[5] **viXra:1611.0368 [pdf]**
*submitted on 2016-11-26 20:05:27*

**Authors:** Edigles Guedes

**Comments:** 7 Pages.

In this paper, I demonstrate one new infinite product for binomial coefficient
and news Euler's and Weierstrass's infinite product for Gamma function among other things.

**Category:** Functions and Analysis

[4] **viXra:1611.0073 [pdf]**
*submitted on 2016-11-05 14:49:21*

**Authors:** Matthew Marko

**Comments:** 13 Pages, English

This algorithm is designed to perform Discrete Fourier Transforms (DFT) to convert temporal data into spectral data. What is unique about this DFT algorithm is that it can produce spectral data at any user-defined resolution; existing DFT methods such as FFT are limited in resolution proportional to the temporal resolution. This algorithm obtains the Fourier Transforms by studying the Coefficient of Determination of a series of artificial sinusoidal functions with the temporal data, and normalizing the variance data into a high-resolution spectral representation of the time-domain data with a finite sampling rate.

**Category:** Functions and Analysis

[3] **viXra:1611.0056 [pdf]**
*replaced on 2016-11-10 10:15:26*

**Authors:** O. P. Ferreira, S. Z. Németh

**Comments:** 12 Pages.

The extended second order cones were introduced by S. Z. Németh and G. Zhang in [S. Z. Németh and G. Zhang. Extended Lorentz cones and variational inequalities on cylinders. J. Optim. Theory Appl., 168(3):756-768, 2016] for solving mixed complementarity problems and variational inequalities on cylinders. R. Sznajder in [R. Sznajder. The Lyapunov rank of extended second order cones. Journal of Global Optimization, 66(3):585-593, 2016] determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. S. Z. Németh and G. Zhang in [S.Z. Németh and G. Zhang. Positive operators of Extended Lorentz cones. arXiv:1608.07455v2, 2016] found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. This note will give formulas for projecting onto the extended second order cones. In the most general case the formula will depend on a piecewise linear equation for one real variable which will be solved by using numerical methods.

**Category:** Functions and Analysis

[2] **viXra:1611.0049 [pdf]**
*submitted on 2016-11-03 23:41:33*

**Authors:** Edigles Guedes

**Comments:** 15 Pages.

In this paper, I demonstrate one infinite product for binomial coefficient, Euler's
and Weierstrass's infinite product for Pochhammer's symbol, limit formula for Pochhammer's
symbol, limit formula for exponential function, Euler's and Weierstrass's infinite product for
Newton's binomial and exponential function, among other things.

**Category:** Functions and Analysis

[1] **viXra:1611.0002 [pdf]**
*replaced on 2017-01-15 23:03:42*

**Authors:** Kenneth C. Johnson

**Comments:** 22 Pages. [v8] Revised Appendix B

This paper generalizes an earlier investigation of linear differential equation solutions via Padé approximation (viXra:1509.0286), for the case of nonhomogeneous equations. Formulas are provided for Padé polynomial orders 1, 2, 3, and 4, for both constant-coefficient and functional-coefficient cases. The scale-and-square algorithm for the constant-coefficient case is generalized for nonhomogeneous equations. Implementation details including step size initialization and tolerance control are discussed.

**Category:** Functions and Analysis