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Book Embedding
In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings into a ''book'', a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie on this boundary line, called the ''spine'', and the edges are required to stay within a single half-plane. The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph. Book thickness is also called pagenumber, stacknumber or fixed outerthickness. Book embeddings have also been used to define several other graph invariants including the pagewidth and book crossing number. Every graph with vertices has book thickness at most \lceil n/2\rceil, and this formula gives the exact book thickness for complete graphs. The graphs with book thickness one are the outerplanar graphs. The graphs with book thickness at most two are the subhamiltonian graphs, which are always planar; more generally, ev ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circular Layout
In graph drawing, a circular layout is a style of drawing that places the vertices of a graph on a circle, often evenly spaced so that they form the vertices of a regular polygon. Applications Circular layouts are a good fit for communications network topologies such as star or ring networks, and for the cyclic parts of metabolic networks. For graphs with a known Hamiltonian cycle, a circular layout allows the cycle to be depicted as the circle, and in this way circular layouts form the basis of the LCF notation for Hamiltonian cubic graphs. A circular layout may be used on its own for an entire graph drawing, but it also may be used as the layout for smaller clusters of vertices within a larger graph drawing, such as its biconnected components, clusters of genes in a gene interaction graph, or natural subgroups within a social network. If multiple vertex circles are used in this way, other methods such as force-directed graph drawing may be used to arrange the clusters. On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graphs
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mihalis Yannakakis
Mihalis Yannakakis ( el, Μιχάλης Γιαννακάκης; born 13 September 1953 in Athens, Greece)Columbia University: CV: Mihalis Yannakakis (accessed 12 November 2009) is professor of computer science at . He is noted for his work in computational complexity, , and other related fields. He won the |
Paul Chester Kainen
Paul Chester Kainen is an American mathematician, an adjunct associate professor of mathematics and director of the Lab for Visual Mathematics at Georgetown University. Kainen is the author of a popular book on the four color theorem, and is also known for his work on book embeddings of graphs. Biography Kainen received his Bachelor of Arts degree from George Washington University in 1966 and was awarded the Ruggles Prize for Excellence in Mathematics. He went on to get his Ph.D. from Cornell University in 1970 with Peter Hilton Peter John Hilton (7 April 1923Peter Hilton, "On all Sorts of Automorphisms", '' The American Mathematical Monthly'', 92(9), November 1985, p. 6506 November 2010) was a British mathematician, noted for his contributions to homotopy theory and ... as his thesis advisor. Kainen's father was the American artist Jacob Kainen. Selected publications *. 2nd ed., Dover, 1986, , . *. References External linksHome pageat GeorgetownPaul Kainen's Page on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamenta Mathematicae
''Fundamenta Mathematicae'' is a peer-reviewed scientific journal of mathematics with a special focus on the foundations of mathematics, concentrating on set theory, mathematical logic, topology and its interactions with algebra, and dynamical systems. Originally it only covered topology, set theory, and foundations of mathematics: it was the first specialized journal in the field of mathematics..... It is published by the Mathematics Institute of the Polish Academy of Sciences. History The journal was conceived by Zygmunt Janiszewski as a means to foster mathematical research in Poland.According to and to the introduction to the 100th volume of the journal (1978, pp=1–2). These two sources cite an article written by Janiszewski himself in 1918 and titled "''On the needs of Mathematics in Poland''". Janiszewski required that, in order to achieve its goal, the journal should not force Polish mathematicians to submit articles written exclusively in Polish, and should be devoted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michigan State University
Michigan State University (Michigan State, MSU) is a public university, public Land-grant university, land-grant research university in East Lansing, Michigan. It was founded in 1855 as the Agricultural College of the State of Michigan, the first of its kind in the United States. It is considered a Public Ivy, or a public institution which offers an academic experience similar to that of an Ivy League university. After the introduction of the Morrill Land-Grant Acts, Morrill Act in 1862, the state designated the college a land-grant institution in 1863, making it the first of the land-grant colleges in the United States. The college became coeducational in 1870. In 1955, the state officially made the college a university, and the current name, Michigan State University, was adopted in 1964. Today, Michigan State has the largest undergraduate enrollment among Michigan's colleges and universities and approximately 634,300 living alums worldwide. The university is a member of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, Unknot, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning. Algebraic structures, with their associated homomorphisms, form mathematical categories. Category theory is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures. Universal algebra is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety of groups''. History Before the nineteenth century, algebra meant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoknot
__NOTOC__ A pseudoknot is a nucleic acid secondary structure containing at least two stem-loop structures in which half of one stem is intercalated between the two halves of another stem. The pseudoknot was first recognized in the turnip yellow mosaic virus in 1982. Pseudoknots fold into knot-shaped three-dimensional conformations but are not true topological knots. Prediction and identification The structural configuration of pseudoknots does not lend itself well to bio-computational detection due to its context-sensitivity or "overlapping" nature. The base pairing in pseudoknots is not well nested; that is, base pairs occur that "overlap" one another in sequence position. This makes the presence of pseudoknots in RNA sequences more difficult to predict by the standard method of dynamic programming, which use a recursive scoring system to identify paired stems and consequently, most cannot detect non-nested base pairs. The newer method of stochastic context-free grammars su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nucleic Acid Secondary Structure
Nucleic acid secondary structure is the basepairing interactions within a single nucleic acid polymer or between two polymers. It can be represented as a list of bases which are paired in a nucleic acid molecule. The secondary structures of biological DNAs and RNAs tend to be different: biological DNA mostly exists as fully base paired double helices, while biological RNA is single stranded and often forms complex and intricate base-pairing interactions due to its increased ability to form hydrogen bonds stemming from the extra hydroxyl group in the ribose sugar. In a non-biological context, secondary structure is a vital consideration in the nucleic acid design of nucleic acid structures for DNA nanotechnology and DNA computing, since the pattern of basepairing ultimately determines the overall structure of the molecules. Fundamental concepts Base pairing In molecular biology, two nucleotides on opposite complementary DNA or RNA strands that are connected via hydrogen b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
