scholarly journals A classification of all 1-Salem graphs

2014 ◽  
Vol 17 (1) ◽  
pp. 582-594 ◽  
Author(s):  
Lee Gumbrell ◽  
James McKee
Keyword(s):  
New York ◽  
Minimal Polynomial ◽  
Cyclotomic Polynomials ◽  
Critical Interval ◽  
Combinatorial Structures ◽  
Induced Subgraph ◽  
Combinatorial Objects ◽  
Salem Number ◽  
Bipartite Case

AbstractOne way to study certain classes of polynomials is by considering examples that are attached to combinatorial objects. Any graph $G$ has an associated reciprocal polynomial $R_{G}$, and with two particular classes of reciprocal polynomials in mind one can ask the questions: (a) when is $R_{G}$ a product of cyclotomic polynomials (giving the cyclotomic graphs)? (b) when does $R_{G}$ have the minimal polynomial of a Salem number as its only non-cyclotomic factor (the non-trivial Salem graphs)? Cyclotomic graphs were classified by Smith (Combinatorial structures and their applications, Proceedings of Calgary International Conference, Calgary, AB, 1969 (eds R. Guy, H. Hanani, H. Saver and J. Schönheim; Gordon and Breach, New York, 1970) 403–406); the maximal connected ones are known as Smith graphs. Salem graphs are ‘spectrally close’ to being cyclotomic, in that nearly all their eigenvalues are in the critical interval $[-2,2]$. On the other hand, Salem graphs do not need to be ‘combinatorially close’ to being cyclotomic: the largest cyclotomic induced subgraph might be comparatively tiny.We define an $m$-Salem graph to be a connected Salem graph $G$ for which $m$ is minimal such that there exists an induced cyclotomic subgraph of $G$ that has $m$ fewer vertices than $G$. The $1$-Salem subgraphs are both spectrally close and combinatorially close to being cyclotomic. Moreover, every Salem graph contains a $1$-Salem graph as an induced subgraph, so these $1$-Salem graphs provide some necessary substructure of all Salem graphs. The main result of this paper is a complete combinatorial description of all $1$-Salem graphs: in the non-bipartite case there are $25$ infinite families and $383$ sporadic examples.

Galois theory of Salem polynomials

2009 ◽  
Vol 148 (1) ◽  
pp. 47-54 ◽  
Author(s):  
CHRISTOS CHRISTOPOULOS ◽  
JAMES MCKEE
Keyword(s):  
Unit Circle ◽  
Galois Group ◽  
Galois Theory ◽  
Minimal Polynomial ◽  
Galois Groups ◽  
Cyclotomic Polynomials ◽  
Salem Number ◽  
Extra Structure

AbstractLet f(x) ∈ [x] be a monic irreducible reciprocal polynomial of degree 2d with roots r1, 1/r1, r2, 1/r2, . . ., rd, 1/rd. The corresponding trace polynomial g(x) of degree d is the polynomial whose roots are r1 + 1/r1, . . ., rd + 1/rd. If the Galois groups of f and g are Gf and Gg respectively, then Gg ≅ Gf/N, where N is isomorphic to a subgroup of C2d. In a naive sense, the generic case is Gf ≅ C2d ⋊ Sd, with N ≅ C2d and Gg ≅ Sd. When f(x) has extra structure this may be reflected in the Galois group, and it is not always true even that Gf ≅ N ⋊ Gg. For example, for cyclotomic polynomials f(x) = Φn(x) it is known that Gf ≅ N ⋊ Gg if and only if n is divisible either by 4 or by some prime congruent to 3 modulo 4.In this paper we deal with irreducible reciprocal monic polynomials f(x) ∈ [x] that are ‘close’ to being cyclotomic, in that there is one pair of real positive reciprocal roots and all other roots lie on the unit circle. With the further restriction that f(x) has degree at least 4, this means that f(x) is the minimal polynomial of a Salem number. We show that in this case one always has Gf ≅ N ⋊ Gg, and moreover that N ≅ C2d or C2d−1, with the latter only possible if d is odd.

VI.—New Classifications of the Brachiopoda

1893 ◽  
Vol 10 (7) ◽  
pp. 318-323
Author(s):  
Agnes Crane
Keyword(s):  
New York ◽  
History Of ◽  
The Individual

Mr. Charles Schuchert of Newhaven, Conn., U.S.A., has recently published in the “American Geologist” (Vol. xi.No. 3) an important and highly suggestive “Classification of the Brachiopoda,” based on the history of the class (Chronogenesis) and the ontogeny of the individual. It embodies the latest results of the remarkable investigations on the Palaeozoic forms of Prof. James Hall and Mr. J. M. Clarke, who have thrown so much light on the evolution of genera among the Brachiopoda in the eighth volume of “The Palæontology of New York” (Part I. Brachiopoda, 1892).

Automated Text Classification of News Articles: A Practical Guide

2020 ◽  
Vol 29 (1) ◽  
pp. 19-42 ◽  
Author(s):  
Pablo Barberá ◽  
Amber E. Boydstun ◽  
Suzanna Linn ◽  
Ryan McMahon ◽  
Jonathan Nagler
Keyword(s):  
New York ◽  
New York Times ◽  
Fixed Number ◽  
Machine Learning Algorithms ◽  
Supervised Machine Learning ◽  
Methodological Choices ◽  
Units Of Analysis ◽  
Human Validation ◽  
Corpus Selection

Automated text analysis methods have made possible the classification of large corpora of text by measures such as topic and tone. Here, we provide a guide to help researchers navigate the consequential decisions they need to make before any measure can be produced from the text. We consider, both theoretically and empirically, the effects of such choices using as a running example efforts to measure the tone of New York Times coverage of the economy. We show that two reasonable approaches to corpus selection yield radically different corpora and we advocate for the use of keyword searches rather than predefined subject categories provided by news archives. We demonstrate the benefits of coding using article segments instead of sentences as units of analysis. We show that, given a fixed number of codings, it is better to increase the number of unique documents coded rather than the number of coders for each document. Finally, we find that supervised machine learning algorithms outperform dictionaries on a number of criteria. Overall, we intend this guide to serve as a reminder to analysts that thoughtfulness and human validation are key to text-as-data methods, particularly in an age when it is all too easy to computationally classify texts without attending to the methodological choices therein.

scholarly journals Considering context and dynamics: A classification of transit-orientated development for New York City

2020 ◽  
Vol 85 ◽  
pp. 102711
Author(s):  
Yunzhe Liu ◽  
Alex Singleton ◽  
Daniel Arribas-Bel
Keyword(s):  
New York ◽  
New York City ◽  
York City

Classification of cardiomyopathies

2018 ◽  
Author(s):  
Eloisa Arbustini ◽  
Valentina Favalli ◽  
Alessandro Di Toro ◽  
Alessandra Serio ◽  
Jagat Narula
Keyword(s):  
New York ◽  
American Heart Association ◽  
American Heart ◽  
Heart Association ◽  
Genetic Mutation ◽  
Phenotypic Data ◽  
Septal Thickness ◽  
New York Heart Association ◽  
European Society Of Cardiology

For over 50 years, the definition and classification of cardiomyopathies have remained anchored in the concept of ventricular dysfunction and myocardial structural remodelling due to unknown cause. The concept of idiopathic was first challenged in 2006, when the American Heart Association classification subordinated the phenotype to the aetiology. Cardiomyopathies were classified as genetic, acquired, and mixed. In 2008, the European Society of Cardiology proposed a phenotype-driven classification that separated familial (genetic) from non-familial (non-genetic) forms of cardiomyopathy. Both classifications led the way to a precise phenotypic and aetiological description of the disease and moved away from the previously held notion of idiopathic disease. In 2013, the World Heart Federation introduced a descriptive and flexible nosology—the MOGE(S) classification—describing the morphofunctional (M) phenotype of cardiomyopathy, the involvement of additional organs (O), the familial/genetic (G) origin, and the precise description of the (a)etiology including genetic mutation, if applicable (E); reporting of functional status such as American College of Cardiology/American Heart Association stage and New York Heart Association classification (S) was left optional. MOGE(S) is a bridge between the past and the future. It allows description of comprehensive phenotypic data, all genetic and non-genetic causes of cardiomyopathy, and incorporates description of familial clustering in a genetic disease. MOGE(S) is the instrument of precision diagnosis for cardiomyopathies. The addition of the early and unaffected phenotypes to the (M) descriptor outlines the clinical profile of an early affected family member; the examples include non-dilated hypokinetic cardiomyopathy in dilated cardiomyopathy and septal thickness (13–14 mm) in hypertrophic cardiomyopathy classes.

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