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Thesis PlanB (M.S.)--University of Wisconsin--Stout, 2007. / Includes bibliographical references.

Some inequalities of H. J. Ryser with combinatorial applications are generalized. Let f be a non-negative concave symmetric function on v-tuples of non-negative reals. If f has the property that when θa + (1- θ)b ∈ G[subscript f] = f[power -1] ({t:t > 0}), 0 < θ < 1, then f(θa + (1- θ)b) = θf(a) + (1-θ)f(b), then we say that f is strictly concave. (Similarly, if f is convex and has the property just mentioned, then we say that f is strictly convex). Let H be a non-negative hermitian matrix with eigenvalues λ₁, ..., λ[subscript v], where λ₁ ≧ ... ≧λ[subscript e] > λ[subscript e+1] = … = λ[subscript v] = 0. Let h be an integer, 1 < h, such that e ≦h ≦ v and define k and λ by k = trace (H)/h, λ[subscript h] ≦k + (h-1) λ ≦λ₁. Define the matrix B of order h by B = (k- λ)I + λJ, where I is the identity matrix all of whose entries are 1's. Let B₀ = B ∔ 0, where the matrix B₀ of order v is the direct sum of the matrix B of order h and the (v-h)-order zero matrix. Let f(H) denote f(λ₁, … , λ[subscript v]). Then we prove theorems of the following nature. THEOREM: The matrices H and B₀ satisfy f(H) ≦ f(B₀). If f is strictly concave and if (λ₁, ..., λ[subscript v]) ∈ G[subscript f] then equality holds if and only if H and B₀ have the same eigenvalues. If f is strictly concave and if for some integer z, G[subscript f] is the set of non-negative vectors with at least z positive coordinates and if k + (h-1) λ ≠ 0 and z ≦ h or k + (h-1)λ = 0 and z < h, then f(H) = f(B₀) if and only if H and B₀ have the same eigenvalues. If f is convex a similar theorem with the inequality reversed can be proved. We discuss various choices of the function f and indicate some applications of the results to some combinatorial problems. / Science, Faculty of / Mathematics, Department of / Graduate

Let Ω be an arbitrary open set in IR³. Let || • || denote the L²(Ω) norm, and let [formula omitted] denote the completion of [formula omitted] in the Dirichlet norm || ∇•||. The pointwise bound [forumula omitted] is established for all functions [formula omitted] with Δ u є L² (Ω). The constant [formula omitted] is shown to be the best possible. Previously, inequalities of this type were proven only for bounded smooth domains or convex domains, with constants depending on the regularity of the boundary. A new method is employed to obtain this sharp inequality. The key idea is to estimate the maximum value of the quotient ⃒u(x)⃒/ || ∇u || ½ || Δ u || ½, where the point x is fixed, and the function u varies in the span of a finite number of eigenfunctions of the Laplacian. This method admits generalizations to other elliptic operators and other domains. The inequality is applied to study the initial-boundary value problem for Burgers' equation: [formula omitted] in arbitrary domains, with initial data in [formula omitted]. New a priori estimates are obtained. Adapting and refining known theory for Navier-Stokes equations, the existence and uniqueness of bounded smooth solutions are established. As corollaries of the inequality and its proof, pointwise bounds are given for eigenfunctions of the Laplacian in terms of the corresponding eigenvalues in two- and three-dimensional domains. / Science, Faculty of / Mathematics, Department of / Graduate

In the study of the spaces (formula omitted) of functions for which the pth powers of all the derivatives up to order ℓ are summable in the domain Ω⊂R, it has been found that there are mutual relations between various spaces. These relations were developed under the name "embedding theorems". The first embedding theorem (for spaces (formula omitted) were proved by Sobolev [3]*. Subsequently these spaces became known as Sobolev Spaces. However, in the study of existence of solutions for well-posed boundary value problems, there arose the necessity to consider spaces of distributions: an example is the space dual to (formula omitted). For a thorough development of distributions see L. Schwarz's texts [4]. Furthermore, the classes of Sobolev spaces had to be widened to fractional values of ℓ, the latter spaces being particularly useful in the study of non-linear problems. This thesis follows the development of generalized Sobolev spaces as in Volevich and Panayakh [1]. In section I we prove the basic theorems in this formulation. In section II, the existence of a function is proved using the formulation of section II. The proof of the proposition in which a modification has been made was given by Agranovich and Vishik [2]. The proposition is essential to the applications of Sobolev spaces to differential operators. The result states that ll u ll µ ≤ constant ll u,Ω ll µ for (formula omitted) for the particular case when the weighting function is (formula omitted) and Ω is a half-line. (For definitions see section I). Section III is devoted to a brief comparison of this formulation of Sobolev spaces to other approaches. / Science, Faculty of / Mathematics, Department of / Graduate

Two aspects of the theory of barotropic modons are examined in this thesis. First, sufficient neutral stability conditions are derived in the form of an integral constraint for westward and eastward-travelling modons. It is shown that eastward-travelling and westward-travelling modons are neutrally stable to perturbations in which the energy is contained mainly in spectral components with wavenumber magnitudes (|ƞ|) satisfying |ƞ|<κ and |ƞ|>κ, respectively, where κ is the modon wavenumber. These results imply that when κ/|ƞ|>1 the slope of the neutral stability curve proposed by McWilliams et al.(l98l) for eastward-travelling modons must begin to increase as κ/|ƞ| increases. The neutral stability condition is computed with mesoscale wavenumber eddy energy spectra representative of the atmosphere and ocean. Eastward-travelling atmospheric modons are neutrally stable to the observed seasonally- and annually-averaged atmospheric eddies. The neutral stability of westward-travelling atmospheric modons and oceanic modons cannot be inferred on the basis of the observed wavenumber eddy energy spectra for the atmosphere and ocean. Second, a leading order perturbation theory is developed to describe the propagation of barotropic modons in a slowly varying medium. Two problems are posed and solved. A perturbation solution is obtained describing the propagation of an eastward-travelling modon modulated by a weak bottom Ekman boundary layer. The results predict that the modon radius and translation speed decay exponentially and that the modon wavenumber increases exponentially, resulting in an exponential amplitude decay in the streamfunction and vorticity. These results agree with the numerical solution of McWilliams et al.(l98l). A leading order perturbation theory is also developed describing modon propagation over slowly varying topography. Nonlinear hyperbolic equations are derived to describe the evolution of the slowly varying modon radius, translation speed and wavenumber for arbitrary finite-amplitude topography. To leading order, the modon is unaffected by meridional gradients in topography. Analytical perturbation solutions for the modon radius, translation speed and wavenumber are obtained for small-amplitude topography. The perturbations take the form of westward and eastward-travelling transients and a stationary component proportional to the topography. The general solution is applied to ridge-like and escarpment-like topographic configurations. / Science, Faculty of / Mathematics, Department of / Graduate

"This paper is a study of finite systems of homogenous linear inequalities, homogeneous linear equations, and nonhomogeneous equations. To each inequality or equation in one system there corresponds a nonnegative or unrestricted variable in the other and conversely. The array of coefficients in one system is the negative transpose in the other system. This duality furnishes the foundation for the duality of matrix games and linear programming"--Introduction. / Typescript. / "June, 1959." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: Paul J. McCarthy, Professor Directing Paper. / Includes bibliographical references (leaves 38-39).

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