Dimension prints and the avoidance of sets for flow solutions of non-autonomous ordinary differential equations

We provide a criterion for a generalised flow solution of a non-autonomous ordinary differential equation to avoid a subset of the phase space. This improves on that established by Aizenman for the autonomous case, where avoidance is guaranteed if the underlying vector field is sufficiently regular and the subset has sufficiently small box-counting dimension. We define the r-codimension print of a subset $S\subset \R^{n}\times [0,T]$, which is a subset of $(0,\infty]^{2}$ that encodes the dimension of S in a way that distinguishes spatial and temporal detail. We prove that the subset S is avoided by a generalised flow solution with underlying vector field in $L^{p}(0, T; L^{q}(R^{n}))$ if the Holder conjugates (q^{*}; p^{*}) are in the r-codimension print of S

Strict inequality in the box-counting dimension product formulas

We supplement the well known upper and lower box-counting product inequalities to give a new product formula for subsets of metric spaces. We develop a procedure for constructing sets so that the upper and lower box-counting dimensions of these sets and their product can take arbitrary values satisfying the above product formula. In particular we illustrate how badly behaved both the lower and upper box-counting dimensions can be on taking products

Expectations of linear functions with respect to truncazted multinormal distributions, with applications for uncertainty analysis in environmental modelling

Uncertainty can hamper the stringency of commitments under cap and trade schemes. We assess how well intensity targets, where countries' permit allocations are indexed to future realised GDP, can cope with uncertainties in a post-Kyoto international greenhouse emissions trading scheme. We present some empirical foundations for intensity targets and derive a simple rule for the optimal degree of indexation to GDP. Using an 18-region simulation model of a 2020 global cap-and-trade treaty under multiple uncertainties and endogenous commitments, we estimate that optimal intensity targets could achieve global abatement as much as 20 per cent higher than under absolute targets, and even greater increases in welfare measures. The optimal degree of indexation to GDP would vary greatly between countries, including super-indexation in some advanced countries, and partial indexation for most developing countries. Standard intensity targets (with one-toone indexation) would also improve the overall outcome, but to a lesser degree and not in all cases. Although target indexation is no magic wand for a future global climate treaty, gains from reduced cost uncertainty might justify increased complexity, framing issues and other potential downsides of intensity targets.linear functions, truncazted multinormal distributions, uncertainty analysis, environmental modelling

“Do You Know Why That’s Funny?” Connecting the Scholarship of Humor to the Practice of After-Dinner Speaking

Forensic educators have a unique opportunity to connect students with centuries of scholarship, yet it remains unclear how coaches utilize communication research to aid students in constructing events. This article questions how studies of humor can enhance connections between the forensic student and the broader field of research. Through applying theories of humor to the practice of After- Dinner Speaking (ADS), this paper indicates studies of humor in classical and contemporary scholarship provide useful frameworks in the construction of ADS, and offers suggestions for making more explicit connections between theory, pedagogy, and practice

HST Observations of Gravitationally Lensed Features in the Rich Cluster Ac114

Deep Hubble Space Telescope images of superlative resolution obtained for the distant rich cluster AC114 (z=0.31) reveal a variety of gravitational lensing phenomena for which ground-based spectroscopy is available. We present a luminous arc which is clearly resolved by HST and appears to be a lensed z=0.64 sub-L star spiral galaxy with a detected rotation curve. Of greatest interest is a remarkably symmetrical pair of compact blue images separated by 10 arcsec and lying close to the cluster cD. We propose that these images arise from a single very faint background source gravitationally lensed by the cluster core. Deep ground-based spectroscopy confirms the lensing hypothesis and suggests the source is a compact star forming system at a redshift z=1.86. Taking advantage of the resolved structure around each image and their very blue colours, we have identified a candidate third image of the same source roughly 50 arcsec away. The angular separation of the three images is much larger than previous multiply-imaged systems and indicates a deep gravitational potential in the cluster centre. Resolved multiply-imaged systems, readily recognised with HST, promise to provide unique constraints on the mass distribution in the cores of intermediate redshift clusters.Comment: submitted to ApJ, 6 pages (no figures), uuencoded Postscript, compressed TAR of Postscript figures available via anonymous ftp in users/irs/figs/ac114_figs.tar.gz on astro.caltech.edu. PAL-IRS-

Generalised Cantor sets and the dimension of products

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of ‘equi-homogeneity’ of a set, which requires a uniformity in the cardinality of local covers at all length-scales and at all points, and we show that a large class of homogeneous Moran sets have this property. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set ‘attains’ these dimensions (analogous to ‘s-sets’ when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that for any α ∈ (0, 1) and any β, γ ∈ (0, 1) such that β + γ ≥ 1 we can construct two generalised Cantor sets C and D such that dimBC = αβ, dimBD = α γ, and dimAC = dimAD = dimA (C × D) = dimB (C × D) = α

On the regularity of Lagrangian trajectories corresponding to suitable weak solutions of the Navier-Stokes equations

The putative singular set S in space-time of a suitable weak solution u of the 3D Navier–Stokes equations has box-counting dimension no greater than 5/3. This allows one to prove that almost all trajectories avoid S. Moreover, for each point x that does not belong to S, one can find a neighbourhood U of x such that the function u is continuous on U and space derivatives of u are bounded on every compact subset of U. It follows that almost all Lagrangian trajectories corresponding to u are C^{1} functions of time (Robinson & Sadowski, Nonlinearity 2009). We recall the main idea of the proof, give examples that clarify in what sense the uniqueness of trajectories is considered, and make some comments on how this result might be improved