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Olefin metathesis is a valuable synthetic tool, widely used in several fields of science. Due to the importance of this transformation several contributions have been made in this field in order to understand mechanistic aspects, reactivity and applicability of this process. In this topic, ruthenium indenylidene complexes have shown great activity and stability in metathesis, making them very valuable pre-catalysts. However, several aspects of these pre-catalysts have not been evaluated yet. For example, even though reports of active second generation ruthenium indenylidene complexes bearing bulky N-heterocyclic carbenes are present in the literature, no studies have been done to understand how steric hindrance affects the process. For these reasons, [RuCl₂(IPr*)(PPh₃)(3-phenylindenylidene)] (IPr*-PPh₃) and [RuCl₂(IPr*)(Py)(3-phenylindenylidene)] (IPr*-Py), bearing the very bulky ligand, IPr* have been synthesised and compared with [RuCl₂(IPr)(PPh₃)(3-phenylindenylidene)] (IPr-PPh₃) and the new [RuCl₂(IPr)(Py)(3-phenylindenylidene)] (IPr-Py). Another important aspect, presented in this thesis, is the investigation of the stability of indenylidene pre-catalysts in alcohol solvents. Surprisingly, several different decomposition processes occur depending on the starting complex and the alcohol used. Mechanistic investigation into this decomposition, allowed us to develop a better understanding of this process, and to predict the decomposition product based on the environment. In particular, this study revealed that [RuCl(η⁵-3-phenylindenyl)(PPh₃)₂] (Eta-5) is accessed from [RuCl₂(3-phenylindenylidene)(PPh₃)₂] (M₁₀) via a novel indenylidene to η⁵-indenyl rearrangement. This formal decomposition product has been found to be active in at least 20 different catalytic transformations, rendering it a versatile catalytic tool.

Palladium catalysed asymmetric hydroxy- and alkoxycarbonylation reactions of alkenes have the potential to deliver valuable chiral carboxylic acid and ester building blocks from cheap feedstocks: alkenes, carbon monoxide and water (alcohols in the case of alkoxycarbonylation). Despite the attractive nature of these reactions, extensive research has so far been unable to produce effective catalysts which are capable of controlling both regio- and enantioselectivity. Building on exciting recent results involving the use of highly enantioselective palladium catalysts derived from Phanephos-type ligands, this research focuses on paracyclophane-diphosphines and their use in asymmetric hydroxy- and alkoxycarbonylation reactions. An investigation into reaction conditions analysed several factors, including solvents, CO-pressure, acidic additives and halide sources, to provide optimal activity and selectivities. Two novel electron-poor paracyclophane-diphosphines and their mono- and di-palladium complexes were synthesised and shown to provide exceptional levels of regioselectivity while maintaining high levels of asymmetric induction. These are the first such examples of hydroxy- or alkoxycarbonylation catalysts to facilitate simultaneous control over both regio- and enantioselectivity. The most effective catalyst was used to promote the reactions of a selection of aryl alkenes and was shown to be tolerant of several different functional groups. A selection of non-symmetric paracyclophane-diphosphine ligands and their palladium complexes were also synthesised and assessed for their performance in hydroxy- and alkoxycarbonylation. We also report the use of Phanephos-type ligands to promote the highly enantioselective hydroxycarbonylation of N-(p-toluenesulfonyl)-3-pyrroline to deliver a chiral proline derivative in high ee.

We study the free centre-by-metabelian Lie ring, that is, the free Lie ring with the property that the second derived ideal is contained in the centre. We exhibit explicit generating sets for the homogeneous components and the fine homogeneous components of the second derived ideal. Each of these components is a direct sum of a free abelian group and a (possibly trivial) elementary abelian $2$-group. Our generating sets are such that some of their elements generate the torsion subgroup while the remaining ones freely generate a free abelian group. A key ingredient of our approach is the determination of the dimensions of the corresponding homogeneous components of the free centre-by-metabelian Lie algebra over fields of characteristic other than $2$. For this we exploit a $6$-term exact sequence of modules over a polynomial ring that is originally defined over the integers, but turns into a sequence whose terms are projective modules after tensoring with a suitable field. Our results correct a partly erroneous theorem in the literature. Moreover, we study the product of three homogeneous components of a free Lie algebra. Let $L$ be a free Lie algebra of finite rank over a field and let $L_n$ denote the degree $n$ homogeneous component of $L$. Formulae for the dimension of the subspaces $[L_n,L_m]$ for all $n$ and $m$ were obtained by Ralph St\"{o}hr and Michael Vaughan-Lee. Formulae for the dimension of the subspaces of the form $[L_n,L_m,L_k]$ under certain conditions on $n,m$ and $k$ were obtained by Nil Mansuro\u{g}lu and Ralph St\"{o}hr. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field. For example, the dimension of $[L_2,L_2,L_1]$ over fields of characteristic $2$ is different from the dimension over fields of characteristic other than $2$.

Μια πολλαπλότητα Riemann (M, g) ονομάζεται Einstein αν έχει σταθερή καμπυλότητα Ricci. Είναι γνωστό ότι αν (M=G/K, g) είναι μια συμπαγής ομογενής πολλαπλότητα Riemann, τότε οι G-αναλλοίωτες μετρικές Einstein μοναδιαίου όγκου, είναι τα κρίσιμα σημεία του συναρτησοειδούς ολικής βαθμωτής καμπυλότητας περιορισμένο στο χώρο των G-αναλλοίωτων μετρικών με όγκο 1. Για μια G-αναλλοίωτη μετρική Riemann η εξίσωση Einstein ανάγεται σε ένα σύστημα αλγεβρικών εξισώσεων. Οι θετικές πραγματικές λύσεις του συστήματος αυτού είναι ακριβώς οι G-αναλλοίωτες μετρικές Einstein που δέχεται η πολλαπλότητα Μ. Μια σημαντική οικογένεια συμπαγών ομογενών χώρων αποτελείται από τις γενικευμένες πολλαπλότητες σημαιών. Κάθε τέτοιος χώρος είναι μια τροχιά της συζυγούς αναπαράστασης μιας συμπαγούς, συνεκτικής, ημι-απλής ομάδας Lie G. Πρόκειται για ομογενείς πολλαπλότητες της μορφής G/C(S), όπου C(S) είναι ο κεντροποιητής ενός δακτυλίου S στην G. Κάθε τέτοιος χώρος δέχεται ένα πεπερασμένο πλήθος από G-αναλλοίωτες μετρικές Kahler-EInstein. Στην παρούσα διατριβή ταξινομούμε όλες τις πολλαπλότητες σημαιών G/K που αντιστοιχούν σε μια απλή ομάδα Lie G, των οποίων η ισοτροπική αναπαράσταση διασπάται σε 2 ή 4 μη αναγώγιμους και μη ισοδύναμους Ad(K)-αναλλοίωτους προσθετέους. Για κάθε τέτοιο χώρο λύνουμε την αναλλοίωτη εξίσωση Εinstein, και παρουσιάζουμε την αναλυτική μορφή νέων G-αναλλοίωτων μετρικών Einstein. Στις περισσότερες περιπτώσεις παρουσιάζουμε την πλήρη ταξινόμηση των αναλλοίωτων μετρικών Einstein. Επίσης εξετάζουμε το ισομετρικό πρόβλημα. Για την κατασκευή της εξίσωσης Einstein σε κάποιες πολλαπλότητες σημαιών με 4 ισοτροπικούς προσθετέους χρησιμοποιούμε την νηματοποίηση συστροφής που δέχεται κάθε πολλαπλότητα σημαιών επί ενός ισοτροπικά μη αναγώγιμου συμμετρικού χώρου συμπαγούς τύπου. Αυτή η μέθοδος είναι καινούργια και μπορεί να εφαρμοστεί και σε άλλες πολλαπλότητες σημαιών. / A Riemannian manifold (M, g) is called Einstein, if it has constant Ricci curvature. It is well known that if (M=G/K, g) is a compact homogeneous Riemannian manifold, then the G-invariant \tl{Einstein} metrics of unit volume, are the critical points of the scalar curvature function restricted to the space of all G-invariant metrics with volume 1. For a G-invariant Riemannian metric the Einstein equation reduces to a system of algebraic equations. The positive real solutions of this system are the $G$-invariant Einstein metrics on M. An important family of compact homogeneous spaces consists of the generalized flag manifolds. These are adjoint orbits of a compact semisimple Lie group. Flag manifolds of a compact connected semisimple Lie group exhaust all compact and simply connected homogeneous Kahler manifolds and are of the form G/C(S), where C(S) is the centralizer (in G) of a torus S in G. Such homogeneous spaces admit a finite number of G-invariant complex structures, and for any such complex structure there is a unique compatible G-invariant Kahler-Einstein metric. In this thesis we classify all flag manifolds M=G/K of a compact simple Lie group G, whose isotropy representation decomposes into 2 or 4, isotropy summands. For these spaces we solve the (homogeneous) Einstein equation, and we obtain the explicit form of new G-invariant Einstein metrics. For most cases we give the classification of homogeneous Einstein metrics. We also examine the isometric problem. For the construction of the Einstein equation on certain flag manifolds with four isotropy summands, we apply for first time the twistor fibration of a flag manifold over an isotropy irreducible symmetric space of compact type. This method is new and it can be used also for other flag manifolds. For flag manifolds with two isotropy summands, we use the restricted Hessian and we characterize the new Einstein metrics as local minimum points of the scalar curvature function restricted to the space of G-invariant Riemannian metrics of volume 1. We mention that the classification of flag manifolds with two isotropy summands gives us new examples of homogeneous spaces, for which the motion of a charged particle under the electromagnetic field, and the geodesics curves, are completely determined.

Regionalization of hydrometeorological variables such as rainfall and temperature is necessary for various applications related to water resources planning and management. Sampling variability and randomness associated with the variables, as well as non-availability and paucity of data pose a challenge in modelling the variables. This challenge can be addressed by using stochastic models that utilize information from hydrometeorologically similar locations for modelling the variables. A set of locations that are hydrometeorologically similar are referred to as homogeneous region or pooling group and the process of identifying a homogeneous region is referred to as regionalization. The thesis concerns development of new approaches to regionalization of (i) extreme rainfall,(ii) maximum and minimum temperatures, and (iii) rainfall together with maximum and minimum temperatures. Regionalization of extreme rainfall and frequency analysis based on resulting regions yields quantile estimates that find use in design of water control (e.g., barrages, dams, levees) and conveyance structures (e.g., culverts, storm sewers, spillways) to mitigate damages that are likely due to floods triggered by extreme rainfall, and land-use planning and management. Regionalization based on both rainfall and temperature yield regions that could be used to address a wide spectrum of problems such as meteorological drought analysis, agricultural planning to cope with water shortages during droughts, downscaling of precipitation and temperature. Conventional approaches to regionalization of extreme rainfall are based extensively on statistics derived from extreme rainfall. Therefore delineated regions are susceptible to sampling variability and randomness associated with extreme rainfall records, which is undesirable. To address this, the idea of forming regions by considering attributes for regionalization as seasonality measure and site location indicators (which could be determined even for ungauged locations) is explored. For regionalization, Global Fuzzy c-means (GFCM) cluster analysis based methodology is developed in L-moment framework. The methodology is used to arrive at a set of 25 homogeneous extreme rainfall regions over India considering gridded rainfall records at daily scale, as there is dearth of regionalization studies on extreme rainfall in India Results are compared with those based on commonly used region of influence (ROI) approach that forms site-specific regions for quantile estimation, but lacks ability to delineate a geographical area into a reasonable number of homogeneous regions. Gridded data constitute spatially averaged rainfall that might originate from a different process (more synoptic) than point rainfall (more convective). Therefore to investigate utility of the developed GFCM methodology in arriving at meaningful regions when applied to point rainfall data, the methodology is applied to daily rainfall records available for 1032 gauges in Karnataka state of India. The application yielded 22 homogeneous extreme rainfall regions. Experiments carried out to examine utility of GFCM and ROI based regions in arriving at quantile estimates for ungauged sites in the study area reveal that performance of GFCM methodology is fairly close to that of ROI approach. Errors were marginally lower in the case of GFCM approach in analysis with observed point rainfall data over Karnataka, while its converse was noted in the case of analysis with gridded rainfall data over India. Neither of the approaches (CA, ROI) was found to be consistent in yielding least error in quantile estimates over all the sites. The existing approaches to regionalization of temperature are based on temperature time series or their related statistics, rather than attributes effecting temperature in the study area. Therefore independent validation of the delineated regions for homogeneity in temperature is not possible. Another drawback of the existing approaches is that they require adequate number of sites with contemporaneous temperature records for regionalization, because the delineated regions are susceptible to sampling variability and randomness associated with the temperature records that are often (i) short in length, (ii) limited over contemporaneous time period and (iii) spatially sparse. To address these issues, a two-stage clustering approach is developed to arrive at regions that are homogeneous in terms of both monthly maximum and minimum temperatures ( and ). First-stage of the approach involves (i) identifying a common set of possible predictors (LSAVs) influencing and over the entire study area, and (ii) using correlations of those predictors with and along with location indicators (latitude, longitude and altitude) as the basis to delineate sites in the study area into hard clusters through global k-means clustering algorithm. The second stage involves (i) identifying appropriate LSAVs corresponding to each of the first-stage clusters, which could be considered as potential predictors, and (ii) using the potential predictors along with location indicators (latitude, longitude and altitude) as the basis to partition each of the first-stage clusters into homogeneous temperature regions through global fuzzy c-means clustering algorithm. A set of 28 homogeneous temperature regions was delineated over India using the proposed approach. Those regions are shown to be effective when compared to an existing set of 6 temperature regions over India for which inter-site cross-correlations were found to be weak and negative for several months, which is undesirable. Effectiveness of the newly formed regions is demonstrated. Utility of the proposed maxTminT homogeneous temperature regions in arriving at PET estimates for ungauged locations within the study area was demonstrated. The estimates were found to be better when compared to those based on the existing regions. The existing approaches to regionalization of hydrometeorological variables are based on principal components (PCs)/ statistics/indices determined from time-series of those variables at monthly and seasonal scale. An issue with use of PCs for regionalization is that they have to be extracted from contemporaneous records of hydrometeorological variables. Therefore delineated regions may not be effective when the available records are limited over contemporaneous time period. A drawback associated with the use of statistics/indices is that they (i) may not be meaningful when data exhibit nonstationarity and (ii) do not encompass complete information in the original time series. Consequently the resulting regions may not be effective for the desired purpose. To address these issues, a new approach is proposed. It considers information extracted from wavelet transformations of the observed multivariate hydrometeorological time series as the basis for regionalization by global fuzzy c-means clustering procedure. The approach can account for dynamic variability in the time series and its nonstationarity (if any). Effectiveness of the proposed approach in forming homogeneous hydrometeorological regions is demonstrated by application to India, as there are no prior attempts to form such regions over the country. The investigations resulted in identification of 29 regions over India, which are found to be effective and meaningful. Drought Severity-Area-Frequency (SAF) curves are developed for each of the newly formed regions considering the drought index to be Standardized Precipitation Evapotranspiration Index (SPEI).

Many practical cases of fracture can be considered as one-dimensional, that is, propagating in one dimension and characterised by opening (mode I) and shearing (mode II) action only with no tearing (mode III) action. A double cantilever beam (DCB) represents the most fundamental one-dimensional fracture problem. There has however been considerable confusion in calculating its mixed-mode energy release rate (ERR) partition. In this work, new and completely analytical mixed-mode partition theories are developed for one-dimensional fractures in isotropic homogeneous and laminated composite DCBs, based on linear elastic fracture mechanics (LEFM) and using the Euler and Timoshenko beam theories. They are extended to isotropic homogeneous and laminated composite straight beam structures and isotropic homogeneous plates based on the Kirchhoff-Love and Mindlin-Reissner plate theories. They are also extended to non-rigid elastic interfaces for isotropic homogeneous DCBs. A new approach is used, based on orthogonal pure fracture modes. Two sets of orthogonal pairs of pure modes are found. They are distinct from each other in the present Euler beam and Kirchhoff-Love plate partition theories and coincide on the first set in the present Timoshenko beam and Mindlin-Reissner plate partition theories. After the two sets of pure modes are shown to be unique and orthogonal, they are used to partition mixed modes. Interaction is found between the mode I and mode II modes of the first set in the present Euler beam and Kirchhoff-Love plate partition theories. This alters the ERR partition but does not affect the total ERR. There is no interaction in the present Timoshenko beam or Mindlin-Reissner plate partition theories. The theories distinguish between local and global ERR partitions. Local pureness is defined with respect to the crack tip. Global pureness is defined with respect to the entire region mechanically affected by the crack. It is shown that the global ERR partition using any of the present partition theories or two-dimensional elasticity is given by the present Euler beam or Kirchhoff-Love plate partition theories. The present partition theories are extensively validated using the finite element method (FEM). The present beam and plate partition theories are in excellent agreement with results from the corresponding FEM simulations. Approximate 'averaged partition rules' are also established, based on the average of the two present beam or plate partition theories. They give close approximations to the partitions from two-dimensional elasticity. The propagation of mixed-mode interlaminar fractures in laminated composite beams is investigated using experimental results from the literature and various partition theories. The present Euler beam partition theory offers the best and most simple explanation for all the experimental observations. It is in excellent agreement with the linear failure locus and is significantly closer than other partition theories. It is concluded that its excellent performance is either due to the failure of materials generally being based on global partitions or due to the through-thickness shear effect being negligibly small for the specimens tested. The present partition theories provide an excellent tool for studying interfacial fracture and delamination. They are readily applicable to a wide-range of engineering structures and will be a valuable analytical tool for many practical applications.