Eigenvalue distributions from impacts on a ring

We consider the collision dynamics produced by three beads with masses (m ₁, m ₂, m ₃) sliding without friction on a ring, where the masses are scaled so that m ₁ = 1/ɛ, m ₂ = 1, m ₃ = 1 − ɛ, for 0 ⩽ ɛ ⩾ 1. The singular limits ɛ = 0 and ɛ = 1 correspond to two equal mass beads colliding on the ring with a wall, and without a wall respectively. In both these cases, all solutions are periodic and the eigenvalue distributions (around the unit circle) associated with the products of collision matrices are discrete. We then numerically examine the regime which parametrically connects these two states, i.e. 0 < ɛ < 1, and show that the eigenvalue distribution is generically uniform around the unit circle, which implies that the dynamics are no longer periodic. By a sequence of careful numerical experiments, we characterize how the uniform spectrum collapses from continuous to discrete in the two singular limits ɛ → 0 and ɛ → 1 for an ensemble of initial velocities sampled uniformly on a fixed energy surface. For the limit ɛ → 0, the distribution forms Gaussian peaks around the discrete limiting values ± 1, ± i, with variances that scale in power law form as σ ² ∼ αɛ ^β. By contrast, the convergence in the limit ɛ → 1 to the discrete values ±1 is shown to follow a logarithmic power-law σ ² ∼ log(ɛ ^β).     

Global behavior of the branch of positive solutions for nonlinear Sturm–Liouville problems

We consider the nonlinear Sturm–Liouville problem

On the moments of elements of continued fractions for some rational numbers

Let p be a prime and let 1 ≤ a ≤ p − 1. In the paper, an asymptotics for the sum over a of the moments of order α (0 < α < 1) of the sequence of elements of the expansion of a/p into a continued fraction is obtained. As a corollary, an upper bound for the number of those a whose expansions contain at least one element larger than log ^λ p (λ > 1) is derived. Note that in the case considered, the set of elements has no limiting distribution as p → ∞, which is in contrast with the case of rational fractions b/c, where (b, c) = 1 and b² + c² ≤ R² (R → ∞). Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 13–22.     

New asymptotic formula for eigenvalues of nonlinear Sturm-Liouville problems

We study the nonlinear Sturm-Liouville problem

Scaling limits in the second Painlevé transcendent

By the isomonodromy deformation method, the asymptotics of the general solution for the second Painlevé equation y_xx=2y³+xy−α asReα→∞ are described for any x. The corresponding formulas are also presented. Bibliography: 23 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 60–101.     

An answer to a question by J. Milnor

We consider two commuting automorphismsT ₁,T ₂ of the Lebesque space (M, M, μ) such thath _m,n=h(T ₁ ^m T ₂ ⁿ )<∞ whereh is the measure-theoretic entropy. Under additional assumptions we show the existence of the limits lim (1/m)h _m,n wherem→∞,n→∞,m/n→ω and ω is an irrational number.     

Boundary values of holomorphic functions, singular unitary representations ofO(p,q), and their limits asq→∞

Let Ω be a bounded circular domain in ℂ ^N , let M be a submanifold in the boundary of Ω, and let H be a Hilbert space of holomorphic functions in Ω. We show that, under certain conditions stated in terms of the reproducing kernel of the space H, the restriction operator to the submanifold M is well defined for all functions from H. We apply this result to constructing a family of “singular” unitary representations of the groups SO(p,q). The singular representations arise as discrete components of the spectrum in the decomposition of irreducible unitary highest weight representations of the groups U(p,q) restricted to the subgroups SO(p,q). Another property of the singular representations is that they persist in the limit as q→∞. Bibliography: 70 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 9–91. Translated by B. Bekker.     

Cesàro-type operators on Hardy, BMOA and bloch spaces

Let H ^p, p ∈ (0, ∞], BMOA and B ^a, a ∈ (0, ∞) be the classical p-Hardy, analytic BMO(∂) (bounded mean oscillation on the unit circle) and a-Bloch space on the unit disk. In this paper, we prove that the Cesàro-type operator: C ^α, α ∈ (−1, ∞) is bounded on H ^p, p ∈ (0, ∞) and on B ^a, a ∈ (1, ∞), but, unbounded on H ^∞, BMOA and B ^a, a ∈ (0, 1]. In particular, we give an answer to the Stempak’s open problem.     

Maximality of analytic operator algebras

LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH ^∞(α) be the associated analytic subalgebra; i.e.H ^∞(α)={X ∈M: sp_∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH ^∞(α) isH ^∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M ^α = Ci)H ^∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH ^∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α).     

Weak generators of the algebral ∞ and the commutant of a model operator

Let B be a Blaschke product with simple zeros in the unit disk, let Λ be the set of its zeros, and let ϕ∈H^∞. It is known that ϕ+BH^∞ is a weak^* generator of the algebra H^∞/BH^∞ if (for B that satisfy the Carleson condition (C)) and only if the sequence ϕ(Λ) is a weak^* generator of the algebra l^∞. In this paper, we show that for any Blaschke product B with simple zeros that does not satisfy condition (C), there exists B=B₁·…·B_N, where N ∈ℕ, and B₁, …, B_N are Blaschke products satisfying condition (C), there exists a function ϕ∈H^∞ such that ϕ(Λ) is a weak^* generator of the algebra l^∞, and ϕ+BH^∞ is not a weak^* generator of the algebra H^∞/BH^∞. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 73–85. Translated by M. F. Gamal'.     

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

Limit Distributions of Eigenvalues for Random Block Toeplitz and Hankel Matrices

Block Toeplitz and Hankel matrices arise in many aspects of applications. In this paper, we will research the distributions of eigenvalues for some models and get the semicircle law. Firstly we will give trace formulas of block Toeplitz and Hankel matrix. Then we will prove that the almost sure limit g_T^(m)\gamma_{T}^{(m)} (g_H^(m))(\gamma_{H}^{(m)}) of eigenvalue distributions of random block Toeplitz (Hankel) matrices exist and give the moments of the limit distributions where m is the order of the blocks. Then we will prove the existence of almost sure limit of eigenvalue distributions of random block Toeplitz and Hankel band matrices and give the moments of the limit distributions. Finally we will prove that g_T^(m)\gamma_{T}^{(m)} (g_H^(m))(\gamma_{H}^{(m)}) converges weakly to the semicircle law as m→∞.     

Asymptotic behavior of the solution to the two-dimensional stationary problem of flow past a body far from it

In the exterior domain Ω⊂ℝ² we consider the two-dimensional Navier-stokes system Δu-▽p=(u,▽)u, div u=0 whose solution possesses a finite Dirichlet integral and satisfies the condition lim_|x|→∞ u(x)=(1, 0). For this solution, we establish the estimate |u(x)−(1, 0)|≤c|x| ^−α, where α>1/4. This estimate implies an asymptotic expression for the solution indicating the presence of a track behind the body. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 246–253, February, 1999.     

A functional limit theorem for the position of a particle in the Lorentz model

We consider a particle moving through a medium under a constant external field. The medium consists of immobile spherical obstacles of equal radii randomly distributed in ℝ³. When the particle collides with an obstacle, it reflects inelastically, with restitution coefficient α ∈, (0, 1). We study the asymptotics of X(t), the position of the particle at time t, as t → ∞. The main result is a functional limit theorem for X(t). Its proof is based on the functional CLT for Markov chains. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 328, 2005, pp. 42–68. An erratum to this article is available at .     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

Bernstein and Gelfand widths of certain classes of analytic functions. II

The Gelfand widths of the unit ball of H²(ν) (the weighted Hardy space) with respect to the metric of the space L_∞(T_r) are considered (here T_r is the circle of radius r centered at the origin), as well as the Bernstein widths of the unit ball of H^∞ with respect to the metric of the space L₂(T_r, μ). Asymptotic formulas for the widths in question are established for arbitrary measures ν, μ. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 134–140. Translated by S. V. Kislyakov.     

Controllability in a filled domain for the multidimensional wave equation with a singular boundary control

In accordance with the demands of the so-called local approach to inverse problems, the set of “waves” u^f (·, T) is studied, where u^f (x,t) is the solution of the initial boundary-value problem u_tt−Δu=0 in Ω×(0,T), u|_t<0=0, u|_∂Ω×(0,T)=f, and the (singular) control f runs over the class L²((0,T); H^−m (∂Ω)) (m>0). The following result is established. Let Ω^T={x ∈ Ω : dist(x, ∂Ω)<T)} be a subdomain of Ω ⊂ ℝⁿ (diam Ω<∞) filled with waves by a final instant of time t=T, let T_*=inf{T : Ω^T=Ω} be the time of filling the whole domain Ω. We introduce the notation D_m=Dom((−Δ)^m/2), where (−Δ) is the Laplace operator, Dom(−Δ)=H²(Ω)∩H ₀ ¹ (Ω);D_−m=(D_m)′;D_−m(Ω^T)={y∈D_−m:supp y ⋐ Ω^T. If T<T., then the reachable set R _m ^T ={u^t(·, T): f ∈ L₂((0,T), H^−m (∂Ω))} (∀m>0), which is dense in D_−m(Ω^T), does not contain the class C ₀ ^∞ (Ω^T). Examples of a ∈ C ₀ ^∞ , a ∈ R _m ^T , are presented. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 210, 1994, pp. 7–21. Translated by T. N. Surkova.     

Viscous and inviscid magneto-hydrodynamics equations

In this paper we study the classical problem in turbulence for the magneto-hydrodynamics (MHD) equations: whether the solutions (u ^(v),B ^(v)) of the viscous MHD equations tend to the solutions (u ⁽⁰⁾,B ^(v)) of the inviscid MHD equations as the Reynolds numbersRe, Rm → ∞. As a preparation we first derive bounds for ||(u ⁽⁰⁾,B ⁽⁰⁾(t)||H ^m) (m ≥3) in terms of deformation tensor related quantities (0.1) {ie251-1} We then show that asRe → ∞ andRm → ∞, the difference {ie-251-2} {ie-251-3} converges to zero uniformly int as long as the quantities in (0.1) remain finite. The convergence rates are explicit. Supported by the NSF grant DMS 9304580 at IAS.     

The maximal (C, α, β) operator of two-parameter walsh-fourier series

The two-parameter dyadic martingale Hardy spacesH _p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from H_p to L_p (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H ₁ ^# , L₁), where the Hardy space H ₁ ^# is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H ₁ ^# converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_p whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈H_p, the (C, α, β) means converge to f in H_p norm. The same results are proved for the conjugate (C, α, β) means, too.