Addendum to “A Proof of the Newton-Cotes Quadrature Formulas with Error Term”

1971 ◽  
Vol 78 (9) ◽  
pp. 988-988
Author(s):  
D. R. Hayes ◽  
L. Rubin
1970 ◽  
Vol 77 (10) ◽  
pp. 1065-1072 ◽  
Author(s):  
D. R. Hayes ◽  
L. Rubin

Author(s):  
Gradimir Milovanovic

Some selected Ostrowski type inequalities and a connection with numerical integration are studied in this survey paper, which is dedicated to the memory of Professor D. S. Mitrinovic, who left us 25 years ago. His significant inuence to the development of the theory of inequalities is briefly given in the first section of this paper. Beside some basic facts on quadrature formulas and an approach for estimating the error term using Ostrowski type inequalities and Peano kernel techniques, we give several examples of selected quadrature formulas and the corresponding inequalities, including the basic Ostrowski's inequality (1938), inequality of Milovanovic and Pecaric (1976) and its modifications, inequality of Dragomir, Cerone and Roumeliotis (2000), symmetric inequality of Guessab and Schmeisser (2002) and asymmetric in-equality of Franjic (2009), as well as four point symmetric inequalites by Alomari (2012) and a variant with double internal nodes given by Liu and Park (2017).


1970 ◽  
Vol 77 (10) ◽  
pp. 1065 ◽  
Author(s):  
D. R. Hayes ◽  
L. Rubin

2018 ◽  
Vol 481 (2) ◽  
pp. 136-137
Author(s):  
V. Chubarikov ◽  
Keyword(s):  

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1799
Author(s):  
Irene Gómez-Bueno ◽  
Manuel Jesús Castro Díaz ◽  
Carlos Parés ◽  
Giovanni Russo

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.


2020 ◽  
pp. 016327872098559
Author(s):  
Michael T. McKay ◽  
Frank C. Worrell ◽  
Jon C. Cole

The Adolescent and Adult Time Inventory–Time Attitudes Scale (AATI-TA) measures emotional engagement with the past, present, and future, and scores have been shown to relate meaningfully to health outcomes. For past, present, and future, five items are used to assess both positive and negative attitudes. Although evidence for the hypothesized six-factor solution has been widely reported, some studies have indicated problems with the Future Negative items. Given that a large and growing literature has emerged on the six-factor AATI-TA, and that AATI-TA scores have shown much better and more consistent fit than other temporal psychology measures, we sought to investigate the future negative factor in detail. Secondary analyses were performed on two datasets. The first was a University convenience sample ( N = 410) and the second was an adolescent sample ( N = 1,612). Confirmatory factor analyses revealed that the fit for the five Future Negative items was poor. Modification indices suggested that a correlated error term between Items 4 and 10 would result in good fit, and this was indeed the case. Models without Item 4 or Item 10 also yielded acceptable fit. Analyses using all four operationalizations of Future Negative (original scale, without Item 4 or Item 10, or with the correlated error between Items 4 and 10) to predict symptoms of anxiety and depression, and emotional self-efficacy revealed minor differences in the predictive validity coefficients. Potential ways forward, including a correlated error term or the dropping or replacement of Item 10, are discussed.


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