Bounded variation and the strength of Helly's selection theorem

We analyze the strength of Helly's selection theorem HST, which is the most important compactness theorem on the space of functions of bounded variation. For this we utilize a new representation of this space intermediate between $L_1$ and the Sobolev space W1,1, compatible with the, so called, weak* topology. We obtain that HST is instance-wise equivalent to the Bolzano-Weierstra\ss\ principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice

Some models of crack growth in brittle materials

This work is devoted to the study of models of fractures growth in brittle elastic materials; it collects the results obtained during my Ph.D., that are contained in [77, 76, 78]. We consider quasi-static rate-independent models, as well as rate-dependent ones and the case in which the first ones are limits of the second ones when certain physical parameters vanish. The term quasistatic means that, at each instant, the system is assumed to be in equilibrium with respect to its time-dependent data; this setting is typical of systems whose internal time scale is much smaller than that of the loadings. By rate-independent system we mean that, if the time-dependent data are rescaled by a strictly monotone increasing function, then the system reacts by rescaling the solutions in the same manner

Wave front tracking in systems of conservation laws

summary:This paper contains several recent results about nonlinear systems of hyperbolic conservation laws obtained through the technique of Wave Front Tracking

Glosarium Matematika

273 p.; 24 cm

Unreliable and resource-constrained decoding

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student submitted PDF version of thesis.Includes bibliographical references (p. 185-213).Traditional information theory and communication theory assume that decoders are noiseless and operate without transient or permanent faults. Decoders are also traditionally assumed to be unconstrained in physical resources like material, memory, and energy. This thesis studies how constraining reliability and resources in the decoder limits the performance of communication systems. Five communication problems are investigated. Broadly speaking these are communication using decoders that are wiring cost-limited, that are memory-limited, that are noisy, that fail catastrophically, and that simultaneously harvest information and energy. For each of these problems, fundamental trade-offs between communication system performance and reliability or resource consumption are established. For decoding repetition codes using consensus decoding circuits, the optimal tradeoff between decoding speed and quadratic wiring cost is defined and established. Designing optimal circuits is shown to be NP-complete, but is carried out for small circuit size. The natural relaxation to the integer circuit design problem is shown to be a reverse convex program. Random circuit topologies are also investigated. Uncoded transmission is investigated when a population of heterogeneous sources must be categorized due to decoder memory constraints. Quantizers that are optimal for mean Bayes risk error, a novel fidelity criterion, are designed. Human decision making in segregated populations is also studied with this framework. The ratio between the costs of false alarms and missed detections is also shown to fundamentally affect the essential nature of discrimination. The effect of noise on iterative message-passing decoders for low-density parity check (LDPC) codes is studied. Concentration of decoding performance around its average is shown to hold. Density evolution equations for noisy decoders are derived. Decoding thresholds degrade smoothly as decoder noise increases, and in certain cases, arbitrarily small final error probability is achievable despite decoder noisiness. Precise information storage capacity results for reliable memory systems constructed from unreliable components are also provided. Limits to communicating over systems that fail at random times are established. Communication with arbitrarily small probability of error is not possible, but schemes that optimize transmission volume communicated at fixed maximum message error probabilities are determined. System state feedback is shown not to improve performance. For optimal communication with decoders that simultaneously harvest information and energy, a coding theorem that establishes the fundamental trade-off between the rates at which energy and reliable information can be transmitted over a single line is proven. The capacity-power function is computed for several channels; it is non-increasing and concave.by Lav R. Varshney.Ph.D

2012-2013, University of Memphis bulletin

University of Memphis bulletin containing the graduate catalog for 2012-2013.https://digitalcommons.memphis.edu/speccoll-ua-pub-bulletins/1432/thumbnail.jp

2014-2015, University of Memphis bulletin

University of Memphis bulletin containing the graduate catalog for 2014-2015.https://digitalcommons.memphis.edu/speccoll-ua-pub-bulletins/1433/thumbnail.jp