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

Boundary value problems for elliptic partial differential operators on bounded domains

For a symmetric operator or relation A with infinite deficiency indices in a Hilbert space we develop an abstract framework for the description of symmetric and self-adjoint extensions A_Θ of A as restrictions of an operator or relation T which is a core of the adjoint A^*. This concept is applied to second order elliptic partial differential operators on smooth bounded domains, and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

LHCb trigger streams optimization

The LHCb experiment stores around $10^{11}$ collision events per year. A typical physics analysis deals with a final sample of up to $10^7$ events. Event preselection algorithms (lines) are used for data reduction. Since the data are stored in a format that requires sequential access, the lines are grouped into several output file streams, in order to increase the efficiency of user analysis jobs that read these data. The scheme efficiency heavily depends on the stream composition. By putting similar lines together and balancing the stream sizes it is possible to reduce the overhead. We present a method for finding an optimal stream composition. The method is applied to a part of the LHCb data (Turbo stream) on the stage where it is prepared for user physics analysis. This results in an expected improvement of 15% in the speed of user analysis jobs, and will be applied on data to be recorded in 2017.Comment: Submitted to CHEP-2016 proceeding

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 Conditions for Singular Perturbations of Self-Adjoint Operators

Let A:D(A)\subseteq\H\to\H be an injective self-adjoint operator and let \tau:D(A)\to\X, X a Banach space, be a surjective linear map such that \|\tau\phi\|_\X\le c \|A\phi\|_\H. Supposing that \text{\rm Range} (\tau')\cap\H' =\{0\}, we define a family $A^\tau_\Theta$ of self-adjoint operators which are extensions of the symmetric operator $A_{|\{\tau=0\}.}$. Any $\phi$ in the operator domain $D(A^\tau_\Theta)$ is characterized by a sort of boundary conditions on its univocally defined regular component \phireg, which belongs to the completion of D(A) w.r.t. the norm \|A\phi\|_\H. These boundary conditions are written in terms of the map $\tau$, playing the role of a trace (restriction) operator, as \tau\phireg=\Theta Q_\phi, the extension parameter $\Theta$ being a self-adjoint operator from X' to X. The self-adjoint extension is then simply defined by A^\tau_\Theta\phi:=A \phireg. The case in which $A\phi=T*\phi$ is a convolution operator on LD, T a distribution with compact support, is studied in detail.Comment: Revised version. To appear in Operator Theory: Advances and Applications, vol. 13