Solving Two Conjectures regarding Codes for Location in Circulant Graphs

Identifying and locating-dominating codes have been widely studied in circulant graphs of type $C_n(1,2, \ldots, r)$, which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs $C_n(1,3)$. They showed that the smallest cardinality of a locating-dominating code in $C_n(1,3)$ is at least $\lceil n/3 \rceil$ and at most $\lceil n/3 \rceil + 1$ for all $n \geq 9$. Moreover, they proved that the lower bound is strict when $n \equiv 0, 1, 4 \pmod{6}$ and conjectured that the lower bound can be increased by one for other $n$. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in $C_n(1,3)$ is at least $\lceil 4n/11 \rceil$ and at most $\lceil 4n/11 \rceil + 1$ for all $n \geq 11$. Furthermore, they proved that the lower bound is attained for most of the lengths $n$ and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs

Partition games

We introduce CUT, the class of 2-player partition games. These are NIM type games, played on a finite number of heaps of beans. The rules are given by a set of positive integers, which specifies the number of allowed splits a player can perform on a single heap. In normal play, the player with the last move wins, and the famous Sprague-Grundy theory provides a solution. We prove that several rulesets have a periodic or an arithmetic periodic Sprague-Grundy sequence (i.e. they can be partitioned into a finite number of arithmetic progressions of the same common difference). This is achieved directly for some infinite classes of games, and moreover we develop a computational testing condition, demonstrated to solve a variety of additional games. Similar results have previously appeared for various classes of games of take-and-break, for example octal and hexadecimal; see e.g. Winning Ways by Berlekamp, Conway and Guy (1982). In this context, our contribution consists of a systematic study of the subclass `break-without-take'

Analysis of Aftershock Parameters for the Alaskan Subduction Zone Tectonic Region

Forecasting how many earthquakes will occur following a potentially damaging earthquake helps the public and emergency operators stay safe and make informed decisions. The U.S. Geological Survey issues aftershock forecasts following potentially damaging earthquakes, using models to predict the number of earthquakes that should occur within the next day, week, month, and year with 95% confidence to reflect the uncertainty in aftershock behavior. The USGS considers the forecast to be successful when the number of earthquakes observed within the forecasted time period is within the 95% confidence interval. For aftershock sequences that occur along the forearc of the Alaskan subduction zone, the observations consistently lie within this broad range of success, however the forecasts systematically over-predict the number of aftershocks. This suggests that the parameters which drive the forecast model’s rate of decay needs to change to reflect the decreased observed aftershock activity. I analyzed seventeen earthquake sequences in the Alaska forearc region and have developed a set of parameters that may improve early-sequence aftershock forecasting in the region. I stacked the 17 sequences and used maximum likelihood estimation to determine the new parameters and their uncertainty. Following the lessons of Page et al. (BSSA, 2016), I combined the inter-sequence variability and the uncertainty of the parameters from the stacked sequences to produce the total standard deviation for a new forecast model

Optimal bounds on codes for location in circulant graphs

Identifying and locating-dominating codes have been studied widely in circulant graphs of type Cn(1,2,3,...,r) over the recent years. In 2013, Ghebleh and Niepel studied locating-dominating and identifying codes in the circulant graphs Cn(1,d) for d=3 and proposed as an open question the case of d>3. In this paper we study identifying, locating-dominating and self-identifying codes in the graphs Cn(1,d), Cn(1,d-1,d) and Cn(1,d-1,d,d+1). We give a new method to study lower bounds for these three codes in the circulant graphs using suitable grids. Moreover, we show that these bounds are attained for infinitely many parameters n and d. In addition, new approaches are provided which give the exact values for the optimal self-identifying codes in Cn(1,3) and Cn(1,4)

Effective detection of proteins following electrophoresis using extracts of locally available food species

Procedures in life sciences research laboratories often require chemicals and plasticware that are costly, toxic or pose a risk to the environment. Therefore, sustainable alternatives would be of interest, provided that they generate suitable data quality. Coomassie blue and silver staining are the most widely used methods for detecting proteins following electrophoresis in the laboratory. However, their use presents challenges in terms of safety and waste management. In the current study, aqueous extracts were prepared from a series of common food species and evaluated as alternative stains for protein detection. Beets, blueberries, purple cabbage, raspberries and strawberries were employed to stain identical proteins separated under the same conditions in electrophoresis gels. Extracts of the first two species resulted in protein bands that were detectable through visible light transillumination, whereas extracts from all five species generated specific protein bands under ultraviolet light. The raspberry-derived extract was selected for further study based on the brightness of the fluorescent protein bands and minimal background staining. For both bovine serum albumin and lysozyme at 2.5 μg and 0.5 μg protein per band, the mean signal intensities obtained with raspberry extract staining were just below half of those obtained with Coomassie blue. Furthermore, the mean intensities using raspberry extract were equivalent to those obtained using Coomassie blue in the detection of 0.1 μg protein. Therefore, raspberry could be used to produce an effective stain for the routine laboratory analysis of proteins

Mortality and pulmonary complications in patients undergoing surgery with perioperative SARS-CoV-2 infection: an international cohort study

Background: The impact of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) on postoperative recovery needs to be understood to inform clinical decision making during and after the COVID-19 pandemic. This study reports 30-day mortality and pulmonary complication rates in patients with perioperative SARS-CoV-2 infection. Methods: This international, multicentre, cohort study at 235 hospitals in 24 countries included all patients undergoing surgery who had SARS-CoV-2 infection confirmed within 7 days before or 30 days after surgery. The primary outcome measure was 30-day postoperative mortality and was assessed in all enrolled patients. The main secondary outcome measure was pulmonary complications, defined as pneumonia, acute respiratory distress syndrome, or unexpected postoperative ventilation. Findings: This analysis includes 1128 patients who had surgery between Jan 1 and March 31, 2020, of whom 835 (74·0%) had emergency surgery and 280 (24·8%) had elective surgery. SARS-CoV-2 infection was confirmed preoperatively in 294 (26·1%) patients. 30-day mortality was 23·8% (268 of 1128). Pulmonary complications occurred in 577 (51·2%) of 1128 patients; 30-day mortality in these patients was 38·0% (219 of 577), accounting for 81·7% (219 of 268) of all deaths. In adjusted analyses, 30-day mortality was associated with male sex (odds ratio 1·75 [95% CI 1·28–2·40], p\textless0·0001), age 70 years or older versus younger than 70 years (2·30 [1·65–3·22], p\textless0·0001), American Society of Anesthesiologists grades 3–5 versus grades 1–2 (2·35 [1·57–3·53], p\textless0·0001), malignant versus benign or obstetric diagnosis (1·55 [1·01–2·39], p=0·046), emergency versus elective surgery (1·67 [1·06–2·63], p=0·026), and major versus minor surgery (1·52 [1·01–2·31], p=0·047). Interpretation: Postoperative pulmonary complications occur in half of patients with perioperative SARS-CoV-2 infection and are associated with high mortality. Thresholds for surgery during the COVID-19 pandemic should be higher than during normal practice, particularly in men aged 70 years and older. Consideration should be given for postponing non-urgent procedures and promoting non-operative treatment to delay or avoid the need for surgery. Funding: National Institute for Health Research (NIHR), Association of Coloproctology of Great Britain and Ireland, Bowel and Cancer Research, Bowel Disease Research Foundation, Association of Upper Gastrointestinal Surgeons, British Association of Surgical Oncology, British Gynaecological Cancer Society, European Society of Coloproctology, NIHR Academy, Sarcoma UK, Vascular Society for Great Britain and Ireland, and Yorkshire Cancer Research

Résolution de quelques problèmes d'optimisation dans les graphes et les jeux combinatoires

I studied three optimization problems on graphs and combinatorial games.First, identifying codes were studied : vertices couteract faults. Identifying codes help locate the fault to repare it. We focused on circulant graphs by embedding them on infinite grids.Then, the marking and the coloring games were studied : two player games were one player wants to build something (a proper coloration or a proper marking) and the other wants to prevent the first player from doing so. For the marking game we studied the evolution of the strategy when modifying the graph. For the coloring game we defined a new edge-wise decomposition of graphs and we defined a new strategy on this decomposition that improves known results on planar graphs.In the end, I studied pure breaking games : two players take turns to break a heap of tokens in a given number of non-empty heaps. We focused on winning strategies for the game starting with a unique heap on n tokens. These games seem, on first sight, to be all regular : we showed this is the case for some of them and we gave a test to study one game at a time. Only one of these games does not seem to be regular, its behavior remains a mystery.To sum up, I studied three bilateral problems that use different methods and have different purposes in combinatoricsJ'ai étudié trois problèmes d'optimisation dans les graphes et les jeux combinatoires.Tout d'abord, les codes identifiants dans les graphes où les sommets font faces à des failles: les codes cherchent à repérer les failles pour les réparer. On s'est intéressé aux codes identifiants dans les graphes circulants en utilisant des plongements de ces graphes dans des grilles infinies.Ensuite, j'ai étudié le jeu de marquage de sommets et le jeu de coloration d'arêtes: ici deux joueurs se font face, le premier cherche à construire une coloration correcte (ou un marquage correct) et le deuxième cherche à l'en empêcher. Pour le jeu de marquage on s'est intéressé aux changements de stratégie gagnante lorsqu'on modifie le graphe. Pour le jeu de coloration d'arêtes on a donné une stratégie gagnante pour le premier joueur pourvu que le graphe considéré admette une certaine décomposition sur les arêtes. On améliore notamment des résultats sur les graphes planaires.Enfin j'ai étudié les jeux à tas purement de casse: deux joueurs à tour de rôle prennent un tas et le cassent en un certain nombre de tas non vides. On s'intéresse aux stratégies gagnantes lorsque les joueurs jouent sur un unique tas contenant n jetons. Ces jeux de pure casse semblent, à l'oeil nu, être réguliers. On a montré que c'est effectivement le cas pour certains et on a donné un test qui permet de déterminer la régularité cas par cas. Un seul cas ne semble pas correspondre à cette régularité: son comportement reste un mystère.En conclusion, je me suis intéressé à trois problèmes bilatéraux qui utilisent différentes méthodes et qui remplissent des propos différents dans le domaine de la combinatoir

Solving Two Conjectures regarding Codes for Location in Circulant Graphs

Identifying and locating-dominating codes have been widely studied in circulant graphs of type $C_n(1,2, \ldots, r)$, which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs $C_n(1,3)$. They showed that the smallest cardinality of a locating-dominating code in $C_n(1,3)$ is at least $\lceil n/3 \rceil$ and at most $\lceil n/3 \rceil + 1$ for all $n \geq 9$. Moreover, they proved that the lower bound is strict when $n \equiv 0, 1, 4 \pmod{6}$ and conjectured that the lower bound can be increased by one for other $n$. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in $C_n(1,3)$ is at least $\lceil 4n/11 \rceil$ and at most $\lceil 4n/11 \rceil + 1$ for all $n \geq 11$. Furthermore, they proved that the lower bound is attained for most of the lengths $n$ and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs