Counting Non-Convex 5-Holes in a Planar Point Set

Let S be a set of n points in the general position, that is, no three points in S are collinear. A simple k-gon with all corners in S such that its interior avoids any point of S is called a k-hole. In this paper, we present the first algorithm that counts the number of non-convex 5-holes in S. To our best knowledge, prior to this work there was no known algorithm in the literature except a trivial brute force algorithm. Our algorithm runs in time O(T+Q), where T denotes the number of 3-holes, or empty triangles, in S and Q that denotes the number of non-convex 4-holes in S. Note that T+Q ranges from Ω(n2) to O(n3), while its expected number is Θ(n2logn) when the points in S are chosen uniformly and independently at random from a convex and bounded body in the plane.

FINANCIAL STABILITY AS A FINANCIAL SECURITY INDICATOR OF AN ENTERPRISE

The system diagnostics of enterprise financial security developed by the authors are based on taking into account the combined effect of the main elements of the financial stability management process. On the basis of the justification of the interdependence of the main components of an enterprise’s financial security (on the one hand, the types of financial stability and the liquidity of the balance sheet, on the other hand, their correlative effect on the level of financial security) the authors proposed a model for its evaluation. It has been proposed that the type of financial stability of an enterprise should be determined on the basis of the identification of the financial situation in accordance with the scale developed on the basis of the values of the main financial stability ratios. The type of liquidity on the balance sheet is based on a comparison of liquidity-based items of assets with maturities. The unified impact of types of financial stability and balance sheet liquidity on the level of financial security became the basis for the development a matrix for diagnostics the general position of financial security of the enterprise. Based on the established relationship between the degrees of financial stability and liquidity of an enterprise on the one hand, and the level of financial security of operating activities on the other, a model has been developed to assess the level of financial security of the enterprise’s operating activities. It has been proposed that the financial stability and liquidity of an enterprise should be determined on the basis of a three-tiered indicator by classifying financial situations within the established indicator scale: depending on the priority of selecting funds to finance the tangible portion of a negotiable asset and the sufficiency and composition of a negotiable asset to meet current liabilities. On this basis, a diagnostic matrix of the financial security position of the enterprise’s operational activities has been developed. The interconnection of the positions of the financial security of the enterprise and the unification of its level enabled the authors to develop a matrix of zones of the general position of the financial security of an enterprise where, depending on the combination of financial security levels, zones are distinguished from absolute financial security to financial danger. The testing of each element of the proposed enterprise financial security diagnostic’s system on the materials of a selected group of enterprises of the oil-and-fat industry confirms the practical significance of the developed tools in the process of managing their general financial security. Keywords: financial security, financial security level, financial security position, financial security of operating activities, financial stability, liquidity, oil-and-fat enterprises. JEL Classification G30, M20, Q14 Formulas: 14; fig.:5; tabl.: 4; bibl.: 22.

Isometry Invariant Shape Recognition of Projectively Perturbed Point Clouds by the Mergegram Extending 0D Persistence

Rigid shapes should be naturally compared up to rigid motion or isometry, which preserves all inter-point distances. The same rigid shape can be often represented by noisy point clouds of different sizes. Hence, the isometry shape recognition problem requires methods that are independent of a cloud size. This paper studies stable-under-noise isometry invariants for the recognition problem stated in the harder form when given clouds can be related by affine or projective transformations. The first contribution is the stability proof for the invariant mergegram, which completely determines a single-linkage dendrogram in general position. The second contribution is the experimental demonstration that the mergegram outperforms other invariants in recognizing isometry classes of point clouds extracted from perturbed shapes in images.

Linear General Position (i.e. Arcs) for Zero-Dimensional Schemes Over a Finite Field

We extend some of the usual notions of projective geometry over a finite field (arcs and caps) to the case of zero-dimensional schemes defined over a finite field Fq. In particular we prove that for our type of zero-dimensional arcs the maximum degree in any r-dimensional projective space is r(q + 1) and (if either r = 2 or q is odd) all the maximal cases are projectively equivalent and come from a rational normal curve.

Galen and the Formal Cause

This paper focuses on Galen’s views about the formal cause against the wider background of his account of causation. In his works on the pulse, Galen provides an in-depth discussion of issues such as the theory of causes and the problem of definition. While Galen is inclined to incorporate Peripatetic doctrines and vocabulary, he neglects Aristotle’s formal cause both in his account of causes and in that of definition. Further parallels outside the corpus on the pulse confirm Galen’s reserved attitude towards the (Peripatetic) formal cause. In his work That the Faculties of the Soul Follow the Mixtures of the Body, however, Galen refers favourably to the Peripatetic theory of hylomorphic form. More precisely, Galen equates the Peripatetic en-mattered form with the mixture of homoeomerous bodies (the ratio or proportion of the elementary qualities in them). Here I will attempt to provide an explanation of Galen’s mixed attitude towards the formal cause and I will argue that his general position is consistent after all. In addition to this, I will try to set Galen’s position against the background of the Peripatetic debates on the status of form.

An issue of legal personality of Kyivan Rus’ in Ukraininan law science

Problem setting. Legal personality of Kyivan Rus’ is very complex issue, which contains elements such as contract law, law of war, ambassadorial law, general position of the state in the international relations sphere. The condition of Kyivan Rus’ in medieval history can be determined only after analyzing researches of well-known scientists, who had different opinions on this subject. In spite of the fact that a lot of researches were made on this topic, there is no clear and unambiguous answer to the question: “was Kyivan Rus’ independent and equal subject of international law?” In our opinion, this topic is actual even nowadays, because without an analysis of the issue it is impossible to form a modern understanding of Ukrainian statehood and its features in different periods. Analysis of recent researches and publications. Valuable contribution to the research of Kyivan Rus’ position in international relations sphere were made by lots of scientists in areas of Ukrainian history and history of international law such as O. Zadoroznyi, P. Tolochko, O. Butkevich, A. Dmitriev, Y. Dmitriev, M. Kotlyar, V. Pashuto, D. Feldman, V. Butkevich, I. Shekera, O. Pavlenko etc. Target of research is to analyze and compare opinions of different authors on the issue of determining Kyivan Rus’ as legal entity of international law. To achieve this target these tasks have to be solved: to research and analyze modern scientists’ studies about the position of Kyivan Rus’ in international law sphere in medieval period; to compare scientists’ views on legal personality of the state and come to a certain conclusion on this issue. Article’s main body. In this article author analyzes different periods of Kyivan Rus’ existence, general position of the state in international relations sphere and opinions of different scientists on this subject. Also, the article provides a comparison of scientists’ views on the topic of legal personality of international law of Kyivan Rus’. Conclusions. Kyivan state in different periods of its existence was in various international legal positions. Despite of the fact, that features of international law of Kyivan Rus’ is a topic for controversy, Ukrainian and foreign scientists came to the conclusion that Rus’ was full-fledged subject of international law and after its collapse it revived in the Principality of Galicia-Volhynia, Zaporozhian Sich and the Cossack Hetmanate, Ukrainian People’s Republic, UkSSR (as independent UN member) and modern independent Ukraine.

The Chromatic Number of Two Families of Generalized Kneser Graphs Related to Finite Generalized Quadrangles and Finite Projective 3-Spaces

Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective space $\mathrm{PG}(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\geqslant 43$ we determine the largest independent set of $\Gamma$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\geqslant 47$, this information is then used to show that $\Gamma$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove similar results for the graph that is built in the same way on the chambers of the generalized quadrangle.

Sheaves of maximal intersection and multiplicities of stable log maps

A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.