Generalized solutions in PDEs and the Burgers' equation

In many situations, the notion of function is not sufficient and it needs to be extended. A classical way to do this is to introduce the notion of weak solution; another approach is to use generalized functions. Ultrafunctions are a particular class of generalized functions that has been previously introduced and used to define generalized solutions of stationary problems in [4,7,9,11,12]. In this paper we generalize this notion in order to study also evolution problems. In particular, we introduce the notion of Generalized Ultrafunction Solution (GUS) for a large family of PDEs, and we confront it with classical strong and weak solutions. Moreover, we prove an existence and uniqueness result of GUS's for a large family of PDEs, including the nonlinear Schroedinger equation and the nonlinear wave equation. Finally, we study in detail GUS's of Burgers' equation, proving that (in a precise sense) the GUS's of this equation provide a description of the phenomenon at microscopic level

A model problem for ultrafunctions

IIn this article. we show that non-Archimedean mathematics (NAM), namely mathematics which uses infinite and infinitesimal numbers, is useful to model some physical problems which cannot be described by the usual mathematics. The problem which we will consider here is the minimization of the functional E(u, q) = 1/2 integral(Omega)vertical bar del u(x)vertical bar(2) dx + u(q). When Omega subset of R-N is a bounded open set and u is an element of C-0(2) (Omega), this problem has no solution since inf E (u, q) = -infinity. On the contrary, as we will show, this problem is well posed in a suitable non-Archimedean frame. More precisely, we apply the general ideas of NAM and some of the techniques of Non Standard Analysis to a new notion of generalized functions, called ultrafunctions, which are a particular class of functions based on a Non-Archimedean field. In this class of functions, the above problem is well posed and it has a solution

Generalized solutions of variational problems and applications

Ultrafunctions are a particular class of generalized functions defined on a hyperreal field that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach

A mixed-valence diruthenium(ii,iii) complex endowed with high stability: from experimental evidence to theoretical interpretation

We herein report the synthesis and multi-technique characterization of [Ru2Cl((2-phenylindol-3-yl)glyoxyl-l-leucine-l-phenylalanine)4], a novel diruthenium(ii,iii) complex obtained by reacting [Ru2(μ-O2CCH3)4Cl] with a dual indolylglyoxylyl dipeptide anticancer agent. We soon realised that the compound is very stable under several different conditions including aqueous buffers or organic solvents. It is also completely unreactive toward proteins. The high stability is also suggested by cellular experiments in a glioblastoma cell line. Indeed, while the parent ligand exerts high cytotoxic effects in the low μM range, the complex is completely non-cytotoxic against the same line, most probably because of the lack of ligand release. To investigate the reasons for such high stability, we carried out DFT calculations that are fully consistent with the experimental findings. The results highlight that the stability of [Ru2Cl((2-phenylindol-3-yl)glyoxyl-l-leucine-l-phenylalanine)4] relies on the nature of the ligand, including its steric hindrance that prevents the reaction of any nucleophilic group with the Ru2 core. Ligand displacement is the key step to allow reactivity with the biological targets of metal-based prodrugs. Accordingly, we discuss the implications of some important aspects that should be considered when active molecules are chosen as ligands for the synthesis of paddle-wheel-like complexes with medicinal applications. This journal i

Hypernatural Numbers as Ultrafilters

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers

Spread pattern of the first dengue epidemic in the city of Salvador, Brazil

<p>Abstract</p> <p>Background</p> <p>The explosive epidemics of dengue that have been occurring in various countries have stimulated investigation into new approaches to improve understanding of the problem and to develop new strategies for controlling the disease. The objective of this study was to evaluate the characteristics of diffusion of the first dengue epidemic that occurred in the city of Salvador in 1995.</p> <p>Methods</p> <p>The epidemiological charts and records of notified cases of dengue in Salvador in 1995 constituted the source of data. The cases of the disease were georeferenced according to census areas (spatial units) and epidemiological weeks (temporal unit). Kernel density estimation was used to identify the pattern of spatial diffusion using the R-Project computer software program.</p> <p>Results</p> <p>Of the 2,006 census areas in the city, 1,400 (70%) registered cases of dengue in 1995 and the spatial distribution of these records revealed that by the end of 1995 practically the entire city had been affected by the virus, with the largest concentration of cases occurring in the western region, composed of census areas with a high population density and predominantly horizontal residences compared to the eastern region of the city, where there is a predominance of vertical residential buildings.</p> <p>Conclusion</p> <p>The pattern found in this study shows the characteristics of the classic process of spreading by contagion that is common to most infectious diseases. It was possible to identify the epicenter of the epidemic from which centrifugal waves of the disease emanated. Our results suggest that, if a more agile control instrument existed that would be capable of rapidly reducing the vector population within a few days or of raising the group immunity of the population by means of a vaccine, it would theoretically be possible to adopt control actions around the epicenter of the epidemic and consequently reduce the incidence of the disease in the city. This finding emphasizes the need for further research to improve the technology available for the prevention of this disease.</p

A generalization of Gauss&#8217; divergence theorem

This paper is devoted to the proof Gauss' divegence theorem in the framework of "ultrafunctions". They are a new kind of generalized functions, which have been introduced recently by Benci in 2013 and developed by the authors. Their peculiarity is that they are based on a non-Archimedean field, namely on a field which contains infinite and infinitesimal numbers. Ultra functions have been introduced to provide generalized solutions to equations which do not have any solutions, not even among the distributions

A model problem for ultrafunctions

IIn this article. we show that non-Archimedean mathematics (NAM), namely mathematics which uses infinite and infinitesimal numbers, is useful to model some physical problems which cannot be described by the usual mathematics. The problem which we will consider here is the minimization of the functional E(u, q) = 1/2 integral(Omega)vertical bar del u(x)vertical bar(2) dx + u(q). When Omega subset of R-N is a bounded open set and u is an element of C-0(2) (Omega), this problem has no solution since inf E (u, q) = -infinity. On the contrary, as we will show, this problem is well posed in a suitable non-Archimedean frame. More precisely, we apply the general ideas of NAM and some of the techniques of Non Standard Analysis to a new notion of generalized functions, called ultrafunctions, which are a particular class of functions based on a Non-Archimedean field. In this class of functions, the above problem is well posed and it has a solution

Carbonic Anhydrase Activators for Neurodegeneration: An Overview

Carbonic anhydrases (CAs) are a family of ubiquitous metal enzymes catalyzing the reversible conversion of CO2 and H2O to HCO3- with the release of a proton. They play an important role in pH regulation and in the balance of body fluids and are involved in several functions such as homeostasis regulation and cellular respiration. For these reasons, they have been studied as targets for the development of agents for treating several pathologies. CA inhibitors have been used in therapy for a long time, especially as diuretics and for the treatment of glaucoma, and are being investigated for application in other pathologies including obesity, cancer, and epilepsy. On the contrary, CAs activators are still poorly studied. They are proposed to act as additional (other than histidine) proton shuttles in the rate-limiting step of the CA catalytic cycle, which is the generation of the active hydroxylated enzyme. Recent studies highlight the involvement of CAs activation in brain processes essential for the transmission of neuronal signals, suggesting CAs activation might represent a potential therapeutic approach for the treatment of Alzheimer's disease and other conditions characterized by memory impairment and cognitive problems. Actually, some compounds able to activate CAs have been identified and proposed to potentially resolve problems related to neurodegeneration. This review reports on the primary literature regarding the potential of CA activators for treating neurodegeneration-related diseases