Skip to content
Free planNo credit card requiredStart writing
AI for Research home
AI for Research home
Dashboard
Sign inGet started

Search 250 million research papers

Read the abstract, see who wrote it and who cites it, and get the citation in APA, MLA, Chicago or Harvard.

About 4,630 papers for “Mathematical notation”

Mathematical notation.

George G. Vega Yon (8182833), Duncan C. Thomas (7811444), John Morrison (91443) et al. · Figshare · 2021

Cited by 0Open access

Mathematical Notation

Dean Keith Simonton · Cambridge University Press eBooks · 2004

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Cited by 0

Mathematical notation.

Agustin Lage-Castellanos (4229272), Giancarlo Valente (635297), Elia Formisano (351283) et al. · Figshare · 2019

Cited by 0Open access

Mathematical Notation

2016

Extract iffif and only if Ḋthe time derivative dD∕dt of quantity D Σsummation Eexpectation operator, e.g., Epx=∑xipi if outcomes x have probability p⁠. Pr[e]probability of event e mTtranspose of matrix m. For example, if m=acbd⁠, then mT=abcd v⋅u=vuTdot product ∑i=1nviui of row vectors v, u λeigenvalue: solution of the characteristic equation |J-λI|=0 B∘AHadamard product of matrices, defined by entry-by-entry multiplication si=NiNfraction (or share) of given population using strategy (or trait) i sthe state vector (s1,…,sn) Sthe state space, consisting of row vectors whose components si are non-negative and sum to 1 Ni′the number of progeny of i-strategists wi,Wifitness: ability of a strategy (or trait) to increase its share; typically Wi=Ni′Ni⁠. Upper (lower) case indicates discrete- (continuous-) time applications. In two population application ui and vi are also used to...

Cited by 0

Mathematical notation.

János Abonyi (10106040), Ádám Ipkovich (15285785), Gyula Dörgő (13919393) et al. · Figshare · 2023

<div><p>Non-negative matrix factorization (NMF) efficiently reduces high dimensionality for <i>many</i>-objective ranking problems. In multi-objective optimization, as long as only three or four conflicting viewpoints are present, an optimal solution can be determined by finding the Pareto front. When the number of the objectives increases, the multi-objective problem evolves into a <i>many</i>-objective optimization task, where the Pareto front becomes oversaturated. The key idea is that NMF aggregates the objectives so that the Pareto front can be applied, while the Sum of Ranking Differences (SRD) method selects the objectives that have a detrimental effect on the aggregation, and validates the findings. The applicability of the method is illustrated by the ranking of 1176 universities based on 46 variables of the CWTS Leiden Ranking 2020 database. The performance of NMF is compared to principal component analysis (PCA) and sparse non-negative matrix factorization-based solutions. The results illustrate that PCA incorporates negatively correlated objectives into the same principal component. On the contrary, NMF only allows non-negative correlations, which enable the proper use of the Pareto front. With the combination of NMF and SRD, a non-biased ranking of the universities based on 46 criteria is established, where Harvard, Rockefeller and Stanford Universities are determined as the first three. To evaluate the ranking capabilities of the methods, measures based on Relative Entropy (RE) and Hypervolume (HV) are proposed. The results confirm that the sparse NMF method provides the most informative ranking. The results highlight that academic excellence can be improved by decreasing the proportion of unknown open-access publications and short distance collaborations. The proportion of gender indicators barely correlate with scientific impact. More authors, long-distance collaborations, publications that have more scientific impact and citations on average highly influence the university ranking in a positive direction.</p></div>

Cited by 0Open access

Mathematical notation.

Océane Dubois (21989812), Agnès Roby-Brami (4057294), Ross Parry (5418143) et al. · Figshare · 2025

This file contains the details of all mathematical notations used in this article. (PDF)

Cited by 0Open access

Mathematical notation

T. M. Flett · 1980

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Cited by 0

Mathematical Notation

Pertti Lounesto · Cambridge University Press eBooks · 2001

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Cited by 0

Forms; mathematical notation; LATEX

Jonathan R. D. Kuhn · 2015

An interactive workbook, the class notes for an introductory Statistics course devel-oped for both Internet and classroom students, is described in this paper. This inter-active workbook is presented on the Internet in Portable Document Format (PDF) because this format allows easy inclusion of mathematical symbols. A statistical experiment reveals that this interactive workbook provides enough information for Internet students to perform at a level comparable to classroom students1. 1

Cited by 0

Mathematical Preliminaries and Notation

David Poole, Alan K. Mackworth · 2010

This appendix gives some definitions of fundamental mathematical concepts that are used in AI, but are traditionally taught in other courses. It also introduces some notation and data structures that are used in various parts of the book.

Cited by 0

Summary of mathematical notation.

Kyle C. McGovern (17420005), Michelle Pistner Nixon (17420008), Justin D. Silverman (6889319) · Figshare · 2023

<div><p>By applying Differential Set Analysis (DSA) to sequence count data, researchers can determine whether groups of microbes or genes are differentially enriched. Yet sequence count data suffer from a <i>scale limitation</i>: these data lack information about the scale (i.e., size) of the biological system under study, leading some authors to call these data compositional (i.e., proportional). In this article, we show that commonly used DSA methods that rely on normalization make strong, implicit assumptions about the unmeasured system scale. We show that even small errors in these <i>scale assumptions</i> can lead to positive predictive values as low as 9%. To address this problem, we take three novel approaches. First, we introduce a sensitivity analysis framework to identify when modeling results are robust to such errors and when they are suspect. Unlike standard benchmarking studies, this framework does not require ground-truth knowledge and can therefore be applied to both simulated and real data. Second, we introduce a statistical test that provably controls Type-I error at a nominal rate despite errors in scale assumptions. Finally, we discuss how the impact of scale limitations depends on a researcher’s scientific goals and provide tools that researchers can use to evaluate whether their goals are at risk from erroneous scale assumptions. Overall, the goal of this article is to catalyze future research into the impact of scale limitations in analyses of sequence count data; to illustrate that scale limitations can lead to inferential errors in practice; yet to also show that rigorous and reproducible scale reliant inference is possible if done carefully.</p></div>

Cited by 0Open access