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Intersecting codes are linear codes where every two nonzero codewords have non-trivially intersecting support. In this article we expand on the theory of this family of codes, by showing that nondegenerate intersecting codes correspond to sets of points (with multiplicites) in a projective space that are not contained in two hyperplanes. This correspondence allows the use of geometric arguments to demonstrate properties and provide constructions of intersecting codes. We improve on existing bounds on their length and provide explicit constructions of short intersecting codes. Finally, generalizing a link between coding theory and the theory of the Davenport constant (a combinatorial invariant of finite abelian groups), we provide new asymptotic bounds on the weighted $2$-wise Davenport constant. These bounds then yield results on factorizations in rings of algebraic integers and related structures., Comment: 31 pages

Let $G$ be a finite abelian group with $\exp(G)$ the exponent of $G$. Then $\mathsf W(G)$ denotes the set of cross numbers of minimal zero-sum sequences over $G$ and $\mathsf w(G)$ denotes the set of all cross numbers of non-trivial zero-sum free sequences over $G$. It is clear that $\mathsf W(G)$ and $\mathsf w(G)$ are bounded subsets of $\frac{1}{\exp(G)}\mathbb{N}$ with maximum $ \mathsf K(G)$ and $\mathsf k(G)$, respectively (here $\mathsf{K}(G)$ and $\mathsf{k}(G)$ denote the large and the small cross number of $G$, respectively). We give results on the structure of $\mathsf W(G)$ and $\mathsf w(G)$. We first show that both sets contain long arithmetic progressions and that only close to the maximum there might be some gaps. Then, we provide groups for which $\mathsf W(G)$ and $\mathsf w(G)$ actually are arithmetic progressions, and argue that this is rather a rare phenomenon. Finally, we provide some results in case there are gaps.

Spatial and spatiotemporal volatility models are a class of models designed to capture spatial dependence in the volatility of spatial and spatiotemporal data. Spatial dependence in the volatility may arise due to spatial spillovers among locations; that is, if two locations are in close proximity, they can exhibit similar volatilities. In this paper, we aim to provide a comprehensive review of the recent literature on spatial and spatiotemporal volatility models. We first briefly review time series volatility models and their multivariate extensions to motivate their spatial and spatiotemporal counterparts. We then review various spatial and spatiotemporal volatility specifications proposed in the literature along with their underlying motivations and estimation strategies. Through this analysis, we effectively compare all models and provide practical recommendations for their appropriate usage. We highlight possible extensions and conclude by outlining directions for future research.

In this paper, using the shrinkage-based approach for portfolio weights and modern results from random matrix theory we construct an effective procedure for testing the efficiency of the expected utility (EU) portfolio and discuss the asymptotic behavior of the proposed test statistic under the high-dimensional asymptotic regime, namely when the number of assets $p$ increases at the same rate as the sample size $n$ such that their ratio $p/n$ approaches a positive constant $c\in(0,1)$ as $n\to\infty$. We provide an extensive simulation study where the power function and receiver operating characteristic curves of the test are analyzed. In the empirical study, the methodology is applied to the returns of S\&P 500 constituents., Comment: 27 pages, 5 figures, 2 tables

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. We consider the system $\mathcal L (H)$ of all sets of lengths of $H$ and study when $\mathcal L (H)$ contains or is contained in a system $\mathcal L (H')$ of a Krull monoid $H'$ with finite class group $G'$, prime divisors in all classes and Davenport constant $\mathsf D (G')=\mathsf D (G)$. Among others, we show that if $G$ is either cyclic of order $m \ge 7$ or an elementary $2$-group of rank $m-1 \ge 6$, and $G'$ is any group which is non-isomorphic to $G$ but with Davenport constant $\mathsf D (G')=\mathsf D (G)$, then the systems $\mathcal L (H)$ and $\mathcal L (H')$ are incomparable., Comment: European J. Combinatorics, to appear

In time-series analyses, particularly for finance, generalized autoregressive conditional heteroscedasticity (GARCH) models are widely applied statistical tools for modelling volatility clusters (i.e., periods of increased or decreased risk). In contrast, it has not been considered to be of critical importance until now to model spatial dependence in the conditional second moments. Only a few models have been proposed for modelling local clusters of increased risks. In this paper, we introduce novel spatial GARCH and exponential GARCH processes in a unified spatial and spatiotemporal GARCH-type model, which also covers all previously proposed spatial ARCH models as well as time-series GARCH models. For this common modelling framework, estimators are derived based on nonlinear least squares and on the maximum-likelihood approach. In addition to the theoretical contributions of this paper, we suggest a model selection strategy that is verified by a series of Monte Carlo simulation studies. Eventually, the use of the unified model is demonstrated by an empirical example that focuses on real estate prices from 1995 to 2014 across the ZIP-Code areas of Berlin. A spatial autoregressive model is applied to the data to illustrate how locally varying model uncertainties can be captured by the spatial GARCH-type models.

Let H be a Krull monoid with finite class group G and suppose that every class contains a prime divisor. Then sets of lengths in H have a well-defined structure which just depends on the class group G. With methods from additive combinatorics we establish a characterization of those class groups G guaranteeing that all sets of lengths are (almost) arithmetical progressions., Comment: To appear in Rev. Mat. Iberoamericana. The content of this preprint was initially disseminated as part (mostly Chapter 4) of an early version of the manuscript ''A characterization of class groups via sets of lengths'' (hal-01131955, version 1 and arXiv:1503.04679v1) by the same authors

The inverse problem associated to the Erd\H{o}s-Ginzburg-Ziv constant and the $\eta$-constant is solved for finite abelian groups of the form $C_2 \oplus C_2 \oplus C_{2n}$ where $n \ge 2$ is an integer., Comment: 16 pages

For a finite abelian group $(G,+, 0)$ the Harborth constant $\mathsf{g}(G)$ is the smallest integer $k$ such that each squarefree sequence over $G$ of length $k$, equivalently each subset of $G$ of cardinality at least $k$, has a subsequence of length $\exp(G)$ whose sum is $0$. In this paper, it is established that $\mathsf{g}(G)= 3n + 3$ for prime $n \neq 3$ and $\mathsf{g}(C_3 \oplus C_9)= 13$.

Given a malfunctioning system, sequential diagnosis aims at identifying the root cause of the failure in terms of abnormally behaving system components. As initial system observations usually do not suffice to deterministically pin down just one explanation of the system's misbehavior, additional system measurements can help to differentiate between possible explanations. The goal is to restrict the space of explanations until there is only one (highly probable) explanation left. To achieve this with a minimal-cost set of measurements, various (active learning) heuristics for selecting the best next measurement have been proposed. We report preliminary results of extensive ongoing experiments with a set of selection heuristics on real-world diagnosis cases. In particular, we try to answer questions such as "Is some heuristic always superior to all others?", "On which factors does the (relative) performance of the particular heuristics depend?" or "Under which circumstances should I use which heuristic?", Comment: This work was presented at the International Workshop on Principles of Diagnosis 2018 (DX-2018) and a version of this work was formally published as "Patrick Rodler, Wolfgang Schmid. On the impact and proper use of heuristics in test-driven ontology debugging. RuleML+RR, 2018."

We determine the exact value of the $\eta$-constant and the multiwise Davenport constants for finite abelian groups of rank three having the form $G \simeq C_2 \oplus C_{n_2} \oplus C_{n_3}$ with $2 \mid n_2 \mid n_3$. Moreover, we determine the Erd\H{o}s-Ginzburg-Ziv constant of these groups under the assumption that $n_2/2$ has Property D or $n_2 = n_3$., Comment: 17 pages

The paper solves the problem of optimal portfolio choice when the parameters of the asset returns distribution, like the mean vector and the covariance matrix are unknown and have to be estimated by using historical data of the asset returns. The new approach employs the Bayesian posterior predictive distribution which is the distribution of the future realization of the asset returns given the observable sample. The parameters of the posterior predictive distributions are functions of the observed data values and, consequently, the solution of the optimization problem is expressed in terms of data only and does not depend on unknown quantities. In contrast, the optimization problem of the traditional approach is based on unknown quantities which are estimated in the second step leading to a suboptimal solution. We also derive a very useful stochastic representation of the posterior predictive distribution whose application leads not only to the solution of the considered optimization problem, but provides the posterior predictive distribution of the optimal portfolio return used to construct a prediction interval. A Bayesian efficient frontier, a set of optimal portfolios obtained by employing the posterior predictive distribution, is constructed as well. Theoretically and using real data we show that the Bayesian efficient frontier outperforms the sample efficient frontier, a common estimator of the set of optimal portfolios known to be overoptimistic., Comment: 21 pages, 5 figures

We explicitly determine those sets of nonnegative integers which occur as sets of lengths in all numerical monoids., Comment: Banach Center Publications, to appear. arXiv admin note: substantial text overlap with arXiv:1710.04388

Model-Based Diagnosis deals with the identification of the real cause of a system's malfunction based on a formal system model and observations of the system behavior. When a malfunction is detected, there is usually not enough information available to pinpoint the real cause and one needs to discriminate between multiple fault hypotheses (called diagnoses). To this end, Sequential Diagnosis approaches ask an oracle for additional system measurements. This work presents strategies for (optimal) measurement selection in model-based sequential diagnosis. In particular, assuming a set of leading diagnoses being given, we show how queries (sets of measurements) can be computed and optimized along two dimensions: expected number of queries and cost per query. By means of a suitable decoupling of two optimizations and a clever search space reduction the computations are done without any inference engine calls. For the full search space, we give a method requiring only a polynomial number of inferences and show how query properties can be guaranteed which existing methods do not provide. Evaluation results using real-world problems indicate that the new method computes (virtually) optimal queries instantly independently of the size and complexity of the considered diagnosis problems and outperforms equally general methods not exploiting the proposed theory by orders of magnitude.

In this study, we construct two tests for the weights of the global minimum variance portfolio (GMVP) in a high-dimensional setting, namely, when the number of assets $p$ depends on the sample size $n$ such that $\frac{p}{n}\to c \in (0,1)$ as $n$ tends to infinity. In the case of a singular covariance matrix with rank equal to $q$ we assume that $q/n\to \tilde{c}\in(0, 1)$ as $n\to\infty$. The considered tests are based on the sample estimator and on the shrinkage estimator of the GMVP weights. We derive the asymptotic distributions of the test statistics under the null and alternative hypotheses. Moreover, we provide a simulation study where the power functions and the receiver operating characteristic curves of the proposed tests are compared with other existing approaches. We observe that the test based on the shrinkage estimator performs well even for values of $c$ close to one., Comment: 16 pages, 10 figures (final version, accepted for publication in IEEE Transactions of Signal Processing)

We show that for every finite nonempty set L of integers greater than or equal to 2 there are a numerical monoid H and a squarefree element a $\in$ H whose set of lengths L(a) is equal to L., Comment: Forum Math., to appear

We carry out a detailed investigation of congruence half-factorial Krull monoids with finite cyclic class group and related problems. Specifically, we determine precisely all relatively large values that can occur as a minimal distance of a Krull monoid with finite cyclic class group, as well as the exact distribution of prime divisors over the ideal classes in these cases. Our results apply to various classical objects, including maximal orders and certain semi-groups of modules. In addition, we present applications to quantitative problems in factorization theory. More specifically, we determine exponents in the asymptotic formulas for the number of algebraic integers whose sets of lengths have a large difference.

Many wind speed forecasting approaches have been proposed in literature. In this paper a new statistical approach for jointly predicting wind speed, wind direction and air pressure is introduced. The wind direction and the air pressure are important to extend the forecasting accuracy of wind speed forecasts. A good forecast for the wind direction helps to bring the turbine into the predominant wind direction. We combine a multivariate seasonal time varying threshold autoregressive model with interactions (TVARX) with a threshold seasonal autoregressive conditional heteroscedastic (TARCHX) model. The model includes periodicity, conditional heteroscedasticity, interactions of different dependent variables and a complex autoregressive structure with non-linear impacts. In contrast to ordinary likelihood estimation approaches, we apply a high-dimensional shrinkage technique instead of a distributional assumption for the dependent variables. The iteratively re-weighted least absolute shrinkage and selection operator (LASSO) method allows to capture conditional heteroscedasticity and a comparatively fast computing time. The proposed approach yields accurate predictions of wind speed, wind direction and air pressure for a short-term period. Prediction intervals up to twenty-four hours are presented.

In this work we present strategies for (optimal) measurement selection in model-based sequential diagnosis. In particular, assuming a set of leading diagnoses being given, we show how queries (sets of measurements) can be computed and optimized along two dimensions: expected number of queries and cost per query. By means of a suitable decoupling of two optimizations and a clever search space reduction the computations are done without any inference engine calls. For the full search space, we give a method requiring only a polynomial number of inferences and guaranteeing query properties existing methods cannot provide. Evaluation results using real-world problems indicate that the new method computes (virtually) optimal queries instantly independently of the size and complexity of the considered diagnosis problems.

We consider the estimation of the multi-period optimal portfolio obtained by maximizing an exponential utility. Employing Jeffreys' non-informative prior and the conjugate informative prior, we derive stochastic representations for the optimal portfolio weights at each time point of portfolio reallocation. This provides a direct access not only to the posterior distribution of the portfolio weights but also to their point estimates together with uncertainties and their asymptotic distributions. Furthermore, we present the posterior predictive distribution for the investor's wealth at each time point of the investment period in terms of a stochastic representation for the future wealth realization. This in turn makes it possible to use quantile-based risk measures or to calculate the probability of default. We apply the suggested Bayesian approach to assess the uncertainty in the multi-period optimal portfolio by considering assets from the FTSE 100 in the weeks after the British referendum to leave the European Union. The behaviour of the novel portfolio estimation method in a precarious market situation is illustrated by calculating the predictive wealth, the risk associated with the holding portfolio, and the default probability in each period., Comment: 38 pages, 5 figures

In many model-based diagnosis applications it is impossible to provide such a set of observations and/or measurements that allow to identify the real cause of a fault. Therefore, diagnosis systems often return many possible candidates, leaving the burden of selecting the correct diagnosis to a user. Sequential diagnosis techniques solve this problem by automatically generating a sequence of queries to some oracle. The answers to these queries provide additional information necessary to gradually restrict the search space by removing diagnosis candidates inconsistent with the answers. During query computation, existing sequential diagnosis methods often require the generation of many unnecessary query candidates and strongly rely on expensive logical reasoners. We tackle this issue by devising efficient heuristic query search methods. The proposed methods enable for the first time a completely reasoner-free query generation while at the same time guaranteeing optimality conditions, e.g. minimal cardinality or best understandability, of the returned query that existing methods cannot realize. Hence, the performance of this approach is independent of the (complexity of the) diagnosed system. Experiments conducted using real-world problems show that the new approach is highly scalable and outperforms existing methods by orders of magnitude.

In this paper, we introduce a new spatial model that incorporates heteroscedastic variance depending on neighboring locations. The proposed process is regarded as the spatial equivalent to the temporal autoregressive conditional heteroscedasticity (ARCH) model. We show additionally how the introduced spatial ARCH model can be used in spatiotemporal settings. In contrast to the temporal ARCH model, in which the distribution is known given the full information set of the prior periods, the distribution is not straightforward in the spatial and spatiotemporal setting. However, it is possible to estimate the parameters of the model using the maximum-likelihood approach. Via Monte Carlo simulations, we demonstrate the performance of the estimator for a specific spatial weighting matrix. Moreover, we combine the known spatial autoregressive model with the spatial ARCH model assuming heteroscedastic errors. Eventually, the proposed autoregressive process is illustrated using an empirical example. Specifically, we model lung cancer mortality in 3108 U.S. counties and compare the introduced model with two benchmark approaches.

Let $H$ be an atomic monoid. The set of distances $\Delta (H)$ of $H$ is the set of all $d \in \mathbb{N}$ with the following property: there are irreducible elements $u\_1, \ldots, u\_k, v\_1 \ldots, v\_{k+d}$ such that $u\_1 \cdot \ldots \cdot u\_k=v\_1 \cdot \ldots \cdot v\_{k+d}$ but $u\_1 \cdot \ldots \cdot u\_k$ cannot be written as a product of $\ell$ irreducible elements for any $\ell \in \mathbb{N}$ with $k\lt \ell \lt k+d$. It is well-known (and easy to show) that, if $\Delta (H)$ is nonempty, then $\min \Delta (H) = \gcd \Delta (H)$. In this paper we show conversely that for every finite nonempty set $\Delta \subset \mathbb{N}$ with $\min \Delta = \gcd \Delta$ there is a finitely generated Krull monoid $H$ such that $\Delta (H)=\Delta$., Comment: Revised version. To appear in Journal of Algebra

Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far). Under reasonable algebraic finiteness assumptions, sets of lengths of transfer Krull monoids and of weakly Krull monoids satisfy the Structure Theorem for Sets of Lengths. In spite of this common feature we demonstrate that systems of sets of lengths for a variety of classes of weakly Krull monoids are different from the system of sets of lengths of any transfer Krull monoid.

Some of the fundamental notions related to sets of lengths of Krull monoids with finite class group are discussed, and a survey of recent results is given. These include the elasticity and related notions, the set of distances, and the structure theorem for sets of lengths. Several open problems are mentioned.

For a finite abelian group $(G,+)$ the Harborth constant is defined as the smallest integer $\ell$ such that each squarefree sequence over $G$ of length $\ell$ has a subsequence of length equal to the exponent of $G$ whose terms sum to $0$. The plus-minus weighted Harborth constant is defined in the same way except that the existence of a plus-minus weighted subsum equaling $0$ is required, that is, when forming the sum one can chose a sign for each term. The inverse problem associated to these constants is the problem of determining the structure of squarefree sequences of maximal length that do not yet have such a zero-subsum. We solve the inverse problems associated to these constant for certain groups, in particular for groups that are the direct sum of a cyclic group and a group of order two. Moreover, we obtain some results for the plus-minus weighted Erd\H{o}s--Ginzburg--Ziv constant.

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u\_1 \cdot \ldots \cdot u\_k$, with irreducibles $u\_1, \ldots u\_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf L (a)$ of all possible $k$ is called the set of lengths of $a$. It is well-known that the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ depends only on the class group $G$. In the present paper we study the inverse question asking whether or not the system $\mathcal L (H)$ is characteristic for the class group. Consider a further Krull monoid $H'$ with class group $G'$ such that every class contains a prime divisor and suppose that $\mathcal L (H) = \mathcal L (H')$. We show that, if one of the groups $G$ and $G'$ is finite and has rank at most two, then $G$ and $G'$ are isomorphic (apart from two well-known pairings)., Comment: The current version is close to the one to appear in J. Korean Math. Soc., yet it contains a detailed proof of Proposition 2.4. The content of Chapter 4 of the first version had been split off and is presented in ' A characterization of Krull monoids for which sets of lengths are (almost) arithmetical progressions' by the same authors (see hal-01976941 and arXiv:1901.03506)

We consider two families of weighted zero-sum constants for finite abelian groups. For a finite abelian group $( G , + )$, a set of weights $W \subset \mathbb{Z}$, and an integral parameter $m$, the $m$-wise Davenport constant with weights $W$ is the smallest integer $n$ such that each sequence over $G$ of length $n$ has at least $m$ disjoint zero-subsums with weights $W$. And, for an integral parameter $d$, the $d$-constrained Davenport constant with weights $W$ is the smallest $n$ such that each sequence over $G$ of length $n$ has a zero-subsum with weights $W$ of size at most $d$. First, we establish a link between these two types of constants and several basic and general results on them. Then, for elementary $p$-groups, establishing a link between our constants and the parameters of linear codes as well as the cardinality of cap sets in certain projective spaces, we obtain various explicit results on the values of these constants.

Let $H$ be a Krull monoid with class group $G$ and suppose that each class contains a prime divisor. Then every element $a \in H$ has a factorization into irreducible elements, and the set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. We consider the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths, and we characterize (in terms of the class group $G$) when $\mathcal L (H)$ is additively closed under set addition., Comment: Revista Matem{\'a}tica Iberoamericana, to appear

We estimate the global minimum variance (GMV) portfolio in the high-dimensional case using results from random matrix theory. This approach leads to a shrinkage-type estimator which is distribution-free and it is optimal in the sense of minimizing the out-of-sample variance. Its asymptotic properties are investigated assuming that the number of assets $p$ depends on the sample size $n$ such that $\frac{p}{n}\rightarrow c\in (0,+\infty)$ as $n$ tends to infinity. The results are obtained under weak assumptions imposed on the distribution of the asset returns, namely it is only required the fourth moments existence. Furthermore, we make no assumption on the upper bound of the spectrum of the covariance matrix. As a result, the theoretical findings are also valid if the dependencies between the asset returns are described by a factor model which appears to be very popular in financial literature nowadays. This is also well-documented in a numerical study where the small- and large-sample behavior of the derived estimator are compared with existing estimators of the GMV portfolio. The resulting estimator shows significant improvements and it turns out to be robust to the deviations from normality., Comment: 38 pages inc. 16 figures. Revised and corrected version