17,120 research outputs found
Cyclotomic Constructions of Skew Hadamard Difference Sets
We revisit the old idea of constructing difference sets from cyclotomic
classes. Two constructions of skew Hadamard difference sets are given in the
additive groups of finite fields using unions of cyclotomic classes of order
, where is a prime and a positive integer. Our main tools
are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A
Strongly Regular Graphs From Unions of Cyclotomic Classes
We give two constructions of strongly regular Cayley graphs on finite fields
\F_q by using union of cyclotomic classes and index 2 Gauss sums. In
particular, we obtain twelve infinite families of strongly regular graphs with
new parameters.Comment: 17 pages; to appear in J. Combin. Theory (B
Semi-regular Relative Difference Sets with Large Forbidden Subgroups
Motivated by a connection between semi-regular relative difference sets and
mutually unbiased bases, we study relative difference sets with parameters
in groups of non-prime-power orders. Let be an odd prime. We
prove that there does not exist a relative difference set in any
group of order , and an abelian relative difference set can
only exist in the group . On the other hand, we
construct a family of non-abelian relative difference sets with parameters
, where is an odd prime power greater than 9 and
(mod 4). When is a prime, , and 1 (mod 4), the
non-abelian relative difference sets constructed here are
genuinely non-abelian in the sense that there does not exist an abelian
relative difference set with the same parameters
Asymptotic Laplacian-Energy-Like Invariant of Lattices
Let denote the Laplacian eigenvalues of
with vertices. The Laplacian-energy-like invariant, denoted by , is a novel topological index. In this paper, we
show that the Laplacian-energy-like per vertex of various lattices is
independent of the toroidal, cylindrical, and free boundary conditions.
Simultaneously, the explicit asymptotic values of the Laplacian-energy-like in
these lattices are obtained. Moreover, our approach implies that in general the
Laplacian-energy-like per vertex of other lattices is independent of the
boundary conditions.Comment: 6 pages, 2 figure
A New Code for Nonlinear Force-Free Field Extrapolation of the Global Corona
Reliable measurements of the solar magnetic field are still restricted to the
photosphere, and our present knowledge of the three-dimensional coronal
magnetic field is largely based on extrapolation from photospheric magnetogram
using physical models, e.g., the nonlinear force-free field (NLFFF) model as
usually adopted. Most of the currently available NLFFF codes have been
developed with computational volume like Cartesian box or spherical wedge while
a global full-sphere extrapolation is still under developing. A
high-performance global extrapolation code is in particular urgently needed
considering that Solar Dynamics Observatory (SDO) can provide full-disk
magnetogram with resolution up to . In this work, we present a
new parallelized code for global NLFFF extrapolation with the photosphere
magnetogram as input. The method is based on magnetohydrodynamics relaxation
approach, the CESE-MHD numerical scheme and a Yin-Yang spherical grid that is
used to overcome the polar problems of the standard spherical grid. The code is
validated by two full-sphere force-free solutions from Low & Lou's
semi-analytic force-free field model. The code shows high accuracy and fast
convergence, and can be ready for future practical application if combined with
an adaptive mesh refinement technique.Comment: Accepted by ApJ, 26 pages, 10 figure
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