We present a new algorithm for fast matrix multiplication using tensor decompositions which have special features. Thanks to these features we obtain exponents lower than what the rank of the tensor decomposition suggests. In particular for $6\times 6$ matrix multiplication we reduce the exponent of the recent algorithm by Moosbauer and Poole from $2.8075$ to $2.8016$, while retaining a reasonable leading coefficient.
AbstractCorrosion is an electrochemical phenomenon. It can occur via different modes of attack, each having its own mechanisms, and therefore there are multiple metrics for evaluating corrosion resistance. In corrosion resistant alloys (CRAs), the rate of localized corrosion can exceed that of uniform corrosion by orders of magnitude. Therefore, instead of uniform corrosion rate, more complex electrochemical parameters are required to capture the salient features of corrosion phenomena. Here, we collect a database with an emphasis on metrics related to localized corrosion. The six sections of the database include data on various metal alloys with measurements of (1) pitting potential, Epit, (2) repassivation potential, Erp, (3) crevice corrosion potential, Ecrev, (4) pitting temperature, Tpit, (5) crevice corrosion temperature, Tcrev, and (6) corrosion potential, Ecorr, corrosion current density, icorr, passivation current density, ipass, and corrosion rate. The experimental data were collected from 85 publications and include Al- and Fe-based alloys, high entropy alloys (HEAs), and a Ni-Cr-Mo ternary system. This dataset could be used in the design of highly corrosion resistant alloys.
Peter Makaula, Sekeleghe Amos Kayuni, Kondwani Chidzammbuyo Mamba
et al.
Abstract Background Mass drug administration is one of the key interventions recommended by WHO to control certain NTDs. With most support from donors, health workers distribute antihelminthic drugs annually in Malawi. Mean community coverage of MDA from 2018 to 2020 was high at 87% for praziquantel and 82% for albendazole. However, once donor support diminishes sustaining these levels will be challenging. This study intended to compare the use of the community-directed intervention approach with the standard practice of using health workers in delivery of MDA campaigns. Methods This was a controlled implementation study carried out in three districts, where four health centres and 16 villages in each district were selected and randomly assigned to intervention and control arms which implemented MDA campaigns using the CDI approach and the standard practice, respectively. Cross-sectional and mixed methods approach to data collection was used focusing on quantitative data for coverage and knowledge levels and qualitative data to assess perceptions of health providers and beneficiaries at baseline and follow-up assessments. Quantitative and qualitative data were analyzed using IBM SPSS software version 26 and NVivo 12 for Windows, respectively. Results At follow-up, knowledge levels increased, majority of the respondents were more knowledgeable about what schistosomiasis was (41%-44%), its causes (41%-44%) and what STH were (48%-64%), while knowledge on intermediate host for schistosomiasis (19%-22%), its types (9%-13%) and what causes STH (15%-16%) were less known both in intervention and control arm communities. High coverage rates for praziquantel were registered in intervention (83%-89%) and control (86%-89%) communities, intervention (59%-79) and control (53%-86%) schools. Costs for implementation of the study indicated that the intervention arm used more resources than the control arm. Health workers and community members perceived the use of the CDI approach as a good initiative and more favorable over the standard practice. Conclusions The use of the CDI in delivery of MDA campaigns against schistosomiasis and STH appears feasible, retains high coverages and is acceptable in intervention communities. Despite the initial high costs incurred, embedding into community delivery platforms could be considered as a possible way forward addressing the sustainability concern when current donor support wanes. Trial registration Pan-African Clinical Trials Registry PACTR202102477794401, date: 25/02/2021.
A new runtime environment for the execution of recursive matrix algorithms on a supercomputer with distributed memory is proposed. It is designed both for dense and sparse matrices. The environment ensures decentralized control of the computation process. As an example of a block recursive algorithm, the Cholesky factorization of a symmetric positive definite matrix in the form of a block dichotomous algorithm is described. The results of experiments with different numbers of cores are presented, demonstrating good scalability of the proposed solution.
We present a new algorithm for computing the characteristic polynomial of an arbitrary endomorphism of a finite Drinfeld module using its associated crystalline cohomology. Our approach takes inspiration from Kedlaya's p-adic algorithm for computing the characteristic polynomial of the Frobenius endomorphism on a hyperelliptic curve using Monsky-Washnitzer cohomology. The method is specialized using a baby-step giant-step algorithm for the particular case of the Frobenius endomorphism, and in this case we include a complexity analysis that demonstrates asymptotic gains over previously existing approaches
Consider a sparse multivariate polynomial f with integer coefficients. Assume that f is represented as a "modular black box polynomial", e.g. via an algorithm to evaluate f at arbitrary integer points, modulo arbitrary positive integers. The problem of sparse interpolation is to recover f in its usual sparse representation, as a sum of coefficients times monomials. For the first time we present a quasi-optimal algorithm for this task.
Creative telescoping is an algorithmic method initiated by Zeilberger to compute definite sums by synthesizing summands that telescope, called certificates. We describe a creative telescoping algorithm that computes telescopers for definite sums of D-finite functions as well as the associated certificates in a compact form. The algorithm relies on a discrete analogue of the generalized Hermite reduction, or equivalently, a generalization of the Abramov-Petkovšek reduction. We provide a Maple implementation with good timings on a variety of examples.
Polynomial factorization and root finding are among the most standard themes of computational mathematics. Yet still, little has been done for polynomials over quaternion algebras, with the single exception of Hamiltonian quaternions for which there are known numerical methods for polynomial root approximation. The sole purpose of the present paper is to present a polynomial factorization algorithm for division quaternion algebras over number fields, together with its adaptation for root finding.
This extended abstract was written to accompany an invited talk at the 2021 SC-Square Workshop, where the author was asked to give an overview of SC-Square progress to date. The author first reminds the reader of the definition of SC-Square, then briefly outlines some of the history, before picking out some (personal) scientific highlights.
This paper aims to initialize a dynamical aspect of symbolic integration by studying stability problems in differential fields. We present some basic properties of stable elementary functions and D-finite power series that enable us to characterize three special families of stable elementary functions involving rational functions, logarithmic functions, and exponential functions. Some problems for future studies are proposed towards deeper dynamical studies in differential and difference algebra.
In this paper, we present a deterministic algorithm to find a strong generic position for an algebraic space curve. We modify our existing algorithm for computing the topology of an algebraic space curve and analyze the bit complexity of the algorithm. It is $\tilde{\mathcal {O}} (N^{20})$, where $N=\max\{d,τ\}$, $d, τ$ are the degree bound and the bit size bound of the coefficients of the defining polynomials of the algebraic space curve. To our knowledge, this is the best bound among the existing work. It gains the existing results at least $N^2$.
Gérard Duchamp, Nihar Gargava, Hoang Ngoc Minh
et al.
In this short note, we give a localized version of the basic triangle theorem, first published in 2011 (see [4]) in order to prove the independence of hyperlogarithms over various function fields. This version provides direct access to rings of scalars and avoids the recourse to fraction fields as that of meromorphic functions for instance.
Проведен геометрический и топологический анализ металлооксида с минимальным известным содержанием кислорода CsO, образующегося из кислородсодержащего расплава металлического Cs. Для определения кластеров-прекурсоров кристаллических структур использованы специальные алгоритмы разложения структурных графов на кластерные субструктуры (пакет программ ToposPro). Определены участвующие в самосборке кристаллических структур кластеры-прекурсоры: трехоктаэдрические кластеры CsO, октаэдрические кластеры Cs, тетраэдрические кластеры Cs. Реконструированы симметрийный и топологический коды процессов самосборки кристаллических структур из кластеров-прекурсоров в виде: первичная цепь микрослой микрокаркас.
In this paper, we propose two new deterministic interpolation algorithms for a sparse multivariate polynomial given as a standard black-box by introducing new Kronecker type substitutions. Let $f\in \RB[x_1,\dots,x_n]$ be a sparse black-box polynomial with a degree bound $D$. When $\RB=\C$ or a finite field, our algorithms either have better bit complexity or better bit complexity in $D$ than existing deterministic algorithms. In particular, in the case of deterministic algorithms for standard black-box models, our second algorithm has the current best complexity in $D$ which is the dominant factor in the complexity.
Changbo Chen, Svyatoslav Covanov, Farnam Mansouri
et al.
We propose a new algorithm for multiplying dense polynomials with integer coefficients in a parallel fashion, targeting multi-core processor architectures. Complexity estimates and experimental comparisons demonstrate the advantages of this new approach.
We propose a differential analog of the notion of integral closure of algebraic function fields. We present an algorithm for computing the integral closure of the algebra defined by a linear differential operator. Our algorithm is a direct analog of van Hoeij's algorithm for computing integral bases of algebraic function fields.
The aim of this article is twofold: on the one hand it is intended to serve as a gentle introduction to the topic of creative telescoping, from a practical point of view; for this purpose its application to several problems is exemplified. On the other hand, this chapter has the flavour of a survey article: the developments in this area during the last two decades are sketched and a selection of references is compiled in order to highlight the impact of creative telescoping in numerous contexts.
We state and analyze a generalization of the "truncation trick" suggested by Gourdon and Sebah to improve the performance of power series evaluation by binary splitting. It follows from our analysis that the values of D-finite functions (i.e., functions described as solutions of linear differential equations with polynomial coefficients) may be computed with error bounded by 2^(-p) in time O(p*(lg p)^(3+o(1))) and space O(p). The standard fast algorithm for this task, due to Chudnovsky and Chudnovsky, achieves the same time complexity bound but requires Θ(p*lg p) bits of memory.
We continue to investigate which polynomials can possibly occur as factors in the denominators of rational solutions of a given partial linear difference equation. In an earlier article we had introduced the distinction between periodic and aperiodic factors in the denominator, and we gave an algorithm for predicting the aperiodic ones. Now we extend this technique towards the periodic case and present a refined algorithm which also finds most of the periodic factors.