Signed graph embedding: when everybody can sit closer to friends than enemies

Signed graphs are graphs with signed edges. They are commonly used to represent positive and negative relationships in social networks. While balance theory and clusterizable graphs deal with signed graphs to represent social interactions, recent empirical studies have proved that they fail to reflect some current practices in real social networks. In this paper we address the issue of drawing signed graphs and capturing such social interactions. We relax the previous assumptions to define a drawing as a model in which every vertex has to be placed closer to its neighbors connected via a positive edge than its neighbors connected via a negative edge in the resulting space. Based on this definition, we address the problem of deciding whether a given signed graph has a drawing in a given $\ell$-dimensional Euclidean space. We present forbidden patterns for signed graphs that admit the introduced definition of drawing in the Euclidean plane and line. We then focus on the $1$-dimensional case, where we provide a polynomial time algorithm that decides if a given complete signed graph has a drawing, and constructs it when applicable

Student and School Indicators for Youth in California's Central Valley

Provides a statistical portrait of elementary, secondary, and postsecondary education in the Central Valley. Examines trends in school resources, course enrollment, and student achievement in the region, and compares to trends in the rest of the state

String rewriting for Double Coset Systems

In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solved using our approach. Even when both enumerative and rewriting techniques are present, our rewriting methods will be competitive because they i) do not require the preliminary calculation of cosets; and ii) as with single coset problems, there are many examples for which rewriting is more effective than enumeration. Automata provide the means for identifying expressions for normal forms in infinite situations and we show how they may be constructed in this setting. Further, related results on logged string rewriting for monoid presentations are exploited to show how witnesses for the computations can be provided and how information about the subgroups and the relations between them can be extracted. Finally, we discuss how the double coset problem is a special case of the problem of computing induced actions of categories which demonstrates that our rewriting methods are applicable to a much wider class of problems than just the double coset problem.Comment: accepted for publication by the Journal of Symbolic Computatio

School Leadership Training under Globalisation: Comparisons of the UK, the US and Norway

In common the three countries see a need for increased quality of schooling as necessary because of globalisation. Leadership is crucial to achieve quality. However, there are distinct critiques in all countries fearing ineffective bureaucratization. There is resistance among education researchers towards the market orientation and the application of the language of business. Universities have played a conservative role. In terms of differences, the UK is uniformby its centrally organised National College, while the US with over 500 programmes and no national coordination shows complexity, if not chaos. Norway, with its National Network gives much freedom to individual institutions, although the diversity leads to tensions when the municipalities now can choose the training providers. All three nations are attempting to ‘reframe and reform’. Some educators think the defining factors will be quality of performance and quality of collaboration, while others believe that there must be a shift from focus on performance to focus on learning.globalisation, school leadership, training, policy

Solid-state NMR study of polymorphism in pharmaceuticals

This thesis is concerned with attempts to establish solid-state NMR as a complementary technique to X-ray crystallography for providing information on crystal structure, i.e. information regarding conformation, packing and inter- or intra-molecular interactions. Methods of assigning the solid-state NMR spectrum have been used which rely upon (^1)H-(^13)C dipolar interactions and upon the shielding anisotropy experienced by the nuclei. Single pulse experiments have also been used to identify solvate molecules. Of particular importance, the anisotropy and asymmetry have been found for a number of steroids, and the values used to assign the high frequency region of the spectrum. This has enabled chemical shifts to be linked to the hydrogen bonds that are present. Thus, the mode of hydrogen bonding can be predicted in forms of cortisone acetate for which the crystal structure is not available. Chemical shifts have also been linked with the conformation of the D ring. In this way, important crystallographic information has been gleaned from the solid-state NMR spectrum. The effect of the formation of hydrogen bonds upon the shielding tensor components of carbonyl and ester carbons has been studied, and found to cause a high frequency shift in δ22, which is probably oriented along the C=0 bonds. This leads to a high frequency shift in the isotropic chemical shift. The effect of the spinning speed upon the centreband intensity of a peak as the anisotropy and asymmetry are varied has been investigated. It has been found that at low spiiming speeds, the centreband intensity does not vary in a simple fashion. Thus it is recommended that full shielding tensor analysis is performed. The effect of the spinning speed on the sample temperature has also been studied. Results show that at low spinning speeds (up to 2 kHz) there is sample cooling as the spinning speed increases (attributed to the Joule -Thompson effect). But at higher spinning speeds (above 2 kHz) then there is sample heating as the spinning speed increases. The importance of this result is discussed. The effect of quadrupolar nuclei (35C1 and (^14)N) upon the spectrum of a spin-(^1)(_2) nucleus ((^13)C) has been studied. In favourable cases, the anisotropy in the spin-spin coupling constant has been found, whilst in other cases, the sign and magnitude of the quadrupolar coupling constant have been obtained. The broadening of resonances of carbon atoms adjacent to quadrupolar nuclei has been valuable in assigning the solid-state NMR spectra. The broadening effect has also been observed at more distant sites. It has not been possible to distinguish the effects of (^37)C1 and (^35)C1 on the carbon spectra of chlorine containing steroids