Computing the Hessenberg matrix associated with a self-similar measure

We introduce in this paper a method to calculate the Hessenberg matrix of a sum of measures from the Hessenberg matrices of the component measures. Our method extends the spectral techniques used by G. Mantica to calculate the Jacobi matrix associated with a sum of measures from the Jacobi matrices of each of the measures. We apply this method to approximate the Hessenberg matrix associated with a self-similar measure and compare it with the result obtained by a former method for self-similar measures which uses a fixed point theorem for moment matrices. Results are given for a series of classical examples of self-similar measures. Finally, we also apply the method introduced in this paper to some examples of sums of (not self-similar) measures obtaining the exact value of the sections of the Hessenberg matrix.

Approximate entropy of network parameters

We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We first define a purely structural entropy obtained by computing the approximate entropy of the so-called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erdös-Rényi networks. By using classical results of Pincus, we show that our entropy measure often converges with network size to a certain binary Shannon entropy. As a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs allow us to naturally associate with a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.

The Kramer sampling theorem revisited

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings; all these generalizations are illustrated with examples. All the different situations along the paper share a basic approach: the functions to be sampled are obtaining by duality in a separable Hilbert space H through an H -valued kernel K defined on an appropriate domain.

On the approximate parametrization problem of algebraic curves.

The problem of parameterizing approximately algebraic curves and surfaces is an active research field, with many implications in practical applications. The problem can be treated locally or globally. We formally state the problem, in its global version for the case of algebraic curves (planar or spatial), and we report on some algorithms approaching it, as well as on the associated error distance analysis.

Laminar counterflow parallel-plate heat exchangers: Exact and approximate solutions

Multilayered, counterflow, parallel-plate heat exchangers are analyzed numerically and theoretically. The analysis, carried out for constant property fluids, considers a hydrodynamically developed laminar flow and neglects longitudinal conduction both in the fluid and in the plates. The solution for the temperature field involves eigenfunction expansions that can be solved in terms of Whittaker functions using standard symbolic algebra packages, leading to analytical expressions that provide the eigenvalues numerically. It is seen that the approximate solution obtained by retaining the first two modes in the eigenfunction expansion provides an accurate representation for the temperature away from the entrance regions, specially for long heat exchangers, thereby enabling simplified expressions for the wall and bulk temperatures, local heat-transfer rate, overall heat-transfer coefficient, and outlet bulk temperatures. The agreement between the numerical and theoretical results suggests the possibility of using the analytical solutions presented herein as benchmark problems for computational heat-transfer codes.

Rational Hausdorff Divisors: A New approach to the Approximate Parametrization of Curves.

When applying computational mathematics in practical applications, even though one may be dealing with a problem that can be solved algorithmically, and even though one has good algorithms to approach the solution, it can happen, and often it is the case, that the problem has to be reformulated and analyzed from a different computational point of view. This is the case of the development of approximate algorithms. This paper frames in the research area of approximate algebraic geometry and commutative algebra and, more precisely, on the problem of the approximate parametrization.

Approximate solution for frequency synchronization in a finite-size Kuramoto model

Peer reviewed

Complex mtDNA constitutes an approximate 620-kb insertion on Arabidopsis thaliana chromosome 2: Implication of potential sequencing errors caused by large-unit repeats

Previously conducted sequence analysis of Arabidopsis thaliana (ecotype Columbia-0) reported an insertion of 270-kb mtDNA into the pericentric region on the short arm of chromosome 2. DNA fiber-based fluorescence in situ hybridization analyses reveal that the mtDNA insert is 618 ± 42 kb, ≈2.3 times greater than that determined by contig assembly and sequencing analysis. Portions of the mitochondrial genome previously believed to be absent were identified within the insert. Sections of the mtDNA are repeated throughout the insert. The cytological data illustrate that DNA contig assembly by using bacterial artificial chromosomes tends to produce a minimal clone path by skipping over duplicated regions, thereby resulting in sequencing errors. We demonstrate that fiber-fluorescence in situ hybridization is a powerful technique to analyze large repetitive regions in the higher eukaryotic genomes and is a valuable complement to ongoing large genome sequencing projects.

A Quantile-Based Probabilistic Mean Value Theorem

For non-negative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows us to construct new distributions with support (0, 1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the “expected reversed proportional shortfall order”, and a new characterization of random lifetimes involving the reversed hazard rate function.

A four moments theorem for Gamma limits on a Poisson chaos

This paper deals with sequences of random variables belonging to a fixed chaos of order q generated by a Poisson random measure on a Polish space. The problem is investigated whether convergence of the third and fourth moment of such a suitably normalized sequence to the third and fourth moment of a centred Gamma law implies convergence in distribution of the involved random variables. A positive answer is obtained for q = 2 and q = 4. The proof of this four moments theorem is based on a number of new estimates for contraction norms. Applications concern homogeneous sums and U-statistics on the Poisson space.

Lithology and approximate sub-bottom depths and ages of the transition of soft to firm ooze and of firm ooze to chalk at DSDP Leg 89 and 90 Sites

Leg 90 recovered approximately 3705 m of core at eight sites lying at middle bathyal depths (1000-2200 m) (Sites 587 to 594) in a traverse from subtropical to subantarctic latitudes in the southwest Pacific region, chiefly on Lord Howe Rise in the Tasman Sea. This chapter summarizes some preliminary lithostratigraphic results of the leg and includes data from Site 586, drilled during DSDP Leg 89 on the Ontong-Java Plateau that forms the northern equatorial point of the latitudinal traverse. The lithofacies consist almost exclusively of continuous sections of very pure (>95% CaCO3) pelagic calcareous sediment, typically foraminifer-bearing nannofossil ooze (or chalk) and nannofossil ooze (or chalk), which is mainly of Neogene age but extends back into the Eocene at Sites 588, 592, and 593. Only at Site 594 off southeastern New Zealand is there local development of hemipelagic sediments and several late Neogene unconformities. Increased contents of foraminifers in Leg 90 sediments, notably in the Quaternary interval, correspond to periods of enhanced winnowing by bottom currents. Significant changes in the rates of sediment accumulation and in the character and intensity of sediment bioturbation within and between sites probably reflect changes in calcareous biogenic productivity as a result of fundamental paleoceanographic events in the region during the Neogene. Burial lithification is expressed by a decrease in sediment porosity from about 70 to 45% with depth. Concomitantly, microfossil preservation slowly deteriorates as a result of selective dissolution or recrystallization of some skeletons and the progressive appearance of secondary calcite overgrowths, first about discoasters and sphenoliths, and ultimately on portions of coccoliths. The ooze/chalk transition occurs at about 270 m sub-bottom depth at each of the northern sites (Sites 586 to 592) but is delayed until about twice this depth at the two southern sites (Sites 593 and 594). A possible explanation for this difference between geographic areas is the paucity of discoasters and sphenoliths at the southern sites; these nannofossil elements provide ideal nucleation sites for calcite overgrowths. Toward the bottom of some holes, dissolution seams and flasers appear in recrystallized chalks. The very minor terrigenous fraction of the sediment consists of silt- through clay-sized quartz, feldspar, mica, and clay minerals (smectite, illite, kaolinite, and chlorite), supplied as eolian dust from the Australian continent and by wind and ocean currents from erosion on South Island, New Zealand. Changes in the mass accumulation rates of terrigenous sediment and in clay mineral assemblages through time are related to various external controls, such as the continued northward drift of the Indo-Australian Plate, the development of Antarctic ice sheets, the increased desertification of the Australian continent after 14 m.y. ago, and the progressive increase in tectonic relief of New Zealand through the late Cenozoic. Disseminated glass shards and (altered) tephra layers occur in Leg 90 cores. They were derived from major silicic eruptions in North Island, New Zealand, and from basic to intermediate explosive volcanism along the Melanesian island chains. The tephrostratigraphic record suggests episodes of increased volcanicity in the southwest Pacific centered near 17, 13, 10, 5 and 1 m.y. ago, especially in the middle and early late Miocene. In addition, submarine basaltic volcanism was widespread in the southeast Tasman Sea around the Eocene/Oligocene boundary, possibly related to the propagation of the Southeast Indian Ridge through western New Zealand as a continental rift system.

Sperner's theorem on maximal-sized antichains and its generation /

Includes bibliographical references.

Theorem proving with abstraction, part II /

Supported in part by the National Science Foundation under grant MCS 77-22830.