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.     

Photoproduction of the Charged Top-Pions at the LHeC

The top triangle moose (TTM) model, which can be seen as the deconstructed version of the topcolor-assisted technicolor (TC2) model, predicts the existence of the charged top-pions πt± in low energy spectrum. In the context of this model, we consider photoproduction of πt± via the subprocesses γb→tπ-and γ→πt+ at the large hadron-electron collider (LHeC), in which high energy photon beams are generated by using the Compton backscatting method. We find that, as long as the charged top-pions are not too heavy, they can be abundantly produced via γb collision.     

On a Conjecture of Erdős,Gallai, and Tuza

Erd?s, Gallai, and Tuza posed the following problem: given an n‐vertex graph G, let denote the smallest size of a set of edges whose deletion makes G triangle‐free, and let denote the largest size of a set of edges containing at most one edge from each triangle of G. Is it always the case that ? We have two main results. We first obtain the upper bound , as a partial result toward the Erd?s–Gallai–Tuza conjecture. We also show that always , where m is the number of edges in G; this bound is sharp in several notable cases.     

One-Relator Products Induced from Generalized Triangle Groups

Abstract

We study the quotient of a free product of groups by the normal closure of a word that is contained in a certain type of 2-generator subgroup, and hence can be expressed in the form of a pushout involving a generalized triangle group. Using the theory of generalized triangle groups, we extend the range of conditions under which a number of results for one-relator products of groups, such as the Freiheitssatz and the solubility of the word problem, are known to hold.     

An improved randomized approximation algorithm for maximum triangle packing

This paper deals with the maximum triangle packing problem. For this problem, Hassin and Rubinstein gave a randomized polynomial-time approximation algorithm that achieves an expected ratio of for any constant ?>0. By modifying their algorithm, we obtain a new randomized polynomial-time approximation algorithm for the problem which achieves an expected ratio of 0.5257(1−?) for any constant ?>0.     

Extremal problems on triangle areas in two and three dimensions

The study of extremal problems on triangle areas was initiated in a series of papers by Erd?s and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles.In the plane, our main result is an O(n44/19)=O(n^2.3158) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O(n7/3) from 1992. We also make progress in a number of important special cases. We show that: (i) For points in convex position, there exist n-element point sets that span Ω(nlogn) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most ; there exist n-element point sets (for arbitrarily large n) that span (6/π²−o(1))n² minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O(n); this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O(n); this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span Ω(nlogn) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds).In 3-space we prove an O(n17/7β(n))=O(n^2.4286) upper bound on the number of unit-area triangles spanned by n points, where β(n) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O(n8/3), is an old result of Erd?s and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n²+O(n), and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω(n4/3) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O(n4/3+ε), for any ε>0. (iii) Every set of n points, not all on a line, determines at least Ω(n2/3/β(n)) triangles of distinct areas, which share a common side.     

用预期效用理论单位三角形解释共同比率效应

本文提供一个新的视角来观察共同比率效应.这个悖论可以用期望效用理论和预期效用理论来解释.特别地,Machina(1987)在期望效用理论单位三角形比较一对彩票,共同比率效应形成一个悖论.在预期效用理论单位三角形中,效用函数无差异曲线保持平行,但是彩票的连线向内收敛,从而说明共同比率效应在预期效用理论可能是合理的,而不再是一个悖论.     

Infrared diode laser spectroscopy of the ν = 2 band of AlF

A vibrational–rotational spectrum of the ν = 2 transitions of a high-temperature molecule AlF was observed between 1490 and 1586 cm⁻¹ with a diode laser spectrometer. Measurements were made on the ν = 3–1, 4–2, 5–3 and 8–6 bands at a temperature of 900 °C. Measured spectral lines were fitted to effective band constants ν₀, B_ν and D_ν for each band. Present measurements were made with only one Pb-salt laser diode. Physical significance of the effective band constants is discussed.