Quantum scalar field in quantum gravity: the propagator and Lorentz invariance in the spherically symmetric case

We recently studied gravity coupled to a scalar field in spherical symmetry using loop quantum gravity techniques. Since there are local degrees of freedom one faces the "problem of dynamics". We attack it using the "uniform discretization technique". We find the quantum state that minimizes the value of the master constraint for the case of weak fields and curvatures. The state has the form of a direct product of Gaussians for the gravitational variables times a modified Fock state for the scalar field. In this paper we do three things. First, we verify that the previous state also yields a small value of the master constraint when one polymerizes the scalar field in addition to the gravitational variables. We then study the propagators for the polymerized scalar field in flat space-time using the previously considered ground state in the low energy limit. We discuss the issue of the Lorentz invariance of the whole approach. We note that if one uses real clocks to describe the system, Lorentz invariance violations are small. We discuss the implications of these results in the light of Horava's Gravity at the Lifshitz point and of the argument about potential large Lorentz violations in interacting field theories of Collins et. al.Comment: Dedicated to Josh Goldberg, to appear in special issue of Gen. Rel. Grav., 14 pages RevTex. We expanded the section on Lorentz invarianc

Custom sandwich pairs

AbstractFor many equations arising in practice, the solutions are critical points of functionals. In previous papers we have shown that there are pairs of subsets, called sandwich pairs, that can produce critical points even though they do not separate the functional. All that is required is that the functional be bounded from above on one of the sets and bounded from below on the other, with no relationship needed between the bounds. This provides a distinct advantage in applications. The present paper discusses the situation in which one cannot find sandwich pairs for which the functional is bounded below on one set and bounded above on the other. We develop a method which can deal with such situations and apply it to problems in partial differential equations

Sparse Image Representation with Epitomes

Sparse coding, which is the decomposition of a vector using only a few basis elements, is widely used in machine learning and image processing. The basis set, also called dictionary, is learned to adapt to specific data. This approach has proven to be very effective in many image processing tasks. Traditionally, the dictionary is an unstructured "flat" set of atoms. In this paper, we study structured dictionaries which are obtained from an epitome, or a set of epitomes. The epitome is itself a small image, and the atoms are all the patches of a chosen size inside this image. This considerably reduces the number of parameters to learn and provides sparse image decompositions with shiftinvariance properties. We propose a new formulation and an algorithm for learning the structured dictionaries associated with epitomes, and illustrate their use in image denoising tasks.Comment: Computer Vision and Pattern Recognition, Colorado Springs : United States (2011

Molecular typing and evolution of Salmonella enterica serovar Typhimurium

Salmonella enterica serovar Typhimurium is a common cause of salmonellosis among humans and animals worldwide. In Australia, Typhimurium is responsible for over half of the salmonellosis cases. The Anderson phage-typing scheme is the primary means of long-term surveillance of Typhimurium outbreak isolates, and has played an important role in epidemiology. However, there exist quite a number of strains of Typhimurium that cannot be defined by the phage-typing scheme. Furthermore, the knowledge of evolutionary relationships among isolates of different phage types is still very limited and the genetic basis of phage type variation remains largely unknown. To address these issues, this study focused on molecular typing and evolution of Typhimurium. Fluorescent amplified-fragment length polymorphism (AFLP) was applied to 46 Typhimurium isolates comprising nine phage types in Australia using the restriction enzymes MseI and EcoRI and MseI +1 / EcoRI +1 primer pair combinations. The selected phage types, DT9, DT135, DT64, DT44, DT126, DT12a, DT1, DT141 and DT108, have been dominant or frequent phage types in animal and human infections in Australia in recent years. AFLP in the present study showed a very good discrimination power with Simpson index of diversity of 0.98, 35 different AFLP patterns were observed in the 46 isolates studied. The tree based on AFLP patterns showed good correlation with phage type, grouped most Typhimurium isolates by phage type, and differentiated all nine phage types. Furthermore, 84 phage-type specific polymorphic AFLP fragments, for which presence or absence correlated with phage type (including 25 with one exception to phage-type specificity) were observed in the 46 strains studied. Eighteen phage-type specific AFLP fragments were cloned and sequenced. Sixteen are of known genes or have a homologue in the databases. It was found a predominance of phage and plasmid genes rather than mutational changes in the AFLP fragments studied. Of the 18 cloned and sequenced AFLP fragments, only four relate to mutational changes in the S. enterica chromosome, the other 14 comprise DNA of mobile elements: nine are phage related, three are plasmid related and two are gain of DNA from unknown origin. Twelve of the 18 sequenced phage-type specific AFLP markers are polymorphic because the DNA is present or absent as indicated by Southern hybridization. Two of these markers were successfully used in preliminary PCR-based typing of 30 DT9 and 29 DT135 isolates from worldwide collections. 27 of the 30 DT9 isolates and all DT135 isolates tested were correctly categorized. The results implied a good potential to use the sequence of these fragments as the basis for a multiplex PCR or a microarray based molecular 'phage' typing method for Typhimurium. This thesis also studied the molecular evolutionary relationships among the same set of 46 Typhimurium isolates using mutational changes detected by AFLP, or analysis of intergenic regions and their flanking genes in genome sequences. The complete genome sequence of Typhimurium LT2 was analysed by computer modelled AFLP. The polymorphic AFLP fragments, which matched with the modelled LT2 AFLP fragments, were amplified and sequenced by LT2 genome based primers to determine the changes. Forty-nine intergenic regions with higher pairwise differences between LT2 and Typhi CT18 were amplified and sequenced using LT2 genome based primers for one isolate of each phage type. 51 polymorphic sites were detected consisting of 18 in AFLP fragments and 33 in intergenic regions or their flanking genes. PCR-RFLP (restriction fragment length polymorphism) and SNaPshot were used to further investigate the distribution of the single nucleotide polymorphisms (SNPs) detected in intergenic regions in all isolates studied. Of the 18 mutational changes detected in AFLP fragments, eight were indels (insertions / deletions) and ten single base substitutions. Of the eight indels, four were in genes, three in intergenic regions, and one covered adjacent intergenic and coding regions. The four indels in genes all caused frameshift mutations, including three single base indels and one 19 bp deletion. Of the ten substitutions, one was in an intergenic region and nine in genes comprising three synonymous and six non-synonymous substitutions. Of the 33 polymorphic sites detected from sequences of 23 intergenic regions and their flanking genes, one was IS200 insertion and 32 single nucleotide polymorphisms (SNPs), of which 30 were single base substitutions and two were single base indels. Nine of the 33 variations were found in the flanking genes, which were all single base substitutions comprising four synonymous, four non-synonymous substitutions and one non-sense mutation. More non-synonymous than synonymous substitutions were found for those in coding regions within Typhimurium, indicating that slightly deleterious intraspecies mutations can be fixed within clones, such as various lineages of Typhimurium. The 51 polymorphic sites, which were inferred from sequences of both mutation related AFLP fragments, and intergenic regions and their flanking genes, gave a single phylogenetic tree of the 46 Typhimurium isolates studied. All sequences involved were compared with the homologous sequences in the available S. enterica genome sequences for serovars Typhi, Paratyphi A, Gallinarum, Enteritidis and Pullorum and this enabled the determination of the direction of the mutational changes in the isolates studied and the root of the phylogenetic tree. There were only two events inferred to have occurred twice, the remaining 49 polymorphisms can be explained by a single event. The data indicated that Typhimurium has a very strong clonal structure with a very low level of recombination over the time for diversification of Typhimurium as majority of clonal variations are from point mutations rather than recombination. The phylogenetic tree based on mutational changes showed that most Typhimurium isolates of a given phage type are in the same evolutionary group, but that some phage types appear to have arisen more than once. Comparison of the phylogenetic tree with AFLP data gave examples of unrelated isolates of a given phage type having common AFLP fragments comprising plasmid or phage genes, supporting the view that phage type can be determined by presence of specific phages or plasmids. The mutation-based tree showed that six of the nine phage types studied appeared to have a single origin, at least for the isolates studied. It also found that DT1 and DT44 had two independent origins even for the limited set of strains used. The distribution of DT12a isolates into two groups could be explained that the group of three DT12a isolates were derived from the other group of four DT12a isolates, where the root of the tree might be. The data also confirmed that DT64 arose from DT9. The phylogenetic tree that was generated based on essentially mutational changes provides clear relationships of the closely related Typhimurium isolates with high level of consistency and reasonable confidence. This study provided one of the few analyses of relationships of isolates within a clone. Matching actual AFLP with computer modeled AFLP and sequencing intergenic regions provide very good new strategies to identify mutational polymorphisms and to study the molecular evolutionary relationships in the closely related isolates

Can we allow linear dependencies in the dictionary in the sparse synthesis framework?

Signal recovery from a given set of linear measurements using a sparsity prior has been a major subject of research in recent years. In this model, the signal is assumed to have a sparse representation under a given dictionary. Most of the work dealing with this subject has focused on the reconstruction of the signal's representation as the means for recovering the signal itself. This approach forced the dictionary to be of low coherence and with no linear dependencies between its columns. Recently, a series of contributions that focus on signal recovery using the analysis model find that linear dependencies in the analysis dictionary are in fact permitted and beneficial. In this paper we show theoretically that the same holds also for signal recovery in the synthesis case for the l0- synthesis minimization problem. In addition, we demonstrate empirically the relevance of our conclusions for recovering the signal using an l1-relaxation.Comment: 2 figures, to appear in ICASSP 201