Adjoints of composition operators on Hardy spaces of the half-plane

Building on techniques used in the case of the disc, we use a variety of methods to develop formulae for the adjoints of composition operators on Hardy spaces of the upper half-plane. In doing so, we prove a slight extension of a known necessary condition for the boundedness of such operators, and use it to provide a complete classification of the bounded composition operators with rational symbol. We then consider some specific examples, comparing our formulae with each other, and with other easily deduced formulae for simple cases.     

The Dirac delta function in two settings of Reverse Mathematics

The program of Reverse Mathematics (Simpson 2009) has provided us with the insight that most theorems of ordinary mathematics are either equivalent to one of a select few logical principles, or provable in a weak base theory. In this paper, we study the properties of the Dirac delta function (Dirac 1927; Schwartz 1951) in two settings of Reverse Mathematics. In particular, we consider the Dirac Delta Theorem, which formalizes the well-known property ${\int_\mathbb{R}f(x)\delta(x)\,dx=f(0)}$ of the Dirac delta function. We show that the Dirac Delta Theorem is equivalent to weak K?nig’s Lemma (see Yu and Simpson in Arch Math Log 30(3):171–180, 1990) in classical Reverse Mathematics. This further validates the status of WWKL₀ as one of the ‘Big’ systems of Reverse Mathematics. In the context of ERNA’s Reverse Mathematics (Sanders in J Symb Log 76(2):637–664, 2011), we show that the Dirac Delta Theorem is equivalent to the Universal Transfer Principle. Since the Universal Transfer Principle corresponds to WKL, it seems that, in ERNA’s Reverse Mathematics, the principles corresponding to WKL and WWKL coincide. Hence, ERNA’s Reverse Mathematics is actually coarser than classical Reverse Mathematics, although the base theory has lower first-order strength.     

Observation of Berry’s Geometric Phase by Neutron Interferometry

On the 25th anniversary of Berry’s historic papers on the geometric phase, I discuss here our neutron interferometry experiment in which this phase is clearly separated from the dynamical phase. The connection of this experiment to the observation of the sign reversal of the wave function of a fermion during a 2π precession in a magnetic field by three groups independently in 1975 is discussed.     

The porous silicon morphology effect on the growth of electrodeposited FeNi alloy

In this work, we report on cathodic deposition of FeNi thin films into porous silicon (PS) formed on n-type Si. Macroporous and mesoporous silicon layers were formed at constant potential or current density. The electrodeposited thin films were characterized by Energy Dispersive Spectroscopy (EDS) and X-Ray Diffraction (XRD). The magnetic properties of the FeNi layers were investigated by hysteresis loops measurements. SEM images of the FeNi films indicate that tubular and granular forms were obtained depending on the porous silicon surface morphology. Moreover, the FeNi films compositions are found to depend on the porous silicon microstructure. Finally, the XRD spectra of the deposited films show the presence of FeNi (111) and FeNi (220) peaks. The FeNi (111) peak has been shown for all polarization potentials, in agreement with results reported in the literature.     

Preparation and characterization of iodinated poly(vinyl alcohol) microfibril

Iodination of poly(vinyl alcohol) (PVA) microfibril, which was obtained from saponification of poly(vinyl pivalate), was conducted before and after zone drawing at various conditions. The resulting PVA microfibrils were characterized by differential scanning calorimeter and scanning electron microscopy. Surface morphologies of these PVA microfibrils showed some differences between PVA microfibril iodinated after and before drawing. Crude shapes of PVA microfibrils iodinated after drawing indicated that iodine decreased the structural regularity severely. On the other hand, PVA microfibrils iodinated before drawing showed relatively ordered surfaces. This was ascribed to the enhanced molecular ordering of PVA microfibrils due on zone drawing. Iodinated PVA microfibrils showed a decrease in crystal melting temperature of about 100°C compared to the untreated sample. PVA microfibrils drawn after iodination showed relatively higher crystal melting temperature than those of microfibrils iodinated after drawing. These results were considered as the proofs of the changes in crystalline lattice of the PVA microfibrils. Effects, of drawing temperature on sublimation of iodine were also evaluated.     

Characterization of pyrene end-labeled poly(ethylene glycol) by high resolution MALDI time-of-flight mass spectrometry

We report the characterization of a sample of poly(ethylene glycol) (PEG, M _n = 3841, M _w/M _n = 1.01), and its derivative end-labeled with pyrenebutyrate groups, using high resolution MALDI time-of-flight mass spectrometry. A matrix of 2-(4-hydroxyphenyl-azo)benzoic acid containing a trace of either sodium chloride or potassium chloride was employed for laser desorption. Peaks due to the sodium or potassium cationized polymers were obtained, equally spaced at 44 mass units apart. For the pyrenebutyrate diester, the analysis shows that 80 ± 2% of the chains were doubly labeled, with the ramaining chains containing only a single pyrene group. Molecular weight determinations for both sets of samples were entirely consistent with size-exclusion chromatography measurements, but were obtained with greater accuracy and less ambiguity about the influence of the end groups on the hydrodynamic radius of the polymer.