In 2007, Andrews introduced Durfee symbols and k-marked Durfee symbols so as to give a combinatorial interpretation for the symmetrized moment function $\eta _{2k}(n)$ of ranks of partitions. He also considered the relations between odd Durfee symbols and the mock theta function $\omega (q)$, and proved that the $2k$th moment function $\eta _{2k}^0(n)$ of odd ranks of odd Durfee symbols counts $(k+1)$-marked odd Durfee symbols of n. In this paper, we first introduce the definition of symmetrized positive odd rank moments $\eta _k^{0+}(n)$ and prove that for all $1\leq i\leq k+1$, $\eta _{2k-1}^{0+}(n)$ is equal to the number of $(k+1)$-marked odd Durfee symbols of n with the ith odd rank equal to zero and $\eta _{2k}^{0+}(n)$ is equal to the number of $(k+1)$-marked Durfee symbols of n with the ith odd rank being positive. Then we calculate the generating functions of $\eta _{k}^{0+}(n)$ and study its asymptotic behavior. Finally, we use Wright’s variant of the Hardy–Ramanujan circle method to obtain an asymptotic formula for $\eta _{k}^{0+}(n)$.

Let $\mathcal {O}(\pi )$ denote the number of odd parts in an integer partition $\pi$. In 2005, Stanley introduced a new statistic $\operatorname {srank}(\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$, where $\pi '$ is the conjugate of $\pi$. Let $p(r,\,m;n)$ denote the number of partitions of $n$ with srank congruent to $r$ modulo $m$. Generating function identities, congruences and inequalities for $p(0,\,4;n)$ and $p(2,\,4;n)$ were then established by a number of mathematicians, including Stanley, Andrews, Swisher, Berkovich and Garvan. Motivated by these works, we deduce some generating functions and inequalities for $p(r,\,m;n)$ with $m=16$ and $24$. These results are refinements of some inequalities due to Swisher.

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

We show that there are biases in the number of appearances of the parts in two residue classes in the set of ordinary partitions. More precisely, let $p_{j,k,m} (n)$ be the number of partitions of n such that there are more parts congruent to j modulo m than parts congruent to k modulo m for $m \geq 2$ . We prove that $p_{1,0,m} (n)$ is in general larger than $p_{0,1,m} (n)$ . We also obtain asymptotic formulas for $p_{1,0,m}(n)$ and $p_{0,1,m}(n)$ for $m \geq 2$ .

The aim of this paper is to find a concrete bound for the error involved when approximating the nth Hermite function (in the oscillating range) by an asymptotic formula due to D. Dominici. This bound is then used to study the accuracy of certain approximations to Hermite expansions and to Fourier transforms. A way of estimating an unknown probability density is proposed.

The Hardy operator $T_a$ on a tree $\Gamma$ is defined by \[ (T_af)(x):=v(x) \int^x_a f(t)u(t)\,dt \quad \mbox{for } a, x\in \Gamma. \] Properties of $T_a$ as a map from $L^p(\Gamma)$ into itself are established for $1\le p \le \infty$. The main result is that, with appropriate assumptions on $u$ and $v$, the approximation numbers $a_n(T_a)$ of $T_a$ satisfy \begin{equation*} \tag{$*$} \lim_{n\rightarrow \infty} na_n(T_a) = \alpha_p\int_{\Gamma}|uv|\,dt \end{equation*} for a specified constant $\alpha_p$ and $1 p<\infty$. This extends results of Naimark, Newman and Solomyak for $p=2$. Hitherto, for $p\neq 2$, $(*)$ was unknown even when $\Gamma$ is an interval. Also, upper and lower estimates for the $l^q$ and weak-$l^q$ norms of $\{a_n(T_a)\}$ are determined.

2000 Mathematical Subject Classification: 47G10, 47B10.