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## On the linear classification of even and odd permutation matrices and the complexity of computing the permanent

The problem of linear classification of the parity of permutation matrices is studied. This problem is related to the analysis of complexity of a class of algorithms designed for computing the permanent of a matrix that generalizes the Kasteleyn algorithm. Exponential lower bounds on the magnitude of the coefficients of the functional that classifies the even and odd permutation matrices in the case of the field of real numbers and similar linear lower bounds on the rank of the classifying map for the case of the field of characteristic 2 are obtained.

It was shown that in some knot theories the crucial role is played by parity, i.e. a function on crossings valued in {0,1} and behaving nicely with respect to Reidemeister moves. Any parity allows one to construct functorial mappings from knots to knots, to refine many invariants and to prove minimality theorems for knots. In the present paper, we generalise the notion of parity and construct parities with coefficients from an abelian group rather than Z2 and investigate them for different knot theories. For some knot theories we show that there is the universal parity, i.e. such a parity that any other parity factors through it. We realise that in the case of flat knots all parities originate from homology groups of underlying surfaces and, at the same time, allow one to “localise” the global homological information about the ambient space at crossings. We prove that there is only one non-trivial parity for free knots, the Gaussian parity. At the end of the paper we analyse the behaviour of some invariants constructed for some modifications of parities.

An invariant subcode of a linear block code under the permutation is introduced. The concept of invariant subcode has two types of applications. The first type is decoding of linear block codes given the group of symmetry. The second type is the attack the McEliece cryptosystem based on codes correcting errors. Several examples illustrating the concept are presented.

The present monograph is devoted to low-dimensional topology in the context of two thriving theories: parity theory and theory of graph-links, the latter being an important generalization of virtual knot theory constructed by means of intersection graphs. Parity theory discovered by the second-named author leads to a new perspective in virtual knot theory, the theory of cobordisms in two-dimensional surfaces, and other new domains of topology. Theory of graph-links highlights a new combinatorial approach to knot theory.

The classical Turán theorem determines the minimum number of edges in a graph on n vertices with independence number α. We consider unit-distance graphs on the Euclidean plane, i.e., graphs G= (V, E) with V⊂ R2 and E= {{x, y} : | x- y| = 1} , and show that the minimum number of edges in a unit-distance graph on n vertices with independence number α⩽ λn, λ∈[14,27], is bounded from below by the quantity 19-50λ3n, which is several times larger than the general Turán bound and is tight at least for λ=27. © 2016, Springer Science+Business Media New York.

The Human Fertility Database (HFD) was created in response to the growing need of freely available, high-quality, internationally comparable population-level fertility data. In the midst of discussions about the shift to later childbearing and low fertility levels in industrialized countries, the lack of proper fertility data became particularly apparent. A number of other sources of fertility data exist, but accessibility, scope and comparability of these data vary widely, and data documentation is often inadequate or absent. Officially launched in 2009, the HFD is a collaborative endeavour of the Max Planck Institute for Demographic Research (MPIDR) in Rostock (Germany) and the Vienna Institute of Demography / Wittgenstein Centre for Demography and Global Human Capital (Austria). The development of the HFD was inspired by the success of the Human Mortality Database (HMD) that had become an important resource of detailed mortality data. By providing free access to detailed fertility data which meet the highest quality standards, the HFD not only fills the gap in the availability of comparable fertility data, but also promotes more sophisticated analyses and methodological advances.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The geographic information system (GIS) is based on the first and only Russian Imperial Census of 1897 and the First All-Union Census of the Soviet Union of 1926. The GIS features vector data (shapefiles) of allprovinces of the two states. For the 1897 census, there is information about linguistic, religious, and social estate groups. The part based on the 1926 census features nationality. Both shapefiles include information on gender, rural and urban population. The GIS allows for producing any necessary maps for individual studies of the period which require the administrative boundaries and demographic information.

Existing approaches suggest that IT strategy should be a reflection of business strategy. However, actually organisations do not often follow business strategy even if it is formally declared. In these conditions, IT strategy can be viewed not as a plan, but as an organisational shared view on the role of information systems. This approach generally reflects only a top-down perspective of IT strategy. So, it can be supplemented by a strategic behaviour pattern (i.e., more or less standard response to a changes that is formed as result of previous experience) to implement bottom-up approach. Two components that can help to establish effective reaction regarding new initiatives in IT are proposed here: model of IT-related decision making, and efficiency measurement metric to estimate maturity of business processes and appropriate IT. Usage of proposed tools is demonstrated in practical cases.