Eigenfunctions of the Laplacian and associated Ruelle operator

Let $\Gamma$ be a co-compact Fuchsian group of isometries on the Poincar\'e disk \DD and $\Delta$ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction $f$ of $\Delta$, equivariant by $\Gamma$ with real eigenvalue $\lambda=-s(1-s)$, where $s={1/2}+ it$, admits an integral representation by a distribution \dd_{f,s} (the Helgason distribution) which is equivariant by $\Gamma$ and supported at infinity \partial\DD=\SS^1. The geodesic flow on the compact surface \DD/\Gamma is conjugate to a suspension over a natural extension of a piecewise analytic map T:\SS^1\to\SS^1, the so-called Bowen-Series transformation. Let $\ll_s$ be the complex Ruelle transfer operator associated to the jacobian $-s\ln |T'|$. M. Pollicott showed that \dd_{f,s} is an eigenfunction of the dual operator $\ll_s^*$ for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction $\psi_{f,s}$ of $\ll_s$ for the eigenvalue 1, given by an integral formula \psi_{f,s} (\xi)=\int \frac{J(\xi,\eta)}{|\xi-\eta|^{2s}} \dd_{f,s} (d\eta), \noindent where $J(\xi,\eta)$ is a $\{0,1\}$-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface \DD/\Gamma

Geochemical Endmembers preserved in Gale Crater: A tale of two mudstones and their compositional differences according to ChemCam

Gale crater contains two fine-grained mudstone sedimentary units: The Sheepbed mudstone member, and the Murray formation mud-stones. These mudstones formed as part of an ancient fluviolacustrine system. The NASA Curiosity rover has analysed these mudstone units using the Chemistry and Camera (ChemCam), Alpha Particle X-ray Spectrometer (APXS) and Chemistry and Mineralogy (CheMin) onboard instrument suites. Subsequent mineralogical analyses have uncovered a wide geochemical and mineralogical diversity across and within these two mudstone formations. This study aims to determine the principal cause (alteration or source region) of this geochemical variation through a statistical analysis of the ChemCam dataset up to sol 1482, including the lower to middle Murray formation

Post-critical set and non existence of preserved meromorphic two-forms

We present a family of birational transformations in $CP_2$ depending on two, or three, parameters which does not, generically, preserve meromorphic two-forms. With the introduction of the orbit of the critical set (vanishing condition of the Jacobian), also called ``post-critical set'', we get some new structures, some "non-analytic" two-form which reduce to meromorphic two-forms for particular subvarieties in the parameter space. On these subvarieties, the iterates of the critical set have a polynomial growth in the \emph{degrees of the parameters}, while one has an exponential growth out of these subspaces. The analysis of our birational transformation in $CP_2$ is first carried out using Diller-Favre criterion in order to find the complexity reduction of the mapping. The integrable cases are found. The identification between the complexity growth and the topological entropy is, one more time, verified. We perform plots of the post-critical set, as well as calculations of Lyapunov exponents for many orbits, confirming that generically no meromorphic two-form can be preserved for this mapping. These birational transformations in $CP_2$, which, generically, do not preserve any meromorphic two-form, are extremely similar to other birational transformations we previously studied, which do preserve meromorphic two-forms. We note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned, but drastically different results as far as a more ``probabilistic'' approach of dynamical systems is concerned (Lyapunov exponents). With these examples we see that the existence of a preserved meromorphic two-form explains most of the (numerical) discrepancy between the topological and probabilistic approach of dynamical systems.Comment: 34 pages, 7 figure

Canalization of the evolutionary trajectory of the human influenza virus

Since its emergence in 1968, influenza A (H3N2) has evolved extensively in genotype and antigenic phenotype. Antigenic evolution occurs in the context of a two-dimensional 'antigenic map', while genetic evolution shows a characteristic ladder-like genealogical tree. Here, we use a large-scale individual-based model to show that evolution in a Euclidean antigenic space provides a remarkable correspondence between model behavior and the epidemiological, antigenic, genealogical and geographic patterns observed in influenza virus. We find that evolution away from existing human immunity results in rapid population turnover in the influenza virus and that this population turnover occurs primarily along a single antigenic axis. Thus, selective dynamics induce a canalized evolutionary trajectory, in which the evolutionary fate of the influenza population is surprisingly repeatable and hence, in theory, predictable.Comment: 29 pages, 5 figures, 10 supporting figure

Can Weakness in End-Range Plantar Flexion After Achilles Tendon Repair Be Prevented?

Background: Disproportionate end-range plantar flexion weakness, decreased passive stiffness, and inability to perform a heel rise on a decline after Achilles tendon repair are thought to reflect increased tendon compliance or tendon lengthening. Since this was first noted, we have performed stronger repairs and avoided stretching into dorsiflexion for the first 12 weeks after surgery. Hypothesis: Using stronger repairs and avoiding stretching into dorsiflexion would eliminate end-range plantar flexion weakness and normalize passive stiffness. Study Design: Case series; Level of evidence, 4. Methods: Achilles repairs with epitendinous augmentation were performed on 18 patients. Plantar flexion torque, dorsiflexion range of motion (ROM), passive joint stiffness, and standing single-legged heel rise on a decline were assessed at 43 ± 24 months after surgery (range, 9 months to 8 years). Maximum isometric plantar flexion torque was measured at 20° and 10° of dorsiflexion, neutral position, and 10° and 20° of plantar flexion. Passive dorsiflexion ROM was measured with a goniometer. Passive joint stiffness was computed from the increase in passive torque from 10° to 20° of dorsiflexion. Tendon thickness was measured by use of digital calipers. Plantar flexion electromyographic (EMG) data were recorded during strength and functional tests. Analysis of variance and chi-square tests were used to assess weakness and function. Results: Marked weakness was evident on the involved side at 20° of plantar flexion (deficit, 26% ± 18%; Conclusion: The use of stronger repair techniques and attempts to limit tendon elongation by avoiding dorsiflexion stretching did not eliminate weakness in end-range plantar flexion. EMG data confirmed that end-range weakness was not due to neural inhibition. Physiological changes that alter the force transmission capability of the healing tendon may be responsible for this continued impairment. This weakness has implications for high-demand jumping and sprinting after Achilles tendon repair

Forming a Mogi Doughnut in the Years Prior to and Immediately Before the 2014 M8.1 Iquique, Northern Chile, Earthquake

Asperities are patches where the fault surfaces stick until they break in earthquakes. Locating asperities and understanding their causes in subduction zones is challenging because they are generally located offshore. We use seismicity, interseismic and coseismic slip, and the residual gravity field to map the asperity responsible for the 2014M8.1 Iquique, Chile, earthquake. For several years prior to the mainshock, seismicity occurred exclusively downdip of the asperity. Two weeks before the mainshock, a series of foreshocks first broke the upper plate then the updip rim of the asperity. This seismicity formed a ring around the slip patch (asperity) that later ruptured in the mainshock. The asperity correlated both with high interseismic locking and a circular gravity low, suggesting that it is controlled by geologic structure. Most features of the spatiotemporal seismicity pattern can be explained by a mechanical model in which a single asperity is stressed by relative plate motion

Comments on MHV Tree Amplitudes for Conformal Supergravitons from Topological B-Model

We use the twistor-string theory on the B-model of CP^{3|4} to compute the maximally helicity violating(MHV) tree amplitudes for conformal supergravitons. The correlator of a bilinear in the affine Kac-Moody current(Sugawara stress-energy tensor) can generate these amplitudes. We compare with previous results from open string version of twistor-string theory. We also compute the MHV tree amplitudes for both gravitons and gluons from the correlators between stress-energy tensor and current.Comment: 27p

Embeddings of SL(2,Z) into the Cremona group

Geometric and dynamic properties of embeddings of SL(2,Z) into the Cremona group are studied. Infinitely many non-conjugate embeddings which preserve the type (i.e. which send elliptic, parabolic and hyperbolic elements onto elements of the same type) are provided. The existence of infinitely many non-conjugate elliptic, parabolic and hyperbolic embeddings is also shown. In particular, a group G of automorphisms of a smooth surface S obtained by blowing-up 10 points of the complex projective plane is given. The group G is isomorphic to SL(2,Z), preserves an elliptic curve and all its elements of infinite order are hyperbolic.Comment: to appear in Transformation Group