Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds

The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [29], and we provide a characterization for singularities as well as a deformability criterion.

Characteristic directions of two-dimensional biholomorphisms

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

The experiment on assessment of point position accuracy at the manual digitizing of a raster map

The article gives a part of an experiment on determining the most common errors in the digitization of objects of different topological types, namely considering errors in the setting of points. The experiment was performed in 2015–2017 on the basis of the Kursk State University, the number of participants was 63 people. Each of them digitized 30 points in four scales of the display (that is, 120 points from a subject). Then the directions and absolute values of the errors were determined. According to these initial data it was established that the widespread model of errors in the setting of points, defined by two-dimensional normal distribution, in many cases does not correspond to reality (was not observed with more than half of the subjects). The hypothesis of the equability of error directions was proved even to a lesser degree, since the majority had its own characteristic direction of error (most often this was displacement of the position of the point from the true to the southeast, least of all to the northwest). 45 out of 63 subjects demonstrated an uneven distribution of directions of setting points around the true position. The absolute value of the allowed errors in meters is proportional to the scale of the display, but is unchanged in pixels of the screen. The median error is approximately 1 pix, the 99th quantile of the distribution of error values corresponds to 4,1 pix. The results of the experiment make it possible to choose in advance the optimum scale of the digitization of the raster (not too large and not too shallow) in order to achieve the required quality of vectorization.

Anthropocentric and theocentric spirituality as an object of psychological research

The characteristic direction of psychological and theological interpretations of spirituality is very important. The traditional psychological approach to the spiritual sphere is characterised by reductionism, which consists in reducing spiritual experiences to mental experiences, or even biological processes. The studies in the field of religion psychology led to distinguish between two types of spirituality. The first one is theocentric spirituality, where human being places God in the centre of his interest and life in general. The second type of spirituality is anthropocentric spirituality, focused on human being, his own aspirations, preferences and needs. Both types of spirituality have certain value. Their close characteristics includes sources of inspiration, purpose, presented image of God, as well as understanding of spirituality and manner of realizing spiritual life. In order to distinguish between two types of spirituality, anthropocentric and theocentric, in practice, a proper research method – Range of Theocentric and Anthropocentric Spirituality (SDT – DA) had to be developed. The individuals with theocentric spirituality displayed a higher level of stability and emotional balance, better social adjustment, higher sense of duty and attachment to acceptable social standards, deeper and more satisfactory contacts with other human beings, more trust and openness towards others, as well as higher trust to themselves and to God. Such individuals are better at handling difficulties and have optimistic attitude to life.

THE NON-DICRITICAL ORDER AND ATTRACTING DOMAINS OF HOLOMORPHIC MAPS TANGENT TO THE IDENTITY

We study the local dynamics of holomorphic maps f in Cn tangent to the identity at a fixed point p with a non-degenerate characteristic direction [v]. In [M. Hakim, Analytic transformation of (Cp, 0) tangent to the identity, Duke Math. J.92 (1998) 403–428], n - 1 invariants αj, 1 ≤ j ≤ n - 1, called the directors, were associated to [v] and it was shown that if Re αj > 0 for all j then f has an attracting domain at p tangent to [v]. In this paper, we study the case Re αj = 0 for some j. With the help of a new invariant μ called the non-dicritical order, we show that f has an attracting domain at p tangent to [v] if μ ≥ 1. We also study the "spiral domains" when μ = 0. For n = 2, we show that f has an attracting domain at p tangent to [v] if and only if either the director α > 0 or μ ≥ 1.

ATTRACTING DOMAINS OF MAPS TANGENT TO THE IDENTITY WHOSE ONLY CHARACTERISTIC DIRECTION IS NON-DEGENERATE

Let f be a holomorphic germ on ℂ2 that fixes the origin and is tangent to the identity. Assume that f has a non-degenerate characteristic direction [v]. Hakim gave conditions that guarantee the existence of attracting domains along [v], however, when f has only one characteristic direction, these conditions are not satisfied. We prove that when [v] is unique, the existence results still hold. In particular, there is a domain Ω whose points converge to the origin along [v] and, on Ω, f is conjugate to a translation. Furthermore, if f is a global automorphism, the corresponding domain of attraction is a Fatou–Bieberbach domain.