Multigraded linear series and recollement

Given a scheme Y equipped with a collection of globally generated vector bundles E1,…,En, we study the universal morphism from Y to a fine moduli space M(E) of cyclic modules over the endomorphism algebra of E:=OY⊕E1⊕⋯⊕En. This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup G⊂GL(2,k), every sub-minimal partial resolution of A2k/G is isomorphic to a fine moduli space M(EC) where EC is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid's recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution

Genetic algorithms for feature selection and weighting

Abstract Automated techniques to optimise the retrieval of relevant cases in a CBR system are desirable as a way to reduce the expensive knowledge acquisition phase. This paper concentrates on feature selection methods that assist in indexing the case-base, and feature weighting methods that improve the similarity-based selection of relevant cases. Two main types of method are presented: filter methods use no feedback from the learning algorithm that will be applied; wrapper methods incorporate feedback and hence take account of learning bias. Wrapper methods based on Genetic Algorithms have been found to deliver the best results with a tablet design application, but these generic methods are flexible about the criterion to be optimised, and should be applicable to a wide variety of problems. Introduction The majority of CBR systems rely on a good case-base organisation, an effective index and a (possibly knowledge intensive) similarity matching to select cases, that can then be used to solve a problem, see Many CBR tools provide standard means of constructing indexes. Isoft's ReCall is typical in using a C4.5 [Quinlan 1993] generated decision tree, constructed from the cases in the case-base, as the index. However, induction algorithms like C4.5 apply a greedy selection approach and so the features used by the index are not always the optimal ones. This is a particular problem when the cases contain many features irrelevant to the problem solving The cases identified by the index are next ranked according to their similarity to the new problem. The simplest similarity metric is Euclidean distance between normalised feature vectors. However, a "useful" (from the point of view of solving a problem) similarity should take account of the relative importances of various features. Certainly in a situation where many features are irrelevant to the problem to be solved, a simple similarity measure is insufficient. This problem can be partially solved by identifying and removing irrelevant features as before. However, a more flexible method assigns weights to the features to indicate their relative importance to the problem solving. Although the selection of the relevant features can usually be done quite accurately by an expert, feature weighting can only be done approximately by an expert, often by categorising the relevance as one from a small set of possible degrees of relevance. Therefore, applying an automated algorithm to find feature weights is attractive. Section 2 reviews feature selection and weighting methods. Our tablet formulation problem domain is introduced in Section 3

Mastering the Master Space

Supersymmetric gauge theories have an important but perhaps under-appreciated notion of a master space, which controls the full moduli space. For world-volume theories of D-branes probing a Calabi-Yau singularity X the situation is particularly illustrative. In the case of one physical brane, the master space F is the space of F-terms and a particular quotient thereof is X itself. We study various properties of F which encode such physical quantities as Higgsing, BPS spectra, hidden global symmetries, etc. Using the plethystic program we also discuss what happens at higher number N of branes. This letter is a summary and some extensions of the key points of a longer companion paper arXiv:0801.1585.Comment: 10 pages, 1 Figur

Instantons, Quivers and Noncommutative Donaldson-Thomas Theory

We construct noncommutative Donaldson-Thomas invariants associated with abelian orbifold singularities by analysing the instanton contributions to a six-dimensional topological gauge theory. The noncommutative deformation of this gauge theory localizes on noncommutative instantons which can be classified in terms of three-dimensional Young diagrams with a colouring of boxes according to the orbifold group. We construct a moduli space for these gauge field configurations which allows us to compute its virtual numbers via the counting of representations of a quiver with relations. The quiver encodes the instanton dynamics of the noncommutative gauge theory, and is associated to the geometry of the singularity via the generalized McKay correspondence. The index of BPS states which compute the noncommutative Donaldson-Thomas invariants is realized via topological quantum mechanics based on the quiver data. We illustrate these constructions with several explicit examples, involving also higher rank Coulomb branch invariants and geometries with compact divisors, and connect our approach with other ones in the literature.Comment: 95 pages, 5 figures; v2: clarifying comments added, discussions using tilting strengthened, references added and updated; v3: minor corrections, final version to be published in Nuclear Physics

Resilience and Leadership in Dangerous Contexts

Daniel B. Cnossen was born and raised in Topeka, growing up on a small farm. He spent his childhood reading, running, playing sports, and working on the farm. Cnossen enrolled in the United States Naval Academy in 1998. He had never before seen the ocean and did not know how to swim, but he asked his new friends at the academy to teach him; he would often skip lunch to spend time in the pool. Cnossen joined the Navy triathlon team to strengthen his swimming. By his senior year, he had been elected captain of the team, which he helped lead to a national championship. After graduation, he headed to San Diego to undergo training as a Navy SEAL. Cnossen served several tours overseas. On September 6, 2009, less than thirty-six hours on the ground in Kandahar, Afghanistan, he activated a landmine, losing both legs and suffering internal injuries. Lieutenant Cnossen is now back in the United States facing new challenges. He is doing so with the same dedication and enthusiasm that he used to surmount previous challenges. He is positive and appreciative of his friends and family, and he is happy to be alive. Described by some as stoic, Cnossen is seen by those who know him best as soft-spoken and humble. No one as full of curiosity, zest, and humor as he is could be described as stoic. As Cnossen began his rehabilitation, he noted that now he would be able to do even more pull-ups. While at the Walter Reed Army Medical Center, Cnossen and other troops were visited by President Barack Obama. As the president was leaving, he noticed a copy of War and Peace on Cnossen\u27s bedside table. The two men joked that merely lifting the book would be another form of physical therapy

Resilience and Leadership in Dangerous Contexts

Daniel B. Cnossen was born and raised in Topeka, growing up on a small farm. He spent his childhood reading, running, playing sports, and working on the farm. Cnossen enrolled in the United States Naval Academy in 1998. He had never before seen the ocean and did not know how to swim, but he asked his new friends at the academy to teach him; he would often skip lunch to spend time in the pool. Cnossen joined the Navy triathlon team to strengthen his swimming. By his senior year, he had been elected captain of the team, which he helped lead to a national championship. After graduation, he headed to San Diego to undergo training as a Navy SEAL. Cnossen served several tours overseas. On September 6, 2009, less than thirty-six hours on the ground in Kandahar, Afghanistan, he activated a landmine, losing both legs and suffering internal injuries. Lieutenant Cnossen is now back in the United States facing new challenges. He is doing so with the same dedication and enthusiasm that he used to surmount previous challenges. He is positive and appreciative of his friends and family, and he is happy to be alive. Described by some as stoic, Cnossen is seen by those who know him best as soft-spoken and humble. No one as full of curiosity, zest, and humor as he is could be described as stoic. As Cnossen began his rehabilitation, he noted that now he would be able to do even more pull-ups. While at the Walter Reed Army Medical Center, Cnossen and other troops were visited by President Barack Obama. As the president was leaving, he noticed a copy of War and Peace on Cnossen\u27s bedside table. The two men joked that merely lifting the book would be another form of physical therapy

Quiver GIT for varieties with tilting bundles

In the setting of a variety X admitting a tilting bundle T we consider the problem of constructing X as a quiver GIT quotient of the algebra A:=EndX(T)opA:=EndX(T)op . We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of X and the closed points of a quiver representation moduli functor for A=EndX(T)opA=EndX(T)op then X is indeed a fine moduli space for this moduli functor, and we prove this result without any assumptions on the singularities of X. As an application we consider varieties which are projective over an affine base such that the fibres are of dimension 1, and the derived pushforward of the structure sheaf on X is the structure sheaf on the base. In this situation there is a particular tilting bundle on X constructed by Van den Bergh, and our result allows us to reconstruct X as a quiver GIT quotient for an easy to describe stability condition and dimension vector. This result applies to flips and flops in the minimal model program, and in the situation of flops shows that both a variety and its flop appear as moduli spaces for algebras produced from different tilting bundles on the variety. We also give an application to rational surface singularities, showing that their minimal resolutions can always be constructed as quiver GIT quotients for specific dimension vectors and stability conditions. This gives a construction of minimal resolutions as moduli spaces for all rational surface singularities, generalising the G-Hilbert scheme moduli space construction which exists only for quotient singularities

Lathe converted for grinding aspheric surfaces

A standard overarm tracing lathe converted by the addition of an independently driven diamond grinding wheel is used for grinding aspheric surfaces. The motion of the wheel is controlled by the lathe air tracer following the template which produces the desired aspheric profile

MartiTracks: A Geometrical Approach for Identifying Geographical Patterns of Distribution

Panbiogeography represents an evolutionary approach to biogeography, using rational cost-efficient methods to reduce initial complexity to locality data, and depict general distribution patterns. However, few quantitative, and automated panbiogeographic methods exist. In this study, we propose a new algorithm, within a quantitative, geometrical framework, to perform panbiogeographical analyses as an alternative to more traditional methods. The algorithm first calculates a minimum spanning tree, an individual track for each species in a panbiogeographic context. Then the spatial congruence among segments of the minimum spanning trees is calculated using five congruence parameters, producing a general distribution pattern. In addition, the algorithm removes the ambiguity, and subjectivity often present in a manual panbiogeographic analysis. Results from two empirical examples using 61 species of the genus Bomarea (2340 records), and 1031 genera of both plants and animals (100118 records) distributed across the Northern Andes, demonstrated that a geometrical approach to panbiogeography is a feasible quantitative method to determine general distribution patterns for taxa, reducing complexity, and the time needed for managing large data sets

Surface Rupture of the November 2002 M7.9 Denali Fault Earthquake, Alaska, and Comparison to Other Strike-Slip Ruptures

On November 3, 2002, a moment-magnitude (Mw) 7.9 earthquake produced 340 km of surface rupture on the Denali fault and two related faults in central Alaska. The rupture, which proceeded from west to east, began with a 40-km-long break on a previously unknown thrust fault. Estimates of surface slip on this thrust were 3-6 m. Next came the principal surface break, along 220 km of the Denali fault. There, right-lateral offset averaged almost 5 m and increased eastward to a maximum of nearly 9 m. Finally, slip turned southeastward onto the Totschunda fault, where dextral offsets up to 3 m continued for another 70 km. This three-part rupture ranks among the longest documented strike-slip events of the past two centuries. The surface-slip distribution supports and clarifies models of seismological and geodetic data that indicated initial thrusting followed by rightlateral strike slip, with the largest moment release near the east end of the Denali fault. The Denali fault ruptured beneath the Trans-Alaska oil pipeline. The pipeline withstood almost 6 m of lateral offset, because engineers designed it to survive such offsets based on pre-construction geological studies. The Denali fault earthquake was typical of large-magnitude earthquakes on major intracontinental strike-slip faults, in the length of the rupture, the multiple fault strands that ruptured, and the variable slip along strike