The Addition Formulas for the Sine and Cosine

1941 ◽  
Vol 48 (10) ◽  
pp. 688
Author(s):  
E. J. McShane
Keyword(s):  
Addition Formulas

scholarly journals Sample Sizes Needed for Specified Margins of Relative Error in the Estimates of the Repeatability and Reproducibility Standard Deviations

2005 ◽  
Vol 88 (5) ◽  
pp. 1503-1510 ◽  
Author(s):  
Foster D McClure ◽  
Jung K Lee
Keyword(s):  
Standard Deviation ◽  
Sample Size ◽  
Relative Error ◽  
Analytical Method ◽  
Actual Error ◽  
Repeatability Standard Deviation ◽  
Repeatability And Reproducibility ◽  
Standard Deviations ◽  
True Values ◽  
Addition Formulas

Abstract Sample size formulas are developed to estimate the repeatability and reproducibility standard deviations (sr and sR) such that the actual error in (sr and sR) relative to their respective true values, σr and σR, are at predefined levels. The statistical consequences associated with AOAC INTERNATIONAL required sample size to validate an analytical method are discussed. In addition, formulas to estimate the uncertainties of (sr and sR) were derived and are provided as supporting documentation.Formula for the Number of Replicates Required for a Specified Margin of Relative Error in the Estimate of the Repeatability Standard Deviation

scholarly journals Truncated-Exponential-Based Appell-Type Changhee Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1588
Author(s):  
Tabinda Nahid ◽  
Parvez Alam ◽  
Junesang Choi
Keyword(s):  
Recurrence Relations ◽  
Explicit Representation ◽  
Integral Inequalities ◽  
Exponential Polynomials ◽  
Integral Formulas ◽  
Open Questions ◽  
Changhee Polynomials ◽  
Natural Way ◽  
Addition Formulas

The truncated exponential polynomials em(x) (1), their extensions, and certain newly-introduced polynomials which combine the truncated exponential polynomials with other known polynomials have been investigated and applied in various ways. In this paper, by incorporating the Appell-type Changhee polynomials Chn*(x) (10) and the truncated exponential polynomials in a natural way, we aim to introduce so-called truncated-exponential-based Appell-type Changhee polynomials eCn*(x) in Definition 1. Then, we investigate certain properties and identities for these new polynomials such as explicit representation, addition formulas, recurrence relations, differential and integral formulas, and some related inequalities. We also present some integral inequalities involving these polynomials eCn*(x). Further we discuss zero distributions of these polynomials by observing their graphs drawn by Mathematica. Lastly some open questions are suggested.

Research on Debonding Formula Model of RC Beams Strengthened with CFRP Sheets

2011 ◽  
Vol 368-373 ◽  
pp. 1038-1041
Author(s):  
An Hong Bao ◽  
Zhen Yu Qiu ◽  
Peng Wang
Keyword(s):  
Mechanical Properties ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Concrete Beams ◽  
Rc Beams ◽  
Element Analysis ◽  
Cfrp Sheets ◽  
The Finite Element Analysis ◽  
Fiber Plate ◽  
Addition Formulas

Debonding of concrete occurs when the interface principal stress reaches the ultimate tensile strength. We propose the use of carbon fiber plate attached to the beam bottom, which makes finite element analysis of the mechanical properties of debonding concrete beams more reasonable. In addition, formulas of this theory are given and applied in the finite element analysis. Finally, it is shown by a number of experimental results.

The Addition Formulas in Trigonometry

1942 ◽  
Vol 49 (5) ◽  
pp. 326
Author(s):  
A. S. Householder
Keyword(s):  
Addition Formulas
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