Zeros of Weakly Holomorphic Modular Forms of Levels 2 and 3

Let M-k(#)(N) be the space of weakly holomorphic modular forms for Gamma(0)(N) that are holomorphic at all cusps except possibly at infinity. We study a canonical basis for M-k(#)(2) and M-k(#)(3) and prove that almost all modular forms in this basis have the property that the majority of their zeros in a fundamental domain lie on a lower boundary arc of the fundamental domain.

Agrin defines polarized distribution of orthogonal arrays of particles in astrocytes

Accumulating evidence indicates that agrin, a heparan sulphate proteoglycan of the extracellular matrix, plays a role in the organization and maintenance of the blood-brain barrier. This evidence is based on the differential effects of agrin isoforms on the expression and distribution of the water channel protein, aquaporin-4 (AQP4), on the swelling capacity of cultured astrocytes of neonatal mice and on freeze-fracture data revealing an agrin-dependent clustering of orthogonal arrays of particles (OAPs), the structural equivalent of AQP4. Here, we show that the OAP density in agrin-null mice is dramatically decreased in comparison with wild-types, by using quantitative freeze-fracture analysis of astrocytic membranes. In contrast, anti-AQP4 immunohistochemistry has revealed that the immunoreactivity of the superficial astrocytic endfeet of the agrin-null mouse is comparable with that in wild-type mice. Moreover, in vitro, wild-type and agrin-null astrocytes cultured from mouse embryos at embryonic day 19.5 differ neither in AQP4 immunoreactivity, nor in OAP density in freeze-fracture replicas. Analyses of brain tissue samples and cultured astrocytes by reverse transcription with the polymerase chain reaction have not demonstrated any difference in the level of AQP4 mRNA between wild-type astrocytes and astrocytes from agrin-null mice. Furthermore, we have been unable to detect any difference in the swelling capacity between wild-type and agrin-null astrocytes. These results clearly demonstrate, for the first time, that agrin plays a pivotal role for the clustering of OAPs in the endfoot membranes of astrocytes, whereas the mere presence of AQP4 is not sufficient for OAP clustering.

Linear instability of orthogonal compressible leading-edge boundary layer flow

Instability analysis of compressible orthogonal swept leading-edge boundary layer flow was performed in the context of BiGlobal linear theory. 1, 2 An algorithm was developed exploiting the sparsity characteristics of the matrix discretizing the PDE-based eigenvalue problem. This allowed use of the MUMPS sparse linear algebra package 3 to obtain a direct solution of the linear systems associated with the Arnoldi iteration. The developed algorithm was then applied to efficiently analyze the effect of compressibility on the stability of the swept leading-edge boundary layer and obtain neutral curves of this flow as a function of the Mach number in the range 0 ≤ Ma ≤ 1. The present numerical results fully confirmed the asymptotic theory results of Theofilis et al. 4 Up to the maximum Mach number value studied, it was found that an increase of this parameter reduces the critical Reynolds number and the range of the unstable spanwise wavenumbers.

A positivity result in the theory of Macdonald polynomials

We outline here a proof that a certain rational function Cn(q, t), which has come to be known as the “q, t-Catalan,” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because Cn(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection

Robust fitting of Zernike polynomials to noisy point clouds defined over connected domains of arbitrary shape

A new method for fitting a series of Zernike polynomials to point clouds defined over connected domains of arbitrary shape defined within the unit circle is presented in this work. The method is based on the application of machine learning fitting techniques by constructing an extended training set in order to ensure the smooth variation of local curvature over the whole domain. Therefore this technique is best suited for fitting points corresponding to ophthalmic lenses surfaces, particularly progressive power ones, in non-regular domains. We have tested our method by fitting numerical and real surfaces reaching an accuracy of 1 micron in elevation and 0.1 D in local curvature in agreement with the customary tolerances in the ophthalmic manufacturing industry.

The Zeros of Riemann Zeta Partial Sums Yield Solutions to f(x) + f(2x) + · · · + f(nx) = 0

This paper proves that every zero of any n th , n ≥ 2, partial sum of the Riemann zeta function provides a vector space of basic solutions of the functional equation f(x)+f(2x)+⋯+f(nx)=0,x∈R . The continuity of the solutions depends on the sign of the real part of each zero.

Line and area orthogonality of Jacobi polynomials,

"Supported in part by contract U.S. AEC AT(11-1) 1469 and grant NSF-6J-217".

Parallel methods and bounds of evaluating polynomials.

Bibliography: p. 24.

Modeling Optical Response of Thin Films: Choice of the Refractive Index Dispersion Law

Determination of the so-called optical constants (complex refractive index N, which is usually a function of the wavelength, and physical thickness D) of thin films from experimental data is a typical inverse non-linear problem. It is still a challenge to the scientific community because of the complexity of the problem and its basic and technological significance in optics. Usually, solutions are looked for models with 3-10 parameters. Best estimates of these parameters are obtained by minimization procedures. Herein, we discuss the choice of orthogonal polynomials for the dispersion law of the thin film refractive index. We show the advantage of their use, compared to the Selmeier, Lorentz or Cauchy models.

Parallel Class Intersection Matrices of Orthogonal Resolutions

This work was partially supported by the Bulgarian National Science Fund under Contract No MM 1405. Part of the results were announced at the Fifth International Workshop on Optimal Codes and Related Topics (OCRT), White Lagoon, June 2007, Bulgaria

Discriminant Sets of Families of Hyperbolic Polynomials of Degree 4 and 5

∗ Research partially supported by INTAS grant 97-1644

On Representations of Algebraic Polynomials by Superpositions of Plane Waves

* The author was supported by NSF Grant No. DMS 9706883.

Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function

2000 Mathematics Subject Classification: 26A33, 33C45

Estimate for the Number of Zeros of Abelian Integrals on Elliptic Curves

2000 Mathematics Subject Classification: Primary 34C07, secondary 34C08.