The Fréchet derivative L f ( A , E ) of the matrix function plays an important role in many different applications, including condition number estimation and network analysis. We present several Krylov subspace methods for computing low-rank approximations when direction term is rank one (which can easily be extended to general low rank). analyze convergence resulting both Hermitian non-Hermiti...

We prove that the smallest degree of an apolar 0-dimensional scheme to a general cubic form in n+ 1 variables is at most 2n+ 2, when n ≥ 8, and therefore smaller than the rank of the form. When n = 8 we show that the bound is sharp, i.e. the smallest degree of an apolar subscheme is 18.

Abstract. For any function f in L∞(D), let Tf denote the corresponding Toeplitz operator the Bergman space A(D). A recent result of D. Luecking shows that if Tf has finite rank then f must be the zero function. Using a refined version of this result, we show that if all except possibly one of the functions f1, . . . , fm are radial and Tf1 · · · Tfm has finite rank, then one of these functions ...

We numerically verify the Conjecture of Birch and SwinnertonDyer concerning the analytic and geometric rank of an elliptic curve. An algorithm (based on the work of Cremona) is developed in the PARI/GP language for computing the order of vanishing of the L-function for any (non-singular) curve. The analytic rank outputs for several families of curves are compared with readily available data on ...

In this paper we consider a level l Appell function, and find, for all odd l, a partial differential equation it satisfies. For l = 3 this recovers the Rank-Crank PDE, found by Atkin and Garvan, and for l = 5 we get a similar PDE found by Garvan.

In this article, we study the Beilinson–Bloch–Kato conjecture for motives associated to Rankin–Selberg products of conjugate self-dual automorphic representations, within framework Gan–Gross–Prasad conjecture. We show that if central critical value L-function does not vanish, then Bloch–Kato Selmer group with coefficients in a favorable field corresponding motive vanishes. also class constructe...