough to relax to one atmosphere and radiatively transfer back to the ground state. For analysing the pressure induced thermal population of the excited state, the mechanoluminescence(ML) and high pressure photoluminescence(PL) of several organic and inorganic crystals were measured. The study indicated that usual pressure coefficient of energy shift of the order of 50–100 cm⁻¹/kbar and the stress at the crack-tip of the order of 5–10 kbar, are not sufficient to cause the thermal population of the excited state. If by any means the product of pressure coefficient and stress at the mobile crack-tip can be increased by 50 to 100 times, then the thermal population of the excited states may take place. Using the pressure coefficient of energy shift and the difference in ML and PL spectra, and using independently the change in relative intensities of the vibronic peaks, the pressure at the emitting mechanoluminescent crystal sites is evaluated and it is found to be of the order of several kbar which varies from crystal to crystal.     

Minimum Cost Compromise Mixed Allocation

When more than one (say p) characteristics in multivariate stratified population are defined on each unit of the population, the individual optimum allocations may differ widely and can not be used practically. Moreover, there may be a situation such that no standard allocation is advisable to all the strata, for one reason or another. In such a situation, Clark and Steel (J R Stat Soc, Ser D Stat 49(2):197–207, 2000) suggested that different allocations may be used for different groups of strata having some common characteristics for double sampling in stratification. Later on, Ahsan et al. (Aligarh J Stat 25:87–97, 2005) used the same concept in univariate stratified sampling. They minimized the variance of the stratified sample mean for a fixed cost to obtain an allocation and called this allocation “mixed allocation”. In the present paper, a “compromise mixed allocation” is worked out for the fixed precisions of the estimates of the p-population means of a multivariate stratified population. A numerical example is also presented.     

A comprehensive study on the oscillation frequency of spherical fullerenes in carbon nanotubes under different system parameters

Amongst possible new nanomechanical devices created based on carbon nanostructures, high-frequency nanoscale oscillators, or the so-called gigahertz oscillators have attracted much attention. In this paper, the oscillatory behavior of spherical fullerenes inside carbon nanotubes is thoroughly investigated. To this end, the continuum approximation together with Lennard-Jones potential is used to evaluate the van der Waals potential energy and interaction force. The equation of motion is directly solved based on the actual force distribution between the two nanostructures, without any simplifying assumption. A semi-analytical expression is obtained for the oscillation frequency into which the effect of initial conditions is incorporated. Thereafter, this newly derived expression is utilized in order to present a comprehensive study on the effects of different system variables such as geometrical parameters and initial conditions on the oscillation frequency. Based upon these studies, some new features of such oscillations have been revealed.     

Iterative algorithms for systems of extended regularized nonconvex variational inequalities and fixed point problems

This paper points out some fatal errors in the equivalent formulations used in Noor 2011 [Noor MA. Projection iterative methods for solving some systems of general nonconvex variational inequalities. Applied Analysis. 2011;90:777–786] and consequently in Noor 2009 [Noor MA. System of nonconvex variational inequalities. Journal of Advanced Research Optimization. 2009;1:1–10], Noor 2010 [Noor MA, Noor KI. New system of general nonconvex variational inequalities. Applied Mathematics E-Notes. 2010;10:76–85] and Wen 2010 [Wen DJ. Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Analysis. 2010;73:2292–2297]. Since these equivalent formulations are the main tools to suggest iterative algorithms and to establish the convergence results, the algorithms and results in the aforementioned articles are not valid. It is shown by given some examples. To overcome with the problems in these papers, we consider a new system of extended regularized nonconvex variational inequalities, and establish the existence and uniqueness result for a solution of the aforesaid system. We suggest and analyse a new projection iterative algorithm to compute the unique solution of the system of extended regularized nonconvex variational inequalities which is also a fixed point of a nearly uniformly Lipschitzian mapping. Furthermore, the convergence analysis of the proposed iterative algorithm under some suitable conditions is studied. As a consequence, we point out that one can derive the correct version of the algorithms and results presented in the above mentioned papers.     

Predictor–Corrector Methods for General Regularized Nonconvex Variational Inequalities

This paper is devoted to the study of a new class of nonconvex variational inequalities, named general regularized nonconvex variational inequalities. By using the auxiliary principle technique, a new modified predictor–corrector iterative algorithm for solving general regularized nonconvex variational inequalities is suggested and analyzed. The convergence of the iterative algorithm is established under the partially relaxed monotonicity assumption. As a consequence, the algorithm and results presented in the paper overcome incorrect algorithms and results existing in the literature.