A congruence involving products of qq-binomial coefficients

In this paper we establish a $q$-analogue of a congruence of Sun concerning the products of binomial coefficients modulo the square of a prime.Comment: 9 page

Modification of Einstein A Coefficient in Dissipative Gas Medium

Spontaneous radiation in dissipative gas medium such as plasmas is investigated by Langevin equations and the modified Weisskopf-Wigner approximation. Since the refractive index of gas medium is expected to be nearly unity, we shall first neglect the medium polarization effect. We show that absorption in plasmas may in certain case modify the Einstein A coefficient significantly and cause a pit in the A coefficient-density curves for relatively low temperature plasmas and also a pit in the A coefficient-temperature curves. In the next, the effect of medium polarization is taken into account in addition. To our surprise, its effect in certain case is quite significant. The dispersive curves show different behaviors in different region of parameters

Hybrid graded element model for nonlinear functionally graded materials

A hybrid graded element model is developed in this article for solving the heat conduction problem of nonlinear functionally graded materials (FGMs), whose material properties not only vary spatially but also are temperature dependent. In the proposed approach, both Kirchhoff transformation and iterative method are introduced to deal with the nonlinear term in the heat conduction equation of nonlinear FGMs. Then, the graded element is formulated based on two sets of independent temperature fields. One is the intra-element temperature field, which is defined within the element domain and constructed by a linear combination of fundamental solutions; the other is the frame field, which is defined on the element boundary only and used as the boundary interpolation functions of the element to ensure the field continuity over the inter-element boundary. This model can simulate the graded material properties naturally due to the inherent properties of fundamental solutions, which are employed in constructing the graded element. Moreover, a multi-subdomain method is developed to deal with the problem with different materials. Finally, the performance of the proposed method is assessed by several benchmark examples. The results are in excellent agreement with the analytical solutions

(E)-2-(Cyclo­hexyl­methyl­ene)succinic acid

The title compound, C11H16O4, crystallizes with three molecules in the asymmetric unit. The cyclo­hexane ring adopts a chair conformation. Inter­molecular O—H⋯O hydrogen bonds are observed and these help to establish the crystal packing