First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

Optimal convergence estimates for the trace of the polynomial L2-projection operator on a simplex

In this paper we study convergence of the L2-projection onto the space of polynomials up to degree p on a simplex in Rd, d >= 2. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in Rd from [P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163].

Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L2 and tensorized View the MathML source simultaneously on a standard k-dimensional cube. In the special case k=2 the suggested approximation operator is also optimal in L2 and tensorized H1 (without essential boundary conditions). This allows to construct an optimal sparse p-version FEM with sparse piecewise continuous polynomial splines, reducing the number of unknowns from O(p2), needed for the full tensor product computation, to View the MathML source, required for the suggested sparse technique, preserving the same optimal convergence rate in terms of p. We apply this result to an elliptic differential equation and an elliptic integral equation with random loading and compute the covariances of the solutions with View the MathML source unknowns. Several numerical examples support the theoretical estimates.

Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

Eigenvalues of a one-dimensional Dirac operator pencil

We study the spectrum of a one-dimensional Dirac operator pencil, with a coupling constant in front of the potential considered as the spectral parameter. Motivated by recent investigations of graphene waveguides, we focus on the values of the coupling constant for which the kernel of the Dirac operator contains a square integrable function. In physics literature such a function is called a confined zero mode. Several results on the asymptotic distribution of coupling constants giving rise to zero modes are obtained. In particular, we show that this distribution depends in a subtle way on the sign variation and the presence of gaps in the potential. Surprisingly, it also depends on the arithmetic properties of certain quantities determined by the potential. We further observe that variable sign potentials may produce complex eigenvalues of the operator pencil. Some examples and numerical calculations illustrating these phenomena are presented.

Differential effect of beetroot bread on postprandial DBP according to Glu298Asp polymorphism in the eNOS gene: a pilot study.

Our objective was to investigate whether the presence of Glu298Asp polymorphism in the endothelial NO synthase (eNOS) gene differentially affects the postprandial blood pressure response to dietary nitrate-rich beetroot bread. A randomised, single-blind, controlled, crossover acute pilot study was performed in 14 healthy men (mean age: 34±9 years) who were retrospectively genotyped for Glu298Asp polymorphism (7GG; T carriers 7). Volunteers were randomised to receive 200 g beetroot-enriched bread (1.1 mmol nitrate) or control bread (no beetroot; 0.01 mmol nitrate) on two separate occasions 10 days apart. Baseline and incremental area under the curve of blood pressure and NOx (nitrate/nitrite) were measured for a 6-h postprandial period. A treatment × genotype interaction was observed for diastolic blood pressure (P<0.02), which was significantly lower in T carriers (P<0.01) after consumption of beetroot bread compared with control bread. No significant differences were observed in the GG group. The beneficial diastolic blood pressure reduction was observed only in the T carriers of the Glu298Asp polymorphism in the eNOS gene after consumption of nitrate-rich beetroot bread. These data require confirmation in a larger population group.

Differential interference effects of negative emotional states on subsequent semantic and perceptual processing

Past studies have revealed that encountering negative events interferes with cognitive processing of subsequent stimuli. The present study investigates whether negative events affect semantic and perceptual processing differently. Presentation of negative pictures produced slower reaction times than neutral or positive pictures in tasks that require semantic processing, such as natural or man-made judgments about drawings of objects, commonness judgments about objects, and categorical judgments about pairs of words. In contrast, negative picture presentation did not slow down judgments in subsequent perceptual processing (e.g., color judgments about words, size judgments about objects). The subjective arousal level of negative pictures did not modulate the interference effects on semantic or perceptual processing. These findings indicate that encountering negative emotional events interferes with semantic processing of subsequent stimuli more strongly than perceptual processing, and that not all types of subsequent cognitive processing are impaired by negative events.

Differential brain activity during emotional versus nonemotional reversal learning

The ability to change an established stimulus–behavior association based on feedback is critical for adaptive social behaviors. This ability has been examined in reversal learning tasks, where participants first learn a stimulus–response association (e.g., select a particular object to get a reward) and then need to alter their response when reinforcement contingencies change. Although substantial evidence demonstrates that the OFC is a critical region for reversal learning, previous studies have not distinguished reversal learning for emotional associations from neutral associations. The current study examined whether OFC plays similar roles in emotional versus neutral reversal learning. The OFC showed greater activity during reversals of stimulus–outcome associations for negative outcomes than for neutral outcomes. Similar OFC activity was also observed during reversals involving positive outcomes. Furthermore, OFC activity is more inversely correlated with amygdala activity during negative reversals than during neutral reversals. Overall, our results indicate that the OFC is more activated by emotional than neutral reversal learning and that OFC's interactions with the amygdala are greater for negative than neutral reversal learning.

Coalescence of particles by differential sedimentation

We consider a three dimensional system consisting of a large number of small spherical particles, distributed in a range of sizes and heights (with uniform distribution in the horizontal direction). Particles move vertically at a size-dependent terminal velocity. They are either allowed to merge whenever they cross or there is a size ratio criterion enforced to account for collision efficiency. Such a system may be described, in mean field approximation, by the Smoluchowski kinetic equation with a differential sedimentation kernel. We obtain self-similar steady-state and time-dependent solutions to the kinetic equation, using methods borrowed from weak turbulence theory. Analytical results are compared with direct numerical simulations (DNS) of moving and merging particles, and a good agreement is found.

Cardiac protein kinases: the cardiomyocyte kinome and differential kinase expression in human failing hearts

Aims. Protein kinases are potential therapeutic targets for heart failure, but most studies of cardiac protein kinases derive from other systems, an approach that fails to account for specific kinases expressed in the heart and the contractile cardiomyocytes. We aimed to define the cardiomyocyte kinome (i.e. the protein kinases expressed in cardiomyocytes) and identify kinases with altered expression in human failing hearts. Methods and Results. Expression profiling (Affymetrix microarrays) detected >400 protein kinase mRNAs in rat neonatal ventricular myocytes (NVMs) and/or adult ventricular myocytes (AVMs), 32 and 93 of which were significantly upregulated or downregulated (>2-fold), respectively, in AVMs. Data for AGC family members were validated by qPCR. Proteomics analysis identified >180 cardiomyocyte protein kinases, with high relative expression of mitogen-activated protein kinase cascades and other known cardiomyocyte kinases (e.g. CAMKs, cAMP-dependent protein kinase). Other kinases are poorly-investigated (e.g. Slk, Stk24, Oxsr1). Expression of Akt1/2/3, BRaf, ERK1/2, Map2k1, Map3k8, Map4k4, MST1/3, p38-MAPK, PKCδ, Pkn2, Ripk1/2, Tnni3k and Zak was confirmed by immunoblotting. Relative to total protein, Map3k8 and Tnni3k were upregulated in AVMs vs NVMs. Microarray data for human hearts demonstrated variation in kinome expression that may influence responses to kinase inhibitor therapies. Furthermore, some kinases were upregulated (e.g. NRK, JAK2, STK38L) or downregulated (e.g. MAP2K1, IRAK1, STK40) in human failing hearts. Conclusions. This characterization of the spectrum of kinases expressed in cardiomyocytes and the heart (cardiomyocyte and cardiac kinomes) identified novel kinases, some of which are differentially expressed in failing human hearts and could serve as potential therapeutic targets.

Physiological and morphological adaptations of herbaceous perennial legumes allow differential access to sources of varyingly soluble phosphate

The aim of this study was to investigate the capacity of three perennial legume species to access sources of varyingly soluble phosphorus (P) and their associated morphological and physiological adaptations. Two Australian native legumes with pasture potential (Cullen australasicum and Kennedia prostrata) and Medicago sativa cv. SARDI 10 were grown in sand under two P levels (6 and 40 µg P g−1) supplied as Ca(H2PO4)2·H2O (Ca-P, highly soluble, used in many fertilizers) or as one of three sparingly soluble forms: Ca10(OH)2(PO4)6 (apatite-P, found in relatively young soils; major constituent of rock phosphate), C6H6O24P6Na12 (inositol-P, the most common form of organic P in soil) and FePO4 (Fe-P, a poorly-available inorganic source of P). All species grew well with soluble P. When 6 µg P g−1 was supplied as sparingly soluble P, plant dry weight (DW) and P uptake were very low for C. australasicum and M. sativa (0.1–0.4 g DW) with the exception of M. sativa supplied with apatite-P (1.5 g). In contrast, K. prostrata grew well with inositol-P (1.0 g) and Fe-P (0.7 g), and even better with apatite-P (1.7 g), similar to that with Ca-P (1.9 g). Phosphorus uptake at 6 µg P g−1 was highly correlated with total root length, total rhizosphere carboxylate content and total rhizosphere acid phosphatase (EC 3.1.3.2) activity. These findings provide strong indications that there are opportunities to utilize local Australian legumes in low P pasture systems to access sparingly soluble soil P and increase perennial legume productivity, diversity and sustainability.

Differential roles of the PKC novel isoforms, PKCδ and PKCε, in mouse and human platelets

Background Increasing evidence suggests that individual isoforms of protein kinase C (PKC) play distinct roles in regulating platelet activation. Methodology/Principal Findings In this study, we focus on the role of two novel PKC isoforms, PKCδ and PKCε, in both mouse and human platelets. PKCδ is robustly expressed in human platelets and undergoes transient tyrosine phosphorylation upon stimulation by thrombin or the collagen receptor, GPVI, which becomes sustained in the presence of the pan-PKC inhibitor, Ro 31-8220. In mouse platelets, however, PKCδ undergoes sustained tyrosine phosphorylation upon activation. In contrast the related isoform, PKCε, is expressed at high levels in mouse but not human platelets. There is a marked inhibition in aggregation and dense granule secretion to low concentrations of GPVI agonists in mouse platelets lacking PKCε in contrast to a minor inhibition in response to G protein-coupled receptor agonists. This reduction is mediated by inhibition of tyrosine phosphorylation of the FcRγ-chain and downstream proteins, an effect also observed in wild-type mouse platelets in the presence of a PKC inhibitor. Conclusions These results demonstrate a reciprocal relationship in levels of the novel PKC isoforms δ and ε in human and mouse platelets and a selective role for PKCε in signalling through GPVI.

Differential roles for the adapters Gads and LAT in platelet activation by GPVI and CLEC-2.

The results demonstrate that Gads plays a key role in linking the adapter LAT to SLP-76 in response to weak activation of GPVI and CLEC-2 whereas LAT is required for full activation over a wider range of agonist concentrations. These results reveal the presence of a Gads-independent pathway of platelet activation downstream of LAT.

Differential uptake, partitioning and transfer of Cd and Zn in the soil-pea plant-aphid system

The biomagnification of trace metals during transfer from contaminated soil to higher trophic levels may potentially result in the exposure of predatory arthropods to toxic concentrations of these elements. This study examined the transfer of Cd and Zn in a soil−plant−arthropod system grown in series of field plots that had received two annual applications of municipal biosolids with elevated levels of Cd and Zn. Results showed that biosolids amendment significantly increased the concentration of Cd in the soil and the shoots of pea plants and the concentration of Zn in the soil, pea roots, shoots, and pods. In addition, the ratio of Cd to Zn concentration showed that Zn was preferentially transferred compared to Cd through all parts of the system. As a consequence, Zn was biomagnified by the system whereas Cd was biominimized. Cd and Zn are considered to exhibit similar behaviors in biological systems. However, the Cd/Zn ratios demonstrated that in this system, Cd is much less labile in the root−shoot−pod and shoot−aphid pathways than Zn.