The Probability Content of Cones in Isotropic Random Fields

This paper provides computable representations for the evaluation of the probability content of cones in isotropic random fields. A decomposition of quadratic forms in spherically symmetric random vectors is obtained and a representation of their moments is derived in terms of finite sums. These results are combined to obtain the distribution function of quadratic forms in spherically symmetric or central elliptically contoured random vectors. Some numerical examples involving the sample serial covariance are provided. Ratios of quadratic forms are also discussed.     

On splitting of totally singular quadratic forms

In this note we completely study the standard splitting of quasi-Pfister forms and their neighbors, and we include some general results on standard splitting towers of totally singular quadratic forms. The author was supported by the European research network HPRN-CT-2002-00287 “AlgebraicK-Theory, Linear Algebraic Groups and Related Structures”.     

Speciation studies of nickel and chromium in wastewater from an electroplating plant

A speciation scheme involving the use of flame atomic absorption spectrometry (FAAS) and differential pulse adsorptive cathodic stripping voltammetry (DPAdCSV) techniques was applied to studies of nickel and chromium in wastewater from a nickel-chrome electroplating plant. Dimethylglyoxime (DMG) and diethylenetriaminepentaacetic acid (DTPA) were employed as complexing agents for adsorptive voltammetric determination of Ni and Cr, respectively. Cr(III) and Cr(VI) were determined by exploiting differences in their reactivity towards DTPA at HMDE. Total dissolved metal content was in the range 2906-3141 and 30.7-31.2 mg l⁻¹ for Ni and Cr, respectively. A higher percentage of the metal was present as labile species (mean value of 67.9% for Ni and 79.8% for Cr) suggesting that strongly binding ligands are not ubiquitous in the sample. About 77.8% of Cr was found to exist in the higher oxidization state, Cr(IV). Results on effect of dilution on lability of the metal forms in the sample using DPAdCSV showed slight peak shifts to a more negative (cathodic) value by −0.036 V for Ni and −0.180 V for Cr with a dilution factor of 100, while peak intensity (cathodic current) remained fairly constant.     

The twelfth moment of central values of Hecke series

Let H_j(s) be the Hecke L-function attached to the Maass wave form for the jth eigenvalue of the hyperbolic Laplacian acting in the Hilbert space of automorphic functions for the full modular group. The following mean value estimate for the central values is proved:

     

Infrared Variables for the SU(3) Yang–Mills Field

We generalize the self-dual parameterization of the SU(2) Yang–Mills field proposed by Niemi and Faddeev for describing the infrared limit of the theory to the case of the gauge group SU(3). We demonstrate that the duality property intrinsic to the SU(2) gauge field cannot be transferred automatically to the higher-rank group case. We interpret the algebraic structures appearing in the Lagrangian for the new compact variables in terms of the group products SU(2)³.     

Knot signature functions are independent

A Seifert matrix is a square integral matrix satisfying

To such a matrix and unit complex number there corresponds a signature,

Let denote the set of unit complex numbers with positive imaginary part. We show that is linearly independent, viewed as a set of functions on the set of all Seifert matrices.

If is metabolic, then unless is a root of the Alexander polynomial, . Let denote the set of all unit roots of all Alexander polynomials with positive imaginary part. We show that is linearly independent when viewed as a set of functions on the set of all metabolic Seifert matrices.

To each knot one can associate a Seifert matrix , and induces a knot invariant. Topological applications of our results include a proof that the set of functions is linearly independent on the set of all knots and that the set of two-sided averaged signature functions, , forms a linearly independent set of homomorphisms on the knot concordance group. Also, if is the root of some Alexander polynomial, then there is a slice knot whose signature function is nontrivial only at and . We demonstrate that the results extend to the higher-dimensional setting.

     

A generalization of a theorem of Rankin and Swinnerton-Dyer on zeros of modular forms

Rankin and Swinnerton-Dyer (1970) prove that all zeros of the Eisenstein series in the standard fundamental domain for lie on . In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc . Using this result we prove a speculation of Ono, namely that the zeros of the unique ``gap function" in , the modular form with the maximal number of consecutive zero coefficients in its -expansion following the constant , has zeros only on . In addition, we show that the -invariant maps these zeros to totally real algebraic integers of degree bounded by a simple function of weight .

     

The length of harmonic forms on a compact Riemannian manifold

We study -dimensional Riemannian manifolds with harmonic forms of constant length and first Betti number equal to showing that they are -step nilmanifolds with some special metrics. We also characterize, in terms of properties on the product of harmonic forms, the left-invariant metrics among them. This allows us to clarify the case of equality in the stable isosytolic inequalities in that setting. We also discuss other values of the Betti number.

     

Elliptic Apostol sums and their reciprocity laws

We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter having positive imaginary part. When , these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable . We obtain a reciprocity law for these sums, which gives rise to new relations between certain modular forms (of one variable).