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

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

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

In this paper, we investigate the 2-adic valuations of the Stirling numbers $S(n, k)$ of the second kind. We show that $v_2(S(4i, 5))=v_2(S(4i+3, 5))$ if and only if $i\not\equiv 7\pmod {32}$. This confirms a conjecture of Amdeberhan, Manna and Moll raised in 2008. We show also that $v_2(S(2^n+1, k+1))= s_2(n)-1$ for any positive integer $n$, where $s_2(n)$ is the sum of binary digits of $n$. It proves another conjecture of Amdeberhan, Manna and Moll.Comment: 9 pages. To appear in International Journal of Number Theor

The manipulated left-handedness in a rare-earth-ion-doped optical fiber by the incoherent pumping field

The left-handedness was demonstrated by the simulation with a three-level quantum system in an E3+r -dopped ZrF4-BaF2-LaF3-AlF3-NaF (ZBLAFN) optical fiber. And the left-handedness can be regulated by the incoherent pumping field. Our scheme may provide a solid candidate other than the coherent atomic vapour for left-handedness, and may extend the application of the rare-earth-ion-doped optical fiber in metamaterials and of the incoherent pumping light field in quantum optics.Comment: 5 pages, 7 figure

Continuously tunable electronic structure of transition metal dichalcogenides superlattices

Two dimensional transition metal dichalcogenides (TMDC) have very interesting properties for optoelectronic devices. In this work we theoretically investigate and predict that superlattices comprised of MoS$_{2}$ and WSe$_{2}$ multilayers possess continuously tunable electronic structure having direct band gap. The tunability is controlled by the thickness ratio of MoS$_{2}$ versus WSe$_{2}$ of the superlattice. When this ratio goes from 1:2 to 5:1, the dominant K-K direct band gap is continuously tuned from 0.14 eV to 0.5 eV. The gap stays direct against -0.6% to 2% in-layer strain and up to -4.3% normal-layer compressive strain. The valance and conduction bands are spatially separated. These robust properties suggest that MoS$_{2}$ and WSe$_{2}$ multilayer superlattice should be an exciting emerging material for infrared optoelectronics.Comment: 5 pages, 4 figures and 1 tabl

The Extension for Mean Curvature Flow with Finite Integral Curvature in Riemannian Manifolds

We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature on a finite time interval $[0,T)$ can be extended over time $T$. Moreover, we show that the condition is optimal in some sense.Comment: 13 page

Extend Mean Curvature Flow with Finite Integral Curvature

In this note, we first prove that the solution of mean curvature flow on a finite time interval $[0,T)$ can be extended over time $T$ if the space-time integration of the norm of the second fundamental form is finite. Secondly, we prove that the solution of certain mean curvature flow on a finite time interval $[0,T)$ can be extended over time $T$ if the space-time integration of the mean curvature is finite. Moreover, we show that these conditions are optimal in some sense.Comment: 13 page

Implications of equalities among the elements of CKM and PMNS matrices

Investigating the CKM matrix in different parametrization schemes, it is noticed that those schemes can be divided into a few groups where the sine values of the CP phase for each group are approximately equal. Using those relations, several approximate equalities among the elements of CKM matrix are established. Assuming them to be exact, there are infinite numbers of solutions and by choosing special values for the free parameters in those solutions, several textures presented in literature are obtained. The case can also be generalized to the PMNS matrix for the lepton sector. In parallel, several mixing textures are also derived by using presumed symmetries, amazingly, some of their forms are the same as what we obtained, but not all. It hints existence of a hidden symmetry which is broken in the practical world. The nature makes its own selection on the underlying symmetry and the way to break it, while we just guess what it is.Comment: 11 pages, Submitted to 'Chinese Physics C

Secrecy Rate Maximization for Intelligent Reflecting Surface Assisted Multi-Antenna Communications

We investigate transmission optimization for intelligent reflecting surface (IRS) assisted multi-antenna systems from the physical-layer security perspective. The design goal is to maximize the system secrecy rate subject to the source transmit power constraint and the unit modulus constraints imposed on phase shifts at the IRS. To solve this complicated non-convex problem, we develop an efficient alternating algorithm where the solutions to the transmit covariance of the source and the phase shift matrix of the IRS are achieved in closed form and semi-closed forms, respectively. The convergence of the proposed algorithm is guaranteed theoretically. Simulations results validate the performance advantage of the proposed optimized design