Let K be a global field of char p and let Fq be the algebraic closure of Fp in K. For an elliptic curve E/K with non-constant j-invariant, the L-function L(T,E/K) is a polynomial in 1+T ·Z[T ]. For any N > 1 invertible in K and finite subgroup T ⊂ E(K) of order N , we compute the mod N reduction of L(T,E/K) and determine an upper-bound for the order of vanishing at 1/q, the so-called analytic r...

We study the l0-Low Rank Approximation Problem, where the goal is, given anm×nmatrix A, to output a rank-k matrix A for which ‖A′ −A‖0 is minimized. Here, for a matrix B, ‖B‖0 denotes the number of its non-zero entries. This NP-hard variant of low rank approximation is natural for problems with no underlying metric, and its goal is to minimize the number of disagreeing data positions. We provid...

For a noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the joint rank-k numerical range associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint rank k-numerical range are obtained and their implications to quantum computing are discussed. It is shown that for a given k if the dimension of the un...

A bimatrix game (A, B) is called a game of rank k if the rank of the matrix A + B is at most k. We consider the problem of enumerating the Nash equilibria in (non-degenerate) games of rank 1. In particular, we show that even for games of rank 1 not all equilibria can be reached by a Lemke– Howson path and present a parametric simplex-type algorithm for enumerating all Nash equilibria of a non-d...

This note, attempts to further Pan’s second-order quasi-Newton methods([1]). To complement the numerical implementation, the linear convergence of a rank-one second-order update and the least change property are presented.

We describe a class of rank-2 graphs whose C∗-algebras are AT algebras. For a subclass which we call rank-2 Bratteli diagrams, we compute the K-theory of the C∗-algebra. We identify rank-2 Bratteli diagrams whose C∗-algebras are simple and have real-rank zero, and characterise the K-invariants achieved by such algebras. We give examples of rank-2 Bratteli diagrams whose C∗-algebras contain as f...

For odd m, I write down tensors in C m ⊗C m ⊗C m of border rank at least 2m − 1, showing the non-triviality of the Young-flattening equations of [6] that vanish on the matrix multiplication tensor. I also study the border rank of the tensors of [1] and [3]. I show the tensors T 2 k ∈ C k ⊗C 2 k ⊗C 2 k , of [1], despite having rank equal to 2 k+1 − 1, have border rank equal to 2 k. I show the eq...

For redundant manipulators, the inverse kinematics equations are typically solved by taking the pseudoinverse of the Jacobian, or by adding enough constraints to algebraically remove the redundancy. Many popular measures of redundancy also depend on computing the algebraic rank or determinant of the Jacobian. We present fast, parallelizable algorithms to dynamically update all the pseudo-invers...

For a global field K and an elliptic curve Eη over K(T), Silverman's specialization theorem implies rank(Eη(K(T))) ≤ rank(Et(K)) for all but finitely many t ∈ P 1 (K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the exampl...

For a global field K and an elliptic curve Eη over K(T), Silverman's specialization theorem implies rank(Eη(K(T))) ≤ rank(Et(K)) for all but finitely many t ∈ P 1 (K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the exampl...