87,122 research outputs found
Binomial coefficients, Catalan numbers and Lucas quotients
Let be an odd prime and let be integers with and . In this paper we determine
mod for ; for example,
where is the Jacobi symbol, and is the Lucas
sequence given by , and for
. As an application, we determine modulo for any integer , where denotes the
Catalan number . We also pose some related conjectures.Comment: 24 pages. Correct few typo
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Precision Extrusion Deposition of Polycaprolactone/Hydroxyapatite Tissue Scaffolds
Freeform fabrication provides an effective process tool to manufacture advanced tissue scaffolds
with specific designed properties. Our research focuses on using a novel Precision Extrusion
Deposition (PED) process technique to directly fabricate Polycaprolactone (PCL) and composite
PCL/ Hydroxyapatite (HA) tissue scaffolds. The scaffold morphology and the mechanical
properties were evaluated using SEM and mechanical testing. In vitro biological studies were
conducted to investigate the cellular responses of the composite scaffolds. Results and
characterizations demonstrate the viability of the PED process as well as the good mechanical
property, structural integrity, controlled pore size, pore interconnectivity, and the biological
compatibility of the fabricated scaffolds.Mechanical Engineerin
Variational study of the Holstein polaron
The paper deals with the ground and the first excited state of the polaron in
the one dimensional Holstein model. Various variational methods are used to
investigate both the weak coupling and strong coupling case, as well as the
crossover regime between them. Two of the methods, which are presented here for
the first time, introduce interesting elements to the understanding of the
nature of the polaron. Reliable numerical evidence is found that, in the strong
coupling regime, the ground and the first excited state of the self-trapped
polaron are well described within the adiabatic limit. The lattice vibration
modes associated with the self-trapped polarons are analyzed in detail, and the
frequency softening of the vibration mode at the central site of the small
polaron is estimated. It is shown that the first excited state of the system in
the strong coupling regime corresponds to the excitation of the soft phonon
mode within the polaron. In the crossover regime, the ground and the first
excited state of the system can be approximated by the anticrossing of the
self-trapped and the delocalized polaron state. In this way, the connection
between the behavior of the ground and the first excited state is qualitatively
explained.Comment: 11 pages, 4 figures, PRB 65, 14430
Strategy for designing broadband epsilon-near-zero metamaterial with loss compensation by gain media
A strategy is proposed to design the broadband gain-doped epsilon-near-zero
(GENZ) metamaterial. Based on the Milton representation of effective
permittivity, the strategy starts in a dimensionless spectral space, where the
effective permittivity of GENZ metamaterial is simply determined by a pole-zero
structure corresponding to the operating frequency range. The physical
structure of GENZ metamaterial is retrieved from the pole-zero structure via a
tractable inverse problem. The strategy is of great advantage in practical
applications and also theoretically reveals the cancellation mechanism
dominating the broadband near-zero permittivity phenomenon in the spectral
space
Super congruences and Euler numbers
Let be a prime. We prove that
, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of
combinatorial nature. We also formulate many conjectures concerning super
congruences and relate most of them to Euler numbers or Bernoulli numbers.
Motivated by our investigation of super congruences, we also raise a conjecture
on 7 new series for , and the constant
(with (-) the Jacobi symbol), two of which are
and
\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$
Hidden Markov model tracking of continuous gravitational waves from a neutron star with wandering spin
Gravitational wave searches for continuous-wave signals from neutron stars
are especially challenging when the star's spin frequency is unknown a priori
from electromagnetic observations and wanders stochastically under the action
of internal (e.g. superfluid or magnetospheric) or external (e.g. accretion)
torques. It is shown that frequency tracking by hidden Markov model (HMM)
methods can be combined with existing maximum likelihood coherent matched
filters like the F-statistic to surmount some of the challenges raised by spin
wandering. Specifically it is found that, for an isolated, biaxial rotor whose
spin frequency walks randomly, HMM tracking of the F-statistic output from
coherent segments with duration T_drift = 10d over a total observation time of
T_obs = 1yr can detect signals with wave strains h0 > 2e-26 at a noise level
characteristic of the Advanced Laser Interferometer Gravitational Wave
Observatory (Advanced LIGO). For a biaxial rotor with randomly walking spin in
a binary orbit, whose orbital period and semi-major axis are known
approximately from electromagnetic observations, HMM tracking of the
Bessel-weighted F-statistic output can detect signals with h0 > 8e-26. An
efficient, recursive, HMM solver based on the Viterbi algorithm is
demonstrated, which requires ~10^3 CPU-hours for a typical, broadband (0.5-kHz)
search for the low-mass X-ray binary Scorpius X-1, including generation of the
relevant F-statistic input. In a "realistic" observational scenario, Viterbi
tracking successfully detects 41 out of 50 synthetic signals without spin
wandering in Stage I of the Scorpius X-1 Mock Data Challenge convened by the
LIGO Scientific Collaboration down to a wave strain of h0 = 1.1e-25, recovering
the frequency with a root-mean-square accuracy of <= 4.3e-3 Hz
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