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Publisher Copyright: © 2023 Wiley-VCH GmbH. For (Formula presented.) and variable exponents (Formula presented.) and (Formula presented.) with values in [1, ∞], let the variable exponents (Formula presented.) be defined by (Formula presented.) The Riesz–Thorin–type interpolation theorem for variable Lebesgue spaces says that if a linear operator T acts boundedly from the variable Lebesgue space (Formula presented.) to the variable Lebesgue space (Formula presented.) for (Formula presented.), then (Formula presented.) where C is an interpolation constant independent of T. We consider two different modulars (Formula presented.) and (Formula presented.) generating variable Lebesgue spaces and give upper estimates for the corresponding interpolation constants Cmax and Csum, which imply that (Formula presented.) and (Formula presented.), as well as, lead to sufficient conditions for (Formula presented.) and (Formula presented.). We also construct an example showing that, in many cases, our upper estimates are sharp and the interpolation constant is greater than one, even if one requires that (Formula presented.), (Formula presented.) are Lipschitz continuous and bounded away from one and infinity (in this case, (Formula presented.)). authorsversion published

Funding Information: For the second and third authors, this work was partially supported through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020, financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was by supported by Fidelidade-Companhia de Seguras, S.A. Funding Information: This work was published with financial support from Fidelidade-Companhia de Seguras, S.A. to which the authors express their warmest acknowledgment. The authors express gratitude to the three referees for their comments, corrections and questions that led to a revised and better version of this work. Publisher Copyright: © 2023 by the authors. We present an analysis on variability Bitcoin characteristics that help to quantitatively differentiate Bitcoin from the state-owned traditional currencies and the asset Gold. We provide a detailed study on returns of exchange rates—against the Swiss Franc—of several traditional currencies together with Bitcoin and Gold; for that purpose, we define a distance between currencies by means of the spectral densities of the ARMA models of the returns of the exchange rates, and we present the computed matrix of the distances between the chosen currencies. A statistical analysis of these matrix distances is further proposed, which shows that the distance between Bitcoin and any other currency or Gold is not comparable to any of the distances between currencies or between currencies and Gold and not involving Bitcoin. This result shows that Bitcoin is essentially different from the traditional currencies and from Gold, at least in what concerns the structure of its variance and auto-covariances. publishersversion published

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2C1A01100526). Publisher Copyright: © 2022 by the authors. The Behrens–Fisher problem occurs when testing the equality of means of two normal distributions without the assumption that the two variances are equal. This paper presents approaches based on the exact and near-exact distributions for the test statistic of the Behrens–Fisher problem, depending on different combinations of even or odd sample sizes. We present the exact distribution when both sample sizes are odd and the near-exact distribution when one or both sample sizes are even. The near-exact distributions are based on a finite mixture of generalized integer gamma (GIG) distributions, used as an approximation to the exact distribution, which consists of an infinite series. The proposed tests, based on the exact and the near-exact distributions, are compared with Welch’s t-test through Monte Carlo simulations, in particular for small and unbalanced sample sizes. The results show that the proposed approaches are competent solutions to the Behrens–Fisher problem, exhibiting precise sizes and better powers than Welch’s approach for those cases. Numerical studies show that the Welch’s t-test tends to be a bit more conservative than the test statistics based on the exact or near-exact distribution, in particular when sample sizes are small and unbalanced, situations in which the proposed exact or near-exact distributions obtain higher powers than Welch’s t-test. publishersversion published

Funding Information: For the first author this work was partially supported through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020 financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was by supported by Fidelidade-Companhia de Seguros, S.A. to which the authors express their warmest acknowledgment. Funding Information: This work was published with financial support from by the New University of Lisbon. The authors express gratitude to the comments, corrections, and questions of the referees that led to a revised and better version of this work. Publisher Copyright: © 2023 by the authors. The structure of the random matrices introduced in this work is given by deterministic matrices—the skeletons of the random matrices—built with an algorithm of matrix substitutions with entries in a finite field of integers modulo some prime number, akin to the algorithm of one dimensional automatic sequences. A random matrix has the structure of a given skeleton if to the same number of an entry of the skeleton, in the finite field, it corresponds a random variable having, at least, as its expected value the correspondent value of the number in the finite field. Affine matrix substitutions are introduced and fixed point theorems are proven that allow the consideration of steady states of the structure which are essential for an efficient observation. For some more restricted classes of structured random matrices the parameter estimation of the entries is addressed, as well as the convergence in law and also some aspects of the spectral analysis of the random operators associated with the random matrix. Finally, aiming at possible applications, it is shown that there is a procedure to associate a canonical random surface to every random structured matrix of a certain class. publishersversion published

Funding Information: Funding: For the second author, this work was done under partial financial support of RFBR (Grant No. 19-01-00451). For the first, third and fourth authors this work was partially supported through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020 financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was funded by the insurance company Future Healthcare. Funding Information: Acknowledgments: This work was published with finantial support from the insurance company Future Healthcare. The authors would like to thank Future Healthcare for this generous support and for their interest in the development of models for disease and health insurance problems in Portugal. The authors wish to express their gratitude to JLS, former publisher at McGraw-Hill Book Company, for his extensive revision of the English language in this work. Funding Information: For the second author, this work was done under partial financial support of RFBR (Grant No. 19-01-00451). For the first, third and fourth authors this work was partially supported through the project of the Centro de Matem?tica e Aplica??es, UID/MAT/00297/2020 financed by the Funda??o para a Ci?ncia e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was funded by the insurance company Future Healthcare. Publisher Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. Made available in DSpace on 2022-02-12T23:27:16Z (GMT). No. of bitstreams: 0 Previous issue date: 2021-12-15 publishersversion published

Grant n. 19-01-00451 UID/MAT/00297/2020 We propose a stochastic algorithm for global optimisation of a regular function, possibly unbounded, defined on a bounded set with regular boundary; a function that attains its extremum in the boundary of its domain of definition. The algorithm is determined by a diffusion process that is associated with the function by means of a strictly elliptic operator that ensures an adequate maximum principle. In order to preclude the algorithm to be trapped in a local extremum, we add a pure random search step to the algorithm. We show that an adequate procedure of parallelisation of the algorithm can increase the rate of convergence, thus superseding the main drawback of the addition of the pure random search step. publishersversion published

Submitted by Isabel Melo (imelo@sa.isel.pt) on 2020-05-13T18:05:16Z No. of bitstreams: 1 Ranks_TMQuinteiro.pdf: 404654 bytes, checksum: db135bf332f766653a2b0e9c4e96ef20 (MD5) Made available in DSpace on 2020-05-13T18:05:16Z (GMT). No. of bitstreams: 1 Ranks_TMQuinteiro.pdf: 404654 bytes, checksum: db135bf332f766653a2b0e9c4e96ef20 (MD5) Previous issue date: 2019-04-19 info:eu-repo/semantics/publishedVersion

Funding Information: This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/00297/2020 (Centro de Matemática e Aplicações). Publisher Copyright: © 2021, Springer Nature Switzerland AG. The synchronization phenomenon was reported for the first time by Christiaan Huygens, when he noticed the strange tendency of a couple of clocks to synchronise their movements. More recently this phenomena was shown to be ubiquitous in nature and it is broadly studied by its applications, for example in biological cycles. We consider the problem of synchronization of a general network of linearly coupled oscillators, not necessarily identical. In this case the existence of a linear synchronization space is not expected, so we present an approach based on the proof of the existence of a synchronization manifold, the so-called generalised synchronization. Based on some results developed by R. Smith and on Wazewski’s principle, a general theory on the existence of invariant manifolds that attract the solutions of the system that are bounded in the future, is presented. Applications and estimates on parameters for the existence of synchronization are presented for several examples: systems of coupled pendulum type equations, coupled Lorenz systems of equations, and oscillators coupled through a medium, among many others. authorsversion published

In this work, we derived new asymptotic results for multinomial models. To obtain these results, we started by studying limit distributions in models with a compact parameter space. This restriction holds since the key parameter whose components are the probabilities of the possible outcomes have non-negative components that add up to 1. Based on these results, we obtained confidence ellipsoids and simultaneous confidence intervals for models with normal limit distributions. We then studied the covariance matrices of the limit normal distributions for the multinomial models. This was a transition between the previous general results and on the inference for multinomial models in which we considered the chi-square tests, confidence regions and non-linear statistics—namely log-linear models with two numerical applications to those models. Namely, our approach overcame the hierarchical restrictions assumed to analyse the multidimensional contingency table. publishersversion published

In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions., The authors acknowledge the support of the Center for Mathematics and Applications (CMA)—FCT-NOVA, Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, and CMAT—Centre of Mathematics—University of Minho. The first author acknowledges Fundação para a Ciência e Tecnologia (Portuguese Foundation for Science and Technology) within Projects UIDB/04621/2020 and UIDP/04621/2020. The second author acknowledges the Fundação para a Ciência e a Tecnologia through Project UIDB/00297/2020 (Centro de Matemática e Aplicações). The third author acknowledges the funding by Fundação para a Ciência e Tecnologia through Projects UIDB/00013/2020 and UIDP/00013/2020

National Research Foundation, South Africa (SRUG190308422768 nr. 120839), the South African NRF SARChI Research Chair in Computational and Methodological Statistics (UID: 71199), the South African DST-NRF-MRC SARChI Research Chair in Biostatistics (Grant no. 114613), the Research Development Programme at UP 296/2019, and the project UIDB/00297/2020 at UNL. This paper introduces an enriched -μ distribution which may act as fading model with its origins via the scale mixture construction. The distribution's characteristics are visited and its feasibility as a fading candidate in wireless communications systems is investigated. The analysis of the system reliability and some performance measures of wireless communications systems over this enriched -μ fading candidate are illustrated. Computable representations of the Laplace transform for this scale mixture construction are also provided. The derived expressions are explored via numerical investigations. Tractable results are computed in terms of the Meijer G-function. This unified scale mixture approach allows access to previously unconsidered underlying models that may yield improved fits to experimental data in practice. publishersversion published

This work is a contribution to the Project UIDB/04035/2020, funded by FCT - Fundacao para a Ciencia e a Tecnologia, Portugal. In this short note, we show that an assertion presented in the main result of Chen Xu, Mengmei Xi, Xuejun Wang and Hao Xia’s paper, “Lr convergence for weighted sums of extended negatively dependent random variables”, is false. A reformulation of this statement is announced making it valid. publishersversion published

We show that the Hardy-Littlewood maximal operator is bounded on a reflexive variable Lebesgue space $L^{p(\cdot)}$ over a space of homogeneous type $(X,d,\mu)$ if and only if it is bounded on its dual space $L^{p'(\cdot)}$, where $1/p(x)+1/p'(x)=1$ for $x\in X$. This result extends the corresponding result of Lars Diening from the Euclidean setting of $\mathbb{R}^n$ to the setting of spaces of homogeneous type $(X,d,\mu)$., Comment: Accepted for publication in Studia Mathematica

A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup $$\mathscr {TF}_{n}$$ of all order-preserving transformations on an n-element zig-zag-ordered set. We determine the rank of $$\mathscr {TF}_{n}$$ and provide a minimal generating set for $$ \mathscr {TF}_{n}$$ . Moreover, a formula for the number of idempotents in $$\mathscr {TF}_{n}$$ is given.

For both authors, this work was partially supported by the Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matematica e Aplicacoes) and the project PTDC/MHC-FIL/2583/2014. The first author was also funded by the FCT project PTDC/MAT-PUR/31174/2017. In 1900, Hilbert gave a lecture at the International Congress of Mathematicians in Paris, for which he prepared 23 problems that mathematicians should solve during the twentieth century. It was found that there was a note on a 24th problem focusing on the problem of simplicity of proofs. One of the lines of research that was generated from this problem was the identification of proofs. In this article, we present a possible method for exploring the identification of proofs based on the membership problem original from the theory of polynomial rings. To show this, we start by giving a complete worked-out example of a membership problem, that is the problem of checking if a given polynomial belongs to an ideal generated by finitely many polynomials. This problem can be solved by considering Gröbner bases and the corresponding reductions. Each reduction is a simplification of the polynomial and it corresponds to a rewriting step. In proving that a polynomial is a member of an ideal, a rewriting process is used, and many different such processes can be considered. To better illustrate this, we consider a graph where each rewriting step corresponds to an edge, and thus a path corresponds to a rewriting process. In this paper, we consider the identification of paths, within the context of the membership problem, to propose a criterion of identification of proofs. authorsversion published

Funding Information: Funding: For the second author, this work was done under partial financial support of RFBR (Grant n. 19-01-00451). For the first and third author this work was partially supported through the project of the Centro de Matemática e Aplicações, UID/MAT/00297/2020 financed by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology). The APC was funded by the insurance company Fidelidade. We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times. publishersversion published

Given a triangular array $\left\{X_{n,k}, \, 1 \leqslant k \leqslant n, n \geqslant 1 \right\}$ of random variables satisfying $\mathbb{E} \lvert X_{n,k} \rvert^{p} < \infty$ for some $p \geqslant 1$ and sequences $\{b_{n} \}$, $\{c_{n} \}$ of positive real numbers, we shall prove that $\sum_{n=1}^\infty c_n \mathbb{E} \left[ |\sum_{k=1}^n (X_{n,k} - \mathbb{E} \, X_{n,k})| / b_n - \varepsilon \right]_+^p < \infty$, where $x_+ = \max(x,0)$. Our results are announced in a general setting, allowing us to obtain the convergence of the series in question under various types of dependence., 14 pages

In this paper we derive a general expression for integer powers of real upper and lower anti-bidiagonal matrices with constant anti-diagonals using Chebyshev polynomials. An explicit formula for the inverse of these matrices is also provided. published

Crystal graphs, in the sense of Kashiwara, carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. In the particular case of the crystal graph for the $q$-analogue of the special linear Lie algebra $\mathfrak{sl}_{n}$, this monoid is the celebrated plactic monoid, whose elements can be identified with Young tableaux. The crystal graph and the so-called Kashiwara operators interact beautifully with the combinatorics of Young tableaux and with the Robinson--Schensted--Knuth correspondence and so provide powerful combinatorial tools to work with them. This paper constructs an analogous `quasi-crystal' structure for the hypoplactic monoid, whose elements can be identified with quasi-ribbon tableaux and whose connection with the theory of quasi-symmetric functions echoes the connection of the plactic monoid with the theory of symmetric functions. This quasi-crystal structure and the associated quasi-Kashiwara operators are shown to interact just as neatly with the combinatorics of quasi-ribbon tableaux and with the hypoplactic version of the Robinson--Schensted--Knuth correspondence. A study is then made of the interaction of the crystal graph of the plactic monoid and the quasi-crystal graph for the hypoplactic monoid. Finally, the quasi-crystal structure is applied to prove some new results about the hypoplactic monoid., Comment: 55 pages. Minor revision to fix typos, add references, and discuss an open question

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz-Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces., Comment: 18 pages

A Mixed Integer Linear Programming model for the design and planning of green supply chains is developed. Strategic and tactical decisions are taken, namely on facility location and capacity installation, supplier selection, technology selection, transportation network definition, supply planning, and product recovery. The aim of this work is to study the use of environmental indicators in these decisions while accounting for profit objectives. Different objective functions concerning environmental aspects are implemented. ReCiPe quantifies the environmental performance of the supply chain and combinations of ReCiPe's midpoint categories allow a deeper analysis of the impact of these categories in strategic and tactical decisions. The goal is to understand if focusing on selected categories affects supply chain decisions and overall supply chain environmental impact. Net Present Value quantifies the economic performance and is used for lexicographic optimization. The model is applied to a case-study and important managerial insights are obtained. From a holistic point of view, it answers the question: how should supply chain environmental impact be assessed? From a case-study perspective, insights are obtained regarding what type of improvements should be implemented to reduce the environmental impact and how this would affect supply chain strategic and tactical decisions, along with economic performance.

We give a combinatorial characterization of conjugacy in the sylvester monoid (the monoid of binary search trees), showing that conjugacy is decidable for this monoid. We then prove that conjugacy is undecidable in general for homogeneous monoids and even for multihomogeneous monoids., Comment: 16 pages, 4 figures. Minor revisions and improvements

Sem PDF. Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) through Centro de Matematica e Aplicacoes (PEst-OE/MAT/UI0297/2014) We will discuss orthogonal models and error orthogonal models and their algebraic structure, using as background, commutative Jordan algebras. The role of perfect families of symmetric matrices will be emphasized, since they will play an important part in the construction of the estimators for the relevant parameters. Perfect families of symmetric matrices form a basis for the commutative Jordan algebra they generate. When normality is assumed, these perfect families of symmetric matrices will ensure that the models have complete and sufficient statistics. This will lead to uniformly minimum variance unbiased estimators for the relevant parameters. published