The capacitated standard response fire protection siting problem: Deterministic and probabilistic models

We propose and present computational experience on a model that sites capacitated stations, engine companies and truck companies in such a way that the population or calls covered by the standard response is maximized. The standard response to a fire alarm, as defined by the Insurance Services Office (ISO), consists typically of at least three engine companiesand at least two trucks companies, located within standard distances (different for engines and trucks) from every demand point. The model utilized limits the capacity of each station toC _j vehicles which could be either engines or trucks or a mixture. The possibility that at the time a call comes in the full standard response will not be available within the applicable distance standards is considered as well in a probabilistic standard response model.     

Deterministic Diffusion

Ukrainian Mathematical Journal - We present a series of definitions and properties of lifting dynamical systems (LDS) corresponding to the notion of deterministic diffusion. We give heuristic...     

Stochastic and Deterministic Bundles

We consider a bundle determined by a classifying map with skeleton smooth in the Chen — Souriau sense. We show that the stochastic classifying map is homotopic to a deterministic classifying map on the Holder loop space. In honor of Prof. A. Skorokhod for his 75th birthday __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 9, pp. 1284–1288, September, 2005.     

Constraint Selection and Deterministic Annealing

The deterministic annealing optimization method is related to homotopy methods of optimization, but is oriented towards global optimization: specifically, it tries to tune a penalty parameter, thought of as ``temperature', in such a way as to reach a global optimum. Optimization by deterministic annealing is based on thermodynamics, in the same sense that simulated annealing is based on statistical mechanics. It is claimed to be very fast and effective, and is popular in significant engineering applications. The language used to describe it is usually that of statistical physics and there has been relatively little attention paid by the optimization community; this paper in part attempts to overcome this barrier by describing deterministic annealing in more familiar terms.The main contribution of this paper is to show explicitly that that constraints can be handled in the context of deterministic annealing by using constraint selection functions, a generalization of penalty and barrier functions. Constraint selection allows embedding of discrete problems into (non-convex) continuous problems.We also show how an idealized version of deterministic annealing can be understood in terms of bifurcation theory, which clarifies limitations of its convergence properties.     

Deterministic mean-variance-optimal consumption and investment

In dynamic optimal consumption–investment problems one typically aims to find an optimal control from the set of adapted processes. This is also the natural starting point in case of a mean-variance objective. In contrast, we solve the optimization problem with the special feature that the consumption rate and the investment proportion are constrained to be deterministic processes. As a result we get rid of a series of unwanted features of the stochastic solution including diffusive consumption, satisfaction points and consistency problems. Deterministic strategies typically appear in unit-linked life insurance contracts, where the life-cycle investment strategy is age dependent but wealth independent. We explain how optimal deterministic strategies can be found numerically and present an example from life insurance where we compare the optimal solution with suboptimal deterministic strategies derived from the stochastic solution.     

Generalized discrepancies on the sphere

Generalized discrepancies are a class of discrepancies introduced in the seminal paper [1] to measure uniformity of points over the unit sphere in ℝ³. However, convergence to 0 of this quantity has been shown only in the case of spherical t –designs. In the following, we completely characterize sequences for which convergence to 0 of D (𝒫_N ; A ) holds. The interest of this result is that, when evaluating uniformity on the sphere, generalized discrepancies are much simpler to compute than the well-known spherical cap discrepancy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Minimal discrepancies of toric singularities

The main purpose of this paper is to prove that minimal discrepancies ofn-dimensional toric singularities can accumulate only from above and only to minimal discrepancies of toric singularities of dimension less thann. I also prove that some lower-dimensional minimal discrepancies do appear as such limit.     

Statistical properties of generalized discrepancies

When testing that a sample of points in the unit hypercube comes from a uniform distribution, the Kolmogorov-Smirnov and the Cramér-von Mises statistics are simple and well-known procedures. To encompass these measures of uniformity, Hickernell introduced the so-called generalized -discrepancies. These discrepancies can be used in numerical integration through Monte Carlo and quasi-Monte Carlo methods, design of experiments, uniformity testing and goodness-of-fit tests. The aim of this paper is to derive the statistical asymptotic properties of these statistics under Monte Carlo sampling. In particular, we show that, under the hypothesis of uniformity of the sample of points, the asymptotic distribution is a complex stochastic integral with respect to a pinned Brownian sheet. On the other hand, if the points are not uniformly distributed, then the asymptotic distribution is Gaussian.

     

Deterministic diffusion of particles

The diffusion of the particles is described in terms of a mean motion with a speed equal to the osmotic velocity associated with the diffusion process. Three numerical schemes are presented. The first two are based on the approximation of the gradient on an irregular mesh. The third is derived from a finite-element approach. Voronoi diagrams are used to handle the irregular grid of the particles. The convergence of the schemes is studied numerically, by comparing the results with the exact solution. Applications to the Fokker-Planck equation and to the problem of disposing particles according to a given probability distribution are presented.     

Deterministic flow-demand location problems

In flow-covering (interception) models the focus is on the demand for service that originates from customers travelling in the network (not for the purpose of obtaining the service). In contrast, in traditional location models a central assumption is that the demand for service comes from customers residing at nodes of the network. In this paper we combine these two types of models. The paper presents four new problems. Two of the four deal with the problem of locating m facilities so as to maximize the total number of potential customers covered by the facilities (where coverage does not necessarily imply the actual consumption of service). In the two other problems the attention is directed to the consumption of service and thus the criteria is to maximize (minimize) the number of actual users (distance travelled). It is shown in the paper that all four problems have similar structure to other known location problems.     

Deterministic sparse FFT algorithms

In this paper we consider sparse signals which are known to vanish outside a support interval of length bounded by m < N. For the case that m is known, we propose a deterministic algorithm of complexity for reconstruction of x from its discrete Fourier transform (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Deterministic Aperiodic Tile Sets

Wang tiles are square tiles with colored edges. We construct an aperiodic set of Wang tiles that is strongly deterministic in the sense that any two adjacent edges of a tile determine the tile uniquely. Consequently, the tiling group of this set is not hyperbolic and it acts discretely and co-compactly on a CAT(0) space. Submitted: February 1998, revised: October 1998.     

Fast Deterministic Processor Allocation

Interval allocation has been suggested as a possible formalization for the PRAM of the (vaguely defined) processor allocation problem, which is of fundamental importance in parallel computing. The interval allocation problem is, given n nonnegative integers x₁, . . ., x_n, to allocate n nonoverlapping subarrays of sizes x₁, . . ., x_n from within a base array of O(Σⁿ_j=1x_j) cells. We show that interval allocation problems of size n can be solved in O((log log n)³) time with optimal speedup on a deterministic CRCW PRAM. In addition to a general solution to the processor allocation problem, this implies an improved deterministic algorithm for the problem of approximate summation. For both interval allocation and approximate summation, the fastest previous deterministic algorithms have running times of Θ(log n/log log n). We describe an application to the problem of computing the connected components of an undirected graph. Finally we show that there is a nonuniform deterministic algorithm that solves interval allocation problems of size n in O(log log n) time with optimal speedup.     

Deterministic Minimax Impulse Control

We prove the uniqueness of the viscosity solution of an Isaacs quasi-variational inequality arising in an impulse control minimax problem, motivated by an application in mathematical finance.     

Logic and probabilistic systems

Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set of the sentences that get limit probability 1 is a theory, in fact can be a complete set. We prove incompleteness results for this setting, by showing for instance that for every there are true sentences that get limit probability less than . No set as above can contain the set of all true sentences, although we exhibit some containing all the true sentences. We also consider an approach based on the notions of inner probability and outer probability, and we compare this approach with the one based on trial and error probabilistic functions. Although the two approaches are shown to be different, we single out an important case in which they are equivalent. Received March 20, 1995     

Some Deterministic and Random Vector Equilibrium Problems

In this paper, some generalized concepts about semicontinuity and convexity for vector-valued bifunctions are introduced. A new existence theorem for vector equilibrium problems is proved. Further, a random version of this result is considered. Applications to cone saddle points, random vector optimization, random vector variational inequalities, and random vector best approximation problems are finally given.