Mathematics

This note presents an analytic technique for proving the linear independence of certain small subsets of real numbers over the rational numbers. The applications of this test produce simple linear independence proofs for the subsets of triples {1,e,π} , {1,e, π −1 } , and {1, π r , π s } , where 1≤r<s are fixed integers.

There is given a characterization of hyperbolic secant distribution by the independence of linear forms with random coefficients. We provide a characterization by the identic distribution property. Keywords: hyperbolic secant distribution; characterization of distributions; linear forms; random coefficients.

The Ulam spiral inspired us to calculate and present Lissajous curves where the orthogonally added functions are a finite sum of sinus and cosines functions with consecutive prime number frequencies.

Trisecting an angle has been proved to be impossible by Euclidean Geometry, using only straight edge and compass. However, there is a method using Origami (paper folding) procedure to trisect an angle. The algebraic analysis of the same gives us a method of finding trisection using a locus of a point of intersection of two circles. The algebraic analysis and the equation for the locus of the point of intersection of two circles leading to trisection without any measurements is described here. The proof of trisection is exactly same as that of the Origami procedure.

A class of log-trigonometric integrals are evaluated in terms of elliptic functions. From this, by using the elliptic integral singular values, one can obtain closed form evaluations of integrals such as ∫ 0 π/2 ln(cosh x 3 – √ +cos ln(2cosx) 3 – √ )dx= π 2 8 3 – √ − π 4 ln(1+ 3 – √ )+ 13π 24 ln2.

In this paper, we prove logarithmically complete monotonicity properties of certain ratios of the k -gamma function. As a consequence, we deduce some inequalities involving the k -gamma and k -trigamma functions.

Both the ellipse and the hyperbola are geometric places that can be defined by establishing a relationship between points P of the plane and two fixed points A and B (which are its foci F ′ =A and F=B ). Given two points A and B of the plan (which we no longer call the foci F and F ′ ), we are going to present three geometric places associated with A and B other than the ellipse and the hyperbola.

The Riemann Hypothesis is the main open problem of Number Theory and several scientists are trying to solve this problem. In this regard, in a recent work [8], a difference equation has been proposed that calculates the nth non-trivial zero in the critical range. In this work, we seek to optimize this estimation by calculating Lyapunov numbers for this non-linear map in order to seek the best value for the bifurcation parameter. Analytical results are presented.

We study the algebraic structure of the eigenvalues of a Hamiltonian that corresponds to a many-body fermionic system. As the Hamiltonian is quadratic in fermion creation and/or annihilation operators, the system is exactly integrable and the complete single fermion excitation energy spectrum is constructed using the non-interacting fermions that are eigenstates of the quadratic matrix related to the system Hamiltonian. Connection to the Riemann Hypothesis is discussed.

The article presents a matrix differential operator and a pseudoinverse matrix differential operator for finding a particular solution to nonhomogeneous linear ordinary differential equations (ODE) with constant coefficients with special types of the right-hand side. Calculation requires the determination of an inverse or pseudoinverse matrix. If the matrix is singular, the Moore-Penrose pseudoinverse matrix is used for the calculation, which is simply calculated as the inverse submatrix of the considered matrix. It is shown that block matrices are effectively used to calculate a particular solution.