Advances in sequential data assimilation and numerical weather forecasting: an Ensemble Transform Kalman-Bucy Filter, a study on clustering in deterministic Ensemble Square Root Filters, and a test of a new time stepping scheme in an atmospheric model

This dissertation deals with aspects of sequential data assimilation (in particular ensemble Kalman filtering) and numerical weather forecasting. In the first part, the recently formulated Ensemble Kalman-Bucy (EnKBF) filter is revisited. It is shown that the previously used numerical integration scheme fails when the magnitude of the background error covariance grows beyond that of the observational error covariance in the forecast window. Therefore, we present a suitable integration scheme that handles the stiffening of the differential equations involved and doesn’t represent further computational expense. Moreover, a transform-based alternative to the EnKBF is developed: under this scheme, the operations are performed in the ensemble space instead of in the state space. Advantages of this formulation are explained. For the first time, the EnKBF is implemented in an atmospheric model. The second part of this work deals with ensemble clustering, a phenomenon that arises when performing data assimilation using of deterministic ensemble square root filters in highly nonlinear forecast models. Namely, an M-member ensemble detaches into an outlier and a cluster of M-1 members. Previous works may suggest that this issue represents a failure of EnSRFs; this work dispels that notion. It is shown that ensemble clustering can be reverted also due to nonlinear processes, in particular the alternation between nonlinear expansion and compression of the ensemble for different regions of the attractor. Some EnSRFs that use random rotations have been developed to overcome this issue; these formulations are analyzed and their advantages and disadvantages with respect to common EnSRFs are discussed. The third and last part contains the implementation of the Robert-Asselin-Williams (RAW) filter in an atmospheric model. The RAW filter is an improvement to the widely popular Robert-Asselin filter that successfully suppresses spurious computational waves while avoiding any distortion in the mean value of the function. Using statistical significance tests both at the local and field level, it is shown that the climatology of the SPEEDY model is not modified by the changed time stepping scheme; hence, no retuning of the parameterizations is required. It is found the accuracy of the medium-term forecasts is increased by using the RAW filter.

Positive solutions of nonlinear problems involving the square root of the Laplacian

We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type.

The Brezis-Nirenberg type problem involving the square root of the Laplacian

We establish existence and non-existence results to the Brezis-Nirenberg type problem involving the square root of the Laplacian in a bounded domain with zero Dirichlet boundary condition.

Square Root Finding In Graphs

Abstract: Root and root finding are concepts familiar to most branches of mathematics. In graph theory, H is a square root of G and G is the square of H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Graph square is a basic operation with a number of results about its properties in the literature. We study the characterization and recognition problems of graph powers. There are algorithmic and computational approaches to answer the decision problem of whether a given graph is a certain power of any graph. There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four. The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five. We settle the conjecture that recognition of square of graphs with girth 5 is NP-complete. This result is providing the complete dichotomy theorem for square root finding problem.

Hemodynamic Performance of Edwards Intuity Valve in a Compliant Aortic Root Model

Numerous designs of bioprosthetic valves exist. The sutureless surgical valve is a newer design concept which combines elements of the transcatheter valve technology with surgical valves. This design aims at shorter and easier implantation. It was the aim of this study to perform hemodynamic and kinematic measurements for this type of valves to serve as a baseline for following studies which investigate the effect of the aortic root on the valve performance. To this end, the Edwards Intuity aortic valve was investigated in a new in vitro flow loop mimicking the left heart. The valve was implanted in a transparent, compliant aortic root model, and the valve kinematics was investigated using a high speed camera together with synchronized hemodynamic measurements of pressures and flows. The valve closure was asynchronous (one by one leaflet), and the valve started to close before the deceleration of the fluid. The aortic root model showed a dilation of the sinuses which was different to the ascending aorta, and the annulus was found to move towards the left ventricle during diastole and towards the aorta during systole.

SemQuaRE - An extension of the SQuaRE quality model for the evaluation of semantic technologies

To correctly evaluate semantic technologies and to obtain results that can be easily integrated, we need to put evaluations under the scope of a unique software quality model. This paper presents SemQuaRE, a quality model for semantic technologies. SemQuaRE is based on the SQuaRE standard and describes a set of quality characteristics specific to semantic technologies and the quality measures that can be used for their measurement. It also provides detailed formulas for the calculation of such measures. The paper shows that SemQuaRE is complete with respect to current evaluation trends and that it has been successfully applied in practice.

Continued-product algorithms for division, logarithm, exponential and square root in radix ten /

Typescript.

Table of quarter-squares of all whole numbers from 1 to 200000, for simplifying multiplication, squaring, and extraction of the square-root, and to render the results of these operations more certain ...

Mode of access: Internet.

Tables of [square root sign] 1 - r² and 1 - r² for use in partial correlation and in trigonometry,

Photo-lithoprint reproduction. Ann Arbor, Mich., Edwards Bros., 1947.

Mathematics for the practical engineer. Engineers' study course from Power; a mathematical manual for the power plant engineer. Contains simplified lessons for home-study in fractions, ratio and proportion, percentage, involution and evolution, square root, cube root, formulas, logarithms, mensuration and plane trigonometry,

Mode of access: Internet.

Ein Beitrag zur Mannigfaltigkeitslehre. Mannigfaltigkeiten deren Linienelemente auf die Form ds = f([square root symbol][Sigma]x²k̳)[square root symbol][Sigma]dx²k̳ gebracht werden können.

In formula in title, k̳ is subscript directly beneath superscript 2.