Lectures on topology of words

We discuss a topological approach to words introduced by the author in [Tu2]–[Tu4]. Words on an arbitrary alphabet are approximated by Gauss words and then studied up to natural modifications inspired by the Reidemeister moves on knot diagrams. This leads us to a notion of homotopy for words. We introduce several homotopy invariants of words and give a homotopy classification of words of length five. Based on notes by Eri Hatakenaka, Daniel Moskovich, and Tadayuki Watanabe     

On the variety question for wRRA

In this paper we consider a question of Jónsson [6] whether the class of weakly representable relation algebras is a variety. We prove that the class is closed under taking homomorphic images provided that a certain embedding condition obtains. Received June 21, 2005; accepted in final form October 17, 2006.     

A note on quasi-Gorenstein rings

In this paper, after giving a criterion for a Noetherian local ring to be quasi-Gorenstein, we obtain some sufficient conditions for a quasi-Gorenstein ring to be Gorenstein. In the course, we provide a slight generalization of a theorem of Evans and Griffith. Received: 16 February 2008     

A note on holomorphic approximation in Banach spaces

Given a complex Banach space X and a holomorphic function f on its unit ball B, we discuss the problem whether f can be approximated, uniformly on smaller balls, by functions g holomorphic on all of X. Research partially supported by NSF grant DMS0700281.     

A note on the congruences of the nakano superlattice and some properties of the associated quotients

In this paper we explore theNakano superlattice (H, ⊔, ⊓), where ⊔, ⊓ are the Nakanohyperoperations x⊔y={z:x∨z=y∨z=x∨y},x⊓y={z:x∧z=y∧z=x∧y}. In particular, we study the properties of congruences on the Nakano superlattice and the associated quotients. New hyperoperations are introduced on the quotient and their properties studied.     

A note on Markov type constants

We prove that, if a geodesic metric space has Markov type 2 with constant 1, then it is an Alexandrov space of nonnegative curvature. The same technique provides a lower bound of the Markov type 2 constant of a space containing a tripod or a branching point. Received: 28 November 2007     

A note on character sums

We investigate certain character sums and prove some discrepancy-type inequalities for incomplete sums.      

A note on planar semimodular lattices

In an earlier paper, we constructed all finite, planar, semimodular lattices in three simple steps from the direct product of two finite chains. In this note we prove that one of the three steps can be eliminated. The research of the first author was supported by the NSERC of Canada.     

A note on primes in short intervals

This paper is concerned with the number of primes in short intervals. We prove that , for θ > 1/2, with the assumption of an heuristic hypothesis weaker than the Lindel?f hypothesis. Received: 8 October 2007, Revised: 14 April 2008     

A note on the generalized rank reduction

We give constructive necessary and sufficient conditions for the validity of the generalized rank reduction formula of Ouellette [12]:

A note on the eigenvalue ratio of vibrating strings

For vibrating strings with symmetric single-well densities, it is known that the ratio λ₂/λ₁1 is maximized when the density is constant. In this note, we extend this result to a class of symmetric densities.      

A note on a spanning 3-tree

A tree T is called a k-tree, if the maximum degree of T is at most k. In this paper, we prove that if G is an n-connected graph with independence number at most n + m + 1 (n≥1,n≥m≥0), then G has a spanning 3-tree T with at most m vertices of degree 3.     

A generalization of an integral of Ramanujan

This paper considers a generalization of an integral introduced by S. Ramanujan in his third notebook. Ramanujan’s integral is itself a version of the dilogarithm,

A note on the instability of embeddings of Cauchy-Riemann manifolds

We prove that there are compact strictly pseudoconvex CR manifolds, embedded into some Euclidean space, that admit small deformations that are also embeddable but their embeddings cannot be chosen close to the original embedding. Both authors were partially supported by NSF grants.     

A note on generalized connectedness

This note is devoted to study the preservation of connectedness under the basic operators in generalized topological spaces. Some characterizations of generalized connectedness are given. As an application, we generalize some results in topological spaces. Supported in part by the NSFC (No. 10571151, 10671173).     

A Construction of Subsets of the Reals which have a Similarity Decomposition

We construct a subset of the set R of real numbers of cardinality |R| which has a similarity decomposition, and which has an ordertype < that of R. Seymour Ginsburg had posed the question whether there exist sets with another ordertype than that of R which also have a similarity decomposition.      

Pro-p groups of rank 3 and the question of Iwasawa

For a prime p > 3 we determine all pro-p groups that satisfy d(G) = d(H) = 3 for all open subgroups H of G. Received: 29 July 2008     

A sharp transition for the two-dimensional Ising percolation

We show that the percolation transition for the two-dimensional Ising model is sharp. Namely, we show that for every reciprocal temperature >0, there exists a critical valueh _c () of external magnetic fieldh such that the following two statements hold.

Morse and Hedlund’s Skew Sturmian Words Revisited

For any infinite word r over A = {a, b} we associate two infinite words min(r), max(r) such that any prefix of min(r) (max(r), respectively) is the lexicographically smallest (greatest, respectively) among the factors of r of the same length. We prove that (min(r); max(r)) = (as, bs) for some infinite word s if and only if r is a proper Sturmian word or an ultimately periodic word of a particular form. This result is based on a lemma concerning sequences of infinite words. Received July 11, 2005