Structural and Spectral Investigations of Transition Metal Complexes Of Di-2-Pyridyl Ketone N (4), N (4)-Disubstituted Thiosemicarbazones

The study deals with structural and spectral investigations of transition metal complexes of di-2-pyridyl ketone N(4),N(4)-disubstituted thiosemicarbazones. The main objective and scope of the work deals with di-2-pyridyl ketone N(4),N(4)-disubstituted thiosemicarbazones are quardridentate NNNS donor ligands. To chosen this ligand for study because, the ligands are prepared and characterized for the first time, since there are two pyridyl nitorgens, dimmers and polymers of complexes may result leading to interesting structural aspects. The work includes the preparation of the thiosemicarbzones and their structural and spectral studies, synthesis and spectral characterization of complexes of copper(II),,nickel(II),manganese(II), dioxovanadium(V),cobalt(III),zinc(II),cadmium(II) of the ligand HL, synthesis and spectral characterization of complexes of copper(II),manganese(II), of the ligand HL and the development of X-ray quality crystals and its X-ray diffraction studies. The structural characterization techniques are elemental analysis, conductivity measurements, magnetic measurements, electronic spectroscopy, H NMR spectroscopy, Infrared spectroscopy and X-ray crystallography.

Studies in Fuzzy Commutative Algebra

The main objective of this thesis was to extend some basic concepts and results in module theory in algebra to the fuzzy setting.The concepts like simple module, semisimple module and exact sequences of R-modules form an important area of study in crisp module theory. In this thesis generalising these concepts to the fuzzy setting we have introduced concepts of ‘simple and semisimple L-modules’ and proved some results which include results analogous to those in crisp case. Also we have defined and studied the concept of ‘exact sequences of L-modules’.Further extending the concepts in crisp theory, we have introduced the fuzzy analogues ‘projective and injective L-modules’. We have proved many results in this context. Further we have defined and explored notion of ‘essential L-submodules of an L-module’. Still there are results in crisp theory related to the topics covered in this thesis which are to be investigated in the fuzzy setting. There are a lot of ideas still left in algebra, related to the theory of modules, such as the ‘injective hull of a module’, ‘tensor product of modules’ etc. for which the fuzzy analogues are not defined and explored.

Geometric Algebra and Einstein’s Electron

This thesis entitled Geometric algebra and einsteins electron: Deterministic field theories .The work in this thesis clarifies an important part of Koga’s theory.Koga also developed a theory of the electron incorporating its gravitational field, using his substitutes for Einstein’s equation.The third chapter deals with the application of geometric algebra to Koga’s approach of the Dirac equation. In chapter 4 we study some aspects of the work of mendel sachs (35,36,37,).Sachs stated aim is to show how quantum mechanics is a limiting case of a general relativistic unified field theory.Chapter 5 contains a critical study and comparison of the work of Koga and Sachs. In particular, we conclude that the incorporation of Mach’s principle is not necessary in Sachs’s treatment of the Dirac equation.

Investigations on the Structural, Spectral and Magnetic Interactions of Transition Metal Complexes of Multidentate Ligands from Di-2-pyridyl ketone and N(4)-substituted Thiosemicarbazides

The present work deals with the investigations on sthe structural spectral and magnetic interactions of transition metal complexes of multidentate ligands from D1-2-pyridyl ketone and N(4)-Substituted thiosemicarbazides.Thiosemicarbazones are thiourea derivatives with the general formula R2N— C(S)—NH—N=CR2. In the solution state, the thiosemicarbazones exhibit the thionethiol tautomerism similar to the keto-enol tautomerism, and in solution state the thiol form predominates and a deprotonation at the thiolate group in alcoholic medium enhances the coordination abilities ofthe thiosemicarbazones.The magnetochemistry of metal complexes of di-2-pyridyl ketone is a current hot subject of research, which mainly owes to the excellent structural diversity of the complexes ranging from cubanes to clusters, with promising ferromagnetic outputs.Only few efforts were aimed at the magnetochemistry of metal complexes of thiosemicarbazones, and that too were concerned with the complexes of bisttltioscinicarbazones). However, as far as the monothiosemicarbazones are concerned, the magnetochemistry of transition metal complexes of di-2-pyridyl ketone thiosemicarbazones turned up quite unexplored. Consequently, an investigation into it appeared novel and promising to us and that prompted this study, which can be regarded as the initial step towards exploring the magnetochemistry of thiosemicarbazone complexes, especially of di-2-pyridyl ketone derivatives.We could successfully isolate single crystals suitable for X-ray diffraction for the first three ligands. To conclude, we have synthesized some new thiosemicarbazones and their transition metal complexes and studied their structural, spectral and magnetic attributes. Some ofthe complexes revealed interesting stereochemistries and possible bridging characteristics with spectroscopic evidences. Unfortunately, single crystal Xray diffraction studies could not be carried out for many of these interesting compounds due to the lack of availability of suitable quality single crystals. However, the magnetic studies provided support for the proposed stereochemistry giving evidences for their magnetically concentrated nature. The magnetic susceptibilities measured at six different temperatures in the 80-298 K range are fitted into different magnetic equations, which provided an idea about the magnetic behavior of the compounds under study. Some of the copper, oxovanadium, nickel and cobalt complexes are found to possess anomalous magnetic moments, i.e., they revealed no regular gradation with temperature. However, some other copper complexes are observed to be antiferromagnetic, due to super-exchange pathways. The manganese complexes and one of the cobalt complexes are also observed to be antiferromagnetic in nature. However, some nickel complexes have turned up to be ferromagnetic. Accordingly, the versatile stereoehemistry and magnetic behavior of the complexes studied, prompt us to conclude that the transition metal complexes of di-2-pyridyl ketone thiosemicarbazones are promising systems for potential magnetic applications.

Some Problems In Algebra And Topology A Study Of Fuzzy Convexity With Special Reference To Separation Properties

This thesis is a study of abstract fuzzy convexity spaces and fuzzy topology fuzzy convexity spaces No attempt seems to have been made to develop a fuzzy convexity theoryin abstract situations. The purpose of this thesis is to introduce fuzzy convexity theory in abstract situations

Copper(II) complexes derived from di-2-pyridyl ketone-N4-phenyl-3-semicarbazone: Synthesis and spectral studies

Five copper(II) complexes [CuLCl]2·CuCl2·4H2O (1), [CuLOAc] (2), [CuLNO3]2 (3), [CuLN3] (4) and [CuLNCS]·3/2H2O (5) of di-2-pyridyl ketone-N4-phenyl-3-semicarbazone (HL) were synthesized and characterized by elemental analyses and electronic, infrared and EPR spectral techniques. In all these complexes the semicarbazone undergoes deprotonation and coordinates through enolate oxygen, azomethine and pyridyl nitrogen atoms. All the complexes are EPR active due to the presence of an unpaired electron. EPR spectra of all the complexes in DMF at 77K suggest axial symmetry and the presence of half field signals for the complexes 1 and 3 indicates dimeric structures

Synthesis and Properties of Polyurethanes of Hexamethylene Di-Isocyanate with Multifunctional Chromophores in the Main Chain

Various polyurethanes containing photoactive bis(azo) and bis(o-nitrobenzyl) groups in the main chain were synthesized by polyaddition reactions of diols such as bis(4-hydroxyphenylazo)-2,20-dinitrodiphenylmethane, 4-hydroxy-3-methylphenylazo- 40-hydroxyphenylazo-2,20-dinitrodiphenylmethane and bis(4-hydroxy-3- methylphenylazo)-2,20-dinitrodiphenylmethane with hexamethylene di-isocyanate (HMDI), in dimethyl acetamide (DMAc) in the presence of di-n-butyltin dilaurate (DBTDL) as catalyst. All of them were characterized by IR, UV-vis, 1H NMR and 13C NMR spectra and also by thermogravimetric analysis (TGA), differential scanning calorimetry (DSC) and gel permeation chromatography (GPC).

Versatile binding properties of a di-2-pyridyl ketone nicotinoylhydrazone ligand: Crystal structure of a Cu(II) complex

Six new copper complexes of di-2-pyridyl ketone nicotinoylhydrazone (HDKN) have been synthesized. The complexes have been characterized by a variety of spectroscopic techniques and the structure of [Cu(DKN)2]·H2O has been determined by single crystal X-ray diffraction. The compound [Cu(DKN)2]·H2O crystallized in the monoclinic space group P21 and has a distorted octahedral geometry. The IR spectra revealed the presence of variable modes of chelation for the investigated ligand. The EPR spectra of compounds [Cu2(DKN)2( -N3)2] and [Cu2(DKN)2( -NCS)2] in polycrystalline state suggest a dimeric structure as they exhibited a half field signal, which indicate the presence of a weak interaction between two Cu(II) ions in these complexes

Fourier-Matrizen und Ringe mit Basis

Bei der Bestimmung der irreduziblen Charaktere einer Gruppe vom Lie-Typ entwickelte Lusztig eine Theorie, in der eine sogenannte Fourier-Transformation auftaucht. Dies ist eine Matrix, die nur von der Weylgruppe der Gruppe vom Lie-Typ abhängt. Anhand der Eigenschaften, die eine solche Fourier- Matrix erfüllen muß, haben Geck und Malle ein Axiomensystem aufgestellt. Dieses ermöglichte es Broue, Malle und Michel füur die Spetses, über die noch vieles unbekannt ist, Fourier-Matrizen zu bestimmen. Das Ziel dieser Arbeit ist eine Untersuchung und neue Interpretation dieser Fourier-Matrizen, die hoffentlich weitere Informationen zu den Spetses liefert. Die Werkzeuge, die dabei entstehen, sind sehr vielseitig verwendbar, denn diese Matrizen entsprechen gewissen Z-Algebren, die im Wesentlichen die Eigenschaften von Tafelalgebren besitzen. Diese spielen in der Darstellungstheorie eine wichtige Rolle, weil z.B. Darstellungsringe Tafelalgebren sind. In der Theorie der Kac-Moody-Algebren gibt es die sogenannte Kac-Peterson-Matrix, die auch die Eigenschaften unserer Fourier-Matrizen besitzt. Ein wichtiges Resultat dieser Arbeit ist, daß die Fourier-Matrizen, die G. Malle zu den imprimitiven komplexen Spiegelungsgruppen definiert, die Eigenschaft besitzen, daß die Strukturkonstanten der zugehörigen Algebren ganze Zahlen sind. Dazu müssen äußere Produkte von Gruppenringen von zyklischen Gruppen untersucht werden. Außerdem gibt es einen Zusammenhang zu den Kac-Peterson-Matrizen: Wir beweisen, daß wir durch Bildung äußerer Produkte von den Matrizen vom Typ A(1)1 zu denen vom Typ C(1) l gelangen. Lusztig erkannte, daß manche seiner Fourier-Matrizen zum Darstellungsring des Quantendoppels einer endlichen Gruppe gehören. Deswegen ist es naheliegend zu versuchen, die noch ungeklärten Matrizen als solche zu identifizieren. Coste, Gannon und Ruelle untersuchen diesen Darstellungsring. Sie stellen eine Reihe von wichtigen Fragen. Eine dieser Fragen beantworten wir, nämlich inwieweit rekonstruiert werden kann, zu welcher endlichen Gruppe gegebene Matrizen gehören. Den Darstellungsring des getwisteten Quantendoppels berechnen wir für viele Beispiele am Computer. Dazu müssen unter anderem Elemente aus der dritten Kohomologie-Gruppe H3(G,C×) explizit berechnet werden, was bisher anscheinend in noch keinem Computeralgebra-System implementiert wurde. Leider ergibt sich hierbei kein Zusammenhang zu den von Spetses herrührenden Matrizen. Die Werkzeuge, die in der Arbeit entwickelt werden, ermöglichen eine strukturelle Zerlegung der Z-Ringe mit Basis in bekannte Anteile. So können wir für die meisten Matrizen der Spetses Konstruktionen angeben: Die zugehörigen Z-Algebren sind Faktorringe von Tensorprodukten von affinen Ringe Charakterringen und von Darstellungsringen von Quantendoppeln.

Versuche zur Synthese ferrocendiylverbrückter Di(amido)-Chelatkomplexe von Gruppe-4-Metallen

Die Polymerisation von α-Olefinen mit Derivaten von Gruppe-4-Metallocenen ist von großem technologischen Interesse. In den letzten Jahren hat sich bei der Suche nach metallocenalternativen Präkatalysatoren u.a. aufgrund der theoretischen Arbeiten von Ziegler gezeigt, dass Di(amido)-Chelatkomplexe mit Gruppe-4-Metallen vielversprechende Spezies für die α-Olefinpolymerisation darstellen. Im Rahmen der vorliegenden Arbeit sollten die stereoelektronischen Eigenschaften solcher Komplexe durch Arylgruppen mit sterisch anspruchsvollen Alkylsubstituenten beeinflusst werden. Weitere interessante Eigenschaften sollten durch die die Stickstoffatome verbrückende Ferroceneinheit erzielt werden, da diese als molekulares Kugellager und redoxaktive Schaltereinheit fungieren kann. Die Di(arylamino)ferrocenligandvorstufen Fe[(C5H4)NHPh]2, Fe[(C5H4)NH(2,6-C6H3Me2)]2 und Fe[(C5H4)NH(2,4,6-i-Pr3C6H2)]2 konnten durch Hartwig-Buchwald-artige Kreuzkupplung von 1,1´-Diaminoferrocen mit dem jeweiligen Arylbromid erhalten werden. Dagegen misslangen über diese Syntheseroute zahlreiche Versuche zur Synthese von Derivaten mit Substituenten in meta-Position des Arylringes. Die Darstellung der Titan- und Zirkoniumchelatkomplexe gelang durch Metathesereaktion der Di(arylamino)ferrocene mit M(NMe2)4 bzw. M(CH2Ph)4 (M = Ti, Zr), die unter Eliminierung von 2 Äquivalenten HNMe2 bzw. Toluol ablaufen. Dabei zeigte sich, dass bei sterisch anspruchsvollen Di(arylamino)ferrocenligandsystemen keine Metathesereaktion mit Ti(NMe2)4 möglich ist, was auch für analoge Reaktionen mit Ti(CH2Ph)4 zu erwarten ist. Ganz anders sind dagegen die Verhältnisse in der Zirkoniumchemie. Hier konnten durch Umsetzung von Fe[(C5H4)NH(2,4,6-i-Pr3C6H2)]2 mit Zr(NMe2)4 bzw. Zr(CH2Ph)4 die Komplexe [{Fe[C5H4(NC6H2-2,4,6-i-Pr3)]2}Zr(NMe2)2] und [{Fe[C5H4(NC6H2-2,4,6-i-Pr3)]2}Zr(CH2Ph)2] dargestellt werden. Hier findet sich eine senkrechte Anordnung der Arylringe zur Chelatringebene, die die nach Ziegler günstige Orbitalüberlappung ermöglicht, die zu einer besonders hohen katalytischen Aktivität dieser Komplexe in der Ethylenpolymerisation führen sollte. Nach üblicher Aktivierung zeigen diese Komplexe jedoch nur niedrige Aktivitäten in der Ethylenpolymerisation. Ob strukturelle Parameter für dieses Ergebnis verantwortlich sind, oder sogar Defizite im Ziegler-Modell vorliegen, sollte Gegenstand zukünftiger Untersuchungen sein.

On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

An introduction to ordinary differential equations by computer algebra-systems

We report on an elementary course in ordinary differential equations (odes) for students in engineering sciences. The course is also intended to become a self-study package for odes and is is based on several interactive computer lessons using REDUCE and MATHEMATICA . The aim of the course is not to do Computer Algebra (CA) by example or to use it for doing classroom examples. The aim ist to teach and to learn mathematics by using CA-systems.

Root parametrized differential equations

The present thesis is about the inverse problem in differential Galois Theory. Given a differential field, the inverse  problem asks which linear algebraic groups can be realized as differential Galois groups of Picard-Vessiot extensions of this field.   In this thesis we will concentrate on the realization of the classical groups as differential Galois groups. We introduce a method for a very general realization of these groups. This means that we present for the classical groups of Lie rank $l$ explicit linear differential equations where the coefficients are differential polynomials in $l$ differential indeterminates over an algebraically closed field of constants $C$, i.e. our differential ground field is purely differential transcendental over the constants.   For the groups of type $A_l$, $B_l$, $C_l$, $D_l$ and $G_2$ we managed to do these realizations at the same time in terms of Abhyankar's program 'Nice Equations for Nice Groups'. Here the choice of the defining matrix is important. We found out that an educated choice of $l$ negative roots for the parametrization together with the positive simple roots leads to a nice differential equation and at the same time defines a sufficiently general element of the Lie algebra. Unfortunately for the groups of type $F_4$ and $E_6$ the linear differential equations for such elements are of enormous length. Therefore we keep in the case of $F_4$ and $E_6$ the defining matrix differential equation which has also an easy and nice shape.   The basic idea for the realization is the application of an upper and lower bound criterion for the differential Galois group to our parameter equations and to show that both bounds coincide. An upper and lower bound criterion can be found in literature. Here we will only use the upper bound, since for the application of the lower bound criterion an important condition has to be satisfied. If the differential ground field is $C_1$, e.g., $C(z)$ with standard derivation, this condition is automatically satisfied. Since our differential ground field is purely differential transcendental over $C$, we have no information whether this condition holds or not.   The main part of this thesis is the development of an alternative lower bound criterion and its application. We introduce the specialization bound. It states that the differential Galois group of a specialization of the parameter equation is contained in the differential Galois group of the parameter equation. Thus for its application we need a differential equation over $C(z)$ with given differential Galois group. A modification of a result from Mitschi and Singer yields such an equation over $C(z)$ up to differential conjugation, i.e. up to transformation to the required shape. The transformation of their equation to a specialization of our parameter equation is done for each of the above groups in the respective transformation lemma.

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