On the edit distance function of the random graph

Given a hereditary property of graphs $\mathcal{H}$ and a $p\in [0,1]$ , the edit distance function $\textrm{ed}_{\mathcal{H}}(p)$ is asymptotically the maximum proportion of edge additions plus edge deletions applied to a graph of edge density p sufficient to ensure that the resulting graph satisfies $\mathcal{H}$ . The edit distance function is directly related to other well-studied quantities such as the speed function for $\mathcal{H}$ and the $\mathcal{H}$ -chromatic number of a random graph. Let $\mathcal{H}$ be the property of forbidding an Erdős–Rényi random graph $F\sim \mathbb{G}(n_0,p_0)$ , and let $\varphi$ represent the golden ratio. In this paper, we show that if $p_0\in [1-1/\varphi,1/\varphi]$ , then a.a.s. as $n_0\to\infty$ , \begin{align*} {\textrm{ed}}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{-\log(1-p_0)}, \frac{1-p}{-\log p_0} \right\}. \end{align*} Moreover, this holds for $p\in [1/3,2/3]$ for any $p_0\in (0,1)$ . A primary tool in the proof is the categorization of p-core coloured regularity graphs in the range $p\in[1-1/\varphi,1/\varphi]$ . Such coloured regularity graphs must have the property that the non-grey edges form vertex-disjoint cliques.

A Unified Approach to the Decomposition Theorems in Baer $$*$$-Rings

The aim of the paper is to generalize decomposition theorems showed in Bagheri-Bardi et al. (Linear Algebra Appl 583:102–118, 2019; Linear Algebra Appl 539:117–133, 2018) by a unified approach. We show a general decomposition theorem with respect to a hereditary property. Then the vast majority of decompositions known in the algebra of Hilbert space operators is generalized to elements of Baer $$*$$ ∗ -rings by this theorem. The theorem yields also results which are new in the algebra of bounded Hilbert space operators. Additionally, the model of summands in Wold–Słociński decomposition is given in Baer $$*$$ ∗ -rings.

Fuzzy pairwise regular bitopological spaces in quasi-coincidence sense

This paper introduces three notions of fuzzy pairwise regular between bitopological spaces in quasi-coincidence sense. Then, we investigate some relations between ours and other counterparts. We observe that all these concepts are preserved under one-one, onto, fuzzy closed, fuzzy open, and fuzzy continuous mappings. Also, the hereditary property is satisfied by these concepts. Journal of Bangladesh Academy of Sciences, Vol. 44, No. 2, 139-143, 2020

Some new separation axioms in fuzzy soft topological spaces

In this paper, a new form of separation axioms called r-fuzzy soft Ti;(i = 0,1,2,3,4), r-fuzzy soft regular and r-fuzzy soft normal axioms are introduced in a fuzzy soft topological space based on the paper Ayg?no?lu et al. [7]. Also, the relations of these axioms with each other are investigated with the help of examples. Furthermore, some fuzzy soft invariance properties, namely fuzzy soft topological property and hereditary property are specified.

The institutions of fiduciary (trust) management and testamentary foundations in Russia and Germany: comparative analysis

The study was aimed to consider the existing institutions of fiduciary management of hereditary property and testamentary foundations in Russian and German legislation, to compare the institute of testamentary foundations with the fiduciary management of hereditary property in the legislation of the Russian Federation, to identify similarities and differences between Russian testamentary foundations and German (posthumous) testamentary foundations. Such research methods as comparative legal analysis, synthesis, analysis, formal-logical and other methods of scientific research have been employed within the study conducted. The article presents the results of a comparative legal analysis of Russian and German legislations and points out potential challenges in Russian law enforcement practice. Comparative legal analysis and comparison of the current Russian legislation with the experience of legislative regulation of (posthumous) testamentary foundations in the Federal Republic of Germany allowed to reveal a tendency of adopting the positive experience of German law.

Theoretic and Practical Aspects of Protection of the Right of Ownership in the Hereditary Relations

The relevance of the subject matter lies in the fact that inheritance is one of the most common grounds for acquiring property by individuals. Considering the fact that the heirs are often relatives of the testator, to avoid disputes between them, the law should contain an effective mechanism for resolving relations between heirs over the redistribution of inheritance or change of the order of inheritance, and a mechanism to protect the rights and interests of heirs in case of disputes. The purpose of this study is to identify gaps and inconsistencies in civil legislation and case law in the study of the main ways to protect the rights of heirs in hereditary relations, and ways to resolve them. It is noted that in the presence of disputes between the heirs, it is not the protection of property rights that is carried out, because the heirs have not yet acquired the right of ownership, but the protection of the right to inheritance, according to which they will be able to acquire ownership of the inherited property. There is a lack of a particular list of ways to protect the rights of heirs in the legislation of Ukraine, which has a negative impact on judicial practice, as they often use inappropriate methods of protection. The study analyses the case law of hereditary disputes and identifies the main mistakes that courts make in resolving such cases. Particular attention is focused on the study of such methods of protection as the recognition of the certificate of inheritance as invalid, the hereditary recognition of the property that belonged to the deceased, but was not part of the inheritance. The study investigates the moment of ownership of the hereditary property of the heirs and a critical analysis of the provisions of Article 1268 of the Civil Code, which determine the moment from which the inheritance belongs to the heir – namely from the moment of opening the inheritance. There is a conflict between the rules of Article 1268 and Article 3 of the Law of Ukraine "On state registration of real rights to immovable property and their encumbrances" in terms of establishing the moment of ownership of immovable property by inheritance

Failure rate properties of parallel systems

We study failure rate monotonicity and generalised convex transform stochastic ordering properties of random variables, with an emphasis on applications. We are especially interested in the effect of a tail-weight iteration procedure to define distributions, which is equivalent to the characterisation of moments of the residual lifetime at a given instant. For the monotonicity properties, we are mainly concerned with hereditary properties with respect to the iteration procedure providing counterexamples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterion, based on the analysis of the sign variation of a suitable function. This criterion is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.