Distribution theory and fundamental solutions of differential operators

The objective of this dissertation is to study the theory of distributions and some of its applications. Certain concepts which we would include in the theory of distributions nowadays have been widely used in several fields of mathematics and physics. It was Dirac who first introduced the delta function as we know it, in an attempt to keep a convenient notation in his works in quantum mechanics. Their work contributed to open a new path in mathematics, as new objects, similar to functions but not of their same nature, were being used systematically. Distributions are believed to have been first formally introduced by the Soviet mathematician Sergei Sobolev and by Laurent Schwartz. The aim of this project is to show how distribution theory can be used to obtain what we call fundamental solutions of partial differential equations.

Fixed Points of Closed and Compact Composite Sequences of Operators and Projectors in a Class of Banach Spaces

Some results on fixed points related to the contractive compositions of bounded operators in a class of complete metric spaces which can be also considered as Banach's spaces are discussed through the paper. The class of composite operators under study can include, in particular, sequences of projection operators under, in general, oblique projective operators. In this paper we are concerned with composite operators which include sequences of pairs of contractive operators involving, in general, oblique projection operators. The results are generalized to sequences of, in general, nonconstant bounded closed operators which can have bounded, closed, and compact limit operators, such that the relevant composite sequences are also compact operators. It is proven that in both cases, Banach contraction principle guarantees the existence of unique fixed points under contractive conditions.

Approximate Solutions by Truncated Taylor Series Expansions of Nonlinear Differential Equations and Related Shadowing Property with Applications

This paper investigates the errors of the solutions as well as the shadowing property of a class of nonlinear differential equations which possess unique solutions on a certain interval for any admissible initial condition. The class of differential equations is assumed to be approximated by well-posed truncated Taylor series expansions up to a certain order obtained about certain, in general nonperiodic, sampling points t(i) is an element of [t(0), t(J)] for i = 0, 1, . . . , J of the solution. Two examples are provided.

Accuracy in Copy Number Calling by qPCR and PRT : A Matter of DNA

7 p.

What mechanism of niche segregation allows the coexistence of sympatric sibling rhinolophid bats?

Introduction: Our purpose was to assess how pairs of sibling horseshoe bats coexists when their morphology and echolocation are almost identical. We collected data on echolocation, wing morphology, diet, and habitat use of sympatric Rhinolophus mehelyi and R. euryale. We compared our results with literature data collected in allopatry with similar protocols and at the same time of the year (breeding season). Results:Echolocation frequencies recorded in sympatry for R. mehelyi (mean = 106.8 kHz) and R. euryale (105.1 kHz) were similar to those reported in allopatry (R. mehelyi 105–111 kHz; R. euryale 101–109 kHz). Wing parameters were larger in R. mehelyi than R. euryale for both sympatric and allopatric conditions. Moths constitute the bulk of the diet of both species in sympatry and allopatry, with minor variation in the amounts of other prey. There were no inter-specific differences in the use of foraging habitats in allopatry in terms of structural complexity, however we found inter-specific differences between sympatric populations: R. mehelyi foraged in less complex habitats. The subtle inter-specific differences in echolocation frequency seems to be unlikely to facilitate dietary niche partitioning; overall divergences observed in diet may be explained as a consequence of differential prey availability among foraging habitats. Inter-specific differences in the use of foraging habitats in sympatry seems to be the main dimension for niche partitioning between R. mehelyi and R. euryale, probably due to letter differences in wing morphology. Conclusions: Coexistence between sympatric sibling horseshoe bats is likely allowed by a displacement in spatial niche dimension, presumably due to the wing morphology of each species, and shifts the niche domains that minimise competition. Effective measures for conservation of sibling/similar horseshoe bats should guarantee structural diversity of foraging habitats.

Using the system dynamics paradigm in teaching and learning technological university subjets

2nd International Conference on Education and New Learning Technologies