The property that a graph has an embedding in the projective plane is closed under taking minors. Thus by the well known Graph Minor theorem of Robertson and Seymour, there exists a finite list of minor-minimal graphs, call it L, such that a given graph G is projective planar if and only if G does not contain any graph isomorphic to a member of L as a minor. Glover, Huneke and Wang found 35 graphs in L, and Archdeacon proved that those are all the members of L, but Archdeacon's proof never appeared in any refereed journal. In this thesis we develop a modern approach and technique for finding the list L, independent of previous work. Our approach is based on conditioning on the connectivity of a member of L. Assume G is a member of L. If G is not 3-connected then the structure of G is well understood. In the case that G is 3-connected, the problem breaks down into two main cases, either G has an internal separation of order three or G is internally 4-connected. In this thesis we find the set of all 3-connected minor minimal non-projective planar graphs with an internal 3-separation. For proving our main result, we use a technique which can be considered as a variation and generalization of the method that Robertson, Seymour and Thomas used for non-planar extension of planar graphs. Using this technique, besides our main result, we also classify the set of minor minimal obstructions for a-, ac-, abc-planarity for rooted graphs. (A rooted graph (G,a,b,c) is a-planar if there exists a split of the vertex a to a' and a' in G such that the new graph G' obtained by the split has an embedding in a disk such that the vertices a', b, a', c are on the boundary of the disk in the order listed. We define b- and c-planarity analogously. We say that the rooted graph (G,a,b,c) is ab-planar if it is a-planar or b-planar, and we define abc-planarity analogously.)

Most AI approaches for fault isolation handle only the behavioral modes OK and NOT OK. To be able to isolate faults in components with generalized behavioral modes, a new framework is needed. By introducing domain logic and assigning the behavior of a component to a behavioral mode domain, efficient representation and calculation of diagnostic information is made possible. Diagnosing components with generalized behavioral modes also requires extending familiar characterizations. The characterizations candidate, generalized kernel candidate and generalized minimal candidate are introduced and it is indicated how these are deduced. It is concluded that neither the full candidate representation nor the generalized kernel candidate representation are conclusive enough. The generalized minimal candidate representation focuses on the interesting diagnostic statements to a large extent. If further focusing is needed, it is satisfactory to present the minimal candidates which have a probability close to the most probable minimal candidate. The performance of the fault isolation algorithm is very good, faults are isolated as far as it is possible with the provided diagnostic information.

In 2010, Prof. Chang and Prof. Lee applied Lagrange interpolation formula to decode a class of binary cyclic codes, but they did not provide an effective way to calculate the Lagrange interpolation formula. In this thesis, we use the least common multiple of polynomials to compute it effectively. Let E be an extension field of degree m over F = F_p and £] be a primitive nth root of unity in E. For a nonzero element r in E, the minimal polynomial of r over F is denoted by m_r(x). Then, let Min (r, F) denote the least common multiple of m_r£]^i(x) for i = 0, 1,..., n-1 and be called the generalized minimal polynomial of over F. For any binary quadratic residue code mentioned in this thesis, the set of all its correctable error patterns can be partitioned into root sets of some generalized minimal polynomials over F. Based on this idea, we can develop an effective method to calculate the Lagrange interpolation formula.

The increasing infection of Cryptococcus species and Tricosporon asahii emerged in clinical patients who were immunocompromised. They usually induce lung, skin, brain and systemic infection. Morbidity and mortality of immunocompromised patients are higher than normal healthy people. Cryptococcus neoformans var. grubii ¡]serotype A¡^ infections were reported in clinical cases predominantly and they were isolated from birds¡¦ droppings in large amount. Cryptococcus neoformans var. gattii ¡]serotype B, C¡^ had a natural life in plants, especially Eucalypticus trees. Isolations from other trees were reported increasingly in the tropical and subtropical areas. Comparing to Cryptococcus species, Tricosporon asahii is the normal mycoses of soil. In this study, we performed an environmental investigation concerning Cryptococcus species and Tricosporon asahii in Southern Taiwan. 120 droppings of racing pigeons and 114 samples from Eucalypticus trees were obtained. The results revealed that 30 Cryptococcus neoformans were isolated from racing pigeons¡¦ droppings ¡]25%¡^, as well as 4 Cryptococcus laurentii ¡]3.3%¡^ and 2 Cryptococcus albidus ¡]1.7%¡^. In addition, 25 Tricosporon asahii ( 20.8% ) were isolated from droppings of racing pigeons. But, none of Cryptococcus species or Tricosporon asahii is isolated from Eucalypticus trees ¡]0%¡^. All of Cryptococcus neoformans isolated from pigeons¡¦ droppings were var. grubii ¡]serotype A¡^ and their drug susceptibility tests showed sensitive to Amphotericin B ¡]minimal inhibitory concentration ¡Ø0.25£gg/ml¡^ and Fluconazole ¡]minimal inhibitory concentration 2£gg/ml¡^ and Flucytosine ¡]minimal inhibitory concentration ¡Ø1£gg/ml¡^. To sum up, both Cryptococcus species and Tricosporon asahii were isolated from droppings of racing pigeons in our study, especially Tricosporon asahii in large amount. Opportunistic infection caused by these species should be given more attention to racing pigeons which have close contact with human . Intensive investigation and surveillance should be carried out in the future to provide an information for the control and prevention of diseases.

This paper studies minimally-supported D-optimal designs for polynomial regression model with logarithmically concave (log-concave) weight functions. Many commonly used weight functions in the design literature are log-concave. We show that the determinant of information matrix of minimally-supported design is a log-concave function of ordered support points and the D-optimal design is unique. Therefore, the numerically D-optimal designs can be determined e¡Óciently by standard constrained concave programming algorithms.

The selection and modification of kernel functions is a very important problem in the field of support vector learning. However, the kernel function of a support vector machine has great influence on its performance. The kernel function projects the dataset from the original data space into the feature space, and therefore the problems which couldn¡¦t be done in low dimensions could be done in a higher dimension through the transform of the kernel function. In this thesis, we adopt the FCM clustering algorithm to group data patterns into clusters, and then use a statistical approach to calculate the standard deviation of each pattern with respect to the other patterns in the same cluster. Therefore we can make a proper estimation on the distribution of data patterns and assign a proper standard deviation for each pattern. The standard deviation is the same as the variance of a radial basis function. Then we have the origin data patterns and the variance of each data pattern for support vector learning. Experimental results have shown that our approach can derive better kernel functions than other methods, and also can have better learning and generalization abilities.

The research in the thesis investigates the use of minimal path techniques to track and detect cracks, modeled as curves, in critical infrastructure like pavements and bridges. We developed a novel minimal path algorithm to detect curves with complex topology that may have both closed cycles and open sections using an arbitrary point on the curve as the sole input. Specically, we applied the novel algorithm to three problems: semi-automatic crack detection, detection of continuous cracks for crack sealing applications and detection of crack growth in structures like bridges. The current state of the art minimal path techniques only work with prior knowledge of either both terminal points or one terminal point plus total length of the curve. For curves with multiple branches, all terminal points need to be known. Therefore, we developed a new algorithm that detects curves and relaxes the necessary user input to one arbitrary point on the curve. The document presents the systematic development of this algorithm in three stages. First, an algorithm that can detect open curves with branches was formulated. Then this algorithm was modied to detect curves that also have closed cycles. Finally, a robust curve detection algorithm was devised that can increase the accuracy of curve detection. The algorithm was applied to crack images and the results of crack detection were validated against the ground truth. In addition, the algorithm was also used to detect features like catheter tube and optical nerves in medical images. The results demonstrate that the algorithm is able to accurately detect objects that can be modeled as open curves.

Let e > 3 and mo,... ,me_i be positive integers with gcd(m0,... ,me_i) = 1, which form an almost arithmetic sequence, i.e., some e - 1 of these form an arithmetic progression. We further assume that m0,... ,mc_1 generate F := Σ e-1 I=0 Nmi minimally. Note that any three integers and also any arithmetic progression form an almost arithmetic sequence. We assume that 0 < m0 < • • • < me-2 form an arithmetic progression and n := mc-i is arbitrary Put p := e - 2. Let K be a field and XQ) ... ,Xj>, Y,T be indeterminates. Let p denote the kernel of the if-algebra homomorphism η: K[XQ, ..., XV) Y) -* K^T], defined by r){Xi) = Tm\.. .η{Xp) = Tmp, η](Y) = Tn. Then, p is the defining ideal for the affine monomial curve C in A^, defined parametrically by Xo = Trr^)...)Xv = T^}Y = T*. Furthermore, p is a homogeneous ideal with respect to the gradation on K[X0)... ,XP,F], given by wt(Z0) = mo, • • •, wt(Xp) = mp, wt(Y) = n. Let 4 := K[XQ> ...,XP) Y)/p denote the coordinate ring of C. With the assumption ch(K) = 0, in Chapter 1 we have derived an explicit formula for μ(DerK(A)), the minimal number of generators for the A-module DerK(A), the derivation module of A. Furthermore, since type(A) = μ(DerK(A)) — 1 and the last Betti number of A is equal to type(A), we therefore obtain an explicit formula for the last Betti number of A as well A minimal set of binomial generatorsG for the ideal p had been explicitly constructed by PatiL In Chapter 2, we show that the set G is a Grobner basis with respect to grevlex monomial ordering on K[X0)..., Xp, Y]. As an application of this observation, in Chapter 3 we obtain an explicit minimal free resolution for affine monomial curves in A4K defined by four coprime positive integers mo,.. m3, which form a minimal arithmetic progression. (Please refer the pdf file forformulas)

<p>Most AI approaches for fault isolation handle only the behavioral modes OK and NOT OK. To be able to isolate faults in components with generalized behavioral modes, a new framework is needed. By introducing domain logic and assigning the behavior of a component to a behavioral mode domain, efficient representation and calculation of diagnostic information is made possible. </p><p>Diagnosing components with generalized behavioral modes also requires extending familiar characterizations. The characterizations candidate, generalized kernel candidate and generalized minimal candidate are introduced and it is indicated how these are deduced. </p><p>It is concluded that neither the full candidate representation nor the generalized kernel candidate representation are conclusive enough. The generalized minimal candidate representation focuses on the interesting diagnostic statements to a large extent. If further focusing is needed, it is satisfactory to present the minimal candidates which have a probability close to the most probable minimal candidate. </p><p>The performance of the fault isolation algorithm is very good, faults are isolated as far as it is possible with the provided diagnostic information.</p>

Ce travail se situe à la fois en Géométrie Algébrique et l'Algèbre Commutative. La premier partie de cette thèse est consacrée à l'anneau de Rees (blow-up ring) et la fibre spéciale d'un idéal de réseau de codimenson 2 dans un anneau de polynômes. Dans le cas où l'idéal est engendré par trois ou quatre éléments, une présentation explicite de l'anneau de Rees est donnée. Dans le cas général, nous définissons le graphe de syzygies de l'idéal, et l'étudions combinatoirement. Nous obtenons : 1/ La dimension de la fibre spéciale est 2 ou 3. 2/ Si l'idéal n'est pas une intersection complète, alors la fibre spéciale est Cohen--Macaulay de dimension 3, réduite, de degré minimal, i.e. la fibre spéciale a des propriétés géométriques remarquables. Une présentation explicite de la fibre spéciale est aussi donnée. 3/ L'anneau de Rees est Cohen--Macaulay, et engendré par des formes de degré au plus 2. La deuxième partie de la thèse est consacrée aux idéaux simpliciaux, introduits par M. Morales. En étudiant des propriétés combinatoires, nous donnons une large classe d'idéaux binômiaux simpliciaux pour lesquels le nombre de réduction est 1.