A priori bounds and multiplicity for fully nonlinear equations with quadratic growth in the gradient

We consider fully nonlinear uniformly elliptic equations with quadratic growth in the gradient, such as

$?F(x,u,Du,D^{2}u)=\lambda c(x)u+〈M(x)Du,Du〉+h(x)$

in a bounded domain with a Dirichlet boundary condition; here $\lambda \in \mathbb{R}$, $c,h\in L^p(\mathrm{\Omega })$, $p>n\ge 1$, $c?0$ and the matrix M satisfies $0<{\mu }_{1}I\le M\le {\mu }_{2}I$. Recently this problem was studied in the “coercive” case $\lambda c\le 0$, where uniqueness of solutions can be expected; and it was conjectured that the solution set is more complex for noncoercive equations. This conjecture was verified in 2015 by Arcoya, de Coster, Jeanjean and Tanaka for equations in divergence form, by exploiting the integral formulation of the problem. Here we show that similar phenomena occur for general, even fully nonlinear, equations in nondivergence form. We use different techniques based on the maximum principle.We develop a new method to obtain the crucial uniform a priori bounds, which permit to us to use degree theory. This method is based on basic regularity estimates such as half-Harnack inequalities, and on a Vázquez type strong maximum principle for our kind of equations.     

Tutorial review on validation of liquid chromatography–mass spectrometry methods: Part II

This is the part II of a tutorial review intending to give an overview of the state of the art of method validation in liquid chromatography mass spectrometry (LC–MS) and discuss specific issues that arise with MS (and MS–MS) detection in LC (as opposed to the “conventional” detectors). The Part II starts with briefly introducing the main quantitation methods and then addresses the performance related to quantification: linearity of signal, sensitivity, precision, trueness, accuracy, stability and measurement uncertainty. The last section is devoted to practical considerations in validation. With every performance characteristic its essence and terminology are addressed, the current status of treating it is reviewed and recommendations are given, how to handle it, specifically in the case of LC–MS methods.     

Spatiotemporal patterns of stable isotopes and hydrochemistry in springs and river flow of the upper Karkheh River Basin,Iran

Karst springs of the Zagros Mountains contribute a significant amount to agricultural and human water demands of western and south-western Iran. For an adequate management of available water resources in semi-arid and arid regions, sufficient hydrological monitoring is needed, and hydro-chemical and isotope hydrological data provide important additional information. About 350 water samples were collected from precipitation, river water, and karst springs of the upper part of the Karkheh River Basin (20,895 km²) located between 33°35^′ and 34°55^′ North and 46°22^′ and 49°10^′ East with elevations ranging from 928 to 3563 m above sea level. Sampling was conducted in monthly time resolution from August 2011 to July 2012. All samples were analysed for hydro-chemical parameters (pH, electrical conductivity, and major ions) and stable isotopes (deuterium, oxygen-18). Isotope values of precipitation indicate a local meteoric water line (Zagros MWL δ²H=6.8 δ¹⁸O+10.1; R²=0.99) situated between the Mediterranean MWL and Global MWL. Spring and river water isotope values vary between?7.1 and?4.1 ‰, and?38 and?25 ‰ for δ¹⁸O and δ²H, respectively, responding to winter snowmelt and evaporation. This work implements stable isotopes and hydro-chemical information of springs and river water to understand hydrological and hydro-geological interrelations in karstic semi-arid areas and helps to improve the current water resources management practices of western Iran.     

Relationships between the canonical ascending and descending central series of ideals of an associative algebra

We consider the canonical descending and ascending central series of ideals of an associative algebra. In particular, we prove that some ideal in the descending central series is finite-dimensional if and only if some ideal in the ascending central series is finite-codimensional. This result is the associative algebra analogue of results due to Reinhold Baer and Philip Hall in group theory and Ian Stewart in Lie algebra. We also prove various related results.     

Blow‐up phenomena for a semilinear parabolic equation with weighted inner absorption under nonlinear boundary flux

Blow‐up phenomena for a nonlinear divergence form parabolic equation with weighted inner absorption term are investigated under nonlinear boundary flux in a bounded star‐shaped region. We assume some conditions on weight function and nonlinearities to guarantee that the solution exists globally or blows up at finite time. Moreover, by virtue of the modified differential inequality, upper and lower bounds for the blow‐up time of the solution are derived in higher dimensional spaces. Three examples are presented to illustrate applications of our results. Copyright © 2016 John Wiley & Sons, Ltd.     

Omega‐3 Fatty Acid Grafted PAMAM‐Paclitaxel Conjugate Exhibits Enhanced Anticancer Activity in Upper Gastrointestinal Cancer Cells

Upper Gastrointestinal Cancers (UGCs) are a leading cause of cancer‐related deaths worldwide. Paclitaxel (PTX) is frequently used for the treatment of UGCs; however, low bioavailability, reduced solubility, and dose‐dependent toxicity impede its therapeutic use. PAMAMG_4.0‐NH₂‐DHA is synthesized by linking amine‐terminated fourth‐generation poly(amidoamine) (PAMAMG_4.0‐NH₂) dendrimers with omega‐3 fatty acid docosahexaenoic acid (DHA). Next, PAMAMG_4.0‐NH₂‐DHA‐PTX (DHATX) and PAMAMG_4.0‐NH₂‐PTX (PAX) conjugates are synthesized by subsequent covalent binding of PTX with PAMAMG_4.0‐NH₂‐DHA and PAMAMG_4.0‐NH₂, respectively. ¹H‐NMR and MALDI‐TOF analyses are performed to confirm conjugation of DHA to PAMAMG_4.0‐NH₂ and PTX to PAMAMG_4.0‐NH₂‐DHA. The cell viability, clonogenic cell survival, and flow cytometry analyses are used to determine the anticancer activity of PTX, PAX, and DHATX in UGC cell lines. The in vitro data indicate that treatment with DHATX is significantly more potent than PTX or PAX at inhibiting cellular proliferation, suppressing long‐term survival, and inducing cell death in UGC cells.

     

Polarization tensors of planar domains as functions of the admittivity contrast

(Electric) polarization tensors describe part of the leading order term of asymptotic voltage perturbations caused by low volume fraction inhomogeneities of the electrical properties of a medium. They depend on the geometry of the support of the inhomogeneities and on their admittivity contrast. Corresponding asymptotic formulas are of particular interest in the design of reconstruction algorithms for determining the locations and the material properties of inhomogeneities inside a body from measurements of current flows and associated voltage potentials on the body’s surface. In this work, we consider the two-dimensional case only and provide an analytic representation of the polarization tensor in terms of spectral properties of the double layer integral operator associated with the support of simply connected conductivity inhomogeneities. Furthermore, we establish that an (infinitesimal) simply connected inhomogeneity has the shape of an ellipse, if and only if the polarization tensor is a rational function of the admittivity contrast with at most two poles whose residues satisfy a certain algebraic constraint. We also use the analytic representation to provide a proof of the so-called Hashin–Shtrikman bounds for polarization tensors; a similar approach has been taken previously by Golden and Papanicolaou and Kohn and Milton in the context of anisotropic composite materials.     

具有N-各向异性拉普拉斯算子的超线性问题的先验界

通过改进Brezis和Merle的方法,结合Moser-Trudinger不等式,移动平面方法及比较原理,得到了方程-Q_Nu=f(u),u∈W_0~(1,N)(Ω)的正解的先验界,其中Ω是R~N中的一个有界光滑区域,非线性项f至多具有指数型增长.     

A note on functional a posteriori estimates for elliptic optimal control problems

In this work, new results on functional type a posteriori estimates for elliptic optimal control problems with control constraints are presented. More precisely, we derive new, sharp, guaranteed, and fully computable lower bounds for the cost functional in addition to the already existing upper bounds. Using both, the lower and the upper bounds, we arrive at two‐sided estimates for the cost functional. We prove that these bounds finally lead to sharp, guaranteed and fully computable upper estimates for the discretization error in the state and the control of the optimal control problem. First numerical tests are presented confirming the efficiency of the a posteriori estimates derived. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 403–424, 2017     

B-矩阵线性互补问题的误差界估计

利用严格对角占优M-矩阵的逆矩阵的无穷大范数的范围,给出了B-矩阵线性互补问题误差界新的估计式.相应数值算例表明了结果的有效性.