###### A Lexical Approach to Estimating Environmental Goods and Services Output in the Construction Sector via Soft Classification of Enterprise Activity Descriptions Using Latent Dirichlet Allocation

## Abstract

The research question addressed here is whether the semantic value implicit in environmental terms in an activity description text string, can be translated into economic value for firms in the construction sector. We address this question using a relatively new applied statistical method called Latent Dirichlet Allocation (LDA). We first identify a satellite register of firms in construction sector that engage in some form of environmental work. From these we construct a vocabulary of meaningful words. Then, for each firm in turn on this satellite register we take its activity description text string and process this string with LDA. This *softly-classifies* the descriptions on the satellite register into just seven environmentally relevant topics. With this seven-topic classification we proceed to extract a statistically meaningful weight of evidence associated with environmental terms in each activity description. This weight is applied to the associated firm’s overall output value recorded on our national Business Register to arrive at a supply side estimate of the firm’s EGSS value. On this basis we find the EGSS estimate for construction in Ireland in 2013 is about EURO 229m. We contrast this estimate with estimates from other countries obtained by demand side methods and show it compares satisfactorily, thereby enhancing its credibility. Our method also has the advantage that it provides a breakdown of EGSS output by EU environmental classifications (CEPA/CReMA) as these align closely to discovered topics. We stress the success of this application of LDA relies greatly on our small vocabulary which is constructed directly from the satellite register.

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*n*-TH Root Selections in Fields

## Abstract

In this work we generalize the results of [9] to the higher level case: we define *n*-th root selections in fields of characteristic ≠ 2, that is subgroups of the multiplicative group of a field whose existence is equivalent to the existence of a partial inverse of the *x* ↦ *x ^{n}* function, provide necessary and sufficient conditions for such a subgroup to exist, study their existence under field extensions, and give some structural results describing the behaviour of maximal n-th root selection fields.

###### New Year Mathematical Card or V Points Mathematical Constant

## Abstract

The article describes an attempt to define a new mathematical constant - the probability of obtaining a hyperbola or an ellipse when throwing five random points on a plane.

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A Note on Multiplicative (Generalized) (*α*, *β*)-Derivations in Prime Rings

## Abstract

Let *R* be a prime ring with center *Z*(*R*). A map *G* : *R* →*R* is called a multiplicative (generalized) (*α*, *β*)-derivation if *G*(*xy*)= *G*(*x*)*α*(*y*)+*β*(*x*)*g*(*y*) is fulfilled for all *x*; *y* ∈ *R*, where *g* : *R* → *R* is any map (not necessarily derivation) and *α*; *β* : *R* → *R* are automorphisms. Suppose that *G* and *H* are two multiplicative (generalized) (*α*, *β*)-derivations associated with the mappings *g* and *h*, respectively, on *R* and *α*, *β* are automorphisms of *R*. The main objective of the present paper is to investigate the following algebraic identities: (*i*) *G*(*xy*) + *α*(*xy*) = 0, (*ii*) *G*(*xy*) + *α*(*yx*) = 0, (*iii*) *G*(*xy*) + *G*(*x*)*G*(*y*) = 0, (*iv*) *G*(*xy*) = *α*(*y*) ○ *H*(*x*) and (*v*) *G*(*xy*) = [*α*(*y*), *H*(*x*)] for all *x*, *y* in an appropriate subset of *R*.

###### On a New One Parameter Generalization of Pell Numbers

## Abstract

In this paper we present a new one parameter generalization of the classical Pell numbers. We investigate the generalized Binet’s formula, the generating function and some identities for *r*-Pell numbers. Moreover, we give a graph interpretation of these numbers.

###### On Explainable Fuzzy Recommenders and their Performance Evaluation

## Abstract

This paper presents a novel approach to the design of explainable recommender systems. It is based on the Wang–Mendel algorithm of fuzzy rule generation. A method for the learning and reduction of the fuzzy recommender is proposed along with feature encoding. Three criteria, including the Akaike information criterion, are used for evaluating an optimal balance between recommender accuracy and interpretability. Simulation results verify the effectiveness of the presented recommender system and illustrate its performance on the MovieLens 10M dataset.

###### On Jacobsthal and Jacobsthal-Lucas Hybrid Numbers

## Abstract

The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we consider special kinds of hybrid numbers, namely the Jacobsthal and the Jacobsthal-Lucas hybrid numbers and we give some their properties.

###### On the Convergence of Sigmoidal Fuzzy Grey Cognitive Maps

## Abstract

Fuzzy cognitive maps (FCMs) are recurrent neural networks applied for modelling complex systems using weighted causal relations. In FCM-based decision-making, the inference about the modelled system is provided by the behaviour of an iteration. Fuzzy grey cognitive maps (FGCMs) are extensions of fuzzy cognitive maps, applying uncertain weights between the concepts. This uncertainty is expressed by the so-called grey numbers. Similarly as in FCMs, the inference is determined by an iteration process which may converge to an equilibrium point, but limit cycles or chaotic behaviour may also turn up. In this paper, based on the grey connections between the concepts and the parameters of the sigmoid threshold function, we give sufficient conditions for the existence and uniqueness of fixed points of sigmoid FGCMs.

###### The Phase–Space Approach to time Evolution of Quantum States in Confined Systems: the Spectral Split–Operator Method

## Abstract

Using the phase space approach, we consider the quantum dynamics of a wave packet in an isolated confined system with three different potential energy profiles. We solve the Moyal equation of motion for the Wigner function with the highly efficient spectral split-operator method. The main aim of this study is to compare the accuracy of the employed algorithm through analysis of the total energy expectation value, in terms of deviation from its exact value. This comparison is performed for the second and fourth order factorizations of the time evolution operator.

###### Probing for Informal Work Activity

## Abstract

The Current Population Survey (CPS) is the source of official US labor force statistics. The wording of the CPS employment questions may not always cue respondents to include informal work in their responses, especially when providing proxy reports about other household members. In a survey experiment conducted using a sample of Amazon Mechanical Turk respondents, additional probing identified a substantial amount of informal work activity not captured by the CPS employment questions, both among those with no employment and among those categorized as employed based on answers to the CPS questions. Among respondents providing a proxy report for another household member, the share identifying additional work was systematically greater among those receiving a detailed probe that offered examples of types of informal work than among those receiving a simpler global probe. Similar differences between the effects of the detailed and the global probe were observed when respondents answered for themselves only among those who had already reported multiple jobs. The findings suggest that additional probing could improve estimates of employment and multiple job holding in the CPS and other household surveys, but that the nature of the probe is likely to be important.