A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

Paracyclic and quasi-cyclic groups

Summary For a finite group G put l_g= min {[G: ,]¦xG}and .Call G paracyclic iff l_GG It is proved that any metacyclic group is paracyclic and that l_G2_G whenever ¦G¦=p expG (p prime).Moreover all non-paracyclic groups satisfying the last condition are classified.Research partially supported by GNSAGA of CNR.     

On the isomorphisms and automorphism groups of circulants

Denote byC _n(S) the circulant graph (or digraph). LetM be a minimal generating element subset ofZ _n, the cyclic group of integers modulon, and In this paper, we discuss the problems about the automorphism group and isomorphisms ofC _n(S). When M S , we determine the automorphism group ofC _n(S) and prove that for any T if and only ifT = S, where is an integer relatively prime ton. The automorphism groups and isomorphisms of some other types of circulant graphs (or digraphs) are also considered. In the last section of this paper, we give a relation between the isomorphisms and the automorphism groups of circulants.     

More on limit theorems for iterates of probability measures on semigroups and groups

Summary In this paper, we continue earlier works of one of the authors on vague convergence of the sequence _k,n= _k+1 *...* _n, where _n is a sequence of probability measures on semigroups or groups. Typical results in this paper are: Theorem. Let S be a locally compact noncompact second countable group such that being the support of a probability measure on S. Suppose there exists an open set V with compact closure such that x ^–1 Vx=V for every xS. Then for all compact sets K, sup{ ⁿ (Kx): xS0 as n. Theorem. Let S be an at most countable discrete group. Let _n be a sequence of probability measures on S. Then for all nonnegative integers k, the sequence _k,n converges vaguely to some probability measure if and only if there exists a finite subgroup G such that the series and for any proper subgroup G of G and any choice of elements g_n in S, the series . A sufficient condition for the vague convergence of the sequence _k,n to a probability measure is that (i) there exists a finite subgroup G such that and (ii) _n(e)>s>0 for all n, e being the identity.The author was supported by NSF grant MCS77-03639     

Properties of a jointly ergodic action of the direct product of two groups

An ergodic action a of the direct product of and , not isomorphic to a product of actions of and G, is constructed, such that the actions of and G separately are not ergodic. The actions of on its ergodic components are metrically isomorphic if and only if these components are taken into one another by the action of G. Finally, the centralizerC (×G) is such thatC (×G)/(×G)₂.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 684–688, May, 1991.     

[FIA] B − Gruppen und Hypergruppen

The spaceG _B of -orbits of an [FIA] _B ^– groupG is a commutative hypergroup (=commutative convo in Jewett's terminology). Its space of hypergroup characters is a hypergroup itself (see [3]) which can be identified withE (G, B) (the extreme points ofB-invariant positive definite continuous functionsp:G withp (e)=1), endowed with the topology of compact convergence. We determine the structure space ofL ¹ (E(G, B), _PL ) where _PL is the Plancherel measure (which is the Haar measure ofE(G, B)) and characterize the convolution algebrasM (E(G, B)) andL ¹ (E(G, B)) via the inverse Fourier-Stieltjes transformation.     

Dividierbarkeit und Typenbedingungen in verallgemeinerten nilpotenten Gruppen

In this paper we generalize theorems of S. N. ernikov, G. Baumslag and R. B. Warfield. S. N. ernikov showed that a hypercentral groupG is -divisible whenever,G/G' is -divisible. His theorem is identical to a special case of TheoremA. G. Baumslag andR. B. Warfield proved that the commutator subgroup _k of a nilpotent groupG is -divisible wheneverG (Baumslag [1], 14.5) orG/Z _k-1 (Warfield [8], 4.13) is -divisible. We show this implication for hypercentral and the torsion-free weakly nilpotent groups. In TheoremD similar statements for the typesets ofG/Z _k-1 and _k are proved; hereG is nilpotent or torsion-free weakly nilpotent. The commutator formulas of Sect. 3 are essential for the proof of the theorems. These formulas seem to be new and are interesting in their own right.     

Hoeffding's inequalities: A counterexample

A sequence : ₀ satisfiesHoeffding's inequality of order n if wheneverX ₁,...,X _n are independent nonnegative integer-valued elementary random variables and are independent identically distributed nonnegative integer-valued elementary random variables, the common distribution of which is the average of those ofX ₁,...,X _n. We show that for each integerm greater than 2 there exists a sequence satisfying Hoeffding's inequality of every order greater thanm but not that of orderm. This answers a question raised by Berg, Christensen, and Ressel.     

Automatic structures,rational growth,and geometrically finite hyperbolic groups

Summary We show that the set of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic groupG is dense in the product of the sets over all maximal parabolic subgroupsP. The set of equivalence classes of biautomatic structures onG is isomorphic to the product of the sets over the cusps (conjugacy classes of maximal parabolic subgroups) ofG. Each maximal parabolicP is a virtually abelian group, so and were computed in [NS1].We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics forG is regular. Moreover, the growth function ofG with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are.Oblatum 14-VI-1993 & 4-I-1994Both authors acknowledge support from the NSF for this research.     

A stability theorem for a class of linear evolution systems

Dynamical systems of the form ,u(0)=u ₀, where,B is a densely defined linear operator mapping its domainD (B ) into — the infinitesimal generator of a semigroup of operatorsT (t, B) of classC ₀ — are investigated, such that for each solutionu to , whereP is the spectral eigenprojection onto the null space ofB.It is shown that under some general hypotheses concerning spectral properties ofB the above stability condition is equivalent with the following situation: There exist (i) a normal generating coneK such thatT(t;B)KK fort0 and (ii) a strictly positive element in the dual coneK such that , whereB denotes the dual ofB. Condition (ii) implies the so called total concentration time preservation, i. e. .     

Dimension Theory and Nonstable K1 of Quadratic Modules

Employing Bak's dimension theory, we investigate the nonstable quadratic K-group K _1,2n (A, ) = G _2n (A, )/E _2n (A, ), n 3, where G _2n (A, ) denotes the general quadratic group of rank n over a form ring (A, ) and E _2n (A, ) its elementary subgroup. Considering form rings as a category with dimension in the sense of Bak, we obtain a dimension filtration G _2n (A, ) G _2n ⁰(A, ) ; G _2n ¹(A, ) ... E _2n (A, ) of the general quadratic group G _2n (A, ) such that G _2n (A, )/G _2n ⁰(A, ) is Abelian, G _2n ⁰(A, ) G _2n ¹(A, ) ... is a descending central series, and G _2n ^d(A)(A, ) = E _2n (A, ) whenever d(A) = (Bass–Serre dimension of A) is finite. In particular K _1,2n (A, ) is solvable when d(A) < .     

Complex homogeneous spaces of real groups with top homology in codimension two

SupposeX=G/H is a connected homogeneous complex manifold, whereG is a Lie group andH is a closed subgroup. Assume (X) and letG/H G/I be the holomorphic reduction ofX. If the top nonvanishing homology group ofX with coefficients in ₂ is in codimension two, then either a complex Lie group acts tansitively onG/I (see [3]) orG/I is biholomorphic to the unit disk.Partially supported by NSERC Grant A3494 and by the Deutsche Forschungsgemeinschaft     

Transvection free groups and invariants of polynomial tensor exterior algebras

Let :GGl(n, ) be a representation of a finite groupG over a field such that the ring of invariants is a polynomial algebra . It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of ), the invariants ofG acting on the tensor product of a polynomial and an exterior algebra are given by ,d denoting the exterior derivative. We show that in the modular case, the ring of invariants in is of this form if and only if is a polynomial algebra and all pseudoreflections in (G) are diagonalizable.     

Capacity theory without duality

Summary The paper develops a theory of capacity for a Borel right process without duality assumptions. The basic tool in this approach is a stationary process ralative to an excessive measure.IfP _t )t0 denotes the semigroup of the process on the state spaceE and ifm is an excessive measure onE, then there exists a processY = (Y _t ) _t onE with random birth and death and a -finite measureQ _m such thatY is stationary underQ _m and Markov with respect to (P _t ).For a setB inE the hitting (resp. last exit) time ofY is denoted by _B (resp. _B ), andB is called transient (resp. cotransient) ifQ _m ( _B =)= 0 (resp.Q _m ( _B = – )=0. The main theorem then states that for a both transient and contransient setB the distributions of _B and _B underQ _m are the same. For suchB the capacity is denfined byC(B):=Q _m ( _B [0, 1] and the cocapacity by (B):=Q _m ( _B [0, 1], and it is shown that these definitions in fact generalize previous definitions under duality assumptions.Without duality assumption there is no representation of the capacitary potential in terms of a capacitary measure, but there exists a cocapacitary entrance law _t ^B which generalizes the notion of a cocapacitary measure such that (B)= _t ^B (1).The paper contains investigations of transience and cotransience, a decomposition of the cocapacitrary entrance law, some remarks on left versions, and furthermore a generalization of Spitzer's asymptotic formula.Research supported in part by NSF Grant DMS 8419377Research carried out while visiting University of California, San Diego, during Spring 1985     

Hamiltonian cycles in 1-tough graphs

For a graphG, let ₃ = min{ _i=1 ³ d(u_i): {u₁, u₂, u₃} is an independent set ofG} and = min{ _i=1 ³ d(u_i) – is an independent set ofG}. In this paper, we shall prove the following result: LetG be a 1-tough graph withn vertices such that ₃ n and – 4. ThenG is hamiltonian. This generalizes a result of Fassbender [2], a result of Flandrin, Jung and Li [3] and a result of Jung [5].Supported in part by das promotionsstipendium nach dem NaFöG and the Post-Doctoral Foundation of China.     

On π-Solvable and Locally π-Solvable Groups with Factorization

We prove that, in a locally -solvable group G = AB with locally normal subgroups A and B, there exist pairwise-permutable Sylow - and p-subgroups A , A _p and B , B _p , p , of the subgroups A and B, respectively, such that A B is a Sylow -subgroup of the group G and, for an arbitrary nonempty set ,

On the facial structure of the set covering polytope

Given a bipartite graphG = (V, U, E), a cover ofG is a subset with the property that each nodeu U is adjacent to at least one nodev D. If a positive weightc _v is associated with each nodev V, the covering problem (CP) is to find a cover ofG having minimum total weight.In this paper we study the properties of the polytopeQ(G) ^|V| , the convex hull of the incidence vectors of all the covers inG. After discussing some general properties ofQ(G) we introduce a large class of bipartite graphs with special structure and describe several types of rank facets of the associated polytopes.Furthermore we present two lifting procedures to derive valid inequalities and facets of the polytopeQ(G) from the facets of any polytopeQ(G) associated with a subgraphG ofG. An example of the application of the theory to a class of hard instances of the covering problem is also presented.     

Win probabilities and simple majorities in probabilistic voting situations

This paper considers an election between candidatesA andB in which (1) voters may be uncertain about which candidate they will vote for, and (2) the winner is to be determined by a lottery betweenA andB that is based on their vote totals. This lottery is required to treat voters equally, to treat candidates equally, and to respond nonnegatively to increased support for a candidate. The set _n of all such lottery rules based on a total ofn voters is the convex hull of aboutn/2 basic lottery rules which include the simple majority rule. For odd values ofn 3 let , and for even values ofn 4 let . With the average of then voters probabilities of voting forA, it is shown that within _n the simple majority rule maximizes candidateA's overall win probability whenever , and that(n) is the smallest number for which this is true. Similarly, the simple majority rule maximizesB's overall win probability whenever (the average of the voters probabilities of voting forB) is as large as(n). This research was supported by the National Science Foundation, Grant SOC 75-00941.     

Covering properties and Følner conditions for locally compact groups

Summary LetG be a locally compact group with left Haar measurem _G on the Borel sets IB(G) (generated by open subsets) and write |E|=m _G (E). Consider the following geometric conditions on the groupG.(FC If >0 and compact setKG are given, there is a compact setU with 0<|U|< and |x U U|/|U|< for allxK.(A) If >0 and compact setKG, which includes the unit, are given there is a compact setU with 0<|U|< and |K U U|/|U|<.HereA B=(A/B)(B/A) is the symmetric difference set; by regularity ofm _G it makes no difference if we allowU to be a Borel set. It is well known that (A)(FC) and it is known that validity of these conditions is intimately connected with amenability ofG: the existence of a left invariant mean on the spaceCB(G) of all continuous bounded functions. We show, for arbitrary locally compact groupsG, that (amenable)(FC)(A). The proof uses a covering property which may be of interest by itself: we show that every locally compact groupG satisfies.(C) For at least one setK, with int(K)Ø and compact, there is an indexed family {x J}G such that {Kx } is a covering forG whose covering index at each pointg (the number of J withgKx ) is uniformly bounded throughoutG.     

Codings of graphs with binary edge labels

LetG(V,E) be a graph. A mappingf:E{0,1} ^m is called a (binary) coding ofG, if the induced mapping , assigns different vectors to the vertices. For the Boolean sum,f is called aB-code, and for the mod 2 sum anM-code. Letm _B (G) resp.m _M (G) be the smallest lengthm for whichB-codes resp.M-codes are possible. Trivially,m _B (G),m _M (G) log₂|V|. Improving results of Z. Tuza we showm _B (G)log₂|V| + 1,m _M (G)log₂|V|+4.