Oscillatory behavior of linear matrix Hamiltonian systems

We establish some new oscillation criteria for the matrix linear Hamiltonian system X ′ = A (t)X + B (t)Y, Y ′ = C (t)X –A *(t)Y by using a new function class X and monotone functionals on a suitable matrix space. In doing so, many existing results are generalized and improved. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

C-Cosine Functions and the Abstract Cauchy Problem, II

For a bounded linear injectionCon a Banach spaceXand a closed linear operatorA : D(A) X → Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t) = Au(t),t R,u(0) = Cx,u′(0) = Cy, has a unique strong solution for everyx,y D(A) if and only if (2)A₁ = AD(A²) generates aC₁-cosine function onX₁(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC₁ = CX₁. Under the assumption thatAis densely defined andC⁻¹AC = A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t) = Av(t) + C(x + ty) + ∫^t₀ Cg(r) dr,t R,v(0) = v′(0) = 0, has a unique strong solution for everyg L¹_locandx, y X; (5) the problemw″(t) = Aw(t) + Cg(t),t R,w(0) = Cx,w′(0) = Cy, has a unique weak solution for everyg L¹_locandx, y X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A + Bis also a generator.     

Viability for nonlinear multi-valued reaction–diffusion systems

In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, ${\mathcal{K}}In this paper we consider a nonlinear evolution reaction–diffusion system governed by multi-valued perturbations of m-dissipative operators, generators of nonlinear semigroups of contractions. Let X and Y be real Banach spaces, K{\mathcal{K}} be a nonempty and locally closed subset in \mathbbR ×X×Y, A:D(A) í X\rightsquigarrow X, B:D(B) í Y\rightsquigarrow Y{\mathbb{R} \times X\times Y,\, A:D(A)\subseteq X\rightsquigarrow X, B:D(B)\subseteq Y\rightsquigarrow Y} two m-dissipative operators, F:K ? X{F:\mathcal{K} \rightarrow X} a continuous function and G:K \rightsquigarrow Y{G:\mathcal{K} \rightsquigarrow Y} a nonempty, convex and closed valued, strongly-weakly upper semi-continuous (u.s.c.) multi-function. We prove a necessary and a sufficient condition in order that for each (t,x,h) ? K{(\tau,\xi,\eta)\in \mathcal{K}}, the next system

{ lc u￠(t) ? Au(t)+F(t,u(t),v(t))    t 3 tv￠(t) ? Bv(t)+G(t,u(t),v(t))    t 3 tu(t)=x,    v(t)=h, \left\{ \begin{array}{lc} u'(t)\in Au(t)+F(t,u(t),v(t))\quad t\geq\tau \\ v'(t)\in Bv(t)+G(t,u(t),v(t))\quad t\geq\tau \\ u(\tau)=\xi,\quad v(\tau)=\eta, \end{array} \right.      4. Some extensions of the Kantorovich inequality and statistical applications      C. G. Khatri  C. Radhakrishna Rao 《Journal of multivariate analysis》1981,11(4):498-505 Kantorovich gave an upper bound to the product of two quadratic forms, (X′AX) (X′A−1X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants |X′AX| |X′A−1X| where X is n × k matrix such that X′X = Ik. In this paper we determine the bounds for the traces and determinants of matrices of the type X′AYY′A−1X, X′B2X(X′BCX)−1 X′C2X(X′BCX)−1 where X and Y are n × k matrices such that X′X = Y′Y = Ik and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation.      5. A generalization of a theorem on biseparating maps      Karim Boulabiar  Gerard Buskes  Melvin Henriksen   《Journal of Mathematical Analysis and Applications》2003,280(2):334-349 In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that (a)=ωψ(a) for all aA. We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.      6. Correlations of functions of normal variables      Sam Gutmann 《Journal of multivariate analysis》1978,8(4):573-578 Let (X1, X2,…, Xk, Y1, Y2,…, Yk) be multivariate normal and define a matrix C by Cij = cov(Xi, Yj). If (i) (X1,…, Xk) = (Y1,…, Yk) and (ii) C is symmetric positive definite, then 0 < varf(X1,…, Xk) < ∞ corr(f(X1,…, Xk),f(Y1,…, Yk)) > 0. Condition (i) is necessary for the conclusion. The sufficiency of (i) and (ii) follows from an infinite-dimensional version, which can also be applied to a pair of jointly normal Brownian motions.      7. On Ulyanov inequalities in Banach spaces and semigroups of linear operators      W. Trebels  U. Westphal   《Journal of Approximation Theory》2009,160(1-2):154 Let X,Y be Banach spaces and {T(t):t≥0} be a consistent, equibounded semigroup of linear operators on X as well as on Y; it is assumed that {T(t)} satisfies a Nikolskii type inequality with respect to X and Y:T(2t)fY(t)T(t)fX. Then an abstract Ulyanov type inequality is derived between the (modified) K-functionals with respect to (X,DX((-A)α)) and (Y,DY((-A)α)),α>0, where A is the infinitesimal generator of {T(t)}. Particular choices of X,Y are Lorentz–Zygmund spaces, of {T(t)} are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case , are partly covered.      8. Absolute continuity of information channels      Hisaharu Umegaki 《Journal of multivariate analysis》1974,4(4):382-400 Let [X, v, Y] be an abstract information channel with the input X = (X, ) and the output Y = (Y, ) which are measurable spaces, and denote by L(Y) = L(Y, ) the Banach space of all bounded signed measures with finite total variation as norm. The channel distribution ν(·,·) is considered as a function defined on (X, ) and valued in L(Y). It will be proved that, if the measurable space (Y, ) is countably generated, then the is a strongly measurable function from X into L(Y) if and only if there exists a probability measure μ on (Y, ) which dominates every measure ν(x, ·) (x X). Furthermore, under this condition, the Radon-Nikodym derivative ν(x, dy)/μ(dy) is jointly measurable with respect to the product measure space (X, , m) (Y, , μ) where m is any but fixed probability measure of (X, ). As an application, it will be shown that the channel given as above is uniformly approximated by channels of Hibert-Schmidt type.      9. Continuity of spectrum and approximate point spectrum on operator matrices      S. Sánchez-Perales  S.V. Djordjevi? 《Journal of Mathematical Analysis and Applications》2011,378(1):289-294 Let X and Y be given Banach spaces. For A∈B(X), B∈B(Y) and C∈B(Y,X), let MC be the operator defined on X⊕Y by . In this paper we give conditions for continuity of τ at MC through continuity of τ at A and B, where τ can be equal to the spectrum or approximate point spectrum.      10. Canonical forms of Borel-measurable mappings Δ: [ω] →      Hans Jürgen Prmel  Bernd Voigt 《Journal of Combinatorial Theory, Series A》1985,40(2) We show that for every Borel-measurable mapping Δ: [ω]ω → there exists A [ω]ω and there exists a continuous mapping Γ: [A]ω → [A]ω with Γ(X) X such that for all X, Y [A]ω it follows that Δ(X) = Δ(Y) if Γ(X) = Γ(Y). In a sense, this is generalization of the Erdös-Rado canonization theorem      11. Some cardinal invariants on the space      Süleyman nal  etin Vural 《Topology and its Applications》2005,150(1-3):255-265 Let Cα(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X such that closed under finite unions. We define two properties (E1) and (E2) on the triple (α,X,Y) which yield new equalities and inequalities between some cardinal invariants on Cα(X,Y) and some cardinal invariants on the spaces X, Y such as: Theorem If Y is an equiconnected space with a base consisting of φ-convex sets, then for each fC(X,Y), χ(f,Cα(X,Y))=αa(X).we(f(X)).Corollary Let Y be a noncompact metric space and let the triple (α,X,Y) satisfy (E1). The following are equivalent: (i) Cα(X,Y) is a first-countable space. (ii) π-character of the space Cα(X,Y) is countable. (iii) Cα(X,Y) is of pointwise countable type. (iv) There exists a compact subset K of Cα(X,Y) such that π-character of K in the space Cα(X,Y) is countable. (v) αa(X)0. (vi) Cα(X,Y) is metrizable. (vii) Cα(X,Y) is a q-space. (viii) There exists a sequence of nonempty open subset of Cα(X,Y) such that each sequence with gnOn for each nω, has a cluster point in Cα(X,Y). Keywords: Function space; Network; Character; Equiconnected; Arens number      12. C-semigroups and strongly continuous semigroups      Ralph deLaubenfels 《Israel Journal of Mathematics》1993,81(1-2):227-255 We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA| Y , the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖[Im(C)]≡‖C −1 x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B| X andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true. We construct fractional powers of generators of boundedC-semigroups. We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was supported by an Ohio University Research Grant.      13. Global solvability for abstract semilinear evolution equations      Hirokazu Oka  Naoki Tanaka 《Mathematische Nachrichten》2010,283(8):1171-1193 The Cauchy problem for the abstract semilinear evolution equation u′(t) = Au (t) + B (u (t)) + C (u (t)) is discussed in a general Banach space X. Here A is the so‐called Hille‐Yosida operator in X, B is a differentiable operator from D (A) into X, and C is a locally Lipschitz continuous operator from D (A) into itself. A vectorvalued functional defined only on X is used and appropriate conditions on the nonlinear operators B and C are imposed so that a vector‐valued functional defined on the domain of the operator A may be constructed in order to specify the growth of a global solution. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)      14. Chaos caused by a strong-mixing measure-preserving transformation      Jincheng Xiong  Ercai Chen 《中国科学A辑(英文版)》1997,40(3):253-260 The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer i such that limi→∞ f r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.      15. On the location of fixed points on pairs of spaces      Helga Schirmer 《Topology and its Applications》1988,30(3) Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).      16. Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups      MiaoLI QuanZHENG 《数学学报(英文版)》2004,20(5):821-828 Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.      17. Special uniform approximations of continuous vector-valued functions. Part II: special approximations in CX(T)CY(S)      Vlad Timofte   《Journal of Approximation Theory》2003,123(2):270-275 In this paper, which is a continuation of Timofte (J. Approx. Theory 119 (2002) 291–299, we give special uniform approximations of functions from CXY(T×S) and C∞(T×S,XY) by elements of the tensor products CX(T)CY(S), respectively C0(T,X)C0(S,Y), for topological spaces T,S and Γ-locally convex spaces X,Y (all four being Hausdorff).      18. Domains of attraction of nonnormal operator-stable laws      Mark M. Meerschaert 《Journal of multivariate analysis》1986,19(2) A sequence of independent, identically distributed random vectors X1, X2, X3,… is said to belong to the domain of attraction of a random vector Y is there exist linear operators An and constant vectors bn such that An(X1,…, Xn)+bn converges in distribution to Y. We present a simple, necessary, and sufficient condition for the existence of such An, Bn in the case where Y has no normal component.      19. Linear programming and operator means      T. D. Morley 《Advances in Applied Mathematics》1989,10(4) The shorted operator, the geometric mean, and the cascade limit are all examples of operations that are of the form sup{X¦C + K X ≥ 0}, where K X denotes the Kronecker product of the matrix K with the matrix X, K is a given n by n self-adjoint matrix, and C is a given positive semidefinite matrix. The supremum is taken with respect to the partial order generated by the positive semidefinite matrices. In all of the above examples the matrix K has exactly one negative eigenvalue. We show by linear programming techniques that if K has this property, and Xmax = sup{X¦C + K X ≥ 0}, then (Xmaxc, c) = inf tr(AY), subject to: −∑i,j = 1nkijYij ≥ cc*, Y = {Yij)i,j = 1n ≥ 0} In the case of the geometric mean A#B of two positive semidefinite matrices, this implies the new result that (A#Bc, c) = inf{tr(AY11 + BY22¦Y12 + Y21 ≥ cc*, Y ≥ 0}.      20. The Jacobian conjecture,the <Emphasis Type="Italic">d</Emphasis>-inversion approximation and its natural boundary      R. Peretz 《Journal of Mathematical Sciences》2010,168(3):428-436 Let F ? \mathbbC[ X, Y ]2 F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} be an étale map of degree deg F = d. An étale map G ? \mathbbC[ X,Y ]2 G \in \mathbb{C}{\left[ {X,Y} \right]^2} is called a d-inverse approximation of F if deg G ≤ d and F ◦ G =(X + A(X, Y), Y + B(X, Y)) and G ◦ F =(X + C(X, Y), Y + D(X, Y)), where the orders of the four polynomials A, B, C, and D are greater than d. It is a well-known result that every \mathbbC2 {\mathbb{C}^2} -automorphism F of degree d has a d-inverse approximation, namely, F −1. In this paper, we prove that if F is a counterexample of degree d to the two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles.

Some extensions of the Kantorovich inequality and statistical applications

Kantorovich gave an upper bound to the product of two quadratic forms, (X′AX) (X′A⁻¹X), where X is an n-vector of unit length and A is a positive definite matrix. Bloomfield, Watson and Knott found the bound for the product of determinants |X′AX| |X′A⁻¹X| where X is n × k matrix such that X′X = I_k. In this paper we determine the bounds for the traces and determinants of matrices of the type X′AYY′A⁻¹X, X′B²X(X′BCX)⁻¹ X′C²X(X′BCX)⁻¹ where X and Y are n × k matrices such that X′X = Y′Y = I_k and A, B, C are given matrices satisfying some conditions. The results are applied to the least squares theory of estimation.     

A generalization of a theorem on biseparating maps

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that

The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that

(a)=ωψ(a) for all aA.

We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.     

Correlations of functions of normal variables

Let (X₁, X₂,…, X_k, Y₁, Y₂,…, Y_k) be multivariate normal and define a matrix C by C_ij = cov(X_i, Y_j). If (i) (X₁,…, X_k) = (Y₁,…, Y_k) and (ii) C is symmetric positive definite, then 0 < varf(X₁,…, X_k) < ∞ corr(f(X₁,…, X_k),f(Y₁,…, Y_k)) > 0. Condition (i) is necessary for the conclusion. The sufficiency of (i) and (ii) follows from an infinite-dimensional version, which can also be applied to a pair of jointly normal Brownian motions.     

On Ulyanov inequalities in Banach spaces and semigroups of linear operators

Let X,Y be Banach spaces and {T(t):t≥0} be a consistent, equibounded semigroup of linear operators on X as well as on Y; it is assumed that {T(t)} satisfies a Nikolskii type inequality with respect to X and Y:T(2t)f_Y(t)T(t)f_X. Then an abstract Ulyanov type inequality is derived between the (modified) K-functionals with respect to (X,D_X((-A)^α)) and (Y,D_Y((-A)^α)),α>0, where A is the infinitesimal generator of {T(t)}. Particular choices of X,Y are Lorentz–Zygmund spaces, of {T(t)} are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case , are partly covered.     

Absolute continuity of information channels

Let [X, v, Y] be an abstract information channel with the input X = (X, ) and the output Y = (Y, ) which are measurable spaces, and denote by L(Y) = L(Y, ) the Banach space of all bounded signed measures with finite total variation as norm. The channel distribution ν(·,·) is considered as a function defined on (X, ) and valued in L(Y). It will be proved that, if the measurable space (Y, ) is countably generated, then the is a strongly measurable function from X into L(Y) if and only if there exists a probability measure μ on (Y, ) which dominates every measure ν(x, ·) (x X). Furthermore, under this condition, the Radon-Nikodym derivative ν(x, dy)/μ(dy) is jointly measurable with respect to the product measure space (X, , m) (Y, , μ) where m is any but fixed probability measure of (X, ). As an application, it will be shown that the channel given as above is uniformly approximated by channels of Hibert-Schmidt type.     

Continuity of spectrum and approximate point spectrum on operator matrices

Let X and Y be given Banach spaces. For A∈B(X), B∈B(Y) and C∈B(Y,X), let MC be the operator defined on X⊕Y by . In this paper we give conditions for continuity of τ at MC through continuity of τ at A and B, where τ can be equal to the spectrum or approximate point spectrum.     

Canonical forms of Borel-measurable mappings Δ: [ω] →

We show that for every Borel-measurable mapping Δ: [ω]^ω → there exists A [ω]^ω and there exists a continuous mapping Γ: [A]^ω → [A]^ω with Γ(X) X such that for all X, Y [A]^ω it follows that Δ(X) = Δ(Y) if Γ(X) = Γ(Y). In a sense, this is generalization of the Erdös-Rado canonization theorem     

Some cardinal invariants on the space

Let C_α(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X such that closed under finite unions. We define two properties (E1) and (E2) on the triple (α,X,Y) which yield new equalities and inequalities between some cardinal invariants on C_α(X,Y) and some cardinal invariants on the spaces X, Y such as: Theorem If Y is an equiconnected space with a base consisting of φ-convex sets, then for each fC(X,Y), χ(f,C_α(X,Y))=αa(X).w_e(f(X)).Corollary Let Y be a noncompact metric space and let the triple (α,X,Y) satisfy (E1). The following are equivalent:

(i) C_α(X,Y) is a first-countable space.

(ii) π-character of the space C_α(X,Y) is countable.

(iii) C_α(X,Y) is of pointwise countable type.

(iv) There exists a compact subset K of C_α(X,Y) such that π-character of K in the space C_α(X,Y) is countable.

(v) αa(X)₀.

(vi) C_α(X,Y) is metrizable.

(vii) C_α(X,Y) is a q-space.

(viii) There exists a sequence of nonempty open subset of C_α(X,Y) such that each sequence with g_nO_n for each nω, has a cluster point in C_α(X,Y).

C-semigroups and strongly continuous semigroups

We show that, whenA generates aC-semigroup, then there existsY such that [M(C)] →Y →X, andA| _Y , the restriction ofA toY, generates a strongly continuous semigroup, where ↪ means “is continuously embedded in” and ‖x‖_[Im(C)]≡‖C ⁻¹ x‖. There also existsW such that [C(W)] →X →W, and an operatorB such thatA=B| _X andB generates a strongly continuous semigroup onW. If theC-semigroup is exponentially bounded, thenY andW may be chosen to be Banach spaces; in general,Y andW are Frechet spaces. If ρ(A) is nonempty, the converse is also true. We construct fractional powers of generators of boundedC-semigroups. We would like to thank R. Bürger for sending preprints, and the referee for pointing out reference [37]. This research was supported by an Ohio University Research Grant.     

Global solvability for abstract semilinear evolution equations

The Cauchy problem for the abstract semilinear evolution equation u^′(t) = Au (t) + B (u (t)) + C (u (t)) is discussed in a general Banach space X. Here A is the so‐called Hille‐Yosida operator in X, B is a differentiable operator from D (A) into X, and C is a locally Lipschitz continuous operator from D (A) into itself. A vectorvalued functional defined only on X is used and appropriate conditions on the nonlinear operators B and C are imposed so that a vector‐valued functional defined on the domain of the operator A may be constructed in order to specify the growth of a global solution. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chaos caused by a strong-mixing measure-preserving transformation

The chaos caused by a strong-mixing preserving transformation is discussed and it is shown that for a topological spaceX satisfying the second axiom of countability and for an outer measurem onX satisfying the conditions: (i) every non-empty open set ofX ism-measurable with positivem-measure; (ii) the restriction ofm on Borel σ-algebra ℬ(X) ofX is a probability measure, and (iii) for everyY⊂X there exists a Borel setB⊂ℬ(X) such thatB⊃Y andm(B) =m(Y), iff:X→X is a strong-mixing measure-preserving transformation of the probability space (X, ℬ(X),m), and if {m}, is a strictly increasing sequence of positive integers, then there exists a subsetC⊂X withm (C) = 1, finitely chaotic with respect to the sequence {m _i}, i.e. for any finite subsetA ofC and for any mapF:A→X there is a subsequencer _i such that lim_i→∞ f ^r i(a) =F(a) for anya ∈A. There are some applications to maps of one dimension. the National Natural Science Foundation of China.     

On the location of fixed points on pairs of spaces

Let f: (X, A)→(X, A) be an admissible selfmap of a pair of metrizable ANR's. A Nielsen number of the complement Ñ(f; X, A) and a Nielsen number of the boundary ñ(f; X, A) are defined. Ñ(f; X, A) is a lower bound for the number of fixed points on C1(X - A) for all maps in the homotopy class of f. It is usually possible to homotope f to a map which is fixed point free on Bd A, but maps in the homotopy class of f which have a minimal fixed point set on X must have at least ñ(f; X, A) fixed points on Bd A. It is shown that for many pairs of compact polyhedra these lower bounds are the best possible ones, as there exists a map homotopic to f with a minimal fixed point set on X which has exactly Ñ(f; X - A) fixed points on C1(X−A) and ñ(f; X, A) fixed points on Bd A. These results, which make the location of fixed points on pairs of spaces more precise, sharpen previous ones which show that the relative Nielsen number N(f; X, A) is the minimum number of fixed points on all of X for selfmaps of (X, A), as well as results which use Lefschetz fixed point theory to find sufficient conditions for the existence of one fixed point on C1(X−A).     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

Special uniform approximations of continuous vector-valued functions. Part II: special approximations in CX(T)CY(S)

In this paper, which is a continuation of Timofte (J. Approx. Theory 119 (2002) 291–299, we give special uniform approximations of functions from C_XY(T×S) and C_∞(T×S,XY) by elements of the tensor products C_X(T)C_Y(S), respectively C₀(T,X)C₀(S,Y), for topological spaces T,S and Γ-locally convex spaces X,Y (all four being Hausdorff).     

Domains of attraction of nonnormal operator-stable laws

A sequence of independent, identically distributed random vectors X₁, X₂, X₃,… is said to belong to the domain of attraction of a random vector Y is there exist linear operators A_n and constant vectors b_n such that A_n(X₁,…, X_n)+b_n converges in distribution to Y. We present a simple, necessary, and sufficient condition for the existence of such A_n, B_n in the case where Y has no normal component.     

Linear programming and operator means

The shorted operator, the geometric mean, and the cascade limit are all examples of operations that are of the form sup{X¦C + K X ≥ 0}, where K X denotes the Kronecker product of the matrix K with the matrix X, K is a given n by n self-adjoint matrix, and C is a given positive semidefinite matrix. The supremum is taken with respect to the partial order generated by the positive semidefinite matrices. In all of the above examples the matrix K has exactly one negative eigenvalue. We show by linear programming techniques that if K has this property, and X_max = sup{X¦C + K X ≥ 0}, then (X_maxc, c) = inf tr(AY), subject to: −∑_{i,j = 1}ⁿk_ijY_ij ≥ cc^*, Y = {Y_ij)_{i,j = 1}ⁿ ≥ 0} In the case of the geometric mean A#B of two positive semidefinite matrices, this implies the new result that (A#Bc, c) = inf{tr(AY₁₁ + BY₂₂¦Y₁₂ + Y₂₁ ≥ cc^*, Y ≥ 0}.     

The Jacobian conjecture,the <Emphasis Type="Italic">d</Emphasis>-inversion approximation and its natural boundary

Let F ? \mathbbC[ X, Y ]² F \in \mathbb{C}{\left[ {X,\,Y} \right]^2} be an étale map of degree deg F = d. An étale map G ? \mathbbC[ X,Y ]² G \in \mathbb{C}{\left[ {X,Y} \right]^2} is called a d-inverse approximation of F if deg G ≤ d and F ◦ G =(X + A(X, Y), Y + B(X, Y)) and G ◦ F =(X + C(X, Y), Y + D(X, Y)), where the orders of the four polynomials A, B, C, and D are greater than d. It is a well-known result that every \mathbbC² {\mathbb{C}^2} -automorphism F of degree d has a d-inverse approximation, namely, F ⁻¹. In this paper, we prove that if F is a counterexample of degree d to the two-dimensional Jacobian conjecture, then F has no d-inverse approximation. We also give few consequences of this result. Bibliography: 18 titles.