Effectiveness of Hindman's theorem for bounded sums

We consider the strength and effective content of restricted versions of Hindman's Theorem in which the number of colors is specified and the length of the sums has a specified finite bound. Let $\mathsf{HT}^{\leq n}_k$ denote the assertion that for each $k$-coloring $c$ of $\mathbb{N}$ there is an infinite set $X \subseteq \mathbb{N}$ such that all sums $\sum_{x \in F} x$ for $F \subseteq X$ and $0 < |F| \leq n$ have the same color. We prove that there is a computable $2$-coloring $c$ of $\mathbb{N}$ such that there is no infinite computable set $X$ such that all nonempty sums of at most $2$ elements of $X$ have the same color. It follows that $\mathsf{HT}^{\leq 2}_2$ is not provable in $\mathsf{RCA}_0$ and in fact we show that it implies $\mathsf{SRT}^2_2$ in $\mathsf{RCA}_0$. We also show that there is a computable instance of $\mathsf{HT}^{\leq 3}_3$ with all solutions computing $0'$. The proof of this result shows that $\mathsf{HT}^{\leq 3}_3$ implies $\mathsf{ACA}_0$ in $\mathsf{RCA}_0$

A polychromatic Ramsey theory for ordinals

The Ramsey degree of an ordinal α is the least number n such that any colouring of the edges of the complete graph on α using finitely many colours contains an n-chromatic clique of order type α. The Ramsey degree exists for any ordinal α < ω ω . We provide an explicit expression for computing the Ramsey degree given α. We further establish a version of this result for automatic structures. In this version the ordinal and the colouring are presentable by finite automata and the clique is additionally required to be regular. The corresponding automatic Ramsey degree turns out to be greater than the set theoretic Ramsey degree. Finally, we demonstrate that a version for computable structures fails

Dynamic properties and optical phase conjugation of two-photon pumped ultrashort blue stimulated emission in a chromophore solution

The dynamic properties of two-photon pumped blue lasing (~470 nm) in the solution of an organic chromophore [2-acetyl-6-(dimethylamino)naphthalene], excited by ~160-fs laser pulses at ~775 nm, have been studied. Both the forward and backward stimulated emission are enhanced by feedback from the reflection at the two optical windows of the solution filled cuvette. Under current experimental conditions, the lasing wavelengths in the forward and backward directions were almost the same, but both blueshifted compared to the fluorescence peak wavelength of the sample solution. The temporal behavior of the lasing output was recorded by a high-speed streak camera system. The multipulse structure and spectral properties of the output lasing are semiquantitatively explained. In addition, excellent optical phase-conjugation properties of the backward stimulated emission were observed; the aberration influences from an aberrator on the backward lasing beam were automatically removed

Polystyrene/clay nanocomposites by atom transfer radical nitroxide coupling chemistry

An efficient protocol, atom transfer radical nitroxide coupling chemistry, for the preparation of polymer/clay nanocomposites via grafting-onto strategy with well-defined polymer, which were synthesized via atom transfer radical polymerization, has been described. The radical coupling, taking place between the clay layers, not only leads to attached polymer chains but also to successful nanocomposite formation with highly exfoliated morphology. Copyright © 2012 Wiley Periodicals, Inc

On FPL configurations with four sets of nested arches

The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a,b,c,d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a formula as a sum of determinants. This is made quite explicit when min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates the numbers of configurations with b=d.Comment: 22 pages, TeX, 16 figures; a new formula for a generating function adde

Actin: its cumbersome pilgrimage through cellular compartments

In this article, we follow the history of one of the most abundant, most intensely studied proteins of the eukaryotic cells: actin. We report on hallmarks of its discovery, its structural and functional characterization and localization over time, and point to present days’ knowledge on its position as a member of a large family. We focus on the rather puzzling number of diverse functions as proposed for actin as a dual compartment protein. Finally, we venture on some speculations as to its origin

Partial Recursive Functions and Finality

Abstract. We seek universal categorical conditions ensuring the representability of all partial recursive functions. In the category Pfn of sets and partial functions, the natural numbers provide both an initial algebra and a final coalgebra for the functor 1 + −. We recount how finality yields closure of the partial functions on natural numbers under Kleene’s µ-recursion scheme. Noting that Pfn is not cartesian, we then build on work of Paré and Román, obtaining weak initiality and finality conditions on natural numbers algebras in monoidal categories that ensure the (weak) representability of all partial recursive functions. We further obtain some positive results on strong representability. All these results adapt to Kleisli categories of cartesian categories with natural numbers algebras. However, in general, not all partial recursive functions need be strongly representable.

Distinct Molecular Evolutionary Mechanisms Underlie the Functional Diversification of the Wnt and TGFβ Signaling Pathways

The canonical Wnt pathway is one of the oldest and most functionally diverse of animal intercellular signaling pathways. Though much is known about loss-of-function phenotypes for Wnt pathway components in several model organisms, the question of how this pathway achieved its current repertoire of functions has not been addressed. Our phylogenetic analyses of 11 multigene families from five species belonging to distinct phyla, as well as additional analyses employing the 12 Drosophila genomes, suggest frequent gene duplications affecting ligands and receptors as well as co-evolution of new ligand–receptor pairs likely facilitated the expansion of this pathway’s capabilities. Further, several examples of recent gene loss are visible in Drosophila when compared to family members in other phyla. By comparison the TGFβ signaling pathway is characterized by ancient gene duplications of ligands, receptors, and signal transducers with recent duplication events restricted to the vertebrate lineage. Overall, the data suggest that two distinct molecular evolutionary mechanisms can create a functionally diverse developmental signaling pathway. These are the recent dynamic generation of new genes and ligand–receptor interactions as seen in the Wnt pathway and the conservative adaptation of ancient pre-existing genes to new roles as seen in the TGFβ pathway. From a practical perspective, the former mechanism limits the investigator’s ability to transfer knowledge of specific pathway functions across species while the latter facilitates knowledge transfer