Subgroup growth of lattices in semisimple Lie groups

We give very precise bounds for the congruence subgroup growth of arithmetic groups. This allows us to determine the subgroup growth of irreducible lattices of semisimple Lie groups. In the most general case our results depend on the Generalized Riemann Hypothesis for number fields but we can state the following unconditional theorem: Let $G$ be a simple Lie group of real rank at least 2, different than D_4(\bbc), and let $\Gamma$ be any non-uniform lattice of $G$. Let $s_n(\Gamma)$ denote the number of subgroups of index at most $n$ in $\Gamma$. Then the limit $\lim\limits_{n\to \infty} \frac{\log s_n(\Gamma)}{(\log n)^2/ \log \log n}$ exists and equals a constant $\gamma(G)$ which depends only on the Lie type of $G$ and can be easily computed from its root system.Comment: 34 page

Convergence and multiplicities for the Lempert function

Given a domain $\Omega \subset \mathbb C$, the Lempert function is a functional on the space Hol (\D,\Omega) of analytic disks with values in $\Omega$, depending on a set of poles in $\Omega$. We generalize its definition to the case where poles have multiplicities given by local indicators (in the sense of Rashkovskii's work) to obtain a function which still dominates the corresponding Green function, behaves relatively well under limits, and is monotonic with respect to the indicators. In particular, this is an improvement over the previous generalization used by the same authors to find an example of a set of poles in the bidisk so that the (usual) Green and Lempert functions differ.Comment: 24 pages; many typos corrected thanks to the referee of Arkiv for Matemati

Кинетика 99mTc-МИБИ в опухоли молочной железы по данным математического моделирования

Метою роботи є якісний опис кінетики 99mTc-МІБІ в пухлині молочної залози для оцінки ступеня хіміорезистентності.Аналіз кінетики 99mTc-МІБІ в пухлині молочної залози здійснювався по даним математичного моделювання. Розроблена математична модель показує, що основними параметрами мамосцинтиграфії з 99mTc-МІБІ, які необхідно розглядати для оцінки хіміорезистентності пухлини є: ефективний кровоток пухлини, інтенсивність виведення радіофармпрепарату (РФП) з клітини, кліренс крові від РФП, рівень вимивання РФП з пухлини.Objective: qualitative description of the kinetics of 99mTc-MIBI in the breast tumor to assess chemoresistance.Analysis of the kinetics of 99mTc-MIBI in breast tumors is according to mathematical modeling. The mathematical model shows that the main parameters mammostsintigrafy with 99mTc-MIBI to be considered for evaluation of tumor chemoresistance are effective tumor blood flow, rate of excretion radiopharmaceutical with cells, clearance of blood from the radiopharmaceutical, the level of radiopharmaceutical washout from the tumor.Целью работы является качественное описание кинетики 99mTc-МИБИ в опухоли молочной железы для оценки степени химиорезистентности. Анализ кинетики 99mTc-МИБИ в опухоли молочной железы производится по данным математического моделирования. Разработанная математическая модель показывает, что основными параметрами маммосцинтиграфии с 99mTc-МИБИ, которые необходимо рассматривать для оценки химиорезистентности опухоли являются: эффективный опухолевый кровоток, интенсивность выведения радифармпрепарата (РФП) из клетки, клиренс крови от РФП, уровень вымывания РФП из опухоли

Self-consistent tilted-axis-cranking study of triaxial strongly deformed bands in 158^{158}Er at ultrahigh spin

Stimulated by recent experimental discoveries, triaxial strongly deformed (TSD) states in $^{158}$Er at ultrahigh spins have been studied by means of the Skyrme-Hartree-Fock model and the tilted-axis-cranking method. Restricting the rotational axis to one of the principal axes -- as done in previous cranking calculations -- two well-defined TSD minima in the total Routhian surface are found for a given configuration: one with positive and another with negative triaxial deformation $\gamma$. By allowing the rotational axis to change direction, the higher-energy minimum is shown to be a saddle point. This resolves the long-standing question of the physical interpretation of the two triaxial minima at a very similar quadrupole shape obtained in the principal axis cranking approach. Several TSD configurations have been predicted, including a highly deformed band expected to cross lesser elongated TSD bands at the highest spins. Its transitional quadrupole moment $Q_t \approx 10.5$\,eb is close to the measured value of $\sim$11\,eb; hence, it is a candidate for the structure observed in experiment.Comment: 5 pages, 5 figure

Elliptic Thermal Correlation Functions and Modular Forms in a Globally Conformal Invariant QFT

Global conformal invariance (GCI) of quantum field theory (QFT) in two and higher space-time dimensions implies the Huygens' principle, and hence, rationality of correlation functions of observable fields (see Commun. Math. Phys. 218 (2001) 417-436; hep-th/0009004). The conformal Hamiltonian $H$ has discrete spectrum assumed here to be finitely degenerate. We then prove that thermal expectation values of field products on compactified Minkowski space can be represented as finite linear combinations of basic (doubly periodic) elliptic functions in the conformal time variables (of periods 1 and $\tau$) whose coefficients are, in general, formal power series in $q^{1/2}=e^{i\pi\tau}$ involving spherical functions of the "space-like" fields' arguments. As a corollary, if the resulting expansions converge to meromorphic functions, then the finite temperature correlation functions are elliptic. Thermal 2-point functions of free fields are computed and shown to display these features. We also study modular transformation properties of Gibbs energy mean values with respect to the (complex) inverse temperature $\tau$ ($Im(\tau)=\beta/(2\pi)>0$). The results are used to obtain the thermodynamic limit of thermal energy densities and correlation functions.Comment: LaTex. 56 pages. The concept of global conformal invariance set in a historical perspective (new Sect. 1.1 in the Introduction), references added; minor corrections in the rest of the pape

Jacobi Identity for Vertex Algebras in Higher Dimensions

Vertex algebras in higher dimensions provide an algebraic framework for investigating axiomatic quantum field theory with global conformal invariance. We develop further the theory of such vertex algebras by introducing formal calculus techniques and investigating the notion of polylocal fields. We derive a Jacobi identity which together with the vacuum axiom can be taken as an equivalent definition of vertex algebra.Comment: 35 pages, references adde

Proper holomorphic mappings between symmetrized ellipsoids

We characterize the existence of proper holomorphic mappings in the special class of bounded $(1,2,...,n)$-balanced domains in $\mathbb{C}^n$, called the symmetrized ellipsoids. Using this result we conclude that there are no non-trivial proper holomorphic self-mappings in the class of symmetrized ellipsoids. We also describe the automorphism groupof these domains.Comment: 10 pages, some modification

Exhausting domains of the symmetrized bidisc

We show that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.Comment: 6 page