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Moment sequences and their applicationsLi, Xiaoguang 20 October 2005 (has links)
In this dissertation, we first present a unified treatment of compact moment problems, both the truncated and full moment cases. Second, we define the lower and upper functions V±(ð₁,... ð <sub>n</sub>) on the convex hull of the curve Γ<sub>n</sub> = {(t,.·.,t<sup>n</sup>): t ∈ [0,1] } for each positive integer n. Explicit formulas of these functions are derived and applied to the study of the subnormal completion problem in operator theory. Last, we show that certain power functions are the building blocks of completely positive functions; by our definition, these functions are the continuous functions on the interval [0, 1] that map each Hausdorff moment sequence of a probability measure into another one. / Ph. D.
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Analytical determination of autocorrelation and noise power density spectrum of randomly modulated pulse width square wavesPark, Hen Suh January 1966 (has links)
The Wiener theory of the minimum mean square error ·criterion is well furnished by knowing the autocorrelation function of the input to the linear system. This input signal is generally an additive mixture of a piecewise continuous message and a noise.
The problem considered in this paper is the determination of the autocorrelation function and also their power density spectrum of the noise component for the random base and height modulated square wave whose leading edges are periodic functions of time.
We note that the adopted probability density function for heights of random square wave have Gamma-Distribution Density Function. In addition to this distribution function, we consider the rectangular and Beta-density function on the base of square waves. In fact, the leading edges of most periodic-random base function can be simply described by using the rectangular and Beta-Density function.
Another matter under consideration is the visualization of the variations noise power density spectrum immersed in the masked signal (mixtured signal) with respect to the variance σ² and s² of Gamma and Beta-distribution, respectively. / M.S.
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On some semi-linear equations related to phase transitions: Rigidity of global solutions and regularity of free boundariesZhang, Chilin January 2024 (has links)
In this thesis, we study minimizers of the energy functional 𝐽 (𝑢,Ω) = ∫_Ω |∇𝑢|²/2 + 𝑊(𝑢) 𝑑𝑥 for two different potentials 𝑊(𝑢).
In the first part we consider the Allen-Cahn energy, where 𝑊(𝑢) = (1 − 𝑢²)² is a doublewell potential which is relevant in the theory of phase transitions and minimal interfaces. We investigate the rigidity properties of global minimizers in low dimensions. In particular we extend a result of Savin on the De Giorgi’s conjecture to include minimizers that are not necessarily bounded, and that can have subquadratic growth at infinity.
In the second part we consider potentials of the type 𝑊(𝑢) = 𝑢⁺ which appear in obstacletype free boundary problems. We establish higher order estimates and the analyticity of the regular part of the free boundary. Our method relies on developing higher order boundary Harnack estimates iteratively and deducing them from Schauder estimates for certain elliptic equations with degenerate weights.
Finally we consider similar regularity questions of the free boundary in the Signorini problem which also known as the thin obstacle problem. We develop 𝐶²^𝛼 estimates of the free boundary under sharp assumptions on the coefficients and the data.
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A DFT study of the catalytic hydrocyanation of ethylene with nickel complexesHeydenrych, Greta 12 1900 (has links)
Dissertation (PhD)--University of Stellenbosch, 2005. / ENGLISH ABSTRACT: DFT calculations employing the B3LYP functional were done to investigate the mechanism for the Ni-catalyzed hydrocyanation of ethylene as proposed by Tolman. Although this reaction is an important industrial process, its mechanism has never been studied computationally, apart from calculations pertaining to ligand tailoring.
This study comprises a detailed configurational analysis of each step of the reaction cycle, charge decomposition analysis of pertinent species and analysis of the activation barriers involved at each step. A model ligand, PH3, is employed, due to its electronic similarity to the experimental ligand most widely used, P(O-o-tolyl)3, and its small size, which makes it amenable for calculations at this level.
It was found that oxidative addition of HCN to the precursor complex (ethylene)NiL2 (L=PH3) can take place in one step and that it is the rate-determining step in the gas phase. The resulting adduct has H+ (which becomes a hydride) and CN- coordinated in the cis configuration. Ligand dissociation yields three configurations of (ethylene)-NiHCNL, of which only two can participate in the catalytic cycle. It is shown that this is because migration-insertion of ethylene into the Ni-H bond takes place before, or concomitant with, association of a second ethylene molecule, contrary to expectation. This path therefore requires that ethylene and hydrogen are coordinated in the cis configuration, something only possible for two of the three isomers of (ethylene)NiHCNL. The calculations support the mechanism of associative reductive elimination and shows that elimination can only take place if the ethyl and cyanide groups are in the cis configuration.
Analysis of the energetic profile of the reaction shows that entropy effects play a very important role in the propagation of the cycle, at least in the gas phase.
Preliminary work on the effect of Lewis acids the catalytic cycle is presented, with structural and energetic analysis.
An important general conclusion is that the standard way of representing the energy profile of reactions where intermolecular transitions (as opposed to intramolecular transitions only) take place can be misleading. It will be argued that the implicit assumption that two species which are minimum energy structures on distinct potential energy surfaces will also be an energy minimum on one potential energy surface skews the energy profile of the reaction. The consequence of this is that care must be taken in representing energy profiles for reactions where more than one distinct species participates. / AFRIKAANSE OPSOMMING: Die meganisme, soos deur Tolman voorgestel, van die Ni-gekataliseerde hidrosianering van etileen word ondersoek met behulp van Kohn-Sham elektrondigtheidsteorie (density finctional theory, DFT) berekeninge waarin die B3LYP-funksionaal gebruik word. Alhoewel die reaksie ‘n belangrike proses is in die industrie, is die volle me-ganisme nog nooit met behulp van berekeninge bestudeer nie. Daar is egter wel al werk gedoen aangaande sekere aspekte van die reaksie, byvoorbeeld ligandontwerp.
Hierdie studie behels ‘n noukeurige konfigurasionele analise van elke stap van die reaksie-siklus, ladingsverdelingsanalise (charge decomposition analysis, CDA) van sekere belangrike spesies asook die analise van die energiestappe betrokke by elke stap. Fosfien is gekies as ‘n modelligand, omdat dit elektronies ooreenstem met P(O-o-toliel)3, die ligand wat meestal in eksperimentele werk gebruik is. Die klein grootte van fosfien maak dit ook geskik vir be-rekeninge op hierdie vlak.
Daar is bevind dat die oksidatiewe addisie van HCN aan die voorgangerkompleks (etileen)-NiL2 (L=PH3) in een stap kan plaasvind en in die gasfase snelheidsbepalend is. Die adduk (ettileen)NiHCNL2 bevat H+ (wat ‘n hidried word) en CN- in die cis-posisie relatief tot mekaar. Liganddissosiasie lewer drie isomere van (etileen)NiHCNL. Daar is bevind dat slegs twee van dié isomere aan die katalitiese reaksie kan deelneem, omdat die migrasie-inplasing (migration-insertion) van etileen in die Ni-H-binding voor, of saam met, die assosiasie van ‘n tweede etileen-molekule plaasvind. Dit is slegs moontlik indien waterstof en etileen cis teenoor mekaar is, wat geld vir twee van die isomere. Die meganisme van assosiatiewe reduktiewe eliminasie word deur die berekeninge gerugsteun. Voorts blyk dit vanuit die berekeninge dat die etiel- en sianiedgroepe cis teenoor mekaar moet wees voordat reduktiewe eliminasie van propionitriel kan plaasvind.
Analise van die energetiese profiel van die reaksie toon dat entropie-effekte ‘n belangrike rol speel in die voortsetting van die reaksie in die gasfase.
Die invloed van Lewissure op die katalitiese siklus word, met behulp van strukturele en energetiese analise bespreek.
‘n Belangrike algemene gevolgtrekking is dat die standaardvoorstelling van die energetiese profiel van reaksies waarin intermolekulêre oorgange (teenoor slegs intramolekulêre oorgange) voorkom, misleidend kan wees. Dit word gestel dat die implisiete aanname dat twee spesies wat minumum-energiestrukture verteenwoordig op twee verskillende potensiële energie-oppervlaktes ook ‘n minimum-energiestruktuur voorstel op een potensiële energie-oppervlakte, die energieprofiel skeeftrek. Gevolgtrekkings vanuit hierdie energieprofiele van reaksies waar meer as een onderskeibare spesie deelneem, moet dus met omsigtigheid gemaak word.
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Time-resolved resonance Raman investigation of selected para-substituted phenylnitrenium ions and the 2-fluorenylnitrenium ionreaction with guanosineChan, Pik-ying., 陳碧瑩. January 2005 (has links)
published_or_final_version / abstract / Chemistry / Doctoral / Doctor of Philosophy
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Time-resolved resonance raman and density functional theory studies ofselected arylnitrenium ions and their reactions with guanosinederivatives and aryl azidesXue, Jiadan., 薛佳丹. January 2008 (has links)
published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy
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Time-resolved resonance raman and density functional theory studies ofthe photochemistry of (S)-ketoprofenChuang, Yung-ping., 莊蓉萍. January 2008 (has links)
published_or_final_version / Chemistry / Master / Master of Philosophy
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Density functional theory studies of selected hydrogen bond assisted chemical reactionsGuo, Zhen, 郭臻 January 2009 (has links)
published_or_final_version / Chemistry / Doctoral / Doctor of Philosophy
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A complementary study of perovskites : combining diffraction, solid-state NMR and first principles DFT calculationsJohnston, Karen Elizabeth January 2010 (has links)
Perovskites, ABX₃, and their associated solid-solutions are a particularly important and attractive area of research within materials chemistry. Owing to their structural and compositional flexibility and potential physical properties they are one of the largest classes of materials currently under investigation. This thesis is concerned with the synthesis and structural characterisation of several perovskite-based materials using a combined approach of high-resolution synchrotron X-ray and neutron powder diffraction (NPD), solid-state Nuclear Magnetic Resonance (NMR) and first-principles Density Functional Theory (DFT) calculations. Initial investigations concentrated on room temperature NaNbO₃, a perovskite widely debated in the literatue. Published crystallographic data indicate NaNbO₃ possesses two crystallographically distinct Na sites in space group Pbcm. Whilst some of our materials appear in agreement with this (notably a commercially purchased sample) many of our laboratory-synthesised samples of NaNbO₃ routinely comprise of two phases, which we show to be the antiferroelectric Pbcm and polar P2₁ma polymorphs. Several different synthetic methods were utilised during this investigation and the quantity of each phase present was found to vary as a function of preparative method. ²³Na, ⁹³Nb and ¹⁷O DFT calculations were used in conjunction with experiment to aid in spectral analysis, assignment and interpretation. In addition, ab initio random structure searching (AIRSS) was utilised in an attempt to predict the most stable phases of NaNbO₃. This proved to be both successful and highly informative. A series of NaNbO₃-related solid-solutions, namely K[subscript(x)]Na[subscript(1-x)]NbO₃ (KNN), Li[subscript(x)]Na[subscript(1-x)]NbO₃ (LNN) and Na[subscript(1-x)]Sr[subscript(x/2)]□[subscript(x/2)]NbO₃ (SNN) have also been synthesised and characterised. The substitution of K⁺ , Li⁺ and Sr²⁺ cations onto the A site appears to produce the same polar P2₁ma phase initially identified in the room temperature NaNbO₃ investigation. The abrupt change in cation size in the KNN and LNN series, and the introduction of vacancies in the SNN series, is thought to result in a structural distortion which, in turn, causes the formation of the P2₁ma phase. A low temperature synchrotron X-ray powder diffraction study (12 < T < 295 K) was completed for a sample of NaNbO₁ composed of the P2₁ma polymorph (~90%) and a small quantity of the Pbcm phase (~10%). A region of phase coexistence was identified between the P2₁ma, R3c and Pbcm phases over a relatively large temperature range. Full conversion of the P2₁ma phase to the low temperature R3c phase was not possible and, consistently, the P2₁ma phase was the most abundant phase present. Factors such as structural, strain, crystallite size and morphology are thought to be crucial in determining the exact phases of NaNbO₃ produced, both at low and room temperature. The solid-solution La[subscript(1-x)]Y[subscript(x)]ScO₃ was also investigated. Compositions x = 0, 0.2, 0.4, 0.6, 0.8 and 1 were successfully synthesised and characterised. Refined high-resolution NPD data indicates that an orthorhombic structure, in space group Pbnm, was retained throughout the solid-solution. Using ⁴⁵Sc and ⁸⁹Y MAS NMR each sample was found to exhibit disorder, believed to result from both a distribution of quadrupole and chemical shifts. NMR parameters were calculated for several model Sc and Y compounds using DFT methods to determine the feasibility and accuracy of ⁴⁵Sc and ⁸⁹Y DFT calculations. These proved successful and subsequent calculations were completed for the end members LaScO₃ and YScO₃. DFT calculations were also utilised to gain insight into the disorder exhibited in the La[subscript(1-x)]Y[subscript(x)]ScO₃ solid-solution.
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Assessment of density functional methods for computing structures and energies of organic and bioorganic moleculesCao, Jie January 2011 (has links)
The work in this thesis mainly focuses on the assessment of density functional methods for computing structures and energies of organic and bioorganic molecules. Previous studies found dramatic conformational and stability changes from B3LYP to MP2 geometry optimization for some Tyr-Gly conformers. Possible reasons could be large intramolecular basis set superposition errors (BSSEs) in the MP2 calculations and the lack of dispersion in the B3LYP calculations. The fragmentation method and three kinds of rotation methods were used to investigate intramolecular BSSE. It is concluded that the rotation method cannot be used to correct intramolecular BSSE along a rotation profile. Another methodology is to employ modern density functionals. We focused on M06-L with the Tyr-Gly conformer ‘book6’. Potential energy profiles were determined by computing the energy for geometries optimized at various fixed values of a distance that controls the degree of foldedness of the structure. M06-L manifested itself as a very promising method to investigate the potential energy surface of small peptides containing aromatic residues. To predict Tyr-Gly structures, 108 potential conformers were created with a Fortran program. The geometry optimizations were done using M06-L/6-31G(d) and M05-2X/6-31+G(d). Two schemes were employed and the most stable conformers were compared to the 20 stable conformers found by B3LYP. Both schemes found 10 conformers similar to one of the B3LYP stable conformers, as well as several newly found conformers. The study of a missing B3LYP stable conformer showed that the possible reason of missing conformers may be the lack in dispersion in B3LYP theory. To study the hydration effect, we studied the conformations of neutral and zwitterionic 3-fluoro-γ-aminobutyric acid (3F-GABA) in solution using different solvation models, mainly the explicit water molecule models. Zwitterionic forms of 3F-GABA are preferred in solution. M06-2X performs better in calculating transition energy profiles than MP2.
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