In this study we use some known convolution sums to find the representation number for each of the three octonary quadratic forms [Formula: see text], [Formula: see text] and [Formula: see text].
In this paper, the convolution sums [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] are evaluated for all [Formula: see text] and their evaluations are used to determine the number of representation of a positive integer [Formula: see text] by the forms [Formula: see text] and [Formula: see text]
In this study, one-dimensional transient wave propagation in multilayered functionally graded media is investigated. The multilayered medium consists of N different layers of functionally graded materials (FGMs), i.e., it is assumed that the stiffness and the density of each layer are varying continuously in the direction perpendicular to the layering, but isotropic and homogeneous in the other two directions. The top surface of the layered medium is subjected to a uniform dynamic in-plane time-dependent normal stress; whereas, the lower surface of the layered medium is assumed free of surface tractions or fixed. Moreover, the multilayered medium is assumed to be initially at rest and its layers are assumed to be perfectly bonded to each other. The method of characteristics is employed to obtain the solutions of this initial-boundary-value problem. The numerical results are obtained and displayed in curves showing the variation of the normal stress component with time. These curves reveal clearly the scattering effects caused by the reflections and refractions of waves at the boundaries and at the interfaces of the layers. The curves also display the effects of non-homogeneity in the wave profiles. The curves further show that the numerical technique applied in this study is capable of predicting the sharp variations in the field variables in the neighborhood of the wave fronts. By suitably adjusting the material constants, solutions for the case of isotropic, homogeneous and linearly elastic multilayered media and for some special cases including two different functionally graded layers are also obtained. Furthermore, solutions for some special cases are compared with the existing solutions in the literature; very good agreement is found.
Let N(a1, …, a4; n) denote the number of representations of an integer n by the form [Formula: see text]. In this paper we derive formulae for N(1, 1, 1, 2; n) and N(1, 2, 2, 2; n). These formulae are given in terms of σ3(n).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.