976 resultados para Filtering techniques
Resumo:
The study of codes, classically motivated by the need to communicate information reliably in the presence of error, has found new life in fields as diverse as network communication, distributed storage of data, and even has connections to the design of linear measurements used in compressive sensing. But in all contexts, a code typically involves exploiting the algebraic or geometric structure underlying an application. In this thesis, we examine several problems in coding theory, and try to gain some insight into the algebraic structure behind them.
The first is the study of the entropy region - the space of all possible vectors of joint entropies which can arise from a set of discrete random variables. Understanding this region is essentially the key to optimizing network codes for a given network. To this end, we employ a group-theoretic method of constructing random variables producing so-called "group-characterizable" entropy vectors, which are capable of approximating any point in the entropy region. We show how small groups can be used to produce entropy vectors which violate the Ingleton inequality, a fundamental bound on entropy vectors arising from the random variables involved in linear network codes. We discuss the suitability of these groups to design codes for networks which could potentially outperform linear coding.
The second topic we discuss is the design of frames with low coherence, closely related to finding spherical codes in which the codewords are unit vectors spaced out around the unit sphere so as to minimize the magnitudes of their mutual inner products. We show how to build frames by selecting a cleverly chosen set of representations of a finite group to produce a "group code" as described by Slepian decades ago. We go on to reinterpret our method as selecting a subset of rows of a group Fourier matrix, allowing us to study and bound our frames' coherences using character theory. We discuss the usefulness of our frames in sparse signal recovery using linear measurements.
The final problem we investigate is that of coding with constraints, most recently motivated by the demand for ways to encode large amounts of data using error-correcting codes so that any small loss can be recovered from a small set of surviving data. Most often, this involves using a systematic linear error-correcting code in which each parity symbol is constrained to be a function of some subset of the message symbols. We derive bounds on the minimum distance of such a code based on its constraints, and characterize when these bounds can be achieved using subcodes of Reed-Solomon codes.
Resumo:
The advent of molecular biology has had a dramatic impact on all aspects of biology, not least applied microbial ecology. Microbiological testing of water has traditionally depended largely on culture techniques. Growing understanding that only a small proportion of microbial species are culturable, and that many microorganisms may attain a viable but non-culturable state, has promoted the development of novel approaches to monitoring pathogens in the environment. This has been paralleled by an increased awareness of the surprising genetic diversity of natural microbial populations. By targeting gene sequences that are specific for particular microorganisms, for example genes that encode diagnostic enzymes, or species-specific domains of conserved genes such as 16S ribosomal RNA coding sequences (rrn genes), the problems of culture can be avoided. Technical developments, notably in the area of in vitro amplification of DNA using the polymerase chain reaction (PCR), now permit routine detection and identification of specific microorganisms, even when present in very low numbers. Although the techniques of molecular biology have provided some very powerful tools for environmental microbiology, it should not be forgotten that these have their own drawbacks and biases in sampling. For example, molecular techniques are dependent on efficient lysis and recovery of nucleic acids from both vegetative forms and spores of microbial species that may differ radically when growing in the laboratory compared with the natural environment. Furthermore, PCR amplification can introduce its own bias depending on the nature of the oligonucleotide primers utilised. However, despite these potential caveats, it seems likely that a molecular biological approach, particularly with its potential for automation, will provide the mainstay of diagnostic technology for the foreseeable future.