Boundary relations and generalized resolvents of symmetric operators

The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint exit space extensions of a, not necessarily densely defined, symmetric operator, in terms of maximal dissipative (in \dC_+) holomorphic linear relations on the parameter space (the so-called Nevanlinna families). The new notion of a boundary relation makes it possible to interpret these parameter families as Weyl families of boundary relations and to establish a simple coupling method to construct the generalized resolvents from the given parameter family. The general version of the coupling method is introduced and the role of boundary relations and their Weyl families for the Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page

Application of “piercing the corporate veil” doctrine in the Ukrainian law

Purpose: In the article, authors develop a structure of applying the gaps of corporate law and the possibility of restricting all possible structures of the legal field in Ukraine. The functioning of corporate law is always exercised according to the principle of the company’s greatest possible involvement in the employee’s everyday life. There is always differentiation emerging, which determines to what extent the existence of corporate spirit and ethics are needed within the society. Design/Methodology/Approach: The method of comparative law was used as the subject of the study, which enabled us to compare the customary rules of law with specific corporate law rules. Additionally, it is appropriate to apply the historical method, which fully reflects that the article elaborates the historical aspect of the development of the studied phenomenon as well as the formation of the holistic component. Findings: The article implements the aspects of managing the legal regulation of corporate law on the basis of modernizing separate provisions of the legal area of a social environment. Practical Implications: The perspectives of applying the corporate law provisions in the state’s economic development can be defined as the conclusions of the study. Originality/Value: The authors clearly demonstrate the obligation to implement the provision that stipulates that the corporate law, in case its principles are violated, has still to be oriented at understanding the specificity of its application in commercial institutions.peer-reviewe

Boundary triplets for skew-symmetric operators and the generation of strongly continuous semigroups

We give a self-contained and streamlined exposition of a generation theorem for C0-semigroups based on the method of boundary triplets. We apply this theorem to port-Hamiltonian systems where we discuss recent results appearing in stability and control theory. We give detailed proofs and require only a basic knowledge of operator and semigroup theory.Comment: 19 page

Bitangential interpolation in generalized Schur classes

Bitangential interpolation problems in the class of matrix valued functions in the generalized Schur class are considered in both the open unit disc and the open right half plane, including problems in which the solutions is not assumed to be holomorphic at the interpolation points. Linear fractional representations of the set of solutions to these problems are presented for invertible and singular Hermitian Pick matrices. These representations make use of a description of the ranges of linear fractional transformations with suitably chosen domains that was developed in a previous paper.Comment: Second version, corrected typos, changed subsection 5.6, 47 page

The Low-Molecular-Weight Ligands of the Gonadotropin Receptors as the New Generation of the Regulators of the Reproductive Functions and Steroidogenesis

In clinic, the luteinizing (LH) and follicle-stimulating (FSH) hormones and human chorionic gonadotropin (hCG) are used to treat reproductive dysfunctions and in assisted reproductive technology. They are the αβ-heterodimeric complexes and specifically bind to ectodomain of G protein-coupled LH and FSH receptors. This leads to activation of many signaling cascades; some of which are responsible for steroidogenesis, folliculogenesis, and spermatogenesis, while the others, such as β-arrestin pathways, trigger the downregulation of gonadotropin receptors. A low selectivity of the intracellular signaling of gonadotropins and a large number of their isoforms are the main causes of undesirable effects of gonadotropins, limiting their clinical applications. Unlike gonadotropins, the low-molecular-weight (LMW) ligands interact with an allosteric site located in the transmembrane domain of the LH and FSH receptors and selectively activate the certain signaling pathway, preventing a number of side effects of gonadotropins. The LMW ligands are characterized by activity of the full and inverse agonists and neutral antagonists, as well as the positive and negative modulators, and they have the in vivo activity, including when administered orally. This review focuses on the advances in the development of LMW allosteric ligands of the LH and FSH receptors and the prospects for their use in reproductive medicine

Experimental Appropriateness Verification of K. Gorodetsky`s Mathematical Model for Losses Determinination in Hydrostatic Transmissions for Modern Hydrolic Machines

There were experimentally obtained dependences of efficiency coefficients for two hydrostatic transmissions of various sizes and performance on the basis of axial-piston hydraulic machines under various loading conditions. The experimental data is compared with the calculated one obtained using the mathematical model of K. Gorodetsky. The object of study was the hydrostatic monoblock transmission manufactured in 1999 and the hydrostatic transmission made in 2012 with a spaced hydraulic pump and hydraulic motor

Update of the Unitarity Triangle Analysis

We present the status of the Unitarity Triangle Analysis (UTA), within the Standard Model (SM) and beyond, with experimental and theoretical inputs updated for the ICHEP 2010 conference. Within the SM, we find that the general consistency among all the constraints leaves space only to some tension (between the UTA prediction and the experimental measurement) in BR(B -> tau nu), sin(2 beta) and epsilon_K. In the UTA beyond the SM, we allow for New Physics (NP) effects in (Delta F)=2 processes. The hint of NP at the 2.9 sigma level in the B_s-\bar B_s mixing turns out to be confirmed by the present update, which includes the new D0 result on the dimuon charge asymmetry but not the new CDF measurement of phi_s, being the likelihood not yet released.Comment: 4 pages, 2 figures, Proceedings of the 35th International Conference of High Energy Physics - ICHEP2010 (July 22-28, 2010, Paris

The Role of Apelin in the Functioning of the Reproductive System

Adipokine apelin through the apelin receptors activates a wide range of signaling cascades in the target cells and controls their growth, differentiation, apoptosis, and energy metabolism. In the recent years, the evidence has been obtained that all components of the hypothalamic-pituitary-gonad axis, in which apelin and its receptor are expressed, are targets of apelin. In the hypothalamus, apelin modulates the activity of the melanocortin and ghrelin systems and indirectly affects the production of gonadoliberin. In the ovaries, it controls the growth and maturation of the follicles, stimulates the angiogenesis, and affects the basal and stimulated by the other factors steroidogenic activity in follicular cells. The changes in the apelin signaling system are closely associated with dysfunctions of the female reproductive system, such as polycystic ovary syndrome, endometriosis, and cancer. Information on the regulation of the male reproductive system by apelin is limited to animal studies showing the effect of apelin on the hypothalamic components of the gonad axis. The participation of apelin in the regulation of the reproductive system opens up the broad opportunities for the development of new approaches for the correction of abnormalities in this system and for the treatment of infertility