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In many fields of science, engineering, and economics large amounts of data are stored and there is a need to analyze these data in order to extract information for various purposes. Data mining is a general concept involving different tools for performing this kind of analysis. The development of mathematical models and efficient algorithms is of key importance. In this thesis we discuss algorithms for the reduced rank regression problem and algorithms for the computation of the best multilinear rank approximation of tensors. The first two papers deal with the reduced rank regression problem, which is encountered in the field of state-space subspace system identification. More specifically the problem is \[ \min_{\rank(X) = k} \det (B - X A)(B - X A)\tp, \] where $A$ and $B$ are given matrices and we want to find $X$ under a certain rank condition that minimizes the determinant. This problem is not properly stated since it involves implicit assumptions on $A$ and $B$ so that $(B - X A)(B - X A)\tp$ is never singular. This deficiency of the determinant criterion is fixed by generalizing the minimization criterion to rank reduction and volume minimization of the objective matrix. The volume of a matrix is defined as the product of its nonzero singular values. We give an algorithm that solves the generalized problem and identify properties of the input and output signals causing a singular objective matrix. Classification problems occur in many applications. The task is to determine the label or class of an unknown object. The third paper concerns with classification of handwritten digits in the context of tensors or multidimensional data arrays. Tensor and multilinear algebra is an area that attracts more and more attention because of the multidimensional structure of the collected data in various applications. Two classification algorithms are given based on the higher order singular value decomposition (HOSVD). The main algorithm makes a data reduction using HOSVD of 98--99 \% prior the construction of the class models. The models are computed as a set of orthonormal bases spanning the dominant subspaces for the different classes. An unknown digit is expressed as a linear combination of the basis vectors. The resulting algorithm achieves 5\% in classification error with fairly low amount of computations. The remaining two papers discuss computational methods for the best multilinear rank approximation problem \[ \min_{\cB} \| \cA - \cB\| \] where $\cA$ is a given tensor and we seek the best low multilinear rank approximation tensor $\cB$. This is a generalization of the best low rank matrix approximation problem. It is well known that for matrices the solution is given by truncating the singular values in the singular value decomposition (SVD) of the matrix. But for tensors in general the truncated HOSVD does not give an optimal approximation. For example, a third order tensor $\cB \in \RR^{I \x J \x K}$ with rank$(\cB) = (r_1,r_2,r_3)$ can be written as the product \[ \cB = \tml{X,Y,Z}{\cC}, \qquad b_{ijk}=\sum_{\lambda,\mu,\nu} x_{i\lambda} y_{j\mu} z_{k\nu} c_{\lambda\mu\nu}, \] where $\cC \in \RR^{r_1 \x r_2 \x r_3}$ and $X \in \RR^{I \times r_1}$, $Y \in \RR^{J \times r_2}$, and $Z \in \RR^{K \times r_3}$ are matrices of full column rank. Since it is no restriction to assume that $X$, $Y$, and $Z$ have orthonormal columns and due to these constraints, the approximation problem can be considered as a nonlinear optimization problem defined on a product of Grassmann manifolds. We introduce novel techniques for multilinear algebraic manipulations enabling means for theoretical analysis and algorithmic implementation. These techniques are used to solve the approximation problem using Newton and Quasi-Newton methods specifically adapted to operate on products of Grassmann manifolds. The presented algorithms are suited for small, large and sparse problems and, when applied on difficult problems, they clearly outperform alternating least squares methods, which are standard in the field.

This study focusses on modelling important aspects of the evolution of pulsar wind nebulae using two different approaches. The first uses a hydrodynamic model to simulate the morphological evolution of a spherically-symmetric composite supernova remnant that is expanding into a homogeneous interstellar medium. In order to extend this model, a magnetic field is included in a kinematic fashion, implying that the reaction of the fluid on the magnetic field is taken into account, while neglecting any counter-reaction of the field on the fluid. This approach is valid provided that the ratio of electromagnetic to particle energy in the nebula is small, or equivalently, for a large plasma β environment. This model therefore allows one to not only calculate the evolution of the convection velocity but also, for example, the evolution of the average magnetic field. The second part of this study focusses on calculating the evolution of the energy spectra of the particles in the nebula using a number of particle evolution models. The first of these is a spatially independent temporal evolution model, similar to the models that can be found in the literature. While spatially independent models are useful, a large part of this study is devoted to developing spatially dependent models based on the Fokker-Planck transport equation. Two such models are developed, the first being a spherically-symmetric model that includes the processes of convection, diffusion, adiabatic losses, as well as the non-thermal energy loss processes of synchrotron radiation and inverse Compton scattering. As the magnetic field geometry can lead to the additional transport process of drift, the previous model is extended to an axisymmetric geometry, thereby allowing one to also include this process. / PhD (Space Physics), North-West University, Potchefstroom Campus, 2014

The general aim of my dissertation was to investigate the role of plant interactions in driving population dynamics. Both theoretical and empirical approaches were employed. All my studies were conducted on the basis of metabolic scaling theory (MST), because the complex, spatially and temporally varying structures and dynamics of ecological systems are considered to be largely consequences of biological metabolism. However, MST did not consider the important role of plant interactions and was found to be invalid in some environmental conditions. Integrating the effects of plant interactions and environmental conditions into MST may be essential for reconciling MST with observed variations in nature. Such integration will improve the development of theory, and will help us to understand the relationship between individual level process and system level dynamics. As a first step, I derived a general ontogenetic growth model for plants which is based on energy conservation and physiological processes of individual plant. Taking the mechanistic growth model as basis, I developed three individual-based models (IBMs) to investigate different topics related to plant population dynamics: 1. I investigated the role of different modes of competition in altering the prediction of MST on plant self-thinning trajectories. A spatially-explicit individual-based zone-of-influence (ZOI) model was developed to investigate the hypothesis that MST may be compatible with the observed variation in plant self-thinning trajectories if different modes of competition and different resource availabilities are considered. The simulation results supported my hypothesis that (i) symmetric competition (e.g. belowground competition) will lead to significantly shallower self-thinning trajectories than asymmetric competition as predicted by MST; and (ii) individual-level metabolic processes can predict population-level patterns when surviving plants are barely affected by local competition, which is more likely to be in the case of asymmetric competition. 2. Recent studies implied that not only plant interactions but also the plastic biomass allocation to roots or shoots of plants may affect mass-density relationship. To investigate the relative roles of competition and plastic biomass allocation in altering the mass-density relationship of plant population, a two-layer ZOI model was used which considers allometric biomass allocation to shoots or roots and represents both above- and belowground competition simultaneously via independent ZOIs. In addition, I also performed greenhouse experiment to evaluate the model predictions. Both theoretical model and experiment demonstrated that: plants are able to adjust their biomass allocation in response to environmental factors, and such adaptive behaviours of individual plants, however, can alter the relative importance of above- or belowground competition, thereby affecting plant mass-density relationships at the population level. Invalid predictions of MST are likely to occur where competition occurs belowground (symmetric) rather than aboveground (asymmetric). 3. I introduced the new concept of modes of facilitation, i.e. symmetric versus asymmetric facilitation, and developed an individual-based model to explore how the interplay between different modes of competition and facilitation changes spatial pattern formation in plant populations. The study shows that facilitation by itself can play an important role in promoting plant aggregation independent of other ecological factors (e.g. seed dispersal, recruitment, and environmental heterogeneity). In the last part of my study, I went from population level to community level and explored the possibility of combining MST and unified neutral theory of biodiversity (UNT). The analysis of extensive data confirms that most plant populations examined are nearly neutral in the sense of demographic trade-offs, which can mostly be explained by a simple allometric scaling rule based on MST. This demographic equivalence regarding birth-death trade-offs between different species and functional groups is consistent with the assumptions of neutral theory but allows functional differences between species. My initial study reconciles the debate about whether niche or neutral mechanisms structure natural communities: the real question should be when and why one of these factors dominates. A synthesis of existing theories will strengthen future ecology in theory and application. All the studies presented in my dissertation showed that the approaches of individual-based and pattern-oriented modelling are promising to achieve the synthesis.

In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.

Ce mémoire porte sur l’étude des équations d’évolution sur des variétés à coubure non nulle, plus particulièrement l’équation des ondes sur les espaces symétriques riemanniens de type non compact.Des propriétés de dispersion des solutions du problème de Cauchy homogène sont démontrées. Ces propriétés sont ensuite utilisées pour établir des estimations dites estimations de Strichartz. L’examen de ces estimées permet de déduire que le problème de Cauchy non linéaire avec des non-linéarités de type puissance est globalement bien posé pour des données initiales petites et localement bien posé pour des données arbitraires.Après un chapitre introductif dédié aux définitions, propriétés algébriques et géométriques des espaces symétriques et à quelques aspects élémentaires d’analyse harmonique sphérique sur ces espaces, un article est présenté : Wave equation on Riemannian symmetric spaces. Cet article contient nos résultats principaux. Dans le dernier chapitre nous présentons en détail deux problèmes ouverts qui prolongent nos travaux. Il s’agit respectivement d’établir le lien entre le comportement asymptotique des estimées et les orbites nilpotentes, et l’étude de l’équation des ondes pour les formes différentielles sur les espaces symétriques. / In this memoir we study evolution equations on curved manifolds. In particular we are interested in the wave equation on Riemannian symmetric spaces of the noncompact type.Dispersive properties of solutions of homogeneous Cauchy problem are proved. These properties are then used to establish Strichartz-type estimates. A closer study of these estimates shows that the nonlinear Cauchy problem with power-like nonlinearities is globally well posed for small initial data and locally well posed for arbitrary initial data.The first chapter is devoted to definitions, algebraic and geometric properties of symmetric spaces and to few elementary aspects of spherical analysis on these spaces. Then our main results are represented in an article : Wave equation on Riemannian symmetric spaces. In the last chapter we present in detail two open problems for future work. One issue is to establish a link between the asymptotic behavior of the estimates and nilpotent orbits, while another issue is the study of wave equation for differential forms on symmetric spaces.

Cette thèse est consacrée à l’étude des représentations modulaires des algèbres d’Ariki-Koike, et des liens avec la théorie des cristaux et des bases canoniques de Kashiwara via le théorème de catégorification d’Ariki. Dans un premier temps, on étudie, grâce à des outils combinatoires, les matrices de décomposition de ces algèbres en généralisant les travaux de Geck et Jacon. On classifie entièrement les cas d’existence et de non-existence d’ensembles basiques, en construisant explicitement ces ensembles lorsqu’ils existent. On explicite ensuite les isomorphismes de cristaux pour les représentations de Fock de l’algèbre affine quantique Uq(sle). On construit alors un isomorphisme particulier, dit canonique, qui permet entre autres une caractérisation non-récursive de n’importe quelle composante connexe du cristal. On souligne également les liens avec la combinatoire des mots sous-jacente à la structure cristalline des espaces de Fock, en décrivant notamment un analogue de la correspondance de Robinson-Schensted-Knuth pour le type A affine. / This thesis is devoted to the study of modular representations of Ariki-Koike algebras, and of the connections with Kashiwara’s crystal and canonical bases theory via Ariki’s categorification theorem. First, we study, using combinatorial tools, the decomposition matrices associated to these algebras, generalising the works of Geck and Jacon. We fully classify the cases of existence and non-existence of canonical basic sets, and we explicitely construct these sets when they exist. Next, we make explicit the crystal isomorphisms for Fock spaces representations of the quantum affine algebra Uq(sle). We then construct of a particular isomorphism, so-called canonical, which gives, inter alia, a non-recursive description of any connected component of the crystal. We also stress the links with the combinatorics of words underlying the crystal structure of Fock spaces, by describing notably an analogue of the Robinson-Schensted-Knuth correspondence for affine type A.

Cette thèse est consacrée au problème d’évolution des théories de gravité modifiée : après avoir rappelé ce qu’il en est pour la Relativité Générale (RG), nous exposons le formalisme n + 1 des théories ƒ(R), Brans-Dicke et tenseur-scalaire et redémontrons un résultat connu : le problème de Cauchy est bien posé pour ces théories, et les équations de contrainte se réduisent à celles de la RG avec un champ de matière. Puis nous effectuons la même décomposition n + 1 pour les théories de Lovelock et, ce qui est nouveau, ƒ(Lovelock). Nous étudions ensuite les équations de contrainte des théories de Lovelock et montrons qu’elles sont, dans le cas conformément plat et symétrique en temps, la prescription d’une somme de σk-courbures. Afin de résoudre cette équation de prescription, nous introduisons une nouvelle famille de polynômes semi-symétriques homogènes et développons des résultats de concavité pour ces polynômes. Nous énonçons une conjecture qui, si elle était avérée, nous permettrait de résoudre l’équation de prescription dans de nombreux cas : ∀ P;Q ∈ ℝ[X], avec deg P = deg Q = p, P et Q sont scindés => p ∑ k=0 P(k) Q(p-k) est scindé / This thesis is devoted to the evolution problem for modified gravity theories. After having explained this problem for General Relativity (GR), we present the n + 1 formalism for ƒ(R) theories, Brans-Dicke and scalar-tensor theories. We recall a known result: the Cauchy problem for these theories is well-posed, and the constraint equations are reduced to those of GR with a matter field. Then we proceed to the same n+1 decomposition for Lovelock and ƒ(Lovelock) theories, the latter being an original result. We show that in the locally conformally flat timesymmetric case, they can be written as the prescription of a sum of σk-curvatures. In order to solve the prescription equation, we introduce a new family of homogeneous semisymmetric polynomials and prove some concavity results for those polynomials. We express the following conjecture: if this is true, we are able to solve the prescription equation in many cases. ∀ P;Q ∈ ℝ[X], avec deg P = deg Q = p, P and Q are real-rooted => p ∑ k=0 P(k) Q(p-k) is real-rooted:

Neste trabalho apresentamos um estudo das reações de spallation a energias interme- diárias em núcleos actinídeos e pré-actinídeos. Para esta finalidade foi utilizado o modelo de Monte Carlo CRISP (Colaboração Rio-São Paulo), que neste estudo foi importante na reprodução da distribuição de massa de produtos residuais e as seções de choque de fissão e espalação. Estes observáveis são importantes para o estudo de Reatores Hibridos ADS considerado como dispositivos promissores para a transmutação de resíduos nucle- ares. Os modelos físicos necessários para uma correta simulação de dados experimentais foram já implementadas no CRISP, como o modelo de evaporação para emissão de par- tículas descrito por Weisskopf de 1937, e para fissão o clássico modelo de Bohr/Wheeler de 1939. Para a obtenção da distribuição dos fragmentos de massa de fissão o CRISP conta também com um modelo baseado na parametrização multimodal de fissão, que si- mula os processos de fissão simétrica e assimétrica predominantes em altas e baixas ener- gias, respectivamente. Os resultados obtidos do CRISP depois da aplicação dos modelos mencionados, foram os rendimentos de massa dos fragmentos residuais, os quais foram analisadas para o cálculo da seção de choque de fissão e espalação mediante uma fórmula implementada no modelo. Com o resultado se fez o gráfico da distribuição de massa para cada uma das reações analisadas. Uma das reações estudadas foi a reação induzida por fótons de Bremsstrahlung com energias máximas de 50 e 3500 MeV em um alvo de 181 Ta, calculando a distribuição de massa de fissão e espalação, mostrando bons resultados de acordo com os dados experimentais. Nas reações induzidas por prótons foram calcula- das as seções de choque de fissão e espalação assim como sua respectiva distribuição de massa dos produtos residuais. Neste caso estudamos duas reações, sendo: a reação p (1 GeV) + 208 Pb, e a reação de p (660 MeV) + 238 U. Para a primeira reação com chumbo os resultados do CRISP foram comparados com dados experimentais, e também com os resultados obtidos do modelo MCNPX-Bertini do trabalho de Baylac-Domengetroy de 2003, que simulou a mesma reação com chumbo. Obtendo-se melhores resultados com o CRISP mas com uma superestimação de dados no final da distribuição calculada. No caso do urânio, foi necessário usar a chamada fissão superassimétrica porque a distribuição de massa experimental é mais complexa e o modelo multimodal clássico não é suficiente para sua correta simulação. Foi também estudado as reações induzidas por dêuterons usando o modelo CRISP, mostrando os resultados da distribuição de massa para 197 Au e 208 Pb com algumas limitações do modelo para este tipo de reações. / In this work we present a study of the spallation reactions by intermediate energies in actinide and pre-actinide nuclei. For this purpose we used the Monte Carlo model CRISP (Rio-São Paulo Collaboration), for our study was important in the reproduction of the mass distribution of waste products and the total fission and spallation cross secti- ons. These observables are important for the study of Accelerator Driven System Reac- tors (ADS) considered as promising devices for the transmutation of nuclear waste. The physical models needed for a correct simulation of experimental data were already imple- mented in CRISP, such as the evaporation model for emission of particles described by Weisskopf in 1937, and the classical Bohr/Wheeler model in 1939, for fission. To obtain the fragment mass distribution for fission, CRISP has a model based on multimodal fis- sion parameter, which simulates the processes called symmetric and asymmetric fission that are predominant at high and low energies respectively. The CRISP results, obtai- ned after the application of the above mentioned models, were the mass yield of residual fragments, which were analyzed to calculate the fission and spallation cross section using a formula that was implemented in the CRISP model. With these result was obtained the mass distribution for each reaction analyzed. One of the reactions studied was a re- action induced by Bremsstrahlung photons with endpoint energies of 50 MeV and 3500 in a target 181 Ta, calculating the fission and spallation mass distribution, showing good results according the experimental data. In the reactions induced by protons were cal- culated fission and spallation cross sections as well as their respective mass distribution of the residual products. In this case we study two reactions, as follows: p (1 GeV) + 208 , and p (660 MeV) + 238 U. For the first reaction with lead, the results of CRISP were compared with experimental data and with results obtained of MCNPX-Bertini model of Baylac-Domengetroy work in 2003, simulated the same reaction with lead. Obtaining better results with CRISP, but with data-overestimated at the end of calculated distribu- tion. For uranium it was necessary to use the called superasymmetric fission, because the experimental mass distribution is more complicated and the classical model is not suffi- cient for a correct simulation. Has been also studied the reactions induced by deuterons using the CRISP model, showing the mass distribution

Ferroelectric materials are very promising for a variety of applications such as high-permittivity capacitors, ferroelectric memories, pyroelctric sensors, piezoelectric and electrostrictive transducers and electro-optic devices, etc. In the area of ferroelectric ceramics, lead-based compounds, which include lead zirconatetitanate (PZT) solid solutions, occupy an important place because of their superior physical properties. However, due to the toxicity of lead, there is an increasing concern over recycling and disposing of the devices made out of these compounds, which has compelled the researchers around the globe to search for lead-free compounds with promising piezo and ferroelectric properties. Ferroelectric materials that belong to Aurivillius family of oxides have become increasingly important from the perspective of industrial applications because of their high Curie-temperatures, high resistivity, superior polarization fatigue resistanceand stable piezoelectric properties at high temperatures. These bismuth layer-structured ferroelectrics (BLSF) comprise an intergrowth of [Bi2O2]2+ layers and [An+1Bn O3n+1]2- pseudo-perovskite units, where ‘n’ represents the number of perovskite-like layers stacked along the c-axis. ‘A’ stands for a mono-, di- or trivalent ions or a combination of them, ‘B’ represents a small ion with high valencysuch as Ti4+, Nb5+, Ta5+or a combination of them.Ferroelectricity in the orthorhombic phase of these compounds was generally attributed to the cationic displacement along the polar a-axis and the tilting of octahedra around the a- and c-axes. Sr2Bi4Ti5O18(SBT) is ann = 5 member of the Aurivillius family and possess promising ferroelectric and piezoelectric properties that could be exploited for a wide range of applications, including ferroelectric random access memories (FeRAM), piezoelectric actuators, transducers and transformers. Reports in the literaturereveal that the ferroelectricand piezoelectric properties of these oxides can be tuned depending on synthesis routes vis-a-vis micro-structural aspects (texture, grain size) and site specific dopant substitutions.In the present study, textured SBT ceramics were fabricated using pre-reacted precursors and their anisotropic dielectric, piezoelectric and ferroelectric properties were demonstrated. Grain size tunability with regard to their physical properties was accomplished in the ceramics, fabricated using fine powders obtained from citrate assisted sol-gel synthesis. The grain size dependent second harmonic generation activity of SBT ceramics was investigated. Enhancement in the piezoelectric and ferroelectric properties of SBT ceramics was achieved by substituting A site ions (Sr2+) with a combination of Na+ and Bi3+. From the perspective of non-linear optical device applications, physical properties associated with the SBT crystallized in a transparent lithium borate glass matrix were studied. The results obtained in the present investigations are organized as follows, Chapter 1 gives a brief exposure to the field of ferroelectrics. The emphasis has been on the ferroelectric oxides belonging to the Aurivillius family. Structural aspects and the underlying phenomena associated with ferroelectricity in these compounds are discussed. A brief introduction to the glasses, thermodynamic aspects of glass formation and fabrication of glass- ceramics are included. Basic principles involved in the non-linear optical activities are highlighted. Chapter 2 describes the various experimental techniques that were employed to synthesize and characterize the materials under investigation. The experimental details pertaining to the measurement of various physical properties are included. Chapter 3 deals with the fabrication of Sr2Bi4Ti5O18 ceramics using the pre-reacted Bi4Ti3O12 and SrTiO3 powders viasolid-state reaction route. These in stoichiometric ratio were uniaxially pressed and sintered at 1130oC for 3 h resulting in textured Sr2Bi4Ti5O18 ceramics. The obtained dense ceramics exhibited crystallographic anisotropy with prominent c-axis oriented grains (Lotgering factor of 0.62) parallel to the uniaxially pressed direction. The resultant anisotropy in the ceramics was attributed to the reactive template-like behavior of Bi4Ti3O12 that was used as a precursor to fabricate Sr2Bi4Ti5O18 ceramics. Dielectric, ferro and piezoelectric properties measured on the ceramics in the direction perpendicular to the uniaxially pressed axis were found to be superior to that measured in the parallel direction. Chapter 4 reports the details pertaining to the synthesis of strontium bismuth titanate (Sr2Bi4Ti5O18) powders comprising crystallites of average sizes in the range of 94–1400 nm via citrate-assisted sol-gel route. X-ray powder diffraction, Transmission Electron Microscopy (TEM) and Raman spectroscopy were employed for the structural studies. A crystallite size-dependent variation in the lattice parameters and the shift in the Raman vibration modes were observed. Second harmonic signal (532 nm) intensity of the Sr2Bi4Ti5O18 powders increased with the increase in the average crystallite size and the maximum intensity obtained in the reflection mode was 1.4 times as high as that of the powdered KH2PO4. Piezo force microscopic analyses carried out on an isolated crystallite of size 74 nm, established its single domain nature with the coercive field as high as 347 kV/cm. There was a systematic increase in the d33 value with an increase in the size of the crystallite and a high piezoelectric coefficient of ~27 pm/V was obtained from an isolated crystallite of size 480 nm. Chapter 5 illustrates the details concerning the fabrication of Sr2Bi4Ti5O18(SBT) ceramics with different grain sizes (93 nm–1.42 μm) using nano-crystalline powders synthesized via citrate assisted sol-gel method. The grain growth in these powder compacts was found to be controlled via the grain boundary curvature mechanism, associated with anactivation energy of 181.9 kJ/mol. Interestingly with a decrease in grain size there was an increase in the structural distortion which resulted in a shift of Curie-temperature (phase transition) towards higher temperatures than that of conventional bulk ceramics. Extended Landau phenomenological theory for the ferroelectric particles was invoked to explain experimentally observed size dependent phase transition temperature and the critical size for SBT is predicted to be 11.3 nm. Grain size dependent dielectric, ferroelectric and piezoelectric properties of the SBT ceramics were studied and the samples comprising average grain size of 645 nm exhibited superior physical properties that include remnant polarization (2Pr) = 16.4 μC cm-2, coercive field (Ec) = 38 kV cm-1, piezoelectric coefficient (d33) = 22 pC N-1 and planar electromechanical coupling coefficient (kp) = 14.8 %. In Chapter 6, the studies pertaining to the fabrication of Sr(2-x)(Na0.5Bi0.5)xBi4Ti5O18 (SNBT) ceramics for various x values (0, 0.1, 0.25, 0.3, 0.4 and 0.5), using fine powders synthesized via sol-gel route are dealt with. X-ray powder diffraction, transmission electron microscopy and Raman spectroscopic studies were carried out to confirm composition dependent structural changes taking place in the SNBT ceramics. Scanning electron microscopic studies carried out on ceramics revealed that dopants played an important role in inhibiting the grain growth. Dielectric constants of the ceramics were found to decrease with an increase in ‘x’. The increase in Curie temperature with increase in ‘x’ is attributed to the decrease in the tolerance factor. Particularly,x = 0.3 composition of the SNBT ceramics exhibited better piezo and ferroelectric properties with a higher Curie-temperature (569 K). The piezoelectric coefficient (d33) and the planar electromechanical coupling coefficient (kp) of SNBT(x = 0.3) were enhanced by 25% and 42% respectively as compared to that of the undoped ceramics. Chapter 7 deals with the glasses in the system (100 –x) {Li2O + 2B2O3} ─x {2SrO + 2Bi2O3 +5TiO2} (where, x = 10, 25 and 35) fabricated via conventional melt-quenching technique. The amorphous and glassy characteristics of the samples were confirmed respectively using X-ray diffraction (XRD) and differential scanning calorimetric (DSC) methods. All the compositions under investigation exhibited two distinct crystallization peaks (exothermic peaks in the DSC traces): the first peak at ~ 545 °C and the second at ~610 °C that were found to be associated with the crystallization of the phases (as confirmed from the XRD studies) Sr2Bi4Ti5O18 (SBT)and Li2B4O7 (LBO) respectively. Non-isothermal crystallization kinetics (using modified Ozawa-type plots) for SBT crystallization in the LBO glass matrix for the compositions x = 10 and 35, indicated three dimensional growth of the crystallites from pre-existing nuclei present in the as-quenched samples and their effective activation energies for crystallization were found to be around 686 ± 85 kJ/mol and 365 ± 53 kJ/mol, respectively. The optical band gap of the as-quenched glasses for the composition x = 35 was 2.52 eV, is less than that of the composition x = 10 (2.91 eV). The Urbach energies for the as-quenched glasses of compositions x = 10, 25 and 35 were found to be 118 ± 2 meV, 119 ± 2 meV and 192 ± 1 meV respectively.The glasses associated with the composition x = 35, on controlled heat-treatment at 515 °C for various durations (1―20 h), yielded glass-ceramics comprising SBT nano-crystals (18―28 nm) embedded in the LBO glass matrix. Compressive strain in the nano-crystallites of SBT, analyzed using Williamson-Hall method was found to decrease with an increase in the crystallite size. The second harmonic generation signal (532 nm) intensity emanating from glass-nanocrystal composites comprising 22.1 nm SBT crystallites was nearly 0.3 times that of a KDP single crystal. Although each chapter is provided with conclusions and a list of references, thesis ends with a separate summary and conclusions.