Infinite dimensional integrals beyond Monte Carlo methods: yet another approach to normalized infinite dimensional integrals

An approach to (normalized) infinite dimensional integrals, including normalized oscillatory integrals, through a sequence of evaluations in the spirit of the Monte Carlo method for probability measures is proposed. in this approach the normalization through the partition function is included in the definition. For suitable sequences of evaluations, the ("classical") expectation values of cylinder functions are recoveredComment: Submitted as a communication in the ICMSQUARE conference, september 201

The role of endothelial cilia in post-embryonic vascular development

BACKGROUND: Cilia are essential for morphogenesis and maintenance of many tissues. Loss-of-function of cilia in early zebrafish development causes a range of vascular defects including cerebral haemorrhage and reduced arterial vascular mural cell coverage. In contrast, loss of endothelial cilia in mice has little effect on vascular development. We therefore used a conditional rescue approach to induce endothelial cilia ablation after early embryonic development and examined the effect on vascular development and mural cell development in post-embryonic, juvenile and adult zebrafish. RESULTS: ift54(elipsa) mutant zebrafish are unable to form cilia. We rescued cilia formation and ameliorated the phenotype of ift54 mutants using a novel Tg(ubi:loxP-ift54-loxP-myr-mcherry,myl7:EGFP)sh488 transgene expressing wildtype ift54 flanked by recombinase sites, then used a Tg(kdrl:cre)s898 transgene to induce endothelial-specific inactivation of ift54 at post-embryonic ages. Fish without endothelial ift54 function could survive to adulthood and exhibited no vascular defects. Endothelial inactivation of ift54 did not affect development of tagln-positive vascular mural cells around either the aorta or the caudal fin vessels, nor formation of vessels after tailfin resection in adult animals. CONCLUSIONS: Endothelial cilia are not essential for development and remodelling of the vasculature in juvenile and adult zebrafish when inactivated after embryogenesis

The role of Sox6 in zebrafish muscle fiber type specification

Background The transcription factor Sox6 has been implicated in regulating muscle fiber type-specific gene expression in mammals. In zebrafish, loss of function of the transcription factor Prdm1a results in a slow to fast-twitch fiber type transformation presaged by ectopic expression of sox6 in slow-twitch progenitors. Morpholino-mediated Sox6 knockdown can suppress this transformation but causes ectopic expression of only one of three slow-twitch specific genes assayed. Here, we use gain and loss of function analysis to analyse further the role of Sox6 in zebrafish muscle fiber type specification. Methods The GAL4 binary misexpression system was used to express Sox6 ectopically in zebrafish embryos. Cis-regulatory elements were characterized using transgenic fish. Zinc finger nuclease mediated targeted mutagenesis was used to analyse the effects of loss of Sox6 function in embryonic, larval and adult zebrafish. Zebrafish transgenic for the GCaMP3 Calcium reporter were used to assay Ca2+ transients in wild-type and mutant muscle fibres. Results Ectopic Sox6 expression is sufficient to downregulate slow-twitch specific gene expression in zebrafish embryos. Cis-regulatory elements upstream of the slow myosin heavy chain 1 (smyhc1) and slow troponin c (tnnc1b) genes contain putative Sox6 binding sites required for repression of the former but not the latter. Embryos homozygous for sox6 null alleles expressed tnnc1b throughout the fast-twitch muscle whereas other slow-specific muscle genes, including smyhc1, were expressed ectopically in only a subset of fast-twitch fibers. Ca2+ transients in sox6 mutant fast-twitch fibers were intermediate in their speed and amplitude between those of wild-type slow- and fast-twitch fibers. sox6 homozygotes survived to adulthood and exhibited continued misexpression of tnnc1b as well as smaller slow-twitch fibers. They also exhibited a striking curvature of the spine. Conclusions The Sox6 transcription factor is a key regulator of fast-twitch muscle fiber differentiation in the zebrafish, a role similar to that ascribed to its murine ortholog

A Rigorous Approach to the Feynman-Vernon Influence Functional and its Applications. I

A rigorous representation of the Feynman-Vernon influence functional used to describe open quantum systems is given, based on the theory of infinite dimensional oscillatory integrals. An application to the case of the density matrices describing the Caldeira-Leggett model of two quantum systems with a quadratic interaction is treated

Time separation as a hidden variable to the Copenhagen school of quantum mechanics

The Bohr radius is a space-like separation between the proton and electron in the hydrogen atom. According to the Copenhagen school of quantum mechanics, the proton is sitting in the absolute Lorentz frame. If this hydrogen atom is observed from a different Lorentz frame, there is a time-like separation linearly mixed with the Bohr radius. Indeed, the time-separation is one of the essential variables in high-energy hadronic physics where the hadron is a bound state of the quarks, while thoroughly hidden in the present form of quantum mechanics. It will be concluded that this variable is hidden in Feynman's rest of the universe. It is noted first that Feynman's Lorentz-invariant differential equation for the bound-state quarks has a set of solutions which describe all essential features of hadronic physics. These solutions explicitly depend on the time separation between the quarks. This set also forms the mathematical basis for two-mode squeezed states in quantum optics, where both photons are observable, but one of them can be treated a variable hidden in the rest of the universe. The physics of this two-mode state can then be translated into the time-separation variable in the quark model. As in the case of the un-observed photon, the hidden time-separation variable manifests itself as an increase in entropy and uncertainty.Comment: LaTex 10 pages with 5 figure. Invited paper presented at the Conference on Advances in Quantum Theory (Vaxjo, Sweden, June 2010), to be published in one of the AIP Conference Proceedings serie

Decomposing generalized measurements into continuous stochastic processes

One of the broadest concepts of measurement in quantum theory is the generalized measurement. Another paradigm of measurement--arising naturally in quantum optics, among other fields--is that of continuous-time measurements, which can be seen as the limit of a consecutive sequence of weak measurements. They are naturally described in terms of stochastic processes, or time-dependent random variables. We show that any generalized measurement can be decomposed as a sequence of weak measurements with a mathematical limit as a continuous stochastic process. We give an explicit construction for any generalized measurement, and prove that the resulting continuous evolution, in the long-time limit, collapses the state of the quantum system to one of the final states generated by the generalized measurement, being decomposed, with the correct probabilities. A prominent feature of the construction is the presence of a feedback mechanism--the instantaneous choice weak measurement at a given time depends on the outcomes of earlier measurements. For a generalized measurement with $n$ outcomes, this information is captured by a real $n$-vector on an $n$-simplex, which obeys a simple classical stochastic evolution.Comment: 9 pages, LaTeX, name changed, typos correcte

G-Brownian Motion as Rough Paths and Differential Equations Driven by G-Brownian Motion

The present paper is devoted to the study of sample paths of G-Brownian motion and stochastic differential equations (SDEs) driven by G-Brownian motion from the view of rough path theory. As the starting point, we show that quasi-surely, sample paths of G-Brownian motion can be enhanced to the second level in a canonical way so that they become geometric rough paths of roughness 2 < p < 3. This result enables us to introduce the notion of rough differential equations (RDEs) driven by G-Brownian motion in the pathwise sense under the general framework of rough paths. Next we establish the fundamental relation between SDEs and RDEs driven by G-Brownian motion. As an application, we introduce the notion of SDEs on a differentiable manifold driven by GBrownian motion and construct solutions from the RDE point of view by using pathwise localization technique. This is the starting point of introducing G-Brownian motion on a Riemannian manifold, based on the idea of Eells-Elworthy-Malliavin. The last part of this paper is devoted to such construction for a wide and interesting class of G-functions whose invariant group is the orthogonal group. We also develop the Euler-Maruyama approximation for SDEs driven by G-Brownian motion of independent interest

Brownian bridges to submanifolds

We introduce and study Brownian bridges to submanifolds. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. We use the formula to derive lower bounds, an asymptotic relation and derivative estimates. We also see a connection to hypersurface local time. This work is motivated by the desire to extend the analysis of path and loop spaces to measures on paths which terminate on a submanifold