Renormalization Group Functional Equations

Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories, and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling {\sigma} functions, and lead to exact functional relations for the local flow {\beta} functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of {\sigma} are sometimes not true fixed points under continuous changes in scale, and zeroes of {\beta} do not necessarily signal fixed points of the flow, but instead may only indicate turning points of the trajectories.Comment: A physical model with a limit cycle added as section IV, along with reference

The Schr\"oder functional equation and its relation to the invariant measures of chaotic maps

The aim of this paper is to show that the invariant measure for a class of one dimensional chaotic maps, $T(x)$, is an extended solution of the Schr\"oder functional equation, $q(T(x))=\lambda q(x)$, induced by them. Hence, we give an unified treatment of a collection of exactly solved examples worked out in the current literature. In particular, we show that these examples belongs to a class of functions introduced by Mira, (see text). Moreover, as a new example, we compute the invariant densities for a class of rational maps having the Weierstrass $\wp$ functions as an invariant one. Also, we study the relation between that equation and the well known Frobenius-Perron and Koopman's operators.Comment: 9 page

Around a problem of Nicole Brillouët–Belluot

We determine nontrivial intervals I ⊂ (0,+∞), numbers α ∈ R and continuous bijections f : I → I such that f(x)f−1(x) = xα for every x ∈ I

Stationary states of an electron in periodic structures in a constant uniform electrical field

On the basis of the transfer matrix technique an analytical method to investigate the stationary states, for an electron in one-dimensional periodic structures in an external electrical field, displaying the symmetry of the problem is developed. These solutions are shown to be current-carrying. It is also shown that the electron spectrum for infinite structures is continuous, and the corresponding wave functions do not satisfy the symmetry condition of the problem.Comment: 10 pages (Latex), no figures, in the revised variant some mistakes in the English text are corrected and also the first two paragraphs in the Conclusion are refined (Siberian physical-technical institute at the Tomsk state university, Tomsk, Russia

On the flow map for 2D Euler equations with unbounded vorticity

In Part I, we construct a class of examples of initial velocities for which the unique solution to the Euler equations in the plane has an associated flow map that lies in no Holder space of positive exponent for any positive time. In Part II, we explore inverse problems that arise in attempting to construct an example of an initial velocity producing an arbitrarily poor modulus of continuity of the flow map.Comment: http://iopscience.iop.org/0951-7715/24/9/013/ for published versio

Bone Morphogenic Proteins are Immunoregulatory Cytokines Controlling FOXP3+ T\u3csub\u3ereg\u3c/sub\u3e Cells

Bone morphogenic proteins (BMPs) are members of the transforming growth factor β (TGF-β) cytokine family promoting differentiation, homeostasis, and self-renewal of multiple tissues. We show that signaling through the bone morphogenic protein receptor 1α (BMPR1α) sustains expression of FOXP3 in Treg cells in peripheral lymphoid tissues. BMPR1α signaling promotes molecular circuits supporting acquisition and preservation of Treg cell phenotype and inhibiting differentiation of pro-inflammatory effector Th1/Th17 CD4+ T cell. Mechanistically, increased expression of KDM6B (JMJD3) histone demethylase, an antagonist of the polycomb repressive complex 2, underlies lineage-specific changes of T cell phenotypes associated with abrogation of BMPR1α signaling. These results reveal that BMPs are immunoregulatory cytokines mediating maturation and stability of peripheral FOXP3+ regulatory T cells (Treg cells) and controlling generation of iTreg cells. Thus, we establish that BMPs, a large cytokine family, are an essential link between stromal tissues and the adaptive immune system involved in sustaining tissue homeostasis by promoting immunological tolerance

Wedge states in string field theory

The wedge states form an important subalgebra in the string field theory. We review and further investigate their various properties. We find in particular a novel expression for the wedge states, which allows to understand their star products purely algebraically. The method allows also for treating the matter and ghost sectors separately. It turns out, that wedge states with different matter and ghost parts violate the associativity of the algebra. We introduce and study also wedge states with insertions of local operators and show how they are useful for obtaining exact results about convergence of level truncation calculations. These results help to clarify the issue of anomalies related to the identity and some exterior derivations in the string field algebra.Comment: 40 pages, 9 figures, v3: section 3.3 rewritten, few other corrections, set in JHEP styl

On a functional equation involving iterates and powers

We present a complete list of all continuous solutions f : (0,+∞)→(0,+∞) of the equation f 2(x) = γ [f (x)]αxβ, where α, β and γ > 0 are given real numbers