Semilinear mixed problems on Hilbert complexes and their numerical approximation

Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010), 281-354] that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract language of Hilbert complexes. In another recent article [arXiv:1005.4455], we extended the Arnold-Falk-Winther framework by analyzing variational crimes (a la Strang) on Hilbert complexes. In particular, this gave a treatment of finite element exterior calculus on manifolds, generalizing techniques from surface finite element methods and recovering earlier a priori estimates for the Laplace-Beltrami operator on 2- and 3-surfaces, due to Dziuk [Lecture Notes in Math., vol. 1357 (1988), 142-155] and later Demlow [SIAM J. Numer. Anal., 47 (2009), 805-827], as special cases. In the present article, we extend the Hilbert complex framework in a second distinct direction: to the study of semilinear mixed problems. We do this, first, by introducing an operator-theoretic reformulation of the linear mixed problem, so that the semilinear problem can be expressed as an abstract Hammerstein equation. This allows us to obtain, for semilinear problems, a priori solution estimates and error estimates that reduce to the Arnold-Falk-Winther results in the linear case. We also consider the impact of variational crimes, extending the results of our previous article to these semilinear problems. As an immediate application, this new framework allows for mixed finite element methods to be applied to semilinear problems on surfaces.Comment: 22 pages; v2: major revision, particularly sharpening of error estimates in Section

Effects of isoflavones (soy phyto-estrogens) on serum lipids: a meta-analysis of randomized controlled trials

OBJECTIVES: To determine the effects of isoflavones (soy phyto-estrogens) on serum total cholesterol (TC), low density lipoprotein cholesterol (LDL), high density lipoprotein cholesterol (HDL) and triglyceride (TG). METHODS: We searched electronic databases and included randomized trials with isoflavones interventions in the forms of tablets, isolated soy protein or soy diets. Review Manager 4.2 was used to calculate the pooled risk differences with fixed effects model. RESULTS: Seventeen studies (21 comparisons) with 853 subjects were included in this meta-analysis. Isoflavones tablets had insignificant effects on serum TC, 0.01 mmol/L (95% CI: -0.17 to 0.18, heterogeneity p = 1.0); LDL, 0.00 mmol/L (95% CI: -0.14 to 0.15, heterogeneity p = 0.9); HDL, 0.01 mmol/L (95% CI: -0.05 to 0.06, heterogeneity p = 1.0); and triglyceride, 0.03 mmol/L (95% CI: -0.06 to 0.12, heterogeneity p = 0.9). Isoflavones interventions in the forms of isolated soy protein (ISP), soy diets or soy protein capsule were heterogeneous to combine. CONCLUSIONS: Isoflavones tablets, isolated or mixtures with up to 150 mg per day, seemed to have no overall statistical and clinical benefits on serum lipids. Isoflavones interventions in the forms of soy proteins may need further investigations to resolve whether synergistic effects are necessary with other soy components

One year soy protein supplementation has positive effects on bone formation markers but not bone density in postmenopausal women

BACKGROUND: Although soy protein and its isoflavones have been reported to reduce the risk of osteoporosis in peri- and post-menopausal women, most of these studies are of short duration (i.e. six months). The objective of this study was to examine if one year consumption of soy-containing foods (providing 25 g protein and 60 mg isoflavones) exerts beneficial effects on bone in postmenopausal women. METHODS: Eighty-seven eligible postmenopausal women were randomly assigned to consume soy or control foods daily for one year. Bone mineral density (BMD) and bone mineral content (BMC) of the whole body, lumbar (L1-L4), and total hip were measured using dual energy x-ray absorptiometry at baseline and after one year. Blood and urine markers of bone metabolism were also assessed. RESULTS AND DISCUSSION: Sixty-two subjects completed the one-year long study. Whole body and lumbar BMD and BMC were significantly decreased in both the soy and control groups. However, there were no significant changes in total hip BMD and BMC irrespective of treatment. Both treatments positively affected markers of bone formation as indicated by increased serum bone-specific alkaline phosphatase (BSAP) activity, insulin-like growth factor-I (IGF-I), and osteocalcin (BSAP: 27.8 and 25.8%, IGF-I: 12.8 and 26.3%, osteocalcin: 95.2 and 103.4% for control and soy groups, respectively). Neither of the protein supplements had any effect on urinary deoxypyridinoline excretion, a marker of bone resorption. CONCLUSION: Our findings suggest that although one year supplementation of 25 g protein per se positively modulated markers of bone formation, this amount of protein was unable to prevent lumbar and whole body bone loss in postmenopausal women

Convergence and quasi-optimality of an adaptive finite element method for controlling L2 errors

In this paper, a contraction property is proved for an adaptive finite element method for controlling the global L 2 error on convex polyhedral domains. Furthermore, it is shown that the method converges in L 2 with the best possible rate. The method that is analyzed is the standard adaptive method except that, if necessary, additional refinements are made to keep the meshes sufficiently mildly graded. This modification does not compromise the quasi-optimality of the resulting algorithm