Vertex identifying codes for the n-dimensional lattice

An $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and different. Here, we provide an overview on codes for the $n$-dimensional lattice, discussing the case of 1-identifying codes, constructing a sparse code for the 4-dimensional lattice as well as showing that for fixed $n$, the minimum density of an $r$-identifying code is $\Theta(1/r^{n-1})$.Comment: 10p

On Vertex Identifying Codes For Infinite Lattices

PhD Thesis--A compilation of the papers: "Lower Bounds for Identifying Codes in Some Infinite Grids", "Improved Bounds for r-identifying Codes of the Hex Grid", and "Vertex Identifying Codes for the n-dimensional Lattics" along with some other resultsComment: 91p

Improved Bounds for rr-Identifying Codes of the Hex Grid

For any positive integer $r$, an $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and pairwise distinct. For a finite graph, the density of a code is $|C|/|V(G)|$, which naturally extends to a definition of density in certain infinite graphs which are locally finite. We find a code of density less than $5/(6r)$, which is sparser than the prior best construction which has density approximately $8/(9r)$.Comment: 12p

Associations of moderate to vigorous physical activity and sedentary behavior with depressive and anxiety symptoms in self-isolating people during the COVID-19 pandemic:A cross-sectional survey in Brazil

This is a cross-sectional study evaluating the associations of self-reported moderate to vigorous physical activity, and sedentary behavior with depressive, anxiety, and co-occurring depressive and anxiety symptoms (D&A) in self-isolating Brazilians during the COVID-19 pandemic. Depressive and anxiety symptoms were collected using the Beck Depression and Anxiety Inventories (BDI and BAI). Among the 937 participants (females=72.3%), those performing ≥30 min/day of moderate to vigorous or ≥15 min/day of vigorous physical activity had lower odds of prevalent depressive, anxiety, and co-occurring D&A symptoms. Those spending ≥10 h/day sedentary were more likely to have depressive symptoms.status: publishe

Factors associated with regular physical activity participation among people with severe mental ill health

Purpose People with severe mental ill health (SMI) are less physically active and more sedentary than the general population. There is limited research investigating the correlates of physical activity (PA) in people with SMI impeding development of successful interventions. This study aimed to assess the factors associated with regular participation of PA among a large sample of people with SMI. Methods The data for this study were collected from the ‘Lifestyle Health and Wellbeing’ (HWB) cohort that collected data through self-administered questionnaire from participants with SMI. Self-reported participation in regular PA was the main outcome variable. Potential predictors of PA were grouped as demographic, biological, psychological and behavioural variables. Multivariable logistic regressions were conducted considering PA participation as the dependent variable adjusted for possible correlated predictors. Results In total, 3,287 people with SMI (mean (SD) age 47.7 (14.58) years, 59% male) were included; 38% reported undertaking regular PA and 61% wanted to undertake more physical activity. Multivariable logistic regressions showed that the following factors were associated with undertaking more regular PA: being male, aged 18-65 years, having a body mass index between 18.5 and 30 kg/m2, having better self-perceived general health condition, not having a health problem that limits activity, giving higher importance to maintain a healthy lifestyle, and eating more fruit and vegetables. Conclusions Having a better self-perceived general health and placing importance on maintaining a healthy lifestyle were important predictors of regular PA. Lifestyle interventions targeting increased PA among people with SMI should be shaped by their health perception and informed by their needs

On Vertex Identifying Codes For Infinite Lattices

For any position integer r, an r-identifying code on a graph G is a set C which is a subset of V(G) such that the intersection of the radius-r closed neighborhood with C is nonempty and pairwise distinct. For a finite graph, the density of a code is |C|/|V(G)|, which extends naturally to a definition of density on certain infinite graphs which are locally finite. This thesis explores the concept of density on certain infinite graphs, each of which have a representation on an n-dimensional lattice and finds some new bounds for these densities.</p