Bijections for Entringer families

Andr\'e proved that the number of alternating permutations on $\{1, 2, \dots, n\}$ is equal to the Euler number $E_n$. A refinement of Andr\'e's result was given by Entringer, who proved that counting alternating permutations according to the first element gives rise to Seidel's triangle $(E_{n,k})$ for computing the Euler numbers. In a series of papers, using generating function method and induction, Poupard gave several further combinatorial interpretations for $E_{n,k}$ both in alternating permutations and increasing trees. Kuznetsov, Pak, and Postnikov have given more combinatorial interpretations of $E_{n,k}$ in the model of trees. The aim of this paper is to provide bijections between the different models for $E_{n,k}$ as well as some new interpretations. In particular, we give the first explicit one-to-one correspondence between Entringer's alternating permutation model and Poupard's increasing tree model.Comment: 19 page

Metal-insulator transition for the almost Mathieu operator

We prove that for Diophantine \om and almost every \th, the almost Mathieu operator, (H_{\omega,\lambda,\theta}\Psi)(n)=\Psi(n+1) + \Psi(n-1) + \lambda\cos 2\pi(\omega n +\theta)\Psi(n), exhibits localization for \lambda > 2 and purely absolutely continuous spectrum for \lambda < 2. This completes the proof of (a correct version of) the Aubry-Andr\'e conjecture.Comment: 17 pages, published versio

Expression of Smooth Muscle Myosin Heavy Chains and Unloaded Shortening in Single Smooth Muscle Cells

The functional significance of the variable expression of the smooth muscle myosin heavy chain (SM-MHC) tail isoforms, SM1 and SM2, was examined at the mRNA level (which correlates with the protein level) in individual permeabilized rabbit arterial smooth muscle cells (SMCs). The length of untethered single permeabilized SMCs was monitored during unloaded shortening in response to increased Ca2+ (pCa 6.0), histamine (1 μM), and phenylephrine (1 μM). Subsequent to contraction, the relative expression of SM1 and SM2 mRNAs from the same individual SMCs was determined by reverse transcription-polymerase chain reaction amplification and densitometric analysis. Correlational analyses between the SM2-to-SM1 ratio and unloaded shortening in saponin- and α-toxin-permeabilized SMCs (n = 28) reveal no significant relationship between the SM-MHC tail isoform ratio and unloaded shortening velocity. The best correlations between SM2/SM1 and the contraction characteristics of untethered vascular SMCs were with the minimum length attained following contraction (n = 20 andr = 0.72 for α-toxin,n = 8 andr = 0.78 for saponin). These results suggest that the primary effect of variable expression of the SM1 and SM2 SM-MHC tail isoforms is on the cell final length and not on shortening velocity

Gorenstein dimension of modules over homomorphisms

Given a homomorphism of commutative noetherian rings R --> S and an S-module N, it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals sup{m | Tor^R_m(E,N) \noteq 0} where E is the injective hull of the residue field of R. This result is analogous to a theorem of Andr\'e on flat dimension.Comment: 14 pp. To appear in J. Pure Appl. Algebra. Also available from http://www.math.unl.edu/~lchristensen3/index.htm

Strong characterizing sequences of countable groups

Andr\'as Bir\'o and Vera S\'os prove that for any subgroup $G$ of \T generated freely by finitely many generators there is a sequence $A\subset \N$ such that for all \beta \in \T we have ($\|.\|$ denotes the distance to the nearest integer) $$\beta\in G \Rightarrow \sum_{n\in A} \| n \beta\| < \infty,\quad \quad \quad \beta\notin G \Rightarrow \limsup_{n\in A, n \to \infty} \|n \beta\| > 0.$$ We extend this result to arbitrary countable subgroups of \T. We also show that not only the sum of norms but the sum of arbitrary small powers of these norms can be kept small. Our proof combines ideas from the above article with new methods, involving a filter characterization of subgroups of \T