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.
In this paper a direct, constructive proof of the equivalence of the Normal Algorithm and Turing machine using the Turing machines NAS (Normal Algorithm Simulator) and NAC (Normal Algorithm Converter) is presented. The Turing machine NAS can simulate any particular Normal Algorithm, and NAG can convert the quintuples of a...
In this paper, we are concerned with the very general notion of
a universal algebra. A universal algebra essentially consists of a set
A together with a possibly infinite set of finitary operations on. A.
Generally, these operations are related by means of equations, yielding
different algebraic structures such as...
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 classical theory of elasticity and plasticity does
not recognize explicitly the existence of a "transition
zone" between elastic and plastic states, which instead,
makes extensive use of ad-hoc, semi-empirical laws, such as
yield conditions, at the "yield surface" to match both the
extreme states. In the present investigation, it...
This study undertakes to determine the existence or nonexistence
of an implication in either direction between any two out of
nine different modes of convergence, with the use of any subset of a
set of ten auxiliary hypotheses. The functions are real finite-valued
measurable functions defined on an arbitrary abstract...
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...
This paper records a study of two quadratic number fields. In the first field, denoted by Ra[[square root] 11], the unique factorization theorem holds. In the second field, denoted by Ra[[square root] 10], it is demonstrated that the unique factorization theorem does not hold and therefore ideals are introduced to...
The electromagnetic field in a cone of arbitrary slant
height with a symmetrically placed time harmonic ring source is
studied. Through the use of the modified Helmholtz equation as
an intermediate, we obtain the solution of the semi-infinite
cone directly from the finite cone. To demonstrate the need
for the...
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...
A technique of differentiation with respect to the distance to
the boundary of an outer parallel-body is applied to known measures of
sets of p-dimensional linear spaces which intersect a general convex
body in n-dimensional euclidean space in order to obtain an appropriate
definition of the measures of sets of...
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.
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...
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...
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...
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...
The electromagnetic field produced by a line current oriented
parallel to the edges of two perfectly conducting parallel half
planes is considered. Maxwell's equations reduce to a single wave
equation involving only one component of the electric field. Moreover
the value of the field is zero on the two half...
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...
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...
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.
This paper makes available practical algorithms and their associated
FORTRAN IV computer programs for finding the roots of
polynomial equations.
The purpose of this paper is to examine effective algorithms
for solving polynomial algebraic equations in one unknown on a digital
computer. The advent of high - speed digital computing...
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 background for this paper is the use of quadrature formulas
for the solution of ordinary differential equations. If we know the
values of the dependent variable for which we are solving, and its
derivative, at several equally spaced points, i.e., at values of the independent
variable separated by equal...
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 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.
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.
The Schwarz-Christoffel transformation is used to map simply connected polygons onto the upper half plane and can be applied to problems in which the validity of Laplace's equation is assumed. However the direct integration of the resulting complex integral is often not possible and some approximate method must be used....
This thesis is concerned with the detailed derivation
of Sheppard's Correction Formula relating the true to the
raw moments by the use of Euler-MacLaurin formula. Special
moment relationships have been derived using different
special functions, whose application is made to derive correction
formula for the semi-invariants. Special investigation
is carried...
In this thesis we consider computer techniques for inverting
n X n matrices and linear Fredholm integral operators of the
second kind. We develop techniques which allow us to prove the
existence of and find approximations to inverses for the above
types of operators. In addition, we are able to...
It is well -known that a topology for a space can be described in
terms of neighborhood systems, closed sets, closure operator or convergence
as well as open sets. In fact, it is also possible to describe
a topology in terms of interior operator or boundary operator. This
paper is...
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...
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...
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...
This paper defines and discusses some of the separation
axioms of topological spaces. In the cases considered,
a search is made for sets of conditions which
would be equivalent in a space satisfying a given separation
axiom to the existence of a family of real valued,
continuous functions which separates...
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...
The work in this thesis falls into two parts. The first part
presents a rigorous theory of probability densities and correlation
functions within the framework of the exponential spaces of D. S.
Carter. The second part extends this discussion to include infinite
systems.
The first part begins by considering an...
In a discrete review inventory process, when the demand
forms a stochastically convergent sequence of random variables,
it seems reasonable that the optimal stationary (s, S) inventory
policy will be a function of the limiting demand and cost structure
only. The intent of this paper is to provide a rigorous...
A theory of straight line triangulations of points in the plane is developed. A basic transformation is presented, and it is shown that any triangulation may be transformed into any other triangulation which has the same boundary by a finite sequence of the basic transformations. The proof of the transformation...
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...
A geometric condition on differentiable maps is given which is
equivalent to the set of critical values being nowhere dense. In particular,
the geometric condition is satisfied for radially ductile maps.
On the other hand it is proved that the induced map on the de Rham
complex will be a...
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...
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,...
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....
In this paper we use the set of all positive integers as a sample space whose probability density function is unknown. Then a generalization of the probability distribution of the most significant digits of the set of all physical constants is obtained on the strength of (i) a very general...
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...
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 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...
General techniques for constructing Scheduling Algorithms
are described as well as their applications to two diverse problems,
namely the scheduling at a university and the imitation of the activity
of a taxonomist in microbiology.
The objective of this paper is to seek the general solution of the
complete Navier-Stokes equations governing heterogeneous, time-dependent, incompressible, viscous, laminar fluid dynamics through
digital simulation. In particular, two-layered stratified flows are
treated by using a numerical algorithm called the Marker-and-Cell
extended method (MACE), which is an extension of...
This thesis examines various net or finite difference methods
for solving parabolic partial differential equations in one space variable
with constant coefficients. Included in this investigation are
explicit, implicit and multi-step methods of varying orders of accuracy.
These methods are compared with respect to accuracy, speed,
efficiency, stability, simplicity of...
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.
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.
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...
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...