Functions of nearly maximal Gowers-Host-Kra norms on Euclidean spaces

Let $k\geq 2, n\geq 1$ be integers. Let $f: \mathbb{R}^{n} \to \mathbb{C}$. The $k$th Gowers-Host-Kra norm of $f$ is defined recursively by \begin{equation*} \| f\|_{U^{k}}^{2^{k}} =\int_{\mathbb{R}^{n}} \| T^{h}f \cdot \bar{f} \|_{U^{k-1}}^{2^{k-1}} \, dh \end{equation*} with $T^{h}f(x) = f(x+h)$ and $\|f\|_{U^1} = | \int_{\mathbb{R}^{n}} f(x)\, dx |$. These norms were introduced by Gowers in his work on Szemer\'edi's theorem, and by Host-Kra in ergodic setting. It's shown by Eisner and Tao that for every $k\geq 2$ there exist $A(k,n)< \infty$ and $p_{k} = 2^{k}/(k+1)$ such that $\| f\|_{U^{k}} \leq A(k,n)\|f\|_{p_{k}}$, with $p_{k} = 2^{k}/(k+1)$ for all $f \in L^{p_{k}}(\mathbb{R}^{n})$. The optimal constant $A(k,n)$ and the extremizers for this inequality are known. In this exposition, it is shown that if the ratio $\| f \|_{U^{k}}/\|f\|_{p_{k}}$ is nearly maximal, then $f$ is close in $L^{p_{k}}$ norm to an extremizer

Bayesian sequential D-D optimal model-robust designs.

Alphabetic optimal design theory assumes that the model for which the optimal design is derived is usually known. However in real-life applications, this assumption may not be credible, as models are rarely known in advance. Therefore, optimal designs derived under the classical approach may be the best design but for the wrong assumed model. In this paper, we extend Neff's (1996) Bayesian two-stage approach to design experiments for the general linear model when initial knowledge of the model is poor. A Bayesian optimality procedure that works well under model uncertainty is used in the first stage and the second stage design is then generated from an optimality procedure that incorporates the improved model knowledge from the first stage. In this way, a Bayesian D-D optimal model robust design is developed. Results show that the Bayesian D-D optimal design is superior in performance to the classical one-stage D-optimal and the one-stage Bayesian D-optimal designs. We also investigate through a simulation study the ratio of sample sizes for the two stages and the minimum sample size desirable in the first stage.Applications; D-D optimality; Knowledge; Model; Two-stage procedure; Posterior probabilities;

cDNA Cloning Demonstrates the Expression of Pregnancy-Specific Glycoprotein Genes, a Subgroup of the Carcinoembryonic Antigen Gene Family, in Fetal Liver

The pregnancy-specific glycoprotein (PSG) genes constitute a subgroup of the carcinoembryonic antigen (CEA) gene family. Here we report the cloning of four cDNAs coding for different members of the PSG family from a human fetal liver cDNA library. They are derived from three closely related genes (PSG1, PSG4 and PSG6). Two of the cDNA clones represent splice variants of PSG1 (PSG1a, PSG1d) differing in their C-terminal domain and 3′-untranslated regions. All encoded proteins show the same domain arrangement (N-RA1-RA2-RB2-C). Transcripts of the genes PSG1 and PSG4 could be detected in placenta by hybridization with gene-specific oligonucleotides. Expression of cDNA in a mouse and monkey cell line shows that the glycosylated PSG1a protein has a Mr of 65–66 kD and is released from the transfected cells. Sequence comparisons in the C-terminal domain and the 3′-untranslated regions of CEA/PSG-like genes suggests a complex splicing pattern to exist for various gene family members and a common evolutionary origin of these region

Waves in the Skyrme--Faddeev model and integrable reductions

In the present article we show that the Skyrme--Faddeev model possesses nonlinear wave solutions, which can be expressed in terms of elliptic functions. The Whitham averaging method has been exploited in order to describe slow deformation of periodic wave states, leading to a quasi-linear system. The reduction to general hydrodynamic systems have been considered and it is compared with other integrable reductions of the system.Comment: 16 pages, 5 figure

Exotic galilean symmetry and non-commutative mechanics

Some aspects of the "exotic" particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.Comment: Review paper. Published versio