Multiplicativity of the gamma factors of Rankin–Selberg integrals for SO 2l × GL n

Let SO _2l be the special even orthogonal group, split or quasi–split, defined over a local non–Archimedian field. The Rankin–Selberg method for a pair of generic representations of SO _2l × GL _n constructs a family of integrals, which are used to define γ and L-factors. Here we prove full multiplicative properties for the γ-factor, namely that it is multiplicative in each variable. As a corollary, the γ-factor is identical with Shahidi’s standard γ-factor.     

Poisson approximation for non-backtracking random walks

Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. The authors of [3] studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length n on a high-girth n-vertex regular expander, is typically (1+o(1)))log n/log log n, as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes. In this work, we answer the above question by combining a generalized form of Brun’s sieve with some extensions of the ideas in [3]. Let N _t denote the number of vertices visited precisely t times by a non-backtracking random walk of length n on a regular n-vertex expander of fixed degree and girth g. We prove that if g = ω(1), then for any fixed t, N _t /n is typically 1/et! + o(1). Furthermore, if g = Ω(log log n), then N _t /n is typically 1+o(1)/et! niformly on all t ≤ (1 − o(1)) log n/log log n and 0 for all t ≥ (1 + o(1)) log n/log log n. In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables.     

The distance between superspecial abelian varieties with real multiplication

Let L be a totally real field of strict class number one and let OL be its ring of integers. Let p be a rational prime which is unramified in L. We consider the distance between two superspecial abelian varieties with real multiplication in characteristic p, where by “distance” we mean the minimal degree of an OL-isogeny. We give upper and lower bounds on the distance between superspecial abelian varieties with real multiplication by L in characteristic p in terms of p and the degree and discriminant of L.     

Uniformly cross intersecting families

Let A and B denote two families of subsets of an n-element set. The pair (A,B) is said to be ℓ-cross-intersecting iff |A∩B|=ℓ for all A∈ A and B∈B. Denote by P _e (n) the maximum value of |A||B| over all such pairs. The best known upper bound on P _e (n) is Θ(2 ⁿ ), by Frankl and R?dl. For a lower bound, Ahlswede, Cai and Zhang showed, for all n ≥ 2ℓ, a simple construction of an ℓ-cross-intersecting pair (A,B) with |A||B| = $ \left( {{*{20}c} {2\ell } \\ \ell \\ } \right) $ \left( {\begin{array}{*{20}c} {2\ell } \\ \ell \\ \end{array} } \right) 2 ^n−2ℓ = Θ(2 ⁿ /$ \sqrt \ell $ \sqrt \ell ), and conjectured that this is best possible. Consequently, Sgall asked whether or not P _e (n) decreases with ℓ.     

The welfare effects of public opinion polls

This paper presents an experimental study of the effects of polls on voters’ welfare. The analysis shows that polls have a different effect on closely divided and lopsided divided electorates. The data show that in closely divided electorates (and only for these electorates) the provision of information on the voters’ distribution of preferences significantly raises the participation of subjects supporting the slightly larger team relative to the smaller team. This causes a substantial increase on the frequency of electoral victories of the larger team. As a consequence, we observe a steep decrease in the welfare of the members of the smaller team because they vote more often and yet they loose the elections more frequently. Polls are detrimental to aggregate welfare in closely divided electorates because the decrease in the payoffs of the minority is stronger than the increase in the payoffs of the majority. In lopsided divided electorates polls don’t have a significant different effect on the voters’ turnout conditional on their team size. We do observe an increase on the frequency of electoral victories of the larger team after the provision of information, but this is in part due to smaller teams’ members voting less frequently and saving the participation costs. As a consequence, while polls have a negative effect on the relative payoffs of the minority for these electorates as well, they have a positive effect on total welfare.     

There Are Not Too Many Magic Configurations

A finite planar point set P is called a magic configuration if there is an assignment of positive weights to the points of P such that, for every line l determined by P, the sum of the weights of all points of P on l equals 1. We prove a conjecture of Murty from 1971 and show that if a set of n points P is a magic configuration, then P is in general position, or P contains n−1 collinear points, or P is a special configuration of 7 points. The research by Rom Pinchasi was supported by a Grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

The two-neutron transfer reactions (O, O) with O, C and Mg targets

Excitation functions, angular distributions and differential ranges were measured for the ²⁶Mg(¹⁸O, ¹⁶O)²⁸Mg reaction at ¹⁸O beam energies of 20–45 MeV. Excitation functions only were measured for the reactions ¹⁴C(¹⁸O, ¹⁹O)¹³C, ¹⁴C(¹⁸O, ¹⁶O)¹⁶C, ¹⁴C(¹⁸O, ²⁰O)¹²C, ¹⁴C(¹⁸O, ¹⁵N)¹⁷N and ¹⁸O(¹⁸O, ¹⁹O)¹⁷O, ¹⁸O(¹⁸O, ¹⁶O)²⁰O, ¹⁸O(¹⁸O, ¹⁵N)²¹F at ¹⁸O beam energies of 13–41 MeV. We have identified these as direct reactions in which a single neutron, a two-neutron cluster, a deuteron and a triton are transferred between projectile and target.

The cross sections for two-neutron transfer reactions were found to be relatively high and those for the ¹⁸O+¹⁸O and the ¹⁴C+¹⁸O reactions were higher than the ones of single-neutron transfers over most of the energy range.

Attempts were made to apply the theory of Buttle and Goldfarb for single-neutron transfer to the case of two-neutron transfer in the ²⁶Mg(¹⁸O, ¹⁶O)²⁸Mg reaction below the Coulomb barrier. It is shown that for those reactions for which the assumptions, implicit in the model, are valid, good agreement is obtained with experiment. We also tried to apply the diffraction model of Dar and Kozlovsky to the calculation of the angular distribution of these reactions. A good fit to the experimental results could be obtained if quite different sets of parameters were used in the calculations for the two bombarding energies.     

Declining valuations in sequential auctions

We analyze an independent private values model where a number of objects are sold in sequential first- and second-price auctions. Bidders have unit demand and their valuation for an object is decreasing in the rank number of the auction in which it is sold. We derive efficient equilibria if prices are announced after each auction or if no information is given to bidders. We show that the sequence of prices constitutes a supermartingale. Even if we correct for the decrease in valuations for objects sold in later auctions we find that average prices are declining.Received June 2004We are grateful to Christian Groh, Wolfgang Köhler and Benny Moldovanu for helpful suggestions. Financial support from the German Science Foundation through SFB 504 and SFB/TR 15 at the University of Mannheim and the University of Bonn is gratefully acknowledged.     

Long term behavior of lithographically prepared in vitro neuronal networks

We measured the long term spontaneous electrical activity of neuronal networks with different sizes, grown on lithographically prepared substrates and recorded with multi-electrode-array technology. The time sequences of synchronized bursting events were used to characterize network dynamics. All networks exhibit scale-invariant Lévy distributions and long-range correlations. These observations suggest that different-size networks self-organize to adjust their activities over many time scales. As predictions of current models differ from our observations, this calls for revised models.