Random Matrix Theory and the Fourier Coefficients of Half-Integral Weight Forms

Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the centre of the critical strip are used to motivate a series of conjectures concerning the value-distribution of the Fourier coefficients of half-integral weight modular forms related to these L-functions. Our conjectures may be viewed as being analogous to the Sato-Tate conjecture for integral weight modular forms. Numerical evidence is presented in support of them.Comment: 28 pages, 8 figure

Discretisation for odd quadratic twists

The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the Shafarevich-Tate group). We discuss a couple of models and present data on this question.Comment: To appear in the Proceedings of the INI Workshop on Random Matrix Theory and Elliptic Curve

Structural tailoring of select fiber composite structures

A multidisciplinary design process for aerospace propulsion composite structures was formalized and embedded into computer codes. These computer codes are streamlined to obtain tailored designs for select composite structures. The codes available are briefly described with sample cases to illustrate their applications. The sample cases include aircraft engine blades, propfans (turboprops), flat, and cylindrical panels. Typical results illustrate that the use of these codes enable the designer to obtain designs which meet all the design requirements with maximum benefits in efficiency, noise, weight or thermal distortions

Probabilistic structural analysis to quantify uncertainties associated with turbopump blades

A probabilistic study of turbopump blades has been in progress at NASA Lewis Research Center for over the last two years. The objectives of this study are to evaluate the effects of uncertainties in geometry and material properties on the structural response of the turbopump blades to evaluate the tolerance limits on the design. A methodology based on probabilistic approach was developed to quantify the effects of the random uncertainties. The results indicate that only the variations in geometry have significant effects

Autocorrelation of Random Matrix Polynomials

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions

Sliding friction between an elastomer network and a grafted polymer layer: the role of cooperative effects

We study the friction between a flat solid surface where polymer chains have been end-grafted and a cross-linked elastomer at low sliding velocity. The contribution of isolated grafted chains' penetration in the sliding elastomer has been early identified as a weakly velocity dependent pull-out force. Recent experiments have shown that the interactions between the grafted chains at high grafting density modify the friction force by grafted chain. We develop here a simple model that takes into account those interactions and gives a limit grafting density beyond which the friction no longer increases with the grafting density, in good agreement with the experimental dataComment: Submitted to Europhys. Letter

Dynamic multilateral markets

We study dynamic multilateral markets, in which players' payoffs result from intra-coalitional bargaining. The latter is modeled as the ultimatum game with exogenous (time-invariant) recognition probabilities and unanimity acceptance rule. Players in agreeing coalitions leave the market and are replaced by their replicas, which keeps the pool of market participants constant over time. In this infinite game, we establish payoff uniqueness of stationary equilibria and the emergence of endogenous cooperation structures when traders experience some degree of (heterogeneous) bargaining frictions. When we focus on market games with different player types, we derive, under mild conditions, an explicit formula for each type's equilibrium payoff as the market frictions vanish

Conditioned place preference and locomotor activity in response to methylphenidate, amphetamine and cocaine in mice lacking dopamine D4 receptors

Methylphenidate (MP) and amphetamine (AMPH) are the most frequently prescribed medications for the treatment of attention-deficit/hyperactivity disorder (ADHD). Both drugs are believed to derive their therapeutic benefit by virtue of their dopamine (DA)-enhancing effects, yet an explanation for the observation that some patients with ADHD respond well to one medication but not to the other remains elusive. The dopaminergic effects of MP and AMPH are also thought to underlie their reinforcing properties and ultimately their abuse. Polymorphisms in the human gene that codes for the DA D4 receptor (D4R) have been repeatedly associated with ADHD and may correlate with the therapeutic as well as the reinforcing effects of responses to these psychostimulant medications. Conditioned place preference (CPP) for MP, AMPH and cocaine were evaluated in wild-type (WT) mice and their genetically engineered littermates, congenic on the C57Bl/6J background, that completely lack D4Rs (knockout or KO). In addition, the locomotor activity in these mice during the conditioning phase of CPP was tested in the CPP chambers. D4 receptor KO and WT mice showed CPP and increased locomotor activity in response to each of the three psychostimulants tested. D4R differentially modulates the CPP responses to MP, AMPH and cocaine. While the D4R genotype affected CPP responses to MP (high dose only) and AMPH (low dose only) it had no effects on cocaine. Inasmuch as CPP is considered an indicator of sensitivity to reinforcing responses to drugs these data suggest a significant but limited role of D4Rs in modulating conditioning responses to MP and AMPH. In the locomotor test, D4 receptor KO mice displayed attenuated increases in AMPH-induced locomotor activity whereas responses to cocaine and MP did not differ. These results suggest distinct mechanisms for D4 receptor modulation of the reinforcing (perhaps via attenuating dopaminergic signalling) and locomotor properties of these stimulant drugs. Thus, individuals with D4 receptor polymorphisms might show enhanced reinforcing responses to MP and AMPH and attenuated locomotor response to AMPH.Fil: Thanos, P. K.. NIAAA Intramural Program; Estados Unidos. Brookhaven National Laboratory; Estados Unidos. Universidad de Buenos Aires; ArgentinaFil: Bermeo, C.. Brookhaven National Laboratory; Estados UnidosFil: Rubinstein, Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires; ArgentinaFil: Suchland, K. L.. Oregon Health & Science University; Estados UnidosFil: Wang, G. J.. Brookhaven National Laboratory; Estados UnidosFil: Grandy, David K.. Oregon Health & Science University; Estados UnidosFil: Volkow, N. D.. NIAAA Intramural Program; Estados Unido

The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow

We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible fluid. Our starting point is the Stokes or steady Navier-Stokes equations set in a bounded domain with the disjoint union of N balls of radius 1/N removed, and with a no-slip boundary condition for the fluid at the surface of each ball. The large N limit of the fluid velocity field is governed by the same (Navier-)Stokes equations in the whole domain, with an additional term (Brinkman's force) that is (minus) the total drag force exerted by the fluid on the particle system. This can be seen as a generalization of Allaire's result in [Arch. Rational Mech. Analysis 113 (1991), 209-259] who treated the case of motionless, periodically distributed balls. Our proof is based on slightly simpler, though similar homogenization techniques, except that we avoid the periodicity assumption and use instead the phase-space empirical measure for the particle system. Similar equations are used for describing the fluid phase in various models for sprays