Allen AI christened its system ELMo, for Embeddings from Language Models.
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These embeddings can be stored and referenced by other models to infer identity.
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I look and embeddings and I think, 'Oh my heck, this is phenomenal!
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The system, named DeepText, is based on a machine learning concept called word embeddings.
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"Word embeddings are very popular, especially when you don't have much data," De-Arteaga told Motherboard.
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He drew upon an idea, called "word embeddings," that had been around for more than 10 years.
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That is, we can turn any text into a column of numbers, and those are called neural embeddings.
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If you have ever used Spotify's Discover Weekly, you have run head on into embeddings, you just didn't know it.
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But a neural network pretrained with word embeddings is still blind to the meaning of words at the sentence level.
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It was created by Google researchers a couple of years ago to create word embeddings in order to inform natural language processing.
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Essentially, word embeddings find parallels in usage, according to María De-Arteaga, a Carnegie Mellon researcher specializing in machine learning and public policy.
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" It was an engineer named Joel Grus who came up with "ELMo" to stand for "Embeddings from Language Models," he says, and the name "instantly stuck.
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Simplified sample word embeddings highlighting separate word vectors in Spanish and English for "soccer" Simplified sample word embeddings highlighting separate word vectors in Spanish and English for "soccer" Previously it's led to the company essentially translating foreign languages to English and then running English classifiers on them, but this has been a rough solution due to translation errors, but perhaps more importantly the solution has been far too slow.
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Artificial intelligence models are trained with millions of facial images and — depending on the use case — are designed to generate mathematical representations of a human face (called embeddings).
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ELMo ("Embeddings from Language Models"), however, lets the system handle polysemy with ease; as evidence of its utility, it was awarded best paper honors at NAACL last week.
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Instead of pretraining just the first layer of a network with word embeddings, the researchers began training entire neural networks on a broader basic task called language modeling.
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Earlier this year, the chief scientist of the text analytics company Luminoso, Rob Speer, built an algorithm based on word embeddings to try to understand the sentiment of text posts.
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Word embeddings are essentially vectors that allow text classifiers to approach human language in a more context-driven way, highlighting the interrelatedness of words to eventually derive shared meaning or intent.
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"The ability of neural networks to learn interpretable word embeddings, say, does not remotely suggest that they are the right kind of tool for a human-level understanding of the world," Hewitt writes.
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Embeddings are the reason Google search is so powerful and chatbots can follow a conversation––their insights into how words are used by real people are fundamental to a NLP's understanding of a language.
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"Once I had a model that could translate between names and their embeddings, I could generate new names, blend existing names together, do arithmetic on names, and more," he said in a Medium post describing how he made it.
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Known as word embeddings, this dictionary encoded associations between words as numbers in a way that deep neural networks could accept as input — akin to giving the person inside a Chinese room a crude vocabulary book to work with.
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The company's Applied Machine Learning team has spent the past year working on a technology called multilingual embeddings which it says could significantly improve the speed at which its natural language processing tech is able to operate across foreign languages.
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In category theory, there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable under pullbacks. Ideally the class of all embedded subobjects of a given object, up to isomorphism, should also be small, and thus an ordered set.
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A graph is said to be intrinsically linked if, no matter how it is embedded, the embedding is always linked. Although linkless and flat embeddings are not the same, the graphs that have linkless embeddings are the same as the graphs that have flat embeddings..
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These knowledge graph embeddings allow them to be connected to machine learning methods that require feature vectors like word embeddings. This can complement other estimates of conceptual similarity.
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For example, a path between two smooth embeddings is a smooth isotopy.
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Isolates can be nested, and may be placed within embeddings and overrides.
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A relatively 'easy' result is to prove that any two embeddings of a 1-manifold into R4 are isotopic. This is proved using general position, which also allows to show that any two embeddings of an -manifold into are isotopic. This result is an isotopy version of the weak Whitney embedding theorem. Wu proved that for , any two embeddings of an -manifold into are isotopic.
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In bioinformatics problems involving the folding structure of RNA, single-page book embeddings represent classical forms of nucleic acid secondary structure, and two-page book embeddings represent pseudoknots. Other applications of book embeddings include abstract algebra and knot theory. There are several open problems concerning book thickness. It is unknown whether the book thickness of an arbitrary graph can be bounded by a function of the book thickness of its subdivisions.
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Applications include embeddings,Gross, Tucker 1987 computing genus distribution,Gross 2011 and Hamiltonian decompositions.
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Embeddings of classical logic into intuitionistic logic are related to continuation passing style translation.
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2, 235–273.Donaldson, S.K. Scalar curvature and projective embeddings. I. J. Differential Geom.
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Toroidal embeddings have recently led to advances in algebraic geometry, in particular resolution of singularities.
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They use this to explain some properties of word embeddings, including their use to solve analogies.
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Mazur, Barry, On embeddings of spheres., Bull. Amer. Math. Soc. 65 1959 59-65\. of this theorem.
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However, integer-distance straight line embeddings are known to exist for cubic graphs.. raised the question of whether every graph with a linkless embedding in three-dimensional Euclidean space has a linkless embedding in which all edges are represented by straight line segments, analogously to Fáry's theorem for two-dimensional embeddings.
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Sinclair's initial forms part of the name of the GNRS conjecture on metric embeddings of minor-closed graph families.
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Sanjeev Arora, Yingyu Liang, and Tengyu Ma. "A simple but tough-to-beat baseline for sentence embeddings.", 2016; openreview:SyK00v5xx.
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An extension of word2vec to construct embeddings from entire documents (rather than the individual words) has been proposed. This extension is called paragraph2vec or doc2vec and has been implemented in the C, Python and Java/Scala tools (see below), with the Java and Python versions also supporting inference of document embeddings on new, unseen documents.
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In the area of mathematics known as differential topology, the disc theorem of states that two embeddings of a closed k-disc into a connected n-manifold are ambient isotopic provided that if k = n the two embeddings are equioriented. The disc theorem implies that the connected sum of smooth oriented manifolds is well defined.
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Even though the applicability of lexical chains is diverse, there is little work exploring them with recent advances in NLP, more specifically with word embeddings. In, lexical chains are built using specific patterns found on WordNet and used for learning word embeddings. Their resulting vectors, are validated in the document similarity task. Gonzales et al.
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The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space. Isometric means preserving the length of every path. For instance, bending without stretching or tearing a page of paper gives an isometric embedding of the page into Euclidean space because curves drawn on the page retain the same arclength however the page is bent. The first theorem is for continuously differentiable (C1) embeddings and the second for analytic embeddings or embeddings that are smooth of class Ck, 3 ≤ k ≤ ∞.
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It can also be used to construct low dimensional embeddings, which can be useful for a variety of machine learning applications.
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Unicode 6.3 recognized that directional embeddings usually have too strong an effect on their surroundings and are thus unnecessarily difficult to use.
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Furthermore, the analogue of Tutte's spring theorem applies in this case. Toroidal graphs also have book embeddings with at most 7 pages.
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Each such field is a representative of an equivalence class of (essentially distinct) field embeddings for K and L in some extension M.
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Although the GNRS conjecture remains unsolved, it has been proven for some minor-closed graph families that bounded-distortion embeddings exist. These include the series-parallel graphs and the graphs of bounded circuit rank, the graphs of bounded pathwidth, the 2-clique-sums of graphs of bounded size, and the k-outerplanar graphs. In contrast to the behavior of metric embeddings into \ell_1 spaces, every finite metric space has embeddings into \ell_2 with stretch arbitrarily close to one by the Johnson–Lindenstrauss lemma, and into \ell_\infty spaces with stretch exactly one by the tight span construction.
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In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis since the 1980s, without being referred to by any name E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74 (1983), 441–448.(Lemma 6),V.
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Surgery on normal maps allows one to systematically kill elements in the relative homotopy groups by representing them as embeddings with trivial normal bundle.
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Thus, for instance, de Bruijn graphs, graphs of bounded treewidth, planar graphs, and proper minor-closed graph families have three-dimensional embeddings of linear volume. .
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The term tensor sketch was coined in 2013 describing a technique by Rasmus Pagh from the same year. Originally it was understood using the fast Fourier transform to do fast convolution of count sketches. Later research works generalized it to a much larger class of dimensionality reductions via Tensor random embeddings. Tensor random embeddings were introduced in 2010 in a paperKasiviswanathan, Shiva Prasad, et al.
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In their formulation, the graph's vertices must be placed along the spine of the book, and each edge must lie in a single page... Important milestones in the later development of book embeddings include the proof by Mihalis Yannakakis in the late 1980s that planar graphs have book thickness at most four, and the discovery in the late 1990s of close connections between book embeddings and bioinformatics.
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The notion of a book, as a topological space, was defined by C. A. Persinger and Gail Atneosen in the 1960s.. See also . As part of this work, Atneosen already considered embeddings of graphs in books. The embeddings he studied used the same definition as embeddings of graphs into any other topological space: vertices are represented by distinct points, edges are represented by curves, and the only way that two edges can intersect is for them to meet at a common endpoint. In the early 1970s, Paul C. Kainen and L. Taylor Ollmann developed a more restricted type of embedding that came to be used in most subsequent research.
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A less combinatorial example is the operad of little intervals: The space A(n) consists of all embeddings of n disjoint intervals into the unit interval.
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In case the two given continuous functions f and g from the topological space X to the topological space Y are embeddings, one can ask whether they can be connected 'through embeddings'. This gives rise to the concept of isotopy, which is a homotopy, H, in the notation used before, such that for each fixed t, H(x, t) gives an embedding. A related, but different, concept is that of ambient isotopy. Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. For example, the map from the interval [−1, 1] into the real numbers defined by f(x) = −x is not isotopic to the identity g(x) = x.
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Representing words considering their context through fixed size dense vectors (word embeddings) has become one of the most fundamental blocks in several NLP systems. Even though most of traditional word embedding techniques conflate words with multiple meanings into a single vector representation, they still can be used to improve WSD. In addition to word embeddings techniques, lexical databases (e.g., WordNet, ConceptNet, BabelNet) can also assist unsupervised systems in mapping words and their senses as dictionaries.
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The seven graphs in the Petersen family. No matter how these graphs are embedded into three-dimensional space, some two cycles will have nonzero linking number. In two dimensions, only the planar graphs may be embedded into the Euclidean plane without crossings, but in three dimensions, any undirected graph may be embedded into space without crossings. However, a spatial analogue of the planar graphs is provided by the graphs with linkless embeddings and knotless embeddings.
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Some graphs, such as the complete bipartite graph K_{2,3}, have convex drawings but not strictly convex drawings. A combinatorial characterization for the graphs with convex drawings is known, and they can be recognized in linear time, but the grid dimensions needed for their drawings and an efficient algorithm for constructing small convex grid drawings of these graphs are not known in all cases. Convex drawings should be distinguished from convex embeddings, in which each vertex is required to lie within the convex hull of its neighboring vertices. Convex embeddings can exist in dimensions other than two, do not require their graph to be planar, and even for planar embeddings of planar graphs do not necessarily force the outer face to be convex.
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Sentence embedding is the collective name for a set of techniques in natural language processing (NLP) where sentences are mapped to vectors of real numbers Paper Summary: Evaluation of sentence embeddings in downstream and linguistic probing tasks Oren Barkan, Noam Razin, Itzik Malkiel, Ori Katz, Avi Caciularu, Noam Koenigstein. "Scalable Attentive Sentence-Pair Modeling via Distilled Sentence Embedding". AAAI 2020; arxiv:1908.05161. The Current Best of Universal Word Embeddings and Sentence Embeddings Daniel Cer, Yinfei Yang, Sheng-yi Kong, Nan Hua, Nicole Limtiaco, Rhomni St. John, Noah Constant, Mario Guajardo-Cespedes, Steve Yuan, Chris Tar, Yun-Hsuan Sung, Brian Strope: “Universal Sentence Encoder”, 2018; arXiv:1803.11175. Ledell Wu, Adam Fisch, Sumit Chopra, Keith Adams, Antoine Bordes: “StarSpace: Embed All The Things!”, 2017; arXiv:1709.03856.
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First, a set of candidate entities is chosen using string matching, acronyms, and known aliases. Then the best link among the candidates is chosen with a ranking support vector machine (SVM) that uses linguistic features. Recent systems, such as the one proposed by Tsai et al., employ word embeddings obtained with a skip-gram model as language features, and can be applied to any language as long as a large corpus to build word embeddings is provided.
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Embeddings are codings of categorical values into high-dimensional real-valued (sometimes complex- valued) vector spaces, usually in such a way that ‘similar’ values are assigned ‘similar’ vectors, or with respect to some other kind of criterion making the vectors useful for the respective application. A common special case are word embeddings, where the possible values of the categorical variable are the words in a language and words with similar meanings are to be assigned similar vectors.
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The graphs that do not have knotless embeddings (that is, they are intrinsically knotted) include K7 and K3,3,1,1.; . However, there also exist minimal forbidden minors for knotless embedding that are not formed (as these two graphs are) by adding one vertex to an intrinsically linked graph.. One may also define graph families by the presence or absence of more complex knots and links in their embeddings,; . or by linkless embedding in three-dimensional manifolds other than Euclidean space.
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If a circular embedding exists, it might not be on a surface of minimal genus: Nguyen Huy Xuong described a biconnected toroidal graph none of whose circular embeddings lie on a torus.
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Robertson's irreducible apex graph, showing that the YΔY- reducible graphs have additional forbidden minors beyond those in the Petersen family. A minor of a graph G is another graph formed from G by contracting and removing edges. As the Robertson–Seymour theorem shows, many important families of graphs can be characterized by a finite set of forbidden minors: for instance, according to Wagner's theorem, the planar graphs are exactly the graphs that have neither the complete graph K5 nor the complete bipartite graph K3,3 as minors. Neil Robertson, Paul Seymour, and Robin Thomas used the Petersen family as part of a similar characterization of linkless embeddings of graphs, embeddings of a given graph into Euclidean space in such a way that every cycle in the graph is the boundary of a disk that is not crossed by any other part of the graph.. Horst Sachs had previously studied such embeddings, shown that the seven graphs of the Petersen family do not have such embeddings, and posed the question of characterizing the linklessly embeddable graphs by forbidden subgraphs.. Robertson et al.
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Let \alpha, \beta be two quasi- isometric embeddings of [0, +\infty[ into X ("quasi-geodesic rays"). They are considered equivalent if and only if the function t \mapsto d(\alpha(t), \beta(t)) is bounded on [0, +\infty[. If the space X is proper then the set of all such embeddings modulo equivalence with its natural topology is homeomorphic to \partial X as defined above. A similar realisation is to fix a basepoint and consider only quasi-geodesic rays originating from this point.
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A constant function is both monotone and antitone; conversely, if f is both monotone and antitone, and if the domain of f is a lattice, then f must be constant. Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are order embeddings (functions for which x ≤ y if and only if f(x) ≤ f(y)) and order isomorphisms (surjective order embeddings).
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If (E, M) is a factorization system, then the morphisms in M may be regarded as the embeddings, especially when the category is well powered with respect to M. Concrete theories often have a factorization system in which M consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article. As usual in category theory, there is a dual concept, known as quotient. All the preceding properties can be dualized.
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Restricting the standard absolute value on R to F gives an archimedean absolute value on F; such an absolute value is also referred to as a real place of F. On the other hand, the roots of factors of degree two are pairs of conjugate complex numbers, which allows for two conjugate embeddings into C. Either one of this pair of embeddings can be used to define an absolute value on F, which is the same for both embeddings since they are conjugate. This absolute value is called a complex place of F.Cohn, Chapter 11 §C p. 108Conrad If all roots of f above are real (respectively, complex) or, equivalently, any possible embedding F ⊂ C is actually forced to be inside R (resp. C), F is called totally real (resp.
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In particular, used a construction for book embeddings that keep the degree of each vertex within each page low, as part of a method for embedding graphs into a three-dimensional grid of low volume..
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Hassler Whitney (March 23, 1907 – May 10, 1989) was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, characteristic classes, and geometric integration theory.
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From 2003 to 2007 he was president of Istituto Nazionale di Alta Matematica Francesco Severi. In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley (Equivariant embeddings of homogeneous spaces).
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Regular 3D embeddings of the Klein bottle fall into three regular homotopy classes (four if one paints them). The three are represented by # The "traditional" Klein bottle # Left handed figure-8 Klein bottle # Right handed figure-8 Klein bottle The traditional Klein bottle embedding is achiral. The figure-8 embedding is chiral (the pinched torus embedding above is not regular as it has pinch points so it's not relevant in this section). The three embeddings above cannot be smoothly transformed into each other in three dimensions.
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When this happens, correspondingly, all dual graphs are isomorphic. Because different embeddings may lead to different dual graphs, testing whether one graph is a dual of another (without already knowing their embeddings) is a nontrivial algorithmic problem. For biconnected graphs, it can be solved in polynomial time by using the SPQR trees of the graphs to construct a canonical form for the equivalence relation of having a shared mutual dual. For instance, the two red graphs in the illustration are equivalent according to this relation.
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"Iterated Forcing and Elementary Embeddings". In Handbook of Set Theory, Springer, pp. 775–883, esp. pp. 814ff. Silver proved the consistency of Chang's conjecture using the Silver collapse (which is a variation of the Levy collapse).
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The question of whether K6 has a linkless or flat embedding was posed within the topology research community in the early 1970s by . Linkless embeddings were brought to the attention of the graph theory community by , who posed several related problems including the problem of finding a forbidden graph characterization of the graphs with linkless and flat embeddings; Sachs showed that the seven graphs of the Petersen family (including K6) do not have such embeddings. As observed, linklessly embeddable graphs are closed under graph minors, from which it follows by the Robertson–Seymour theorem that a forbidden graph characterization exists. The proof of the existence of a finite set of obstruction graphs does not lead to an explicit description of this set of forbidden minors, but it follows from Sachs' results that the seven graphs of the Petersen family belong to the set.
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In algebraic number theory, the narrow class group of a number field K is a refinement of the class group of K that takes into account some information about embeddings of K into the field of real numbers.
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An injectively immersed submanifold that is not an embedding. If M is compact, an injective immersion is an embedding, but if M is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.
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After all words are annotated and disambiguated, they can be used as a training corpus in any standard word embedding technique. In its improved version, MSSA can make use of word sense embeddings to repeat its disambiguation process iteratively.
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A Seifert surface bounded by a set of Borromean rings; these surfaces can be used as tools in geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
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At Stockholm University, Hans Rådström suggested that Enflo consider Hilbert's fifth problem in the spirit of functional analysis.Rådström had himself published several articles on Hilbert's fifth problem from the point of view of semigroup theory. Rådström was also the (initial) advisor of Martin Ribe, who wrote a thesis on metric linear spaces that need not be locally convex; Ribe also used a few of Enflo's ideas on metric geometry, especially "roundness", in obtaining independent results on uniform and Lipschitz embeddings (Benyamini and Lindenstrauss). This reference also describes results of Enflo and his students on such embeddings.
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We take two knots, K1 and K2, in three-dimensional space. A knot is an embedding of a one-dimensional space, the "loop of string" (or the circle), into this space, and this embedding gives a homeomorphism between the circle and its image in the embedding space. The intuitive idea behind the notion of knot equivalence is that one can deform one embedding to another through a path of embeddings: a continuous function starting at t = 0 giving the K1 embedding, ending at t = 1 giving the K2 embedding, with all intermediate values corresponding to embeddings. This corresponds to the definition of isotopy.
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Dirichlet's unit theorem provides a description of the structure of the multiplicative group of units O× of the ring of integers O. Specifically, it states that O× is isomorphic to G × Zr, where G is the finite cyclic group consisting of all the roots of unity in O, and r = r1 + r2 − 1 (where r1 (respectively, r2) denotes the number of real embeddings (respectively, pairs of conjugate non- real embeddings) of K). In other words, O× is a finitely generated abelian group of rank r1 + r2 − 1 whose torsion consists of the roots of unity in O.
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Marcelo Osvaldo Magnasco is a biophysicist and a professor at The Rockefeller University. He is known for his work on thermal ratchets as models of biological motors, auditory biophysics, bailout embeddings,Julyan H. E. Cartwright, Marcelo O. Magnasco, and Oreste Piro "Bailout embeddings, targeting of invariant tori, and the control of Hamiltonian chaos", Phys. Rev. E 65, 045203(R) (2002)., neural coding, other studies of biological networks such as leaf venation, and for placing the date of the solar eclipse mentioned in the Odyssey on April 16, 1178 B.C. together with Constantino Baikouzis of the National University of La Plata.
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According to Dirichlet's unit theorem, the torsionfree unit rank r of an algebraic number field K with r1 real embeddings and r2 pairs of conjugate complex embeddings is determined by the formula r = r1 \+ r2 − 1. Hence a totally real cubic field K with r1 = 3, r2 = 0 has two independent units ε1, ε2 and a complex cubic field K with r1 = r2 = 1 has a single fundamental unit ε1. These fundamental systems of units can be calculated by means of generalized continued fraction algorithms by Voronoi, which have been interpreted geometrically by Delone and Faddeev.
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An arc diagram of the Goldner–Harary graph. In order to create a planar diagram, two triangles of the graph have been subdivided into four by the dashed red line, causing one of the graph edges to extend both above and below the line. Book embedding has also been frequently applied in the visualization of network data. Two of the standard layouts in graph drawing, arc diagrams and circular layouts, can be viewed as book embeddings, and book embedding has also been applied in the construction of clustered layouts, simultaneous embeddings, and three-dimensional graph drawings.
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An embedded graph uniquely defines cyclic orders of edges incident to the same vertex. The set of all these cyclic orders is called a rotation system. Embeddings with the same rotation system are considered to be equivalent and the corresponding equivalence class of embeddings is called combinatorial embedding (as opposed to the term topological embedding, which refers to the previous definition in terms of points and curves). Sometimes, the rotation system itself is called a "combinatorial embedding".... An embedded graph also defines natural cyclic orders of edges which constitutes the boundaries of the faces of the embedding.
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Therefore circle graphs capture various aspects of this routing problem. Colorings of circle graphs may also be used to find book embeddings of arbitrary graphs: if the vertices of a given graph G are arranged on a circle, with the edges of G forming chords of the circle, then the intersection graph of these chords is a circle graph and colorings of this circle graph are equivalent to book embeddings that respect the given circular layout. In this equivalence, the number of colors in the coloring corresponds to the number of pages in the book embedding.
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The application of semicircles to edge layout in book embeddings was already made by , but the explicit connection of arc diagrams with two-page book embeddings seems to be due to . For instance, a maximal planar graph has such an embedding if and only if it contains a Hamiltonian cycle. Therefore, a non-Hamiltonian maximal planar graph such as the Goldner–Harary graph cannot have a planar embedding with one semicircle per edge. Testing whether a given graph has a crossing- free arc diagram of this type (or equivalently, whether it has pagenumber two) is NP-complete.
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"Phylogenetic inference for function-valued traits: speech sound evolution." Trends in ecology & evolution 27.3 (2012): 160-166.. and meaning. e.g. Hamilton, William L., Jure Leskovec, and Dan Jurafsky. "Diachronic word embeddings reveal statistical laws of semantic change." arXiv preprint arXiv:1605.09096 (2016).
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Murphy graduated from the University of Nevada, Reno in 2007, the first in her family to earn a college degree. She completed her doctorate at Stanford University in 2012; her dissertation, Loose Legendrian Embeddings in High Dimensional Contact Manifolds, was supervised by Yakov Eliashberg.
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Every planar graph G has an embedding G_\phi such that G_\phi contains a good spanning tree. A good spanning tree and a suitable embedding can be found from G in linear-time. Not all embeddings of G contain a good spanning tree.
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In B(2,3). each vertex is visited once, whereas in B(2,4), each edge is traversed once. Embeddings resembling this one can be used to show that the binary De Bruijn graphs have queue number 2 and that they have book thickness at most 5.
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Remak's analysis may have influenced John von Neumann, who was a fellow lecturer in Berlin, but most of it has not been translated into English and it remains little known and appreciated in the English-speaking world.Kurz and Salvadori, pp 40-46. In 1932 Remak published a paper giving a lower bound for the regulator of an algebraic number field in terms of the numbers r1 and r2 of real embeddings and pairs of complex embeddings. He went on to investigate relations between the regulator and the discriminant of an algebraic number field, isolating an important class of CM-fields ("fields with unit defect").
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An elementary embedding of a structure N into a structure M of the same signature σ is a map h: N → M such that for every first-order σ-formula φ(x1, …, xn) and all elements a1, …, an of N, :N \models φ(a1, …, an) if and only if M \models φ(h(a1), …, h(an)). Every elementary embedding is a strong homomorphism, and its image is an elementary substructure. Elementary embeddings are the most important maps in model theory. In set theory, elementary embeddings whose domain is V (the universe of set theory) play an important role in the theory of large cardinals (see also Critical point).
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In combinatorial mathematics, rotation systems (also called combinatorial embeddings) encode embeddings of graphs onto orientable surfaces, by describing the circular ordering of a graph's edges around each vertex. A more formal definition of a rotation system involves pairs of permutations; such a pair is sufficient to determine a multigraph, a surface, and a 2-cell embedding of the multigraph onto the surface. Every rotation scheme defines a unique 2-cell embedding of a connected multigraph on a closed oriented surface (up to orientation preserving topological equivalence). Conversely, any embedding of a connected multigraph G on an oriented closed surface defines a unique rotation system having G as its underlying multigraph.
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Linkless embeddings started being studied within the algorithms research community in the late 1980s through the works of and . Algorithmically, the problem of recognizing linkless and flat embeddable graphs was settled once the forbidden minor characterization was proven: an algorithm of can be used to test in polynomial time whether a given graph contains any of the seven forbidden minors.The application of the Robertson–Seymour algorithm to this problem was noted by . This method does not construct linkless or flat embeddings when they exist, but an algorithm that does construct an embedding was developed by , and a more efficient linear time algorithm was found by .
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The first encoder takes positional information and embeddings of the input sequence as its input, rather than encodings. The positional information is necessary for the Transformer to make use of the order of the sequence, because no other part of the Transformer makes use of this.
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Explicit formatting characters, also referred to as "directional formatting characters", are special Unicode sequences that direct the algorithm to modify its default behavior. These characters are subdivided into "marks", "embeddings", "isolates", and "overrides". Their effects continue until the occurrence of either a paragraph separator, or a "pop" character.
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Other equivalent representations for cellular embeddings include the ribbon graph, a topological space formed by gluing together topological disks for the vertices and edges of an embedded graph, and the graph-encoded map, an edge-colored cubic graph with four vertices for each edge of the embedded graph.
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In theoretical computer science and metric geometry, the GNRS conjecture connects the theory of graph minors, the stretch factor of embeddings, and the approximation ratio of multi-commodity flow problems. It is named after Anupam Gupta, Ilan Newman, Yuri Rabinovich, and Alistair Sinclair, who formulated it in 2004.
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The standard notation r1 and r2 for the number of real and complex embeddings is used, respectively (see below). Calculating the archimedean places of F is done as follows: let x be a primitive element of F, with minimal polynomial f (over Q). Over R, f will generally no longer be irreducible, but its irreducible (real) factors are either of degree one or two. Since there are no repeated roots, there are no repeated factors. The roots r of factors of degree one are necessarily real, and replacing x by r gives an embedding of F into R; the number of such embeddings is equal to the number of real roots of f.
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A linkless embedding is an embedding of the graph with the property that any two cycles are unlinked; a knotless embedding is an embedding of the graph with the property that any single cycle is unknotted. The graphs that have linkless embeddings have a forbidden graph characterization involving the Petersen family, a set of seven graphs that are intrinsically linked: no matter how they are embedded, some two cycles will be linked with each other.. A full characterization of the graphs with knotless embeddings is not known, but the complete graph is one of the minimal forbidden graphs for knotless embedding: no matter how is embedded, it will contain a cycle that forms a trefoil knot..
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"Topics in parallel computing: Embeddings and simulations of INs: Optimal embedding of tori into meshes". Every link in the folded torus network is very short—almost as short as the nearest-neighbor links in a simple grid interconnect—and therefore low-latency."The 3D Torus architecture and the Eurotech approach".
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A metric space X is a real tree if for any pair of points x, y \in X all (topological) embeddings \sigma of the segment [0,1] into X such that \sigma(0) = x, \, \sigma(1) = y have the same image (which is then a geodesic segment from x to y).
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Another equivalent definition of queue layouts involves embeddings of the given graph onto a cylinder, with the vertices placed on a line in the cylinder and with each edge wrapping once around the cylinder. Edges that are assigned to the same queue are not allowed to cross each other, but crossings are allowed between edges that belong to different queues.. Queue layouts were defined by , by analogy to previous work on book embeddings of graphs, which can be defined in the same way using stacks in place of queues. As they observed, these layouts are also related to earlier work on sorting permutations using systems of parallel queues, and may be motivated by applications in VLSI design and in communications management for distributed algorithms.
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No real examples of degree 4 have been recorded. In spoken language, multiple center-embeddings even of degree 2 are so rare as to be practically non- existing (Karlsson 2007). Center embedding is the focus of a science fiction novel, Ian Watson's The Embedding, and plays a part in Ted Chiang's Story of Your Life.
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Some techniques that combine lexical databases and word embeddings are presented in AutoExtend and Most Suitable Sense Annotation (MSSA). In AutoExtend, they present a method that decouples an object input representation into its properties, such as words and their word senses. AutoExtend uses a graph structure to map words (e.g. text) and non-word (e.g.
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Recent attempts at detecting DGA domain names with deep learning techniques have been extremely successful, with F1 scores of over 99%. These deep learning methods typically utilize LSTM and CNN architectures, though deep word embeddings have shown great promise for detecting dictionary DGA. However, these deep learning approaches can be vulnerable to adversarial techniques.
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Topological tameness may be viewed as a property of the ends of the manifold, namely, having a local product structure. An analogous statement is well known in two dimensions, i.e. for surfaces. However, as the example of Alexander horned sphere shows, there are wild embeddings among 3-manifolds, so this property is not automatic.
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Moses Samson Charikar is an Indian computer scientist who works as a professor at Stanford University. He was previously a professor at Princeton University. The topics of his research include approximation algorithms, streaming algorithms, and metric embeddings. He is known for the creation of the SimHash algorithm used by Google for near duplicate detection. .
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Singh has described Hate5six.com as a "democratic, community-controlled system", by having viewers vote for what video will be uploaded through Patreon. Patrons also vote for charities that 8.56% of the proceeds go to. In 2017, Singh expanded the website to include SAGE (Sage Analyzes Graph Embeddings), an artificial intelligence system that recommends bands based on listening patterns.
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Luminoso continues to actively conduct research in natural language processing and word embeddings and regularly participates in evaluations such as SemEval. At SemEval 2017, Luminoso participated in Task 2, measuring the semantic similarity of word pairs within and across five languages. Its solution outperformed all competing systems in every language pair tested, with the exception of Persian.
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In mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the theory of graph minors and topological embeddings. The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and Paul Seymour. Its proof is very long and involved.
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Haroske is the author of the book Envelopes and Sharp Embeddings of Function Spaces (Chapman & Hall, 2007). With Hans Triebel she also wrote Distributions, Sobolev Spaces, Elliptic Equations (EMS Textbooks in Mathematics, European Mathematical Society, 2008). She is one of the editors of Function Spaces, Differential Operators and Nonlinear Analysis: The Hans Triebel Anniversary Volume (Springer Basel AG, 2003).
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A circle graph, the intersection graph of chords of a circle. For book embeddings with a fixed vertex order, finding the book thickness is equivalent to coloring a derived circle graph. Finding the book thickness of a graph is NP-hard. This follows from the fact that finding Hamiltonian cycles in maximal planar graphs is NP-complete.
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The class of subhamiltonian graphs (but not this name for them) was introduced by , who proved that these are exactly the graphs with two-page book embeddings.. Subhamiltonian graphs and Hamiltonian augmentations have also been applied in graph drawing to problems involving embedding graphs onto universal point sets, simultaneous embedding of multiple graphs, and layered graph drawing.
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Regular skew polyhedra can also be constructed in dimensions higher than 4 as embeddings into regular polytopes or honeycombs. For example, the regular icosahedron can be embedded into the vertices of the 6-demicube; this was named the regular skew icosahedron by H. S. M. Coxeter. The dodecahedron can be similarly embedded into the 10-demicube.
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In mathematical order theory, an order embedding is a special kind of monotone function, which provides a way to include one partially ordered set into another. Like Galois connections, order embeddings constitute a notion which is strictly weaker than the concept of an order isomorphism. Both of these weakenings may be understood in terms of category theory.
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Paul Chester Kainen is an American mathematician, an adjunct associate professor of mathematics and director of the Lab for Visual Mathematics at Georgetown University.Mathematics faculty listing, Georgetown U., retrieved 2012-07-05. 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.
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In complex geometry, ramified covering spaces are used to model Riemann surfaces, and to analyze maps between surfaces, such as by the Riemann–Hurwitz formula. In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of isometric embeddings, isometric immersions, and Riemannian submersions; a basic result is the Nash embedding theorem.
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The concept of duality can be extended to graph embeddings on two-dimensional manifolds other than the plane. The definition is the same: there is a dual vertex for each connected component of the complement of the graph in the manifold, and a dual edge for each graph edge connecting the two dual vertices on either side of the edge. In most applications of this concept, it is restricted to embeddings with the property that each face is a topological disk; this constraint generalizes the requirement for planar graphs that the graph be connected. With this constraint, the dual of any surface-embedded graph has a natural embedding on the same surface, such that the dual of the dual is isomorphic to and isomorphically embedded to the original graph.
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For instance, the complete graph is a toroidal graph: it is not planar but can be embedded in a torus, with each face of the embedding being a triangle. This embedding has the Heawood graph as its dual graph.. The same concept works equally well for non- orientable surfaces. For instance, can be embedded in the projective plane with ten triangular faces as the hemi-icosahedron, whose dual is the Petersen graph embedded as the hemi-dodecahedron.. Even planar graphs may have nonplanar embeddings, with duals derived from those embeddings that differ from their planar duals. For instance, the four Petrie polygons of a cube (hexagons formed by removing two opposite vertices of the cube) form the hexagonal faces of an embedding of the cube in a torus.
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One of the original motivations for studying book embeddings involved applications in VLSI design, in which the vertices of a book embedding represent components of a circuit and the wires represent connections between them. Book embedding also has applications in graph drawing, where two of the standard visualization styles for graphs, arc diagrams and circular layouts, can be constructed using book embeddings. In transportation planning, the different sources and destinations of foot and vehicle traffic that meet and interact at a traffic light can be modeled mathematically as the vertices of a graph, with edges connecting different source-destination pairs. A book embedding of this graph can be used to design a schedule that lets all the traffic move across the intersection with as few signal phases as possible.
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A symmetric embedding of the Nauru graph on a genus-4 surface, with six dodecagonal faces. The Nauru graph has two different embeddings as a generalized regular polyhedron: a topological surface partitioned into edges, vertices, and faces in such a way that there is a symmetry taking any flag (an incident triple of a vertex, edge, and face) into any other flag.. One of these two embeddings forms a torus, so the Nauru graph is a toroidal graph: it consists of 12 hexagonal faces together with the 24 vertices and 36 edges of the Nauru graph. The dual graph of this embedding is a symmetric 6-regular graph with 12 vertices and 36 edges. The other symmetric embedding of the Nauru graph has six dodecagonal faces, and forms a surface of genus 4.
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Minimizing the number of crossings is NP-complete, but may be approximated with an approximation ratio of where is the number of vertices.. Minimizing the one- page or two-page crossing number is fixed-parameter tractable when parameterized by the cyclomatic number of the given graph, or by a combination of the crossing number and the treewidth of the graph... Heuristic methods for reducing the crossing complexity have also been devised, based e.g. on a careful vertex insertion order and on local optimization. Two-page book embeddings with a fixed partition of the edges into pages can be interpreted as a form of clustered planarity, in which the given graph must be drawn in such a way that parts of the graph (the two subsets of edges) are placed in the drawing in a way that reflects their clustering. Two-page book embedding has also been used to find simultaneous embeddings of graphs, in which two graphs are given on the same vertex set and one must find a placement for the vertices in which both graphs are drawn planarly with straight edges.. Book embeddings with more than two pages have also been used to construct three- dimensional drawings of graphs.
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There are several equivalent statements of the Smale conjecture. One is that the component of the unknot in the space of smooth embeddings of the circle in 3-space has the homotopy-type of the round circles, equivalently, O(3). Another equivalent statement is that the group of diffeomorphisms of the 3-ball which restrict to the identity on the boundary is contractible.
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His research is on algebraic topology and differential topology. In work with Ib Madsen, he resolved the Mumford Conjecture about rational characteristic classes of surface bundles in the limit as the genus tends to infinity.Allen Hatcher, A Short Exposition of the Madsen–Weiss Theorem Building on earlier work of Thomas Goodwillie, he developed Embedding Calculus, a Calculus of functors for embeddings of manifolds.
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Different trees will, in general, correspond to different Shabat polynomials, as will different embeddings or colorings of the same tree. Up to normalization and linear transformations of its argument, the Shabat polynomial is uniquely determined from a coloring of an embedded tree, but it is not always straightforward to find a Shabat polynomial that has a given embedded tree as its dessin d'enfant.
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NMT departs from phrase-based statistical approaches that use separately engineered subcomponents. Neural machine translation (NMT) is not a drastic step beyond what has been traditionally done in statistical machine translation (SMT). Its main departure is the use of vector representations ("embeddings", "continuous space representations") for words and internal states. The structure of the models is simpler than phrase-based models.
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Volić's research is in algebraic topology. He is the author of over thirty articles and two books and has delivered more than two hundred lectures in some twenty countries. He has contributed to the fields of calculus of functors, spaces of embeddings and immersions, configuration space integrals, finite type invariants, Milnor invariants, rational homotopy theory, topological data analysis, and social choice theory.
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He also formulated Fraïssé's conjecture on order embeddings, and introduced the notion of compensor in the theory of posets.Petits posets : dénombrement, représentabilité par cercles et compenseurs, Roland Fraïssé and Nik Lygeros Comptes Rendus de l'Académie des Sciences, Série I 313 (1991), no. 7, 417—420 Most of his career was spent as Professor at the University of Provence in Marseille, France.
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The right-handed trefoil knot. In geometric topology a basic type are embeddings, of which knot theory is a central example, and generalizations such as immersions, submersions, covering spaces, and ramified covering spaces. Basic results include the Whitney embedding theorem and Whitney immersion theorem. Riemann surface for the function f(z) = , shown as a ramified covering space of the complex plane.
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For philosophical reasons, it is important to note that it is not necessary to work in ZFC or any related system to carry out this proof. A common argument against the use of NFU as a foundation for mathematics is that the reasons for relying on it have to do with the intuition that ZFC is correct. It is sufficient to accept TST (in fact TSTU). In outline: take the type theory TSTU (allowing urelements in each positive type) as a metatheory and consider the theory of set models of TSTU in TSTU (these models will be sequences of sets T_i (all of the same type in the metatheory) with embeddings of each P(T_i) into P_1(T_{i+1}) coding embeddings of the power set of T_i into T_{i+1} in a type-respecting manner).
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Codimension also has some clear meaning in geometric topology: on a manifold, codimension 1 is the dimension of topological disconnection by a submanifold, while codimension 2 is the dimension of ramification and knot theory. In fact, the theory of high- dimensional manifolds, which starts in dimension 5 and above, can alternatively be said to start in codimension 3, because higher codimensions avoid the phenomenon of knots. Since surgery theory requires working up to the middle dimension, once one is in dimension 5, the middle dimension has codimension greater than 2, and hence one avoids knots. This quip is not vacuous: the study of embeddings in codimension 2 is knot theory, and difficult, while the study of embeddings in codimension 3 or more is amenable to the tools of high-dimensional geometric topology, and hence considerably easier.
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One formulation of the conjecture involves embeddings of the shortest path distances of weighted undirected graphs into \ell_1 spaces, real vector spaces in which the distance between two vectors is the sum of their coordinate differences. If an embedding maps all pairs of vertices with distance d to pairs of vectors with distance in the range [cd,Cd] then its stretch factor or distortion is the ratio C/c; an isometry has stretch factor one, and all other embeddings have greater stretch factor. The graphs that have an embedding with at most a given distortion are closed under graph minor operations, operations that delete vertices or edges from a graph or contract some of its edges. The GNRS conjecture states that, conversely, every nontrivial minor-closed family of graphs can be embedded into an \ell_1 space with bounded distortion.
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Traditional knot theory models a knot as a simple closed loop in three-dimensional space. Such a knot has no thickness or physical properties such as tension or friction. Physical knot theory incorporates more realistic models. The traditional model is also studied but with an eye toward properties of specific embeddings ("conformations") of the circle. Such properties include ropelength and various knot energies (O’Hara 2003).
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The primary interface to rasdaman is the query language. Embeddings into C++ and Java APIs allow invocation of queries, as well as client-side convenience functions for array handling. Arrays per se are delivered in the main memory format of the client language and processor architecture, ready for further processing. Data format codecs allow to retrieve arrays in common raster formats, such as CSV, PNG, and NetCDF.
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To improve the breaking resistance clay boards are often embedded in a hessian skin on the backside or similar embeddings. By introducing the clay panels, the building material loam can also be used in dry construction. Clay wallboards are a sustainable alternative to gypsum plasterboards, suitable for drywall applications for interior walls and ceilings. It can be applied to either timber or metal studwork.
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The "override" directional formatting characters allow for special cases, such as for part numbers (e.g. to force a part number made of mixed English, digits and Hebrew letters to be written from right to left), and are recommended to be avoided wherever possible. As is true of the other directional formatting characters, "overrides" can be nested one inside another, and in embeddings and isolates.
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Every Möbius ladder is a nonplanar apex graph, meaning that it cannot be drawn without crossings in the plane but removing one vertex allows the remaining graph to be drawn without crossings. Möbius ladders have crossing number one, and can be embedded without crossings on a torus or projective plane. Thus, they are examples of toroidal graphs. explores embeddings of these graphs onto higher genus surfaces.
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This result is an isotopy version of the strong Whitney embedding theorem. As an isotopy version of his embedding result, Haefliger proved that if is a compact -dimensional -connected manifold, then any two embeddings of into are isotopic provided . The dimension restriction is sharp: Haefliger went on to give examples of non-trivially embedded 3-spheres in (and, more generally, -spheres in ). See further generalizations.
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Its dual is not a simple graph, since each face shares three edges with four other faces, but a multigraph. This dual can be formed from the graph of a regular octahedron by replacing each edge with a bundle of three parallel edges. The set of faces of any one of these two embeddings is the set of Petrie polygons of the other embedding.
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Nominal unification is efficiently decidable. This fact led to the development of alphaProlog, a Prolog-like logic programming language with facilities for binding names in terms, where Prolog's standard first-order unification algorithm is replaced with nominal unification. Nominal term embeddings may be seen as alternatives to de Bruijn encodings and higher-order abstract syntax, where the latter uses the simply typed lambda calculus as a metalanguage.
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For example, it follows that any closed oriented Riemannian surface can be C1 isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space (for small \epsilon there is no such C2-embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε−2). And, there exist C1 isometric embeddings of the hyperbolic plane in R3.
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The classic path addition method of Hopcroft and Tarjan. was the first published linear-time planarity testing algorithm in 1974. An implementation of Hopcroft and Tarjan's algorithm is provided in the Library of Efficient Data types and Algorithms by Mehlhorn, Mutzel and Näher . In 2012, Taylor Alt URL extended this algorithm to generate all permutations of cyclic edge-order for planar embeddings of biconnected components.
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Journal of Molecular Biology. pp. 431-448. ISSN 0022-2836 that can be determined mathematically.Self-Assembly of Viral Capsids via a Hamiltonian Paths Approach: The Case of Bacteriophage MS2 Some different viruses require different 3-dimensional models, and so Twarock continued to examine the mathematics and biology at play.R.Twarock, M. Valiunas, & E. Zappa (2015) Orbits of crystallographic embeddings of non-crystallographic groups and applications to virology.
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36 The invariants of a quadratic forms over a local field are precisely the dimension, discriminant and Hasse invariant.Serre (1973) p.39 For quadratic forms over a number field, there is a Hasse invariant ±1 for every finite place. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.
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Ernst Hellinger studied integral equations, the infinite system of equations, real functions and continued fractions. A type of integral which he introduced in his dissertation became known as "the Hellinger integral", used for defining the Hellinger distance. Hellinger distance has been used to process natural language and learning word embeddings. In addition, the Hilbert–Hellinger theory of forms in infinitely many variables profoundly influenced mathematical analysis.
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70 (2005), no. 3, 453–472. The approximability of a Kähler metric by Kähler metrics induced from projective embeddings is also relevant to Yau's picture of the Yau-Tian-Donaldson conjecture, as indicated above. In a highly technical 2008 article, Xiuxiong Chen and Tian studied the regularity theory of certain complex Monge-Ampère equations, with applications to the study of the geometry of extremal Kähler metrics.
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If true, this would imply a conjecture of W. T. Tutte that every bridgeless graph has a nowhere-zero 5-flow. A stronger type of embedding than a circular embedding is a polyhedral embedding, an embedding of a graph on a surface in such a way that every face is a simple cycle and every two faces that intersect do so in either a single vertex or a single edge. (In the case of a cubic graph, this can be simplified to a requirement that every two faces that intersect do so in a single edge.) Thus, in view of the reduction of the cycle double cover conjecture to snarks, it is of interest to investigate polyhedral embeddings of snarks. Unable to find such embeddings, Branko Grünbaum conjectured that they do not exist, but disproved Grünbaum's conjecture by finding a snark with a polyhedral embedding.
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For each context window, MSSA calculates the centroid of each word sense definition by averaging the word vectors of its words in WordNet's glosses (i.e., short defining gloss and one or more usage example) using a pre-trained word embeddings model. These centroids are later used to select the word sense with the highest similarity of a target word to its immediately adjacent neighbors (i.e., predecessor and successor words).
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The actual projective embeddings are complicated (see equations defining abelian varieties) when n > 1, and are really coextensive with the theory of theta-functions of several complex variables (with fixed modulus). There is nothing as simple as the cubic curve description for n = 1. Computer algebra can handle cases for small n reasonably well. By Chow's theorem, no complex torus other than the abelian varieties can 'fit' into projective space.
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A nonobtuse triangle mesh is composed of a set of triangles in which no angle is obtuse, i.e. greater than 90°. If each (triangle) face angle is strictly less than 90°, then the triangle mesh is said to be acute. The immediate benefits of a nonobtuse or acute mesh include more efficient and more accurate geodesic computation using fast marching, and guaranteed validity for planar mesh embeddings via discrete harmonic maps.
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In this way, Tutte embeddings can be used to find Schlegel diagrams of every convex polyhedron. For every 3-connected planar graph G, either G itself or the dual graph of G has a triangle, so either this gives a polyhedral representation of G or of its dual; in the case that the dual graph is the one with the triangle, polarization gives a polyhedral representation of G itself..
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When Gn = U(n) (the unitary group) this construction is called Gelfand Zeitlin basis. Since the representations of U(n) are the same as algebraic representations of GL(n) so we also obtain a basis of any algebraic irreducible representation of GL(n). However one should be aware that the constructed basis isn't canonical as it depends of the choice of the embeddings U(i) ⊂ U(i+1).
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Kalai is known for his algorithm for generating random factored numbers (see Bach's algorithm), for efficiently learning learning mixtures of Gaussians, for the Blum-Kalai-Wasserman algorithm for learning parity with noise, and for the intractability of the folk theorem in game theory. More recently, Kalai is known for identifying and reducing gender bias in word embeddings, which are a representation of words commonly used in AI systems.
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Gain graphs used in topological graph theory as a means to construct graph embeddings in surfaces are known as "voltage graphs" (Gross 1974; Gross and Tucker 1977). The term "gain graph" is more usual in other contexts, e.g., biased graph theory and matroid theory. The term group-labelled graph has also been used, but it is ambiguous since "group labels" may be--and have been--treated as weights.
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Yau had proposed that, rather than holomorphic vector fields on the manifold itself, it should be relevant to study the deformations of projective embeddings of Kähler manifolds under holomorphic vector fields on projective space. This idea was modified by Tian, introducing the notion of K-stability and showing that any Kähler-Einstein manifold must be K-stable. Simon Donaldson, in 2002, modified and extended Tian's definition of K-stability.
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Developments in biomedical text mining have incorporated identification of biological entities with named entity recognition, or NER. Names and identifiers for biomolecules such as proteins and genes, chemical compounds and drugs, and disease names have all been used as entities. Most entity recognition methods are supported by pre-defined linguistic features or vocabularies, though methods incorporating deep learning and word embeddings have also been successful at biomedical NER.
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In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces.J.L. Gross and T.W. Tucker, Topological graph theory, Wiley Interscience, 1987 It also studies immersions of graphs. Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges intersecting.
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The edge, or boundary, of a Möbius strip is homeomorphic (topologically equivalent) to a circle. Under the usual embeddings of the strip in Euclidean space, as above, the boundary is not a true circle. However, it is possible to embed a Möbius strip in three dimensions so that the boundary is a perfect circle lying in some plane. For example, see Figures 307, 308, and 309 of "Geometry and the imagination".
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One advantage of defining varieties over arbitrary fields through the theory of schemes is that such definitions are intrinsic and free of embeddings into ambient affine n-space. A k-algebraic set is a separated and reduced scheme of finite type over Spec(k). A k-variety is an irreducible k-algebraic set. A k-morphism is a morphism between k-algebraic sets regarded as schemes over Spec(k).
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Force-directed methods in graph drawing date back to the work of , who showed that polyhedral graphs may be drawn in the plane with all faces convex by fixing the vertices of the outer face of a planar embedding of the graph into convex position, placing a spring-like attractive force on each edge, and letting the system settle into an equilibrium.. Because of the simple nature of the forces in this case, the system cannot get stuck in local minima, but rather converges to a unique global optimum configuration. Because of this work, embeddings of planar graphs with convex faces are sometimes called Tutte embeddings. The combination of attractive forces on adjacent vertices, and repulsive forces on all vertices, was first used by ;. additional pioneering work on this type of force-directed layout was done by .. The idea of using only spring forces between all pairs of vertices, with ideal spring lengths equal to the vertices' graph-theoretic distance, is from ..
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Graph embeddings also offer a convenient way to predict links. Graph embedding algorithms, such as Node2vec, learn an embedding space in which neighboring nodes are represented by vectors so that vector similarity measures, such as dot product similarity, or euclidean distance, hold in the embedding space. These similarities are functions of both topological features and attribute- based similarity. One can then use other machine learning techniques to predict edges on the basis of vector similarity.
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The notion of a sheaf and sheafification of a presheaf date to early category theory, and can be seen as the linear form of the calculus of functors. The quadratic form can be seen in the work of André Haefliger on links of spheres in 1965, where he defined a "metastable range" in which the problem is simpler. This was identified as the quadratic approximation to the embeddings functor in Goodwillie and Weiss.
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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.
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In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions. Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies.
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The front façade, oriented toward Auersperg Street, was designed as a combination of brick and stone embeddings, some of them archaeological remains from the place. It was modelled in the manner of the Italian palazzo, similar to the house of the Italian architect Federico Zuccari. The handles of the main door end with a little head of Pegasus. The symbiosis of the fragile glass and massive walls in other parts of exterior is entirely original.
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The complete graph K6, the Petersen graph, and the other five graphs in the Petersen family do not have linkless embeddings. Every graph minor of a linklessly embeddable graph is again linklessly embeddable, as is every graph that can be reached from a linklessly embeddable graph by a Y-Δ transform. The linklessly embeddable graphs have the Petersen family graphs as their forbidden minors,. and include the planar graphs and apex graphs.
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In 1964, Keldysh was made a full professor at Moscow State University and in 1966 she published the book Topological embeddings into Euclidean space to help her students understand her lectures. She lectured until 1974, when she resigned in protest of the expulsion of one of her students. At the time, Novikov was ill and he died in January 1975. She died one year and one month later, on 16 February 1976 in Moscow.
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The reasons for successful word embedding learning in the word2vec framework are poorly understood. Goldberg and Levy point out that the word2vec objective function causes words that occur in similar contexts to have similar embeddings (as measured by cosine similarity) and note that this is in line with J. R. Firth's distributional hypothesis. However, they note that this explanation is "very hand-wavy" and argue that a more formal explanation would be preferable. Levy et al.
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In the mathematical field of order theory, an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that either of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections.; .
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Between 2012 and 2014 Bah was a postdoctoral researcher at the École Polytechnique Fédérale de Lausanne, where he continued his investigations into compressed sensing. He was a member of the Laboratory for Information and Inference Systems. In 2014 Bah joined the University of Texas at Austin, where he worked on signal processing, machine learning and sampling strategies in high- dimensional data. He developed a matrix for dimensionality reduction that uses bi-Lipschitz embeddings, which can exploit data redundancy.
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E.H. Roelfzema is the pen name of Erik Hazelhoff Roelfzema Jr. (February 17, 1947, The Hague - February 11, 2010, Kockengen), Dutch artist, writer, poet, lyricist and musician. In his art he used encaustic techniques, as well as polymer resin (glass panels), and monomer resin (embeddings). He travelled extensively, and from 1970 until 1990 he lived in Ahualoa, Hawaii, working as a farmer and fisherman, and surfing. He returned to the Netherlands in 1990 and married photographer Patricia Steur.
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A k-tuple point (double, triple, etc.) of an immersion is an unordered set } of distinct points with the same image . If M is an m-dimensional manifold and N is an n-dimensional manifold then for an immersion in general position the set of k-tuple points is an -dimensional manifold. Every embedding is an immersion without multiple points (where ). Note, however, that the converse is false: there are injective immersions that are not embeddings.
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The notion of transversality of a pair of submanifolds is easily extended to transversality of a submanifold and a map to the ambient manifold, or to a pair of maps to the ambient manifold, by asking whether the pushforwards of the tangent spaces along the preimage of points of intersection of the images generate the entire tangent space of the ambient manifold.Guillemin and Pollack 1974, p.28. If the maps are embeddings, this is equivalent to transversality of submanifolds.
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In contemporary mathematics the term knot is sometimes used to describe a more general phenomenon related to embeddings. Given a manifold with a submanifold , one sometimes says can be knotted in if there exists an embedding of in which is not isotopic to . Traditional knots form the case where and or . The Schoenflies theorem states that the circle does not knot in the 2-sphere: every topological circle in the 2-sphere is isotopic to a geometric circle.
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Let K be an algebraic number field, and let OK be its ring of integers. Let b1, ..., bn be an integral basis of OK (i.e. a basis as a Z-module), and let {σ1, ..., σn} be the set of embeddings of K into the complex numbers (i.e. injective ring homomorphisms K → C). The discriminant of K is the square of the determinant of the n by n matrix B whose (i,j)-entry is σi(bj).
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This fundamental equivalence between rotation systems and 2-cell-embeddings was first settled in a dual form by Heffter and extensively used by Ringel during the 1950s. Independently, Edmonds gave the primal form of the theorem and the details of his study have been popularized by Youngs. The generalization to the whole set of multigraphs was developed by Gross and Alpert. Rotation systems are related to, but not the same as, the rotation maps used by Reingold et al.
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GloVe can be used to find relations between words like synonyms, company-product relations, zip codes and cities, etc. It is also used by the SpaCy model to build semantic word embeddings/feature vectors while computing the top list words that match with distance measures such as Cosine Similarity and Euclidean distance approach. It was also used as the word representation framework for the online and offline systems designed to detect psychological distress in patient interviews.
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In particular, all dual graphs, for all the different planar embeddings of , have isomorphic graphic matroids.. For nonplanar surface embeddings, unlike planar duals, the dual graph is not generally an algebraic dual of the primal graph. And for a non-planar graph , the dual matroid of the graphic matroid of is not itself a graphic matroid. However, it is still a matroid whose circuits correspond to the cuts in , and in this sense can be thought of as a generalized algebraic dual of .. The duality between Eulerian and bipartite planar graphs can be extended to binary matroids (which include the graphic matroids derived from planar graphs): a binary matroid is Eulerian if and only if its dual matroid is bipartite.. The two dual concepts of girth and edge connectivity are unified in matroid theory by matroid girth: the girth of the graphic matroid of a planar graph is the same as the graph's girth, and the girth of the dual matroid (the graphic matroid of the dual graph) is the edge connectivity of the graph..
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In this case, the embedded metric is a finite metric space, whose distances are shortest path lengths in a graph, and the metric into which is embedded is the Euclidean plane. When the graph and its embedding are fixed, but the graph edge weights can vary, the stretch factor is minimized when the weights are exactly the Euclidean distances between the edge endpoints. Research in this area has focused on finding sparse graphs for a given point set that have low stretch factor.. The Johnson–Lindenstrauss lemma asserts that any finite metric space with points can be embedded into a Euclidean space of dimension with distortion , for any constant , where the constant factor in the -notation depends on the choice of .. This result, and related methods of constructing low-distortion metric embeddings, are important in the theory of approximation algorithms. A major open problem in this area is the GNRS conjecture, which (if true) would characterize the families of graphs that have bounded-stretch embeddings into \ell_1 spaces as being all minor-closed graph families.
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To motivate the (conjectural) motivic Galois group, fix a field k and consider the functor :finite separable extensions K of k → non- empty finite sets with a (continuous) transitive action of the absolute Galois group of k which maps K to the (finite) set of embeddings of K into an algebraic closure of k. In Galois theory this functor is shown to be an equivalence of categories. Notice that fields are 0-dimensional. Motives of this kind are called Artin motives.
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David Arends Gieseker (born 23 November 1943 in Oakland, California)biographical information from American Men and Women of Science, Thomson Gale 2004 is an American mathematician, specializing in algebraic geometry. Gieseker received his bachelor's degree in 1965 from Reed College and his master's degree from Harvard University in 1967. In 1970 he received his Ph.D. under Robin Hartshorne with thesis Contributions to the Theory of Positive Embeddings in Algebraic Geometry. Gieseker became a professor at the University of California, Los Angeles in 1975.
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Using word embedding as an RNN input layer allows the network to parse sentences and phrases using an effective compositional vector grammar. A compositional vector grammar can be thought of as probabilistic context free grammar (PCFG) implemented by an RNN. Recursive auto-encoders built atop word embeddings can assess sentence similarity and detect paraphrasing. Deep neural architectures provide the best results for constituency parsing, sentiment analysis, information retrieval, spoken language understanding, machine translation, contextual entity linking, writing style recognition, Text classification and others.
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Area: Theory and Methods Best Paper Award: Martin Emms and Hector-Hugo Franco-Penya. "ON ORDER EQUIVALENCES BETWEEN DISTANCE AND SIMILARITY MEASURES ON SEQUENCES AND TREES" Best Student Paper: Anna C. Carli, Mario A. T. Figueiredo, Manuele Bicego and Vittorio Murino. "GENERATIVE EMBEDDINGS BASED ON RICIAN MIXTURES" Area: Applications Best Paper Award: Laura Antanas, Martijn van Otterlo, José Oramas, Tinne Tuytelaars and Luc De Raedt. "A RELATIONAL DISTANCE-BASED FRAMEWORK FOR HIERARCHICAL IMAGE UNDERSTANDING" Best Student Paper: Laura Brandolini and Marco Piastra.
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Alexander also proved that the theorem does hold in three dimensions for piecewise linear/smooth embeddings. This is one of the earliest examples where the need for distinction between the categories of topological manifolds, differentiable manifolds, and piecewise linear manifolds became apparent. Now consider Alexander's horned sphere as an embedding into the 3-sphere, considered as the one-point compactification of the 3-dimensional Euclidean space R3. The closure of the non-simply connected domain is called the solid Alexander horned sphere.
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An interesting aspect of Haefliger's work is that the isotopy classes of embeddings of in form a group, with group operation given by the connect sum, provided the co-dimension is greater than two. Haefliger based his work on Stephen Smale's h-cobordism theorem. One of Smale's theorems is that when one deals with knots in co-dimension greater than two, even inequivalent knots have diffeomorphic complements. This gives the subject a different flavour than co-dimension 2 knot theory.
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The space(s) of Schwartz distributions can be embedded into the simplified algebra by (component-wise) convolution with any element of the algebra having as representative a δ-net, i.e. a family of smooth functions \varphi_\varepsilon such that \varphi_\varepsilon\to\delta in D' as ε → 0\. This embedding is non-canonical, because it depends on the choice of the δ-net. However, there are versions of Colombeau algebras (so called full algebras) which allow for canonical embeddings of distributions.
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Manifold alignment can be used to find linear (feature-level) projections, or nonlinear (instance-level) embeddings. While the instance- level version generally produces more accurate alignments, it sacrifices a great degree of flexibility as the learned embedding is often difficult to parameterize. Feature-level projections allow any new instances to be easily embedded in the manifold space, and projections may be combined to form direct mappings between the original data representations. These properties are especially important for knowledge-transfer applications.
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If a graph is planar, this is the only way to embed it flatly and linklessly into space: every flat embedding can be continuously deformed to lie on a flat plane. And conversely, every nonplanar linkless graph has multiple linkless embeddings. An apex graph. If the planar part of the graph is embedded on a flat plane in space, and the apex vertex is placed above the plane and connected to it by straight line segments, the resulting embedding is flat.
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Coates and Ng note that certain variants of k-means behave similarly to sparse coding algorithms. In a comparative evaluation of unsupervised feature learning methods, Coates, Lee and Ng found that k-means clustering with an appropriate transformation outperforms the more recently invented auto-encoders and RBMs on an image classification task. K-means also improves performance in the domain of NLP, specifically for named-entity recognition; there, it competes with Brown clustering, as well as with distributed word representations (also known as neural word embeddings).
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Even worse, the search engine might produce spurious matches (or false positives (FP)), such as retrieving documents referring to "France" as a country. Many approaches orthogonal to entity linking exist to retrieve documents similar to an input document. For example, latent semantic analysis (LSA) or comparing document embeddings obtained with doc2vec. However, these techniques do not allow the same fine-grained control that is offered by entity linking, as they will return other documents instead of creating high-level representations of the original one.
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Shape descriptors can be classified by their invariance with respect to the transformations allowed in the associated shape definition. Many descriptors are invariant with respect to congruency, meaning that congruent shapes (shapes that could be translated, rotated and mirrored) will have the same descriptor (for example moment or spherical harmonic based descriptors or Procrustes analysis operating on point clouds). Another class of shape descriptors (called intrinsic shape descriptors) is invariant with respect to isometry. These descriptors do not change with different isometric embeddings of the shape.
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Komlós, G. N. Sárközy, E. Szemerédi: Blow-up Lemma, "Combinatorica", 17 (1), 1997, pp. 109-123J. Komlós, G. N. Sárközy, E. Szemerédi: An algorithmic version of the Blow-up Lemma, "Random Structures and Algorithms", 12, 1998, pp. 297-312 in which, together with János Komlós and Endre Szemerédi he proved that the regular pairs in Szemerédi regularity lemma behave like complete bipartite graphs under the correct conditions. The lemma allowed for deeper exploration into the nature of embeddings of large sparse graphs into dense graphs.
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For a plane graph G, twice the evaluation of the Tutte polynomial at the point (3,3) equals the sum over weighted Eulerian orientations in the medial graph of G, where the weight of an orientation is 2 to the number of saddle vertices of the orientation (that is, the number of vertices with incident edges cyclically ordered "in, out, in out"). Since the Tutte polynomial is invariant under embeddings, this result shows that every medial graph has the same sum of these weighted Eulerian orientations.
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Laver's theorem, in order theory, states that order embedding of countable total orders is a well-quasi-ordering. That is, for every infinite sequence of totally-ordered countable sets, there exists an order embedding from an earlier member of the sequence to a later member. This result was previously known as Fraïssé's conjecture, after Roland Fraïssé, who conjectured it in 1948; Richard Laver proved the conjecture in 1971. More generally, Laver proved the same result for order embeddings of countable unions of scattered orders.
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A projective spherical variety is a Mori dream space. Spherical embeddings are classified by so-called colored fans, a generalization of fans for toric varieties; this is known as Luna-Vust Theory. In his seminal paper, develops a framework to classify complex spherical subgroups of reductive groups; he reduces the classification of spherical subgroups to wonderful subgroups. He works out completely the case of groups of type A and conjectures that the combinatorial objects (homogeneous spherical data) he introduces indeed provide a combinatorial classification of spherical subgroups.
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The disadvantage of such a definition is that not all varieties come with natural embeddings into projective space. For example, under this definition, the product is not a variety until it is embedded into the projective space; this is usually done by the Segre embedding. However, any variety that admits one embedding into projective space admits many others by composing the embedding with the Veronese embedding. Consequently, many notions that should be intrinsic, such as the concept of a regular function, are not obviously so.
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Inclusion maps tend to be homomorphisms of algebraic structures; thus, such inclusion maps are embeddings. More precisely, given a substructure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for some binary operation , to require that :\iota(x\star y)=\iota(x)\star \iota(y) is simply to say that is consistently computed in the sub-structure and the large structure. The case of a unary operation is similar; but one should also look at nullary operations, which pick out a constant element.
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Here the point is that closure means such constants must already be given in the substructure. Inclusion maps are seen in algebraic topology where if is a strong deformation retract of , the inclusion map yields an isomorphism between all homotopy groups (that is, it is a homotopy equivalence). Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects (which is to say, objects that have pullbacks; these are called covariant in an older and unrelated terminology) such as differential forms restrict to submanifolds, giving a mapping in the other direction.
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Word2vec is a group of related models that are used to produce word embeddings. These models are shallow, two-layer neural networks that are trained to reconstruct linguistic contexts of words. Word2vec takes as its input a large corpus of text and produces a vector space, typically of several hundred dimensions, with each unique word in the corpus being assigned a corresponding vector in the space. Word vectors are positioned in the vector space such that words that share common contexts in the corpus are located close to one another in the space.
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In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of Aronszajn and Panitchpakdi (1956; see e.g. ) that these two different types of definitions are equivalent.
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The "isolate" directional formatting characters signal that a piece of text is to be treated as directionally isolated from its surroundings. As of Unicode 6.3, these are the formatting characters that are being encouraged in new documents – once target platforms are known to support them. These formatting characters were introduced after it became apparent that directional embeddings usually have too strong an effect on their surroundings and are thus unnecessarily difficult to use. Unlike the legacy 'embedding' directional formatting characters, 'isolate' characters have no effect on the ordering of the text outside their scope.
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Some proofs produced by the school are not considered satisfactory because of foundational difficulties. These included frequent use of birational models in dimension three of surfaces that can have non-singular models only when embedded in higher-dimensional projective space. In order to avoid these issues, a sophisticated theory of handling a linear system of divisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings in projective space). Many modern techniques were found, in embryonic form, and in some cases the articulation of these ideas exceeded the available technical language.
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For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures. The properties of the spin representations depend, in a subtle way, on the dimension and signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms, which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.
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A pointed pseudotriangulation is a planar straight-line drawing of a graph, with the properties that the outer face is convex, that every bounded face is a pseudotriangle, a polygon with only three convex vertices, and that the edges incident to every vertex span an angle of less than 180 degrees. The graphs that can be drawn as pointed pseudotriangulations are exactly the planar Laman graphs.. However, Laman graphs have planar embeddings that are not pseudotriangulations, and there are Laman graphs that are not planar, such as the utility graph K3,3.
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In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational numbers. The name of these embeddings is French for a "child's drawing"; its plural is either dessins d'enfant, "child's drawings", or dessins d'enfants, "children's drawings". A dessin d'enfant is a graph, with its vertices colored alternately black and white, embedded in an oriented surface that, in many cases, is simply a plane. For the coloring to exist, the graph must be bipartite.
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The authors examined keyword search, the recommendation system, and the recommendation system with social network working separately and compares the results in terms of search quality. The results show that a personalized search engine with the recommendation system produces better quality results than the standard search engine, and that the recommendation system with social network even improves more. Recent paper “Search personalization with embeddings” shows that a new embedding model for search personalization, where users are embedded on a topical interest space, produces better search results than strong learning-to-rank models.
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However handling these face- based orders is less straightforward, since in some cases some edges may be traversed twice along a face boundary. For example this is always the case for embeddings of trees, which have a single face. To overcome this combinatorial nuisance, one may consider that every edge is "split" lengthwise in two "half- edges", or "sides". Under this convention in all face boundary traversals each half-edge is traversed only once and the two half-edges of the same edge are always traversed in opposite directions.
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There is a precise relation: the lift of a Legendrian submanifold in a symplectization of a contact manifold is a Lagrangian submanifold. The simplest example of Legendrian submanifolds are Legendrian knots inside a contact three-manifold. Inequivalent Legendrian knots may be equivalent as smooth knots; that is, there are knots which are smoothly isotopic where the isotopy cannot be chosen to be a path of Legendrian knots. Legendrian submanifolds are very rigid objects; typically there are infinitely many Legendrian isotopy classes of embeddings which are all smoothly isotopic.
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These are submanifolds whose dimensions are one half that of space time, and such that the pullback of the Kähler form to the submanifold vanishes. The worldvolume theory on a stack of N D2-branes is the string field theory of the open strings of the A-model, which is a U(N) Chern–Simons theory. The fundamental topological strings may end on the D2-branes. While the embedding of a string depends only on the Kähler form, the embeddings of the branes depends entirely on the complex structure.
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Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate studies at Princeton University, from where he received his Ph.D. in mathematics in 1959 after completing a doctoral dissertation titled "On embeddings of spheres." He then became a Junior Fellow at Harvard University from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard.
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In mathematics, specifically the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field. If K is a number field of degree d then there are d distinct embeddings of K into C. We let KC be the image of K in the product Cd, considered as equipped with the usual Hermitian inner product. If c denotes complex conjugation, let KR denote the subspace of KC fixed by c, equipped with a scalar product. This is the Minkowski space of K.
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It is also possible to find a simultaneous embedding with fixed edges for any pair of a planar graph and a tree... It is an open question whether the existence of a simultaneous embedding with fixed edges for two given graphs can be tested in polynomial time. However, for three or more graphs, the problem is NP- complete. When simultaneous embeddings with fixed edges do exist, they can be found in polynomial time for pairs of outerplanar graphs, and for Biconnected graphs, i.e. pairs of graphs whose intersection is biconnected....
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3 Based on these two results, he conjectured that in fact every connected graph with a planar cover is projective.; , Conjecture 4, p. 4 As of 2013, this conjecture remains unsolved. It is also known as Negami's "1-2-∞ conjecture", because it can be reformulated as stating that the minimum ply of a planar cover, if it exists, must be either 1 or 2. K1,2,2,2, the only possible minimal counterexample to Negami's conjecture Like the graphs with planar covers, the graphs with projective plane embeddings can be characterized by forbidden minors.
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A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of in , especially in the case . The branch of mathematics that studies knots is known as knot theory, and has many simple relations to graph theory.
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Vertex addition methods work by maintaining a data structure representing the possible embeddings of an induced subgraph of the given graph, and adding vertices one at a time to this data structure. These methods began with an inefficient O(n2) method conceived by Lempel, Even and Cederbaum in 1967.. It was improved by Even and Tarjan, who found a linear- time solution for the s,t-numbering step,. and by Booth and Lueker, who developed the PQ tree data structure. With these improvements it is linear- time and outperforms the path addition method in practice.
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In fact, there exist graphs that are formed by adding one edge to a planar subgraph, where the optimal drawing has only two crossings but where fixing the planar embedding of the subgraph forces a linear number of crossings to be created. As a compromise between finding the optimal planarization of a planar subgraph plus one edge, and keeping a fixed embedding, it is possible to search over all embeddings of the planarized subgraph and find the one that minimizes the number of crossings formed by the new edge..
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Cross-stitch counted-thread embroidery Knitted mathematical objects include the Platonic solids, Klein bottles and Boy's surface. The Lorenz manifold and the hyperbolic plane have been crafted using crochet.}.. Knitted and crocheted tori have also been constructed depicting toroidal embeddings of the complete graph K7 and of the Heawood graph.. The crocheting of hyperbolic planes has been popularized by the Institute For Figuring; a book by Daina Taimina on the subject, Crocheting Adventures with Hyperbolic Planes, won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year..
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For a dozen years before becoming the White House Domestic Policy Adviser (1994 to 2006), Zinsmeister was editor-in-chief of The American Enterprise, a national magazine covering politics, business, and culture. Zinsmeister was an embedded journalist during the 2003 invasion of Iraq, and then served three additional months-long embeddings with combat units during the insurgency stage of the war. He shot a documentary film about soldiers in Iraq, called "WARRIORS", which was funded by the Corporation for Public Broadcasting and nationally broadcast by PBS. He wrote three books of Iraq reporting.
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Manifest background independence is primarily an aesthetic rather than a physical requirement. It is analogous and closely related to requiring in differential geometry that equations be written in a form that is independent of the choice of charts and coordinate embeddings. If a background-independent formalism is present, it can lead to simpler and more elegant equations. However, there is no physical content in requiring that a theory be manifestly background- independent – for example, the equations of general relativity can be rewritten in local coordinates without affecting the physical implications.
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Indyk's research focuses primarily on computational geometry in high-dimensions, streaming algorithms, and computational learning theory. He has made a range of contributions to these fields, particularly in the study of low-distortion embeddings, algorithmic coding theory, and geometric and combinatorial pattern matching. He has also made contributions to the theory of compressed sensing. His work on algorithms for computing the Fourier transform of signals with sparse spectra faster than the Fast Fourier transform algorithm was selected by MIT Technology Review as a TR10 Top 10 Emerging Technology in 2012.
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Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low- dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle easy, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture; see Gluck twists. The distinction is because surgery theory works in dimension 5 and above (in fact, it works topologically in dimension 4, though this is very involved to prove), and thus the behavior of manifolds in dimension 5 and above is controlled algebraically by surgery theory. In dimension 4 and below (topologically, in dimension 3 and below), surgery theory does not work, and other phenomena occur.
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He proved with his coauthors essentially that a huge class of semidefinite programming algorithms for the famous vertex cover problem will not achieve a solution of value less than the value of the optimal solution times a factor of two. With Nati Linial and Michael Saks, he showed how to embed trees into Euclidean metrics with low distortion. And in a later result, he showed how to do JL-style embeddings that preserved not only distances, but also higher order volumes. He died in a climbing accident in Alaska on May 29, 2010, leaving behind three children and a wife..
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Both of these functors are, in fact, right inverses to U (meaning that UD and UI are equal to the identity functor on Set). Moreover, since any function between discrete or between indiscrete spaces is continuous, both of these functors give full embeddings of Set into Top. Top is also fiber-complete meaning that the category of all topologies on a given set X (called the fiber of U above X) forms a complete lattice when ordered by inclusion. The greatest element in this fiber is the discrete topology on X, while the least element is the indiscrete topology.
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It was concerned with the smooth embeddings of an m-manifold in Rn. In 1969 Porteous published Topological Geometry with Van Nostrand Reinhold and Company. It was reviewed in Mathematical Reviews by J. Eells, who interpreted it as a three-term textbook for a sequence in abstract algebra, geometric algebra, and differential calculus in Euclidean and Banach spaces and on manifolds. Eells says "Surely this book is the product of substantial thought and care, both from the standpoints of consistent mathematical presentation and of student's pedagogical requirements." In 1981 a second edition was published with Cambridge University Press.
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Although the Rado graph is universal for induced subgraphs, it is not universal for isometric embeddings of graphs, where an isometric embedding is a graph isomorphism which preserves distance. The Rado graph has diameter two, and so any graph with larger diameter does not embed isometrically into it. has described a family of universal graphs for isometric embedding, one for each possible finite graph diameter; the graph in his family with diameter two is the Rado graph. The Henson graphs are countable graphs (one for each positive integer ) that do not contain an -vertex clique, and are universal for -clique-free graphs.
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An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. It is often useful to embed topological spaces in compact spaces, because of the special properties compact spaces have. Embeddings into compact Hausdorff spaces may be of particular interest. Since every compact Hausdorff space is a Tychonoff space, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. In fact, the converse is also true; being a Tychonoff space is both necessary and sufficient for possessing a Hausdorff compactification.
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The dual graph of this embedding has four vertices forming a complete graph with doubled edges. In the torus embedding of this dual graph, the six edges incident to each vertex, in cyclic order around that vertex, cycle twice through the three other vertices. In contrast to the situation in the plane, this embedding of the cube and its dual is not unique; the cube graph has several other torus embeddings, with different duals. Many of the equivalences between primal and dual graph properties of planar graphs fail to generalize to nonplanar duals, or require additional care in their generalization.
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Spark NLP is an open-source text processing library for advanced natural language processing for the Python, Java and Scala programming languages. The library is built on top of Apache Spark and its Spark ML library for speed and scalability and on top of TensorFlow for deep learning training & inference functionality. Its goal is to provide an API for natural language processing pipelines that implement state-of-the-art academic research results as production-grade, scalable, and trainable software. The library offers pre- trained neural network models, pipelines, and embeddings, as well as support for training custom models.
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Smale's graduate adviser Raoul Bott at first told Smale that the result was obviously wrong . His reasoning was that the degree of the Gauss map must be preserved in such "turning"—in particular it follows that there is no such turning of S1 in R2. But the degrees of the Gauss map for the embeddings f and −f in R3 are both equal to 1, and do not have opposite sign as one might incorrectly guess. The degree of the Gauss map of all immersions of S2 in R3 is 1, so there is no obstacle.
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Clifford algebras exhibit a 2-fold periodicity over the complex numbers and an 8-fold periodicity over the real numbers, which is related to the same periodicities for homotopy groups of the stable unitary group and stable orthogonal group, and is called Bott periodicity. The connection is explained by the geometric model of loop spaces approach to Bott periodicity: their 2-fold/8-fold periodic embeddings of the classical groups in each other (corresponding to isomorphism groups of Clifford algebras), and their successive quotients are symmetric spaces which are homotopy equivalent to the loop spaces of the unitary/orthogonal group.
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A three-page book embedding of the complete graph . Because it is not a planar graph, it is not possible to embed this graph without crossings on fewer pages, so its book thickness is three. 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.
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In universal algebra and model theory, a class of structures K is said to have the joint embedding property if for all structures A and B in K, there is a structure C in K such that both A and B have embeddings into C. It is one of the three properties used to define the age of a structure. A first-order theory has the joint embedding property if the class of its models of has the joint embedding property Chang, C. C.; Keisler, H. Jerome (2012). Model Theory (Third edition ed.). Dover Publications. pp.
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In mathematics, Vopěnka's principle is a large cardinal axiom. The intuition behind the axiom is that the set-theoretical universe is so large that in every proper class, some members are similar to others, with this similarity formalized through elementary embeddings. Vopěnka's principle was first introduced by Petr Vopěnka and independently considered by H. Jerome Keisler, and was written up by . According to , Vopěnka's principle was originally intended as a joke: Vopěnka was apparently unenthusiastic about large cardinals and introduced his principle as a bogus large cardinal property, planning to show later that it was not consistent.
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The Dedekind–MacNeille completion is self-dual: the completion of the dual of a partial order is the same as the dual of the completion.. The Dedekind–MacNeille completion of has the same order dimension as does itself.This result is frequently attributed to an unpublished 1961 Harvard University honors thesis by K. A. Baker, "Dimension, join-independence and breadth in partially ordered sets". It was published by . In the category of partially ordered sets and monotonic functions between partially ordered sets, the complete lattices form the injective objects for order-embeddings, and the Dedekind–MacNeille completion of is the injective hull of ..
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Suppose that are a set of generators for the unit group modulo roots of unity. If is an algebraic number, write for the different embeddings into or , and set to 1 or 2 if the corresponding embedding is real or complex respectively. Then the matrix whose entries are , has the property that the sum of any row is zero (because all units have norm 1, and the log of the norm is the sum of the entries in a row). This implies that the absolute value of the determinant of the submatrix formed by deleting one column is independent of the column.
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Each decoder consists of three major components: a self-attention mechanism, an attention mechanism over the encodings, and a feed-forward neural network. The decoder functions in a similar fashion to the encoder, but an additional attention mechanism is inserted which instead draws relevant information from the encodings generated by the encoders. Like the first encoder, the first decoder takes positional information and embeddings of the output sequence as its input, rather than encodings. Since the transformer should not use the current or future output to predict an output though, the output sequence must be partially masked to prevent this reverse information flow.
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The algorithm described above performs a "one-step" alignment, finding embeddings for both data sets at the same time. A similar effect can also be achieved with "two- step" alignments , following a slightly modified procedure: # Project each input data set to a lower-dimensional space independently, using any of a variety of dimension reduction algorithms. # Perform linear manifold alignment on the embedded data, holding the first data set fixed, mapping each additional data set onto the first's manifold. This approach has the benefit of decomposing the required computation, which lowers memory overhead and allows parallel implementations.
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The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves are formal linear combinations of points on the curve. Classical work on algebraic curves related these to intrinsic data, such as the regular differentials on a compact Riemann surface, and to extrinsic properties, such as embeddings of the curve into projective space. While divisors on higher- dimensional varieties continue to play an important role in determining the structure of the variety, on varieties of dimension two or more there are also higher codimension cycles to consider.
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In mathematics, the Johnson–Lindenstrauss lemma is a result named after William B. Johnson and Joram Lindenstrauss concerning low-distortion embeddings of points from high-dimensional into low-dimensional Euclidean space. The lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly preserved. The map used for the embedding is at least Lipschitz, and can even be taken to be an orthogonal projection. The lemma has uses in compressed sensing, manifold learning, dimensionality reduction, and graph embedding.
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A topological graph is also called a drawing of a graph. An important special class of topological graphs is the class of geometric graphs, where the edges are represented by line segments. (The term geometric graph is sometimes used in a broader, somewhat vague sense.) The theory of topological graphs is an area of graph theory, mainly concerned with combinatorial properties of topological graphs, in particular, with the crossing patterns of their edges. It is closely related to graph drawing, a field which is more application oriented, and topological graph theory, which focuses on embeddings of graphs in surfaces (that is, drawings without crossings).
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Outside of computer science, CPS is of more general interest as an alternative to the conventional method of composing simple expressions into complex expressions. For example, within linguistic semantics, Chris Barker and his collaborators have suggested that specifying the denotations of sentences using CPS might explain certain phenomena in natural language. In mathematics, the Curry–Howard isomorphism between computer programs and mathematical proofs relates continuation-passing style translation to a variation of double- negation embeddings of classical logic into intuitionistic (constructive) logic. Unlike the regular double-negation translation, which maps atomic propositions p to ((p → ⊥) → ⊥), the continuation passing style replaces ⊥ by the type of the final expression.
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More generally, every planar graph of minimum degree at least three either has an edge of total degree at most 12, or at least 60 edges that (like the edges in the triakis icosahedron) connect vertices of degrees 3 and 10. If all triangular faces of a polyhedron are vertex-disjoint, there exists an edge with smaller total degree, at most eight. Generalizations of the theorem are also known for graph embeddings onto surfaces with higher genus. The theorem cannot be generalized to all planar graphs, as the complete bipartite graphs K_{1,n-1} and K_{2,n-2} have edges with unbounded total degree.
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By Steinitz's theorem, these graphs are exactly the polyhedral graphs, the graphs of convex polyhedra. A planar graph is 3-vertex-connected if and only if its dual graph is 3-vertex-connected. More generally, a planar graph has a unique embedding, and therefore also a unique dual, if and only if it is a subdivision of a 3-vertex-connected planar graph (a graph formed from a 3-vertex-connected planar graph by replacing some of its edges by paths). For some planar graphs that are not 3-vertex- connected, such as the complete bipartite graph , the embedding is not unique, but all embeddings are isomorphic.
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Nirenberg's Ph.D. thesis provided a resolution of the Weyl problem and Minkowski problem of differential geometry. The former asks for the existence of isometric embeddings of positively curved Riemannian metrics on the two-dimensional sphere into three-dimensional Euclidean space, while the latter asks for closed surfaces in three- dimensional Euclidean space of prescribed Gaussian curvature. The now-standard approach to these problems is through the theory of the Monge-Ampère equation, which is a fully nonlinear elliptic partial differential equation. Nirenberg made novel contributions to the theory of such equations in the setting of two-dimensional domains, building on the earlier 1938 work of Charles Morrey.
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Canonical inner models at the level of subcompact cardinals satisfy the square principle at all but subcompact cardinals. (Existence of such models has not yet been proved, but in any case the square principle can be forced for weaker cardinals.) Quasicompactness is one of the strongest large cardinal properties that can be witnessed by current inner models that do not use long extenders. For current inner models, the elementary embeddings included are determined by their effect on P(κ) (as computed at the stage the embedding is included), where κ is the critical point. This prevents them from witnessing even a κ+ strongly compact cardinal κ.
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There are some variations of Reinhardt cardinals, forming a hierarchy of hypotheses asserting the existence of elementary embeddings V\to V. J3: There is a nontrivial elementary embedding j: V\to V J2: There is a nontrivial elementary embedding j: V\to V and DC\lambda holds, where \lambda is the least fixed-point above the critical point. J1: There is a cardinal \kappa such that for every ordinal \alpha, there is an elementary embedding j : V\to V with j(\kappa)>\alpha and having critical point \kappa. Each of J1 and J2 immediately imply J3. A cardinal \kappa as in J1 is known as a super Reinhardt cardinal.
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More recently, it refers largely to the use of nonlinear partial differential equations to study geometric and topological properties of spaces, such as submanifolds of Euclidean space, Riemannian manifolds, and symplectic manifolds. This approach dates back to the work by Tibor Radó and Jesse Douglas on minimal surfaces, John Forbes Nash Jr. on isometric embeddings of Riemannian manifolds into Euclidean space, work by Louis Nirenberg on the Minkowski problem and the Weyl problem, and work by Aleksandr Danilovich Aleksandrov and Aleksei Pogorelov on convex hypersurfaces. In the 1980s fundamental contributions by Karen Uhlenbeck,Jackson, Allyn. (2019). Founder of geometric analysis honored with Abel Prize Retrieved 20 March 2019.
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In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three- dimensional analogue of the planar graphs.. Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding. Flat embeddings are automatically linkless, but not vice versa.
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Gowers's construction for the lower bound of Szemerédi's regularity lemma first introduced a weaker version of this lemma, restricted to bipartite graphs, in order to prove Szemerédi's theorem,. and in he proved the full lemma.. Extensions of the regularity method to hypergraphs were obtained by Rödl and his collaborators... and Gowers... János Komlós, Gábor Sárközy and Endre Szemerédi later (in 1997) proved in the blow-up lemma that the regular pairs in Szemerédi regularity lemma behave like complete bipartite graphs under the correct conditions. The lemma allowed for deeper exploration into the nature of embeddings of large sparse graphs into dense graphs. The first constructive version was provided by Alon, Duke, Lefmann, Rödl and Yuster.
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M11 has a sharply 4-transitive permutation representation on 11 points, whose point stabilizer is sometimes denoted by M10, and is a non-split extension of the form A6.2 (an extension of the group of order 2 by the alternating group A6). This action is the automorphism group of a Steiner system S(4,5,11). The induced action on unordered pairs of points gives a rank 3 action on 55 points. M11 has a 3-transitive permutation representation on 12 points with point stabilizer PSL2(11). The permutation representations on 11 and 12 points can both be seen inside the Mathieu group M12 as two different embeddings of M11 in M12, exchanged by an outer automorphism.
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K-theory was invented in the late 1950s by Alexander Grothendieck in his study of intersection theory on algebraic varieties. In the modern language, Grothendieck defined only K0, the zeroth K-group, but even this single group has plenty of applications, such as the Grothendieck–Riemann–Roch theorem. Intersection theory is still a motivating force in the development of (higher) algebraic K-theory through its links with motivic cohomology and specifically Chow groups. The subject also includes classical number-theoretic topics like quadratic reciprocity and embeddings of number fields into the real numbers and complex numbers, as well as more modern concerns like the construction of higher regulators and special values of L-functions.
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The use of the phrase "arc diagram" for this kind of drawings follows the use of a similar type of diagram by to visualize the repetition patterns in strings, by using arcs to connect pairs of equal substrings. However, this style of graph drawing is much older than its name, dating back to the work of and , who used arc diagrams to study crossing numbers of graphs. An older but less frequently used name for arc diagrams is linear embeddings. write that arc diagrams "may not convey the overall structure of the graph as effectively as a two-dimensional layout", but that their layout makes it easy to display multivariate data associated with the vertices of the graph.
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A different method uses an inductive construction of 3-connected graphs to incrementally build planar embeddings of every 3-connected component of G (and hence a planar embedding of G itself). The construction starts with K4 and is defined in such a way that every intermediate graph on the way to the full component is again 3-connected. Since such graphs have a unique embedding (up to flipping and the choice of the external face), the next bigger graph, if still planar, must be a refinement of the former graph. This allows to reduce the planarity test to just testing for each step whether the next added edge has both ends in the external face of the current embedding.
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More generally, any algebraic curve of genus one, for example the intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked point to act as the identity. Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and this correspondence is also a group isomorphism. Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem.
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In 1986, Hamilton and Michael Gage applied Hamilton's Nash-Moser theorem and well-posedness result for parabolic equations to prove the well-posedness for mean curvature flow; they considered the general case of a one-parameter family of immersions of a closed manifold into a smooth Riemannian manifold. Then, they specialized to the case of immersions of the circle into the two-dimensional Euclidean space , which is the simplest context for curve shortening flow. Using the maximum principle as applied to the distance between two points on a curve, they proved that if the initial immersion is an embedding, then all future immersions in the mean curvature flow are embeddings as well. Furthermore, convexity of the curves is preserved into the future.
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In the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the other without breaking it. Such a distortion is an example of an ambient isotopy. More precisely, let N and M be manifolds and g and h be embeddings of N in M. A continuous map :F:M \times [0,1] \rightarrow M is defined to be an ambient isotopy taking g to h if F0 is the identity map, each map Ft is a homeomorphism from M to itself, and F1 ∘ g = h.
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Here a modulus (or ray divisor) is a formal finite product of the valuations (also called primes or places) of K with positive integer exponents. The archimedean valuations that might appear in a modulus include only those whose completions are the real numbers (not the complex numbers); they may be identified with orderings on K and occur only to exponent one. The modulus m is a product of a non- archimedean (finite) part mf and an archimedean (infinite) part m∞. The non- archimedean part mf is a nonzero ideal in the ring of integers OK of K and the archimedean part m∞ is simply a set of real embeddings of K. Associated to such a modulus m are two groups of fractional ideals.
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The book thickness, pagenumber, or stack number of is the minimum number of pages required for a book embedding of . Another parameter that measures the quality of a book embedding, beyond its number of pages, is its pagewidth. This is the maximum number of edges that can be crossed by a ray perpendicular to the spine within a single page. Equivalently (for book embeddings in which each edge is drawn as a monotonic curve), it is the maximum size of a subset of edges within a single page such that the intervals defined on the spine by pairs of endpoints of the edges all intersect each other... It is crucial for these definitions that edges are only allowed to stay within a single page of the book.
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The Goldner–Harary graph provides an example of a planar graph that does not have book thickness two: it is a maximal planar graph, so it is not possible to add any edges to it while preserving planarity, and it does not have a Hamiltonian cycle. Because of this characterization by Hamiltonian cycles, graphs that have two-page book embeddings are also known as subhamiltonian graphs.. All planar graphs whose maximum degree is at most four have book thickness at most two.. Planar 3-trees have book thickness at most three.. More generally, all planar graphs have book thickness four. It has been claimed by Mihalis Yannakakis in 1986. that there exist some planar graphs that have book thickness exactly four.
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A directed acyclic graph must be planar in order to have an upward planar drawing, but not every planar acyclic graph has such a drawing. Among the planar directed acyclic graphs with a single source (vertex with no incoming edges) and sink (vertex with no outgoing edges), the graphs with upward planar drawings are the st-planar graphs, planar graphs in which the source and sink both belong to the same face of at least one of the planar embeddings of the graph. More generally, a graph G has an upward planar drawing if and only if it is directed and acyclic, and is a subgraph of an st-planar graph on the same vertex set., pp. 111–112; , 6.1 "Inclusion in a Planar st-Graph", pp.
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If U is Urysohn universal and X is any separable metric space, then there exists an isometric embedding f:X → U. (Other spaces share this property: for instance, the space l∞ of all bounded real sequences with the supremum norm admits isometric embeddings of all separable metric spaces ("Fréchet embedding"), as does the space C[0,1] of all continuous functions [0,1]→R, again with the supremum norm, a result due to Stefan Banach.) Furthermore, every isometry between finite subsets of U extends to an isometry of U onto itself. This kind of "homogeneity" actually characterizes Urysohn universal spaces: A separable complete metric space that contains an isometric image of every separable metric space is Urysohn universal if and only if it is homogeneous in this sense.
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In mathematics, Jean-Pierre Serre conjectured the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group. Namely, he conjectured that if G is such a group over a perfect field F of cohomological dimension at most 2, then the Galois cohomology set H1(F, G) is zero. A converse of the conjecture holds: if the field F is perfect and if the cohomology set H1(F, G) is zero for every semisimple simply connected algebraic group G then the p-cohomological dimension of F is at most 2 for every prime p. The conjecture holds in the case where F is a local field (such as p-adic field) or a global field with no real embeddings (such as Q()).
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Many important families of combinatorial structures, such as the acyclic orientations of a graph or the adjacencies between regions in a hyperplane arrangement, can be represented as partial cube graphs. An important special case of a partial cube is the skeleton of the permutohedron, a graph in which vertices represent permutations of a set of ordered objects and edges represent swaps of objects adjacent in the order. Several other important classes of graphs including median graphs have related definitions involving metric embeddings . A flip graph is a graph formed from the triangulations of a point set, in which each vertex represents a triangulation and two triangulations are connected by an edge if they differ by the replacement of one edge for another.
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In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map is an embedding is often indicated by the use of a "hooked arrow" (); thus: f : X \hookrightarrow Y. (On the other hand, this notation is sometimes reserved for inclusion maps.) Given X and Y, several different embeddings of X in Y may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain X with its image f(X) contained in Y, so that .
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In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a closure operator). In a concrete category, an embedding is a morphism ƒ: A → B which is an injective function from the underlying set of A to the underlying set of B and is also an initial morphism in the following sense: If g is a function from the underlying set of an object C to the underlying set of A, and if its composition with ƒ is a morphism ƒg: C → B, then g itself is a morphism. A factorization system for a category also gives rise to a notion of embedding.
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In 1969, Branko Grünbaum conjectured that every 3-regular graph with a polyhedral embedding on any two-dimensional oriented manifold such as a torus must be of class one. In this context, a polyhedral embedding is a graph embedding such that every face of the embedding is topologically a disk and such that the dual graph of the embedding is simple, with no self-loops or multiple adjacencies. If true, this would be a generalization of the four color theorem, which was shown by Tait to be equivalent to the statement that 3-regular graphs with a polyhedral embedding on a sphere are of class one. However, showed the conjecture to be false by finding snarks that have polyhedral embeddings on high-genus orientable surfaces.
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The face cycles of these embeddings generate a proper subset of all Eulerian subgraphs. The homology group H_2(S,\Z_2) of the given surface S characterizes the Eulerian subgraphs that cannot be represented as the boundary of a set of faces. Mac Lane's planarity criterion uses this idea to characterize the planar graphs in terms of the cycle bases: a finite undirected graph is planar if and only if it has a sparse cycle basis or 2-basis, a basis in which each edge of the graph participates in at most two basis cycles. In a planar graph, the cycle basis formed by the set of bounded faces is necessarily sparse, and conversely, a sparse cycle basis of any graph necessarily forms the set of bounded faces of a planar embedding of its graph..
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"Structures métriques pour les variétés riemanniennes", edited by Lafontaine and Pierre Pansu, 1981.M. Gromov, Groups of Polynomial growth and Expanding Maps, Publications mathematiques I.H.É.S., 53, 1981 This distance measures how far two compact metric spaces are from being isometric. If X and Y are two compact metric spaces, then dGH (X, Y) is defined to be the infimum of all numbers dH(f(X), g(Y)) for all metric spaces M and all isometric embeddings f : X → M and g : Y → M. Here dH denotes Hausdorff distance between subsets in M and the isometric embedding is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no compact Riemannian manifold admits such an embedding into Euclidean space of the same dimension.
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The dual graph depends on how the primal graph is embedded: different planar embeddings of a single graph may lead to different dual graphs. Matroid duality is an algebraic extension of planar graph duality, in the sense that the dual matroid of the graphic matroid of a planar graph is isomorphic to the graphic matroid of the dual graph. A kind of geometric duality also occurs in optimization theory, but not one that reverses dimensions. A linear program may be specified by a system of real variables (the coordinates for a point in Euclidean space a system of linear constraints (specifying that the point lie in a halfspace; the intersection of these halfspaces is a convex polytope, the feasible region of the program), and a linear function (what to optimize).
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If a cactus is connected, and each of its vertices belongs to at most two blocks, then it is called a Christmas cactus. Every polyhedral graph has a Christmas cactus subgraph that includes all of its vertices, a fact that plays an essential role in a proof by that every polyhedral graph has a greedy embedding in the Euclidean plane, an assignment of coordinates to the vertices for which greedy forwarding succeeds in routing messages between all pairs of vertices.. In topological graph theory, the graphs whose cellular embeddings are all planar are exactly the subfamily of the cactus graphs with the additional property that each vertex belongs to at most one cycle. These graphs have two forbidden minors, the diamond graph and the five-vertex friendship graph.
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The "pearls" of the title include theorems, proofs, problems, and examples in graph theory. It has ten chapters; after an introductory chapter on basic definitions, the remaining chapters material on graph coloring; Hamiltonian cycles and Euler tours; extremal graph theory; subgraph counting problems including connections to permutations, derangements, and Cayley's formula; graph labelings; planar graphs, the four color theorem, and the circle packing theorem; near-planar graphs; and graph embeddings on topological surfaces. The book also includes several unsolved problems such as the Oberwolfach problem on covering complete graphs by cycles, the characterization of magic graphs, and ringel's "earth-moon" problem on coloring biplanar graphs. Despite its subtitle promising "a comprehensive introduction" to graph theory, many important topics in graph theory are not covered, with the selection of topics reflecting author Ringel's research interests.
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The nature of the multiple points classifies immersions; for example, immersions of a circle in the plane are classified up to regular homotopy by the number of double points. At a key point in surgery theory it is necessary to decide if an immersion of an m-sphere in a 2m-dimensional manifold is regular homotopic to an embedding, in which case it can be killed by surgery. Wall associated to f an invariant μ(f) in a quotient of the fundamental group ring Z[1(N)] which counts the double points of f in the universal cover of N. For , f is regular homotopic to an embedding if and only if by the Whitney trick. One can study embeddings as "immersions without multiple points", since immersions are easier to classify.
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Sanjeev Arora (born January 1968) is an Indian American theoretical computer scientist who is best known for his work on probabilistically checkable proofs and, in particular, the PCP theorem. He is currently the Charles C. Fitzmorris Professor of Computer Science at Princeton University, and his research interests include computational complexity theory, uses of randomness in computation, probabilistically checkable proofs, computing approximate solutions to NP-hard problems, geometric embeddings of metric spaces, and theoretical machine learning (especially deep learning). He received a B.S. in Mathematics with Computer Science from MIT in 1990 and received a Ph.D. in Computer Science from the University of California, Berkeley in 1994 under Umesh Vazirani. Earlier, in 1986, Sanjeev Arora had topped the IIT JEE but transferred to MIT after 2 years at IIT Kanpur.
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In mathematics, specifically in combinatorial commutative algebra, a convex lattice polytope P is called normal if it has the following property: given any positive integer n, every lattice point of the dilation nP, obtained from P by scaling its vertices by the factor n and taking the convex hull of the resulting points, can be written as the sum of exactly n lattice points in P. This property plays an important role in the theory of toric varieties, where it corresponds to projective normality of the toric variety determined by P. Normal polytopes have popularity in algebraic combinatorics. These polytopes also represent the homogeneous case of the Hilbert bases of finite positive rational cones and the connection to algebraic geometry is that they define projectively normal embeddings of toric varieties.
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In some cases, a graph may be embedded in space in such a way that, for each cycle in the graph, one can find a disk bounded by that cycle that does not cross any other feature of the graph. In this case, the cycle must be unlinked from all the other cycles disjoint from it in the graph. The embedding is said to be flat if every cycle bounds a disk in this way.. A similar definition of a "good embedding" appears in ; see also and . A flat embedding is necessarily linkless, but there may exist linkless embeddings that are not flat: for instance, if G is a graph formed by two disjoint cycles, and it is embedded to form the Whitehead link, then the embedding is linkless but not flat.
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Avner Magen (March 30, 1968 – May 29, 2010) was an associate professor of computer science at the University of Toronto whose research focused on the theory of metric embeddings, discrete geometry and computational geometry. He completed his undergraduate and graduate studies at the Hebrew University of Jerusalem, and received his Ph.D. in Computer Science in 2002, under the supervision of Nati Linial.. He held a postdoctoral fellowship at NEC Research in Princeton, New Jersey, from 2000 until 2002. He joined the University of Toronto in 2002, first as a postdoctoral fellow, and then as an assistant professor in 2004. He was promoted to associate professor in 2009.. His major contributions include an algorithm for approximating the weight of the Euclidean minimum spanning tree in sublinear time, and finding a tight integrality gap for the vertex cover problem using the Frankl–Rödl graphs.
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The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: If Y is a topological space, and fi : Xi → Y is a continuous map for each i ∈ I, then there exists precisely one continuous map f : X → Y such that the following set of diagrams commute: Characteristic property of disjoint unions This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map f : X → Y is continuous iff fi = f o φi is continuous for all i in I. In addition to being continuous, the canonical injections φi : Xi → X are open and closed maps. It follows that the injections are topological embeddings so that each Xi may be canonically thought of as a subspace of X.
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However, if a graph can be made Hamiltonian by the addition of vertices and edges it can also be made Hamiltonian by the addition of edges alone, so this extra freedom does not change the definition.For instance in a 2003 technical report "Book embeddings of graphs and a theorem of Whitney", Paul Kainen defines subhamiltonian graphs to be subgraphs of planar Hamiltonian graphs, without restriction on the vertex set of the augmentation, but writes that "in the definition of subhamiltonian graph, one can require that the extension only involve the inclusion of new edges." In a subhamiltonian graph, a subhamiltonian cycle is a cyclic sequence of vertices such that adding an edge between each consecutive pair of vertices in the sequence preserves the planarity of the graph. A graph is subhamiltonian if and only if it has a subhamiltonian cycle.
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Other pairs of graph types that always admit a simultaneous embedding, but that might need larger grid sizes, include a wheel graph and a cycle graph, a tree and a matching, or a pair of graphs both of which have maximum degree two. However, pairs of planar graphs and a matching, or of a tree and a path exist, that have no simultaneous geometric embedding... Testing whether two graphs admit a simultaneous geometric embedding is NP-hard.. More precisely, it is complete for the existential theory of the reals. The proof of this result also implies that for some pairs of graphs that have simultaneous geometric embeddings, the smallest grid on which they can be drawn has doubly exponential size.. . When a simultaneous geometric embedding exists, it automatically is also a simultaneous embedding with fixed edges.
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He defined new generalized homology and cohomology theories called bordism and cobordism, and pointed out that many of the deep results on cobordism of manifolds found by René Thom, C. T. C. Wall, and others could be naturally reinterpreted as statements about these cohomology theories. Some of these cohomology theories, in particular complex cobordism, turned out to be some of the most powerful cohomology theories known. He introduced the J-group J(X) of a finite complex X, defined as the group of stable fiber homotopy equivalence classes of sphere bundles; this was later studied in detail by J. F. Adams in a series of papers, leading to the Adams conjecture. With Hirzebruch he extended the Grothendieck–Riemann–Roch theorem to complex analytic embeddings, and in a related paper they showed that the Hodge conjecture for integral cohomology is false.
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If is restricted to C^k(K;U) then the following induced map is a homeomorphism (and thus a TVS-isomorphism): :C^k(K;U) \to C^k(K;V) and thus the next two maps (which like the previous map are defined by f\mapsto I(f)) are topological embeddings: :C^k(K;U) \to C^k(V), :C^k(K;U)\to C_c^k(V). (the topology on C_c^k(V) is the canonical LF topology, which is defined later). Using C_c^k(U) i f\mapsto I(f)\in C_c^k(V) we identify C_c^k(U) with its image in C_c^k(V) \subset C^k(V). Since C^k(K;U)\subset C_c^k(U), through this identification C^k(K;U) can also be considered as a subset of C^k(V).
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Of particular interest are those embeddings where the image of X is dense in K; these are called Hausdorff compactifications of X. Given any embedding of a Tychonoff space X in a compact Hausdorff space K the closure of the image of X in K is a compactification of X. In the same 1930 article where Tychonoff defined completely regular spaces, he also proved that every Tychonoff space has a Hausdorff compactification. Among those Hausdorff compactifications, there is a unique "most general" one, the Stone–Čech compactification βX. It is characterised by the universal property that, given a continuous map f from X to any other compact Hausdorff space Y, there is a unique continuous map g from βX to Y that extends f in the sense that f is the composition of g and j.
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In other words, it is believed to be independent (see large cardinal for a discussion). It is usually formulated as follows: :0† exists if and only if there exists a non-trivial elementary embedding j : L[U] → L[U] for the relativized Gödel constructible universe L[U], where U is an ultrafilter witnessing that some cardinal κ is measurable. If 0† exists, then a careful analysis of the embeddings of L[U] into itself reveals that there is a closed unbounded subset of κ, and a closed unbounded proper class of ordinals greater than κ, which together are indiscernible for the structure (L,\in,U), and 0† is defined to be the set of Gödel numbers of the true formulas about the indiscernibles in L[U]. Solovay showed that the existence of 0† follows from the existence of two measurable cardinals.
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In linguistics, center embedding is the process of embedding a phrase in the middle of another phrase of the same type. This often leads to difficulty with parsing which would be difficult to explain on grammatical grounds alone. The most frequently used example involves embedding a relative clause inside another one as in: : A man that a woman loves \Rightarrow : A man that a woman that a child knows loves \Rightarrow : A man that a woman that a child that a bird saw knows loves \Rightarrow : A man that a woman that a child that a bird that I heard saw knows loves In theories of natural language parsing, the difficulty with multiple center embedding is thought to arise from limitations of the human short term memory. In order to process multiple center embeddings, we have to store many subjects in order to connect them to their predicates.
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Other topics covered through the book include the rigid geometric chirality of tree-like molecular structures such as tartaric acid, and the stronger topological chirality of molecules that cannot be deformed into their mirror image without breaking and re-forming some of their molecular bonds. It discusses results of Flapan and Jonathan Simon on molecules with the molecular structure of Möbius ladders, according to which every embedding of a Möbius ladder with an odd number of rungs is chiral while Möbius ladders with an even number of rungs have achiral embeddings. It uses the symmetries of graphs, in a result that the symmetries of certain graphs can always be extended to topological symmetries of three-dimensional space, from which it follows that non-planar graphs with no self-inverse symmetry are always chiral. It discusses graphs for which every embedding is topologically knotted or linked.
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In Western art history, mise en abyme is a formal technique in which an image contains a smaller copy of itself, in a sequence appearing to recur infinitely; "recursive" is another term for this. The modern meaning of the phrase originates with the author André Gide who used it to describe self- reflexive embeddings in various art forms and to describe what he sought in his own work.Medieval mise-en-abyme: the object depicted within itself (collection of papers) As examples, Gide cites both paintings such as Las Meninas by Diego Velázquez and literary forms such as William Shakespeare's use of the "play within a play" device in Hamlet, where a theatrical company presents a performance for the characters that illuminates a thematic aspect of the play itself. This use of the phrase mise en abyme was picked up by scholars and popularized in the 1977 book Le récit spéculaire.
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Whether for interpolation, denoising, or extrapolation, their innovation accounts for dynamic and/or nonlinear interdependencies of nodal processes. These are instrumental in practice to predict partially observed dynamic processes over communication networks; to estimate IP traffic and map anomalies in such networks; to infer functions over brain networks, as well as regulatory processes by leveraging genetic perturbations on gene networks; and even track cascades over social networks under smooth or switching dynamics. To cope with large-scale graphs, they further developed canonical correlation analysis tools for graph data; data adaptive active sampling strategies; node embeddings with adaptive similarities; and random walk driven adaptive diffusions that can outperform state-of-the-art graph convolutional neural networks. Giannakis and collaborators have also contributed to the resurgence of artificial intelligence (AI), and specifically to the areas of crowdsourcing, ensemble learning, interactive learning, and the associated performance analyses.
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One direction of Steinitz's theorem (the easier direction to prove) states that the graph of every convex polyhedron is planar and 3-connected. As shown in the illustration, planarity can be shown by using a Schlegel diagram: if one places a light source near one face of the polyhedron, and a plane on the other side, the shadows of the polyhedron edges will form a planar graph, embedded in such a way that the edges are straight line segments. The 3-connectivity of a polyhedral graph is a special case of Balinski's theorem that the graph of any k-dimensional convex polytope is k-connected.. The other, more difficult, direction of Steinitz's theorem states that every planar 3-connected graph is the graph of a convex polyhedron. There are three standard approaches for this part: proofs by induction, lifting two-dimensional Tutte embeddings into three dimensions using the Maxwell–Cremona correspondence, and methods using the circle packing theorem to generate a canonical polyhedron.
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An alternative notation is to say that a homotopy between two continuous functions f, g: X \to Y is a family of continuous functions h_t: X \to Y for t \in [0,1] such that h_0 = f and h_1 = g, and the map (x, t) \mapsto h_t(x) is continuous from X \times [0,1] to Y. The two versions coincide by setting h_t(x) = H(x,t). It is not sufficient to require each map h_t(x) to be continuous.Path homotopy and separately continuous functions The animation that is looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into . X is the torus, Y is , f is some continuous function from the torus to R3 that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts; g is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape.
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In a maximal planar graph, the book thickness is two if and only if a Hamiltonian cycle exists. Therefore, it is also NP-complete to test whether the book thickness of a given maximal planar graph is two. If an ordering of the vertices of a graph along the spine of an embedding is fixed, then a two-page embedding (if it exists) can be found in linear time, as an instance of planarity testing for a graph formed by augmenting the given graph with a cycle connecting the vertices in their spine ordering. claimed that finding three-page embeddings with a fixed spine ordering can also be performed in polynomial time although his writeup of this result omits many details.. However, for graphs that require four or more pages, the problem of finding an embedding with the minimum possible number of pages remains NP-hard, via an equivalence to the NP-hard problem of coloring circle graphs, the intersection graphs of chords of a circle.
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Roughly, the Whitney trick allows one to "unknot" knotted spheres – more precisely, remove self-intersections of immersions; it does this via a homotopy of a disk – the disk has 2 dimensions, and the homotopy adds 1 more – and thus in codimension greater than 2, this can be done without intersecting itself; hence embeddings in codimension greater than 2 can be understood by surgery. In surgery theory, the key step is in the middle dimension, and thus when the middle dimension has codimension more than 2 (loosely, 2½ is enough, hence total dimension 5 is enough), the Whitney trick works. The key consequence of this is Smale's h-cobordism theorem, which works in dimension 5 and above, and forms the basis for surgery theory. A modification of the Whitney trick can work in 4 dimensions, and is called Casson handles – because there are not enough dimensions, a Whitney disk introduces new kinks, which can be resolved by another Whitney disk, leading to a sequence ("tower") of disks.
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An (n+1)-dimensional cobordism is a quintuple (W; M, N, i, j) consisting of an (n+1)-dimensional compact differentiable manifold with boundary, W; closed n-manifolds M, N; and embeddings i\colon M \hookrightarrow \partial W, j\colon N \hookrightarrow\partial W with disjoint images such that :\partial W = i(M) \sqcup j(N)~. The terminology is usually abbreviated to (W; M, N).The notation "(n+1)-dimensional" is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional cobordism" refers to a 5-dimensional cobordism between 4-dimensional manifolds or a 6-dimensional cobordism between 5-dimensional manifolds. M and N are called cobordant if such a cobordism exists. All manifolds cobordant to a fixed given manifold M form the cobordism class of M. Every closed manifold M is the boundary of the non-compact manifold M × [0, 1); for this reason we require W to be compact in the definition of cobordism.
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In general topology, an embedding is a homeomorphism onto its image.. . More explicitly, an injective continuous map f : X \to Y between topological spaces X and Y is a topological embedding if f yields a homeomorphism between X and f(X) (where f(X) carries the subspace topology inherited from Y). Intuitively then, the embedding f : X \to Y lets us treat X as a subspace of Y. Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image f(X) is neither an open set nor a closed set in Y. For a given space Y, the existence of an embedding X \to Y is a topological invariant of X. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.
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A homotopy between two embeddings of the torus into R3: as "the surface of a doughnut" and as "the surface of a coffee mug". This is also an example of an isotopy. Formally, a homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H: X \times [0,1] \to Y from the product of the space X with the unit interval [0, 1] to Y such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second parameter of H as time then H describes a continuous deformation of f into g: at time 0 we have the function f and at time 1 we have the function g. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from f to g as the slider moves from 0 to 1, and vice versa.
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The problem of assigning edges to the two pages in a compatible way can be formulated as either an instance of 2-satisfiability, or as a problem of testing the bipartiteness of the circle graph whose vertices are the basepairs and whose edges describe crossings between basepairs. Alternatively and more efficiently, as show, a bi-secondary structure exists if and only if the diagram graph of the input (a graph formed by connecting the bases into a cycle in their sequence order and adding the given basepairs as edges) is a planar graph. This characterization allows bi-secondary structures to be recognized in linear time as an instance of planarity testing. used the connection between secondary structures and book embeddings as part of a proof of the NP-hardness of certain problems in RNA secondary structure comparison.. And if an RNA structure is tertiary rather than bi-secondary (that is, if it requires more than two pages in its diagram), then determining the page number is again NP-hard..
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In 2006, Streinu won the Grigore Moisil Award of the Romanian Academy for her work with Ciprian Borcea using complex algebraic geometry to show that every minimally rigid graph with fixed edge lengths has at most 4n different embeddings into the Euclidean plane, where n denotes the number of distinct vertices of the graph.. In 2010, Streinu won the David P. Robbins Prize of the American Mathematical Society for her combinatorial solution to the carpenter's rule problem. In this problem, one is given an arbitrary simple polygon with flexible vertices and rigid edges, and must show that it can be manipulated into a convex shape without ever introducing any self-crossings. Streinu's solution augments the input to form a pointed pseudotriangulation, removes one convex hull edge from this graph, and shows that this edge removal provides a single degree of freedom allowing the polygon to be made more convex one step at a time... In 2012 she became a fellow of the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 2013-08-05.
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Informally, imagine terrain Y, and its part X, such that wherever in Y you place a sharpshooter, and an apple at another place in Y, and then let the sharpshooter fire, the bullet will go through the apple and will always hit a point of X, or at least it will fly arbitrarily close to points of X – then we say that Y is aimed at X. A priori, it may seem plausible that for a given X the superspaces Y that aim at X can be arbitrarily large or at least huge. We will see that this is not the case. Among the spaces which aim at a subspace isometric to X, there is a unique (up to isometry) universal one, Aim(X), which in a sense of canonical isometric embeddings contains any other space aimed at (an isometric image of) X. And in the special case of an arbitrary compact metric space X every bounded subspace of an arbitrary metric space Y aimed at X is totally bounded (i.e. its metric completion is compact).
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After a third chapter relating the crossing number to graph parameters including skewness, bisection width, thickness, and (via the Albertson conjecture) the chromatic number, the final chapter of part I concerns the computational complexity of finding minimum-crossing graph drawings, including the results that the problem is both NP-complete and fixed-parameter tractable. In the second part of the book, two chapters concern the rectilinear crossing number, describing graph drawings in which the edges must be represented as straight line segments rather than arbitrary curves, and Fáry's theorem that every planar graph can be drawn without crossings in this way. Another chapter concerns 1-planar graphs and the associated local crossing number, the smallest number such that the graph can be drawn with at most crossings per edge. Two chapters concern book embeddings and string graphs, and two more chapters concern variations of the crossing number that count crossings in different ways, for instance by the number of pairs of edges that cross or that cross an odd number of times.
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The root discriminant of a number field, K, of degree n, often denoted rdK, is defined as the n-th root of the absolute value of the (absolute) discriminant of K. The relation between relative discriminants in a tower of fields shows that the root discriminant does not change in an unramified extension. The existence of a class field tower provides bounds on the root discriminant: the existence of an infinite class field tower over Q() where m = 3·5·7·11·19 shows that there are infinitely many fields with root discriminant 2 ≈ 296.276. If we let r and 2s be the number of real and complex embeddings, so that n = r + 2s, put ρ = r/n and σ = 2s/n. Set α(ρ, σ) to be the infimum of rdK for K with (r', 2s') = (ρn, σn). We have (for all n large enough) : \alpha(\rho,\sigma) \ge 60.8^\rho 22.3^\sigma and on the assumption of the generalized Riemann hypothesis : \alpha(\rho,\sigma) \ge 215.3^\rho 44.7^\sigma .
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The projective line P(A) over a ring A can also be identified as the space of projective modules in the module A \oplus A. An element of P(A) is then a direct summand of A \oplus A. This more abstract approach follows the view of projective geometry as the geometry of subspaces of a vector space, sometimes associated with the lattice theory of Garrett BirkhoffBirkhoff and Maclane (1953) Survey of modern algebra, pp 293–8, or 1997 AKP Classics edition, pp 312–7 or the book Linear Algebra and Projective Geometry by Reinhold Baer. In the case of the ring of rational integers Z, the module summand definition of P(Z) narrows attention to the U[m, n], m coprime to n, and sheds the embeddings which are a principle feature of P(A) when A is topological. The 1981 article by W. Benz, Hans- Joachim Samaga, & Helmut Scheaffer mentions the direct summand definition. In an article "Projective representations: projective lines over rings"A Blunck & H Havlicek (2000) "Projective representations: projective lines over rings", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 70:287–99, .
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The first response is that these mirror fermions might exist and have very large masses. The second response, stated in the original paper and in his latest work, is that there is not a single embedding of gravitational spin(1,3) in e8, but three embeddings related by triality, with respect to which the 64− contains a second generation of physical fermions, and the third generation of fermions is contained within spin(12,4). The algebraic breakdown of the 248-dimensional e8 Lie algebra relevant to E8 Theory is :e8 = spin(4,4) + spin(8) + 8V ⊗ 8V \+ 8+ ⊗ 8+ \+ 8− ⊗ 8− This decomposition, attributed to Bertram Kostant, relies on the triality isomorphism between eight-dimensional vectors, 8v, positive- chiral spinors, 8+, and negative-chiral spinors, 8−, relating to the division algebra of the octonions. Within this decomposition, the strong force su(3) embeds in spin(8), three triality-related gravitational spin(1,3)’s embed in spin(4,4), the three generations of 60 fermions embed in 8V ⊗ 8V \+ 8+ ⊗ 8+ \+ 8− ⊗ 8−, and the gravitational frame, Higgs, and electroweak bosons embed throughout, with 18 colored X bosons remaining as new predicted particles.
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The dual polyhedron has a dual graph, a graph with one vertex for each face of the polyhedron and with one edge for every two adjacent faces. The same concept of planar graph duality may be generalized to graphs that are drawn in the plane but that do not come from a three-dimensional polyhedron, or more generally to graph embeddings on surfaces of higher genus: one may draw a dual graph by placing one vertex within each region bounded by a cycle of edges in the embedding, and drawing an edge connecting any two regions that share a boundary edge. An important example of this type comes from computational geometry: the duality for any finite set of points in the plane between the Delaunay triangulation of and the Voronoi diagram of . As with dual polyhedra and dual polytopes, the duality of graphs on surfaces is a dimension-reversing involution: each vertex in the primal embedded graph corresponds to a region of the dual embedding, each edge in the primal is crossed by an edge in the dual, and each region of the primal corresponds to a vertex of the dual.
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The linking number of two closed curves in three-dimensional space is a topological invariant of the curves: it is a number, defined from the curves in any of several equivalent ways, that does not change if the curves are moved continuously without passing through each other. The version of the linking number used for defining linkless embeddings of graphs is found by projecting the embedding onto the plane and counting the number of crossings of the projected embedding in which the first curve passes over the second one, modulo 2.. The projection must be "regular", meaning that no two vertices project to the same point, no vertex projects to the interior of an edge, and at every point of the projection where the projections of two edges intersect, they cross transversally; with this restriction, any two projections lead to the same linking number. The linking number of the unlink is zero, and therefore, if a pair of curves has nonzero linking number, the two curves must be linked. However, there are examples of curves that are linked but that have zero linking number, such as the Whitehead link.
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