Approximations to complementary error function by method of least squares

The work considers rods with a constant cross-section. The deformation law of each layer of the rod is adopted as an approximation by a polynomial of the second order. The method of determining the coefficients of the indicated polynomial and the limit deformations under compression and tension of the material of each layer is described with the presence of three traditional characteristics: modulus of elasticity, limit stresses at compression and tension. On the basis of deformation diagrams of the concrete grades B10, B30, B50 under tension and compression, these coefficients are determined by the method of least squares. The deformation diagrams of these concrete grades are compared on the basis of the approximations obtained by the limit values and the method of least squares, and it is found that these diagrams approximate quite well the real deformation diagrams at deformations close to the limit. The main problem in this work is to determine if the rod is able withstand the applied loads, before intensive cracking processes in concrete. So as a criterion of the conditional limit state this work adopts the maximum permissible deformation value under tension or compression corresponding to the points of transition to a falling branch on the deformation diagram level in one or more layers of the rod. The Kirchhoff-Lyav classical kinematic hypotheses are assumed to be valid for the rod deformation. The cases of statically determinable and statically indeterminable problems of bend of the rod are considered. It is shown that in the case of statically determinable loadings, the general solution of the problem comes to solving a system of three nonlinear algebraic equations which roots can be obtained with the necessary accuracy using the well-developed methods of computational mathematics. The general solution of the problem for statically indeterminable problems is reduced to obtaining a solution to a system of three nonlinear differential equations for three functions - deformation and curvatures. The Bubnov-Galerkin method is used to approximate the solution of this equation on the segment along the length of the rod, and specific examples of its application to the Maple system of symbolic calculations are considered.

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Algorithm 181: complementary error function—large X

Communications of the ACM ◽

10.1145/366604.366638 ◽

1963 ◽

Vol 6 (6) ◽

pp. 315

Author(s):

Henry C. Thacher

Keyword(s):

Error Function ◽

Complementary Error Function

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Functional ratings in sports

Journal of Quantitative Analysis in Sports ◽

10.1515/jqas-2020-0001 ◽

2020 ◽

Vol 16 (3) ◽

pp. 183-191

Author(s):

Brad Lowery ◽

Abigail Slater ◽

Kaison Thies

Keyword(s):

Least Squares ◽

Method Of Least Squares ◽

Sports Teams ◽

New Model ◽

College Basketball ◽

Division 1

AbstractIn this paper, we present a new model for ranking sports teams. Our model uses all scoring data from all games to produce a functional rating by the method of least squares. The functional rating can be interpreted as a team average point differential adjusted for strength of schedule. Using two team’s functional ratings we can predict the expected point differential at any time in the game. We looked at three variations of our model accounting for home-court advantage in different ways. We use the 2018–2019 NCAA Division 1 men’s college basketball season to test the models and determined that home-court advantage is statistically important but does not differ between teams.

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