The characterization of zero-sum (mod 2) bipartite Ramsey numbers

Let G be a bipartite graph, with k|e(G). The zero-sum bipartite Ramsey number B(G, Z_k) is the smallest integer t such that in every Z_k-coloring of the edges of K_t,t, there is a zero-sum mod k copy of G in K_t,t. In this article we give the first proof that determines B(G, Z₂) for all possible bipartite graphs G. In fact, we prove a much more general result from which B(G, Z₂) can be deduced: Let G be a (not necessarily connected) bipartite graph, which can be embedded in K_n,n, and let F be a field. A function f : E(K_n,n) → F is called G-stable if every copy of G in K_n,n has the same weight (the weight of a copy is the sum of the values of f on its edges). The set of all G-stable functions, denoted by U(G, K_n,n, F) is a linear space, which is called the K_n,n uniformity space of G over F. We determine U(G, K_n,n, F) and its dimension, for all G, n and F. Utilizing this result in the case F = Z₂, we can compute B(G, Z₂), for all bipartite graphs G. © 1998 John Wiley & Sons, Inc. J. Graph Theory 29: 151–166, 1998     

A simple map with no prime factors

An ergodic measure-preserving transformationT of a probability space is said to be simple (of order 2) if every ergodic joining λ ofT with itself is eitherμ×μ or an off-diagonal measureμ _S , i.e.,μ _S (A×B)=μ(A∩S ^;−n ;B) for some invertible, measure preservingS commuting withT. Veech proved that ifT is simple thenT is a group extension of any of its non-trivial factors. Here we construct an example of a weakly mixing simpleT which has no prime factors. This is achieved by constructing an action of the countable Abelian group ℤ⊕G, whereG=⊕ _i=1 ^∞ ℤ₂, such that the ℤ-subaction is simple and has centralizer coinciding with the full ℤ⊕G-action.     

Total colorings of graphs of order 2n having maximum degree 2n−2

Let χ _t (G) and †(G) denote respectively the total chromatic number and maximum degree of graphG. Yap, Wang and Zhang proved in 1989 that ifG is a graph of orderp having †(G)≥p−4, then χ _t (G≤Δ(G)+2. Hilton has characterized the class of graphG of order 2n having †(G)=2n−1 such that χ _t (G=Δ(G)+2. In this paper, we characterize the class of graphsG of order 2n having †(G)=2n−2 such that χ _t (G=Δ(G)+2 Research supported by National Science Council of the Republic of China (NSC 79-0208-M009-15)     

Bands of invariantly extensible measures

Given two σ-algebrasU ⊂A, invariant under a fixed semigroupG of transformations, the following subsetC of the lattice coneM (U) _G ofG-invariant finite measures onU is shown to be (the positive part of) a band inM (U) _G : AG-invariant measure μ belongs toC iff the setexM Bμ) _G of extremalG-invariant extensions of μ toB is non-empty and eachG-invariant extensionv of μ admits a barycentric decompositionv=→v′ρ(dv′) with some representing probability ρ onexM U μ) _G .—Any band of extensible measures allows to study the corresponding extension problem locally.     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

Abelian 2-class field towers over the cyclotomic {\mathbb {Z}_2}-extensions of imaginary quadratic fields

For the cyclotomic \mathbb Z₂{\mathbb Z_2}-extension k _∞ of an imaginary quadratic field k, we consider whether the Galois group G(k _∞) of the maximal unramified pro-2-extension over k _∞ is abelian or not. The group G(k _∞) is abelian if and only if the nth layer of the \mathbb Z₂{\mathbb {Z}_2}-extension has abelian 2-class field tower for all n ≥ 1. The purpose of this paper is to classify all such imaginary quadratic fields k in part by using Iwasawa polynomials.     

On normal subgroups in the periodic products of S.I. Adian

A subgroup H of a given group G is called a hereditarily factorizable subgroup (HF subgroup) if each congruence on H can be extended to some congruence on the entire group G. An arbitrary group G ₁ is an HF subgroup of the direct product G ₁ × G ₂, as well as of the free product G ₁ * G ₂ of groups G ₁ and G ₂. In this paper a necessary and sufficient condition is obtained for a factor Gi of Adian’s n-periodic product Π _i∈I ⁿ G _i of an arbitrary family of groups {G _i } _i∈I to be an HF subgroup. We also prove that for each odd n ≥ 1003 any noncyclic subgroup of the free Burnside group B(m, n) contains an HF subgroup isomorphic to the group B(∞, n) of infinite rank. This strengthens the recent results of A.Yu. Ol’shanskii and M. Sapir, D. Sonkin, and S. Ivanov on HF subgroups of free Burnside groups. This result implies, in particular, that each (noncyclic) subgroup of the group B(m, n) is SQ-universal in the class of all groups of period n. Moreover, it turns out that any countable group of period n is embedded in some 2-generated group of period n, which strengthens the previously obtained result of V. Obraztsov. At the end of the paper we prove that the group B(m, n) is distinguished as a direct factor in any n-periodic group in which it is contained as a normal subgroup.     

A new characterization of L 2(q) by its noncommuting graph

Let G be a non-abelian group and associate a non-commuting graph ∇(G) with G as follows: the vertex set of ∇(G) is G\Z(G) with two vertices x and y joined by an edge whenever the commutator of x and y is not the identity. In this short paper we prove that if G is a finite group with ∇(G) ≅ ∇(M), where M = L ₂(q) (q = p ⁿ , p is a prime), then G ≅ M.      

Automorphism Groups of 2-Valent Connected Cayley Digraphs on Regular <Emphasis Type="Italic">p</Emphasis>-Groups

 Let X=Cay(G,S) be a 2-valent connected Cayley digraph of a regular p-group G and let G _R be the right regular representation of G. It is proved that if G _R is not normal in Aut(X) then X≅[2K ₁ ] with n>1, Aut(X) ≅Z ₂ wrZ _2n , and either G=Z _2n+1 =<a> and S={a,a ²ⁿ⁺¹ }, or G=Z _2n ×Z ₂ =<a>×<b> and S={a,ab}. Received: May 26, 1999 Final version received: June 19, 2000     

Bounding the number of connected components of a real algebraic set

For every polynomial mapf=(f ₁,…,f _k): ℝ ⁿ →ℝ ^k , we consider the number of connected components of its zero set,B(Z _f) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree off _i), and thek-tuple (Δ₁,...,Δ₄), Δ _k being the Newton polyhedron off _i respectively. Our aim is to boundB(Z _f) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μ _d (n)=d(2d−1) ⁿ⁻¹. Considered as a polynomial ind, μ _d (n) has leading coefficient equal to 2 ⁿ⁻¹. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μ _d (n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)n ^k−1 dⁿ, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections.     

Balanced Convex Partitions of Measures in ℝ2

 Let n≥2 be an integer and let μ₁ and μ₂ be measures in ℝ² such that each μ _i is absolutely continuous with respect to the Lebesgue measure and μ₁(ℝ²)=μ₂(ℝ²)=n. Let u≠0 be a vector on the plane. We show that if μ₁(B)=μ₂(B)=n for some bounded domain B, then there exist positive integers n ₁,n ₂ with n ₁+n ₂=n and disjoint open half-planes D ₁,D ₂ such that , μ₁(D ₁)=μ₂(D ₁)=n ₁ and μ₁(D ₂)=μ₂(D ₂)=n ₂; or there exist positive integers n ₁,n ₂,n ₃ with n ₁+n ₂+n ₃=n and disjoint open convex domains D ₁,D ₂,D ₃ such that , μ₁(D ₁)=μ₂(D ₁)=n ₁, μ₁(D ₂)= μ₂(D ₂)=n ₂, μ₁(D ₃)=μ₂(D ₃)=n ₃ and such that the ray is parallel to u. We also show a similar result for partitioning of point sets on the plane. Received: November 24, 1999 Final version received: February 9, 2001     

The theory of stationary point processes

An axiomatic formulation is presented for point processes which may be interpreted as ordered sequences of points randomly located on the real line. Such concepts as forward recurrence times and number of points in intervals are defined and related in set-theoretic Note that for α∈A,G ^α may not coverG _α as a convex subgroup and so we cannot use Theorem 1.1 to prove this result. Moreover, all that we know about theG ^α/G_α is that each is an extension of a trivially ordered subgroup by a subgroup ofR. ItB is a plenary subset ofA, then there exists av-isomorphism μ ofG intoV(B, G ^β/G_β), but whether or not μ is ano-isomorphism is not known. Presently at Massachusetts Institute of Technology, Lexington, Mass., U.S.A. This work was supported by the National Aeronautics and Space Administration under research grant NsG-2-59     

A Local Analysis of Imprimitive Symmetric Graphs

Let Г be a G-symmetric graph admitting a nontrivial G-invariant partition . Let Г be the quotient graph of Г with respect to . For each block B ∊ , the setwise stabiliser G_B of B in G induces natural actions on B and on the neighbourhood Г (B) of B in Г . Let G_(B) and G_[B] be respectively the kernels of these actions. In this paper we study certain “local actions" induced by G_(B) and G_[B], such as the action of G_[B] on B and the action of G_(B) on Г (B), and their influence on the structure of Г. Supported by a Discovery Project Grant (DP0558677) from the Australian Research Council and a Melbourne Early Career Researcher Grant from The University of Melbourne.     

The ranks of central unit groups of integral group rings of alternating groups

Let G be a finite group and U(Z(Z G)) be the group of units of the center Z(Z G) of the integral group ring Z G (the central unit group of the ring Z G). The purpose of the present work is to study the ranks r _n of groups U(Z(ZAn)), i.e., of central unit groups of integral group rings of alternating groups A _n . We shall find all values n for r _n = 1 and propose an approach on how to describe the groups U(Z(ZAn)) in these cases, and we will present some results of calculations of r _n for n ≤ 600.     

Gruppi Policiclici Dotati di un Automorfismo Uniforme di Ordinepq

An automorphismϕ of a groupG is said to be uniform il for everyg ∈G there exists anh ∈G such thatG=h ⁻¹ h ^ρ . It is a well-known fact that ifG is finite, an automorphism ofG is uniform if and only if it is fixed-point-free. In [7] Zappa proved that if a polycyclic groupG admits an uniform automorphism of prime orderp thenG is a finite (nilpotent)p′-group. In this paper we continue Zappa’s work considering uniform automorphism of orderpg (p andq distinct prime numbers). In particular we prove that there exists a constantμ (depending only onp andq) such that every torsion-free polycyclic groupG admitting an uniform automorphism of orderpq is nilpotent of class at mostμ. As a consequence we prove that if a polycyclic groupG admits an uniform automorphism of orderpq thenZ _μ (G) has finite index inG.

Al professore Guido Zappa per il suo 90⁰ compleanno     

Super Connectivity and Super Edge Connectivity of the Mycielskian of a Graph

Mycielski introduced a new graph transformation μ(G) for graph G, which is called the Mycielskian of G. A graph G is super connected or simply super-κ (resp. super edge connected or super-λ), if every minimum vertex cut (resp. minimum edge cut) isolates a vertex of G. In this paper, we show that for a connected graph G with |V(G)| ≥ 2, μ(G) is super-κ if and only if δ(G) < 2κ(G), and μ(G) is super-λ if and only if G\ncong K₂{G\ncong K_2}.     

Dissipation of Convolution Powers in a Metric Group

In contrast to what is known about probability measures on locally compact groups, a metric group G can support a probability measure μ which is not carried on a compact subgroup but for which there exists a compact subset C⊆G such that the sequence μ ⁿ (C) fails to converge to zero as n tends to ∞. A noncompact metric group can also support a probability measure μ such that supp μ=G and the concentration functions of μ do not converge to zero. We derive a number of conditions which guarantee that the concentration functions in a metric group G converge to zero, and obtain a sufficient and necessary condition in order that a probability measure μ on G satisfy lim  _n→∞ μ ⁿ (C)=0 for every compact subset C⊆G. Supported by an NSERC Grant.     

Weighted Bergman spaces¶associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H _ℂ\G _ℂ. Then there exist certain G-invariant Stein subdomains Ξ of H _ℂ\G _ℂ. The Haar measure on H _ℂ\G _ℂ gives rise to a G-invariant measure on Ξ. With respect to this measure one can define the Bergman space B ²(Ξ) of square integrable holomorphic functions on Ξ. The group G acts unitarily on the Hilbert space B ²(Ξ) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B ²(Ξ), i.e., the disintegration formula for the left regular representation into irreducibles. Received: Received: 23 November 1998     

Adaptive policies for time-varying stochastic systems under discounted criterion

We consider a class of time-varying stochastic control systems, with Borel state and action spaces, and possibly unbounded costs. The processes evolve according to a discrete-time equation x _{n + 1}=G _n (x _n , a _n , ξ_n), n=0, 1, … , where the ξ_n are i.i.d. ℜ^k-valued random vectors whose common density is unknown, and the G _n are given functions converging, in a restricted way, to some function G _∞ as n→∞. Assuming observability of ξ_n, we construct an adaptive policy which is asymptotically discounted cost optimal for the limiting control system x _n+1=G _∞ (x _n , a _n , ξ_n).     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.