Hybrid kernel estimates of space–time earthquake occurrence rates using the epidemic-type aftershock sequence model

The following steps are suggested for smoothing the occurrence patterns in a clustered space–time process, in particular the data from an earthquake catalogue. First, the original data is fitted by a temporal version of the ETAS model, and the occurrence times are transformed by using the cumulative form of the fitted ETAS model. Then the transformed data (transformed times and original locations) is smoothed by a space–time kernel with bandwidth obtained by optimizing a naive likelihood cross-validation. Finally, the estimated intensity for the original data is obtained by back-transforming the estimated intensity for the transformed data. This technique is used to estimate the intensity for earthquake occurrence data for associated with complex sequences of events off the East Coast of Tohoku district, northern Japan. The intensity so obtained is compared to the conditional intensity estimated from a full space–time ETAS model for the same data.     

What is the alternating series test?

<正>We learned all the convergence tests before,which all terms are positive.Please try to use your brain to imagine how to deal with series whose terms are not always positive.The most simplest series are alternating series,whose terms alternate in sign.     

Objective Bayesian analysis of the Frechet stress–strength model

Several reference priors and a general form of matching priors are derived for a stress–strength system, and it is concluded that none of the reference priors is a matching prior. The study shows that the matching prior performs better than Jeffreys prior and reference priors in meeting the target coverage probabilities.     

On the classical Maki–Thompson rumour model in continuous time

In this paper, the Maki–Thompson model is slightly refined in continuous time, and a new general solution is obtained for each dynamics of spreading of a rumour. It is derived an equation for the size of a stochastic rumour process in terms of transitions. We give new lower and upper bounds for the proportion of total ignorants who never learned a rumour and the proportion of total stiflers who either forget the rumour or cease to spread the rumour when the rumour process stops, under general initial conditions. Simulation results are presented for the analytical solutions. The model and these numerical results are capable to explain the behaviour of the dynamics of any other dynamical system having interactions similar to the ones in the stochastic rumour process and requiring numerical interpretations to understand the real phenomena better. The numerical process in the differential equations of the model is investigated by using error-estimates. The estimated error is calculated by the Runge–Kutta method and found either negligible or zero for a relatively small size of the population. This pioneering paper introduces a new mathematical method into Operations research, motivated by various areas of scientific, social and daily life, it presents numerical computations, discusses structural frontiers and invites the interested readers to future research.     

Rankine–Hugoniot conditions obtained by using the space–time Hamilton action

In the quadri–dimensional space–time, the variation of Hamilton’s action is a powerful tool to study the process equations for conservative fluid media. In this framework, Hamilton’s principle allows to obtain equation of motions, equation of energy but also Rankine–Hugoniot conditions. The variational method may be a versatile key to obtain the shock-wave conditions for complex media when the equations of processes are not expressed by linear or quasi-linear differential equations.

     

Asymptotic analysis of the Guyer–Krumhansl–Stefan model for nanoscale solidification

Nanoscale solidification is becoming increasingly relevant in applications involving ultra-fast freezing processes and nanotechnology. However, thermal transport on the nanoscale is driven by infrequent collisions between thermal energy carriers known as phonons and is not well described by Fourier’s law. In this paper, the role of non-Fourier heat conduction in nanoscale solidification is studied by coupling the Stefan condition to the Guyer–Krumhansl (GK) equation, which is an extension of Fourier’s law, valid on the nanoscale, that includes memory and non-local effects. A systematic asymptotic analysis reveals that the solidification process can be decomposed into multiple time regimes, each characterised by a non-classical mode of thermal transport and unique solidification kinetics. For sufficiently large times, Fourier’s law is recovered. The model is able to capture the change in the effective thermal conductivity of the solid during its growth, consistent with experimental observations. The results from this study provide key quantitative insights that can be used to control nanoscale solidification processes.     

Multi–regime models for nonlinear nonstationary time series

Nonlinear nonstationary models for time series are considered, where the series is generated from an autoregressive equation whose coefficients change both according to time and the delayed values of the series itself, switching between several regimes. The transition from one regime to the next one may be discontinuous (self-exciting threshold model), smooth (smooth transition model) or continuous linear (piecewise linear threshold model). A genetic algorithm for identifying and estimating such models is proposed, and its behavior is evaluated through a simulation study and application to temperature data and a financial index.     

Cramér–Karhunen–Loève representation and harmonic principal component analysis of functional time series

We develop a doubly spectral representation of a stationary functional time series, and study the properties of its empirical version. The representation decomposes the time series into an integral of uncorrelated frequency components (Cramér representation), each of which is in turn expanded in a Karhunen–Loève series. The construction is based on the spectral density operator, the functional analogue of the spectral density matrix, whose eigenvalues and eigenfunctions at different frequencies provide the building blocks of the representation. By truncating the representation at a finite level, we obtain a harmonic principal component analysis of the time series, an optimal finite dimensional reduction of the time series that captures both the temporal dynamics of the process, as well as the within-curve dynamics. Empirical versions of the decompositions are introduced, and a rigorous analysis of their large-sample behaviour is provided, that does not require any prior structural assumptions such as linearity or Gaussianity of the functional time series, but rather hinges on Brillinger-type mixing conditions involving cumulants.     

Profitability analysis using IDEA–DA framework

The purpose of this paper is to propose a framework of imprecise data envelopment analysis–discriminant analysis (IDEA–DA) to deal with the profitability analysis of 31 chain stores belonging to a hairdressing company, known as company \(L\) , in central Taiwan. Return on investment (ROI) is used as the measure of profitability. Five predictor variables are adopted to establish the discriminant function. Five classification types (i.e., 2-group, 3-group, 4-group, 5-group and 6-group) are included in the experiments. The results show that the proposed discriminant function is a robust one with respect to the rankings obtained under different classification types; moreover, the 4-group type is suitable to classify the ROI and it acquires a satisfactory apparent rate. Hence, the 4-group type is suggested in practice work. The study shows the proposed IDEA–DA is an effective tool for company \(L\) to conduct the profitability analysis. By firstly discriminating the ROI data collected, consequently showing the aspects for improving ROI, and then use this tool at certain check points in the next year to predict the ROI levels of the stores for the upcoming whole year. For the store which is predicted to be group member with a low or very low ROI level, timely reforms could be made for improving the ROI level of the whole year. The proposed model can also be used to deal with the similar problems for the other chain-store systems which possess the analogue characteristic.     

On a model for the Navier–Stokes equations using magnetization variables

It is known that in a classical setting, the Navier–Stokes equations can be reformulated in terms of so-called magnetization variables w that satisfy

(1)${?}_{t}w+(\mathbb{P}w?\mathrm{?})w+{(\mathrm{?}\mathbb{P}w)}^{?}w?\mathrm{\Delta }w=0,$

and relate to the velocity u via a Leray projection $u=\mathbb{P}w$. We will prove the equivalence of these formulations in the setting of weak solutions that are also in $L^{\infty }(0,T;H^{1/2})\cap L^2(0,T;H^{3/2})$ on the 3-dimensional torus.Our main focus is the proof of global well-posedness in $H^{1/2}$ for a new variant of (1), where $\mathbb{P}w$ is replaced by w in the second nonlinear term:

(2)${?}_{t}w+(\mathbb{P}w?\mathrm{?})w+\frac{1}{2}\mathrm{?}|w{|}^2?\mathrm{\Delta }w=0.$

This is based on a maximum principle, analogous to a similar property of the Burgers equations.     

On the inventory model with variable lead time and price–quantity discount

In this paper, we propose a mixed integer optimization approach for solving the inventory problem with variable lead time, crashing cost, and price–quantity discount. A linear programming relaxation based on piecewise linearization techniques is derived for the problem. It first converts non-linear terms into the sum of absolute terms, which are then linearized by goal programming techniques and linearization approaches. The proposed method can eliminate the complicated multiple-step solution process used in the traditional inventory models. In addition, the proposed model allows constraints to be added by the inventory decision-maker as deemed appropriate in real-world situations.     

A posteriori error analysis using the constitutive law for the Crouzeix–Raviart element

In this work, we study the error in the approximation of the solution of elliptic partial differential equations obtained with the nonconforming finite elements method; we adopt the error in a constitutive law approach.     

Quantitative stability analysis for vector problems of 0–1 programming

We consider multiple objective 0–1 programming problems in the situation where parameters of objective functions and linear constraints are exposed to independent perturbations. We study quantitative characteristics of stability (stability radii) of problem solutions. An approach to deriving formulae and estimations of stability radii is presented. This approach is applied to stability analysis of the linear 0–1 programming problem and problems with two types of nonlinear objective functions: linear absolute value and quadratic.     

On the local in time well-posedness of an elliptic–parabolic ferroelectric phase-field model

We consider a state-of-the-art ferroelectric phase-field model arising from the engineering area in recent years, which is mathematically formulated as a coupled elliptic–parabolic differential system. We utilize the maximal parabolic regularity theory to show the local in time well-posedness of the ferroelectric problem in both 2D and 3D spaces, which is sharp in the sense that the local solution is unique and a blow-up criterion is present. The well-posedness result will firstly be proved under some general assumptions. Afterwards we give sufficient geometric and regularity conditions which will guarantee the fulfillment of the imposed assumptions.     

Slack-adjusted DEA for time series analysis: Performance measurement of Japanese electric power generation industry in 1984–1993

Using a new slack-adjusted data envelopment analysis (SA-DEA) model which explicitly incorporates an influence of slacks into its efficiency measurement, this study discusses a use of various efficiencies and index measures for DEA dynamic analysis. An analytical formulation to determine the type of return to scale (RTS) is proposed for the new DEA model. This paper mathematically discusses when multiple solutions occur on RTS and how to deal with such a difficulty. As an important case study, this paper applies the proposed DEA approach to examine the performance of Japanese electric power generation companies from 1984 to 1993. Two policy implications are suggested for guiding the Japanese electric power industry.     

An improved local convergence analysis for Newton–Steffensen-type method

We provide a local convergence analysis for Newton–Steffensen-type algorithm for solving nonsmooth perturbed variational inclusions in Banach spaces. Under new center–conditions and the Aubin continuity property, we obtain the linear local convergence of Newton–Steffensen method. Our results compare favorably with related obtained in (Argyros and Hilout, 2007 submitted; Hilout in J. Math. Anal. Appl. 339:753–761, 2008).     

Analysis and synthesis of switched nonlinear systems using the T–S fuzzy model

In this paper, the methods based on Lyapunov stability theorem to study the stability and switching law design for the T–S fuzzy switched systems with state-driven switching method are presented. Furthermore, these methods can be applied to cases when all individual systems are unstable. The PDC is employed to design fuzzy controllers from the T–S fuzzy models. The stabilization analysis is reduced to a problem of finding a common Lyapunov function for a set of linear matrix inequalities. Finally, a numerical example and an illustrative example based on the chemical process example are given to show the merits of the proposed approach, respectively.     

Simulation of Student–Lévy processes using series representations

Lévy processes have become very popular in many applications in finance, physics and beyond. The Student–Lévy process is one interesting special case where increments are heavy-tailed and, for 1-increments, Student t distributed. Although theoretically available, there is a lack of path simulation techniques in the literature due to its complicated form. In this paper we address this issue using series representations with the inverse Lévy measure method and the rejection method and prove upper bounds for the mean squared approximation error. In the numerical section we discuss a numerical inversion scheme to find the inverse Lévy measure efficiently. We extend the existing numerical inverse Lévy measure method to incorporate explosive Lévy tail measures. Monte Carlo studies verify the error bounds and the effectiveness of the simulation routine. As a side result we obtain series representations of the so called inverse gamma subordinator which are used to generate paths in this model.