Measure boundary value problem for semilinear elliptic equations with critical Hardy potentials

Let \Omega\subset\BBR^N be a bounded $C^2$ domain and \CL_\gk=-\Gd-\frac{\gk}{d^2} the Hardy operator where d=\dist (.,\prt\Gw) and 0<\gk\leq\frac{1}{4}. Let \ga_{\pm}=1\pm\sqrt{1-4\gk} be the two Hardy exponents, \gl_\gk the first eigenvalue of \CL_\gk with corresponding positive eigenfunction \phi_\gk. If $g$ is a continuous nondecreasing function satisfying \int_1^\infty(g(s)+|g(-s)|)s^{-2\frac{2N-2+\ga_+}{2N-4+\ga_+}}ds<\infty, then for any Radon measures \gn\in \GTM_{\phi_\gk}(\Gw) and \gm\in \GTM(\prt\Gw) there exists a unique weak solution to problem P_{\gn,\gm}: \CL_\gk u+g(u)=\gn in \Gw, u=\gm on \prt\Gw. If $g(r)=|r|^{q-1}u$ ($q>1$) we prove that, in the subcritical range of $q$, a necessary and sufficient condition for solving P_{0,\gm} with \gm>0 is that \gm is absolutely continuous with respect to the capacity associated to the Besov space B^{2-\frac{2+\ga_+}{2q'},q'}(\BBR^{N-1}). We also characterize the boundary removable sets in terms of this capacity. In the subcritical range of $q$ we classify the isolated singularities of positive solutions

AG codes from the second generalization of the GK maximal curve

The second generalized GK maximal curves $\mathcal{GK}_{2,n}$ are maximal curves over finite fields with $q^{2n}$ elements, where $q$ is a prime power and $n \geq 3$ an odd integer, constructed by Beelen and Montanucci. In this paper we determine the structure of the Weierstrass semigroup $H(P)$ where $P$ is an arbitrary $\mathbb{F}_{q^2}$-rational point of $\mathcal{GK}_{2,n}$. We show that these points are Weierstrass points and the Frobenius dimension of $\mathcal{GK}_{2,n}$ is computed. A new proof of the fact that the first and the second generalized GK curves are not isomorphic for any $n \geq 5$ is obtained. AG codes and AG quantum codes from the curve $\mathcal{GK}_{2,n}$ are constructed; in some cases, they have better parameters with respect to those already known

Generalized k-core percolation on correlated and uncorrelated multiplex networks

It has been recognized that multiplexes and interlayer degree correlations can play a crucial role in the resilience of many real-world complex systems. Here we introduce a multiplex pruning process that removes nodes of degree less than ki and their nearest neighbors in layer i for i=1,...,m, and establish a generic framework of generalized k-core (Gk-core) percolation over interlayer uncorrelated and correlated multiplex networks of m layers, where k=(k1,...,km) and m is the total number of layers. Gk-core exhibits a discontinuous phase transition for all k owing to cascading failures. We have unraveled the existence of a tipping point of the number of layers, above which the Gk-core collapses abruptly. This dismantling effect of multiplexity on Gk-core percolation shows a diminishing marginal utility in homogeneous networks when the number of layers increases. Moreover, we have found the assortative mixing for interlayer degrees strengthens the Gk-core but still gives rise to discontinuous phase transitions as compared to the uncorrelated counterparts. Interlayer disassortativity on the other hand weakens the Gk-core structure. The impact of correlation effect on Gk-core tends to be more salient systematically over k for heterogenous networks than homogeneous ones

Adenovirus-mediated suppression of hypothalamic glucokinase affects feeding behavior

Glucokinase (GK), the hexokinase involved in glucosensing in pancreatic β-cells, is also expressed in arcuate nucleus (AN) neurons and hypothalamic tanycytes, the cells that surround the basal third ventricle (3V). Several lines of evidence suggest that tanycytes may be involved in the regulation of energy homeostasis. Tanycytes have extended cell processes that contact the feeding-regulating neurons in the AN, particularly, agouti-related protein (AgRP), neuropeptide Y (NPY), cocaine- and amphetamine-regulated transcript (CART) and proopiomelanocortin (POMC) neurons. In this study, we developed an adenovirus expressing GK shRNA to inhibit GK expression in vivo. When injected into the 3V of rats, this adenovirus preferentially transduced tanycytes. qRT-PCR and Western blot assays confirmed GK mRNA and protein levels were lower in GK knockdown animals compared to the controls. In response to an intracerebroventricular glucose injection, the mRNA levels of anorexigenic POMC and CART and orexigenic AgRP and NPY neuropeptides were altered in GK knockdown animals. Similarly, food intake, meal duration, frequency of eating events and the cumulative eating time were increased, whereas the intervals between meals were decreased in GK knockdown rats, suggesting a decrease in satiety. Thus, GK expression in the ventricular cells appears to play an important role in feeding behavior.Fil: Uranga, Romina Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Bahía Blanca. Instituto de Investigaciones Bioquímicas de Bahía Blanca. Universidad Nacional del Sur. Instituto de Investigaciones Bioquímicas de Bahía Blanca; Argentina. Universidad de Concepción; ChileFil: Millán, Carola. Universidad de Concepción; Chile. Universidad Adolfo Ibañez; ChileFil: Barahona, María José. Universidad de Concepción; ChileFil: Recabal, Antonia. Universidad de Concepción; ChileFil: Salgado, Magdiel. Universidad de Concepción; ChileFil: Martinez, Fernando. Universidad de Concepción; ChileFil: Ordenes, Patricio. Universidad de Concepción; ChileFil: Elizondo Vega, Roberto. Universidad de Concepción; ChileFil: Sepúlveda, Fernando. Universidad de Concepción; ChileFil: Uribe, Elena. Universidad de Concepción; ChileFil: García Robles, María de los Ángeles. Universidad de Concepción; Chil

Verification of Gyrokinetic codes: theoretical background and applications

In fusion plasmas the strong magnetic field allows the fast gyro-motion to be systematically removed from the description of the dynamics, resulting in a considerable model simplification and gain of computational time. Nowadays, the gyrokinetic (GK) codes play a major role in the understanding of the development and the saturation of turbulence and in the prediction of the subsequent transport. Naturally, these codes require thorough verification and validation. Here we present a new and generic theoretical framework and specific numerical applications to test the faithfulness of the implemented models to theory and to verify the domain of applicability of existing GK codes. For a sound verification process, the underlying theoretical GK model and the numerical scheme must be considered at the same time, which has rarely been done and therefore makes this approach pioneering. At the analytical level, the main novelty consists in using advanced mathematical tools such as variational formulation of dynamics for systematization of basic GK code's equations to access the limits of their applicability. The verification of numerical scheme is proposed via the benchmark effort. In this work, specific examples of code verification are presented for two GK codes: the multi-species electromagnetic ORB5 (PIC) and the radially global version of GENE (Eulerian). The proposed methodology can be applied to any existing GK code. We establish a hierarchy of reduced GK Vlasov-Maxwell equations implemented in the ORB5 and GENE codes using the Lagrangian variational formulation. At the computational level, detailed verifications of global electromagnetic test cases developed from the CYCLONE Base Case are considered, including a parametric $\beta$-scan covering the transition from ITG to KBM and the spectral properties at the nominal $\beta$ value.Comment: 16 pages, 2 Figures, APS DPP 2016 invited pape

Maximal curves from subcovers of the GK-curve

For every $q=n^3$ with $n$ a prime power greater than $2$, the GK-curve is an $\mathbb F_{q^2}$-maximal curve that is not $\mathbb F_{q^2}$-covered by the Hermitian curve. In this paper some Galois subcovers of the GK curve are investigated. We describe explicit equations for some families of quotients of the GK-curve. New values in the spectrum of genera of $\mathbb F_{q^2}$-maximal curves are obtained. Finally, infinitely many further examples of maximal curves that cannot be Galois covered by the Hermitian curve are provided