We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements G(A) is abelian such ...that all prime divisors of the order of G(A) are > 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.
This volume contains the proceedings of the AMS Special Session on Algebraic and Combinatorial Structures in Knot Theory and the AMS Special Session on Spatial Graphs, both held from October 24-25, ...2015, at California State University, Fullerton, CA.Included in this volume are articles that draw on techniques from geometry and algebra to address topological problems about knot theory and spatial graph theory, and their combinatorial generalizations to equivalence classes of diagrams that are preserved under a set of Reidemeister-type moves.The interconnections of these areas and their connections within the broader field of topology are illustrated by articles about knots and links in spatial graphs and symmetries of spatial graphs in $S^3$ and other 3-manifolds.
We give a necessary condition for a subset of $\mathbb R^3$ to be the set of critical points of some smooth function. In particular we obtain that for example neither the Whitehead continuum nor the ...p-adic solenoid are such critical sets.
A key justification for using nonrandomized experiments is that, with proper adjustment, their results can well approximate results from randomized experiments. This hypothesis has not been ...consistently supported by empirical studies; however, previous methods used to study this hypothesis have confounded assignment method with other study features. To avoid these confounding factors, this study randomly assigned participants to be in a randomized experiment or a nonrandomized experiment. In the randomized experiment, participants were randomly assigned to mathematics or vocabulary training; in the nonrandomized experiment, participants chose their training. The study held all other features of the experiment constant; it carefully measured pretest variables that might predict the condition that participants chose, and all participants were measured on vocabulary and mathematics outcomes. Ordinary linear regression reduced bias in the nonrandomized experiment by 84-94% using covariate-adjusted randomized results as the benchmark. Propensity score stratification, weighting, and covariance adjustment reduced bias by about 58-96%, depending on the outcome measure and adjustment method. Propensity score adjustment performed poorly when the scores were constructed from predictors of convenience (sex, age, marital status, and ethnicity) rather than from a broader set of predictors that might include these.
Please see the online supplements for a Letter to the Editor.
We give a one-parameter family of Bridgeland stability conditions on the derived category of a smooth projective complex surface $S$ and describe "wall-crossing behavior'' for objects with the same ...invariants as $\mathcal O_C(H)$ when $H$ generates Pic$(S)$ and $C \in |H|$. If, in addition, $S$ is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus' stable pairs for curves embedded in the moduli spaces.
For obtaining causal inferences that are objective, and therefore have the best chance of revealing scientific truths, carefully designed and executed randomized experiments are generally considered ...to be the gold standard. Observational studies, in contrast, are generally fraught with problems that compromise any claim for objectivity of the resulting causal inferences. The thesis here is that observational studies have to be carefully designed to approximate randomized experiments, in particular, without examining any final outcome data. Often a candidate data set will have to be rejected as inadequate because of lack of data on key covariates, or because of lack of overlap in the distributions of key covariates between treatment and control groups, often revealed by careful propensity score analyses. Sometimes the template for the approximating randomized experiment will have to be altered, and the use of principal stratification can be helpful in doing this. These issues are discussed and illustrated using the framework of potential outcomes to define causal effects, which greatly clarifies critical issues.
The grain boundary character distribution strongly affects the properties of polycrystalline materials. Grain boundaries of similar characters form networks, whose topological invariants can be ...considered as distribution descriptors. Understanding the evolution of such descriptors during severe plastic deformations (SPD) can elucidate the evolution of properties and underpin the design of processing routes for target behaviour. For topological analysis of grain boundary networks, polycrystalline materials are considered here as polyhedral complexes with grain boundaries classified into two types — low-angle and high-angle. Changes of grain boundary types are calculated using sub-grain rotations, which reflects the physical mechanism of microstructure evolution during SPD. A non-physical approach, by direct conversions of low-angle to high-angle boundaries, is also explored as a reference to demonstrate the impact of the physical constraint imposed by rotations. Reported is the discovery of topological phase transitions in the grain boundary networks which might take place during severe plastic deformations of different copper alloys. Depending on the evolution approach, the transitions correspond to zeros of the Euler characteristic, or of the logarithm of the inverse connectivity, of the grain boundary network. The relations between these transitions and the fraction of high-angle grain boundaries, and between the fraction of high-angle grain boundaries and plastic strain obtainable experimentally, provide new perspectives for grain boundary engineering and network design. Determining the dominant evolution mechanism and critical accumulated strain for a given material and processing route requires further experimental studies of triple junction evolution.
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