The notion of a fuzzy metric space due to George and Veeramani has many advantages in analysis since many notions and results from classical metric space theory can be extended and generalized to the setting of fuzzy metric spaces, for instance: the notion of completeness, completion of spaces as well as extension of maps. The layout of the dissertation is as follows: Chapter 1 provide the necessary background in the context of metric spaces, while chapter 2 presents some concepts and results from classical metric spaces in the setting of fuzzy metric spaces. In chapter 3 we continue with the study of fuzzy metric spaces, among others we show that: the product of two complete fuzzy metric spaces is also a complete fuzzy metric space. Our main contribution is in chapter 4. We introduce the concept of a standard fuzzy pseudo metric space and present some results on fuzzy metric identification. Furthermore, we discuss some properties of t-nonexpansive maps. / Mathematical Sciences / M. Sc. (Mathematics)

Pratima Sadangi in a Ph.D. thesis submitted to Utkal University proved results on degree of approximation of functions by operators associated with their Fourier series. In this dissertation, we consider degree of approximation of functions in Hα,ρ by different operators. In Chapter 1 we mention basic definitions needed for our work. In Chapter 2 we discuss different methods of summation. In Chapter 3 we define the Hα,ρ metric and present the degree of approximation problem relating to Fourier series and conjugate series of functions in the Hα,ρ metric using Karamata (Κλ) means. In Chapter 4 we present the degree of approximation of an integral associated with the conjugate series by the Euler, Borel and (e,c) means of a series analogous to the Hardy-Littlewood series in the Hα,ρ metric. In Chapter 5 we propose problems to be solved in the future.

No description available.

A new coding scheme called "turbo coding" has generated tremendous interest in channel coding of digital communication systems due to its high error correcting capability. Two key innovations in turbo coding are parallel concatenated encoding and iterative decoding. A soft-in soft-out component decoder can be implemented using the maximum a posteriori (MAP) or the maximum likelihood (ML) decoding algorithm. While the MAP algorithm offers better performance than the ML algorithm, the computation is complex and not suitable for hardware implementation. The log-MAP algorithm, which performs necessary computations in the logarithm domain, greatly reduces hardware complexity. With the proliferation of the battery powered devices, power dissipation, along with speed and area, is a major concern in VLSI design. In this thesis, we investigated a low-power design of a turbo decoder based on the log-MAP algorithm. Our turbo decoder has two component log-MAP decoders, which perform the decoding process alternatively. Two major ideas for low-power design are employment of a variable number of iterations during the decoding process and shutdown of inactive component decoders. The number of iterations during decoding is determined dynamically according to the channel condition to save power. When a component decoder is inactive, the clocks and spurious inputs to the decoder are blocked to reduce power dissipation. We followed the standard cell design approach to design the proposed turbo decoder. The decoder was described in VHDL, and then synthesized to measure the performance of the circuit in area, speed and power. Our decoder achieves good performance in terms of bit error rate. The two proposed methods significantly reduce power dissipation and energy consumption. / Master of Science

This doctoral thesis is devoted to investigate the problem of establishing connections between Domain Theory and the theory of fuzzy metric spaces, in the sense of Kramosil and Michalek, by means of the notion of a formal ball, and then constructing topological and computational models for (complete) fuzzy metric spaces. The antecedents of this research are mainly the well-known articles of A. Edalat and R. Heckmann [A computational model for metric spaces, Theoret- ical Computer Science 193 (1998), 53-73], and R. Heckmann [Approximation of metric spaces by partial metric spaces, Applied Categorical Structures 7 (1999), 71-83], where the authors obtained nice and direct links between Do- main Theory and the theory of metric spaces - two crucial tools in the study of denotational semantics - by using formal balls. Since every metric induces a fuzzy metric (the so-called standard fuzzy metric), the problem of extending Edalat and Heckmann's works to the fuzzy framework arises in a natural way. In our study we essentially propose two di erent approaches. For the rst one, valid for those fuzzy metric spaces whose continuous t-norm is the minimum, we introduce a new notion of fuzzy metric completeness (the so-called standard completeness) that allows us to construct a (topological) model that includes the classical theory as a special case. The second one, valid for those fuzzy metric spaces whose continuous t-norm is greater or equal than the Lukasiewicz t-norm, allows us to construct, among other satisfactory results, a fuzzy quasi-metric on the continuous domain of formal balls whose restriction to the set of maximal elements is isometric to the given fuzzy metric. Thus we obtain a computational model for complete fuzzy metric spaces. We also prove some new xed point theorems in complete fuzzy metric spaces with versions to the intuitionistic case and the ordered case, respec- tively. Finally, we discuss the problem of extending the obtained results to the asymmetric framework. / Ricarte Moreno, L. (2013). Topological and Computational Models for Fuzzy Metric Spaces via Domain Theory [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/34670

Spherical, Euclidean, and hyperbolic simplices can be characterized by the dihedral angles on their codimension-two faces. These characterizations analyze the Gram matrix, a matrix with entries given by cosines of dihedral angles. Hyperideal hyperbolic simplices are non-compact generalizations of hyperbolic simplices wherein the vertices lie outside hyperbolic space. We extend recent characterization results to include fully general hyperideal simplices. Our analysis utilizes the Gram matrix, however we use inversive distances instead of dihedral angles to accommodate fully general hyperideal simplices.For two-dimensional triangulations, an angle structure is an assignment of three face angles to each triangle. An angle structure permits a globally consistent scaling provided the faces can be simultaneously scaled so that any two contiguous faces assign the same length to their common edge. We show that a class of symmetric Euclidean angle structures permits globally consistent scalings. We develop a notion of virtual scaling to accommodate spherical and hyperbolic triangles of differing curvatures and show that a class of symmetric spherical and hyperbolic angle structures permit globally consistent virtual scalings.The double tetrahedron is a triangulation of the three-sphere obtained by gluing two congruent tetrahedra along their boundaries. The pentachoron is a triangulation of the three-sphere obtained from the boundary of the 4-simplex. As piecewise flat manifolds, the geometries of the double tetrahedron and pentachoron are determined by edge lengths that gives rise to a notion of a metric. We study notions of Einstein metrics on the double tetrahedron and pentachoron. Our analysis utilizes Regge's Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds.A notion of conformal structure on a two dimensional piecewise flat manifold is given by a set of edge constants wherein edge lengths are calculated from the edge constants and vertex based parameters. A conformal variation is a smooth one parameter family of the vertex parameters. The analysis of conformal variations often involves the study of degenerating triangles, where a face angle approaches zero. We show for a conformal variation that remains weighted Delaunay, if the conformal parameters are bounded then no triangle degenerations can occur.

Problems of population genetics are magnified by problems of big data. My dissertation spans the disciplines of computer science and population genetics, leveraging computational approaches to biological problems to address issues in genomics research. In this dissertation I develop more efficient metric search algorithms. I also show that vast majority of the genomes of great apes are impacted by the forces of natural selection. Finally, I introduce a heuristic to identify neutralomes—regions that are evolving with minimal selective pressures—and use these neutralomes for inferences on effective population size in great apes. We begin with a formal and far-reaching problem that impacts a broad array of disciplines including biology and computer science; the 𝑘-nearest neighbors problem in generalized metric spaces. The 𝑘-nearest neighbors (𝑘-NN) problem is deceptively simple. The problem is as follows: given a query q and dataset D of size 𝑛, find the 𝑘-closest points to q. This problem can be easily solved by algorithms that compute 𝑘th order statistics in O(𝑛) time and space. It follows that if D can be ordered, then it is perhaps possible to solve 𝑘-NN queries in sublinear time. While this is not possible for an arbitrary distance function on the points in D, I show that if the points are constrained by the triangle inequality (such as with metric spaces), then the dataset can be properly organized into a dispersion tree (Appendix A). Dispersion trees are a hierarchical data structure that is built around a large dispersed set of points. Dispersion trees have construction times that are sub-quadratic (O(𝑛¹·⁵ log⁡ 𝑛)) and use O(𝑛) space, and they use a provably optimal search strategy that minimizes the number of times the distance function is invoked. While all metric data structures have worst-case O(𝑛) search times, dispersion trees have average-case search times that are substantially faster than a large sampling of comparable data structures in the vast majority of spaces sampled. Exceptions to this include extremely high dimensional space (d>20) which devolve into near-linear scans of the dataset, and unstructured low-dimensional (d<6) Euclidean spaces. Dispersion trees have empirical search times that appear to scale as O(𝑛ᶜ) for 0<c<1. As solutions to the 𝑘-NN problem are in general too slow to be used effectively in the arena of big data in genomics, it is my hope that dispersion trees may help lift this barrier. With source-code that is freely available for academic use, dispersion trees may be useful for nearest neighbor classification problems in machine learning, fast read-mapping against a reference genome, and as a general computational tool for problems such clustering. Next, I turn to problems in population genomics. Genomic patterns of diversity are a complex function of the interplay between demographics, natural selection and mechanistic forces. A central tenet of population genetics is the neutral theory of molecular evolution which states the vast majority of changes at the molecular level are (relatively) selectively neutral; that is, they do not effect fitness. A corollary of the neutral theory is that the frequency of most alleles in populations are dictated by neutral processes and not selective processes. The forces of natural selection impact not just the site of selection, but linked neutral sites as well. I proposed an empirical assessment of the extents of linked selection in the human genome (Appendix B). Recombination decouples sites of selection from the genomic background, thus it serves to mitigate the effects of linked selection. I use two metrics on recombination, both the minimum genetic distance to genes and local rates of recombination, to parse the effects of linked selection into selection from genic and nongenic sources in the human genome. My empirical assessment shows profound linked selective effects from nongenic sources, with these effects being greater than that of genic sources on the autosomes, as well as generally greater effects on the X chromosome than on the autosomes. I quantify these trends using multiple linear regression, and then I model the effects of linked selection to conserved elements across the whole of the genome. Places predicted to be neutral by my model do not, unlike the vast majority of the genome, show these linked selective effects. This demonstrates that linkage to these regulatory elements, and not some other mechanistic force, accounts for our findings. Further, neutrally evolving regions are extremely rare (~1%) in the genome, and despite generally larger linked selective effects on the X chromosome, the size of this “neutralome” is proportionally larger on the X chromosome than on the autosomes. To account for this and to extend my findings to other great apes I improve on my procedure to find neutralomes, and apply this procedure to the genome of humans, Nigerian chimpanzees, bonobos, and western lowland gorillas (Appendix C). In doing so I show that like humans, these other apes are also enormously impacted by linked selection, with their neutralomes being substantially smaller than the neutralomes of humans. I then use my genomic predictions on neutrality to see how the landscape of linked selection changes across the X chromosome and the autosomes in regions close to, and far from, genes. While I had previously demonstrated the linked selective forces near genes are stronger on the X chromosome than on the autosomes in these taxa, I show that regions far from genes show the opposite; regions far from genes show more selection from noncoding targets on the autosomes than on the X chromosome. This finding is replicated across our great ape samples. Further, inferences on the relative effective population size of the X chromosome and the autosomes both near and far from genes can be biased as a result.

Nos livros das Odes Horácio empregou treze esquemas métricos distribuídos por poemas de temas diversos. A influência da tradição eólica representada pelos dois musicistas de Lesbos, Safo e Alceu, pautou a maioria das escolhas temáticas e formais do poeta. Odes compostas em metros asclepiadeus e jônicos kataV stivcon, em estrofes sáficas, alcaicas e asclepiadéias e em dísticos de formação variada (cola datílicos, sáficos, jâmbicos e trocaicos), mostram ritmos próprios, capazes de imprimir, no ânimo do ouvinte, sensações diferenciadas, de acordo com a natureza da seqüência métrica empregada. A teoria do ethos métrico leva em consideração o conceito da conveniência (Prevpon, decorum): conteúdo e forma em harmonia na criação poética. Os efeitos impressivos das medidas gregas, naturalizadas por Horácio, que deu feição datílica aos versos eólicos, fixou quantidades livres e disciplinou as estrofes, devem-se ao caráter psicagógico dos metros, herdado da antiga teoria musical. Desde Platão e Aristóteles, passando por Cícero, Demétrio, Dionísio de Halicarnasso, Longino e Quintiliano, registramse a preocupação de classificar os metros de acordo com sua adequação a cada tipo de composição e a censura de seu uso indiscriminado na prosa e na poesia. A análise rítmico-semântica de algumas odes de Horácio revelou o zelo do poeta em combinar forma e conteúdo e em selecionar palavras de composição sonora e formação métrica em harmonia com o sentido. Nas Odes a musicalidade do ritmo métrico tem implicações semânticas, realçando a expressão textual. / It has been observed in Horace\'s Odes books that he has employed thirteen metrical schemes distributed among thematic different poems. Aeolic tradition influence, represented by the two Lesbian musicians Sappho and Alcaeus, was responsible for most of the formal and thematic choices of the poet. Odes written in asclepiadean meters and ionic kataV stivcon, in sapphic, alcaic end asclepiadean strophes and in various distics (cola datctylics, sapphics, iambics and trochaics) show their own rhythms, which are able to impress different sensations to their listeners, according to the nature of the metrical sequence used. The theory of metrical ethos considers the convenience concept (Prevpon, decorum): subject and form harmonically living in poetic creation. The impressive effects of greek measures, (which were naturalized by Horace, gave dactylic features to the aeolic verses, fixed free amounts and regulated the strophes), are due to the psychagogic character of the meters, inherited by the old musical theory. Since Plato, Aristotle, Cicero, Demetrius, Dionysius of Halicarnassus, Longinus and Quintilianus, there is a worry at classifying meters according to their adequacy to each kind of composition and disapproval of its nonrestrictive use in prose and poetry. The rhythmic-semantics analysis of some odes from Horace revealed the poet care to combine form and subject and to select sonorous words and metrics in harmony with sense. In Odes, metrics rhythm musicality has semantic implications that highlight the textual expression.

Nous montrons la validité du formalisme multifractal pour les mesures aléatoires faiblement Gibbs portées par l’ attracteur associé à une dynamique aléatoire C¹ codée par un sous-shift de type fini aléatoire, et expansive en moyenne. Nous établissons également des loi de type 0-∞ pour les mesures de Hausdorff et de packing généralisées des ensembles de niveau de la dimension locale, et calculons les dimensions de Hausdorff et de packing des ensembles de points en lesquels la dimension inférieure locale et la dimension supérieure locale sont prescrites. Lorsque l’attracteur est un ensemble de Cantor de mesure de Lebesgue nulle, nous montrons la validité du formalisme multifractal pour les mesures discrètes obtenues comme inverses de ces mesures faiblement Gibbs. / We establish the validity of the multifractal formalism for random weak Gibbs measures supported on the attractor associated with a C¹ random dynamics coded by a random subshift of finite type, and expanding in the mean. We also prove a 0-∞ law for the generalized Hausdorff and packing measures of the level sets of the local dimension, and we compute the Hausdorff and packing dimensions of the sets of points at which the lower and upper local dimensions are prescribed. In the case that the attractor is a Cantor set of zero Lebesgue measure, we prove the validity of the multifractal formalism for the discrete measures obtained as inverse of these weak Gibbs measures.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1982. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE / Bibliography: leaves 106-109. / by Rae Michael Andrew Shortt. / Ph.D.