Finite Product Research Articles

Brouwer's fixed-point theorem on the finite product of compact linearly ordered topological spaces

Brouwer's fixed-point theorem on the finite product of compact linearly ordered topological spaces

Sums of Finite Product of Chebyshev Polynomials of Third and Fourth Kind and Fibonacci and Lucas Numbers

Objective: Classical orthogonal polynomials are extensively used for the numerical solution of differential equations and numerical analysis. Various generalization problems involving Chebyshev polynomials have been studied and one such area is the classical linearization problem. In this paper, we will consider a generalized classical linearization problem and study some identities representing the sums of finite products of the 3rd and 4th kinds of Chebyshev polynomials, Fibonacci numbers, and Lucas numbers as a linear sum of the Chebyshev polynomials of 2nd kind and their derivatives. Methods: Differential calculus, combinatorial, and elementary algebraic computations are employed to yield the results. Findings: A new set of identities expressing the sums of finite products of the 3rd and 4th kinds of Chebyshev polynomials, Fibonacci numbers, and Lucas numbers as a linear sum of the 2nd kinds of Chebyshev polynomials and their derivatives are studied. Novelty: Earlier studies have established the sums of finite products of the Lucas numbers, Fibonacci numbers, Fibonacci and Pell polynomials, and Chebyshev polynomials of the 1st and 2nd kinds as a linear sum of other orthogonal polynomials. However, representations of sums of finite products of the 3rd and 4th kinds of Chebyshev polynomials, Fibonacci numbers, and Lucas numbers as a linear sum of the 2nd kinds of Chebyshev polynomials and their derivatives have not been studied, which is the prime focus of this manuscript. Keywords: Linearization, Chebyshev Polynomials, Fibonacci Numbers, Lucas numbers, Orthogonality

Holomorphic factorization of mappings into the symplectic group

It is shown that any symplectic 2n\times 2n matrix, whose entries are complex holomorphic functions on a reduced Stein space, can be decomposed into a finite product of elementary symplectic matrices if and only if it is null-homotopic. Moreover, if this is the case, the number of factors can be bounded by a constant depending only on n and the dimension of the space.

The categories of L-convex spaces and L-convergence spaces: extensionality and productivity of quotient maps

Based on a complete residuated lattice L, we show that the category of L-convex spaces is not extensional and is closed under the formation of finite products of quotient maps. Then we propose the concept of (preconcave, concave) L-convergence spaces via L-co-Scott closed sets and prove that the category of concave L-convergence spaces is isomorphic to that of L-concave spaces. Finally, we investigate the categorical properties of L-convergence spaces and show that it is extensional and closed under the formation of finite products of quotient maps.

Set-Valued Recursions Arising from Vantage-Point Trees

Abstract We study vantage-point trees constructed using an independent sample from the uniform distribution on a fixed convex body K in $$(\mathbb {R}^d,\Vert \cdot \Vert )$$ ( R d , ‖ · ‖ ) , where $$\Vert \cdot \Vert $$ ‖ · ‖ is an arbitrary norm on $$\mathbb {R}^d$$ R d . We prove that a sequence of sets, associated with the left boundary of a vantage-point tree, forms a recurrent Harris chain on the space of convex bodies in $$(\mathbb {R}^d,\Vert \cdot \Vert )$$ ( R d , ‖ · ‖ ) . The limiting object is a ball polyhedron, that is, an a.s. finite intersection of closed balls in $$(\mathbb {R}^d,\Vert \cdot \Vert )$$ ( R d , ‖ · ‖ ) of possibly different radii. As a consequence, we derive a limit theorem for the length of the leftmost path of a vantage-point tree.

On Nilpotent Minimum logics defined by lattice filters and their paraconsistent non-falsity preserving companions

Abstract Nilpotent Minimum logic (NML) is a substructural algebraizable logic that is a distinguished member of the family of systems of Mathematical Fuzzy logic, and at the same time it is the axiomatic extension with the prelinearity axiom of Nelson and Markov’s Constructive logic with strong negation. In this paper our main aim is to characterize and axiomatize paraconsistent variants of NML and its extensions defined by (sets of) logical matrices over linearly ordered NM-algebra with lattice filters as designated values, with special emphasis on those that only exclude the falsum truth-value, called non-falsity preserving logics. We also consider turning these non-falsity preserving logics into Logics of Formal Inconsistency by expanding them with a consistency operator, and we axiomatize them as well. Finally, we provide a full description of the logics defined by finite products of matrices over finite NM-chains.

Two new generators of Archimedean copulas with their properties

. In this article, two new generators of bivariate Archimedean copulas are proposed. Several useful properties, such as the Kendall’s τ rank correlation coefficient and lower and upper tail dependence coefficients are derived. This article suggests methods of generating new Archimedean generators from existing Archimedean generators. The motivation comes from Najjari (2018) who conjectures that a finite product of a number of Archimedean generators is again an Archimedean generator. Applying Najjari’s conjecture, two new Archimedean copulas are proposed, along with their important properties. In selecting the initial generators, we have utilized the list of Archimedean generators given in Nelsen (2006).

On z-elements of multiplicative lattices

The aim of this paper is to investigate further properties of z-elements in multiplicative lattices. We utilize z-closure operators to extend several properties of z-ideals to z-elements and introduce various distinguished subclasses of z-elements, such as z-prime, z-semiprime, z-primary, z-irreducible, and z-strongly irreducible elements, and study their properties. We provide a characterization of multiplicative lattices where z-elements are closed under finite products and a representation of z-elements in terms of z-irreducible elements in z-Noetherian multiplicative lattices.

A multivariate finite horizon production control chart for monitoring the food production process

ABSTRACT Finite horizon production (FHP) processes are prevailing in the food industry. However, monitoring these multivariate FHP processes poses a significant challenge for statistical process control, primarily due to inadequate information about the distribution of multivariate observations. In this paper, a multivariate FHP (MFHP) control chart is proposed to monitor the food production processes with a limited number of reference samples, where the statistic of the proposed chart is derived from the distance from the monitored samples to support vectors of a support vector machine (SVM) classifier. The control limit of the statistic is designed from a conditional perspective, taking into account the variability generated across different reference samples. An enhanced cache-optimized sequential minimal optimization algorithm (ECSMOA) is proposed to speed up the computation of the control limit. Extensive numerical analysis indicates that the proposed chart is more effective and robust in preserving a desired in-control performance and guaranteeing an out-of-control performance. Particularly, the variance of performance measures was reduced by approximately 87.09 % . Finally, an industrial example based on whole wheat bread production processes is given to demonstrate the implementation of the proposed chart.

LEF-groups and computability of reversible endomorphisms of symbolic varieties

Fix a group G and let X be an algebraic variety over an algebraically closed field k of characteristic zero. We investigate the invertibility of algebraic cellular automata, namely, G-equivariant uniformly continuous self-maps [Formula: see text] whose local defining maps are induced by morphisms of algebraic varieties [Formula: see text] where [Formula: see text] is a finite memory set. When G is locally embeddable into finite groups (LEF), we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata and thus computable in polynomial time. Generalizations are also obtained for finite product Hopfian pointed object alphabets in concrete categories. Moreover, we prove that for algebraic cellular automata, the notions of reversibility and injectivity are equivalent whenever G is surjunctive and the field k is, additionally, uncountable of arbitrary characteristic.

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.

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.

The binary products of algebras with genetic realization

Abstract The problem of the existence of finite products in the category of algebras with genetic realization is considered.

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.