Piotr Mikusiński

The two-sided laplace transform for boehmians

An extension of the two-sided Laplace transform to a space of Boehmians is denned. The extension preserves operational properties of the transform and applies to generalized functions with unbounded support which are locally of infinite order. A large class of analytic functions which are transforms of Boehmians is described. An application to differential equations is presented.

Probabilistic Interpretations of Copulas and Their Convex Sums

The copula C contains valuable information about the type of dependence that exists between random variables having C as their copula. It is this dependence, captured in the copula, which is of interest in this paper. One might think of a copula as a canonical representative of all those distribution functions H that correspond to random variables X and Y which have a specific sort of relationship to each other. It would be nice to have a catalog of copulas which specifies the sort of dependence each copula captures. With such a long term goal in mind, we give probabilistic interpretations for certain copulas, i.e., we give necessary and sufficient conditions on the random variables X and Y in order that C should be their copula. Also we give probabilistic interpretations for convex sums (mixtures) of copulas.

On Approximation of Copulas

One can hope to prove interesting properties of copulas by verifying them first for some class of “simple” copulas then invoking a limit process. The most commonly used limit for copulas is the uniform limit. However the uniform limit is not completely satisfactory in that, on the one hand, any copula can be approximated arbitrarily closely by invertible copulas and, on the other hand, the * operation of Darsow, Nguyen, and Olsen is not jointly continuous with respect to the uniform limit. We show that certain approximations of copulas lead naturally to a stronger convergence than uniform convergence and give a short proof of the associativity of * as an application.

On flexibility of boehmians

Two problems concerning Boehmians are discussed: local comparison of Boehmians and regular operators and composition of Boehmians and diffeomorphisms.

The Radon transform of Boehmians

The Radon transform, which enables one to reconstruct a function of N variables from the knowledge of its integrals over all hyperplanes of di- mension N - 1 , has been extended to Schwartz distributions by several people including Gelfand, Graev, and Vilenkin, who extended it to tempered distribu- tions. In this paper we extend the Radon transform to a space of Boehmians. Boehmians are defined as sequences of convolution quotients and include Schwartz distributions and regular Mikusinski operators. Our extension of the Radon transform includes generalized functions of infinite order with compact support. The technique used in this paper is based on algebraic properties of the Radon transform and its convolution structure rather than on their analytic properties. Our results do not contain nor are contained in those obtained by Gelfand et al.

Investigating the relationship between adverse events and infrastructure development in an active war theater using soft computing techniques

We investigate the relationship between adverse events and infrastructure development investments in an active war theater.We develop soft computing techniques (ANN, FIS, and ANFIS) for estimating the number of adverse events based on the occurrence of economic improvement projects.The performance of each model was investigated and compared to all other models using the calculated mean absolute percentage error (MAPE) values.When the model accuracy was calculated based on the MAPE for each of the models, ANN had better predictive accuracy than FIS and ANFIS models, as demonstrated by experimental results.The sensitivity analysis results show that the importance of economic development projects varied based on the specific regions and time period. The purpose of this paper is to investigate the relationship between adverse events and infrastructure development investments in an active war theater by using soft computing techniques including fuzzy inference systems (FIS), artificial neural networks (ANNs), and adaptive neuro-fuzzy inference systems (ANFIS) where the accuracy of the predictions is directly beneficial from an economic and humanistic point of view. Fourteen developmental and economic improvement projects were selected as independent variables. A total of four outputs reflecting the adverse events in terms of the number of people killed, wounded or hijacked, and the total number of adverse events has been estimated.The results obtained from analysis and testing demonstrate that ANN, FIS, and ANFIS are useful modeling techniques for predicting the number of adverse events based on historical development or economic project data. When the model accuracy was calculated based on the mean absolute percentage error (MAPE) for each of the models, ANN had better predictive accuracy than FIS and ANFIS models, as demonstrated by experimental results. For the purpose of allocating resources and developing regions, the results can be summarized by examining the relationship between adverse events and infrastructure development in an active war theater, with emphasis on predicting the occurrence of events. We conclude that the importance of infrastructure development projects varied based on the specific regions and time period.

Doubly stochastic measures, topology, and latticework hairpins

Abstract We investigate topologies which can be put on the set of doubly stochastic measures on the unit square and show equivalence of all the standard ones and others considered here. This topology has a very nice characterization in terms of copulas. In terms of this topology we establish a simple relationship between a metric distance between two doubly stochastic measures concentrated on hairpins and the sup-norm distance between their mass spreaders, show the denseness of polygonal hairpins in the set of hairpins, and show the nowhere denseness of latticework hairpins in the set of hairpins.

The Fourier transform of Lévy measures on a semigroup

Using completely positive-definite functions defined on a commutative semigroup, we construct a space of pseudoquotients that can be identified, via a Fourier transform, with a class of Lévy measures. As an application we obtain a generalization of the Lévy-Khinchin representation for completely alternating functions.

On De Graaf spaces of pseudoquotients

A space of pseudoquotients

Application of diffusion maps to identify human factors of self-reported anomalies in aviation.

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