The exponential trapezoidal method for semilinear integro-differential equations

The exponential trapezoidal rule is proposed and analyzed for the numerical integration of semilinear integro-differential equations. Although the method is implicit, the numerical solution is easily obtained by standard fixed-point iteration, making its implementation straightforward. Second-order convergence in time is shown in an abstract Hilbert space framework under reasonable assumptions on the problem. Numerical experiments illustrate the proven order of convergence

Explicit exponential Runge-Kutta methods for semilinear integro-differential equations

The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. By expanding the errors of the numerical method in terms of the solution, we derive order conditions that form the basis of our error bounds for integro-differential equations. The order conditions are further used for constructing numerical methods. The convergence analysis is performed in a Hilbert space setting, where the smoothing effect of the resolvent family is heavily used. For the linear case, we derive the order conditions for general order $p$ and prove convergence of order $p$, whenever these conditions are satisfied. In the semilinear case, we consider in addition spatial discretization by a spectral Galerkin method, and we require locally Lipschitz continuous nonlinearities. We derive the order conditions for orders one and two, construct methods satisfying these conditions and prove their convergence. Finally, some numerical experiments illustrating our theoretical results are given

Relationship between Estradiol and Antioxidant Enzymes Activity of Ischemic Stroke

Some evidence suggests the neuroprotection of estrogen provided by the antioxidant activity of this compound. The main objective of this study was to determine the level of estradiol and its correlation with the activity of antioxidant enzymes, total antioxidant status and ferritin from ischemic stroke subjects. The study population consisted of 30 patients with acute ischemic stroke and 30 controls. There was no significant difference between estradiol in stroke and control group. The activity of superoxide dismutase and level of ferritin was higher in stroke compared with control group (P < .05, P < .001, resp.). There was no significant correlation between estradiol and glutathione peroxidase, glutathione reductase, catalase, total antioxidant status, and ferritin in stroke and control groups. We observed inverse correlation between estradiol with superoxide dismutase in males of stroke patients (r = −0.54, P = .029). Our results supported that endogenous estradiol of elderly men and women of stroke or control group has no antioxidant activity

Matrix metalloproteinase-2 C-735T and its interaction with matrix metalloproteinase-7 A-181G polymorphism are associated with the risk of preeclampsia: influence on total antioxidant capacity and blood pressure

Matrix metalloproteinase (MMP) -2 C-735 T and MMP-7 A-181 G genotypes were studied in 144 pregnant patients with mild and severe preeclampsia and 103 healthy pregnant women. Significantly higher frequencies of CT and TT genotypes in patients compared to controls increased the risk of preeclampsia by 2.42 and 3.13 times, respectively. In severe preeclamptic women in the presence of MMP-2 CT the level of total antioxidant capacity was significantly lower than MMP-2 CC genotype. Also, in the presence of MMP-2 CT + TT blood pressure was significantly increased compared to CC genotype in all the patients. The combined presence of MMP-2 T and the MMP-7 A alleles compared to MMP-2 C and MMP-7 A alleles significantly increased the risk of preeclampsia by 3.08-fold. Our findings demonstrate an association between the MMP-2 C-735 T polymorphism with blood pressure and the risk of preeclampsia. Also, in the presence of polymorphism total antioxidant capacity level decreased in severe preeclampsia.Impact statement What is already known on this subject: Matrix metalloproteinases (MMPs) including MMP-2 might be involved in the pathogenesis of preeclampsia through alteration of invasive ability of trophoblastic cells and abnormal placentation. In one available study the absence of association between MMP-2 C-735T polymorphism with gestational hypertension or preeclampsia has been reported. What the results of this study add: We found that the presence of MMP-2 C-735T polymorphism increased the risk of preeclampsia and there was a significantly lower level of total antioxidant capacity in the presence of the polymorphism in severe preeclampsia. Also, we found significantly higher systolic and diastolic blood pressures in the presence of MMP-2 C-735T polymorphism. We detected a synergism between the MMP-2 T and the MMP-7 A alleles that increased the risk of preeclampsia. What the implications are of these findings for clinical practice and/or further research: New findings of our study are involvement of lower activity MMP-2 -735 T allele and its synergism with MMP-7 A allele, low promoter activity allele, in the pathogenesis of preeclampsia through possible impairment of placentation and also by decreased total antioxidant capacity and increased blood pressure. Further association studies of the role of MMP-2 polymorphism and MMP-2 activity in relation to oxidative stress parameters and blood pressure could elucidate the role of MMP-2 and MMP-7 in the pathogenesis of preeclampsia