Theory of impedance networks: The two-point impedance and LC resonances

We present a formulation of the determination of the impedance between any two nodes in an impedance network. An impedance network is described by its Laplacian matrix L which has generally complex matrix elements. We show that by solving the equation L u_a = lambda_a u_a^* with orthonormal vectors u_a, the effective impedance between nodes p and q of the network is Z = Sum_a [u_{a,p} - u_{a,q}]^2/lambda_a where the summation is over all lambda_a not identically equal to zero and u_{a,p} is the p-th component of u_a. For networks consisting of inductances (L) and capacitances (C), the formulation leads to the occurrence of resonances at frequencies associated with the vanishing of lambda_a. This curious result suggests the possibility of practical applications to resonant circuits. Our formulation is illustrated by explicit examples.Comment: 21 pages, 3 figures; v4: typesetting corrected; v5: Eq. (63) correcte

Spanning Trees on Graphs and Lattices in d Dimensions

The problem of enumerating spanning trees on graphs and lattices is considered. We obtain bounds on the number of spanning trees $N_{ST}$ and establish inequalities relating the numbers of spanning trees of different graphs or lattices. A general formulation is presented for the enumeration of spanning trees on lattices in $d\geq 2$ dimensions, and is applied to the hypercubic, body-centered cubic, face-centered cubic, and specific planar lattices including the kagom\'e, diced, 4-8-8 (bathroom-tile), Union Jack, and 3-12-12 lattices. This leads to closed-form expressions for $N_{ST}$ for these lattices of finite sizes. We prove a theorem concerning the classes of graphs and lattices ${\cal L}$ with the property that $N_{ST} \sim \exp(nz_{\cal L})$ as the number of vertices $n \to \infty$, where $z_{\cal L}$ is a finite nonzero constant. This includes the bulk limit of lattices in any spatial dimension, and also sections of lattices whose lengths in some dimensions go to infinity while others are finite. We evaluate $z_{\cal L}$ exactly for the lattices we considered, and discuss the dependence of $z_{\cal L}$ on d and the lattice coordination number. We also establish a relation connecting $z_{\cal L}$ to the free energy of the critical Ising model for planar lattices ${\cal L}$.Comment: 28 pages, latex, 1 postscript figure, J. Phys. A, in pres

Spanning trees on the Sierpinski gasket

We obtain the numbers of spanning trees on the Sierpinski gasket $SG_d(n)$ with dimension $d$ equal to two, three and four. The general expression for the number of spanning trees on $SG_d(n)$ with arbitrary $d$ is conjectured. The numbers of spanning trees on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4$ are also obtained.Comment: 20 pages, 8 figures, 1 tabl

Pumping Speed Measurement and Analysis for the Turbo Booster Pump

This study applies testing apparatus and a computational approach to examine a newly designed spiral-grooved turbo booster pump (TBP), which has both volume type and momentum transfer type vacuum pump functions, and is capable of operating at optimum discharge under pressures from approximately 1000 Pa to a high vacuum. Transitional flow pumping speed is increased by a well-designed connecting element. Pumping performance is predicted and examined via two computational approaches, namely the computational fluid dynamics (CFD) method and the direct simulation Monte Carlo (DSMC) method. In CFD analysis, comparisons of measured and calculated inlet pressure in the slip and continuum flow demonstrate the accuracy of the calculation. Meanwhile, in transition flow, the continuum model of CFD is unsuitable for calculating such rarefied gas. The pumping characteristics for a full 3D model on a rotating frame in transition and molecular regimes thus are simulated using the DSMC method and then confirmed experimentally. However, when the Knudsen number is in the range 0.5 < Kn < 0.1, neither CFD computation nor DSMC simulation is suitable for analyzing the pumping speed of the turbo booster pump. In this situation, the experimental approach is the most appropriate and effective method for analyzing pumping speed. Moreover, the developed pump is tested using assessment systems constructed according to ISO and JVIS-005 standards, respectively. Comparisons are also made with other turbo pumps. The compared results show that the turbo booster pump presented here has good foreline performance

Influence of realistic parameters on state-of-the-art LWFA experiments

We examine the influence of non-ideal plasma-density and non-Gaussian transverse laser-intensity profiles in the laser wakefield accelerator analytically and numerically. We find that the characteristic amplitude and scale length of longitudinal density fluctuations impacts on the final energies achieved by electron bunches. Conditions that minimize the role of the longitudinal plasma density fluctuations are found. The influence of higher order Laguerre-Gaussian laser pulses is also investigated. We find that higher order laser modes typically lead to lower energy gains. Certain combinations of higher order modes may, however, lead to higher electron energy gains.Comment: 16 pages, 6 figures; Accepted for publication in Plasma Physics and Controlled Fusio

Two-frequency shell model for hypernuclei and meson-exchange hyperon-nucleon potentials

A two-frequency shell model is proposed for investigating the structure of hypernuclei starting with a hyperon-nucleon potential in free space. In a calculation using the folded-diagram method for Λ¹⁶O, the Λ single particle energy is found to have a saturation minimum at an oscillator frequency ħωΛ≈10MeV, for the Λ orbit, which is considerably smaller than ħωN=14MeV for the nucleon orbit. The spin-dependence parameters derived from the Nijmegen NSC89 and NSC97f potentials are similar, but both are rather different from those obtained with the Jülich-B potential. The ΛNN three-body interactions induced by ΛN-ΣN transitions are important for the spin parameters, but relatively unimportant for the low-lying states of Λ¹⁶O.Yiharn Tzeng, S. Y. Tsay Tzeng, T. T. S. Kuo, T.-S.H. Lee, and V. G. D. Stok

Dimer coverings on the Sierpinski gasket with possible vacancies on the outmost vertices

We present the number of dimers $N_d(n)$ on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three, four or five, where one of the outmost vertices is not covered when the number of vertices $v(n)$ is an odd number. The entropy of absorption of diatomic molecules per site, defined as $S_{SG_d}=\lim_{n \to \infty} \ln N_d(n)/v(n)$, is calculated to be $\ln(2)/3$ exactly for $SG_2(n)$. The numbers of dimers on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4,5$ are also obtained exactly. Their entropies are equal to $\ln(6)/7$, $\ln(28)/12$, $\ln(200)/18$, respectively. The upper and lower bounds for the entropy are derived in terms of the results at a certain stage for $SG_d(n)$ with $d=3,4,5$. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of $S_{SG_d}$ with $d=3,4,5$ can be evaluated with more than a hundred significant figures accurate.Comment: 35 pages, 20 figures and 1 tabl

Probabilistic Bisimulation: Naturally on Distributions

In contrast to the usual understanding of probabilistic systems as stochastic processes, recently these systems have also been regarded as transformers of probabilities. In this paper, we give a natural definition of strong bisimulation for probabilistic systems corresponding to this view that treats probability distributions as first-class citizens. Our definition applies in the same way to discrete systems as well as to systems with uncountable state and action spaces. Several examples demonstrate that our definition refines the understanding of behavioural equivalences of probabilistic systems. In particular, it solves a long-standing open problem concerning the representation of memoryless continuous time by memory-full continuous time. Finally, we give algorithms for computing this bisimulation not only for finite but also for classes of uncountably infinite systems