On the invertibility of the operator <Emphasis Type="Italic">d/dt</Emphasis> + <Emphasis Type="Italic">A</Emphasis> in certain functional spaces

We prove that the operator d/dt + A constructed on the basis of a sectorial operator A with spectrum in the right half-plane of ℂ is continuously invertible in the Sobolev spaces W _p ¹ (ℝ, D _α), α ≥ 0. Here, D _α is the domain of definition of the operator A ^α and the norm in D _α is the norm of the graph of A ^α. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1020–1025, August, 2007.     

Completely positive maps and*-isomorphism ofC *-algebras II

LetA, B be unitalC ^*-algebras,D _A ¹ the set of all completely positive maps ϕ fromA toM _n (C), with Tr ϕ(I)≤1(n≥3). If Ψ is an α-invariant affine homeomorphism betweenD _A ¹ andD _B ¹ with Ψ (0)=0, thenA is^*-isomorphic toB. Obtained results can be viewed as non-commutative Kadison-Shultz theorems. This work is supported by the National Natural Science Foundation of China.     

Nondegenerate cohomology pairing for transitive Lie algebroids,characterization

The Evens-Lu-Weinstein representation (Q _A , D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q _A ^or , D^or) by tensoring by orientation flat line bundle, Q _A ^or =Q_A⊗or (M) and D ^or=D⊗∂ _A ^or . It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q _A ^or , D^or) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂ _A ^or ) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.     

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).     

Some Simple Estimates for the Singular Values of Matrices

Abstract We first provide a simple estimate for ||A~(-1)||_∞ and ||A~(-1)||_1 of a strictly diagonally dominant matrixA. On the Basis of the result, we obtain an estimate for the smallest singular value of A. Secondly, by scalingwith a positive diagonal matrix D, we obtain some simple estimates for the smallest singular value of an H-matrix, which is not necessarily positive definite. Finally, we give some examples to show the effectiveness ofthe new bounds.     

泛函方程组的解析解

§ 1　IntroductionThe Feigenbaum functional equation plays an importantrole in the theory concerninguniversal properties of one-parameter families of maps of the interval that has the formf2 (λx) +λf(x) =0 ,0 <λ=-f(1 ) <1 ,f(0 ) =1 ,(1 .1 )where f is a map ofthe interval[-1 ,1 ] into itself.Lanford[1 ] exhibited a computer-assist-ed proof for the existence of an even analytic solution to Eq.(1 .1 ) .It was shown in[2 ]that Eq.(1 .1 ) does not have an entire solution.Si[3] discussed the it…     

Generalized Smash Products

In this paper．we study the ring #(D．B)and obtain two very interesting results. First we prove in Theorem 3 that the category of rational left BU-modules is equivalent to both the category of #-rational left modules and the category of all(B．D)-Hopf modules BM^D．Cai and Chen have proved this result in the case B=D=A．Secondly they have proved that if A has a nonzero left integral then A#A^*rat is a dense subring of Endk(A)．We prove that #(A,A) is a dense subring of Endk(Q),where Q is a certain subspace of #(A．A)under the condition that the antipode is bijective(see Theorem18).This condition is weaker than the condition that A has a nonzero integral.It is well known the antipode is bijective in case A has a nonzero integral．Furthermore if A has nonzero left integral,Q can be chosen to be A(see Corollary 19)and #(A,A)is both left and right primitive.Thus A#A^*rat #(A,A)-Endk(A)．Moreover we prove that the left singular ideal of the ring #(A,A)is zero．A corollary of this is a criterion for A with nonzero left integral to be finite-dimensional,namely the ring #(A,A)has a finite uniform dimension．     

Global-in-time Uniform Convergence for Linear Hyperbolic-Parabolic Singular Perturbations

We consider the Cauchy problem εu^″ε ＋ δu′ε ＋ Auε = 0, uε（0） = uo, u′ε（0） = ul, where ε 〉 0, δ 〉 0, H is a Hilbert space, and A is a self-adjoint linear non-negative operator on H with dense domain D（A）. We study the convergence of （uε） to the solution of the limit problem ,δu＇ ＋ Au = 0, u（0） = u0. For initial data （u0, u1） ∈ D（A1/2）× H, we prove global-in-time convergence with respect to strong topologies. Moreover, we estimate the convergence rate in the case where （u0, u1）∈ D（A3/2） ∈ D（A1/2）, and we show that this regularity requirement is sharp for our estimates. We give also an upper bound for ｜u′ε（t）｜ which does not depend on ε.     

Isoperimetric inequalities on minimal submanifolds of space forms

For a domainU on a certaink-dimensional minimal submanifold ofS ⁿ orH ⁿ, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k ^k ω _k M (D) ^k-1 ≤Vol(∂D) ^k , where ω _k is the volume of the unit ball ofR ^k . Also, we prove that ifD is any domain on a minimal surface inS ₊ ⁿ (orH ⁿ, respectively), thenD satisfies an isoperimetric inequality2π A≤L ²+A² (2π A≤L²−A² respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH ⁿ, then(k−1) Vol(U)≤Vol(∂U). Supported in part by KME and GARC     

Spectra of Bipartite P- and Q-Polynomial Association Schemes

 Let Y=(X,{R _i }_0≤i≤D) denote a symmetric association scheme with D≥3, and assume Y is not an ordinary cycle. Suppose Y is bipartite P-polynomial with respect to the given ordering A ₀, A ₁,…, A _D of the associate matrices, and Q-polynomial with respect to the ordering E ₀, E ₁,…,E _D of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i)–(iv). (i)

Reducible polar representations

In this paper the reducible polar representations of the compact connected Lie groups are classified. It turns out that there only exist “interesting” reducible polar representations of Lie groups of the types A ₃, A ₃×T ¹, B ₃, B ₃×T ¹, D ₄, D ₄×T ¹ and D ₄×A ₁. Up to equivalence, there is just one such representation of the first four Lie groups, there are three reducible polar representations of D ₄ and six of D ₄×T ¹ and D ₄×A ₁, respectively. From this follows immediately the classification of the compact connected subgroups of SO(n) which act transitively on products of spheres. Received: 28 April 2000     

The Dichotomy Between Traces on d-sets F in R^n and the Density of D（R^n／Г） in Function Spaces

A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.     

On the restricted arc-connectivity of s-geodetic digraphs

For a strongly connected digraph D the minimum ,cardinality of an arc-cut over all arc-cuts restricted arc-connectivity λ′（D） is defined as the S satisfying that D - S has a non-trivial strong component D1 such that D - V（D1） contains an arc. Let S be a subset of vertices of D. We denote by w＋（S） the set of arcs uv with u ∈ S and v S, and by w-（S） the set of arcs uv with u S and v ∈ S. A digraph D = （V, A） is said to be λ′-optimal if λ′（D） =ξ′（D）, where ξ′（D） is the minimum arc-degree of D defined as ξ（D） = min {ξ′（xy） ： xy ∈ A}, and ξ′（xy） = min（｜ω＋（{x,y}）｜, ｜w-（{x,y}）｜, ｜w＋（x） ∪ w- （y） ｜, ｜w- （x） ∪ω＋ （y）｜}. In this paper a sufficient condition for a s-geodetic strongly connected digraph D to be λ′-optimal is given in terms of its diameter. Furthermore we see that the h-iterated line digraph Lh（D） of a s-geodetic digraph is λ′-optimal for certain iteration h.     

A characterization of going-down-ring pairs of commutative rings

SoientA ⊆B des anneaux (commutaifs et unitaires). On dit que (A,B) est une paire d’anneaux de going-down siD est un anneau de going-down pour tout anneauD tel queA ⊆D ⊆B. On preuve que (A,B) est une paire d’anneaux de going-down si et seulement siA[b ₁,b ₂] est un anneau de going-down pour toutb ₁,b ₂ εB.     

On directional convexity

Motivated by problems from calculus of variations and partial differential equations, we investigate geometric properties of D-convexity. A function f: R ^d → R is called D-convex, where D is a set of vectors in R ^d, if its restriction to each line parallel to a nonzero v ∈ D is convex. The D-convex hull of a compact set A ⊂ R ^d, denoted by co^D(A), is the intersection of the zero sets of all nonnegative D-convex functions that are zero on A. It also equals the zero set of the D-convex envelope of the distance function of A. We give an example of an n-point set A ⊂ R ² where the D-convex envelope of the distance function is exponentially close to zero at points lying relatively far from co ^D(A), showing that the definition of the D-convex hull can be very nonrobust. For separate convexity in R ³ (where D is the orthonormal basis of R ³), we construct arbitrarily large finite sets A with co ^D(A) ≠ A whose proper subsets are all equal to their D-convex hull. This implies the existence of analogous sets for rank-one convexity and for quasiconvexity on 3 × 3 (or larger) matrices. This research was supported by Charles University Grants No. 158/99 and 159/99.     

On almost-divided domains

Let B be a domain, Q a maximal ideal of B, π: B → B/Q the canonical surjection, D a subring of B/Q, and A:=π ⁻¹(D). If both B and D are almost-divided domains (resp., n-divided domains), then A = B × _B/Q D is an almost-divided domain (resp., an n-divided domain); the converse holds if B is quasilocal. If 2 ≤ d ≤ ∞, an example is given of an almost-divided domain of Krull dimension d which is not a divided domain.      

On <Emphasis Type="Italic">α</Emphasis>-regular archimedean <Emphasis Type="Italic">f</Emphasis>-rings

For a commutative ring A with identity, and for infinite cardinals α as well as the symbol ∞, which indicates the situation in which there are no cardinal restrictions, one defines A to be α-regular if for each subset D of A, with |D| < α and de = 0, for any two distinct d, e ∈ D, there is an s ∈ A such that d ² s = d, for each d ∈ D, and if xd = 0, for each d ∈ D, then xs = 0     

Codepth Two and Related Topics

A depth two extension A | B is shown to be weak depth two over its double centralizer V _A (V _A (B)) if this is separable over B. We consider various examples and non-examples of depth one and two properties. Depth two and its relationship to direct and tensor product of algebras as well as cup product of relative Hochschild cochains is examined. Section 6 introduces a notion of codepth two coalgebra homomorphism g : C → D, dual to a depth two algebra homomorphism. It is shown that the endomorphism ring of bicomodule endomorphisms End ^D C ^D forms a right bialgebroid over the centralizer subalgebra g ^* : D ^* → C ^* of the dual algebra C ^*. Dedicated to Daniel Kastler on his eightieth birthday.     

On the boundedness of one recurrent sequence in a banach space

We establish necessary and sufficient conditions under which a sequence x ₀ = y ₀ , x _n+1 = Ax _n  + y _n+1 , n ≥ 0, is bounded for each bounded sequence { y_n :n \geqslant 0 } ì { x ? è_{n = 1}^￥ D( Aⁿ ) |sup_{n \geqslant 0} || Aⁿ x || < ￥ }\left\{ {y_n :n \geqslant 0} \right\} \subset \left\{ {\left. {x \in \bigcup\nolimits_{n = 1}^\infty {D\left( {A^n } \right)} } \right|\sup _{n \geqslant 0} \left\| {A^n x} \right\| < \infty } \right\}, where A is a closed operator in a complex Banach space with domain of definition D(A) .