Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V, not the zero vector v is an eigenvector of T if T is a scalar multiple of v; this condition can be written as the equation T = λ v, where λ is a scalar in the field F, known as the eigenvalue, characteristic value, or characteristic root associated with the eigenvector v. If the vector space V is finite-dimensional the linear transformation T can be represented as a square matrix A, the vector v by a column vector, rendering the above mapping as a matrix multiplication on the left-hand side and a scaling of the column vector on the right-hand side in the equation A v = λ v. There is a direct correspondence between n-by-n square matrices and linear transformations from an n-dimensional vector space to itself, given any basis of the vector space.
For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction, stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations; the prefix eigen- is adopted from the German word eigen for "proper", "characteristic". Utilized to study principal axes of the rotational motion of rigid bodies and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, matrix diagonalization. In essence, an eigenvector v of a linear transformation T is a non-zero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue.
This condition can be written as the equation T = λ v, referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar. For example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex; the Mona Lisa example pictured at right provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point; the linear transformation in this example is called a shear mapping. Points in the top half are moved to the right and points in the bottom half are moved to the left proportional to how far they are from the horizontal axis that goes through the middle of the painting; the vectors pointing to each point in the original image are therefore tilted right or left and made longer or shorter by the transformation. Notice that points along the horizontal axis do not move at all. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation because the mapping does not change its direction.
Moreover, these eigenvectors all have an eigenvalue equal to one because the mapping does not change their length, either. Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can take many forms. For example, the linear transformation could be a differential operator like d d x, in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as d d x e λ x = λ e λ x. Alternatively, the linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices that are referred to as eigenvectors. If the linear transformation is expressed in the form of an n by n matrix A the eigenvalue equation above for a linear transformation can be rewritten as the matrix multiplication A v = λ v, where the eigenvector v is an n by 1 matrix. For a matrix and eigenvectors can be used to decompose the matrix, for example by diagonalizing it. Eigenvalues and eigenvectors give rise to many related mathematical concepts, the prefix eigen- is applied liberally when naming them: The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation.
The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace or characteristic space of T. If the set of eigenvectors of T form a basis of the domain of T this basis is called an eigenbasis. Eigenvalues are introduced in the context of linear algebra or matrix theory. However, they arose in the study of quadratic forms and differential equations. In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the pri
Paul Seymour (mathematician)
Paul Seymour is a professor at Princeton University. His research interest is in discrete mathematics graph theory, he was responsible for important progress on regular matroids and unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs. Many of his recent papers are available from his website, he won a Sloan Fellowship in 1983, the Ostrowski Prize in 2004. He received an honorary doctorate from the University of Waterloo in 2008 and one from the Technical University of Denmark in 2013. Seymour was born in Plymouth, England, he was a day student at Plymouth College, studied at Exeter College, gaining a BA degree in 1971, D. Phil in 1975. From 1974–1976 he was a college research fellow at University College of Swansea, returned to Oxford for 1976–1980 as a Junior Research Fellow at Merton College, with the year 1978–79 at University of Waterloo, he became an associate and a full professor at Ohio State University, Ohio, between 1980 and 1983, where he began research with Neil Robertson, a fruitful collaboration that continued for many years.
From 1983 until 1996, he was at Bellcore, New Jersey. He was an adjunct professor at Rutgers University from 1984–1987 and at the University of Waterloo from 1988–1993, he became professor at Princeton University in 1996. He is Editor-in-Chief for the Journal of Graph Theory, he married Shelley MacDonald of Ottawa in 1979, they have two children and Emily. The couple separated amicably in 2007, his brother Leonard W. Seymour is Professor of gene therapy at Oxford University. Combinatorics in Oxford in the 1970s was dominated by matroid theory, due to the influence of Dominic Welsh and Aubrey William Ingleton. Much of Seymour's early work, up to about 1980, was on matroid theory, included three important matroid results: his D. Phil. Thesis on matroids with the max-flow min-cut property. There were several other significant papers from this period: a paper with Welsh on the critical probabilities for bond percolation on the square lattice. In 1980 he moved to Ohio State University, began work with Neil Robertson.
This led to Seymour's most important accomplishment, the so-called "Graph Minors Project", a series of 23 papers, published over the next thirty years, with several significant results: the graph minors structure theorem, that for any fixed graph, all graphs that do not contain it as a minor can be built from graphs that are of bounded genus by piecing them together at small cutsets in a tree structure. In about 1990 Robin Thomas began to work with Seymour, their collaboration resulted in several important joint papers over the next ten years: a proof of a conjecture of Sachs, characterizing by excluded minors the graphs that admit linkless embeddings in 3-space. In 2000 the trio were supported by the American Institute of Mathematics to work on the strong perfect graph conjecture, a famous open question, raised by Claude Berge in the early 1960s. Seymour's student Maria Chudnovsky joined them in 2001, in 2002 the four jointly proved the conjecture. Seymour continued to work with Chudnovsky, obtained several more results about induced subgraphs, in particular a polynomial-time algorithm to test whether a graph is perfect, a general description of all claw-free graphs
In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph K5 nor the complete bipartite graph K3,3; the Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs, preserved by deletions and edge contractions. For every fixed graph H, it is possible to test whether H is a minor of an input graph G in polynomial time. Other results and conjectures involving graph minors include the graph structure theorem, according to which the graphs that do not have H as a minor may be formed by gluing together simpler pieces, Hadwiger's conjecture relating the inability to color a graph to the existence of a large complete graph as a minor of it. Important variants of graph minors include the topological minors and immersion minors.
An edge contraction is an operation which removes an edge from a graph while merging the two vertices it used to connect. An undirected graph H is a minor of another undirected graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, deleting some isolated vertices; the order in which a sequence of such contractions and deletions is performed on G does not affect the resulting graph H. Graph minors are studied in the more general context of matroid minors. In this context, it is common to assume that all graphs are connected, with self-loops and multiple edges allowed; this point of view has the advantage that edge deletions leave the rank of a graph unchanged, edge contractions always reduce the rank by one. In other contexts it makes more sense to allow the deletion of a cut-edge, to allow disconnected graphs, but to forbid multigraphs. In this variation of graph minor theory, a graph is always simplified after any edge contraction to eliminate its self-loops and multiple edges.
A function f is referred to as "minor-monotone". In the following example, graph H is a minor of graph G: H. G; the following diagram illustrates this. First construct a subgraph of G by deleting the dashed edges, contract the gray edge: It is straightforward to verify that the graph minor relation forms a partial order on the isomorphism classes of undirected graphs: it is transitive, G and H can only be minors of each other if they are isomorphic because any nontrivial minor operation removes edges or vertices. A deep result by Neil Robertson and Paul Seymour states that this partial order is a well-quasi-ordering: if an infinite list G1, G2... of finite graphs is given there always exist two indices i < j such that Gi is a minor of Gj. Another equivalent way of stating this is that any set of graphs can have only a finite number of minimal elements under the minor ordering; this result proved a conjecture known as Wagner's conjecture, after Klaus Wagner. In the course of their proof and Robertson prove the graph structure theorem in which they determine, for any fixed graph H, the rough structure of any graph which does not have H as a minor.
The statement of the theorem is itself long and involved, but in short it establishes that such a graph must have the structure of a clique-sum of smaller graphs that are modified in small ways from graphs embedded on surfaces of bounded genus. Thus, their theory establishes fundamental connections between graph minors and topological embeddings of graphs. For any graph H, the simple H-minor-free graphs must be sparse, which means that the number of edges is less than some constant multiple of the number of vertices. More if H has h vertices a simple n-vertex simple H-minor-free graph can have at most O edges, some Kh-minor-free graphs have at least this many edges. Thus, if H has h vertices H-minor-free graphs have average degree O and furthermore degeneracy O. Additionally, the H-minor-free graphs have a separator theorem similar to the planar separator theorem for planar graphs: for any fixed H, any n-vertex H-minor-free graph G, it is possible to find a subset of O vertices whose removal splits G into two subgraphs with at most 2n/3 vertices per subgraph.
Stronger, for any fixed H, H-minor-free graphs have treewidth O. The Hadwiger conjecture in graph theory proposes that if a graph G does not contain a minor isomorphic to the complete graph on k vertices G has a proper coloring with k − 1 colors
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the Journal Citation Reports, the journal has a 2013 impact factor of 0.840. Proceedings of the American Mathematical Society publishes articles from all areas of pure and applied mathematics, including topology, analysis, number theory, logic and statistics; this journal is indexed in the following databases: Mathematical Reviews Zentralblatt MATH Science Citation Index Science Citation Index Expanded ISI Alerting Services CompuMath Citation Index Current Contents / Physical, Chemical & Earth Sciences. Bulletin of the American Mathematical Society Memoirs of the American Mathematical Society Notices of the American Mathematical Society Journal of the American Mathematical Society Transactions of the American Mathematical Society Official website Proceedings of the American Mathematical Society on JSTOR
Not to be confused with László M. Lovász, a different combinatorial mathematician who works with Jacob Fox. László Lovász is a Hungarian mathematician, best known for his work in combinatorics, for which he was awarded the Wolf Prize and the Knuth Prize in 1999, the Kyoto Prize in 2010, he is the current president of the Hungarian Academy of Sciences. He served as president of the International Mathematical Union between January 1, 2007 and December 31, 2010. Lovász was born on March 1948 in the city of Budapest, his father was a surgeon. When Lovász was 14 he found a mathematical article written by Paul Erdős. One year he became acquainted with Erdős, they talked about mathematics and other subjects. This experience inspired Lovász in searching for more knowledge. In high school, Lovász won gold medals at the International Mathematical Olympiad. Lovász received his Candidate of Sciences degree in 1970 at the Hungarian Academy of Sciences, his advisor was Tibor Gallai. Until 1975, Lovász worked at Eötvös Loránd University, between 1975–1982, he led the Department of Geometry at the University of Szeged.
In 1982, he returned to the Eötvös University. The former and current scientists of the department include György Elekes, András Frank, József Beck, Éva Tardos, András Hajnal, Lajos Pósa, Miklós Simonovits, Tamás Szőnyi. Lovász was a professor at Yale University during the 1990s and was a collaborative member of the Microsoft Research Center until 2006, he returned to Eötvös Loránd University, where he was the director of the Mathematical Institute. In 2014 he was elected the President of the Hungarian Academy of Sciences. Lovász is married to Katalin Vesztergombi. Lovász was awarded the Brouwer Medal in 1993, the Wolf Prize in 1999, the Bolyai prize in 2007 and Hungary's Széchenyi Grand Prize, he received the Advanced Grant of the European Research Council. He was elected foreign member of the Royal Netherlands Academy of Arts and Sciences and Royal Swedish Academy of Sciences, honorary member of the London Mathematical Society, he received the Kyoto Prize for Basic Science. In 2012 he became a fellow of the American Mathematical Society.
Lovász is listed as an ISI cited researcher. Topological combinatorics Lovász conjecture Erdős–Faber–Lovász conjecture Lovász local lemma Lenstra–Lenstra–Lovász lattice basis reduction algorithm Geometry of numbers Perfect graph theorem Greedoid Bell number Lovász number Graph limit Lovász's Homepage László Lovász at the Mathematics Genealogy Project "László Lovász's results". International Mathematical Olympiad
Alexander Schrijver is a Dutch mathematician and computer scientist, a professor of discrete mathematics and optimization at the University of Amsterdam and a fellow at the Centrum Wiskunde & Informatica in Amsterdam. Since 1993 he has been co-editor in chief of the journal Combinatorica. Schrijver earned his Ph. D. in 1977 from the Vrije Universiteit in Amsterdam, under the supervision of Pieter Cornelis Baayen. He worked for the Centrum Wiskunde & Informatica in pure mathematics from 1973 to 1979, was a professor at Tilburg University from 1983 to 1989. In 1989 he rejoined the Centrum Wiskunde & Informatica, in 1990 he became a professor at the University of Amsterdam. In 2005, he stepped down from management at CWI and instead became a CWI Fellow. Schrijver was one of the winners of the Delbert Ray Fulkerson Prize of the American Mathematical Society in 1982 for his work with Martin Grötschel and László Lovász on applications of the ellipsoid method to combinatorial optimization, he won the INFORMS Frederick W. Lanchester Prize in 1986 for his book Theory of Linear and Integer Programming, again in 2004 for his book Combinatorial Optimization: Polyhedra and Efficiency.
In 2003, he won the George B. Dantzig Prize of the Mathematical Programming Society and SIAM for "deep and fundamental research contributions to discrete optimization". In 2006, he was a joint winner of the INFORMS John von Neumann Theory Prize with Grötschel and Lovász for their work in combinatorial optimization, in particular for their joint work in the book Geometric Algorithms and Combinatorial Optimization showing the polynomial-time equivalence of separation and optimization. In 2008, his work with Adri Steenbeek on scheduling the Dutch train system was honored with INFORMS' Franz Edelman Award for Achievement in Operations Research and the Management Sciences, he won the SIGMA prize of the Dutch SURF foundation for a mathematics education project. In 2015 he won the highest distinction within Operations Research in Europe. In 2005 Schrijver won the Spinoza Prize of the NWO, the highest scientific award in the Netherlands, for his research in combinatorics and algorithms. In the same year he became a Knight of the Order of the Netherlands Lion.
In 2002, Schrijver received an honorary doctorate from the University of Waterloo in Canada, in 2011 he received another one from Eötvös Loránd University in Hungary. Schrijver became a member of the Royal Netherlands Academy of Arts and Sciences in 1995, he became a corresponding member of the North Rhine-Westphalia Academy for Sciences and Arts in 2005, joined the German Academy of Sciences Leopoldina in 2006, was elected to the Academia Europaea in 2008. In 2012 he became a fellow of the American Mathematical Society. Theory of Linear and Integer Programming Geometric Algorithms and Combinatorial Optimization Combinatorial Optimization Combinatorial Optimization: Polyhedra and Efficiency
ArXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, astronomy, electrical engineering, computer science, quantitative biology, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month. ArXiv was made possible by the compact TeX file format, which allowed scientific papers to be transmitted over the Internet and rendered client-side. Around 1990, Joanne Cohn began emailing physics preprints to colleagues as TeX files, but the number of papers being sent soon filled mailboxes to capacity. Paul Ginsparg recognized the need for central storage, in August 1991 he created a central repository mailbox stored at the Los Alamos National Laboratory which could be accessed from any computer.
Additional modes of access were soon added: FTP in 1991, Gopher in 1992, the World Wide Web in 1993. The term e-print was adopted to describe the articles, it began as a physics archive, called the LANL preprint archive, but soon expanded to include astronomy, computer science, quantitative biology and, most statistics. Its original domain name was xxx.lanl.gov. Due to LANL's lack of interest in the expanding technology, in 2001 Ginsparg changed institutions to Cornell University and changed the name of the repository to arXiv.org. It is now hosted principally with eight mirrors around the world, its existence was one of the precipitating factors that led to the current movement in scientific publishing known as open access. Mathematicians and scientists upload their papers to arXiv.org for worldwide access and sometimes for reviews before they are published in peer-reviewed journals. Ginsparg was awarded a MacArthur Fellowship in 2002 for his establishment of arXiv; the annual budget for arXiv is $826,000 for 2013 to 2017, funded jointly by Cornell University Library, the Simons Foundation and annual fee income from member institutions.
This model arose in 2010, when Cornell sought to broaden the financial funding of the project by asking institutions to make annual voluntary contributions based on the amount of download usage by each institution. Each member institution pledges a five-year funding commitment to support arXiv. Based on institutional usage ranking, the annual fees are set in four tiers from $1,000 to $4,400. Cornell's goal is to raise at least $504,000 per year through membership fees generated by 220 institutions. In September 2011, Cornell University Library took overall administrative and financial responsibility for arXiv's operation and development. Ginsparg was quoted in the Chronicle of Higher Education as saying it "was supposed to be a three-hour tour, not a life sentence". However, Ginsparg remains on the arXiv Scientific Advisory Board and on the arXiv Physics Advisory Committee. Although arXiv is not peer reviewed, a collection of moderators for each area review the submissions; the lists of moderators for many sections of arXiv are publicly available, but moderators for most of the physics sections remain unlisted.
Additionally, an "endorsement" system was introduced in 2004 as part of an effort to ensure content is relevant and of interest to current research in the specified disciplines. Under the system, for categories that use it, an author must be endorsed by an established arXiv author before being allowed to submit papers to those categories. Endorsers are not asked to review the paper for errors, but to check whether the paper is appropriate for the intended subject area. New authors from recognized academic institutions receive automatic endorsement, which in practice means that they do not need to deal with the endorsement system at all. However, the endorsement system has attracted criticism for restricting scientific inquiry. A majority of the e-prints are submitted to journals for publication, but some work, including some influential papers, remain purely as e-prints and are never published in a peer-reviewed journal. A well-known example of the latter is an outline of a proof of Thurston's geometrization conjecture, including the Poincaré conjecture as a particular case, uploaded by Grigori Perelman in November 2002.
Perelman appears content to forgo the traditional peer-reviewed journal process, stating: "If anybody is interested in my way of solving the problem, it's all there – let them go and read about it". Despite this non-traditional method of publication, other mathematicians recognized this work by offering the Fields Medal and Clay Mathematics Millennium Prizes to Perelman, both of which he refused. Papers can be submitted in any of several formats, including LaTeX, PDF printed from a word processor other than TeX or LaTeX; the submission is rejected by the arXiv software if generating the final PDF file fails, if any image file is too large, or if the total size of the submission is too large. ArXiv now allows one to store and modify an incomplete submission, only finalize the submission when ready; the time stamp on the article is set. The standard access route is through one of several mirrors. Sev