Sum Of Classes Research Articles

DIFFUSE γ-RAY EMISSION FROM UNRESOLVED BL LAC OBJECTS

Blazars represent the most abundant class of high-energy extragalactic $\gamma$-ray sources. The subset of blazars known as BL Lac objects is on average closer to Earth and characterized by harder spectra at high energy than the whole sample. The fraction of BL Lacs that is too dim to be detected and resolved by current $\gamma$-ray telescopes is therefore expected to contribute to the high-energy isotropic diffuse $\gamma$-ray background (IGRB). The IGRB has been recently measured over a wide energy range by the Large Area Telescope (LAT) on board the Gamma-ray Space Telescope ({\it Fermi}). We present a new prediction of the diffuse $\gamma$-ray flux due to the unresolved BL Lac blazar population. The model is built upon the spectral energy distribution and the luminosity function derived from the fraction of BL Lacs detected (and spectrally characterized) in the $\gamma$-ray energy range. We focus our attention on the ${\cal O}(100)$ GeV energy range, predicting the emission up to the TeV scale and taking into account the absorption on the extragalactic background light. In order to better shape the BL Lac spectral energy distribution, we combine the {\it Fermi}-LAT data with Imaging Atmospheric Cerenkov Telescopes measurements of the most energetic sources. Our analysis is carried on separately for low- and intermediate-synchrotron-peaked BL Lacs on one hand, and high-synchrotron-peaked BL Lacs on the other one: we find in fact statistically different features for the two. The diffuse emission from the sum of both BL Lac classes increases from about 10$\%$ of the measured IGRB at 100 MeV to $\sim$100$\%$ of the data level at 100 GeV. At energies greater than 100 GeV, our predictions naturally explain the IGRB data, accommodating their softening with increasing energy. Uncertainties are estimated to be within of a factor of two of the best-fit flux up to 500 GeV.

Class product and character product

For each finite group G, the product in the group ring of all the conjugacy class sums is a positive integer multiple of the sum of the elements in a special coset of the commutator subgroup G′, as Brauer and Wielandt first observed in the case G′ = G. We show that the corresponding special element G! in A := G/G′ is the product of B! over specified subgroups B of A. Somewhat analogously, the product of all the irreducible characters of G, restricted to the center Z of G, is a multiple of a special linear character !G of Z, and !G is the product of !(Z/Y) over specified subgroups Y of Z.

On certain subclasses of multivalent analytic functions

In the present paper the authors introduce two new subclasses of multivalent analytic functions. Distortion inequalities and partial sums of functions in these classes are given.

Bounded max-colorings of graphs

In a bounded max-coloring of a vertex/edge weighted graph, each color class is of cardinality at most b and of weight equal to the weight of the heaviest vertex/edge in this class. The bounded max-vertex/edge-coloring problems ask for such a coloring minimizing the sum of all color classes' weights. These problems generalize the well known max-coloring problems by taking into account the number of available resources (i.e., the cardinality of color classes) in practical applications. In this paper we present complexity results and approximation algorithms for the bounded max-coloring problems on general graphs, bipartite graphs and trees.

Co-2-plex vertex partitions

This paper studies co-k-plex vertex partitions and more specifically co-2-plex vertex partitions. Co-$$k$$k-plexes and $$k$$k-plexes were first introduced in 1978 in the context of social network analysis. However, the study of co-k-plex vertex partitions or decomposing a graphs into degree bounded subgraphs can be at least dated back to the work of Lovasz (Studia Sci Math Hung 1:237---238, 1966). In this paper, we derive analogues for well-known results on the chromatic number, and present two algorithms for constructing co-2-plex vertex partitions. The first algorithm minimizes the number of partition classes while the second algorithm minimizes a weighted sum of the partition classes, where the weight of a partition class depends on the level of adjacency among its vertices.

On Minimal Fuzzy Ideals of Semigroups

The present paper contains the sufficient condition of a fuzzy semigroup to be a fuzzy group using fuzzy points. The existence of a fuzzy kernel in semigroup is explored. It has been shown that every fuzzy ideal of a semigroup contains every minimal fuzzy left and every minimal fuzzy right ideal of semigroup. The fuzzy kernel is the class sum of minimal fuzzy left (right) ideals of a semigroup. Every fuzzy left ideal of a fuzzy kernel is also a fuzzy left ideal of a semigroup. It has been shown that the product of minimal fuzzy left ideal and minimal fuzzy right ideal of a semigroup forms a group. The representation of minimal fuzzy left (right) ideals and also the representation of intersection of minimal fuzzy left ideal and minimal fuzzy right ideal are shown. The fuzzy kernel of a semigroup is basically the class sum of all the minimal fuzzy left (right) ideals of a semigroup. Finally the sufficient condition of fuzzy kernel to be completely simple semigroup has been proved.

On divisibility of exponential sums of polynomials of special type over fields of characteristic 2

We study the divisibility by eight of exponential sums of several classes of functions over finite fields of characteristic two. For the binary classical Kloosterman sums K(a) over such fields we give a simple recurrent algorithm for finding the largest k, such that 2 k divides the Kloosterman sum K(a). This gives a simple description of zeros of such Kloosterman sums.

On the cluster category of a marked surface without punctures

We study in this paper the cluster category C(S,M) of a marked surface (S,M). We explicitly describe the objects in C(S,M) as direct sums of homotopy classes of curves in (S,M) and one-parameter families related to closed curves in (S,M). Moreover, we describe the Auslander-Reiten structure of the category C(S,M) in geometric terms and show that the objects without self-extensions in C(S,M) correspond to curves in (S,M) without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation.

Conjugacy Classes, Class Sums and Character Tables for Hopf Algebras

We extend the notion of conjugacy classes and class sums from finite groups to semisimple Hopf algebras and show that the conjugacy classes are obtained from the factorization of H as irreducible left D(H)-modules. For quasitriangular semisimple Hopf algebras H, we prove that the product of two class sums is an integral combination of the class sums up to d −2 where d = dim H. We show also that in this case the character table is obtained from the S-matrix associated to D(H). Finally, we calculate explicitly the generalized character table of D(kS 3), which is not a character table for any group. It moreover provides an example of a product of two class sums which is not an integral combination of class sums.

Finding sums for an infinite class of alternating series

Calculus II students know that many alternating series are convergent by the Alternating Series Test. However, they know few alternating series (except geometric series and some trivial ones) for which they can find the sum. In this article, we present a method that enables the students to find sums for infinitely many alternating series in the following form

Basis of a Humeomics Science: Chemical Fractionation and Molecular Characterization of Humic Biosuprastructures

We propose a mild stepwise fractionation of molecular components of a humic acid (HA) suprastructure and their structural identification by advanced analytical methods. This procedure may be the basis of a "Humeomics" approach to characterize natural humic molecules and clarify their relations with ecosystems functions. Sequential fractionation included: (1) organic solvent extraction, (2) transesterification with boron trifluoride in methanol (BF(3)-CH(3)OH), (3) methanolic alkaline hydrolysis (KOH-CH(3)OH), and (4) cleavage of ether and glycosidic bonds with HI. Structural identification of initial and final material, separated organo-soluble and hydrosoluble fractions, and subfractions was conducted by GC-MS, HPSEC-ESI-MS (high-resolution, Orbitrap), and solid- and liquid-state NMR. GC-MS revealed in organosoluble unbound fractions the presence of both saturated and unsaturated, linear and branched, alkanoic, hydroxyalkanoic and alkandioic acids, n-alkanes, and n-alkanols. These components decreased progressively in fractions obtained after weak and strong ester cleavage. Unsubstituted alkanoic acids with variable chain length were ubiquitously detected in all fractions, thereby suggesting their fundamental function in the architecture of humic suprastructures. An important role in differentiating supramolecular associations should also be attributed to substituted alkanoic acids that were detected in variable amounts in different fractions. The content of aromatic acids and steroids was only noticed in the latter fractions. HPSEC-ESI-MS of initial and final solid fractions showed similar compounds, as indicated by GC-MS, whereas the hydrosoluble fraction after transesterification revealed fewer of these compounds but noticeable nitrogen-containing acids. A large amount of "cyclic" acids were identified by MS empirical formula in initial HA, and, to a lesser extent, in the final fractionation residue as well as in the hydrosoluble fraction. The predominant alkyl NMR signals in organosoluble extracts and those of CH-N, CH-O, and O-CH-O groups in hydrosoluble fraction confirmed mass spectrometry results. Homo- and heterocorrelated liquid-state NMR spectra indicated spin systems interactions varying with separated fractions. Solid-state and dipolar-dephasing NMR spectra of final residue showed predominance of sp(2) carbons, 66% of which were quaternary carbons, and a significant increase in conformational rigidity with respect to initial HA. Separated fractions accounted for 60% of initial HA weight, and losses were attributed to hydration water, liberated volatile compounds, and decarboxylation. Quantization of analytes showed that the sum of compound classes in separated fractions was greater than that for the initial HA, thereby showing that stepwise fractionation increased significantly the analytical identification of humic molecules. Our results suggest this "Humeomics" approach as a valid path for mapping humic molecular composition and assess humus origin and formation.

A Robust Automatic Merging Possibilistic Clustering Method

Krishnapuram and Keller [IEEE Trans. Fuzzy Syst., vol. 1, no. 2, pp. 98-110, May 1993] proposed a possibilistic c-means (PCM) clustering by relaxing the constraint in fuzzy c-means (FCMs) that the memberships of a data point across classes sum to 1. The PCM algorithm has a tendency to produce coincident clusters. This can be a merit of PCM as a good mode-seeking algorithm if initials and parameters are suitably chosen. However, the performance of PCM heavily depends on initializations and parameter selection. In this paper, we propose a mechanism of robust automatic merging. We then create an automatic merging possibilistic clustering method (AM-PCM), where the proposed algorithm does not only solve these parameter-selection and initialization problems but also obtains an optimal cluster number. The proposed AM-PCM algorithm first uses all data points as initial cluster centers and then automatically merges these surrounding points around each cluster mode such that it can self-organize data groups according to the original data structure. The AM-PCM can exhibit the robustness to parameter, noise, cluster number, different volumes, and initializations. The computational complexity of AM-PCM is also analyzed. Comparisons between AM-PCM and other clustering methods are made. Some numerical data and real datasets are used to show these good aspects of AM-PCM. Experimental results and comparisons actually demonstrate that the proposed AM-PCM is an effective and parameter-free robust clustering algorithm.

Central Units, Class Sums and Characters of the Symmetric Group

In the search for central units of a group algebra, we look at the class sums of the group algebra of the symmetric group S n in characteristic zero, and we show that they are units in very special instances.

Structure constants related to symmetric Hopf algebras

Let k be an algebraically closed field of characteristic 0. In this paper we continue our study of structure constants for semisimple Hopf algebras H whose character algebra is commutative, and for non-semisimple factorizable ribbon Hopf algebras. This is done from the point of view of symmetric algebras, such as group algebras. In particular we consider general fusion rules which are structure constants associated to products of irreducible characters and structure constants associated to generalizations of class sums and conjugacy classes. Our methods are reminiscent on one hand of the methods employed when analyzing tilting modules for certain quantum groups and on the other hand of “splitting modules” for certain Drinfeld doubles. The family of irreducible characters is divided in two according to the vanishing of their quantum dimension. The fusion rules on the character algebra are computed with respect to this division.

Lessons learned in content architecture harmonization and metadata models

PurposeThis paper aims to review key content, architecture, and metadata model decisions and strategies in creation of a publication portal (on DVD to start), based on a 30+ year series of flagship reports from the World Bank.Design/methodology/approachThe paper describes and analyzes key considerations and aspects of the project, including content architecture, content analysis, DTD selection, retrospective conversion, vendor management, design of metadata architectures, use of automated profiling methods, user‐information behavior, and search architectures supporting complex content architectures. It includes the challenges of applying an institutionally based taxonomy required to express subject‐matter responsibilities and relationships within the World Bank.FindingsThe team learned that the metadata behavior and architecture (inheritance, relationships, variations) are more complex than simple links between parent and child objects. The project also reinforced the importance of comprehensive and dynamic topic taxonomy for classifying content that is both historical and current. The approach to defining classes for each full report (parent) will be likely to change, given what has been learned. The team would recommend that parts be classified and the sum of the part classes be assigned to the whole report. As a result of this exploratory work, the Bank's approach to classification and indexing of report series is changing from a top‐down to a bottom‐up inheritance.Originality/valueThe study provides insights into both general and World Bank‐specific challenges in creating a publication portal and derives some best practices for content architecture, metadata architecture, and use of automated profiling methods.

Elimination of 5S DNA unit classes in newly formed allopolyploids of the generaAegilopsandTriticum

Two classes of 5S DNA units, namely the short (containing units of 410 bp) and the long (containing units of 500 bp), are recognized in species of the wheat (the genera Aegilops and Triticum) group. While every diploid species of this group contains 2 unit classes, the short and the long, every allopolyploid species contains a smaller number of unit classes than the sum of the unit classes of its parental species. The aim of this study was to determine whether the reduction in these unit classes is due to the process of allopolyploidization, that is, interspecific or intergeneric hybridization followed by chromosome doubling, and whether it occurs during or soon after the formation of the allopolyploids. To study this, the number and types of unit classes were determined in several newly formed allotetraploids, allohexaploids, and an allooctoploid of Aegilops and Triticum. It was found that elimination of unit classes of 5S DNA occurred soon (in the first 3 generations) after the formation of the allopolyploids. This elimination was reproducible, that is, the same unit classes were eliminated in natural and synthetic allopolyploids having the same genomic combinations. No further elimination occurred in the unit classes of the 5S DNA during the life of the allopolyploid. The genetic and evolutionary significance of this elimination as well as the difference in response to allopolyploidization of 5S DNA and rDNA are discussed.

Approximating the max-edge-coloring problem

The max-edge-coloring problem is a natural weighted generalization of the classical edge-coloring problem arising in the domain of communication systems. In this problem each color class is assigned the weight of the heaviest edge in this class and the objective is to find a proper edge-coloring of the input graph minimizing the sum of all color classes’ weights. We present new approximation results, that improve substantially the known ones, for several variants of the problem with respect to the class of the underlying graph. In particular, we deal with variants which are known to be NP-hard (general and bipartite graphs) or are proven to be NP-hard in this paper (complete graphs with bi-valued edge weights) or whose complexity question still remains open (trees).

SWISS MADE: Standardized WithIn Class Sum of Squares to evaluate methodologies and dataset elements.

Contemporary high dimensional biological assays, such as mRNA expression microarrays, regularly involve multiple data processing steps, such as experimental processing, computational processing, sample selection, or feature selection (i.e. gene selection), prior to deriving any biological conclusions. These steps can dramatically change the interpretation of an experiment. Evaluation of processing steps has received limited attention in the literature. It is not straightforward to evaluate different processing methods and investigators are often unsure of the best method. We present a simple statistical tool, Standardized WithIn class Sum of Squares (SWISS), that allows investigators to compare alternate data processing methods, such as different experimental methods, normalizations, or technologies, on a dataset in terms of how well they cluster a priori biological classes. SWISS uses Euclidean distance to determine which method does a better job of clustering the data elements based on a priori classifications. We apply SWISS to three different gene expression applications. The first application uses four different datasets to compare different experimental methods, normalizations, and gene sets. The second application, using data from the MicroArray Quality Control (MAQC) project, compares different microarray platforms. The third application compares different technologies: a single Agilent two-color microarray versus one lane of RNA-Seq. These applications give an indication of the variety of problems that SWISS can be helpful in solving. The SWISS analysis of one-color versus two-color microarrays provides investigators who use two-color arrays the opportunity to review their results in light of a single-channel analysis, with all of the associated benefits offered by this design. Analysis of the MACQ data shows differential intersite reproducibility by array platform. SWISS also shows that one lane of RNA-Seq clusters data by biological phenotypes as well as a single Agilent two-color microarray.

Gene expression in human sepsis: what have we learned?

In the previous issue of Critical Care, Tang and colleagues offer a very novel systematic review of 12 studies of gene expression in blood of human sepsis. The review concludes that there is no discernable transition from a pro- to an anti-inflammatory expression phenotype in blood in human sepsis. The authors found that upregulation of pathogen recognition receptors and signal transduction pathways was a consistent theme in expression studies. The review by Tang and colleagues has strengths, including defined screening criteria, broad literature review, strict inclusion criteria, and transparent methods for assessing strengths and weaknesses of studies. There are other issues to consider. First, one source of variation in gene expression studies in sepsis is variability in time from onset of sepsis to time of blood draw. Another source of variation is differences between tissues in gene expression at the same time in sepsis. Whole blood is a mélange of tissues (a variety of leukocytes); therefore, one assesses a weighted sum of all leukocyte classes. About half of the studies assessed peripheral mononuclear cells. A third great source of variable gene expression in sepsis is heterogeneity of causes and microbiology of sepsis. Only one study compared Gram-positive with Gram-negative sepsis. Only three studies confirmed microarray data with an independent measurement of expression. One interpretation is that two of three studies of early sepsis found increased expression of pro-inflammatory genes. In late sepsis, three of six studies found increased expression of pro-inflammatory genes whereas three of six studies found decreased expression of pro-inflammatory genes. The balance of pro- to anti-inflammatory gene expression is difficult to quantify. Sample size is highly variable in studies (n = 12 to 176). These limitations require a leap of faith to suggest that the paradigm of sepsis as a pro-inflammatory phenotype that shifts to an anti-inflammatory phenotype is flawed: the absence of evidence in expression studies is not the same as having well-conducted studies with clear negative evidence.

SWISS MADE: Standardized WithIn Class Sum of Squares to Evaluate Methodologies and Dataset Elements

SWISS MADE: Standardized WithIn Class Sum of Squares to Evaluate Methodologies and Dataset Elements