Moment inequalities and central limit properties of isotropic convex bodies

The object of our investigations are isotropic convex bodies , centred at the origin and normed to volume one, in arbitrary dimensions. We show that a certain subset of these bodies – specified by bounds on the second and fourth moments – is invariant under forming ‘expanded joinsrsquo;. Considering a body K as above as a probability space and taking , we define random variables on K. It is known that for subclasses of isotropic convex bodies satisfying a ‘concentration of mass property’, the distributions of these random variables are close to Gaussian distributions, for high dimensions n and ‘most’ directions . We show that this ‘central limit property’, which is known to hold with respect to convergence in law, is also true with respect to -convergence and -convergence of the corresponding densities. Received: 21 March 2001 / in final form: 17 October 2001 / Published online: 4 April 2002     

Finite groups with an almost regular automorphism of order four

P. Shumyatsky’s question 11.126 in the “Kourovka Notebook” is answered in the affirmative: it is proved that there exist a constant c and a function of a positive integer argument f(m) such that if a finite group G admits an automorphism ϕ of order 4 having exactly m fixed points, then G has a normal series G ⩾ H ⩽ N such that |G/H| ⩽ f(m), the quotient group H/N is nilpotent of class ⩽ 2, and the subgroup N is nilpotent of class ⩽ c (Thm. 1). As a corollary we show that if a locally finite group G contains an element of order 4 with finite centralizer of order m, then G has the same kind of a series as in Theorem 1. Theorem 1 generalizes Kovács’ theorem on locally finite groups with a regular automorphism of order 4, whereby such groups are center-by-metabelian. Earlier, the first author proved that a finite 2-group with an almost regular automorphism of order 4 is almost center-by-metabelian. The proof of Theorem 1 is based on the authors’ previous works dealing in Lie rings with an almost regular automorphism of order 4. Reduction to nilpotent groups is carried out by using Hall-Higman type theorems. The proof also uses Theorem 2, which is of independent interest, stating that if a finite group S contains a nilpotent subgroup T of class c and index |S: T | = n, then S contains also a characteristic nilpotent subgroup of class ⩽ c whose index is bounded in terms of n and c. Previously, such an assertion has been known for Abelian subgroups, that is, for c = 1. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 575–602, September–October, 2006.     

Subgroups of free profinite groups and large subfields of\mathop Q\limits^ \sim

We prove that many subgroups of free profinite groups are free, and use this to give new examples of pseudo-algebraically closed subfields of satisfying Hilbert’s Irreducibility Theorem, and to solve problems posed by M. Jarden and A. Macintyre. We also find a subfield of which does not satisfy Hilbert’s Irreducibility Theorem, but all of whose proper finite extensions do. The first author was supported by NSF grant MCS76-11625.     

A desarguesian theorem for algebraic combinatorial geometries

The points of an algebraic combinatorial geometry are equivalence classes of transcendentals over a fieldk; two transcendentals represent the same point when they are algebraically dependent overk. The points of an algebraically closed field of transcendence degree two (three) overk are the lines (resp. planes) of the geometry. We give a necessary and sufficient condition for two coplanar lines to meet in a point (Theorem 1) and prove the converse of Desargues’ theorem for these geometries (Theorem 2). A corollary: the “non-Desargues” matroid is non-algebraic. The proofs depend on five properties (or postulates). The fifth of these is a deep property first proved by Ingleton and Main [3] in their paper showing that the Vámos matroid is non-algebraic.     

Diamond theorem for a finitely generated free profinite group

We extend Haran’s Diamond Theorem to closed subgroups of a finitely generated free profinite group. This gives an affirmative answer to Problem 25.4.9 of Fried and Jarden (in Field Arithmetic, 2nd edn, Springer, Berlin Heidelberg, New York, 2005).     

Divisibility in O-primitive lattice-ordered permutation groups

We show that divisibility fails in a wide spectrum of periodic lattice-ordered permutation groups. The methods are applied to the six previously known ‘pathological’ doubly transitive groups represented on ℝ or on ℝ⁺, and only two of these groups turn out to be divisible.     

A combination theorem for Veech subgroups of the mapping class group

In this paper we prove a combination theorem for Veech subgroups of the mapping class group analogous to the first Klein–Maskit combination theorem for Kleinian groups in which two Fuchsian subgroups are amalgamated along a parabolic subgroup. As a corollary, we construct subgroups of the mapping class group (for all genera at least 2), which are isomorphic to non-abelian closed surface groups in which all but one conjugacy class (up to powers) is pseudo-Anosov. Received: October 2004 Revision: April 2005 Accepted: April 2005 C.J.L.’s work was partially supported by an NSF postdoctoral fellowship. A.W.R’s work was partially supported by an NSF grant.     

Extending the Ehresmann-Schein-Nambooripad Theorem

We extend the ‘∨-premorphisms’ part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (J. Algebra 141:422–462, 1991) for the ‘morphisms’ part. However, it is so-called ‘∧-premorphisms’ which have proved useful in recent years in the study of partial actions. We therefore obtain an Ehresmann-Schein-Nambooripad-type theorem for (ordered) ∧-premorphisms in the case of two-sided restriction semigroups and inductive categories. As a corollary, we obtain such a theorem in the inverse case.     

On profinite groups with finite abelianizations

Profinite groups with finite p-abelianizations arise in various contexts: group theory, number theory and geometry. Using Ph. Furtw?ngler’s transfer vanishing theorem it will be proved that a finitely generated profinite group Ĝ with this property satisfies 〚Ĝ〛) = 0 (Thm. A). As a consequence one finds that a hereditarily just-infinite non-virtually cyclic pro-p group has only one end (Cor. B). Applied to 3-dimensional Poincaré duality groups, Theorem A yields a generalization of A. Reznikov’s theorem on 3-dimensional co-compact hyperbolic lattices violating W. Thurston’s conjecture (Thm. C).     

Random trees under CH

We extend Jensen’s Theorem that Souslin’s Hypothesis is consistent with CH, by showing that the statement Souslin’s Hypothesis holds in any forcing extension by a measure algebra is consistent with CH. We also formulate a variation of the principle (*) (see [AT97], [Tod00]) for closed sets of ordinals, and show its consistency relative to the appropriate large cardinal hypothesis. Its consistency with CH would extend Silver’s Theorem that, assuming the existence of an inaccessible cardinal, the failure of Kurepa’s Hypothesis is consistent with CH, by its implication that the statement Kurepa’s Hypothesis fails in any forcing extension by a measure algebra is consistent with CH.     

E-free objects and e-locality for completely regular semigroups

We prove that the e-variety CR(H), of all completely regular semigroups whose subgroups belong to some group variety H, is e-local; that is, every regular, locally completely regular semigroupoid [with subgroups fromH] divides a completely regular semigroup [with subgroups from H], in a ‘regular’ way. In a future paper with P.G. Trotter, this theorem will be applied to semidirect products of e-varieties and to e-free E-solid regular semigroups. A key role in the proof is played by the e-free semigroups in the e-variety CR(H). We provide a solution to the ‘word problem’ in these semigroups, in the style of that for free completely regular semigroups given by Kadourek and Polàk. The solution is derived from the author's work on free products of completely regular semigroups. Communicated by F. Pastijn The author is indebted to the Australian Research Council and to National Science Foundation grant INT-8913404 for their support of this research.     

A Ramsey bound on stable sets in Jordan pillage games

Jordan (J Econ Theory 131(1):26–44, 2006) defined ‘pillage games’, a class of cooperative games whose dominance operator is represented by a ‘power function’ satisfying coalitional and resource monotonicity axioms. In this environment, he proved that stable sets must be finite. We provide a graph theoretical interpretation of the problem which tightens the finite bound to a Ramsey number. We also prove that the Jordan pillage axioms are independent.     

Cohomological stratification of diagram algebras

The class of cellularly stratified algebras is defined and shown to include large classes of diagram algebras. While the definition is in combinatorial terms, by adding extra structure to Graham and Lehrer’s definition of cellular algebras, various structural properties are established in terms of exact functors and stratifications of derived categories. The stratifications relate ‘large’ algebras such as Brauer algebras to ‘smaller’ ones such as group algebras of symmetric groups. Among the applications are relative equivalences of categories extending those found by Hemmer and Nakano and by Hartmann and Paget, as well as identities between decomposition numbers and cohomology groups of ‘large’ and ‘small’ algebras.     

PB∞-subgroups of Butler groups

It is well known [BF] that in the constructible universe (V=L) the class ofB ₁-groups is closed under prebalanced subgroups. A similar attempt at the increasing countable unions of prebalanced subgroups has been done in [B]. Here we give an affirmative answer to a similar question concerning the so-calledP B ^∞-subgroups. This research was initiated while the first author was on a short visit at the University of Colorado, Colorado Springs, in February 1994 and supported by Charles University grant GAUK 45/94.     

Weak Keys in MST 1

The public key cryptosystem MST₁ has been introduced by Magliveras et al. [12] (Public Key Cryptosystems from Group Factorizations. Jatra Mountain Mathematical Publications). Its security relies on the hardness of factoring with respect to wild logarithmic signatures. To identify ‘wild-like’ logarithmic signatures, the criterion of being totally-non-transversal has been proposed. We present tame totally-non-transversal logarithmic signatures for the alternating and symmetric groups of degree ≥ 5. Hence, basing a key generation procedure on the assumption that totally-non-transversal logarithmic signatures are ‘wild like’ seems critical. We also discuss the problem of recognizing ‘weak’ totally-non-transversal logarithmic signatures, and demonstrate that another proposed key generation procedure based on permutably transversal logarithmic signatures may produce weak keys. Communicated by: P. Wild     

The Blackwell and Dubins Theorem and Rényi’s Amount of Information Measure: Some Applications

This survey paper provides first for an overview of how quantum-like concepts could be used in macroscopic environments like economics. The paper then argues for the use of the concept of a quantum mechanical wave function as an ‘information wave function’. A rationale is provided on why such interpretation is reasonable. After having defined the ‘information wave function’, Ψ(q), we argue how | Ψ(q)| ² can be interpreted as a Radon-Nikodym derivative. We consider how we can connect, using the | Ψ(q)| ², the Blackwell and Dubins (Ann. Math. Stat. 33:882–886, 1961) Theorem with Rényi’s (Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, 1961) measure of quantity of information. We also define ‘ambiguity of information’ and ‘multi-sourced information’.     

Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.     

On the first cohomology of arithmetic groups

We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p ², p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices. Received: 14 September 2000 / Accepted: 6 June 2001     

On some classes of regular semigroups

An idempotent e of a semigroup S is called right [left] principal (B.R. Srinivasan, [2]) if fef=fe [fef=ef] for every idempotent f of S. Say that S has property (LR) [(LR₁)] if every ℒ-class of S contains atleast [exactly] one right principal idempotent. There and six further properties obtained by replacing, ‘ℒ-class’ by ‘ℛ-class’ and/or ‘right principal’ by ‘left principal’ are examined. If S has (LR₁), the set of right principal elementsa of S (aa′ is right principal for some inversea′ ofa) is an inverse subsemigroup of S, generalizing a theorem of Srinivasan [2] for weakly inverse semigroups. It is shown that the direct sum of all dual Schützenberger representations of an (LR) semigroup is faithful (cf[1], Theorem 3.21, p. 119). Finally, necessary and sufficient conditions are given on a regular subsemigroup S of the full transformation semigroup on a set in order that S has each of the properties (LR), (LR₁), etc.