A general wavelet-based profile decomposition in the critical embedding of function spaces

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(\phi_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by G\'erard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0<s=d/2-d/p$, and then generalized by Jaffard to the case where $X$ is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces $X$ and $Y$ that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of $X$ and $Y$ satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older and BMO spaces.Comment: 24 page

Sparse Deterministic Approximation of Bayesian Inverse Problems

We present a parametric deterministic formulation of Bayesian inverse problems with input parameter from infinite dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a generalized polynomial chaos representation of the posterior density with respect to the prior measure, given noisy observational data. We analyze the sparsity of the posterior density in terms of the summability of the input data's coefficient sequence. To this end, we estimate the fluctuations in the prior. We exhibit sufficient conditions on the prior model in order for approximations of the posterior density to converge at a given algebraic rate, in terms of the number $N$ of unknowns appearing in the parameteric representation of the prior measure. Similar sparsity and approximation results are also exhibited for the solution and covariance of the elliptic partial differential equation under the posterior. These results then form the basis for efficient uncertainty quantification, in the presence of data with noise

Hopewell Newsletter- June 2010

1. Small Scale Geoarchaeological Investigations of Earthen Wall Construction at the Hopeton Earthworks (33RO 26) Several geoarchaeological and geophysical investigations have been conducted at the Hopeton Earthworks (33RO26) and met with good success (Figure 1). As such, soil cores excavated in the summer of 2007 were studied using a two-pronged methodology of soil profile characterization and magnetic susceptibility testing. These techniques were employed to closely examine three sections of earthen wall at different locations at Hopeton as a means of understanding the soil composition of the walls and assessing possible uniformity in wall construction across the site. The following focuses solely on the geoarchaeological study of the selected locations. 2. The Role of Geophysics at Hopewell Culture National Historical Park Hopewell Culture National Historical Park is best known for its earthwork complexes built by the Hopewell during the Middle Woodland period. The park was originally established in 1923 to preserve the Mound City site after its use as a World War I training camp. In the past two decades, the park has added four additional earthworks—Hopeton Earthworks, Hopewell Mound Group, Seip Earthworks, and High Bank Works—and recently had its boundaries increased to include Spruce Hill. The recent growth in land located within the park has provided opportunities for archaeological research, most of which used some sort of geophysical technique. 3. Recent Investigations at the Mound City Group Ohio Hopewell earthworks have been studied extensively, both in historic and in modern times. Mound excavations during historic and modern times have revealed much about the construction and use of mound space. More recent research at Hopewell sites in Ohio has focused on non-mound areas both within and adjacent to the earthworks. This article focuses on current research conducted outside of the enclosure at the Mound City Group. 4. Clues to the Relationship of the Riverbank Site (33RO1059) to other Ohio Hopewell Sites Through Instrumental Neutron Activation Analysis on Pottery Excavations were conducted at the Riverbank Site (33RO1059), located by the Hopewell Site (33RO27), in 2004 and 2006 to gather data from the site before it is eroded away by the Paint Creek and to improve understanding of the role of small sites located near large Hopewell earthworks. Instrumental Neutron Activation Analysis (INAA) was conducted on pottery from the Riverbank Site to help clarify its relationship with other nearby sites. Fifteen sherds from the site were sent to the University of Missouri Research Reactor (MURR) for INAA, and the results were compared to the larger database of Hopewell ceramics. Based on the results, which show that the fifteen sherds formed a unique trace element compositional group, the site likely represents a shortterm occupation, possibly for a pilgrimage to the Hopewell Site, and the ceramics were likely either locally constructed for use at the site or constructed elsewhere to carry items on the way to Hopewell. 5. Feature Finds from the Riverbank Site, 33RO1059 Located just south and east of the Hopewell site’s Square Enclosure by about 225 meters is the Riverbank site, 33RO1059, which represents multiple occupations spanning the Archaic through historic periods, and includes a significant Middle Woodland component. Particularly noteworthy are two impressive pit features that were encountered in 2006 during a data recovery project undertaken by the Midwest Archeological Center. These pits, Features 7 and 8, would have been contemporaneous with Hopewell activities at the nearby earthwork complex, and the material culture recovered from them provides an intriguing glimpse of activities outside of the earthwork walls. 6. The Initial Phase of the Magnetic Investigations of the Mound City Group (32RO 32) at the Hopewell Culture National Historical Park, Ross County, Ohio The Mound City Group (Site 32RO32) covers approximately 13 acres. It contained at least 23 mounds when Squier and Davis made the first study of the mound complex in 1846. In the summer of 2009, the Midwest Archeological Center conducted the initial phase of a magnetic survey of the entire site. The magnetic survey in the southern portion of the site covered 11,200 m2 or 2.77 acres. The magnetic data indicated the presence of numerous magnetic anomalies associated with the Hopewell occupation and with the World War I training facility of Camp Sherman

Precision Tests of the Standard Model

30 páginas, 11 figuras, 11 tablas.-- Comunicación presentada al 25º Winter Meeting on Fundamental Physics celebrado del 3 al 8 de MArzo de 1997 en Formigal (España).Precision measurements of electroweak observables provide stringent tests of the Standard Model structure and an accurate determination of its parameters. An overview of the present experimental status is presented.This work has been supported in part by CICYT (Spain) under grant No. AEN-96-1718.Peer reviewe

Conditioning bounds for traveltime tomography in layered media

This paper revisits the problem of recovering a smooth, isotropic, layered wave speed profile from surface traveltime information. While it is classic knowledge that the diving (refracted) rays classically determine the wave speed in a weakly well-posed fashion via the Abel transform, we show in this paper that traveltimes of reflected rays do not contain enough information to recover the medium in a well-posed manner, regardless of the discretization. The counterpart of the Abel transform in the case of reflected rays is a Fredholm kernel of the first kind which is shown to have singular values that decay at least root-exponentially. Kinematically equivalent media are characterized in terms of a sequence of matching moments. This severe conditioning issue comes on top of the well-known rearrangement ambiguity due to low velocity zones. Numerical experiments in an ideal scenario show that a waveform-based model inversion code fits data accurately while converging to the wrong wave speed profile

Concentration analysis and cocompactness

Loss of compactness that occurs in may significant PDE settings can be expressed in a well-structured form of profile decomposition for sequences. Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where $X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a set of surjective isometries on both $X$ and $Y$. A profile decomposition is a representation of a bounded sequence in $X$ as a sum of elementary concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any sequence in $X$ that develops no $D$-concentrations has a subsequence convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this property is called $D$-cocompact, a property weaker than, but related to, compactness. We survey known cocompact imbeddings and their role in profile decompositions

Relativistic separable dual-space Gaussian Pseudopotentials from H to Rn

We generalize the concept of separable dual-space Gaussian pseudopotentials to the relativistic case. This allows us to construct this type of pseudopotential for the whole periodic table and we present a complete table of pseudopotential parameters for all the elements from H to Rn. The relativistic version of this pseudopotential retains all the advantages of its nonrelativistic version. It is separable by construction, it is optimal for integration on a real space grid, it is highly accurate and due to its analytic form it can be specified by a very small number of parameters. The accuracy of the pseudopotential is illustrated by an extensive series of molecular calculations

Discovery of the Transiting Planet Kepler-5B

We present 44 days of high duty cycle, ultra precise photometry of the 13th magnitude star Kepler-5 (KIC 8191672, T(eff) = 6300 K, log g = 4.1), which exhibits periodic transits with a depth of 0.7%. Detailed modeling of the transit is consistent with a planetary companion with an orbital period of 3.548460 +/- 0.000032 days and a radius of 1.431(-0.052)(+0.041) R(J). Follow-up radial velocity measurements with the Keck HIRES spectrograph on nine separate nights demonstrate that the planet is more than twice as massive as Jupiter with a mass of 2.114(-0.059)(+0.056) M(J) and a mean density of 0.894 +/- 0.079 g cm(-3).NASA's Science Mission DirectorateAstronom

The K2-ESPRINT Project. I. Discovery of the Disintegrating Rocky Planet K2-22b with a Cometary Head and Leading Tail

We present the discovery of a transiting exoplanet candidate in the K2 Field-1 with an orbital period of 9.1457 hr: K2-22b. The highly variable transit depths, ranging from $\sim$0\% to 1.3\%, are suggestive of a planet that is disintegrating via the emission of dusty effluents. We characterize the host star as an M-dwarf with $T_{\rm eff} \simeq 3800$ K. We have obtained ground-based transit measurements with several 1-m class telescopes and with the GTC. These observations (1) improve the transit ephemeris; (2) confirm the variable nature of the transit depths; (3) indicate variations in the transit shapes; and (4) demonstrate clearly that at least on one occasion the transit depths were significantly wavelength dependent. The latter three effects tend to indicate extinction of starlight by dust rather than by any combination of solid bodies. The K2 observations yield a folded light curve with lower time resolution but with substantially better statistical precision compared with the ground-based observations. We detect a significant "bump" just after the transit egress, and a less significant bump just prior to transit ingress. We interpret these bumps in the context of a planet that is not only likely streaming a dust tail behind it, but also has a more prominent leading dust trail that precedes it. This effect is modeled in terms of dust grains that can escape to beyond the planet's Hill sphere and effectively undergo `Roche lobe overflow,' even though the planet's surface is likely underfilling its Roche lobe by a factor of 2.Comment: 22 pages, 16 figures. Final version accepted to Ap