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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 28 Nov 2023 18:51:57 +0000Contributions to the Theory of Functionals
https://resolver.caltech.edu/CaltechTHESIS:12222021-155415955
Authors: Martin, Robert Samuel
Year: 1932
DOI: 10.7907/9rtv-x659
<p>Part 1. Polynomials and analytics in vector space</p>
<p>Part. 2. Various theorems on the representation of functional forms and transformations by means of Stieltjes integrals</p>https://thesis.library.caltech.edu/id/eprint/14459Analytic Functions in General Analysis
https://resolver.caltech.edu/CaltechTHESIS:12132017-090831567
Authors: Taylor, Angus Ellis
Year: 1936
DOI: 10.7907/V5AY-TP78
<p>The theory of functions of a complex variable is distinguished
from the theory of functions of a real variable by its simplicity - a
simplicity is directly traceable to the complexity of the variable.
Two of the remarkable simplicities of the theory are, first, that from
the assumption that f(z) is differentiable throughout the neighborhood
of a point z = z<sub>0</sub> follows the existence of all higher derivatives and the
convergence of the Taylor's series for f(z); and secondly, that we are
able to classify in simple terms the possible singularities of an
analytic function.</p>
<p>It is the purpose of this work to generalize, insofar as is
possible, the basic theorems of the classical theory, and to investigate
in what measure the simplicities mentioned above are preserved when the
arguments and function values lie in a Banach space. Of the three principally
recognized points of view which are used in developing the theory
of analytic functions we have used mainly the one due to Cauchy, which
finds its natural extension in the ideas of Gateaux concerning differentials.
Much of the work which we present was sketched in a memoir of
Gateaux on functionals of continuous functions.* In addition we have developed
the "Weierstrassian" properties of analytic functions, using as a foundation
the notion of polynomial as set forth by R.S. Martin.**
Finally, a brief section is devoted to a generalization of the Cauchy-Riemann
equations. Nothing has been done with the implicitly suggested
theory of pairs of conjugate harmonic functions, however.</p>
<p>The study of differentials leads to an important result showing
the relation of the Fréchet and Gateaux concepts of a differential.</p>
<p>The classification of singular points is a most difficult problem.
We have dealt completely with removable singularities, and showed to
some extent the departures from classical theory which are caused by the
generalization here undertaken. A more detailed investigation should be
carried out in special cases.</p>
<p>I freely express my admiration for the treaties of Professor
W.F. Osgood, Lehrbuch der Funktionentheorie, to which I have had constant
recourse in the writing of this thesis. Many of the proofs are directly
carried over, with only the slight changes made necessary by the abstract
nature of the quantities in hand.</p>
<p>To professor A.D. Michal I am indebted for encouragement and advice
at all times.</p>https://thesis.library.caltech.edu/id/eprint/10605Analysis in Linear Topological Spaces
https://resolver.caltech.edu/CaltechTHESIS:07052023-222228608
Authors: Paxson, Edwin Woolman
Year: 1937
DOI: 10.7907/j0h4-j979
<p>During the past five years the study of linear
topological spaces, those non-metrisable spaces intermediate
between the pure spaces of ensemble topology and the normed
spaces of General Analysis, has received much attention,
particularly under the hands of the Russian and Polish schools
headed by Kolmogoroff and Tychonoff. Interest in the slightly
weaker spaces of topological group type has also been greatly
stimulated by the search for strong purely topological foundations
of group theory by Schreier, von Dantzig and others,
and for an abstract formulation of continuous group theory as
in the work of Michal and Elconin.</p>
<p>Because of the implied necessity of theories in the
large for situations such as these, it was felt desirable to
determine what analytic entities could be defined, while preserving
as large a portion of the usual properties as possible,
directly for the base spaces, without the rigid local intermediary
of the norm.</p>
<p>I should like to express here my appreciation of the
sustained assistance and advice of Professor A.D. Michal in the
development of this thesis.</p>https://thesis.library.caltech.edu/id/eprint/16134A General Differential Geometry with Two Types of Linear Connection
https://resolver.caltech.edu/CaltechETD:etd-08262008-145426
Authors: Wyman, Max
Year: 1940
DOI: 10.7907/92JW-1918
The object of this thesis was the study of a differential geometry for a Hausdorff space endowed with an affine linear connection and a non-holonomic linear connection. The coordinate spaces were taken to be Banach spaces. In Chapter II we define the notion of a non-holonomic contravariant vector field, and by means of the non-holonomic linear connection introduce the operation of covariant differentiation. It was then found that many of the formal tensor theorems carried over to such spaces.
For certain types of Hausdorff space it is possible to develop a normal representation theory, and by means of it to obtain normal non-holonomic vector forms. This then enables us to generalize the Michal-Hyers replacement theorem for differential invariants.
Chapter IV is concerned with the determination of nonholonomic linear connections. This leads to the consideration of interspace adjoints for linear functions.
In the main the results obtained in this thesis are generalizations of results obtained for finite dimensional spaces by A.D. Michal and J. L. Botsford. However the projective theory developed in Chapter V is new for spaces of finite dimension.
https://thesis.library.caltech.edu/id/eprint/3234Differential Geometry of a Space with a Two-Point Differential Metric
https://resolver.caltech.edu/CaltechETD:etd-08262008-151031
Authors: Lass, Harry
Year: 1948
DOI: 10.7907/8JJQ-1Y83
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
In this thesis we have generalized the Riemannian line element […] to the case where […] is a function of two points, x1, x2, and we consider the differential geometry of the line element […].
The extremalizing of L = […] leads to a pair of curves […], called dyodesics, these curves being obvious generalizations of the geodesics of Riemannian geometry. A projective geometry of these paths is then investigated.
We then introduce a concept of parallel displacement of vectors relative to two paths […] which is directly analagous to parallel displacement in a Riemannian space. Parallel displacement is found to depend in a very natural way on six fundamental two-point tensors, the vanishing of these tensors implying that the space is flat, and for this case the dyodesics take the simple forms […] for special coordinate systems.
From the definition of parallel displacement arises a method for generating new two-point tensor fields by a process equivalent to covariant differentiation in Riemannian geometry. Parallel vector fields and ennuples of vectors are then introduced. It is shown that the ennuples […], […], form parallel vector fields for the metric space […]. We then define parallel displacement in sub-spaces and introduce a generalized covariant differentiation process, this last enabling us to develop second fundamental forms for hyper-surfaces. It is found that special and important types of coordinate systems may be set up independently at the points M1, and M2. These coordinates enable us to generate new tensors by a method of extension. An equivalence problem is then studied.
Finally, a line element […] is introduced for two masses at M1, M2, the […] satisfying […], the T's corresponding to the Ricci tensor of Riemannian geometry. The dyodesics obtained for this space approximate the Einstein solution for the one body problem when the mass of the particle at M, is small compared with that at M2. The motion for two equal masses differs from that obtained by Robertson in his solution of the equations of motion obtained by Einstein, Infeld, and Hoffman. The difference lies in the yet undetermined periastron effect for double stars.https://thesis.library.caltech.edu/id/eprint/3235Quadratic Differential Equations in Banach Spaces and Analytic Functionals
https://resolver.caltech.edu/CaltechETD:etd-06242004-135659
Authors: Reed, Irving Stoy
Year: 1949
DOI: 10.7907/TEVV-6V18
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
In Chapter I the preliminary concepts are introduced and applied to the quadratic differential equations, [...], in a complete normed linear ring. In Chapters II and V the functional equation, [...] is studied under suitable assumptions and shown to be related to the Frechet differential of the above differential equation as a functional of [...]. In Chapter IV the more general functional equation, [...], where T is an endomorphism, is treated.
The existence of the Frechet differential with respect to [...] is shown in Chapter II by means of a power series for the solution of [...] where Q is a trilinear function. The solution of [...] is obtained in Chapter VI in the terminology of Chapter IV. Moreover, the solution is shown to satisfy unquely the differential system, [...], [...] and to possess a generalized Taylor series expansion. The above equation is generalized with similar results to [...], where T is a trilinear function.
Chapter VII is concerned with examples of Chapter VI. For instance, the solution of the matrix differential equation, [...], [...], is treated both as a function of [...] and as an analytic functional of the [...] continuous functions [...].https://thesis.library.caltech.edu/id/eprint/2710Contributions to Tensor Analysis
https://resolver.caltech.edu/CaltechETD:etd-10022006-144129
Authors: Guy, William Thomas
Year: 1951
DOI: 10.7907/XC95-GY46
This thesis treats two separate problems. The first concerns the transverse vibrations of a beam and of a thin rectangular flat plate. These vibrations are associated with a function space which has the properties of a generalized "Riemannian" function space. The geodesics of this space are shown to play a role analogous to that played by the geodesics of the configuration space in the classical treatment of the finite dimensional case. Part I is introductory and treats a few aspects of the vibrations of beams with various end conditions under a change of parameter. Part II develops the integro-differential equation for the thin rectangular flat plate. The associated function space and its geodesics are then studied in some detail. The space is found to be not one of constant Riemannian curvature. An example is worked out to illustrate the ideas, and an extension is suggested. The second problem (part III) considers the equations of motion of hydrodynamics of viscous flow with moving axes. Use is made of the space of a kinetic metric introduced by McVittie, who considered non-viscous flow only. The Newtonian equations are obtained by taking certain approximations. The equations of motion in terms of the vorticity tensor are developed. Two examples are discussed illustrating the theory, one concerning instability necessary for tropical cyclones.https://thesis.library.caltech.edu/id/eprint/3867A Type of Pseudo-Norm
https://resolver.caltech.edu/CaltechTHESIS:10132017-091346134
Authors: Diamond, Robert James
Year: 1951
DOI: 10.7907/EDZW-V217
<p>In mathematical literature, the term pseudo-norm has no one specific
definition but is used for functionals satisfying some but not all of the
postulates for a norm. The notion of such functionals or "pseudo-norms"
is common in the study of linear topological spaces,<sup>1)</sup> which, from one point
of view, may be regarded as generalizations of normed linear spaces. The
particular type of pseudo-norm considered in this thesis is the triangular
norm of Menger's "generalized vector space".<sup>2)</sup> Menger noticed that only
the triangle property of the norm was necessary in order to obtain certain
results in the calculus of variations, and thought that a linear space with
a generalized triangular "distance" might prove to be a fruitful concept.</p>
<p>We first consider spaces (type K, see text) which are more specialized
than those treated by Menger. In this thesis, spaces of the latter type
are termed "spaces of type G". Apart from the intrinsic interest of type K
spaces, certain aspects of their theory are applied in Chapter IV to the
treatment of spaces of type G.</p>
<p>In Chapter I a space of type K is defined, the independence of the
pseudo-norm postulates is established, and the question of the continuity
of the pseudo-norm is treated. In Chapter II the notion of equivalence
classes leads to a vector space of type K/Z, the existence of which depends
only on the presence of a pseudo-norm in K. The more general spaces of
type G are then introduced. A metric topology defined in terms of the
pseudo-norm is discussed in Chapter III and functionals linear with respect
to this topology are considered.</p>
<p>The question of the Gâteaux differentiability of the pseudo-norm is
taken up in Chapter IV and a connection is established between this
property and the existence of functionals linear in the topology of the
pseudo-norm.</p>
<p>Chapter V investigates connections between the pseudo-norm and
ordering relations in a real vector space. Conditions are found under which a
partial ordering can be defined in terms of a given pseudo-norm, and
conversely.</p>
<p>_____________________________________________________________________________________</p>
<p>1). See, for example, Hyers, Ref. 7; LaSalle, Ref. 9; von Neumann, Ref. 16;
Wehausen, Ref. 19.</p>
<p>2). Menger, Ref. 13, p 96.</p>https://thesis.library.caltech.edu/id/eprint/10513Relaxation Phenomena and the Origin of Earthquakes
https://resolver.caltech.edu/CaltechTHESIS:10132017-094352043
Authors: Lieber, Paul
Year: 1951
DOI: 10.7907/S59Y-Q172
<p>In this thesis an attempt is made to demonstrate in
accordance with known physical principles that significant
changes in the macroscopic equilibrium of the earth can be
attributed to rate processes which do not call upon the
existence of macroscopic gradients and disturbances in the earth.
Such processes and their relation to known mechanisms of
plastic deformation and to the experimentally established
behavior of materials under high pressures are critically
evaluated. This evaluation is carried out in the light of some
well-established concepts of statistical mechanics and modern
physics. In so doing specific methods for producing seismic
disturbances which are based upon known mechanisms of plastic
deformation and rupture become indicated. It is also shown that
under very high pressures, pressure and temperature can complement
each other in producing disturbances associated with
polymorphic transitions of materials leading to a reduction in
symmetry of their lattice structure.</p>
<p>A specific mechanism for producing and repeating earthquakes
at shallow and intermediate depths is proposed. This mechanism is
based upon the existence of a visco-elastic surface layer supported
by a plastic material embodied with stress
relaxing properties.</p>
<p>The effect of such a configuration upon the propagation
of Love Waves is investigated quantitatively. This investigation
shows that the plastic sub-layer would explain the observed selection
of the period of Love Waves.</p>https://thesis.library.caltech.edu/id/eprint/10514A General Similarity Theory of Partial Differential Equations and its Use in the Solution of Problems in Aeronautics
https://resolver.caltech.edu/CaltechETD:etd-03192009-091419
Authors: Morgan, Antony John Andrew
Year: 1951
DOI: 10.7907/9142-KZ49
A general similarity theory of systems of partial differential equations of any order in any number of independent variables is developed with the aid of the theory of continuous one-parameter groups of transformations. The theory is illustrated by means of several known examples of similarity equations, previously given without motivation, in Hydrodynamics. With the aid of the theory two new examples of similarity equations, one in Elasticity and one in Fluid Mechanics, have been found; these are discussed in the text.
https://thesis.library.caltech.edu/id/eprint/1024Transformation of Linear Spaces and Linear Operators by Inverse Reversion
https://resolver.caltech.edu/CaltechTHESIS:10112017-134914663
Authors: Elconin, Victor
Year: 1952
DOI: 10.7907/RV02-JW45
<p>This thesis develops a new method for transforming
and extending the classes of operators and operands which appear in
certain linear operations in such a way that restrictions on the
ranges and domains of the operands and on the algebraic manipulation
of the operators are reduced and removed. In particular, the
method leads to a complete rationalization of the P operators and
impulse 'functions' employed by Heaviside, Dirac and others in the
analysis of certain linear systems.</p>
<p>In this method, the operators A of a primary class K
are, in effect, first reversed, forming A<sup>*</sup>, then inverted,
forming A<sup>*-1</sup>, the inverse reverse of A, and these operators are
utilized to effect the remaining transformations and class
extensions. The method is therefore epitomized by the phrase
inverse reversion.</p>https://thesis.library.caltech.edu/id/eprint/10505Complex Function Theory for Functions with Arguments and Values in Locally Convex Linear Topological Spaces
https://resolver.caltech.edu/CaltechTHESIS:10042017-091604604
Authors: Bernholtz, Benjamin
Year: 1952
DOI: 10.7907/NSNX-7J43
In Chapter I a brief introduction to the basic notions of
locally convex linear topological spaces is given. In Chapter II,
a theory of analytic functions is developed for functions of a
complex variable with values in a sequentially complete locally
convex complex linear topological space (l.t.s). The theory is
sketched of continuous linear functions on one sequentially complete
locally convex complex l.t.s. to a second such space. In the same
spirit some theorems relating to functions of several complex
variables taking their values in a sequentially complete locally
convex complex l.t.s. are developed. In Chapter III, functions on
one sequentially complete locally convex complex l.t.s to a second
such space are studied and in particular notions of differentiability
and analyticity. An analogue of the Cauchy-Riemann theory of functions
of a complex variable is discussed.https://thesis.library.caltech.edu/id/eprint/10481