Abstract:
Based on the theories of Ramanujan's elliptic functions and the (p, k)-parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins' cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.

Abstract:
We show that the $q$-derangement numbers satisfy a ratio monotone property, which is analogous to the log-concavity and is stronger than the spiral property and the unimodality.

Abstract:
We present an approach to proving the 2-log-convexity of sequences satisfying three-term recurrence relations. We show that the Apery numbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4 and the large Schroder numbers are all 2-log-convex. Numerical evidence suggests that all these sequences are k-log-convex for any $k\geq 1$ possibly except for a constant number of terms at the beginning.

Abstract:
The Boros-Moll polynomials $P_m(a)$ arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that $P_m(a)$ is 2-log-concave for any $m\geq 2$. Let $d_i(m)$ be the coefficient of $a^i$ in $P_m(a)$. We also show that the sequence $\{i (i+1)(d_i^{\,2}(m)-d_{i-1}(m)d_{i+1}(m))\}_{1\leq i \leq m}$ is log-concave. This leads another proof of Moll's minimum conjecture.

Abstract:
Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove nonterminating basic hypergeometric series identities. This observation enables us to extend the q-WZ method to identities on infinite series. As examples, we will give the q-WZ pairs for some classical identities such as the q-Gauss sum, the $_6\phi_5$ sum, Ramanujan's $_1\psi_1$ sum and Bailey's $_6\psi_6$ sum.

Abstract:
In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this paper, we show that the Boros-Moll polynomials possess the ratio monotone property which implies the log-concavity and the spiral property. We conclude with a conjecture which is stronger than Moll's conjecture on the $\infty$-log-concavity.

Abstract:
Let $d_i(m)$ denote the coefficients of the Boros-Moll polynomials. Moll's minimum conjecture states that the sequence $\{i(i+1)(d_i^2(m)-d_{i-1}(m)d_{i+1}(m))\}_{1\leq i \leq m}$ attains its minimum with $i=m$. This conjecture is a stronger than the log-concavity conjecture proved by Kausers and Paule. We give a proof of Moll's conjecture by utilizing the spiral property of the sequence $\{d_i(m)\}_{0\leq i \leq m}$, and the log-concavity of the sequence $\{i!d_i(m)\}_{0\leq i \leq m}$.

Abstract:
Let $\bar{p}(n)$ denote the number of overpartitions of $n$. It was conjectured by Hirschhorn and Sellers that $\bar{p}(40n+35)\equiv 0\ ({\rm mod\} 40)$ for $n\geq 0$. Employing 2-dissection formulas of quotients of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for $\bar{p}(40n+35)$ modulo 5. Using the $(p, k)$-parametrization of theta functions given by Alaca, Alaca and Williams, we give a proof of the congruence $\bar{p}(40n+35)\equiv 0\ ({\rm mod\} 5)$. Combining this congruence and the congruence $\bar{p}(4n+3)\equiv 0\ ({\rm mod\} 8)$ obtained by Hirschhorn and Sellers, and Fortin, Jacob and Mathieu, we give a proof of the conjecture of Hirschhorn and Sellers.

Abstract:
We introduce the notion of interlacing log-concavity of a polynomial sequence $\{P_m(x)\}_{m\geq 0}$, where $P_m(x)$ is a polynomial of degree m with positive coefficients $a_{i}(m)$. This sequence of polynomials is said to be interlacing log-concave if the ratios of consecutive coefficients of $P_m(x)$ interlace the ratios of consecutive coefficients of $P_{m+1}(x)$ for any $m\geq 0$. Interlacing log-concavity is stronger than the log-concavity. We show that the Boros-Moll polynomials are interlacing log-concave. Furthermore we give a sufficient condition for interlacing log-concavity which implies that some classical combinatorial polynomials are interlacing log-concave.

Abstract:
A mutant Aspergillus sp. CJ 22-326-14 with higher production of chitosanase was selected by serial mutation procedure, and the optimization of medium components was performed using one-factor-at-a-time combined with orthogonal array design method. The results showed that chitosan, wheat bran and (NH4)2SO4 were the most important components by one-factor-at-a-time experiments. And the optimal concentrations were obtained by orthogonal array design method as (g/L): chitosan 20, wheat bran 10, (NH4) 2SO4 2, KH2PO4 2 and MgSO4·7H2O 0.5. After the strain improvement and optimization of the media, the chitosanase activity from Aspergillus sp. CJ 22-326-14 reached to 3.92 U/mL in 250 mL flasks, which was nearly 4-fold increase of chitosanase production at the parent lowest level.