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The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known, see G. Lettl, A. Pethő and P. Voutier, Simple families of Thue inequalities, Trans. Amer. Math. Soc. 351 (1999) 1871–1894, On the arithmetic of simplest sextic fields and related Thue equations, in Number Theory: Diophantine, Computational and Algebraic Aspects, eds. K. Győry et al. (de Gruyter, Berlin, 1998), pp. 331–348. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper, we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.

We derive a Karhunen–Loeve expansion of the Gauss process $$ {B}_t-g(t){\int}_0^1{g}^{\hbox{'}}(u)\mathrm{d}{B}_u,t\in \left[0,1\right] $$ , where (Bt)t ∈ [0, 1] is a standardWiener process, and g : [0, 1] → ℝ is a twice continuously differentiable function with g(0) = 0 and $$ {\int}_0^1{\left(g\hbox{'}(u)\right)}^2\mathrm{d}u=1 $$ . This process is an important limit process in the theory of goodness-of-fit tests. We formulate two particular cases with the functions $$ g(t)=\left(\sqrt{2}/\pi \right)\sin \left(\pi t\right),t\in \left[0,1\right] $$ , and g(t) = t, t ∈ [0, 1]. The latter corresponds to the Wiener bridge over [0, 1] from 0 to 0.

The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevic means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where $$f,g,h,k:I\rightarrow \mathbb {R}$$ are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and $$t,s\in \mathbb {R}_+$$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $$f,g,h,k:I\rightarrow \mathbb {R}$$ are four times continuously differentiable.

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In this paper we deal with composite rational functions having zeros and poles forming consecutive elements of an arithmetic progression. We also correct a result published earlier related to composite rational functions having a fixed number of zeros and poles.

The conditional maxima of independent Poisson random variables are studied. A triangular array of row-wise independent Poisson random variables is considered. If condition is given for the ...

In this paper we deal with two quantum relative entropy preserver problems on the cones of positive (either positive definite or positive semidefinite) operators. The first one is related to a quantum R\'enyi relative entropy like quantity which plays an important role in classical-quantum channel decoding. The second one is connected to the so-called maximal $f$-divergences introduced by D. Petz and M. B. Ruskai who considered this quantity as a generalization of the usual Belavkin-Staszewski relative entropy. We emphasize in advance that all the results are obtained for finite dimensional Hilbert spaces.

In 1987 Harris proved (Proc Am Math Soc 101(4):637–643, 1987)—among others—that for each $$1\le p

Rudas, Clogg, and Lindsay (1994, J. R Stat Soc. Ser. B, 56, 623) introduced the so-called mixture index of fit, also known as pi-star (π*), for quantifying the goodness of fit of a model. It is the lowest proportion of ‘contamination’ which, if removed from the population or from the sample, makes the fit of the model perfect. The mixture index of fit has been widely used in psychometric studies. We show that the asymptotic confidence limits proposed by Rudas et al. (1994, J. R Stat Soc. Ser. B, 56, 623) as well as the jackknife confidence interval by Dayton (2003, Br. J. Math. Stat. Psychol., 56, 1) perform poorly, and propose a new bias-corrected point estimate, a bootstrap test and confidence limits for pi-star. The proposed confidence limits have coverage probability much closer to the nominal level than the other methods do. We illustrate the usefulness of the proposed method in practice by presenting some practical applications to log-linear models for contingency tables.

A. Rapcsak obtained necessary and sufficient conditions for the projective Finsler metrizability in terms of a second order partial differential system. In this paper we investigate the integrability of the Rapcsak system and the extended Rapcsak system, by using the Spencer version of the Cartan-Kahler theorem. We also consider the extended Rapcsak system completed with the curvature condition. We prove that in the non-isotropic case there is a nontrivial Spencer cohomology group in the sequences determining the 2-acyclicity of the symbol of the corresponding differential operator. Therefore the system is not integrable and higher order obstruction exists.

Let p be a prime number and K be the finite field of p elements, i.e.K=GF(p). Further let G be an elementary abelian p-group of order pm. Then the group algebra K[G] is modular. We consider K[G] as an ambient space and the ideals of K[G] as linear codes. A basis of a linear space is called visible, if there exists a member of the basis with the minimum (Hamming) weight of the space. The group algebra approach enables us to find some linear codes with a visible basis in the Jacobson radical of K[G]. These codes can be generated by monomials(Drensky & Lakatos, 1989). For p>2, some of our monomial codes have better parameters than the Generalized ReedMuller codes. In the last part of the paper we determine the automorphism groups of some of the introduced codes.

Our aim in this article is to establish weighted Hardy-type inequalities in a broad family of means. In other words, for a fixed vector of weights $(\lambda_{n})_{n=1}^{\infty}$ and a weighted mean $\mathscr{M}$ , we search for the smallest extended real number $C$ such that ¶ \[\sum_{n=1}^{\infty}\lambda_{n}\mathscr{M}((x_{1},\ldots ,x_{n}),(\lambda_{1},\ldots,\lambda_{n}))\le C\sum_{n=1}^{\infty}\lambda_{n}x_{n}\quad \text{for all }x\in \ell_{1}(\lambda).\] The main results provide a complete answer in the case when $\mathscr{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, this is the case if $\mathscr{M}$ is symmetric, concave, and the sequence $(\frac{\lambda_{n}}{\lambda_{1}+\cdots+\lambda_{n}})_{n=1}^{\infty}$ is nonincreasing. In addition, we prove that if $\mathscr{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $\lambda$ is the constant vector.

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Accurate and reliable probabilistic forecasts of hydrological quantities like runoff or water level are beneficial to various areas of society. Probabilistic state-of-the-art hydrological ensemble prediction models are usually driven with meteorological ensemble forecasts. Hence, biases and dispersion errors of the meteorological forecasts cascade down to the hydrological predictions and add to the errors of the hydrological models. The systematic parts of these errors can be reduced by applying statistical post-processing. For a sound estimation of predictive uncertainty and an optimal correction of systematic errors, statistical post-processing methods should be tailored to the particular forecast variable at hand. Former studies have shown that it can make sense to treat hydrological quantities as bounded variables. In this paper, a doubly truncated Bayesian model averaging (BMA) method, which allows for flexible post-processing of (multi-model) ensemble forecasts of water level, is introduced. A case study based on water level for a gauge of river Rhine, reveals a good predictive skill of doubly truncated BMA compared both to the raw ensemble and the reference ensemble model output statistics approach.

In this paper we study conjugate parallelisms and their conformal changes on Finsler manifolds. We provide sufficient conditions for a Finsler manifold endowed with two conjugate (resp. conformally conjugate) covering parallelisms to become a Berwald (resp. Wagner) manifold. As an application for Lie groups, we give a new proof for a theorem of Latifi and Razavi about bi-invariant Finsler functions being Berwald. By introducing the concept of a conformal change of a parallelism, we also obtain a conceptual proof of a theorem of Hashiguchi and Ichijyō: the class of generalized Berwald manifolds is closed under conformal change.

Diophantine problems involving recurrence sequences have a long history and is an actively studied topic within number theory. In this paper, we connect to the field by considering the equation \begin{align*} B_mB_{m+d}\dots B_{m+(k-1)d}=y^\ell \end{align*} in positive integers $m,d,k,y$ with $\gcd(m,d)=1$ and $k\geq 2$, where $\ell\geq 2$ is a fixed integer and $B=(B_n)_{n=1}^\infty$ is an elliptic divisibility sequence, an important class of non-linear recurrences. We prove that the above equation admits only finitely many solutions. In fact, we present an algorithm to find all possible solutions, provided that the set of $\ell$-th powers in $B$ is given. (Note that this set is known to be finite.) We illustrate our method by an example., Comment: To appear in Bulletin of Australian Math Society

In this paper we study the invariant metrizability and projective metrizability problems for the special case of the geodesic spray associated to the canonical connection of a Lie group. We prove that such canonical spray is projectively Finsler metrizable if and only if it is Riemann metrizable. This result means that this structure is rigid in the sense that considering left-invariant metrics, the potentially much larger class of projective Finsler metrizable canonical sprays, corresponding to Lie groups, coincides with the class of Riemann metrizable canonical sprays. Generalisation of these results for geodesic orbit spaces are given., final version, accepted by MJOM

In this paper, skew-morphisms, which are extensively studied in graph theory, are considered in the setting of matrix algebras. Different properties of skew-morphisms are obtained and their classification in some specific cases is given.

Ensemble model output statistics (EMOS) is a statistical tool for post-processing forecast ensembles of weather variables obtained from multiple runs of numerical weather prediction models in order to produce calibrated predictive probability density functions (PDFs). The EMOS predictive PDF is given by a parametric distribution with parameters depending on the ensemble forecasts. We propose an EMOS model for calibrating wind speed forecasts based on weighted mixtures of truncated normal (TN) and log-normal (LN) distributions where model parameters and component weights are estimated by optimizing the values of proper scoring rules over a rolling training period. The new model is tested on wind speed forecasts of the 50 member European Centre for Medium-Range Weather Forecasts ensemble, the 11 member Aire Limit\'ee Adaptation dynamique D\'eveloppement International-Hungary Ensemble Prediction System ensemble of the Hungarian Meteorological Service and the eight-member University of Washington mesoscale ensemble, and its predictive performance is compared to that of various benchmark EMOS models based on single parametric families and combinations thereof. The results indicate improved calibration of probabilistic and accuracy of point forecasts in comparison with the raw ensemble and climatological forecasts. The mixture EMOS model significantly outperforms the TN and LN EMOS methods, moreover, it provides better calibrated forecasts than the TN-LN combination model and offers an increased flexibility while avoiding covariate selection problems., Comment: 24 pages, 7 tables, 10 figures

Let A be a finite set, and let a symmetric binary relation be given on A . The goal of correlation clustering is to find a partition of A , with minimal conflicts with respect to the relation given. In this paper we investigate correlation clustering of subsets of the positive integers, based upon a relation defined by the help of the greatest common divisor.

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In this paper we describe the structure of all isometries of the space of self-adjoint traceless operators on a finite dimensional Hilbert space under the metrics coming from the operator norm or the Schatten norms. We prove that up to a translation and multiplication by −1, they are unitary or antiunitary similarity transformations.

Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and automorphisms are additive functions satisfying some further functional equations as well. It is an important question that how these morphisms can be characterized among additive mappings in general. The paper contains some multivariate characterizations of higher order derivations. The univariate characterizations are given as consequences by the diagonalization of the multivariate formulas. The method allows us to refine the process of computing the solutions of univariate functional equations of the form \[ \sum_{k=1}^{n}x^{p_{k}}f_{k}(x^{q_{k}})=0, \] where $p_k$ and $q_k$ ($k=1, \ldots, n$) are given nonnegative integers and the unknown functions $f_{1}, \ldots, f_{n}\colon R\to R$ are supposed to be additive on the ring $R$. It is illustrated by some explicit examples too. As another application of the multivariate setting we use spectral analysis and spectral synthesis in the space of the additive solutions to prove that such a functional equation characterizes derivations of higher order. The results are uniformly based on the investigation of the multivariate version of the functional equations., This paper is dedicated to the 65th birthday of Professor L\'aszl\'o Sz\'ekelyhidi

Two dimensional Chebyshev systems, quoted also as regular pairs, induce convex structures both in an axiomatic and in an algebraic way. The aim of this note is to link these structures, by showing that they coincide.

A partial order is called semilinear iff the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable existentially closed semilinear order, which we denote by S2. We study the reducts of S2, that is, the relational structures with the same domain as S2 all of whose relations are first-order definable in S2. Our main result is a classification of the model-complete cores of the reducts of S2. From this, we also obtain a classification of reducts up to first-order interdefinability, which is equivalent to a classification of all closed permutation groups that contain the automorphism group of S2., Comment: 25 pages

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Summary Environmental epidemiological studies of the health effects of air pollution frequently utilize the generalized additive model (GAM) as the standard statistical methodology, considering the ambient air pollutants as explanatory covariates. Although exposure to air pollutants is multi-dimensional, the majority of these studies consider only a single pollutant as a covariate in the GAM model. This model restriction may be because the pollutant variables do not only have serial dependence but also interdependence between themselves. In an attempt to convey a more realistic model, we propose here the hybrid generalized additive model–principal component analysis–vector auto-regressive (GAM–PCA–VAR) model, which is a combination of PCA and GAMs along with a VAR process. The PCA is used to eliminate the multicollinearity between the pollutants whereas the VAR model is used to handle the serial correlation of the data to produce white noise processes as covariates in the GAM. Some theoretical and simulation results of the methodology proposed are discussed, with special attention to the effect of time correlation of the covariates on the PCA and, consequently, on the estimates of the parameters in the GAM and on the relative risk, which is a commonly used statistical quantity to measure the effect of the covariates, especially the pollutants, on population health. As a main motivation to the methodology, a real data set is analysed with the aim of quantifying the association between respiratory disease and air pollution concentrations, especially particulate matter PM10, sulphur dioxide, nitrogen dioxide, carbon monoxide and ozone. The empirical results show that the GAM–PCA–VAR model can remove the auto-correlations from the principal components. In addition, this method produces estimates of the relative risk, for each pollutant, which are not affected by the serial correlation in the data. This, in general, leads to more pronounced values of the estimated risk compared with the standard GAM model, indicating, for this study, an increase of almost 5.4% in the risk of PM10, which is one of the most important pollutants which is usually associated with adverse effects on human health.

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We give necessary and sufficient conditions for the existence of primitive algebraic integers with index A in totally complex bicyclic biquadratic number fields where A is an odd prime or a positive rational integer at most 10. We also determine all these elements and prove that there are infinitely many totally complex bicyclic biquadratic number fields containing elements with index A.

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We prove a finiteness result for the number of solutions of a Diophantine equation of the form $$u_n u_{n+1}\cdots u_{n+k}\pm 1 =\pm u_m^2$$ , where $$\{ u_n\}_{n\ge 1}$$ is a binary recurrent sequence whose characteristic equation has roots which are real quadratic units.

Let $K$ be a number field. In the terminology of Nagell a unit $\varepsilon$ of $K$ is called {\it exceptional} if $1-\varepsilon$ is also a unit. The existence of such a unit is equivalent to the fact that the unit equation $\varepsilon_1+\varepsilon_2+\varepsilon_3=0$ is solvable in units $\varepsilon_1,\varepsilon_2,\varepsilon_3$ of $K$. Numerous number fields have exceptional units. They have been investigated by many authors, and they have important applications. In this paper we deal with a generalization of exceptional units. We are interested in the smallest integer $k$ with $k\geq 3$, denoted by $\ell(K)$, such that the unit equation $\varepsilon_1+\dots+\varepsilon_k=0$ is solvable in units $\varepsilon_1,\dots,\varepsilon_k$ of $K$. If no such $k$ exists, we set $\ell(K)=\infty$. Apart from trivial cases when $\ell(K)=\infty$, we give an explicit upper bound for $\ell(K)$. We obtain several results for $\ell(K)$ in number fields of degree at most $4$, cyclotomic fields and general number fields of given degree. We prove various properties of $\ell(K)$, including its magnitude, parity as well as the cardinality of number fields $K$ with given degree and given odd resp. even value $\ell(K)$. Finally, as an application, we deal with certain arithmetic graphs, namely we consider the representability of cycles. We conclude the paper by listing some problems and open questions., Comment: We expand the proof of Theorem 2.4. Although the original proof is correct we feel that it is worth to give more explanation in one of the cases

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We aim to characterise those transformations on the set of density operators (which are the mathematical representatives of the states in quantum information theory) that preserve a so-called generalised entropy ofonefixed convex combination of operators. The characterisation strengthens a recent result of Karder and Petek where the preservation of the same quantity was assumed forallconvex combinations.

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Let G be a finite, nilpotent group. The computational complexity of the equation solvability problem over G is known to be in P. The complexity is understood in the length of the equation over G. So far the fastest algorithm to decide whether or not an equation of length n has a solution over G was running in O ( n | G | | G | ⋯ | G | | G | ) time. Here the height of the tower is the nilpotency class of G. We prove that one can decide in O ( n 1 2 | G | 2 log ⁡ | G | ) time whether an equation of length n has a solution over G. The key ingredient of the proof is to represent group expressions using the polycyclic presentation of p-groups.

In this paper, we give the solution of a problem formulated in Kominek and Sikorska (Aequationes Math 90:107–121, 2016) in connection with the functional equation $$\begin{aligned} f(xy)-f(x)-f(y)=g(x+y)-g(x)g(y). \end{aligned}$$ Our result can also be interpreted in the way that, under some additional condition, the logarithmic and the exponential Cauchy equations are strongly alien.

The purpose of this paper is to show that functions that derivate the two-variable product function and one of the exponential, trigonometric or hyperbolic functions are also standard derivations. The more general problem considered is to describe finite sets of differentiable functions such that derivations with respect to this set are automatically standard derivations.

We show that the largest character degree of an alternating group $A_n$ with $n\geq 5$ can be bounded in terms of smaller degrees in the sense that \[ b(A_n)^2