In case of the PDE's the concept of solving by separation of variables has a well defined meaning. One seeks a solution in a form of a product or sum and tries to build the general solution out of these particular solutions. There are also known systems of second order ODE's describing potential motions and certain rigid bodies that are considered to be separable. However, in those cases, the concept of separation of variables is more elusive; no general definition is given. In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible.

During development of the mammalian nervous system, neural stem cells generate neurons first and glia second, thereby allowing the initial establishment of neuronal circuitry, and subsequent matching of glial numbers and position to that circuitry. Multiple molecular mechanisms act in concert to control neural precursor expansion prior to neurogenesis, and to allow for an exponential generation of neurons while ensuring the maintenance of sufficient precursors to produce later-born neurons, glial cells and adult neural stem cells. Throughout cortical development, these processes are regulated in part by the precursors’ environment as well as intrinsic changes in precursors and their modes of division, which regulate the fate of daughter cells and the balance between self-renewal and differentiation. In the first part of this thesis, the protein tyrosine phosphatase SHP-2 was identified as a novel signaling protein that regulates the neurogenic to gliogenic switch by potentiating neurogenic signals and suppressing gliogenic signals until the appropriate developmental time point for astrogenesis, providing one mechanism whereby precursors integrate conflicting environmental cues. A Noonan Syndrome (NS)-associated activated SHP-2 mutation causes perturbations in neural cell genesis, which may contribute to the mild mental retardation and learning disabilities observed in NS patients. In the second part of this thesis, a novel Rho-regulatory pathway which includes the Rho-GEF Lfc and its negative regulator Tctex-1 were also found to regulate neurogenesis, potentially by directing mitotic spindle orientation during precursor divisions, thereby regulating the symmetric and asymmetric nature of radial precursor divisions.

The biosynthesis of wax components containing secondary functional groups was investigated in the current study. Two fundamentally different pathways were proposed to introduce the secondary functional groups. One pathway involves hydroxylation of elongated substrates. Wax components characterized by two functional groups located on or near the centre of the carbon chain, nonacosane-14,15-diol, -14,16-diol and -13,15-diol as well as corresponding ketols were identified for the first time in Arabidopsis stem wax. The alkanediols and ketols were dominated by the C-14,15 isomers. The absence of alkanediols and ketols in Arabidopsis mah1 mutants that are deficient in secondary alcohol biosynthesis confirmed the biosynthetic relationship between secondary alcohols and alkanediols/ketols (Chapter 3). In pea (Pisum sativum) leaf wax, two novel compound classes were identified as primary/secondary alcohols dominated by octacosane-1,14-diol and secondary/secondary alkanediols hentriacontane-9,16-diol, -8,15-diol and -10,17-diol. Co-localization of the secondary/secondary alkanediols and hentriacontan-15-ol and -16-ol pointed to a biosynthetic relationship (Chapter 4). The diverse structures of compounds identified in the current study suggested that hydroxylases can use substrates other than alkanes. The predominance of isomers within homologues indicated a regiospecificity of the hydroxylases involved in wax biosynthesis. In addition to hydroxylation, secondary functional groups could also be introduced through elongation of carbon chains. Homologous series of 5-hydroxyaldehydes (C₂₄ and C₂₆-C₃₆) and 1,5-alkanediols (C₂₈-C₃₈) were identified in yew (Taxus baccata) needle wax. The relative position of both functional groups suggested that these two compound classes are biosynthetically related and their secondary functional groups are introduced during elongation (Chapter 5). The results of incubation of ¹⁴C-labeled malonyl-CoA and acyl-CoAs with different chain lengths in the presence of California poppy (Eschscholzia californica) microsomes provided the first evidence to support the elongation hypothesis. The results indicated that a carbonyl group rather than a hydroxyl group is introduced during elongation. To provide molecular tools for further investigations of the hypothetical pathway, three full length cDNAs encoding putative KCSs were cloned and one of them, PKCSI, was functionally characterized.

Understanding the origin of protein folds, and the mechanism by which evolution has generated them, is a critically important step on a path towards rational protein design. Modifying existing proteins and designing our own novel folds and functions is a lofty but achievable goal, for which there are many foreseeable rewards. It is believed that modern proteins may have arisen from a primordial set of peptide precursors, which were initially only pseudo-stable or stable only as complexes with RNA, and later were able to self-assemble into multimeric complexes that resembled modern folds. In order to experimentally examine the feasibility of this theory, an attempt was made at reconstructing the evolutionary path of a beta-trefoil. The beta-trefoil is a naturally abundant fold or superfold, possessing pseudo-threefold symmetry, and usually having a sugar-binding function. It has been proposed that such a fold could arise from the triplication of just one small peptide on the order of 40-50 amino acids in length. The evolutionary path of a ricin, a family within the beta-trefoils known to possess a carbohydrate binding function was the chosen template for evolutionary modelling. It was desirable to have a known function associated with this design, such that it would be possible to determine if not only the fold, but also the function, could be reconstructed. A small peptide of 47 amino acids was designed and expressed. This peptide not only trimerized as expected, but possessed the carbohydrate binding function it was predicted to have. In an evolutionary model of the early protein world, the gene for this peptide would undergo duplication and later, triplication, eventually resulting in a completely symmetrical beta-trefoil, which would represent the first modern beta-trefoil fold. Such a completely symmetrical protein was also designed and expressed by triplicating the gene for the aforementioned small peptide. This hypothetical first modern beta-trefoil is: well folded, stable, soluble, and appears to adopt a beta-trefoil fold. Together these results demonstrate that an evolutionary model of early life: that proteins first existed as self-assembling modular peptides, and subsequent to gene duplications or fusions, as what we now recognize as modern folds, is experimentally consistent and not only generates stable structures, but those with function, which of course is a prime requisite of evolution. Moreover the results show that it may be possible to use this modular nature of protein folding to design our own proteins and predict the structure of others.

This thesis looked at three encryption applications that all use the symmetric encryption algorithms AES, Twofish and Serpent but differ in their implementation and how this difference would illustrate itself in performance benchmarks depending on the type of storage device that they were used on. Three mechanical hard drives and one solid state drive were used in the performance benchmarks which measured a variety of different disk operations across the three encryption applications and their algorithms. From the benchmarks performance charts were produced which showed that DiskCryptor had the best performance when using a solid state drive and that TrueCrypt had the best performance when using mechanical hard drives. By choosing DiskCryptor as the encryption application when using a solid state drive a performance increase of 38.9% compared to BestCrypt and 28.4% compared to TrueCrypt was achieve when using the AES algorithm. It was also shown that Twofish was overall the best performing algorithm. The primary conclusion that can be drawn from this thesis is that it is important to choose the right encryption application depending on the type of storage device used in order to get the best performance possible.

Understanding the origin of protein folds, and the mechanism by which evolution has generated them, is a critically important step on a path towards rational protein design. Modifying existing proteins and designing our own novel folds and functions is a lofty but achievable goal, for which there are many foreseeable rewards. It is believed that modern proteins may have arisen from a primordial set of peptide precursors, which were initially only pseudo-stable or stable only as complexes with RNA, and later were able to self-assemble into multimeric complexes that resembled modern folds. In order to experimentally examine the feasibility of this theory, an attempt was made at reconstructing the evolutionary path of a beta-trefoil. The beta-trefoil is a naturally abundant fold or superfold, possessing pseudo-threefold symmetry, and usually having a sugar-binding function. It has been proposed that such a fold could arise from the triplication of just one small peptide on the order of 40-50 amino acids in length. The evolutionary path of a ricin, a family within the beta-trefoils known to possess a carbohydrate binding function was the chosen template for evolutionary modelling. It was desirable to have a known function associated with this design, such that it would be possible to determine if not only the fold, but also the function, could be reconstructed. A small peptide of 47 amino acids was designed and expressed. This peptide not only trimerized as expected, but possessed the carbohydrate binding function it was predicted to have. In an evolutionary model of the early protein world, the gene for this peptide would undergo duplication and later, triplication, eventually resulting in a completely symmetrical beta-trefoil, which would represent the first modern beta-trefoil fold. Such a completely symmetrical protein was also designed and expressed by triplicating the gene for the aforementioned small peptide. This hypothetical first modern beta-trefoil is: well folded, stable, soluble, and appears to adopt a beta-trefoil fold. Together these results demonstrate that an evolutionary model of early life: that proteins first existed as self-assembling modular peptides, and subsequent to gene duplications or fusions, as what we now recognize as modern folds, is experimentally consistent and not only generates stable structures, but those with function, which of course is a prime requisite of evolution. Moreover the results show that it may be possible to use this modular nature of protein folding to design our own proteins and predict the structure of others.

The character theory of the symmetric group is a powerful method of studying enu- merative questions about factorizations of permutations, which arise in areas including topology, geometry, and mathematical physics. This method relies on having an encoding of the enumerative problem in the centre Z(n) of the algebra C[S_n] spanned by the symmetric group S_n. This thesis develops methods to deal with permutation factorization problems which cannot be encoded in Z(n). The (p,q,n)-dipole problem, which arises in the study of connections between string theory and Yang-Mills theory, is the chief problem motivating this research. This thesis introduces a refinement of the (p,q,n)-dipole problem, namely, the (a,b,c,d)- dipole problem. A Join-Cut analysis of the (a,b,c,d)-dipole problem leads to two partial differential equations which determine the generating series for the problem. The first equation determines the series for (a,b,0,0)-dipoles, which is the initial condition for the second equation, which gives the series for (a,b,c,d)-dipoles. An analysis of these equa- tions leads to a process, recursive in genus, for solving the (a,b,c,d)-dipole problem for a surface of genus g. These solutions are expressed in terms of a natural family of functions which are well-understood as sums indexed by compositions of a binary string. The combinatorial analysis of the (a,b,0,0)-dipole problem reveals an unexpected fact about a special case of the (p,q,n)-dipole problem. When q=n−1, the problem may be encoded in the centralizer Z_1(n) of C[S_n] with respect to the subgroup S_{n−1}. The algebra Z_1(n) has many combinatorially important similarities to Z(n) which may be used to find an explicit expression for the genus polynomials for the (p,n−1,n)-dipole problem for all values of p and n, giving a solution to this case for all orientable surfaces. Moreover, the algebraic techniques developed to solve this problem provide an alge- braic approach to solving a class of non-central problems which includes problems such as the non-transitive star factorization problem and the problem of enumerating Z_1- decompositions of a full cycle, and raise intriguing questions about the combinatorial significance of centralizers with respect to subgroups other than S_{n−1}.

Sampling theory studies the equivalence between continuous and discrete representations of information. This equivalence is ubiquitously used in communication engineering and signal processing. For example, it allows engineers to store continuous signals as discrete data on digital media. The classical sampling theorem, also known as the theorem of Whittaker-Shannon-Kotel'nikov, enables one to perfectly and stably reconstruct continuous signals with a constant bandwidth from their discrete samples at a constant Nyquist rate. The Nyquist rate depends on the bandwidth of the signals, namely, the frequency upper bound. Intuitively, a signal's `information density' and `effective bandwidth' should vary in time. Adjusting the sampling rate accordingly should improve the sampling efficiency and information storage. While this old idea has been pursued in numerous publications, fundamental problems have remained: How can a reliable concept of time-varying bandwidth been defined? How can samples taken at a time-varying Nyquist rate lead to perfect and stable reconstruction of the continuous signals? This thesis develops a new non-Fourier generalized sampling theory which takes samples only as often as necessary at a time-varying Nyquist rate and maintains the ability to perfectly reconstruct the signals. The resulting Nyquist rate is the critical sampling rate below which there is insufficient information to reconstruct the signal and above which there is redundancy in the stored samples. It is also optimal for the stability of reconstruction. To this end, following work by A. Kempf, the sampling points at a Nyquist rate are identified as the eigenvalues of self-adjoint extensions of a simple symmetric operator with deficiency indices (1,1). The thesis then develops and in a sense completes this theory. In particular, the thesis introduces and studies filtering, and yields key results on the stability and optimality of this new method. While these new results should greatly help in making time-variable sampling methods applicable in practice, the thesis also presents a range of new purely mathematical results. For example, the thesis presents new results that show how to explicitly calculate the eigenvalues of the complete set of self-adjoint extensions of such a symmetric operator in the Hilbert space. This result is of interest in the field of functional analysis where it advances von Neumann's theory of self-adjoint extensions.

The field of Computational Fluid Dynamics (CFD) is in a continual state of advancement due to new numerical techniques, optimization of existing codes, and the increase in memory and processing speeds of computers. In this thesis, the solution technique for a pre-existing Navier-Stokes flow solver is adapted from an explicit Runge Kutta method to a Lower-Upper Symmetric Gauss-Seidel (LU-SGS) implicit time integration method. Explicit time integration methods were originally used in CFD codes because these methods require less memory. Information needed to advance the flow in time is localized to each grid point. These explicit methods are, however, restricted by time step sizes due to stability criteria. In contrast, implicit methods are unaffected by a large time step sizes but are restricted by memory requirements due to the complexities of unstructured grids. The implementation of LU-SGS performs grid re-ordering for unstructured meshes because of the coupling of grid points in the integration method's solution. The explicit and implicit flow solvers were tested for inviscid flows in incompressible, compressible, and transoinc flow regimes. The results found by comparing the implicit and explicit algorithms revealed a significant speed up in convergence to steady state by the LU-SGS method in terms of iteration number and CPU time per iteration.

The binary response experiments are often used in many areas. In many investigations, different kinds of optimal designs are discussed under an assumed model. There are also some discussions on optimal designs for discriminating models. The main goal in this work is to find an optimal design with two support points which minimizes the maximal probability differences between possible models from two types of symmetric location and scale families. It is called the minimum bias two-points design, or the $mB_2$ design in short here. D- and A-efficiencies of the $mB_2$ design obtained here are evaluated under an assumed model. Furthermore, when the assumed model is incorrect, the biases and the mean square errors in evaluating the true probabilities are computed and compared with that by using the D- and A-optimal designs for the incorrectly assumed model.