Novel algorithms for the network lifetime problem in wireless settings

One of the major concerns in wireless ad-hoc networks design is energy efficiency. Wireless devices are typically equipped with a limited energy supply sufficient only for a limited amount of time which is reversely proportional to the transmission power of the device. The network lifetime is defined as the time the first device runs out of its initial energy charge. In this paper we study the maximum network lifetime problem for broadcast and data gathering in wireless settings. We provide polynomial time approximation algorithms, with guaranteed performance bounds while considering omnidirectional and unidirectional transmissions. We also consider an extended variant of the maximum lifetime problem, which simultaneously satisfies additional constraints, such as bounded hop-diameter and degree of the routing tree, and minimizing the total energy used in a single transmission. Finally, we evaluate the performance of some of our algorithms through simulations.     

Enumeration and randomized constructions of hypertrees

Over 30 years ago, Kalai proved a beautiful d‐dimensional analog of Cayley's formula for the number of n‐vertex trees. He enumerated d‐dimensional hypertrees weighted by the squared size of their (d ? 1)‐dimensional homology group. This, however, does not answer the more basic problem of unweighted enumeration of d‐hypertrees, which is our concern here. Our main result, Theorem 1.4, significantly improves the lower bound for the number of d‐hypertrees. In addition, we study a random 1‐out model of d‐complexes where every (d ? 1)‐dimensional face selects a random d‐face containing it, and show that it has a negligible d‐dimensional homology.     

Q-switching an all-fiber laser using acousto-optic null coupler

A new method for Q-switching an all-fiber laser is presented. It is based on induced acoustic long period grating operating on a null coupler, which acts as acoustically controlled tunable output coupler. Q-switching is achieved by switching on and off the acoustic wave in a burst mode, thereby generating laser pulses that are ~400 times shorter than the acoustically controlled coupler’s rise time. Output pulse energy of 22 μJ and temporal width of ~100 ns were measured at a wavelength of 1.54 μm.     

Statistics on wreath products and generalized binomial-stirling numbers

Various statistics on wreath products are defined via canonical words, “colored” right to left minima and “colored” descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of binomial-Stirling type. These extended Stirling numbers determine (via matrix inversion) dual systems, which are also shown to have combinatorial realizations within the wreath product. The above setting also gives rise to a MacMahon-type equi-distribution theorem over subsets with prescribed statistics. Partially supported by Minerva Grant No. 8441 and by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272. Partially supported by EC's IHRP Programme, within the Research Training Network “Algebraic Combinatorics in Europe”, grant HPRN-CT-2001-00272.     

No directed fractal percolation in zero area

We consider the fractal percolation process on the unit square with fixed decimation parameterN and level-dependent retention parameters {p _k}; that is, for allk ⩾ 1, at thek th stage every retained square of side lengthN ¹⁻ k is partitioned intoN ² congruent subsquares, and each of these is retained with probabilityp _k. independent of all others. We show that if Π_k p _k =0 (i.e., if the area of the limiting set vanishes a.s.), then a.s. the limiting set contains no directed crossings of the unit square (a directed crossing is a path that crosses the unit square from left to right, and moves only up, down, and to the right).     

Time-reversal symmetry violation and the oscillating universe

The expressions for the fractional number ofK ⁰'s and¯K ⁰'s in a neutral kaon beam are discussed with reference to time-reversal asymmetry. The suggested relation between the sign of Re ( is the Lee-WuT-violation parameter) and the cosmological arrow of time ifCPT is broken is further clarified.Research sponsored by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under AFOSR grant number EOOAR-68-0010, through the European Office of Aerospace Research.     

The design of a Tort law to control accidents

In this paper we analyse a two-stage game involving the government and n agents who engage in a single activity (driving). The government establishes the legal policy setting and the agents proceed to play a non-cooperative game of incomplete information with a risk of accident in which their behavioral strategy is their level of care. We examine the Nash-equilibrium conditions for single-activity accidents between heterogeneous agents, ‘good’ drivers or ‘bad’ drivers allowing a variable damage function and a liability rule defined on the cube. The relative desirability for society of alternative equilibria and the conditions under which they can obtain are discussed. The constraints which circumscribe the ability of the government to induce an equilibrium involving careful driving are demonstrated. It transpires that when the proportion of good drivers increases, it becomes more difficult to sustain a careful equilibrium whereas an equilibrium of reckless behavior becomes easier to sustain. Various extensions of the models are also presented.     

Quasisymmetric embeddings, the observable diameter, and expansion properties of graphs

It is shown that the edges of any n-point vertex expander can be replaced by new edges so that the resulting graph is an edge expander, and such that any two vertices that are joined by a new edge are at distance in the original graph. This result is optimal, and is shown to have various geometric consequences. In particular, it is used to obtain an alternative perspective on the recent algorithm of Arora et al. [Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, 2004, pp. 222-231.] for approximating the edge expansion of a graph, and to give a nearly optimal lower bound on the ratio between the observable diameter and the diameter of doubling metric measure spaces which are quasisymmetrically embeddable in Hilbert space.     

New Coins From Old: Computing With Unknown Bias

Suppose that we are given a function f : (0, 1)→(0,1) and, for some unknown p∈(0, 1), a sequence of independent tosses of a p-coin (i.e., a coin with probability p of “heads”). For which functions f is it possible to simulate an f(p)-coin? This question was raised by S. Asmussen and J. Propp. A simple simulation scheme for the constant function f(p)≡1/2 was described by von Neumann (1951); this scheme can be easily implemented using a finite automaton. We prove that in general, an f(p)-coin can be simulated by a finite automaton for all p ∈ (0, 1), if and only if f is a rational function over ℚ. We also show that if an f(p)-coin can be simulated by a pushdown automaton, then f is an algebraic function over ℚ; however, pushdown automata can simulate f(p)-coins for certain nonrational functions such as . These results complement the work of Keane and O’Brien (1994), who determined the functions f for which an f(p)-coin can be simulated when there are no computational restrictions on the simulation scheme. * Supported by a Miller Fellowship. † Supported in part by NSF Grant DMS-0104073 and by a Miller Professorship. ‡ This work is supported under a National Science Foundation Graduate Research Fellowship.