Finite Product Research Articles

Fundamentals of a healthy and sustainable diet

BackgroundA healthy and sustainable diet is a prerequisite for population and planetary health. The evidence of associations between dietary patterns and health outcomes has now been synthesised to inform more than 100 national dietary guidelines. Yet, people select foods, not whole dietary patterns, even in the context of following specific diets such as a Mediterranean diet, presenting challenges to researchers, policymakers and practitioners wanting to translate dietary guideline recommendations into food-level selection guidance for citizens. Understanding the fundamentals that underpin healthy and sustainable diets provides a scientific basis for helping navigate these challenges. This paper’s aim is to describe the fundamentals of a healthy and sustainable diet.ResultsThe scientific rationale underpinning what is a healthy and sustainable diet is universal. Everyone shares a physiological need for energy and adequate amounts, types and combinations of nutrients. People source their energy and nutrient needs from foods that are themselves sourced from food systems. The physiological need and food systems’ sustainability have been shaped through evolutionary and ecological processes, respectively. This physiological need can be met, and food systems’ sustainability protected, by following three interlinked dietary principles: (i) Variety – to help achieve a nutritionally adequate diet and help protect the biodiversity of food systems. (ii) Balance – to help reduce risk of diet-related non-communicable diseases and excessive use of finite environmental resources and production of greenhouse gas emissions. (iii) Moderation – to help achieve a healthy body weight and avoid wasting finite environmental resources used in providing food surplus to nutritional requirements.ConclusionThe fundamentals of a healthy and sustainable diet are grounded in evolutionary and ecological processes. They are represented by the dietary principles of variety, balance and moderation and can be applied to inform food-level selection guidance for citizens.

Correction: Preimages under linear combinations of iterates of finite Blaschke products

Correction: Preimages under linear combinations of iterates of finite Blaschke products

Stochastic Entropy Production Associated with Quantum Measurement in a Framework of Markovian Quantum State Diffusion

The reduced density matrix that characterises the state of an open quantum system is a projection from the full density matrix of the quantum system and its environment, and there are many full density matrices consistent with a given reduced version. Without a specification of relevant details of the environment, the time evolution of a reduced density matrix is therefore typically unpredictable, even if the dynamics of the full density matrix are deterministic. With this in mind, we investigate a two-level open quantum system using a framework of quantum state diffusion. We consider the pseudorandom evolution of its reduced density matrix when subjected to an environment-driven process that performs a continuous quantum measurement of a system observable, invoking dynamics that asymptotically send the system to one of the relevant eigenstates. The unpredictability is characterised by a stochastic entropy production, the average of which corresponds to an increase in the subjective uncertainty of the quantum state adopted by the system and environment, given the underspecified dynamics. This differs from a change in von Neumann entropy, and can continue indefinitely as the system is guided towards an eigenstate. As one would expect, the simultaneous measurement of two non-commuting observables within the same framework does not send the system to an eigenstate. Instead, the probability density function describing the reduced density matrix of the system becomes stationary over a continuum of pure states, a situation characterised by zero further stochastic entropy production. Transitions between such stationary states, brought about by changes in the relative strengths of the two measurement processes, give rise to finite positive mean stochastic entropy production. The framework investigated can offer useful perspectives on both the dynamics and irreversible thermodynamics of measurement in quantum systems.

A Method for Predicting the Remaining Service Life of Finite Data Products Based on Gaussian Stochastic Processes

There is a problem of poor predictive performance when predicting the remaining service life of products. Therefore, this paper proposes a finite data product remaining service life prediction method based on Gaussian stochastic processes. Analyze the types of product failure and degradation under limited data, and determine the factors affecting the degradation performance of product life through constant stress accelerated degradation tests, step stress accelerated degradation tests, and sequential stress accelerated aging tests; using the Wiener process to analyze the performance degradation characteristics of products, setting product failure thresholds, defining the remaining life of the product based on the degradation process, and using equal interval partitioning methods to determine the product health index, and clarifying the product failure process under limited data; by calculating the mathematical expected value, the numerical characteristics of the remaining life of the product are defined. Variance and moments are used to determine the predicted second-order central moment and origin moment. Based on the monotonic increasing characteristics of the inverse Gaussian process, the probability distribution of the random variable of product failure is determined after the Gaussian process. Under limited data, the actual working time of the product and the degradation amount at the time scale are clarified, and a preset threshold is set for the degradation amount. Build a residual life prediction model for product degradation and output the prediction results. The experimental results show that this method can accurately predict the remaining service life of the product and has good predictive performance.

Supply Chain Management in Smart City Manufacturing Clusters: An Alternative Approach to Urban Freight Mobility with Electric Vehicles

The development of green production types such as personalized production and shared manufacturing, which use additive technologies in city multifloor manufacturing clusters (CMFMCs), has led to an increase in last-mile parcel delivery (LMPD) activity. This study investigates the integration of electric vehicles and crowdshipping systems into smart CMMCs to improve urban logistics operations related to the distribution of products to consumers. The aim of this study is to improve the LMPD performance of these integrated systems and to provide alternative solutions for sustainable city logistics using the potential of crowdshipping and vehicle sharing fleets (VSFs) in the city logistics nodes (CLNs) of CMFMCs. The issues presented by the loading–unloading operations and sustainable crowdshipping scenarios for LMPD in CMFMCs are considered. This paper presents a new performance evaluation model for crowdshipping LMPD in CMFMCs using VSFs. The case study shows that the proposed model enables the analysis of LMPD performance in CMFMCs, taking into account their finite production capacity, and that it facilitates the planning of cargo turnover and the structure of VSFs consisting of e-bicycles, e-cars, and e-light commercial vehicles (e-LCVs). The model is verified based on a case study for sustainable LMPD scenarios using VSFs. The proposed model enables the planning of both short- and long-term logistics operations with the specified performance indicator of VSF usage in CMFMCs. The validity of using the integrated potential of crowdshipping and vehicle sharing services for LMPD under demand uncertainty in CMFMCs is discussed. This study should prove useful for decision-making and planning processes related to LMPD in CMFMCs and large cities.

When Heavy Tails Disrupt Statistical Inference

Heavy tails (HT) arise in many applications and their presence can disrupt statistical inference, yet the HT statistical literature requires a theoretical background most practicing statisticians lack. We provide an overview of the influence of HT on the performance of basic statistical methods and useful theorems aimed at the practitioner encountering HT in an applied setting. Higher or even lower product moments (i.e., variance, skewness, etc.) can be infinite for some HT populations, yet all L-moments are always finite, given that the mean exists, thus, the theory of L-moments is uniquely suited to all HT distributions and data. We document how L-kurtosis, (a kurtosis measure based on the fourth L-moment) provides a general and practical heaviness index for contrasting tail heaviness across distributions and datasets and how a single L-moment diagram can document both the prevalence and impact of HT distributions and data across disciplines and datasets. Surprisingly, the theory of L-moments, an extension and evolution of probability weighted moments, has been largely overlooked by the literature on HT distributions that exhibit infinite moments. Experiments reveal L-kurtosis ranges under which various HT distributions result in mild to severe disruption to the bootstrap, the central limit theorem (CLT), and the law of large numbers, even for distributions which exhibit finite product moments.

The efficiency of CUSUM schemes for monitoring the multivariate coefficient of variation in short runs process

Current monitoring technologies emphasize and address the issue of monitoring high-volume production processes. The high flexibility and diversity of current industrial production processes make monitoring technology for small batch processes even more important. In multivariate process monitoring, a broader applicability exists in multivariate coefficients of variation (MCV) based monitoring schemes due to the lower restriction of the process. In view of the effectiveness of MCV monitoring and with the aim to achieve further performance improvement of current MCV monitoring schemes in a finite horizon production, we additionally introduce two one-sided cumulative sum (CUSUM) MCV schemes. In the case of deterministic and random shifts, the design parameters of the proposed schemes are obtained via an optimization procedure designed by the Markov chain method and the corresponding performance is analysed based on different run length (RL) characteristics, including the mean and the standard deviation. Simulation comparisons with existing exponentially weighted moving average (EWMA) MCV schemes show that the proposed CUSUM MCV schemes are more efficient in monitoring most of the shifts, including the deterministic and random shifts. Finally, to demonstrate the benefits of the new monitoring schemes, a comprehensive case study on monitoring a steel sleeve manufacturing process is conducted.

Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.

Stone Pseudovarieties

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.

On the Fourier Coefficients of Powers of a Finite Blaschke Product

Abstract Given a finite Blaschke product $B$ we prove asymptotically sharp estimates on the $\ell ^{\infty }$-norm of the sequence of the Fourier coefficients of $B^{n}$ as $n$ tends to $\infty $. This norm decays as $n^{-1/N}$ for some $N\ge 3$. Furthermore, for every $N\ge 3$, we produce explicitly a finite Blaschke product $B$ with decay $n^{-1/N}$. As an application we construct a sequence of $n\times n$ invertible matrices $T$ with arbitrary spectrum in the unit disk and such that the quantity $|\det{T}|\cdot \|T^{-1}\|\cdot \|T\|^{1-n}$ grows as a power of $n$. This is motivated by Schäffer’s question on norms of inverses.

Two new algebraic structures more general than weak rings

In 1994, Bergelson and Hindman established a remarkable result for IP* sets: for any IP* set A and any sequence l in ℕ, there exists a sum subsystem l′ of l such that all finite sums and finite products of l′ are contained in A. There are many studies about this result, where Hindman and Strauss extended it to general weak rings. In this paper, we introduce two algebraic structures - adequate partial weak rings and adequate poor rings, which are more general than weak rings. And we establish that result in both structures. Moreover, we also obtain Hindman theorem with two operations in these two structures.

Geodesic planes in a geometrically finite end and the halo of a measured lamination

Recent works [22,23,3,33] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifold M of infinite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of M, giving a complete classification of their closures in M.In particular, we show that the behavior is different depending on whether exotic roofs exist or not. Here an exotic roof is a geodesic plane contained in an end E of M, which limits on the convex core boundary ∂E, but cannot be separated from the core by a support plane of ∂E.A necessary condition for the existence of exotic roofs is the existence of exotic rays for the bending lamination. Here an exotic ray is a geodesic ray that has a finite intersection number with a measured lamination L but is not asymptotic to any leaf nor eventually disjoint from L. We establish that exotic rays exist if and only if L is not a multicurve. The proof is constructive, and the ideas involved are important in the construction of exotic roofs.We also show that the existence of geodesic rays satisfying a stronger condition than being exotic, phrased only in terms of the hyperbolic surface ∂E and the bending lamination, is sufficient for the existence of exotic roofs. As a result, we show that geometrically finite ends with exotic roofs exist in every genus. Moreover, in genus 1, when the end is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.

Some results on the total zero-divisor graph of a commutative ring

PurposeThe purpose of this paper is to characterize a commutative ring R with identity which is not an integral domain such that ZT(R), the total zero-divisor graph of R is connected and to determine the diameter and radius of ZT(R) whenever ZT(R) is connected. Also, the purpose is to generalize some of the known results proved by Duric et al. on the total zero-divisor graph of R.Design/methodology/approachWe use the methods from commutative ring theory on primary decomposition and strong primary decomposition of ideals in commutative rings. The structure of ideals, the primary ideals, the prime ideals, the set of zero-divisors of the finite direct product of commutative rings is used in this paper. The notion of maximal Nagata prime of the zero-ideal of a commutative ring is also used in our discussion.FindingsFor a commutative ring R with identity, ZT(R) is the intersection of the zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R. The zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R are well studied. Hence, we determine necessary and sufficient condition so that ZT(R) agrees with the zero-divisor graph of R (respectively, agrees with the total graph induced by the set of non-zero zero-divisors of R). If Z(R) is an ideal of R, then it is noted that ZT(R) agrees with the zero-divisor graph of R. Hence, we focus on rings R such that Z(R) is not an ideal of R. We are able to characterize R such that ZT(R) is connected under the assumptions that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. With the above assumptions, we are able to determine the domination number of ZT(R).Research limitations/implicationsDuric et al. characterized Artinian rings R such that ZT(R) is connected. In this paper, we extend their result to rings R such that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. As an Artinian ring is isomorphic to the direct product of a finite number of Artinian local rings, we try to characterize R such that ZT(R) is connected under the assumption that R is ta finite direct product of rings R1, R2, … Rn with Z(Ri) is an ideal of Ri for each i between 1 to n. Their result on domination number of ZT(R) is also generalized in this paper. We provide several examples to illustrate our results proved.Practical implicationsThe implication of this paper is that the existing result of Duric et al. is applicable to large class of commutative rings thereby yielding more examples. Moreover, the results proved in this paper make us to understand the structure of commutative rings better. It also helps us to learn the interplay between the ring-theoretic properties and the graph-theoretic properties of the graph associated with it.Originality/valueThe results proved in this paper are original and they provide more insight into the structure of total zero-divisor graph of a commutative ring. This paper provides several examples. Not much work done in the area of total zero-divisor graph of a commutative ring. This paper is a contribution to the area of graphs and rings and may inspire other researchers to study the total zero-divisor graph in further detail.

State complexity bounds for projection, shuffle, up- and downward closure and interior on commutative regular languages

We consider the state complexity of projection, shuffle, up- and downward closure and interior on commutative regular languages. We deduce the state complexity bound n|Σ| for upward closure and downward interior, and (2nm)|Σ|, (nm)|Σ|, (n+m−1)|Σ| and (n+m−2)|Σ| for the shuffle on commutative regular, group, aperiodic and finite languages, respectively, with state complexities n and m over the alphabet Σ. We do not know whether these bounds are sharp. For projection, downward closure and upward interior, we give the sharp bound n. Our results are obtained by using the index and period vectors of a regular language, which we introduce in the present work and investigate w.r.t. the above operations and also union and intersection. Furthermore, we characterize the commutative aperiodic and commutative group languages in terms of these parameters and prove that a commutative regular language equals a finite union of shuffle products of commutative finite and commutative group languages.

Direct and ordinal products realized by triangular norm operators with no zero divisors

In this note we continue the work of Chon, as well as Mezzomo, Bedregal, and Santiago, by studying algebraic operations on fuzzy posets and bounded fuzzy lattices. We first prove that fuzzy posets are closed under finite direct products whenever the triangular norm realizing the product construction has no zero divisors. This result is then extended to the case of bounded fuzzy lattices. Some immediate consequences are then obtained within the setting of direct products realized by triangular norms with no nilpotent elements as well as strictly monotone and cancellative triangular norms. We then introduce a triangular norm based construction of ordinal products and similarly show that fuzzy posets are closed under ordinal products whenever the triangular norm realizing the product construction has no zero divisors.

Decidable objects and molecular toposes

We study several sufficient conditions for the molecularity/local-connectedness of geometric morphisms. In particuar, we show that if $\mathcal{S}$ is a Boolean topos then, for every hyperconnected essential geometric morphism ${p : \mathcal{E} \rightarrow \mathcal{S}}$ such that the leftmost adjoint $p_!$ preserves finite products, $p$ is molecular and ${p^* : \mathcal{S} \rightarrow \mathcal{E}}$ coincides with the full subcategory of decidable objects in $\mathcal{E}$. We also characterize the reflections between categories with finite limits that induce molecular maps between the respective presheaf toposes. As a corollary we establish the molecularity of certain geometric morphisms between Gaeta toposes.

On finite groups with the Magnus Property

AbstractWe investigate finite groups with the Magnus Property (MP), where a group is said to have the MP if whenever two elements have the same normal closure, then they are conjugate or inverse conjugate. In particular, we observe that a finite MP group is solvable, determine the finite primitive MP groups, and determine all the possible orders of the chief factors of a finite MP group. We also determine the MP finite direct products of finite primitive groups, as well as the MP crown‐based powers of a finite monolithic primitive group.

Arbitrary finite intersections of doubling measures and applications

We make major progress on a folkloric conjecture in analysis by constructing a measure on the real line which is doubling on all n-adic intervals for any finite list of n∈N, yet not doubling overall. In particular, we extend previous results in the area, including those of Boylan-Mills-Ward and Anderson-Hu, by using a wide array of substantially new ideas. In addition, we provide several nontrivial applications to reverse Hölder weights, Ap weights, Hardy spaces, BMO and VMO function classes, and connect our results with key principles and conjectures across number theory.

Words of analytic paraproducts on Hardy and weighted Bergman spaces

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by Tgf(z)=∫0zf(ζ)g′(ζ)dζ, Sgf(z)=∫0zf′(ζ)g(ζ)dζ, and Mgf(z)=g(z)f(z). We are concerned with the study of the boundedness of operators in the algebra Ag generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in Ag are finite linear combinations of finite products (words) of Tg,Sg,Mg which may involve a large amount of cancellations to be understood. The results in [1] show that boundedness of operators in a fairly large subclass of Ag can be characterized by one of the conditions g∈H∞, or gn belongs to BMOA or the Bloch space, for some integer n>0. However, it is also proved that there are many operators, even single words in Ag whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in Ag in terms of a “fractional power” of the symbol g, that only depends on the number of appearances of each of the letters Tg,Sg,Mg in the given word.

Recombination of stable sampling sets and stable interpolation sets in functional quasinormed spaces

This contribution is aimed in obtaining new results in combining stable sampling sets (respectively, stable interpolation sets) of a given quasinormed space in order to obtain other new ones. We apply these results to Paley-Wiener spaces and. In addition, we study the problem of obtaining a generator system of a given quasinormed space, and obtain conditions for a finite product of subsets of a given quasinormed space to be a generator system, using the interpolation and sampling theory for quasinormed spaces of functions.