Gradual degradation in the supercurrent state of a chaotic superconducting tunnel junction under time-varying perturbation

We report on the degradation of the zero-voltage supercurrent generated in a Josephson tunnel junction residing in an asymmetric potential of the ratchet type, and driven by a quasiperiodic external signal having incommensurable frequencies with irrational ratio ω₂/ω₁ equals to the Golden Mean. In the underdamped regime and via computing the current–voltage (I–V) characteristic curves, we demonstrate that the disappearance of the superconducting state can be correlated to chaotic behaviour, where dynamical phase fluctuations and symmetry breakings associated with the potential and modulating signal are substantially taking place.     

On the self matching properties of [jτ]

The graph of [jτ] against j displays self matching in that if we displace this graph by a distance of F_i, then it is found that the displaced graph matches the original graph except at certain isolated points represented by an interesting Fibonacci function. From this it is shown that the frequency of mismatches is the unexpectedly simple expression 1/(τⁱ). The results are proved using lemmas, based on Zeckendorf sums, which have an appeal of their own. These also give simplified solutions to the recurrence of Downey and Griswold. Similar results apply with the Golden Sequence whose jth term is [(j+1)τ]−[jτ].     

Complex Fibonacci and Lucas Numbers,Continued Fractions,and the Square Root of the Golden Ratio (Condensed Version)

The connections between the Golden Ratio namely (1 + √5)/2, a simple continued fraction, and Fibonacci and Lucas numbers, are familiar. The Fibonacci and Lucas numbers have many fascinating properties. We now point out that the square root of the Golden Ratio is the real part of a simple periodic continued fraction but using (complex) Gaussian integers a + ib instead of the natural integers. This fact provokes a definition and a study of complex Fibonacci and Lucas numbers, and the study again turns out to have a rich theoretic structure. A fuller account will appear in The Fibonacci Quarterly.     

The power of style: Differential operator scaling in the lexical compression of sequences generated by psychological,content-free computer tasks

We define and demonstrate the usefulness of a power law on binary sequences of finite length, computed as the changing measure of their lexical compression under the action of the algorithm with increasing sample length. This method successfully discriminates between the binary series from a fair coin, Golden Mean generator, and one containing the peak hierarchy of the Thue-Morse sequence. It also differentiates between the patterns of computer screen mouse cursor orbits produced by human subjects with subclinical obsessive-compulsive and borderline personalities (as defined by DSM-III-R, Axis II criteria) during the completion of content-free tasks. © 1997 John Wiley & Sons, Inc.     

Notes on the Rescaled Sasaki Type Metric on the Cotangent Bundle

Let (M,g) be an n-dimensional Riemannian manifold and T^*M be its cotangent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compatible paracomplex structures on T^*M. Second, we construct locally decomposable Golden Riemannian structures on T^*M. Finally we investigate curvature properties of T^*M.     

A fascinating polynomial sequence arising from an electrostatics problem on the sphere

A positive unit point charge approaching from infinity a perfectly spherical isolated conductor carrying a total charge of +1 will eventually cause a negatively charged spherical cap to appear. The determination of the smallest distance ρ(d) (d is the dimension of the unit sphere) from the point charge to the sphere where still all of the sphere is positively charged is known as Gonchar’s problem. Using classical potential theory for the harmonic case, we show that 1+ρ(d) is equal to the largest positive zero of a certain sequence of monic polynomials of degree 2d?1 with integer coefficients which we call Gonchar polynomials. Rather surprisingly, ρ(2)?is the Golden ratio and ρ(4) the lesser known Plastic number. But Gonchar polynomials have other interesting properties. We discuss their factorizations, investigate their zeros and present some challenging conjectures.     

A note on Chrystal’s equation

The exact, explicit form of the transcendental solution of Chrystal’s equation, a first order nonlinear ordinary differential equation (ODE) of degree two, is derived in terms of the Lambert W-function. It is shown that this case of the general solution is dual-valued over a finite interval and that, for a special case of the coefficients, its zeros involve the Golden ratio. Additionally, a number of applications involving special cases of this ODE are noted and the main properties of the Lambert W-function are briefly reviewed.     

上证指数序中的Fibonacci数字和黄金比率

考察了上证指数,验证了在艾略特波浪模式下,黄金比率和Fibonacci数字的存在性.对在貌似随机波动的股价运动模式中存在其它的运动模式提供了又一个实例.     

Speculative dynamics with bounded rationality learning

We study a linear model for a future market characterized by the presence of different classes of traders. In the market there are three classes of traders: rational traders, feedback traders and fundamentalist traders. Each class of traders is described by a trading strategy and by an information set about the fundamental. The analysis is developed under bounded rationality, rational traders forming expectations do not know the “true” model but believe in a misspecified model. The convergence of the learning activity to the Rational Expectations Equilibria of the model is analyzed. Two different learning mechanisms are studied: the Ordinary Least Squares algorithm and the Least Mean Squares algorithm. The main goal of the study is to analyze how the presence of different classes of traders in the market affects the robustness of the Rational Expectations Equilibria of the model with respect to bounded rationality learning. Moreover we verify the claim that bubbles and erratic behavior in the stock price dynamics may arise because of learning non-convergence to Rational Expectations Equilibria. The results show that if the Ordinary Least Squares algorithm is used by the agents to update beliefs, convergence to one of the two Rational Expectations Equilibria of the model is ensured only if there are positive feedback traders in the market. On the contrary, the Least Mean Squares algorithm guarantees convergence to the Rational Expectations Equilibria given an appropriate initial belief.     

Ergodic theorems for coset spaces

We study in this paper the validity of the Mean Ergodic Theorem along left Følner sequences in a countable amenable group G. Although the Weak Ergodic Theorem always holds along any left Følner sequence in G, we provide examples where the Mean Ergodic Theorem fails in quite dramatic ways. On the other hand, if G does not admit any ICC quotients, e.g., if G is virtually nilpotent, then the Mean Ergodic Theorem holds along any left Følner sequence. In the case when a unitary representation of a countable amenable group is induced from a unitary representation of a “sufficiently thin” subgroup, we show that the Mean Ergodic Theorem holds for the induced representation along any left Følner sequence. Furthermore, we show that every countable (infinite) amenable group L embeds into a countable (not necessarily amenable) group G which admits a unitary representation with the property that for any left Følner sequence (F_n) in L, there exists a sequence (s_n) in G such that the Mean (but not the Weak) Ergodic Theorem fails in a rather strong sense along the (right-translated) sequence (F_ns_n) in G. Finally, we provide examples of countable (not necessarily amenable) groups G with proper, infinite-index subgroups H, so that the Pointwise Ergodic Theorem holds for averages along any strictly increasing and nested sequence of finite subsets of the coset G/H.     

The Solution of Travelling Salesman Problems Based on Industrial Data

This paper partially replicates a comparison of travelling salesman heuristics carried out by Golden et al. more than a decade ago. It differs in two important ways, however. (1) We consider two heuristics—k-OPT and the space filling curve technique—which were developed after the original comparison. These new techniques appear to add little to the quality of solutions to the test problems utilized here. (2) Instead of test problems using geographical data, ours are based on programs for parts produced by a numerically controlled machine. Because of the different characteristics of these test problems we reach somewhat different conclusions concerning the efficacy of the different procedures.     

Gradual degradation in the supercurrent state of a chaotic superconducting tunnel junction under time-varying perturbation

We report on the degradation of the zero-voltage supercurrent generated in a Josephson tunnel junction residing in an asymmetric potential of the ratchet type, and driven by a quasiperiodic external signal having incommensurable frequencies with irrational ratio ω₂/ω₁ equals to the Golden Mean. In the underdamped regime and via computing the current–voltage (I–V) characteristic curves, we demonstrate that the disappearance of the superconducting state can be correlated to chaotic behaviour, where dynamical phase fluctuations and symmetry breakings associated with the potential and modulating signal are substantially taking place.     

Mean ergodic theorem in symmetric spaces

We investigate the validity of the Mean Ergodic Theorem in symmetric Banach function spaces E. The assertion of that theorem always holds when E is separable, whereas the situation is more delicate when E is non-separable. To describe positive results in the latter setting, we use the connections with the theory of singular traces.     

Hierarchical entropy analysis for biological signals

We develop a hierarchical entropy (HE) method to quantify the complexity of a time series based on hierarchical decomposition and entropy analysis. The proposed method is applied to the Gaussian white noise and the 1/f noise. We prove that the difference frequency components of the Gaussian white noise with the same scale factor have the same value of entropies, and the values decline as the scale factor increases. We also apply the HE method to the 1/f noise, and prove mathematically that a lower frequency component of a 1/f noise is also a 1/f noise and verify numerically that a higher frequency component of a 1/f random vector is approximately equal to a Gaussian random vector. The theoretical results are confirmed by numerical results. Moreover, we show that the HE method is an efficient method to analyze heartbeat signals by applying it to the cardiac interbeat interval time series of healthy young and elderly subjects, congestive heart failure (CHF) subjects and atrial fibrillation (AF) subjects.     

Measuring consensus in decision making: an application to maritime command and control

An important component of maritime command and control is the information that has value to Royal Navy commanders in making campaign decisions. Studies aimed at identifying information requirements generally do so in a wargame context with several test subjects assuming command roles. An important adjunct to this work then is the assessment of how closely the subjects agree that the proposed information set is indeed valuable. In this paper, we focus on assessing the degree to which the test subjects participating in a MoD sponsored maritime command and control study agreed to a proposed set of information elements deemed valuable to taking combat decisions. The methodology involves the simulation of naval combat. The participants (former Royal Navy Captains and Admirals) are asked to choose a course of action that best accomplishes a stated mission. Information is provided on request. Each pair of participants is characterized by its set of information requirements. The objectives are (1) to define an overall information set that minimizes disagreement among the participants in some way; and (2) to develop a metric that assesses the amount of disagreement among the participants. Two metrics are presented: a mean consensus and a median consensus.     

On the Andrews–Schur proof of the Rogers–Ramanujan identities

In this article, we study one of Andrews’ proofs of the Rogers–Ramanujan identities published in 1970. His proof inspires connections to some famous formulas discovered by Ramanujan. During the course of study, we discovered identities such as $$\sum_{n\geq0}\frac{q^{n^2}}{(q;q)_n}=\frac{1}{\sqrt{5}}\Biggl(\beta \prod_{n=1}^{\infty}\frac{1}{1+\alpha q^{n/5}+q^{2n/5}}-\alpha \prod_{n=1}^{\infty}\frac{1}{1+\beta q^{n/5}+q^{2n/5}}\Biggr),$$ where β=?1/α is the Golden Ratio.     

Combining evolutionary and stochastic gradient techniques for system identification

In the present contribution, a novel method combining evolutionary and stochastic gradient techniques for system identification is presented. The method attempts to solve the AutoRegressive Moving Average (ARMA) system identification problem using a hybrid evolutionary algorithm which combines Genetic Algorithms (GAs) and the Least Mean Squares LMS algorithm. More precisely, LMS is used in the step of the evaluation of the fitness function in order to enhance the chromosomes produced by the GA. Experimental results demonstrate that the proposed method manages to identify unknown systems, even in cases with high additive noise. Furthermore, it is observed that, in most cases, the proposed method finds the correct order of the unknown system without using a lot of a priori information, compared to other system identification methods presented in the literature. So, the proposed hybrid evolutionary algorithm builds models that not only have small MSE, but also are very similar to the real systems. Except for that, all models derived from the proposed algorithm are stable.     

A mathematical programming approach to clusterwise regression model and its extensions

The clusterwise regression model is used to perform cluster analysis within a regression framework. While the traditional regression model assumes the regression coefficient (β) to be identical for all subjects in the sample, the clusterwise regression model allows β to vary with subjects of different clusters. Since the cluster membership is unknown, the estimation of the clusterwise regression is a tough combinatorial optimization problem. In this research, we propose a “Generalized Clusterwise Regression Model” which is formulated as a mathematical programming (MP) problem. A nonlinear programming procedure (with linear constraints) is proposed to solve the combinatorial problem and to estimate the cluster membership and β simultaneously. Moreover, by integrating the cluster analysis with the discriminant analysis, a clusterwise discriminant model is developed to incorporate parameter heterogeneity into the traditional discriminant analysis. The cluster membership and discriminant parameters are estimated simultaneously by another nonlinear programming model.