Two-point resistance of an m×n resistor network with an arbitrary boundary and its application in RLC network

A rectangular m × n resistor network with an arbitrary boundary is investigated, and a general resistance formula between two nodes on an arbitrary axis is derived by the Recursion-Transform(RT) method, a problem that has never been resolved before, for the Green's function technique and the Laplacian matrix approach are inapplicable to it. To have the exact solution of resistance is important but it is difficult to obtain under the condition of arbitrary boundary. Our result is directly expressed in a single summation and mainly composed of characteristic roots, which contain both finite and infinite cases. Further, the current distribution is given explicitly as a byproduct of the method. Our framework can be effectively applied to RLC networks. As an application to the LC network, we find that our formulation leads to the occurrence of resonances at h_1= 1-cosφ_i-sinφ_icotnφ_i. This somewhat curious result suggests the possibility of practical applications of our formulae to resonant circuits.     

Strategies for reducing large fMRI data sets for independent component analysis

In independent component analysis (ICA), principal component analysis (PCA) is generally used to reduce the raw data to a few principal components (PCs) through eigenvector decomposition (EVD) on the data covariance matrix. Although this works for spatial ICA (sICA) on moderately sized fMRI data, it is intractable for temporal ICA (tICA), since typical fMRI data have a high spatial dimension, resulting in an unmanageable data covariance matrix. To solve this problem, two practical data reduction methods are presented in this paper. The first solution is to calculate the PCs of tICA from the PCs of sICA. This approach works well for moderately sized fMRI data; however, it is highly computationally intensive, even intractable, when the number of scans increases. The second solution proposed is to perform PCA decomposition via a cascade recursive least squared (CRLS) network, which provides a uniform data reduction solution for both sICA and tICA. Without the need to calculate the covariance matrix, CRLS extracts PCs directly from the raw data, and the PC extraction can be terminated after computing an arbitrary number of PCs without the need to estimate the whole set of PCs. Moreover, when the whole data set becomes too large to be loaded into the machine memory, CRLS-PCA can save data retrieval time by reading the data once, while the conventional PCA requires numerous data retrieval steps for both covariance matrix calculation and PC extractions. Real fMRI data were used to evaluate the PC extraction precision, computational expense, and memory usage of the presented methods.     

Field quantization for chaotic resonators with overlapping modes

Feshbach's projector technique is employed to quantize the electromagnetic field in optical resonators with an arbitrary number of escape channels. We find spectrally overlapping resonator modes coupled due to the damping and noise inflicted by the external radiation field. For wave chaotic resonators the mode dynamics is determined by a non-Hermitean random matrix. Upon including an amplifying medium, our dynamics of open-resonator modes may serve as a starting point for a quantum theory of random lasing.     

Distinguishing N Photons in an Entangled State from N Separate Photons

We present a multimode model to describe an arbitrary N-photon state. In general, the N photons can be distinguished through their temporal modes. From this model, we can find the criterion for the N photons in an indistinguishable state of a single temporal mode. We find that simple multi-photon detection scheme cannot distinguish N photons in different temporal modes and only a multi-photon interference experiment can accomplish the goal. We apply the theory to the four-photon case in the process of parametric down-conversion. We will present the results for various four-photon interference schemes and identify a quantity to characterize how well the four photons are indistinguishable.     

Analytic Structure and Power-Series Expansion of the Jost Matrix

For the Jost-matrix that describes the multi-channel scattering, the momentum dependencies at all the branching points on the Riemann surface are factorized analytically. The remaining single-valued matrix functions of the energy are expanded in the power-series near an arbitrary point in the complex energy plane. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This makes it possible to obtain an analytic expression for the Jost-matrix (and therefore for the S-matrix) near an arbitrary point on the Riemann surface (within the domain of its analyticity) and thus to locate the resonant states as the S-matrix poles. This approach generalizes the standard effective-range expansion that now can be done not only near the threshold, but practically near an arbitrary point on the Riemann surface of the energy. Alternatively, The semi-analytic (power-series) expression of the Jost matrix can be used for extracting the resonance parameters from experimental data. In doing this, the expansion coefficients can be treated as fitting parameters to reproduce experimental data on the real axis (near a chosen center of expansion E ₀) and then the resulting semi-analytic matrix S(E) can be used at the nearby complex energies for locating the resonances. Similarly to the expansion procedure in the three-dimensional space, we obtain the expansion for the Jost function describing a quantum system in the space of two dimensions (motion on a plane), where the logarithmic branching point is present.     

Analysis of arbitrary index profile planar optical waveguides and multilayer nonlinear structures: a simple finite difference algorithm

we present here a simple numerical method to obtain the mode effective indices as well as field distributions of modes of any arbitrary profile planar optical waveguide. The method is based on the solutions of scalar and semivectorial Helmoltz’s equation by finite difference algorithm and devised with a field convergence technique. This approach is quite general and can be applied straightforwardly to calculate the guided as well as quasi- or leaky modes of any arbitrary planar structure without the need to solve any eigenvalue equation or any complex matrix formalism. Besides providing the ease of application, the algorithm is particularly useful for waveguides with any graded index profile or with irregular multilayered structure and multilayered waveguides with a localised arbitrary nonlinear film. The performance of our method is verified against typical problems with analytical solutions or having results known otherwise, and is shown to yield results with good accuracy.     

Quasi-resonant Modes of Massive Scalar Fields in Schwarzschild–de Sitter Space-Time

Recent research of massive fields quasinormal modes suggested that the arbitrary long living modes can be exist. Using different orders of WKB method, we study the massive scalar fields quasinormal modes of Schwarzschild–de Sitter black holes. It is shown that the WKB method can not applied for large massive scalar fields directly in asymptotic flat space-time but can fit well in de Sitter space-time. We prove the non-existence of QRMs in de Sitter space-time and find that the real parts of QNMs increase linearly and the imaginary parts approach to special values as the mass of scalar fields increase.     

Silicon Photonics Rectangular Universal Interferometer

Universal multiport photonic interferometers that can implement any arbitrary unitary transformation between input and output optical modes are essential to support advanced optical functions. Integrated versions of these components can be implemented by means of either a fixed triangular or a fixed rectangular arrangement of the same components. We propose the implementation of a fixed rectangular universal interferometer using a reconfigurable hexagonal waveguide mesh circuit. A suitable adaptation synthesis algorithm tailored to this mesh configuration is provided and the experimental demonstration of a rectangular multiport interferometer by means of a fabricated silicon photonics chip is reported. The 7‐hexagonal cell chip can implement 2 × 2, 3 × 3 and 4 × 4 arbitrary unitary transformations. The proposed hexagonal waveguide mesh operates in a similar way as a Field Programmable Gate Array (FPGA) in electronics. We believe that this work represents an important step‐forward towards fully programmable and integrable multiport interferometers.     

Prescriptions for gravitational initial data as judged by a simple radiating model

Several methods of prescribing initial data for gravitational and matter fields, which are intended to eliminate extraneous radiation that is not produced by the matter source, are analysed in a simple exactly soluble radiating model. The model consists of an harmonic oscillator coupled to a scalar field along future light cones of Minkowski space time. In particular we analyze the asymptotic regime of the oscillator and find it is characterized essentially by two distinct decay modes. They differ in the way they behave both in the limit of small coupling constant and in a certain Newtonian limit. As a criterion to select initial data for the field with no extra radiation, we require that these initial data sets should put the oscillator from the start into the asymptotic regime. The underlying hypothesis here is that initial transients result from excitation of the oscillator by incoming radiation. We then see that the requirement of a uniform Newtonian limit leads to unique data for the scalar field for each arbitrary data set for the oscillator. We further find that this unique data set indeed satisfies our criterion.     

Structure formation in turbulence as an instability of effective quantum plasma

Structure formation in turbulence can be understood as an instability of “plasma” formed by fluctuations serving as effective particles. These “particles” are quantumlike in the sense that their wavelengths are non-negligible compared to the sizes of background coherent structures. The corresponding “kinetic equation” describes the Wigner matrix of the turbulent field, and the coherent structures serve as collective fields. This formalism is usually applied to manifestly quantumlike or scalar waves. Here, we show how to systematically extend it to more complex systems using compressible Navier–Stokes turbulence as an example. In this case, the fluctuation Hamiltonian is a five-dimensional matrix operator and diverse modulational modes are present. As an illustration, we calculate these modes for a sinusoidal shear flow and find two modulational instabilities. One of them is specific to supersonic flows, and the other one is a Kelvin–Helmholtz-type instability that is a generalization of the known zonostrophic instability. Our calculations are readily extendable to other types of turbulence, for example, magnetohydrodynamic turbulence in plasma.     

Determination of Optical Waveguide Refractive-Index Profiles with the Inverse Analytic Transfer Matrix Method

In this paper, we present an inverse analytic transfer matrix (IATM) method to predict the refractive index profiles from the measurement of mode indices based on the analytic transfer matrix (ATM) method, which can exactly determine the modal characteristics of a planar optical waveguide with arbitrary index profiles. IATM method has been proved to be universally reliable not only in slowly changing profiles but also the profiles changing rapidly in which inverse WKB method would find difficult through numerical analysis for several typical refractive index profiles.     

Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox

The concept of steering was introduced by Schr?dinger in 1935 as a generalization of the Einstein-Podolsky-Rosen paradox for arbitrary pure bipartite entangled states and arbitrary measurements by one party. Until now, it has never been rigorously defined, so it has not been known (for example) what mixed states are steerable (that is, can be used to exhibit steering). We provide an operational definition, from which we prove (by considering Werner states and isotropic states) that steerable states are a strict subset of the entangled states, and a strict superset of the states that can exhibit Bell nonlocality. For arbitrary bipartite Gaussian states we derive a linear matrix inequality that decides the question of steerability via Gaussian measurements, and we relate this to the original Einstein-Podolsky-Rosen paradox.     

New critical matrix models and generalized universality

We study a class of one-matrix models with an action containing nonpolynomial terms. By tuning the coupling constants in the action to criticality we obtain that the eigenvalue density vanishes as an arbitrary real power at the origin, thus defining a new class of multicritical matrix models. The corresponding microscopic scaling law is given and possible applications to the chiral phase transition in QCD are discussed. For generic coupling constants off-criticality we prove that all microscopic correlation functions at the origin of the spectrum remain in the known Bessel universality class. An arbitrary number of Dirac mass terms can be included and the corresponding massive universality is maintained as well. We also investigate the critical behavior at the edge of the spectrum: there, in contrast to the behavior at the origin, we find the same critical exponents as derived from matrix models with a polynomial action.     

Thermodynamics of Restricted Boltzmann Machines and Related Learning Dynamics

We investigate the thermodynamic properties of a restricted Boltzmann machine (RBM), a simple energy-based generative model used in the context of unsupervised learning. Assuming the information content of this model to be mainly reflected by the spectral properties of its weight matrix W, we try to make a realistic analysis by averaging over an appropriate statistical ensemble of RBMs. First, a phase diagram is derived. Otherwise similar to that of the Sherrington–Kirkpatrick (SK) model with ferromagnetic couplings, the RBM’s phase diagram presents a ferromagnetic phase which may or may not be of compositional type depending on the kurtosis of the distribution of the components of the singular vectors of W. Subsequently, the learning dynamics of the RBM is studied in the thermodynamic limit. A “typical” learning trajectory is shown to solve an effective dynamical equation, based on the aforementioned ensemble average and explicitly involving order parameters obtained from the thermodynamic analysis. In particular, this let us show how the evolution of the dominant singular values of W, and thus of the unstable modes, is driven by the input data. At the beginning of the training, in which the RBM is found to operate in the linear regime, the unstable modes reflect the dominant covariance modes of the data. In the non-linear regime, instead, the selected modes interact and eventually impose a matching of the order parameters to their empirical counterparts estimated from the data. Finally, we illustrate our considerations by performing experiments on both artificial and real data, showing in particular how the RBM operates in the ferromagnetic compositional phase.     

基于切延迟的椭圆反射腔的吸引子研究

本文使用转移矩阵的方法,引入椭圆角转换函数,使椭圆问题得到简化,推导出十分简单的切延迟椭圆反射的迭代公式,这样非常有利于理论分析.切延迟椭圆反射腔映射系统(TD-ERCS)在切延迟1单位时存在吸引子,利用该公式,对其吸引子形成的原因及稳定性做了理论分析,发现圆的吸引子与椭圆不尽相同;同时发现椭圆有两个不动线,但只有一个是稳定的.本文还发现,随着椭圆压缩因子μ的减小,对于任意的切延迟因子m,相邻两次迭代数据间的相关性增强,这说明将该系统用作密码系统,椭圆压缩因子μ不能太小,同时混沌系统本身要求μ不能太大,否则降低安全度. 关键词： 混沌 切延迟 TD-ERCS 吸引子     

光学无源谐振腔的矩阵理论(柱坐标)(Ⅱ)——轴对称稳定光学无源谐振腔的计算

本文的目的是探讨应用文献[1]中提出的光学无源谐振腔的矩阵方程,解决腔中振荡横模的具体计算方法问题,并讨论其精度。为了便于与已有公式较可靠的结果相比,我们挑选了由两面对称的球面反射镜组成的、菲涅耳数N≤1的各种稳定腔作为计算对象;还分别用数学试验法和文献[1]中得出的误差上限公式,求出所得结果的误差。结果表明:对于基横模,所得结果与文献[3,6]的结果在报道的精度内很好地符合;对于包括高阶模在内的各阶横模,文献[1]中所得公式的误差上限都是正确的,但往往大大偏高。计算结果实际上往往具有高得多的精度。文中对g=0,0.5,0.8,0.9,0.95,菲涅耳数N=1的稳定腔,列出了l=0,p=0,1,2,3,4,5的各阶横模的本征值;绘出了l=0,1,2,p=0,1,2各阶横模场的相对振幅与位相分布曲线。还绘出了上述诸类稳定腔中N=1/π时TEM₀₀,TEM₀₁模的场分布曲线。从所得结果中得出了一些新的规律,并进行了某些讨论。本文结果表明:用这种矩阵理论,确实可以较为方便地一次求得模损耗不极近于1的、角模数l任意给定而径模数p不同的、所有各阶横模的性质,并能给出所得结果的误差上限,保证其具有相当高的精度。由于所用坐标系关系,本方法仅适用于具有理想轴对称性的谐振腔。 关键词：     

光学无源谐振腔的矩阵理论(柱坐标)(Ⅰ)——自洽场矩阵方程

本文讨论了文献中提出的根据标量光波传播矩阵理论的光学无源谐振腔自洽场矩阵方程。提出了对自洽场性质的新认识、元模转换概念和自洽场元模结构分析方法。在普遍情况下,严格证明了上述矩阵方程可截取为有穷阶以求近似解。给出了确定上述有穷阶矩阵方程本征值误差上限的严格公式。还给出了估算由该方程导出的所有结果的计算误差上限的较为方便的公式与方法。本文提出的光学无源谐振腔的矩阵理论较方便于各阶横模,包括那些模损耗相当近于1的高阶横模的计算。作者认为这个理论还应该较适合于复杂谐振腔的分析和计算。由于所用坐标系的关系,本文所提出的理论仅适用于理想轴对称性系统。 关键词：     

任意坐标系下非圆正规光波导的一般解及应用

给出了任意坐标系下非圆正规光纤随传输距离z指数变化的一般解，它具有exp｛i(+ΔβV)z｝E0(x,y)的形式，而V是由主轴偏角θ惟一确定的矩阵，称为主轴矩阵.还给出了相应的亥姆霍兹方程，并应用于研究有微小变形的二层弱导非圆光纤，和解释光纤纵向不均匀性对双折射的抑制效应. 关键词： 非圆正规光纤 偏振主轴 偏振模色散(PMD)     

Toward a scenario with complementary stochastic and deterministic information in financial fluctuations

We present the results of our analysis of time series for a collection of 345 stocks listed in S&P 500, to show that integrated information on collective fluctuations in financial data can be revealed quantitatively on two aspects, focusing on either the stochastic or the deterministic content of the data. The latter is obtained by relating the fluctuations in high frequency data of one-day moving averages (HF1MA) for the prices of individual stocks analogously to the displacements for Brownian motion for the tracer particles. It has been shown in a previous study of data for each month over the years 1996–1999 [11], that the kinetic parameters carry effectively the market-specific information. In an attempt to extend such a many-particle scenario, we pay attention in this study to the stock-stock cross correlations and decompose the fluctuations into the Karhunan–Lo$\stackrel{`}{\mathrm{e}}$ve expansions, to find the general features of the collective modes in their time-wise as well as the stock-wise components, comparing the results for the time series of original prices and those of HF1MA. We found robust patterns of time-wise correlations in the eigenmodes, which may be analyzed further to find market-specific information.     

Multiple Andreev reflections as a transport problem in energy space

We present an approach for analysing the dc current in voltage biased quantum superconducting junctions. By separating terms from different n -particle processes, we find that the n -particle current can be mapped on the problem of wave transport through a potential structure with n barriers. We discuss the relation between resonances in such structures and the subgap structures in the current–voltage characteristics. At zero temperature we find, exactly, that only processes creating real excitations contribute to the current. Our results are valid for a general SXS junction, where the X region is an arbitrary nonsuperconducting region described by an energy-dependent transfer matrix.