On the sphericity of the subgroup PSL2(ℝ) in the group of diffeomorphisms of the circle

It is shown that the group PSL₂(?) is a spherical subgroup in the group of C³-diffeomorphisms of the circle.     

On the constants in the estimates of the rate of convergence in von Neumann’s ergodic theorem

We study the rate of convergence in von Neumann’s ergodic theorem. We obtain constants connecting the power rate of convergence of ergodic means and the power singularity at zero of the spectral measure of the corresponding dynamical system (these concepts are equivalent to each other). All the results of the paper have obvious exact analogs for wide-sense stationary stochastic processes.     

Global indeterminacy of the equilibrium in the Chamley model of endogenous growth in the vicinity of a Bogdanov–Takens bifurcation

This paper studies the dynamics implied by the Chamley (1993) model, a variant of the two-sector model with an implicit characterization of the learning function. We first show that under some “regularity” conditions regarding the learning function, the model has (a) one steady state, (b) no steady states or (c) two steady states (one saddle and one non-saddle). Moreover, via the Bogdanov–Takens theorem, we prove that for critical regions of the parameters space, the dynamics undergoes a particular global phenomenon, namely the homoclinic bifurcation. Because these findings imply the existence of a continuum of equilibrium trajectories, all departing from the same initial value of the predetermined variable, the model exhibits global indeterminacy.     

One-loop calculation of the β-function in the modified formulation of the Yang-Mills theory

In the one-loop approximation for the modified Yang-Mills theory, we calculate contributions to the two- and three-point Green’s functions of the gauge field A _μ , to the Green’s function of the anticommuting fields e and b, and to the vertex function Γ _Aeb. We find the renormalization constants Z ₁ and Z ₂ and the corresponding constants \(\bar Z_1\) and \(\bar Z_2\) for the anticommuting fields and show that they satisfy the same equation as in the standard Yang-Mills theory. We demonstrate that the β-functions in the standard and modified theories coincide.     

A General Theorem of the Alternative in the Context of Set-Valued Function

1.IntroductionTheclassicalworkOiltheoremsofthealternativeforsingle-valuedfunctionbyG..d..II]andMotzkin[2]wereextendbymanyresearchersinvariousways(suchasJeyakumar['l'],Lily],andTamminen[6]),whereconditionsofconvexityaremoreorlessrelaxed.Met's..rk[:]givesemphasistotheoremsofthealternativeinthesettingofset-valuedfunctionwhereconditionofconvexityissettobecone-convex.Thispaperaimsatprovidinganunifiedextensionoftheabove-mentionedtheorems,andgivesoutageneraltheoremofthealternativeinthecontextofset…     

Existence of the Self-Similar Solutions in the Heat Flow of Harmonic Maps

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Existence of stationary solutions of the Navier–Stokes equations in two dimensions in the presence of a wall

We consider the problem of a body moving within an incompressible fluid at constant speed parallel to a wall, in an otherwise unbounded domain. This situation is modeled by the incompressible Navier–Stokes equations in an exterior domain in a half space, with appropriate boundary conditions on the wall, the body, and at infinity. Here we prove existence of stationary solutions for this problem for the simplified situation where the body is replaced by a source term of compact support.     

Reducing conjugacy in the full diffeomorphism group of ℝ to conjugacy in the subgroup of orientation-preserving maps

Let Diffeo = Diffeo(ℝ) denote the group of infinitely differentiable diffeomorphisms of the real line ℝ, under the operation of composition, and let Diffeo⁺ be the subgroup of diffeomorphisms of degree +1, i.e., orientation-preserving diffeomorphisms. We show how to reduce the problem of determining whether or not two given elements f, g ∈ Diffeo are conjugate in Diffeo to associated conjugacy problems in the subgroup Diffeo⁺. The main result concerns the case when f and g have degree −1, and specifies (in an explicit and verifiable way) precisely what must be added to the assumption that their (compositional) squares are conjugate in Diffeo⁺, in order to ensure that f is conjugated to g by an element of Diffeo⁺. The methods involve formal power series and results of Kopell on centralisers in the diffeomorphism group of a half-open interval. Bibliography: 4 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 231–237.     

The stationary Oseen equations in an exterior domain: An approach in weighted Sobolev spaces

In this work, we study the linearized Navier–Stokes equations in an exterior domain of R³${\mathbb{R}}^3$ at the steady state, that is, the Oseen equations. We are interested in the existence and the uniqueness of weak, strong and very weak solutions in L^p$L^p$-theory which makes our work more difficult. Our analysis is based on the principle that linear exterior problems can be solved by combining their properties in the whole space R³${\mathbb{R}}^3$ and the properties in bounded domains. Our approach rests on the use of weighted Sobolev spaces.     

A theorem on measures in dimension 2 and applications to vortex sheets

We find conditions under which measures belong to H⁻¹(R²)$H^{-1}({\mathbb{R}}^2)$. Next we show that measures generated by the Prandtl, Kaden as well as Pullin spirals, objects considered by physicists as incompressible flows generating vorticity, satisfy assumptions of our theorem, thus they are (locally) elements of H⁻¹(R²)$H^{-1}({\mathbb{R}}^2)$. Moreover, as a by-product, we prove an embedding of the space of Morrey type measures in H⁻¹$H^{-1}$.     

On removable sets for convex functions

In the present article we provide a sufficient condition for a closed set F∈R^d$F\in {\mathbb{R}}^d$ to have the following property which we call c  -removability: Whenever a continuous function f:R^d→R$f:{\mathbb{R}}^d\to \mathbb{R}$ is locally convex on the complement of F  , it is convex on the whole R^d${\mathbb{R}}^d$. We also prove that no generalized rectangle of positive Lebesgue measure in R²${\mathbb{R}}^2$ is c-removable. Our results also answer the following question asked in an article by Jacek Tabor and Józef Tabor (2010) [5]: Assume the closed set F⊂R^d$F\subset {\mathbb{R}}^d$ is such that any locally convex function defined on R^d?F${\mathbb{R}}^d?F$ has a unique convex extension on R^d${\mathbb{R}}^d$. Is F   necessarily intervally thin (a notion of smallness of sets defined by their “essential transparency” in every direction)? We prove the answer is negative by finding a counterexample in R²${\mathbb{R}}^2$.     

Asymptotic profiles for wave equations with strong damping

We consider the Cauchy problem in Rⁿ${\mathbf{R}}^n$ for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted L^1,1(Rⁿ)$L^{1,1}({\mathbf{R}}^n)$ data by using a method introduced in [9] and/or [10].     

Remark on the stability of the large stationary solutions to the Navier–Stokes equations under the general flux condition

Consider stationary weak solutions of the Navier–Stokes equations in a bounded domain in R³${\mathbb{R}}^3$ under the nonhomogeneous boundary condition. We give a new approach for the stability of the stationary flow in the L²$L^2$-framework. Furthermore, we give some examples of stable solutions which may be large in L³(Ω)$L^3\left(\Omega \right)$ or W^1,3/2(Ω)$W^{1,3/2}\left(\Omega \right)$.     

Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data

The low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data is rigorously justified in the whole space R³${\mathbb{R}}^3$. First, the uniform-in-Mach-number estimates of the solutions in a Sobolev space are established on a finite time interval independent of the Mach number. Then the low Mach number limit is proved by combining these uniform estimate with a theorem due to Métivier and Schochet (2001) [45] for the Euler equations that gives the local energy decay of the acoustic wave equations.     

On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy

In this paper, we present a uniqueness result for solutions to the Gross–Pitaevskii hierarchy on the three-dimensional torus, under the assumption of an a priori spacetime bound. We show that this a priori bound is satisfied for factorized solutions to the hierarchy which come from solutions of the nonlinear Schrödinger equation. In this way, we obtain a periodic analogue of the uniqueness result on R³${\mathbb{R}}^3$ previously proved by Klainerman and Machedon [75], except that, in the periodic setting, we need to assume additional regularity. In particular, we need to work in the Sobolev class H^α$H^{\alpha }$ for α>1$\alpha >1$. By constructing a specific counterexample, we show that, on T³${\mathbb{T}}^3$, the existing techniques from the work of Klainerman and Machedon approach do not apply in the endpoint case α=1$\alpha =1$. This is in contrast to the known results in the non-periodic setting, where these techniques are known to hold for all α?1$\alpha ?1$. In our analysis, we give a detailed study of the crucial spacetime estimate associated to the free evolution operator. In this step of the proof, our methods rely on lattice point counting techniques based on the concept of the determinant of a lattice. This method allows us to obtain improved bounds on the number of lattice points which lie in the intersection of a plane and a set of radius R, depending on the number-theoretic properties of the normal vector to the plane. We are hence able to obtain a sharp range of admissible Sobolev exponents for which the spacetime estimate holds.     

Harness processes and harmonic crystals

In the Hammersley harness processes the R$\mathbb{R}$-valued height at each site i∈Z^d$i\in {\mathbb{Z}}^d$ is updated at rate 1 to an average of the neighboring heights plus a centered random variable (the noise). We construct the process “a la Harris” simultaneously for all times and boxes contained in Z^d${\mathbb{Z}}^d$. With this representation we compute covariances and show L²$L^2$ and almost sure time and space convergence of the process. In particular, the process started from the flat configuration and viewed from the height at the origin converges to an invariant measure. In dimension three and higher, the process itself converges to an invariant measure in L²$L^2$ at speed t^1−d/2$t^{1-d/2}$ (this extends the convergence established by Hsiao). When the noise is Gaussian the limiting measures are Gaussian fields (harmonic crystals) and are also reversible for the process.