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## Regularity for the optimal compliance problem with length penalization

We study the regularity and topological structure of a compact connected set $S$ minimizing the “compliance" functional with a length penalization. The compliance here is the work of the force applied to a membrane which is attached along the set $S$. This shape optimization problem, which can be interpreted as that of finding the best location for attaching a membrane subject to a given external force, can be seen as an elliptic PDE version of the minimal average distance problem. We prove that minimizers in the given region consist of a finite number of smooth curves which meet only at triple points with angles of 120 degrees, contain no loops, and possibly touch the boundary of the region only tangentially. The proof uses, among other ingredients, some tools from the theory of free discontinuity problems (monotonicity formula, flatness improving estimates, blow-up limits), but adapted to the specific problem of min-max type studied here, which constitutes a significant difference with the classical setting and may be useful also for similar other problems.

Given two probability measures μ and ν we consider a mass transportation mapping T satisfying 1) T sends μ to ν , 2) T has the form T=ϕ∇ϕ|∇ϕ| , where ϕ is a function with convex sublevel sets. We prove a change of variables formula for T . We also establish Sobolev estimates for ϕ , and a new form of the parabolic maximum principle. In addition, we discuss relations to the Monge-Kantorovich problem, curvature flows theory, and parabolic nonlinear PDE's.

The book proposes and substantiates the idea of developing an interdisciplinary concept of anti-corruption compliance in the Russian Federation, based on the use of the achievements of the sciences of international law, criminal law, criminal procedure law, criminalistics and investigative activities.

The authors propose an own definition of the anti-corruption compliance and analyse main reasons for implementation of anti-corruption compliance in companies. Special attention is paid to distinctive features of legal regulation of the anti-corruption compliance in Russia and in foreign countries. The book contributes to the integration of main scientific definitions and concepts from criminal law and criminalistics, such as corruption crime, corruption offences and corruption risks into the concept of anti-corruption compliance system.

According to the authors, the interdisciplinary concept of anti-corruption compliance should create a solid basis for developing standards and recommendations for Russian companies and guidelines for judges and law enforcement officers.

The author analyzes incentives for Russian companies to implement anti-corruption compliance programmes, place of anti-corruption compliance in corporate structure, main components of the Code of Conduct and anti-corruption compliance programme.

The article deals with the solution of the problems of the formation and use of applied ontologies as a

Abstract: The purpose of this article - the forming of theoretical foundations, analysis of working tools and technologies for the construction of an information system that allows to optimize the scanning of IS for compliance with information security standards, the analysis of the data and the definition of the relationship of the various requirements of the relevant standards. Such information system will allow to optimize business processes for is management, to reduce the number of configuration errors, to reduce the influence of the human factor in the construction of is complexes. The article presents a study on the use of standards profiles when checking the company's compliance with external and internal standards of information security.

The article analyzes the moral considerations, legal and market incentives for organizations to implement anti-corruption compliance. It also assesses the role of anti-corruption compliance in prevention of corruption offences.

50% patients with chronic disease shows unintentional and intentional medication non-adherence. Medication non-adherence causes 4 to 10% of preventable healthcare losses. SMS reminders are associated with significant improvement in adherence rates. The current paper tests whether the question-behavior effect (QBE) could improve SMS reminders effectiveness. Preliminary RCT (N=40) shows positive, but not significant effect of QBE/SMS combination on medication adherence rates (ASD=0.249 (95% C. I. -0.061 – 0.560)), which corresponds to the predicted effect size (ASD=0.166 (95% C. I. 0.044 – 0.276)). Additional research is required for confirmation of QBE/SMS combination effectiveness.

The book is the result of an extensive research conducted by professors of five academic institutions from Brazil, Russia, India, China and South Africa - the BRICS countries. The purpose of this work is to analyze the anti-corruption and anti-money laundering legislation in each of these countries, showing the similarities and differencies in terms of domestic legal frames and the international guidelines.

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.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.