A fuzzy process of individuation

ABSTRACT

It is shown that an aspect of the process of individuation may be thought of as a fuzzy set. The process of individuation has been interpreted as a two-valued problem in the history of philosophy. In this work, I intend to show that such a process in its psychosocial aspect is better understood in terms of a fuzzy set, characterized by a continuum membership function. According to this perspective, species and their members present different degrees of individuation. Such degrees are measured from the membership function of the psychosocial process of individuation. Thus, a social analysis is suggested by using this approach in human societies.     

A reliability semi‐Markov model involving geometric processes

We consider a semi‐Markov process that models the repair and maintenance of a repairable system in steady state. The operating and repair times are independent random variables with general distributions. Failures can be caused by an external source or by an internal source. Some failures are repairable and others are not. After a repairable failure, the system is not as good as new and our model reflects that. At a non‐repairable failure, the system is replaced by a new one. We assume that external failures occur according to a Poisson process. Moreover, there is an upper limit N of repairs, it is replaced by a new system at the next failure, regardless of its type. Operational and repair times are affected by multiplicative rates, so they follow geometric processes. For this system, the stationary distribution and performance measures as well as the availability and the rate of occurrence of different types of failures in stationary state are calculated. Copyright © 2002 John Wiley & Sons, Ltd.     

A computer‐aided methodology for the optimization of electrostatic separation processes in recycling

The rapid growth of technological products has led to an increasing volume of waste electrical and electronic equipments (WEEE), which could represent a valuable source of critical raw materials. However, current mechanical separation processes for recycling are typically poorly operated, making it impossible to modify the process parameters as a function of the materials under treatment, thus resulting in untapped separation potentials. Corona electrostatic separation (CES) is one of the most popular processes for separating fine metal and nonmetal particles derived from WEEE. In order to optimize the process operating conditions (i.e., variables) for a given multi‐material mixture under treatment, several technological and economical criteria should be jointly considered. This translates into a complex optimization problem that can be hardly solved by a purely experimental approach. As a result, practitioners tend to assign process parameters by few experiments based on a small material sample and to keep these parameters fixed during the process life‐cycle. The use of computer experiments for parameter optimization is a mostly unexplored area in this field. In this work, a computer‐aided approach is proposed to the problem of optimizing the operational parameters in CES processes. Three metamodels, developed starting from a multi‐body simulation model of the process physics, are presented and compared by means of a numerical and simulation study. Our approach proves to be an effective framework to optimize the CES process performance. Furthermore, by comparing the predicted response surfaces of the metamodels, additional insight into the process behavior over the operating region is obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Bernoulli Cluster Field: Voronoi Tessellations

A new point process is proposed which can be viewed either as a Boolean cluster model with two cluster modes or as a p-thinned Neyman-Scott cluster process with the retention of the original parent point. Voronoi tessellation generated by such a point process has extremely high coefficients of variation of cell volumes as well as of profile areas and lengths in the planar and line induced tessellations. An approximate numerical model of tessellation characteristics is developed for the case of small cluster size; its predictions are compared with the results of computer simulations. Tessellations of this type can be used as models of grain structures in steels.     

A phenomenon of waiting time in phase change problems driven by radiative heat transfer

In the paper (J. Food Process Eng. 2008; in press) we emphasized that during a phase change process in which the heat input is driven by a radiation transfer mechanism, a peculiar phenomenon may occur, characterized by a temporary stop of the increase of the boundary temperature due to a sudden change of the heat transfer coefficient upon phase transition. This time interval is needed to allow the thermal properties of the surface to evolve toward a state that is compatible with the heat intake rate corresponding to the new phase. The occurrence of the waiting time is motivated and studied for a general one‐dimensional Stefan problem. Then an application is presented to the much complicated problem considered in (J. Food Process Eng. 2008; in press), namely, the model for frying process. Copyright © 2008 John Wiley & Sons, Ltd.     

Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators

A stochastic variational inequality is proposed to model a white noise excited elasto-plastic oscillator. The solution of this inequality is essentially a continuous diffusion process for which a governing diffusion equation is obtained to study the evolution in time of its probability distribution. The diffusion equation is degenerate, but using the fact that the degeneracy occurs on a bounded region we are able to show the existence of a unique solution satisfying the desired properties. We prove the ergodic properties of the process and characterize the invariant measure. Our approach relies on extending Khasminskii’s method (Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980), which in the present context leads to the study of degenerate Dirichlet problems with nonlocal boundary conditions. This research was partially supported by a grant from CEA, Commissariat à l’énergie atomique and by the National Science Foundation under grant DMS-0705247.     

Three dimensional distribution of Brownian motion extrema

This paper describes the joint distributions of minima, maxima and endpoint values for a three dimensional (3D) Wiener process. In particular, the results provide the joint cumulative distributions for the maxima and/or minima of the components of the process. The method of images is used to derive explicit expressions for the densities; the analysis can only be carried out for special correlation structures and requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used in several applied fields such as financial mathematics to obtain analytical expressions for prices of options for the 3D geometric Brownian motion process.     

Simulation Study for the Clan of Ancestors in a Perfect Simulation Scheme of a Continuous One-Dimensional Loss Network

Perfect simulation of a one-dimensional loss network on ℝ with length distribution π and cable capacity C is performed using the clan of ancestors method. Previous works estimated the region of convergence of this scheme using a domination by a branching process. In this work, we show that the domination by the branching process is far from sharp and that there is room for improvement. Moreover, we derive an empirical relation concerning the critical value using simulation studies on the number of rectangles present in the clan of ancestors.      

A Characteristic Martingale Related to the Counting Process of Records

Let (X _n ) be a sequence of nonnegative, integrable, independent and identically distributed random variables, with common distribution function F. We consider the problem of finding all distribution functions F such that N _n −cM _n is a discrete time martingale, where N _n is the counting process of upper records, M _n =max {X ₁,…,X _n } is the process of partial maxima and c is a positive constant. We solve the problem by explicitly giving the solution with finite support and using this for constructing the solution for the general case by a limiting process. We show that the set of solutions can be parameterized by their support and the mass at the leftmost point of the support.     

Some properties of convex hulls generated by homogeneous Poisson point processes in an unbounded convex domain

LetC(A) be the convex hull generated by a Poisson point process in an unbounded convex setA. A representation ofAC(A) as the union of curvilinear triangles with independent areas is established. In the case whenA is a cone the properties of the representation are examined more completely. It is also indicated how to simulateC(A) directly without first simulating the process itself.     

Yarn motion during unwinding from packages

We study the motion of yarn modelled as a one-dimensional inelastic string. In textile production, the yarn is being withdrawn from cross-wound packages in warping and weft insertion. During unwinding, there appear forces in the yarn that are approximately proportional to the square of the unwinding velocity. The yarn tension is not constant, but it oscillates within some interval. This is especially noticeable in over-end unwinding from a static cross-wound package. Even when the yarn is not strongly stressed, so that the tension never exceeds a few percent of the breaking strength, the yarn can still break sometimes. The production process requires as large warping and weaving speeds as possible; therefore, it is necessary to improve our understanding of the cross-wound package unwinding and to find the necessary modifications of the yarn unwinding process. In addition to empirical tests, it has proved useful to study yarn unwinding by mathematical modelling and computer simulations. We state the equations of motion that describe the yarn unwinding and develop a mathematical model that permits to simulate the process of unwinding.     

Generalized Inv-Log-Gamma-G processes

Abstract

Gamma processes belong to subordinators for which very small jumps occurs infinitely many times in any finite time interval but their sums are finite. Here we consider their novel and important modifications with a nice application potential. A generalization of fractional kth lower record value process defined in Bieniek and Szynal, called Inverse-Log-Gamma-G process is investigated. Explicit relation with the Gamma process is presented and conditional, posterior and finite dimensional distributions are derived. The results are obtained by appropriate transformations of known stochastic processes. In contrast with the regression this allows us to describe the finite dimensional distributions of the processes of interest and in this way to make their full characterization.     

The Gerber-Shiu function and the generalized Cramér-Lundberg model

In this paper we extend some results in Cramér [7] by considering the expected discounted penalty function as a generalization of the infinite time ruin probability. We consider his ruin theory model that allows the claim sizes to take positive as well as negative values. Depending on the sign of these amounts, they are interpreted either as claims made by insureds or as income from deceased annuitants, respectively. We then demonstrate that when the events’ arrival process is a renewal process, the Gerber-Shiu function satisfies a defective renewal equation. Subsequently, we consider some special cases such as when claims have exponential distribution or the arrival process is a compound Poisson process and annuity-related income has Erlang(n, β) distribution. We are then able to specify the parameter and the functions involved in the above-mentioned defective renewal equation.     

Rates of convergence of diffusions with drifted Brownian potentials

We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.

     

The Gaussian approximation for multi-color generalized Friedman’s urn model

The generalized Friedman’s urn model is a popular urn model which is widely used in many disciplines.In particular,it is extensively used in treatment allocation schemes in clinical trials.In this paper,we show that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a multi-color generalized Friedman’s urn model with both homogeneous and non-homogeneous generating matrices.The Gaussian process is a solution of ...     

Coagulation and fragmentation with discrete mass loss

A nonlinear integro-differential equation that models a coagulation and multiple fragmentation process in which discrete fragmentation mass loss can occur is examined using the theory of strongly continuous semigroups of operators. Under the assumptions that the coagulation kernel K is bounded and the fragmentation rate function a satisfies a linear growth condition, global existence and uniqueness of solutions that lose mass in accordance with the model are established. In the case when no coagulation is present and the fragmentation process is governed by power-law kernels, an explicit formula is given for the substochastic semigroup associated with the resulting mass-loss fragmentation equation.     

Optimal Selling of an Asset with Jumps Under Incomplete Information

ABSTRACT

We study the optimal liquidation strategy of an asset with price process satisfying a jump diffusion model with unknown jump intensity. It is assumed that the intensity takes one of two given values, and we have an initial estimate for the probability of both of them. As time goes by, by observing the price fluctuations, we can thus update our beliefs about the probabilities for the intensity distribution. We formulate an optimal stopping problem describing the optimal liquidation problem. It is shown that the optimal strategy is to liquidate the first time the point process falls below (goes above) a certain time-dependent boundary.     

Block Orderings for Tensor-Product Grids in Two and Three Dimensions

We consider two-line and two-plane orderings for a convection–diffusion model problem in two and three dimensions, respectively. These strategies are aimed at introducing dense diagonal blocks, at the price of a slight increase of the bandwidth of the matrix, compared to natural lexicographic ordering. Comprehensive convergence analysis is performed for the block Jacobi scheme. We then move to consider a two-step preconditioning technique, and analyze the numerical properties of the linear systems that are solved in each step of the iterative process. For the 3-dimensional problem this approach is a viable alternative to the Incomplete LU approach, and may be easier to implement in parallel environments. The analysis is illustrated and validated by numerical examples.     

A central limit theorem for normalized products of random matrices

This note concerns the asymptotic behavior of a Markov process obtained from normalized products of independent and identically distributed random matrices. The weak convergence of this process is proved, as well as the law of large numbers and the central limit theorem. This work was supported by the PSF Organization under Grant No. 2005-7-02, and by the Consejo Nacional de Ciencia y Tecnología under Grants 25357 and 61423.