On congruence equations arising from suborbital graphs

In this paper we deal with congruence equations arising from suborbital graphs of the normalizer of Γ_0(m) in PSL(2,R) . We also propose a conjecture concerning the suborbital graphs of the normalizer and the related congruence equations. In order to prove the existence of solution of an equation over prime finite field, this paper utilizes the Fuchsian group action on the upper half plane and Farey graphs properties

Real-world efficacy and safety of Ledipasvir plus Sofosbuvir and Ombitasvir/Paritaprevir/Ritonavir +/- Dasabuvir combination therapies for chronic hepatitis C: A Turkish experience

Background/Aims: This study aimed to evaluate the real-life efficacy and tolerability of direct-acting antiviral treatments for patients with chronic hepatitis C (CHC) with/without cirrhosis in the Turkish population.Material and Methods: A total of 4,352 patients with CHC from 36 different institutions in Turkey were enrolled. They received ledipasvir (LDV) and sofosbuvir (SOF)+/- ribavirin (RBV) ombitasvir/paritaprevir/ritonavir +/- dasabuvir (PrOD)+/- RBV for 12 or 24 weeks. Sustained virologic response (SVR) rates, factors affecting SVR, safety profile, and hepatocellular cancer (HCC) occurrence were analyzed.Results: SVR12 was achieved in 92.8% of the patients (4,040/4,352) according to intention-to-treat and in 98.3% of the patients (4,040/4,108) according to per-protocol analysis. The SVR12 rates were similar between the treatment regimens (97.2%-100%) and genotypes (95.6%-100%). Patients achieving SVR showed a significant decrease in the mean serum alanine transaminase (ALT) levels (50.90 +/- 54.60 U/L to 17.00 +/- 14.50 U/L) and model for end-stage liver disease (MELD) scores (7.51 +/- 4.54 to 7.32 +/- 3.40) (p<0.05). Of the patients, 2 were diagnosed with HCC during the treatment and 14 were diagnosed with HCC 37.0 +/- 16.0 weeks post-treatment. Higher initial MELD score (odds ratio [OR]: 1.92, 95% confidence interval [CI]: 1.22-2.38; p=0.023]), higher hepatitis C virus (HCV) RNA levels (OR: 1.44, 95% CI: 1.31-2.28; p=0.038), and higher serum ALT levels (OR: 1.38, 95% CI: 1.21-1.83; p=0.042) were associated with poor SVR12. The most common adverse events were fatigue (12.6%), pruritis (7.3%), increased serum ALT (4.7%) and bilirubin (3.8%) levels, and anemia (3.1%).Conclusion: LDV/SOF or PrOD +/- RBV were effective and tolerable treatments for patients with CHC and with or without advanced liver disease before and after liver transplantation. Although HCV eradication improves the liver function, there is a risk of developing HCC.Turkish Association for the Study of The Liver (TASL

On the Action of Γ0(N)\Gamma ^0(N) on Q^\hat{\mathbb{Q}}

In this paper we examine $\Gamma ^0(N)$-orbits on $\hat{\mathbb{Q}}$and the suborbital graphs for $\Gamma ^0(N)$. Each such suborbitalgraph is a disjoint union of subgraphs whose vertices form a blockof imprimitivity for $\Gamma ^0(N)$. Moreover, these subgraphs areshown to be vertex $\Gamma ^0(N)$-transitive and edge $\Gamma^0(N)$-transitive. Finally, necessary and sufficient conditions forbeing self-paired edge are provided

On Suborbital Graphs for the Extended Modular Group (Gamma)over-cap

WOS: 000326105900016In this paper, we show that the extended modular group acts on transitively and imprimitively. Then the number of orbits of on is calculated and compared with the number of orbits of on . Especially, we obtain the graphs of on , for each and each unit , then we determine the suborbital graph . We also give the edge conditions in and the necessary and sufficient conditions for a circuit to be triangle in (F) over cap (u,N)

On the Action of Γ0(N)\Gamma ^0(N) on Q^\hat{\mathbb{Q}}

In this paper we examine $\Gamma ^0(N)$-orbits on $\hat{\mathbb{Q}}$and the suborbital graphs for $\Gamma ^0(N)$. Each such suborbitalgraph is a disjoint union of subgraphs whose vertices form a blockof imprimitivity for $\Gamma ^0(N)$. Moreover, these subgraphs areshown to be vertex $\Gamma ^0(N)$-transitive and edge $\Gamma^0(N)$-transitive. Finally, necessary and sufficient conditions forbeing self-paired edge are provided

On conjugacy of elliptic elements and circuits in suborbital graphs of congruence subgroups

WOS: 000300544400004In this paper, we, for the first time, introduce the notion of conjugacy in suborbital graphs. Then we show that the conjugate elliptic elements of the modular group Gamma and of congruence subgroups Gamma(0)(p) give rise to conjugate circuits, corresponding to the related elliptic elements, in the Farey graph F and in the suborbital graph F-u,F-p of the action of Gamma(0)(p), respectively

On the Schlafli differential formula based on edge lengths of tetrahedron in H-3 and S-3

WOS: 000262124300007We obtain a new version of Schlafli differential formula based on edge lengths for the volume of a tetrahedron in hyperbolic and spherical 3-spaces, by using the edge matrix of a hyperbolic( or spherical) tetrahedron and its submatrix

On suborbital graphs of the congruence subgroup gamma_ 0(N)

In this paper we examine some properties of suborbital graphs for the congruence subgroup Γ_0 (N). Then we give necessary and sufficient conditions for graphs to have triangels