Quasiconformal distortion of Riesz capacities and Hausdorff measures in the plane

In this paper we prove the sharp distortion estimates for the quasiconformal mappings in the plane, both in terms of the Riesz capacities from non linear potential theory and in terms of the Hausdorff measures.

Distortion theorems for quasiconformal mappings

L'any 1994, Astala publicà el reconegut teorema de distorió de l'àrea per aplicacions quasiconformes, un resultat innovador que va permetre que n'apareguessin nombrosos més dins d'aquest camp de l'anàlisi durant la darrera dècada. Ens centrem en les conseqüències que té en la distorsió de la mesura de Hausdorff. Seguim la demostració de Lacey, Sawyer i Uriarte-Tuero per la distorsió del contingut de Hausdorff, clarificant-ne alguns punts i canviant l'enfocament per l'acotació de la transformada de Beurling, on prenem les idees d'Astala, Clop, Tolsa, Uriarte-Tuero i Verdera.

Distortion control for delay-sensitive sources

We investigate the problem of finding minimum-distortion policies for streaming delay-sensitive but distortion-tolerant data. We consider cross-layer approaches which exploit the coupling between presentation and transport layers. We make the natural assumption that the distortion function is convex and decreasing. We focus on a single source-destination pair and analytically find the optimum transmission policy when the transmission is done over an error-free channel. This optimum policy turns out to be independent of the exact form of the convex and decreasing distortion function. Then, for a packet-erasure channel, we analytically find the optimum open-loop transmission policy, which is also independent of the form of the convex distortion function. We then find computationally efficient closed-loop heuristic policies and show, through numerical evaluation, that they outperform the open-loop policy and have near optimal performance.

The minimax distortion redundancy in empirical quantizer design

We obtain minimax lower and upper bounds for the expected distortionredundancy of empirically designed vector quantizers. We show that the meansquared distortion of a vector quantizer designed from $n$ i.i.d. datapoints using any design algorithm is at least $\Omega (n^{-1/2})$ awayfrom the optimal distortion for some distribution on a bounded subset of${\cal R}^d$. Together with existing upper bounds this result shows thatthe minimax distortion redundancy for empirical quantizer design, as afunction of the size of the training data, is asymptotically on the orderof $n^{1/2}$. We also derive a new upper bound for the performance of theempirically optimal quantizer.

Dimerization of polyacetylene treated as a spin-Peierls distortion of the Heisenberg Hamiltonian

Extracting a bond-length-dependent Heisenberg-like Hamiltonian from the potential-energy surfaces of the two lowest states of ethylene, it is possible to study the geometry of polyacetylene by minimization of the cohesive energy, using both variational-cluster and Rayleigh-Schrödinger perturbative expansions. The dimerization amplitude is satisfactorily reproduced. Optimizing the variational-cluster-expansion total energy with the equal-bond-length constraint, the barrier to reversal of alternation is obtained. The alternating-to-regular phase transition is treated from the Néel-state starting function and appears to be of second order.

Retrofit Methods for Distortion Cracking Problems in Plate Girder Bridges, TR-436, 2003

This report is formatted to independently present four individual investigations related to similar web gap fatigue problems. Multiple steel girder bridges commonly exhibit fatigue cracking due to out-of-plane displacement of the web near the diaphragm connections. This fatigue-prone web gap area is typically located in negative moment regions of the girders where the diaphragm stiffener is not attached to the top flange. In the past, the Iowa Department of Transportation has attempted to stop fatigue crack propagation in these steel girder bridges by drilling holes at the crack tips. Other nondestructive retrofits have been tried; in a particular case on a two-girder bridge with floor beams, angles were bolted between the stiffener and top flange. The bolted angle retrofit has failed in the past and may not be a viable solution for diaphragm bridges. The drilled hole retrofit is often only a temporary solution, so a more permanent and effective retrofit is required. A new field retrofit has been developed that involves loosening the bolts in the connection between the diaphragm and the girders. Research on the retrofit has been initiated; however, no long-term studies of the effects of bolt loosening have been performed. The intent of this research is to study the short-term effects of the bolt loosening retrofit on I-beam and channel diaphragm bridges. The research also addressed the development of a continuous remote monitoring system to investigate the bolt loosening retrofit on an X-type diaphragm bridge over a number of months, ensuring that the measured strain and displacement reductions are not affected by time and continuous traffic loading on the bridge. The testing for the first three investigations is based on instrumentation of web gaps in a negative moment region on Iowa Department of Transportation bridges with I-beam, channel, and X-type diaphragms. One bridge of each type was instrumented with strain gages and deflection transducers. Field tests, using loaded trucks of known weight and configuration, were conducted on the bridges with the bolts in the tight condition and after implementing the bolt loosening retrofit to measure the effects of loosening the diaphragm bolts. Long-term data were also collected on the X-diaphragm bridge by a data acquisition system that collected the data continuously under ambient truck loading. The collected data were retrievable by an off-site modem connection to the remote data acquisition system. The data collection features and ruggedness of this system for remote bridge monitoring make it viable as a pilot system for future monitoring projects in Iowa. Results indicate that loosening the diaphragm bolts reduces strain and out-of-plane displacement in the web gap, and that the reduction is not affected over time by traffic or environmental loading on the bridge. Reducing the strain in the web gap allows the bridge to support more cycles of loading before experiencing fatigue, thus increase the service life of the bridge. Two-girder floor beam bridges may also exhibit fatigue cracking in girder webs.

Beyond Value-at-Risk : GlueVaR Distortion Risk Measures

We propose a new family of risk measures, called GlueVaR, within the class of distortion risk measures. Analytical closed-form expressions are shown for the most frequently used distribution functions in financial and insurance applications. The relationship between Glue-VaR, Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR) is explained. Tail-subadditivity is investigated and it is shown that some GlueVaR risk measures satisfy this property. An interpretation in terms of risk attitudes is provided and a discussion is given on the applicability in non-financial problems such as health, safety, environmental or catastrophic risk management

The connection between distortion risk measures and ordered weighted averaging operators

Distortion risk measures summarize the risk of a loss distribution by means of a single value. In fuzzy systems, the Ordered Weighted Averaging (OWA) and Weighted Ordered Weighted Averaging (WOWA) operators are used to aggregate a large number of fuzzy rules into a single value. We show that these concepts can be derived from the Choquet integral, and then the mathematical relationship between distortion risk measures and the OWA and WOWA operators for discrete and finite random variables is presented. This connection offers a new interpretation of distortion risk measures and, in particular, Value-at-Risk and Tail Value-at-Risk can be understood from an aggregation operator perspective. The theoretical results are illustrated in an example and the degree of orness concept is discussed.