This paper concerns the study of the Bell polynomials and the binomial type sequences. We mainly establish some relations tied to these important concepts. Furthermore, these obtained results are ...exploited to deduce some interesting relations concerning the Bell polynomials which enable us to obtain some new identities for the Bell polynomials. Our results are illustrated by some comprehensive examples.
Air pollution poses a major problem in modern cities, as it has a significant effect in poor quality of life of the general population. Many recent studies link excess levels of major air pollutants ...with health-related incidents, in particular respiratory-related diseases. This introduces the need for city pollution on-line monitoring to enable quick identification of deviations from “normal” pollution levels, and providing useful information to public authorities for public protection. This article considers dynamic monitoring of pollution data (output of multivariate processes) using Kalman filters and multivariate statistical process control techniques. A state space model is used to define the in-control process dynamics, involving trend and seasonality. Distribution-free monitoring of the residuals of that model is proposed, based on binomial-type and generalised binomial-type statistics as well as on rank statistics. We discuss the general problem of detecting a change in pollutant levels that affects either the entire city (globally) or specific sub-areas (locally). The proposed methodology is illustrated using data, consisting of ozone, nitrogen oxides and sulfur dioxide collected over the air-quality monitoring network of Athens.
Given a commutative ring with identity
R
, many different and interesting operations can be defined over the set
H
R
of sequences of elements in
R
. These operations can also give
H
R
the structure ...of a ring. We study some of these operations, focusing on the binomial convolution product and the operation induced by the composition of exponential generating functions. We provide new relations between these operations and their invertible elements. We also study automorphisms of the Hurwitz series ring, highlighting that some well-known transforms of sequences (such as the Stirling transform) are special cases of these automorphisms. Moreover, we introduce a novel isomorphism between
H
R
equipped with the componentwise sum and the set of the sequences starting with 1 equipped with the binomial convolution product. Finally, thanks to this isomorphism, we find a new method for characterizing and generating all the binomial type sequences.
In this paper, we show that the solution to a large class of "tiling'' problems is given by a polynomial sequence of binomial type. More specifically, we show that the number of ways to place a fixed ...set of polyominos on an $n\times n$ toroidal chessboard such that no two polyominos overlap is eventually a polynomial in $n$, and that certain sets of these polynomials satisfy binomial-type recurrences. We exhibit generalizations of this theorem to higher dimensions and other lattices. Finally, we apply the techniques developed in this paper to resolve an open question about the structure of coefficients of chromatic polynomials of certain grid graphs (namely that they also satisfy a binomial-type recurrence).
A large amount of SPC procedures are based on the assumption that the process subject to monitoring consists of independent observations. Chemical processes as well as many non-industrial processes ...exhibit autocorrelation, for which the above-mentioned control procedures are not suitable. This paper proposes a Phase II control procedure for autocorrelated and possibly locally stationary processes. A time-varying autoregressive (AR) model is proposed, which is capable of dealing with the autocorrelation as well as with local non-stationarities of the temporal process. Such non-stationarities are induced by the time-varying nature of the AR coefficients. The model is optimized during Phase I when it is assured that the process is in control and as a result the model describes accurately the process. The Phase II proposed control procedure is based on a comparison of the current time series model with an alternative model, measuring deviations from it. This comparison is carried out using Bayes factors, which help to establish the in-control or out-of-control state of the process in Phase II. Using the threshold rules of the Bayes factors, we propose a binomial-type control procedure for the monitoring of the process. The methodology of this paper is illustrated using two data sets consisting of temperature measurements at two different stages in the manufacturing of a plastic mould.
This paper introduces a construction principle for generating matrices of digital sequences over a finite field
, which is based on sequences of polynomials and their representations in terms of ...powers of nonconstant polynomials. For the most basic polynomial sequence,
, the representations in terms of powers of linear polynomials yield, within this construction principle, the Pascal matrices, which consist of binomial coefficients and were earlier introduced by Faure for finite prime fields and by Niederreiter for finite field extensions. Generally, for binomial type sequences of polynomials an interesting relation between the generating matrices is worked out, and further examples of generating matrices are given, which contain combinatorial magnitudes as, e.g., binomial coefficients, Stirling numbers of the first kind, Stirling numbers of the second kind, and Bell numbers. Moreover, within this construction principle, explicit constructions of finite-row generating matrices of digital
-sequences are presented, which were so far only known for
equal to
.
Linear models for paired comparisons, the Bradley-Terry model and the Thurstone-Mosteller model in particular, are widely used in sports for ranking and rating purposes. By their formulation, these ...models predict the probability that a player or team defeats another if the playing strengths of the players or teams are known. In this paper, we investigate the prediction accuracy of the two linear models by using them to describe three simple theoretical games which mimic actual sports and whose winning probability, given the playing strength of each player, can be expressed explicitly. A theoretical result is presented, which provides the basis of a linearization method that enables these games to be represented by linear models. The predicted winning probabilities from the linear models are then compared to the actual ones. Comparisons are also made in prediction accuracy between the Bradley-Terry model and the Thurstone-Mosteller model.
A paper by D.L. Reiner researched function sequences of binomial type and established many interesting theorems. But because of an oversight in a lemma, some theorems in his paper need to be revised. ...This paper will revise these theorems and establish some new results on sequences of binomial type.
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