We consider the approximability of the tsp problem in graphs that satisfy a relaxed form of triangle inequality. More precisely, we assume that for some parameter 1, the distances satisfy the inequality dist(x; y) (dist(x; z) + dist(z; y)) for every triple of vertices x, y, and z. We obtain a 4 approximation and also show that for some > 0 it is np-hard to obtain a (1 +) approximation. Our uppe...

Abstract We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. prove that equilateral triangle is local maximiser of lowest eigenvalue among all triangles given area provided negative parameter sufficiently small in absolute value, with smallness depending only. Moreover, using various trial functions, we obtain sufficient conditions for global optimal...

In this paper, we study some controllability and observability properties for a coupled system of time-discrete fourth- second-order parabolic equations. This can be regarded as simplification the well-known stabilized Kumamoto-Sivashinsky equation. Unlike continuous case, prove only relaxed inequality which yields $ \phi(\triangle t) $-controllability result. result tells that cannot reach exa...

A popular question in recreational mathematics is the following: If we drop a spaghetti noodle and it breaks at two random places, what is the probability that we can form a triangle with the three resulting segments? See for example [2, chap.1, sec.6], [3, p.6], [4, p.31], or [7, p.30-36]. This is an elementary problem in geometric probability. Clearly the length of the noodle (or equivalently...

We consider three variants of the problem finding a maximum weight restricted 2-matching in subcubic graph G. (A is any subset edges such that each vertex incident to at most two its edges.) Depending on variant means either triangle-free or square-free both triangle- and square-free. Since computing NP-hard, second third we additionally assume edge-weights are vertex-induced square. While ther...

Triangle inequality violations (TIVs) are the effect of packets between two nodes being routed on the longer direct path between them when a shorter detour path through an intermediary is available. TIVs are a natural, widespread and persistent consequence of Internet routing policies. By exposing opportunities to improve the delay between two nodes, TIVs can help myriad applications that seek ...