58,490 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
Projection Measurement of the Maximally Entangled N-Photon State for a Demonstration of N-Photon de Broglie Wavelength
We construct a projection measurement process for the maximally entangled
N-photon state (the NOON-state) with only linear optical elements and
photodetectors. This measurement process will give null result for any N-photon
state that is orthogonal to the NOON state. We examine the projection process
in more detail for N=4 by applying it to a four-photon state from type-II
parametric down-conversion. This demonstrates an orthogonal projection
measurement with a null result. This null result corresponds to a dip in a
generalized Hong-Ou-Mandel interferometer for four photons. We find that the
depth of the dip in this arrangement can be used to distinguish a genuine
entangled four-photon state from two separate pairs of photons. We next apply
the NOON state projection measurement to a four-photon superposition state from
two perpendicularly oriented type-I parametric down-conversion processes. A
successful NOON state projection is demonstrated with the appearance of the
four-photon de Broglie wavelength in the interference fringe pattern.Comment: 8 pages, 3 figures, new title, some content change, replaced Fig.
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.$
Singularities of solution surfaces for quasilinear 1st order partial differential equations
We study singularities of solution surfaces of characteristic Cauchy problem for quasilinear first order partial differential equations as an application of the previous result on vector fields near a generic submanifold
Raman spectroscopic determination of the length, strength, compressibility, Debye temperature, elasticity, and force constant of the C-C bond in graphene
From the perspective of bond relaxation and vibration, we have reconciled the
Raman shifts of graphene under the stimuli of the number-of-layer,
uni-axial-strain, pressure, and temperature in terms of the response of the
length and strength of the representative bond of the entire specimen to the
applied stimuli. Theoretical unification of the measurements clarifies that:
(i) the opposite trends of Raman shifts due to number-of-layer reduction
indicate that the G-peak shift is dominated by the vibration of a pair of atoms
while the D- and the 2D-peak shifts involves z-neighbor of a specific atom;
(ii) the tensile strain-induced phonon softening and phonon-band splitting
arise from the asymmetric response of the C3v bond geometry to the C2v
uni-axial bond elongation; (iii) the thermal-softening of the phonons
originates from bond expansion and weakening; and (iv) the pressure- stiffening
of the phonons results from bond compression and work hardening. Reproduction
of the measurements has led to quantitative information about the referential
frequencies from which the Raman frequencies shift, the length, energy, force
constant, Debye temperature, compressibility, elastic modulus of the C-C bond
in graphene, which is of instrumental importance to the understanding of the
unusual behavior of graphene
An Analysis Framework for Inter-User Interference in IEEE 802.15.6 Body Sensor Networks: A Stochastic Geometry Approach
Inter-user interference occurs when multiple body sensor networks (BSNs) are transmitting simultaneously in close proximity to each other. Interference analysis in BSNs is challenging due to the hybrid medium access control (MAC) and the specific channel characteristics of BSNs. This paper presents a stochastic geometry analysis framework for inter-user interference in IEEE 802.15.6 BSNs. An extended Matern point process is proposed to model the complex spatial distribution of the interfering BSNs caused by the hybrid MAC defined in IEEE 802.15.6. We employ stochastic geometry approach to evaluate the performance of BSNs, considering the specific channel characteristics of BSNs in the vicinity of human body. Performance metrics are derived in terms of outage probability and spatial throughput in the presence of inter-user interference. We conduct performance evaluation through extensive simulations and show that the simulation results fit well with the analytic results. Insights are provided on the determination of the interference detection range, the BSN density, and the design of MAC for BSNs
Performance Evaluation of Wearable Sensor Systems: A Case Study in Moderate-Scale Deployment in Hospital Environment
A wearable sensor system enables continuous and remote health monitoring and is widely considered as the next generation of healthcare technology. The performance, the packet error rate (PER) in particular, of a wearable sensor system may deteriorate due to a number of factors, particularly the interference from the other wearable sensor systems in the vicinity. We systematically evaluate the performance of the wearable sensor system in terms of PER in the presence of such interference in this paper. The factors that affect the performance of the wearable sensor system, such as density, traffic load, and transmission power in a realistic moderate-scale deployment case in hospital are all considered. Simulation results show that with 20% duty cycle, only 68.5% of data transmission can achieve the targeted reliability requirement (PER is less than 0.05) even in the off-peak period in hospital. We then suggest some interference mitigation schemes based on the performance evaluation results in the case study
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