421 research outputs found
Primitive geodesic lengths and (almost) arithmetic progressions
In this article, we investigate when the set of primitive geodesic lengths on
a Riemannian manifold have arbitrarily long arithmetic progressions. We prove
that in the space of negatively curved metrics, a metric having such arithmetic
progressions is quite rare. We introduce almost arithmetic progressions, a
coarsification of arithmetic progressions, and prove that every negatively
curved, closed Riemannian manifold has arbitrarily long almost arithmetic
progressions in its primitive length spectrum. Concerning genuine arithmetic
progressions, we prove that every non-compact, locally symmetric, arithmetic
manifold has arbitrarily long arithmetic progressions in its primitive length
spectrum. We end with a conjectural characterization of arithmeticity in terms
of arithmetic progressions in the primitive length spectrum. We also suggest an
approach to a well known spectral rigidity problem based on the scarcity of
manifolds with arithmetic progressions.Comment: v3: 23 pages. To appear in Publ. Ma
Counting coloured planar maps: differential equations
We address the enumeration of q-coloured planar maps counted bythe number of
edges and the number of monochromatic edges. We prove that the associated
generating function is differentially algebraic,that is, satisfies a
non-trivial polynomial differential equation withrespect to the edge variable.
We give explicitly a differential systemthat characterizes this series. We then
prove a similar result for planar triangulations, thus generalizing a result of
Tutte dealing with their proper q-colourings. Instatistical physics terms, we
solvethe q-state Potts model on random planar lattices. This work follows a
first paper by the same authors, where the generating functionwas proved to be
algebraic for certain values of q,including q=1, 2 and 3. It isknown to be
transcendental in general. In contrast, our differential system holds for an
indeterminate q.For certain special cases of combinatorial interest (four
colours; properq-colourings; maps equipped with a spanning forest), we derive
from this system, in the case of triangulations, an explicit differential
equation of order 2 defining the generating function. For general planar maps,
we also obtain a differential equation of order 3 for the four-colour case and
for the self-dual Potts model.Comment: 43 p
Slow light in photonic crystals
The problem of slowing down light by orders of magnitude has been extensively
discussed in the literature. Such a possibility can be useful in a variety of
optical and microwave applications. Many qualitatively different approaches
have been explored. Here we discuss how this goal can be achieved in linear
dispersive media, such as photonic crystals. The existence of slowly
propagating electromagnetic waves in photonic crystals is quite obvious and
well known. The main problem, though, has been how to convert the input
radiation into the slow mode without loosing a significant portion of the
incident light energy to absorption, reflection, etc. We show that the
so-called frozen mode regime offers a unique solution to the above problem.
Under the frozen mode regime, the incident light enters the photonic crystal
with little reflection and, subsequently, is completely converted into the
frozen mode with huge amplitude and almost zero group velocity. The linearity
of the above effect allows to slow light regardless of its intensity. An
additional advantage of photonic crystals over other methods of slowing down
light is that photonic crystals can preserve both time and space coherence of
the input electromagnetic wave.Comment: 96 pages, 12 figure
Investigations of the structural dynamics of the water and proton channels in Photosystem II
Bei der lichtinduzierten Oxidation von Wasser im Photosystem II (PSII) werden zwei wassermoleküle im katalytischen Zyklus des Metallclusters (Mn4CaO5) benötigt, und vier Protonen aus dem Cluster in den Lumen abgegeben. Daher ist es für das Verständnis des Mechanismus´ der Wasseroxidation von entscheidender Bedeutung, die Veränderung der Protonierungszustände am cluster während der Katalyse zu untersuchen. Hierbei sollten sowohl die Wasserkanäle für die Zuführung der Substratwassermoleküle als auch die Transportwege für die Freisetzung der Protonen untersucht werden. Deshalb wurde in meiner ersten Veröffentlichung ein neues Protokoll entwickelt, um einzelne große Kristalle von dPSII mit einer Länge von ~3 mm in der Längsachse zu züchten. Diese Kristalle mit einer Auflösung von ca. 8 Å gemessen. Um eine höhere Auflösung zu erzielen, ist die Verbesserung der Kristallqualität essenziell. Daher wurde in meiner zweiten Veröffentlichung die Struktur des Detergens-Protein-Komplexes von dPSII mit βDM, durch Anwendung von SANS in Kombination mit SAXS untersucht. Die Ergebnisse zeigten, dass βDM eine monomolekulare Schicht um dPSII bildet. Darüber hinaus konnten freie Mizellen von βDM in der Lösung nachgewiesen werden. Damit ist eine weitere Optimierung der βDM-Konzentration in der Proteinlösung erforderlich, um die Bildung von freien Mizellen zu minimieren. In meiner dritten Veröffentlichung wurde die strukturelle Dynamik in den Wasserkanälen, während des S2-S3 Übergangs mit Hilfe der XFEL untersucht. Ein Datensatz mit einer hohen Auflösung von 1,89 Å wurde durch die Zusammenführung von Daten gewonnen, die während des S2-S3 Übergangs gesammelt wurden. In Anbetracht der Analyse der zusammengeführten Daten und der einzelnen Zeitpunkte, die während des S2-S3 Übergangs gesammelt wurden, ist es wahrscheinlich, dass ein Substratwasser durch den O1-Kanal geliefert wird. Im Gegensatz dazu wird ein Proton aus dem Cluster durch den Cl1 Transportweg in Richtung Lumen freigesetzt.The light-induced oxidation of water in Photosystem II (PSII) requires incorporating two water molecules in the catalytic cycle of the active metal cluster (Mn4CaO5). Furthermore, four protons are released towards the bulk from the cluster. Therefore, tracking the change of protonation states at the active catalytic site and the surrounding protein side chains during catalysis and elucidating the pathways of water substrate insertion and proton release are crucial to understanding the water oxidation mechanism. Therefore, in the first study of my work, a new protocol was developed to grow single large dPSIIcc crystals with a length of ~3 mm in the long axis. These crystals, soaked in D2O containing buffer, diffracted to about 8 Å resolution. Improving the crystal quality is crucial for achieving a better resolution. Consequently, in the second study of my work, the structure of the detergent-protein complex of βDM-dPSIIcc has been investigated by applying SANS in combination with SAXS. The results showed that βDM is forming a monomolecular layer around the dimeric PSII core complex (dPSIIcc). Moreover, the SAXS data detected a peak assigned to the free micelles of βDM. These results raise the necessity to optimize the βDM concentration in the protein solution to avoid the possible excess of free micelles. In the third study of my work, the structural dynamics in the water channels connecting the cluster to the lumen during the S2 S3 transition were investigated using serial femtosecond XFEL. A high-resolution data set was obtained at a resolution of 1.89 Å by combining data collected at RT. Considering the analysis of the combined data and the individual time points collected during the S2 S3 transition, it is likely that the substrate water insertion into the open coordination site of the Mn1 ion is delivered through the O1 channel. In contrast, a proton from the cluster is released towards the bulk through the Cl1 A channel
Synthesis, characterization and modelling of zinc and silicate co-substituted hydroxyapatite.
Experimental chemistry and atomic modelling studies were performed here to investigate a novel ionic co-substitution in hydroxyapatite (HA). Zinc, silicate co-substituted HA (ZnSiHA) remained phase pure after heating to 1100 °C with Zn and Si amounts of 0.6 wt% and 1.2 wt%, respectively. Unique lattice expansions in ZnSiHA, silicate Fourier transform infrared peaks and changes to the hydroxyl IR stretching region suggested Zn and silicate co-substitution in ZnSiHA. Zn and silicate insertion into HA was modelled using density functional theory (DFT). Different scenarios were considered where Zn substituted for different calcium sites or at a 2b site along the c-axis, which was suspected in singly substituted ZnHA. The most energetically favourable site in ZnSiHA was Zn positioned at a previously unreported interstitial site just off the c-axis near a silicate tetrahedron sitting on a phosphate site. A combination of experimental chemistry and DFT modelling provided insight into these complex co-substituted calcium phosphates that could find biomedical application as a synthetic bone mineral substitute.This work was supported by a NSFGRFP grant (DGE-1042796) (RJF) and a Cambridge International Scholarship (RJF). The modelling work was performed using the Darwin Supercomputer of the University of Cambridge High Performance Computing Service (http://www.hpc.cam.ac.uk/), provided by Dell Inc. using Strategic Research Infrastructure Funding from the Higher Education Funding Council for England and funding from the Science and Technology Facilities Council. HC would like to thank the UK Medical Research Council (Grant number U105960399) for their support.This is the final version of the article. It first appeared from Royal Society Publishing via http://dx.doi.org/10.1098/rsif.2015.019
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