Global Saturation of Regularization Methods for Inverse Ill-Posed Problems

In this article the concept of saturation of an arbitrary regularization method is formalized based upon the original idea of saturation for spectral regularization methods introduced by A. Neubauer in 1994. Necessary and sufficient conditions for a regularization method to have global saturation are provided. It is shown that for a method to have global saturation the total error must be optimal in two senses, namely as optimal order of convergence over a certain set which at the same time, must be optimal (in a very precise sense) with respect to the error. Finally, two converse results are proved and the theory is applied to find sufficient conditions which ensure the existence of global saturation for spectral methods with classical qualification of finite positive order and for methods with maximal qualification. Finally, several examples of regularization methods possessing global saturation are shown.Comment: 29 page

Properties of mass-loading shocks, 2. Magnetohydrodynamics

The one-dimensional magnetohydrodynamics of shocked flows subjected to significant mass loading are considered. Recent observations at comets Giacobini-Zinner and Halley suggest that simple nonreacting MHD is an inappropriate description for active cometary bow shocks. The thickness of the observed cometary shock implies that mass loading represents an important dynamical process within the shock itself, thereby requiring that the Rankine-Hugoniot condition for the mass flux possess a source term. In a formal sense, this renders mass-loading shocks qualitatively similar to combustion shocks, except that mass loading induces the shocked flow to shear. Nevertheless, a large class of stable shocks exist, identified by means of the Lax conditions appropriate to MHD. Thus mass-loading shocks represent a new and interesting class of shocks, which, although found frequently in the solar system, both at the head of comets and, under suitable conditions, upsteam of weakly magnetized and nonmagnetized planets, has not been discussed in any detail. Owing to the shearing of the flow, mass-loading shocks can behave like switch-on shocks regardless of the magnitude of the plasma beta. Thus the behavior of the magnetic field in mass-loading shocks is significantly different from that occurring in nonreacting classical MHD shocks. It is demonstrated that there exist two types of mass-loading fronts for which no classical MHD analogue exists, these being the fast and slow compound mass-loading shocks. These shocks are characterized by an initial deceleration of the fluid flow to either the fast or the slow magnetosonic speed followed by an isentropic expansion to the final decelerated downstream state. Thus these transitions take the flow from a supersonic to a supersonic, although decelerated, downstream state, unlike shocks which occur in classical MHD or gasdynamics. It is possible that such structures have been observed during the Giotto-Halley encounter, and a brief discussion of the appropriate Halley parameters is therefore given, together with a short discussion of the determination of the shock normal from observations. A further interesting new form of mass-loading shock is the “slow-intermediate” shock, a stable shock which possesses many of the properties of intermediate MHD shocks yet which propagates like a slow mode MHD shock. An important property of mass-loading shocks is the large parameter regime (compared with classical MHD) which does not admit simple or stable transitions from a given upstream to a downstream state. This suggests that it is often necessary to construct compound structures consisting of shocks, slip waves, rarefactions, and fast and slow compound waves in order to connect given upstream and downstream states. Thus the Riemann problem is significantly different from that of classical MHD

Mass-loading and parallel magnetized shocks

Recent observations at comets Giacobini-Zinner and Halley suggest that simple non-reacting gas dynamics or MHD is an inappropriate description for the bow shock. The thickness of the observed (sub)shock implies that mass-loading is an important dynamical process within the shock itself, thereby requiring that the Rankine-Hugoniot conditions possess source terms. This leads to shocks with properties similar to those of combustion shocks. We consider parallel magnetized shocks subjected to mass-loading, describe some properties which distinguish them from classical MHD parallel shocks, and establish the existence of a new kind of MHD compound shock. These results will be of importance both to observations and numerical simulations of the comet-solar wind interaction

Generalized Qualification and Qualification Levels for Spectral Regularization Methods

The concept of qualification for spectral regularization methods for inverse ill-posed problems is strongly associated to the optimal order of convergence of the regularization error. In this article, the definition of qualification is extended and three different levels are introduced: weak, strong and optimal. It is shown that the weak qualification extends the definition introduced by Mathe and Pereverzev in 2003, mainly in the sense that the functions associated to orders of convergence and source sets need not be the same. It is shown that certain methods possessing infinite classical qualification, e.g. truncated singular value decomposition (TSVD), Landweber's method and Showalter's method, also have generalized qualification leading to an optimal order of convergence of the regularization error. Sufficient conditions for a SRM to have weak qualification are provided and necessary and sufficient conditions for a given order of convergence to be strong or optimal qualification are found. Examples of all three qualification levels are provided and the relationships between them as well as with the classical concept of qualification and the qualification introduced by Mathe and Perevezev are shown. In particular, spectral regularization methods having extended qualification in each one of the three levels and having zero or infinite classical qualification are presented. Finally several implications of this theory in the context of orders of convergence, converse results and maximal source sets for inverse ill-posed problems, are shown.Comment: 20 pages, 1 figur

Australopithecus afarensis endocasts suggest ape-like brain organization and prolonged brain growth

Human brains are three times larger, are organized differently, and mature for a longer period of time than those of our closest living relatives, the chimpanzees. Together, these characteristics are important for human cognition and social behavior, but their evolutionary origins remain unclear. To study brain growth and organization in the hominin species Australopithecus afarensis more than 3 million years ago, we scanned eight fossil crania using conventional and synchrotron computed tomography. We inferred key features of brain organization from endocranial imprints and explored the pattern of brain growth by combining new endocranial volume estimates with narrow age at death estimates for two infants. Contrary to previous claims, sulcal imprints reveal an ape-like brain organization and no features derived toward humans. A comparison of infant to adult endocranial volumes indicates protracted brain growth in A. afarensis, likely critical for the evolution of a long period of childhood learning in hominins

Real-TIme and Networked Displays: Data Streams in Art Applications

In a world that has become vastly more complicated, multi-layered information systems have an increasingly compelling impact. Access to complex, interactive data is necessary to informed thought and global action. Utilizing computer modeling, aesthetically-driven prototypes for displaying numerical data provide fresh viewpoints, promoting deeper awareness of living institutions and developing trends. Visually compelling information lends clarity to the grand cycles of nature and human creativity, while revealing intimate perspectives on daily life. This versatile project emphasizes artistically sophisticated live data streams. Evolving artworks allow for a multitude of display options; images can be projected onto architectural or sculptural forms, displayed in buildings, terminals, and offices, and accessed via PDA\u27s and the web. Artists, scientists, and other experts will research data to be utilized in the creation of responsive visualizations. The displays seek to provide the technological infrastructure to 1) sustain the ability for creation of evolving information forms as modes of presentation advance 2) allow for live visualization of dynamic data streams relevant on local, regional and global levels, and 3) open up the possibility for imaginative uses of data to experts pursuing different methodologies. A robust, scalable framework for adapting data visualizations to various platforms will be implemented through utilization of open, standards-based software libraries. A central server will aggregate public syndication feeds of data related to pertinent subject areas. Through a web syndication protocol, streaming information will be used as the source of data for display. Server resources open to the public as well as research communities can aid in the study of current issues through artistically and contextually accurate visualization. The project will help detect long-term patterns in the global environment! , enhance sensitivity to the quietly functioning aspects of our surroundings, and open expanded avenues for collaboration between the arts and sciences

Simulator study of the low-speed handling qualities of a supersonic cruise arrow-wing transport configuration during approach and landing

A fixed-based simulator study was conducted to determine the low-speed flight characteristics of an advanced supersonic cruise transport having an arrow wing, a horizontal tail, and four dry turbojets with variable geometry turbines. The primary piloting task was the approach and landing. The statically unstable (longitudinally) subject configuration has unacceptable low-speed handling qualities with no augmentation. Therefore, a hardened stability augmentation system is required to achieve acceptable handling qualities, should the normal operational stability and control augmentation system fail. In order to achieve satisfactory handling qualities, considerable augmentation was required

Concentric left ventricular remodeling and aortic stiffness: A comparison of obesity and hypertension

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