Cauchy Function Research Articles

Content similarity matching for video sequence identification

To manage large database system with video, effective video indexing and retrieval are required. A large number of video retrieval algorithms have been presented for frame-wise user query or video content query, whereas a few video identification algorithms have been proposed for video sequence query. In this paper, we propose an effective video identification algorithm for video sequence query that employs the Cauchy function of histograms between successive frames and the modified Hausdorff distance. To effectively match the video sequences with a low computational load, we make use of the key frames extracted by the cumulative Cauchy function and compare the set of key frames using the modified Hausdorff distance. Experimental results with several color video sequences show that the proposed algorithm for video identification yields remarkably higher performance than conventional algorithms such as Euclidean metric, and directed divergence methods.

Positiveness conditions for the Cauchy function for differential equations with distributed delay

We obtain unimprovable sufficient conditions for the positivity of the Cauchy function for differential equation with distributed delay. Based on these conditions, we study some asymptotic properties of solutions of the Hutchinson-Wright equation, the Lasota-Wazevska equation, and the Nicholson equation.

Cauchy Estimation for Linear Scalar Systems

An estimation paradigm is presented for scalar discrete linear systems entailing additive process and measurement noises that have Cauchy probability density functions (pdf). For systems with Gaussian noises, the Kalman filter has been the main estimation paradigm. However, many practical system uncertainties that have impulsive character, such as radar glint, are better described by stable non-Gaussian densities, for example, the Cauchy pdf. Although the Cauchy pdf does not have a well defined mean and does have an infinite second moment, the conditional density of a Cauchy random variable, given its linear measurements with an additive Cauchy noise, has a conditional mean and a finite conditional variance, both being functions of the measurement. For a single measurement, simple expressions are obtained for the conditional mean and variance, by deriving closed form expressions for the infinite integrals associated with the minimum variance estimation problem. To alleviate the complexity of the multi-stage estimator, the conditional pdf is represented in a special factored form. A recursion scheme is then developed based on this factored form and closed form integrations, allowing for the propagation of the conditional mean and variance over an arbitrary number of time stages. In simulations, the performance of the newly developed scalar discrete-time Cauchy estimator is significantly superior to a Kalman filter in the presence of Cauchy noise, whereas the Cauchy estimator deteriorates only slightly compared to the Kalman filter in the presence of Gaussian noise. Remarkably, this new recursive Cauchy conditional mean estimator has parameters that are generated by linear difference equations with stochastic coefficients, providing computational efficiency.

Estimates for the Cauchy function of linear singular differential equations and some applications

For higher-order linear singular equations, we find sharp estimates for the Cauchy function and its partial derivatives. We use these estimates to study the properties of solutions of singular differential inequalities and obtain optimal sufficient conditions for the unique solvability of the linear singular Cauchy problem.

Using simulated annealing to schedule a flowshop manufacturing cell with sequence-dependent family setup times

This paper proposes a simulated annealing-based meta-heuristic to minimise makespan in a flowshop manufacturing cell with sequence-dependent family setup times. To escape from local minima, Cauchy function – rather than the Boltzmann function – is used during the annealing process. The effectiveness and efficiency of the proposed simulated annealing-based meta-heuristic is compared against the existing heuristics on a benchmark problem dataset used in earlier studies. These computational results show that the proposed simulated annealing-based meta-heuristic is highly effective as compared to the state-of-the-art meta-heuristics for this problem on the same benchmark instances.

Maximum Principles and Boundary Value Problems for First-Order Neutral Functional Differential Equations

We obtain the maximum principles for the first-order neutral functional differential equation where , and are linear continuous operators, and are positive operators, is the space of continuous functions, and is the space of essentially bounded functions defined on . New tests on positivity of the Cauchy function and its derivative are proposed. Results on existence and uniqueness of solutions for various boundary value problems are obtained on the basis of the maximum principles.

The fixed sign property of solutions and stability of linear differential equations with varying distributed delay

We study one class of linear differential equations with varying distributed delay. We obtain an effective (in terms of parameters of the initial problem) criterion for the positivity of the Cauchy function of this class of equations. Based on this result, we establish effective criteria for the exponential stability of equations under consideration.

Performance evaluation of flexible manufacturing systems under uncertain and dynamic situations

The present era demands the efficient modelling of any manufacturing system to enable it to cope with unforeseen situations on the shop floor. One of the complex issues affecting the performance of manufacturing systems is the scheduling of part types. In this paper, the authors have attempted to overcome the impact of uncertainties such as machine breakdowns, deadlocks, etc., by inserting slack that can absorb these disruptions without affecting the other scheduled activities. The impact of the flexibilities in this scenario is also investigated. The objective functions have been formulated in such a manner that a better trade-off between the uncertainties and flexibilities can be established. Consideration of automated guided vehicles (AGVs) in this scenario helps in the loading or unloading of part types in a better manner. In the recent past, a comprehensive literature survey revealed the supremacy of random search algorithms in evaluating the performance of these types of dynamic manufacturing system. The authors have used a metaheuristic known as the quick convergence simulated annealing (QCSA) algorithm, and employed it to resolve the dynamic manufacturing scenario. The metaheuristic encompasses a Cauchy distribution function as a probability function that helps in escaping the local minima in a better manner. Various machine breakdown scenarios are generated. A ‘heuristic gap’ is measured, and it indicates the effectiveness of the performance of the proposed methodology with the varying problem complexities. Statistical validation is also carried out, which helps in authenticating the effectiveness of the proposed approach. The efficacy of the proposed approach is also compared with deterministic priority rules.

On the generation of Tikhonov variates

A novel, simple and efficient method for the generation of Tikhonov (a.k.a. von Mises) random variates is proposed. In the proposed method, circular variates of a prescribed Tikhonov distribution pT(x;alpha,xi) are generated via the transformation of variates selected randomly, on a one-for-one basis, from a bank of K distinct Cauchy and Gaussian generators. The mutually exclusive probabilities of sampling from each of the Cauchy or Gaussian generators, as well as the variance and half-width parameters that specify the latter, are derived directly from the Cauchy, Gaussian and Tikhonov characteristic functions, all of which are either known or given in closed form. The proposed random mixture technique is extremely efficient in that a single pair of uniform random numbers is consumed in the generation of each Tikhonov (or von Mises) sample, regardless of the prescribed concentration and centrality parameters (alpha, xi), all requiring neither the rejection of samples, nor the repetitive evaluation of computationally demanding functions. Additional attractive features of the method are as follows. By construction, the first (dominant) N circular moments of Tikhonov variates generated with the proposed random mixture technique are the ones that best approximate their corresponding theoretical values, with errors measured exactly. The exact distribution of generated Tikhonov variates is determined analytically, and its (Kullback-Leibler) divergence to the exact Tikhonov PDF is shown also analytically to be negligible. Finally, the technique establishes a connection between Tikhonov and Gaussian variates which can be exploited, e.g., in the generation of piecewise-continuous pseudo-random functions with Tikhonov-distributed outcomes.

Modeling the planning and scheduling across the outsourcing supply chain: a Chaos-based fast Tabu–SA approach

Planning and Scheduling are the interrelated manufacturing functions and should be solved simultaneously to achieve the real motives of integration in manufacturing. In this paper, we have addressed the advanced integrated planning and scheduling problem in a rapidly changing environment, where the selection of outsourcing machine/operation, meeting the customers (single or multiple) due date, minimizing the makespan are the main objectives while satisfying several technological constraints. We developed a mixed integer programming model for integrated planning and scheduling across the outsourcing supply chain and showed how such models can be used to make strategic decisions. It is a computationally complex and mathematically intractable problem to solve. In this paper, a Chaos-based fast Tabu-simulated annealing (CFTSA) incorporating the features of SA, Tabu and Chaos theory is proposed and applied to solve a large number of problems with increased complexity. In CFTSA algorithm, five types of perturbation schemes are developed and Cauchy probability function is used to escape from local minima and achieve the optimal/near optimal solution in a lesser number of iterations. An intensive comparative study shows the robustness of proposed algorithm. Percentage Heuristic gap is used to show the effectiveness and two ANOVA analyses are carried out to show the consistency and accuracy of the proposed approach.

Uniqueness theorems for Cauchy integrals

If $\mu$ is a finite complex measure in the complex plane $\mathbb{C}$ we denote by $C^\mu$ its Cauchy integral defined in the sense of principal value. The measure $\mu$ is called \emph{reflectionless} if it is continuous (has no atoms) and $C^\mu=0$ at $\mu$-almost every point. We show that if $\mu$ is reflectionless and its Cauchy maximal function $C^\mu_*$ is summable with respect to $|\mu|$ then $\mu$ is trivial. An example of a reflectionless measure whose maximal function belongs to the weak $L^1$ is also constructed, proving that the above result is sharp in its scale. We also give a partial geometric description of the set of reflectionless measures on the line and discuss connections of our results with the notion of sets of finite perimeter in the sense of De Giorgi.

О связи фундаментального решения и функции Коши линейных дифференциально-разностных уравнений нейтрального типа

О связи фундаментального решения и функции Коши линейных дифференциально-разностных уравнений нейтрального типа

О положительности функции Коши одного интегро-дифференциального уравнения

О положительности функции Коши одного интегро-дифференциального уравнения

Impacts of prior mis-specification on Bayesian fisheries stock assessment

One of the key features of a Bayesian stock assessment is that the modeller needs to provide knowledge on model parameters. Priors summarise modellers’ understanding of model parameters and are often defined by a probability distribution function. Priors are often mis-specified with arbitrary and unrealistic accuracy and precision in perceiving the state of nature for the parameters as a result of our limited understanding of fisheries ecosystems. Commonly used probability functions such as normal distribution functions tend to be sensitive to prior mis-specification, resulting in large uncertainty and/or errors in Bayesian stock assessment. Fat-tailed functions such as the Cauchy distribution function have been found to be robust to prior mis-specification. Using the Maine sea urchin fishery as an example, we evaluated the impacts of mis-specification in defining the prior distributions on Bayesian stock assessment. The present study suggests that the quantification of priors with a Cauchy distribution tends to be robust to the prior mis-specification. Given our limited understanding of fisheries a function such as the Cauchy distribution function that is robust to prior mis-specification tends to be more desirable. Future studies should explore the use of other fat-tailed distribution functions for quantifying priors in fisheries stock assessment.

Neural network modeling of survival dynamics of holometabolous insects: A case study

Survival process and mortality distribution of holometabolous insects were hard to be described by mechanistic models due to their distinctive development stages in the life cycle. Neural networks are flexible approximators for linear or nonlinear ecological systems. This study aimed to evaluate the effectiveness and performance of BP ANN (feed-forward backpropagation artificial neural network) and conventional models in modeling the survival process and mortality distribution of a holometabolous insect, Spodoptera litura F. (Lepidoptera: Noctuidae). Training data on survival process and mortality distribution of S. litura were recorded under different temperatures. BP ANN, three empirical models, five probabilistic density functions, a multi-stage based dynamic model, and a trend surface model were used to modeling the time changing and temperature dependent relationships of the insect. Overall performances were compared among these models. The results demonstrated that BP ANN could be effectively used to model the survival process and mortality distribution of S. litura. It exhibited the best performance in the one-dimensional and two-dimensional simulations. Some features in survival process of the holometabolous insect, like the static stages for egg and pupa, were better simulated by BP ANN. Cross validation and prediction analysis demonstrated that BP ANN was also a better and robust predictor for the survival process of S. litura. Among the probabilistic density functions such as normal function, logarithm-normal function, Cauchy function, and chi-squared function, the performance of Cauchy density function was similar to BP ANN in the simulation of mortality distribution, and chi-squared function was the worst. The multi-stage based dynamic model yielded the worst performance in survival process simulation. It was proved that the network settings, such as the number of hidden layers and hidden neurons influenced the performance of BP ANN.

Stability analysis of nonautonomous difference systems with delaying arguments

We present sufficient conditions for the stability of the nonautonomous difference system x ( k + 1 ) = ∑ j = 0 m A j ( k ) x ( k − j ) , k ∈ Z + , with m ⩾ 1 , when the ( n × n ) -matrices A j ( ⋅ ) are slowly varying coefficients. The proposed approach is based on the generalization of the “freezing” method for ordinary differential equations. The stability conditions are formulated in terms of the corresponding Cauchy's function.

Further Results on the Smoothability of Cauchy Hypersurfaces and Cauchy Time Functions

Recently, folk questions on the smoothability of Cauchy hypersurfaces and time functions of a globally hyperbolic spacetime M, have been solved. Here we give further results, applicable to several problems: (1) Any compact spacelike acausal submanifold H with boundary can be extended to a spacelike Cauchy hypersurface S. If H were only achronal, counterexamples to the smooth extension exist, but a continuous extension (in fact, valid for any compact achronal subset K) is still possible. (2) Given any spacelike Cauchy hypersurface S, a Cauchy temporal function T (i.e., a smooth function with past-directed timelike gradient everywhere, and Cauchy hypersurfaces as levels) with S equal to one of the levels, is constructed -thus, the spacetime splits orthogonally as $R \times S$ in a canonical way. Even more, accurate versions of this result are obtained if the Cauchy hypersurface S were non-spacelike (including non-smooth, or achronal but non-acausal).

Non-oscillation of the first-order differential equations with unbounded memory for stabilization by control signal

In this paper we shall demonstrate that problems of chaos stabilization can be reduced to the analysis of a corresponding stability of differential equations with unbounded memory. In order to obtain stability results a special monotone technique, based on the positivity of the Cauchy function, is used. New results on stabilization of linear and nonlinear systems are proposed.

Ferroelectric 90° Domain Evaluation in Tetragonal Pb(Mg1/3Nb2/3)O3–PbTiO3 Ceramics

The domain structure of ferroelectrics changes during poling has a direct influence on the macroscopic properties of the materials. The intensity variation of the different X‐ray diffraction (XRD) pattern profiles was used to identify the percentage of 90° domain reorientation in the tetragonal phase of Pb(Mg1/3Nb2/3)O3–PbTiO3 (PMN–PT) ceramics after poling. The results are consistent with the change of piezoelectric properties. In addition, by using XRD patterns, a spatial distribution of polarization in a well‐poled 0.62PMN–0.38PT ceramics has been determined and was found to be best described by the Cauchy function W00l (ϕ)=1/(1+0.023ϕ2).

Aberration reduction by multiple relays of an incoherent image

Consider a generally aberrated one-dimensional (1D) optical pupil P illuminated by quasi-monochromatic light of mean wavelength lambda. In past work it was found that, if the pupil's intensity point-spread function (psf) is multiply convolved with itself, as in an imaging relay system, and then ideally (stigmatically) demagnified, the resulting psf s(x) approaches a fixed Cauchy form s(x) = deltax( pi2x2 + deltax2)(-1), which is independent of the aberrations of the pupil. Here deltax is the Nyquist sampling interval given by deltax = lambdaf/2 with f the f/number of the pupil. This Cauchy form for this intensity psf s(x) also manifestly lacks sidelobes. The overall questions that we examine are how far do these effects carry over to the case of a circular, two-dimensional (2D) pupil, and to what extent do practical imaging considerations compromise the theoretical results? It is found that, in the presence of spherical aberration of all orders, the resulting theoretical psf of a large number of self-convolutions approaches a "circular" Cauchy form, S(r) = 2deltar[pi2r2 + (4deltar/pi)2](-3/2), where deltar is the Nyquist sampling interval lambdaf/2 with f the f/number of the (now) circular pupil. Thus, for these aberrations the 1D effect does carry over to the 2D case: The output psf does not depend on the aberrations and completely lacks sidelobes. However, when all aberrations are generally present, the output psf s(r, theta) does depend on the aberrations, although its azimuthal average over theta still preserves the circular Cauchy form, as a superposition of Cauchy functions. Imaging requirements for achieving these ideal effects are briefly discussed as well as probability laws for photons that are implied by the above-mentioned PSF's s(x) and S(r). Real-time super resolution is not attained, since the stigmatic imaging demanded of the demagnification step requires the use of a larger-apertured lens. Rather, the approach achieves significant aberration suppression.