Applications of Powerset Operators, Especially to Matroids

Let $\mathcal{V}$ denote a vector space over an arbitrary field with an inner product. For any collection $\mathcal{S}$ of vectors from $\mathcal{V}$ the collection of all vectors orthogonal to each vector in $\mathcal{S}$ is a subspace, denoted as $\mathcal{S}^{\perp_v}$ and called the \textit{orthogonal complement} of $\mathcal{S}$. One of the fundamental theorems of vector space theory states that, $(\mathcal{S}^{\perp_v})^{\perp_v}$ is the subspace \textit{spanned} by $\mathcal{S}$. Thus the ``spanning\u27\u27 operator on the subsets of a vector space is the square of the ``orthogonal complement\u27\u27 operator. In matroid theory, the orthogonal complement of a matroid $M$ is also well-defined and similarly results in another matroid. Although this new matroid is more commonly referred to as the `dual matroid\u27, denoted as $M^*$, and typically formed using a very different approach. There is an interesting relation between the circuits of a matroid $M$ and the cocircuits of $M$ (the circuits of its dual matroid $M^*$) which aligns much more closely to the orthogonal complement of a vector space. We expand on this relation to define a powerset operator: $(\phantom{S})^*$. Given $\mathcal{S} \subseteq \mathcal{P}(E)$, we denote $\mathcal{S}^*$ to be the minimal sets of $\{X \subseteq E \colon X \text{ is nonempty, } |X \cap A| \neq 1 \text{ for each } A \in \mathcal{S}\}$. We call this powerset operator the \textbf{circuit duality operator}. Unlike the vector space orthogonal complement operator, this circuit duality operator may not behave as nicely when applied to collections that do not correspond to a matroid. This thesis is an investigation into the development of tools and additional operators to help understand the collections of sets that result in a matroid under one or more applications of the circuit duality operator

Ideal Clutters

Let E be a finite set of elements, and let C be a family of subsets of E called members. We say that C is a clutter over ground set E if no member is contained in another. The clutter C is ideal if every extreme point of the polyhedron { x>=0 : x(C) >= 1 for every member C } is integral. Ideal clutters are central objects in Combinatorial Optimization, and they have deep connections to several other areas. To integer programmers, they are the underlying structure of set covering integer programs that are easily solvable. To graph theorists, they are manifest in the famous theorems of Edmonds and Johnson on T-joins, of Lucchesi and Younger on dijoins, and of Guenin on the characterization of weakly bipartite graphs; not to mention they are also the set covering analogue of perfect graphs. To matroid theorists, they are abstractions of Seymour’s sums of circuits property as well as his f-flowing property. And finally, to combinatorial optimizers, ideal clutters host many minimax theorems and are extensions of totally unimodular and balanced matrices. This thesis embarks on a mission to develop the theory of general ideal clutters. In the first half of the thesis, we introduce and/or study tools for finding deltas, extended odd holes and their blockers as minors; identically self-blocking clutters; exclusive, coexclusive and opposite pairs; ideal minimally non-packing clutters and the τ = 2 Conjecture; cuboids; cube-idealness; strict polarity; resistance; the sums of circuits property; and minimally non-ideal binary clutters and the f-Flowing Conjecture. While the first half of the thesis includes many broad and high-level contributions that are accessible to a non-expert reader, the second half contains three deep and technical contributions, namely, a character- ization of an infinite family of ideal minimally non-packing clutters, a structure theorem for ±1-resistant sets, and a characterization of the minimally non-ideal binary clutters with a member of cardinality three. In addition to developing the theory of ideal clutters, a main goal of the thesis is to trigger further research on ideal clutters. We hope to have achieved this by introducing a handful of new and exciting conjectures on ideal clutters

Understanding the Attitude of Generation Z Consumers Towards Advertising Avoidance on the Internet

One of the biggest challenges faced by marketers today is to comprehend the reasons behind people’s avoidance towards advertisements worldwide, and how that can be managed. Although many researchers have explored the subject in both traditional and contemporary marketing communication mediums, there is no evidence of studies conducted in the context of Bangladeshi market, with specific concentration on Generation Z consumers. This generation constitutes a significant portion of the entire population in the country, indicating that a major share of current and potential customers belong to this age group. Intriguingly, even though they are characterized as highly tech-savvy customers, they are also more likely to avoid online advertisements, making the marketing efforts of organizations ineffective. Therefore, this study investigates the determinants that cause the Generation Z consumers in Bangladesh to avoid advertisements on the Internet. The collected data from 280 respondents were analyzed through descriptive statistics using SPSS.24, followed by confirmatory factor analysis (CFA) and structure equation modeling (SEM), which were performed with the help of AMOS.17 to eventually test the hypotheses developed for this study. The findings indicate goal impediment, privacy concern, ad clutter, and negative experiences are positively related to advertising avoidance online. Keywords: advertising avoidance, goal impediment, privacy concern, ad clutter, negative experience, and generation-z consumer

Effects of spatial resolution on radar-based precipitation estimation using sub-kilometer X-band radar measurements

Known for the ability to observe precipitation at spatial resolution higher than rain gauge networks and satellite products, weather radars allow us to measure precipitation at spatial resolutions of 1 kilometer (typical resolution for operational radars) and a few hundred meters (often used in research activities). In principle, we can operate a weather radar at resolution higher than 100m and the expectation is that radar data at higher spatial resolution can provide more information. However, there is no systematic research about whether the additional information is noise or useful data contributing to the quantitative precipitation estimation. In order to quantitatively investigate the changes, as either benefits or drawbacks, caused by increasing the spatial resolution of radar measurements, we set up an X-band radar field experiment from May to October in 2017 in the Stuttgart metropolitan region. The scan strategy consists of two quasi-simultaneous scans with a 75-m and a 250-m radial resolution respectively. They are named as the fine scan and the coarse scan, respectively. Both scans are compared to each other in terms of the radar data quality and their radar-based precipitation estimates. The primary results from these comparisons between the radar data of these two scans show that, in contrast to the coarse scan, the fine scan data are characterized with losses of weak echoes, are more subjected to external signals and second-trip echoes (drawback), are more effective in removing non-meteorological echoes (benefit), are more skillful in delineating convective storms (benefit), and show a better agreement with the external reference data (benefit)