Division algebras of degree 4 and 8 with involution

We develop necessary and sufficient conditions for central simple algebras to have involutions of the first kind, and to be tensor products of quaternion subalgebras. The theory is then applied to give an example of a division algebra of degree 8 with involution (of the first kind), without quaternion subalgebras, answering an old question of Albert; another example is a division algebra of degree 4 with involution (*) has no (*)-invariant quaternion subalgebras. The research of the second author is supported by the Anshel Pfeffer Chair. The third author would like to express his gratitude to Professor J. Tits for many stimulating conversations.     

Local degree distributions: Examples and counterexamples

In several scale free graph models the asymptotic degree distribution and the characteristic exponent change when only a smaller set of vertices is considered. After recalling the sufficient conditions for the existence of asymptotic local degree distribution [1], several random graph models are presented that satisfy these assumptions. We show the necessity of the main conditions by constructing counterexamples.     

The similarity degree of an operator algebra, II

For every integer , there is a unital closed subalgebra with similarity degree equal precisely to d, in the sense of our previous paper. This means that for any unital homomorphism we have with independent of u, and the exponent d in this estimate cannot be improved. The proof that the degree is larger than crucially uses an upper bound for the norms of certain Gaussian random matrices due to Haagerup and Thorbj?rnsen. We also include several complements to our previous publications on the same subject. Received: 25 April, 1999; Accepted: 23 June, 1999     

Division of the Dickson algebra by the Steinberg unstable module

We compute the division of the Dickson algebra by the Steinberg unstable module in the category of unstable modules over the mod-2 Steenrod algebra.     

Division algebras of prime degree with infinite genus

The genus gen(D) of a finite-dimensional central division algebra D over a field F is defined as the collection of classes [D′] ∈ Br(F), where D′ is a central division F-algebra having the same maximal subfields as D. For any prime p, we construct a division algebra of degree p with infinite genus. Moreover, we show that there exists a field K such that there are infinitely many nonisomorphic central division K-algebras of degree p and any two such algebras have the same genus.     

Two divergence-rate counterexamples

An example is constructed of a pair of ergodic processes for which the upper and lower divergence rates are not equal. An example is also constructed of a pair of ergodic processes for which the limiting rate is 0, yet the processes are not equal. In both cases it is shown how the techniques will yieldB-process counterexamples.Partially supported by NSF Grant DMS-9024240.From 9/1 to 12/1 each year: Mathematics Institute, POB 127, 1364 Budapest, Hungary. 011-361-1-177-175. At other time: Department of Mathematics, University of Toledo, Toledo, Ohio 43606. (419) 537-2069.     

Superalgebras and counterexamples

Dedicated to the 70th birthday of Anatolii Illarionovich Shirshov.     

Two counterexamples in normalization

In this paper, we offer a correction to Commutative Algebra by Oscar Zariski and Pierre Samuel by noting two erroneous theorems and offering counterexamples for each.

     

The lower and upper bounds of the degree distribution of a graded algebra

For a graded algebra,the minimal projective resolution often reveals amounts of information.All generated degrees of modules in the minimal resolution of the trivial module form a sequence,which can be called the degree distribution of the algebra.We try to find lower and upper bounds of the degree distribution,introduce the notion of(s,t)-(homogeneous) determined algebras and construct such algebras with the aid of algebras with pure resolutions.In some cases,the Ext-algebra of an(s,t)-(homogeneous) determined algebra is finitely generated.     

Family of counterexamples to King's conjecture

In this short Note we present an infinite family of arbitrary high dimensional counterexamples to the King's conjecture.     

Domination graphs: examples and counterexamples

We give a counterexample to a conjecture of Dahlhaus et al. [3] claiming that the Special Quadratic Consensus Method yields a polynomial-time recognition for domination graphs, and discuss several new properties of domination graphs.