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In the minimum eigenvalue problem, we are given a collection of vectors and the goal is to pick a subset B to maximize the minimum eigenvalue of the matrix formed by the sum of their outer products. We give a -time randomized algorithm that finds an assignment subject to a partition constraint whose minimum eigenvalue is at least $$1-\epsilon$$ times the optimum, with high probability. As a byproduct, we also get a simple algorithm for an algorithmic version of Kadison-Singer problem. 

An in-silico exercise was developed for a general chemistry laboratory course at St. Bonaventure University in which students examined potential energy surfaces, molecular orbital diagrams, and how bond orders and Lewis structures are connected. Pre- and post-assessment data suggests that, though students learned from the exercise, they are not connecting the concepts of bond order, Lewis structures, and resonance. There was a statistically significant improvement in the assessment scores before and after the laboratory experiment, and there was no statistical difference between the post-assessment and the follow-up assessment, which occurred after students completed the lab report 1 week after the initial experiment. The data suggest an improved understanding of computational chemistry concepts as well as improvement in the individual concepts of resonance, Lewis structures, and bond orders. However, an assessment question connecting these concepts did not show an improvement. An additional questionnaire was conducted to explore this discrepancy. This study indicates that more investigation is necessary with regard to students’ ability to make logical connections among bond orders, Lewis structures, and resonance. 

Post-Wilsonian physics views theories not as isolated points but elements of bigger universality classes, with effective theories emerging in the infrared. This paper makes initial attempts to apply this viewpoint to homogeneous geometries on group manifolds, and complexity geometry in particular. We observe that many homogeneous metrics on low-dimensional Lie groups have markedly different short-distance properties, but nearly identical distance functions at longer distances. Using Nielsen's framework of complexity geometry, we argue for the existence of a large universality class of definitions of quantum complexity, each linearly related to the other, a much finer-grained equivalence than typically considered in complexity theory. We conjecture that at larger complexities, a new effective metric emerges that describes a broad class of complexity geometries, insensitive to various choices of 'ultraviolet' penalty factors. Finally we lay out a broader mathematical program of classifying the effective geometries of right-invariant group manifolds. 

Let g \mathfrak {g} be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g \mathfrak {g} -modules Y ( χ , η ) Y(\chi , \eta ) introduced by Kostant. We prove that the set of all contravariant forms on Y ( χ , η ) Y(\chi , \eta ) forms a vector space whose dimension is given by the cardinality of the Weyl group of g \mathfrak {g} . We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules M ( χ , η ) M(\chi , \eta ) introduced by McDowell.