Random matrix theory and symmetric spaces

In this review we discuss the relationship between random matrix theories and symmetric spaces. We show that the integration manifolds of random matrix theories, the eigenvalue distribution, and the Dyson and boundary indices characterizing the ensembles are in strict correspondence with symmetric spaces and the intrinsic characteristics of their restricted root lattices. Several important results can be obtained from this identification. In particular the Cartan classification of triplets of symmetric spaces with positive, zero and negative curvature gives rise to a new classification of random matrix ensembles. The review is organized into two main parts. In Part I the theory of symmetric spaces is reviewed with particular emphasis on the ideas relevant for appreciating the correspondence with random matrix theories. In Part II we discuss various applications of symmetric spaces to random matrix theories and in particular the new classification of disordered systems derived from the classification of symmetric spaces. We also review how the mapping from integrable Calogero--Sutherland models to symmetric spaces can be used in the theory of random matrices, with particular consequences for quantum transport problems. We conclude indicating some interesting new directions of research based on these identifications.Comment: 161 pages, LaTeX, no figures. Revised version with major additions in the second part of the review. Version accepted for publication on Physics Report

Serum magnesium and copper levels in Bangladeshi women with gestational diabetes mellitus

Background and objectives: Alteration of magnesium (Mg) and copper (Cu) concentrations in blood has been observed in normal pregnancy as well as in gestational diabetes mellitus (GDM). The present study was aimed to evaluate the serum Mg and Cu levels in Bangladeshi women with GDM in their second and third trimester of pregnancy. Methods: The study was conducted at Mymensingh Medical College Hospital from July 2013 to June 2014. Pregnant women, in their second and third trimester, attending the outpatient department of Obstetrics and Gynecology and the Department of Endocrinology of Mymensingh Medical College Hospital were enrolled by purposive sampling technique. GDM was diagnosed on the basis of oral glucose tolerance test (OGTT) as defined in WHO criteria 2013. Blood glucose was estimated by enzymatic GOD-PAP colorimetric method. The cut off value for fasting plasma glucose level was ≥6.1 mmol/L or ≥7.8 mmol/L 2 hours after glucose load. Serum Cu was estimated by 3, 5-DiBr-PAESA method and Mg by Xylidyl Blue-I Method as per manufacturer’s instruction. Results: A total of 172 pregnant women in their second and third trimester were enrolled. Out of 172 participants, 86 had GDM and 86 were normoglycemic (control). The mean age of GDM and control groups was 28.6±3.2 years and 27.3±3.1 years respectively. The BMI was 26.4±1.5 m/kg2 and 26.3±1.3 m/kg2. Serum Mg level was significantly low (p< 0.001) in 2nd and 3rd trimesters in GDM cases (1.39±0.26 mg/dl and 0.93±0.15 mg/dl) compared to control group (1.67±0.3 mg/dl and 1.67±0.31mg/dl). On the contrary, serum Cu levels in GDM cases were significantly (p<0.002) higher in both trimesters (224±333.8 µg/dl and 243.91±6.89 µg/dl) compared to those without GDM (220.1±7.6 µg/dl and 234.9±4.6 µg/dl). There was significant (p<0.001) increase of serum Cu levels in 3rd trimester compared to 2nd trimester in both GDM and non GDM cases. Conclusion: There was distinct alteration of serum Mg and Cu levels in GDM compared to normal pregnancy. IMC J Med Sci 2017; 11(1): 25-2