Giving a knot, there are three rules to help us finding the Kauffman bracket polynomial. Choosing knot’s orientation, then applying the Seifert algorithm to find the Euler characteristic and genus of its surface. Finally finding the relationship of the Kauffman bracket polynomial and the genus of the alternating links is the main goal of this paper.

FERRAZ, Carlos Handrey Araújo. Dinâmica de redes Booleanas aleatórias na presença de agente danificador. 2007. 99 f. Tese (Doutorado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2007. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-05-05T20:04:53Z No. of bitstreams: 1 2007_tese_chaferraz.pdf: 1298388 bytes, checksum: 78cd24083b62d39536e51a28bb85dd73 (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-05-07T14:26:22Z (GMT) No. of bitstreams: 1 2007_tese_chaferraz.pdf: 1298388 bytes, checksum: 78cd24083b62d39536e51a28bb85dd73 (MD5) / Made available in DSpace on 2015-05-07T14:26:22Z (GMT). No. of bitstreams: 1 2007_tese_chaferraz.pdf: 1298388 bytes, checksum: 78cd24083b62d39536e51a28bb85dd73 (MD5) Previous issue date: 2007 / Nós realizamos simulações de computador em autômatos de Kauffman em diversos grafos tais como redes quadradas regulares e agregados de percolação invasiva afim de investigar transições de fase, entropia total, distribuição radial do dano total médio (expoente dinâmico $z$) e velocidade de propagação do dano quando se introduz um agente danificador no sistema, apelidado o "homem estranho". A despeito do aumento na eficiência de danificação, nós não observamos qualquer mudança apreciável no limiar de transição para o caos tanto para o caso de rede quadrada como para o caso de mundo pequeno quando o homem estranho é adicionado em comparação a quando pequenos danos iniciais são inseridos ao sistema. A velocidade de propagação da nuvem de dano até tocar as bordas do sistemas tanto para o caso de rede quadrada como para o caso de mundo pequeno obedece uma lei de potência, com um expoente crítico de velocidade $alpha$ que depende fortemente do tipo de rede. Particularmente, nós temos estudado o espalhamento do dano quando algumas conexões são removidas na rede quadrada e quando se considera agregados especiais de percolação invasiva (agregados de alta saturação de borda, HBSC). A velocidade de propagação nestes sistemas é bastante sensível ao grau de diluição na rede quadrada e ao grau de saturação de borda em agregados de percolação invasiva. Finalmente, esperamos que estes e outros cálculos mais elaborados sejam de ajuda para que se possam entender problemas mais gerais relacionados a propagação de defeitos simples em sistemas complexos bem descritos por autômatos celulares.

NÃs realizamos simulaÃÃes de computador em autÃmatos de Kauffman em diversos grafos tais como redes quadradas regulares e agregados de percolaÃÃo invasiva afim de investigar transiÃÃes de fase, entropia total, distribuiÃÃo radial do dano total mÃdio (expoente dinÃmico $z$) e velocidade de propagaÃÃo do dano quando se introduz um agente danificador no sistema, apelidado o "homem estranho". A despeito do aumento na eficiÃncia de danificaÃÃo, nÃs nÃo observamos qualquer mudanÃa apreciÃvel no limiar de transiÃÃo para o caos tanto para o caso de rede quadrada como para o caso de mundo pequeno quando o homem estranho à adicionado em comparaÃÃo a quando pequenos danos iniciais sÃo inseridos ao sistema. A velocidade de propagaÃÃo da nuvem de dano atà tocar as bordas do sistemas tanto para o caso de rede quadrada como para o caso de mundo pequeno obedece uma lei de potÃncia, com um expoente crÃtico de velocidade $alpha$ que depende fortemente do tipo de rede. Particularmente, nÃs temos estudado o espalhamento do dano quando algumas conexÃes sÃo removidas na rede quadrada e quando se considera agregados especiais de percolaÃÃo invasiva (agregados de alta saturaÃÃo de borda, HBSC). A velocidade de propagaÃÃo nestes sistemas à bastante sensÃvel ao grau de diluiÃÃo na rede quadrada e ao grau de saturaÃÃo de borda em agregados de percolaÃÃo invasiva. Finalmente, esperamos que estes e outros cÃlculos mais elaborados sejam de ajuda para que se possam entender problemas mais gerais relacionados a propagaÃÃo de defeitos simples em sistemas complexos bem descritos por autÃmatos celulares.

Thesis (Ph.D.)--University of Delaware, 2008. / Principal faculty advisor: Ann E. Gibson, Dept. of Art History. Includes bibliographical references.

This dissertation focused on the longitudinal analysis of business start-ups using three waves of data from the Kauffman Firm Survey. The first essay used the data from years 2004-2008, and examined the simultaneous relationship between a firm’s capital structure, human resource policies, and its impact on the level of innovation. The firm leverage was calculated as, debt divided by total financial resources. Index of employee well-being was determined by a set of nine dichotomous questions asked in the survey. A negative binomial fixed effects model was used to analyze the effect of employee well-being and leverage on the count data of patents and copyrights, which were used as a proxy for innovation. The paper demonstrated that employee well-being positively affects the firm's innovation, while a higher leverage ratio had a negative impact on the innovation. No significant relation was found between leverage and employee well-being. The second essay used the data from years 2004-2009, and inquired whether a higher entrepreneurial speed of learning is desirable, and whether there is a linkage between the speed of learning and growth rate of the firm. The change in the speed of learning was measured using a pooled OLS estimator in repeated cross-sections. There was evidence of a declining speed of learning over time, and it was concluded that a higher speed of learning is not necessarily a good thing, because speed of learning is contingent on the entrepreneur's initial knowledge, and the precision of the signals he receives from the market. Also, there was no reason to expect speed of learning to be related to the growth of the firm in one direction over another. The third essay used the data from years 2004-2010, and determined the timing of diversification activities by the business start-ups. It captured when a start-up diversified for the first time, and explored the association between an early diversification strategy adopted by a firm, and its survival rate. A semi-parametric Cox proportional hazard model was used to examine the survival pattern. The results demonstrated that firms diversifying at an early stage in their lives show a higher survival rate; however, this effect fades over time.

Made available in DSpace on 2016-10-13T14:10:04Z (GMT). No. of bitstreams: 1 Leonardo Yuji Tamura.pdf: 2018327 bytes, checksum: dca70d285f6272ea72951fe189bab096 (MD5) Previous issue date: 2016-03-22 / The aim of this work is to deepen the understanding of the process of adaptation of firms in the international environment. For such it relied on studies that faced the firm as a complex adaptive system and utilized the Stuart Kauffman s NK model. The NK model was originally conceived to study biological phenomena but has been applied by scholars in strategy and organizational studies since 1997. To enable the use of the NK model for the study of multinational firms, the model was extended to embrace specific international business concepts such as the gaining of competitive advantage in different countries from the adaptation of the firm s internal characteristics to the local environment. The chosen methodology is based on the paradigm of agent-based modeling and simulation. Accordingly, the firms were modeled as autonomous agents that search for the optimization of their competitive advantage by the means of the adaptation process. This approach allowed the study the emergent properties of the system from the agents interaction and behavior. The results of the simulation showed that gaining competitive advantage from the firm s attributes in different countries enabled the emergence of new viable organizational forms. It was also noted that one organizational form that does not provides optimal competitive advantage in a particular country may still be viable in a global context. Another result was the emergence of the complexity catastrophe, which is the degradation of the competitive advantage resulted from the addition of conflicting constraints. Such conflicting constraints are a result of the simultaneous optimization of the competitive advantage in many countries in many different ways due to the possibility of local adaptation. / O objetivo deste trabalho é aprofundar o entendimento sobre o processo de adaptação de empresas no ambiente internacional. Para isto foram utilizados estudos que encararam a empresa como um sistema adaptativo complexo e utilizaram o modelo NK de Stuart Kauffman. O modelo NK foi originalmente concebido para o estudo de fenômenos biológicos, mas vem sendo aplicado por acadêmicos em trabalhos de estratégia e organizações desde 1997. Para que fosse possível utilizar o modelo NK para o estudo de empresas multinacionais, o modelo foi estendido para abarcar conceitos específicos de negócios internacionais como, por exemplo, a obtenção de vantagem competitiva em diferentes países a partir da adaptação de caraterísticas internas da empresa ao ambiente local. A metodologia utilizada foi baseada no paradigma da modelagem e simulação baseado em agentes. Com base neste paradigma as empresas foram modeladas como agentes autônomos que, por meio de um processo de adaptação buscam otimizar sua vantagem competitiva. Esta abordagem permite estudar propriedades emergentes do sistema a partir da interação e do comportamento dos agentes. Os resultados das simulações mostraram que a obtenção de vantagem competitiva a partir dos atributos organizacionais da empresa em diversos países possibilita o surgimento de novas formas organizacionais viáveis. Também se observou que uma forma organizacional que não propicia vantagem competitiva ótima em um país específico, ainda pode ser viável num contexto global. Outro resultado obtido foi a emergência da catástrofe da complexidade, que é a degradação da vantagem competitiva em decorrência da adição de restrições conflitantes. Tais restrições conflitantes são resultado da necessidade de otimizar simultaneamente a vantagem competitiva em diversos países de diversas formas diferentes devido à possibilidade de adaptação local.

The purpose of this thesis is to define a “local” version of Ozsváth and Szabó’s Heegaard Floer homology HFL^ for links in the 3-sphere, i.e. a Heegaard Floer homology HFT^ for tangles in the 3-ball. The decategorification of HFL^ is the classical Alexander polynomial for links; likewise, the decategorification of HFT^ gives a local version ∇ˢ of the Alexander polynomial. In the first chapter of this thesis, we give a purely combinatorial definition of this polynomial invariant ∇ˢ via Kauffman states and Alexander codes and investigate some of its properties. As an application, we show that the multivariate Alexander polynomial is mutation invariant. In the second chapter, we define HFT^ in two slightly different, but equivalent ways: One is via Juhász’s sutured Floer homology, the other by imitating the construction of HFL^. We then state a glueing theorem in terms of Zarev’s bordered sutured Floer homology, which endows HFT^ with additional structure. As an application, we show that any two links related by mutation about a (2,−3)-pretzel tangle have the same δ-graded link Floer homology. This result relies on a computer calculation. In the third and last chapter, we specialise to 4-ended tangles. In this case, we give a reformulation of HFT^ with a glueing structure in terms of (what we call) peculiar modules. Together with a glueing theorem, we can easily recover oriented and unoriented skein relations for HFL^. Our peculiar modules also enjoy some symmetry relations, which support a conjecture about δ-graded mutation invariance of HFL^. However, stronger symmetries would be needed to actually prove this conjecture. Finally, we explore the relationship between peculiar modules and twisted complexes in the wrapped Fukaya category of the 4-punctured sphere. There are four appendices, some of which might be of independent interest: In the first appendix, we describe a general construction of dg categories which unifies all algebraic structures used in this thesis, in particular type A and type D modules from bordered theory. In the second appendix, we prove a generalised version of Kauffman’s clock theorem, which plays a major role for our decategorified invariants. The last two appendices are manuals for two Mathematica programs. The first is a tool for computing the generators of HFT^ and the decategorified tangle invariant ∇ˢ. The second allows us to compute bordered sutured Floer homology using nice diagrams.

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.

Cette thèse se divise en trois parties qui peuvent être toutes regroupées autour d'une même bannière : l'étude de structures algébriques reliées aux algèbres de type Askey–Wilson. Alors que dans la première partie on s'efforce d'obtenir des interprétations duales (au sens de Howe) de ces algèbres, dans les autres parties on étudie des généralisations de ces algèbres. Des dégénérations de l'algèbre de Sklyanin, générées par des blocs plus fondamentaux que ceux générant les algèbres de type Askey–Wilson, sont étudiées dans la deuxième partie et des généralisations de plus haut rang des algèbres de type Askey–Wilson sont étudiées dans la troisième partie. Dans la première partie, en invoquant la dualité de Howe, deux interprétations duales sont obtenues pour les algèbres de Racah, Bannai–Ito, Askey–Wilson, Higgs, Hahn, \(q\)-Hahn et dual \(-1\) Hahn. La façon dont la dualité de Howe opère est rendue explicite par l'examen de processus de réduction dimensionnelle. Un modèle superintégrable 2D de mécanique quantique superconforme dont l'algèbre de symétrie est celle de type dual \(-1\) Hahn est également introduit et solutionné. Dans la deuxième partie, des algèbres générées par des opérateurs de contiguïté et d'échelle encodant des propriétés de familles de polynômes sont étudiées. Ces opérateurs appartiennent à la classe des opérateurs de Sklyanin–Heun, qui peuvent être définis sur plusieurs grilles diverses. On découvre qu'ils génèrent des dégénérations de l'algèbre de Sklyanin. On démontre que les représentations irréductibles de dimension finie de ces algèbres ont pour base des familles de para-polynômes. Les grilles linéaires, quadratiques, exponentielles et d'Askey–Wilson sont étudiées et mènent respectivement aux polynômes orthogonaux des familles de para-Krawtchouk, para-Racah, \(q\)-para-Krawtchouk et \(q\)-para-Racah. Enfin, la façon dont les polynômes de para-Krawtchouk et d'autres familles de polynômes orthogonaux sont reliées aux représentations tridiagonales du plan de Jordan déformé est présentée. Dans la dernière partie, on explore des généralisations à plus haut rang pour les algèbres de Racah et Askey–Wilson. Pour ce faire, on étudie les réalisations de ces algèbres en termes de Casimirs intermédiaires. Le rôle de la matrice \(R\) tressée est élucidé : celle-ci permet de relier divers Casimirs intermédiaires entre eux par conjugaison. Un isomorphisme entre l'algèbre de skein du crochet de Kauffman de la sphère à 4 trous et l'algèbre engendrée par les Casimir intermédiaires dans \(U_q(\mathfrak{sl}_2)^{\otimes 3}\) est présenté et permet d'interpréter de façon diagrammatique la conjugaison par la matrice \(R\) tressée mentionnée ci-haut. Finalement, une présentation du centralisateur \(Z_n(\mathfrak{sl}_2)\) de \(U(\mathfrak{sl}_2)\) dans \(U(\mathfrak{sl}_2)^{\otimes n}\) par générateurs et relations est obtenue et on montre que ce centralisateur est isomorphe à un quotient (obtenu explicitement) de l'algèbre de Racah de plus haut rang \(R(n)\). / This thesis is divided in three parts which all orbit around the same theme: the study of algebraic structures related to the algebras of Askey–Wilson type. In the first part we obtain two interpretations that are dual in the sense of Howe for the algebras of Askey–Wilson type. Meanwhile, the other two parts are concerned with generalizations of these algebras. In the second part, we study degenerations of the Sklyanin algebra, which are built out of generators that are more fundamental than those of the Askey–Wilson algebra. In the last part, generalizations of the Askey–Wilson type algebras to higher rank are studied. In the first part, dual interpretations are obtained for the Racah, Bannai–Ito, Askey–Wilson, Higgs, Hahn, \(q\)-Higgs and dual \(-1\) Hahn algebras by invoking Howe duality. The way that this Howe duality operates is made explicit through the examination of a dimensional reduction procedure. A 2D superintegrable superconformal quantum mechanics model, whose symmetry algebra is the one of dual \(-1\) Hahn type, is also introduced and solved. In the second part, we study algebras that are generated by contiguity and ladder operators that encode properties of families of orthogonal polynomials. We show that these operators belong to the Sklyanin–Heun class of operators, which can be defined for various grids. We also show how their algebraic relations correspond to those of degenerations of the Sklyanin algebra. Then, we show how various families of para-polynomials support finite-dimensional irreducible representations of these degenerate algebras. From the linear, quadratic, exponential and Askey–Wilson grids, we are respectively led to the para-Krawtchouk, para-Racah, \(q\)-para-Krawtchouk and \(q\)-para-Racah polynomials. Later, we connect the para-Krawtchouk polynomials (and other families of orthogonal polynomials) to tridiagonal representations of the deformed Jordan plane. In the final part, we explore higher rank generalizations of the Racah and Askey–Wilson algebras. To that end, their realizations in terms of intermediate Casimir elements are studied. The role of the braided \(R\)-matrix is understood as follows: it connects various intermediate Casimir elements through conjugation. We obtain an isomorphism between the Kauffman bracket skein algebra of the four-punctured sphere and the algebra generated by the intermediate Casimir elements in \(U_q(\mathfrak{sl}_2)^{\otimes3}\). This leads to a diagrammatic interpretation of the conjugation by the braided \(R\)-matrix mentioned in the above. Lastly, a presentation of the centralizer \(Z_n(\mathfrak{sl}_2)\) of \(U(\mathfrak{sl}_2)\) in \(U(\mathfrak{sl}_2)^{\otimes n}\) by generators and relations is obtained and we show that this centralizer is isomorphic to a quotient (which we provide explicitly) of the higher rank Racah algebra \(R(n)\).

This dissertation examines the role of public investments in inducing small firms to develop risky, early-stage technologies. It contributes to expanding our understanding of the consequences of research, innovation, and entrepreneurship policies and programs by investigating in more depth the effect of the Small Business Innovation Research (SBIR) program on the innovation effort, ability to attract external capital, and other metrics of post-entry performance of small business start-ups using a new sample and estimation approach. This study integrated the Kauffman Firm Survey from the Ewing Marion Kauffman Foundation with the SBIR recipient dataset from the U.S. Small Business Administration and used advances in the micro-econometrics of program evaluation to empirically construct the counterfactual outcomes of SBIR recipients. We found empirical evidence of the input additionality effect of the SBIR program. The treatment effects analyses also found a significant positive effect of SBIR on innovation propensity and employment. However, it appears that public co-financing of commercial R&D has crowded-out privately financed R&D of small business start-ups in the United States. A dollar of SBIR subsidy decreased firm-financed R&D by about $0.16. Contrary to prior SBIR studies, we did not find any significant “halo effect” or “certification effect” of receiving an SBIR award on attracting external capital. What we discovered is a different certification effect of the SBIR program: SBIR grantees are more likely to attract external patents. This finding confirms that innovation requires a portfolio of internal and external knowledge assets as theorized by David Teece and his colleagues.