A nullstellensatz for sequences over F_p

Let p be a prime and let A=(a_1,...,a_l) be a sequence of nonzero elements in F_p. In this paper, we study the set of all 0-1 solutions to the equation a_1 x_1 + ... + a_l x_l = 0. We prove that whenever l >= p, this set actually characterizes A up to a nonzero multiplicative constant, which is no longer true for l < p. The critical case l=p is of particular interest. In this context, we prove that whenever l=p and A is nonconstant, the above equation has at least p-1 minimal 0-1 solutions, thus refining a theorem of Olson. The subcritical case l=p-1 is studied in detail also. Our approach is algebraic in nature and relies on the Combinatorial Nullstellensatz as well as on a Vosper type theorem.Comment: 23 page

Advanced water iodinating system

Potable water stores aboard manned spacecraft must remain sterile. Suitable sterilization techniques are needed to prevent microbial growth. The development of an advanced water iodinating system for possible application to the shuttle orbiter and other advanced spacecraft, is considered. The AWIS provides a means of automatically dispensing iodine and controlling iodination levels in potable water stores. In a recirculation mode test, simulating application of the AWIS to a water management system of a long term six man capacity space mission, noniodinated feed water flowing at 32.2 cu cm min was iodinated to 5 + or - ppm concentrations after it was mixed with previously iodinated water recirculating through a potable water storage tank. Also, the AWIS was used to successfully demonstrate its capability to maintain potable water at a desired I2 concentration level while circulating through the water storage tank, but without the addition of noniodinated water

On Karatsuba's Problem Concerning the Divisor Function τ(n)\tau(n)

We study an asymptotic behavior of the sum \sum\limits_{n\le x}\frac{\D \tau(n)}{\D \tau(n+a)}. Here $\tau(n)$ denotes the number of divisors of $n$ and $a\ge 1$ is a fixed integer.Comment: 32 page

Open circular billiards and the Riemann hypothesis

A comparison of escape rates from one and from two holes in an experimental container (e.g. a laser trap) can be used to obtain information about the dynamics inside the container. If this dynamics is simple enough one can hope to obtain exact formulas. Here we obtain exact formulas for escape from a circular billiard with one and with two holes. The corresponding quantities are expressed as sums over zeroes of the Riemann zeta function. Thus we demonstrate a direct connection between recent experiments and a major unsolved problem in mathematics, the Riemann hypothesis.Comment: 5 pages, 4 embedded postscript figures; v2: more explicit on how the Reimann Hypothesis arises from a comparison of one and two hole escape rate

A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra

A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method.Comment: 10 pages. This version has been revised according to a referee's suggestions. The additions include a discussion of the (lower) density of the set N_p, and the results of more extensive machine computations. Note that the title has also changed. To appear in Israel J. Mat

The H=xp model revisited and the Riemann zeros

Berry and Keating conjectured that the classical Hamiltonian H = xp is related to the Riemann zeros. A regularization of this model yields semiclassical energies that behave, in average, as the non trivial zeros of the Riemann zeta function. However, the classical trajectories are not closed, rendering the model incomplete. In this paper, we show that the Hamiltonian H = x (p + l_p^2/p) contains closed periodic orbits, and that its spectrum coincides with the average Riemann zeros. This result is generalized to Dirichlet L-functions using different self-adjoint extensions of H. We discuss the relation of our work to Polya's fake zeta function and suggest an experimental realization in terms of the Landau model.Comment: 5 pages, 3 figure

The history, evolution, and future of big data &amp; analytics:A bibliometric analysis of its relationship to performance in organizations

Big data and analytics (BDA) are gaining momentum, particularly in the practitioner world. Research linking BDA to improved organizational performance seems scarce and widely dispersed though, with the majority focused on specific domains and/or macro-level relationships. In order to synthesize past research and advance knowledge of the potential organizational value of BDA, the authors obtained a data set of 327 primary studies and 1252 secondary cited papers. This paper reviews this body of research, using three bibliometric methods. First, it elucidates its intellectual foundations via co-citation analysis. Second, it visualizes the historical evolution of BDA and performance research and its substreams through algorithmic historiography. Third, it provides insights into the field's potential evolution via bibliographic coupling. The results reveal that the academic attention for the BDA-performance link has been increasing rapidly. The study uncovered ten research clusters that form the field's foundation. While research seems to have evolved following two main, isolated streams, the past decade has witnessed more cross-disciplinary collaborations. Moreover, the study identified several research topics undergoing focused development, including financial and customer risk management, text mining and evolutionary algorithms. The review concludes with a discussion of the implications for different functional management domains and the gaps for both research and practice.</p

The current progress of the ALICE Ring Imaging Cherenkov Detector

Recently, the last two modules (out of seven) of the ALICE High Momentum Particle Identification detector (HMPID) were assembled and tested. The full detector, after a pre-commissioning phase, has been installed in the experimental area, inside the ALICE solenoid, at the end of September 2006. In this paper we review the status of the ALICE/HMPID project and we present a summary of the series production of the CsI photo-cathodes. We describe the key features of the production procedure which ensures high quality photo-cathodes as well as the results of the quality assessment performed by means of a specially developed 2D scanner system able to produce a detailed map of the CsI photo-current over the entire photo-cathode surface. Finally we present our recent R&D efforts toward the development of a novel generation of imaging Cherenkov detectors with the aim to identify, in heavy ions collisions, hadrons up to 30 GeV/c.Comment: Presented at the Imaging-2006 Conference, Stockholm, Sweden, June 200