###### Inverse Ambiguous Functions and Automorphisms on Finite Groups

## Abstract

If *G* is a finite group, then a bijective function *f* : *G → G* is inverse ambiguous if and only if *f*(*x*)^{−1} = *f*
^{−1}(*x*) for all *x ∈ G*. We give a precise description when a finite group admits an inverse ambiguous function and when a finite group admits an inverse ambiguous automorphism.

###### Isomorphism Theorems for Coalgebras

## Abstract

Let *F* be an endofunctor of a category *C*. We prove isomorphism theorems for *F* -coalgebras under condition that the underlying category *C* is exact; that is, regular with exact sequences. Also, *F* is not assumed to preserve pullbacks.

###### Left Derivable Maps at Non-Trivial Idempotents on Nest Algebras

## Abstract

Let *Alg* 𝒩 be a nest algebra associated with the nest 𝒩 on a (real or complex) Banach space 𝕏. Suppose that there exists a non-trivial idempotent *P* ∈ *Alg* 𝒩 with range *P* (𝕏) ∈ 𝒩, and *δ* : *Alg* 𝒩 → *Alg* 𝒩 is a continuous linear mapping (generalized) left derivable at *P*, i.e. *δ* (*ab*) = *aδ* (*b*) + *bδ* (*a*) (*δ* (*ab*) = *aδ*(*b*) + *bδ*(*a*) *− baδ*(*I*)) for any *a, b* ∈ *Alg* 𝒩 with *ab* = *P*, where *I* is the identity element of *Alg* 𝒩. We show that is a (generalized) Jordan left derivation. Moreover, in a strongly operator topology we characterize continuous linear maps on some nest algebras *Alg* 𝒩 with the property that *δ* (*P* ) = 2*Pδ* (*P* ) or *δ* (*P* ) = 2*P δ* (*P* ) *− Pδ* (*I*) for every idempotent *P* in *Alg* 𝒩.

###### 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.

######
*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.

######
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.