相关论文: Elementary Matrix Decomposition and The Computatio…

We study the unipotent Albanese map appearing in the non-abelian Chabauty method of Minhyong Kim. In particular we explore the explicit computation of the $p$-adic de Rham period map $j^{dr}_n$ on elliptic and hyperelliptic curves over…

An improved PV-reduction method for one-loop integrals with auxiliary vector $R$ has been proposed in \cite{Feng:2021enk,Hu:2021nia}. It has also been shown that the new method is a self-completed method in \cite{Feng:2022uqp}. Analytic…

Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality,…

We comment on two randomized algorithms for constructing low-rank matrix decompositions. Both algorithms employ the Subsampled Randomized Hadamard Transform [14]. The first algorithm appeared recently in [9]; here, we provide a novel…

CUR matrix decomposition is a randomized algorithm that can efficiently compute the low rank approximation for a given rectangle matrix. One limitation with the existing CUR algorithms is that they require an access to the full matrix A for…

We give several generalisations of the Deuring reduction criterion for elliptic curves to abelian varieties of higher dimension. In particular the Newton polygon of the reduction of an abelian variety A with complex multiplication by F at a…

Expander decompositions of graphs have significantly advanced the understanding of many classical graph problems and led to numerous fundamental theoretical results. However, their adoption in practice has been hindered due to their…

In this work, we focus on the Neumann-Neumann method (NNM), which is one of the most popular non-overlapping domain decomposition methods. Even though the NNM is widely used and proves itself very efficient when applied to discrete problems…

Using Semaev's summation polynomials, we derive a new equation for the $\mathbb{F}_q$-rational points of the trace zero variety of an elliptic curve defined over $\mathbb{F}_q$. Using this equation, we produce an optimal-size representation…

The matrix completion problem consists of finding or approximating a low-rank matrix based on a few samples of this matrix. We propose a new algorithm for matrix completion that minimizes the least-square distance on the sampling set over…

We present an algebraic method for constructing a highly effective coarse grid correction to accelerate domain decomposition. The coarse problem is constructed from the original matrix and a small set of input vectors that span a low-degree…

This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality…

Class groups of real quadratic fields represent fundamental structures in algebraic number theory with significant computational implications. While Stark's conjecture establishes theoretical connections between special units and class…

We present normal forms for elliptic curves over a field of characteristic $2$ analogous to Edwards normal form, and determine bases of addition laws, which provide strikingly simple expressions for the group law. We deduce efficient…

Deterministic recursive algorithms for the computation of matrix triangular decompositions with permutations like LU and Bruhat decomposition are presented for the case of commutative domains. This decomposition can be considered as a…

We derive simplified normal forms for an area-preserving map in a neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points appear in generic two-parameter families of area-preserving maps. We also derive a simplified…

A modeling methodology and matrix formalism is presented that permits analysis of arbitrarily complex interferometric waveguide systems, including polarization and backreflection effects. Considerable improvement results from separation of…

We use an invariant-theoretic method to compute certain twists of the modular curves X(n) for n=7,9,11. Searching for rational points on these twists enables us to find non-trivial pairs of n-congruent elliptic curves over Q, i.e. pairs of…

We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory…

A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation…