Statistics of subgroups of the modular group

We count the finitely generated subgroups of the modular group [Formula: see text]. More precisely, each such subgroup [Formula: see text] can be represented by its Stallings graph [Formula: see text], we consider the number of vertices of [Formula: see text] to be the size of [Formula: see text] and we count the subgroups of size [Formula: see text]. Since an index [Formula: see text] subgroup has size [Formula: see text], our results generalize the known results on the enumeration of the finite index subgroups of [Formula: see text]. We give asymptotic equivalents for the number of finitely generated subgroups of [Formula: see text], as well as of the number of finite index subgroups, free subgroups and free finite index subgroups. We also give the expected value of the isomorphism type of a size [Formula: see text] subgroup and prove a large deviation statement concerning this value. Similar results are proved for finite index and for free subgroups. Finally, we show how to efficiently generate uniformly at random a size [Formula: see text] subgroup (respectively, finite index subgroup, free subgroup) of [Formula: see text].

Peripheral Uric Acid as a Biomarker in Individuals With Bipolar Disorder: A Case Control Study

Background: At present, no well-established biomarkers were ever found to distinguish unipolar disorder (UD) and bipolar disorder (BD). This study aimed to explore whether uric acid (UA) could be a biomarker to distinguish UD and BD. Methods: Peripheral UA of 119 patients with BD in acute stage (AS) and 77 in remission stage (RS), and 95 patients with UD in AS and 61 in RS were measured, so were 180 healthy controls. Differences in continuous variables among groups were assessed by the independent samples t-test and one-way analysis of variance. The chi-square test was applied to categorical data such as gender. Results: UA levels in BD group were higher than UD and HC groups regardless of the AS or RS, while differences in UA levels between UD group and HC group were not significant. Differences of UA levels between BD-M (bipolar mania/hypomania) and BD-D (bipolar depression) subgroups were not significant, and UA levels of BD-M and BD-D subgroups were higher than UD and HC groups. Only in UD group, UA levels of drug-use subgroup were higher than drug-naïve/free subgroup, but differences disappeared when analyzed stratified by sex; whether in drug-use or drug-naïve/free subgroup, differences of UA levels between BD-M and BD-D groups were not significant.Conclusion: The study suggests UA levels may be a biomarker of BD to distinguish from UD.

Peripheral Uric Acid as a Biomarker in Individuals with Bipolar Disorder

Background: At present, no well-established biomarkers were ever found to distinguish unipolar disorder (UD) and bipolar disorder (BD). This study aimed to explore whether uric acid (UA) could be a biomarker to distinguish UD and BD. Methods: Peripheral UA of 119 patients with BD in acute stage (AS) and 77 in remission stage (RS), and 95 patients with UD in AS and 61 in RS were measured, so were 180 healthy controls. Results: UA levels in BD group were higher than UD and HC groups regardless of the AS or RS, while differences in UA levels between UD group and HC group were not significant. Differences of UA levels between BD-M and BD-D subgroups were not significant, and UA levels of BD-M and BD-D subgroups were higher than UD and HC groups. Only in UD group, UA levels of drug-use subgroup were higher than drug-naïve/free subgroup, but differences disappeared when analyzed stratified by sex; whether in drug-use or drug-naïve/free subgroup, differences of UA levels between BD-M and BD-D groups were not significant.Conclusion: The study suggests UA levels may be a biomarker of BD to distinguish from UD.

Larazotide Acetate for Treatment of Celiac Disease: A Systematic Review and Meta-Analysis of Randomized Controlled Trials

Purpose: The standard of care for treatment of celiac disease (CD) is a stringent lifetime gluten-free diet (GFD), which is very challenging. Larazotide acetate (AT-1001) is an anti-zonulin which functions as a gut permeability regulator for treatment of CD. We endeavored to conduct a systematic review and meta-analysis of all randomized controlled trials (RCTs) which studied the efficacy and safety of larazotide acetate in patients with CD. Methods: We examined four databases from inception to 20-August-2020 using related keywords. We identified all relevant RCTs and judged their risk of bias. We pooled continuous outcomes as mean difference and dichotomous outcomes as risk ratio with 95% confidence interval under fixed-effects meta-analysis model. Results: Four RCTs met our eligibility criteria, comprising 626 patients (larazotide acetate, n=465, placebo, n=161). Three and two studies reported outcomes of patients undergoing gluten challenge and GFD, respectively. For change in lactulose-to-mannitol ratio, the overall effect estimate did not reveal a significant difference between larazotide acetate and placebo groups. For change in total gastrointestinal symptom rating scale (GSRS), subgroup analysis showed that larazotide acetate significantly yielded better symptomatic improvement in the gluten challenge but not gluten free subgroup. Similar finding was found for change in celiac-disease GSRS (CD-GSRS) favoring the gluten challenge over gluten free subgroup. When compared to placebo, larazotide acetate favorably reduced the adverse event (AE) of gluten-related diarrhea in patients who underwent gluten challenge. Other AEs were comparable between both treatment groups. Conclusions: Larazotide acetate is well-endured and superior to placebo in alleviating gastrointestinal symptoms.

Kummer Theory for Number Fields and the Reductions of Algebraic Numbers II

Let K be a number field, and let G be a finitely generated and torsion-free subgroup of K×. For almost all primes p of K, we consider the order of the cyclic group (G mod 𝔭), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if ℓe is a prime power and a is a multiple of ℓ (and a is a multiple of 4 if ℓ =2), then the density of primes 𝔭 of K such that the order of (G mod 𝔭) is congruent to a modulo ℓe only depends on a through its ℓ-adic valuation.

Peripheral Non-enzymatic Antioxidants in Patients with Schizophrenia：A Case-control Study

Background: Recent studies show that oxidative stress is associated with the pathogenesis of schizophrenia. There are two major types of antioxidant systems in vivo, namely enzymatic antioxidants and non-enzymatic antioxidants. This study investigated differences of non-enzymatic antioxidants between schizophrenia patients and healthy controls. Methods: Peripheral UA, ALB, and TBIL of 107 schizophrenic patients in the acute stage and 101 in the remission stage were measured respectively, so were 273 healthy controls. Results: The levels of UA ( P =0.020) and TBIL ( P <0.001) of schizophrenic patients in acute stage were higher than those of healthy controls, while the level of ALB ( P <0.001) was lower. Similar results were detected form schizophrenic patients in the remission stage. Schizophrenic patients in acute stage were divided into antipsychotics-use subgroup (n=56) and antipsychotics-naïve/free subgroup (n=51). The level of UA ( P =0.001) in the antipsychotics-use subgroup was higher than that in the antipsychotics-naïve/free subgroup, while the level of TBIL ( P =0.002) was lower than that in antipsychotics-naïve/free subgroup. 77 schizophrenic patients in the acute stage were followed up, and there was no significant difference in the level of UA before and after treatment, but levels of ALB ( P <0.001) and TBIL ( P <0.001) decreased significantly after the treatment. Conclusion: This study demonstrated that the dysfunction of the peripheral non-enzymatic anti-oxidation system might be involved in the pathogenesis of schizophrenia.

Free subgroups in maximal subgroups of SLn(D)

Given a non-commutative finite-dimensional [Formula: see text]-central division ring [Formula: see text], [Formula: see text] a subnormal subgroup of [Formula: see text] and [Formula: see text] a non-abelian maximal subgroup of [Formula: see text], then either [Formula: see text] contains a non-cyclic free subgroup or there exists a non-central maximal normal abelian subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is a subfield of [Formula: see text], [Formula: see text] is Galois and [Formula: see text], also [Formula: see text] is a finite simple group with [Formula: see text].

Peripheral Non-enzymatic Antioxidants in Patients with Schizophrenia：A Case-control Study

Background: Recent studies show that oxidative stress is associated with the pathogenesis of schizophrenia. There are two major types of antioxidant systems in vivo, namely enzymatic antioxidants and non-enzymatic antioxidants. This study investigated differences of non-enzymatic antioxidant between schizophrenia patients and healthy controls. Methods: Peripheral UA, ALB and TBIL of 107 schizophrenic patients in acute stage and 101 in remission stage were measured respectively, so were 273 healthy controls. Results: The levels of UA ( P =0.020) and TBIL ( P <0.001) of schizophrenic patients in acute stage were higher than those of healthy controls, while the level of ALB ( P <0.001) was lower. Similar results were detected form schizophrenic patients in remission stage. Schizophrenic patients in acute stage were divided into antipsychotics-use subgroup (n=56) and antipsychotics-naïve/free subgroup (n=51). The level of UA ( P =0.001) in antipsychotics-use subgroup was higher than that in antipsychotics-naïve/free subgroup, while the level of TBIL ( P =0.002) was lower than that in antipsychotics-naïve/free subgroup. 77 schizophrenic patients in acute stage were followed up, and there was no significant difference in level of UA before and after treatment, but levels of ALB ( P <0.001) and TBIL ( P <0.001) decreased significantly after the treatment. Conclusion: This study demonstrated that the dysfunction of peripheral non-enzymatic anti-oxidation system might be involved in the pathogenesis of schizophrenia. Keywords: Schizophrenia; Uric acid; Albumin; Total bilirubin

Peripheral Non-enzymatic Antioxidants in Patients with Schizophrenia：A Case-control Study

Background: Recent studies show that oxidative stress is associated with the pathogenesis of schizophrenia. There are two major types of antioxidant systems in vivo, namely enzymatic antioxidants and non-enzymatic antioxidants. This study investigated differences of non-enzymatic antioxidant between schizophrenia patients and healthy controls. Methods: Peripheral UA, ALB and TBIL of 107 schizophrenic patients in acute stage and 101 in remission stage were measured respectively, so were 273 healthy controls. Results: The levels of UA (P=0.020) and TBIL (P<0.001) of schizophrenic patients in acute stage were higher than those of healthy controls, while the level of ALB (P<0.001) was lower. Similar results were detected form schizophrenic patients in remission stage. Schizophrenic patients in acute stage were divided into antipsychotics-use subgroup (n=56) and antipsychotics-naïve/free subgroup (n=51). The level of UA (P=0.001) in antipsychotics-use subgroup was higher than that in antipsychotics-naïve/free subgroup, while the level of TBIL (P=0.002) was lower than that in antipsychotics-naïve/free subgroup. 77 schizophrenic patients in acute stage were followed up, and there was no significant difference in level of UA before and after treatment, but levels of ALB (P<0.001) and TBIL (P<0.001) decreased significantly after the treatment. Conclusion: This study demonstrated that the dysfunction of peripheral non-enzymatic anti-oxidation system might be involved in the pathogenesis of schizophrenia. Keywords: Schizophrenia; Uric acid; Albumin; Total bilirubin