Best approximation

The present paper is  concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex
subset  of a locally convex space  and taking values in a locally convex space ordered by a normal cone.
One proves also equi-Lipschitz properties for  pointwise bounded families of continuous convex mappings, provided the source space  is barrelled.
Some results  on Lipschitz properties  of continuous convex functions defined on metrizable topological vector spaces are included as well.

Relations between the off thermal equilibrium dynamical process of on-line learning and the thermally equilibrated off-line learning are studied for potential gradient descent learning. The approach of Opper to study online Bayesian algorithms is extended to potential based or maximum likelihood learning. We look at the on-line learning algorithm that best approximates the off-line algorithm in the sense of least Kullback-Leibler information loss. It works by updating the weights along the gradient of an effective potential different from the parent off-line potential. The interpretation of this off equilibrium dynamics holds some similarities to the cavity approach of Griniasty. We are able to analyze networks with non-smooth transfer functions and transfer the smoothness requirement to the potential.

This paper studies how to optimally embed a general metric, represented by a graph, into a target space while preserving the relative magnitudes of most distances. More precisely, in an ordinal embedding, we must preserve the relative order between pairs of distances (which pairs are larger or smaller), and not necessarily the values of the distances themselves. The relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop polynomial-time constant-factor approximation algorithms for minimizing the relaxation in an embedding of an unweighted graph into a line metric and into a tree metric. These two basic target metrics are particularly important for representing a graph by a structure that is easy to understand, with applications to visualization, compression, clustering, and nearest-neighbor searching. Along the way, we improve the best known approximation factor for ordinally embedding unweighted trees into the line down to 2. Our results illustrate an important contrast to optimal-distortion metric embeddings, where the best approximation factor for unweighted graphs into the line is O(n 1/2), and even for unweighted trees into the line the best isÕ(n 1/3). We also show that Johnson-Lindenstrauss-type dimensionality reduction is possible with ordinal relaxation and 1 metrics (and p metrics with 1 ≤ p ≤ 2), unlike metric embedding of 1 metrics.

This paper studies how to optimally embed a general metric, represented by a graph, into a target space while preserving the relative magnitudes of most distances. More precisely, in an ordinal embedding, we must preserve the relative order between pairs of distances (which pairs are larger or smaller), and not necessarily the values of the distances themselves. The relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop polynomial-time constant-factor approximation algorithms for minimizing the relaxation in an embedding of an unweighted graph into a line metric and into a tree metric. These two basic target metrics are particularly important for representing a graph by a structure that is easy to understand, with applications to visualization, compression, clustering, and nearest-neighbor searching. Along the way, we improve the best known approximation factor for ordinally embedding unweighted trees into the line down to 2. Our results illustrate an important contrast to optimal-distortion metric embeddings, where the best approximation factor for unweighted graphs into the line is O(n 1/2), and even for unweighted trees into the line the best isÕ(n 1/3). We also show that Johnson-Lindenstrauss-type dimensionality reduction is possible with ordinal relaxation and 1 metrics (and p metrics with 1 ≤ p ≤ 2), unlike metric embedding of 1 metrics.

The objective in Data Grids is to reduce access and file (replica) transfer latencies, as well as to avoid single site congestion by the numerous requesters. To facilitate access and transfer of the data, the files of the Data Grid are distributed across the multiple sites. The effectiveness of a replica selection strategy in data grids depends on its ability to serve the requirement posed by the users' jobs. Most jobs are required to be executed at a specific execution time. To achieve the QoS perceived by the users, response time metrics should ...

Conserved segments in DNA or protein sequences are strong candidates for functional elements and thus appropriate methods for computing them need to be developed and compared. We describe five methods and computer programs for finding highly conserved blocks within previously computed multiple alignments, primarily for DNA sequences. Two of the methods are already in common use; these are based on good column agreement and high information content. Three additional methods find blocks with minimal evolutionary change, blocks that differ in at most k positions per row from a known center sequence and blocks that differ in at most k positions per row from a center sequence that is unknown a priori. The center sequence in the latter two methods is a way to model potential binding sites for known or unknown proteins in DNA sequences. The efficacy of each method was evaluated by analysis of three extensively analyzed regulatory regions in mammalian β β β β-globin gene clusters and the control region of bacterial arabinose operons. Although all five methods have quite different theoretical underpinnings, they produce rather similar results on these data sets when their parameters are adjusted to best approximate the experimental data. The optimal parameters for the method based on information content varied little for different regulatory regions of the β β β β-globin gene cluster and hence may be extrapolated to many other regulatory regions. The programs based on maximum allowed mismatches per row have simple parameters whose values can be chosen a priori and thus they may be more useful than the other methods when calibration against known functional sites is not available.

Rare Earth Element (REE), Ba/Ca, Sr/Ca and δ 18 O PO4 analyses have been performed on the ganoine and isopedine of different types of isolated scales (anterior, middle, posterior flank scales and dorsal body margin scales) from juvenile lepisosteids from the Fontllonga-3 ...

An inclusion compound comprising β-cyclodextrin (β-CD) and diferrocenyldimethylsilane, Fc–SiMe 2 –Fc [Fc = (η 5 -C 5 H 5 )Fe(η 5 -C 5 H 4 )], has been prepared and characterized in the solid state by powder X-ray diffraction (XRD), thermogravimetric analysis (TGA) and ...

Purpose: To study various standardized uptake value (SUV)-based approaches to ascertain the best strategy for delineating metabolic tumor volumes (MTV). Methods and Materials: Twenty-two consecutive previously treated esophageal cancer patients with positron emission tomography (PET) imaging and computed tomography (CT)-based radiotherapy plans were studied.

Approximately 500-Ma-old orthogneisses are widespread in the eastern part of the Variscan belt and are commonly interpreted to have intruded mica-schist series of assumed Neoproterozoic age. New SHRIMP zircon ages of quartzofeldspathic metavolcanogenic rocks of the mica schist series in the eastern part of the Karkonosze-Izera Massif (SW Poland) indicate that they are late Cambrian/early Ordovician rather than Neoproterozoic in age, based on the zircon age spectra distributed mainly between ca. 500 and 660 Ma (with a few Proterozoic inherited minimum ages of ca. 970 and 1,825 Ma). Younger zircon dates, dispersed between ca. 412 and 464 Ma, are interpreted as a result of Pb-loss likely caused by subsequent metamorphism. Consequently, the felsic metavolcanogenic rocks appear to be roughly contemporaneous with the intrusion of ca. 500-Ma-old orthogneiss protoliths (with the pooled concordia age of 487 ± 8 Ma interpreted as the best approximation of the protolith intrusive age). Field relationships, petrological and geochemical features of the felsic and mafic rocks studied support a model in which the accompanying mica schist series are not the original country rocks to the ca. 500 Ma granite intrusions, and indicate that their recent close proximity is the result of tectonic juxtaposition. However, both the mica schists enclosing the bimodal metavolcanic rocks, and the orthogneisses, are interpreted to represent a Cambro-Ordovician passive continental margin sequence being part of the Saxothuringian domain. They are tectonically overlain to the east by HP/T metamorphic units, comprising MORB-type metaigneous rocks, and delineating a tectonic suture separating the Saxothuringian block in the west from an assumed continental block (Tepla-Barrandian) to the south-east.

To approximate complex data, we propose new type of low-dimensional "principal object": principal cubic complex. This complex is a generalization of linear and non- linear principal manifolds and includes them as a particular case. To construct such an object, we combine the method of topological grammars with the minimization of elastic energy defined for its embedment into multidimensional data space.

Given a positive definite covariance matrix Σ, we strive to construct an optimal approximate factor analysis model HH^ +D, with H having a prescribed number of columns and D>0 diagonal. The optimality criterion we minimize is the I-divergence between the corresponding normal laws. Lifting the problem into a properly chosen larger space enables us to derive an alternating minimization algorithm à la Csiszár-Tusnády for the construction of the best approximation. The convergence properties of the algorithm are studied, with special attention given to the case where D is singular.

We present new combinatorial approximation algorithms for k-set cover. Previous approaches are based on extending the greedy algorithm by efficiently handling small sets. The new algorithms further extend them by utilizing the natural idea of computing large packings of elements into sets of large size. Our results improve the previously best approximation bounds for the k-set cover problem for all values of k ≥ 6. The analysis technique could be of independent interest; the upper bound on the approximation factor is obtained by bounding the objective value of a factor-revealing linear program.

We present new combinatorial approximation algorithms for the k-set cover problem. Previous approaches are based on extending the greedy algorithm by efficiently handling small sets. The new algorithms further extend these approaches by utilizing the natural idea of computing large packings of elements into sets of large size. Our results improve the previously best approximation bounds for the k-set cover problem for all values of k ≥ 6. The analysis technique used could be of independent interest; the upper bound on the approximation factor is obtained by bounding the objective value of a factor-revealing linear program.

The aim of this paper is to obtain new coincidence and common fixed point theorems by using Lipschitz type conditions of hybrid maps (not necessarily continuous) on a metric space. As applications, we demonstrate the existence of common fixed points from the set of best approximations. Our work sets analogues, unifies and improves various known results existing in the literature.

High intensity high current proton accelerators are finding increasing use of H negative ion sources due to its unique advantages in dealing with the space charge problem at the time of injection in the rapid cycling proton synchrotron. The multi-cusp magnetic field configuration has been found most suitable technique for plasma confinement and invariably used for generation of H- ions through volume production. This arrangement has the capability to generate uniform and dense plasma, which can produce high negative ion current density in the presence of appropriate filter magnetic field. Two most important parameters of multi-pole configuration are the cusp field strength and the central field free region (Usually taken as |B | < 10 Gauss) as they determine the leak width and the uniform density plasma volume in the ion source. The field intensity calculated by Liberman [1] in the approximation, showed that the magnetic field decay exponentially inside the plasma chamber, simila...