The unramified computation of Rankin-Selberg integrals for SO2l × GLn

Let SO _2l be the special orthogonal group, either split or quasi-split over a number field, and 1 < l < n. We compute the local integral, where data are unramified, derived from the global Rankin-Selberg construction for SO _2l × GL _n . In the general case, the local integral is difficult to compute directly, so instead it is transformed to an integral related to a construction for SO _2n+1×GL _n , which carries a Bessel model on SO _2n+1. For the quasisplit case, when l = n − 1 we are able to compute the local integral, by a modification of our recently introduced approach using invariant theory. This leads to another proof of our result for 1 < l < n, as well as a new proof of a known result regarding the unramified Bessel function.     

Critical Ising on the Square Lattice Mixes in Polynomial Time

The Ising model is widely regarded as the most studied model of spin-systems in statistical physics. The focus of this paper is its dynamic (stochastic) version, the Glauber dynamics, introduced in 1963 and by now the most popular means of sampling the Ising measure. Intensive study throughout the last three decades has yielded a rigorous understanding of the spectral-gap of the dynamics on everywhere except at criticality. While the critical behavior of the Ising model has long been the focus for physicists, mathematicians have only recently developed an understanding of its critical geometry with the advent of SLE, CLE and new tools to study conformally invariant systems.     

Spectra of lifted Ramanujan graphs

A random n-lift of a base-graph G is its cover graph H on the vertices [n]×V(G), where for each edge uv in G there is an independent uniform bijection π, and H has all edges of the form (i,u),(π(i),v). A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan.Let G be a graph with largest eigenvalue λ₁ and let ρ be the spectral radius of its universal cover. Friedman (2003) [12] proved that every “new” eigenvalue of a random lift of G is with high probability, and conjectured a bound of ρ+o(1), which would be tight by results of Lubotzky and Greenberg (1995) [15]. Linial and Puder (2010) [17] improved Friedman?s bound to . For d-regular graphs, where λ₁=d and , this translates to a bound of O(d2/3), compared to the conjectured .Here we analyze the spectrum of a random n-lift of a d-regular graph whose nontrivial eigenvalues are all at most λ in absolute value. We show that with high probability the absolute value of every nontrivial eigenvalue of the lift is . This result is tight up to a logarithmic factor, and for λ?d2/3−ε it substantially improves the above upper bounds of Friedman and of Linial and Puder. In particular, it implies that a typical n-lift of a Ramanujan graph is nearly Ramanujan.     

Small vertex‐transitive and Cayley graphs of girth six and given degree: an algebraic approach

We examine the existing constructions of the smallest known vertex‐transitive graphs of a given degree and girth 6. It turns out that most of these graphs can be described in terms of regular lifts of suitable quotient graphs. A further outcome of our analysis is a precise identification of which of these graphs are Cayley. We also investigate higher level of transitivity of the smallest known vertex‐transitive graphs of a given degree and girth 6 and relate their constructions to near‐difference sets. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:265‐284, 2011     

Canonical subgroups over Hilbert modular varieties

We obtain new results on the geometry of Hilbert modular varieties in positive characteristic and morphisms between them. Using these results and methods of rigid geometry, we develop a theory of canonical subgroups for abelian varieties with real multiplication. To cite this article: E.Z. Goren, P.L Kassaei, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Olefin metathesis catalyst bearing a chelating phosphine ligand

An improved synthetic procedure for the complex (SPY-5-34)-dichloro-(κ²(C,P)-diphenyl-(2-benzylidene)-phosphine)-(1,3-bis(2,4,6-trimethylphenyl)-4,5-dihydro-imidazol-2-ylidene)-ruthenium (2) was elaborated and the title compound was tested as latent initiator in Ring Opening Metathesis Polymerization (ROMP) and as catalyst for Ring-Closing Metathesis (RCM) at elevated temperatures. While not particularly suited as latent initiator for ROMP, exhibiting a switching temperature of only 42 °C in the polymerization of a typical norbornene derivative, 2 shows an appealing performance in RCM of α,ω-dienes at higher temperatures.     

Flexural vibration patterning using an array of actuators

This paper describes a method to create unique vibration patterns by means of an array of actuators acting on the boundaries of the controlled structure. The method builds, in an iterative process, a black-box model by employing a series of probing external excitation vectors. Once a model has been identified, an optimization stage seeks the dynamic force-vector that generates the best approximation for a specified deformation pattern. Among the obtainable periodic responses are standing or travelling flexural waves of single or multi-frequencies in one or two spatial dimensions, as well as rotating, vortex-like, flexural waves in 2D. The paper describes the force-tuning process through which dozens of actuators can be modified automatically until the response, measured at hundreds of sensed locations, complies with the anticipated response. The excitation that yields the desired response is calculated using an over-determined set of least-squares equations that approximates the inverse of the identified model. The precise iterative tuning process can overcome some nonlinearity, manifested by additional harmonics in the response, by injecting forces that nullify their effect. Methods for wavelength optimization and wave identification in 1D and 2D are also discussed. The proposed tuning method showed good results in numerical simulation and in real world experiments.