A family of extremal hypergraphs for Ryser's conjecture

Ryser's Conjecture states that for any $r$-partite $r$-uniform hypergraph, the vertex cover number is at most $r{-}1$ times the matching number. This conjecture is only known to be true for $r\leq 3$ in general and for $r\leq 5$ if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. $r$-partite hypergraphs whose cover number is $r-1$ times its matching number. Aside from a few sporadic examples, the set of uniformities $r$ for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order $r-1$ exists. We produce a new infinite family of $r$-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order $r-2$ exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the {\em Ryser poset} of extremal intersecting $r$-partite $r$-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in $\sqrt{r}$. This provides further evidence for the difficulty of Ryser's Conjecture

Matchings and covers of multipartite hypergraphs

Koenig’s theorem is a classic result in combinatorics which states that for every bipartite graph G, the cover number of G (denoted by τ (G)) is equal to its matching number (denoted by ν(G)). The theorem’s importance stems from its many applications in various areas of mathematics, such as optimisation theory and algorithmic analysis. Ryser’s Conjecture for multipartite hypergraphs is a proposed generalisation of Koenig’s theorem made in the 1970s. It asserts that for every r-partite hypergraph H, we have the following inequality: τ (H) ≤ (r − 1)ν(H). The conjecture is only known to be true for tripartite hypergraphs and a few other special cases. In the first part of this thesis, we present algorithms that – for a given r – are able to prove or disprove the conjecture in the case of r-partite intersecting hypergraphs. Moreover, for a given r, the algorithms can also be used to enumerate all r-partite extremal hypergraphs to Ryser’s Conjecture. Extremal hypergraphs are r-partite hypergraphs for which the cover number is exactly r − 1 times the matching number. The second part of this thesis focuses on the case of 4-partite hypergraphs. It is motivated by a recent result on Ryser’s Conjecture for tripartite hypergraphs. The result classifies all tripartite extremal hypergraphs, and implies that if H is a tripartite extremal hypergraph, then it must contain ν(H) vertex-disjoint tripartite intersecting extremal hypergraphs. This result leads to the natural question of whether a similar characterisation of r-partite extremal hypergraphs is possible for other values of r? In particular, for the first open case of Ryser’s Conjecture, the case of r = 4. We shed some light on this question, by first classifying all 4-partite intersecting extremal hypergraphs. We then present a list of 4-partite extremal hypergraphs with matching number equal to two, such that none of them contain two vertex-disjoint 4-partite extremal hypergraphs. Our result shows that a straightforward characterisation of 4-partite extremal hypergraphs is not possible in terms of vertex-disjoint intersecting extremal hypergraphs. However, we still conjecture an analogue to the classification of tripartite extremal hypergraphs, which is not contradicted by our extremal examples. In the final part of the thesis we focus on intersecting extremal hypergraphs to Ryser’s Conjecture. Apart from a few sporadic constructions in the literature, there is only one known family of r-partite extremal hypergraphs, which comes from finite projective planes. The family contains an r-partite extremal hypergraph to Ryser’s Conjecture, whenever a finite projective plane of order r − 1 exists. Our contribution is to first calculates bounds on the sparsest possible extremal hypergraphs for small values of r. We then prove the existence of a new family of extremal hypergraphs to Ryser’s Conjecture. The new family contains an r-partite intersecting extremal hypergraph to Ryser’s Conjecture, whenever a finite projective plane of order r − 2 exists. Moreover we are able to show via a number theoretic argument, that there are infinitely many cases for which our new family contains an extremal hypergraph, when the currently known family of extremal hypergraphs is known not to contain one

Intersecting extremal constructions in Ryser's Conjecture for r-partite hypergraphs

Ryser's Conjecture states that for any r-partite r-uniform hypergraph the vertex cover number is at most r−1 times the matching number. This conjecture is only known to be true for r≤3. For intersecting hypergraphs, Ryser's Conjecture reduces to saying that the edges of every r-partite intersecting hypergraph can be covered by r−1 vertices. This special case of the conjecture has only been proven for r≤5. It is interesting to study hypergraphs which are extremal in Ryser's Conjecture i.e, those hypergraphs for which the vertex cover number is exactly r−1 times the matching number. There are very few known constructions of such graphs. For large r the only known constructions come from projective planes and exist only when r−1 is a prime power. Mansour, Song and Yuster studied how few edges a hypergraph which is extremal for Ryser's Conjecture can have. They defined f(r) as the minimum integer so that there exist an r-partite intersecting hypergraph H with τ(H)=r−1 and with f(r) edges. They showed that f(3)=3,f(4)=6, f(5)=9, and 12≤f(6)≤15. In this paper we focus on the cases when r=6 and 7. We show that f(6)=13 improving previous bounds. We also show that f(7)≤22, giving the first known extremal hypergraphs for the r=7 case of Ryser's Conjecture. These results have been obtained independently by Aharoni, Barat, and Wanless

Pulse versus daily oral Alfacalcidol treatment of secondary hyperparathyroidism in hemodialysis patients: a randomized controlled trial.

Secondary hyperparathyroidism is a common complication of chronic kidney disease and is managed using vitamin D replacement therapy. Very few studies have examined the effectiveness of pulse alfacalcidol therapy in comparison to daily oral alfacalcidol therapy in suppressing serum parathyroid hormone (PTH) levels in hemodialysis patients. The aim of this randomized controlled trial was to replicate the findings of prior studies comparing effectiveness of pulse oral alfacalcidol therapy versus daily oral alfacalcidol therapy in suppressing PTH after 13 weeks of therapy using a Palestinian sample of hemodialysis patients, and to identify demographic and biomedical characteristics of patients that are independently associated with PTH levels. One hundred and sixty-seven patients completed the study, 88 in the daily group and 79 in the pulse group. The pulse group had more clinically significant reduction in mean PTH level by 75 pg/dL at 13 weeks than the daily group, but this was not statistically significant. The effect of alfacalcidol therapy on metabolism of phosphate and corrected calcium levels was comparable in both groups, and pulse therapy was not associated with increased risk of hypercalcemia and hyperphosphatemia. Serum PTH levels were independently and inversely associated with older age and diabetes. Switching daily alfacalcidol therapy to thrice-weekly alfacalcidol pulse therapy seems safe and convenient, especially for hemodialysis patients with poor compliance with treatment. This study also highlights the importance of monitoring and preventing malnutrition in hemodialysis patients and maintaining optimal glycemic control in diabetic hemodialysis patients

The effects of Ramadan fasting on clinical and biochemical markers among hemodialysis patients: A prospective cohort study.

Ramadan fasting is compulsory for all healthy adult Muslims. Although sick people are exempted from Ramadan fasting, some patients such as hemodialysis patients prefer to fast during Ramadan. The effect of Ramadan fasting on clinical outcomes and biochemical markers among hemodialysis patients is not clear. The aim of this study was to examine the effects of daily Ramadan fasting and partial Ramadan fasting on key biochemical and clinical markers among hemodialysis patients as compared to hemodialysis patients who chose not to fast during Ramadan. A prospective cohort study of 269 end stage renal disease patients were recruited from the hemodialysis unit in An-Najah National University Hospital, Nablus, Palestine. The participants were divided into three cohorts based on their plans for fasting during Ramadan in May 2018; Ramadan fasting group (RFG), Ramadan partial fasting group (RPFG) and Ramadan not-fasting group (RNFG). Key clinical and biochemical markers were measured before, during and after Ramadan. After adjustment for diabetic and hypertension status and other sociodemographic variables, RFG had higher mean inter-dialytic weight gain (IDWG) by 0.62 kg than RNFG (95% confidence interval (CI) 0.26, 0.99). RPFG also had slight increase in mean IDWG than RNFG by 0.35 kg (95% CI 0.11, 0.60). Additionally, RFG and RPFG had significant increase in mean serum potassium as compared to RNFG. Diabetes was independently associated with increased IDWG by 0.48 kg (0.25, 0.72). Diabetes and hypertension were associated with some independent changes in biochemical markers, but these were clinically negligible. Our findings suggest that Ramadan fasting (fully or partially) is tolerable by hemodialysis patients and is not associated with important clinical complications. However, these patients should be made aware of the potential risk of fluid overload and hyperkalemia, if they decide to fast during Ramadan. Thus, they should be closely monitored and instructed to adhere to their dietary and fluid intake allowances. Further prospective cohort studies with comprehensive dietary measures and information on adverse clinical outcomes may provide more evidence about the tolerability and safety of Ramadan fasting by hemodialysis patients.The publication of this article was funded by the Qatar National Library. The funder had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

A family of extremal hypergraphs for Ryser's conjecture

Ryser's Conjecture states that for any r-partite r-uniform hypergraph, the vertex cover number is at most r−1 times the matching number. This conjecture is only known to be true for r≤3 in general and for r≤5 if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. r-partite hypergraphs whose cover number is r−1 times its matching number. Aside from a few sporadic examples, the set of uniformities r for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order r−1 exists. We produce a new infinite family of r-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order r−2 exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the Ryser poset of extremal intersecting r-partite r-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in r. This provides further evidence for the difficulty of Ryser's Conjecture

The Alpha-Synuclein Gene (SNCA) is a Genomic Target of Methyl-CpG Binding Protein 2 (MeCP2)—Implications for Parkinson’s Disease and Rett Syndrome

Mounting evidence suggests a prominent role for alpha-synuclein (a-syn) in neuronal cell function. Alterations in the levels of cellular a-syn have been hypothesized to play a critical role in the development of Parkinson’s disease (PD); however, mechanisms that control expression of the gene for a-syn (SNCA) in cis and trans as well as turnover of a-syn are not well understood. We analyzed whether methyl-CpG binding protein 2 (MeCP2), a protein that specifically binds methylated DNA, thus regulating transcription, binds at predicted binding sites in intron 1 of the SNCA gene and regulates a-syn protein expression. Chromatin immunoprecipitation (ChIP) and electrophoretic mobility-shift assays (EMSA) were used to confirm binding of MeCP2 to regulatory regions of SNCA. Site-specific methylation and introduction of localized mutations by CRISPR/Cas9 were used to investigate the binding properties of MeCP2 in human SK-N-SH neuroblastoma cells. The significance of MeCP2 for SNCA regulation was further investigated by overexpressing MeCP2 and mutated variants of MeCP2 in MeCP2 knockout cells. We found that methylation-dependent binding of MeCP2 at a restricted region of intron 1 of SNCA had a significant impact on the production of a-syn. A single nucleotide substitution near to CpG1 strongly increased the binding of MeCP2 to intron 1 of SNCA and decreased a-syn protein expression by 60%. In contrast, deletion of a single nucleotide closed to CpG2 led to reduced binding of MeCP2 and significantly increased a-syn levels. In accordance, knockout of MeCP2 in SK-N-SH cells resulted in a significant increase in a-syn production, demonstrating that SNCA is a genomic target for MeCP2 regulation. In addition, the expression of two mutated MeCP2 variants found in Rett syndrome (RTT) showed a loss of their ability to reduce a-syn expression. This study demonstrates that methylation of CpGs and binding of MeCP2 to intron 1 of the SNCA gene plays an important role in the control of a-syn expression. In addition, the changes in SNCA regulation found by expression of MeCP2 variants carrying mutations found in RTT patients may be of importance for the elucidation of a new molecular pathway in RTT, a rare neurological disorder caused by mutations in MECP2

The Chemical Composition of Achillea wilhelmsii C. Koch and Its Desirable Effects on Hyperglycemia, Inflammatory Mediators and Hypercholesterolemia as Risk Factors for Cardiometabolic Disease

This study was done to identify the content compounds of Achillea wilhelmsii (A. wilhelmsii) and to evaluate its hypoglycemic and anti-hypercholesterolemic activity and effect on inflammatory mediators. The extracts and fractions of A. wilhelmsii were thoroughly analyzed using high performance liquid chromatography (HPLC), and the total content of phenols and flavonoids was determined. The hypoglycemic activity was evaluated in vivo using alloxan-induced diabetic mice. The effect upon inflammatory mediators was evaluated in vitro using the human monocytic leukemia cell line (THP-1). The anti-hypercholesterolemic activity was evaluated in vitro using the 3-hydroxy-3-methylglutaryl-CoA (HMG-CoA) reductase assay kit. The water extract (WE)-treated group showed the highest reduction in the fasting blood glucose levels (FBGL). The chloroform fraction (CF) and ethyl acetate fraction (EAF) both showed a significant ability to reduce the secretion of tumor necrosis factor alpha (TNF-α). The EAF, however, also attenuated the levels of matrix metalloproteinase-2 (MMP-2) and matrix metalloproteinase-9 (MMP-9). The CF showed the most significant 3-hydroxy-3-methylglutaryl-CoA reductase (HMGR) inhibition activity. The five main compounds in the CF were isolated and identified. Out of the five compounds in the CF, 1β,10β-epoxydesacetoxymatricarin (CP1) and leucodin (CP2) showed the highest anti-hypercholesterolemic potential. A molecular docking study provided corresponding results

The Relationship Between Cutaneous Anthrax and Melanogenesis: A Toxic Affair

The severe and often lethal disease Anthrax has been known since antiquity, and is caused by the spore-forming, gram-positive bacterium, Bacillus anthracis (B. a). After spores enter through an open wound, the host develops cutaneous Anthrax, which is identified by the development of typical black colored lesions, eschars. The name Anthrax itself derives from the Greek word for coal. Interestingly, the cause of the black eschars is not understood nor has their role in anthrax pathogenesis or host defense been adequately investigated. We set out to understand the cellular mechanism responsible for the formation of these black lesions, as well as understanding their role in disease progression. Bacillus anthracis secretes two toxic factors, Edema Factor (EF) and Lethal factor (LF), both paralyze the immune response in the early phase of infection and promote alarming symptoms in the later stages of the disease. Edema Factor is a potent Adenylate Cyclase, that leads to an uncontrolled rise in cAMP concentration, and is responsible for edema; whereas lethal toxin is a zinc metalloprotease that cleaves MAPKK ultimately inhibiting cell growth and division. From experiments in Drosophila melanogaster transgenically expressing anthrax toxins, we find that cAMP signaling promotes pigmentation through a defined pathway mediated by Protein Kinase A (PKA). PKA is known to cause specific phosphorylation and activation of Tyrosine Hydroxylase (TH) the rate-limiting enzyme required for melanin synthesis. Examination of Tyrosine Hydroxylase in Drosophila melanogaster in response to EF expression shows a direct relationship with pigmentation and revealed novel phenotypes of Anthrax modeled in this insect system. We propose that a similar mechanism is at play in human cells, and that EF and LF may act similarly in melanocytes, potentially leading to uncontrolled melanogenesis in anthrax-infected skin tissues. Ultimately, this leads to the question as to whether these black cutaneous lesions are a product of host defense aimed at controlling the spread of the pathogen, or whether they are a part of Bacillus anthracis virulence strategy