Algorithms for ham-sandwich cuts

Given disjoint setsP ₁,P ₂, ...,P _d inR ^d withn points in total, ahamsandwich cut is a hyperplane that simultaneously bisects theP _i . We present algorithms for finding ham-sandwich cuts in every dimensiond>1. Whend=2, the algorithm is optimal, having complexityO(n). For dimensiond>2, the bound on the running time is proportional to the worst-case time needed for constructing a level in an arrangement ofn hyperplanes in dimensiond−1. This, in turn, is related to the number ofk-sets inR ^d−1 . With the current estimates, we get complexity close toO(n ^3/2 ) ford=3, roughlyO(n ^8/3 ) ford=4, andO(n ^d−1−a(d) ) for somea(d)>0 (going to zero asd increases) for largerd. We also give a linear-time algorithm for ham-sandwich cuts inR ³ when the three sets are suitably separated. A preliminary version of the results of this paper appeared in [16] and [17]. Part of this research by J. Matoušek was done while he was visiting the School of Mathematics, Georgia Institute of Technology, Atlanta, and part of his work on this paper was supported by a Humboldt Research Fellowship. W. Steiger expresses gratitude to the NSF DIMACS Center at Rutgers, and his research was supported in part by NSF Grants CCR-8902522 and CCR-9111491.     

A rigid graph for every set

A graph G is called rigid if the identical mapping V(G)→V(G) is the only homomorphism G→G. In this note we give a simple construction of a rigid oriented graph on every set. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 108–110, 2002     

Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions

In this paper we deal with nonlinear second-order boundary value problems with impulses. The impulsive functions depend implicitly on the different considered variables and the boundary value conditions are nonlinear. In both cases functional dependence on the solution is allowed. The existence results follow from the existence of a pair of well-ordered lower and upper solutions.     

On Nash Equilibria for Noncooperative Games Governed by the Burgers Equation

Existence of a Nash equilibrium in a noncooperative game governed by the one-dimensional Burgers equation, proposed in the case of pointwise controls in Ref. 1, is proved under data qualifications that guarantee the diffusion term in the Burgers’ equation to be dominant enough with respect to the uniform convexity of the payoffs. This work was partly supported by Grants 201/03/0934 (GA čR) and MSM 0021620839 (MšMT čR). Inspiring discussions with Angel M. Ramos are acknowledged.     

Plasmaneutrino spectrum

The spectrum of the neutrinos produced in the massive photon and longitudinal plasmon decay process has been computed with four levels of approximation for the dispersion relations. Some analytical formulae in the limit cases are derived. Interesting conclusions related to previous calculations of the energy loss in stars are presented. The high energy tail of the neutrino spectrum is shown to be proportional to exp(-E/kT), where E is the neutrino energy and kT is the temperature of the plasma. PACS 97.90.+j; 97.60.-s; 95.55.Vj; 52.27.Ep     

IR finite one-loop box scalar integral with massless internal lines

The IR finite one-loop box scalar integral with massless internal lines has been recalculated. The result is very compact, simple and valid for arbitrary values of the relevant kinematic variables. It is given in terms of only two dilogarithms and a few logarithms, all of very simple arguments. Received: 31 January 2002 / Published online: 26 April 2002     

Current and shot noise in two capacitively coupled single electron transistors with an atomic sized spacer

The currents and their fluctuations in two capacitively coupled single electron transistors are determined in the limit of sequential tunnelling. Our considerations are restricted to the case when the islands (dots) of the transistors are atomic-sized, which means each of them has only one single electronic level available for the tunnelling processes. The Coulomb interactions of accumulated charges on the both single electron transistors lead to the effect of the negative differential resistance. An enhancement of the current shot-noise was also found. Spectral decomposition analysis indicated the two main contributions to the shot-noise: low- and high-frequency fluctuations. It was found that the low frequency fluctuations (polarization noise) are responsible for a strong enhancement of the current noise. Received 9 October 2001 / Received in final form 8 March 2002 Published online 9 July 2002     

On the numerical solution of axisymmetric domain optimization problems by dual finite element method

Shape optimization of an axisymmetric three-dimensional domain with an elliptic boundary value state problem is solved. Since the cost functional is given in terms of the cogradient of the solution, a dual finite element method based on the minimum of complementary energy principle is used. © 1994 John Wiley & Sons, Inc.     

Crystallization-induced asymmetric transformation. Application to conjugate addition of benzylamine to amides of benzoylacrylic acid

Adducts of the conjugate addition of benzylamine to enantiopure amides of aroylacrylic acid possess high enantiomeric and diastereomeric purity. A high degree of stereoselectivity has been achieved by means of crystallization-induced asymmetric transformation. A practical synthesis leading to dipeptides containing homophenylalanine is depicted.