Multiplicity One Theorems, S-Version

We proved three theorems of $S$-version of the mulyiplicity one

Grunwald- Wang Theorem, an Effective Version

The main purpose of this note is to establish an effective version of the Grunwald--Wang Theorem, which asserts that given a family of local characters $\chi^{v}$ of $K_{v}^{*}$ of exponent $m$ where $v \in S$ for a finite set $S$ of primes of $K$, there exists a global character $\chi$ of the idele class group $C_{K}$ of exponent $m$ (unless some special case occurs, when it is $2 m$) whose component at $v$ is $\chi^{v}$. The effectiveness problem for this theorem is to bound the norm $N (\chi)$ of the conductor of $\chi$ in terms of $K$, $m$, $S$ and $N (\chi^{v})$. The Kummer case (when $K$ contains $\mu_{m}$) is easy since it is almost an application of the Chinese Remainder Theorem. In this note, we solve this problem completely in general case, and give three versions of bound, one is with \GRH, and the other two are unconditional bounds. These effective results have some interesting applications in concrete situations. To give a simple example, if we fix $p$ and $l$, one gets a good least upper bound for $N$ such that $p$ is not an $l$--th power mod $N$. One also gets the least upper bound for $N$ such that $l^{r} \mid \phi (N)$ and $p$ is not an $l$--th power mod $N$. Some part of this note is adopted (with some revision) from my unpublished thesis (2001) (\cite{Wang2001})

Search for Higgs, Leptoquarks, and Exotics at Tevatron

This paper reviews some of the most recent results from the CDF and DZERO experiments on the searches for Standard Model and Non-Standard Model Higgs bosons, and other new phenomena at the Tevatron. Both experiments examine data from proton anti-proton collision at sqrt{s} = 1.96 TeV, of integrated luminosity ~200$ pb**-1 (per experiment), to search for Higgs predicted in the Standard Model and beyond Standard Model, supersymmetric particles in the Gauge Mediated Symmetry Breaking scenario, leptoquarks, and excited electrons. No signal was observed, and limits on the signatures and models are derived.Comment: Contribution to the proceedings of the XXXIX Recontres de Moriond on Electroweak Interactions and Unified Theorie