Analytic cell decomposition and analytic motivic integration

The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over \FF_q((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields $K$ with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over $K$. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of \emph{analytic} motivic integration and \emph{analytic} motivic constructible functions in the line of R. Cluckers and F. Loeser [\emph{Fonctions constructible et int\'egration motivic I}, Comptes rendus de l'Acad\'emie des Sciences, {\bf 339} (2004) 411 - 416]

Control of germ-band retraction in Drosophila by the zinc-finger protein HINDSIGHT

Drosophila embryos lacking hindsight gene function have a normal body plan and undergo normal germ-band extension. However, they fail to retract their germ bands. hindsight encodes a large nuclear protein of 1920 amino acids that contains fourteen C2H2-type zinc fingers, and glutamine-rich and proline-rich domains, suggesting that it functions as a transcription factor. Initial embryonic expression of hindsight RNA and protein occurs in the endoderm (midgut) and extraembryonic membrane (amnioserosa) prior to germ-band extension and continues in these tissues beyond the completion of germ-band retraction. Expression also occurs in the developing tracheal system, central and peripheral nervous systems, and the ureter of the Malpighian tubules. Strikingly, hindsight is not expressed in the epidermal ectoderm which is the tissue that undergoes the cell shape changes and movements during germ-band retraction. The embryonic midgut can be eliminated without affecting germ-band retraction. However, elimination of the amnioserosa results in the failure of germ-band retraction, implicating amnioserosal expression of hindsight as crucial for this process. Ubiquitous expression of hindsight in the early embryo rescues germ-band retraction without producing dominant gainof-function defects, suggesting that hindsight’s role in germ-band retraction is permissive rather than instructive. Previous analyses have shown that hindsight is required for maintenance of the differentiated amnioserosa (Frank, L. C. and Rushlow, C. (1996) Development 122, 1343-1352). Two classes of models are consistent with the present data. First, hindsight’s function in germ-band retraction may be limited to maintenance of the amnioserosa which then plays a physical role in the retraction process through contact with cells of the epidermal ectoderm. Second, hindsight might function both to maintain the amnioserosa and to regulate chemical signaling from the amnioserosa to the epidermal ectoderm, thus coordinating the cell shape changes and movements that drive germ-band retraction

Quantum Analogue Computing

We briefly review what a quantum computer is, what it promises to do for us, and why it is so hard to build one. Among the first applications anticipated to bear fruit is quantum simulation of quantum systems. While most quantum computation is an extension of classical digital computation, quantum simulation differs fundamentally in how the data is encoded in the quantum computer. To perform a quantum simulation, the Hilbert space of the system to be simulated is mapped directly onto the Hilbert space of the (logical) qubits in the quantum computer. This type of direct correspondence is how data is encoded in a classical analogue computer. There is no binary encoding, and increasing precision becomes exponentially costly: an extra bit of precision doubles the size of the computer. This has important consequences for both the precision and error correction requirements of quantum simulation, and significant open questions remain about its practicality. It also means that the quantum version of analogue computers, continuous variable quantum computers (CVQC) becomes an equally efficient architecture for quantum simulation. Lessons from past use of classical analogue computers can help us to build better quantum simulators in future.Comment: 10 pages, to appear in the Visions 2010 issue of Phil. Trans. Roy. Soc.

RNA localization in development

Cytoplasmic RNA localization is an evolutionarily ancient mechanism for producing cellular asymmetries. This review considers RNA localization in the context of animal development. Both mRNAs and non-protein-coding RNAs are localized in Drosophila, Xenopus, ascidian, zebrafish, and echinoderm oocytes and embryos, as well as in a variety of developing and differentiated polarized cells from yeast to mammals. Mechanisms used to transport and anchor RNAs in the cytoplasm include vectorial transport out of the nucleus, directed cytoplasmic transport in association with the cytoskeleton, and local entrapment at particular cytoplasmic sites. The majority of localized RNAs are targeted to particular cytoplasmic regions by cis-acting RNA elements; in mRNAs these are almost always in the 3'-untranslated region (UTR). A variety of trans-acting factors—many of them RNA-binding proteins—function in localization. Developmental functions of RNA localization have been defined in Xenopus, Drosophila, and Saccharomyces cerevisiae. In Drosophila, localized RNAs program the antero-posterior and dorso-ventral axes of the oocyte and embryo. In Xenopus, localized RNAs may function in mesoderm induction as well as in dorso-ventral axis specification. Localized RNAs also program asymmetric cell fates during Drosophila neurogenesis and yeast budding

Grid Diagrams for Lens Spaces and Combinatorial Knot Floer Homology

Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov and Alexander gradings in terms of combinatorial data on the grid diagram. Motivated by existing results for the Floer homology of knots in S^3 and the similarity of the combinatorics presented here, we conjecture that a certain family of knots is characterized by their Floer homology. Coupled with work of the third author, an affirmative answer to this would prove the Berge conjecture, which catalogs the knots in S^3 admitting lens space surgeries.Comment: 27 pages, 8 figures; Expositional improvements, corrected normalization of A grading in proof of Lemma 4.1

Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility

Word equations are a crucial element in the theoretical foundation of constraint solving over strings, which have received a lot of attention in recent years. A word equation relates two words over string variables and constants. Its solution amounts to a function mapping variables to constant strings that equate the left and right hand sides of the equation. While the problem of solving word equations is decidable, the decidability of the problem of solving a word equation with a length constraint (i.e., a constraint relating the lengths of words in the word equation) has remained a long-standing open problem. In this paper, we focus on the subclass of quadratic word equations, i.e., in which each variable occurs at most twice. We first show that the length abstractions of solutions to quadratic word equations are in general not Presburger-definable. We then describe a class of counter systems with Presburger transition relations which capture the length abstraction of a quadratic word equation with regular constraints. We provide an encoding of the effect of a simple loop of the counter systems in the theory of existential Presburger Arithmetic with divisibility (PAD). Since PAD is decidable, we get a decision procedure for quadratic words equations with length constraints for which the associated counter system is \emph{flat} (i.e., all nodes belong to at most one cycle). We show a decidability result (in fact, also an NP algorithm with a PAD oracle) for a recently proposed NP-complete fragment of word equations called regular-oriented word equations, together with length constraints. Decidability holds when the constraints are additionally extended with regular constraints with a 1-weak control structure.Comment: 18 page

Polynomial differential equations compute all real computable functions on computable compact intervals

In the last decade, the eld of analog computation has experienced renewed interest. In particular, there have been several attempts to un- derstand which relations exist between the many models of analog com- putation. Unfortunately, most models are not equivalent. It is known that Euler's Gamma function is computable according to computable analysis, while it cannot be generated by Shannon's General Purpose Analog Computer (GPAC). This example has often been used to argue that the GPAC is less powerful than digital computation. However, as we will demonstrate, when computability with GPACs is not restricted to real-time generation of functions, we obtain two equiva- lent models of analog computation. Using this approach, it has been shown recently that the Gamma func- tion becomes computable by a GPAC [1]. Here we extend this result by showing that, in an appropriate framework, the GPAC and computable analysis are actually equivalent from the computability point of view, at least in compact intervals. Since GPACs are equivalent to systems of polynomial di erential equations then we show that all real computable functions over compact intervals can be de ned by such models

The general purpose analog computer and computable analysis are two equivalent paradigms of analog computation

In this paper we revisit one of the rst models of analog computation, Shannon's General Purpose Analog Computer (GPAC). The GPAC has often been argued to be weaker than computable analysis. As main contribution, we show that if we change the notion of GPACcomputability in a natural way, we compute exactly all real computable functions (in the sense of computable analysis). Moreover, since GPACs are equivalent to systems of polynomial di erential equations then we show that all real computable functions can be de ned by such models

Vgl1, a multi-KH domain protein, is a novel component of the fission yeast stress granules required for cell survival under thermal stress

Multiple KH-domain proteins, collectively known as vigilins, are evolutionarily highly conserved proteins that are present in eukaryotic organisms from yeast to metazoa. Proposed roles for vigilins include chromosome segregation, messenger RNA (mRNA) metabolism, translation and tRNA transport. As a step toward understanding its biological function, we have identified the fission yeast vigilin, designated Vgl1, and have investigated its role in cellular response to environmental stress. Unlike its counterpart in Saccharomyces cerevisiae, we found no indication that Vgl1 is required for the maintenance of cell ploidy in Schizosaccharomyces pombe. Instead, Vgl1 is required for cell survival under thermal stress, and vgl1Δ mutants lose their viability more rapidly than wild-type cells when incubated at high temperature. As for Scp160 in S. cerevisiae, Vgl1 bound polysomes accumulated at endoplasmic reticulum (ER) but in a microtubule-independent manner. Under thermal stress, Vgl1 is rapidly relocalized from the ER to cytoplasmic foci that are distinct from P-bodies but contain stress granule markers such as poly(A)-binding protein and components of the translation initiation factor eIF3. Together, these observations demonstrated in S. pombe the presence of RNA granules with similar composition as mammalian stress granules and identified Vgl1 as a novel component that required for cell survival under thermal stress