Lower spectral radius and spectral mapping theorem for suprema preserving mappings

We study Lipschitz, positively homogeneous and finite suprema preserving mappings defined on a max-cone of positive elements in a normed vector lattice. We prove that the lower spectral radius of such a mapping is always a minimum value of its approximate point spectrum. We apply this result to show that the spectral mapping theorem holds for the approximate point spectrum of such a mapping. By applying this spectral mapping theorem we obtain new inequalites for the Bonsall cone spectral radius of max type kernel operators.Comment: arXiv admin note: text overlap with arXiv:1612.0175

Barabanov norms, Lipschitz continuity and monotonicity for the max algebraic joint spectral radius

We present several results describing the interplay between the max algebraic joint spectral radius (JSR) for compact sets of matrices and suitably defined matrix norms. In particular, we extend a classical result for the conventional algebra, showing that the JSR can be described in terms of induced norms of the matrices in the set. We also show that for a set generating an irreducible semigroup (in a cone-theoretic sense), a monotone Barabanov norm always exists. This fact is then used to show that the max algebraic JSR is locally Lipschitz continuous on the space of compact irreducible sets of matrices with respect to the Hausdorff distance. We then prove that the JSR is Hoelder continuous on the space of compact sets of nonnegative matrices. Finally, we prove a strict monotonicity property for the max algebraic JSR that echoes a fact for the classical JSR

Polynomial convolutions in max-plus algebra

Recently, in a work that grew out of their exploration of interlacing polynomials, Marcus, Spielman and Srivastava and then Marcus studied certain combinatorial polynomial convolutions. These convolutions preserve real-rootedness and capture expectations of characteristic polynomials of unitarily invariant random matrices, thus providing a link to free probability. We explore analogues of these types of convolutions in the setting of max-plus algebra. In this setting the max-permanent replaces the determinant, the maximum is the analogue of the expected value and real-rootedness is replaced by full canonical form. Our results resemble those of Marcus et al., however, in contrast to the classical setting we obtain an exact and simple description of all roots.Comment: 27 page

Interval multi-linear systems for tensors in the max-plus algebra and their application in solving the job shop problem

summary:In this paper, we propose the notions of the max-plus algebra of the interval tensors, which can be used for the extension of interval linear systems to interval multi-linear systems in the max-plus algebra. Some properties and basic results of interval multi-linear systems in max-plus algebra are derived. An algorithm is developed for computing a solution of the multi-linear systems in the max-plus algebra. Necessary and sufficient conditions for the interval multi-linear systems for weak solvability over max-plus algebra are obtained as well. Also, some examples are given for illustrating the obtained results. Moreover, we briefly sketch how our results can be used in the max-plus algebraic system theory for synchronized discrete event systems

The Analytic Hierarchy Process, Max Algebra and Multi-objective Optimisation

The Analytic Hierarchy Process (AHP) is widely used for decision making involving multiple criteria. Elsner and van den Driessche introduced a max-algebraic approach to the single criterion AHP. We extend this to the multi-criteria AHP, by considering multi-objective generalisations of the single objective optimisation problem solved in these earlier papers. We relate the existence of globally optimal solutions to the commutativity properties of the associated matrices; we relate min-max optimal solutions to the generalised spectral radius; and we prove that Pareto optimal solutions are guaranteed to exist.Comment: 1 figur

Liability of the online platform operator: case of Airbnb

Napredek sodobne tehnologije je omogočil transakcije med posamezniki (»P2P«) v večjem obsegu kot kdaj koli prej in na trgu se je pojavila kopica spletnih tržnic za deljenje storitev med posamezniki, ki delujejo v različnih sektorjih. Ta magistrska naloga se osredotoča na platforme za ponujanje nastanitev, ki posameznikom omogočajo oglaševanje stanovanj potencialnim gostom po celem svetu in s tem ponujajo alternativno možnost obiskovalcem, ki iščejo nekaj drugega kot zgolj hotelsko sobo. Cilj je predstaviti trenutno ureditev odškodninske odgovornosti upravljavcev spletnih platform in potrebo po prilagoditvi prava Evropske unije in držav članic, ki bo upoštevala spreminjajočo strukturo trga. Avtor pritrjuje stališču Evropske komisije, da je treba spletne platforme prvenstveno obravnavati kot novo priložnost na trgu, vendar pa je določena mera regulacije neizogibna. Trenutna ureditev sloni predvsem na Direktivi o elektronskem poslovanju, ki predvideva določene obveznosti, ki bi jih platforme, ki ponujajo nastanitve, morale upoštevati, saj jih lahko obravnavamo kot »ponudnike storitev informacijske družbe«. Toda določbe direktive glede odškodninske odgovornosti upravljavca platforme so pomanjkljive in zastarele. Razmah spletnih posredniških platform je sprožil zakonodajne pobude v številnih državah članicah EU. Tudi EU posveča vedno večjo pozornost ureditvi delovanja tovrstnih platform. Vendar pa je moč občutiti določeno mero odpora proti regulaciji, še posebej Evropska komisija skuša poudariti, da je treba poiskati pravo razmerje med regulacijo in inovacijo. Pomemben korak k ustrezni ureditvi spletnih posredniških platform je naredila skupina evropskih strokovnjakov, ki je pripravila Osnutek direktive o spletnih posredniških platformah, ki lahko služi kot temelj za nadaljnjo zakonodajo. Slovenija je v svojo zakonodajo implementirala Direktivo o elektronskem poslovanju in se tako v večji meri ujema s trenutno ureditvijo v Evropski uniji. Pozornost pa bi bilo treba posvetiti tudi Zakonu o nepremičninskem posredovanju, ki bi se ob ustrezni razlagi lahko uporabil za urejanje delovanja spletnih platform, ki posredujejo pri ponujanju nepremičnin.Advance of the modern technology has enabled peer-to-peer (“P2P”) transactions on a larger scale than ever before and a wave of online marketplaces has emerged for “P2P” exchange services in various sectors. This thesis focuses on accommodation platforms, which enable individuals to advertise their apartments to potential guests worldwide and by that provide an alternative to visitors who are looking for something else than a hotel room. The objective is to reflect upon the present regulation of the liability of the platform operator and the need to adjust EU and national law to account the changing market structure. The author agrees with the position of the European Commission that online platforms should primarily be seen as an opportunity, however, a certain regulatory approach is inevitable. E-Commerce Directive provides certain obligations which should also be upheld by accommodation platforms, as the later can be regarded as “information society service providers”. However, the directive is scarce regarding the liability of the platform operator and its provisions are outdated. The upsurge of online intermediary platforms has triggered regulatory initiatives in several EU member states. Recently, regulators at the European level have also started paying attention. However, a certain reluctance towards regulation can be observed, especially European Commission is trying to emphasize the importance of finding the right balance between regulation and innovation. An important step towards regulation of online intermediary platforms was made by a group of European legal scholars, who prepared a Discussion Draft of a Directive on Online Intermediary Platforms, which may serve as foundation for future legislation. Slovenia has implemented the E-Commerce Directive and thus largely corresponds to the current situation in the European Union. However, attention should also be devoted to Real Estate Agencies Act, which could, with appropriate interpretation, apply to online accommodation platforms as well