Raúl Fierro

Bio: Raúl Fierro is an academic researcher from Pontifical Catholic University of Valparaíso. The author has contributed to research in topics: Fixed point & Epidemic model. The author has an hindex of 7, co-authored 38 publications receiving 213 citations. Previous affiliations of Raúl Fierro include Valparaiso University.

A Non-Central Version of the Birnbaum-Saunders Distribution for Reliability Analysis

Abstract: The Birnbaum-Saunders distribution has largely been applied to material fatigue and reliability studies to relate the time until failure to some type of cumulative damage. This damage is produced by the growth of a dominant crack in material specimens, the propagation of which is due to cyclic patterns of stress. However, the Birnbaum-Saunders model was constructed under restrictive conditions that may not be valid for certain applications. In particular, this model was developed assuming that the crack extension in each cycle has a constant s-mean over time. In this article, we assume the crack extensions in each cycle have a non-constant s -mean. This leads us to introduce a non-central version of the Birnbaum-Saunders distribution. Specifically, a comprehensive description of the properties and characteristics of this new model is presented. The suitability of this non-central distribution is shown by using real, and simulated data.

Well-posedness of fixed point problems

Abstract: We state conditions for well-posedness of a fixed point problem for single and multivalued operators, where these operators are deterministic or random. These results are applied to families of different generalized contractions.

The Hawkes process with different exciting functions and its asymptotic behavior

Abstract: The standard Hawkes process is constructed from a homogeneous Poisson process and uses the same exciting function for different generations of offspring. We propose an extension of this process by considering different exciting functions. This consideration may be important in a number of fields; e.g. in seismology, where main shocks produce aftershocks with possibly different intensities. The main results are devoted to the asymptotic behavior of this extension of the Hawkes process. Indeed, a law of large numbers and a central limit theorem are stated. These results allow us to analyze the asymptotic behavior of the process when unpredictable marks are considered.

Fixed Point Theorems for Random Lower Semi-Continuous Mappings

Abstract: Let X, d be a metric space and S a closed and nonempty subset of X. Denote by 2 resp., C X the family of all nonempty resp., nonempty and closed subsets of X. A mapping T : S → 2 is said to satisfy condition P if, for every closed ball B of S with radius r ≥ 0 and any sequence {xn} in S for which d xn, B → 0 and d xn, T xn → 0 as n → ∞, there exists x0 ∈ B such that x0 ∈ T x0 where d x, B inf{d x, y : y ∈ B}. If Ω is any nonempty set, we say that the operator T : Ω × S → 2 satisfies condition P if for each ω ∈ Ω, the mapping T ω, · : S → 2 satisfies condition P . We should observe that this latter condition is related to a condition that was originally introduced by Petryshyn 1 for single-valued operators, in order to prove existence of fixed points. However, in our case, the condition is used to prove the measurability of a certain operator. On the other hand, in the year 2001, Shahzad cf. 2 using an idea of Itoh cf. 3 , see also 4 , proved that under a somewhat more restrictive condition, named condition A , the following result.

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Fixed Point Theory

Abstract: Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

An Introduction To The Theory Of Point Processes

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