Modularity of a certain Calabi-Yau threefold and combinatorial congruences

We prove a certain Calabi-Yau threefold is modular in the sense that the number of points on its reduction modulo p is expressed in terms of the pth coefficient of a weight 4 newform in S₄(Γ₀(6)). We also give a mod p² combinatorial expression for these coefficients. 2000 Mathematics Subject Classification: 11G25, 11G40     

Three Spectrophotometric Methods for the Determination of Oxomemazine Hydrochloride in Bulk and in Pharmaceutical Formulations Using Bromocresol Green,Congo Red,and Methyl Orange

Abstract

Three simple, sensitive, and highly accurate spectrophotometric methods have been developed for the determination of oxomemazine hydrochloride (OXO‐HCl) in bulk and in pharmaceutical formulations. These methods are based on the formation of yellow ion‐pair complexes between the examined drug and bromocresol green (BCG), congo red (CR), and methyl orange (MO) as reagents in universal buffer solution of pH 3.0, 5.5, and 3.5, respectively. The formed complexes were extracted with chloroform and measured at 413, 495, and 484 nm, respectively for the three systems. The best conditions of the reaction were studied and optimized. Beer's law was obeyed in the concentration ranges 2.0–18.0, 2.0–14.0, and 2.0–16.0 µg ml^?1 with molar absorptivity of 4.1×10⁴, 1.1×10⁴, and 3.5×10⁴ mol^?1cm^?1, for the BCG, CR, and MO methods, respectively. Sandell's sensitivity, correlation coefficient, detection, and quantification limits are also calculated. The proposed methods have been applied successfully for the analysis of the drug in pure and in its dosage forms. No interference was observed from common pharmaceutical excipients and additives. Statistical comparison of the results with those obtained by HPLC method shows excellent agreement and indicates no significant difference in accuracy and precision.     

Fractional Nambu Mechanics

The fractional generalization of Nambu mechanics is constructed by using the differential forms and exterior derivatives of fractional orders. The generalized Pfaffian equations are obtained and one example is investigated in details. On leave of absence from Institute of Space Sciences, P.O. Box, MG-23, 76900, Magurele-Bucharest, Romania.     

Weierstrass points on X 0(p ℓ) and arithmetic properties of Fourier coefficients of cusp forms

Generally it is unknown, whether or not ∞ is a Weierstrass point on the modular curve X ₀(N) if N is squarefree. A classical result of Atkin and Ogg states that ∞ is not a Weierstrass point on X ₀(N), if N=pM with p prime, p ∤ M and the genus of X ₀(M) zero. We use results of Kohnen and Weissauer to show that there is a connection between this question and the p-adic valuation of cusp forms under the Atkin–Lehner involution. This gives, in a sense, a generalization of Ogg’s Theorem in some cases.      

Invariants of Directed Spaces

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.      

STRONG SUBORDINATION OF SYMMETRIC DIRICHLET FORMS

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On the number of Fourier coefficients that determine a Hilbert modular form

We estimate the number of Fourier coefficients that determine a Hilbert modular cusp form of arbitrary weight and level. The method is spectral (Rayleigh quotient) and avoids the use of the maximum principle.

     

Mechanical theorem proving in differential geometry

An automated reasoning method, based on Wu’s method and calculus of differential forms, is proposed for mechanical theorem proving in local theory of space surfaces in differential geometry. The method has been used to simplify one of Chem’s theorems: “The non-trivial families of isometric surfaces having the same principal curvatures are W-surfaces.” Some other theorems are also tested by this method. The proofs are generally simpler than those in differential geometry textbooks. Project supported partially by the National Natural Science Foundation of China.     

A remark on congruences for coefficients of Faber polynomials

Faber polynomials appear when weight zero Hecke operators act on the modular j-invariant. They are polynomials in j with rational integral coefficents. Using the theory of p-adic modular forms we establish some congruences and divisibilities for these coefficients. 2000 Mathematics Subject Classification Primary—11F33 Supported by NSF grant DMS-0501225.