Quantitative recurrence properties in conformal iterated function systems

Let $\Lambda$ be a countable index set and $S=\{\phi_i: i\in \Lambda\}$ be a conformal iterated function system on $[0,1]^d$ satisfying the open set condition. Denote by $J$ the attractor of $S$. With each sequence $(w_1,w_2,...)\in \Lambda^{\mathbb{N}}$ is associated a unique point $x\in [0,1]^d$. Let $J^\ast$ denote the set of points of $J$ with unique coding, and define the mapping $T:J^\ast \to J^\ast$ by $Tx= T (w_1,w_2, w_3...) = (w_2,w_3,...)$. In this paper, we consider the quantitative recurrence properties related to the dynamical system $(J^\ast, T)$. More precisely, let $f:[0,1]^d\to \mathbb{R}^+$ be a positive function and $$R(f):=\{x\in J^\ast: |T^nx-x|<e^{-S_n f(x)}, \ {\text{for infinitely many}}\ n\in \mathbb{N}\},$$ where $S_n f(x)$ is the $n$th Birkhoff sum associated with the potential $f$. In other words, $R(f)$ contains the points $x$ whose orbits return close to $x$ infinitely often, with a rate varying along time. Under some conditions, we prove that the Hausdorff dimension of $R(f)$ is given by $\inf\{s\ge 0: P(T, -s(f+\log |T'|))\le 0\}$, where $P$ is the pressure function and $T'$ is the derivative of $T$. We present some applications of the main theorem to Diophantine approximation.Comment: 25 page

Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential

Let $((0,1], T)$ be the doubling map in the unit interval and $\varphi$ be the Saint-Petersburg potential, defined by $\varphi(x)=2^n$ if $x\in (2^{-n-1}, 2^{-n}]$ for all $n\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\_n(x)=\varphi(x)+\cdots+\varphi(T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\frac{1}{n\log n}S\_n(x)$ converges to $\frac{1}{\log 2}$ in probability. We determine the Hausdorff dimension of the level set $\{x: \lim\_{n\to\infty}S\_n(x)/n=\alpha\} \ (\alpha>0)$, as well as that of the set $\{x: \lim\_{n\to\infty}S\_n(x)/\Psi(n)=\alpha\} \ (\alpha>0)$, when $\Psi(n)=n\log n, n^a$ or $2^{n^\gamma}$ for $a>1$, $\gamma>0$. The fast increasing Birkhoff sum of the potential function $x\mapsto 1/x$ is also studied.Comment: 17 page

Combined effect of ultrasound-guided percutaneous abdominal paracentesis drainage and ulinastatin on severe acute pancreatitis

Purpose: To study the clinical effect of a combination of ultrasound-guided percutaneous abdominal paracentesis drainage (APD) and ulinastatin on severe acute pancreatitis (AP).Methods: A total of 94 patients with severe AP in Intensive Care Unit, Jiaozhou Central Hospital, Qingdao, from December 2017 to December 2018 were selected as the research subjects. They were divided into control and study groups, with 47 patients in each group. Patients in the control group underwent laparotomy drainage, while patients in the study group underwent ultrasound-guidedpercutaneous APD. Patients in both groups received ulinastatin perfusion. Subsequently, clinical effectsand other relevant indicators were determined.Results: Overall response was significantly higher in the study group than in the control group (p &lt; 0.05). The times taken for disappearance of postoperative symptoms, normalization of serum amylase level, and hospitalization were significantly shorter in the study group than in the control group (p &lt; 0.05). For every indicator, the study group exhibited more benefits after than before treatment; however, post-treatment levels of blood glucose, hemodiastase and urinary amylase were better than those in the control group (p &lt; 0.05). Incidence of postoperative complications was lower in the study group than in control group (p &lt; 0.05).Conclusion: The combination of ultrasound-guided percutaneous APD with ulinastatin produces marked beneficial effects on severe AP patients. It facilitates the remission of adverse symptoms, and enhances&nbsp; normalization of indicator levels. Moreover, it displays low incidence of complications, better prognosis and recovery, and absence of post-operation infections

Fuzzy Search for Multiple Chinese Keywords in Cloud Environment

With the continuous development of cloud computing and big data technology, the use of cloud storage is more and more extensive, and a large amount of data is outsourced for public cloud servers, and the security problems that follow are gradually emerging. It can not only protect the data privacy of users, but also realize efficient retrieval and use of data, which is an urgent problem for cloud storage. Based on the existing fuzzy search and encrypted data fuzzy search schemes, this paper uses the characteristics of fuzzy sounds and polysemy that are unique to Chinese, and realizes the synonym construction of keywords through Chinese Pinyin and Chinese-English translation, and establishes the fuzzy word and synonym set of keywords. This paper proposes a Chinese multi-keyword fuzzy search scheme in a cloud environment, which realizes the fuzzy search of multiple Chinese keywords and protects the private key by using a pseudo-random function. Finally, the safety analysis and system experiments verify that the scheme has high security, good practicability, and high search success rate

The motion characteristics of a cylinder vehicle in the oblique water-exit process

The hydrodynamic model of a vehicle exiting the water surface obliquely has been analyzed. The analyzed object is a cylinder vehicle and its motion characteristics. Two methods have been used to simulate the water-exit process under the same conditions: Numerical Simulation Method (NSM) and Theoretical Model Solution Method (TMSM). The comparison results of the two methods can validate the hydrodynamic model founded in this paper. Different initial angles and different initial velocities have been simulated by this hydrodynamic model and the numerical simulation has been analyzed. The analysis reveals the rule of change of altitude and position of the vehicle in the water-exit process, and its motion after it exits the water surface. This paper explains why it is more difficult for the vehicle to exit the water obliquely than vertically. The results show that the hydrodynamic model of the water exiting vehicle can be used to research the exiting water motion characteristics. The models simulate the physics of motion realistically and this hydrodynamic model can be used as a foundation for the future research of the stability and control of a vehicle exiting the water