Semigroups on Frechet spaces and equations with infinite delays

In this paper, we show existence and uniqueness of a solution to a functional differential equation with infinite delay. We choose an appropriate Frechet space so as to cover a large class of functions to be used as initial functions to obtain existence and uniqueness of solutions.

Designing a blended learning model for primary school language learning

In Finland, there is a desperate need for flexible, reliable and functional multi-e-learning settings for pupils aged 11-13. Southern Finland has several ongoing e-learning projects, but none that develop a multiple setting, with learning and teaching occurring between more than two schools. In 2006, internet connections were not broadband and data transfer was mainly audio data. Connections and technical problems occurred, which were an obstacle to multi-e-learning. Internet connections today enable web-based learning in major parts of
Lapland and by 2015, broadband will reach even the remotest villages up north. Therefore, it is important to research the possibilities of multi-e-learning and to build collaborative, learner-centred, versatile network models for primary school-aged pupils. The resulting model will facilitate distance learning to extend education to rural, sparsely populated areas, and it will give a model of using mobile devices in language portfolios. This will promote regional equality and prevent exclusion. Working with portfolios provides the opportunity to develop mobility from a pedagogical point of view. It is important to study the pros and cons of mobile devices in producing artefacts on portfolios in e-learning and language learning settings.
The current study represents a design-based research approach. The design research approach includes two important aspects concerning the current research: ‘a teacher as researcher’ aspect, which means there is the possibility to be strongly involved in developing processes and an obstacle-aspect, which means that problems while developing, are seen as a
promoter in evolving the designed model, as apposed to negative results.

Designing a blended learning model for primary school language learning

In Finland, there is a desperate need for flexible, reliable and functional multi-e-learning settings for pupils aged 11-13. Southern Finland has several ongoing e-learning projects, but none that develop a multiple setting, with learning and teaching occurring between more than two schools. In 2006, internet connections were not broadband and data transfer was mainly audio data. Connections and technical problems occurred, which were an obstacle to multi-e-learning. Internet connections today enable web-based learning in major parts of Lapland and by 2015, broadband will reach even the remotest villages up north. Therefore, it is important to research the possibilities of multi-e-learning and to build collaborative, learner-centred, versatile network models for primary school-aged pupils. The resulting model will facilitate distance learning to extend education to rural, sparsely populated areas, and it will give a model of using mobile devices in language portfolios. This will promote regional equality and prevent exclusion. Working with portfolios provides the opportunity to develop mobility from a pedagogical point of view. It is important to study the pros and cons of mobile devices in producing artefacts on portfolios in e-learning and language learning settings. The current study represents a design-based research approach. The design research approach includes two important aspects concerning the current research: ‘a teacher as researcher’ aspect, which means there is the possibility to be strongly involved in developing processes and an obstacle-aspect, which means that problems while developing, are seen as a promoter in evolving the designed model, as apposed to negative results.

Mechanical and Thermal Properties of Eva Blended with Biodegradable Ethyl Cellulose

In this study, biodegradable blend of Poly (Ethylene-co-Vinyl Acetate) (EVA) and Ethyl Cellulose (EC) were prepared. Ethylene vinyl alcohol (EVOH) copolymer was used as an interfacial compatibilizer to enhance adhesion between EVA and EC. The melt blended compatibilized biocomposites were examined for mechanical and thermal properties as per the ASTM standards. It has been found that the EC has a reinforcing effect on EVA leading to enhanced tensile strength and also impart biodegradability. Thus, a high loading of 50% EC could be added without compromising Much on the mechanical properties. Analysis of the tensile data using predictive theories showed an enhanced interaction of the dispersed phase (EC) and the matrix (EVA). The compatibilizing effects of EVOH on these blends were confirmed by the significant improvement in the mechanical properties comparable with neat EVA as also observed by SEM microscopy. The TGA thermograms exhibits two-stage degradation and as EC content increases, the onset temperature for thermal degradation reduces. (C) 2009 Wiley Periodicals, Inc. J Appl Polym Sci 116: 1044-1056, 2010

Support theorems on R-n and non-compact symmetric spaces

We consider convolution equations of the type f * T = g, where f, g is an element of L-P (R-n) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T, we show that f is compactly supported, provided g is. Similar results are proved for non-compact symmetric spaces as well. (C) 2010 Elsevier Inc. All rights reserved.

Chemistry in Confined Spaces: High-Energy Conformer of a Piperidine Derivative is Favored Within a Water-Soluble Capsuleplex

Propyloxy-substituted piperidine in solution adopts a conformation in which its alkoxy group is equatorially positioned Surprisingly, two conformers of it that do not interconvert in the NMR time scale at room temperature have been found within an octa-acid capsule The serendipitous finding of the axial conformer of propyloxy-substituted piperidine within a supramolecular capsule highlights the value of confined spaces in physical organic chemistry.

On the relation between rotation increments in different tangent spaces

In computational mechanics, finite rotations are often represented by rotation vectors. Rotation vector increments corresponding to different tangent: spaces are generally related by a linear operator, known as the tangential transformation T. In this note, we derive the higher order terms that are usually left out in linear relation. The exact nonlinear relation is also presented. Errors via the linearized T are numerically estimated. While the concept of T arises out of the nonlinear characteristics of the rotation manifold, it has been derived via tensor analysis in the context of computational mechanics (Cardona and Geradin, 1988). We investigate the operator T from a Lie group perspective, which provides a better insight and a 1-1 correspondence between approaches based on tensor analysis and the standard matrix Lie group theory. (C) 2010 Elsevier Ltd. All rights reserved.

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

Various Tb theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón-Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operator T on some function space, one needs only to test it on some suitable function b. The main object of this dissertation is to prove very general Tb theorems. The dissertation consists of four research articles and an introductory part. The framework is general with respect to the domain (a metric space), the measure (an upper doubling measure) and the range (a UMD Banach space). Moreover, the used testing conditions are weak. In the first article a (global) Tb theorem on non-homogeneous metric spaces is proved. One of the main technical components is the construction of a randomization procedure for the metric dyadic cubes. The difficulty lies in the fact that metric spaces do not, in general, have a translation group. Also, the measures considered are more general than in the existing literature. This generality is genuinely important for some applications, including the result of Volberg and Wick concerning the characterization of measures for which the analytic Besov-Sobolev space embeds continuously into the space of square integrable functions. In the second article a vector-valued extension of the main result of the first article is considered. This theorem is a new contribution to the vector-valued literature, since previously such general domains and measures were not allowed. The third article deals with local Tb theorems both in the homogeneous and non-homogeneous situations. A modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg is extended to cover the case of upper doubling measures. This technique is also used in the homogeneous setting to prove local Tb theorems with weak testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele. This gives a completely new and direct proof of such results utilizing the full force of non-homogeneous analysis. The final article has to do with sharp weighted theory for maximal truncations of Calderón-Zygmund operators. This includes a reduction to certain Sawyer-type testing conditions, which are in the spirit of Tb theorems and thus of the dissertation. The article extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators. New techniques are introduced to overcome the difficulties introduced by the non-linearity of maximal truncations.

A maximal operator, Carleson's embedding, and tent spaces for vector-valued functions

This thesis is concerned with the area of vector-valued Harmonic Analysis, where the central theme is to determine how results from classical Harmonic Analysis generalize to functions with values in an infinite dimensional Banach space. The work consists of three articles and an introduction. The first article studies the Rademacher maximal function that was originally defined by T. Hytönen, A. McIntosh and P. Portal in 2008 in order to prove a vector-valued version of Carleson's embedding theorem. The boundedness of the corresponding maximal operator on Lebesgue-(Bochner) -spaces defines the RMF-property of the range space. It is shown that the RMF-property is equivalent to a weak type inequality, which does not depend for instance on the integrability exponent, hence providing more flexibility for the RMF-property. The second article, which is written in collaboration with T. Hytönen, studies a vector-valued Carleson's embedding theorem with respect to filtrations. An earlier proof of the dyadic version assumed that the range space satisfies a certain geometric type condition, which this article shows to be also necessary. The third article deals with a vector-valued generalizations of tent spaces, originally defined by R. R. Coifman, Y. Meyer and E. M. Stein in the 80's, and concerns especially the ones related to square functions. A natural assumption on the range space is then the UMD-property. The main result is an atomic decomposition for tent spaces with integrability exponent one. In order to suit the stochastic integrals appearing in the vector-valued formulation, the proof is based on a geometric lemma for cones and differs essentially from the classical proof. Vector-valued tent spaces have also found applications in functional calculi for bisectorial operators. In the introduction these three themes come together when studying paraproduct operators for vector-valued functions. The Rademacher maximal function and Carleson's embedding theorem were applied already by Hytönen, McIntosh and Portal in order to prove boundedness for the dyadic paraproduct operator on Lebesgue-Bochner -spaces assuming that the range space satisfies both UMD- and RMF-properties. Whether UMD implies RMF is thus an interesting question. Tent spaces, on the other hand, provide a method to study continuous time paraproduct operators, although the RMF-property is not yet understood in the framework of tent spaces.

Analogues of the Wiener-Tauberian and Schwartz theorems for radial functions on symmetric spaces

We prove a Wiener Tauberian theorem for the L-1 spherical functions on a semisimple Lie group of arbitrary real rank. We also establish a Schwartz-type theorem for complex groups. As a corollary we obtain a Wiener Tauberian type result for compactly supported distributions.

Toeplitz Operators with Special Symbols on Segal-Bargmann Spaces

We study the boundedness of Toeplitz operators on Segal-Bargmann spaces in various contexts. Using Gutzmer's formula as the main tool we identify symbols for which the Toeplitz operators correspond to Fourier multipliers on the underlying groups. The spaces considered include Fock spaces, Hermite and twisted Bergman spaces and Segal-Bargmann spaces associated to Riemannian symmetric spaces of compact type.