The ping-pong lemma was introduced by Klein in the late 1800s to show that certain subgroups of isometries of hyperbolic 3-space are free and remains one of very few tools that certify when a pair of group elements generate a free subgroup or semigroup. Quantitatively applying the ping-pong lemma to more general group actions on metric spaces requires a blend of understanding the large-scale global geometry of the underlying space with local combinatorial and dynamical behavior of the action. In the 1980s, Gromov publish a sequence of seminal works introducing several metric notions of non-positive curvature in group theory where he asked which finitely generated groups have uniform exponential growth. We give an overview of various developments of non-positive curvature in group theory and past results related to building free semigroups in the setting of non-positive curvature. We highlight joint work with Radhika Gupta and Kasia Jankiewicz and with Carolyn Abbott and Davide Spriano that extends these tools and techniques to show several groups with that act on cube complexes and many hierarchically hyperbolic groups have uniform exponential growth. / Mathematics

We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L2 inner product. We examine this L2 projection and an H1 projection induced by the H1 semi-inner product. The H1 projection has an advantage over the L2 projection in that it retains tangential Dirichlet boundary conditions for the flow. Both the H1 and L2 projections are solutions to saddle point problems that are efficiently solved by a preconditioned Uzawa algorithm. / Ph. D.

An open queueing network model (QNM) is proposed for wormhole-routed hypercubes with finite buffers and deterministic routing subject to a compound Poisson arrival process (CPP) with geometrically distributed batches or, equivalently, a generalised exponential (GE) interarrival time distribution. The GE/G/1/K queue and appropriate GE-type flow formulae are adopted, as cost-effective building blocks, in a queue-by-queue decomposition of the entire network. Consequently, analytic expressions for the channel holding time, buffering delay, contention blocking and mean message latency are determined. The validity of the analytic approximations is demonstrated against results obtained through simulation experiments. Moreover, it is shown that the wormholerouted hypercubes suffer progressive performance degradation with increasing traffic variability (burstiness).

This research is motivated by identified gaps in contemporary planning practices and production processes within firms. Relying solely on experiential knowledge has proven limiting, necessitating a more systematic approach. Previous instances of data anomalies, particularly ongoing challenges in achieving satisfactory delivery reliability, have underlined the need for deeper insights into underlying patterns. The objectives of this study are: • To identify and analyze specific obstacles and challenges affecting load balance and delivery security in Borl.nge's production system. • To explore various methods or strategies aimed at enhancing the process of generating reliable capacity forecasting methods. Both primary and secondary research methods were employed. Primary methods included interviews and the development of a forecast model, while secondary studies encompassed the latest research in the field. The thesis revealed five primary factors hindering capacity attainment: 1. WIP(work in progress)/ slabs material shortages disrupt production flow and escalate costs due to the need for external sourcing of slabs. 2. Transport issues, including incorrect internal deliveries and the weather conditions, pose challenges. 3. Personnel shortages hinder the efficient utilization of production capacity. 4. Machine breakdowns result in production interruptions, leading to capacity loss and inefficiency. 5. Inventory problems, such as insufficient capacity and poor management, impede smooth production operations. Additionally, the second objective was addressed by implementing exponential smoothing for capacity planning forecasts. By updating forecasts every 13 weeks, this study improves the production forecast.

To analyze a communication system without the aid of simulation, the channel noise for the simulation must be assumed to be normal. The assumption is often valid, but the normal distribution may not be able to model the channel noise adequately in some environments. This thesis will explore the generalized exponential distribution for better noise modeling and robustness testing in communication system. When using the generalized exponential distribution for the channel noise, the analysis will become analytically intractable, and simulation becomes mandatory. To generate the noise with the distribution, the rejection method can be used. However, since the distribution can take on different shapes, finding the appropriate Upper Bounding Function (UBF) for the method is very difficult. Thus, two modified versions of the rejection method will be examined. They are the Transformed Density Rejection (TDR) and Ratio-of-Uniform (RoU) method; their quality, efficient, trade-offs, etc will be discussed. Choosing TDR, a simulation of a BPSK communication system will be performed. With the simulation, it can further ascertain that the random variates generated by TDR can be used to model the channel noise and to test the robustness of a communication system. / Master of Science

The development of an economical method for determining accurately group constants of hexagonal and rectangular cells is considered in this dissertation. The mathematical model constructed for this purpose has the capability to characterize the group constants for the entire range of the neutron spectrum. Furthermore, this model is also rigorous enough to predict the group constants with the required accuracy for a specific range of interest in the energy spectrum and for a variety of energy group configurations. The model is implemented separately for the fast and thermal energy regions. These regions are subsequently coupled via the source term. The construction of the model for the fast energy range has been pursued by implementing the transport equation specialized in a two-region cell. The regions are coupled via the escape probability functions. The model for the thermal energy range has been attained by implementing the appropriate Nelkin and Honeck amplitude functions within the kernels of the transport equation. The Nelkin amplitude function is utilized for treating light water moderated systems, and the Honeck amplitude function relates to heavy water moderated systems. The group constants calculated with the economical model have been benchmarked with those computed by the VIM Monte Carlo code. The values obtained for the group constants agree within 1-2% with those computed by VIM for the fast energy region. The agreements for the thermal energy region are within 2-3%. The CPU running time of the implemented model is about 3 1/2 minutes for a four group configuration. On the other hand a typical VIM run comprising 25,000 neutron histories and a four-group structure expends about 30 minuts of CPU time for light water moderated systems. Moreover, similar VIM runs utilizing heavy water as moderator require over one hour of CPU time. Therefore, the implemented model makes utilization of computer resources with a cost advantage of a factor of 10 or better as compared to VIM. This economical benefit of the implemented model enables it to be coupled directly with fuel depletion codes, whereby the group constants and the fuel isotopics are updated at relatively short time intervals. On the other hand, the coupling of VIM with burnup codes would result in prohibitively expensive CPU costs. / Ph. D.

Exponential smoothing algorithms are very attractive for the practical world such as in industry. When considering bivariate exponential smoothing methods, in addition to the properties of univariate methods, additional properties give insight to relationships between the two components of a process, and also to the overall structure of the model. It is important to study these properties, but even with the merits the bivariate exponential smoothing algorithms have, exponential smoothing algorithms are nonstatistical/nonstochastic and to study the properties within exponential smoothing may be worthless. As an alternative approach, the (bivariate) ARIMA and the structural models which are classes of statistical models, are shown to generalize the exponential smoothing algorithms. We study these properties within these classes as they will have implications on exponential smoothing algorithms. Forecast properties are studied using the state space model and the Kalman filter. Comparison of ARIMA and structural model completes the study. / Mathematical Sciences / M. Sc. (Statistics)

The work done in this thesis is to examine various methods for curve fitting. Linear least squares and non-linear least squares will be described and compared, and the Newton method, Gauss--Newton method and Levenberg--Marquardt method will be applied to example problems. / Syftet med denna uppsats är att beskriva och använda olika metoder för kurvanpassning, det vill säga att passa matematiska funktioner till data. De metoder som undersöks är Newtons metod, Gauss--Newton metoden och Levenberg--Marquardt metoden. Även skillnaden mellan linjär minsta kvadrat anpassning och olinjär minsta kvadrat anpassning. Till sist tillämpas Newton, Gauss Newton och Levenberg--Marquardt metoderna på olika exempel.

Durante o desenvolvimento do sistema nervoso, as oscilações da concentração de cálcio intracelular livre resultam na proliferação celular, migração e diferenciação neuronal. Nesta tese foram investigadas a participação dos receptores ionotrópicos purinérgicos dos tipos P2X2 e P2X4 seletivos ao influxo de cálcio durante a diferenciação neuronal in vitro das células de carcinoma embrionário murino P19. Identificamos o padrão diferencial de expressão de receptores purinérgicos nas células indiferenciadas e neurônios P19. O receptor P2X4 é expresso durante toda a diferenciação neuronal e o receptor P2X2 é detectado na fase tardia da diferenciação em neurônios. Através de ensaios farmacológicos, foi possível identificar a participação dos receptores metabotropicos P2Y e do receptor P2X4 na formação dos corpos embriônicos, na proliferação celular e ou na determinação do fenótipo de progenitor neural. Durante a maturação neuronal os receptores P2X2 e P2Y1 participam da determinação do fenótipo neuronal glutamatérgico NMDA e os receptores P2X2 e P2Y2 no fenótipo neuronal colinérgico. A ausência de inibidores específicos e seletivos aos receptores purinérgicos levou-nos a empregar a técnica SELEX (Systematic Evolution of Ligands by EXponential enrichment) a fim de identificar inibidores seletivos aos receptores P2X2 e P2X4. A técnica envolve a utilização da biblioteca combinatória randômica de RNA 2\'- F pirimidina modificadas resistentes a nucleases. Após 9 ciclos de seleção in vitro de SELEX (ciclo 9-P2X4), as sequências selecionadas mostraram-se seletivas a ligação somente ao receptor P2X4 e não aos receptores P2X2 ou P2X7 através de ensaios de ligação radioligante-receptor. Por patch clamping na configuração whole cell recording identificou-se que além de seletividade ao receptor, que a aplicação do RNA ciclo 9- P2X4 promoveu inibição da corrente ativada pelo ATP somente nos receptores P2X4 e não em P2X2 em celulas 1321N1 astrocitoma transfectadas. A incubação do RNA ciclo 9-P2X4 na concentração de 200 nM com as células no estágio indiferenciado inibiu a formação dos corpos embriônicos. Já utilização de 25 nM, resultou em mudanças morfológicas nas células diferenciadas. Estes dados corroboram com os dados farmacológicos que identificaram a participação do receptor P2X4 na diferenciação precoce. Após 11 ciclos P2X2 de seleção, identificou-se sequências com especificidade de ligação aos receptores P2X2. Aptâmeros, moleculas de RNA com sequência identificada e com alta afinidade ao alvo da seleção, foram isolados de ambas as bibliotecas, ciclo 9 P2X4 e ciclo 11 P2X2. A co-aplicação destes aptâmeros e ATP em ensaios de whole-cell recording resultou na inibição de 30 a 80% da corrente ativada pelo ATP nos receptores P2X2 ou P2X4. Estes testes em células PC12 de rato, que expressa os receptores endógenos, resultou em inibição da corrente ativada pelo ATP de modo semelhante. Além de termos desenvolvido aptâmeros como ferramentas para elucidar as funções dos receptores P2X2 e P2X4 durante o desenvolvimento, diferenciação, em processos fisiológicos e patológicos, estas moléculas resistentes a nucleases são as primeiras identificadas capazes de reconhecer, discernir e inibir dois subtipos de receptores purinérgicos sendo promissores para utilização terapêutica. / During the development of the nervous system, oscillations of intracellular calcium concentrations activate programs of gene expression resulting in proliferation, migration and neuronal differentiation of embryonic cells. In this thesis, the participation of ionotropic P2X2 and P2X4 receptor subtypes, whose receptor channels are highly permeable for calcium influx in the cells, was studied during the process of neuronal differentiation. We have identified differential gene expression of purinergic receptors in undifferentiated and neuronal-differentiated P19 cells. P2X4 receptor expression was present along neuronal differentiation of P19 cells, whereas P2X2 receptor expression was only detected when P19 cells became neurons. Based on purinergic receptor pharmacology we have determined the participation of P2X4 receptors in addition to metabotropic P2Y2 receptors in the formation of embryonic bodies as prerequisites for phenotype determination of P19 neural progenitor cells. Final neuronal maturation of P19 cells in the presence or absence of agonists or antagonists of purinergic receptors implicated the involvement of P2X2, P2Y1, and P2Y2 in the determination of the final neuronal phenotype, such as expression of NMDA-glutamate and cholinergic receptors. In order to further evaluate the functions of these P2X receptors and due to the absence of specific inhibitors for these receptor subtypes, we have used the SELEX technique (Systematic Evolution of Ligands by EXponential enrichment) to select for specific inhibitors for P2X2 and P2X4 receptors. The 2\' -F-pyrimidine modified, nuclease- resistant combinatorial SELEX RNA pool enriched with inhibitors of P2X4 receptors following nine cycles of in vitro selection (cycle 9-P2X4) specifically interacted with P2X4 receptors and not with P2X2 or P2X7 receptors as verified in radioligand-receptor binding studies. Moreover, whole-cell recording measurements using astrocytoma cells expressing recombinant rat P2X2 or P2X4 receptors showed inhibition of P2X4 but not of P2X2 receptors by the selected RNA molecules. RNA molecules selected in vitro in 11 reiterative SELEX cycles using the P2X2 receptor as target specifically bound to membrane extracts containing recombinant P2X2 receptors. From both selected RNA libraries (against P2X4 and P2X2 receptors) aptamers, as RNA molecules with identified sequences and high-affinity binding, were identified by cloning and DNA sequencing. The presence of these aptamers in whole-cell recording experiments resulted in 30-80% inhibition of ATP-induced receptor activity and did not provoke any inhibitory effects on P2X receptors which had not been used as selection target. The activity of the aptamers selected using recombinant receptors as targets in inhibiting wild-type P2X4 or P2X2 receptors was verified in whole-cell recording experiments with PC12 cells which endogenously express both receptor subtypes. In addition of having developed aptamers as tools to elucidate P2X2 and P2X4 receptor functions during neuronal differentiation, these nuclease-resistant aptamers are suitable for in vivo use and may turn into therapeutics in the inhibition of purinergic receptor participation in pathophysiological conditions.

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