Nonlinear resonances of a semi-infinite cable on a nonlinear elastic foundation

The Multiple Time Scale (MTS) method is applied to the study of nonlinear resonances of a semi-infinite cable resting on a nonlinear elastic foundation, subject to a constant uniformly distributed load and to a linear viscous damping force. The zero order solution provides the static displacement, which is governed by a nonlinear equation which has been solved in closed form. The first order solution provides the linear resonances, which are seen to be functions of the nonlinearity parameter and of the static displacement at the finite boundary only. Although the first-order governing equation is linear, it has non constant coefficients and cannot be solved in closed form, so that a numerical solution is considered; the eigenfrequencies obtained in this way are also compared with the approximate eigenvalues obtained by the WKB method. At the second order of the MTS expansion, we see that the solution is independent of the intermediate time scale; some additional terms are present, including a time-independent shift of the average position of the oscillations. Finally, the nonlinear frequency–amplitude response curves, which are investigated in detail and which represent the main result of this work, are obtained from the solvability condition at the third order.     

Average densities of frequency bands of wavelets

A function is called a wavelet if its integral translations and dyadic dilations form an orthonormal basis for L ²(?). The support of the Fourier transform of a wavelet is called its frequency band. In this paper, we study the relation between diameters and measures of frequency bands of wavelets, precisely say, we study the ratio of the measure to the diameter. This reflects the average density of the frequency band of a wavelet. In particular, for multiresolution analysis (MRA) wavelets, we do further research. First, we discuss the relation between diameters and measures of frequency bands of scaling functions. Next, we discuss the relation between frequency bands of wavelets and the corresponding scaling functions. Finally, we give the precise estimate of the measure of frequency bands of wavelets. At the same time, we find that when the diameters of frequency bands tend to infinity, the average densities tend to zero.     

The small free oscillations of a strip

The refined equations of the free oscillations of a rod-strip, constructed previously in a first approximation by reducing the two-dimensional equations to one-dimensional equations by using trigonometric basis functions and satisfying the static boundary conditions on the boundary surfaces are analysed. These equations, the solutions of which are obtained for the case of hinge-supported end sections of the rod, are split into two independent systems of equations. The first of these describe non-classical fixed longitudinal-transverse forms of free oscillations, which are accompanied by a distortion of the plane form of the cross section. It is shown that the oscillation frequencies corresponding to them depend considerably on Poisson's ratio and the modulus of elasticity in the transverse direction, while for a rod of average thickness for the same value of the frequency parameter (the tone) they may be considerably lower than the frequencies corresponding to the classical longitudinal forms of free oscillations, which are performed while preserving the plane form of the cross sections. The second system of equations describes transverse flexural-shear forms of free oscillations, whose frequencies decrease as the transverse shear modulus decreases. They are practically equivalent in quality and content to the similar equations of well-known versions of the refined theories, but, unlike them, when the number of the tone increases and the relative thickness parameter decreases they lead to the solutions of the classical theory of rods.     

KAM-tori near an analytic elliptic fixed point

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector ω ₀ is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that ω ₀ has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency ω ₀. This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.     

Generalized linear models for dependent frequency and severity of insurance claims

Traditionally, claim counts and amounts are assumed to be independent in non-life insurance. This paper explores how this often unwarranted assumption can be relaxed in a simple way while incorporating rating factors into the model. The approach consists of fitting generalized linear models to the marginal frequency and the conditional severity components of the total claim cost; dependence between them is induced by treating the number of claims as a covariate in the model for the average claim size. In addition to being easy to implement, this modeling strategy has the advantage that when Poisson counts are assumed together with a log-link for the conditional severity model, the resulting pure premium is the product of a marginal mean frequency, a modified marginal mean severity, and an easily interpreted correction term that reflects the dependence. The approach is illustrated through simulations and applied to a Canadian automobile insurance dataset.     

On summing permutations and some statistical properties

A closed form expression is obtained for the sum of all permutations of n objects taken r at a time. The average and variance of the permutations are derived and are shown to be proportional to the average and variance of the objects themselves. The proportionality constant is a function of only r, n and the base b and is independent of the actual objects considered. Previous results aimed at determining the sum of permutations are shown to be very specific cases of the current development.     

Diffusion approximation of the Wright-Fisher model of population genetics: Single-locus two alleles

We investigate an autoregressive diffusion approximation method applied to the Wright-Fisher model in population genetics by considering a Markov chain with Bernoulli distributed independent variables. The use of an autoregressive diffusion method and an averaged allelic frequency process lead to an Orn-stein-Uhlenbeck diffusion process with discrete time. The normalized averaged frequency process possesses independent allele frequency indicators with constant conditional variance at equilibrium. In a monoecious diploid population of size N with r generations, we consider the time to equilibrium of averaged allele frequency in a single-locus two allele pure sampling model.     

Universal properties of stable states of a free energy model with small parameters

We consider the problem of minimizing functionals of the form , for small , subject to the constraint . (Here is typically a double well potential.) We study structural properties of minimizers that are shared by all minimizers of the problem, for sufficiently small values of . It is shown that the average ‘mass’ and ‘energy’ of minimizers, over intervals of length , is almost uniformly distributed (depending on ), for all sufficiently small . Here is a constant depending on the degree of ‘uniformity’, but independent of . Received May 5, 1996 / Accepted October 28, 1996     

The Effects of Combining Observation Periods in Time Series

This paper considers what happens when 'monthly' data, which can be modelled by a linear transform function together with a noise or error term, are aggregated to form 'annual' data. It is assumed that in the monthly model noise and input are independent, and it is shown that if the parameters of the annual model are so chosen that the input and noise are uncorrelated at all lags, then the parameters are functions of the structure generating the input series. However, if the annual noise and input are uncorrelated, the resulting model leads to the same estimate of gain and average lag as the monthly model. It is pointed out that this is at variance with reported empirical studies where annual models lead to much greater average lags. An example is given to show that the explanation may lie in the over-simplification of annual models. It is frequently assumed that a monthly Koyck model implies a similar annual model. This is not so, and the omission of a lagged term in the input series accounts for the bias in the average lag.     

Necessary conditions for distributed-parameter systems with controls of fewer variables than state variables

Necessary conditions are proved for multi-dimensional control systems, in which the state is a function of several independent variables in a fixed domain, while the controlu is a function of only some of the independent variables (that is,u is a lower-dimensional control). The cost functional can be taken in the usual Lagrange form of a multiple integral or in Mayer form. The state equations are a system of partial differential equations in explicit form with assigned boundary conditions, and constraints may be imposed on the values of the control variables.The author wishes to thank Professor L. Cesari for his many helpful comments and assistance in the preparation of this paper and Professor L. DeBoer for his interest and encouragement. This work was sponsored by the United States Air Force under Grants Nos. AF-AFOSR-69-1767-A and AFOSR-69-1662.     

A generalization of the Lipschitz summation formula and some applications

The Lipschitz formula is extended to a two-variable form. While the theorem itself is of independent interest, we justify its existence further by indicating several applications in the theory of modular forms.

     

FIFO Versus LIFO Issuing Policies for Stochastic Perishable Inventory Systems

We consider an inventory system for perishable items in which the arrival times of the items to be stored and the ones of the demands for those items form independent Poisson processes. The shelf lifetime of every item is finite and deterministic. Every demand is for a single item and is satisfied by one of the items on the shelf, if available. A demand remains unsatisfied if it arrives at an empty shelf. The aim of this paper is to compare two issuing policies: under FIFO (‘first in, first out’) any demand is satisfied by the item with the currently longest shelf life, while under LIFO (‘last in, first out’) always the youngest item on the shelf is assigned first. We determine the long-run net average profit as a function of the system parameters under each of the two policies, taking into account the revenue earned from satisfied demands, the cost of shelf space, penalties for unsatisfied demands, and the purchase cost of incoming items. The analytical results are used in several numerical examples in which the optimal input rate and the maximum expected long-run average profit under FIFO and under LIFO are determined and compared. We also provide a sensitivity analysis of the optimal solution for varying parameter values.     

The change-of-variance function for dependent data

Summary The infinitesimal stability of the asymptotic variance is considered forM-estimators of a location parameter when the nominal sample with i.i.d. data is contaminated by a possibly dependent process. It is shown that the resulting change-of-variance function can be expressed as a sum of two terms, one corresponding contamination of the univariate distribution, and one to contamination of the bivariate distributions. A change-of-variance sensitivity is introduced, the form of which is closely related to the average patch length of the outliers. Finally, optimalV-robust and mostV-robust score functions are derived. The resulting family of estimators is the same as for independent data in the general case, but the truncation point approaches zero when dependency is accounted for. For redescending score-functions, the family of estimators is changed.This paper was written under support by the Swedish Board for Technical Development, contract 712-89-1073     

Equivalence Transformations of Quasilinear First Order Systems and Reduction to Autonomous and Homogeneous Form

Classes of 2×2 first order quasilinear partial differential equations involving arbitrary continuously differentiable functions that can be mapped into autonomous and homogeneous form through equivalence transformations are considered. Equivalence transformations are point transformations of independent and dependent variables of differential equations involving arbitrary elements. The transformations act on the arbitrary elements as point transformations of an augmented space of independent, dependent variables and additional variables representing values taken by the arbitrary elements. Projecting the admitted symmetries into the space determined by the independent and dependent variables, we determine some finite transformations mapping the system into autonomous and homogeneous form. Some physical applications are considered and a comparison with reduction of quasilinear first order systems to autonomous and homogeneous form through Lie point symmetries is discussed.     

Analysis of Alternating-Priority Queueing Models with (Cross) Correlated Switchover Times

This paper analyzes a single server queueing system in which service is alternated between two queues and the server requires a (finite) switchover time to switch from one queue to the other. The distinction from classical results is that the sequence of switchover times from each of the queues need not be i.i.d. nor independent from each other; each sequence is merely required to form a stationary ergodic sequence. With the help of stochastic recursive equations explicit expressions are derived for a number of performance measures, most notably for the average delay of a customer and the average queue lengths under different service disciplines. With these expressions a comparison is made between the service disciplines and the influence of correlation is studied. Finally, through a number of examples it is shown that the correlation can significantly increase the mean delay and the average queue lengths indicating that the correlation between switchover times should not be ignored. This has important implications for communication systems in which a common communication channel is shared amongst various users and where the time between consecutive data transfers is correlated (for example in ad-hoc networks). In addition to this a number of notational mistakes in well-known existing literature are pointed out. AMS subject classification: 68M20, 60J85 A shorter version of this work has been published in the proceedings of IEEE Infocom 2005. This work was partly sponsored by the EURONGI network of excellence.     

Optimal repair–replacement policies for a system with two types of failures

In this paper, the optimal replacement problem is investigated for a system with two types of failures. One type of failure is repairable, which is conducted by a repairman when it occurs, and the other is unrepairable, which leads to a replacement of the system at once. The repair of the system is not “as good as new”. The consecutive operating times of the system after repair form a decreasing geometric process, while the repair times after failure are assumed to be independent and identically distributed. Replacement policy N is adopted, where N is the number of repairable failures. The system will be replaced at the Nth repairable failure or at the unrepairable failure, whichever occurs first. Two replacement models are considered, one is based on the limiting availability and the other based on the long-run average cost rate of the system. We give the explicit expressions for the limiting availability and the long-run average cost rate of the system under policy N, respectively. By maximizing the limiting availability A(N) and minimizing the long-run average cost rate C(N), we theoretically obtain the optimal replacement policies N^∗ in both cases. Finally, some numerical simulations are presented to verify the theoretical results.     

基于联合估计法的债券信用利差期限结构拟合及应用

基于2016~2018年月度数据,通过独立估计的单曲线样条模型和SV 模型、联合估计的多曲线样条模型和SV模型拟合公司债信用利差期限结构,进而对模型拟合效果进行比较,讨论模型在宏观经济预测中的应用,得到以下结论:(1)拟合模型的函数形式是导致理论信用利差期限结构曲线翻折的原因。样条模型和SV模型拟合的信用利差曲线形状完全不同,且模型函数变动引起的误差变动大于曲线变动引起的误差变动。(2)联合估计模型可以修正独立估计模型的人为扭曲形式。多曲线模型的结果更接近实际信用利差,误差波动性明显减小,曲线更为平滑,且联合估计的多曲线样条模型优于独立估计的单曲线样条模型、独立估计的SV 模型和联合估计的SV模型。(3)公司债信用利差期限结构在一定程度上蕴含了市场对未来宏观经济的预期信息,且在短期内预测结果随先行期限延长而改善。因此,宏观经济政策制定者需关注信用利差和期限结构模型拟合研究,重视对信用利差期限结构的深度信息挖掘,从而提高中国宏观政策制定者调控手段的前瞻性和有效性。     

Statistics of a parametric oscillator in a thermal reservoir

A parametric Bose oscillator evolving from a state in the form of a signal-noise superposition is considered. The region of values of its frequency, the temperature, the optical pumping, and the time for which the occupation numbers satisfy sub-Poisson statistics is found. It is noted that the sub-Poisson effect is delayed by thermal fluctuations in the long-wavelength region and suppressed by quantum fluctuations in the short-wavelength region, the suppression taking the form of a frequency threshold condition at low temperatures.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 158–165, July, 1992.     

Basic randomness of nature and ether-drift experiments

We re-consider the idea that quantum fluctuations might reflect the existence of an ‘objective randomness’, i.e. a basic property of the vacuum state which is independent of any experimental accuracy of the observations or limited knowledge of initial conditions. Besides being responsible for the observed quantum behavior, this might introduce a weak, residual form of ‘noise’ which is intrinsic to natural phenomena and could be important for the emergence of complexity at higher physical levels. By adopting Stochastic Electro Dynamics as a heuristic model, we are driven to a picture of the vacuum as a form of highly turbulent ether, which is deep-rooted into the basic foundational aspects of both quantum physics and relativity, and to search for experimental tests of this scenario. An analysis of the most precise ether-drift experiments, operating both at room temperature and in the cryogenic regime, shows that, at present, there is some ambiguity in the interpretation of the data. In fact the average amplitude of the signal has precisely the magnitude expected, in a ‘Lorentzian’ form of relativity, from an underlying stochastic ether and, as such, might not be a spurious instrumental effect. This puzzle, however, should be solved in a next future with the use of new cryogenically cooled optical resonators whose stability should improve by about two orders of magnitude. In these new experimental conditions, the persistence of the present amplitude would represent a clean evidence for the type of random vacuum we are envisaging.     

Examination of the 8th grade students' TIMSS mathematics success in terms of different variables1

The aim of this study is to determine how the TIMSS mathematics success of the 8th grade students differentiates according to the school type, gender, mathematics report mark, parents' education level, cognitive domains and cognitive domains by gender. Relational survey method was used in the study. Six-hundred fifty two 8th grade students studying in the same city in Turkey participated in this study. In this study, a 45 question test that was made up by choosing TIMSS 2011 mathematics questionnaire was used as a data collection tool. Quantitative data analysis methods were used in the data analysis, frequency, percentage, average, standard deviation, independent sample test, one-way analysis of variance and post-hoc tests were applied to data by using SPSS packaged software. At the end of the study, it was determined that the school type, mathematics school mark, parents' education level and cognitive domains influenced the students' TIMSS mathematics success but their gender was a neutral element. Moreover, it was seen that schools which are really successful in national exams are more successful in TIMSS exam; students whose mathematics school marks are 5 and whose parents graduated from university are more successful in TIMSS exams than others.