r-Maximal major subsets

The question of which r.e. setsA possess major subsetsB which are alsor-maximal inA (A⊂_rm B) arose in attempts to extend Lachlan’s decision procedure for the αε-theory of ℰ^*, the lattice of r.e. sets modulo finite sets, and Soare’s theorem thatA andB are automorphic if their lattice of supersets ℒ^*(A) and ℒ^*(B) are isomorphic finite Boolean algebras. We characterize the r.e. setsA with someB⊂_rm A as those with a Δ₃ function that for each recursiveR _i specifiesR _i or as infinite on and to be preferred in the construction ofB. There are r.e.A andB with ℒ^*(A) and ℒ^*(B) isomorphic to the atomless Boolean algebra such thatA has anrm subset andB does not. Thus 〈ℰ^*,A〉 and 〈ℰ^*,B〉 are not even elementarily equivalent. In every non-zero r.e. degree there are r.e. sets with and withoutrm subsets. However the classF of degrees of simple sets with norm subsets satisfies . The authors were partially supported by NSF Grants MCS 76-07258, MCS 77-04013 and MCS 77-01965 respectively.     

On a superatomic Boolean algebra which is not generated by a well-founded sublattice

Let b denote the unboundedness number of ω^ω. That is, b is the smallest cardinality of a subset such that for everyg∈ω^ω there isf ∈ F such that {n: g(n) ≤ f(n)}is infinite. A Boolean algebraB is wellgenerated, if it has a well-founded sublatticeL such thatL generatesB. We show that it is consistent with ZFC that , and there is a Boolean algebraB such thatB is not well-generated, andB is superatomic with cardinal sequence 〈ℵ₀, ℵ₁, ℵ₁, 1〉. This result is motivated by the fact that if the cardinal sequence of a Boolean algebraB is 〈ℵ₀, ℵ₀, λ, 1〉, andB is not well-generated, then λ≥b.     

Ratio limit theorems for Markov processes

Convergence of andμP ⁿ(B)/μP ⁿ(a) is established for a certain class of Markov operators,P, whereμ is a measure andB is a subset ofA. The results are proved under certain conditions onP and the setA.     

Strong monotonicity of operator functions

LetA, B be bounded selfadjoint operators on a Hilbert space. We will give a formula to get the maximum subspace such that is invariant forA andB, and . We will use this to show strong monotonicity or strong convexity of operator functions. We will see that when 0≤A≤B, andB−A is of finite rank,A ^t ≤B ^t for somet>1 if and only if the null space ofB−A is invariant forA.     

TC semigroups and inflations

An algebraA satisfiesTC (the term condition) if for any and anyn + 1-ary termp.TC algebras have been extensively studied. We previously determined the structure of allTC semigroups. We use this result to show that ifS is aTC semigroup thenS _E = {a ε S | ax is an idempotent for somex ε S} is an inflation ofS _Reg (the set of regular elements ofS) andS _Reg ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup. As a corollary of this result, we show thatS is a semisimpleTC semigroup iffS ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup.     

Interpolation of compactness using Aronszajn-Gagliardo functors

We prove that ifT: A ₀ →B ₀ andT: A ₁ →B ₁ both are compact, then is also compact, whereF is the minimal or the maximal functor in the sense of Aronszajn-Gagliardo. We also derive some results for ordered couples. Supported in part by Ministerio de Educación y Ciencia (SAB-87-0172).     

An additivity principle for Goldie rank

LetA be a noetherian ring. In generalA will not admit a classical Artinian ring of quotients. Yet a problem in enveloping algebras leads one to consider the possible embedding ofA in a prime ringB which is finitely generated as a left and a rightA module. Under certain additional technical assumptions, it is shown that the setS of regular elements ofA is regular inB and is an Ore set in bothA andB withS ⁻¹ A andS ⁻¹ B Artinian. This enables one to establish the following additivity principle for Goldie rank. Let {P ₁,P ₂, …P ₁} be the set of minimal primes ofA. Then under the above conditions it is shown that there exist positive integersz ₁,z ₂, …,z, such that , where rk denotes Goldie rank. This applies to the study of primitive ideals in the enveloping algebra of a complex semisimple Lie algebra. This paper was written while the authors were guests of the Institute for Advanced Studies, The Hebrew University of Jerusalem. The first author was on leave of absence from the Centre Nationale de la Recherche Scientifique, France.     

A theorem on arrangements of lines in the plane

LetA be an arrangement ofn lines in the plane. IfR ₁, …,R _r arer distinct regions ofA, andR _i is ap _i-gon (i=1, …,r) then we show that . Further we show that for allr this bound is the best possible ifn is sufficiently large. Financial support for this research was provided by the Carnegie Trust for the Universities of Scotland.     

THE FUNDAMENTAL GROUP OF THE AUTOMORPHISM GROUP OF A NONCOMMUTATIVE TORUS

Assume that each completely irrational noncommutative torus is realized as an inductive limit of circle algebras, and that for a completely irrational noncommutative torus Aw of rank m there are a completely irrational noncommutative torus Aρ of rank m and a positive integer d such that tr(Aw)=1/d.tr(Aρ).It is proved that the set of all C^*-algebras of sections of locally trivial C^*-algebra bundles over S^2 with fibres Aω has a group sturcture,denoted by π1^s(Aut(Aω)),which is isomorphic to Zif Ed>1 and {0} if d>1.Let Bcd be a cd-homogeneous C^*-algebra over S^2×T^2 of which no non-trivial matrix algebra can be factored out.The spherical noncommutative torus Sρ^cd is defined by twisting C^*(T2×Z^m-2) in Bcd ×C^*(Z^m-3) by a totally skew multiplier ρ on T^2×Z^m-2。It is shown that Sρ^cd×Mρ∞ is isomorphic to C(S^2)×C^*(T^2×Z^m-2,ρ）× Mcd(C)×Mρ∞ if and only if the set of prime factors of cd is a subset of the set of prime factors of p.     

Obstructions to the extension of partial maps

One of the most important problems in topology is the minimization (in some sense) of obstructions to extending a partial map , i.e., of a subsetF ⊂ Z/A such thatf can be globally extended to its complement. It is shown that ifZ is a fixed metric space with dimZ ≤ n andp, q ≥−1 are fixed numbers, then obstructions to extending all partial maps can be concentrated in preselected fairly thin subsets ofZ. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 803–812, December 1997 Translated by O. V. Sipacheva     

Some embeddings of infinite-dimensional spaces

It is shown that ifA is a weakly infinite-dimensional subset of a metric spaceR then aG _δ setB ofR exists such thatA⊆B andB is weakly infinite-dimensional. A similar result holds for a set having strong transfinite inductive dimension. As a consequence each weakly infinite-dimensional metric space possesses a weakly infinite-dimensional complete metric extension. A similar result holds also for a space having strong transfinite inductive dimension.     

Almost semirecursive sets

LetA be a subset of , and leta∉A. The setA is said to be almost semirecursive, if there is a two-place general recursive functionf such thatf(x, y)ε{x, y, a}∧({x, y}⊆A⇌f(x, y)εA) for all . Among other facts, it is proved that ifA and are almost semirecursive sets, thenA is a semirecursive set, and that there exists a wsr^*-set that is neither a wsr-nor an almost semirecursive set. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 188–193, August, 1999.     

Hypergraphs in which all disjoint pairs have distinct unions

Let ℓ be a set-system ofr-element subsets on ann-element set,r≧3. It is proved that if |ℓ|>3.5 then ℓ contains four distinct membersA, B, C, D such thatA∪B=C∪D andA∩B=C∩D=0.     

Admissibility spectra throughω 1

Jensen showed that any countable sequenceA ofA-admissibles is the initial part of the admissibility spectrum of a real. We considerω ₁-long sequences, to be realized byB ⊆ω ₁. The problem is similar to finding a club subset of a stationary set. We investigate when such aB can be forced and when one is already inV.     

A note on injectiveC(T)-spaces and the Amir boundary

We sovle in the negative a problem of Wolfe ifC(T _A ) is an injective Banach space wheneverC(T) is injective,T compact, andT _A is the Amir boundary ofT (i.e., the complement of the maximal open extremally disconnected subset ofT). In particular, we findT such thatC(T) is aP ₃-space andT _A ∼βN\N. The author’s research was partially supported by a grant of MEN, Poland.     

Exact constants in inequalities of the jackson type for quadrature formulas

We prove that if is the error of a simple quadrature formula and ω(ε, δ)₁ is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: whereD _r is the Bernoulli kernel.     

Galois theory for Hopf algebroids

An extensionB⊂A of algebras over a commutative ringk is anH-extension for anL-bialgebroidH ifA is anH-comodule algebra andB is the subalgebra of its coinvariants. It isH-Galois if the canonical mapA ⊗_B A →A ⊗_L H is an isomorphism or, equivalently, if the canonical coringA ⊗_L H:A is a Galois coring. In the case of Hopf algebroid anyH _R-extension is shown to be also anH _L-extension. If the antipode is bijective then also the notions ofH _R-Galois extensions and ofH _L-Galois extensions are proven to coincide. Results about bijective entwining structures are extended to entwining structures over non-commutative algebras in order to prove a Kreimer-Takeuchi type theorem for a finitely generated projective Hopf algebroidH with bijective antipode. It states that anyH-Galois extensionB⊂A is projective, and ifA isk-flat then already the surjectivity of the canonical map implies the Galois property. The Morita theory, developed for corings by Caenepeel, Vercruysse and Wang is applied to obtain equivalent criteria for the Galois property of Hopf algebroid extensions. This leads to Hopf algebroid analogues of results for Hopf algebra, extensions by Doi and, in the case of Frobenius Hopf algebroids, by Cohen, Fishman and Montgomery.

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Sunto Un'estensioneB(A di algebre su un anello commutativok è unaH-estensione per unL-bialgebroideH seA è unaH-comodulo algebra eB è la sottoalgebra dei suoi coinvarianti. Essa èH-Galois se l'applicazione canonicaA ⊗_A B→A ⊗_L H è un isomorfismo o, equivalentemente, se il coanello canonicoA ⊗_L H:A è un coanello di Galois. Nel caso di un algebroide di Hopf si dimostra che ogniH _R-estensione è unaH _L-estensione. Se l'antipode è biiettivo allora si dimostra che anche le nozioni di estensioniH _R-Galois eH _L-Galois coincidono. I risultati per le strutture biiettive entwining sono estesi alle strutture entwining su algebre non commutative, al fine di dimostrare un teorema simile al Teorema dii Kreimer-Takeuchi per un Hopf algebroideH proiettivo finitamento generato con antipode biiettivo. Il teorema afferma che ogni estensioneH-GaloisB ⊂A è proiettiva e seA èk-piatto allora la suriettività dell'applicazione canonica è sufficiente a garantire la proprietà di Galois. La teoria di Morita, sviluppata per i coanelli da Caenepeel, Vercruysse e Wang, viene applicata per ottenere criteri equivalenti per la proprietà di Galois per estensioni di algebroidi di Hopf. Questo conduce a risultati analoghi, per algebroidi di Hopf, a quelli ottenuti da Doi per estensioni di algebre di Hopf e da Cohen Fishman e Montgomery nel caso degli algebroidi di Hopf Frobenius.

     

Potenzieren und Wurzelziehen in rationalen Quaternionenalgebren

LetF be a field not of characteristic 2 andQ =F +F i +F j +F k the quaternion algebra overF whereij = -ji =k andi ² = α andj ² = β with 0 ≠ α, β ∈F fixed. (IfF = ℝ and α = β = - 1 thenQ is the division algebra of the Hamilton quaternions.) IfF = ℚ and Q is a division algebra then by embedding certain quadratic number fields inQ we derive an efficient formula to compute the powers of any quaternion. This formula is even true in general and reads as follows. If a, a₁, a₂, a₃ ∈F andn ∈ ℕ then where ω ig a square root of αa₁ ² + βa ₂ ² - αβa ₃ ² in or overF and andA ₀ =na ^n-1. With the help of this formula and related ones we are able to solve the equationX ⁿ =q for arbitrary quaternionsq and positive integers n in case ofF = ℝ and hence in case ofF ⊂ ℝ as well. IfF = ℝ then the total number of all solutions equals 0, 1, 2, 4,n or ∞. (4 is possible even whenn < 4.) In case ofF = ℚ, which we are primarily interested in, there are always either at most six or infinitely many solutions. Further, for everyq ≠ 0 there is at most one solution provided thatn is odd and not divisible by 3. The questions when there are infinitely many solutions and when there are none can always be decided by checking simple conditions on the radicandq ifF = ℝ. ForF = ℚ the two questions are comprehensively investigatet in a natural connection with ternary and quaternary quadratic rational forms. Finally, by applying some of our theorems on powers and roots of quate-rions we also obtain several nice results in matrix theory. For example, for every k ∈ ℤ the mappingA ↦A ^k on the group of all nonsingular 2-by-2 matrices over ℚ is injective if and only ifk is odd and not divisible by 3.

     

Nonlinear volterra integrodifferential equations in a Banach space

We study the Cauchy problem associated with the Volterra integrodifferential equation u\left( t \right) \in Au\left( t \right) + \int {_0^1 B\left( {t - s} \right)u\left( s \right)ds + f\left( t \right),} u\left( 0 \right) = u_0 \in D\left( A \right), whereA is anm-dissipative non-linear operator (or more generally, anm-D(ω) operator), defined onD(A) ⊂X, whereX is a real reflexive Banach space. We show that ifB is of the formB=FA+K, whereF, K :X →D(D _s), whereD _s is the differentiation operator, withF bounded linear andK andD _sK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy problem.