In this article, we focus our attention on (q,h)-Gauss?s binomial formula from which discover the additive property of (q; h)-exponential functions. We state (q,h)-analogue Gauss?s in terms proper polynomials T(q,h) own essential properties similar to ordinary polynomials. present (q,h)-Taylor series and analyze conditions for its convergence. introduce a new (q,h)-analytic exponential function...

Is it possible that the lengths of two sides of a primitive Heron triangle have a c ommon factor? The triangle with sides 9, 65, 70 and area 252 shows that this is possible. But notice that the common factor 5 is a prime of the form 4+1. Surprisingly, however, such a common factor cannot be a prime of the form 4 ; 1. Heron's name is familiar to anyone who has used the formula = p s(s ; a)(s ; b...

An efficient and secure group communication in Mobile Adhoc Network (MANETs) has brought attention to many security issues. In this paper, we propose Energy Efficient Dynamic Core based Multicast Routing Protocol to establish the path between the nodes that participate in group communication. During the routing process each node computes a Pythagorean Triple and constructs a Pythagorean Triple ...

This paper presents a method for determining all of the co-prime right angle triangles in the Euclidean field by looking at the intersection of the Pythagorean and Platonic right angle triangles and the corresponding lattice that this produces. The co-prime properties of each lattice point representing a unique right angle triangle are then considered. This paper proposes a conjunction between ...

Application of the local bell-modulated cosine bases of Coifman and Meyer to the problem of scattering by PEC cylinders leads to a sparse Method of Moments interaction matrix. To optimize the sparsity representation it is imperative to find bell functions with nice Fourier radiation properties. Here we pose the problem of bell optimization in terms of Chebyshev’s inequality, resulting in a fami...

The dually flat structure introduced by Amari–Nagaoka is highlighted in information geometry and related fields. In practical applications, however, the underlying pseudo-Riemannian metric may often be degenerate, such an excellent geometric rarely defined on entire space. To fix this trouble, present paper, we propose a novel generalization of for certain class singular models from viewpoint L...

This paper studies forward and reverse projections for the Rényi divergence of order α ∈ (0,∞) on α-convex sets. The forward projection on such a set is motivated by some works of Tsallis et al. in statistical physics, and the reverse projection is motivated by robust statistics. In a recent work, van Erven and Harremoës proved a Pythagorean inequality for Rényi divergences on α-convex sets und...

Let \((a, b, c)\) be a primitive Pythagorean triple parameterized as \(a=u^2-v^2, b=2uv, c=u^2+v^2\), where \(u>v>0\) are co-prime and not of the same parity. In 1956, L. Jesmanowicz conjectured that for any positive integer \(n\), Diophantine equation \((an)^x+(bn)^y=(cn)^z\) has only solution \((x,y,z)=(2,2,2)\). this connection we call \((x,y,z)\ne (2,2,2)\) with \(n>1\) exceptional...