### Article

## On closure operators related to maximal tricliques in tripartite hypergraphs

Triadic Formal Concept Analysis (3FCA) was introduced by Lehman and

Wille almost two decades ago. Many researchers work in Data Mining and

Formal Concept Analysis using the notions of closed sets, Galois and closure

operators, and closure systems. However, a proper closure operator for enu-

meration of triconcepts, i.e. maximal triadic cliques of tripartite hypergraphs,

was not introduced. In this paper, we show that the previously introduced

operators for obtaining triconcepts and maximal connected and complete sets

(MCCSs) are not always consistent and provide the reader with a definition of

valid closure operator and associated set system. Moreover, we study the diffi-

culties of related problems from order-theoretic and combinatorial point view as

well as provide the reader with justifications of the complexity classes of these

problems.

This paper presents several definitions of “optimal patterns” in triadic data and results of experimental comparison of five triclustering algorithms on real-world and synthetic datasets. The evaluation is carried over such criteria as resource efficiency, noise tolerance and quality scores involving cardinality, density, coverage, and diversity of the patterns. An ideal triadic pattern is a totally dense maximal cuboid (formal triconcept). Relaxations of this notion under consideration are: OAC-triclusters; triclusters optimal with respect to the least-square criterion; and graph partitions obtained by using spectral clustering. We show that searching for an optimal tricluster cover is an NP-complete problem, whereas determining the number of such covers is #P-complete. Our extensive computational experiments lead us to a clear strategy for choosing a solution at a given dataset guided by the principle of Pareto-optimality according to the proposed criteria.

Analysis of polyadic data (for example *n*-ary relations) becomes a popular task nowadays. While several data mining techniques exist for dyadic contexts, their extensions to triadic case are not obvious. In this work, we study development of ideas of Formal Concept Analysis for processing three-dimensional data, namely OAC-triclustering (from Object, Attribute, Condition). We consider several similar methods, study relations between their outputs and organize them in an ordered structure.

Triclustring Toolbox is a collection of triclustering methods consolidated into a single interface. It provides access to both box- and prime-based OAC (Object-Attribute-Condition) triclustering, Spectral triclustering and features implementations of DataPeeler and Trias. The application also contains algorithms for mining triclusters of similar values: NOAC and Tri-K-Means. Quality of triclusters is measured in terms of density, diversity, coverage, and variance, if applicable. Formats for input and output data of all the methods are universal, which makes comparison and interpretation of the results easier. The code is written in C# (.Net 4.5) and runs on Windows. Triclustring Toolbox was used to provide experimental results in several articles on triclustering.

Analysis of polyadic data (for example, multi-way tensors and n-ary relations) becomes more and more popular task nowadays. While several datamining techniques exist for (numeric) dyadic contexts, their extensions to the triadic case are not obvious, if possible at all. In this work, we study development of the ideas of Formal Concept Analysis for processing three-dimensional data, namely the so called OAC-triclustering (from Object, Attribute, Condition). Among several known methods, we have reasonably selected the most effective one and used it to propose an algorithm NOAC-triclustering for mining triclusters of similar values in real-valued triadic contexts. We have also proposed a second simple algorithm, Tri-K-Means, based on clustering algorithm K-Means, for the purpose of comparison. The experimental part demonstrates application of the algorithms to both computer-generated and real-world data.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.