@article{4200,
abstract = {Zebrafish embryos homozygous for the masterblind (mb1) mutation exhibit a striking phenotype in which the eyes and telencephalon are reduced or absent and diencephalic fates expand to the front of the brain. Here we show that mb1(-/-) embryos carry an amino-acid change at a conserved site in the Wnt pathway scaffolding protein, Axin1. The amino-acid substitution present in the mbl allele abolishes the binding of Axin to Gsk3 and affects Tcf-dependent transcription. Therefore, Gsk3 activity may be decreased in mbl(-/-) embryos and in support of this possibility, overexpression of either wild-type Axin1 or Gsk3 beta can restore eye and telencephalic fates to mb1(-/-) embryos. Our data reveal a crucial role for Axin1-dependent inhibition of the Wnt pathway in the early regional subdivision of the anterior neural plate into telencephalic, diencephalic, and eye-forming territories.},
author = {Heisenberg, Carl-Philipp J and Houart, Corinne and Take Uchi, Masaya and Rauch, Gerd and Young, Neville and Coutinho, Pedro and Masai, Ichiro and Caneparo, Luca and Concha, Miguel and Geisler, Robert and Dale, Trevor and Wilson, Stephen and Stemple, Derek},
journal = {Genes and Development},
number = {11},
pages = {1427 -- 1434},
publisher = {Cold Spring Harbor Laboratory Press},
title = {{A mutation in the Gsk3-binding domain of zebrafish Masterblind/Axin1 leads to a fate transformation of telencephalon and eyes to diencephalon}},
doi = {10.1101/gad.194301},
volume = {15},
year = {2001},
}
@article{8522,
abstract = {For diffeomorphisms of smooth compact manifolds, we consider the problem of how fast the number of periodic points with period $n$grows as a function of $n$. In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for $C^2$ or smoother diffeomorphisms. In the present work we show that, by contrast, for a measure-theoretic notion of genericity we call ``prevalence'', the growth is not much faster than exponential. Specifically, we show that for each $\delta > 0$, there is a prevalent set of ( $C^{1+\rho}$ or smoother) diffeomorphisms for which the number of period $n$ points is bounded above by $\operatorname{exp}(C n^{1+\delta})$ for some $C$ independent of $n$. We also obtain a related bound on the decay of the hyperbolicity of the periodic points as a function of $n$. The contrast between topologically generic and measure-theoretically generic behavior for the growth of the number of periodic points and the decay of their hyperbolicity shows this to be a subtle and complex phenomenon, reminiscent of KAM theory.},
author = {Kaloshin, Vadim and Hunt, Brian R.},
issn = {1079-6762},
journal = {Electronic Research Announcements of the American Mathematical Society},
keywords = {General Mathematics},
number = {4},
pages = {17--27},
publisher = {American Mathematical Society},
title = {{A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I}},
doi = {10.1090/s1079-6762-01-00090-7},
volume = {7},
year = {2001},
}
@article{8524,
abstract = {A number α∈R is diophantine if it is not well approximable by rationals, i.e. for some C,ε>0 and any relatively prime p,q∈Z we have |αq−p|>Cq−1−ε. It is well-known and is easy to prove that almost every α in R is diophantine. In this paper we address a noncommutative version of the diophantine properties. Consider a pair A,B∈SO(3) and for each n∈Z+ take all possible words in A, A -1, B, and B - 1 of length n, i.e. for a multiindex I=(i1,i1,…,im,jm) define |I|=∑mk=1(|ik|+|jk|)=n and \( W_n(A,B ) = \{W_{\cal I}(A,B) = A^{i_1} B^{j_1} \dots A^{i_m} B^{j_m}\}_{|{\cal I|}=n \).¶Gamburd—Jakobson—Sarnak [GJS] raised the problem: prove that for Haar almost every pair A,B∈SO(3) the closest distance of words of length n to the identity, i.e. sA,B(n)=min|I|=n∥WI(A,B)−E∥, is bounded from below by an exponential function in n. This is the analog of the diophantine property for elements of SO(3). In this paper we prove that s A,B (n) is bounded from below by an exponential function in n 2. We also exhibit obstructions to a “simple” proof of the exponential estimate in n.},
author = {Kaloshin, Vadim and Rodnianski, I.},
issn = {1016-443X},
journal = {Geometric And Functional Analysis},
number = {5},
pages = {953--970},
publisher = {Springer Nature},
title = {{Diophantine properties of elements of SO(3)}},
doi = {10.1007/s00039-001-8222-8},
volume = {11},
year = {2001},
}
@article{8521,
abstract = {We continue the previous article's discussion of bounds, for prevalent diffeomorphisms of smooth compact manifolds, on the growth of the number of periodic points and the decay of their hyperbolicity as a function of their period $n$. In that article we reduced the main results to a problem, for certain families of diffeomorphisms, of bounding the measure of parameter values for which the diffeomorphism has (for a given period $n$) an almost periodic point that is almost nonhyperbolic. We also formulated our results for $1$-dimensional endomorphisms on a compact interval. In this article we describe some of the main techniques involved and outline the rest of the proof. To simplify notation, we concentrate primarily on the $1$-dimensional case.},
author = {Kaloshin, Vadim and Hunt, Brian R.},
issn = {1079-6762},
journal = {Electronic Research Announcements of the American Mathematical Society},
keywords = {General Mathematics},
number = {5},
pages = {28--36},
publisher = {American Mathematical Society},
title = {{A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II}},
doi = {10.1090/s1079-6762-01-00091-9},
volume = {7},
year = {2001},
}