An Erd\"os--R\'ev\'esz type law of the iterated logarithm for order statistics of a stationary Gaussian process

Let $\{X(t):t\in\mathbb R_+\}$ be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, $\mathbb E X(t) = 0$, $\mathbb E X^2(t) = 1$ and correlation function satisfying (i) $r(t) = 1 - C|t|^{\alpha} + o(|t|^{\alpha})$ as $t\to 0$ for some $0\le\alpha\le 2, C>0$, (ii) $\sup_{t\ge s}|r(t)|0$ and (iii) $r(t) = O(t^{-\lambda})$ as $t\to\infty$ for some $\lambda>0$. For any $n\ge 1$, consider $n$ mutually independent copies of $X$ and denote by $\{X_{r:n}(t):t\ge 0\}$ the $r$th smallest order statistics process, $1\le r\le n$. We provide a tractable criterion for assessing whether, for any positive, non-decreasing function $f$, $\mathbb P(\mathscr E_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text{i.o.})$ equals 0 or 1. Using this criterion we find that, for a family of functions $f_p(t)$, such that $z_p(t)=\mathbb P(\sup_{s\in[0,1]}X_{r:n}(s)>f_p(t))=\mathscr C(t\log^{1-p} t)^{-1}$, $\mathscr C>0$, $\mathbb P(\mathscr E_{f_p})= 1_{\{p\ge 0\}}$. Consequently, with $\xi_p (t) = \sup\{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}$, for $p\ge 0$, $\lim_{t\to\infty}\xi_p(t)=\infty$ and $\limsup_{t\to\infty}(\xi_p(t)-t)=0$ a.s.. Complementary, we prove an Erd\"os-R\'ev\'esz type law of the iterated logarithm lower bound on $\xi_p(t)$, i.e., $\liminf_{t\to\infty}(\xi_p(t)-t)/h_p(t) = -1$ a.s., $p>1$, $\liminf_{t\to\infty}\log(\xi_p(t)/t)/(h_p(t)/t) = -1$ a.s., $p\in(0,1]$, where $h_p(t)=(1/z_p(t))p\log\log t$

On the unitarity of higher-dervative and nonlocal theories

We consider two simple models of higher-derivative and nonlocal quantu systems.It is shown that, contrary to some claims found in literature, they can be made unitary.Comment: 8 pages, no figure

Performance Evaluation of Road Traffic Control Using a Fuzzy Cellular Model

In this paper a method is proposed for performance evaluation of road traffic control systems. The method is designed to be implemented in an on-line simulation environment, which enables optimisation of adaptive traffic control strategies. Performance measures are computed using a fuzzy cellular traffic model, formulated as a hybrid system combining cellular automata and fuzzy calculus. Experimental results show that the introduced method allows the performance to be evaluated using imprecise traffic measurements. Moreover, the fuzzy definitions of performance measures are convenient for uncertainty determination in traffic control decisions.Comment: The final publication is available at http://www.springerlink.co

The Efficacy of Renal Replacement Therapy for Rewarming of Patients in Severe Accidental Hypothermia-Systematic Review of the Literature.

Renal replacement therapy (RRT) can be used to rewarm patients in deep hypothermia. However, there is still no clear evidence for the effectiveness of RRT in this group of patients. This systematic review aims to summarize the rewarming rates during RRT in patients in severe hypothermia, below or equal to 32 °C. This systematic review was registered in the PROSPERO International Prospective Register of Systematic Reviews (identifier CRD42021232821). We searched Embase, Medline, and Cochrane databases using the keywords hypothermia, renal replacement therapy, hemodialysis, hemofiltration, hemodiafiltration, and their abbreviations. The search included only articles in English with no time limit, up until 30 June 2021. From the 795 revised articles, 18 studies including 21 patients, were selected for the final assessment and data extraction. The mean rate of rewarming calculated for all studies combined was 1.9 °C/h (95% CI 1.5-2.3) and did not differ between continuous (2.0 °C/h; 95% CI 0.9-3.0) and intermittent (1.9 °C/h; 95% CI 1.5-2.3) methods (p > 0.9). Based on the reviewed literature, it is currently not possible to provide high-quality recommendations for RRT use in specific groups of patients in accidental hypothermia. While RRT appears to be a viable rewarming strategy, the choice of rewarming method should always be determined by the specific clinical circumstances, the available resources, and the current resuscitation guidelines

Körner’s septum (petrosquamosal lamina): the anatomical variant or clinical problem?

Körner’s septum (KS) or petrosquamosal lamina is a bony lamina beginning at the articular fossa, extending above the middle ear, and running inferiorly and laterally to the facial nerve canal as it proceeds to the mastoid apex. This septum marks the junction of petrous and squamous bones. The paper presents details of the anatomical structure of KS, which is most often present at the level of the head of the malleus and/or the anterior semicircular canal. Attention is paid to embryological aspects of temporal bone development that lead to the formation of KS. Two imaging techniques most frequently used to diagnose KS are described, high resolution computed tomography (HRCT) and cone-beam computed tomography. Also presented is a case report of a 6-year-old patient suffering from chronic otitis media who developed a cholesteatoma due to presence of KS, illustrated with HRCT images and intraoperative capture. The authors describe diagnostic difficulties associated with this anatomical variant in the middle ear. The article also discusses the more frequent occurrence of this clinical problem in ears operated on due to chronic inflammation, retraction pocket or tympanosclerosis in comparison to healthy ears

Gaussian queues in light and heavy traffic

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. The setting considered is that of a centered Gaussian process $X\equiv\{X(t):t\in\mathbb R\}$ with stationary increments and variance function $\sigma^2_X(\cdot)$, equipped with a deterministic drift $c>0$, reflected at 0: $$Q_X^{(c)}(t)=\sup_{-\infty<s\le t}(X(t)-X(s)-c(t-s)).$$ We study the resulting stationary workload process $Q^{(c)}_X\equiv\{Q_X^{(c)}(t):t\ge0\}$ in the limiting regimes $c\to 0$ (heavy traffic) and $c\to\infty$ (light traffic). The primary contribution is that we show for both limiting regimes that, under mild regularity conditions on the variance function, there exists a normalizing function $\delta(c)$ such that $Q^{(c)}_X(\delta(c)\cdot)/\sigma_X(\delta(c))$ converges to a non-trivial limit in $C[0,\infty)$

Development of the interatrial wall during the ontogenesis of foetuses and children up to one year of age

Background: The foramen ovale, present in foetal interatrial septum, plays an important role during foetal life. During delivery, foramen ovale closes and becomes fossa ovalis, starting the pulmonary circulation. The aim of our study was to describe the growth of the interatrial wall and changes in location of the foramen ovale, and fossa ovalis during the ontogenesis in the human hearts.Materials and methods: The study was performed on post-mortem material obtained from 92 human hearts from 22nd week of foetal life up to 1 year of age, fixed in a 4% formalin solution.Results: The interatrial wall size in the studied development period was greater in the horizontal than in the vertical dimension. During ontogenesis up to 1 year old, the anterior and inferior parts of the interatrial wall increased their shares considerably by 8% and 6%, respectively. The percentage participation of foramen ovale in the interatrial wall construction in the foetal period formed more than 50% of its size and fairly decreased reaching in infants about 39%.Conclusions: Our study demonstrated that during ontogenesis, from the foetal period to infancy, the parts of the interatrial wall increase their dimensions unevenly. The foramen ovale growth is smaller, compared to the rest of the interatrial wall development. On the basis of our data we can assume that the foramen ovale centre tends to be found in the postero-inferior quadrant of the interatrial wall (foetuses) and in postero-superior quadrant of the interatrial wall — in infants