Local laws for non-Hermitian random matrices

The product of m ∈ N independent random square matrices whose elements are independent identically distributed random variables with zero mean and unit variance is considered. It is known that, as the size of the matrices increases to infinity, the empirical spectral measure of the normalized eigenvalues of the product converges with probability 1 to the distribution of the mth power of the random variable uniformly distributed on the unit disk of the complex plane. In particular, in the case of m = 1, the circular law holds. The purpose of this paper is to prove the validity of the local circular law (as well as its generalization to the case of any fixed m) in the case where the distribution of the matrix elements has finite absolute moment of order 4 + δ,δ > 0,. Recent results of Bourgade, Yau, and Yin, of Tao and Vu, and of Nemish are generalized.     

On Eigenvalues of the Transition Matrix of Some Count-Data Markov Chains

We analyze the eigenstructure of count-data Markov chains. Our main focus is on so-called CLAR(1) models, which are characterized by having a linear conditional mean, and also on the case of a finite range, where the second largest eigenvalue determines the speed of convergence of the forecasting distributions. We derive a lower bound for the second largest eigenvalue, which often (but not always) even equals this eigenvalue. This becomes clear by deriving the complete set of eigenvalues for several specific cases of CLAR(1) models.     

Measuring centrality by a generalization of degree

Network analysis has emerged as a key technique in communication studies, economics, geography, history and sociology, among others. A fundamental issue is how to identify key nodes, for which purpose a number of centrality measures have been developed. This paper proposes a new parametric family of centrality measures called generalized degree. It is based on the idea that a relationship to a more interconnected node contributes to centrality in a greater extent than a connection to a less central one. Generalized degree improves on degree by redistributing its sum over the network with the consideration of the global structure. Application of the measure is supported by a set of basic properties. A sufficient condition is given for generalized degree to be rank monotonic, excluding counter-intuitive changes in the centrality ranking after certain modifications of the network. The measure has a graph interpretation and can be calculated iteratively. Generalized degree is recommended to apply besides degree since it preserves most favourable attributes of degree, but better reflects the role of the nodes in the network and has an increased ability to distinguish among their importance.     

Groups with the Weak Maximal Condition on Non-permutable Subgroups

Let G be a group with the weak maximal condition on non-permutable subgroups. We prove that if G is a generalized radical group then G is either quasihamiltonian or a soluble-by-finite minimax group.     

On distributional solutions of local and nonlocal problems of porous medium type

We present a theory of well-posedness and a priori estimates for bounded distributional (or very weak) solutions of

(0.1)${?}_{t}u?{\mathfrak{L}}^{\sigma ,\mu }[\phi (u)]=g(x,t)\text{in}{\mathbb{R}}^N\times (0,T),$

where φ is merely continuous and nondecreasing, and ${\mathfrak{L}}^{\sigma ,\mu }$ is the generator of a general symmetric Lévy process. This means that ${\mathfrak{L}}^{\sigma ,\mu }$ can have both local and nonlocal parts like, e.g., ${\mathfrak{L}}^{\sigma ,\mu }=\mathrm{\Delta }?{(?\mathrm{\Delta })}^{\frac{1}{2}}$. New uniqueness results for bounded distributional solutions to this problem and the corresponding elliptic equation are presented and proven. A key role is played by a new Liouville type result for ${\mathfrak{L}}^{\sigma ,\mu }$. Existence and a priori estimates are deduced from a numerical approximation, and energy-type estimates are also obtained.     

Relative Reduction and Buchberger’s Algorithm in Filtered Free Modules

In this paper we develop a relative Gröbner basis method for a wide class of filtered modules. Our general setting covers the cases of modules over rings of differential, difference, inversive difference and difference–differential operators, Weyl algebras and multiparameter twisted Weyl algebras (the last class of rings includes the classes of quantized Weyl algebras and twisted generalized Weyl algebras). In particular, we obtain a Buchberger-type algorithm for constructing relative Gröbner bases of filtered free modules.     

A Logical Analysis of the Anselm’s Unum Argumentum (from <Emphasis Type="BoldItalic">Proslogion</Emphasis>)

Anselm of Cantorbery wrote Proslogion (1077–1078), where is formulated the famous ‘Unum argumentum’ about the existence of God. This argument was been disputed and criticized by numerous logicians from an extensional view point. The classical predicate logic is not able to give a formal frame to develop an adequate analysis of this argument. According to us, this argument is not an ontological proof; it analyses the meaning of the “quo nihil maius cogitari posit”, a characterization of God, and establish, by absurd, that “quod non posit cogitare non esse” by using the hypothesis: “to think in re” is taller than “to think in solo intelectu”. We discuss this relation and the difference between the meanings of the elementary predicates ‘to be in re’, ‘to be in intellectu’ and ‘to be in solo intellectu’. We propose a new logical approach of this ‘Unum argumentum’ of Anselm by using Curry’s Combinatory Logic (1958, 1973). Indeed, Combinatory Logic is an abstract applicative formalism of operators applied to operands; in this formalism, the predicates, viewed as specific operators, can be composed and can be transformed, by an intrinsic way, into more complex predicates, by means of abstract operators, called “combinators”, studied by Combinatory Logic. We show that this formalism is a logical frame where it becomes possible to discuss and to formulate cognitive representations of the meanings of predicates used inside of the ‘Unum argumentum’ and to explain how the argument runs.     

Enantioselective Liquid Chromatographic Separations Using Macrocyclic Glycopeptide-Based Chiral Selectors

Numerous chemical compounds of high practical importance, such as drugs, fertilizers, and food additives are being commercialized as racemic mixtures, although in most cases only one of the isomers possesses the desirable properties. As our understanding of the biological actions of chiral compounds has improved, the investigation of the pharmacological and toxicological properties has become more and more important. Chirality has become a major issue in the pharmaceutical industry; therefore, there is a continuous demand to extend the available analytical methods for enantiomeric separations and enhance their efficiency. Direct liquid chromatography methods based on the application of chiral stationary phases have become a very sophisticated field of enantiomeric separations by now. Hundreds of chiral stationary phases have been commercialized so far. Among these, macrocyclic glycopeptide-based chiral selectors have proved to be an exceptionally useful class of chiral selectors for the separation of enantiomers of biological and pharmacological importance. This review focuses on direct liquid chromatography-based enantiomer separations, applying macrocyclic glycopeptide-based chiral selectors. Special attention is paid to the characterization of the physico-chemical properties of these macrocyclic glycopeptide antibiotics providing detailed information on their applications published recently.     

Quality-by-Design-Based Development of n-Propyl-Gallate-Loaded Hyaluronic-Acid-Coated Liposomes for Intranasal Administration

The present study aimed to develop n-propyl gallate (PG)-encapsulated liposomes through a novel direct pouring method using the quality-by-design (QbD) approach. A further aim was to coat liposomes with hyaluronic acid (HA) to improve the stability of the formulation in nasal mucosa. The QbD method was used for the determination of critical quality attributes in the formulation of PG-loaded liposomes coated with HA. The optimized formulation was determined by applying the Box–Behnken design to investigate the effect of composition and process variables on particle size, polydispersity index (PDI), and zeta potential. Physiochemical characterization, in vitro release, and permeability tests, as well as accelerated stability studies, were performed with the optimized liposomal formulation. The optimized formulation resulted in 90 ± 3.6% encapsulation efficiency, 167.9 ± 3.5 nm average hydrodynamic diameter, 0.129 ± 0.002 PDI, and −33.9 ± 4.5 zeta potential. Coated liposomes showed significantly improved properties in 24 h in an in vitro release test (>60%), in vitro permeability measurement (420 μg/cm²) within 60 min, and also in accelerated stability studies compared to uncoated liposomes. A hydrogen-peroxide-scavenging assay showed improved stability of PG-containing liposomes. It can be concluded that the optimization of PG-encapsulated liposomes coated with HA has great potential for targeting several brain diseases.     

Interspecific Variation between the American Quercus virginiana and Mediterranean Quercus Species in Terms of Seed Nutritional Composition,Phytochemical Content,and Antioxidant Activity

This study aimed to evaluate a complete nutritional composition in the seeds Quercus virginiana to compare this nutritional composition with three Mediterranean Quercus species. We analyzed the seed morphometry, proximate composition, phytochemicals, and antioxidant capacity. The seed of Q. virginiana presented the smaller seed size and weight, while Q. suber presented the highest values. Moreover, Q. virginiana seeds showed the highest amounts of sugar and total lipids, digestibility, energy, palmitic acid, and stearic acid. On the other hand, Q. virginiana seeds showed the lowest values of linoleic acid. Moreover, Q. coccifera seeds presented the highest total phenolics and flavonoids contents and antioxidant activity. The clustering analysis revealed a significant similarity in seed morphometry and nutritional composition between the Mediterranean Q. ilex and Q. suber, grouping with the American Q. virginiana, but to a considerable distance; by contrast, the Mediterranean Q. coccifera was the most distant in the clustering analysis. The content of phenolics and flavonoids and digestibility value were the variables that contributed to the separation to a greater extent in the clustering of the four species. The nutritional and biological activity assessment of plant seed may be considered as an essential mission to find new sustainable sources and novel chemical agents. In this sense, Quercus seeds may be an alternative and a competitive food source for the agri-food industry.