For a family of compact Riemann surfaces Xt of genus g > 1, parameterized by the Schottky space Sg, we define a natural basis of H(Xt, ω n Xt ) which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n = 1. We introduce a holomorphic function F (n) on Sg which generalizes the classical product ∏ ∞ m=1 (1 − q) for n = 1 and g = 1. W...

The irreducible representations of the symmetric groups and their Iwahori-Hecke algebras have been classified and constructed by James [6] and Dipper and James [2], yet simple properties of these modules, such as their dimensions, are still not known. Every irreducible representation of these algebras is constructed by quotienting out the radical of a bilinear form on a particular type of modul...

In this paper, we solve a problem posed by the late Rodica Simion regarding type B Gram determinants, cf. [5]. We present this in a fashion influenced by the work of W.B.R.Lickorish on Witten-Reshetikhin-Turaev invariants of 3-manifolds. We will give a history of this problem in a sequel paper in which we also plan to address other related questions by Simion [6, 5] and connect the problem to F...

We develop methods that construct an optimal set of vectors with a specified inner product structure, from a given set of vectors in a complex Hilbert space. The optimal vectors are chosen to minimize the sum of the squared norms of the errors between the constructed vectors and the given vectors. Four special cases are considered. In the first, the constructed vectors are orthonormal. In the s...

We propose hashing to facilitate efficient kernels. This generalizes previous work using sampling and we show a principled way to compute the kernel matrix for data streams and sparse feature spaces. Moreover, we give deviation bounds from the exact kernel matrix. This has applications to estimation on strings and graphs.

a = semimajor axis b = magnetic field vector e = eccentricity e = orbital element vector e · = matrix exponential f = true anomaly h = angular momentum magnitude i = inclination J = cost Jn = nth zonal harmonic coefficient M = mean anomaly m = mass m = magnetic dipole vector N − 1 = number of thrusts n = mean motion p = semilatus rectum q = electrical charge r = orbit radius magnitude r = posit...

The issue of Cα-degree reduction of triangular Bézier surfaces is exposed. It is anticipated that both triangular Bézier surfaces are Cα-continuous at the vertices. The Euclidean norm as well as the L2−norm is used. The final solutions are given in terms of the matrix of degree raising, the Gram matrix, and the Bézier points. Moreover, it is shown that the solutions using both norms are equival...

Selecting appropriate inputs for systems described by complex networks is an important but difficult problem that largely remains open in the field of control networks. Recent work has proposed two methods energy efficient input selection; a gradient-based heuristic and greedy approximation algorithm. We propose here alternative method selection based on analytic solution controllability Gramia...

The berization of aane systems via dual Gramian techniques, that was developed in previous papers of the authors, is applied here for the study of aane frames which have an aane dual system. Gramian techniques are also used to verify whether a dual pair of aane frames is a also a pair of bi-orthogonal Riesz bases. A general method for a painless derivation of a dual pair of aane frames from an ...

[1] We introduce a new approach to the joint inversion of multimodal geophysical data using Gramian spaces of model parameters and Gramian constraints, computed as determinants of the corresponding Gram matrices of the multimodal model parameters and/or their attributes. We demonstrate that this new approach is a generalized technique that can be applied to the simultaneous joint inversion of a...