On the right spread order of convolutions of heterogeneous exponential random variables

A sufficient condition for comparing convolutions of heterogeneous exponential random variables in terms of right spread order is established. As a consequence, it is shown that a convolution of heterogeneous independent exponential random variables is more skewed than that of homogeneous exponential random variables in the sense of NBUE order. This gives a new insight into the distribution theory of convolutions of independent random variables. A sufficient condition is also derived for comparing such convolutions in terms of Lorenz order.     

Complex Stable Sums of Complex Stable Random Variables

A definition of complex stable random variables is presented which includes earlier definitions as special cases. The class of complex stable random variables is characterized and is shown to be a subclass of the operator stable random variables. The exact conditions under which a sum of independent complex stable random variables is again complex stable are also found.     

A stopping game in a stochastic and fuzzy environment

A zero-sum stopping game for a sequence of fuzzy-valued random variables is discussed. The fuzzy random variables are estimated by probabilistic expectation and fuzzy expectation. A saddle point is given for the stopping game.     

Tests for multiple monotone symmetric conglutinations of variables in Boolean functions

A linear lower and a quadratic upper bound on the Shannon function for the length of a detecting test for multiple monotone symmetric conglutinations are proved. Additionally, the exact value of the Shannon function for the length of a diagnostic test for multiple monotone symmetric conglutinations is found. Some of the variables of a Boolean function are said to conglutinate if each of them is replaced by a function of these variables. A conglutination is called multiple if there are several groups of conglutinated variables.     

Optimal investment with two-factor uncertainty

This paper presents a real options model to value the option to invest in a project contingent on two stochastic factors. A general sensitivity analysis is conducted highlighting the importance of the variance and correlation between the two variables. A higher correlation is shown to increase always the values of the trigger, the active project and the option. The impact of uncertainty is more complex and depends on the assumption about which variables adjust and the correlation between the variables and the market.     

A heuristic algorithm for a chance constrained stochastic program

A chance constrained stochastic program is considered that arises from an application to college enrollments and in which the objective function is the expectation of a linear function of the random variables. When these random variables are independent and normally distributed with mean and variance that are linear in the decision variables, the deterministic equivalent of the problem is a nonconvex nonlinear knapsack problem. The optimal solution to this problem is characterized and a greedy-type heuristic algorithm that exploits this structure is employed. Computational results show that the algorithm performs well, especially when the normal random variables are approximations of binomial random variables.     

Almost sure convergence of sums of maxima and minima of positive random variables

For a sequence of independent and identically distributed positive random variables, the almost sure convergence of sums of maxima (when suitably normalized) to appropriate constants is proved for both bounded and unbounded random variables. A similar result is also proved for sums of minima of such variables.     

A central limit theorem for fuzzy random variables

Fuzzy random variables have been proposed to treat situations in which both random behavior and fuzzy perception must be considered. A definition of independence is given for fuzzy random variables, as well as a notion of fuzzy Gaussian random variables. It is shown that a sum or mean of independent fuzzy random variables converges in the limit to a fuzzy Gaussian random variable, thus providing a fuzzy analogue of the central limit theorem of classical probability theory.     

Nonnormal deterministic equivalents and a transformation in stochastic mathematical programming

This paper considers random variables of the continuous type in a stochastic programming problem and presents (1) a general approach to the development of deterministic equivalents of constraints to be satisfied within certain probability limits, and (2) a deterministic transformation of a stochastic programming problem with random variables in the objective function. Deterministic equivalents are developed for constraints containing uniform random variables, but the approach used can be applied to other types of continuous random variables, as well. When the random variables appear in the objective function, a deterministic transformation of the stochastic programming problem is obtained to yield a closed-form solution without resort to a Monte Carlo computer simulation. Extension of this approach to stochastic problems with discrete random variables and integer decision variables is discussed briefly. A numerical example is presented.     

The number of centered forms for a polynomial

A general centered form for polynomials is defined. This centered form is based on the concept of an arrangement of the variables of the polynomial. A formula for the number of arrangements of a term and of a polynomial is given. Some examples are computed showing that even polynomials with moderately many variables may have a very large number of possible centered forms.     

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.     

On the mean number of trials until the last trials satisfy a given condition

Random variables are collected one at a time until the last k variables satisfy a given condition. The mean waiting time until this happens is studied and a general lemma is given. If the condition is satisfied by many possible stopping sequences the probability distribution of the k last variables is discussed. A fairly general treatment is given for the case when k = 2. Two other special cases are mentioned, viz. the case when the condition is defined by order relations between the last variables, and the case when the variables are discrete.     

A branch-and-cut procedure for the Udine Course Timetabling problem

A branch-and-cut procedure for the Udine Course Timetabling problem is described. Simple compact integer linear programming formulations of the problem employ only binary variables. In contrast, we give a formulation with fewer variables by using a mix of binary and general integer variables. This formulation has an exponential number of constraints, which are added only upon violation. The number of constraints is exponential. However, this is only with respect to the upper bound on the general integer variables, which is the number of periods per day in the Udine Course Timetabling problem.     

On a property of sums of independent variables

A result is obtained for sums of independent random variables. On the basis of the properties of a sum of random variables we obtain a conclusion with regard to the properties of the terms.     

A linear approximation model for the parameter design problem

We consider the problem of minimizing the variance of a nonlinear function of several random variables, where the decision variables are the mean values of these random variables. This problem arises in the context of robust design of manufactured products and/or manufacturing processes, where the decision variables are the set points for various design components. A nonlinear programming (NLP) model is developed to obtain approximate solutions for this problem. This model is based on a Taylor series expansion of the given function, up to its linear term. A case study pertaining to the design of a coil spring is discussed, and its corresponding NLP model is developed. Using this model, a non-inferior frontier curve is obtained that shows the trade-off between the minimum mass of the coil spring and the minimum variance of its performance characteristic. Results of applying the model to three other case studies are also presented.     

On a Simpler Basis for the Pseudovariety EDS

A basis for the pseudovariety EDS which involves two distinct variables is computed. This improves upon a presently known basis for EDS that involves three distinct variables.     

A local limit theorem for independent random variables

A local limit theorem is given for independent noninteger random variables under a condition which is more general than one previously given, and which reduces, in the case of identically distributed random variables, to a well-known result.     

A statistic for measuring the balance of a sample

Consider a finite population which has many auxiliary variables. A statistic, which is a function of the moments of the auxiliary variables, is proposed to measure the balance of a sample. The mean and variance of this statistic are derived.     

Super Jaulent-Miodek hierarchy and its super Hamiltonian structure,conservation laws and its self-consistent sources

A super Jaulent-Miodek hierarchy and its super Hamiltonian structures are constructed by means of a kind of Lie super algebras and super trace identity. Moreover, the self-consistent sources of the super Jaulent-Miodek hierarchy is presented based on the theory of self-consistent sources. Further- more, the infinite conservation laws of the super Jaulent-Miodek hierarchy are also obtained. It is worth noting that as even variables are boson variables, odd variables are fermi variables in the spectral problem, the commutator is different from the ordinary one.     

The solution and sensitivity of a general optimal control problem

A solution method for the general optimal control problem is presented. This can be used to solve optimal control problems for which the system dynamics are not necessarily described by differential or difference equations. Having obtained the solution it is of theoretical and practical interest to investigate the sensitivity of the optimal trajectory to perturbations. Indicators of this sensitivity are the adjoint variables derived in the maximum principle. A method of deriving the adjoint variables from the solution is described. To illustrate the solution method and the determination of the adjoint variables a problem in the urban housing market is used.