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Pages
 Title
 GENERALIZED QUOTIENT RINGS AND ZERODIVISORPRESERVING MODULES.
 Creator
 RICHMAN, FRED., The University of Chicago
 Date Issued
 1963, 1963
 PURL
 http://purl.flvc.org/fcla/dt/40283
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 PISOT SEQUENCES AND PISOTVIJAYARAGHAVAN NUMBERS.
 Creator
 DELEON, MORRIS JACK., The Pennsylvania State University
 Date Issued
 1968, 1968
 PURL
 http://purl.flvc.org/fcla/dt/40325
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 ON HYPERSINGULAR INTEGRALS WITH COMPLEX HOMOGENEITY.
 Creator
 Sagher, Yoram, The University of Chicago
 Date Issued
 1967, 1967
 PURL
 http://purl.flvc.org/fcla/dt/40282
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 FINITE GROUPS ADMITTING FIXEDPOINTFREE AUTOMORPHISMS.
 Creator
 HOFFMAN, FREDERICK., University of Virginia
 Date Issued
 1964, 1964
 PURL
 http://purl.flvc.org/fcla/dt/40291
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 ON THE MONOTONE EXTENSION PROPERTY.
 Creator
 MECH, WILLIAM PAUL., University of Illinois at UrbanaChampaign
 Date Issued
 1970, 1970
 PURL
 http://purl.flvc.org/fcla/dt/40437
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 SOME RESULTS IN COMBINATORIAL MATRIX THEORY.
 Creator
 Levow, Roy B., University of Pennsylvania
 Date Issued
 1969, 1969
 PURL
 http://purl.flvc.org/fcla/dt/40425
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 Some problems in operator theory and the geometry of Banach spaces.
 Creator
 Hoim, Terje., Kent State University
 Abstract/Description

One of the fundamental problems in Operator Theory is the Invariant Subspace Problem asking whether every bounded linear operator on an infinite dimensional complex Banach space admits a closed nontrivial invariant subspace. The study of invariant subspaces can be seen as a study of particular properties of orbits of operators. We study the orbits of a class of isometries of L1[0, 1]. Every isometry of Lp, 1 ≤ p < infinity, p ≠ 2, can be written as Tf = h(f ∘ tau). When tau is not...
Show moreOne of the fundamental problems in Operator Theory is the Invariant Subspace Problem asking whether every bounded linear operator on an infinite dimensional complex Banach space admits a closed nontrivial invariant subspace. The study of invariant subspaces can be seen as a study of particular properties of orbits of operators. We study the orbits of a class of isometries of L1[0, 1]. Every isometry of Lp, 1 ≤ p < infinity, p ≠ 2, can be written as Tf = h(f ∘ tau). When tau is not measure preserving, we show that the set of functions f in L1[0, 1] for which the orbit of f under the isometry T is equivalent to the usual canonical basis of l1 is an open dense set. A similar problem is also studied for other classical Banach spaces., In 1996 P. Enflo introduced the concept of extremal vectors and their connection to the Invariant Subspace Problem. We continue studying the properties and behaviour of backward minimal vectors, give some new formulas and improve results from papers by S. Ansari and P. Enflo., Finally we turn to the application of Functional Analysis and the geometry of Banach spaces to mathematical economics. We study the simplest form of economic activity called a pure exchange economy together with the existence of equilibrium prices problem. More precisely, we study how well the equilibrium price for the subeconomy En approximates the equilibrium price for a larger economy EN when these two economies have the same distribution of agents' characteristics.
Show less  Date Issued
 2000, 2000
 PURL
 http://purl.flvc.org/fcla/dt/40933
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 Generic Galois extensions for groups of order p('3).
 Creator
 Blue, Meredith Patricia., The University of Texas at Austin
 Abstract/Description

The existence of a generic Galois extension for a given group G over a field F is equivalent being able to lift the extension over local rings as well as pull the extension back to dense subfields of a complete field. The extensions are intimately related to Noether's problem, a method of attack for the Inverse Galois problem. In 1987 David Saltman showed the existence of generic Galois extensions for groups of order p3 for p odd over a ground field containing pth roots of unity. In this...
Show moreThe existence of a generic Galois extension for a given group G over a field F is equivalent being able to lift the extension over local rings as well as pull the extension back to dense subfields of a complete field. The extensions are intimately related to Noether's problem, a method of attack for the Inverse Galois problem. In 1987 David Saltman showed the existence of generic Galois extensions for groups of order p3 for p odd over a ground field containing pth roots of unity. In this paper Saltman's result is extended to include base fields without a pth root of unity. In particular generic extensions axe shown to exist over Q .
Show less  Date Issued
 2000, 2000
 PURL
 http://purl.flvc.org/fcla/dt/40935
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 On spectral properties of positive operators.
 Creator
 Zhang, XiaoDong., California Institute of Technology
 Abstract/Description

This thesis deals with the spectral behavior of positive operators and related ones on Banach lattices. We first study the spectral properties of those positive operators that satisfy the socalled condition (c). A bounded linear operator T on a Banach space is said to satisfy the condition (c) if it is invertible and if the number 0 is in the unbounded connected component of its resolvent set $\rho(T)$. By using techniques in complex analysis and in operator theory, we prove that if T is a...
Show moreThis thesis deals with the spectral behavior of positive operators and related ones on Banach lattices. We first study the spectral properties of those positive operators that satisfy the socalled condition (c). A bounded linear operator T on a Banach space is said to satisfy the condition (c) if it is invertible and if the number 0 is in the unbounded connected component of its resolvent set $\rho(T)$. By using techniques in complex analysis and in operator theory, we prove that if T is a positive operator satisfying the condition (c) on a Banach lattice E then there exists a positive number a and a positive integer k such that $T\sp k\geq a\cdot I$, where I is the identity operator on E. As consequences of this result, we deduce some theorems concerning the behavior of the peripheral spectrum of positive operators satisfying the condition (c). In particular, we prove that if T is a positive operator with its spectrum contained in the unit circle $\Gamma$ then either $\sigma(T)$ = $\Gamma$ or $\sigma(T)$ is finite and cyclic and consists of kth roots of unity for some k. We also prove that under certain additional conditions a positive operator with its spectrum contained in the unit circle will become an isometry. Another main result of this thesis is the decomposition theorem for disjointness preserving operators. We prove that under some natural conditions if T is a disjointness preserving operator on an order complete Banach lattice E such that its adjoint $T\sp\prime$ is also a disjointness preserving operator then there exists a family of Treducing bands $\{E\sb n:n\geq1\}$ $\cup$ $\{E\sb\infty\}$ of E such that $T\vert\sb{E\sb n}$ has strict period n and that $T\vert\sb{E\sb\infty}$ is aperiodic. We also prove that any disjointness preserving operator with its spectrum contained in a sector of angle less than $\pi$ can be decomposed into a sum of, a central operator and a quasinilpotent operator. Among other things we give some conditions under which an operator T lies in the center of the Banach lattice. Also discussed in this thesis are certain conditions under which a positive operator T with $\sigma(T)$ = $\{1\}$ is greater than or equal to the identity operator I.
Show less  Date Issued
 1991, 1991
 PURL
 http://purl.flvc.org/fcla/dt/40703
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 ON ABSOLUTELY TORSIONFREE RINGS AND KERNEL FUNCTORS.
 Creator
 VIOLAPRIOLI, JORGE E., Rutgers The State University of New Jersey  New Brunswick
 Date Issued
 1973, 1973
 PURL
 http://purl.flvc.org/fcla/dt/40474
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 PARTIAL GEOMETRIC LATTICES (DESIGNS).
 Creator
 MEYEROWITZ, AARON DAVID., Colorado State University
 Abstract/Description

A partial geometric lattice, PGL, is a finite ranked lattice of rank m which, aside from certain nontriviality conditions, satisfies the following condition, as do its intervals: If p is a point and l a line, the number of points q l such that p V q has rank k depends only on the rank of p V l., In case m = 3, this is a partial geometry as defined by Bose. In a PGL, any two intervals bounded by an ispace and an i + sspace contain the same number of i + kspaces. Those numbers determine the...
Show moreA partial geometric lattice, PGL, is a finite ranked lattice of rank m which, aside from certain nontriviality conditions, satisfies the following condition, as do its intervals: If p is a point and l a line, the number of points q l such that p V q has rank k depends only on the rank of p V l., In case m = 3, this is a partial geometry as defined by Bose. In a PGL, any two intervals bounded by an ispace and an i + sspace contain the same number of i + kspaces. Those numbers determine the other parameters of the lattice. As in rank 3, a rank 4 PGL has an association scheme on its points. Explicit expressions are given for the eigenvalues and multiplicities. These are used to search for feasible parameter sets.
Show less  Date Issued
 1984, 1984
 PURL
 http://purl.flvc.org/fcla/dt/40603
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 MODULES OVER ZG, A NONABELIAN GROUP OF ORDER PQ.
 Creator
 Klingler, Lee, The University of Wisconsin  Madison
 Abstract/Description

Let G be a nonabelian group of order pq (p and q primes). In this paper we solve the problem of describing all isomorphism classes of finitely generated left ZGmodules, where ZG is the integral group ring of G. We consider ZGmodules in general and not merely lattices., Our main result is to define a function "class" cl( ) from the category of (finitely generated) left ZGmodules into the finite set of isomorphism classes of fractional (not necessarily full) left ZGmodules in the quotient...
Show moreLet G be a nonabelian group of order pq (p and q primes). In this paper we solve the problem of describing all isomorphism classes of finitely generated left ZGmodules, where ZG is the integral group ring of G. We consider ZGmodules in general and not merely lattices., Our main result is to define a function "class" cl( ) from the category of (finitely generated) left ZGmodules into the finite set of isomorphism classes of fractional (not necessarily full) left ZGmodules in the quotient ring QG of ZG. We define cl( ) in such a way that, for arbitrary finitely generated left ZGmodules M and N, M (TURNEQ) N iff cl(M) = cl(N) and (')M(,t) (TURNEQ) (')N(,t) (tadic completion) for all primes t., With CLS(ZG) the image of the function cl( ), we define an operation "+" on CLS(ZG) in such a way that cl(M (CRPLUS) N) = cl(M) + cl(N). CLS(ZG) forms a semigroup under this operation and decomposes as the disjoint union of a finite collection of subgroups, where each subgroup is itself just a genus of fractional ideals. We show that each of these subgroups (and consequently each genus) has order dividing the order of the locally free class group of ZG., We apply this numerical information to questions of local versus global isomorphism and direct sum decompositions of finitely generated left ZGmodules. In particular (for G as above), we determine those primes t such that the KrullSchmidt theorem holds for finitely generated Z(,t)Gmodules (localization at t). We determine necessary and sufficient conditions that cancellation hold for finitely generated ZGmodules, and we calculate a "power cancellation exponent" e such that M (CRPLUS) X = M (CRPLUS) Y implies X('(e)) = Y('(e)).
Show less  Date Issued
 1984, 1984
 PURL
 http://purl.flvc.org/fcla/dt/40597
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 A computerassisted proof of uniqueness of phase for the hardsquare lattice gas model in two dimensions.
 Creator
 Radulescu, Dan Constantin., Rutgers The State University of New Jersey  New Brunswick
 Abstract/Description

The DobrushinShlosman Theorem was applied to perform a computerassisted proof of uniqueness of phase for the hardsquare gas model in two dimensions, for activities, z, inside the interval $z \in \lbrack 0, 1.50762\rbrack .$ Uniqueness for $z \in\ \lbrack 0, 1\rbrack$ was previously proven by R. L. Dobrushin, J. Kolafa, and S. B. Shlosman using the same approach but different computational methods. Disagreement percolation arguments show uniqueness for $z \in \lbrack 0, 1.255\rbrack $...
Show moreThe DobrushinShlosman Theorem was applied to perform a computerassisted proof of uniqueness of phase for the hardsquare gas model in two dimensions, for activities, z, inside the interval $z \in \lbrack 0, 1.50762\rbrack .$ Uniqueness for $z \in\ \lbrack 0, 1\rbrack$ was previously proven by R. L. Dobrushin, J. Kolafa, and S. B. Shlosman using the same approach but different computational methods. Disagreement percolation arguments show uniqueness for $z \in \lbrack 0, 1.255\rbrack $ nevertheless, the interval $z \in \lbrack 0, 1.50762\rbrack$ seems to be the largest for which uniqueness of phase has been rigorously proven., For the computerassisted proof, lattice cells with dimensions of foursite by foursite, fivesite by fivesite and sixsites by sixsite were used., For lattice cells up to a sixteensite by sixteensite size, some activities for which the criterion fails were also computed. These activities are much smaller (e.g., z = 2.335, for a sixteensite by sixteensite lattice cell) than z = 3.7962 $\pm$ 0.0001, which is the estimated phase transition activity for the hardsquare gas. This suggests that at least for now, the DobrushinShlosman Theorem can not be applied to practically locate a phase transition via a computerassisted proof. Note that the same pattern emerges when applying the criterion to the hardsphere model in three dimensions on the bcc lattice. A lattice cell made of only one cubeeight vertices plus one site in the center of the cubeensures uniqueness on the activity interval (0, 0.17897) which is better than the best rigorous lower bound of 1/9 but still far away from the expected transition activity of about 0.71.
Show less  Date Issued
 1997, 1997
 PURL
 http://purl.flvc.org/fcla/dt/40853
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 Generators of fat point ideals on the projective plane.
 Creator
 Fitchett, Stephanie Ann., The University of Nebraska  Lincoln
 Abstract/Description

This work employs geometric methods to investigate the relationship between the geometry of fat point subschemes of the projective plane and the structure of their defining ideals. In particular, for fat point subschemes supported at any six general points of the projective plane, we determine the number of elements of each degree in a minimal set of homogeneous generators for the defining ideal, called a fat point ideal. This, in turn, implicitly determines a minimal free resolution of the...
Show moreThis work employs geometric methods to investigate the relationship between the geometry of fat point subschemes of the projective plane and the structure of their defining ideals. In particular, for fat point subschemes supported at any six general points of the projective plane, we determine the number of elements of each degree in a minimal set of homogeneous generators for the defining ideal, called a fat point ideal. This, in turn, implicitly determines a minimal free resolution of the ideal. For fat point ideals whose zero locus is a set of seven or eight points in general position, for every degree but one we determine the number of elements in a minimal set of homogeneous generators. Under some additional technical hypotheses which ensure our ability to do necessary computations, for every degree but one we determine bounds on the number of elements in a minimal set of homogeneous generators for fat point ideals with a zero locus of more than eight points. Finally, we compare the bounds we obtain with previously known bounds due to Campanella.
Show less  Date Issued
 1997, 1997
 PURL
 http://purl.flvc.org/fcla/dt/40839
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 EVERY FINITE ABELIAN GROUP IS A BRAUER GROUP.
 Creator
 FORD, TIMOTHY JOE., Colorado State University
 Abstract/Description

Let A be the affine coordinate ring of the plane nodal cubic curve y('2) = x('2)(x+1) over the complex number field. That is, A = (//C){x, y}/(y('2)  x('2)(x+1)). Let n (GREATERTHEQ) 2 be an integer, let f(,n) = zy('n1)  x('n) and let B(,n) be the subring of the localized ring (//C){x, y, z}{1/f(,n)} consisting of all fractions g/f(,n)('r) such that g is a homogeneous polynomial of degree rn, r (GREATERTHEQ) 0. Then we show that the Brauer group of A (CRTIMES)(,(//C)) B(,n) is cyclic of...
Show moreLet A be the affine coordinate ring of the plane nodal cubic curve y('2) = x('2)(x+1) over the complex number field. That is, A = (//C){x, y}/(y('2)  x('2)(x+1)). Let n (GREATERTHEQ) 2 be an integer, let f(,n) = zy('n1)  x('n) and let B(,n) be the subring of the localized ring (//C){x, y, z}{1/f(,n)} consisting of all fractions g/f(,n)('r) such that g is a homogeneous polynomial of degree rn, r (GREATERTHEQ) 0. Then we show that the Brauer group of A (CRTIMES)(,(//C)) B(,n) is cyclic of order n. Thus, every finite cyclic group is the Brauer group of a threedimensional noetherian integral domain. Since every finite abelian group G is a direct sum of cyclic groups we see that any G is the Brauer group of the threedimensional noetherian ring A (CRTIMES)(,(//C)) (B(,n(,1)) (CRPLUS)...(CRPLUS)B(,n(,r))) for suitable choices of n(,i)., Using cohomological techniques we investigate the Brauer group of Y x (,k)' where Y is a scheme over a field k. We give sufficient conditions on Y so that the Brauer group of Y x ' is equal to the Brauer group of Y. In particular, equality holds if Y has dimension one over k and the characteristic of k is zero., Let R be a commutative noetherian connected regular ring. Using excision sequences we show that the sequence of groups, 0 (>) B(R{x}) (>) B{R{x, 1/x}) (>) H(,Z)('3)(X, G(,m)) (>) 0, is exact where X = Proj R{x(,0), x(,1)} and Z = Spec R is the closed subscheme x(,0) = 0. If we further assume that R contains the field of rational numbers, then B(R{x, 1/x}) = B(R) (CRPLUS) Hom(G, / ) where G is the Galois groupof R.
Show less  Date Issued
 1980, 1980
 PURL
 http://purl.flvc.org/fcla/dt/40555
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 THE GOEDEL SPEEDUP PHENOMENON.
 Creator
 Solomon, Martin K., Stevens Institute of Technology
 Date Issued
 1976, 1976
 PURL
 http://purl.flvc.org/fcla/dt/40501
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 SOME PROPERTIES OF DIVISIBILITY IN FIELDS AND INTEGRAL DOMAINS.
 Creator
 BREWER, JAMES WILLIAM., The Florida State University
 Date Issued
 1968, 1968
 PURL
 http://purl.flvc.org/fcla/dt/40315
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 Directsum cancellation of lattices over orders in global fields.
 Creator
 Karr, Ryan Deene., The University of Nebraska  Lincoln
 Abstract/Description

We study the cancellation question for lattices (finitely generated torsionfree modules) over orders in algebraic number fields: Given lattices L, M and N with L ⊕ M ≅ L ⊕ N, when can one conclude that M ≅ N? Some definitive results in the quadratic case were obtained about twenty years ago. Here we concentrate on the case of cubic and higherdegree number fields, where very different techniques are needed. The cubic case appears to be quite difficult, and our results in this case are very...
Show moreWe study the cancellation question for lattices (finitely generated torsionfree modules) over orders in algebraic number fields: Given lattices L, M and N with L ⊕ M ≅ L ⊕ N, when can one conclude that M ≅ N? Some definitive results in the quadratic case were obtained about twenty years ago. Here we concentrate on the case of cubic and higherdegree number fields, where very different techniques are needed. The cubic case appears to be quite difficult, and our results in this case are very incomplete. Perhaps surprisingly, number fields of degree four or more are more tractable, and we have a definitive answer to the cancellation question for a large family of orders in these fields. Our results apply also to the case of algebraic function fields in one variable over a finite field of constants.
Show less  Date Issued
 2002, 2002
 PURL
 http://purl.flvc.org/fcla/dt/40351
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 LOCAL FACTORIZATION OF NONSINGULAR BIRATIONAL MORPHISMS IN DIMENSION GREATER THAN TWO (REGULAR LOCAL RINGS, DESCENDING CHAIN CONDITION, RATIONAL SINGULARITY).
 Creator
 JOHNSTON, BERNARD LAWRENCE., Purdue University
 Abstract/Description

The Local Factorization Theorem of Zariski and Abhyankar implies that between a given pair of 2dimensional regular local rings, S (GREATERTHEQ) R, having the same quotient field, every chain of regular local rings must be finite in length. It is shown that this property extends to every such pair of regular local rings, regardless of dimension. Examples are given to show that this does not hold if "regular" is weakened to various statements, including "Gorenstein", "rational singularity",...
Show moreThe Local Factorization Theorem of Zariski and Abhyankar implies that between a given pair of 2dimensional regular local rings, S (GREATERTHEQ) R, having the same quotient field, every chain of regular local rings must be finite in length. It is shown that this property extends to every such pair of regular local rings, regardless of dimension. Examples are given to show that this does not hold if "regular" is weakened to various statements, including "Gorenstein", "rational singularity", and "normal". More generally, it is shown that the set of ndimensional regular local rings birationally containing an arbitrary integral domain must satisfy the descending chain condition. Some conditions which imply a uniform bound on the lengths of certain chains between two fixed ndimensional regular local rings, as above, are given. Finally, a new class, containing infinitely many minimal regular local overrings containing a fixed regular local ring, is presented.
Show less  Date Issued
 1986, 1986
 PURL
 http://purl.flvc.org/fcla/dt/40632
 Subject Headings
 Mathematics
 Format
 Document (PDF)
 Title
 Algebraic differential equations and nonlinear control systems.
 Creator
 Wang, Yuan, Rutgers The State University of New Jersey  New Brunswick
 Abstract/Description

This dissertation establishes a precise correspondence between realizability of operators defined by convergent generating series and the existence of high order differential equations ("i/o equations") relating derivatives of inputs and outputs., State space models are central to modern nonlinear control theory, since they permit the application of techniques from various mathematics branches such as differential equations, dynamical systems and optimization theory. A natural question is to...
Show moreThis dissertation establishes a precise correspondence between realizability of operators defined by convergent generating series and the existence of high order differential equations ("i/o equations") relating derivatives of inputs and outputs., State space models are central to modern nonlinear control theory, since they permit the application of techniques from various mathematics branches such as differential equations, dynamical systems and optimization theory. A natural question is to decide when a given i/o operator admits a representation by an initialized state space system (the operator is realizable)., To investigate the relation between i/o equations and realizability, we introduce and study the structures of observation spaces, observation algebras and observation fields. In realization theory and many other areas of nonlinear control, the concept of observation space plays a central role. One may define observation spaces in two very different ways. Roughly, one possibility is to take the functions corresponding to derivatives with respect to switching times in piecewise constant controls, and the other is to take highorder derivatives at the final time, if smooth controls are used. It turns out that the existence of algebraic i/o equations is closely related to the finiteness properties of the observation algebra and field associated with the first type of observation space, while realizability is closely related to the finiteness properties of the algebraic objects associated with the other type of observation space. One of the central technical results, given in Chapter 3, shows that the two types of spaces coincide., Based on the results mentioned above, we get our main results: Realizability by singular polynomial systems is equivalent to existence of algebraic i/o equations. We also provide other results relating various special kinds of i/o equations to some specific classes of realizations, for instance, what are called recursive i/o equations are related to realizability by polynomial systems., In Chapter 7, our results relating algebraic i/o equations to realizability by "rational" systems are extended to analytic i/o equations and local realization by analytic systems. By studying properties of meromorphically finitely generated field of functions, together with the application of some known facts in the literature of nonlinear realization, we conclude that the existence of analytic i/o equations implies local realizability by analytic systems.
Show less  Date Issued
 1990, 1990
 PURL
 http://purl.flvc.org/fcla/dt/40695
 Subject Headings
 Mathematics
 Format
 Document (PDF)