Static Data Structure Lower Bounds Imply Rigidity

We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of $t \geq \omega(\log^2 n)$ on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space $(s= (1+\varepsilon)n)$, would already imply a semi-explicit ($\bf P^{NP}\rm$) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial ($t\geq n^{\delta}$) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime $(s=n+o(n))$, we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest

Tight Cell Probe Bounds for Succinct Boolean Matrix-Vector Multiplication

The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC'15]. In recent work, Larsen and Williams [SODA'17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries in $\tilde{O}(n^{7/4})$ time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional $\tilde{O}(n^{7/4})$ bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query time $\tilde{O}(n^{3/2})$ storing just $\tilde{O}(n^{3/2})$ bits on the side. We then complement our data structure with a lower bound showing that any data structure storing $r$ bits on the side, with $n < r < n^2$ must have query time $t$ satisfying $t r = \tilde{\Omega}(n^3)$. For $r \leq n$, any data structure must have $t = \tilde{\Omega}(n^2)$. Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over $\mathbb{F}_2$

Report of the Higher Education Study Commission [to the Governor and the General Assembly of Virginia]

This 1965 Report of the Higher Education Commission, appointed by Governor Albertis S. Harrison, Jr., was created to review higher education in Virginia to be used as a basis for long-range planning by the Commonwealth of Virginia. The Commission was led by Senator Lloyd C. Bird and supported by the staff of the State Council for Higher Education. Divided into eleven sections, this 200-page report details information including geographical location of students, library services, and different instructional services.https://scholarscompass.vcu.edu/vcu_books/1005/thumbnail.jp

Collaborative Development of Open Educational Resources for Open and Distance Learning

Open and distance learning (ODL) is mostly characterised by the up front development of self study educational resources that have to be paid for over time through use with larger student cohorts (typically in the hundreds per annum) than for conventional face to face classes. This different level of up front investment in educational resources, and increasing pressures to utilise more expensive formats such as rich media, means that collaborative development is necessary to firstly make use of diverse professional skills and secondly to defray these costs across institutions. The Open University (OU) has over 40 years of experience of using multi professional course teams to develop courses; of working with a wide range of other institutions to develop educational resources; and of licensing use of its educational resources to other HEIs. Many of these arrangements require formal contracts to work properly and clearly identify IPR and partner responsibilities. With the emergence of open educational resources (OER) through the use of open licences, the OU and other institutions has now been able to experiment with new ways of collaborating on the development of educational resources that are not so dependent on tight legal contracts because each partner is effectively granting rights to the others to use the educational resources they supply through the open licensing (Lane, 2011; Van Dorp and Lane, 2011). This set of case studies examines the many different collaborative models used for developing and using educational resources and explain how open licensing is making it easier to share the effort involved in developing educational resources between institutions as well as how it may enable new institutions to be able to start up open and distance learning programmes more easily and at less initial cost. Thus it looks at three initiatives involving people from the OU (namely TESSA, LECH-e, openED2.0) and contrasts these with the Peer-2-Peer University and the OER University as exemplars of how OER may change some of the fundamental features of open and distance learning in a Web 2.0 world. It concludes that while there may be multiple reasons and models for collaborating on the development of educational resources the very openness provided by the open licensing aligns both with general academic values and practice but also with well established principles of open innovation in businesses

Higher Education and Membership of Voluntary Groups

This research uses the British National Child Development Study to examine the effect of higher education on individual membership of voluntary groups and organizations. Gender differences in the education effects are given emphasis. We apply parametric and nonparametric econometric methods to isolate the influence of confounding variables. There is strong evidence of education endogeneity in the female sample and we observe a negative education effect on women's group membership. Education endogeneity does not cause serious estimation bias in the male sample. Higher education is a significantly positive determinant of men's group membership. Further investigations from a mid-life perspective reveal that the boost of female participation in the workforce and their attitudes towards employment are key factors in the negative association between higher education and women's group membership. Our research provides clues for the divergence in the enrolment in higher education and social participation behavior in Western countries.

Statistical Methods For Detecting Genetic Risk Factors of a Disease with Applications to Genome-Wide Association Studies

This thesis aims to develop various statistical methods for analysing the data derived from genome wide association studies (GWAS). The GWAS often involves genotyping individual human genetic variation, using high-throughput genome-wide single nucleotide polymorphism (SNP) arrays, in thousands of individuals and testing for association between those variants and a given disease under the assumption of common disease/common variant. Although GWAS have identified many potential genetic factors in the genome that affect the risks to complex diseases, there is still much of the genetic heritability that remains unexplained. The power of detecting new genetic risk variants can be improved by considering multiple genetic variants simultaneously with novel statistical methods. Improving the analysis of the GWAS data has received much attention from statisticians and other scientific researchers over the past decade. There are several challenges arising in analysing the GWAS data. First, determining the risk SNPs might be difficult due to non-random correlation between SNPs that can inflate type I and II errors in statistical inference. When a group of SNPs are considered together in the context of haplotypes/genotypes, the distribution of the haplotypes/genotypes is sparse, which makes it difficult to detect risk haplotypes/genotypes in terms of disease penetrance. In this work, we proposed four new methods to identify risk haplotypes/genotypes based on their frequency differences between cases and controls. To evaluate the performances of our methods, we simulated datasets under wide range of scenarios according to both retrospective and prospective designs. In the first method, we first reconstruct haplotypes by using unphased genotypes, followed by clustering and thresholding the inferred haplotypes into risk and non-risk groups with a two-component binomial-mixture model. In the method, the parameters were estimated by using the modified Expectation-Maximization algorithm, where the maximisation step was replaced the posterior sampling of the component parameters. We also elucidated the relationships between risk and non-risk haplotypes under different modes of inheritance and genotypic relative risk. In the second method, we fitted a three-component mixture model to genotype data directly, followed by an odds-ratio thresholding. In the third method, we combined the existing haplotype reconstruction software PHASE and permutation method to infer risk haplotypes. In the fourth method, we proposed a new way to score the genotypes by clustering and combined it with a logistic regression approach to infer risk haplotypes. The simulation studies showed that the first three methods outperformed the multiple testing method of (Zhu, 2010) in terms of average specificity and sensitivity (AVSS) in all scenarios considered. The logistic regression methods also outperformed the standard logistic regression method. We applied our methods to two GWAS datasets on coronary artery disease (CAD) and hypertension (HT), detecting several new risk haplotypes and recovering a number of the existing disease-associated genetic variants in the literature