O-stable ordered groups

An ordered structure M is said to be o-λ-stable if, for every A ⊆ M with |A| ≤ λ and every cut in M, at most λ 1-types over A are consistent with the cut. In the present article, we prove that every o-stable group is abelian. We also study definable subsets and unary functions of o-stable groups.     

When do random subsets decompose a finite group?

Let A, B be two random subsets of a finite group G. We consider the event that the products of elements from A and B span the whole group, i.e. [AB ∪ BA = G]. The study of this event gives rise to a group invariant we call Θ(G). Θ(G) is between 1/2 and 1, and is 1 if and only if the group is abelian. We show that a phase transition occurs as the size of A and B passes √Θ(G)|G| log |G|; i.e. for any ɛ > 0, if the size of A and B is less than (1 − ɛ)√Θ(G)|G| log |G|, then with high probability AB ∪ BA ≠ G. If A and B are larger than (1 + ɛ)√Θ(G)|G| log |G|, then AB ∪ BA = G with high probability.     

A note on Freiman models in Heisenberg groups

Green and Ruzsa recently proved that for any s ≥ 2, any small squaring set A in a (multiplicative) abelian group, i.e., |A·A| < K|A|, has a Freiman smodel: it means that there exists a group G and a Freiman s-isomorphism from A into G such that |G| < f (s,K)|A|.     

Relative purity over noetherian rings

In this note we are going to show that if M is a left module over a left noetherian ring R of the infinite cardinality λ ≥ |R|, then its injective hull E(M) is of the same size. Further, if M is an injective module with |M| ≥ (2^λ)⁺ and K ≤ M is its submodule such that |M/K| ≤ λ, then K contains an injective submodule L with |M/L| ≤ 2^λ. These results are applied to modules which are torsionfree with respect to a given hereditary torsion theory and generalize the results obtained by different methods in author’s previous papers: [A note on pure subgroups, Contributions to General Algebra 12. Proceedings of the Vienna Conference, June 3–6, 1999, Verlag Johannes Heyn, Klagenfurt, 2000, pp. 105–107], [Pure subgroups, Math. Bohem. 126 (2001), 649–652]. This research has been partially supported by the Grant Agency of the Charles University, grant #GAUK 301-10/203115/B-MAT/MFF and also by the institutional grant MSM 113 200 007.     

Singular Integrals with Bilinear Phases

We prove the boundedness from Lp（T2） to itself, 1 〈 p 〈∞, of highly oscillatory singular integrals Sf（x, y） presenting singularities of the kind of the double Hilbert transform on a non-rectangular domain of integration, roughly speaking, defined by ｜y′｜ 〉 ｜x′｜, and presenting phases λ（Ax ＋ By） with 0≤ A, B ≤ 1 and λ≥ 0. The norms of these oscillatory singular integrals are proved to be independent of all parameters A1 B and A involved. Our method extends to a more general family of phases. These results are relevant to problems of almost everywhere convergence of double Fourier and Walsh series.     

(<Emphasis Type="Italic">p, λ</Emphasis>)-Koszul algebras and modules

In this paper, the notions of (p, λ)-Koszul algebra and (p, λ)-Koszul module are introduced. Some criteria theorems for a positively graded algebra A to be (p, λ)-Koszul are given. The notion of weakly (p, λ)-Koszul module is defined as well and let WK _λ ^p (A) denote the category of weakly (p, λ)-Koszul modules. We show that M ∈ WK _λ ^p (A) if and only if it can be approximated by (p, λ)-Koszul submodules, which is equivalent to that G(M) is a (p, λ)-Koszul module, where G(M) denotes the associated graded module of M. As applications, the relationships of the minimal graded projective resolutions of M, G(M) and (p, λ)-Koszul submodules are established. In particular, for a module M ∈ WK _λ ^p (A) we prove that ⊕ _i≥0 Ext _A ⁱ (M,A ₀) ∈ gr ₀(E(A)), we also get as a consequence that the finitistic dimension conjecture is valid in WK _λ ^p (A) under certain conditions.     

Lie jets and symmetries of prolongations of geometric objects

The Lie jet L _θ λ of a field of geometric objects λ on a smooth manifold M with respect to a field θ of Weil A-velocities is a generalization of the Lie derivative L _v λ of a field λ with respect to a vector field v. In this paper, Lie jets L _θ λ are applied to the study of A-smooth diffeomorphisms on a Weil bundle T ^A M of a smooth manifold M, which are symmetries of prolongations of geometric objects from M to T ^A M. It is shown that vanishing of a Lie jet L _θ λ is a necessary and sufficient condition for the prolongation λ ^A of a field of geometric objects λ to be invariant with respect to the transformation of the Weil bundle T ^A M induced by the field θ. The case of symmetries of prolongations of fields of geometric objects to the second-order tangent bundle T ² M are considered in more detail.     

On rational approximations with denominators from thin sets

Given a thin setA we construct a Liouville number α such that for |α −u/υ|, υ ∈A, a nontrivial lower bound can be given.     

Verschiebung and Frobenius operators

Metropolis and Rota introduced the concept of the necklace ring Nr(A) of a commutative ringA. WhenA contains Q as a subring there is a natural bijection γ:Nr(A→1+tA[]. Grothendieck has introduced a ring structure on 1+tA[t] while studyingK-theoretic Chern classes. Nr(A) comes equipped with two families of operatorsF _r,V _r called the Frobenius and Verschiebung operators. Mathematicians studying formal group laws have introduced two families of operators,F _r, andV _r on 1+tA[t]. Metropolis and Rota have not however tried to show that γ preserves, these operators. They transport the operators from Nr(A) to 1+tA[t] using γ. In our present paper we show that γ does preserve all these operators. Part of this work was done while the author was visiting the Institute of Mathematical Sciences, Madras.     

On eigenvalues of symmetric (+1, −1) matrices

To every symmetric matrixA with entries ±1, we associate a graph G(A), and ask (for two different definitions of distance) for the distance ofG(A) to the nearest complete bipartite graph (cbg). Letλ ₁(A),λ ¹ (A) be respectively the algebraically largest and least eigenvalues ofA. The Frobenius distance (see Section 4) to the nearest cbg is bounded above and below by functions ofn −λ ₁ (A), wheren=ord A. The ordinary distance (see Section 1) to the nearest cbg is shown to be bounded above and below by functions ofλ ¹ (A). A curious corollary is: there exists a functionf (independent ofn, and given by (1.1)), such that |λ _i (A) | ≦f(λ ¹(A), whereλ _i (A) is any eigenvalue ofA other thanλ _i (A). This work was supported (in part) by the U.S. Army under contract #DAHC04-C-0023.     

On purifiable subgroups in arbitrary abelian groups

We determine the structure of a p-pure[pure] hull of a p-purifiable [purifiable] subgroup of an arbitrary abelian group. Moreover, we prove that a subgroup A of an abelian group G is purifiable in G if and only if A is p-purifiable in G for every prime p. Using these results, we characterize the groups G for which all subgroups are purifiable in G. Furthermore, we establish several properties of purifiable subgroups.     

Restricted sumsets and a conjecture of Lev

LetA, B, S be finite subsets of an abelian groupG. Suppose that the restricted sumsetC={α+b: α ∈A, b ∈B, and α − b ∉S} is nonempty and somec∈C can be written asa+b witha∈A andb∈B in at mostm ways. We show that ifG is torsion-free or elementary abelian, then |C|≥|A|+|B|−|S|−m. We also prove that |C|≥|A|+|B|−2|S|−m if the torsion subgroup ofG is cyclic. In the caseS={0} this provides an advance on a conjecture of Lev. This author is responsible for communications, and supported by the National Science Fund for Distinguished Young Scholars (No. 10425103) and the Key Program of NSF (No. 10331020) in China.     

On Some Rational Triangles

We prove that every set of n ≥ 3 points in \mathbbR²{\mathbb{R}^2} can be slightly perturbed to a set of n points in \mathbbQ²{\mathbb{Q}^2} so that at least 3(n − 2) of mutual distances between those new points are rational numbers. Some special rational triangles that are arbitrarily close to a given triangle are also considered. Given a triangle ABC, we show that for each ε > 0 there is a triangle A′B′C′ with rational sides and at least one rational median such that |AA′|, |BB′|, |CC′| < ε and a Heronian triangle A′′B′′C′′ with three rational internal angle bisectors such that A^￠￠, B^￠￠, C^￠￠ ? \mathbbQ²{A^{\prime\prime}, B^{\prime\prime}, C^{\prime\prime} \in \mathbb{Q}^2} and |AA′′|, |BB′′|, |CC′′| < ε.     

On thep-rank of Ext

AssumeV =L andλ is regular smaller than the first weakly compact cardinal. Under those circumstances and with arbitrary requirements on the structure of Ext (G, ℤ) (under well known limitations), we construct an abelian groupG of cardinalityλ such that for noG′ ⊆G, |G′| <λ isG/G′ free and Ext (G, ℤ) realizes our requirements. Deceased. Partially supported by NSERC. Partially supported by the United States — Israel Binational Science Foundation. Publication No. 314.     

Infinitary model theory of abelian groups

The paper is a survey of results in the model theory of abelian groups, dealing with two sorts of problems: finding invariants which classify groups up toL _λκ-equivalence; and determining whether certain classes of groups are definable inL _λκ. Research supported by NSF grant GP 43910     

An identity for normal-like operators

Forλεσ(A) (A a bounded linear operator on a Hilbert space) withλ a boundary point of the numerical range, the ‘spectral theory’ forλ is ‘just as ifA were normal’. IfA isnormal-like (the smallest disk containingσ(A) has radiusr=inf _z ‖A − z‖), then also sup {‖Ax‖² − |〈x.Ax〉|²:‖x‖=1}=r ². This research was partially supported by Air Force Contract AF-AFOSR-62-414.     

Generating Uniform Random Vectors in Z p k : The General Case

We consider the rate of convergence of the Markov chain X _n+1=A X _n +B _n (mod p), where A is an integer matrix with nonzero eigenvalues, and {B _n } _n is a sequence of independent and identically distributed integer vectors, with support not parallel to a proper subspace of Q ^k invariant under A. If for all eigenvalues λ _i of A, then n=O((ln p)²) steps are sufficient and n=O(ln p) steps are necessary to have X _n sampling from a nearly uniform distribution. Conversely, if A has the eigenvalues λ _i that are roots of positive integer numbers, |λ ₁|=1 and |λ _i |>1 for all , then O(p ²) steps are necessary and sufficient.      

Counting points in hypercubes and convolution measure algebras

It is shown that ifA andB are non-empty subsets of {0, 1} ⁿ (for somenεN) then |A+B|≧(|A||B|)^α where α=(1/2) log₂ 3 here and in what follows. In particular if |A|=2 ^n-1 then |A+A|≧3 ^n-1 which anwers a question of Brown and Moran. It is also shown that if |A| = 2 ^n-1 then |A+A|=3 ^n-1 if and only if the points ofA lie on a hyperplane inn-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)^α. The above results imply the following improvement of a result of Talagrand [7]: ifX andY are compact subsets ofK (the Cantor set) withm(X),m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))^α wherem is the usual measure onK and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing thatm is not concentrated on any proper Raikov system.     

On cardinal interpolation by Gaussian radial-basis functions: Properties of fundamental functions and estimates for Lebesgue constants

Suppose λ is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal functionL _λ(x)

Smooth 4-folds which contain a P1-bundle¶as an ample divisor

We show that if a smooth projective 4-fold M contains an ample divisor A which is P ¹-bundle π :A→S over a smooth projective surface S, π is extended to a P ²-bundle π :S→S, unless $A$ is isomorphic to P ²×P ¹. Received: 28 September 1998 / Revised version: 16 August 1999