Normalization of sunspot cycles and eigen mode analysis

The smoothed monthly sunspot numbers of the previous 22 complete sunspot cycles are normalized in time domain, and then an eigen mode analysis is carried out to draw the principle factors (or components) in the cycles. The results show that the main characteristics of the solar cycles can be described fairly well by the first 5 eigen modes. The obtained eigen modes are used to predict the declining phase of cycle 23 on the basis of the data prior to its maximum. The prediction indicates that cycle 23 will last for 127 months to December 2006, with the minimum of 6.2.     

A Modal Analysis of Resonance during the Whole-Body Vibration

The mechanism of resonance of a damping system of multi‐degrees of freedom such as the human body and the dependence of resonance on system parameters, particularly on the damping level, are studied in terms of detailed mathematical solutions of both the whole‐body vibrations and the eigen modes for a simple model. It is revealed that resonance would only occur near the eigen frequencies of neutral modes for which the complex eigen frequencies of the corresponding damping modes for the given damping level of the system have not moved far from the starting point (damping‐free case) along the corresponding tracks in the plane of complex eigen frequency yet. The major resonance would occur near the eigen frequency of the neutral mode where the modulus of the characteristic function of the system has the strongest, i.e., the deepest and sharpest, local minimum. For the present model, this neutral mode is the lowest neutral mode. It is found that the resonance and eigen frequencies increase with the stiffness of muscles and decrease with the body mass, with the portion of wobbling mass in the upper body, and with the portion of upper body mass in the whole body. Both the modal analysis and the analysis of the whole‐body vibration show that the phase differences among different parts of the system are still small at the unique or the lowest resonance frequency and increase dramatically only when the frequency of the vibrating source goes beyond the resonance frequency. Thus, some effects of body vibrations, e.g., internal loads, may reach their maximum not at the resonance frequency, but at a frequency somewhat higher than the resonance frequency. This may account for the fact that the frequency ranges for abdominal pain and for lumbosacral pain caused by body vibrations are not exactly the same as the frequency range for major body resonance but shifted to somewhat higher frequency ranges. It is therefore suggested that the frequency used for strength training in terms of vibrating devices should be above 20 Hz in order to avoid not only the major resonance but also the maximal internal loads.     

Limit oscillatory cycles in the single mode flutter of a plate

The development of the single mode flutter of an elastic plate in a supersonic gas flow is investigated in a non-linear formulation. In the case of a small depression in the instability zone, there is a unique limit cycle corresponding to a unique growing mode. Several new non-resonant limit cycles arise when a second increasing mode appears and the domains of their existence and stability are found. Limit cycles with an internal resonance, in which there is energy exchange between the modes, can exist for the same parameters. Relations between the amplitudes of the limit cycles and the parameters of the problem are obtained that enable one to estimate the risk of the onset of flutter.     

Fractional order hybrid system dynamics

Dynamics of multi deformable bodies coupled by standard light fractional order discrete continuum layers is described by coupled partial fractional order differential equations. A mathematical analogy and phenomenological mapping between generalized coordinates of fractional order discrete chain systems and eigen time functions correspond to one of infinite numbers of eigen amplitude functions of dynamics of multi deformable bodies (beams, plates or membranes) coupled by standard light fractional order discrete continuum layers are identified. Eigen fractional order vibration modes are approximately determined analytically and graphically presented. A number of theorems is defined and proofed. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Range and frequency spreadF at Huancayo

The mean occurrence of frequency type of spreadF at Huancayo has been shown to have practically no solar cycle dependence. The occurrence of range type of spreadF is shown to be inversely related to sunspot number. The range spread occurs mostly around 2100 LT for any of the solar epoch, while frequency spread has maximum occurrence between 2300 and 0000 LT in high sunspot years and between 0000 and 0100 LT in low sunspot years. The seasonal variation in the occurrence of either type of spreadF shows minimum in northern solstices (June months) and maximum occurrence in southern solstices (December months). The post-sunset rise ofF layer is most predominant during high sunspot years. These results point out the inadequacy of the theory of spreadF based entirely on the post-sunset upward rise of theF region after sunset.     

Eigen‐frequencies in thin elastic 3‐D domains and Reissner–Mindlin plate models

The eigen‐frequencies of elastic three‐dimensional thin plates are addressed and compared to the eigen‐frequencies of two‐dimensional Reissner–Mindlin plate models obtained by dimension reduction. The qualitative mathematical analysis is supported by quantitative numerical data obtained by the p‐version finite element method. The mathematical analysis establishes an asymptotic expansion for the eigen‐frequencies in power series of the thickness parameter. Such results are new for orthotropic materials and for the Reissner–Mindlin model. The 3‐D and R–M asymptotics have a common first term but differ in their second terms. Numerical experiments for clamped plates show that for isotropic materials and relatively thin plates the Reissner–Mindlin eigen‐frequencies provide a good approximation to the three‐dimensional eigen‐frequencies. However, for some anisotropic materials this is no longer the case, and relative errors of the order of 30 per cent are obtained even for relatively thin plates. Moreover, we showed that no shear correction factor is known to be optimal in the sense that it provides the best approximation of the R–M eigen‐frequencies to their 3‐D counterparts uniformly (for all relevant thicknesses range). Copyright © 2002 John Wiley & Sons, Ltd.     

Tough spiders

Spider graphs are the intersection graphs of subtrees of subdivisions of stars. Thus, spider graphs are chordal graphs that form a common superclass of interval and split graphs. Motivated by previous results on the existence of Hamilton cycles in interval, split and chordal graphs, we show that every 3/2‐tough spider graph is hamiltonian. The obtained bound is best possible since there are (3/2 – ε)‐tough spider graphs that do not contain a Hamilton cycle. © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 23–40, 2007     

Distribution of magnetic shear angle at the ascending phase of CYCLE 23

Using the vector magnetograms observed at Huairou Solar Observing Station of National Astronomical Observatories, the magnetic shear angles of solar active regions at the ascending phase of cycle 23 (1996-2000) were calculated. It is found that the statistical distribution of the magnetic shear angles can be fitted well by Gaussian curves. And the dominant sign of the magnetic shear angles is negative (positive) in the northern (southern) hemisphere. It is consistent with the N-S sign asymmetry of force-free field constant α and current helicity.     

The planar discontinuous piecewise linear refracting systems have at most one limit cycle

In this paper we investigate the limit cycles of planar piecewise linear differential systems with two zones separated by a straight line. It is well known that when these systems are continuous they can exhibit at most one limit cycle, while when they are discontinuous the question about maximum number of limit cycles that they can exhibit is still open. For these last systems there are examples exhibiting three limit cycles.The aim of this paper is to study the number of limit cycles for a special kind of planar discontinuous piecewise linear differential systems with two zones separated by a straight line which are known as refracting systems. First we obtain the existence and uniqueness of limit cycles for refracting systems of focus-node type. Second we prove that refracting systems of focus–focus type have at most one limit cycle, thus we give a positive answer to a conjecture on the uniqueness of limit cycle stated by Freire, Ponce and Torres in Freire et al. (2013). These two results complete the proof that any refracting system has at most one limit cycle.     

Estimating criterion weights using eigenvectors: A comparative study

In addition to Saaty's eigenvector, there are infinitely many eigen weight vectors which can be constructed for any given data of estimated weight ratios. As the judgment of the ratios are dependent on personal experience, learning, situations and state of mind, inconsistencies and degree of easiness or judgment on these individual ratios can be different, we study the properties of the different eigen weight vectors, including that of Saaty and that recently proposed by Cogger and Yu. A general framework for the construction of eigen weight vectors incorporating that confidence in obtaining the individual ratios can be different will be proposed and discussed.     

Self-similar cycles and their local bifurcations in the problem of two weakly coupled oscillators

A system of two non-linear differential equations is considered that simulates the dynamics of two completely identical weakly coupled oscillators both in the case of dissipative and active coupling. The system of normal modes is investigated. All the self-similar periodic solutions, including the asymmetric solutions describing the natural ascillations of oscillators with dissimilar amplitude's, are found analytically. The stability is investigated as well as the local bifurcations of the self-similar cycles when there is a change in stability. In particular, the possibility of the creation of two-dimensional invariant tori is pointed out. In the case of active coupling, it is shown that the basic version of the natural oscillations is a stable antiphase cycle that was observed in Huygens experiments.     

Cycle Range Distributions for Gaussian Processes Exact and Approximative Results

Wave cycles, i.e. pairs of local maxima and minima, play an important role in many engineering fields. Many cycle definitions are used for specific purposes, such as crest–trough cycles in wave studies in ocean engineering and rainflow cycles for fatigue life predicition in mechanical engineering. The simplest cycle, that of a pair of local maximum and the following local minimum is also of interest as a basis for the study of more complicated cycles. This paper presents and illustrates modern computational tools for the analysis of different cycle distributions for stationary Gaussian processes with general spectrum. It is shown that numerically exact but slow methods will produce distributions in almost complete agreement with simulated data, but also that approximate and quick methods work well in most cases. Of special interest is the dependence relation between the cycle average and the cycle range for the simple maximum–minimum cycle and its implication for the range distribution. It is observed that for a Gaussian process with rectangular box spectrum, these quantities are almost independent and that the range is not far from a Rayleigh distribution. It will also be shown that had there been a Gaussian process where exact independence hold then the range would have had an exact Rayleigh distribution. Unfortunately no such Gaussian process exists.This revised version was published online in March 2005 with corrections to the cover date.     

Turing Instabilities at Hopf Bifurcation

Turing–Hopf instabilities for reaction-diffusion systems provide spatially inhomogeneous time-periodic patterns of chemical concentrations. In this paper we suggest a way for deriving asymptotic expansions to the limit cycle solutions due to a Hopf bifurcation in two-dimensional reaction systems and we use them to build convenient normal modes for the analysis of Turing instabilities of the limit cycle. They extend the Fourier modes for the steady state in the classical Turing approach, as they include time-periodic fluctuations induced by the limit cycle. Diffusive instabilities can be properly considered because of the non-catastrophic loss of stability that the steady state shows while the limit cycle appears. Moreover, we shall see that instabilities may appear even though the diffusion coefficients are equal. The obtained normal modes suggest that there are two possible ways, one weak and the other strong, in which the limit cycle generates oscillatory Turing instabilities near a Turing–Hopf bifurcation point. In the first case slight oscillations superpose over a dominant steady inhomogeneous pattern. In the second, the unstable modes show an intermittent switching between complementary spatial patterns, producing the effect known as twinkling patterns.     

Cycle structure of random permutations with cycle weights

We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the parameters, while the distributions of finite cycles are usually independent Poisson random variables.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 109‐133, 2014     

Strange attractors in saddle-node cycles: prevalence and globality

 We consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddle-node cycles. We show that they always exhibit Hénon-like strange attractors for a set of parameter values with positive Lebesgue density at the bifurcation value. In open classes of such families the strange attractors are of global type: their basins contain an a priori defined neighbourhood of the cycle. Furthermore, the bifurcation parameter may also be a point of positive density of hyperbolic dynamics. Oblatum VIII-1993 & 23-II-1995     

Spread-F in the ionosphere over ahmedabad during the years 1954–57

From a study of spread-F or F-scatter at Ahmedabad during the four years 1954–57 of increasing sunspot activity, it was found that the time of its maximum occurrence receded from 03 hr. in low sunspot years to an hour or two before midnight in high sunspot years. This was particularly well seen in the winter and equinoctial months. Also, maximum spread-F activity which was found in summer in sunspot minimum mum years, occurred in equinoxes in maximum sunspot years. The frequency of occurrence of spread-F was found to be a maximum whenhpF₂ was in the range 300–350 km. F-scatter and F₂-stratification were found to be anti-correlated both in their diurnal and seasonal variations. The general trend was towards decreased spread-F with increased sunspot activity. It is concluded that (1) spread-F at Ahmedabad geomagnetic latitude (Φ=13·6° N) undergoes variations similar to those at equatorial stations, more so in high sunspot years, (2) the change-over from low-latitude type to middle-latitude type of variation of spread-F takes place at about geomagnetic latitude 22°, and (3) spread-F at Ahmedabad decreases with increase in magnetic activity, which is the reverse of that observed at high latitudes.     

Nonlinear calibration and data processing of the solar radio burst

The processes of the sudden energy release and energy transfer, and particle accelerations are the most challenge fundamental problems in solar physics as well as in astrophysics. Nowadays, there has been no direct measurement of the plasma parameters and magnetic fields at the coronal energy release site. Under the certain hypothesis of radiation mechanism and transmission process, radio measurement is almost the only method to diagnose coronal magnetic field. The broadband dynamic solar radio spectrometer that has been finished recently in China has higher time and frequency resolutions. Thus it plays an important role during the research of the 23rd solar cycle in China. Sometimes when there were very large bursts, the spectrometer will be overflowed. It needs to take some special process to discriminate the instrument and interference effects from solar burst signals. According to the characteristic of the solar radio broadband dynamic spectrometer, we developed a nonlinear calibration method to deal with the overflow of instrument, and introduced channel-modification method to deal with images. Finally the interference is eliminated with the help of the wavelet method. Here we take the analysis of the well-known solar-terrestrial event on July 14th, 2000 as the example. It shows the feasibility and validity of the method mentioned above. These methods can also be applied to other issues.     

On natural isomorphisms of cycle permutation graphs

Two problems are approached in this paper. Given a permutation onn elements, which permutations onn elements yield cycle permutation graphs isomorphic to the cycle permutation graph yielded by the given permutation? And, given two cycle permutation graphs, are they isomorphic? Here the author deals only with natural isomorphisms, those isomorphisms which map the outer and inner cycles of one cycle permutation graph to the outer and inner cycles of another cycle permutation graph. A theorem is stated which then allows the construction of the set of permutations which yield cycle permutation graphs isomorphic to a given cycle permutation graph by a natural isomorphism. Another theorem is presented which finds the number of such permutations through the use of groups of symmetry of certain drawings of cycles in the plane. These drawings are also used to determine whether two given cycle permutation graphs are isomorphic by a natural isomorphism. These two methods are then illustrated by using them to solve the first problem, restricted to natural isomorphism, for a certain class of cycle permutation graphs.