On Khintchine exponents and Lyapunov exponents of continued fractions

Assume that $x\in [0,1)$ admits its continued fraction expansion $x=[a_1(x), a_2(x),...]$. The Khintchine exponent $\gamma(x)$ of $x$ is defined by $\gamma(x):=\lim\limits_{n\to \infty}\frac{1}{n}\sum_{j=1}^n \log a_j(x)$ when the limit exists. Khintchine spectrum $\dim E_\xi$ is fully studied, where $E_{\xi}:=\{x\in [0,1):\gamma(x)=\xi\} (\xi \geq 0)$ and $\dim$ denotes the Hausdorff dimension. In particular, we prove the remarkable fact that the Khintchine spectrum $\dim E_{\xi}$, as function of $\xi \in [0, +\infty)$, is neither concave nor convex. This is a new phenomenon from the usual point of view of multifractal analysis. Fast Khintchine exponents defined by $\gamma^{\phi}(x):=\lim\limits_{n\to\infty}\frac{1}{\phi(n)} \sum_{j=1}^n \log a_j(x)$ are also studied, where $\phi (n)$ tends to the infinity faster than $n$ does. Under some regular conditions on $\phi$, it is proved that the fast Khintchine spectrum $\dim (\{x\in [0,1]: \gamma^{\phi}(x)= \xi \})$ is a constant function. Our method also works for other spectra like the Lyapunov spectrum and the fast Lyapunov spectrum.Comment: 37 pages, 5 figures, accepted by Ergodic Theory and Dyanmical System

Practice on the Watershed Hydrological Experimental System Reconciling Deterministic and Stochastic Subjects Based on the System Complexity: 1. Theoretical Study

This is the first of a two-part series on the watershed hydrological experimental system (WHES). Since the foundational stage and developmental stage of hydrological basin study with a duration of more than ca. one century, facing with the changing environment and, the declined risk of field study while the catchment hydrology is trapped in a theoretical impasse, a third phase of renovation on hydrological experiments seems ready to come out inevitably. Learned from Chinese decades’ experiences on the field basin study for the question of what is wrong with the status quo, our exploratory idea is reported in this part. From the viewpoint of general system theory based on the paralleled concepts of the ancient Chinese and the Western, it is considered that the adequate method should face the characters of the complex dynamic system instead of previous static, linear system. From the viewpoint of another philosophical paralleled concept of the Middle Way, it should also face the operation and organizing of the mesoscopic systems for the organized complexity. Then, a framework of WHES is suggested with its organization based on the strategy of constrain complexity and add complexity and on the strategy of manipulation including the artificial-natural and controlled-natural objects. Such a trial framework, the Chuzhou WHES, is reported including the suggested critical zone experimental block (CZEB) instead of the experimental basin (EB) in the last decades

Practices on the Watershed Hydrological Experimental System Reconciling Deterministic and Stochastic Subjects Based on the System Complexity: 2. Practice and Test

This is the second of a two-part series on the watershed hydrological experimental system (WHES) aimed at practice and test of it at Chuzhou Hydrology Laboratory. It constitutes both natural and artificial entities of different scales, within which two typical main subjects are reviewed here. First is a natural watershed Nandadish, which is subjected to be a Critical Zone Experimental Block, under manipulation strategy of constrain complexity to compare with the pure natural watersheds, it is the controlled-natural as we termed. Second is an artificial catchment Hydrohill, under the strategy of add complexity to compare with the simple artificial lysimeters, it is the artificial-natural as we termed. The constructions and instrumentations of these experimental catchments are reviewed, especially their renovation version during recent years after a long abandonment. Some results get during the operation of Chuzhou WHES are outlined here as well

Study of J/ψJ/\psi and ψ(3686)→Σ(1385)0Σˉ(1385)0\psi(3686)\rightarrow\Sigma(1385)^{0}\bar\Sigma(1385)^{0} and Ξ0Ξˉ0\Xi^0\bar\Xi^{0}

We study the decays of $J/\psi$ and $\psi(3686)$ to the final states $\Sigma(1385)^{0}\bar\Sigma(1385)^{0}$ and $\Xi^0\bar\Xi^{0}$ based on a single baryon tag method using data samples of $(1310.6 \pm 7.0) \times 10^{6}$ $J/\psi$ and $(447.9 \pm 2.9) \times 10^{6}$ $\psi(3686)$ events collected with the BESIII detector at the BEPCII collider. The decays to $\Sigma(1385)^{0}\bar\Sigma(1385)^{0}$ are observed for the first time. The measured branching fractions of $J/\psi$ and $\psi(3686)\rightarrow\Xi^0\bar\Xi^{0}$ are in good agreement with, and much more precise, than the previously published results. The angular parameters for these decays are also measured for the first time. The measured angular decay parameter for $J/\psi\rightarrow\Sigma(1385)^{0}\bar\Sigma(1385)^{0}$, $\alpha =-0.64 \pm 0.03 \pm 0.10$, is found to be negative, different to the other decay processes in this measurement. In addition, the "12\% rule" and isospin symmetry in the $J/\psi$ and $\psi(3686)\rightarrow\Xi\bar\Xi$ and $\Sigma(1385)\bar{\Sigma}(1385)$ systems are tested.Comment: 11 pages, 7 figures. This version is consistent with paper published in Phys.Lett. B770 (2017) 217-22

Measurement of the proton form factor by studying e+e−→ppˉe^{+} e^{-}\rightarrow p\bar{p}

Using data samples collected with the BESIII detector at the BEPCII collider, we measure the Born cross section of $e^{+}e^{-}\rightarrow p\bar{p}$ at 12 center-of-mass energies from 2232.4 to 3671.0 MeV. The corresponding effective electromagnetic form factor of the proton is deduced under the assumption that the electric and magnetic form factors are equal $(|G_{E}|= |G_{M}|)$. In addition, the ratio of electric to magnetic form factors, $|G_{E}/G_{M}|$, and $|G_{M}|$ are extracted by fitting the polar angle distribution of the proton for the data samples with larger statistics, namely at $\sqrt{s}=$ 2232.4 and 2400.0 MeV and a combined sample at $\sqrt{s}$ = 3050.0, 3060.0 and 3080.0 MeV, respectively. The measured cross sections are in agreement with recent results from BaBar, improving the overall uncertainty by about 30\%. The $|G_{E}/G_{M}|$ ratios are close to unity and consistent with BaBar results in the same $q^{2}$ region, which indicates the data are consistent with the assumption that $|G_{E}|=|G_{M}|$ within uncertainties.Comment: 13 pages, 24 figure

Observation of the ψ(13D2)\psi(1^3D_2) state in e+e−→π+π−γχc1e^+e^-\to\pi^+\pi^-\gamma\chi_{c1} at BESIII

We report the observation of the $X(3823)$ in the process $e^+e^-\to \pi^+\pi^-X(3823) \to \pi^+\pi^-\gamma\chi_{c1}$ with a statistical significance of $6.2\sigma$, in data samples at center-of-mass energies $\sqrt{s}=$4.230, 4.260, 4.360, 4.420 and 4.600~GeV collected with the BESIII detector at the BEPCII electron positron collider. The measured mass of the $X(3823)$ is $(3821.7\pm 1.3\pm 0.7)$~MeV/$c^2$, where the first error is statistical and the second systematic, and the width is less than $16$~MeV at the 90\% confidence level. The products of the Born cross sections for $e^+e^-\to \pi^+\pi^-X(3823)$ and the branching ratio $\mathcal{B}[X(3823)\to \gamma\chi_{c1,c2}]$ are also measured. These measurements are in good agreement with the assignment of the $X(3823)$ as the $\psi(1^3D_2)$ charmonium state.Comment: 7 pages, 3 figures, version to appear in Phys. Rev. Let

Precision measurement of the D∗0D^{*0} decay branching fractions

Using 482 pb$^{-1}$ of data taken at $\sqrt{s}=4.009$ GeV, we measure the branching fractions of the decays of $D^{*0}$ into $D^0\pi^0$ and $D^0\gamma$ to be \BR(D^{*0} \to D^0\pi^0)=(65.5\pm 0.8\pm 0.5)% and \BR(D^{*0} \to D^0\gamma)=(34.5\pm 0.8\pm 0.5)% respectively, by assuming that the $D^{*0}$ decays only into these two modes. The ratio of the two branching fractions is \BR(D^{*0} \to D^0\pi^0)/\BR(D^{*0} \to D^0\gamma) =1.90\pm 0.07\pm 0.05, which is independent of the assumption made above. The first uncertainties are statistical and the second ones systematic. The precision is improved by a factor of three compared to the present world average values

Observation of the Dalitz Decay η′→γe+e−\eta' \to \gamma e^+e^-

We report the first observation of the Dalitz decay $\eta' \to \gamma e^+e^-$, based on a data sample of 1.31 billion $J/\psi$ events collected with the BESIII detector. The $\eta'$ mesons are produced via the $J/\psi \to \gamma \eta'$ decay process. The ratio $\Gamma(\eta' \to \gamma e^+ e^-)/\Gamma(\eta'\to\gamma\gamma)$ is measured to be $(2.13\pm0.09(\text{stat.})\pm0.07(\text{sys.}))\times10^{-2}$. This corresponds to a branching fraction ${\cal B}(\eta' \to \gamma e^+e^-)= (4.69 \pm0.20(\text{stat.})\pm0.23(\text{sys.}))\times10^{-4}$. The transition form factor is extracted and different expressions are compared to the measured dependence on the $e^+e^-$ invariant mass. The results are consistent with the prediction of the Vector Meson Dominance model.Comment: 11 pages,7 figure

Improved measurement of the absolute branching fraction of D+→Kˉ0μ+νμD^{+}\rightarrow \bar K^0 \mu^{+}\nu_{\mu}

By analyzing 2.93 fb$^{-1}$ of data collected at $\sqrt s=3.773$ GeV with the BESIII detector, we measure the absolute branching fraction ${\mathcal B}(D^{+}\rightarrow\bar K^0\mu^{+}\nu_{\mu})=(8.72 \pm 0.07_{\rm stat.} \pm 0.18_{\rm sys.})\%$, which is consistent with previous measurements within uncertainties but with significantly improved precision. Combining the Particle Data Group values of ${\mathcal B}(D^0\to K^-\mu^+\nu_\mu)$, ${\mathcal B}(D^{+}\rightarrow\bar K^0 e^{+}\nu_{e})$, and the lifetimes of the $D^0$ and $D^+$ mesons with the value of ${\mathcal B}(D^{+}\rightarrow\bar K^0 \mu^{+}\nu_{\mu})$ measured in this work, we determine the following ratios of partial widths: $\Gamma(D^0\to K^-\mu^+\nu_\mu)/\Gamma(D^{+}\rightarrow\bar K^0\mu^{+}\nu_{\mu})=0.963\pm0.044$ and $\Gamma(D^{+}\rightarrow\bar K^0 \mu^{+}\nu_{\mu})/\Gamma(D^{+}\rightarrow\bar K^0 e^{+}\nu_{e})=0.988\pm0.033$.Comment: 9 pages; 8 figure

Determination of the number of J/ψJ/\psi events with inclusive J/ψJ/\psi decays

A measurement of the number of $J/\psi$ events collected with the BESIII detector in 2009 and 2012 is performed using inclusive decays of the $J/\psi$ . The number of $J/\psi$ events taken in 2009 is recalculated to be $(223.7\pm1.4)\times 10^6$, which is in good agreement with the previous measurement, but with significantly improved precision due to improvements in the BESIII software. The number of $J/\psi$ events taken in 2012 is determined to be $(1086.9\pm 6.0)\times 10^6$. In total, the number of $J/\psi$ events collected with the BESIII detector is measured to be $(1310.6\pm 7.0)\times 10^6$, where the uncertainty is dominated by systematic effects and the statistical uncertainty is negligible.Comment: 10 pages, 6 figure