The von Neumann Minimax Theorem and its relatives and a study of externality in on-line auctions

This work consists of a theoretical part and an experimental one. The first part provides a simple treatment of the celebrated von Neumann minimax theorem as formulated by Nikaid6 and Sion. It also discusses its relationships with fundamental theorems of convex analysis. The second part is about externality in sponsored search auctions. It shows that in these auctions, advertisers have externality effects on each other which influence their bidding behavior. It proposes Hal R.Varian model and shows how adding externality to this model will affect its properties. In order to have a better understanding of the interaction among advertisers in on-line auctions, it studies the structure of the Google advertisements networ.k and shows that it is a small-world scale-free network.

An Interactive Theorem Prover for First-Order Dynamic Logic

Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs. The method of proving correctness of properties of a computer program using the well-known Hoare Logic can be implemented by utilizing the robustness of dynamic logic. For a very broad range of languages and applications in program veri cation, a theorem prover named KIV (Karlsruhe Interactive Veri er) Theorem Prover has already been developed. But a high degree of automation and its complexity make it di cult to use it for educational purposes. My research work is motivated towards the design and implementation of a similar interactive theorem prover with educational use as its main design criteria. As the key purpose of this system is to serve as an educational tool, it is a self-explanatory system that explains every step of creating a derivation, i.e., proving a theorem. This deductive system is implemented in the platform-independent programming language Java. In addition, a very popular combination of a lexical analyzer generator, JFlex, and the parser generator BYacc/J for parsing formulas and programs has been used.

Determinants of left ventricular mass as measured by Doppler echocardiography in pre-adolescents

This study examined factors contributing to the differences in left ventricular mass as measured by Doppler echocardiography in children. Fourteen boys (10.3 ± 0.3 years of age) and 1 1 girls (10.5 ± 0.4 years of age) participated in the study. Height and weight were measured, and relative body fat was determined from the measurement of skinfold thickness according to Slaughter et al. (1988). Lean Body Mass was then calculated by subtracting the fat mass from the total body mass. Sexual maturation was self-assessed using the stages of sexual maturation by Tanner (1962). Both pubic hair development and genital (penis or breast for boys and girls respectively) development were used to determine sexual maturation. Carotid Pulse pressure was assessed by applanation tomometry in the left carotid artery. Cardiac mass was measured by Doppler Echocardiography. Images of cardiac structures were taken using B-Mode and were then translated to M- Mode. The dimensions at the end diastole were obtained at the onset of the QRS complex of the electrocardiogram in a plane through a standard position. Measurements included: (a) the diameter of the left ventricle at the end diastole was measured from the septum edge to the endocardium mean border, (b) the posterior wall was measured as the distance from to anterior wall to the epicardium surface, and (c) the interventricular septum was quantified as the distance from the surface of the left ventricle border to the right ventricle septum surface. Systolic time measurements were taken at the peak of the T-wave of the electrocardiogram. Each measurement was taken three to five times before averaging. Average values were used to calculate cardiac mass using the following equation (Deveraux et al. 1986). Weekly physical activity metabolic equivalent was calculated using a standardize activity questionnaire (Godin and Shepard, 1985) and peakV02 was measured on a cycloergometer. There were no significant differences in cardiovascular mesurements between boys and girls. Left ventricular mass was correlated (p<0.05) with size, maturation, peakV02 and physical activity metabolic equivalent. In boys, lean body mass alone explained 36% of the variance in left ventricular mass while weight was the single strongest predictor of left ventricular mass (R =0.80) in girls. Lean body mass, genital developemnt and physical activity metabolic equivalent together explained 46% and 81% in boys and girls, respectively. However, the combination of lean body mass, genital development and peakV02 (ml kgLBM^ min"') explained up to 84% of the variance in left ventricular mass in girls, but added nothing in boys. It is concluded that left ventricular mass was not statistically different between pre-adolescent boys and girls suggesting that hormonal, and therefore, body size changes in adolescence have a main effect on cardiac development and its final outcome. Although body size parameters were the strongest correlates of left ventricular mass in this pre-adolescent group of children, to our knowledge, this is the first study to report that sexual maturation, as well as physical activity and fitness, are also strong associated with left ventricular mass in pre-adolescents, especially young females. Arterial variables, such as systolic blood pressure and carotid pulse pressure, are not strong determinants of left ventricular mass in this pre-adolescent group. In general, these data suggest that although there is no gender differences in the absolute values of left ventricular mass, as children grow, the factors that determine cardiac mass differ between the genders, even in the same pre-adolescent age.

Confocal and two-photon microscopy : image enhancement /

Confocal and two-photon microcopy have become essential tools in biological research and today many investigations are not possible without their help. The valuable advantage that these two techniques offer is the ability of optical sectioning. Optical sectioning makes it possible to obtain 3D visuahzation of the structiu-es, and hence, valuable information of the structural relationships, the geometrical, and the morphological aspects of the specimen. The achievable lateral and axial resolutions by confocal and two-photon microscopy, similar to other optical imaging systems, are both defined by the diffraction theorem. Any aberration and imperfection present during the imaging results in broadening of the calculated theoretical resolution, blurring, geometrical distortions in the acquired images that interfere with the analysis of the structures, and lower the collected fluorescence from the specimen. The aberrations may have different causes and they can be classified by their sources such as specimen-induced aberrations, optics-induced aberrations, illumination aberrations, and misalignment aberrations. This thesis presents an investigation and study of image enhancement. The goal of this thesis was approached in two different directions. Initially, we investigated the sources of the imperfections. We propose methods to eliminate or minimize aberrations introduced during the image acquisition by optimizing the acquisition conditions. The impact on the resolution as a result of using a coverslip the thickness of which is mismatched with the one that the objective lens is designed for was shown and a novel technique was introduced in order to define the proper value on the correction collar of the lens. The amoimt of spherical aberration with regard to t he numerical aperture of the objective lens was investigated and it was shown that, based on the purpose of our imaging tasks, different numerical apertures must be used. The deformed beam cross section of the single-photon excitation source was corrected and the enhancement of the resolution and image quaUty was shown. Furthermore, the dependency of the scattered light on the excitation wavelength was shown empirically. In the second part, we continued the study of the image enhancement process by deconvolution techniques. Although deconvolution algorithms are used widely to improve the quality of the images, how well a deconvolution algorithm responds highly depends on the point spread function (PSF) of the imaging system applied to the algorithm and the level of its accuracy. We investigated approaches that can be done in order to obtain more precise PSF. Novel methods to improve the pattern of the PSF and reduce the noise are proposed. Furthermore, multiple soiu'ces to extract the PSFs of the imaging system are introduced and the empirical deconvolution results by using each of these PSFs are compared together. The results confirm that a greater improvement attained by applying the in situ PSF during the deconvolution process.

Developmental variations in the peripheral erythrocytic system of the rainbow trout, Salmo gairdneri

The peripheral circulating erythrocytic system of the rainbow trout, l3 almo gairdner , was examined in vitro in relation differences in the morphology and multiple hemoglobin system organization of adult and juvenile red cells. Cells were separated by velocity sedimentation under unit gravity, a procedure requiring red cell exposure to an incubation medium for periods of at least three hours. Therefore , this must provide an environment in which red cells remain in a condition approximaing normalcy. Previous studies having demonstrated commonly employed media to be ineffective in this regard , a medium was developed through modification of Cortl and saline. One of the principal additions to this me dium , norepinephrine, altered cell regulation of intracellular calcium, magnesium and chloride concentrations. Catecholamine involvement was also suggeste d in the synthes is of hemoglobin . The procedure was found to separtate cells primarily by density and, to a lesser extent, by shape. Characterization of red cells revealed two subpopulations to exist . The first comprised the bulk of the cell population, and were of greater l ength, width, volume and major:minor axis ratio than the smaller population; these were adult cells. The later, juvenile cells were of smaller overall size and were more spherical in shape . Juvenile cells also possessed fewer electrophore tpically distinguishable isomorphs than did adults with only eight of eleven hemoglobin component s typically found With maturation,hemoglobin complement with the development of three more bands. The total complement of the adult cell contained 7 cathodal bands and four anodal hemoglobin isomorphs. Bands acquired with maturation comprised the smallest percentage of the cells hemoglobin. each averaging less than one-percent of the total. Whether these additional bands are derived through degradation and reaggregation of existing components or are the product of pe gQy2 synthesis is not yet known.

The finite bandwidth model for spin fluctuations in Pd

The frequency dependence of the electron-spin fluctuation spectrum, P(Q), is calculated in the finite bandwidth model. We find that for Pd, which has a nearly full d-band, the magnitude, the range, and the peak frequency of P(Q) are greatly reduced from those in the standard spin fluctuation theory. The electron self-energy due to spin fluctuations is calculated within the finite bandwidth model. Vertex corrections are examined, and we find that Migdal's theorem is valid for spin fluctuations in the nearly full band. The conductance of a normal metal-insulator-normal metal tunnel junction is examined when spin fluctuations are present in one electrode. We find that for the nearly full band, the momentum independent self-energy due to spin fluctuations enters the expression for the tunneling conductance with approximately the same weight as the self-energy due to phonons. The effect of spin fluctuations on the tunneling conductance is slight within the finite bandwidth model for Pd. The effect of spin fluctuations on the tunneling conductance of a metal with a less full d-band than Pd may be more pronounced. However, in this case the tunneling conductance is not simply proportional to the self-energy.

Square Root Finding In Graphs

Abstract: Root and root finding are concepts familiar to most branches of mathematics. In graph theory, H is a square root of G and G is the square of H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Graph square is a basic operation with a number of results about its properties in the literature. We study the characterization and recognition problems of graph powers. There are algorithmic and computational approaches to answer the decision problem of whether a given graph is a certain power of any graph. There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four. The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five. We settle the conjecture that recognition of square of graphs with girth 5 is NP-complete. This result is providing the complete dichotomy theorem for square root finding problem.

Edge-choosability of Planar Graphs

According to the List Colouring Conjecture, if G is a multigraph then χ' (G)=χl' (G) . In this thesis, we discuss a relaxed version of this conjecture that every simple graph G is edge-(∆ + 1)-choosable as by Vizing’s Theorem ∆(G) ≤χ' (G)≤∆(G) + 1. We prove that if G is a planar graph without 7-cycles with ∆(G)≠5,6 , or without adjacent 4-cycles with ∆(G)≠5, or with no 3-cycles adjacent to 5-cycles, then G is edge-(∆ + 1)-choosable.

L-Fuzzy Relations in Coq

Heyting categories, a variant of Dedekind categories, and Arrow categories provide a convenient framework for expressing and reasoning about fuzzy relations and programs based on those methods. In this thesis we present an implementation of Heyting and arrow categories suitable for reasoning and program execution using Coq, an interactive theorem prover based on Higher-Order Logic (HOL) with dependent types. This implementation can be used to specify and develop correct software based on L-fuzzy relations such as fuzzy controllers. We give an overview of lattices, L-fuzzy relations, category theory and dependent type theory before describing our implementation. In addition, we provide examples of program executions based on our framework.

Prime Rational Functions and Integral Polynomials

Let f(x) be a complex rational function. In this work, we study conditions under which f(x) cannot be written as the composition of two rational functions which are not units under the operation of function composition. In this case, we say that f(x) is prime. We give sufficient conditions for complex rational functions to be prime in terms of their degrees and their critical values, and we derive some conditions for the case of complex polynomials. We consider also the divisibility of integral polynomials, and we present a generalization of a theorem of Nieto. We show that if f(x) and g(x) are integral polynomials such that the content of g divides the content of f and g(n) divides f(n) for an integer n whose absolute value is larger than a certain bound, then g(x) divides f(x) in Z[x]. In addition, given an integral polynomial f(x), we provide a method to determine if f is irreducible over Z, and if not, find one of its divisors in Z[x].

Student Perspectives of the Context of Recess; Implications for Student Well-Being

With most students in Canada spending approximately 180 days a year in school, averaging more than six hours a day (Morrison & Kirby, 2011), Wei, Szumilas and Kutcher (2011) argue that this places educational institutions in an unique position in terms of influencing the health and well-being of students. This brings forth the need for school environments to be utilized in ways that are conducive to promoting student development. Much of the educational and developmental components embedded within the school system as well as experiences within greatly influence student’s health and well-being. A national statement was made a concerning American children’s education and mental health that is greatly applicable to the Canadian school system. It was stated that schools “must be active partners in the mental health care of our children” because of the “important interplay between emotional health and school success” (Lazarus & Sulkowski, 2011, pp. 15-16). This identifies the need to ensure that all students, as much as possible, are being provided with safe environments and sufficient support in order to encourage positive developmental trajectories of student health and well-being.

Group-invariant solutions of semilinear Schrodinger equations in multi-dimensions

Symmetry group methods are applied to obtain all explicit group-invariant radial solutions to a class of semilinear Schr¨odinger equations in dimensions n = 1. Both focusing and defocusing cases of a power nonlinearity are considered, including the special case of the pseudo-conformal power p = 4/n relevant for critical dynamics. The methods involve, first, reduction of the Schr¨odinger equations to group-invariant semilinear complex 2nd order ordinary differential equations (ODEs) with respect to an optimal set of one-dimensional point symmetry groups, and second, use of inherited symmetries, hidden symmetries, and conditional symmetries to solve each ODE by quadratures. Through Noether’s theorem, all conservation laws arising from these point symmetry groups are listed. Some group-invariant solutions are found to exist for values of n other than just positive integers, and in such cases an alternative two-dimensional form of the Schr¨odinger equations involving an extra modulation term with a parameter m = 2−n = 0 is discussed.

New conserved vorticity integrals for moving surfaces in multi-dimensional fluid flow

For inviscid fluid flow in any n-dimensional Riemannian manifold, new conserved vorticity integrals generalizing helicity, enstrophy, and entropy circulation are derived for lower-dimensional surfaces that move along fluid streamlines. Conditions are determined for which the integrals yield constants of motion for the fluid. In the case when an inviscid fluid is isentropic, these new constants of motion generalize Kelvin’s circulation theorem from closed loops to closed surfaces of any dimension.

Backbone Colouring of Graphs

Consider an undirected graph G and a subgraph of G, H. A q-backbone k-colouring of (G,H) is a mapping f: V(G) {1, 2, ..., k} such that G is properly coloured and for each edge of H, the colours of its endpoints differ by at least q. The minimum number k for which there is a backbone k-colouring of (G,H) is the backbone chromatic number, BBCq(G,H). It has been proved that backbone k-colouring of (G,T) is at most 4 if G is a connected C4-free planar graph or non-bipartite C5-free planar graph or Cj-free, j∈{6,7,8} planar graph without adjacent triangles. In this thesis we improve the results mentioned above and prove that 2-backbone k-colouring of any connected planar graphs without adjacent triangles is at most 4 by using a discharging method. In the second part of this thesis we further improve these results by proving that for any graph G with χ(G) ≥ 4, BBC(G,T) = χ(G). In fact, we prove the stronger result that a backbone tree T in G exists, such that ∀ uv ∈ T, |f(u)-f(v)|=2 or |f(u)-f(v)| ≥ k-2, k = χ(G). For the case that G is a planar graph, according to Four Colour Theorem, χ(G) = 4; so, BBC(G,T) = 4.