Anti-invariant Riemannian submersions from Sasakian manifolds with totally umbilical fibers

The purpose of this paper is to study anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds such that characteristic vector field is vertical or horizontal vector field. We first show that any anti-invariant Riemannian submersions from Sasakian manifold is not a Riemannian submersion with totally umbilical fiber. Then we introduce anti-invariant Riemannian submersions from Sasakian manifolds with totally contact umbilical fibers. We investigate the totally contact geodesicity of fibers of such submersions. Moreover, under this condition, we investigate Ricci curvature of anti-invariant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds.

FOUR-DIMENSIONAL BALL IN A GRAPHIC REPRESENTATION

The paper presents a method for geometric modelling of a four-dimensional ball. For this, the regularities of the change in the shape of the projections of simple geometric images of two-dimensional and three-dimensional spaces during rotation are considered. Rotations of a segment and a circle around an axis are considered; it is shown that during rotation the shape of their projections changes from the maximum value to the degenerate projection. It was found that the set of points of the degenerate projection belongs to the axis of rotation, and each n-dimensional geometric image during rotation forms a body of a higher dimension, that is, one that belongs to (n + 1) -dimensional space. Identified regularities are extended to the four-dimensional space in which the ball is placed. It is shown that the axis of rotation of the ball will be a degenerate projection in the form of a circle, and the ball, when rotating, changes its size from a volumetric object to a flat circle, then increases again, but in the other direction (that is, it turns out), and then in reverse order to its original position. This rotation is more like a deformation, and such a ball of four-dimensional space is a hypersphere. For geometric modelling of the hypersphere and the possibility of its projection image, the article uses the vector model proposed by P.V. Filippov. The coordinate system 0xyzt is defined. The algebraic equation of the hypersphere is given by analogy with the three-dimensional space along certain coordinates of the center a, b, c, d. A variant of hypersection at t = 0 is considered, which confirms by equations obtaining a two-dimensional ball of three-dimensional space, a point (a ball of zero radius), which coincides with the center of the ball, or an imaginary ball. For the variant t = d, the equation of a two-dimensional ball is obtained, in which the radius is equal to R and the coordinates of all points along the 0t axis are equal to d. The variant of hypersection t = k turned out to be interesting, in which the equation of a two-dimensional sphere was obtained, in which the coordinates of all points along the 0t axis are equal to k, and the radius is . Horizontal vector projections of hypersection are constructed for different values of k. It is concluded that the set of horizontal vector projections of hypersections at t = k defines an ellipse.

POSTMODERNISM: DESCRIPTIVE PRACTICES OF CULTURAL REFLECTION

The problem of discreteness and continuity of time and space correlates with the problem of the whole and the general. If ontology is presented as a world of individual things, and metaphysical systems describe the world as a whole, then the philosophy of postmodernism questions and criticizes classical examples of the postulation of integrity. The result of such a worldview disposition is an existential-horizontal vector of goal-setting and expediency as such. The development, or rather, the "habitation" of space in time occurs as its marking - the establishment of marks, leaving traces and cartography. All the monumentality of culture is, thus, a sign-symbolic system that reformats the world in a human-sized format. The temporality of subjectivity is not only an existential tragedy, but also a sacred gift of an essential vision of things and phenomena. Even if in postmodernism this sacredness is legitimized by society and culture, not by the eternity of the universe.

On Lipschitz vector fields and the Cauchy problem in homogeneous groups

We introduce a class of “Lipschitz horizontal” vector fields in homogeneous groups, for which we show equivalent descriptions, e.g., in terms of suitable derivations. We then investigate the associated Cauchy problem, providing a uniqueness result both at equilibrium points and for vector fields of an involutive submodule of Lipschitz horizontal vector fields. We finally exhibit a counterexample to the general well-posedness theory for Lipschitz horizontal vector fields, in contrast with the Euclidean theory.

EVALUATION OF EFFECTIVENESS OF PUBLIC PARTICIPATION PROCESSES

The article talks about the tendency of growth of public participation cases in the current social and political conditions of the country's development. The author compares existing approaches to assessing the effectiveness of public participation processes and highlights that the current assessment methodologies focuses only on researches processes, that related to interaction with the state structures and organizations (vertical vector of public participation forms); practically has no methods that are adapted to analyze the effectiveness of horizontal vector of public participation forms; methods that studies the online public participation process are completely absent. The author pays particular attention to the urgent need to find new methods for assessing the success of public participation in the context of a broader understanding of this phenomenon (vertical and horizontal vectors of public participation), taking into account the newest forms of public participation that are implemented using Internet technologies. The author proposes to create a new tool for studying the effectiveness of public participation process on the basis of one of the existing methods. This new tool provides fourteen indicators and thirty two evaluation criteria, that include such categories as: Process Accessibility, Media Participation, Diversity of Views Represented, Influence on Process, Opportunities for Participation, Integration of users' Concerns, Information Exchange, Project Efficiency, Project Acceptability, Mutual Learning, Mutual Respect, Cost Avoidance, Indirect Cost of Time, Indirect Opportunity Costs, Indirect Costs Associated with Authority and Influence, Indirect Costs Associated with Emotional Issues etc.