We consider conditions under which the SR1 iteration is locally convergent. We apply the result to a pointwise structured SR1 method that has been used in optimal control.

For a prime p and a matrix A ∈ Zn×n, write A as A = p(A quo p)+ (A rem p) where the remainder and quotient operations are applied element-wise. Write the p-adic expansion of A as A = A[0] + pA[1] + p2A[2] + · · · where each A[i] ∈ Zn×n has entries between [0, p − 1]. Upper bounds are proven for the Z-ranks of A rem p, and A quo p. Also, upper bounds are proven for the Z/pZ-rank of A[i] for all ...

Let n ≥ 3. We prove that if p 6= q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank n are incomparable with respect to Borel reducibility.

In this paper we enumerate fuzzy subgroups, up to a natural equivalence, of some finite abelian p-groups of rank two where p is any prime number. After obtaining the number of maximal chains of subgroups, we count fuzzy subgroups using inductive arguments. The number of such fuzzy subgroups forms a polynomial in p with pleasing combinatorial coefficients. By exploiting the order, we label the s...

Different constructions by Cooke, Harper and Zabrodsky and by Cohen and Neisendorfer produce torsion free finite p-local H-spaces of rank l < p−1. The first construction goes through when l = p− 1 and we show the second does as well. However, the space produced need not be an H-space. We give a criterion for when an H-space is obtained. In the special case of rank 2 mod-3 H-spaces, we also give...

In this paper, we provide local and global convergence guarantees for recovering CP (Candecomp/Parafac) tensor decomposition. The main step of the proposed algorithm is a simple alternating rank-1 update which is the alternating version of the tensor power iteration adapted for asymmetric tensors. Local convergence guarantees are established for third order tensors of rank k in d dimensions, wh...

In [9, 10], Hartshorne gives an account of the known vector bundles of "small" rank on P" and also on the punctured spectrum of a regular local ring. Specifically, he remarks on various constructions of rank two vector bundles on P", for n ^ 4, and Horrocks' examples [12] of rank three bundles on P as well. The result of Evans and Griffith [7] on syzygies shows that the existence of rank two in...