On solutions of the Ricci curvature equation and the Einstein equation

We consider the pseudo-Euclidean space (R ⁿ , g), with n ≥ 3 and g _ij = δ _ij ε _i , ε _i = ±1, where at least one ε _i = 1 and nondiagonal tensors of the form T = Σ _ij f _ij dx _i dx _j such that, for i ≠ j, f _ij (x _i , x _j ) depends on x _i and x _j . We provide necessary and sufficient conditions for such a tensor to admit a metric ḡ, conformal to g, that solves the Ricci tensor equation or the Einstein equation. Similar problems are considered for locally conformally flat manifolds. Examples are provided of complete metrics on R ⁿ , on the n-dimensional torus T ⁿ and on cylinders T ^k ×R ^n-k , that solve the Ricci equation or the Einstein equation. Partially supported by CAPES/PROCAD. Partially Supported By Cnpq, Capes/Procad.     

On Representations Associated with Completely n-Positive Linear Maps on Pro-C^＊-Algebras

It is shown that an n × n matrix of continuous linear maps from a pro-C^＊-algebra A to L（H）, which verifies the condition of complete positivity, is of the form [V^＊TijФ（·）V]^n i,where Ф is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and j=1,[Tij]^n i,j=1 is a positive element in the C^＊-algebra of all n×n matrices over the commutant of Ф（A） in L（K）. This generalizes a result of C. Y.Suen in Proc. Amer. Math. Soc., 112（3）, 1991, 709-712. Also, a covariant version of this construction is given.     

Finite Travelling Waves for a Semilinear Degenerate Reaction-Diffusion System

In this paper, the existence and nonexistence of finite travelling waves (FTWs) for a semilinear degenerate reaction-diffusion system is studied, where 0 < α _i < 1, m _ij ≥ 0 and ∑ ^N _{j =1} m _ij > 0, i, j = 1,...,N. Necessary and sufficient conditions on existence and large time behaviours of FTWs of (I) are obtained by using the matrix theory, Schauder's fixed point theorem, and upper and lower solutions method. Received July 3, 1997, Revised June 23, 1999, Accepted March 29, 2000     

Exact controllability for the wave equation with variable coefficients

We consider in this paper the evolution systemy″−Ay=0, whereA =∂ _i(a_ij∂_j) anda _ij ∈C ¹ (ℝ⁺;W ^1,∞ (Ω)) ∩W ^1,∞ (Ω × ℝ⁺), with initial data given by (y ₀,y ₁) ∈L ²(Ω) ×H ⁻¹ (Ω) and the nonhomogeneous conditiony=v on Γ ×]0,T[. Exact controllability means that there exist a timeT>0 and a controlv such thaty(T, v)=y′(T, v)=0. The main result of this paper is to prove that the above system is exactly controllable whenT is “sufficiently large”. Moreover, we obtain sharper estimates onT.     

The nearness problems for symmetric centrosymmetric with a special submatrix constraint

We say that X=[x_ij]_i,j=1ⁿX=[x_{ij}]_{i,j=1}^n is symmetric centrosymmetric if x _ij  = x _ji and x _{n − j + 1,n − i + 1}, 1 ≤ i,j ≤ n. In this paper we present an efficient algorithm for minimizing ||AXA ^T  − B|| where ||·|| is the Frobenius norm, A ∈ ℝ ^m×n , B ∈ ℝ ^m×m and X ∈ ℝ ^n×n is symmetric centrosymmetric with a specified central submatrix [x _ij ] _{p ≤ i,j ≤ n − p} . Our algorithm produces a suitable X such that AXA ^T  = B in finitely many steps, if such an X exists. We show that the algorithm is stable any case, and we give results of numerical experiments that support this claim.     

Fractional Poisson equations and ergodic theorems for fractional coboundaries

For a given contractionT in a Banach spaceX and 0<α<1, we define the contractionT _α=Σ _j=1 ^∞ a _j T ^j , where {a _j } are the coefficients in the power series expansion (1-t)^α=1-Σ _j=1 ^∞ a _j t ^j in the open unit disk, which satisfya _j >0 anda _j >0 and Σ _j=1 ^∞ a _j =1. The operator calculus justifies the notation(I−T) ^α :=I−T _α (e.g., (I−T _1/2)²=I−T). A vectory∈X is called an, α-fractional coboundary for T if there is anx∈X such that(I−T) ^α x=y, i.e.,y is a coboundary forT _α . The fractional Poisson equation forT is the Poisson equation forT _α . We show that if(I−T)X is not closed, then(I−T) ^α X strictly contains(I−T)X (but has the same closure). ForT mean ergodic, we obtain a series solution (converging in norm) to the fractional Poisson equation. We prove thaty∈X is an α-fractional coboundary if and only if Σ _k=1 ^∞ T ^k y/k ^1-α converges in norm, and conclude that lim _n ‖(1/n ^1-α)Σ _k=1 ⁿ T ^k y‖=0 for suchy. For a Dunford-Schwartz operatorT onL ₁ of a probability space, we consider also a.e. convergence. We prove that iff∈(I−T) ^α L ₁ for some 0<α<1, then the one-sided Hilbert transform Σ _k=1 ^∞ T ^k f/k converges a.e. For 1<p<∞, we prove that iff∈(I−T) ^α L _p with α>1−1/p=1/q, then Σ _k=1 ^∞ T ^k f/k ^1/p converges a.e., and thus (1/n ^1/p ) Σ _k=1 ⁿ T ^k f converges a.e. to zero. Whenf∈(I−T) ^1/q L _p (the case α=1/q), we prove that (1/n ^1/p (logn)^1/q )Σ _k=1 ⁿ T ^k f converges a.e. to zero.     

Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type

We study bi-Hamiltonian systems of hydrodynamic type with nonsingular (semisimple) nonlocal bi-Hamiltonian structures. We prove that all such systems of hydrodynamic type are diagonalizable and that the metrics of the bi-Hamiltonian structure completely determine the complete set of Riemann invariants constructed for any such system. Moreover, we prove that for an arbitrary nonsingular (semisimple) nonlocally bi-Hamiltonian system of hydrodynamic type, there exist local coordinates (Riemann invariants) such that all matrix differential-geometric objects related to this system, namely, the matrix (affinor) V_jⁱ(u) of this system of hydrodynamic type, the metrics g ₁ ^ij(u) and g ₂ ^ij(u), the affinor υ_jⁱ(u) = g ₁ ^is(u)g _2,sj(u), and also the affinors (w _1,n)_jⁱ(u) and (w _2,n)_jⁱ(u) of the nonsingular nonlocal bi-Hamiltonian structure of this system, are diagonal in these special “diagonalizing” local coordinates (Riemann invariants of the system). The proof is a natural corollary of the general results of our previously developed theories of compatible metrics and of nonlocal bi-Hamiltonian structures; we briefly review the necessary notions and results in those two theories.     

Small prime solutions of a pair of linear equations in five prime variables

Assume an additional congruent condition on the coefficients. We prove that the pair 5 of linear equations ∑j=1^5 αλjpj = bλ （λ= 1, 2） has solutions in primes pj satisfying pj 〈〈 （｜b1｜＋｜b2｜＋1） maxλ,j ｜αλj｜^2318＋ε. This improves the exponent 79680 without assuming the additional condition of the second author＇s.     

Separation properties forMW-fractals

Corresponding to the irreducible 0–1 matrix (a _ij ) _n×n , take similitude contraction mappingsϕ _ij for eacha _ij =1, ina _ij =1, in R ^d with ratio 0<r _ij <1. There are unique nonempty compact setsF ₁,…,F _n satisfying for each1≤i≤n, F _i. We prove that open set condition holds if and only ifF _i is ans-set for some1≤i≤n, wheres is such that the spectral radius of matrix (r _ij ³ ) _{n x n} is 1. Partly supported by Natural Science Foundation of China, and partly by Natural Science Foundation of Hubei Province     

Near-interpolation

Summary.  A parametric curve fL ₂ ^(m) ([a,b]ℝ ^d ) is a ``near-interpolant' to prescribed data z <sub>ij</sub> ℝ <sup>d</sup> at data sites t <sub>i</sub> [a,b] within tolerances 0<ɛ <sub>ij</sub> ≤∞ if |f <sup>(j−1)</sup> (t <sub>i</sub> )−z <sub>ij</sub> |≤ɛ <sub>ij</sub> for i=1:n and j=1:m, and a ``best near-interpolant' if it also minimizes ∫ _a ^b |f ^(m) |². In this paper optimality conditions are derived for these best near-interpolants. Based on these conditions it is shown that the near-interpolants are actually smoothing splines with weights that appear as Lagrange multipliers corresponding to the constraints. The optimality conditions are applied to the computation of near-interpolants in the last sections of the paper. Received September 4, 2001 / Revised version received July 22, 2002 / Published online October 29, 2002 Mathematics Subject Classification (1991): 41A05, 41A15, 41A29     

Coboundaries of irreducible markov operators onC(K)

LetK be a compact Hausdorff space, and letT be an irreducible Markov operator onC(K). We show that ifgεC(K) satisfies sup _N ‖Σ _j ^=0N T ^j g‖<∞, then (and only then) there existsfεC(K) with (I − T)f=g. Generalizing the result to irreducible Markov operator representations of certain semi-groups, we obtain that bounded cocycles are (continuous) coboundaries. For minimal semi-group actions inC(K), no restriction on the semi-group is needed.     

A Reflexivity Result Concerning Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain Ω which is obtained by removing n closed disks which are centered at λ _j with radius r _j for j = 1, . . . , n from the unit disk. We assume that T is a bounded linear operator on a separable reflexive Banach space whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded. Then either T has a nontrivial hyperinvariant subspace or the WOT-closure of the algebra {f(T) : f is a rational function with poles off [`(W)]{\overline\Omega}} is reflexive.     

Normal forms for stochastic differential equations

Summary.   We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx _t = ∑ _{j =0} ^m f _j (x _t )∘dW _t ^j and dx _t =∑ _{j =0} ^m g _j (x _t )∘dW _t ^j in ℝ ^d with smooth coefficients satisfying f _j (0)=g _j (0)=0 are said to be smoothly equivalent if there is a smooth random diffeomorphism (coordinate transformation) h(ω) with h(ω,0)=0 and Dh(ω,0)=id which conjugates the corresponding local flows,

A question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

On the existence of a field of stress rates in a hardening elastic-plastic medium : PMM vol. 38, n 6, 1974, pp. 1114–1121

The boundary value problem for the stress rates and rates of change fields in the quasi-static motion of a volume V of an elastic-plastic medium [1] consists of finding the pairs σ_ij^., _ij^. related by the governing equations of an appropriate model; here the σ_ij^. should be statically admissible i.e. should satisfy the equations and boundary conditions σ_ij=−X/._i; /.σ_ijn_j|S_p=p_i and _ij should be kinematically admissible i.e. 2/._ij = v_ij + v_ji, where v_i|S_u = u_io Here S_p and S_u are nonintersecting parts of the boundary of the volume V, X_i, p_i, u_i/.o are specified functions. The question of the existence of a solution of this problem reduces to the question of the functional reaching the lower bound in a set of kinematically admissible /._ij^o and statically admissible σ_ij/.^/*. However, its lower bound may not be reached if in the minimization we limit ourselves only to smooth fields. It is proposed to augment the set of admissible fields σ_ij/.^/*,_ij/.^o by closing them in the norm L₂ (for v_i^o this corresponds to closure in the norm II¹). Some properties of the functional I (σ_ij^*,_ij/.) are considered in the augmented set of admissible fields. It is shown that the equivalence of the two problems is conserved, where I (σ_ij^*,_ij⁰ can be minimized in σ_ij^/*,_ij^o or in σ_ij^/*,_ij/.^o, The lower bound is reached in each of three cases, at a single point. From the fact that u_i^o belongs to the Sobolev space W₂⁽¹⁾, there results the absence of surfaces of velocity discontinuity. Variational principles have been used in plasticity theory to construct models [2] and to investigate the existence and properties of solutions [1, 3].     

On split Lie triple systems II

In [4] it is studied that the structure of split Lie triple systems with a coherent 0-root space, that is, satisfying [T ₀, T ₀, T] = 0 and [T ₀, T _α , T ₀] ≠ 0 for any nonzero root α and where T ₀ denotes the 0-root space and T _α the α-root space, by showing that any of such triple systems T with a symmetric root system is of the form T = U + Σ _j I _j with U a subspace of the 0-root space T ₀ and any I _j a well described ideal of T, satisfying [I _j , T, I _k ] = 0 if j ≠ k. It is also shown in [4] that under certain conditions, a split Lie triple system with a coherent 0-root space is the direct sum of the family of its minimal ideals, each one being a simple split Lie triple system, and the simplicity of T is characterized. In the present paper we extend these results to arbitrary split Lie triple systems with no restrictions on their 0-root spaces.     

Limit Laws for Sums of Independent Random Products: the Lattice Case

Let {V _i,j ;(i,j)∈ℕ²} be a two-dimensional array of independent and identically distributed random variables. The limit laws of the sum of independent random products

The second centralizer of a Bernoulli shift is just its powers

If ℐ is a collection of measure preserving transformations of a probability space, byC(ℐ), the centralizer of ℐ, we mean the group of all measure preserving transformationsS such thatTS=ST for allT ∈ ℐ. We show here that ifT is a Bernoulli shift, thenC(C(T))={T ⁱ |i ∈ Z}. The proof is carried out by constructing an action of Z², {T ₁ ⁱ °T ₂ ⁱ |i, j ∈ Z}, whereT ₁ is a Bernoulli shift of arbitrary entropy, but for anyj ≠ 0,C({T ₁,T ₂ ^i} ={T ₁ ⁱ °T ₂ ^k l, k ∈ Z}. The construction is a two-dimensional analogue of Ornstein’s “rank one mixing” transformation.     

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.