Introduction to Integral Discriminants

The simplest partition function, associated with homogeneous symmetric forms S of degree r in n variables, is integral discriminant J_{n|r}(S) = \int e^{-S(x_1 ... x_n)} dx_1 ... dx_n. Actually, S-dependence remains the same if e^{-S} in the integrand is substituted by arbitrary function f(S), i.e. integral discriminant is a characteristic of the form S itself, and not of the averaging procedure. The aim of the present paper is to calculate J_{n|r} in a number of non-Gaussian cases. Using Ward identities -- linear differential equations, satisfied by integral discriminants -- we calculate J_{2|3}, J_{2|4}, J_{2|5} and J_{3|3}. In all these examples, integral discriminant appears to be a generalized hypergeometric function. It depends on several SL(n) invariants of S, with essential singularities controlled by the ordinary algebraic discriminant of S.Comment: 36 pages, 19 figure

Non-Linear Algebra and Bogolubov's Recursion

Numerous examples are given of application of Bogolubov's forest formula to iterative solutions of various non-linear equations: one and the same formula describes everything, from ordinary quadratic equation to renormalization in quantum field theory.Comment: LaTex, 21 page

On Pure Spinor Superfield Formalism

We show that a certain superfield formalism can be used to find an off-shell supersymmetric description for some supersymmetric field theories where conventional superfield formalism does not work. This "new" formalism contains even auxiliary variables in addition to conventional odd super-coordinates. The idea of this construction is similar to the pure spinor formalism developed by N.Berkovits. It is demonstrated that using this formalism it is possible to prove that the certain Chern-Simons-like (Witten's OSFT-like) theory can be considered as an off-shell version for some on-shell supersymmetric field theories. We use the simplest non-trivial model found in [2] to illustrate the power of this pure spinor superfield formalism. Then we redo all the calculations for the case of 10-dimensional Super-Yang-Mills theory. The construction of off-shell description for this theory is more subtle in comparison with the model of [2] and requires additional Z_2 projection. We discover experimentally (through a direct explicit calculation) a non-trivial Z_2 duality at the level of Feynman diagrams. The nature of this duality requires a better investigation

Challenges of beta-deformation

A brief review of problems, arising in the study of the beta-deformation, also known as "refinement", which appears as a central difficult element in a number of related modern subjects: beta \neq 1 is responsible for deviation from free fermions in 2d conformal theories, from symmetric omega-backgrounds with epsilon_2 = - epsilon_1 in instanton sums in 4d SYM theories, from eigenvalue matrix models to beta-ensembles, from HOMFLY to super-polynomials in Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras etc. The main attention is paid to the context of AGT relation and its possible generalizations.Comment: 20 page

РЕЗУЛЬТАТЫ ЭТАПНОГО ХИРУРГИЧЕСКОГО ЛЕЧЕНИЯ ИНФАНТИЛЬНЫХ И ЮВЕНИЛЬНЫХ СКОЛИОЗОВ С ИСПОЛЬЗОВАНИЕМ РАЗЛИЧНЫХ МЕТОДИК

Introduction. The analysis Results of surgical treatment of growing children with infantile and juvenile scoliosis (IS) can the optimal method of treatment select. In young children with significant growth potential spinal fusion may not be the best option as it limits further longitudinal growth of the spine and may to the thoracic insufficiency syndrome result. To address this problem recently several techniques focused, their have advantages and drawbacks.Material and methods. Since 2008 year 127 patients (64 girls, 63 boys) aged (4.5 ± 2.1) years were operated on. In group I 65 patients were operated on using VEPTR (Vertical Expandable Prosthetic Titanium Rib) instrumentation, in group II 42 patients using various spinal instrumentation. 20 patients with congenital kyphosis were excluded. The average follow-up time was (5.6 ± 1.1) years.Results. In group I average value of the primary scoliotic curve before surgery was (74.7 ± 22.9), secondary curve (42.8 ± 16.0), thoracic kyphosis (46.3 ± 27.4), lumbar lordosis (54.6 ± 14). Average value of the primary scoliotic curve after surgery was reduced to (51 ± 20) (correction 31.7%), at followup to (56.5 ± 18.5), secondary curve (31.8 ± 12.8) (25.7%), at follow-up to (32.4 ± 18.4), thoracic kyphosis (36.8 ± 20.8) (20,5%), at follow-up to (41.8 ± 21.0), lumbar lordosis (45.4 ± 12.7) (16,9%), at follow-up to (48.2 ± 11.7) (p < 0.05). Space available for lung before surgery was (84.5 ± 8.7) %, after surgery was (94.8 ± 6.7)%, at follow-up increased to (98.6 ± 5.4) % (p < 0.05). Complications included 11 implant dislocations and 1 infection. In group II average value of the primary scoliotic curve before surgery was (87.6 ± 6.6), secondary curve (47.8 ± 4.6), thoracic kyphosis (61.4 ± 10.4), lumbar lordosis (61.8 ± 4.9). Average value of the primary scoliotic curve after surgery was reduced to 50.6 ± 5.3 (correction 42.3%), at follow-up to (66.1 ± 6.3), secondary curve (24.1 ± 2.9) (49.6%), at follow-up to (37 ± 5.4), thoracic kyphosis (38.8 ± 7.7) (36.8%), at follow-up to (59.4 ± 11.2), lumbar lordosis (47.5 ± 4.1) (23.2%), at follow-up to (64.5 ± 4.5) (p < 0.05). Complications included 23 implant dislocations and 1 infection. No neurological complications.Conclusion. Stage correction fusions using various instrumentation is a method of choice for controlled correction of growing children with IS.Введение. Анализ результатов хирургического лечения сколиотических деформаций позвоночника у активно растущих детей позволяет выбрать оптимальный метод лечения. Стабилизация позвоночника является быть оптимальным вариантом, так как может привести к ограничению его дальнейшего роста и развитию синдрома торакальной недостаточности. На решение этой задачи в последнее время ориентировано несколько методик, имеющих свои достоинства и недостатки.Цель исследования – проанализировать результаты хирургического лечения инфантильных и ювенильных сколиозов с использованием различного инструментария.Материал и методы. В период с 1998 по2014 г. оперировано 127 детей (64 девочки и 63 мальчика) с инфантильными и ювенильными деформациями позвоночника различной этиологии. Операции проводились по двум методикам: первая группа (65 больных) – по методике VEPTR (Vertical Expandable Prosthetic Titanium Rib, США), вторая (42 больных) – с использованием дорсального сегментарного инструментария. Средний возраст начала лечения составил (4,5 ± 2,1) года (I группа), (7,6 ± 2,4) года (II группа). Сроки послеоперационного наблюдения составили (5,6 ± 1,1) года (от 6 мес до 12 лет).Результаты. В группе I средняя величина основной сколиотической дуги перед началом лечения составляла (74,7 ± 22,9)°, противоискривления – (42,8 ± 16)°, грудного кифоза – (46,3±27,4)°, поясничного лордоза – (54,6 ± 14)°. Величина основной сколиотической дуги после операции составляла (51,0 ± 20,0)° (коррекция 31,7%), противоискривления – (31,8 ± 12,8)° (коррекция 25,7%), кифоза – (36,8 ± 20,8)° (коррекция 20,5%), лордоза – (45,4 ± 12,7)° (коррекция 16,9%) (p < 0,05). В конце срока наблюдения величина основной сколиотической дуги составила (56,5 ± 18,5)°, противоискривления – (32,4 ± 18,4)°, кифоза – (41,8 ± 21,0)°, лордоза – (48,2 ± 11,7)° (p < 0,05). При использовании инструментария VEPTR отмечено увеличение отношения пространств доступных для легких в сравнении с исходным – (84,5 ± 8,7), послеоперационным – (94,8 ± 6,7) и значением в конце срока наблюдения – (98,6 ± 5,4) (p < 0,05). У 11 пациентов отмечена нестабильность захватов инструментария, 1 случай нагноения. Во II группе средняя величина основной сколиотической дуги перед началом лечения составляла (87,6 ± 6,6)°, противоискривления – (47,8 ± 4,6)°, грудного кифоза – (61,4 ± 10,4)°, поясничного лордоза – (61,8 ± 4,9)°. Величина основной сколиотической дуги после операции составляла (50,6 ± 5,3)° (коррекция 42,3%), противоискривления – (24,1 ± 2,9)° (коррекция 49,6%), кифоза – (38,8 ± 7,7)° (коррекция 36,8%), лордоза – (47,5 ± 4,1)° (коррекция 23,2%) (p < 0,05). В конце срока наблюдения величина основной сколиотической дуги составила (66,1 ± 6,3)°, противоискривления – (37,0 ± 5,4)°, кифоза – (59,4 ± 11,2)°, лордоза – (64,5 ± 4,5)° (p < 0,05). У 23 пациентов отмечена нестабильность захватов инструментария, 1 случай нагноения. Неврологических осложнений не отмечено.Вывод. При хирургическом лечении инфантильных и ювенильных сколиозов методом выбора являются этапные коррекции с использованием различного инструментария

The matrix model version of AGT conjecture and CIV-DV prepotential

Recently exact formulas were provided for partition function of conformal (multi-Penner) beta-ensemble in the Dijkgraaf-Vafa phase, which, if interpreted as Dotsenko-Fateev correlator of screenings and analytically continued in the number of screening insertions, represents generic Virasoro conformal blocks. Actually these formulas describe the lowest terms of the q_a-expansion, where q_a parameterize the shape of the Penner potential, and are exact in the filling numbers N_a. At the same time, the older theory of CIV-DV prepotential, straightforwardly extended to arbitrary beta and to non-polynomial potentials, provides an alternative expansion: in powers of N_a and exact in q_a. We check that the two expansions coincide in the overlapping region, i.e. for the lowest terms of expansions in both q_a and N_a. This coincidence is somewhat non-trivial, since the two methods use different integration contours: integrals in one case are of the B-function (Euler-Selberg) type, while in the other case they are Gaussian integrals.Comment: 27 pages, 1 figur