In mathematicsthe harmonic series is the divergent infinite series. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music. The divergence of the harmonic series was first proven in the 14th century by Nicole Oresme[1] but this achievement fell into obscurity. Historically, harmonic sequences have had a certain popularity with architects.

This was so particularly in the Baroque period, when architects used them to establish the proportions of floor plansof elevationsand to establish harmonic relationships between both interior and exterior architectural details of churches and palaces. There are several well-known proofs of the divergence of the harmonic series. A few of them are given below. One way to prove divergence is to compare the harmonic series with another divergent series, where each denominator is replaced with the next-largest power of two :.

Each term of the harmonic series is greater than or equal to the corresponding term of the second series, and therefore the sum of the harmonic series must be greater than or equal to the sum of the second series.

However, the sum of the second series is infinite:. It follows by the comparison test that the sum of the harmonic series must be infinite as well. More precisely, the comparison above proves that. This proof, proposed by Nicole Oresme in aroundis considered by many in the mathematical community [ by whom?

It is still a standard proof taught in mathematics classes today. Cauchy's condensation test is a generalization of this argument. It is possible to prove that the harmonic series diverges by comparing its sum with an improper integral.

Specifically, consider the arrangement of rectangles shown in the figure to the right.

### Frank Bidart

Since this area is entirely contained within the rectangles, the total area of the rectangles must be infinite as well. More precisely, this proves that. The generalization of this argument is known as the integral test. The harmonic series diverges very slowly. For example, the sum of the first 10 43 terms is less than In particular.

Leonhard Euler proved both this and also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i. The difference between H n and ln n converges to the Euler—Mascheroni constant.

The difference between any two harmonic numbers is never an integer. This series converges by the alternating series test.G E Farr, Binary functions, degeneracy, and alternating dimaps Discrete Mathematics 5 R Robinson and G E Farr, Structure and recognition of graphs with no 6-wheel subdivision, Algorithmica 55 K Edwards and G Farr, Planarization and fragmentability of some classes of graphs, Discrete Mathematics Accepted version :.

Morgan and G. Also see eigencircle web page and applet.

G E Farr, The complexity of counting colourings of subgraphs of the grid, Combinatorics, Probability and Computing 15 G E Farr, On problems with short certificates. Acta Informatica 31 G E Farr, A generalization of the Whitney rank generating function.

Mathematical Proceedings of the Cambridge Philosophical Society G E Farr, A correlation inequality involving stable set and chromatic polynomials. Journal of Combinatorial Theory Series B 58 G E Farr, The subgraph homeomorphism problem for small wheels, Discrete Mathematics 71 Estivill-Castro and G. Dobbie eds. G Farr, The Imitation Game: is it history, drama or myth? G Farr, Calls for a posthumous pardonIn mathematicsa hypergraph is a generalization of a graph in which an edge can join any number of vertices.

In contrast, in an ordinary graph, an edge connects exactly two vertices. The size of the vertex set is called the order of the hypergraphand the size of edges set is the size of the hypergraph. While graph edges are 2-element subsets of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes.

However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k.

### Graham Farr: selected publications

In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes. So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. A hypergraph is also called a set system or a family of sets drawn from the universal set. Hypergraphs can be viewed as incidence structures. In particular, there is a bipartite "incidence graph" or " Levi graph " corresponding to every hypergraph, and conversely, most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs.

Hypergraphs have many other names. In computational geometrya hypergraph may sometimes be called a range space and then the hyperedges are called ranges. In some literature edges are referred to as hyperlinks or connectors. Special kinds of hypergraphs include: k -uniform onesas discussed briefly above; clutterswhere no edge appears as a subset of another edge; and abstract simplicial complexeswhich contain all subsets of every edge.

The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms. Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called subhypergraphspartial hypergraphs and section hypergraphs. A subhypergraph is a hypergraph with some vertices removed. An alternative term is the restriction of H to A.

The partial hypergraph is a hypergraph with some edges removed. When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involutioni. A connected graph G with the same vertex set as a connected hypergraph H is a host graph for H if every hyperedge of H induces a connected subgraph in G. For a disconnected hypergraph HG is a host graph if there is a bijection between the connected components of G and of Hsuch that each connected component G' of G is a host of the corresponding H'.

The 2-section or clique graphrepresenting graphprimal graphGaifman graph of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge. A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BGand x 1e 1 are connected with an edge if and only if vertex x 1 is contained in edge e 1 in H. Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above.

This bipartite graph is also called incidence graph.

In contrast with ordinary undirected graphs for which there is a single natural notion of cycles and acyclic graphsthere are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special case of ordinary graphs.

A first definition of acyclicity for hypergraphs was given by Claude Berge : [5] a hypergraph is Berge-acyclic if its incidence graph the bipartite graph defined above is acyclic. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph.

This notion of acyclicity is equivalent to the hypergraph being conformal every clique of the primal graph is covered by some hyperedge and its primal graph being chordal ; it is also equivalent to reducibility to the empty graph through the GYO algorithm [7] [8] also known as Graham's algorithma confluent iterative process which removes hyperedges using a generalized definition of ears. However, none of the reverse implications hold, so those four notions are different.GWAS owe their popularity to the expectation that they will make a major impact on diagnosis, prognosis and management of disease by uncovering genetics underlying clinical phenotypes.

The dominant paradigm in GWAS data analysis so far consists of extensive reliance on methods that emphasize contribution of individual SNPs to statistical association with phenotypes. Multivariate methods, however, can extract more information by considering associations of multiple SNPs simultaneously.

Recent advances in other genomics domains pinpoint multivariate causal graph-based inference as a promising principled analysis framework for high-throughput data. Designed to discover biomarkers in the local causal pathway of the phenotype, these methods lead to accurate and highly parsimonious multivariate predictive models.

Using these SNPs we develop two predictive models that can classify cases and disease-free controls with an accuracy of 0. The predictive performance of these models generalizes reasonably well to Swedish subjects from the closely related but not identical Epidemiological Investigation of Rheumatoid Arthritis EIRA cohort with 0.

This suggests that principled multivariate causal and predictive framework for GWAS analysis empowers the community with a new tool for high-quality and more efficient discovery. This article was reviewed by Prof. Anthony Almudevar, Dr. Eugene V. Koonin, and Prof. Marianthi Markatou. Genome-wide association studies GWAS are considered to be one of the primary tools for determining genetic links to disease.

GWAS have been abundant in recent scientific research with more than primary large-scale studies performed in the last 5 years. Each of these studies has genotyped at leastsingle nucleotide polymorphisms SNPs in cohorts that often exceed 1, subjects [ 1 ]. Overall, the reason for popularity of GWAS is the expectation that they will lead to discovery of SNPs implicated in disease and development of predictive models that can facilitate diagnosis, management, and treatment of disease.

Despite recent expansion of genome-wide association studies, methodologies for statistical analysis of the resulting data are still lagging behind. The most dominant paradigm for such analyses is focused on assessing contribution of individual SNPs to statistical association or risk of developing a phenotype [ 2 - 5 ].

Multivariate methods take a step forward by shifting the focus on how combining the individual SNP signals can help classify the phenotypes, and thus uncover additional evidence for possible genetic risk factors. Among multivariate techniques, Bayesian networks, kernel-based classifiers and multivariate regression are making their way as candidate new methodologies for the analysis of GWAS data [ 6 - 11 ].

Of particular relevance to the goals of GWAS are recent multivariate causal graph-based methods that are computationally efficient and can scale well to the dimensionality of GWAS data [ 12 - 14 ]. Local causal biomarkers constitute the Markov boundary and yield the highest accuracy predictions of the phenotype, while other biomarkers do not contribute additional predictive information beyond what is contained in the local causal ones [ 12 - 15 ]. In addition, the set of local causal biomarkers exhibits maximum parsimony, beyond which predictive accuracy is compromised [ 12 - 1416 ].

Graphical representation of the local pathway concept. The local pathway of the phenotype shown with the ash blue colour contains all its direct causes C 1C 2C 3direct effects E 1E 2E 3and direct causes of the direct effects CE 1. This is exactly the Markov boundary of the phenotype.The purpose of the fee is to recover costs associated with the development of data collections included in such sites. Your institution may already be a subscriber. Follow the links above to find out more about the data in these sites and their terms of usage.

Go To: TopReferencesNotes. Data compilation copyright by the U. Secretary of Commerce on behalf of the U. All rights reserved. Wallace, director. View large format table.

Hu, Lu, et al. Ashes and Haken, Ashes, J.

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Structure-retention increments of aliphatic estersJ. Chastrette, Heintz, et al. Germaine and Haken, Germaine, R. Part 1. Simple aliphatic estersJ. Hazzit, Baaliouamer, et al. Food Chem. Rasmussen, Rasmussen, P. Chretien and Dubois, Chretien, J. Garruti, Franco, et al. Scheidig, Czerny, et al.

Wu, Zorn, et al. Tret'yakov, Tret'yakov, K. Jagella and Grosch, Jagella, T. Evaluation of potent odorants of black pepper by dilution and concentration techniquesEur. Food Res. Owens J. Food Agric. Pinto, Guedes, et al. Cantergiani, Brevard, et al.

Tairu, Hofmann, et al. Chemical composition of essential oils of oregano: Origanum syriacum L. Ietswaart, and O. Leffingwell and Alford, Leffingwell, J. Tokitomo, Steihaus, et al. Larsen and Frisvad, Larsen, T.A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. As such, they are the essential basis of all valid deductive inferences particularly in logicmathematics and sciencewhere the process of verification is necessary to determine whether a generalization holds true for any given situation.

Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them.

However, the parts cannot be generalized into a whole—until a common relation is established among all parts. This does not mean that the parts are unrelated, only that no common relation has been established yet for the generalization. The concept of generalization has broad application in many connected disciplines, and might sometimes have a more specific meaning in a specialized context e.

In general, given two related concepts A and B, A is a "generalization" of B equiv. For example, the concept animal is a generalization of the concept birdsince every bird is an animal, but not all animals are birds dogsfor instance. For more, see Specialisation biology.

## Inventor's Paradox

The connection of generalization to specialization or particularization is reflected in the contrasting words hypernym and hyponym. A hypernym as a generic stands for a class or group of equally ranked items, such as the term tree which stands for equally ranked items such as peach and oakand the term ship which stands for equally ranked items such as cruiser and steamer.

In contrast, a hyponym is one of the items included in the generic, such as peach and oak which are included in treeand cruiser and steamer which are included in ship. A hypernym is superordinate to a hyponym, and a hyponym is subordinate to a hypernym. An animal is a generalization of a mammala birda fishan amphibian and a reptile.

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Generalization has a long history in cartography as an art of creating maps for different scale and purpose. Cartographic generalization is the process of selecting and representing information of a map in a way that adapts to the scale of the display medium of the map.

In this way, every map has, to some extent, been generalized to match the criteria of display. This includes small cartographic scale maps, which cannot convey every detail of the real world.

As a result, cartographers must decide and then adjust the content within their maps, to create a suitable and useful map that conveys the geospatial information within their representation of the world. Generalization is meant to be context-specific. That is to say, correctly generalized maps are those that emphasize the most important map elements, while still representing the world in the most faithful and recognizable way. The level of detail and importance in what is remaining on the map must outweigh the insignificance of items that were generalized—so as to preserve the distinguishing characteristics of what makes the map useful and important.

From Wikipedia, the free encyclopedia. Look up generalization in Wiktionary, the free dictionary. There are instances of concept A which are not instances of concept B. See also: Semantic change. Main article: Cartographic generalization. Math Vault. Retrieved Axis Maps. Categories : Generalizations. Namespaces Article Talk. Views Read Edit View history.

Help Community portal Recent changes Upload file. Download as PDF Printable version.The paper considers a project scheduling problem in weighted directed graphs in which arcs represent operations while nodes are identified with starting and finishing endpoints of the operations; arc lengths represent operation durations. The problem is to find the earliest starting times for all operations. This problem generalizes the shortest path problem and the critical path problem. Authors: George M.

Adelson-VelskyEugene Levner. Search Search. Volume 45, Issue 2 May Volume 45, Issue 1 February View PDF. Go to Section. Home Mathematics of Operations Research Vol. George M. Eugene Levner.

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