A superatomic Boolean algebra with few automorphisms

Assuming GCH, we prove that for every successor cardinal μ > ω₁, there is a superatomic Boolean algebra B such that |B| = 2^μ and |Aut B| = μ. Under ◊_ω1, the same holds for μ = ω₁. This answers Monk's Question 80 in [Mo]. Received: 1 January 1998 / Revised version: 18 May 1999 / Published online: 21 December 2000     

The spectrum problem III: Universal theories

We solve the classification problem and essentially the spectrum problem for universal theories (see [6] for discussion of the meaning of this). We first solve it forT such that ifM ₁,M ₂ elementarily extendM ₀ and are independent over it, then overM ₀ ∪M ₁ there is a prime model. This generalizes [2]. This was subsequently used and generalized for countable first order theories. (This will appear in [5].) But note that there the theory is countable and in the case of structure the model is prime over a non-forking tree of models; here the model is generated by the union (and theT not necessarily countable). The universality is used in Theorey.If T is stable and complete then either (A)for every M ₁<M (l=0, 1, 2)models of T, if M ₀ ⊆M ₁,M ₂, {M ₁,M ₂}is independent over M ₀ (i.e. tp(M ₁,M ₂)is finitely satisfiable in M ₀),then the submodel of M which M ₁ ⋃M ₂ generates is an elementary submodel of M, or (B)there is an unstable theory extending the universal part of T (we can replace universal by Σ₂ and slightly more). Conclusion. For any universalT:Either (a) for every modelM ofT there is a treeI with ≦ω levels and submodelsN _η (η ∈I) of power ≦2^|T| (by [5], just ≦|T|) such that (i)M is generated by ∪_ηεl N _η, (ii)η <v⇒⇒N _η, (iii) ifv is an immediate successor ofη then tp(N _v, ⋃{{N _p:ρ ∈I,v≨ϱ}) is finitely satisfiable inN _η (note that asking this just for quantifier-free formulas is enough).Or (b) for every cardinalλ>|T|, there are 2^γ non-isomorphic models for powerλ. Dedicated to Professor Abraham Robinson The author would like to thank John Baldwin for the interesting talks in September 1980 which led to §3 of this work, Rami Grossberg for various corrections, and the BSF and NSF for their partial support. This paper was originally intended to appear in the Proceedings of the Model Theory Year at the Institute for Advanced Studies, The Hebrew University of Jerusalem, September 1980 — August 1981, published in Isr. J. Math., Vol. 49, Nos. 1–3, 1984.     

Holonomy on poisson manifolds and the modular class

(1) It is shown that ifc is real-valued measurable then the Maharam type of (c, P(c),σ) is 2 ^c . This answers a question of D. Fremlin [Fr, (P2f)]. (2) A different construction of a model with a real-valued measurable cardinal is given from that of R. Solovay [So]. This answers a question of D. Fremlin [Fr, (P1)]. (3) The forcing with aκ-complete ideal over a setX, |X| ≥κ cannot be isomorphic to Random × Cohen or Cohen × Random. The result forX=κ was proved in [Gi-Sh1] but, as was pointed out to us by M. Burke, the application of it in [Gi-Sh2] requires dealing with anyX. The application is: ifA _n is a set of reals forn<ω then for some pairwise disjointB _n (forn<ω) we haveB _n ⊆A _n but they have the same outer Lebesgue measure. Partially supported by the Israeli Basic Research Fund. Publ. Number 582.     

Sympletic capacities in two dimensions

Some years ago a class of new symplectic invariants was discovered, the so-called symplectic capacities. In this note we compute the Hofer-Zehnder-capacityc _HZ , which is defined via periodic solutions of Hamiltonian differential equations, for two-dimensional, connected manifoldsM with an area element ω. It turns out thatc _HZ (M,ω) is just the area |∫ _M ω|. Moreover, some examples illustrate the dynamics standing behind the definition ofc _HZ . In the last part we treat the special case of the real plane where also another type of capacities exists, an example of which is Hofer's displacement energy.     

Maxwell equations and the direction of the electromagnetic field

Let M₀ be the Minkowski space, let Λ₂(M₀) be the space of bivectors in M₀, and let G¹ ⊂ Λ₂(M₀) be the manifold of directions of the physical space, consisting of simple bivectors with square −1. A mapping F: U → Λ₂(M₀), U ⊂ ℝ⁴, satisfying the Maxwell equations is regarded as the tensor of an electromagnetic field in vacuum. The field is described on the basis of a special decomposition F = eω + h(*ω), where the mapping ω: U → G¹ is called the direction of the field, and e: U → (0, +∞) and h: U → ℝ are the electric and magnetic coefficients of the field. The Maxwell equations are reformulated in terms of ω, e, and h. Electromagnetic fields whose set of directions is a point or a one-dimensional subset of G¹ are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 118–146.     

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.     

Semi-cosymplectic manifolds

SoitM(Ω, η, ξ,g) une variété à (2m+1)-dimensions presque cosymplectique (i. e. Ω∈Λ² M est de rang 2m et Ω ^m Λη≠0). On définitM comme étant une variété semi-cosymplectique si en termes ded ^ω-cohomologie la paire (Ω, η) satisfait àdη=0,d ^−cη Ω=Ψ∈Λ³ M,c=constant. Dans ce cas le champ vectoriel de structure ξ=b ⁻¹(η) est un champ conforme horizontal et siM est une forme-espace elle est nécessairement du type hyperbolique. Différentes propriétés de cette structure sont étudiés et le cas oùM est une variété para Sasakienne dans le sens large est discuté.     

On exact 2-para Sasakian manifolds

Paracontact and para Sasakian manifoldsM carryingr(1<r≤dimM) Reed vector filds ξ _r have been especially studied by A. Bucki [2], [3], [4]. In the present paper, we consider a (2m+2)-dimensional para Sasakian manifoldM(ϕ, ξ _r , η ^r g), whose Reed convectors η ^r =ξ _r ^b are exact 1-forms and the covariant derivatives of ξ _r are given by ∇ξ _r =f _r dp ^⊺, wheredp ^⊤ means the horizontal component of the soldering formdp andf _r∈C^∞M satisfydf _r =cη ^r ,c=constant. It is proved that such a manifold may be viewed as the local Riemannian productM=M ^⊥×M^⊤, where

Splitting recursively enumerable subalgebras in recursive Boolean algebras

A Boolean algebraB= is recursive ifB is a recursive subset of ω and the operations Λ, v and ┌ are partial recursive. A subalgebraC ofB is recursive an (r.e.) ifC is a recursive (r.e.) subset of B. Given an r.e. subalgebraA, we sayA can be split into two r.e. subalgebrasA ₁ andA ₂ if (A ₁ ∪A ₂)^*=A andA ₁ ∩A ₂={0, 1}. In this paper we show that any nonrecursive r.e. subalgebra ofB can be split into two nonrecursive r.e. subalgebras ofB. This is a natural analogue of the Friedberg's splitting theorem in ω recursion theory.     

An information geometry algorithm for distribution control

In this paper, we consider the problem of distribution control from the viewpoint of information geometry. Different from most existing models used in stochastic control, it is assumed that the control input directly affects the distribution of the system output in probability sense. Here, we set up a new manifold (S), meanwhile the B-spline manifold (B) and the system output manifold (M) can be referred to as its submanifolds. We give an information geometrical algorithm which can be called as geodesic-projection algorithm using the properties of manifold. In the geodesic step, we can obtain the geodesic equation from the initial point V₀ = (ω₁₀, ω₂₀, ··· , ω_(n−1)0) to the specified point V_g = (ω_1g, ω_2g, ··· , ω_(n−1)g) in B. This gives us an optimal trajectory for the points changing along in B. In the projection step, we project the sample points selected from the geodesic onto M. The coordinates of the projections in M give the trajectory of the control input u.     

Instances of the Conjecture of Chang

We consider various forms of the Conjecture of Chang. Part A constitutes an introduction. Donder and Koepke have shown that if ρ is a cardinal such that ρ ≧ ω₁, and (ρ⁺⁺,ρ⁺↠(ρ⁺, ρ), then 0⁺ exists. We obtain the same conclusion in Part B starting from some other forms of the transfer hypothesis. As typical corollaries, we get: Theorem A.Assume that there exists cardinals λ, κ, such that λ ≧ K ⁺ ≧ω₂ and (λ⁺, λ)↠(K ⁺,K. Then 0⁺ exists. Theorem B.Assume that there exists a singularcardinal κ such that(K ⁺,K↠(ω₁, ω₀. Then 0⁺ exists. Theorem C.Assume that (λ ⁺⁺, λ). Then 0⁺ exists (also ifK=ω ₀. Remark. Here, as in the paper of Donder and Koepke, “O⁺ exists” is a matter of saying that the hypothesis is strictly stronger than “L(μ) exists”. Of course, the same proof could give a few more sharps overL(μ), but the interest is in expecting more cardinals, coming from a larger core model. Theorem D.Assume that (λ ⁺⁺, λ)↠(K ⁺, K) and thatK≧ω ₁. Then 0⁺ exists. Remark 2. Theorem B is, as is well-known, false if the hypothesis “κ is singular” is removed, even if we assume thatK≧ω ₂, or that κ is inaccessible. We shall recall this in due place. Comments. Theorem B and Remark 2 suggest we seek the consistency of the hypothesis of the form:K ⁺, K↠(ω_n +1, ω_n), for κ singular andn≧0. 0266 0152 V 3 The consistency of several statements of this sort—a prototype of which is (N _ω+1,N _ω)↠(ω₁, ω₀) —have been established, starting with an hypothesis slightly stronger than: “there exists a huge cardinal”, but much weaker than: “there exists a 2-huge cardinal”. These results will be published in a joint paper by M. Magidor, S. Shelah, and the author of the present paper.     

Rational points of bounded height on Del Pezzo surfaces of degree six

LetK be a number field. Denote byV ₃ a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW ₃ the open complement of the exceptional lines inV ₃. LetN _{W s}(−ω, X) be the number ofK-rational points onW ₃ whose anticanonical heightH _−ω is bounded byX. Manin has conjectured that asymptoticallyN _{W 3}(−ω, X) tends tocX(logX)³, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following theorem: For each number fieldK there exists a constantc _K such thatN _{W 3}(−ω, X)≤c_KX(logX)^3+2r , wherer is the rank of the group of units ofO _K. The constantc _K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN _{W 3}(−ω,X)≤cX^l+ε. The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support.     

Canonical edge-colourings of locally finite graphs

A variety of results on edge-colourings are proved, the main one being the following: ifG is a graph without loops or multiple edges and with maximum degree Δ=Δ(G), and if ν is a given integer 1≦ν≦Δ(G), thenG can be given a proper edge-colouring with the coloursc ₁, ...,c _Δ+1 with the additional property that any edge colouredc _μ with μ≧ν is on a vertex which has on it edges coloured with at least ν − 1 ofc ₁, ...,c _v .     

An extension theorem for separable Banach spaces

We construct a totally disconnected ω^*, norming subsetF of the unit ballB ^* of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B^*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ω^ω), is either isomorphic toC(ω^ω) or toc _u.     

Finitely-presented groups with long lower central series

We construct examples of finitely presented groupsG with the property that, ifG _ω is the intersection of the lower central series ofG, then [G,G _ω ] ≠G _ω .     

Coefficients of nonvanishing functions inH ∞

LesB denote the class of functions analytic in the unit disc ofC which satisfy 0<|f(z)|<1. It is proved that there exists a numberc<1 such that iff∈B and iff(z)=Σ _n=0 ^∞ a _n z ⁿ , then |a _n |<c forn>=1.     

Multipliers of weighted semigroups and associated Beurling Banach algebras

Given a weighted discrete abelian semigroup (S, ω), the semigroup M _ω (S) of ω-bounded multipliers as well as the Rees quotient M _ω (S)/S together with their respective weights [(w)\tilde]\tilde{\omega} and [(w)\tilde]_q\tilde{\omega}_q induced by ω are studied; for a large class of weights ω, the quotient l¹(M_w(S),[(w)\tilde])/l¹(S,w)\ell^1(M_{\omega}(S),\tilde{\omega})/\ell^1(S,{\omega}) is realized as a Beurling algebra on the quotient semigroup M _ω (S)/S; the Gel’fand spaces of these algebras are determined; and Banach algebra properties like semisimplicity, uniqueness of uniform norm and regularity of associated Beurling algebras on these semigroups are investigated. The involutive analogues of these are also considered. The results are exhibited in the context of several examples.     

Invariants forω-categorical,ω-stable theories

In this paper we give a complete solution to the classification problem forω-categorical,ω-stable theories. More explicitly, supposeT isω-categorical,ω-stable with fewer than the maximum number of models in some uncountable power. We associate with each modelM ofT a “simple” invariantI(M), not unlike a vector of dimensions, such thatI(M)=I(N) if and only ifM≅N. The spectrum function,I(−,T), for a first-order theoryT is such that for all infinite cardinals λ,I(λ,T) is the number of nonisomorphic models ofT of cardinality λ. As an application of our “structure theorem” we determine the possible spectrum functions forω-categorical,ω-stable theories.     

Matrices with zero trace

LetM _n(F) denote the algebra ofn-square matrices with elements in a fieldF. In this paper we show that ifM ∈M _n(F) has zero trace thenM=AB−BA for certainA, B ∈ M _n(F), withA nilpotent and traceB=0, apart from some exceptional cases whenn=2 or 3. We also determine whenM=MB−BM for someB ∈ M _n(F). The preparation of this paper was supported in part by the U.S. Air Force under contract AFOSR 698-65.