Lim colim versus colim lim. I

We study a model situation in which direct limit ($\text{colim}$) and inverse limit ($\lim$) do not commute, and offer some computations of their "commutator". The homology of a separable metrizable space $X$ has two well-known approximants: $qH_n(X)$ ("\v{C}ech homology") and $pH_n(X)$ ("\v{C}ech homology with compact supports"), which are not homology theories but are nevertheless interesting as they are $\lim\text{colim}$ and $\text{colim}\lim$ applied to homology of finite simplicial complexes. The homomorphism $\tau_X: pH_n(X)\to qH_n(X)$, which is a special case of the natural map $\text{colim}\lim\to\lim\text{colim}$, need not be either injective (P. S. Alexandrov, 1947) or surjective (E. F. Mishchenko, 1953), but its surjectivity for locally compact $X$ remains an open problem. For locally compact $X$, the dual map in cohomology $pH^n(X)\to qH^n(X)$ is shown to be surjective and its kernel is computed, in terms of $\lim^1$ and a new functor $\lim^1_{\text{fg}}$. The original map $\tau_X$ is surjective and its kernel is computed when $X$ is a "compactohedron", i.e. contains a compactum whose complement is a polyhedron. We also show that $\tau_X$ is surjective for locally compact $X$ assuming (i) two assertions known to be consistent with ZFC and (ii) that all locally compact spaces $X$ satisfy $\lim^2 H_{n+1}(N_\alpha)=0$, where $N_\alpha$ runs over the nerves of all open covers of $X$.Comment: 44 pages, 2 figure

[Review of] Ruthanne Lum McCunn. Sole Survivor

In November, 1942, the British freighter Benlomond was sunk by a German U-boat off the coast of South America with the loss of its entire crew except for a young Chinese steward named Poon Lim. Through his resourcefulness and determination, Lim survived on a wooden raft for 133 days before being picked up by a Brazilian fisherman. Sole Survivor is a fictionalized account of Lim\u27s experience, the longest such ordeal at sea, based largely on interviews with Lim, military and maritime documents, and magazine and newpaper [newspaper] stories

Lim colim versus colim lim. II: Derived limits over a pospace

\v{C}ech cohomology $H^n(X)$ of a separable metrizable space $X$ is defined in terms of cohomology of its nerves (or ANR neighborhoods) $P_\beta$ whereas Steenrod-Sitnikov homology $H_n(X)$ is defined in terms of homology of compact subsets $K_\alpha\subset X$. We show that one can also go vice versa: in a sense, $H^n(X)$ can be reconstructed from $H^n(K_\alpha)$, and if $X$ is finite dimensional, $H_n(X)$ can be reconstructed from $H_n(P_\beta)$. The reconstruction is via a Bousfield-Kan/Araki-Yoshimura type spectral sequence, except that the derived limits have to be "corrected" so as to take into account a natural topology on the indexing set. The corrected derived limits coincide with the usual ones when the topology is discrete, and in general are applied not to an inverse system but to a "partially ordered sheaf". The "correction" of the derived limit functors in turn involves constructing a "correct" (metrizable) topology on the order complex $|P|$ of a partially ordered metrizable space $P$ (such as the hyperspace $K(X)$ of nonempty compact subsets of $X$ with the Hausdorff metric). It turns out that three natural approaches (by using the space of measurable functions, the space of probability measures, or the usual embedding $K(X)\to C(X;\mathbb R)$) all lead to the same topology on $|P|$.Comment: 30 page

The effect of Nafion film on the cathode catalyst layer performance in a low-Pt PEM fuel cell

A single--pore model for performance of the cathode catalyst layer (CCL) in a PEM fuel cell is developed. The model takes into account oxygen transport though the CCL depth and through the thin Nafion film, separating the pore from Pt/C species. Analytical solution to model equations reveals the limiting current density $j_N^{\rm lim}$ due to oxygen transport through the Nafion film. Further, $j_N^{\rm lim}$ linearly depends of the CCL thickness, i.e., the thinner the CCL, the lower $j_N^{\rm lim}$. This result may explain unexpected lowering of low--Pt loaded catalyst layers performance, which has been widely discussing in literature.Comment: 11 page

A comment on "Intergenerational equity: sup, inf, lim sup, and lim inf"

We reexamine the analysis of Chambers (Social Choice and Welfare, 2009), that produces a characterization of a family of social welfare functions in the context of intergenerational equity: namely, those that coincide with either the sup, inf, lim sup, or lim inf rule. Reinforcement, ordinal covariance, and monotonicity jointly identify such class of rules. We show that the addition of a suitable axiom to this three properties permits to characterize each particular rule. A discussion of the respective distinctive properties is provided.Social welfare function; Intergenerational equity; Lim sup ; Lim inf

Passage of Lévy Processes across Power Law Boundaries at Small Times

We wish to characterize when a Lévy process Xt crosses boundaries like tκ, κ > 0, in a one- or two-sided sense, for small times t; thus, we inquire when lim.supt↓0 |Xt|/tκ, lim supt↓0, Xt/tκ and/or lim inft↓0 Xt/tκ are almost surely (a.s.) finite or infinite. Necessary and sufficient conditions are given for these possibilities for all values of κ > 0. This completes and extends a line of research, going back to Blumenthal and Getoor in the 1960s. Often (for many values of κ), when the lim sups are finite a.s., they are in fact zero, but the lim sups may in some circumstances take finite, nonzero, values, a.s. In general, the process crosses one- or two-sided boundaries in quite different ways, but surprisingly this is not so for the case κ = 1/2, where a new kind of analogue of an iterated logarithm law with a square root boundary is derived. An integral test is given to distinguish the possibilities in that case.Supported in part by ARC Grants DP0210572 and DP0664603