An important problem in algebraic combinatorics is finding expansions of products of symmetric functions as sums of symmetric functions. Schur functions form a well-known basis for the ring of symmetric functions. The Littlewood-Richardson rule was introduced to expand the product of two Schur functions as a positive sum of Schur functions. Remmel and Whitney introduced an algorithmic way to fi...

A refractive lens is one of the simplest, most cost-effective and easily available imaging elements. Given a spatially incoherent illumination, can faithfully map every object point to an image in sensor plane, when distances satisfy conditions. However, static limited depth focus, beyond which point-to-point mapping only be obtained by changing either location lens, or sensor. In this study, f...

The problem of restoration in fluorescence 1 microscopy has to deal at the same time with blurring 2 and photon noise. Their combined effects corrupt the 3 image by inserting elements that do not belong to the 4 real object and distort the contrast. This hinders the 5 possibility of using the images for visualization, recog6 nition and analysis using the three-dimensional data. 7 The algorithms...

Acknowledgements We would like to extend particular thanks to Laura Blakeborough of the Home Office Crime and Policing Analysis Unit, who supported the evaluation and provided valuable insights and reflections on the AAR programme and the evaluation. We would also like to thank Anna Richardson and Tonia Davison of the Home Office for their support. Thank you also to staff and stakeholders worki...

ver the past several decades, researchers have suspected and confirmed various links between breakfast consumption and adolescent academic performance, although the reasons for these cognitive consequences are not fully explored nor understood (Dye & Blundell, 2002; Giovannini, Agostoni, & Shamir, 2010; Murphy, 2007). More specifically, it has not yet been determined precisely what roles variou...

INTRODUCTION On March 18, 2009, actress Natasha Richardson died after a head injury. It is possible that the rate of patients presenting with mild head injury and receiving computed tomographies (CTs) may have been influenced by the Richardson event. We hypothesized that there was a statistically significant increase in the rate of census-adjusted head CTs performed for mild trauma after March ...

Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood-Richardson Rule for flag varieties. We discuss three natural generalizations of Richardson varieties which we call projection varieties, intersection varieties, and rank varieties. In many ways, these varieties are more fundamental than Richardson varieties and are more easil...

We point out that the remarkable Knutson and Tao Saturation Theorem [9] and polynomial time algorithms for linear programming [14] have together an important, immediate consequence in geometric complexity theory [15, 16]: The problem of deciding positivity of Littlewood-Richardson coefficients belongs to P ; cf.[10]. Specifically, for GLn(C), positivity of a Littlewood-Richardson coefficient cα...