A database-theoretic variant of Formal Concept Analysis (FCA) defines the concept lattice of a relational structure. The concept lattice naturally extends to a relational concept algebra; this further deepens the connection between FCA and database theory. It seems possible to axiomatically characterize concept lattices of relational structures using the additional concept operations; partial results have been presented in recent work. This requires to be precise about the underlying data model, as even small details can affect the axiomatization. In this article, we give an up-to-date presentation of our primary relational data model.

Conjunctive table algebras are introduced and axiomatically characterized. A conjunctive table algebra is a variant of SPJR algebra (a weaker form of relational algebra), which corresponds to conjunctive queries with equality. The table operations relate to logical operations (e.g. column deletion corresponds to existential quantification). This enables a connection between database theory and algebraic logic, particularly cylindric algebras. A comparison shows which cylindric algebra axioms hold in conjunctive table algebras, which ones are modified, and which ones hold in addition.

The paper introduces orbital concept lattices, which enhance concept lattices of relational structures with a semigroup action that encodes projection, renaming and duplication operations on concept extents, and fuses them with their counterparts on concept intents (formalized by tableau queries). This strengthens the existing connections between this branch of FCA and database theory, and it opens up a new possibility of characterizing such concept lattices by a set of axioms. The orbital semilattices, also introduced in this paper, are a first step in this direction, as they characterize the subsemilattices generated by finite queries, and thereby also enable a connection with algebraic logic.

Everybody knows what musical intervals are, right?! Not so fast. In physics, folks tend to think of frequencies and proportions, while in ancient Greece, they had a monochord to get a grip on intervals. In psychoacoustics, people prefer the concept of difference of pitch, or just say that intervals can be added.In music cognition, we learn that tonal music on the white keys of the piano doesn’t care so much about sound and frequencies but more about the number of steps from one tone to another. Bobby McFerrin demonstrates with an audience how that works for pentatonic music. However, there is an algebraic theory of measurement which gets this all and much more under one umbrella.

In Formal Concept Analysis (FCA), cover- and packing problems occur naturally as an equivalent formulation of the problem to determine the order 2-dimension and breadth of a complete lattice. Furthermore, isolation- and blocking problems provide bounds for these parameters w.r.t. the tensor product of complete lattices.

This article treats the determination of the largest powerset lattice that can be order embedded into a complete lattice \(\mathbb {L}\). That’s a complexity measure for \(\mathbb {L}\) and can be seen as an inner dimension. We show that this embedding problem translates to a set packing problem on the level of formal contexts. From this point of view, similarities to graph theoretic parameters emerge, which lead to a lattice theoretical interpretation of the clique number of simple graphs, in terms of an inner dimension of complete ortholattices. Furthermore, behaviour of the tensor product of complete (ortho)lattices is studied w.r.t. these inner dimensions.

Tin is the chemical element with the largest number of stable isotopes. Its complete proton shell, comparable with the closed electron shells in the chemically inert noble gases, is not a mere precursor to extended stability; since the protons carry the nuclear charge, their spatial arrangement also drives the nuclear electromagnetism. We report high-precision measurements of the electromagnetic moments and isomeric differences in charge radii between the lowest 1/2 + , 3/2 + , and 11/2 − states in 117–131 Sn, obtained by collinear laser spectroscopy. Supported by state-of-the-art atomic-structure calculations, the data accurately show a considerable attenuation of the quadrupole moments in the closed-shell tin isotopes relative to those of cadmium, with two protons less. Linear and quadratic mass-dependent trends are observed. While microscopic density functional theory explains the global behaviour of the measured quantities, interpretation of the local patterns demands higher-fidelity modelling.

Our paper shows that decision trees and random forests can be described via pattern structures. This new perspective leads to a better understanding of random forests and delivers a new way of analyzing complex data with the help of pattern structures.

Scales are a fundamental concept of musical practice around the world. They commonly exhibit symmetry properties that are formally studied using cyclic groups in the field of mathematical scale theory. This paper proposes an axiomatic framework for mathematical scale theory, embeds previous research, and presents the theory of maximally even scales and well-formed scales in a uniform and compact manner. All theorems and lemmata are completely proven in a modern and consistent notation. In particular, new simplified proofs of existing theorems such as the equivalence of non-degenerate well-formedness and Myhill's property are presented. This model of musical scales explicitly formalizes and utilizes the cyclic order relation of pitch classes.

Research shows that pattern structures are a useful tool for analyzing complex data. In our paper, we present a new general framework of using pattern structures as a data mining tool and as an application of the new framework we show a way to handle a classification problem of red wines.

This article treats four optimization problems on posets and applies the results to Formal Concept Analysis. The cover-, packing-, isolationand blocking problem are investigated.

The hyperfine splitting in heavy highly charged ions provide the means to test QED in extremely strong magnetic fields. In order to provide a meaningful test, the splitting has to be measured in H-like and Li-like ions to remove uncertainties from nuclear structure. This has been achieved at the experimental storage ring ESR but a discrepancy to the theoretical prediction of more than 7σ was observed. We report on these measurements as well as on NMR measurements that were performed to solve this issue.

A Penning-trap facility for high-precision mass spectrometry based on a novel detection method has been built. This method consists in measuring motional frequencies of singly-charged ions trapped in strong magnetic fields through the fluorescence photons from laser-cooled 40Ca+ ions, to overcome limitations faced in electronic single-ion detection techniques. The key element of this facility is an open-ring Penning trap coupled upstream to a preparation Penning trap similar to those used at Radioactive Ion Beam facilities. Here we present a full characterization of the trap and demonstrate motional frequency measurements of trapped ions stored by applying external radiofrequency fields in resonance with the ions' eigenmotions, in combination with time-of-flight identification. The infrastructure developed to observe the fluorescence photons from 40Ca+, comprising the 12 laser beams and the optical system to register the image in a high-sensitive CCD sensor, has been proved by taking images of the trapped and cooled 40Ca+ ions. This demonstrates the functionality of the proposed laser-based mass-spectrometry technique, providing a unique platform for precision experiments with implications in different fields of physics.

The LIBELLE experiment performed at the experimental storage ring at the GSI Helmholtz Center for Heavy Ion Research in Darmstadt, Germany, has successfully determined the ground state hyperfine (HFS) splittings in hydrogen-like (Bi-209(82+)) and lithium-like (Bi-209(80+)) bismuth. The study of HFS transitions in highly charged ions enables precision tests of QED in extreme electric and magnetic fields otherwise not attainable in laboratory experiments. Besides the transition wavelengths the time-resolved detection of fluorescence photons following the excitation of the ions by a pulsed laser system also allows the extraction of lifetimes of the upper HFS levels and g-factors of the bound 1s and 2s electrons for both charge states. While the lifetime of the upper HFS state in Bi-209(82+) has already been measured in earlier experiments, an experimental value for lifetime of this state in Bi-209(80+) is reported for the first time in this work.

We have studied the formation and properties of two-species ion Coulomb crystals in the Penning trap of the SpecTrap experiment. These crystals have been formed by injection of admixture ions from an external source into a previously confined and laser-cooled cloud of magnesium ions. This kind of study, performed over a range of the admixture ions' charge-to-mass ratios, indicates the conditions for their sympathetic cooling and the formation of two-species ion crystals. This mechanism allows efficient cooling of the admixed species such as highly charged ions which do not feature suitable laser-cooling transitions, and thus make them accessible to high-resolution laser spectroscopy.

We introduce an order theoretic approach to generalized metrics that covers various concepts of distance. In particular, we point out the role of supermodular mappings on lattices, which we then apply in diverse settings such as comparison of ratings and formal concept lattices.

In this article, we analyze different dimensional concepts of complete (ortho)lattices and their tensor products. The determination of these dimensions can be translated to certain set cover problems and the cardinal product of the complementary underlying formal contexts. To treat this cover problems in a unified manner, we take a more universal approach via the general set cover problem and its product. This yields a sufficient condition for the multiplicativity of various lattice dimensions with respect to the tensor product of complete lattices.

A recent measurement of the hyperfine splitting in the ground state of Li-like ^{208}Bi^{80+} has established a "hyperfine puzzle"-the experimental result exhibits a 7σ deviation from the theoretical prediction [J. Ullmann et al., Nat. Commun. 8, 15484 (2017)NCAOBW2041-172310.1038/ncomms15484; J. P. Karr, Nat. Phys. 13, 533 (2017)NPAHAX1745-247310.1038/nphys4159]. We provide evidence that the discrepancy is caused by an inaccurate value of the tabulated nuclear magnetic moment (μ_{I}) of ^{209}Bi. We perform relativistic density functional theory and relativistic coupled cluster calculations of the shielding constant that should be used to extract the value of μ_{I}(^{209}Bi) and combine it with nuclear magnetic resonance measurements of Bi(NO_{3})_{3} in nitric acid solutions and of the hexafluoridobismuthate(V) BiF_{6}^{-} ion in acetonitrile. The result clearly reveals that μ_{I}(^{209}Bi) is much smaller than the tabulated value used previously. Applying the new magnetic moment shifts the theoretical prediction into agreement with experiment and resolves the hyperfine puzzle.