Browsing by Author "Jones, Frank"

Now showing 1 - 8 of 8
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    A characterization of the tail _-field for certain Markov chains
    (1972) Bachman, Howard Floyd; Jones, Frank
    If C(v,P) is a countable state, recurrent, aperiodic and irreducible Markov Chain with stationary probabilities, the measure of any set of the tail a-field is equal to either zero or one. Although recurrent Markov chains have trivial tail a-fields this is not in general true for transient chains. However the tail g-field for two merging independent Markov chains is trivial. Investigating the invariant measurable sets of 0 leads to the solution of a functional equation of the general form pf = f.
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    A class of symmetric two dimensional logarithmic potentials
    (1993) Shah, Vaishali Satish; Jones, Frank
    This thesis discusses the solution of an elliptical conductor problem in two dimensions. This problem can be easily solved in higher dimensions, but in two dimensions, the potential function is logarithmic near infinity. This thesis also discusses some particular cases of the solution obtained by solving the above problem.
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    An analog to the heat equation in complex space variables
    (1991) Tomlinson, Kathy Adiene; Jones, Frank
    Consider the operator$${\bf P} = {\partial\over\partial t} + \alpha{\partial m\over \partial z m},\qquad \alpha \in {\bf C} - \{0\}$$where $$\partial\over{\partial z}$$ is the usual complex operator:$${\partial\over\partial z} = {1\over 2}\ \left({\partial\over\partial x} - i{\partial\over\partial y}\right).$$When m = 2 and $\alpha$ = $-$1, P bears a remarkable resemblance to the heat operator in one space variable. The "only" difference is that the space variable is now complex. In spite of this superficial similarity, P is quite different from the heat operator. It is neither hypoelliptic nor parabolic. The key result is a formula for a fundamental solution, E. It is obtained formally using Fourier transforms. The formula is a linear combination of Fresnel-like integrals, divided by z and a power of t. It is a $$C\infty$$ function except across t = 0. It has a homogeneity property which is similar to the one the standard fundamental solution for the heat operator possesses. It has a skew-reflection property in the time variable. The proof that E is a fundamental solution is done by applying PE to a test function. It is similar to the standard analogous proof for the heat equation. The main difference is that E is not integrable for fixed non-zero t. Thus we do our calculations with Fourier transforms. This requires making some of the formal arguments in the derivation of E into rigorous ones. The basic tools for this are approximating functions, Cauchy's integral theorem, and Lebesgue's dominated convergence theorem.
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    An elementary proof of the spectral theorem for unbounded operators
    (1965) Bagby, Richard Julian; Jones, Frank
    One of the proofs of the spectral theorem for bounded operators begins by proving that a bounded, positive definite self-adjoint operator on a Hilbert space has a unique positive definite self-adjoint square root. From this result, I will show directly that an unbounded positive definite selfadjoint operator also has a unique square root. From this, I will derive the spectral theorem for unbounded self-adjoint operators. With this approach, the necessary results follow directly from elementary properties of operators on a Hilbert space. The resolution of the identity corresponding to an operator is obtained directly from the operator, rather than from the spectral resolution of related operators.
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    Estimates for singular integrals
    (1964) Lord, Michael Erle; Jones, Frank
    In a paper by Calderon and Zygmund some properties of a certain kind of singular integral are established, and these results are applied to particular fundamental solutions arising in partial differential equations. Jones has considered another class of singular integrals which has application to fundamental solutions of the heat equation. More generally the kernels considered by Jones arise from parabolic differential equations with constant coefficient. This thesis considers a kernel which is a generalization of the kernel treated by Jones, and it has mean value and homogenity properties analogous to those in Jones's paper.
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    On Garabedian's method of solving the wave equation
    (1995) Oehrlein, Chris; Jones, Frank; Wells, R. O.; Anderson, James
    In this thesis, we shall reexamine and provide as clear an exposition as possible of a method presented by P. R. Garabedian which results in an integral formula representation of a solution to the wave equation. The method involves analytically extending a harmonic function of real arguments along a purely imaginary axis in complex space and establishing the validity of the standard integral formula for harmonic functions as a representation of a solution to the Wave equation when one of the arguments is purely imaginary. This is done in the odd dimensional case by integration by parts and an application of the residue theorem, and in the even dimensional case by computing bounds on the integrals.
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    Relations between two classes of singular integrals
    (1964) Tu, Chang-Char; Jones, Frank
    In 1952, Calderdn and Zygmund published a paper in which they discussed a certain kind of singular integral, and used some of their results to establish differential properties of solutions of Poisson's equation. Recently, Jones made some observations on the fundamental solution of the heat equation and gave another class of singular integrals. In this thesis it is shown that a special case of Jones's kernel is of Calderon-Zygmund type and a real, even kernel considered in [2] is a kernel of Jones's type if the coordinate system is suitably chosen. Moreover, for such kernels, the difference of the convolution operators-- defined in the sense of Jones and in the sense of CalderonZygmund, respectively--is a constant multiple of the identity operator.
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    The determination of a coefficient in a parabolic equation cylindrical coordinates
    (1965) Gieszl, Louis Roger; Jones, Frank
    B. F. Jones (Ph.D. Thesis, Rice University, 1961) proved the existence and uniqueness of a solution of a one space variable diffusion equation ut a(t) uxx , where a(t) is an unknown function of time. This article considers the analogous problem for a cylindrical region with symmetry with respect to 8 . In particular, we consider the system (separately for r>1 and r<1) We take the five theorems in Jones' paper as Properties a through e ; and, by taking the appropriate bounds on the function M, we show that L defined by (5) satisfies the five properties. Thus, (1) has a unique solution.