Gruenhage compacta and strictly convex dual norms

We prove that if K is a Gruenhage compact space then C(K)* admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X* is the |.|-closed linear span of K, where K is a Gruenhage compact in the w*-topology and |.| is equivalent to a coarser, w*-lower semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if T is a tree, then C(T)* admits an equivalent, strictly convex dual norm if and only if T is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images

Trees, linear orders and G\^ateaux smooth norms

We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a G\^ateaux smooth norm on C(T), where T is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fr\'echet smooth norms. In addition, we prove that if C(T) admits a G\^ateaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm.Comment: A different version of this paper is to appear in J. London Math. So

Automatic positive semidefinate HAC covariance matrix and GMM estimation

This paper proposes a new class of heteroskedastic and autocorrelation consistent (HAC) covariance matrix estimators. The standard HAC estimation method reweights estimators of the autocovariances. Here we initially smooth the data observations themselves using kernel function–based weights. The resultant HAC covariance matrix estimator is the normalized outer product of the smoothed random vectors and is therefore automatically positive semidefinite. A corresponding efficient GMM criterion may also be defined as a quadratic form in the smoothed moment indicators whose normalized minimand provides a test statistic for the overidentifying moment conditions

The Road to a Climate Change Agreement Runs Through Montreal

The 1987 Montreal Protocol to address ozone layer depletion was a pivotal agreement in the history of global environmental negotiations. It established a process that remains an important precedent for dealing with global environmental problems, including global warming. What made the negotiation of that agreement such an iconic event, and what useful lessons does it hold for climate change negotiators? Richard Smith cites a number offactors that were critical to the success of the Montreal Protocol. For example: (1) the United States played a leadership role from the beginning, including banning the use of chlorofluorocarbons (CFCs) in most aerosols and appointing a chief negotiator with responsibility for developing the U.S. position well before the negotiations began; (2) from the outset all countries that were parties to the agreement, both developed and developing countries, made specific commitments to reduce the production and use of ozone-depleting substances; and (3) the protocol set up a procedure for regularly reviewing and revising its provisions at follow-up meetings, thus accommodating new information rapidly and efficiently. A central lesson that climate change negotiators should learn from the Montreal Protocol is that it set a process in motion, which ultimately led to all parties to the agreement making the necessary commitments to arrest and eventually reverse the deterioration of the stratospheric ozone layer. Clearly, climate change negotiators face a more complex and far-reaching challenge today. The phaseout of ozone-depleting chemicals and related infrastructure involved major industries such as refrigeration, electronics, fire fighting, and aerosols and cost billions of dollars. But reducing greenhouse gas emissions will require fundamentally rethinking the present carbon-based societies and taking steps that will affect virtually every aspect of economic activity. Despite this significant difference in the impact on the economic structure of the countries concerned, however, there are similarities between the two challenges, and climate change negotiators would be well advised to reflect on the Montreal Protocol and the lessons that can be learned from it.

Improved Density and Distribution Function Estimation

Given additional distributional information in the form of moment restrictions, kernel density and distribution function estimators with implied generalised empirical likelihood probabilities as weights achieve a reduction in variance due to the systematic use of this extra information. The particular interest here is the estimation of densities or distributions of (generalised) residuals in semi-parametric models defined by a finite number of moment restrictions. Such estimates are of great practical interest, being potentially of use for diagnostic purposes, including tests of parametric assumptions on an error distribution, goodness-of-fit tests or tests of overidentifying moment restrictions. The paper gives conditions for the consistency and describes the asymptotic mean squared error properties of the kernel density and distribution estimators proposed in the paper. A simulation study evaluates the small sample performance of these estimators. Supplements provide analytic examples to illustrate situations where kernel weighting provides a reduction in variance together with proofs of the results in the paper.Comment: 32 pages, 3 figures, 3 table

Approximation of norms on Banach spaces

Relatively recently it was proved that if $\Gamma$ is an arbitrary set, then any equivalent norm on $c_0(\Gamma)$ can be approximated uniformly on bounded sets by polyhedral norms and $C^\infty$ smooth norms, with arbitrary precision. We extend this result to more classes of spaces having uncountable symmetric bases, such as preduals of the `discrete' Lorentz spaces $d(w,1,\Gamma)$, and certain symmetric Nakano spaces and Orlicz spaces. We also show that, given an arbitrary ordinal number $\alpha$, there exists a scattered compact space $K$ having Cantor-Bendixson height at least $\alpha$, such that every equivalent norm on $C(K)$ can be approximated as above

Smooth and polyhedral approximation in Banach spaces

We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on $c_0(\Gamma)$, where $\Gamma$ is an arbitrary set. We also give a necessary condition for the existence of a polyhedral norm on a weakly compactly generated Banach space, which extends a well-known result of Fonf.Comment: 12 page

Operator machines on directed graphs

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X --> X such that the set A = {x in X : ||R^n(x)|| --> infinity} is non-empty and nowhere dense in X. Moreover, if x in X\A then some subsequence of (R^n(x)) converges weakly to x. This answers in the negative a recent conjecture of Prajitura. The result can be extended to any Banach space containing an infinite-dimensional, complemented subspace with a symmetric basis; in particular, all 'classical' Banach spaces admit such an operator