Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks

Liouville field theory on a sphere is considered. We explicitly derive a differential equation for four-point correlation functions with one degenerate field $V_{-\frac{mb}{2}}$. We introduce and study also a class of four-point conformal blocks which can be calculated exactly and represented by finite dimensional integrals of elliptic theta-functions for arbitrary intermediate dimension. We study also the bootstrap equations for these conformal blocks and derive integral representations for corresponding four-point correlation functions. A relation between the one-point correlation function of a primary field on a torus and a special four-point correlation function on a sphere is proposed

Psychophysiological and sport activity of the student youth as an indicator and determinant of health-preserving culture development

The purpose of the study consisted in exploring specific features of the influence of social activity of student youth on the development of their health-promoting culture. In addition, the study was directed at searching for effective ways of encouraging and developing health-promoting culture among student youth by boosting activities in the sphere of student media. The subject of our study embraced the process of education and development of health-promoting culture among students through the inclusion of the media environment in the activity. Research methods comprised theoretical (analysis, classification, synthesis and generalization) and empirical ones (questionnaire "What is your creative potential"); "Communicative and organizational inclinations". On the basis of the results of the study, the authors of the paper specified and concretized the concepts of general culture and social activity of students through the implementation of creative activity in the media space. An additional education program for students has been designed and tested, the purpose of which is to increase social activity and the development of health-promoting culture in the youth milieu. The designed Program, in addition to the formation of competences in the field of social activity and the development of general culture, information support of events and coverage of the main news among students, is focused on developing students’ professional skills of humanities courses who participate in the media space activities. The significance of the study from a practical point of view lies in the fact that its results can be used in the educational process to boost social activity and develop health-promoting culture of students of higher educational and vocational institutions. The novelty of the study is based on the fact that the designed Program of additional education for students of higher educational and vocational institutions is not only an educational program for mastering educational material but is a practical platform for the implementation of both educational and professional activities. The program comprises methods, technologies and resources that allow students to participate in various events and projects, to express themselves as a creative person, the ability to communicate with society, and organize events. The results of the program implementation raise the general socialization of students; determine the direction of their professional activity; and also provide a foundation for public life in the framework of press services of state bodies and public organizations

The Salts of Copper Octafluoro- and Hexadecafluorophthalocyanines Containing [Cu^II(F₈Pc)^4–]^2– Dianions and [CuF₁₆Pc]⁻ Monoanions

Crystalline anionic salts with copper octafluoro- and hexadecafluorophthalocyanines, (Bu₄N⁺)₂[Cu^II(F₈Pc)^4–]^2–·2C₆H₄Cl₂ (<b>1</b>) and (PPN⁺)₃[CuF₁₆Pc]₃^3–·2C₆H₅CN (<b>2</b>), where PPN⁺ is bis­(triphenylphosphoranylidene)­ammonium and Pc is phthalocyanine, have been obtained. The absence of noticeable absorption in the NIR range and DFT calculations for <b>1</b> indicate that both negative charges are mainly localized on the Pc ligand, and that the [Cu^II(F₈Pc)^4–]^2– dianions are formed without reduction of Cu^II. The magnetic moment of 1.60 μ_B corresponds to the contribution of one <i>S</i> = 1/2 spin per dianion. The spin is localized on the Cu^II atom, which shows an EPR signal characteristic of Cu^II. Dianions are isolated in <b>1</b>, providing only weak magnetic coupling of spins with a Weiss temperature of −4 K. Salt <b>2</b> contains closely packed π–π stacks built of [CuF₁₆Pc]⁻ anions of types I and II, and the interplanar distances are 3.187 and 3.275 Å. According to the DFT calculations, the [CuF₁₆Pc]⁻ anions of types I and II can have different charge distributions, with localization of an extra electron on the copper atoms to form diamagnetic [Cu^I(F₁₆Pc)^2–]⁻ monoanions or delocalization of an extra electron on the F₁₆Pc ligand to form [Cu^II(F₁₆Pc)^•3–]^•– having an <i>S</i> = 1/2 (Cu^II) + 1/2 (F₁₆Pc^•3–) spin state. In fact, at 300 K, the magnetic moment of <b>2</b> of 3.25 μ_B per formula unit is rather close to the contribution from two [Cu^II(F₁₆Pc)^•3–]^•– (calculated μ_eff is 3.46 μ_B). The Weiss temperature of −21.5 K indicates antiferromagnetic coupling of spins, which can be modeled by stronger intermolecular coupling between (F₁₆Pc)^•3– with <i>J</i>₁/<i>k</i>_B = −23.5 K and weaker intramolecular coupling between Cu^II and (F₁₆Pc)^•3– with <i>J</i>₂/<i>k</i>_B = −8.1 K. This interaction is realized in the {[Cu^II(F₁₆Pc)^•3–]^•–}₂ dimers separated by diamagnetic [Cu^I(F₁₆Pc)^2–]⁻ species. In spite of the stacking arrangement of phthalocyanine macrocycles in <b>2</b>, the inhomogeneous charge distribution and nonuniform distances between the macrocycles should suppress electrical conductivity

The Salts of Copper Octafluoro- and Hexadecafluorophthalocyanines Containing [Cu^II(F₈Pc)^4–]^2– Dianions and [CuF₁₆Pc]⁻ Monoanions

Crystalline anionic salts with copper octafluoro- and hexadecafluorophthalocyanines, (Bu₄N⁺)₂[Cu^II(F₈Pc)^4–]^2–·2C₆H₄Cl₂ (<b>1</b>) and (PPN⁺)₃[CuF₁₆Pc]₃^3–·2C₆H₅CN (<b>2</b>), where PPN⁺ is bis­(triphenylphosphoranylidene)­ammonium and Pc is phthalocyanine, have been obtained. The absence of noticeable absorption in the NIR range and DFT calculations for <b>1</b> indicate that both negative charges are mainly localized on the Pc ligand, and that the [Cu^II(F₈Pc)^4–]^2– dianions are formed without reduction of Cu^II. The magnetic moment of 1.60 μ_B corresponds to the contribution of one <i>S</i> = 1/2 spin per dianion. The spin is localized on the Cu^II atom, which shows an EPR signal characteristic of Cu^II. Dianions are isolated in <b>1</b>, providing only weak magnetic coupling of spins with a Weiss temperature of −4 K. Salt <b>2</b> contains closely packed π–π stacks built of [CuF₁₆Pc]⁻ anions of types I and II, and the interplanar distances are 3.187 and 3.275 Å. According to the DFT calculations, the [CuF₁₆Pc]⁻ anions of types I and II can have different charge distributions, with localization of an extra electron on the copper atoms to form diamagnetic [Cu^I(F₁₆Pc)^2–]⁻ monoanions or delocalization of an extra electron on the F₁₆Pc ligand to form [Cu^II(F₁₆Pc)^•3–]^•– having an <i>S</i> = 1/2 (Cu^II) + 1/2 (F₁₆Pc^•3–) spin state. In fact, at 300 K, the magnetic moment of <b>2</b> of 3.25 μ_B per formula unit is rather close to the contribution from two [Cu^II(F₁₆Pc)^•3–]^•– (calculated μ_eff is 3.46 μ_B). The Weiss temperature of −21.5 K indicates antiferromagnetic coupling of spins, which can be modeled by stronger intermolecular coupling between (F₁₆Pc)^•3– with <i>J</i>₁/<i>k</i>_B = −23.5 K and weaker intramolecular coupling between Cu^II and (F₁₆Pc)^•3– with <i>J</i>₂/<i>k</i>_B = −8.1 K. This interaction is realized in the {[Cu^II(F₁₆Pc)^•3–]^•–}₂ dimers separated by diamagnetic [Cu^I(F₁₆Pc)^2–]⁻ species. In spite of the stacking arrangement of phthalocyanine macrocycles in <b>2</b>, the inhomogeneous charge distribution and nonuniform distances between the macrocycles should suppress electrical conductivity