Multivariate Fink type identity and multivariate Ostrowski, comparison of means and Grüss type inequalities

George A. Anastassiou

doi:10.1016/j.mcm.2006.11.008

Generalized multivariate Fink-type identity and some related results on time scales with applications

Journal of Applied Analysis ◽

10.1515/jaa-2019-0020 ◽

2019 ◽

Vol 25 (2) ◽

pp. 189-203

Author(s):

Sabir Hussain ◽

Muhammad Amer Latif

Keyword(s):

Time Scales ◽

Discrete Version ◽

Generalized Polynomials ◽

Arbitrary Time ◽

Type Identity ◽

Grüss Type Inequalities ◽

New Applications

Abstract Generalized Fink-type identity for multi-variables to an arbitrary time scales is obtained, giving some multi-variate Ostrowski, Iyengar and Grüss-type inequalities unifying the corresponding continuous and discrete version. Some new applications to generalized polynomials are also obtained.

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MacWilliams Type Identity for M-Spotty Rosenbloom-Tsfasman Weight Enumerator of Linear Codes over Finite Ring

IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences ◽

10.1587/transfun.e96.a.1496 ◽

2013 ◽

Vol E96.A (6) ◽

pp. 1496-1500

Author(s):

Jianzhang CHEN ◽

Wenguang LONG ◽

Bo FU

Keyword(s):

Linear Codes ◽

Finite Ring ◽

Weight Enumerator ◽

Type Identity

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Some k-fractional extension of Grüss-type inequalities via generalized Hilfer–Katugampola derivative

Advances in Difference Equations ◽

10.1186/s13662-020-03187-7 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Samaira Naz ◽

Muhammad Nawaz Naeem ◽

Yu-Ming Chu

Keyword(s):

Integral Inequality ◽

Fractional Integral ◽

Integral Inequalities ◽

Derivative Operator ◽

Practical Applications ◽

Mathematical Problems ◽

Grüss Type Inequalities

AbstractIn this paper, we prove several inequalities of the Grüss type involving generalized k-fractional Hilfer–Katugampola derivative. In 1935, Grüss demonstrated a fascinating integral inequality, which gives approximation for the product of two functions. For these functions, we develop some new fractional integral inequalities. Our results with this new derivative operator are capable of evaluating several mathematical problems relevant to practical applications.

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Non-negative Independent Factor Analysis for single cell RNA-seq

10.1101/2020.01.31.927921 ◽

2020 ◽

Author(s):

Weiguang Mao ◽

Maziyar Baran Pouyan ◽

Dennis Kostka ◽

Maria Chikina

Keyword(s):

Factor Analysis ◽

Single Cell ◽

Matrix Factorization ◽

Component Analysis ◽

Computational Techniques ◽

Cell Type ◽

Analysis Model ◽

Factor Analysis Model ◽

Type Identity ◽

Wide Range

AbstractMotivationSingle cell RNA sequencing (scRNA-seq) enables transcriptional profiling at the level of individual cells. With the emergence of high-throughput platforms datasets comprising tens of thousands or more cells have become routine, and the technology is having an impact across a wide range of biomedical subject areas. However, scRNA-seq data are high-dimensional and affected by noise, so that scalable and robust computational techniques are needed for meaningful analysis, visualization and interpretation. Specifically, a range of matrix factorization techniques have been employed to aid scRNA-seq data analysis. In this context we note that sources contributing to biological variability between cells can be discrete (or multi-modal, for instance cell-types), or continuous (e.g. pathway activity). However, no current matrix factorization approach is set up to jointly infer such mixed sources of variability.ResultsTo address this shortcoming, we present a new probabilistic single-cell factor analysis model, Non-negative Independent Factor Analysis (NIFA), that combines features of complementary approaches like Independent Component Analysis (ICA), Principal Component Analysis (PCA), and Non-negative Matrix Factorization (NMF). NIFA simultaneously models uni- and multi-modal latent factors and can so isolate discrete cell-type identity and continuous pathway-level variations into separate components. Similar to NMF, NIFA constrains factor loadings to be non-negative in order to increase biological interpretability. We apply our approach to a range of data sets where cell-type identity is known, and we show that NIFA-derived factors outperform results from ICA, PCA and NMF in terms of cell-type identification and biological interpretability. Studying an immunotherapy dataset in detail, we show that NIFA identifies biomedically meaningful sources of variation, derive an improved expression signature for regulatory T-cells, and identify a novel myeloid cell subtype associated with treatment response. Overall, NIFA is a general approach advancing scRNA-seq analysis capabilities and it allows researchers to better take advantage of their data. NIFA is available at https://github.com/wgmao/[email protected]

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