Some results on the palette index of graphs

Given a proper edge coloring $\varphi$ of a graph $G$, we define the palette $S_{G}(v,\varphi)$ of a vertex $v \in V(G)$ as the set of all colors appearing on edges incident with $v$. The palette index $\check s(G)$ of $G$ is the minimum number of distinct palettes occurring in a proper edge coloring of $G$. In this paper we give various upper and lower bounds on the palette index of $G$ in terms of the vertex degrees of $G$, particularly for the case when $G$ is a bipartite graph with small vertex degrees. Some of our results concern $(a,b)$-biregular graphs; that is, bipartite graphs where all vertices in one part have degree $a$ and all vertices in the other part have degree $b$. We conjecture that if $G$ is $(a,b)$-biregular, then $\check{s}(G)\leq 1+\max\{a,b\}$, and we prove that this conjecture holds for several families of $(a,b)$-biregular graphs. Additionally, we characterize the graphs whose palette index equals the number of vertices

Statistical state dynamics of weak jets in barotropic beta-plane turbulence

Zonal jets in a barotropic setup emerge out of homogeneous turbulence through a flow-forming instability of the homogeneous turbulent state (`zonostrophic instability') which occurs as the turbulence intensity increases. This has been demonstrated using the statistical state dynamics (SSD) framework with a closure at second order. Furthermore, it was shown that for small supercriticality the flow-forming instability follows Ginzburg-Landau (G-L) dynamics. Here, the SSD framework is used to study the equilibration of this flow-forming instability for small supercriticality. First, we compare the predictions of the weakly nonlinear G-L dynamics to the fully nonlinear SSD dynamics closed at second order for a wide ranges of parameters. A new branch of jet equilibria is revealed that is not contiguously connected with the G-L branch. This new branch at weak supercriticalities involves jets with larger amplitude compared to the ones of the G-L branch. Furthermore, this new branch continues even for subcritical values with respect to the linear flow-forming instability. Thus, a new nonlinear flow-forming instability out of homogeneous turbulence is revealed. Second, we investigate how both the linear flow-forming instability and the novel nonlinear flow-forming instability are equilibrated. We identify the physical processes underlying the jet equilibration as well as the types of eddies that contribute in each process. Third, we propose a modification of the diffusion coefficient of the G-L dynamics that is able to capture the asymmetric evolution for weak jets at scales other than the marginal scale (side-band instabilities) for the linear flow-forming instability.Comment: 27 pages, 17 figure

A novel fontanelle probe for sensing oxygen saturation in the neonate

Monitoring of blood oxygen saturation (SpO 2 ) of the neonate is essential to the quality of health care provided on a neonatal intensive care unit (NICU). Current sensors are usually placed at the hand or foot, which are dependent of a peripheral blood supply. When the peripheral blood circulation of neonates is compromised conventional peripheral pulse oximeters, in many cases, fail to operate accurately or at all. A new reflectance anterior fontanelle (ANTF) SpO 2 sensor and instrumentation has been developed to investigate SpO 2 s from the neonatal fontanelle. The hypothesis is that perfusion at a central site should be preserved at times of compromised peripheral circulation. Fifteen neonates on an NICU (9 male, 6 female) with a median age of 7 d (IQR = 41.5 d) were selected for monitoring. ANTF photoplethysmographic (PPG) signals were monitored for a maximum period of 2 h. The developed system and custom made sensors were successful at acquiring good quality signals at both wavelengths necessary for pulse oximetry calculations. ANTF SpO 2 s, estimated from the acquired PPGs, were in broad agreement with SpO 2 s obtained from the commercial foot pulse oximeter. A Bland and Altman analysis of the differences between SpO 2 s from the fontanelle PPG sensor and the commercial device show a relatively small mean difference (d = ±2.2%), but with a wide variation (2s = ±17.4%) this observation may be due to the varied levels of ill health patients and is backed up by comparing the commercial device SpO 2 readings at the same moment a blood gas sample was taken (d = 4.8%, 2s = ±15.8%)

Pilot investigation of anterior fontanelle photoplethysmographic signals and their suitability in estimating arterial oxygen saturation

There is a need for more reliable, non-invasive and alternative measurement sites for the monitoring of arterial blood oxygen saturation in critically ill newborns at times of peripheral compromise. A pilot investigation on 14 Intensive Care Unit (ICU) newborns was conducted utilizing custom-made reflectance photoplethysmographic (PPG) sensors placed at the fontanelle and foot. The results suggest that the fontanelle is sensitive to changes in saturation, where saturation values obtained from the custom sensor were compared against commercial pulse oximeter values and results from a blood gas analyzer, however careful placement of the sensor at the fontanelle is an issue that needs further investigation

Esophageal SpO2 measurements from a pediatric burns-patient: A case study

Pulse oximetry is being used in everyday clinical practice in anesthesia utilizing peripheral saturation sensors. However, it may be unreliable in certain clinical situations such as peripheral hypoperfusion. Similar situations occur in burns patients and more importantly burns to extremities which limit the sites available for measurement of peripheral oxygen saturation (SpO2). To overcome these limitations, the esophagus has been investigated as an alternative measurement site, as perfusion may be preferentially preserved centrally. A miniaturized reflectance esophageal saturation (SpO2 probe has been constructed utilizing infrared and red photodiodes and a photodetector. Our case study was aimed at evaluating the reliability of esophageal pulse oximetry in a major burns infant. Measurable photoplethysmographic (PPG) traces and SpO2 values were obtained in the neonatal esophagus. It was found that the esophageal pulse oximeter results were in good agreement with oxygen saturation measurements obtained by a commercial ear lobe pulse oximeter. This study suggests that the esophagus can be used as an alternative site for monitoring arterial blood oxygen saturation by pulse oximetry in burned infants