A simple proof of the Gauss-Bonnet-Chern formula for Finsler manifolds

From the point of view of index theory, we give a simple proof of a Gauss-Bonnet-Chern formula for all Finsler manifolds by the Cartan connection. Based on this, we establish a Gauss-Bonnet-Chern formula for any metric-compatible connection and also derive the Gauss-Bonnet-Chern formula of Lackey

A Gauss-Bonnet-Chern theorem for Finsler vector bundles

In this paper, we give a simple proof of the Gauss-Bonnet-Chern theorem for a real oriented Finsler vector bundle with rank equal to the dimension of the base manifold. As an application, a Gauss-Bonnet-Chern formula for any metric-compatible connection is established on Finsler manifolds

A comparison theorem for Finsler submanifolds and its applications

In this paper, we consider the conormal bundle over a submanifold in a Finsler manifold and establish a volume comparison theorem. As an application, we derive a lower estimate for length of closed geodesics in a Finsler manifold. In the reversible case, a lower bound of injective radius is also obtained

Integral curvature bounds and diameter estimates on Finsler manifolds

In this paper, we study the integral curvatures of Finsler manifolds and prove several Myers type theorems

A Gauss-Bonnet-Chern theorem for complex Finsler manifolds

In this paper, we establish a Gauss-Bonnet-Chern theorem for general closed complex Finsler manifolds

The 2-adic valuations of differences of Stirling numbers of the second kind

Let $m, n, k$ and $c$ be positive integers. Let $\nu_2(k)$ be the 2-adic valuation of $k$. By $S(n,k)$ we denote the Stirling numbers of the second kind. In this paper, we first establish a convolution identity of the Stirling numbers of the second kind and provide a detailed 2-adic analysis to the Stirling numbers of the second kind. Consequently, we show that if $2\le m\le n$ and $c$ is odd, then $\nu_2(S(c2^{n+1},2^m-1)-S(c2^n, 2^m-1))=n+1$ except when $n=m=2$ and $c=1$, in which case $\nu_2(S(8,3)-S(4,3))=6$. This solves a conjecture of Lengyel proposed in 2009.Comment: 20 page

Divisibility by 2 of Stirling numbers of the second kind and their differences

Let $n,k,a$ and $c$ be positive integers and $b$ be a nonnegative integer. Let $\nu_2(k)$ and $s_2(k)$ be the 2-adic valuation of $k$ and the sum of binary digits of $k$, respectively. Let $S(n,k)$ be the Stirling number of the second kind. It is shown that $\nu_2(S(c2^n,b2^{n+1}+a))\geq s_2(a)-1,$ where $0<a<2^{n+1}$ and $2\nmid c$. Furthermore, one gets that $\nu_2(S(c2^{n},(c-1)2^{n}+a))=s_2(a)-1$, where $n\geq 2$, $1\leq a\leq 2^n$ and $2\nmid c$. Finally, it is proved that if $3\leq k\leq 2^n$ and $k$ is not a power of 2 minus 1, then $\nu_2(S(a2^{n},k)-S(b2^{n},k))=n+\nu_2(a-b)-\lceil\log_2k\rceil +s_2(k)+\delta(k),$ where $\delta(4)=2$, $\delta(k)=1$ if $k>4$ is a power of 2, and $\delta(k)=0$ otherwise. This confirms a conjecture of Lengyel raised in 2009 except when $k$ is a power of 2 minus 1.Comment: 23 pages. To appear in Journal of Number Theor

Revised Iterative Solution of Ground State of Double-Well Potential

A revised new iterative method based on Green function defined by quadratures along a single trajectory is developed and applied to solve the ground state of the double-well potential. The result is compared to the one based on the original iterative method. The limitation of the asymptotic expansion is also discussed.Comment: 19 page

Revised Iterative Solution for Groundstate of Schroedinger Equation

A revised iterative method based on Green function defined by quadratures along a single trajectory is proposed to solve the low-lying quantum wave function for Schroedinger equation. Specially a new expression of the perturbed energy is obtained, which is much simpler than the traditional one. The method is applied to solve the unharmonic oscillator potential. The revised iteration procedure gives exactly the same result as those based on the single trajectory quadrature method. A comparison of the revised iteration method to the old one is made using the example of Stark effect. The obtained results are consistent to each other after making power expansion

The universal Kummer congruences

Let $p$ be a prime. In this paper, we present a detailed $p$-adic analysis to factorials and double factorials and their congruences. We give good bounds for the $p$-adic sizes of the coefficients of the divided universal Bernoulli number ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$. Using these we then establish the universal Kummer congruences modulo powers of a prime $p$ for the divided universal Bernoulli numbers ${{\hat B_n}\over n}$ when $n$ is divisible by $p-1$.Comment: 20 pages. To appear in Journal of the Australian Mathematical Societ