Counterexample to Peano's theorem in infinite-dimensionalF′-spaces

LetE be a nonnormable Fréchet space, and letE′ be the space of all continuous linear functionals onE in the strong topology. A continuous mappingf:E′→E′ such that for anyt ₀∈ℝ,x ₀∈E′, the Cauchy problemx=f(x), x(t ₀ )=x ₀ has no solutions is constructed. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 128–137, July, 1997. Translated by V. N. Dubrovsky     

Uniform harmonic approximation with continuous extension to the boundary

Let Ω be an open set in ℝ ⁿ andE be a relatively closed subset of Ω. Further, letC _e(E) be the collection of real-valued continuous functions onE which extend continuously to the closure ofE in ℝ ⁿ . We characterize those pairs (Ω,E) which have the following property: every function inC _e(E) which is harmonic onE ⁰ can be uniformly approximated onE by functions which are harmonic on Ω and whose restrictions toE belong toC _e(E).     

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 brouwer fixed point theorem and tetragon with all vertexes in a surface

LetD be a disc with radiusr in the Euclidean plane ℝ², and letF be a Lipschitz continuous real valued function onD. SupposeA ₁ A ₂₁ A ₃ A ₄ is an isosceles trapezoid with lengths of edges not greater thanr, and ∠A ₁ A ₂₁ A ₃ = α≤π/2 By means of the Brouwer fixed point theorem, it is proved that ifF has a Lipschitz constant λ≤min{1, tgα}, then there exist four coplanar points in the surfaceM = {(x, y, F(x, y))∈ℝ³:(x, y)ℝ} which span a tetragon congruent toA ₁ A ₂₁ A ₃ A ₄. In addition, some further problems are discussed. Project supported by the National Natural Science Foundation of China (Grant No. 19231201).     

Estimates of the periods of periodic points for nonexpansive operators

Suppose thatE is a finite-dimensional Banach space with a polyhedral norm ‖·‖, i.e., a norm such that the unit ball inE is a polyhedron. ℝ ⁿ with the sup norm or ℝ ⁿ with thel ₁-norm are important examples. IfD is a bounded set inE andT:D→D is a map such that ‖T(y)−T(z)‖≤ ‖y−z‖ for ally andz inE, thenT is called nonexpansive with respect to ‖·‖, and it is known that for eachx ∈D there is an integerp=p(x) such that lim _j→∞ T ^jp (x) exists. Furthermore, there exists an integerN, depending only on the dimension ofE and the polyhedral norm onE, such thatp(x)≤N: see [1,12,18,19] and the references to the literature there. In [15], Scheutzow has raised a question about the optimal choice ofN whenE=ℝ ⁿ ,D=K ⁿ , the set of nonnegative vectors in ℝ ⁿ , and the norm is thel ₁-norm. We provide here a reasonably sharp answer to Scheutzow’s question, and in fact we provide a systematic way to generate examples and use this approach to prove that our estimates are optimal forn≤24. See Theorem 2.1, Table 2.1 and the examples in Section 3. As we show in Corollary 2.3, these results also provide information about the caseD=ℝ ⁿ , i.e.,T:ℝ ⁿ →ℝ ⁿ isl ₁-nonexpansive. In addition, it is conjectured in [12] thatN=2 ⁿ whenE=ℝ ⁿ and the norm is the sup norm, and such a result is optimal, if true. Our theorems here show that a sharper result is true for an important subclass of nonexpansive mapsT:(ℝ ⁿ ,‖ · ‖_∞)→(ℝ ⁿ ,‖ · ‖_∞). Partially supported by NSF DMS89-03018.     

Adjoints of semigroups acting on vector-valued function spaces

LetT(t) be the translation group onY=C ₀(ℝ×K)=C ₀(ℝ)⊗C(K),K compact Hausdorff, defined byT(t)f(x, y)=f(x+t, y). In this paper we give several representations of the sun-dialY ^⊙ corresponding to this group. Motivated by the solution of this problem, viz.Y ^⊙=L ¹(ℝ)⊗M(K), we develop a duality theorem for semigroups of the formT ₀(t)⊗id on tensor productsZ⊗X of Banach spaces, whereT ₀(t) is a semigroup onZ. Under appropriate compactness assumptions, depending on the kind of tensor product taken, we show that the sun-dial ofZ⊗X is given byZ ^⊙⊗X*. These results are applied to determine the sun-dials for semigroups induced on spaces of vector-valued functions, e.g.C ₀(Ω;X) andL ^p (μ;X). This paper was written during a half-year stay at the Centre for Mathematics and Computer Science CWI in Amsterdam. I am grateful to the CWI and the Dutch National Science Foundation NWO for financial support.     

Regressions and monotone chains: A ramsey-type extremal problem for partial orders

Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx ₁<x ₂ <...<x _k such thatg(x ₁)≦g(x ₂)≦...≦g(x _k ). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1) ^k . Dedicated to Paul Erdős on his seventieth birthday Research supported in part by the National Science Foundation under ISP-80-11451.     

An ergodic theorem on Banach lattices

E is a Banach lattice that is weakly sequentially complete and has a weak unitu. TLf _n=ϕ means that the infimum of |f _n−ϕ| andu converges strongly to zero.T is a positive contraction operator onE andA _n=(1/n)(I+T+...+T ⁿ⁻¹). Without an additional assumption onE, the “truncated limit” TLA _nf need not exist forf inE. This limit exists for eachf ifE satisfies the following additional assumption (C): For everyf inE ₊ and for every numberα>0, there is a numberβ=β(f, α) such that ifg is inE ₊, ‖g‖≦1, 0≦f′≦f and ‖f′‖>α then ‖f′+g‖≧‖g‖+β. Research of this author is partially supported by NSERC Grant A3974. Research of this author is partially supported by NSF Grant 8301619.     

A family of (n−1)! Diagonal polynomial orders ofN n

Letn>0 be an element of the setN of nonnegative integers, and lets(x)=x ₁+...+x _n , forx=(x ₁, ...,x _n ) N _n . Adiagonal polynomial order inN _n is a bijective polynomialp:N _n N (with real coefficients) such that, for allx,y N _n ,p(x)<p(y) whenevers(x)<s(y). Two diagonal polynomial orders areequivalent if a relabeling of variables makes them identical. For eachn, Skolem (1937) found a diagonal polynomial order. Later, Morales and Lew (1992) generalized this polynomial order, obtaining a family of 2 ^n–2 (n>1) inequivalent diagonal polynomial orders. Here we present, for eachn>0, a family of (n – 1)! diagonal polynomial orders, up to equivalence, which contains the Morales and Lew diagonal orders.     

Absolute continuities of exit measures for superdiffusions

Suppose X= X_t, X_T, P_μis a superdiffusion in ℝ^d with general branching mechanism ψ and general branching rate functionA. We discuss conditions onA to guarantee that the exit measure X_TL of the superdiffusionX from bounded smooth domains in ℝ^d have absolutely continuous states.     

Disjointness in Partially Ordered Vector Spaces

A notion of disjointness in arbitrary partially ordered vector spaces is introduced by calling two elements x and y disjoint if the set of all upper bounds of x + y and −x − y equals the set of all upper bounds of x − y and −x + y. Several elementary properties are easily observed. The question whether the disjoint complement of a subset is a linear subspace appears to be more difficult. It is shown that in directed Archimedean spaces disjoint complements are always subspaces. The proof relies on theory on order dense embedding in vector lattices. In a non-Archimedean directed space even the disjoint complement of a singleton may fail to be a subspace. According notions of disjointness preserving operator, band, and band preserving operator are defined and some of their basic properties are studied.     

Product recurrent properties, disjointness and weak disjointness

Let F be a collection of subsets of ℝ₊ and (X, T) be a dynamical system; x ∈ X is F-recurrent if for each neighborhood U of x, {n ∈ ℝ₊: T ⁿ x ∈ U} ∈ F; x is F-product recurrent if (x, y) is recurrent for any F-recurrent point y in any dynamical system (Y, S). It is well known that x is {infinite}-product recurrent if and only if it is minimal and distal. In this paper it is proved that the closure of a {syndetic}-product recurrent point (i.e., weakly product recurrent point) has a dense minimal points; and a {piecewise syndetic}-product recurrent point is minimal. Results on product recurrence when the closure of an F-recurrent point has zero entropy are obtained.     

Ergodic averages on circles

LetT ₁ andT ₂ be commuting invertible ergodic measure preserving flows on a probability space (X, A, μ). For t = (u,ν) ∈ ℝ², letT ^t =T ₁ ^u T ₂ ^v . LetS ¹ denote the unit circle in ℝ² and σ the rotation invariant unit measure on it. Then, forf∈Lp(X) withp>2, the averagesA _t f(x) = ∫ _s ¹ f(T ^ts x)σ(ds) conver the integral off for a. e.x, ast tends to 0 or infinity. This extends a result of R. Jones [J], who treated the case of three or more dimensions.     

Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

Wavelet expansions of fractal measures

Let μ be a measure on ℝⁿ that satisfies the estimate μ(B _r(x))≤cr ^α for allx ∈ ℝⁿ and allr ≤ 1 (B _r(x) denotes the ball of radius r centered atx. Let ϕ _j,k ^(ɛ) (x)=2 ^nj2ϕ^(ɛ)(2 ^j x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤⁿ, and ∈ ∈E, a finite set, and letP _j (T)=Σ_ɛ,k <T,ϕ _j,k ^(ɛ) >ϕ _j,k ^(ɛ) denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls ^p(dμ) for 1 ≤p ≤ ∞, we show thatP _j(f dμ) ∈ L^p(dx) and ||P _j (fdμ)||L _p(dx)≤c2 ^{j((n-α)/p′))}||f||L _p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P _j (fdμ)||L _p(dx) under more restrictive hypotheses. Communicated by Guido Weiss     

Submanifolds in pseudo-Euclidean spaces satisfying the condition Δx=Ax+B

In this paper we study pseudo-Riemannian submanifolds in ℝ _n ^+k/t satisfying the condition Δx=Ax+B, whereA is an endomorphism of ℝ _n ^+k/t andB is a constant vector in ℝ _n ^+k/t . We give a characterization theorem whenA is a self-adjoint endomorphism. As for hypersurfaces we are able to obtain a classification theorem for any endomorphismA. Supported by a FPI Grant, DGICYT, 1990. Partially supported by a DGICYT Grant No. PS87-0115-C03-03.     

Extremal properties of contraction semigroups onc 0

For any complex Banach spaceX, letJ denote the duality mapping ofX. For any unit vectorx inX and any (C ₀) contraction semigroup (T _t ) _t>0 onX, Baillon and Guerre-Delabriere proved that ifX is a smooth reflexive Banach space and if there isx ^*∈J(x) such that ÷〈(T(t)x, J(x)〈÷→1 ast→∞, then there is a unit vectory∈X which is an eigenvector of the generatorA of (T _t ) _t>0 associated with a purely imaginary eigenvalue. They asked whether this result is still true ifX is replaced byc ₀. In this article, we show the answer is negative Partial results of this paper were obtained when the author attended the International Conference of Convexity at the University of Marne-La-Vallée. He would like to express his gratitude for the kind hospitality offered to him. He would also like to thank Profs. Goldstein and Jamison for their valuable suggestions.     

Retracing the residual curve of a Lyapunov equation solver

Let A∈ℝ ^n×n and let B∈ℝ ^n×p and consider the Lyapunov matrix equation AX+XA ^T +BB ^T =0. If A+A ^T <0, then the extended Krylov subspace method (EKSM) can be used to compute a sequence of low rank approximations of X. In this paper we show how to construct a symmetric negative definite matrix A and a column vector B, for which the EKSM generates a predetermined residual curve.     

Surjective partial differential operators on real analytic functions defined on open convex sets

Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on a covex open set Ω⊂ℝ ⁿ . Let L(P _m ) denote the localizations at ∞ (in the sense of H?rmander) of the principal part P _m . Then Q(x+iτN)≠ 0 for (x,τ)∈ℝ ⁿ ×(ℝ\{ 0}) for any Q∈L(P _m ) if N is a normal to δΩ which is noncharacteristic for Q. Under additional assumptions this implies that P _m must be locally hyperbolic. Received: 24 January 2000     

Automorphisms and isotone self-maps of ordered sets with top and bottom

For an ordered setP letP ^P denote the set of all isotone self-maps on P, that is, all mapsf fromP toP such thatxy impliesf(x)f(y), and let Aut (P) the set of all automorphisms onP, that is, all bijective isotone self-maps inP ^P . We establish an inequality relating ¦P ^P ¦ and ¦Aut(P)¦ in terms of the irreducibles ofP. As a straightforward corollary, we show that Rival and Rutkowski's automorphism conjecture is true for lattices. It is also true for ordered sets with top and bottom whose covering graphs are planar.Supported in part by NSERC (Grant no. A2507).Supported under an NSERC International Research Fellowship.