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.

On some results of A. E. Livingston and coefficient problems for concave univalent functions

We consider functions that map the open unit disc conformally onto the complement of a bounded convex set. We call these functions concave univalent functions. In 1994, Livingston presented a characterization for these functions. In this paper, we observe that there is a minor flaw with this characterization. We obtain certain sharp estimates and the exact set of variability involving Laurent and Taylor coefficients for concave functions. We also present the exact set of variability of the linear combination of certain successive Taylor coefficients of concave functions.

Hermite expansions on R(N) for radial functions

It is proved that the Riesz means S(R)(delta)f, delta > 0, for the Hermite expansions on R(n), n greater-than-or-equal-to 2, satisfy the uniform estimates \\S(R)(delta)f\\p less-than-or-equal-to C \\f\\p for all radial functions if and only if p lies in the interval 2n/(n + 1 + 2delta) < p < 2n/(n - 1 - 2delta).

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.

The Reception of the Maccabean Martyrs : Their Historiographical and Paradigmatic Functions in Antique and Late-antique Jewish and Christian Sources

The study attempts a reception-historical analysis of the Maccabean martyrs. The concept of reception has fundamentally to do with the re-use and interpretation of a text within new texts. In a religious tradition, certain elements become re-circulated and thus their reception may reflect the development of that particular tradition. The Maccabean martyrs first appear in 2 Maccabees. In my study, it is the Maccabean martyr figures who count as the received text; the focus is shifted from the interrelations between texts onto how the figures have been exploited in early Christian and Rabbinic sources. I have divided my sources into two categories and my analysis is in two parts. First, I analyze the reception of the Maccabean martyrs within Jewish and Christian historiographical sources, focusing on the role given to them in the depictions of the Maccabean Revolt (Chapter 3). I conclude that, within Jewish historiography, the martyrs are given roles, which vary between ultimate efficacy and marginal position with regard to making a historical difference. In Christian historiographical sources, the martyrs role grows in importance by time: however, it is not before a Christian cult of the Maccabean martyrs has been established, that the Christian historiographies consider them historically effective. After the first part, I move on to analyze the reception in sources, which make use of the Maccabean martyrs as paradigmatic figures (Chapter 4). I have suggested that the martyrs are paradigmatic in the context of martyrdom, persecution and destruction, on one hand, and in a homiletic context, inspiring religious celebration, on the other. I conclude that, as the figures are considered pre-Christian and biblical martyrs, they function well in terms of Christian martyrdom and have contributed to the development of its ideals. Furthermore, the presentation of the martyr figures in Rabbinic sources demonstrates how the notion of Jewish martyrdom arises from experiences of destruction and despair, not so much from heroic confession of faith in the face of persecution. Before the emergence of a Christian cult of the Maccabean martyrs, their identity is derived namely from their biblical position. Later on, in the homiletic context, their Jewish identity is debated and sometimes reconstructed as fundamentally Christian , despite of their Jewish origins. Similar debate about their identity is not found in the Rabbinic versions of their martyrdom and nothing there indicates a mutual debate between early Christians and Jews. A thematic comparison shows that the Rabbinic and Christian cases of reception are non-reliant on each other but also that they link to one another. Especially the scriptural connections, often made to the Maccabean mother, reveal the similarities. The results of the analyses confirm that the early history of Christianity and Rabbinic Judaism share, at least partly, the same religious environment and intertwining traditions, not only during the first century or two but until Late Antiquity and beyond. More likely, the reception of the Maccabean martyrs demonstrates that these religious traditions never ceased to influence one another.

The Reproducing Kernel DMS-FEM: 3D Shape Functions and Applications to Linear Solid Mechanics

We propose a family of 3D versions of a smooth finite element method (Sunilkumar and Roy 2010), wherein the globally smooth shape functions are derivable through the condition of polynomial reproduction with the tetrahedral B-splines (DMS-splines) or tensor-product forms of triangular B-splines and ID NURBS bases acting as the kernel functions. While the domain decomposition is accomplished through tetrahedral or triangular prism elements, an additional requirement here is an appropriate generation of knotclouds around the element vertices or corners. The possibility of sensitive dependence of numerical solutions to the placements of knotclouds is largely arrested by enforcing the condition of polynomial reproduction whilst deriving the shape functions. Nevertheless, given the higher complexity in forming the knotclouds for tetrahedral elements especially when higher demand is placed on the order of continuity of the shape functions across inter-element boundaries, we presently emphasize an exploration of the triangular prism based formulation in the context of several benchmark problems of interest in linear solid mechanics. In the absence of a more rigorous study on the convergence analyses, the numerical exercise, reported herein, helps establish the method as one of remarkable accuracy and robust performance against numerical ill-conditioning (such as locking of different kinds) vis-a-vis the conventional FEM.

Thermodynamics and phase equilibria in the system Ga---In using multi-parameter functions

A four and a five-parameter functions are used to analyse and interpret the high and low temperature thermodynamic data and phase equilibria in the Ga-In system.

Functions Of Cytochrome-C In Regulation Of Electron-Transfer And Protein Folding

Cytochrome c, a "mobile electron carrier" of the mitochondrial respiratory chain, also occurs in detectable amounts in the cytosol, and can receive electrons from cytochromes present in endoplasmic reticulum and plasma membranes as well as from superoxide and ascorbate. The pigment was found to dissociate from mitochondrial membranes in liver and kidney when rats were subjected to heat exposure and starvation, respectively. Treating cytochrome c with hydroxylamine gives a partially deaminated product with altered redox properties; decreased stimulation of respiration by deficient mitochondria, increased reduction by superoxide, and complete loss of reducibility by plasma membranes. Mitochondria isolated from brown adipose tissue of cold-exposed rats are found to be sub-saturated with cytochrome c. The ability of cytochrome c to reactivate reduced ribonuclease is now reinterpreted as a molecular chaperone role for the hemoprotein.

Affine Hall-Littlewood Functions for A(1)((1)) And Some Constant Term Identities of Cherednik-Macdonald-Mehta Type

We study t-analogs of string functions for integrable highest weight representations of the affine Kac-Moody algebra A(1)((1)). We obtain closed form formulas for certain t-string functions of levels 2 and 4. As corollaries, we obtain explicit identities for the corresponding affine Hall-Littlewood functions, as well as higher level generalizations of Cherednik's Macdonald and Macdonald-Mehta constant term identities.

Waveform synthesis using Walsh functions International Journal of Electronics

A software and a microprocessor based hardware for waveform synthesis using Walsh functions are described. The software is based on Walsh function generation using Hadamard matrices and on the truncated Walsh series expansion for the waveform to be synthesized. The hardware employs six microprocessor controlled programmable Walsh function generators (PWFGs) for generating the first six non-vanishing terms of the truncated Walsh series. Improved approximation to a given waveform may be achieved by employing additional PWFGs.

Molecular Dynamics Simulations of Orientational Relaxation in Dipolar Lattice: Lack of Diffusive Decay for Second and Higher Rank Correlation Functions

Extensive molecular dynamics simulations have been carried out to calculate the orientational correlation functions Cl(t), G(t) = [4n/(21 + l)]Ci=-l (Y*lm(sZ(0)) Ylm(Q(t))) (where Y,,(Q) are the spherical harmonics) of point dipoles in a cubic lattice. The decay of Cl(t) is found to be strikingly different from higher l-correlation functions-the latter do not exhibit diffusive dynamics even in the long time. Both the cumulant expansion expression of Lynden-Bell and the conventional memory function equation provide very good description of the Cl(t) in the short time but fail to reproduce the observed slow, long time decay of c1 (t) .

One protein�many functions

The concept of one enzyme-one activity had influenced biochemistry for over half a century. Over 1000 enzymes are now described. Many of them are highly 'specific'. Some of them are crystallized and their three-dimensional structures determined. They range from 12 to 1000 kDa in molecular weight and possess 124 to several hundreds of amino acids. They occur as single polypeptides or multiple-subunit proteins. The active sites are assembled on these by appropriate tertiary folding of the polypeptide chain, or by interaction of the constituent subunits. The substrate is held by the side-chains of a few amino acids at the active site on the surface, occupying a tiny fraction of the total area. What is the bulk of the protein behind the active site doing? Do all proteins have only one function each? Why not a protein have more than one active site on its large surface? Will we discover more than one activity for some proteins? These newer possibilities are emerging and are finding experimental support. Some proteins purified to homogeneity using assay methods for different activities are now recognized to have the same molecular weight and a high degree of homology of amino acid sequence. Obviously they are identical. They represent the phenomenon of one protein-many functions.

CM defect and Hilbert functions of monomial curves

In this article we consider a semigroup ring R = KGamma] of a numerical semigroup Gamma and study the Cohen- Macaulayness of the associated graded ring G(Gamma) := gr(m), (R) := circle plus(n is an element of N) m(n)/m(n+1) and the behaviour of the Hilbert function H-R of R. We define a certain (finite) subset B(Gamma) subset of F and prove that G(Gamma) is Cohen-Macaulay if and only if B(Gamma) = empty set. Therefore the subset B(Gamma) is called the Cohen-Macaulay defect of G(Gamma). Further, we prove that if the degree sequence of elements of the standard basis of is non-decreasing, then B(F) = empty set and hence G(Gamma) is Cohen-Macaulay. We consider a class of numerical semigroups Gamma = Sigma(3)(i=0) Nm(i) generated by 4 elements m(0), m(1), m(2), m(3) such that m(1) + m(2) = mo m3-so called ``balanced semigroups''. We study the structure of the Cohen-Macaulay defect B(Gamma) of Gamma and particularly we give an estimate on the cardinality |B(Gamma, r)| for every r is an element of N. We use these estimates to prove that the Hilbert function of R is non-decreasing. Further, we prove that every balanced ``unitary'' semigroup Gamma is ``2-good'' and is not ``1-good'', in particular, in this case, c(r) is not Cohen-Macaulay. We consider a certain special subclass of balanced semigroups Gamma. For this subclass we try to determine the Cohen-Macaulay defect B(Gamma) using the explicit description of the standard basis of Gamma; in particular, we prove that these balanced semigroups are 2-good and determine when exactly G(Gamma) is Cohen-Macaulay. (C) 2011 Published by Elsevier B.V.

Derivation and consistency of the partial functions of the ternary system involving interaction coefficients

The logarithm of activity coefficients of the components of the ternary system is derived based on the Maclaurin infinite series, which is expressed in terms of the integral property of the system and subjected to appropriate boundary conditions. The derivation of the functions involves extensive summation of various infinite series pertaining to the first-order interaction coefficients that have been shown completely to remove any truncational error. Since the conventional equations involving interaction coefficients are internally inconsistent, a consistent form of the partial functions is developed in the article using the technique just described. The thermodynamic consistency of the functions based on the Maxwell and the Gibbs-Duhem relations has been established. The derived values of the logarithmic activity coefficients of the components have been found to be in agreement with the thermodynamic data of the Fe-Cr-Ni system at 1873 K and have been found to be independent of the compositional paths.

Spatially dependent zero-frequency response functions and correlation functions in the Kondo model

We present the details of a formalism for calculating spatially varying zero-frequency response functions and equal-time correlation functions in models of magnetic and mixed-valence impurities of metals. The method is based on a combination of perturbative, thermodynamic scaling theory [H. R. Krishna-murthy and C. Jayaprakash, Phys. Rev. B 30, 2806 (1984)] and a nonperturbative technique such as the Wilson renormalization group. We illustrate the formalism for the spin-1/2 Kondo problem and present results for the conduction-spin-density�impurity-spin correlation function and conduction-electron charge density near the impurity. We also discuss qualitative features that emerge from our calculations and discuss how they can be carried over to the case of realistic models for transition-metal impurities.