We propose a differential geometric approach for building families of low-rank covariance matrices, via interpolation on matrix manifolds. In contrast with standard parametric classes, these offer significant flexibility problem-specific tailoring the choice “anchor” matrices interpolation, instance over grid relevant conditions describing underlying stochastic process. The is computationally t...

In this article, we propose a new probabilistic model for the distribution of ranks of elliptic curves in families of fixed Selmer rank, and compare the predictions with previous results, and with the databases of curves over the rationals that we have at our disposal. In addition, we document a phenomenon we refer to as Selmer bias that seems to play an important role in the data and in our mo...

We prove for abelian varieties a global form of Denef and Loeser’s motivic monodromy conjecture. More precisely, we prove that for any abelian C((t))-variety A, its motivic zeta function has a unique pole at Chai’s base change conductor c(A) of A, and that the order of this pole equals one plus the potential toric rank of A. Moreover, we show that for any embedding of Ql in C, the value exp(2πi...

This is the first in a series of papers on rank decompositions of the matrix multiplication tensor. In this paper we: establish general facts about rank decompositions of tensors, describe potential ways to search for new matrix multiplication decompositions, give a geometric proof of the theorem of [3] establishing the symmetry group of Strassen’s algorithm, and present two particularly nice s...

It is shown that under certain circumstances in particular for small datasets the recently proposed citation impact indicators I3(6PR) and R(6,k) behave inconsistently when additional papers or citations are taken into consideration. Three simple examples are presented, in which the indicators fluctuate strongly and the ranking of scientists in the evaluated group is sometimes completely mixed ...

7 Given an n × n matrix, its principal rank characteristic sequence is a sequence of 8 length n + 1 of 0s and 1s where, for k = 0, 1, . . . , n, a 1 in the kth position indicates 9 the existence of a principal submatrix of rank k and a 0 indicates the absence of such 10 a submatrix. The principal rank characteristic sequences for symmetric matrices over 11 various fields are investigated, with ...

For a set of 1D vectors, standard singular value decomposition (SVD) is frequently applied. For a set of 2D objects such as images or weather maps, we form 2DSVD, which computes principal eigenvectors of rowrow and column-column covariance matrices, exactly as in the standard SVD. We study optimality properties of 2DSVD as low-rank approximation and show that it provides a framework unifying tw...