Inverse trigonometric functions

About: Inverse trigonometric functions is a research topic. Over the lifetime, 854 publications have been published within this topic receiving 11141 citations. The topic is also known as: arcus function & antitrigonometric function.

On the closure of the set of functions that can be realized by a given multilayer perceptron

Abstract: Given a multilayer perceptron (MLP) with a fixed architecture, there are functions that can be approximated up to any degree of accuracy, without having to increase the number of the hidden nodes. Those functions belong to the closure F~ of the set F~ of the maps realizable by the MLP. In this paper, we give a list of maps with this property. In particular, it is proven that: 1) rational functions belongs to F~ for networks with inverse tangent activation function; and 2) products of polynomials and exponentials belongs to F~ for networks with sigmoid activation function. Moreover, for a restricted class of MLPs, we prove that the list is complete and give an analytic definition of F~.

A non-standard approach to introduce simple harmonic motion

Abstract: We'll be presenting an approach to solve the equation of simple harmonic mo- tion (SHM) which is non-standard as compared with the usual way of solution presented in textbooks. In addition to help students avoid the unnecessary memorization of formulas to solve physics problems, this approach could help instructors to present the subject in a teaching framework which integrates conceptual and mathematical reasoning, in a systemic way of thinking that will help students to reinforce their quantitative reasoning skills by using mathematical knowledge already familiar to students in a first calculus-based introductory physics course, such as the chain rule for derivatives, inverse trigonometric functions, and integration methods.

Ortho-Skew and Ortho-Sym Matrix Trigonometry

Abstract: This paper introduces some properties of two families of matrices: the Ortho-Skew, which are simultaneously orthogonal and skew-Hermitian, and the real Ortho-Sym matrices, which are orthogonal and symmetric These relationships consist of closed-form compact expressions of trigonometric and hyperbolic functions that show that multiples of these matrices can be interpreted as angles The analogies with trigonometric and hyperbolic functions, such as the periodicity of the trigonometric functions, are all shown Additional expressions are derived from some other functions of matrices such as the logarithm, exponential, inverse, and power functions All these relationships show that the Ortho-Skew and the Ortho-Sym matrices can be respectively considered as matrix extensions of the imaginary and the real units

Full phase anticosine double orthogonal transformation and its improving method for JPEG

Abstract: This invention relates to a total phase inverse cosine double orthogonal transformation method, which defines the transformation as [F]=[V][f][V ] and the inverse transformation as [f]=[V ][F][(V ) ] and uses the improved total phase inverse cosine double orthogonal transformation to replace 2-D scatter cosine transformation to simplify the quantification list so as to save computing time and increase quality of re-built images and coding compression ratio including the following steps: inputting an original image and a bit ratio to be divided into 8x8 pixel blocks and processed by total phase inverse cosine double orthogonal transformation to determine the quantification intervals based on the bit ratio to average the transformation coefficients, carrying out forecast coding to the DC coefficient and AC coefficient scan, variable-length coding, Harfmann entropy coding and outputting the bit sequence of a compressed image.

On a constant in the theory of trigonometric series

Abstract: The note "A constant in the theory of trigonometric series" in the October 1964 issue of Mathematics of Computation provided us with a test for our recently constructed algorithms for the computation of roots of functions, and for numerical quadrature in the presence of singularities. The latter algorithm, utilizing the Gaussian 8-point quadrature formula applied to sub-intervals of variable length, involves a sufficiently small number of ordinates that computational labor and round-off error do not become problems. Use of these algorithms indicated the value ao = .3084438, for the root of the equation fJot2 u cos u du = 0, differing from the reported value, .30483, in the third place. To check this result, we made the transformation u = X4 to weaken the character of the singularity at the origin, and obtained the following table by conventional numerical quadrature, confirming our result: