Anisotropic smoothness classes : from finite element approximation to image models

We propose and study quantitative measures of smoothness which are adapted to anisotropic features such as edges in images or shocks in PDE's. These quantities govern the rate of approximation by adaptive finite elements, when no constraint is imposed on the aspect ratio of the triangles, the simplest examples of such quantities are based on the determinant of the hessian of the function to be approximated. Since they are not semi-norms, these quantities cannot be used to define linear function spaces. We show that they can be well defined by mollification when the function to be approximated has jump discontinuities along piecewise smooth curves. This motivates for using them in image processing as an alternative to the frequently used record variation semi-norm which does not account for the geometric smoothness of the edges.Comment: 24 pages, 2 figure

New coins from old, smoothly

Given a (known) function $f:[0,1] \to (0,1)$, we consider the problem of simulating a coin with probability of heads $f(p)$ by tossing a coin with unknown heads probability $p$, as well as a fair coin, $N$ times each, where $N$ may be random. The work of Keane and O'Brien (1994) implies that such a simulation scheme with the probability $\P_p(N<\infty)$ equal to 1 exists iff $f$ is continuous. Nacu and Peres (2005) proved that $f$ is real analytic in an open set $S \subset (0,1)$ iff such a simulation scheme exists with the probability $\P_p(N>n)$ decaying exponentially in $n$ for every $p \in S$. We prove that for $\alpha>0$ non-integer, $f$ is in the space $C^\alpha [0,1]$ if and only if a simulation scheme as above exists with $\P_p(N>n) \le C (\Delta_n(p))^\alpha$, where \Delta_n(x)\eqbd \max \{\sqrt{x(1-x)/n},1/n \}. The key to the proof is a new result in approximation theory: Let \B_n be the cone of univariate polynomials with nonnegative Bernstein coefficients of degree $n$. We show that a function $f:[0,1] \to (0,1)$ is in $C^\alpha [0,1]$ if and only if $f$ has a series representation $\sum_{n=1}^\infty F_n$ with F_n \in \B_n and $\sum_{k>n} F_k(x) \le C(\Delta_n(x))^\alpha$ for all $x \in [0,1]$ and $n \ge 1$. We also provide a counterexample to a theorem stated without proof by Lorentz (1963), who claimed that if some \phi_n \in \B_n satisfy $|f(x)-\phi_n(x)| \le C (\Delta_n(x))^\alpha$ for all $x \in [0,1]$ and $n \ge 1$, then $f \in C^\alpha [0,1]$.Comment: 29 pages; final version; to appear in Constructive Approximatio

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

3D Structure of Microwave Sources from Solar Rotation Stereoscopy vs Magnetic Extrapolations

We use rotation stereoscopy to estimate the height of a steady-state solar feature relative to the photosphere, based on its apparent motion in the image plane recorded over several days of observation. The stereoscopy algorithm is adapted to work with either one- or two-dimensional data (i.e. from images or from observations that record the projected position of the source along an arbitrary axis). The accuracy of the algorithm is tested on simulated data, and then the algorithm is used to estimate the coronal radio source heights associated with the active region NOAA 10956, based on multifrequency imaging data over 7 days from the Siberian Solar Radio Telescope near 5.7 GHz, the Nobeyama Radio Heliograph at 17 GHz, as well as one-dimensional scans at multiple frequencies spanning the 5.98--15.95 GHz frequency range from the RATAN-600 instrument. The gyroresonance emission mechanism, which is sensitive to the coronal magnetic field strength, is applied to convert the estimated radio source heights at various frequencies, h(f), to information about magnetic field vs. height B(h), and the results are compared to a magnetic field extrapolation derived from photospheric magnetic field observations obtained by Hinode and MDI. We found that the gyroresonant emission comes from the heights exceeding location of the third gyrolayer irrespectively on the magnetic extrapolation method; implications of this finding for the coronal magnetography and coronal plasma physics are discussed.Comment: 26 pages, 13 figures, ApJ accepte

Additive Self Helicity as a Kink Mode Threshold

In this paper we propose that additive self helicity, introduced by Longcope and Malanushenko (2008), plays a role in the kink instability for complex equilibria, similar to twist helicity for thin flux tubes (Hood and Priest (1979), Berger and Field (1984)). We support this hypothesis by a calculation of additive self helicity of a twisted flux tube from the simulation of Fan and Gibson (2003). As more twist gets introduced, the additive self helicity increases, and the kink instability of the tube coincides with the drop of additive self helicity, after the latter reaches the value of $H_A/\Phi^2\approx 1.5$ (where $\Phi$ is the flux of the tube and $H_A$ is additive self helicity). We compare additive self helicity to twist for a thin sub-portion of the tube to illustrate that $H_A/\Phi^2$ is equal to the twist number, studied by Berger and Field (1984), when the thin flux tube approximation is applicable. We suggest, that the quantity $H_A/\Phi^2$ could be treated as a generalization of a twist number, when thin flux tube approximation is not applicable. A threshold on a generalized twist number might prove extremely useful studying complex equilibria, just as twist number itself has proven useful studying idealized thin flux tubes. We explicitly describe a numerical method for calculating additive self helicity, which includes an algorithm for identifying a domain occupied by a flux bundle and a method of calculating potential magnetic field confined to this domain. We also describe a numerical method to calculate twist of a thin flux tube, using a frame parallelly transported along the axis of the tube

Structure and Dynamics of the Sun's Open Magnetic Field

The solar magnetic field is the primary agent that drives solar activity and couples the Sun to the Heliosphere. Although the details of this coupling depend on the quantitative properties of the field, many important aspects of the corona - solar wind connection can be understood by considering only the general topological properties of those regions on the Sun where the field extends from the photosphere out to interplanetary space, the so-called open field regions that are usually observed as coronal holes. From the simple assumptions that underlie the standard quasi-steady corona-wind theoretical models, and that are likely to hold for the Sun, as well, we derive two conjectures on the possible structure and dynamics of coronal holes: (1) Coronal holes are unique in that every unipolar region on the photosphere can contain at most one coronal hole. (2) Coronal holes of nested polarity regions must themselves be nested. Magnetic reconnection plays the central role in enforcing these constraints on the field topology. From these conjectures we derive additional properties for the topology of open field regions, and propose several observational predictions for both the slowly varying and transient corona/solar wind.Comment: 26 pages, 6 figure

Identification of Specific Circular RNA Expression Patterns and MicroRNA Interaction Networks in Mesial Temporal Lobe Epilepsy

Circular RNAs (circRNAs) regulate mRNA translation by binding to microRNAs (miRNAs), and their expression is altered in diverse disorders, including cancer, cardiovascular disease, and Parkinson’s disease. Here, we compare circRNA expression patterns in the temporal cortex and hippocampus of patients with pharmacoresistant mesial temporal lobe epilepsy (MTLE) and healthy controls. Nine circRNAs showed significant differential expression, including circRNA-HOMER1, which is expressed in synapses. Further, we identified miRNA binding sites within the sequences of differentially expressed (DE) circRNAs; expression levels of mRNAs correlated with changes in complementary miRNAs. Gene set enrichment analysis of mRNA targets revealed functions in heterocyclic compound binding, regulation of transcription, and signal transduction, which maintain the structure and function of hippocampal neurons. The circRNA–miRNA–mRNA interaction networks illuminate the molecular changes in MTLE, which may be pathogenic or an effect of the disease or treatments and suggests that DE circRNAs and associated miRNAs may be novel therapeutic targets

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