The Bismut-Elworthy-Li type formulae for stochastic differential equations with jumps

Consider jump-type stochastic differential equations with the drift, diffusion and jump terms. Logarithmic derivatives of densities for the solution process are studied, and the Bismut-Elworthy-Li type formulae can be obtained under the uniformly elliptic condition on the coefficients of the diffusion and jump terms. Our approach is based upon the Kolmogorov backward equation by making full use of the Markovian property of the process.Comment: 29 pages, to appear in Journal of Theoretical Probabilit

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

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

Dominant Topologies in Euclidean Quantum Gravity

The dominant topologies in the Euclidean path integral for quantum gravity differ sharply according on the sign of the cosmological constant. For $\Lambda>0$, saddle points can occur only for topologies with vanishing first Betti number and finite fundamental group. For $\Lambda<0$, on the other hand, the path integral is dominated by topologies with extremely complicated fundamental groups; while the contribution of each individual manifold is strongly suppressed, the ``density of topologies'' grows fast enough to overwhelm this suppression. The value $\Lambda=0$ is thus a sort of boundary between phases in the sum over topologies. I discuss some implications for the cosmological constant problem and the Hartle-Hawking wave function.Comment: 14 pages, LaTeX. Minor additions (computability, relation to ``minimal volume'' in topology); error in eqn (3.5) corrected; references added. To appear in Class. Quant. Gra

On arbitrages arising from honest times

In the context of a general continuous financial market model, we study whether the additional information associated with an honest time gives rise to arbitrage profits. By relying on the theory of progressive enlargement of filtrations, we explicitly show that no kind of arbitrage profit can ever be realised strictly before an honest time, while classical arbitrage opportunities can be realised exactly at an honest time as well as after an honest time. Moreover, stronger arbitrages of the first kind can only be obtained by trading as soon as an honest time occurs. We carefully study the behavior of local martingale deflators and consider no-arbitrage-type conditions weaker than NFLVR.Comment: 25 pages, revised versio

Laplace Operators on Fractals and Related Functional Equations

We give an overview over the application of functional equations, namely the classical Poincar\'e and renewal equations, to the study of the spectrum of Laplace operators on self-similar fractals. We compare the techniques used to those used in the euclidean situation. Furthermore, we use the obtained information on the spectral zeta function to define the Casimir energy of fractals. We give numerical values for this energy for the Sierpi\'nski gasket

Unusual development of light-reflecting pigment cells in intact and regenerating tail in the periodic albino mutant of Xenopus laevis

Unusual light-reflecting pigment cells, “white pigment cells”, specifically appear in the periodic albino mutant (ap/ap) of Xenopus laevis and localize in the same place where melanophores normally differentiate in the wild-type. The mechanism responsible for the development of unusual pigment cells is unclear. In this study, white pigment cells in the periodic albino were compared with melanophores in the wild-type, using a cell culture system and a tail-regenerating system. Observations of both intact and cultured cells demonstrate that white pigment cells are unique in (1) showing characteristics of melanophore precursors at various stages of development, (2) accumulating reflecting platelets characteristic of iridophores, and (3) exhibiting pigment dispersion in response to α-melanocyte stimulating hormone (α-MSH) in the same way that melanophores do. When a tadpole tail is amputated, a functionally competent new tail is regenerated. White pigment cells appear in the mutant regenerating tail, whereas melanophores differentiate in the wild-type regenerating tail. White pigment cells in the mutant regenerating tail are essentially similar to melanophores in the wild-type regenerating tail with respect to their localization, number, and response to α-MSH. In addition to white pigment cells, iridophores which are never present in the intact tadpole tail appear specifically in the somites near the amputation level in the mutant regenerating tail. Iridophores are distinct from white pigment cells in size, shape, blue light-induced fluorescence, and response to α-MSH. These findings strongly suggest that white pigment cells in the mutant arise from melanophore precursors and accumulate reflecting platelets characteristic of iridophores

Evidence for oxygenic photosynthesis half a billion years before the Great Oxidation Event

The early Earth was characterized by the absence of oxygen in the ocean–atmosphere system, in contrast to the well-oxygenated conditions that prevail today. Atmospheric concentrations first rose to appreciable levels during the Great Oxidation Event, roughly 2.5–2.3 Gyr ago. The evolution of oxygenic photosynthesis is generally accepted to have been the ultimate cause of this rise, but it has proved difficult to constrain the timing of this evolutionary innovation. The oxidation of manganese in the water column requires substantial free oxygen concentrations, and thus any indication that Mn oxides were present in ancient environments would imply that oxygenic photosynthesis was ongoing. Mn oxides are not commonly preserved in ancient rocks, but there is a large fractionation of molybdenum isotopes associated with the sorption of Mo onto the Mn oxides that would be retained. Here we report Mo isotopes from rocks of the Sinqeni Formation, Pongola Supergroup, South Africa. These rocks formed no less than 2.95 Gyr ago in a nearshore setting. The Mo isotopic signature is consistent with interaction with Mn oxides. We therefore infer that oxygen produced through oxygenic photosynthesis began to accumulate in shallow marine settings at least half a billion years before the accumulation of significant levels of atmospheric oxygen

The positive transcriptional elongation factor (P-TEFb) is required for neural crest specification

Regulation of gene expression at the level of transcriptional elongation has been shown to be important in stem cells and tumour cells, but its role in the whole animal is only now being fully explored. Neural crest cells (NCCs) are a multipotent population of cells that migrate during early development from the dorsal neural tube throughout the embryo where they differentiate into a variety of cell types including pigment cells, cranio-facial skeleton and sensory neurons. Specification of NCCs is both spatially and temporally regulated during embryonic development. Here we show that components of the transcriptional elongation regulatory machinery, CDK9 and CYCLINT1 of the P-TEFb complex, are required to regulate neural crest specification. In particular, we show that expression of the proto-oncogene c-Myc and c-Myc responsive genes are affected. Our data suggest that P-TEFb is crucial to drive expression of c-Myc, which acts as a ‘gate-keeper’ for the correct temporal and spatial development of the neural crest

Positive and Negative Regulation of Gli Activity by Kif7 in the Zebrafish Embryo

Loss of function mutations of Kif7, the vertebrate orthologue of the Drosophila Hh pathway component Costal2, cause defects in the limbs and neural tubes of mice, attributable to ectopic expression of Hh target genes. While this implies a functional conservation of Cos2 and Kif7 between flies and vertebrates, the association of Kif7 with the primary cilium, an organelle absent from most Drosophila cells, suggests their mechanisms of action may have diverged. Here, using mutant alleles induced by Zinc Finger Nuclease-mediated targeted mutagenesis, we show that in zebrafish, Kif7 acts principally to suppress the activity of the Gli1 transcription factor. Notably, we find that endogenous Kif7 protein accumulates not only in the primary cilium, as previously observed in mammalian cells, but also in cytoplasmic puncta that disperse in response to Hh pathway activation. Moreover, we show that Drosophila Costal2 can substitute for Kif7, suggesting a conserved mode of action of the two proteins. We show that Kif7 interacts with both Gli1 and Gli2a and suggest that it functions to sequester Gli proteins in the cytoplasm, in a manner analogous to the regulation of Ci by Cos2 in Drosophila. We also show that zebrafish Kif7 potentiates Gli2a activity by promoting its dissociation from the Suppressor of Fused (Sufu) protein and present evidence that it mediates a Smo dependent modification of the full length form of Gli2a. Surprisingly, the function of Kif7 in the zebrafish embryo appears restricted principally to mesodermal derivatives, its inactivation having little effect on neural tube patterning, even when Sufu protein levels are depleted. Remarkably, zebrafish lacking all Kif7 function are viable, in contrast to the peri-natal lethality of mouse kif7 mutants but similar to some Acrocallosal or Joubert syndrome patients who are homozygous for loss of function KIF7 alleles