A Fundamental Theorem on the Structure of Symplectic Integrators

I show that the basic structure of symplectic integrators is governed by a theorem which states {\it precisely}, how symplectic integrators with positive coefficients cannot be corrected beyond second order. All previous known results can now be derived quantitatively from this theorem. The theorem provided sharp bounds on second-order error coefficients explicitly in terms of factorization coefficients. By saturating these bounds, one can derive fourth-order algorithms analytically with arbitrary numbers of operators.Comment: 4 pages, no figure

Unearthing the roots of urban sprawl: a critical analysis of form, function and methodology

Urban sprawl is one of the key issues facing cities today. There is a large volume of literature on the topic but despite this there is little agreement as to its characteristics and effects. The paper discusses some of the most contested issues of urban sprawl. It looks at the various definitions of sprawl; examines the effects of sprawl, assessing these in relation to planning and market led approaches; and discusses methodological approaches relating to measures of sprawl in terms of its impacts and forms

Margins and monsters: How some micro cases lead to macro claims

ABSTRACTHow do micro cases lead us to surprising macro claims? Historians often say that the micro level casts light on the macro level. This metaphor of “casting light” suggests that the micro does not illuminate the macro straightforwardly; such light needs to be interpreted. In this essay, I propose and clarify six interpretive norms to guide micro‐to‐macro inferences.I focus on marginal groups and monsters. These are popular cases in social and cultural histories, and yet seem to be unpromising candidates for generalization. Marginal groups are dismissed by the majority as inferior or ill‐fitting; their lives seem intelligible but negligible. Monsters, on the other hand, are somehow incomprehensible to society and treated as such. First, I show that, by looking at how a society identifies a marginal group and interacts with it, we can draw surprising inferences about that society's self‐image and situation. By making sense of a monster's life, we can draw inferences about its society's mentality and intelligibility. These will contest our conception of a macro claim. Second, I identify four risks in making such inferences — and clarify how norms of coherence, challenge, restraint, connection, provocation, and contextualization can manage those risks.My strategy is to analyze two case studies, by Richard Cobb, about a band of violent bandits and a semi‐literate provincial terrorist in revolutionary France. Published in 1972, these studies show Cobb to be an inventive and idiosyncratic historian, who created new angles for studying the micro level and complicated them with his autobiography. They illustrate how a historian's autobiographical, literary, and historiographical interests can mix into a risky, and often rewarding, style

Galois coverings of pointed coalgebras

We introduce the concept of a Galois covering of a pointed coalgebra. The theory developed shows that Galois coverings of pointed coalgebras can be concretely expressed by smash coproducts using the coaction of the automorphism group of the covering. Thus the theory of Galois coverings is seen to be equivalent to group gradings of coalgebras. An advantageous feature of the coalgebra theory is that neither the grading group nor the quiver is assumed finite in order to obtain a smash product coalgebra

Special biserial coalgebras and representations of quantum SL(2)

We develop the theory of special biserial and string coalgebras and other concepts from the representation theory of quivers. These tools are then used to describe the finite dimensional comodules and Auslander-Reiten quiver for the coordinate Hopf algebra of quantum SL(2) at a root of unity. We also describe the stable Green ring and compute quantum dimensions.Comment: previous section 2.2 removed, errors correcte

The Dormant Commerce Clause and State Regulation of the Internet: Are Laws Protecting Minors From Sexual Predators Constitutionally Different Than Those Protecting Minors From Sexually Explicit Materials?

Several states have enacted statutes to protect minors from harmful or obscene materials disseminated over the Internet, as well as from pedophiles seeking to use the Internet to lure them into sexual conduct. State and federal courts have diverged in their analysis of the Dormant Commerce Clause\u27s impact on state regulation in these areas. While state courts have held that the Dormant Commerce Clause does not invalidate state luring statutes, federal courts have been consistent in finding state dissemination statutes unconstitutional. This iBrief summarizes recent state and federal jurisprudence in this area and concludes that state courts have not been successful in distinguishing state luring statutes from federal case law on state dissemination statutes. Therefore, state courts have prematurely aborted the Dormant Commerce Clause examination

Regression adjustments for estimating the global treatment effect in experiments with interference

Standard estimators of the global average treatment effect can be biased in the presence of interference. This paper proposes regression adjustment estimators for removing bias due to interference in Bernoulli randomized experiments. We use a fitted model to predict the counterfactual outcomes of global control and global treatment. Our work differs from standard regression adjustments in that the adjustment variables are constructed from functions of the treatment assignment vector, and that we allow the researcher to use a collection of any functions correlated with the response, turning the problem of detecting interference into a feature engineering problem. We characterize the distribution of the proposed estimator in a linear model setting and connect the results to the standard theory of regression adjustments under SUTVA. We then propose an estimator that allows for flexible machine learning estimators to be used for fitting a nonlinear interference functional form. We propose conducting statistical inference via bootstrap and resampling methods, which allow us to sidestep the complicated dependences implied by interference and instead rely on empirical covariance structures. Such variance estimation relies on an exogeneity assumption akin to the standard unconfoundedness assumption invoked in observational studies. In simulation experiments, our methods are better at debiasing estimates than existing inverse propensity weighted estimators based on neighborhood exposure modeling. We use our method to reanalyze an experiment concerning weather insurance adoption conducted on a collection of villages in rural China.Comment: 38 pages, 7 figure