Since the early 1990s, a classical object in commutative algebra has been the study of binomial ideals. A widely-investigated class of binomial ideals is the one containing those generated by a subset of 2-minors of an (m x n)-matrix of indeterminates. This thesis is devoted to illustrate some algebraic and homological properties of two classes of ideals of 2-minors: binomial edge ideals and polyomino ideals. Binomial edge ideals arise from finite graphs and their appeal results from the fact that their homological properties reflect nicely the combinatorics of the underlying graph. First, we focus on the binomial edge ideals of block graphs. We give a lower bound for their Castelnuovo-Mumford regularity by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present a linear time algorithm to compute Castelnuovo-Mumford regularity and Krull dimension of binomial edge ideals of block graphs. Secondly, we consider some classes of Cohen-Macaulay binomial edge ideals. We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs. Polyomino ideals arise from polyominoes, plane figures formed by joining one or more equal squares edge to edge. It is known that the polyomino ideal of simple polyominoes is prime. We consider multiply connected polyominoes, namely polyominoes with holes, and observe that the non-existence of a certain sequence of inner intervals of the polyomino, called zig-zag walk, gives a necessary condition for the primality of the polyomino ideal. Moreover, by computational approach, we prove that for all polyominoes with rank less than or equal to 14 the above condition is also sufficient. Lastly, we present an infinite class of prime polyomino ideals.

The purpose of this study is to discuss the ideal of public service, what it is and what it should be and how it has evolved in American history. I am concerned that the ideal of public service, a guiding principle for public administrators and government, has been diminished by the emphasis in American public administration on economy, efficiency, and effectiveness. If public sector leaders, such as presidents, over time have changed their focus from discussions of the Constitution or republican principles to other ideas such as taxes and specific government programs, then has the ideal or vision of public service changed? The topic of service, aligned with duty and the responsibility of the citizen, while prevalent in the political theory literature, has been pushed to the back of the public administration literature shelves. Without the ideal of service as a vibrant element of the public administration discourse, the public administration community will have given way to those who see public administration as merely a modified business model. The primary research question posed in this study was: Has the emphasis on economy, efficiency, and effectiveness diminished the ideal of public service? The texts of presidential inaugural addresses were analyzed and reviewed for insights into the ideal of public service by the use of content analysis, using nine criteria that exemplify the ideals of public service. One of the secondary questions posed here was: Has the ideal or vision of public service changed? Discussion of the different criteria illustrated that there has been a change resulting in a different concept of the ideal of public service today. The final question of how and when the ideal of public service changed is answered in the dissertation also, as the results, criterion by criterion, are explained. This study shows that the inaugural addresses captured the ideal of public service dynamically and often eloquently over the course of American history. The changing nature of the role of government, issues of importance to government, and nature of the ideal of public service are on the historical record as set forth in these inaugural addresses. / Ph. D.

Strongly stable ideals are important in algebraic geometry, commutative algebra, and combinatorics. Prompted, for example, by combinatorial approaches for studying Hilbert schemes and the existence of maximal total Betti numbers among saturated ideals with a given Hilbert polynomial, three algorithms are presented. Each of these algorithms produces all strongly stable ideals with some prescribed property: the saturated strongly stable ideals with a given Hilbert polynomial, the almost lexsegment ideals with a given Hilbert polynomial, and the saturated strongly stable ideals with a given Hilbert function. Bounds for the complexity of our algorithms are included. Also included are some applications for these algorithms and some estimates for counting strongly stable ideals with a fixed Hilbert polynomial.

Even though Sweden is perceived to be on the gender equality frontline parenthood is still, according to previous research, influenced by parent ideals which are based on traditional ideas about gender. This is visualized through the fact that women usually spend more time on childcare and household work than men, whereas men mainly provide for the family. Parenthood can therefore be understood as a part of the process of doing gender. We do gender through interaction with others and it is also through interaction that traditional ideas about gender can be maintained and reproduced. In today's society we interact through social media and digital platforms to a large extent, which implies that social science research should study this context more. Based on this argument, this study is designed as a qualitative thematic content analysis which examines how parenthood ideals are visualized through members’ interaction on Familjeliv.se. The aim is to study if traditional ideas about gender are maintained, reproduced and challenged through this interaction. The posts on Familjeliv.se were analyzed using West and Zimmerman’s concept doing gender and Connell’s gender theory. The results show that traditional ideas about gender are both maintained, reproduced and challenged through the discussions on the forum. The users mainly express more equal parent ideals but seem to have difficulty putting this into practice. The conclusion is that the users on Familjeliv.se are doing gender in both normative and different ways. / Trots att Sverige anses vara i framkant gällande jämställdhet präglas föräldraskap, enligt tidigare forskning, till stor del av föräldraideal som baserar sig på traditionella föreställningar om kön. Detta synliggörs genom att kvinnor ägnar mer tid än män åt barnomsorg och hushållsarbete medan män ägnar mer tid åt förvärvsarbete än kvinnor. Föräldraskapet kan således ses som en del i görandet av kön. Görandet av kön sker genom interaktion med andra och det är även på detta sätt som traditionella föreställningar om kön kan upprätthållas och reproduceras. I dagens samhälle interagerar vi människor i stor utsträckning på sociala medier och digitala plattformar, vilket innebär en ny typ av kontext som samhällsforskning bör studera. Baserat på detta resonemang utformades en kvalitativ tematisk innehållsanalys som undersöker hur användare på forumet Familjeliv.se interagerar kring föräldraideal. Syftet är att studera om och i så fall hur traditionella föreställningar om kön upprätthålls, reproduceras och utmanas genom denna interaktion. Diskussionsinläggen på forumet analyserades med utgångspunkt i West och Zimmermans teori om doing gender och Connells genusteori. Resultaten visar att traditionella föreställningar om kön både upprätthålls och reproduceras samt utmanas i diskussionerna på Familjeliv.se. Användarna framställer framför allt mer jämställda föräldraideal men verkar ha svårt att realisera detta i praktiken. Slutsatsen blev således att användarna gör kön på både normativa sätt och annorlunda sätt.

In this research, the women ideals, during the 1930s have been studied based on the content of two women's magazines, Hertha and Husmodern. The purpose of the study was to look in to, how the ideals are expressed in each magazine, how the growing nationalism and the society’s social structure are affected by the design of the ideals, and then concluded in a comparison between the results of the magazines content. The study also had a gender perspective and was based on the theories gendersystem and genderorder. Results showed that ideals of beauty, ideals of housewives, motherhood and knowledge as ideals were indicated in the magazines, in different ways; through advertisements, articles and question columns. There was also a clear difference in which ideals the magazines expressed and also how they were expressed. In summary, the study showed that the prevailing societal changes after the First World War contributed to a process of change among Swedish women and they became increasingly aware of their rights and their independence. The study also showed that beauty- and body ideals became important ideals to follow as a Swedish woman.

<div>The integral closure $\overline{I}$ of an ideal $I$ in a ring $R$ consists of all elements $x \in R$ that are integral over $I$. If $R$ is an algebra over an infinite field $k$, one can define general elements of $I=\left( x_{1},\ldots,x_{n}\right)$ as $x_{\alpha}=\sum_{i=1}^{n}\alpha_{i}x_{i}$ with $(\alpha_{1},\ldots,\alpha_{n})$ belonging to a Zariski-open subset of $k^{n}$. </div><div><br></div><div>We prove that for any ideal $I$ of height at least $2$ in a local, equidimensional excellent algebra over a field of characteristic zero, the integral closure specializes with respect to a general element of $I$. That is, we show that $\overline{I}/(x)=\overline{I/(x)}$. </div><div><br></div><div>In a Noetherian local ring $(R,m)$ of dimension $d$, one has a sequence of ideals approximating the integral closure of $I$ for $I$ an $m$-primary ideal. The ideals </div><div>\[ I \subseteq I_{\{d\}} \subseteq \cdots \subseteq I_{\{1\}} \subseteq I_{\{0\}}=\overline{I}\]</div><div>are the coefficient ideals of $I$. The $i^{\text{th}}$ coefficient ideal $I_{\{i\}}$ of $I$ is the largest ideal containing $I$ and integral over $I$ for which the first $i+1$ Hilbert coefficients of $I$ and $I_{\{i\}}$ coincide. </div><div><br></div><div>With a goal of understanding how coefficient ideals behave under specialization by general elements, we turn to the case of monomial ideals in polynomial rings over a field. A consequence of the specialization of the integral closure is that the $i^{\text{th}}$ coefficient ideal specializes when the $i^{\text{th}}$ coefficient ideal coincides with the integral closure. To this end, we give a formula for first coefficient ideals of $m$-primary monomial ideals generated in one degree in $2$ variables in order to describe when $I_{\{1\}}=\overline{I}$. In the $2$-dimensional case, we characterize the behavior of all coefficient ideals with respect to specialization by general elements. </div><div><br></div><div>In the $d$-dimensional case for $d \geq 3$, we give a characterization of when $I_{\{1\}}=\overline{I}$ for $m$-primary monomial ideals generated in one degree. In the final chapter, we give an application to the core, by characterizing when $\core(I)=\adj(I^{d})$ for such ideals.</div><div><br></div>

<div>This dissertation is based on joint work with Monte Cooper and is broken into two main parts, both of which study the defining ideals of the Rees rings of determinantal and Pfaffian ideals of generic height. In both parts, we attempt to place degree bounds on the defining equations.</div><div> </div><div> The first part of the dissertation consists of Chapters 3 to 5. Let $R = K[x_{1},\ldots,x_{d}]$ be a standard graded polynomial ring over a field $K$, and let $I$ be a homogeneous $R$-ideal generated by $s$ elements. Then there exists a polynomial ring $\mathcal{S} = R[T_{1},\ldots,T_{s}]$, which is also equal to $K[x_{1},\ldots,x_{d},T_{1},\ldots,T_{s}]$, of which the defining ideal of $\mathcal{R}(I)$ is an ideal. The polynomial ring $\mathcal{S}$ comes equipped with a natural bigrading given by $\deg x_{i} = (1,0)$ and $\deg T_{j} = (0,1)$. Here, we attempt to use specialization techniques to place bounds on the $x$-degrees (first component of the bidegrees) of the defining equations, i.e., the minimal generators of the defining ideal of $\mathcal{R}(I)$. We obtain degree bounds by using known results in the generic case and specializing. The key tool are the methods developed by Kustin, Polini, and Ulrich to obtain degree bounds from approximate resolutions. We recover known degree bounds for ideals of maximal minors and submaximal Pfaffians of an alternating matrix. Additionally, we obtain $x$-degree bounds for sufficiently large $T$-degrees in other cases of determinantal ideals of a matrix and Pfaffian ideals of an alternating matrix. We are unable to obtain degree bounds for determinantal ideals of symmetric matrices due to a lack of results in the generic case; however, we develop the tools necessary to obtain degree bounds once similar results are proven for generic symmetric matrices.</div><div> </div><div> The second part of this dissertation is Chapter 6, where we attempt to find a bound on the $T$-degrees of the defining equations of $\mathcal{R}(I)$ when $I$ is a nonlinearly presented homogeneous perfect Gorenstein ideal of grade three having second analytic deviation one that is of linear type on the punctured spectrum. We restrict to the case where $\mathcal{R}(I)$ is not Cohen-Macaulay. This is a natural next step following the work of Morey, Johnson, and Kustin-Polini-Ulrich. Based on extensive computation in Macaulay2, we give a conjecture for the relation type of $I$ and provide some evidence for the conjecture. In an attempt to prove the conjecture, we obtain results about the defining ideals of general fibers of rational maps, which may be of independent interest. We end with some examples where the bidegrees of the defining equations exhibit unusual behavior.</div>

Using cross-cultural and cultural perspectives, this thesis investigated ideals of parenting and the relations between parenting mismatches and youth adjustment in a sample of adolescents from Chinese immigrant families and Canadian non-immigrant families. Asian traditional parenting techniques have been linked to both positive and negative outcomes in Asian immigrant youth. This study sought to clarify these mixed findings by focusing on adolescent interpretations of parenting, specifically their opinions of how parents should behave, which is referred to as parenting ideals. Parental behaviours of warmth, reasoning, monitoring, and harsh discipline were investigated. Results indicated that adolescents from Chinese immigrant families and Canadian adolescents desired similar levels of these parenting behaviours from their fathers and mothers, with the exception of harsh discipline behaviours. Amongst Chinese adolescents, some findings supported the hypothesis that cultural orientation is related to parenting ideals. Parenting mismatches, or discrepancies between actual and ideal parenting, were hypothesized to be related to more depressive symptoms and lower self-esteem. Findings supported this hypothesis in the dimensions of parental warmth and monitoring, but not reasoning. Low endorsement of perceptions, ideals, and mismatches in harsh discipline precluded strong conclusions about this parenting dimension. Directions for future research and implications for parent education and clinical settings were discussed. / Graduate / 0621 / vivienso@uvic.ca

The problem undertaken in this paper is to determine what the algebraic structure of the class of C-sets is, when the notion of sum is to be the "set sum. " While the preliminary work done by Appling took place in the space of additive and bounded real valued functions, the results here are found in the more general setting of a complete lattice ordered group. As a conseque n c e , G . Birkhof f' s book, Lattice Theory, is used as the standard reference for most of the terminology used in the paper. The direction taken is prompted by a paper by W. D. L. Appling, "A Generalization of Absolute Continuity and of an Analogue of the Lebesgue Decomposition Theorem. " Since some of the results obtained provide another approach to a problem originally studied by Nakano, and improved upon by Bernau, reference is made to their work to provide other terminology and examples of alternative approaches to the problem of lateral completion. Thus Chapter I contains a brief history of the notion of C-sets and their relationship to lattice ordered groups, along with a summary of the properties of lattice ordered groups needed for later developments. In addition, several results in the general theory of lattice ordered groups are cited to provide insight into the comparability of the assumptions that will ultimately be made about the groups. Chapter II begins with the axiomatization of the collection of nearest point functions" for the closed A-ideals of the cone of a complete lattice ordered group. The basic results in the chapter establish that the functions defined do indeed characterize the complete A-ideals, and that the maps have a 'nearest point property." The maps are then extended to the entire group and shown to correspond to the "nearest point maps" for a C-set in PAB' Chapter III is devoted to exploring the algebraic structures found in the collection of all closed A-ideal maps, denoted J. J is shown to be a lattice ordered monoid, abelian and complete, containing a maximal group cone P*. It is further shown that the original group cone P is isomorphic to a subset of P*. Chapter IV looks into a rather interesting characterization of P*, one that, in the terminology of Bernau, implies that P* is the cone of the group that is the lateral completion of the original group. A final result is a demonstration that the members of j each have a representation as the sum of an element of P* and an additive element of j.

The primary aim of the research attempted to investigate whether ethnic differences in self-image existed amongst White and African females. It further investigated whether specific factors such as family, peer relations and psychosocial factors (specifically perceptions regarding the media) have had a positive or negative influence on weight. Additionally it explored whether black females may still perceive a fuller figure as being more acceptable or whether changes have occurred within past and/or present ideologies. The study used both qualitative and quantitative analysis. In order to establish whether ethnicity was significantly different from BMI, BSQ, SATAQ-3, t-tests (non-parametric one-way ANOVA comparisons) were performed. Ethnicity was not significantly different in relation to the dependent variables BSQ; overall SATAQ-3 scores and its dimensions; and affirmation, belonging and commitment of the MEIM. However significant differences were found between BMI, the overall MEIM and its subscale ethnic identity. Low and high scores of the MEIM revealed no differences across all measurement instruments used within the study, demonstrating no effect upon BSQ or the SATAQ-3 and its subscales. However there was a significant difference found between ethnic groups with regard to BMI. Fisher’s –z was used to conduct comparisons between the correlations established using Pearson’s Rank Coefficient Correlations. The results indicated that significant relationships did exist between the BSQ, specific subscales of the SATAQ-3, and BMI. Results obtained from Fisher’s-z revealed significant differences on the BSQ and SATAQ-3 (including its subscales) correlations. Multiple Regression was conducted to establish whether BMI, MEIM and SATAQ-3 have had an impact upon the outcome of BSQ. The comparisons revealed that BMI and sociocultural factors may have an impact upon the perceptions of body shape and size. Major themes identified within the qualitative analysis were culture, inherited concepts, media, family, friends and parental (mother) influences amongst others. The results revealed that White females may experience greater body dissatisfaction. African females had stated that their body concept is influenced by ethnicity, while White females communicated media and stereotyped images of white females as being the source of pressure to loose weight.