Some ℵ0-bounded subsets of stone-čech compactifications

We characterize the maximalm-bounded extension of an arbitrary completely regular Hausdorff spaceX. The other principal results are:Theorem. LetX be a locally compact, σ-compact non-compact space with no more than 2^ℵ0 zero-sets. Then assuming the continuum hypothesis,βX − X can be written as the union of 2^2ℵ0 pairwise disjoint, dense ℵ₀-bounded subspaces.Theorem. LetX be a locally compact, σ-compact metric space without isolated points. Then both the set of remote points ofβX and the complement of this set inβX −X are ℵ₀-bounded.     

There are many normal ultrafiltres corresponding to a supercompact cardinal

It is proved that ifκ is supercompact, there are at least (2^• P ^κ(β)^•)⁺normal ultrafilters overP _k (β) and ifV=H.O.D. exactly (2^2• P ^κ(β)^•) normal ultrafilters. This is a part of the author’s Ph.D. thesis prepared under the supervision of Professor Azriel Levy for whose help the author is grateful. In 1966–67, Solovay proved Theorem 1 for the caseβ=κ without the condition of extendability. The same result, under a somewhat weaker assumption was proved by Namba in 1967–68. As noted by Solovay, his proof can be adapted to a generalβ (under weaker assumptions; if |P _k (β) |=β it is only needed that ℵ is 2^β-supercompact). Solovay’s result will be published in [3].     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too.     

The maximal (C,α,β) operator on Hp(T×T)

The two-dimensional classical Hardy space H_p(T×T) on the bidisc are introduced, and it is shown that the maximal operator of the (C,α,β) means of a distribution is bounded from the space H_p(T×T) to L_p(T²) (1/(α+1), 1/(β+1)<p≤∞), and is of weak type (H ₁ ^# (T×T), L₁(T²)), where the Hardy space H ₁ ^# (T×T) is defined by the hybrid maximal function. As a consequence we obtain that the (C, α, β) means of a function f∈H ₁ ^# (T×T)⊃LlogL(T ²) convergs a. e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on the spaces H_p(T×T) whenever 1/(α+1), 1(β+1)<p<∞. Thus, in case f∈H_p(T×T), the (C, α, β) means convergs to f in H_p(T×T) norm whenever (1/(α+1), 1/(β+1)<p<∞). The same results are proved for the conjugate (C, α, β) means, too. This research was made while the author was visiting the Humboldt University in Berlin supported by the Alexander von Humboldt Foundation.     

Notes on combinatorial set theory

We shall prove some unconnected theorems: (1) (G.C.H.) \omega _{\alpha + 1} \to \left( {\omega _\alpha + \xi } \right)_2^2 when ℵ_α is regular, │ξ│⁺<ωα. (2) There is a Jonsson algebra in ℵ_α+n, and \aleph _{a + n} \not \to \left[ {\aleph _{a + n} } \right]_{\aleph _{a + n} }^{n + 1} if 2^{\aleph _{ - - } } = \aleph _{a + n} \cdot (3) If λ>ℵ₀ is a strong limit cardinal, then among the graphs with ≦λ vertices each of valence <λ there is a universal one. (4)(G.C.H.) If f is a set mapping on \omega _{a + 1} (ℵ_α regular) │f(x)∩f(y│<ℵ_α, then there is a free subset of order-type ζ for every ζ<ω_α+1.     

A unified approach for univalent functions with negative coefficients using the Hadamard product

For given analytic functions ϕ(z) = z + Σ _n=2^∞ λ _n z ⁿ , Ψ(z) = z + Σ _n=2^∞ μ with λ _n ≥ 0, μ _n ≥ 0, and λ _n ≥ μ _n and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ _n=2^∞ a _n z ⁿ in U such that f(z)*ψ(z)≠0 and

Utility functions and the ‘lin’ operation for convex sets

For an ℵ₀-dimensional spaceE of alternatives, there is described a preference relation ≿ such that (in a very strong sense)no information about ≿ can be expressed in terms of finite-dimensional linear transformations ofE. The same construction shows that for each countable ordinalβ, E contains a convex coneK such that lin ^β K =E but lin^β K ≡E forα<β.     

On the number of homogeneous models of a given power

It is shown that, For each complete theoryT, the nomberh _T(m) of homogeneous models ofT of powerm is a non-increasing function of uncountabel cardinalsm Moreover, ifh _T(ℵ₀)≦ℵ₀, then the functionh _T is also non-increasing ℵ₀ to ℵ₁. This work was supported in part by NSF contracts GP 4257 and GP5913.     

Some results on consecutive large cardinals II: Applications of radin forcing

Letκ be a 3 huge cardinal in a countable modelV of ZFC, and letA andB be subsets of the successor ordinals <κ so thatA ⋃B={α<κ:α is a successor ordinal}. Using techniques of Gitik, we construct a choiceless modelN _A of ZF of heightκ so thatN _A ╞“ZF+⌍AC _ω+Forα ∈A, ℵ_a is a Ramsey cardinal+Forβ ∈B, ℵ_β is a singular Rowbottom cardinal which carries a Rowbottom filter+Forγ a limit ordinal, ℵ_y is a Jonsson cardinal which carries a Jonsson filter”. The author wishes to express his thanks to the Rutgers Research Council for a Summer Research Fellowship which partially supported this work. The author also wishes to thank Moti Gitik and Bob Mignone for their useful comments concerning the subject matter of this paper.     

Score Lists in Tripartite Hypertournaments

Given non-negative integers l, m, n, α, β and γ with l ≥ α ≥ 1, m ≥ β ≥ 1 and n ≥ γ ≥ 1, an [α,β,γ]-tripartite hypertournament on l + m + n vertices is a four tuple (U, V, W, E), where U, V and W are three sets of vertices with |U| = l , |V| = m and |W| = n, and E is a set of (α + β + γ)-tuples of vertices, called arcs, with exactly α vertices from U, exactly β vertices from V,and exactly γ vertices from W, such that any subset U₁∪ V₁∪ W₁ of U∪ V∪ W, E contains exactly one of the (α + β + γ)! (α + β + γ) − tuples whose entries belong to U₁∪ V₁∪ W₁. We obtain necessary and sufficient conditions for three lists of non-negative integers in non-decreasing order to be the losing score lists or score lists of some [α, β, γ]-tripartite hypertournament. Supported by National Science Foundation of China (No.10501021).     

Decoupling inequalities and interlacement percolation on <Emphasis Type="Italic">G</Emphasis>×ℤ

We study the percolative properties of random interlacements on G×ℤ, where G is a weighted graph satisfying certain sub-Gaussian estimates attached to the parameters α>1 and 2≤β≤α+1, describing the respective polynomial growths of the volume on G and of the time needed by the walk on G to move to a distance. We develop decoupling inequalities, which are a key tool in showing that the critical level u _∗ for the percolation of the vacant set of random interlacements is always finite in our set-up, and that it is positive when α≥1+β/2. We also obtain several stretched exponential controls both in the percolative and non-percolative phases of the model. Even in the case where G=ℤ ^d , d≥2, several of these results are new.     

The number of non-isomorphic models of an unstable first-order theory

It is proved that ifT is an unstable (first-order) theory,λ>|T|+ℵ₀, thenT has exactly 2 ^λ non-isomorphic models of cardinalityλ. In fact we have stronger results: this is true for pseudo-elementary classes, and for almost everyλ≧|T|+ℵ₁. The preparation of this paper was sponsored in part by NSF Grant GP-22937. This work was supported in part by NSF Grant GP-22794.     

Block Matrix Operators and Weak Hyponormalities

We introduce a new model of a block matrix operator M(α, β) induced by two sequences α and β and characterize its p-hyponormality. The model may be viewed as arising from the composition operator C_T on l²₊ : = L²(\mathbb N₀)l^{2}_{+} := L^2(\mathbb {N}_0) induced by a measurable transformation T on the set of nonnegative integers \mathbb N₀\mathbb {N}_0 with point mass measure. Composition operator techniques may then be used to treat the p-hyponormality of M(α, β). Finally, we apply our results to obtain examples of these operators showing the p-hyponormal classes are distinct.     

Exponents of someN-compact spaces

For any topological spaceT, S. Mrówka has defined Exp (T) to be the smallest cardinal κ (if any such cardinals exist) such thatT can be embedded as a closed subset of the productN ^κ of κ copies ofN (the discrete space of cardinality ℵ₀). We prove that forQ, the space of the rationals with the inherited topology, Exp (Q) is equal to a certain covering number, and we show that by modifying some earlier work of ours it can be seen that it is consistent with the usual axioms of set theory including the choice that this number equal any uncountable regular cardinal less than or equal to 2^ℵ ₀. Mrówka has also defined and studied the class ℳ={κ: Exp (N _κ)=κ} whereN _κ is the discrete space of cardinality κ. It is known that the first cardinal not in ℳ must not only be inaccessible but cannot even belong to any of the first ω Mahlo classes. However, it is not known whether every cardinal below 2^ℵ ₀ is contained in ℳ. We prove that if there exists a maximal family of almost-disjoint subsets ofN of cardinality κ, then κ∈ℳ, and we then use earlier work to prove that if it is consistent that there exist cardinals which are not in the first ω Mahlo classes, then it is consistent that there exist such cardinals below 2^ℵ ₀ and that ℳ nevertheless contain all cardinals no greater than 2^ℵ ₀. Finally, we consider the relationship between ℳ and certain “large cardinals”, and we prove, for example, that if μ is any normal measure on a measurable cardinal, then μ(ℳ)=0.     

UNIFORM CONVERGENCE OF CESARO MEANS OF NEGATIVE ORDER OF DOUBLE TRIGONOMETRIC FOURIER SERIES

In this paper we prove that iff ∈ C([-π,π]2) and the function f is bounded partial p-variation for some p ∈ [1, ∞), then the double trigonometric Fourier series of a function f is uniformly (C;-α,-β) summable (α β＜ 1/p,α,β＞ 0) in the sense of Pringsheim. If α β≥ 1/p, then there exists a continuous function f0 of bounded partial double trigonometric Fourier series of fo diverge over cubes.     

Lattice-universal Orlicz spaces on probability spaces

Lattice-universal Orlicz function spacesL ^F _α,β[0, 1] with prefixed Boyd indices are constructed. Namely, given 0<α<β<∞ arbitrary there exists Orlicz function spacesL ^F _α,β[0, 1] with indices α and β such that every Orlicz function spaceL ^G [0, 1] with indices between α and β is lattice-isomorphic to a sublattice ofL ^F _α,β[0, 1]. The existence of classes of universal Orlicz spacesl ^Fα,β(I) with uncountable symmetric basis and prefixed indices α and β is also proved in the uncountable discrete case. Partially supported by BFM2001-1284.     

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）.     

On exponential stability of contraction semigroups

A semigroup [T(t)] on a Hilbert space is exponentially stable if there exist real constants M≥1 and α>0 such that ∥T(t)∥≤Me ^−αt for every t≥0. If [T(t)] is a strongly continuous contraction semigroup, then it is proved that we can set M=1 in the definition of exponential stability if and only if the generator A of [T(t)] is boundedly strict dissipative (just a strict dissipative A is not enough).     

A note on Lagrange interpolation of functions of generalized bounded variation

In this paper we solve a remained problem in [2], whether the following estimate approximation for the classf∈[-1, 1]∩BV by Lagrange interpolation based on the Jacobi abscissas: L _n ^(a,d) (f,x)−f(x)=O(1/n) holds, if α≠β α,β≥−1. The project is supported by the Natural Science Foundation of Zhejiang Province.