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The History of the Calculus

The History of the Calculus

Date: 1945
Creator: Ashburn, Andrew
Description: The purpose of this essay is to trace the development of the concepts of the calculus from their first known appearance, through the formal invention of the method of the calculus in the second half of the seventeenth century, to our own day.
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Comparison of Some Mappings in Topology

Comparison of Some Mappings in Topology

Date: January 1964
Creator: Aslan, Farhad
Description: The main purpose of this paper is the study of transformations in topological space and relationships between special types of transformations.
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Uniqueness Results for the Infinite Unitary, Orthogonal and Associated Groups

Uniqueness Results for the Infinite Unitary, Orthogonal and Associated Groups

Date: May 2008
Creator: Atim, Alexandru Gabriel
Description: Let H be a separable infinite dimensional complex Hilbert space, let U(H) be the Polish topological group of unitary operators on H, let G be a Polish topological group and φ:G→U(H) an algebraic isomorphism. Then φ is a topological isomorphism. The same theorem holds for the projective unitary group, for the group of *-automorphisms of L(H) and for the complex isometry group. If H is a separable real Hilbert space with dim(H)≥3, the theorem is also true for the orthogonal group O(H), for the projective orthogonal group and for the real isometry group. The theorem fails for U(H) if H is finite dimensional complex Hilbert space.
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Contributions to Descriptive Set Theory

Contributions to Descriptive Set Theory

Date: August 2015
Creator: Atmai, Rachid
Description: In this dissertation we study closure properties of pointclasses, scales on sets of reals and the models L[T2n], which are very natural canonical inner models of ZFC. We first characterize projective-like hierarchies by their associated ordinals. This solves a conjecture of Steel and a conjecture of Kechris, Solovay, and Steel. The solution to the first conjecture allows us in particular to reprove a strong partition property result on the ordinal of a Steel pointclass and derive a new boundedness principle which could be useful in the study of the cardinal structure of L(R). We then develop new methods which produce lightface scales on certain sets of reals. The methods are inspired by Jackson’s proof of the Kechris-Martin theorem. We then generalize the Kechris-Martin Theorem to all the Π12n+1 pointclasses using Jackson’s theory of descriptions. This in turns allows us to characterize the sets of reals of a certain initial segment of the models L[T2n]. We then use this characterization and the generalization of Kechris-Martin theorem to show that the L[T2n] are unique. This generalizes previous work of Hjorth. We then characterize the L[T2n] in term of inner models theory, showing that they actually are constructible models over direct limit of ...
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Infinite Planar Graphs

Infinite Planar Graphs

Date: May 2000
Creator: Aurand, Eric William
Description: How many equivalence classes of geodesic rays does a graph contain? How many bounded automorphisms does a planar graph have? Neimayer and Watkins studied these two questions and answered them for a certain class of graphs. Using the concept of excess of a vertex, the class of graphs that Neimayer and Watkins studied are extended to include graphs with positive excess at each vertex. The results of this paper show that there are an uncountable number of geodesic fibers for graphs in this extended class and that for any graph in this extended class the only bounded automorphism is the identity automorphism.
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A Method for Approximating the Distributed Loads of an Airplane by Sets of Point Loads

A Method for Approximating the Distributed Loads of an Airplane by Sets of Point Loads

Date: 1957
Creator: Austin, Charles Wayne
Description: This paper gives the derivation of a method for determining the forces to be applied to these points which will simulate the load distributed over all the airplane.
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Uniformly σ-Finite Disintegrations of Measures

Uniformly σ-Finite Disintegrations of Measures

Date: August 2011
Creator: Backs, Karl
Description: A disintegration of measure is a common tool used in ergodic theory, probability, and descriptive set theory. The primary interest in this paper is in disintegrating σ-finite measures on standard Borel spaces into families of σ-finite measures. In 1984, Dorothy Maharam asked whether every such disintegration is uniformly σ-finite meaning that there exists a countable collection of Borel sets which simultaneously witnesses that every measure in the disintegration is σ-finite. Assuming Gödel’s axiom of constructability I provide answer Maharam's question by constructing a specific disintegration which is not uniformly σ-finite.
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π-regular Rings

π-regular Rings

Date: May 1993
Creator: Badawi, Ayman R.
Description: The dissertation focuses on the structure of π-regular (regular) rings.
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Complemented Subspaces of Bounded Linear Operators

Complemented Subspaces of Bounded Linear Operators

Date: August 2003
Creator: Bahreini Esfahani, Manijeh
Description: For many years mathematicians have been interested in the problem of whether an operator ideal is complemented in the space of all bounded linear operators. In this dissertation the complementation of various classes of operators in the space of all bounded linear operators is considered. This paper begins with a preliminary discussion of linear bounded operators as well as operator ideals. Let L(X, Y ) be a Banach space of all bounded linear operator between Banach spaces X and Y , K(X, Y ) be the space of all compact operators, and W(X, Y ) be the space of all weakly compact operators. We denote space all operator ideals by O.
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Level Curves of the Angle Function of a Positive Definite Symmetric Matrix

Level Curves of the Angle Function of a Positive Definite Symmetric Matrix

Access: Use of this item is restricted to the UNT Community.
Date: December 2009
Creator: Bajracharya, Neeraj
Description: Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(-1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following question: if A and B are commuting positive definite symmetric matrices such that p(A) + p(B) is obtuse, what is the minimal p(S) such that {A, SBS^(-1)} indefinite? In this dissertation we will describe the level curves of the angle function mapping a unit vector x to the angle between x and Ax for a 3 by 3 PDS matrix A, and discuss their interaction with those of a second such matrix.
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