Vector-valued covariant differential operators for the M\"obius transformation

We obtain a family of functional identities satisfied by vector-valued functions of two variables and their geometric inversions. For this we introduce particular differential operators of arbitrary order attached to Gegenbauer polynomials. These differential operators are symmetry breaking for the pair of Lie groups $(SL(2,\mathbb C), SL(2,\mathbb R))$ that arise from conformal geometry.Comment: To appear in Springer Proceedings in Mathematics and Statistic

Nonlinear Differential Equations Satisfied by Certain Classical Modular Forms

A unified treatment is given of low-weight modular forms on \Gamma_0(N), N=2,3,4, that have Eisenstein series representations. For each N, certain weight-1 forms are shown to satisfy a coupled system of nonlinear differential equations, which yields a single nonlinear third-order equation, called a generalized Chazy equation. As byproducts, a table of divisor function and theta identities is generated by means of q-expansions, and a transformation law under \Gamma_0(4) for the second complete elliptic integral is derived. More generally, it is shown how Picard-Fuchs equations of triangle subgroups of PSL(2,R) which are hypergeometric equations, yield systems of nonlinear equations for weight-1 forms, and generalized Chazy equations. Each triangle group commensurable with \Gamma(1) is treated.Comment: 40 pages, final version, accepted by Manuscripta Mathematic

On the ratio of consecutive gaps between primes

In the present work we prove a common generalization of Maynard-Tao's recent result about consecutive bounded gaps between primes and on the Erd\H{o}s-Rankin bound about large gaps between consecutive primes. The work answers in a strong form a 60 years old problem of Erd\"os, which asked whether the ratio of two consecutive primegaps can be infinitely often arbitrarily small, and arbitrarily large, respectively

Representations of integers by certain positive definite binary quadratic forms

We prove part of a conjecture of Borwein and Choi concerning an estimate on the square of the number of solutions to n=x^2+Ny^2 for a squarefree integer N.Comment: 8 pages, submitte

The effect of laser shock peening on hardness and microstructure in a welded marine steel

Residual stress is generally considered as the main criterion in laser shock peening for enhancement of fatigue life. However, changes in material hardness, microstructure and surface roughness can also affect component performance. These three aspects are investigated in this paper for welded marine steel samples subjected to laser peening. After laser peening an increase in hardness was seen across the weld and parent metal, with the local hardness dependent upon the initial hardness of the region before peening. The increase was relatively higher for the weld metal which had lower initial hardness. The local surface displacement profiles reflected the number of laser peening layers applied, and the peening also affected the distortion of the specimen after welding

Distribution of Eigenvalues for the Modular Group

The two-point correlation function of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy-Littlewood method. It is shown that ion the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but they have prominent number-theoretical oscillations at larger scale. The results agree well with numerical simulations.Comment: 72 pages, Latex, the fiogures mentioned in the text are not vital, but can be obtained upon request from the first Autho

Duality in the Quantum Hall Effect - the Role of Electron Spin

At low temperatures the phase diagram for the quantum Hall effect has a powerful symmetry arising from the Law of Corresponding States. This symmetry gives rise to an infinite order discrete group which is a generalisation of Kramers-Wannier duality for the two dimensional Ising model. The duality group, which is a subgroup of the modular group, is analysed and it is argued that there is a quantitative difference between a situation in which the spin splitting of electron energy levels is comparable to the cyclotron energy and one in which the spin splitting is much less than the cyclotron energy. In the former case the group of symmetries is larger than in the latter case. These duality symmetries are used to constrain the scaling functions of the theory and, under an assumption of complex meromorphicity, a unique functional form is obtained for the crossover of the conductivities between Hall states as a function of the external magnetic field. This analytic form is shown to give good agreement with experimental data. The analysis requires a consideration of the way in which longitudinal resistivities are extracted from the experimentally measured longitudinal resistances and a novel method is proposed for determining the correct normalisation for the former.Comment: 22 pages, 8 figures, typeset in LaTe

A dimensionally continued Poisson summation formula

We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When combined with one of the proofs of the Jacobi imaginary transformation of theta functions that does not use the Poisson summation formula, our proof of this generalized Poisson summation formula also provides a new proof of the standard Poisson summation formula for dimensions greater than 2 (with appropriate hypotheses on the function being summed). In general, our methods work to establish the (Voronoi) summation formulae associated with functions satisfying (modular) transformations of the Jacobi imaginary type by means of a density argument (as opposed to the usual Mellin transform approach). In particular, we construct a family of generalized theta series from Jacobi theta functions from which these summation formulae can be obtained. This family contains several families of modular forms, but is significantly more general than any of them. Our result also relaxes several of the hypotheses in the standard statements of these summation formulae. The density result we prove for Gaussians in the Schwartz space may be of independent interest.Comment: 12 pages, version accepted by JFAA, with various additions and improvement

Pulsars as Fantastic Objects and Probes

Pulsars are fantastic objects, which show the extreme states of matters and plasma physics not understood yet. Pulsars can be used as probes for the detection of interstellar medium and even the gravitational waves. Here I review the basic facts of pulsars which should attract students to choose pulsar studies as their future projects.Comment: Invited Lecture on the "First Kodai-Trieste Workshop on Plasma Astrophysics", Kodaikanal Obs, India. Aug.27-Sept.7th, 2007. In: "Turbulence, Dynamos, Accretion Disks, Pulsars and Collective Plasma Processes". Get a copy from: http://www.springerlink.com/content/978-1-4020-8867-

Study of the B^0 Semileptonic Decay Spectrum at the Upsilon(4S) Resonance

We have made a first measurement of the lepton momentum spectrum in a sample of events enriched in neutral B's through a partial reconstruction of B0 --> D*- l+ nu. This spectrum, measured with 2.38 fb**-1 of data collected at the Upsilon(4S) resonance by the CLEO II detector, is compared directly to the inclusive lepton spectrum from all Upsilon(4S) events in the same data set. These two spectra are consistent with having the same shape above 1.5 GeV/c. From the two spectra and two other CLEO measurements, we obtain the B0 and B+ semileptonic branching fractions, b0 and b+, their ratio, and the production ratio f+-/f00 of B+ and B0 pairs at the Upsilon(4S). We report b+/b0=0.950 (+0.117-0.080) +- 0.091, b0 = (10.78 +- 0.60 +- 0.69)%, and b+ = (10.25 +- 0.57 +- 0.65)%. b+/b0 is equivalent to the ratio of charged to neutral B lifetimes, tau+/tau0.Comment: 14 page, postscript file also available at http://w4.lns.cornell.edu/public/CLN