Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions

We establish new existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam. We consider the widely studied boundary conditions corresponding to clamped and hinged ends and many non-local boundary conditions, with a unified approach. Our method is to show that each boundary-value problem can be written as the same type of perturbed integral equation, in the space $C[0,1]$, involving a linear functional $\alpha[u]$ but, although we seek positive solutions, the functional is not assumed to be positive for all positive $u$. The results are new even for the classic boundary conditions of clamped or hinged ends when $\alpha[u]=0$, because we obtain sharp results for the existence of one positive solution; for multiple solutions we seek optimal values of some of the constants that occur in the theory, which allows us to impose weaker assumptions on the nonlinear term than in previous works. Our non-local boundary conditions contain multi-point problems as special cases and, for the first time in fourth-order problems, we allow coefficients of both signs

The central region of the Fornax cluster -- II. Spectroscopy and radial velocities of member and background galaxies

Radial velocities of 94 galaxies brighter than about V_tot = 20 mag in the direction of the central Fornax cluster have been measured. Except for 8 Fornax members, all galaxies lie in the background. Among the 8 members, there are 5 nucleated dwarf ellipticals that are already listed in the FCC (Ferguson 1989, AJ 98, 367). Two of the 3 ``new'' members are very compact and have surface brightnesses comparable to globular clusters, however their luminosities are in the range of dwarf elliptical nuclei. The measured line indices (especially Mg2, H beta, and iron) of the brighter of the compact objects suggest a solar metallicity, whereas the fainter compact object as well as the dE,Ns have line indices that are similar to those of old metal-poor globular clusters (GCs). However, with these data it is not possible to clearly classify the compact objects either as very bright globular clusters, isolated nuclei of dE,Ns, or even compact ellipticals. A background galaxy cluster at z = 0.11 has been found just behind the center of the Fornax cluster. This explains the excess population of galaxies reported in Paper I. The brightest galaxy of the background cluster lies only 1.1 arcmin south of NGC 1399 and is comparable in absolute luminosity with the central Fornax galaxy itself.Comment: 12 pages, LaTeX2e, uses aa.cls, including 9 PostScript figures; accepted for publication in A&AS, also available at http://www.astro.puc.cl/~mhilker/publication.htm

Nonzero solutions of Hammerstein integral equations with discontinuous kernels

AbstractUsing the theory of fixed point index, we establish new results for the existence of nonzero solutions of integral equations of the form u(t)=∫Gk(t,s)f(s,u(s))ds, where G is a compact set in Rn and k changes sign, so positive solutions may not exist, f satisfies Carathéodory conditions and k may be discontinuous. We apply our results to prove the existence of nontrivial solutions of some nonlocal boundary value problems

Positive solutions of BVPs on the half-line involving functional BCs

We study the existence of positive solutions on the half-line of a second order ordinary differential equation subject to functional boundary conditions. Our approach relies on a combination between the fixed point index for operators on compact intervals, a fixed point result for operators on noncompact sets, and some comparison results for principal and nonprincipal solutions of suitable auxiliary linear equations.Comment: 13 page

New results related to a conjecture of Manickam and Singhi

In 1998 Manickam and Singhi conjectured that for every positive integer $d$ and every $n \ge 4d$, every set of $n$ real numbers whose sum is nonnegative contains at least $\binom {n-1}{d-1}$ subsets of size $d$ whose sums are nonnegative. In this paper we establish new results related to this conjecture. We also prove that the conjecture of Manickam and Singhi does not hold for $n=2d+2$

An Appraisal of the Oleocanthal-Rich Extra Virgin Olive Oil (EVOO) and Its Potential Anticancer and Neuroprotective Properties

dietary consumption of olive oil represents a key pillar of the mediterranean diet, which has been shown to exert beneficial effects on human health, such as the prevention of chronic non-communicable diseases like cancers and neurodegenerative diseases, among others. these health benefits are partly mediated by the high-quality extra virgin olive oil (EVOO), which is produced mostly in mediterranean countries and is directly made from olives, the fruit of the olive tree (Olea europaea L.). preclinical evidence supports the existence of antioxidant and anti-inflammatory properties exerted by the polyphenol oleocanthal, which belongs to the EVOO minor polar compound subclass of secoiridoids (like oleuropein). this narrative review aims to describe the antioxidant and anti-inflammatory properties of oleocanthal, as well as the potential anticancer and neuroprotective actions of this polyphenol. based on recent evidence, we also discuss the reasons underlying the need to include the concentrations of oleocanthal and other polyphenols in the EVOO's nutrition facts label. finally, we report our personal experience in the production of a certified organic EVOO with a "protected designation of origin" (PDO), which was obtained from olives of three different cultivars (rotondella, frantoio, and Leccino) harvested in geographical areas located a short distance from one another (villages' names: gorga and camella) within the southern italy "cilento, vallo di diano and alburni national park" of the campania region (province of salerno, Italy)

Wide Field Imaging. I. Applications of Neural Networks to object detection and star/galaxy classification

[Abriged] Astronomical Wide Field Imaging performed with new large format CCD detectors poses data reduction problems of unprecedented scale which are difficult to deal with traditional interactive tools. We present here NExt (Neural Extractor): a new Neural Network (NN) based package capable to detect objects and to perform both deblending and star/galaxy classification in an automatic way. Traditionally, in astronomical images, objects are first discriminated from the noisy background by searching for sets of connected pixels having brightnesses above a given threshold and then they are classified as stars or as galaxies through diagnostic diagrams having variables choosen accordingly to the astronomer's taste and experience. In the extraction step, assuming that images are well sampled, NExt requires only the simplest a priori definition of "what an object is" (id est, it keeps all structures composed by more than one pixels) and performs the detection via an unsupervised NN approaching detection as a clustering problem which has been thoroughly studied in the artificial intelligence literature. In order to obtain an objective and reliable classification, instead of using an arbitrarily defined set of features, we use a NN to select the most significant features among the large number of measured ones, and then we use their selected features to perform the classification task. In order to optimise the performances of the system we implemented and tested several different models of NN. The comparison of the NExt performances with those of the best detection and classification package known to the authors (SExtractor) shows that NExt is at least as effective as the best traditional packages.Comment: MNRAS, in press. Paper with higher resolution images is available at http://www.na.astro.it/~andreon/listapub.htm