Summation of some infinite series by the methods of Hypergeometric functions and partial fractions

In this article we obtain the summations of some infinite series by partial fraction method and by using certain hypergeometric summation theorems of positive and negative unit arguments, Riemann Zeta functions, polygamma functions, lower case beta functions of one-variable and other associated functions. We also obtain some hypergeometric summation theorems for: 8F7[9/2, 3/2, 3/2, 3/2, 3/2, 3, 3, 1; 7/2, 7/2, 7/2, 7/2, 1/2, 2, 2; 1], 5F4[5/3, 4/3, 4/3, 1/3, 1/3; 2/3, 1, 2, 2; 1], 5F4[9/4, 5/2, 3/2, 1/2, 1/2; 5/4, 2, 3, 3; 1], 5F4[13/8, 5/4, 5/4, 1/4, 1/4; 5/8, 2, 2, 1; 1], 5F4[1/2, 1/2, 5/2, 5/2, 1; 3/2, 3/2, 7/2, 7/2; −1], 4F3[3/2, 3/2, 1, 1; 5/2, 5/2, 2; 1], 4F3[2/3, 1/3, 1, 1; 7/3, 5/3, 2; 1], 4F3[7/6, 5/6, 1, 1; 13/6, 11/6, 2; 1] and 4F3[1, 1, 1, 1; 3, 3, 3; −1].

Exploiting the security of N = prqs through approximation of ϕ(N)

This paper reports new techniques that exploit the security of the prime power moduli [Formula: see text] using continued fraction method. Our study shows that the key equation [Formula: see text] can be exploited using [Formula: see text] as good approximation of [Formula: see text]. This enables us to get [Formula: see text] from the convergents of the continued fractions expansion of [Formula: see text] where the bound of the private exponent is [Formula: see text] which leads to the polynomial time factorization of the moduli [Formula: see text]. We further report the polynomial time attacks that can break the security of the generalized prime power moduli [Formula: see text] using generalized system of equation of the form [Formula: see text] and [Formula: see text] by applying simultaneous Diophantine approximations and LLL algorithm techniques where [Formula: see text] and [Formula: see text].

Estimating stand attributes of complex forest types in subtropical mountain environments by combining the shadow fraction method with analyses of high-dynamic-range photographs

Shadow fractions can be overestimated because of topographic shadows, which can occupy a significant area on aerial photographs of mountainous terrain. In this study, we first used high-dynamic-range (HDR) image analysis techniques to extract the original canopy shadow from the topographic shadows on aerial photographs. Subsequently, we applied the shadow fraction method to estimate selected forest attributes (stand height, basal area, and stem volume). In this paper, we discuss the effects of tree shadow fraction normalization, auxiliary spectral information, and forest type on forest attribute estimation. HDR image analysis successfully extracted canopy shadow information from topographic shadows. The tree shadow fraction normalization method had no obvious effect. The shadow fraction enhanced spectral information to estimate stand attributes. Using shadow fractions resulted in better estimates of stand height for mixed-hardwood forest ([Formula: see text] = 0.45), basal area for mixed-hardwood forest ([Formula: see text] = 0.50), and stem volume for conifer–hardwood forest ([Formula: see text] = 0.43). This difference in estimated results is related to the shade patterns produced by stand structures in the different forest types.