Neste trabalho, apresentamos os modelos de regressão spline de nós-livres como uma alternativa aos modelos não lineares utilizados em curvas de crescimento multifásico. Estudaremos o algoritmo de busca cega a través da seção dourada para escolher a melhor quantidade de nós e suas respectivas localidades. O pacote freeknotspline do software livre R foi utilizado para ajustar os modelos propostos. O critério de informação de Akaike foi usado para escolher o melhor modelo para diferentes graus do polinômio associados ao spline. Estudos de simulação foram realizados para entender melhor a posição dos nós, localidade e grau do polinômio relacionado aos modelos de regressão spline de nós-livres e como isto pode afetar a qualidade de ajuste do modelo. Com base no nosso estudo de simulação, propomos uma forma empírica de determinar o numero de nós, deixando que o algoritmo de busca escolha a posição dos nós. A metodologia é aplicada aos dados de crescimento multifásico de vacas da raça Hereford. / In this work, we present the free-knot spline regression models as an alternative to the nonlinear models used in multiphase growth curves. We will study the blind search algorithm through the gold section to choose the best number of knot and their respective locations. The package freeknotspline of the free software R was used to fit the proposed models. The Akaike information criterion was used to choose the best model for different degrees of the polynomial associated with the spline. Simulation studies were performed to better understand the position of the knot, location and degree of the polynomial related to spline regression models of free-knots, and how this may affect the goodness of fit of the model. Based on our simulation study, we propose an empirical way of determining the number of knots, letting the search algorithm choose the position of the knots. The methodology is applied to the multiphase growth data of Hereford breed females.

This thesis covers improvements on the upperbounds for ropelength of a specific class of algebraic knots.

Thesis (M. A. Agriculture (Plant Protection)) -- University of Limpopo, 2019 / Available potato (Solanum tuberosum L.) cultivars do not have any genotype that is resistant to the root-knot (Meloidogyne species) nematodes. Due to the susceptibility of potato cultivars to Meloidogyne species, alternative management strategies had to be researched and developed after the withdrawal of methyl bromide from the agro-chemical markets, amongst which were the cucurbitacin-containing phytonematicides. However, of the available application methods of phytonematicides, the ground leaching technology (GLT) and botinemagation technology were not suitable for use in most high-rainfall potato-producing regions, where production is under rain-fed conditions. The objective of the study, therefore, was to determine whether pre- and post-emergent application of Nemafric-BG phytonematicide would have effects on growth of potato and suppression of M. javanica population densities. Parallel pot trials of pre- and post-emergent application of Nemafric-BL phytonematicide were conducted under greenhouse conditions in autumn (February-April: Experiment 1) 2017 and validated (Experiment 2) in 2018. Each plant was inoculated with 3000 M. javanica eggs and second-stage juveniles (J2). Five treatments, namely, 0, 2, 4, 8 and 16 g concentration of Nemafric-BG phytonematicide, arranged in randomised complete block design, were either applied mixed with seed tubers for pre-emergent or spread on the soil surface after emergence for post-emergent trials. In all cases, plant growth variables were assessed using the Curve-fitting Allelochemical Response Data (CARD) model, whereas nutrient elements (Fe, K, Na and Zn) and nematode variables were assessed using analysis of variance, with data subjected to lines of the best fit. In pre-emergent application trial, plant height (R2 = 0.98) and fresh root mass (R2 = 0.99) exhibited quadratic relations, characterised by density dependent growth patterns with increasing concentrations of Nemafric-BG xv phytonematicide in Experiment 1, similar trends were also observed on plant height (R2 = 0.99) and root mass (R2 = 0.99) in Experiment 2. In contrast, in post-emergent application trial, plant height (R2 = 0.97), fresh root mass (R2 = 0.99) and dry shoot (R2 = 0.98) exhibited quadratic relations in Experiment 1, which ascribed to DDG patterns, similar trends were also observed in Experiment 2 on plant height (R2 = 0.99), fresh root mass (R2 = 0.96) and dry shoot (R2 = 0.99) of potato cv. ꞌMondial G3ꞌ. In pre-emergent application trials, Mean Concentration Stimulation Point (MCSP) = 24.18 and 7.82 g, respectively, in Experiment 1 and Experiment 2, with ∑k being equivalent to 20 and 6 units for potato to the product, respectively, in Experiment 1 and Experiment 2. In contrast, post-emergent application trials, MCSP = 9.87 and 12.10 g, respectively, in Experiment 1 and Experiment 2, whereas the ∑k value for potato to the product was 11 and 6 units, respectively in Experiment 1 and Experiment 2. Increasing concentrations of the phytonematicide significantly (P ≤ 0.05) affected the selected nutrient elements. In pre emergent application trials, K (R2  =  0.96) Na (R2  =  0.90) and Zn (R2  =  0.83) each with increasing Nemafric-BG phytonematicide concentrations exhibited positive quadratic fashion, while Fe (R2  =  0.87) exhibited negative quadratic relations in Experiment 1. In Experiment 2, K (R2  = 0.99), Na (R2  = 0.90) and Zn (R2 =  0.97) contents each in leaf tissues against the increasing concentrations of the phytonematicide exhibited negative quadratic relations, while Fe (R2  = 0.88) exhibited positive quadratic relations. In post emergent trials, Fe (R2 = 0.91, Na (R2 = 0.90) and Zn (R2 = 0.99) contents in leaf tissues against increasing Nemafric-BG phytonematicide concentration exhibited negative quadratic relations, whereas K (R2 = 0.86) exhibited positive quadratic relation in Experiment 1. In Experiment 2, Fe (R2 = 0.93), K (R2 = 0.92), Na ( R2  = 0.79) and Zn (R2 xvi = 0.89) contents in leaf tissues of potato exhibited positive quadratic, respectively. In pre emergent trial for Experiment 1, eggs in roots (R2 = 0.78), J2 in roots (R2 = 0.85), J2 in soil (R2 = 0.97) and Pf (R2 = 0.78) of M. javanica against increasing pre-emergent application concentrations of Nemafric-BG phytonematicide exhibited negative quadratic relations, characterised by DDG patterns. Similar trends were observed on eggs in roots (R2 = 0.82), J2 in roots (R2 = 0.99), J2 in soil (R2 = 0.84) and Pf (R2 = 0.85) in Experiment 2. In contrast, in post-emergent application trial, eggs in roots (R2 = 0.87), J2 in roots (R2 = 0.99), J2 in soil (R2 = 0.91) and Pf (R2 = 0.99) of M. javanica against increasing post emergent application concentrations of Nemafric-BG phytonematicide also exhibited negative quadratic relations in Experiment 1, which ascribed to DDG patterns. Similar trends were also observed on eggs in roots (R2 = 0.72), J2 in roots (R2 = 0.68), J2 in soil (R2 = 0.85) and Pf (R2 = 0.83) in Experiment 2. Results from the study demonstrated that Nemafric-BG phytonematicide stimulated plant growth at lower concentration and the product does not have any detrimental effects in accumulation of nutrient elements in leaf tissues. Therefore, it is concluded, that the product could be applied at the recommended rates of 7.82 and 9.87 g/plant in pre and post-emergent application, respectively, for the management of root-knot nematodes, provided the active ingredient does not accumulate in potato tubers or have any detrimental effects in accumulation of nutrient elements in tubers and temper with nutritional value of potatoes.

<p>  </p> <p>Knot tying boards are low fidelity surgical simulators used to practice tying suture, but devices on the market currently provide no feedback and no way of changing out bands. A simple-to-use knot tying board with interchangeable bands capable of measuring force was designed. This board is comparable in cost to products currently available on the market. The knot tying board was then prototyped and tested. Four MD students completed trials of one-handed and two-handed knot tying with three throws per trail. In testing, the knot tying board was capable of measuring force data, such as peak force during knot tying and the final force the knot exerts on the bands. The device used in conjunction with experienced surgical skills coaches could prove a powerful tool for providing feedback to trainees, and a similar approach could be used with other low fidelity surgical simulators to improve feedback.</p>

Thiostrepton (TS: TS; C72H85N19O18S5) is a thiazoline antibiotic that is effective against Gram-positive bacteria and the malarial parasite, Plasmodium falciparum. Tight binding of TS to the bacterial L11-23S ribosomal RNA (rRNA) complex of the large 50S ribosomal unit inhibits protein biosynthesis. The TS producing organism, Streptomyces azureus, biosynthesizes thiostrepton-resistance methyltransferase (TSR), an enzyme that uses S-adenosyl-L-methionine (AdoMet) as a methyl donor, to modify the TS target site. Methylation of A1067 (Escherichia coli ribosome numbering) by TSR circumvents TS binding. The S. azureus tsr gene was overexpressed in E. coli and the protein purified for biochemical characterization. Although the recombinant protein was produced in a soluble form, its tendency to aggregate made handling a challenge during the initial stages of establishing a purification protocol. Different purification conditions were screened to generate an isolation protocol that yields milligram quantities of protein with little aggregation and sufficient purity for crystallographic studies. Enzymological characterization of TSR was carried out using an assay to monitor AdoMet-dependent ([methyl-3H]-AdoMet) methylation of the rRNA substrate by liquid scintillation counting. During the optimization of assay, it was found that, although this method is frequently employed, it is very time and labour intensive. A scintillation proximity assay was investigated to evaluate whether it could be a method for collecting kinetic data, and was found that further optimization is required. Comparative sequence analysis of TSR has shown it to be a member of the novel Class IV SpoUT family of AdoMet-dependent MTases. Members of this class possess a non-canonical AdoMet binding site containing a deep trefoil knot. Selected SpoUT family proteins were used as templates to develop a TSR homology model for monomeric and dimeric forms. Validation of the homology models was performed with structural validation servers and the model was then used as the basis of ongoing mutagenesis experiments. The X-ray crystal structure of TSR bound with AdoMet (2.45 Å) was elucidated by our collaborators, Drs. Mark Dunstan and Graeme Conn (University of Manchester). This structure confirms TSR MTase’s membership in the SpoUT MTase family with a deep trefoil knot in the catalytic domain. The AdoMet bound in the crystal structure is in an extended conformation not previously observed in SpoUT MTases. RNA docking simulations revealed some features that may be relevant to binding and recognition of TSR to the L11 binding domain of the RNA substrate. Two structure-activity studies were conducted to investigate the TS-rRNA interaction and TS solubility. Computational analyses of TS conformations, molecular orbitals and dynamics provided insight into the possible modes of TS binding to rRNA. Single-site modification of TS was attempted, targeting the dehydroalanine and dehydrobutyrine residues of the antibiotic. These moieties were modified using the polar thiol, 2-mercaptoethanesulfonic acid (2-MESNA). Similar modifications had been previously used to improve solubility and bioavailability of antibiotics. The resulting analogue was structurally characterized (NMR and mass spectrometry) and showed antimicrobial activity against Bacillus subtilis and Staphylococcus aureus.

Thiostrepton (TS: TS; C72H85N19O18S5) is a thiazoline antibiotic that is effective against Gram-positive bacteria and the malarial parasite, Plasmodium falciparum. Tight binding of TS to the bacterial L11-23S ribosomal RNA (rRNA) complex of the large 50S ribosomal unit inhibits protein biosynthesis. The TS producing organism, Streptomyces azureus, biosynthesizes thiostrepton-resistance methyltransferase (TSR), an enzyme that uses S-adenosyl-L-methionine (AdoMet) as a methyl donor, to modify the TS target site. Methylation of A1067 (Escherichia coli ribosome numbering) by TSR circumvents TS binding. The S. azureus tsr gene was overexpressed in E. coli and the protein purified for biochemical characterization. Although the recombinant protein was produced in a soluble form, its tendency to aggregate made handling a challenge during the initial stages of establishing a purification protocol. Different purification conditions were screened to generate an isolation protocol that yields milligram quantities of protein with little aggregation and sufficient purity for crystallographic studies. Enzymological characterization of TSR was carried out using an assay to monitor AdoMet-dependent ([methyl-3H]-AdoMet) methylation of the rRNA substrate by liquid scintillation counting. During the optimization of assay, it was found that, although this method is frequently employed, it is very time and labour intensive. A scintillation proximity assay was investigated to evaluate whether it could be a method for collecting kinetic data, and was found that further optimization is required. Comparative sequence analysis of TSR has shown it to be a member of the novel Class IV SpoUT family of AdoMet-dependent MTases. Members of this class possess a non-canonical AdoMet binding site containing a deep trefoil knot. Selected SpoUT family proteins were used as templates to develop a TSR homology model for monomeric and dimeric forms. Validation of the homology models was performed with structural validation servers and the model was then used as the basis of ongoing mutagenesis experiments. The X-ray crystal structure of TSR bound with AdoMet (2.45 Å) was elucidated by our collaborators, Drs. Mark Dunstan and Graeme Conn (University of Manchester). This structure confirms TSR MTase’s membership in the SpoUT MTase family with a deep trefoil knot in the catalytic domain. The AdoMet bound in the crystal structure is in an extended conformation not previously observed in SpoUT MTases. RNA docking simulations revealed some features that may be relevant to binding and recognition of TSR to the L11 binding domain of the RNA substrate. Two structure-activity studies were conducted to investigate the TS-rRNA interaction and TS solubility. Computational analyses of TS conformations, molecular orbitals and dynamics provided insight into the possible modes of TS binding to rRNA. Single-site modification of TS was attempted, targeting the dehydroalanine and dehydrobutyrine residues of the antibiotic. These moieties were modified using the polar thiol, 2-mercaptoethanesulfonic acid (2-MESNA). Similar modifications had been previously used to improve solubility and bioavailability of antibiotics. The resulting analogue was structurally characterized (NMR and mass spectrometry) and showed antimicrobial activity against Bacillus subtilis and Staphylococcus aureus.

This thesis examines graph polynomials and particularly their complexity. We give short proofs of two results from Gessel and Sagan (1996) which present new evaluations of the Tutte polynomial concerning orientations. A theorem of Massey et al (1997) gives an expression concerning the average size of a forest in a graph. We generalise this result to any simplicial complex. We answer a question posed by Kleinschmidt and Onn (1995) by showing that the language of partitionable simplicial complexes is in NP. We prove the following result concerning the complexity of the Tutte polynomial: Theorem 1. For any fixed k, there exists a polynomial time algorithm A, which will input any graph G, with tree-width at most k, and rational numbers x and y, and evaluate the Tutte polynomial, T(G;x,y). The rank generating function S of a graphic 2-polymatroid was introduced by Oxley and Whittle (1993). It has many similarities to the Tutte polynomial and we prove the following results. Theorem 2. Evaluating S at a fixed point (u,v) is #P-hard unless uv=1 when there is a polynomial time algorithm. Theorem 3. For any fixed k, there exists a polynomial time algorithm A, which will input any graph G, with tree-width at most k, and rational numbers u and v, and evaluate S(G;u,v). We consider a class of graphs $S$, which are those graphs which are obtainable from a graph with no edges using the unsigned version of Reidemeister moves. We examine the relationship between this class and other similarly defined classes such as the delta-wye graphs. There remain many open questions such as whether S contains every graph. However we have an invariant of the moves, based on the Tutte polynomial, which allows us to determine from which graph with no edges, if any, a particular graph can be obtained. Finally we consider a new polynomial on weighted graphs which is motivated by the study of weight systems on chord diagrams. We give three states model and a recipe theorem. An unweighted version of this polynomial is shown to contain as specialisations, a wide range of graph invariants, such as the Tutte polynomial, the polymatroid polynomial of Oxley and Whittle (1993) and the symmetric function generalisation of the chromatic polynomial introduced by Stanley (1995). We close with a discussion of complexity issues proving hardness results for very restricted classes of graphs.

Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many ways of mechanising are: 1 Generating results using automated theorem provers. 2 Interactive theorem proving in a proof assistant which involves a combination of user intervention and automation. In the first part of this thesis, we reformulate the question of equivalence of two Links in first order logic using braid groups. This is achieved by developing a set of axioms whose canonical model is the braid group on infinite strands B∞. This renders the problem of distinguishing knots and links, amenable to implementation in first order logic based automated theorem provers. We further state and prove results pertaining to models of braid axioms. The second part of the thesis deals with formalising knot Theory in Higher Order Logic using the interactive proof assistant -Isabelle. We formulate equivalence of links in higher order logic. We obtain a construction of Kauffman bracket in the interactive proof assistant called Isabelle proof assistant. We further obtain a machine checked proof of invariance of Kauffman bracket.

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