Uniform erosion rates and relief amplitude during glacial cycles in the Southern Alps of New Zealand, as revealed from OSL-thermochronology

Related topics Homogenization methods Applications of homogenization methods Bibliography Index.

Homogenization: Methods and Applications

In this review article, we present in some detail new trends in application of asymptotic techniques to me­ chanical problems. First we consider the various methods which allows for the possibility of extending the perturbation series application space and hence omiting their local character. While applying the asymptotic methods very often the following situation appears: an existence of the asymptotics £ —» 0 implies an exis­ tence of the asymptotics e —> °° (or, in a more general sense, e —> a and e —> b). Therefore, an idea of constructing a single solution valid for a whole interval of parameter e changes is very attractive. In other words, we discuss a problem of asymptotically equivalent function constructions possessing for e —> a and E —> b a known asymptotic behavior. The defined problems are very important from the point of view of both theoretical and applied sciences. In this work, we review the state-of-the-art, by presenting the existing methods and by pointing out their advantages and disadvantages, as well as the fields of their applications. In addition, some new methods are also proposed. The methods are demonstrated on a wide variety of static and dynamic solid mechanics problems and some others involving fluid mechanics. This review article contains 340 references.

New Trends in Asymptotic Approaches: Summation and Interpolation Methods

Let M be a compact manifold and D a Dirac type differential operator on M. Let A be a C ∗ -algebra. Given a bundle W (with connection) of A-modules over M , the operator D can be twisted with this bundle. One can then use a trace on A to define numerical indices of this twisted operator. We prove an explicit formula for these indices. Our result does complement the Mishchenko-Fomenko index theorem valid in the same situation. We establish generalizations of these explicit index formulas if the trace is only defined on a dense and holomorphically closed subalgebra B. As a corollary, we prove a generalized Atiyah L 2 -index theorem if the twist- ing bundle is flat. There are actually many different ways to define these numerical indices. From their construction, it is not clear at all that they coincide. A substantial part of the paper is a complete proof of their equality. In particular, we estab- lish the (well-known but not well-documented) equality of Atiyah's definition of the L 2 -index with a K-theoretic definition. In case A is a von Neumann algebra of type 2, we put special emphasis on the calculation and interpretation of the center valued index. This com- pletely contains all the K-theoretic information about the index of the twisted operator. Some of our calculations are done in the framework of bivariant KK-theory.

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L 2 -index theorems, KK-theory, and connections

We present a new method for the assessment of the most recent cooling and denudation rates using paramagnetic centers in quartz measured by electron spin resonance (ESR) spectroscopy. These centers have a relatively low thermal stability. For cooling rates of 40 o and 1000oC Myr -, effective closure temperatures vary between 55 o and 82oC (Ti center) and 49 o and 64 oC (A1 center), respectively. Samples were collected from two cores that were drilled into the Eldzhurtinskiy Granite, which has an emplacement age of ~2 Ma as measured by U/Pb analyses of zircons. One 1500 rn core was taken from a drill hole into the dome of the granite, a second core of 4000 rn from a drill hole at the base of the Baksan Valley. Our results yield cooling rates of between 160 and 250oC Myr - for the upper core and between 570 o and 600oC Myr - for the lower core; the corresponding denudation rates are ~2.5 (upper core) and 5.5mm a - (lower core). The shape of the temperature profile of the lower core indicates recent erosion. When fitting the temperature data with a two- dimensional heat-transfer model, we obtain a net denudation rate of -10 mm a - and cooling rates in the range of 500oC Myr -, thus confirming the cooling rates estimated by ESR. However, the ESR denudation rates underestimate the erosion rate of the Baksan Valley because the geothermal gradient is not equilibrated between the surface and the depth of the annihilation temperatures, 950 and 1800 rn for the A1 and Ti centers, respectively. We conclude that ESR measurements of paramagnetic centers in quartz will allow the recon- struction of landscape dynamics for the past 10-1000 kyr and that in conjunction with U/Pb, fission track, and Ar/Ar analyses it will be possible to develop dynamic models for Quaternary tectonic movements.

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A new method for the estimation of cooling and denudation rates using paramagnetic centers in quartz: A case study on the Eldzhurtinskiy Granite, Caucasus

$C^*$-algebras and $K$-theory Index theorems The higher signatures Noncommutative differential geometry Bibliography Index.

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$C^*$-Algebras and Elliptic Operators in Differential Topology

Asymptotic expansions, employed in quantum physics as series of perturbation theory, appear as a result of the representation of functional integrals by power series with respect to coupling constant. To derive these series one has to change the order of functional integration and infinite summation. In general, this procedure is incorrect and is responsible for the divergence of the asymptotic expansions. In the present work, we suggest a method of construction of a new perturbation theory. In the framework of this perturbation theory, a convergent series corresponds to any physical quantity represented by a functional integral. The relations between the coefficients of these series and those of the asymptotic expansions are established.

New perturbation theory for quantum field theory: convergent series instead of asymptotic expansions

Determination of the mineral age based on the creation of radiation structure defects depends on the thermal stability of these defects. Accumulation of radiation defects in solids in cooling systems taking into account their time dependent first-order as well as second-order annihilation is analyzed theoretically. Second-order kinetics in comparison with first-order leads to increased duration of a transitional period and to a lower accumulation rate. The contribution of the transitional period is determined by the cooling rate of the geological system and by the production rate of structure defects. Resulting formulas are applied for dating rock-forming quartz from the Eldzhurtinsky granite massif (Great Caucasus, Russia) with the help of electron paramagnetic resonance. It is shown that the EPR age calculated with the formulas derived in this study in comparison with the results of the conventional additive dose technique is closer to the results of 40Ar/39Ar dating. Due to annihilation of radiation defects even at low temperature the EPR dating method can be applied to a limited time range.

Accumulation of structural radiation defects in quartz in cooling systems: basis for dating

In this letter, the method of the approximative calculation of functional integrals with any given accuracy1,2 is developed. Here we use the independence of the integrals on an auxiliary regularization parameter to simplify the calculations. Also we propose a modification of the method that proves to be more convenient for calculations with large values of coupling constants.

Perturbation Theory with Convergent Series for Arbitrary Values of Coupling Constant

We present a new approach to perturbation theory for quantum field theory based on convergent series instead of asymptotic expansions. This approach could be considered as the next step after traditional perturbation theory calculations, which allows more comprehensive use of previously obtained information in finding numerical values with greater accuracy.

Yu. P. Solovyov

Papers

$C^*$-Algebras and Elliptic Operators in Differential Topology

New perturbation theory for quantum field theory: convergent series instead of asymptotic expansions

Accumulation of structural radiation defects in quartz in cooling systems: basis for dating

Perturbation Theory with Convergent Series for Arbitrary Values of Coupling Constant

New Perturbation Theory for Quantum Field Theory: Convergent Series Instead of Asymptotic Expansions