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

p-Adic Schr\"{o}dinger-Type Operator with Point Interactions

A $p$-adic Schr\"{o}dinger-type operator $D^{\alpha}+V_Y$ is studied. $D^{\alpha}$ ($\alpha>0$) is the operator of fractional differentiation and $V_Y=\sum_{i,j=1}^nb_{ij}\delta_{x_i}$ $(b_{ij}\in\mathbb{C})$ is a singular potential containing the Dirac delta functions $\delta_{x}$ concentrated on points $\{x_1,...,x_n\}$ of the field of $p$-adic numbers $\mathbb{Q}_p$. It is shown that such a problem is well-posed for $\alpha>1/2$ and the singular perturbation $V_Y$ is form-bounded for $\alpha>1$. In the latter case, the spectral analysis of $\eta$-self-adjoint operator realizations of $D^{\alpha}+V_Y$ in $L_2(\mathbb{Q}_p)$ is carried out

Unitary boundary pairs for isometric operators in Pontryagin spaces and generalized coresolvents

An isometric operator V in a Pontryagin space H is called standard, if its domain and the range are nondegenerate subspaces in H. A description of coresolvents for standard isometric operators is known and basic underlying concepts that appear in the literature are unitary colligations and characteristic functions. In the present paper generalized coresolvents of non-standard Pontryagin space isometric operators are described. The methods used in this paper rely on a new general notion of boundary pairs introduced for isometric operators in a Pontryagin space setting. Even in the Hilbert space case this notion generalizes the earlier concept of boundary triples for isometric operators and offers an alternative approach to study operator valued Schur functions without any additional invertibility requirements appearing in the ordinary boundary triple approach.Comment: 42 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 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

Імунний статус поросят за застосування клатрохелату Феруму(IV) вагітним свиноматкам

The degree of the physiological development of the body of piglets in the first months after birth is important because, during this period, there is an active formation of many organism systems, including those that provide homeostasis. The resistance of piglets to diseases of various etiologies during life depends on the formation of the immune system. Our study aimed to assess the immune status of piglets born to sows that received injections of solutions of Iron(IV) clatrochelate and cyanocobalamin during pregnancy. To achieve this goal, two groups of newborn piglets-analogs were formed during their retention with suckling sows – control and experimental, 15 animals each. Piglets from 5 sows (3 from each) were injected into the experimental group, which were injected 10 ml of 10 % solution of Iron(IV) clatrochelate and cyanocobalamin solution twice during pregnancy (14 and 7 days before the expected farrowing) intramuscularly. Piglets of the control group, according to the traditional scheme of prevention of iron deficiency anemia on the second day of life, were administered irondextran drug uniferon (200 mg of iron (III) per injection). Piglets and sows were observed for two months; for studies of the content of immunoglobulins in the piglets' serum, the blood samples were taken at 1, 5, 12, and 30 days of life. During the experiment, the indicators of the content of immunoglobulins of classes G, A, M in the serum of piglets of the experimental group and the dynamics of their changes within 30 days after birth did not differ from the control. Two injections of 10 % solution of Iron(IV) clatrochelate in a dose of 10 ml in combination with cyanocobalamin injections at a dose of 500 mg of active substance to pregnant sows 14 and 7 days before the expected farrowing did not adversely affect the immune status of newborn piglets. Furthermore, this has a high prophylactic effect against iron deficiency anemia.Ступінь фізіологічного розвитку організму поросят у перші місяці після народження має важливе значення, оскільки упродовж цього періоду відбувається активне формування багатьох систем організму, зокрема й тих, що забезпечують гомеостаз. Стійкість поросят до захворювань різної етіології упродовж життя залежать саме від становлення імунобіологічної системи. Метою нашої роботи було оцінити імунний статус організму поросят, народжених від свиноматок, яким у період вагітності застосовували ін’єкції розчинів клатрохелату Феруму(IV) і ціанокобаламіну. Для виконання поставленої мети було сформовано 2 групи новонароджених поросят-аналогів у період їх утримання зі свиноматками на підсосі – контрольна та дослідна, по 15 тварин у кожній. У дослідну групу були відібрані поросята від 5-ти свиноматок (по 3 від кожної), яким в період вагітності двічі (за 14 та 7 діб до очікуваного опоросу) внутрішньом’язово вводили по 10 мл 10 % розчину клатрохелату Феруму(IV) та розчин ціанокобаламіну. Поросятам контрольної групи за традиційною схемою профілактики ферумодефіцитної анемії на другу добу життя вводили ферумодекстрановий препарат юніферон (200 мг Феруму(III) на одне введення). Протягом 2 місяців за поросятами та свиноматками здійснювали спостереження; для досліджень вмісту імуноглобулінів у сироватці крові поросят відбирали зразки крові на 1, 5, 12 та 30 доби життя. Упродовж досліду показники вмісту імуноглобулінів класів G, A, М у сироватці крові поросят дослідної групи та динаміка їх змін протягом 30 діб після народження майже не відрізнялась від контролю. Дворазова ін’єкція поросним свиноматкам 10 % розчину клатрохелату Феруму(IV) в дозі 10 мл у поєднанні з ін’єкціями ціанокобаламіну у дозі 500 мкг діючої речовини за 14 та 7 діб до передбачуваного опоросу не впливала негативно на імунний статус новонароджених поросят, забезпечуючи при цьому високий профілактичний ефект щодо ферумодефіцитної анемії

МІКРОБІОЛОГІЧНІ АСПЕКТИ ЛІКУВАННЯ ГНІЙНО- ЗАПАЛЬНИХ ЗАХВОРЮВАНЬ

SUMMARY. Results of diverse studies (antibiotic- and phagoresistance, the ability to biofilm formation) MRSA and MSSA strains of community-acquired pathogens inflammatory processes. Directions of improving diagnosis and treatment of purulent-inflammatory diseases are determined.Key words: purulent-inflammatory diseases, antibiotic resistance, phagosensitivity, MRSA and MSSA strains, biofilm.Проаналізовані результати різнопланових досліджень (антибіотико- та фагорезистентності, здатності до плівкоутворення) MRSA та MSSA позалікарняних штамів-збудників гнійно-запальних процесів. Визначені напрямки удосконалення діагностики та лікування гнійно-запальних захворювань.Ключові слова: гнійно-запальні захворювання, антибіотикорезистентність, фагочутливість, MRSA та MSSA штами, біоплівки

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