On multiple ergodicity of affine cocycles over irrational rotations

Let \({T_\alpha }\) denote the rotation \({T_\alpha }x = x + \alpha \) (mod 1) by an irrational number α on the additive circle T = [0, 1). Let β ₁, …, β _d be d ≥ 1 parameters in [0, 1). One of the goals of this paper is to describe the ergodic properties of the cocycle (taking values in ? ^d+1) generated over \({T_\alpha }\) by the vectorial function Ψ _d+1(x):= (φ(x), φ(x+β ₁), …, φ(x+β _d )), with φ(x) = {x}?½. It was already proved in [LeMeNa03] that Ψ₂ is regular for α with bounded partial quotients. In the present paper we show that Ψ₂ is regular for any irrational α. For higher dimensions, we give sufficient conditions for regularity. While the case d = 2 remains unsolved, for d = 3 we provide examples of non-regular cocycles Ψ₄ for certain values of the parameters β ₁, β ₂, β ₃. We also show that the problem of regularity for the cocycle Ψ _d+1 reduces to the regularity of the cocycles of the form \({\Phi _d} = {({1_{[0,{\beta _j}]}} - {\beta _j})_{j = 1, \ldots ,d}}\) (taking values in ? ^d ). Therefore, a large part of the paper is devoted to the classification problems of step functions with values in ? ^d .     

On a characterization of irreducibility of a non-negative matrix

Let A denote an n×n matrix with all its elements real and non-negative, and let r_i be the sum of the elements in the ith row of A, i=1,…,n. Let B=A?D(r₁,…,r_n), where D(r₁,…,r_n) is the diagonal matrix with r_i at the position (i,i). Then it is proved that A is irreducible if and only if rank B=n?1 and the null space of B^T contains a vector d whose entries are all non-null.     

Hitting time of a moving boundary for a diffusion

We obtain asymptotic estimates for the quantity r = log P[T_f[rang]t] as t → ∞ where T_f = inf\s{s : |X(s)|[rang]f(s)\s} and X is a real diffusion in natural scale with generator a(x) d²(·)/dx² and the ‘boundary’ f(s) is an increasing function. We impose regular variation on a and f and the result is expressed as r = ∫^t₀ λ₁ (f(s) ds(1 + o(1)) where λ₁(f) is the smallest eigenvalue for the process killed at ±f.     

A recurrence among the elements of functions of triangular matrices

A simple relation exists among the elements of φ(T) when φ is an analytic function and T is triangular. This permits the rapid build up of φ(T) from its diagonal. Moreover, exp(B^?1A) can be formed without inverting B.     

HLDER CONTINUOUS SOLUTIONS OF BOUSSINESQ EQUATIONS

We show the existence of dissipative H¨older continuous solutions of the Boussinesq equations. More precise, for any β∈(0,1/5), a time interval [0, T ] and any given smooth energy profile e : [0, T ] →(0, ∞), there exist a weak solution(v, θ) of the 3 d Boussinesq equations such that(v, θ) ∈ Cβ(T~3× [0, T ]) with e(t) =′his T~3|v(x, t)|~2 dx for all t ∈ [0, T ]. Textend the result of [2] about Onsager's conjecture into Boussinesq equation and improve our previous result in [30].     

On Optimal Orientations of Cartesian Products of Trees

. Let d(D) (resp., d(G)) denote the diameter and r(D) (resp., r(G)) the radius of a digraph D (resp., graph G). Let G×H denote the cartesian product of two graphs G and H. An orientation D of G is said to be (r, d)-invariant if r(D)=r(G) and d(D)=d(G). Let {T _i }, i=1,…,n, where n≥2, be a family of trees. In this paper, we show that the graph ∏ _{i =1} ⁿ T _i admits an (r, d)-invariant orientation provided that d(T ₁)≥d(T ₂)≥4 for n=2, and d(T ₁)≥5 and d(T ₂)≥4 for n≥3. Received: July 30, 1997 Final version received: April 20, 1998     

具有奇异性的周期椭圆算子的绝对连续性

该文研究周期椭圆算子sun from(j,l=1) to d D_(jw)(x)a_(jl)D_l+V(x)在R~d(d≥3)中的谱性质,其中A=(a_(jl))是d×d阶的实常值正定矩阵,V(x)和w(x)是关于相同格点的周期标量函数,并且w(x)是正的.利用文中第一作者建立的d-环面上的一致Sobolev不等式,证明了该算子的谱是纯绝对连续的,如果V∈L_(loc)~(2pd/(d+2p))(R~d)且w∈A_(1+α)~(p,∞)(T~d)∩L~∞(T~d)(α0,p≥d),或者V∈L_(loc)~(2d/3)/(R~d),ω∈C~1(T~d),或者V∈L_(loc)~(d/2)(R~d),w∈L_(2,loc)~(d/2)(T~d).     

Companion orthogonal polynomials

We give some properties relating the recurrence relations of orthogonal polynomials associated with any two symmetric distributions dφ₁(x) and d₂(x) such that dφ₂(x) = (1 + kx²)d₁(x). As applications of properties, recurrence relations for many interesting systems of orthogonal polynomials are obtained.     

Phase transition on the Toeplitz algebra of the affine semigroup over the natural numbers

We show that the group of orientation-preserving affine transformations of the rational numbers is quasi-lattice ordered by its subsemigroup N?N^×. The associated Toeplitz C^∗-algebra T(N?N^×) is universal for isometric representations which are covariant in the sense of Nica. We give a presentation of T(N?N^×) in terms of generators and relations, and use this to show that the C^∗-algebra QN recently introduced by Cuntz is the boundary quotient of in the sense of Crisp and Laca. The Toeplitz algebra T(N?N^×) carries a natural dynamics σ, which induces the one considered by Cuntz on the quotient QN, and our main result is the computation of the KMSβ (equilibrium) states of the dynamical system (T(N?N^×),R,σ) for all values of the inverse temperature β. For β∈[1,2] there is a unique KMSβ state, and the KMS₁ state factors through the quotient map onto QN, giving the unique KMS state discovered by Cuntz. At β=2 there is a phase transition, and for β>2 the KMSβ states are indexed by probability measures on the circle. There is a further phase transition at β=∞, where the KMS_∞ states are indexed by the probability measures on the circle, but the ground states are indexed by the states on the classical Toeplitz algebra T(N).     

The construction of stationary two-dimensional Markoff fields with an application to quantum field theory

It is shown that if φ(f)  ∝_R^dφ(y) f(y) dy is a Markoff random field and X_α are multiplicative functionals of φ (with E(X_α) = 1) which converge locally in L₁, then there exists a locally Markoff random field $\text{φ}{}_1$ such that $\text{E(exp(iφ}{}_1\text{(f))) =}\text{lim}{}_{\mathrm{\alpha }}\text{E(X}{}_{\mathrm{<mtext>\alpha </mtext>}}\text{exp}\text{(iφ(φ)))}$. We choose φ to be the two-dimensional generalization of the Ornstein-Uhlenbeck velocity process and take X_α proportional to exp(?λ∝_R² : P(φ(y)) : g_α(y) dy), where: P(φ(y)) : is a regularized even degree polynomial in φ(y). It is then proved that for an appropriate choice of g_α → 1 and small λ, {X_α} does converge locally in L₁ and that the corresponding $\text{φ}{}_1$ is stationary.     

A method of multipliers for mathematical programming problems with equality and inequality constraints

This paper deals with the numerical solution of the general mathematical programming problem of minimizing a scalar functionf(x) subject to the vector constraints φ(x)=0 and ψ(x)≥0. The approach used is an extension of the Hestenes method of multipliers, which deals with the equality constraints only. The above problem is replaced by a sequence of problems of minimizing the augmented penalty function Ω(x, λ, μ,k)=f(x)+λ ^T φ(x)+kφ ^T (x)φ(x) ?μ ^T \(\tilde \psi \) (x)+k \(\tilde \psi \) ^T (x) \(\tilde \psi \) (x). The vectors λ and μ, μ ≥ 0, are respectively the Lagrange multipliers for φ(x) and \(\tilde \psi \) (x), and the elements of \(\tilde \psi \) (x) are defined by \(\tilde \psi \) _(j)(x)=min[ψ_(j)(x), (1/2k) μ_(j)]. The scalark>0 is the penalty constant, held fixed throughout the algorithm. Rules are given for updating the multipliers for each minimization cycle. Justification is given for trusting that the sequence of minimizing points will converge to the solution point of the original problem.     

Multiplicative mappings at some points on matrix algebras

Let M_n be the algebra of all n×n matrices, and let φ:M_n→M_n be a linear mapping. We say that φ is a multiplicative mapping at G if φ(ST)=φ(S)φ(T) for any S,T∈M_n with ST=G. Fix G∈M_n, we say that G is an all-multiplicative point if every multiplicative linear bijection φ at G with φ(I_n)=I_n is a multiplicative mapping in M_n, where I_n is the unit matrix in M_n. We mainly show in this paper the following two results: (1) If G∈M_n with detG=0, then G is an all-multiplicative point in M_n; (2) If φ is an multiplicative mapping at I_n, then there exists an invertible matrix P∈M_n such that either φ(S)=PSP^-1 for any S∈M_n or φ(T)=PT^trP^-1 for any T∈M_n.     

Generalized radix representations and dynamical systems. IV

For r = (r₁,…, r_d) ∈ ?^d the mapping τ_r:?^d →?^d given byτ_r(a₁,…,a_d) = (a₂, …, a_d,−⌊r₁a₁+…+ r_da_d⌋)where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ?^d there exists an integer k > 0 with τ_r^k(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).     

Integer points in backward orbits

A theorem of J. Silverman states that a forward orbit of a rational map φ(z) on P¹(K) contains finitely many S-integers in the number field K when (φ°φ)(z) is not a polynomial. We state an analogous conjecture for the backward orbits using a general S-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map φ(z)=zd, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for zn−β when β≠0 is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for φn(z)−β is bounded independently of n.     

Diameter-girth sufficient conditions for optimal extraconnectivity in graphs

For a connected graph G, the rth extraconnectivity κ_r(G) is defined as the minimum cardinality of a cutset X such that all remaining components after the deletion of the vertices of X have at least r+1 vertices. The standard connectivity and superconnectivity correspond to κ₀(G) and κ₁(G), respectively. The minimum r-tree degree of G, denoted by ξ_r(G), is the minimum cardinality of N(T) taken over all trees T⊆G of order |V(T)|=r+1, N(T) being the set of vertices not in T that are neighbors of some vertex of T. When r=1, any such considered tree is just an edge of G. Then, ξ₁(G) is equal to the so-called minimum edge-degree of G, defined as ξ(G)=min{d(u)+d(v)-2:uv∈E(G)}, where d(u) stands for the degree of vertex u. A graph G is said to be optimally r-extraconnected, for short κ_r-optimal, if κ_r(G)?ξ_r(G). In this paper, we present some sufficient conditions that guarantee κ_r(G)?ξ_r(G) for r?2. These results improve some previous related ones, and can be seen as a complement of some others which were obtained by the authors for r=1.     

On the mean oscillation of the Hessian of solutions to the Monge-Ampère equation

We give interior a priori estimates for the mean oscillation of second derivatives of solutions to the Monge-Ampère equation detD²u=f(x) with zero boundary values, where f(x) is a non-Dini continuous function. If the modulus of continuity of f(x) is φ(r) such that limr→0φ(r)log(1/r)=0, then D²u∈VMO.     

Eigenvectors and cyclic vectors of bilateral weighted shifts. II: Simply invariant subspaces

Let T be an injective bilateral weighted shift onl ² thought as "multiplication by λ" on a space of formal Laurent series L²(β). (a) If L²(β) is contained in a space of quasi-analytic class of functions, then the point spectrum σ_p(T^?) of T^? contains a circle and the cyclic invariant subspaceM _f of T generated by f is simply invariant (i.e., ∩{(T^k M _f)^?: k ≥ 0}= {0}) for each f in L²(β); (b) If L²(β) contains a non-quasi-analytic class of functions (defined on a circle г) of a certain type related with the weight sequence of T, then there exists f in L²(ß) such thatM _f is a non-trivial doubly invariant subspace (i.e., (TM _f)^? =M _f); furthermore, if г ? σ_p(T^*), then σ_p (T^*) = г and f can be chosen so that σ_p([T∣M _f]^*) = г?{α}, for some α ε г. Several examples show that the gap between operators satisfying (a) and operators satisfying (b) is rather small.     

Singular continuous Floquet operator for systems with increasing gaps

Consider the Floquet operator of a time-independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: e^−iH₀Te^{−iκT|φ〉〈φ|}, where T and κ are the period and the coupling constant, respectively. Assume the spectrum of the self-adjoint operator H₀ is pure point, simple, bounded from below and the gaps between the eigenvalues (λ_n) grow like λ_n+1−λ_n∼Cn^d with d?2. Under some hypotheses on the arithmetical nature of the eigenvalues and the vector φ, cyclic for H₀, we prove the Floquet operator of the perturbed system has purely singular continuous spectrum.     

Doubly nonlinear evolution equations governed by time-dependent subdifferentials in reflexive Banach spaces

We prove the existence of solutions of the Cauchy problem for the doubly nonlinear evolution equation: dv(t)/dt+V_∂φt(u(t))∋f(t), v(t)∈H_∂ψ(u(t)), 0<t<T, where H_∂ψ (respectively, V_∂φt) denotes the subdifferential operator of a proper lower semicontinuous functional ψ (respectively, φt explicitly depending on t) from a Hilbert space H (respectively, reflexive Banach space V) into (−∞,+∞] and f is given. To do so, we suppose that V?H≡H^∗?V^∗ compactly and densely, and we also assume smoothness in t, boundedness and coercivity of φt in an appropriate sense, but use neither strong monotonicity nor boundedness of H_∂ψ. The method of our proof relies on approximation problems in H and a couple of energy inequalities. We also treat the initial-boundary value problem of a non-autonomous degenerate elliptic-parabolic problem.     

Necessary and sufficient conditions for boundedness of some commutators with weighted Lipschitz functions

In this paper we show that b∈Lipβ,μ if and only if the commutator [b,T] of the multiplication operator by b and the singular integral operator T is bounded from Lp(μ) to Lq(μ1−q), where 1<p<q<∞, 0<β<1 and 1/q=1/p−β/n. Also we will obtain that b∈Lipβ,μ if and only if the commutator [b,Iα] of the multiplication operator by b and the fractional integral operator Iα is bounded from Lp(μ) to Lr(μ1−(1−α/n)r), where 1<p<∞, 0<β<1 and 1/r=1/p−(β+α)/n with 1/p>(β+α)/n.