Classes of generalized functions with finite type regularities

We introduce and analyze spaces and algebras of generalized functions which correspond to Hölder, Zygmund, and Sobolev spaces of functions. The main scope of the paper is the characterization of the regularity of distributions that are embedded into the corresponding space or algebra of generalized functions with finite type regularities

Optimization of commodity portfolio management

The problem we consider is introduced by Uljarice Bačka, LLC. The core business activities of the company are trade of agriculture commodities, warehousing and distribution and crops production. The main traded goods are: corn, wheat, barely, sunflower, soybean, soybean meal and raw material for crops production: fertilizers, plant protection products, seeds and other. Since a large part of company’s activities relays on corn, predicting the price of that good is of the main interest. In order to make a reasonable predictions, models which incorporate the crucial factors influencing the corn prices are needed. Of course, the important issue is which data are available. Within the data that we obtained, a correlation analysis is performed to point out the relevant parameters. We introduce different methods for obtaining the predictions and provide some numerical results

Isomorphisms of algebras of Colombeau generalized functions

We show that for smooth manifolds X and Y, any isomorphism between the special algebra of Colombeau generalized functions on X, resp. Y is given by composition with a unique Colombeau generalized function from Y to X. We also identify the multiplicative linear functionals from the special algebra of Colombeau generalized functions on X to the ring of Colombeau generalized numbers. Up to multiplication with an idempotent generalized number, they are given by an evaluation map at a compactly supported generalized point on X.Comment: 10 page

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page

Generalized Fourier Integral Operators on spaces of Colombeau type

Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is based on a theory of generalized oscillatory integrals (OIs) whose phase functions as well as amplitudes may be generalized functions of Colombeau type. The mapping properties of these FIOs are studied as the composition with a generalized pseudodifferential operator. Finally, the microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets are investigated. This theory of generalized FIOs is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distributional data

Conservation laws for self-adjoint first order evolution equations

In this work we consider the problem on group classification and conservation laws of the general first order evolution equations. We obtain the subclasses of these general equations which are quasi-self-adjoint and self-adjoint. By using the recent Ibragimov's Theorem on conservation laws, we establish the conservation laws of the equations admiting self-adjoint equations. We illustrate our results applying them to the inviscid Burgers' equation. In particular an infinite number of new symmetries of these equations are found and their corresponding conservation laws are established.Comment: This manuscript has been accepted for publication in Journal of Nonlinear Mathematical Physic

Does α-Amino-β-methylaminopropionic Acid (BMAA) Play a Role in Neurodegeneration?

The association of α-amino-β-methylaminopropionic acid (BMAA) with elevated incidence of amyotrophic lateral sclerosis/Parkinson’s disease complex (ALS/PDC) was first identified on the island of Guam. BMAA has been shown to be produced across the cyanobacterial order and its detection has been reported in a variety of aquatic and terrestrial environments worldwide, suggesting that it is ubiquitous. Various in vivo studies on rats, mice, chicks and monkeys have shown that it can cause neurodegenerative symptoms such as ataxia and convulsions. Zebrafish research has also shown disruption to neural development after BMAA exposure. In vitro studies on mice, rats and leeches have shown that BMAA acts predominantly on motor neurons. Observed increases in the generation of reactive oxygen species (ROS) and Ca2+ influx, coupled with disruption to mitochondrial activity and general neuronal death, indicate that the main mode of activity is via excitotoxic mechanisms. The current review pertaining to the neurotoxicity of BMAA clearly demonstrates its ability to adversely affect neural tissues, and implicates it as a potentially significant compound in the aetiology of neurodegenerative disease. When considering the potential adverse health effects upon exposure to this compound, further research to better understand the modes of toxicity of BMAA and the environmental exposure limits is essential