Victor M. Bogdan

  1. Generalized vectorial Lebesgue and Bochner integration theory.

    Authors: Victor M. Bogdan
    Subjects: Functional Analysis
    Abstract

    This paper contains a development of the Theory of Lebesgue and Bochner
    spaces of summable functions. It represents a synthesis of the results due to
    H. Lebesgue, S. Banach, S. Bochner, G. Fubini, S. Saks, F. Riesz, N. Dunford,
    P. Halmos, and other contributors to this theory.

    The construction of the theory is based on the notion of a measure on a
    prering of sets in any abstract space X. No topological structure of the space
    X is required for the development of the theory.

  2. Representation of fields associated with any moving point mass by means of fundamental fields corresponding to its trajectory in the frame of Einstein's special theory of relativity.

    Authors: Victor M. Bogdan
    Subjects: Mathematical Physics
    Abstract

    Assume that in a Lorentzian frame is given a relativistically admissible
    trajectory of a point mass. An event in such a frame can be described by four
    coordinates, first three representing the position and the last one the time of
    the event. Let G denote the set of all events that do not lie on the
    trajectory.

  3. On Henri Cartan's vectorial mean-value theorem and its applications to Lipschitzian operators and generalized Lebesgue-Bochner-Stieltjes integration theory.

    Authors: Victor M. Bogdan
    Subjects: Functional Analysis
    Abstract

    H. Cartan in his book on differential calculus proved a theorem generalizing
    a Cauchy's mean-value theorem to the case of functions taking values in a
    Banach space.

    Cartan used this theorem in a masterful way to develop the entire theory of
    differential calculus and theory of differential equations in finite and
    infinite dimensional Banach spaces.

  4. Relativistic gravity fields and electromagnetic fields generated by flows of matter.

    Authors: Victor M. Bogdan
    Subjects: Mathematical Physics
    Abstract

    One of the highlight of this note is that the author presents the
    relativistic gravity field that Einstein was looking for. The field is a
    byproduct of the matter in motion. This field can include both the discrete and
    continuous components. In free space the waves produced in this field propagate
    with velocity of light.

    Another highlight is the proof of amended Feynman's formulas for
    electromagnetic potentials. This makes the formulas mathematically complete and
    precise.

  5. Electromagnetic field generated by plasma flows and Feynman and Lienard-Wiechert formulas for a moving point charge.

    Authors: Victor M. Bogdan
    Subjects: Mathematical Physics
    Abstract

    In a fixed Lorentzian frame given is a trajectory $r_2(t,r_0)$ of moving
    plasma such that each line $t\mapsto r_2(t,r_0)$ of flow has derivatives with
    respect to $t$ of order 3. The parameter $r_0$ represents the position of the
    plasma at some initial time $t_0$ and changes in a compact set $F.$ It is
    assumed that the map $(t,r_0)\mapsto r_2(t,r_0)$ is continuous and for fixed
    $t$ the map $r_0\mapsto r_2(t,r_0)$ is one-to-one on $F.$

  6. Electromagnetic field generated by plasma flows and Feynman and Lienard-Wiechert formulas for a moving point charge.

    Authors: Victor M. Bogdan
    Subjects: Mathematical Physics
    Abstract

    In a fixed Lorentzian frame given is a trajectory $r_2(t,r_0)$ of moving
    plasma such that each line $t\mapsto r_2(t,r_0)$ of flow has derivatives with
    respect to $t$ of order 3. The parameter $r_0$ represents the position of the
    plasma at some initial time $t_0$ and changes in a compact set $F.$ It is
    assumed that the map $(t,r_0)\mapsto r_2(t,r_0)$ is continuous and for fixed
    $t$ the map $r_0\mapsto r_2(t,r_0)$ is one-to-one on $F.$

  7. Fields generated by a moving relativistic point mass and mathematical correction to Feynman's law.

    Authors: Victor M. Bogdan
    Subjects: Mathematical Physics
    Abstract

    In this paper the author shows that, after replacing the ordinary derivative
    with respect to time by partial derivative, and treating all quantities
    appearing in Feynman's Law as fields, one can prove that the corrected formula
    generates electromagnetic field satisfying Maxwell equations. The main result
    of the paper is contained in the following Bogdan-Feynman theorem:

  8. The Existence and Uniqueness of Solutions to N-Body Problem of Electrodynamics.

    Authors: Victor M. Bogdan
    Subjects: Mathematical Physics
    Abstract

    Given $n$ charges interacting with each other according to Feynman's law. Let
    $(r_j(t),v_j(t))$ denote the position and velocity of the charge $q_j.$ The
    list $y(t)$ of all such vectors is called a trajectory. A Lipschitzian
    trajectory $x(t), (t\le0),$ with continuous derivative, on which the velocities
    do not exceed some limiting velocity $v<c,$ where $c$ denotes the speed of
    light, is called an initial trajectory. A locally Lipschitzian trajectory
    $y(t)$ is called relativistically admissible if the velocities on it stay below
    the speed of light $c.$

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