Optimising the degree of data smoothing for locally adaptive finite element bivariate smoothing splines

Finite difference and finite element schemes for bivariate thin plate smoothing splines are described. Nested grid SOR iterative methods are known to be able to solve these systems efficiently for large data sets. An iterative Newton procedure for optimising the smoothing parameter to achieve a prescribed residual sum of squares from the data is obtained. It can be added to the SOR iteration with little additional computational cost and is demonstrated on test data to work for a wide range of smoothing parameters. An apparently more accurate version of this procedure, which requires more memory, converges slightly less quickly than the simpler approximation. The simpler method appears to be directly compatible with the SOR iterative method. The Newton method is shown to also work for locally adaptive versions of finite difference smoothing splines. The roughness penalty can be made locally adaptive to respect process-based constraints, such as minimum profile curvature, which depends on the local aspect of the fitted surface. This can be applied to the interpolation of digital elevation models. The weighted residual sum of squares can be made locally adaptive to allow for positional error in data, whether arising from actual data error, or from a finite difference discretisation. This has given rise to an objective method for optimising the grid resolution to the information content of the data

Optimal query complexity for estimating the trace of a matrix

Given an implicit $n\times n$ matrix $A$ with oracle access $x^TA x$ for any $x\in \mathbb{R}^n$, we study the query complexity of randomized algorithms for estimating the trace of the matrix. This problem has many applications in quantum physics, machine learning, and pattern matching. Two metrics are commonly used for evaluating the estimators: i) variance; ii) a high probability multiplicative-approximation guarantee. Almost all the known estimators are of the form $\frac{1}{k}\sum_{i=1}^k x_i^T A x_i$ for $x_i\in \mathbb{R}^n$ being i.i.d. for some special distribution. Our main results are summarized as follows. We give an exact characterization of the minimum variance unbiased estimator in the broad class of linear nonadaptive estimators (which subsumes all the existing known estimators). We also consider the query complexity lower bounds for any (possibly nonlinear and adaptive) estimators: (1) We show that any estimator requires $\Omega(1/\epsilon)$ queries to have a guarantee of variance at most $\epsilon$. (2) We show that any estimator requires $\Omega(\frac{1}{\epsilon^2}\log \frac{1}{\delta})$ queries to achieve a $(1\pm\epsilon)$-multiplicative approximation guarantee with probability at least $1 - \delta$. Both above lower bounds are asymptotically tight. As a corollary, we also resolve a conjecture in the seminal work of Avron and Toledo (Journal of the ACM 2011) regarding the sample complexity of the Gaussian Estimator.Comment: full version of the paper in ICALP 201

Undecidable properties of self-affine sets and multi-tape automata

We study the decidability of the topological properties of some objects coming from fractal geometry. We prove that having empty interior is undecidable for the sets defined by two-dimensional graph-directed iterated function systems. These results are obtained by studying a particular class of self-affine sets associated with multi-tape automata. We first establish the undecidability of some language-theoretical properties of such automata, which then translate into undecidability results about their associated self-affine sets.Comment: 10 pages, v2 includes some corrections to match the published versio

Testing Logselfsimilarity of Soil Particle Size Distribution: Simulation with Minimum Inputs

Particle size distribution (PSD) greatly influences other soil physical properties. A detailed textural analysis is time-consuming and expensive. Soil texture is commonly reported in terms of mass percentages of a small number of size fractions (typically, clay, silt and sand). A method to simulate the PSD from such a poor description or even from the poorest description, consisting in the mass percentages of only two soil size fractions, would be extremly useful for prediction purposes. The goal of this paper is to simulate soil PSDs from the minimum number of inputs, i.e., two and three textural fraction contents, by using a logselfsimilar model and an iterated function system constructed with these data. High quality data on 171 soils are used. Additionally, the characterization of soil texture by entropy-based parameters provided by the model is tested. Results indicate that the logselfsimilar model may be a useful tool to simulate PSD for the construction of pedotransfer functions related to other soil properties when textural information is limited to moderate textural data

Geographical and climatic limits of needle types of one- and two-needled pinyon pines

Aim The geographical extent and climatic tolerances of one- and two-needled pinyon pines (Pinus subsect. Cembroides) are the focus of questions in taxonomy, palaeoclimatology and modelling of future distributions. The identification of these pines, traditionally classified by one- versus two-needled fascicles, is complicated by populations with both one- and two-needled fascicles on the same tree, and the description of two more recently described one-needled varieties: the fallax-type and californiarum-type. Because previous studies have suggested correlations between needle anatomy and climate, including anatomical plasticity reflecting annual precipitation, we approached this study at the level of the anatomy of individual pine needles rather than species. Location Western North America. Methods We synthesized available and new data from field and herbarium collections of needles to compile maps of their current distributions across western North America. Annual frequencies of needle types were compared with local precipitation histories for some stands. Historical North American climates were modeled on a c. 1-km grid using monthly temperature and precipitation values. A geospatial model (ClimLim), which analyses the effect of climate modulated physiological and ecosystem processes, was used to rank the importance of seasonal climate variables in limiting the distributions of anatomical needle types. Results The pinyon needles were classified into four distinct types based upon the number of needles per fascicle, needle thickness and the number of stomatal rows and resin canals. The individual needles fit well into four categories of needle types, whereas some trees exhibit a mixture of two needle types. Trees from central Arizona containing a mixture of Pinus edulis and fallax-type needles increased their percentage of fallax-type needles following dry years. All four needle types occupy broader geographical regions with distinctive precipitation regimes. Pinus monophylla and californiarum-type needles occur in regions with high winter precipitation. Pinus edulis and fallax-type needles are found in regions with high monsoon precipitation. Areas supporting californiarum-type and fallax-type needle distributions are additionally characterized by a more extreme May–June drought. Main conclusions These pinyon needle types seem to reflect the amount and seasonality of precipitation. The single needle fascicle characterizing the fallax type may be an adaptation to early summer or periodic drought, while the single needle of Pinus monophylla may be an adaptation to summer–autumn drought. Although the needles fit into four distinct categories, the parent trees are sometimes less easily classified, especially near their ancestral Pleistocene ranges in the Mojave and northern Sonoran deserts. The abundance of trees with both one- and two-needled fascicles in the zones between P. monophylla, P. edulis and fallax-type populations suggest that needle fascicle number is an unreliable characteristic for species classification. Disregarding needle fascicle number, the fallax-type needles are nearly identical to P. edulis, supporting Little’s (1968) initial classification of these trees as P. edulis var. fallax, while the californiarum-type needles have a distinctive morphology supporting Bailey’s (1987) classification of this tree as Pinus californiarum

Dystonia management: what to expect from the future? The perspectives of patients and clinicians within DystoniaNet Europe

Improved care for people with dystonia presents a number of challenges. Major gaps in knowledge exist with regard to how to optimize the diagnostic process, how to leverage discoveries in pathophysiology into biomarkers, and how to develop an evidence base for current and novel treatments. These challenges are made greater by the realization of the wide spectrum of symptoms and difficulties faced by people with dystonia, which go well-beyond motor symptoms. A network of clinicians, scientists, and patients could provide resources to facilitate information exchange at different levels, share mutual experiences, and support each other's innovative projects. In the past, collaborative initiatives have been launched, including the American Dystonia Coalition, the European Cooperation in Science and Technology (COST-which however only existed for a limited time), and the Dutch DystonieNet project. The European Reference Network on Rare Neurological Diseases includes dystonia among other rare conditions affecting the central nervous system in a dedicated stream. Currently, we aim to broaden the scope of these initiatives to a comprehensive European level by further expanding the DystoniaNet network, in close collaboration with the ERN-RND. In line with the ERN-RND, the mission of DystoniaNet Europe is to improve care and quality of life for people with dystonia by, among other endeavors, facilitating access to specialized care, overcoming the disparity in education of medical professionals, and serving as a solid platform to foster international clinical and research collaborations. In this review, both professionals within the dystonia field and patients and caregivers representing Dystonia Europe highlight important unsolved issues and promising new strategies and the role that a European network can play in activating them.Neurological Motor Disorder

Non-neoclassical up/down asymmetry of impurity emission on Alcator C-Mod

We demonstrate that existing theories are insufficient to explain up/down asymmetries of argon x-ray emission in Alcator C-Mod ohmic plasmas. Instead of the poloidal variation, ñ[subscript z]/〈n[subscript z]〉, being of order the inverse aspect ratio, ϵ, and scaling linearly with B[subscript t][superscript _ over n][subscript e]/I[2 over p], it is observed over 0.8 < r/a < 1.0 to be of order unity and exhibits a threshold behaviour between 3.5 <B[subscript t][superscript _ over n][subscript e]/I[subscript p] < 4.0 (T10[superscript 20] m[superscript −3] MA[superscript −1]). The transition from a poloidally symmetric to asymmetric impurity distribution is shown to occur at densities just below those that trigger a reversal of the core toroidal rotation direction, thought to be linked to the transition between the linear and saturated ohmic confinement regimes. A possible drive is discussed by which anomalous radial transport might sustain the impurity density asymmetry as the ratio of the perpendicular to parallel equilibration times, τ[subscript ⊥,z]/τ[subscript ∥,z], approaches unity. This explanation requires a strong up/down asymmetry in radial flux which, while not observable on C-Mod, has been measured in TEXT and Tore Supra ohmic plasmas.United States. Dept. of Energy (Contract DE-FC02-99ER54512)United States. Dept. of Energy (Fusion Research Postdoctoral Research Program