The play Eguus presents a series of dialectics, opposing forces in dramatic tension. The multi-leveled subjects with which Shaffer works confront each other as thesis and antithesis working towards a tentative synthesis. The contrasts include the conflict of art and science, the Apollonian and Dionysian polarity, and the confrontation of Christianity and paganism. Modern man faces these conflicts and attempts to come to terms with them. These opposites are really paradoxes. They seem to contradict each other, but, in fact, they are not mutually exclusive. Rather than contradicting each other, each aspect of a dialectic influences its counterpoint; both are necessary to make a whole person.

Greatest common divisor domains, Bezout domains, valuation rings, and Prüfer domains are studied. Chapter One gives a brief introduction, statements of definitions, and statements of theorems without proof. In Chapter Two theorems about greatest common divisor domains and characterizations of Bezout domains, valuation rings, and Prüfer domains are proved. Also included are characterizations of a flat overring. Some of the results are that an integral domain is a Prüfer domain if and only if every overring is flat and that every overring of a Prüfer domain is a Prüfer domain.

In the present thesis we analyze different types of additive decompositions of homogeneous polynomials. These problems are usually called Waring-type problems and their story go back to the mid-19th century and, recently, they received the attention of a large community of mathematicians and engineers due to several applications. At the same time, they are related to branches of Commutative Algebra and Algebraic Geometry. The classical Waring problem investigates decompositions of homogeneous polynomials as sums of powers of linear forms. Via Apolarity Theory, the study of these decompositions for a given polynomial F is related to the study of configuration of points apolar to F, namely, configurations of points whose defining ideal is contained in the ``perp'' ideal associated to F. In particular, we analyze which kind of minimal set of points can be apolar to some given polynomial in cases with small degrees and small number of variables. This let us introduce the concept of Waring loci of homogeneous polynomials. From a geometric point of view, questions about additive decompositions of polynomials can be described in terms of secant varieties of projective varieties. In particular, we are interested in the dimensions of such varieties. By using an old result due to Terracini, we can compute these dimensions by looking at the Hilbert series of homogeneous ideal. Hilbert series are very important algebraic invariants associated to homogeneous ideals. In the case of classical Waring problem, we have to look at power ideals, i.e., ideals generated by powers of linear forms. Via Apolarity Theory, their Hilbert series are related to Hilbert series of ideals of fat points, i.e., ideals of configurations of points with some multiplicity. In this thesis, we consider some special configuration of fat points. In general, Hilbert series of ideals of fat points is a very active field of research. We explain how it is related to the famous Fröberg's conjecture about Hilbert series of generic ideals. Moreover, we use Fröberg's conjecture to deduce the dimensions of several secant varieties of particular projective varieties and, then, to deduce results regarding some particular Waring-type problems for polynomials. In this thesis, we mostly work over the complex numbers. However, we also analyze the case of classical Waring decompositions for monomials over the real numbers. In particular, we classify for which monomials the minimal length of a decomposition in sum of powers of linear forms is independent from choosing the ground field as the field of complex or real numbers.

Title: Homomorphic encryption and coding theory Author: Veronika Půlpánová Department: Department of algebra Supervisor: RNDr. Michal Hojsík, Ph.D., Department of algebra Abstract: The current mainstream in fully homomorphic encryption is the appro- ach that uses the theory of lattices. The thesis explores alternative approaches to homomorphic encryption. First we present a code-based homomorphic encrypti- on scheme by Armknecht et. al. and study its properties. Then we describe the family of cryptosystems commonly known as Polly Cracker and identify its pro- blematic aspects. The main contribution of this thesis is the design of a new fully homomorphic symmetric encryption scheme based on Polly Cracker. It proposes a new approach to overcoming the complexity of the simple Polly Cracker - based cryptosystems. It uses Gröbner bases to generate zero-dimensional ideals of po- lynomial rings over finite fields whose factor rings are then used as the rings of ciphertexts. Gröbner bases equip these rings with a multiplicative structure that is easily algorithmized, thus providing an environment for a fully homomorphic cryptosystem. Keywords: Fully homomorphic encryption, Polly Cracker, coding theory, zero- dimensional ideals

I uppsatsen undersöks synen på kropp och ideal som svenska gymnasieungdomar har. I vårt samhälle och på sociala medier publiceras bilder på kroppar, bilderna sprids och skapar ett ideal för hur kroppen ska se ut. Detta kan leda till negativ uppfattning om den egna kroppen. Syftet med undersökningen är att synliggöra och problematisera kroppsideal och normer som finns i samhället och istället skapa en  känsla av tillhörighet, samt att starta en diskussion med gymnasieungdomar. Det är viktigt att tillsammans med gymnasieungdomar prata om kroppsideal och normer för att tillsammans kunna omförhandla ideal och normer. Fyra samtalsgrupper med gymnasieungdomar på två olika skolor i Sverige samt sju personer som hjälpt till som modeller fungerar som källor i projektet och frågeställning om kroppsideal. Frågeställningen lyder; Hur kan normer kring kropp och kroppsideal synliggöras genom samtal med gymnasieelever och konstfacksstudenter? Hur kan dessa samtal ligga till grund för ett konstnärligt arbete? Studiens resultat visar att kroppsliga bilder påverkar gymnasieungdomars syn på kroppsideal. Bland ungdomarna fanns det en kunskap om kroppsidealet. Kunskap om att det inte är hälsosamt samt att normativa idealbilder är sammankopplade med individers självförtroende. Ungdomarna beskrev att ungdomar med ett dåligt självförtroende påverkades mer än vad de som hade ett bra självförtroende.  Mitt gestaltningsarbete består av avgjutna torsos i silikon som gjorts efter samtal med gymnasieungdomar och modeller. Min gestaltning inkluderades åtta stycken silikonkroppar. Personerna som gjutits av är i åldrarna 20-45, men vi alla är vackra oavsett ålder, figur eller könsidentitet. Jag vill med mitt verk påverka vår syn på kroppsideal och skapa en känsla av tillhörighet. Att våga vara stolt över sin egen kropp.

<div>In the first part of this thesis, we study Rees algebras of modules. We investigate their Cohen-Macaulay property and their defining ideal, using <i>generic Bourbaki ideals</i>. These were introduced by Simis, Ulrich and Vasconcelos in [65], in order to characterize the Cohen-Macaulayness of Rees algebras of modules. Thanks to this technique, the problem is reduced to the case of Rees algebras of ideals. Our main results are the following.</div><div><br></div><div><div>In Chapters 3 and 4 we consider a finite module <i>E</i> over a Gorenstein local ring <i>R</i>. In Theorem 3.2.4 and Theorem 4.3.2, we give sufficient conditions for <i>E</i> to be of linear type, while Theorem 4.2.4 provides a sufficient condition for the Rees algebra <i>R(E)</i> of <i>E</i> to be Cohen-Macaulay. These results rely on properties of the residual intersections of a generic Bourbaki ideal <i>I</i> of<i> E</i>, and generalize previous work of Lin (see [46, 3.1 and 3.4]). In the case when <i>E</i> is an ideal, Theorem 4.2.4 had been previously proved independently by Johnson and Ulrich (see [39, 3.1]) and Goto, Nakamura and Nishida (see [20, 1.1 and 6.3]).</div></div><div><br></div><div><div>In Chapter 5, we consider a finite module <i>E</i> of projective dimension one over <i>k</i>[X₁, . . . , X_n]. Our main result, Theorem 5.2.6, describes the defining ideal of <i>R(E)</i>, under the assumption that the presentation matrix φ of <i>E</i> is <i>almost linear</i>, i.e. the entries of all but one column of φ are linear. This theorem extends to modules a known result of Boswell and Mukundan on the Rees algebra of almost linearly presented perfect ideals of height 2 (see [5, 5.3 and 5.7]).</div></div><div><br></div><div><div>The second part of this thesis studies the Cohen-Macaulay property of the special fiber ring<i> F(E)</i> of a module <i>E</i>. In Theorem 6.2.14, we prove that the generic Bourbaki ideals of Simis, Ulrich and Vasconcelos allow to reduce the problem to the case of fiber cones of ideals, similarly as for Rees algebras. We then provide sufficient conditions for <i>F(E)</i> to be Cohen-Macaulay. Our Theorems 6.2.15, 6.1.3 and 6.2.18 are module versions of results proved for the fiber cone of an ideal by Corso, Ghezzi, Polini and Ulrich (see [10, 3.1] and [10, 3.4]) and by Monta˜no (see [47, 4.8]), respectively.</div></div><div><br></div>

For D an integral domain with field of fractions K and E a subset of K, the ring Int (E,D) = {f e K[X]lf (E) C D} of integer-valued polynomials on E has been well studies. In particulare, when E is a finite subset of D, Chapman, Loper, and Smith, as well as Boynton and Klingler, obtained a bound on the number of elements needed to generate a finitely generated ideal of Ing (E, D) in terms of the corresponding bound for D. We obtain analogous results for Int (r) (E, D) - {f e K [X]lf(k) (E) c D for all 0 < k < r} , for finite E and fixed integer r > 1. These results rely on the work of Skolem [23] and Brizolis [7], who found ways to characterize ideals of Int (E, D) from the values of their polynomials at points in D. We obtain similar results for E = D in case D is local, Noetherian, one-dimensional, analytically irreducible, with finite residue field. / by Yuri Villanueva. / Thesis (Ph.D.)--Florida Atlantic University, 2012. / Includes bibliography. / Mode of access: World Wide Web. / System requirements: Adobe Reader.

x, 81 p. : ill. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / We give an exposition of Deligne's tensor category Rep(St) where t is not necessarily an integer. Thereafter, we give a complete description of the blocks in Rep(St) for arbitrary t. Finally, we use our result on blocks to decompose tensor products and classify tensor ideals in Rep(St). / Committee in charge: Victor Ostrik, Chairperson, Mathematics; Daniel Dugger, Member, Mathematics; Jonathan Brundan, Member, Mathematics; Alexander Kleshchev, Member, Mathematics; Michael Kellman, Outside Member, Chemistry

This paper explores the fundamental meanings of deconstruction in fashion design and has the aim to investigate deconstruction in feminine ideal. It also stresses other thoughts of deconstruction in terms of philosophy through Jacques Derrida, architecture and philosopher Peter Eisenman and how deconstruction can be applied to find parallels between its setting and the setting it is compared to. Since the 1960s, deconstruction is a term that has been interpreted within many fields and traversed across different media. Influential Japanese designers have used the term in their works, juxtapositioning them to traditional Western ideas to create clear contrasts between stereotypical and categorised perception and unconventional interpretations. During the 1980s, designers such as Rei Kawakubo and Yohji Yamamoto explored the term to subsequently create a great distress in the fashion field. Their designs were examples of archetypes evoked from the past and presented as newborn strangers or dismantled ghosts. These designers investigated the mechanical functions of each archetype as they sought to find the meaning of each garment to later reinterpret its traditional essence. Also, they questioned the relationship between body and garment, raising thoughts of whether or not the bearer of the garment was personified to the garments traditional significance. The deconstructed element chosen for investigation in this project consists of a personification of the silhouette of the 1870s dress. This personification is discussed in terms of social and moral standards and constrictions as well as the political function of the dress. The fact that you could deconstruct a 1870s dress is clearly a way to take a historical archetype from its traditional meaning and place it into a new context. Similar to Jacques Derrida, the works of deconstruction in fashion design discuss our assumptions of archetypes and whether or not these archetypes can ever lack of historical or individual meaning. The constant dialogue with the past is a catalyst to reinterpret standardisations in fashion design through questioning the conformity of archetypes. / Program: Modedesignutbildningen

This paper determines whether monoid rings with the two-generator property have the strong two-generator property. Dedekind domains have both the two-generator and strong two-generator properties. How common is this? Two cases are considered here: the zero-dimensional case and the one-dimensional case for monoid rings. Each case is looked at to determine if monoid rings that are not PIRs but are two-generated have the strong two-generator property. Full results are given in the zero-dimensional case, however only partial results have been found for the one-dimensional case.