Recent Developments in Homogeneous CR-Hypersurfaces

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup such that $X := G/H$ is Kaehler and the codimension of the top non-vanishing homology group of $X$ with coefficients in $\mathbb Z_2$ is less than or equal to two. We show that $X$ is biholomorphic to a complex homogeneous manifold constructed using well-known basic building blocks, i.e., $\mathbb C, \mathbb C^*$, Cousin groups, and flag manifolds.

Let G be a connected complex Lie group. Then there exists the smallest closed complex subgroup G0 of G such that G/G0 is a Stein group (Morimoto). Moreover G0 is a connected abelian Lie group and every holomorphic function on G0 is a constant. G0 is called an (H, C)-group or a toroidal group. Every connected complex abelian Lie group is isomorphic to the direct product G0 × Cm × C*n, where G0 is an (H,C)-group.

Some recent developments in the theory of the Bergman kernel function: a survey

Beginning with Valiron [262], the value distribution theory of algebroid functions of one complex variable has been studied extensively. A few contributions are listed under references. The Nevalinna theory of meromorphic functions on a complex manifold M or meromorphic mappings from M into a projective variety has been studied (see Griffith-King [75], Stoll [239], Vitter [265]). Here we will concentrate on the value distribution theory and its application of algebroid functions in several complex variables.

Theory of Functions on Complex Manifolds

Colloquium talk in IISc on October 28, 2005 This article is an exposition on the ‘congruence subgroup problem’. This subject is at a common ground shared by group theory and number theory. It deals with certain groups defined arithmetically and their subgroup structure. This is a relatively modern subject having taken off in the mid 1960’s after Klein’s early investigations on the modular group. Apart from interest in its own right, the subject gains importance also because of connections with the theory of automorphic forms which is central to number theory. This exposition is meant to introduce the subject to professional mathematicians working in other areas. At the end, some recent work by the author is mentioned.

This work presents a contemporary treatment of Krein's entire operators with deficiency indices $(1,1)$ and de Branges' Hilbert spaces of entire functions. Each of these theories played a central role in the research of both renown mathematicians. Remarkably, entire operators and de Branges spaces are intimately connected and the interplay between them has had an impact in both spectral theory and the theory of functions. This work exhibits the interrelation between Krein's and de Branges' theories by means of a functional model and discusses recent developments, giving illustrations of the main objects and applications to the spectral theory of difference and differential operators.

Fractional Brownian Motions, Fractional Noises and Applications

593.293.011 In the context of the Green’s function theory, the electronic structure of local defects-vacancies is considered self-consistently in the basis of localized orbitals. The origin and orbital structure of the electronic states in the gap, resonances and antiresonances in the valence band, and defect-induced changes in the electron density are discussed.

Functions Invariant under the Berezin Transform

Using recent development in Poletsky theory of discs, we prove the following result: Let X, Y be two complex manifolds, let Z be a complex analytic space which possesses the Hartogs extension property, let A (resp. B) be a non locally pluripolar subset of X (resp. Y ). We show that every separately holomorphic mapping f : W := (A × Y ) ∪ (X × B) −→ Z extends to a holomorphic mapping f on Ŵ := {(z, w) ∈ X × Y : ω(z, A, X) + ω(w, B, Y ) < 1} such that f = f on W ∩ Ŵ , where ω(·, A, X) (resp. ω(·, B, Y )) is the plurisubharmonic measure of A (resp. B) relative to X (resp. Y ). Generalizations of this result for an N -fold cross are also given. Mathematics Subject Classification (2000): 32D15 (primary); 32D10 (secondary).

Throughout this paper a surface is a 2-dimensional (not necessarily compact) complex manifold. A surface X is homogeneous if a complex Lie group G of holomorphic transformations acts holomorphically and transitively on it. Concisely, X is homogeneous if it can be identified with the left coset space G/H, where if is a closed complex Lie subgroup of G. We emphasize that the assumption that G is a complex Lie group is an essential part of the definition. For example, the 2-dimensional ball B2 is certainly “homogeneous” in the sense that its automorphism group acts transitively. But it is impossible to realize B2 as a homogeneous space in the above sense. The purpose of this paper is to give a detailed classification of the homogeneous surfaces. We give explicit descriptions of all possibilities.

The approach is based on an evolutionary interpretation of the fundamental concepts of cybernetics, general systems theory, information theory, theory of automata, autopoiesis, etc. In the author's view, this enables us to formulate principles of the evolutionary theory of cybernetic systems which is visualised as the theory of structures, functions and evolution of cybernetic or control systems. This suggests a heuristic idea of cybernetic evolution as the higher stage of negentropic universal evolution of open systems.

The recent developments in the performance and miniaturization of uncrewed aircraft systems (UAS) and multispectral imaging sensors provide new tools for the assessment of the spatial and temporal variability of soil properties at sub-meter resolution and at relatively low costs, in comparison to traditional chemical analysis. The accuracy of multispectral data is nevertheless influenced by the anisotropic behaviour of natural surfaces, framed in the general theory of the bidirectional reflectance distribution function (BRDF). Accounting for BRDF effects in multispectral data is paramount before formulating any scientific interpretation. This study presents a semi-empirical spectral normalization methodology for UAS-based multispectral imaging datasets of bare soils to account for the effects of the BRDF, based on the application of an anisotropy factor (ANIF). A dataset of images from 15 flights over bare soil fields in the Belgian loam belt was used to calibrate a model relating the ANIF to a wide range of illumination geometry conditions by using only two angles: relative sensor-pixel-sun zenith and relative sensor-pixel-sun azimuth. The employment of ANIF-corrected images for multispectral orthomosaic generation with photogrammetric software provided spectral maps free of anisotropic-related artefacts in most cases, as assessed by several ad hoc indexes, and was also tested on an independent validation set. Most notably, the standard deviation in the measured reflectance of the same georeferenced point by different pictures decreased from 0.032 to 0.023 (p &lt; 0.05) in the calibration dataset and from 0.037 to 0.030 in the validation dataset. The validation dataset, however, showed the presence of some systematic errors, the causes of which require further investigation.

In the recent developments of the value distribution theory of meromorphic functions, there is an emphasis on the study of deficient values and deficiencies.