The maximal acceleration, Extended Relativistic Dynamics and Doppler type shift for an accelerated source

Based on the generalized principle of relativity and the ensuing symmetry, we have shown that there are only two possible types of transformations between uniformly accelerated systems. The first allowable type of transformation holds if and only if the Clock Hypothesis is true. If the Clock Hypothesis is not true, the transformation is of Lorentz-type and implies the existence of a universal maximal acceleration $a_m$. We present an extension of relativistic dynamics for which all admissible solutions will have have a speed bounded by the speed of light $c$ and the acceleration bounded by $a_m$. An additional Doppler type shift for an accelerated source is predicted. The formulas for such shift are the same as for the usual Doppler shift with $v/c$ replaced by $a/a_m$. The W. K\"{u}ndig experiment of measurement of the transverse Doppler shift in an accelerated system was also exposed to a longtitudal shift due to the acceleration. This experiment, as reanalyzed by Kholmetskii et al, shows that the Clock Hypothesis is not valid. Based on the results of this experiment, we predict that the value of the maximal acceleration $a_m$ is of the order $10^{19}m/s^2$. Moreover, our analysis provides a way to measure experimentally the maximal acceleration with existing technology.Comment: 10 pages, 1 figur

Comment on ``Effective Mass and g-Factor of Four Flux Quanta Composite Fermions"

In a recent Letter, Yeh et al.[Phys. Rev. Lett. 82, 592 (1999)] have shown beautiful experimental results which indicate that the composite fermions with four flux quanta ($^4$CF) behave as fermions with mass and spin just like those with two flux quanta. They observed the collapse of the fractional quantum Hall gaps when the following condition is satisfied with some integer $j$, $g^*\mu_{\rm B}B_{\rm tot} = j \hbar \omega_{\rm c}^*$, where $g^*$ and $\omega_{\rm c}^*$ are the g-factor and the cyclotron frequency of the $^4$CF, respectively. However, in their picture the gap at the Fermi energy remains always finite even if the above condition is satisfied, thus the reason of the collapse was left as a mystery. In this comment it is shown that part of the mystery is resolved by considering the electron-hole symmetry properly.Comment: 2 pages, RevTeX. Minor chang

Stochastic population dynamics under regime switching II

This is a continuation of our paper [Q. Luo, X. Mao, Stochastic population dynamics under regime switching, J. Math. Anal. Appl. 334 (2007) 69-84] on stochastic population dynamics under regime switching. In this paper we still take both white and color environmental noise into account. We show that a sufficient large white noise may make the underlying population extinct while for a relatively small noise we give both asymptotically upper and lower bound for the underlying population. In some special but important situations we precisely describe the limit of the average in time of the population

Comment on "Novel Convective Instabilities in a Magnetic Fluid"

Comment on the paper "Novel Convective Instabilities in a Magnetic Fluid" by W. Luo, T. Du, and J. Huang, Phys. Rev. Lett., v.82, p.4134 (1999).Comment: 1 page, 1 figure, To appear in Phys. Rev. Lett. (2001

Complete Monotonicity of classical theta functions and applications

We produce trigonometric expansions for Jacobi theta functions\\ $\theta_j(u,\tau), j=1,2,3,4$\ where $\tau=i\pi t, t > 0$. This permits us to prove that\ $\log \frac{\theta_j(u, t)}{\theta_j(0, t)}, j=2,3,4$ and $\log \frac{\theta_1(u, t)}{\pi \theta'_1(0, t)}$ as well as $\frac{\frac{\delta\theta_j}{\delta u}}{\theta_j}$ as functions of $t$ are completely monotonic. We also interested in the quotients $S_j(u,v,t) = \frac{\theta_j(u/2,i\pi t)}{\theta_j(u/2,i\pi t)}$. For fixed $u,v$ such that $0\leq u < v < 1$ we prove that the functions $\frac{(\frac{\delta}{\delta t}S_j)}{S_j}$ for $j=1,4$ as well as the functions $-\frac{(\frac{\delta}{\delta t}S_j)}{S_j}$ for $j=2,3$ are completely monotonic for $t \in ]0,\infty[$.\\ {\it Key words and phrases} : theta functions, elliptic functions, complete monotonicity.Comment: 19 page

An extension theorem for separately holomorphic functions with pluripolar singularities

Let $D_j\subset\Bbb C^{n_j}$ be a pseudoconvex domain and let $A_j\subset D_j$ be a locally pluriregular set, $j=1,...,N$. Put $$X:=\bigcup_{j=1}^N A_1\times...\times A_{j-1}\times D_j\times A_{j+1}\times ...\times A_N\subset\Bbb C^{n_1}\times...\times\Bbb C^{n_N}=\Bbb C^n.$$ Let $U\subset\Bbb C^n$ be an open neighborhood of $X$ and let $M\subset U$ be a relatively closed subset of $U$. For $j\in\{1,...,N\}$ let $\Sigma_j$ be the set of all $(z',z'')\in(A_1\times...\times A_{j-1}) \times(A_{j+1}\times...\times A_N)$ for which the fiber $M_{(z',\cdot,z'')}:=\{z_j\in\Bbb C^{n_j}\: (z',z_j,z'')\in M\}$ is not pluripolar. Assume that $\Sigma_1,...,\Sigma_N$ are pluripolar. Put $$X':=\bigcup_{j=1}^N\{(z',z_j,z'')\in(A_1\times...\times A_{j-1})\times D_j \times(A_{j+1}\times...\times A_N)\: (z',z'')\notin\Sigma_j\}.$$ Then there exists a relatively closed pluripolar subset $\hat M\subset\hat X$ of the `envelope of holomorphy' $\hat X\subset\Bbb C^n$ of $X$ such that: $\hat M\cap X'\subset M$, for every function $f$ separately holomorphic on $X\setminus M$ there exists exactly one function $\hat f$ holomorphic on $\hat X\setminus\hat M$ with $\hat f=f$ on $X'\setminus M$, and $\hat M$ is singular with respect to the family of all functions $\hat f$. Some special cases were previously studied in \cite{Jar-Pfl 2001c}.Comment: 19 page