16 research outputs found

    Extremal Problems for Forests in Graphs and Hypergraphs

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    The Turan number, ex_r(n; F), of an r-uniform hypergraph F is the maximum number of hyperedges in an n-vertex r-uniform hypergraph which does not contain F as a subhypergraph. Note that when r = 2, ex_r(n; F) = ex(n; F) which is the Turan number of graph F. We study. Turan numbers in the degenerate case for graphs and hypergraphs; we focus on the case when F is a forest in graphs and hypergraph. In the first chapter we discuss the history of Turan numbers and give several classical results. In the second chapter, we examine the Turan number for forests with path components, forests of path and star components, and forests with all components of order 5. In the third chapter we determine the Turan number of an r-uniform star forest in various hypergraph settings

    On the coloring of k-distance graphs

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    M 105.01: Contemporary Mathematics

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    On the Connectivity of KK-distance Graphs

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    For any kNk \in \mathbb{N}, the kk-distance graph DkGD^{k}G has the same vertex set of GG, and two vertices of DkGD^{k}G are adjacent if they are exactly distance kk apart in the original graph GG. In this paper, we consider the connectivity of DkGD^{k}G and state the conditions for graph GG and integer kk such that the graph DkGD^{k}G is connected

    M 121.00: College Algebra

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    Global, regional, and national burden of disorders affecting the nervous system, 1990–2021: a systematic analysis for the Global Burden of Disease Study 2021

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    BackgroundDisorders affecting the nervous system are diverse and include neurodevelopmental disorders, late-life neurodegeneration, and newly emergent conditions, such as cognitive impairment following COVID-19. Previous publications from the Global Burden of Disease, Injuries, and Risk Factor Study estimated the burden of 15 neurological conditions in 2015 and 2016, but these analyses did not include neurodevelopmental disorders, as defined by the International Classification of Diseases (ICD)-11, or a subset of cases of congenital, neonatal, and infectious conditions that cause neurological damage. Here, we estimate nervous system health loss caused by 37 unique conditions and their associated risk factors globally, regionally, and nationally from 1990 to 2021.MethodsWe estimated mortality, prevalence, years lived with disability (YLDs), years of life lost (YLLs), and disability-adjusted life-years (DALYs), with corresponding 95% uncertainty intervals (UIs), by age and sex in 204 countries and territories, from 1990 to 2021. We included morbidity and deaths due to neurological conditions, for which health loss is directly due to damage to the CNS or peripheral nervous system. We also isolated neurological health loss from conditions for which nervous system morbidity is a consequence, but not the primary feature, including a subset of congenital conditions (ie, chromosomal anomalies and congenital birth defects), neonatal conditions (ie, jaundice, preterm birth, and sepsis), infectious diseases (ie, COVID-19, cystic echinococcosis, malaria, syphilis, and Zika virus disease), and diabetic neuropathy. By conducting a sequela-level analysis of the health outcomes for these conditions, only cases where nervous system damage occurred were included, and YLDs were recalculated to isolate the non-fatal burden directly attributable to nervous system health loss. A comorbidity correction was used to calculate total prevalence of all conditions that affect the nervous system combined.FindingsGlobally, the 37 conditions affecting the nervous system were collectively ranked as the leading group cause of DALYs in 2021 (443 million, 95% UI 378–521), affecting 3·40 billion (3·20–3·62) individuals (43·1%, 40·5–45·9 of the global population); global DALY counts attributed to these conditions increased by 18·2% (8·7–26·7) between 1990 and 2021. Age-standardised rates of deaths per 100 000 people attributed to these conditions decreased from 1990 to 2021 by 33·6% (27·6–38·8), and age-standardised rates of DALYs attributed to these conditions decreased by 27·0% (21·5–32·4). Age-standardised prevalence was almost stable, with a change of 1·5% (0·7–2·4). The ten conditions with the highest age-standardised DALYs in 2021 were stroke, neonatal encephalopathy, migraine, Alzheimer's disease and other dementias, diabetic neuropathy, meningitis, epilepsy, neurological complications due to preterm birth, autism spectrum disorder, and nervous system cancer.InterpretationAs the leading cause of overall disease burden in the world, with increasing global DALY counts, effective prevention, treatment, and rehabilitation strategies for disorders affecting the nervous system are needed

    On the Turan number of path-star forests

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