13,896 research outputs found
The 2-adic valuations of differences of Stirling numbers of the second kind
Let and be positive integers. Let be the 2-adic
valuation of . By 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 and is odd, then except
when and , in which case . 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 and be positive integers and be a nonnegative integer.
Let and be the 2-adic valuation of and the sum of
binary digits of , respectively. Let be the Stirling number of the
second kind. It is shown that where
and . Furthermore, one gets that
, where ,
and . Finally, it is proved that if and is not
a power of 2 minus 1, then
where , if is a power of
2, and otherwise. This confirms a conjecture of Lengyel raised in
2009 except when is a power of 2 minus 1.Comment: 23 pages. To appear in Journal of Number Theor
The universal Kummer congruences
Let be a prime. In this paper, we present a detailed -adic analysis to
factorials and double factorials and their congruences. We give good bounds for
the -adic sizes of the coefficients of the divided universal Bernoulli
number when is divisible by . Using these we
then establish the universal Kummer congruences modulo powers of a prime
for the divided universal Bernoulli numbers when is
divisible by .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
of the second kind. We show that if
and only if . This confirms a conjecture of
Amdeberhan, Manna and Moll raised in 2008. We show also that for any positive integer , where is the sum of
binary digits of . 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 and WSe
multilayers possess continuously tunable electronic structure having direct
band gap. The tunability is controlled by the thickness ratio of MoS
versus WSe 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 and WSe
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 can be extended over
time . 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 can be extended over time 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 can be extended over time 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
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