A poset hierarchy

This article extends a paper of Abraham and Bonnet which generalised the famous Hausdorff characterisation of the class of scattered linear orders. They gave an inductively defined hierarchy that characterised the class of scattered posets which do not have infinite incomparability antichains (i.e. have the FAC). We define a larger inductive hierarchy ^κℌ* which characterises the closure of the class of all κ-well-founded linear orders under inversions, lexicographic sums and FAC weakenings. This includes a broader class of “scattered” posets that we call κ-scattered. These posets cannot embed any order such that for every two subsets of size < κ, one being strictly less than the other, there is an element in between. If a linear order has this property and has size κ it is unique and called ℚ(κ). Partial orders such that for every a < b the set {x: a < x < b} has size ≥ κ are called weakly κ-dense, and posets that do not have a weakly κ-dense subset are called strongly κ-scattered. We prove that ^κℌ* includes all strongly κ-scattered FAC posets and is included in the class of all FAC κ-scattered posets. For κ = ℵ₀ the notions of scattered and strongly scattered coincide and our hierarchy is exactly aug(ℌ) from the Abraham-Bonnet theorem. The authors warmly thank Uri Abraham for his many useful suggestions and comments. Mirna Džamonja thanks EPSRC for their support on an EPSRC Advanced Fellowship.     

The algebra of generalized jacobifields

We study the structure of those vector fields on the tangent bundle of an arbitrary smooth manifold which commute with the geodesic vector field defined by an affine connection. The study is restricted to polylinear fields generated by a pair of symmetric pseudotensor fields of type (k, 1) and (k+1,1), k≥0, defined on the manifold. We establish an isomorphism between the space of infinitesimal automorphisms of fixed type and the space ℌ_k of the solutions of a partial differential equation generalizing the Jacobi equation for the infinitesimal automorphisms of the connection. It is shown that the spaces ℌ_k are finite-dimensional and form a graduated Lie algebra ℌ=⊕ _k=0 ^∞ ℌ_k. These algebras are classified in the case of one-dimensional manifolds. It is proved that if the geodesic vector field is complete, then so are the automorphisms corresponding to covariant constant fields of type (1, 1). Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 222–244. Translated by V. S. Kal’nitskii.     

On banach spaces which containl 1(τ) and types of measures on compact spaces

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyński's about Banach spaces containingL ¹. Let τ be a regular cardinal satisfying the hypothesis that κ^ω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]^τ. 2) A Banach spaceX has a subspace isomorphic tol ¹(τ) if and only ifX ^∗ has a subspace isomorphic toL ¹({0, 1}^τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl ¹ does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl ¹ (ω₁), while the unit ball ofX ^∗ is not weakly^∗ sequentially compact.     

Homogeneity and local symmetry of complex (<Emphasis Type="Italic">κ</Emphasis>, µ)-spaces

We prove that a complex (κ, μ)-space with κ < 1 is a locally homogeneous complex contact metric manifold. Also, a complex (κ, μ)-space has either κ = 1 or is GH-locally symmetric.     

The Spectrum of Annihilation Plus Orthogonal Creation

Let ℌ be a Hilbert space and F(ℌ) be the full Fock space generated by ℌ. For v ∈ ℌ, the (left) creation operator l(v) : F(ℌ) → F(ℌ),f↦ v ⊗ f, and its adjoint, the (left) annihilation operator l(v)^*, are defined. If v, w ∈ℌ are orthonormal, it is proved that the spectrum of the operator l(v) + l(w)^* is purely continuous and conincides with the closed unit disk. This revised version was published online in July 2006 with corrections to the Cover Date.     

On dualL 1-spaces and injective bidual Banach spaces

In a previous paper (Israel J. Math.28 (1977), 313–324), it was shown that for a certain class of cardinals τ,l ¹(τ) embeds in a Banach spaceX if and only ifL ¹([0, 1]^τ) embeds inX ^*. An extension (to a rather wider class of cardinals) of the basic lemma of that paper is here applied so as to yield an affirmative answer to a question posed by Rosenthal concerning dual ℒ₁-spaces. It is shown that ifZ ^* is a dual Banach space, isomorphic to a complemented subspace of anL ¹-space, and κ is the density character ofZ ^*, thenl ¹(κ) embeds inZ ^*. A corollary of this result is that every injective bidual Banach space is isomorphic tol ^∞(κ) for some κ. The second part of this article is devoted to an example, constructed using the continuum hypothesis, of a compact spaceS which carries a homogeneous measure of type ω₁, but which is such thatl ¹(ω₁) does not embed in ℰ(S). This shows that the main theorem of the already mentioned paper is not valid in the case τ = ω₁. The dual space ℰ(S)^* is isometric to , and is a member of a new isomorphism class of dualL ¹-spaces.     

On the weight-spectrum of a compact space

The weight-spectrumSp(w, X) of a spaceX is the set of weights of all infinite closed subspaces ofX. We prove that ifκ>ω is regular andX is compactT ₂ withω(X)≥κ then some λ withκ≤λ≤2^<κ is inSp(ω, X). Under CH this implies that the weight spectrum of a compact space can not omitω ₁, and thus solves problem 22 of [M]. Also, it is consistent with 2^ω=c being anything it can be that every countable closed setT of cardinals less thanc withω ∈ T satisfiesSp(w, X)=T for some separable compact LOTSX. This shows the independence from ZFC of a conjecture made in [AT]. Research supported by OTKA grant no. 1908.     

Reflecting topological properties in continuous images

Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ ⁺ if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ ⁺. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ ⁺ for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ ⁺ for compact spaces.     

More on countably compact,locally countable spaces

Following [5], aT ₃ spaceX is called good (splendid) if it is countably compact, locally countable (andω-fair).G(κ) (resp.S(κ)) denotes the statement that a good (resp. splendid) spaceX with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2 ^ω is big+S(κ) holds unlessω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>_ω+₁+┐G(ω_ω+₁); (iii) the “Chang conjecture” (ω_ω+₁),→(ω ₁,ω) implies ┐S(κ) for allκ≧k≧ω_ω; (iv) ifP addsω ₁ dominating reals toV iteratively then, in , we haveG(λ^ω) for allλ. Research supported by Hungarian National Foundation for Scientific Research grant no. 1805.     

A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions

In this paper the following result is obtained: Suppose f(g,u,v) is nonnegative, continuous in (a, 6) ×R⁺ ×R ⁺ ; f may be singular at κ = a(and/or κ = b) and υ = 0; f is nondecreasing on u for each κ,υ,nonincreasing on υ for each κ,u; there exists a constant q ε (0,1) such that

A characterization of cardinalsκ such that 2λ = 2κ wheneverκ ≦λ < 2κ

We characterize cardinalsκ such that 2^λ = 2^κ wheneverκ ≦λ < 2^κ using ideals in small algebras of sets satisfying certain completeness and saturation conditions. Research of this author was supported by an NSF Grant.     

Integer-valued functions and increasing unions of first countable spaces

We determine those regular cardinals κ with the property that for each increasing κ-chain of first countable spaces there is a compatible first countable topology on the union of the chain. AssumingV=L any such κ must be weakly compact. It is relatively consistent with a supercompact cardinal that each κ>w ₁ has the property. The proofs exploit the connection with interesting families of integer-valued functions. Research of the second author supported by OTKA grant no. 1805. Research of the remaining authors partially supported by NSERC of Canada.     

Spaces hereditarily of κ-type and point κ-type

In this paper we discuss the structure of regular spaces hereditarily of κ-type and point κ-type. In particular, it is shown that a regular spaceX is hereditarily of point κ-type iff for everyp∈X there is a setE (possibly empty) of isolated points ofX such that (1)E∪{p} is compact and (2)X(E∪{p}, X)≤κ.     

On the infimum problem of Hilbert space effects

The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∈ε(H). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A∧B for two effects A. B∈ε(H) are established.     

Some remarks on convex surfaces in simply connected homogeneous three manifolds

Here we extend Hadamards’ Theorem to other homogeneous three manifolds, i.e., we prove that a compact orientable immersed surface Σ in general position whose principal curvatures κ _i , i = 1, 2, satisfy κ _i ≥ τ, is an embedded sphere.     

The consistency strength of “every stationary set reflects”

The consistency strength of a regular cardinal so that every stationary set reflects is the same as that of a regular cardinal with a normal idealI so that everyI-positive set reflects in aI-positive set. We call such a cardinal areflection cardinal and such an ideal areflection ideal. The consistency strength is also the same as the existence of a regular cardinal κ so that every κ-free (abelian) group is κ⁺-free. In L, the first reflection cardinal is greater than the first greatly Mahlo cardinal and less than the first weakly compact cardinal (if any). Research supported by NSERC grant # A8948. Publication # 367. Research partially supported by the BSF.     

Pseudo-coefficients des séries <Emphasis Type="Italic">κ</Emphasis>-discrètes de GL(<Emphasis Type="Italic">n,F</Emphasis>)

Let F be a non-Archimedean locally compact field, n be an integer ≥ 2, κ be a character of F ^× of finite order dividing n, and G be the group GL(n, F). We prove that if π is an irreducible κ-discrete series of G, then there exist some pseudo-coefficients for the twisted character trace(π ∘ A) where A denotes a non-zero intertwining operator between π ⊗ (κ ∘ det) and π.     

Metrizable compacta in the space of continuous functions with the topology of pointwise convergence

We prove that every point-finite family of nonempty functionally open sets in a topological space X has the cardinality at most an infinite cardinal κ if and only if w(X) ≦ κ for every Valdvia compact space Y C _p (X). Correspondingly a Valdivia compact space Y has the weight at most an infinite cardinal κ if and only if every point-finite family of nonempty open sets in C _p (Y) has the cardinality at most κ, that is p(C _p (Y)) ≦ κ. Besides, it was proved that w(Y) = p(C _p (Y)) for every linearly ordered compact Y. In particular, a Valdivia compact space or linearly ordered compact space Y is metrizable if and only if p(C _p (Y)) = ℵ₀. This gives answer to a question of O. Okunev and V. Tkachuk.      

Generalized Symplectic Action and Symplectomorphism Groups of Coadjoint Orbits

Let M be a quantizable symplectic manifold. If ψ_t is a loop in the group {Ham}(M) of Hamiltonian symplectomorphisms of M and A is a 2k-cycle in M, we define a symplectic action κ_A(ψ)∊ U(1) around ψ_t(A), which is invariant under deformations of ψ, and such that κ_A(ψ) depends only on the homology class of A. Using properties of κ_A( ) we determine a lower bound for ♯π₁(Ham(O)), where O is a quantizable coadjoint orbit of a compact Lie group. In particular we prove that ♯π₁(Ham(CPⁿ)) ≥ n+1. Mathematics Subject Classifications (2000): 53D05, 57S05, 57R17, 57T20.     

Universal graphs without instances of CH: Revisited

We give a complete proof of the consistency of the existence of a universal graph of powerλ, whereκ =κ ^<κ <λ = cfλ < 2 ^κ are arbitrary. The author would like to thank the NSF for partially supporting this research, Alice Leonhardt for the beautiful typing, and M. Kojman for proofreading. Publication No. 175A. The proof in the second section of [9] is flawed.