A sharpened Strichartz inequality for radial functions

We prove a new sharpened version of the Strichartz inequality for radial solutions of the Schrödinger equation in two dimensions. We establish an improved upper bound for functions that nearly extremize the inequality, with a negative second term that measures the distance from the initial data to Gaussians.     

On vector variational-like inequality problems

In this paper, we establish some relationships between vector variational-like inequality and vector optimization problems under the assumptions of α-invex functions. We identify the vector critical points, the weakly efficient points and the solutions of the weak vector variational-like inequality problems, under pseudo-α-invexity assumptions. These conditions are more general than those of existing ones in the literature. In particular, this work extends the earlier work of Ruiz-Garzon et al. [G. Ruiz-Garzon, R. Osuna-Gomez, A. Rufian-Lizan, Relationships between vector variational-like inequality and optimization problems, European J. Oper. Res. 157 (2004) 113-119] to a wider class of functions, namely the pseudo-α-invex functions studied in a recent work of Noor [M.A. Noor, On generalized preinvex functions and monotonicities, J. Inequal. Pure Appl. Math. 5 (2004) 1-9].     

On generalized nonlinear variational-like inequality problems

In this work, we consider a generalized nonlinear variational-like inequality problem in the setting of locally convex topological vector spaces and prove the existence of its solutions.     

A new sharpened version of the Neuberg–Pedoe inequality

In this work, we present a new sharpened version of the classical Neuberg–Pedoe inequality. As an application, the following improved Neuberg–Pedoe inequality is derived:

     

On teaching extrema triangle problems using dynamic investigation

An important and interesting area in the study of triangle geometry is the related issue of extrema problems and inequalities. These problems play a significant role in the mathematics study program in high school. In tasks such as these, the difficulty level is high when one does not know in advance what the expected answer is. When one knows what to prove, the difficulty level is lower and most of the effort is aimed at attaining a proof of the expected answer. This can be done using dynamic geometric software. The possibility of making frequent changes to the geometric objects and the ability of dragging objects, contributes to the process of deducing properties, checking hypotheses and generalizing. In this paper, eight investigative tasks in Euclidean geometry are presented together with the applets developed for carrying out the dynamic investigation. Some of the tasks are well known, while others are almost unknown and are worthy of presentation as enrichment for those interested in the subject. The tasks were given to preservice teachers of mathematics as part of an advanced course for integrating technological tools in the teaching of mathematics.     

On the generalized triangle inequality for quasimetrics induced by noncommuting vector fields

For a sufficiently wide class of r-smooth basis vector fields, we obtain necessary and sufficient conditions for some anisotropic metric functions induced by these vector fields to be quasimetrics. These results are applied to the problem of the existence of a nilpotent tangent cone at a distinguished point.     

On a class of inequality problems with lack of compactness

In this paper we prove the existence of at least one solution for a variational-hemivariational inequality on a closed and convex set without using critical point theory. We apply a version of the well-known theorem of Knaster-Kuratowski-Mazurkiewicz. The obtained results can be applied to Schrödinger type problems and to problems with radially symmetric functions.     

On necessary extremum conditions for finite-dimensional problems with inequality constraints

The finite-dimensional optimization problem with equality and inequality constraints is examined. The case where the classical regularity condition is violated is analyzed. Necessary second-order extremum conditions are obtained that are stronger versions of some available results.     

On generalized implicit vector variational inequality problems

In this paper, we introduce and study the generalized implicit vector variational inequality problems with set valued mappings in topological vector spaces. We establish existence theorems for the solution set of these problems be nonempty compact and convex. Our results extend the results by Fang and Huang [ Existence results for generalized implicit vector variational inequalities with multivalued mappings, Indian J. Pure and Appl. Math. 36(2005), 629–640.]     

The matrix valued triangle inequality:quaternion version

The matrix valued triangle inequality is shown to hold for matrices over the quaternions. The history of the inequality and open questions connected with it are described.     

Heuristic matching for graphs satisfying the triangle inequality

Two matching heuristics are presented. The hyper-greedy method runs in time O(n²log n) and produces a matching whose cost is at most 2log₃(1.5n) times optimal. Graphs are given causing this method to achieve nearly this ratio. The factor of two method runs in time O(n²log K), where K is the maximum ratio of edge lengths in the graph, and never requires more than O(n³) time. The factor of two method produces a matching whose cost is at most max(4log₂K, 4log₂n) times optimal, plus lower-order terms. Graphs are given causing this method to achieve a ratio asymptotically equal to $\text{(}\text{log}{}_2\text{n)}\text{2}$.     

Well-posedness for parametric quasivariational inequality problems and for optimization problems with quasivariational inequality constraints

In this article, the concepts of well-posedness and well-posedness in the generalized sense are introduced for parametric quasivariational inequality problems with set-valued maps. Metric characterizations of well-posedness and well-posedness in the generalized sense, in terms of the approximate solutions sets, are presented. Characterization of well-posedness under certain compactness assumptions and sufficient conditions for generalized well-posedness in terms of boundedness of approximate solutions sets are derived. The study is further extended to discuss well-posedness for an optimization problem with quasivariational inequality constraints.     

An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality

We consider the asymmetric traveling salesperson problem with γ-parameterized triangle inequality for γ[1/2,1). That means, the edge weights fulfill w(u,v)γ(w(u,x)+w(x,v)) for all nodes u,v,x. Chandran and Ram [L.S. Chandran, L.S. Ram, Approximations for ATSP with parametrized triangle inequality, in: Proc. 19th Int. Symp. on Theoret. Aspects of Comput. Sci. (STACS), in: Lecture Notes in Comput. Sci., vol. 2285, Springer, Berlin, 2002, pp. 227–237] gave the first constant factor approximation algorithm with polynomial running time for this problem. They achieve performance ratio γ/(1−γ). We devise an approximation algorithm with performance ratio (1+γ)/(2−γ−γ³), which is better for γ[0.5437,1), that is, for the particularly interesting large values of γ.