Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit

We consider the double scaling limit in the random matrix ensemble with an external source \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with two eigenvalues $\pm a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one interval for $0 < a < 1$. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case $a=1$ new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a $3 \times 3$-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.Comment: 36 pages, 9 figure

Interacting Modified Variable Chaplygin Gas in Non-flat Universe

A unified model of dark energy and matter is presented using the modified variable Chaplygin gas for interacting dark energy in a non-flat universe. The two entities interact with each other non-gravitationally which involves a coupling constant. Due to dynamic interaction, the variation in this constant arises that henceforth changes the equations of state of these quantities. We have derived the effective equations of state corresponding to matter and dark energy in this interacting model. Moreover, the case of phantom energy is deduced by putting constraints on the parameters involved.Comment: 9 pages; Accepted for publication in European Physical Journal

Vitamin E-analog Trolox prevents endoplasmic reticulum stress in frozen-thawed ovarian tissue of capuchin monkey (Sapajus apella)

Ovarian fragments were exposed to 0.5 M sucrose and 1 M ethylene glycol (freezing solution; FS) with or without selenium or Trolox. Histological and ultrastructural analyses showed that the percentages of normal follicles in control tissue and in tissue after exposure to FS+50 μM Trolox were similar. Trolox prevented endoplasmic reticulum (ER)-related vacuolization, which is commonly observed in oocytes and stromal tissue after exposure to FS. From the evaluated stress markers, superoxide dismutase 1 (SOD1) was up-regulated in ovarian tissue exposed to FS+10 ng/ml selenium. Ovarian fragments were subsequently frozenthawed in the presence of FS with or without 50 μM Trolox, followed by in vitro culture (IVC). Antioxidant capacity in ovarian fragments decreased after freeze-thawing in Troloxfree FS compared with FS+50 μMTrolox. Although freezing itself minimized the percentage of viable follicles in each solution, Trolox supplementation resulted in higher rates of viable follicles (67 %), even after IVC (61 %). Furthermore, stress markers SOD1 and ERp29 were up-regulated in ovarian tissue frozen-thawed in Trolox-free medium. Relative mRNA expression of growth factors markers was evaluated after freeze-thawing followed by IVC. BMP4, BMP5, CTGF, GDF9 and KL were down-regulated independently of the presence of Trolox in FS but down-regulation was less pronounced in the presence of Trolox. Thus, medium supplementation with 50 μMTrolox prevents ER stress and, consequently, protects ovarian tissue from ER-derived cytoplasmic vacuolization. ERp29 but not ERp60, appears to be a key marker linking stress caused by freezing-thawing and cell vacuolization.http://link.springer.com/journal/441hb201