This paper presents an exposition of the stochastic models
for the Brownian motion. The results of Einstein and Wiener are
presented, together with the Uhlenbeck-Ornstein process which
gives a more realistic model of the Brownian motion of a particle.
Finally, applying a one-one transformation on the forward
Kolmogorov equation we...
The thesis discusses stability of procedures based on linear
computing formulas for numerical integration of an ordinary first-order
differential equation. The theorems are proved: (1) If the
procedure is asymptotically stable it is stable for small positive step
size if the Lipschitz number is negative; (2) Relative stability always
exists...
In his book on abstract algebra, Nathan Jacobson
poses and solves the problem of finding the number of
ways of inserting parentheses in a string of given length
with binary operators. We continue the work of Jacobson
and go beyond it in that we no longer consider one binary
operator...
Properties of Mellin transforms are applied to the summation
of infinite series. A specific example of this application leads to the
establishment of certain properties of a generalization of Lerch's zeta
function.
Nine powerful methods of generating new tables of Mellin
transforms from existing tables of Mellin and other integral...
Computational scheme, equivalence, and Turing machine are
defined. Some computational schemes are examined and shown to
be equivalent to the computational scheme of a Turing machine.
In this paper we consider the behavior of certain surfaces at
certain boundary points. The surfaces under consideration satisfy
a topological definition and are of 2-dimension in 3-dimensional
Euclidean space with the boundary a finite set of straight line
segments. It is shown that the surface of minimum area with...
This thesis treats the problem of enumerating equivalence
classes of Euler paths of full graphs. A full graph
is a complete, unordered, graph with no loops or repeated
edges. Two Euler paths are equivalent if and only if one
can be transformed into the other by a finite sequence of...
The indicated sum of a real scalar and a real or imaginary
vector is called a scator. Either the scalar part or the vector part
may be null. Scators generalize the complex variable to n-space.
The algebra of scators is not generally associative under multiplication
but the commutative and distributive...
An algorithm is described for determining the vertices
and supporting planes (or lines) of the convex hull
of a given set of N distinct points in 3-space. The
method involves determining a finite sequence of convex
hulls, each successive convex hull being a better approximation
to the convex hull of...
The Euler-MacLaurin sum formula has appeared in the titles
of two quite recent papers whose authors were primarily interested
in certain applications. In this paper a somewhat different approach
to the myriad of formulas for summation, integration, differentiation,
etc. , is based on the simple identity which defines the set...
A new procedure is developed for computing a root of algebraic equations with real coefficients and a degree n, where n is 2, 4, 6, 10, 14 or any positive odd integer. A heuristic procedure is added to partially lift the restrictions on the degree n. The procedures are written...
The Cantor set is a compact, totally disconnected, perfect
subset of the real line. In this paper it is shown that two non-empty,
compact, totally disconnected, perfect metric spaces are homeomorphic.
Furthermore, a subset of the real line is homeomorphic
to the Cantor set if and only if it is...
It is
well
known
that
two-terminal
switching
circuits
may
be
represented
by
boolean
formulas.
Thus
the
study
of
certain
switching
circuit
problems
leads
to
the
study of
free
boolean
algebras,
in
particular
to
the
free
boolean
algebra
on
a
countably
infinite
set
of
generators.
An
abstract
characterization
of
this
algebra...
Since the advent of the modern computer there has been great interest in simulation of all types. It is not possible to simulate very large systems on computers. This thesis presents the algorithms necessary to simulate a college with respect to the academic relationship of the student with the college....
Row equivalence, equivalence, and similarity of matrices are studied; some problems concerning an extension of these relations to infinite matrices are discussed.
This thesis contains a collection of properties of the greatest integer part function which were obtained by an extensive literature search. A few original properties are stated and proved and some of the properties which were found unproved in the literature are proved.
One answer to the problem of missing observations in two-way
classification experiments is to insert estimates of missing observations
into deficient cells. Once missing observations have been estimated,
the experimenter may proceed with his analysis using the
familiar normal equations which apply to complete data. This paper
discusses generally the...
The author develops an algorithm for the recursive reduction
of Sylvester's determinant and the determination of the eliminant of
three algebraic equations in three unknowns. The ALGOL 60
language is used for the description of this algorithm.
As indicated by the title, this thesis generalizes the Main Inertia
Theorem of Ostrowski and Schneider [8]. The first three results
concern the formation of a polynomial function f(A, A*, H) so that
the existence of an hermitian H for which f(A, A*, H) is positive
definite is a necessary...
Three special cases of the resection problem of surveying
are examined and solved. The coordinates of unknown points are
found with respect to given points in a rectangular coordinate system.
This is accomplished in the case of (a) one unknown point and three
given points (Snell's problem), (b) two unknown...
The author shows that a necessary and sufficient condition
for a convex polyhedron to be representable as a finite vector sum
of line segments is that each of its faces possesses central symmetry.
The objective of this paper is to describe the design of a real-time system to control two hydroelectric power plants by a digital
computer. The two controlling relations are stated as the continuity
of water flow and the inequality of power generation requirements.
It contains the logical sequence in finding...
This paper is devoted to the study of families of tangent lines
to curves in Euclidean three dimensional space by the medium of a
particular representation for lines.
First, a ring of elements called dual numbers is described,
and a vector space over this ring, whose elements are called dual...
In the study of uniform convergence, one is led naturally to
the question of how uniform convergence on subsets relates to uniform convergence on the whole space. This paper develops theorems on how pointwise convergence relates to uniform convergence
on finite sets, how uniform convergence on finite subsets relates to...
The exponential family of probability distributions is obtained
from σ-finite measures on the real line. We choose the parameter
space to be one-dimensional and the exponent to be linear. Relationships
between the measure, its spectrum, and the parameter space
are examined, moments of the exponential family are studied, and
the...
This work contains a brief history of the four color problem
from 1840 to 1890. This includes Kempe's attempted proof of the
problem as well as maps which illustrate Heawood's discussion of
Kempe's error. The remaining part is a discussion of Kempe's and
Story's work on patching out maps. Story...
The paper concerns itself with the problem of heat transport
in a homogeneous, incompressible, isotropic, semi-infinite solid
whose boundary is nonstationary. The methods of solution in the one
dimensional case are discussed. Numerical examples and a method
for arriving at the solutions published by Bailey and Lakin in their
paper...
This thesis has four main results. First we find a reduction form
for symmetric matrices over fields of characteristic two. This result
parallels the diagonalization theorem for symmetric matrices over
fields of characteristic not two.
Secondly we reduce our reduction form to a canonical form in
perfect fields of characteristic...
G. F. Drukarev has given a method for solving the Fredholm
equations which arise in the study of collisions between electrons
and atoms. He transforms the Fredholm equations into Volterra
equations plus finite algebraic systems. H. Brysk observes that
Drukarev's method applies generally to a Fredholm integral equation
(I-λ G)u...
The author studies the class of rectangular arrangements in
terms of two binary relations on the objects of the arrangement.
He shows how a univalent matrix determines a unique rectangular
arrangement, and how each rectangular arrangement is associated
with one, two, or four distinct matrices, according to the number
of...
Some of the properties of the numbers of two quadratic number
fields are explored. Among these properties is the existence of
unique prime factorization of the integers of the field and the importance
of the concept of ideal numbers in restoring unique factorization
when it does not exist. Some consideration...
In this thesis we consider boundary and initial value problems
associated with the melting of homogeneous cylinders and
spheres. The study includes the temperature distribution in cylinders
and spheres which are perfectly insulated at the surface and
subjected to a heat input respectively along the axis and at the
center....
Proving mathematical theorems usually involves the proof of
an implication, p --> q. Often it is convenient to prove the implication
by proving one which is equivalent to or "stronger" than the original
theorem. Proofs of this type are called indirect proofs.
In Chapter I five forms of indirect proofs...
Let A and B be two subsets of the set of all non-negative
integers with 0 ε A and O ε B. The sum of the sets A and B is
the set C = A + B = {a + b: a ε A, b ε B). For n...
This thesis brings together under one cover a survey of the
history of the real number pi along with a study of the present state
of its theory, application, and calculation.
In
this
paper
we
suppose
that {f[subscript n]}[subscript n ε N] is
a
countable
family
of
mappings
of
a
topological
space
into
itself
and
investigate
the
pointwise
convergence
of
the
sequence
<F[subscript n]> defined
by
either F₁ = f₁, F[subscript n]₊₁ = f[subscript n]₊₁°F[subscript n] n = 1, 2, 3,...
This thesis develops recursion formulas for least-squares
data smoothing with regard to four classes of functions: linear,
quadratic, exponential and trigonometric, which in a linear fashion
involve 2, 3, 2 and 2 parameters respectively. The term recursion
implies here that the estimates for the parameters of the preceding
fit are...
In 1932 A. Ya. Khinchin gave the first partial solution of the celebrated 1931 αβ Conjecture of L.G. Schnirelmann and E. Landau of the density of sums of sets on integers, which was completely proved in 1942 by H.B. Mann.
Khinchin's theorem is proved along with theorems of P. Scherk...
This paper is a classification of 181 problems, each of which involves the evaluation or application of determinants. The problems are those which have appeared in the "Problems and Solutions" sections of the American Mathematical Monthly, Deutsche Mathematiker-Vereinigung Jahresbricht, Mathematics Magazine, and School Science and Mathematics. The problems are classified,...
This report presents a characterization of the quantum mechanical
analog of the Gibbs canonical density. The approach is
based on a method developed by D.S. Carter for the case of classical
statistical mechanics, which considers composite mechanical systems
composed of mechanically and statistically independent components.
After a brief introductory chapter,...
It is well- known that a real number can be defined as an equivalence
class of fundamental rational sequences. In fact, it is also possible
to define a real number as an equivalence class of sequences of
nested closed rational intervals. This paper is devoted to the latter
case.
The computation of symmetric functions can be tedious.
For this reason, the object of this paper is to devise a computer
program so that these symmetric functions can be handled automatically.
The contribution of this investigation is a heuristic for
finding the polynomials proved to exist by the Fundamental Theorem...
This thesis contains a collection of summation formulas for
the greatest integer part function. Proofs are supplied for original
results and for those formulas which are stated without proof in the
literature. References are given for formulas and proofs which
appear in the literature.
Various properties of the integers in two quadratic number
fields will be examined. Among these will be the property of unique
prime factorization. When unique prime factorization breaks down,
as will be the case in one of the quadratic number fields, the concept
of ideal numbers will be introduced.
unique...
A numerical solution to Hodgkin and Huxley's partial differential
system for the propagated action potential is presented. In
addition a three dimensional demonstration of the absolute refractory
period is given. Lastly, theoretical evidence supporting
Rushton's hypothesis is presented.
FORTRAN is readily feasible to translation into ALGOL since
they share many common features. Most of the features that are
unique to FORTRAN can be translated by restricting them somewhat.
The translator will handle explicit declarations of each item
in a block, compensate for the differences in various operators,
compensate...
Consider a transformation group G operating on a space X
and a G- invariant function f defined on a G- invariant subset of
X. By imposing suitable conditions on X, G, f and A, the
author derives sufficient conditions for extending f invariantly to
the whole space, and thus generalizing...
This thesis studies the question of unique factorization in
quadratic integral domains. In the first chapter many general
theorems and definitions from algebraic number theory are introduced.
The second chapter considers an integral domain in which
unique factorization holds. The necessary theorems to prove
unique factorization are developed. The third...
The three important methods of approximation; interpolation,
least- squares, and Chebyshev, are extended into bivariate approximations.
A method of obtaining polynomial approximations for very
general classes of bivariate samples is developed. Bivariate least -
square approximations are reviewed and a method of developing bibariate
orthogonal sequence is derived. A method...
This thesis describes and develops from an axiomatic basis
three principles of logic and some forms of indirect proof. The
three principles are: (1) the law of the excluded middle, (2) the law
of contradiction, and (3) the law of the double negative. Of several
forms of indirect proof, this...
In this paper we present a model of Non-Euclidean geometry
in three dimensions. This will show that the axioms of Non-Euclidean
geometry are consistent if Euclidean geometry and, hence, arithmetic
is consistent. However, the model is incomplete for we have not
included the topic of congruence, the axiom of Archimedes,...
It is well known that every Banach space with a Schauder basis
is separable. However, whether the converse of the above statement
is true is not known.
It is therefore the purpose of this thesis to investigate the question
of under what conditions an arbitrary, separable, Banach space
has a...
Before we apply the laws of probability to hiring practices, a
foundation of basic probability theory will be presented. In this
presentation a number of theorems related to probability will be
proven. These theorems are not necessarily applicable to the problem
which follows; however, they are basic to probability theory...
In this thesis, we study conditions not involving density which
guarantee that a given positive integer is contained in a sum of sets
of nonnegative integers. We survey the literature, give more detailed
proofs of some known theorems, develop some new theorems,
and make some conjectures.
The enclosed thesis presents an algorithm, programmed
in FORTRAN IV, which generates the radii for
the objective of an astronomical refracting telescope.
The results are analyzed by the Trigonometric Ray
Trace, and found to have tolerable amounts of spherical
aberration, chromatic aberration and coma.
In this paper we investigate the Lyapunov mapping
P --> AP + PA *
where A is a positive stable matrix and P is a hermitian
matrix. In particular, for special positive stable A we
characterize the image of the cone of positive definite matrices
under this mapping. In Section...
The system of partial differential equations which governs the
motion of a Newtonian fluid has been known for over a century. Yet,
due to the complexity of the equations, an analytical solution is known
only for a few simple geometries or a few special cases such as very
slow motion....
This paper is the record of an exploration of two quadratic
number fields. The first section is devoted to the field with elements
of the form a +b√3 where a and b are rational numbers.
This field contains an integral domain in which unique factorization
holds. The second section is...