Renormalization group and the <Emphasis Type="Italic">ɛ</Emphasis>-expansion: Representation of the <Emphasis Type="Italic">β</Emphasis>-function and anomalous dimensions by nonsingular integrals

In the framework of the renormalization group and the ɛ-expansion, we propose expressions for the β-function and anomalous dimensions in terms of renormalized one-irreducible functions. These expressions are convenient for numerical calculations. We choose the renormalization scheme in which the quantities calculated using R operations are represented by integrals that do not contain singularities in ɛ. We develop a completely automated calculation system starting from constructing diagrams, determining relevant subgraphs, combinatorial coefficients, etc., up to determining critical exponents. As an example, we calculate the critical exponents of the φ ³ model in the order ɛ ⁴ .     

Critical orbits of symmetric functionals

Let G be a compact Lie group and V a G-module, i.e. a finite-dimensional real vector space on which G acts orthogonally. We are interested in finding G-orbits of critical points of G-invariant C²-functionals f: SV→—, SV the unit sphere of V. Using a generalization of the Borsuk-Ulam theorem by Komiya [15] we give lower bounds for the number of critical orbits with a given orbit type. These results are applied to nonlinear eigenvalue problems which are symmetric with respect to an action of O(3) or a closed subgroup of O(3).     

Monodromy of certain Painlevé–VI transcendents and reflection groups

We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ= and 2α=(2μ-1)² with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space. Oblatum 19-III-1999 & 25-XI-1999?Published online: 21 February 2000     

Critical exponents for groups of isometries

Let Γ be a convex co-compact group of isometries of a CAT(−1) space X and let Γ₀ be a normal subgroup of Γ. We show that, provided Γ is a free group, a sufficient condition for Γ and Γ₀ to have the same critical exponent is that Γ / Γ₀ is amenable.      

Effects of turbulent transfer on critical behavior

Using the field theory renormalization group, we study the critical behavior of two systems subjected to turbulent mixing. The first system, described by the equilibrium model A, corresponds to the relaxational dynamics of a nonconserved order parameter. The second system is the strongly nonequilibrium reaction-diffusion system, known as the Gribov process or directed percolation process. The turbulent mixing is modeled by the stochastic Navier-Stokes equation with a random stirring force with the correlator ∞ δ(t − t′)p ^4−d−y, where p is the wave number, d is the space dimension, and y is an arbitrary exponent. We show that the systems exhibit various types of critical behavior depending on the relation between y and d. In addition to known regimes (original systems without mixing and a passively advected scalar field), we establish the existence of new strongly nonequilibrium universality classes and calculate the corresponding critical dimensions to the first order of the double expansion in y and ɛ = 4 − d (one-loop approximation).     

Stochastic load modelling for electric energy distribution applications

Summary The problem of electric load modelling for low aggregation levels is addressed in this paper. The objective is to obtain good “demand” and “response” behaviour models of any group of loads in an electric energy distribution system for any of the functional applications that are beeing considered in the framework of the Distribution Management Systems, aimed to improve the energy efficiency, reliability and quality of the system. A brief critical revision of the methodologies used for that purpose is in the paper, and the advantages of using approaches where physical knowledge about the load characteristics is used, are stated and demonstrated.     

On the volume functional of compact manifolds with boundary with constant scalar curvature

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.     

Decidability of first-order theories for groups and monoids of integral matrices

We deal with the decidability problem for first-order theories of a complete linear group GL(n,ℤ) of all integral matrices of order n ≥ 3. and of a respective complete linear monoid ML(n,ℤ). It is proved that theories ∀? ∧ GL(3,ℤ). ∃∀∧ GL(3,ℤ). ∀? ∧ ML(3,ℤ), and ∃? ∧ ML(3,ℤ) are critical. and that ∃∀ ∧ νGL(n,ℤ) and ∃∀ ∧ML(n,ℤ) are decidable for any n ≥ 3. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 480–504, July–August, 2000.     

Epsilon-expansion in the N-component ϕ 4 model

The formalism of projection Hamiltonians is applied to the N-component O(N)-invariant ϕ⁴ model in the Euclidean and p-adic spaces. We use two versions of the ε-expansion (with ε = 4 − d and with ε = α − 3d/2, where α is the renormalization group parameter) and evaluate the critical indices ν and η up to the second order of the perturbation theory. The results for the (4− d)-expansion then coincide with the known results obtained via the quantum-field renormalization-group methods. Our calculations give evidence that in dimension three, both expansions describe the same non-Gaussian fixed point of the renormalization group. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 365–384, March, 2006.     

Invariance of Milnor numbers and topology of complex polynomials

We give a global version of Lê-Ramanujam μ-constant theorem for polynomials. Let , , be a family of polynomials of n complex variables with isolated singularities, whose coefficients are polynomials in t. We consider the case where some numerical invariants are constant (the affine Milnor number μ(t), the Milnor number at infinity λ(t), the number of critical values, the number of affine critical values, the number of critical values at infinity). Let n=2, we also suppose the degree of the is a constant, then the polynomials and are topologically equivalent. For we suppose that critical values at infinity depend continuously on t, then we prove that the geometric monodromy representations of the are all equivalent. Received: January 14, 2002     

Student’s t-test for Gaussian scale mixtures

A Student-type test is constructed under a condition weaker than normal. We assume that the errors are scale mixtures of normal random variables and compute the critical values of the suggested s-test. Our s-test is optimal in the sense that if the level is at most α, then the s-test provides the minimum critical values. (The most important critical values are tabulated at the end of the paper.) For α ≤.05, the two-sided s-test is identical with Student’s classical t-test. In general, the s-test is a t-type test, but its degree of freedom should be reduced depending on α. The s-test is applicable for many heavy-tailed errors, including symmetric stable, Laplace, logistic, or exponential power. Our results explain when and why the P-value corresponding to the t-statistic is robust if the underlying distribution is a scale mixture of normal distributions. Bibliography: 24 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 5–19.     

A Canonical Compatible Metric for Geometric Structures on Nilmanifolds

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants.Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory.Communicated by: Nigel Hitchin (Oxford) Mathematics Subject Classifications (2000): Primary: 53D05, 53D55; Secondary: 22E25, 53D20, 14L24, 53C30.     

Large derivatives,backward contraction and invariant densities for interval maps

In this paper, we study the dynamics of a smooth multimodal interval map f with non-flat critical points and all periodic points hyperbolic repelling. Assuming that |Dfⁿ(f(c))|→∞ as n→∞ holds for all critical points c, we show that f satisfies the so-called backward contracting property with an arbitrarily large constant, and that f has an invariant probability μ which is absolutely continuous with respect to Lebesgue measure and the density of μ belongs to L^p for all p<ℓ_max/(ℓ_max-1), where ℓ_max denotes the maximal critical order of f. In the appendix, we prove that various growth conditions on the derivatives along the critical orbits imply stronger backward contraction.     

The Cauchy problem for the 1-D Dirac–Klein–Gordon equation

The Cauchy problem for the Dirac–Klein–Gordon equation are discussed in one space dimension. Time local and global existence for solutions with rough data, especially the solutions for Klein–Gordon equation in the critical and super critical Sobolev norm of [4] are considered. The solutions with general propagation speeds are dealt with.      

Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck’s program of ‘Dessins d’ enfants’, aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show how the study of these varieties leads to new interesting questions in the theory of finite (simple) groups. We would like to thank Fabio Tonoli for helping us with the pictures.     

Critical Points Index for Vector Functions and Vector Optimization

In this work, we study the critical points of vector functions from ℝ ⁿ to ℝ ^m with n≥m, following the definition introduced by Smale in the context of vector optimization. The local monotonicity properties of a vector function around a critical point which are invariant with respect to local coordinate changes are considered. We propose a classification of critical points through the introduction of a generalized Morse index for a critical point, consisting of a triplet of nonnegative integers. The proposed index is based on the sign of an appropriate invariant vector-valued second-order differential.     

Twistor quantization of loop spaces of compact Lie groups

We consider the problem of twistor quantization for the loop space Ω_TG of a compact Lie group G. We show that this problem is solvable in the critical dimension. Dedicated to Vasilii Sergeevich Vladimirov for his 85th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 450–467, December, 2008.     

Renormalization group, operator expansion, and anomalous scaling in a simple model of turbulent diffusion

Using the renormalization group method and the operator expansion in the Obukhov-Kraichnan model that describes the intermixing of a passive scalar admixture by a random Gaussian field of velocities with the correlator 〈v(t,x)v(t′,x)〉−〈v(t,x)v(t′,x′)〉∝δ(t−t′)|x−x′|^ε, we prove that the anomalous scaling in the inertial interval is caused by the presence of “dangerous” composite operators (powers of the local dissipation rate) whose negative critical dimensions determine the anomalous exponents. These exponents are calculated up to the second order of the ε expansion. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 309–314, August, 1999.     

The effect of strongly anisotropic turbulent mixing on critical behavior: Renormalization group analysis of two nonstandard systems

We consider the effect of strongly anisotropic turbulent mixing on the critical behavior of two systems: a φ ³ critical dynamics model describing universal properties of metastable states in the vicinity of a firstorder phase transition and a reaction-diffusion system near the point of a second-order transition between fluctuation and absorption states (a simple epidemic process or the Gribov process). In both cases, we demonstrate the existence of a new strongly nonequilibrium, anisotropic scaling regime (universality class) for which both the mixing and the nonlinearity in the order parameter are relevant. We evaluate the corresponding critical dimensions in the one-loop approximation of the renormalization group.     

Graphs whose critical groups have larger rank

The critical group C(G) of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r(G) be the minimum number of generators (i.e., the rank) of the group C(G) and β(G) be the number of independent cycles of G. In this paper, some forbidden induced subgraphs are given for r(G) = n − 3 and all graphs with r(G) = β(G) = n − 3 are characterized.