Interests Balance of Creditors and Debtors` Families in Bankruptcy Cases

The article is devoted to the legislative norms on bankruptcy of citizens, providing the possibility of exemption from the unbearable debt obligations execution. It is noted that the number of court cases in this category annually increasing. The author studied the current legal regulation and law enforcement practice of consumer bankruptcy relations, taking into account the interests of the debtor's family as a separate community, and analyzed (in the aspect of the problem of abuse of right) the courts approaches on the issues of releasing citizens from the obligations performance; the practice of challenging transactions with the common property made by the debtor and his or her spouse. There is the need to find a balance between the interests of a debtor's family in his bankruptcy case and preventing the debtor from refusing to fulfill his obligations to his creditors. The attention is drawn to the absence of clear criteria for determining the balance of competing interests with a significant number of court disputes on this issue. It was noted that a balance between the interests of creditors and a debtor's family was achieved through retaining the debtor's minimal property to ensure his livelihood. It is concluded that further study of the issue under consideration in the aspect of the implementation of the constitutional and legal principle of family, motherhood and childhood protection in bankruptcy cases.

ON THE MINIMAL PROPERTY OF DE LA VALLÉE POUSSIN’S OPERATOR

Let $X={\mathcal{C}}_{0}(2{\it\pi})$ or $X=L_{1}[0,2{\it\pi}]$. Denote by ${\rm\Pi}_{n}$ the space of all trigonometric polynomials of degree less than or equal to $n$. The aim of this paper is to prove the minimality of the norm of de la Vallée Poussin’s operator in the set of generalised projections ${\mathcal{P}}_{{\rm\Pi}_{n}}(X,\,{\rm\Pi}_{2n-1})=\{P\in {\mathcal{L}}(X,{\rm\Pi}_{2n-1}):P|_{{\rm\Pi}_{n}}\equiv \text{id}\}$.

Minimal invariant spaces in formal topology

A standard result in topological dynamics is the existence of minimal subsystem. It is a direct consequence of Zorn's lemma: given a compact topological space X with a map f: X→X, the set of compact non empty subspaces K of X such that f(K) ⊆ K ordered by inclusion is inductive, and hence has minimal elements. It is natural to ask for a point-free (or formal) formulation of this statement. In a previous work [3], we gave such a formulation for a quite special instance of this statement, which is used in proving a purely combinatorial theorem (van de Waerden's theorem on arithmetical progression).In this paper, we extend our analysis to the case where X is a boolean space, that is compact totally disconnected. In such a case, we give a point-free formulation of the existence of a minimal subspace for any continuous map f: X→X. We show that such minimal subspaces can be described as points of a suitable formal topology, and the “existence” of such points become the problem of the consistency of the theory describing a generic point of this space. We show the consistency of this theory by building effectively and algebraically a topological model. As an application, we get a new, purely algebraic proof, of the minimal property of [3]. We show then in detail how this property can be used to give a proof of (a special case of) van der Waerden's theorem on arithmetical progression, that is “similar in structure” to the topological proof [6, 8], but which uses a simple algebraic remark (Proposition 1) instead of Zorn's lemma. A last section tries to place this work in a wider context, as a reformulation of Hilbert's method of introduction/elimination of ideal elements.