Current Search: Algebra (x)
Pages


Title

A class of rational surfaces with a nonrational singularity explicitly given by a single equation.

Creator

Harmon, Drake., Charles E. Schmidt College of Science, Department of Mathematical Sciences

Abstract/Description

The family of algebraic surfaces X dened by the single equation zn = (y a1x) (y anx)(x 1) over an algebraically closed eld k of characteristic zero, where a1; : : : ; an 2 k are distinct, is studied. It is shown that this is a rational surface with a nonrational singularity at the origin. The ideal class group of the surface is computed. The terms of the ChaseHarrisonRosenberg seven term exact sequence on the open complement of the ramication locus of X ! A2 are computed; the Brauer group...
Show moreThe family of algebraic surfaces X dened by the single equation zn = (y a1x) (y anx)(x 1) over an algebraically closed eld k of characteristic zero, where a1; : : : ; an 2 k are distinct, is studied. It is shown that this is a rational surface with a nonrational singularity at the origin. The ideal class group of the surface is computed. The terms of the ChaseHarrisonRosenberg seven term exact sequence on the open complement of the ramication locus of X ! A2 are computed; the Brauer group is also studied in this unramied setting. The analysis is extended to the surface eX obtained by blowing up X at the origin. The interplay between properties of eX , determined in part by the exceptional curve E lying over the origin, and the properties of X is explored. In particular, the implications that these properties have on the Picard group of the surface X are studied.
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Date Issued

2013

PURL

http://purl.flvc.org/fcla/dt/3360782

Subject Headings

Mathematics, Galois modules (Algebra), Class field theory, Algebraic varieties, Integral equations

Format

Document (PDF)


Title

A study of divisors and algebras on a double cover of the affine plane.

Creator

Bulj, Djordje., Charles E. Schmidt College of Science, Department of Mathematical Sciences

Abstract/Description

An algebraic surface defined by an equation of the form z2 = (x+a1y) ... (x + any) (x  1) is studied, from both an algebraic and geometric point of view. It is shown that the surface is rational and contains a singular point which is nonrational. The class group of Weil divisors is computed and the Brauer group of Azumaya algebras is studied. Viewing the surface as a cyclic cover of the affine plane, all of the terms in the cohomology sequence of Chase, Harrison and Roseberg are computed.

Date Issued

2012

PURL

http://purl.flvc.org/FAU/3355618

Subject Headings

Algebraic number theory, Geometry, Data processing, Noncommutative differential geometry, Mathematical physics, Curves, Algebraic, Commutative rings

Format

Document (PDF)


Title

PUBLIC SCHOOL ADVISORY COMMITTEES: CHARACTERISTICS, CONTRIBUTIONS, AND PERCEPTIONS OF ROLE AND FUNCTIONS.

Creator

CHINN, BEVERLY DUKE, Florida Atlantic University, Logsdon, James D.

Abstract/Description

The purpose of this study is to determine the makeup, organizational structure, and operational aspects of school advisory committees as well as to identify their contributions to local educational programs as reported by principals and advisory committee chairpersons. This study also reports differences in the perceptions of school principals and advisory committee chairpersons as to the role and functions of school advisory committees. Research questions relating to makeup, organizational...
Show moreThe purpose of this study is to determine the makeup, organizational structure, and operational aspects of school advisory committees as well as to identify their contributions to local educational programs as reported by principals and advisory committee chairpersons. This study also reports differences in the perceptions of school principals and advisory committee chairpersons as to the role and functions of school advisory committees. Research questions relating to makeup, organizational structure, operational aspects, and contributions of advisory committees were analyzed. Conclusions: It was concluded that advisory committees are predominately female, as are their chairpersons. Committees usually meet monthly at the school during the evenings. Most committees do not have a constitution and bylaws. School principals and advisory committee chairpersons bad significant differences in perceptions regarding the role and functions of school advisory committees. Dade, Broward, and Palm Beach Counties' respondents had significantly different perceptions regarding the role and functions of school advisory committees. Chairpersons reported that the advisory committees had made contributions in the areas of school safety, maintenance and improvement of the school plant, community and race relations, and articulation between schools. School principals reported that advisory committees had made contributions in the areas of preparation of the school budget, preparation of the annual report of school progress, community and race relations, school safety, and determination of school goals.
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Date Issued

1975

PURL

http://purl.flvc.org/fcla/dt/11659

Subject Headings

Citizens' advisory committees in education, AlgebraProgrammed instruction, AlgebraStudy and teaching (Higher)

Format

Document (PDF)


Title

The effect of curriculum specific computeraided instruction on student achievement in college algebra, a comparative study.

Creator

Cox, Gregg Clayton, Florida Atlantic University, Burgess, Ernest E.

Abstract/Description

This study was designed to determine the effects of curriculum specific computer aided instruction on student achievement in a College Algebra course. Curriculum specific software is microcomputer software which provides both computerized instruction and practice for each topic in the accompanying textbook and is now available for many college mathematics texts. Using methods outlined by Diem (1982) in a previous study, College Algebra students were randomly assigned to one of four groups....
Show moreThis study was designed to determine the effects of curriculum specific computer aided instruction on student achievement in a College Algebra course. Curriculum specific software is microcomputer software which provides both computerized instruction and practice for each topic in the accompanying textbook and is now available for many college mathematics texts. Using methods outlined by Diem (1982) in a previous study, College Algebra students were randomly assigned to one of four groups. Six hypotheses were formulated and tested by comparing both posttest scores and growth quotients for various appropriate groups. At the.05 level of significance the following hypotheses were rejected: There is no significant difference in achievement in learning mathematics between College Algebra students who study linear inequalities using curriculum specific microcomputer drill and practice with traditional lecture and College Algebra students who study linear inequalities using a traditional lecturehomework approach. There is no significant difference in achievement among those receiving curriculum specific microcomputer aided instruction as a result of students' score on the pretest, whether placed in the upper, middle, or lower third of the pretest scores. Implications of the study included the following: (1) The use of curriculum specific computerized drill and practice can significantly increase the mathematics achievement of those students receiving a traditional lecture. (2) There is a significant relationship between a student's pretest score and their level of success when using curriculum specific microcomputer aided instruction. Recommendations for further study included the following: (1) Replication of this experiment investigating different factors such as: (a) Differences related to age, (b) attitude toward computer aided instruction, (c) type of text and software, and (d) differences related to previous computing experience. (2) Research which compares various forms of curriculum specific drill and practice. (3) Development of computerized tutorials which significantly increase student achievement.
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Date Issued

1990

PURL

http://purl.flvc.org/fcla/dt/12250

Subject Headings

AlgebraComputerassisted instruction, AlgebraStudy and teaching (Higher), Computerassisted instruction

Format

Document (PDF)


Title

Integervalued polynomials and pullbacks of arithmetical rings.

Creator

Boynton, Jason, Florida Atlantic University, Klingler, Lee

Abstract/Description

Let D be an integral domain with field of fractions K, and let E be a nonempty finite subset of D. For n > 2, we show that the ngenerator property for D is equivalent to the ngenerator property for Int(E, D), which is equivalent to strong (n + 1)generator property for Int(E, D). We also give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (that is, a ring whose ideals are totally ordered by inclusion), and we give necessary and sufficient...
Show moreLet D be an integral domain with field of fractions K, and let E be a nonempty finite subset of D. For n > 2, we show that the ngenerator property for D is equivalent to the ngenerator property for Int(E, D), which is equivalent to strong (n + 1)generator property for Int(E, D). We also give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (that is, a ring whose ideals are totally ordered by inclusion), and we give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (that is, a ring which is locally a chain ring at every maximal ideal). We characterize all Prufer domains R between D[X] and K[X]such that the conductor C of K[X] into R is nonzero. As an application, we show that for n > 2, such a ring R has the ngenerator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.
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Date Issued

2006

PURL

http://purl.flvc.org/fcla/dt/12221

Subject Headings

Polynomials, Ideals (Algebra), Rings of integers, Categories (Mathematics), Arithmetical algebraic geometry

Format

Document (PDF)


Title

Subgroups of bounded Abelian groups.

Creator

Petroro, Carla., Florida Atlantic University, Schmidmeier, Markus

Abstract/Description

Birkhoff raised the question of how to determine "relative invariants of subgroups" of a given group. Let us consider pairs (A, B ) where B is a finite pnbounded Abelian group and A is a subgroup of B. Maps between pairs (A, B) > (A', B') are morphisms f : B > B' such that f (A) > A'. Classification of such pairs, up to isomorphism, is Birkhoff's famous problem. By the KrullRemakSchmidt theorem, an arbitrary pair (A, B) is a direct sum of indecomposable pairs, and the multiplicities...
Show moreBirkhoff raised the question of how to determine "relative invariants of subgroups" of a given group. Let us consider pairs (A, B ) where B is a finite pnbounded Abelian group and A is a subgroup of B. Maps between pairs (A, B) > (A', B') are morphisms f : B > B' such that f (A) > A'. Classification of such pairs, up to isomorphism, is Birkhoff's famous problem. By the KrullRemakSchmidt theorem, an arbitrary pair (A, B) is a direct sum of indecomposable pairs, and the multiplicities of the indecomposables are determined uniquely. The purpose of this thesis is to describe the decomposition of such pairs, (A, B), explicitly for n = 2 and n = 3. We describe explicitly how an indecomposable pair can possibly embed into a given pair (A, B). This construction gives rise to formulas for the multiplicity of an indecomposable in the direct sum decomposition of the pair (A, B). These decomposition numbers form a full set of relative invariant, as requested by Birkhoff.
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Date Issued

2004

PURL

http://purl.flvc.org/fcla/dt/13118

Subject Headings

Abelian groups, Modules (Algebra), Indecomposable modules, Representations of groups, Algebras, Linear

Format

Document (PDF)


Title

Algebraic and combinatorial aspects of group factorizations.

Creator

Bozovic, Vladimir., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences

Abstract/Description

The aim of this work is to investigate some algebraic and combinatorial aspects of group factorizations. The main contribution of this dissertation is a set of new results regarding factorization of groups, with emphasis on the nonabelian case. We introduce a novel technique for factorization of groups, the socalled free mappings, a powerful tool for factorization of a wide class of abelian and nonabelian groups. By applying a certain group action on the blocks of a factorization, a number...
Show moreThe aim of this work is to investigate some algebraic and combinatorial aspects of group factorizations. The main contribution of this dissertation is a set of new results regarding factorization of groups, with emphasis on the nonabelian case. We introduce a novel technique for factorization of groups, the socalled free mappings, a powerful tool for factorization of a wide class of abelian and nonabelian groups. By applying a certain group action on the blocks of a factorization, a number of combinatorial and computational problems were noted and studied. In particular, we analyze the case of the group Aut(Zn) acting on blocks of factorization of Zn. We present new theoretical facts that reveal the numerical structure of the stabilizer of a set in Zn, under the action of Aut(Zn). New algorithms for finding the stabilizer of a set and checking whether two sets belong to the same orbit are proposed.
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Date Issued

2008

PURL

http://purl.flvc.org/FAU/107805

Subject Headings

Physical measurements, Mapping (Mathematics), Combinatorial enumeration problems, Algebra, Abstract

Format

Document (PDF)


Title

On projected planes.

Creator

Caliskan, Cafer., Charles E. Schmidt College of Science, Department of Mathematical Sciences

Abstract/Description

This work was motivated by the wellknown question: "Does there exist a nondesarguesian projective plane of prime order?" For a prime p 1, determine all subplanes of order p up to collineations, and check whether one of these is nondesarguesian." In this manuscript we use a grouptheoretic methodology to determine the subplane structures of some nondesarguesian planes. In particular, we determine orbit representatives of all proper Qsubplanes both of a VeblenWedderburn (VW) plane of...
Show moreThis work was motivated by the wellknown question: "Does there exist a nondesarguesian projective plane of prime order?" For a prime p < 11, there is only the pappian plane of order p. Hence, such planes are indeed desarguesian. Thus, it is of interest to examine whether there are nondesarguesian planes of order 11. A suggestion by Ascher Wagner in 1985 was made to Spyros S. Magliveras: "Begin with a nondesarguesian plane of order pk, k > 1, determine all subplanes of order p up to collineations, and check whether one of these is nondesarguesian." In this manuscript we use a grouptheoretic methodology to determine the subplane structures of some nondesarguesian planes. In particular, we determine orbit representatives of all proper Qsubplanes both of a VeblenWedderburn (VW) plane of order 121 and of the Hughes plane of order 121, under their full collineation groups. In PI, there are 13 orbits of Baer subplanes, all of which are desarguesian, and approximately 3000 orbits of Fano subplanes. In Sigma , there are 8 orbits of Baer subplanes, all of which are desarguesian, 2 orbits of subplanes of order 3, and at most 408; 075 distinct Fano subplanes. In addition to the above results, we also study the subplane structures of some nondesarguesian planes, such as the Hall plane of order 25, the Hughes planes of order 25 and 49, and the Figueora planes of order 27 and 125. A surprising discovery by L. Puccio and M. J. de Resmini was the existence of a plane of order 3 in the Hughes plane of order 25. We generalize this result, showing that there are subplanes of order 3 in the Hughes planes of order q2, where q is a prime power and q 5 (mod 6). Furthermore, we analyze the structure of the full collineation groups of certain Veblen Wedderburn (VW) planes of orders 25, 49 and 121, and discuss how to recover the planes from their collineation groups.
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Date Issued

2010

PURL

http://purl.flvc.org/FAU/1927609

Subject Headings

Projected planes, Combinatorial designs and configurations, Surfaces, Algebraic, Manifolds (Mathematics)

Format

Document (PDF)


Title

A relational algebra implementation.

Creator

Rivero, Gene Richard., Florida Atlantic University, Solomon, Martin K.

Abstract/Description

Although the Relational Algebra (RA) is a fundamental component of the relational model of database storage and retrieval, it has not been widely implemented in commercial database systems. The Structured Query Language (SQL) has been for some time the most widely implemented relational database language. This thesis provides and describes the first implementation of a particularly powerful version of the relational algebra that includes an nary join operator as well as including the...
Show moreAlthough the Relational Algebra (RA) is a fundamental component of the relational model of database storage and retrieval, it has not been widely implemented in commercial database systems. The Structured Query Language (SQL) has been for some time the most widely implemented relational database language. This thesis provides and describes the first implementation of a particularly powerful version of the relational algebra that includes an nary join operator as well as including the division operator, and nested expressions. With this implementation, that powerful version of the relational algebra can be used as a front end to any SQL92 compliant database management system, such as Oracle, Informix, DB2, Sybase, SQL Server, and Access.
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Date Issued

2003

PURL

http://purl.flvc.org/fcla/dt/13067

Subject Headings

Relation algebras, SQL (Computer program language), Database management, Relational databases

Format

Document (PDF)


Title

THE EFFECTIVENESS OF COMPUTER ASSISTED INSTRUCTION IN COLLEGE ALGEBRA.

Creator

DIEM, DENNIS CHARLTON., Florida Atlantic University, Burgess, Ernest E.

Abstract/Description

This study was designed to determine the extent to which microcomputer instruction affects the learning of mathematics in College Algebra when used as a substitute for traditional methods of instruction. The computer programs involved in the study were designed to teach the student how to find and graph the solution set of linear inequalities with two variables. The lessons were programmed on an Apple II microcomputer and were designed using relatively simple linear branching techniques. The...
Show moreThis study was designed to determine the extent to which microcomputer instruction affects the learning of mathematics in College Algebra when used as a substitute for traditional methods of instruction. The computer programs involved in the study were designed to teach the student how to find and graph the solution set of linear inequalities with two variables. The lessons were programmed on an Apple II microcomputer and were designed using relatively simple linear branching techniques. The subjects involved in the study were enrolled in two sections of College Algebra in an upper division university. The students were randomly assigned to four groups and each group was taught using varying methods of instruction. One group received traditional classroom lecture followed by textbook homework. A second group was exposed to a computer tutorial program followed by textbook homework. A third group received classroom lecture followed by a computer drill and practice program. A fourth group completed both the computer tutorial and the computer drill and practice programs. Prior to exposure to different teaching methods, each group was administered a pretest to determine the extent of their knowledge of the subject matter, establish the randomness assertion, and to determine whether or not the groups were equivalent at the outset. After each group received instruction, a posttest was administered to determine relative levels of achievement. One way between subjects analysis of variance was used with the pretest scores to determine initial differences between the groups. The same statistical procedure was used with the posttest scores. The results of analysis of variance, at the .05 level, indicated that no significant differences in learning took place between the four groups in the study. However, observation of the data seemed to suggest differences which favored the more conventional lecture, homework group. Recommendations for future study included replication of the experiment using the same or modified populations. Additional variables could also be identified such as student attitude, academic background, sex, and age.
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Date Issued

1982

PURL

http://purl.flvc.org/fcla/dt/11807

Subject Headings

AlgebraStudy and teaching (Higher), Computerassisted instruction

Format

Document (PDF)


Title

Finite valuated groups.

Creator

Holroyd, Keiko Ito, Florida Atlantic University, Richman, Fred

Abstract/Description

The concept of valuated groups, simply presented groups, and simultaneous decomposition of an abelian group and a subgroup are discussed. We classify the structure of finite valuated pgroups of order up p^4. With a refinement of a classical theorem on bounded pure subgroups, we also relate the decomposition of a finite valuated pgroup to the simultaneous decomposition of a finite abelian pgroup and a subgroup.

Date Issued

1996

PURL

http://purl.flvc.org/fcla/dt/15320

Subject Headings

Finite groups, Abelian groups, Abelian pgroups, Algebra, Homological

Format

Document (PDF)


Title

On the Loewy structure of the projective indecomposable representations of some stabilizer subgroups of A(8) in characteristic 2.

Creator

Hindman, Peter Blake, Florida Atlantic University, Klingler, Lee

Abstract/Description

Given a module over a ring for which the JordanHolder theorem is valid, the Loewy series is a filtration on the composition factors of the module yielding information on the structure in which they are arranged in the module. We derive subgroups of A8 by considering stabilizers of ntuples derived from partitions of eight letters, and develop their representation theory over a field of characteristic 2, relying heavily on methods of passing information to groups from their subgroups, with...
Show moreGiven a module over a ring for which the JordanHolder theorem is valid, the Loewy series is a filtration on the composition factors of the module yielding information on the structure in which they are arranged in the module. We derive subgroups of A8 by considering stabilizers of ntuples derived from partitions of eight letters, and develop their representation theory over a field of characteristic 2, relying heavily on methods of passing information to groups from their subgroups, with special attention toward obtaining the Loewy structure of their projective indecomposable representations.
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Date Issued

1993

PURL

http://purl.flvc.org/fcla/dt/14971

Subject Headings

Representations of groups, Projective modules (Algebra), Indecomposable modules

Format

Document (PDF)


Title

Pruefer domains, the strong 2generator property, and integervalued polynomials.

Creator

Roth, Heather., Florida Atlantic University, Klingler, Lee

Abstract/Description

We present several results involving three concepts: Prufer domains, the strong 2generator property, and integervalued polynomials. An integral domain D is called a Prufer domain if every nonzero finitely generated ideal of D is invertible. When each 2generated ideal of D has the property that one of its generators can be any arbitrary selected nonzero element of the ideal, we say D has the strong 2generator property . We note that, if D has the strong 2generator property, then D is a...
Show moreWe present several results involving three concepts: Prufer domains, the strong 2generator property, and integervalued polynomials. An integral domain D is called a Prufer domain if every nonzero finitely generated ideal of D is invertible. When each 2generated ideal of D has the property that one of its generators can be any arbitrary selected nonzero element of the ideal, we say D has the strong 2generator property . We note that, if D has the strong 2generator property, then D is a Prufer domain. If Q is the field of fractions of D, and E is a finite nonempty subset of D; we define Int(E, D ) = {f(X) ∈ Q[ X] ∣ f(a) ∈ D for every a ∈ E} to be the ring of integervalued polynomials on D with respect to the subset E. We show that D is a Prufer domain if and only if Int(E, D) is a Prufer domain. Our main theorem is that Int(E, D) has the strong 2generator property if and only if D is a Bezout domain (that is, every finitely generated ideal of D is principal).
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Date Issued

2004

PURL

http://purl.flvc.org/fcla/dt/13151

Subject Headings

Prüfer rings, Rings of integers, Polynomials, Ideals (Algebra), Mathematical analysis

Format

Document (PDF)


Title

The effects of accommodating students' learning styles on academic achievement and attitudes towards algebra.

Creator

Husni, Nabil Afif., Florida Atlantic University, Urich, Ted R.

Abstract/Description

The purpose of this study was to compare the academic achievement and attitude of students who were enrolled in Algebra for College Students taught by traditional lecture methods of instruction with students taught by methods of instruction that matched student learning style preferences. This study included 84 students who were enrolled in Algebra for College Students at Palm Beach Community College. The study was designed to determine whether students' age, gender, GPA, and the number of...
Show moreThe purpose of this study was to compare the academic achievement and attitude of students who were enrolled in Algebra for College Students taught by traditional lecture methods of instruction with students taught by methods of instruction that matched student learning style preferences. This study included 84 students who were enrolled in Algebra for College Students at Palm Beach Community College. The study was designed to determine whether students' age, gender, GPA, and the number of hours worked were correlated with students' attitude gain scores or with their algebra gain scores. Four instruments were used to collect information for this study: (a) an algebra pretest/posttest, (b) a background questionnaire, (c) the Productivity Environmental Preference Survey, and (d) an attitude survey. Reliability was obtained using the SPSS software. The algebra pretest/posttest and the attitude survey had alpha reliability coefficients of 0.7022 and 0.8154 respectively. Twelve hypotheses were developed to determine if there were significant relationships between and among attitudes towards algebra, academic achievement in algebra, and the aforementioned variables. Multiple linear regression was the statistical tool used for data analysis. Each hypothesis was tested at the 0.1/12 = 0.0083 level of significance. Based on the findings, gender played a significant role in this study. Male students who were taught by methods of instruction corresponding to their learning style preferences had slightly higher attitudinal gain scores and consistently higher achievement gain scores than male students who were taught by the traditional lecture method of instruction. On the other hand, female students who were taught by methods of instruction that accommodated their learning style preferences had higher attitudinal gain scores and relatively no change in academic achievement. Additionally, analyses of data collected from male students revealed a significant negative relationship between male students' academic achievement in algebra and the number of hours worked per week. In contrast, analyses of data collected from female students showed a significant positive relationship between female students' academic achievement and number of hours worked per week.
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Date Issued

1997

PURL

http://purl.flvc.org/fcla/dt/12510

Subject Headings

AlgebraStudy and teaching (Higher), Academic achievement

Format

Document (PDF)


Title

Derivation of planar diffeomorphisms from Hamiltonians with a kick.

Creator

Barney, Zalmond C., Charles E. Schmidt College of Science, Department of Mathematical Sciences

Abstract/Description

In this thesis we will discuss connections between Hamiltonian systems with a periodic kick and planar diffeomorphisms. After a brief overview of Hamiltonian theory we will focus, as an example, on derivations of the Hâenon map that can be obtained by considering kicked Hamiltonian systems. We will conclude with examples of Hâenon maps of interest.

Date Issued

2011

PURL

http://purl.flvc.org/FAU/3329833

Subject Headings

Mathematical physics, Differential equations, Partial, Hamiltonian systems, Algebra, Linear, Chaotic behavior in systems

Format

Document (PDF)


Title

Construction of combinatorial designs with prescribed automorphism groups.

Creator

Kolotoglu, Emre., Charles E. Schmidt College of Science, Department of Mathematical Sciences

Abstract/Description

In this dissertation, we study some open problems concerning the existence or nonexistence of some combinatorial designs. We give the construction or proof of nonexistence of some Steiner systems, large sets of designs, and graph designs, with prescribed automorphism groups.

Date Issued

2013

PURL

http://purl.flvc.org/fcla/dt/3360795

Subject Headings

Combinatorial designs and configurations, Finite geometries, Curves, Algebraic, Automorphisms, Mathematical optimization, Steiner systems

Format

Document (PDF)


Title

The existence of minimal logarithmic signatures for classical groups.

Creator

Singhi, Nikhil., Charles E. Schmidt College of Science, Department of Mathematical Sciences

Abstract/Description

A logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a...
Show moreA logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a group G of order Qk j=1 pj mj , the length of any LS for G satises `() Pk j=1mjpj . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS). The MLS conjecture states that every finite simple group has an MLS. If the conjecture is true then every finite group will have an MLS. The conjecture was shown to be true by a number of researchers for a few classes of finite simple groups. However, the problem is still wide open. This dissertation addresses the MLS conjecture for the classical simple groups. In particular, it is shown that MLS's exist for the symplectic groups Sp2n(q), the orthogonal groups O 2n(q0) and the corresponding simple groups PSp2n(q) and 2n(q0) for all n 2 N, prime power q and even prime power q0. The existence of an MLS is also shown for all unitary groups GUn(q) for all odd n and q = 2s under the assumption that an MLS exists for GUn 1(q). The methods used are very general and algorithmic in nature and may be useful for studying all nite simple groups of Lie type and possibly also the sporadic groups. The blocks of logarithmic signatures constructed in this dissertation have cyclic structure and provide a sort of cyclic decomposition for these classical groups.
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Date Issued

2011

PURL

http://purl.flvc.org/FAU/3172943

Subject Headings

Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups

Format

Document (PDF)


Title

On the minimal logarithmic signature conjecture.

Creator

Singhi, Nidhi., Charles E. Schmidt College of Science, Department of Mathematical Sciences

Abstract/Description

The minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups,...
Show moreThe minimal logarithmic signature conjecture states that in any finite simple group there are subsets Ai, 1 i s such that the size jAij of each Ai is a prime or 4 and each element of the group has a unique expression as a product Qs i=1 ai of elements ai 2 Ai. Logarithmic signatures have been used in the construction of several cryptographic primitives since the late 1970's [3, 15, 17, 19, 16]. The conjecture is shown to be true for various families of simple groups including cyclic groups, An, PSLn(q) when gcd(n; q 1) is 1, 4 or a prime and several sporadic groups [10, 9, 12, 14, 18]. This dissertation is devoted to proving that the conjecture is true for a large class of simple groups of Lie type called classical groups. The methods developed use the structure of these groups as isometry groups of bilinear or quadratic forms. A large part of the construction is also based on the Bruhat and Levi decompositions of parabolic subgroups of these groups. In this dissertation the conjecture is shown to be true for the following families of simple groups: the projective special linear groups PSLn(q), the projective symplectic groups PSp2n(q) for all n and q a prime power, and the projective orthogonal groups of positive type + 2n(q) for all n and q an even prime power. During the process, the existence of minimal logarithmic signatures (MLS's) is also proven for the linear groups: GLn(q), PGLn(q), SLn(q), the symplectic groups: Sp2n(q) for all n and q a prime power, and for the orthogonal groups of plus type O+ 2n(q) for all n and q an even prime power. The constructions in most of these cases provide cyclic MLS's. Using the relationship between nite groups of Lie type and groups with a split BNpair, it is also shown that every nite group of Lie type can be expressed as a disjoint union of sets, each of which has an MLS.
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Date Issued

2011

PURL

http://purl.flvc.org/FAU/3172946

Subject Headings

Finite groups, Abelian groups, Number theory, Combinatorial group theory, Mathematical recreations, Linear algebraic groups, Lie groups

Format

Document (PDF)


Title

Quantum Circuits for Cryptanalysis.

Creator

Amento, Brittanney Jaclyn, Steinwandt, Rainer, Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences

Abstract/Description

Finite elds of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these elds can have a signi cant impact on the resource requirements for quantum arithmetic. In particular, we show how the Gaussian normal basis representations and \ghostbit basis" representations can be used to implement inverters with a quantum circuit of depth O(mlog(m)). To the best of our knowledge, this is the rst construction with...
Show moreFinite elds of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these elds can have a signi cant impact on the resource requirements for quantum arithmetic. In particular, we show how the Gaussian normal basis representations and \ghostbit basis" representations can be used to implement inverters with a quantum circuit of depth O(mlog(m)). To the best of our knowledge, this is the rst construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the ItohTsujii algorithm which exploits the property that, in a normal basis representation, squaring corresponds to a permutation of the coe cients. We give resource estimates for the resulting quantum circuit for inversion over binary elds F2m based on an elementary gate set that is useful for faulttolerant implementation. Elliptic curves over nite elds F2m play a prominent role in modern cryptography. Published quantum algorithms dealing with such curves build on a short Weierstrass form in combination with a ne or projective coordinates. In this thesis we show that changing the curve representation allows a substantial reduction in the number of Tgates needed to implement the curve arithmetic. As a tool, we present a quantum circuit for computing multiplicative inverses in F2m in depth O(mlogm) using a polynomial basis representation, which may be of independent interest. Finally, we change our focus from the design of circuits which aim at attacking computational assumptions on asymmetric cryptographic algorithms to the design of a circuit attacking a symmetric cryptographic algorithm. We consider a block cipher, SERPENT, and our design of a quantum circuit implementing this cipher to be used for a key attack using Grover's algorithm as in [18]. This quantum circuit is essential for understanding the complexity of Grover's algorithm.
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Date Issued

2016

PURL

http://purl.flvc.org/fau/fd/FA00004662, http://purl.flvc.org/fau/fd/FA00004662

Subject Headings

Artificial intelligence, Computer networks, Cryptography, Data encryption (Computer science), Finite fields (Algebra), Quantum theory

Format

Document (PDF)


Title

Attitudes of urban high school mathematics teachers toward the mandate requiring algebra for high school graduation.

Creator

Gray, Noel Constantine., Florida Atlantic University, MaslinOstrowski, Patricia

Abstract/Description

The purpose of this study was to determine if urban high school mathematics teachers' attitudes toward the algebra mandate could be predicted by the variables: years of teaching mathematics, college major, highest college degree, gender, and ethnicity. Toward this end, the Attitude Toward the Algebra Mandate Survey (TATAMS) was administered to 98 high school mathematics teachers drawn from a total population of 493 who were employed by the School Board of MiamiDade County, Florida, during...
Show moreThe purpose of this study was to determine if urban high school mathematics teachers' attitudes toward the algebra mandate could be predicted by the variables: years of teaching mathematics, college major, highest college degree, gender, and ethnicity. Toward this end, the Attitude Toward the Algebra Mandate Survey (TATAMS) was administered to 98 high school mathematics teachers drawn from a total population of 493 who were employed by the School Board of MiamiDade County, Florida, during the 19981999 School Year. The study was carried out in June 1999, roughly 20 months after the mandate became effect in Florida. Multiple linear regression analysis was used to test each hypothesis and to provide a model that was. predict of teacher attitudes. Five null hypotheses were formed to determine if there were significant relationships between teacher attitudes toward the algebra mandate and the aforementioned variables. The results of the tests of five null hypotheses showed that the hypotheses that involved years of teaching mathematics and ethnicity was rejected. These five predictor variables accounted for 27% of the variance in teacher attitudes. The inference drawn from the study was that the negative attitudes of veteran White teachers and the positive attitudes of Hispanic teachers toward the mandate appear to have their roots in political and social considerations. Black teachers, on the other hand, have never challenged for the power in the district and are moderate in their attitudes toward the mandate.
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Date Issued

2000

PURL

http://purl.flvc.org/fcla/dt/12623

Subject Headings

Mathematics teachersAttitudes, High schoolsGraduation requirements, MathematicsStudy and teaching, Algebra

Format

Document (PDF)
Pages