Elliptic curve cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on elliptic curve is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation...

We give a combinatorial proof of an identity originally proved by G. E. Andrews in [1]. The identity simplifies a mock theta function first discovered by Rogers. 1 Background and Definitions First we review the basics of partition theory. Definition 1.1. A partition λ of a positive integer n is a finite nonincreasing sequence of positive integers λ1, λ2, . . . , λr such that ∑ r i=1 λi = n. The...

The physiological-structured population models assume that a fixed fraction of energy intake is utilized for individual growth and maintenance while the remaining for adult fertility. The assumption results in two concerns: energy loss for juveniles and a reproduction dilemma for adults. The dilemma results from the possibility that adults have to breed even if metabolic costs fail to be covere...

We construct a matrix model equivalent (exactly, not asymptotically), to the random plane partition model, with almost arbitrary boundary conditions. Equivalently, it is also a random matrix model for a TASEP with arbitrary boundary conditions. Using the known solution of matrix models, this method allows to find the large size asymptotic expansion of plane partitions, to ALL orders. It also al...

A subspace partition Π of V = V (n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. The size of Π is the number of its subspaces. Let σ q (n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρ q (n, t) denote the maximum size of a subspace partition of V in which the sm...

Carbon Spheres were fabricated by hydrothermal method and their structural properties were investigated. Carbon Spheres with average sizes around of 230, 320 and 430 nm were synthesized in different concentration of glucose aqueous solution and different hydrothermal reaction time. The temperature of 180 °C by a hydrothermal reaction was fixed in all of glucose concentration and hydrothermal re...