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10,400 | For further use of self-synchronizing circuit technique SNN accepts neurons can be self-running or synchronized. In contrast to the von Neumann network there are no limitations for topology of neurons for semantic networks. | Semantic neural network | 0.818995 |
10,401 | Semantic neural network (SNN) is based on John von Neumann's neural network and Nikolai Amosov M-Network. There are limitations to a link topology for the von Neumann’s network but SNN accept a case without these limitations. Only logical values can be processed, but SNN accept that fuzzy values can be processed too. ... | Semantic neural network | 0.818995 |
10,402 | This depends upon very crude forms of artificial intelligence that perform reconfiguration by obeying specific rules. An example of this approach is the AN/UYK-43 computer. Formal design philosophies involving active redundancy are required for critical systems where corrective labor is undesirable or impractical to co... | Active redundancy | 0.818994 |
10,403 | Maintenance requires three actions, which usually involve down time and high priority labor costs: Automatic fault detection Automatic fault isolation Automatic reconfigurationActive redundancy eliminates down time and reduces manpower requirements by automating all three actions. This requires some amount of automated... | Active redundancy | 0.818994 |
10,404 | Lab on a Chip is a peer-reviewed scientific journal which publishes original (primary) research and review articles on any aspect of miniaturisation at the micro and nano scale. Lab on a Chip is published twice monthly by the Royal Society of Chemistry (RSC) and the editor-in-chief is Aaron Wheeler. The journal was est... | Lab Chip | 0.818983 |
10,405 | Presented words also tend to be longer (7 or 8 characters) than words presented in other implicit memory tests and the fragments are presented in such a way that only 1 or 2 possibilities for completion exist.An example of a WFC test is as follows: Participants are presented with a list of words including ASSASSIN, EMI... | Indirect tests of memory | 0.81898 |
10,406 | Why there is an instantaneous conservation is the subject of which is the correct interpretation of quantum mechanics. Speculative theories of quantum gravity that combine general relativity with quantum mechanics have their own associated paradoxes that are generally accepted to be artifacts of the lack of a consisten... | Physical paradox | 0.818977 |
10,407 | A significant set of physical paradoxes are associated with the privileged position of the observer in quantum mechanics. Three of the most famous of these are: the double-slit experiment; the EPR paradox and the Schrödinger's cat paradox,all of them proposed as thought experiments relevant to the discussions of the co... | Physical paradox | 0.818977 |
10,408 | The curvature of spacetime at the singularity is infinite which is another way of stating that the theory does not describe the physical conditions at this point. It is hoped that the solution to this paradox will be found with a consistent theory of quantum gravity, something which has thus far remained elusive. A con... | Physical paradox | 0.818977 |
10,409 | This mathematically inelegant but physically useful concept allows for the efficient calculation of the associated physical conditions while conveniently sidestepping the philosophical issue of what actually occurs at the infinitesimally-defined point: a question that physics is as yet unable to answer. Fortunately, a ... | Physical paradox | 0.818977 |
10,410 | Another paradox associated with the causality and the one-way nature of time is Loschmidt's paradox which poses the question how can microprocesses that are time-reversible produce a time-irreversible increase in entropy. A partial resolution to this paradox is rigorously provided for by the fluctuation theorem which r... | Physical paradox | 0.818977 |
10,411 | A set of similar paradoxes occurs within the area of physics involving arrow of time and causality. One of these, the grandfather paradox, deals with the peculiar nature of causality in closed time-like loops. In its most crude conception, the paradox involves a person traveling back in time and murdering an ancestor w... | Physical paradox | 0.818977 |
10,412 | Using the extra element theorem, a circuit element (such as a resistor) can be removed from a circuit, and the desired driving point or transfer function is found. By removing the element that most complicate the circuit (such as an element that creates feedback), the desired function can be easier to obtain. Next, two... | Extra element theorem | 0.818974 |
10,413 | The Extra Element Theorem (EET) is an analytic technique developed by R. D. Middlebrook for simplifying the process of deriving driving point and transfer functions for linear electronic circuits. Much like Thévenin's theorem, the extra element theorem breaks down one complicated problem into several simpler ones. Driv... | Extra element theorem | 0.818974 |
10,414 | Zd(s) is the single-injection driving point impedance "seen" by the extra element. Zn(s) is the double-null-injection driving point impedance "seen" by the extra element. The extra element theorem incidentally proves that any electric circuit transfer function can be expressed as no more than a bilinear function of any... | Extra element theorem | 0.818974 |
10,415 | The (single) extra element theorem expresses any transfer function as a product of the transfer function with that element removed and a correction factor. The correction factor term consists of the impedance of the extra element and two driving point impedances seen by the extra element: The double null injection driv... | Extra element theorem | 0.818974 |
10,416 | In 1998, Mark Westoby proposed a plant ecology strategy scheme (PESS) to explain species distributions based on traits. The dynamic model incorporated a three axes trade-off among specific leaf area (SLA), canopy height at maturity, and seed mass. SLA is defined as the area per unit dry mass of mature leaves, developed... | Plant strategies | 0.818971 |
10,417 | The goal of the L-H-S scheme was to develop an international network that could provide quantifiable comparisons between plant strategies. This started a movement towards incorporating functional traits in plant strategies, and understanding how plant functional traits and environmental factors are related. While Grime... | Plant strategies | 0.818971 |
10,418 | Polysome profiling is a technique in molecular biology that is used to study the association of mRNAs with ribosomes. It is important to note that this technique is different from ribosome profiling. Both techniques have been reviewed and both are used in analysis of the translatome, but the data they generate are at v... | Polysome Profiling | 0.818968 |
10,419 | In silico computer-based modeling technologies have also been applied in: Whole cell analysis of prokaryotic and eukaryotic hosts e.g. E. coli, B. subtilis, yeast, CHO- or human cell lines Discovery of potential cure for COVID-19. Bioprocess development and optimization e.g. optimization of product yields Simulation of... | Biological simulation | 0.818959 |
10,420 | Every Albert form (dimension six form of discriminant −1) is isotropic. The quaternion algebras form a subgroup of the Brauer group of F. Every dimension five form over F is a Pfister neighbour. No biquaternion algebra over F is a division algebra.A nonreal linked field has u-invariant equal to 1,2,4 or 8.: 406 | Linked field | 0.818953 |
10,421 | The field F is linked if any two quaternion algebras over F are linked. : 370 Every global and local field is linked since all quadratic forms of degree 6 over such fields are isotropic. The following properties of F are equivalent:: 342 F is linked. Any two quaternion algebras over F are linked. | Linked field | 0.818953 |
10,422 | {\displaystyle q=\left\langle {-a_{1},-a_{2},a_{1}a_{2},b_{1},b_{2},-b_{1}b_{2}}\right\rangle \ .} It can be regarded as the difference in the Witt ring of the ternary forms attached to the imaginary subspaces of A and B. The quaternion algebras are linked if and only if the Albert form is isotropic. : 70 | Linked field | 0.818953 |
10,423 | Let F be a field of characteristic not equal to 2. Let A = (a1,a2) and B = (b1,b2) be quaternion algebras over F. The algebras A and B are linked quaternion algebras over F if there is x in F such that A is equivalent to (x,y) and B is equivalent to (x,z). : 69 The Albert form for A, B is q = ⟨ − a 1 , − a 2 , a 1 a 2 ... | Linked field | 0.818953 |
10,424 | In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property. | Linked field | 0.818953 |
10,425 | They both have commonalities (e.g., you must steer) and physical differences (e.g., use of feet). One can always specify the abstraction "drive to" and let the implementor decide whether bicycling or driving a car is best. The "wheeled terrestrial transport" function is abstracted and the details of "how to drive" are ... | Hardware abstraction | 0.818951 |
10,426 | However, two noncoplanar lines are called skew lines. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analo... | Parallel (geometry) | 0.818945 |
10,427 | In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean spac... | Parallel (geometry) | 0.818945 |
10,428 | The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line. This can not be proved and must be assumed to be tru... | Parallel (geometry) | 0.818945 |
10,429 | Lewis Carroll), wrote a play, Euclid and His Modern Rivals, in which these texts are lambasted.One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction. According to Wilhelm Killing the idea may be traced... | Parallel (geometry) | 0.818945 |
10,430 | The traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry, so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Eucl... | Parallel (geometry) | 0.818945 |
10,431 | In spherical geometry, all geodesics are great circles. Great circles divide the sphere in two equal hemispheres and all great circles intersect each other. Thus, there are no parallel geodesics to a given geodesic, as all geodesics intersect. Equidistant curves on the sphere are called parallels of latitude analogous ... | Parallel (geometry) | 0.818945 |
10,432 | In non-Euclidean geometry, it is more common to talk about geodesics than (straight) lines. A geodesic is the shortest path between two points in a given geometry. In physics this may be interpreted as the path that a particle follows if no force is applied to it. In non-Euclidean geometry (elliptic or hyperbolic geome... | Parallel (geometry) | 0.818945 |
10,433 | Given models that provide predictions of future sales volume, either as a function of time or price decisions, the firm has certain choices or decisions available to it. Modeling those choices or decisions as an optimization problem provides a means to select the best available set of choices or decisions. In some sett... | Pricing science | 0.818944 |
10,434 | Pricing science is the application of social and business science methods to the problem of setting prices. Methods include economic modeling, statistics, econometrics, mathematical programming. This discipline had its origins in the development of yield management in the airline industry in the 1980s, and has since sp... | Pricing science | 0.818944 |
10,435 | The growth of low-cost carriers offering restriction-free pricing, "name your own price" channels, and auctions all stimulated this interest in applying science to the pricing side of the business. As the applications of scientific methods to these business problems expanded, the discipline of pricing science became mo... | Pricing science | 0.818944 |
10,436 | The methods employed in pricing science may be categorized into two broad areas: 1. forecasting and 2. optimization. The forecasting problem reflects the fact that the pricing decisions are intended to affect purchase events over some future time horizon. The optimization problem reflects the mathematical complexity re... | Pricing science | 0.818944 |
10,437 | Instead of optimizing the offers available in response to very dynamic capacity, these business-to-business applications provided the means to optimize offers based on the particular characteristics of the transaction being contemplated and the customer. Applications have included business services providers, industria... | Pricing science | 0.818944 |
10,438 | Pricing science has its roots in the development of yield management programs developed by the airline industry shortly after deregulation of the industry in the early 1980s. These programs provided model-based support to answer the central question faced by deregulated airlines: "How many bookings should I accept, for... | Pricing science | 0.818944 |
10,439 | Many optimization problems are formulated as constrained or unconstrained mathematical programs, either linear programs (LP) or mixed integer programs (MIP), for which many solution techniques and commercial solvers are available. | Pricing science | 0.818944 |
10,440 | In some businesses, pricing decisions are supported using forecasting and optimization methods executed on an as needed basis using general purpose analytic tools. In this setting, when periodic, or ad hoc decisions are made, analysis of historical transaction data sets is performed. This approach is often seen in larg... | Pricing science | 0.818944 |
10,441 | In statistics and information theory, the expected formation matrix of a likelihood function L ( θ ) {\displaystyle L(\theta )} is the matrix inverse of the Fisher information matrix of L ( θ ) {\displaystyle L(\theta )} , while the observed formation matrix of L ( θ ) {\displaystyle L(\theta )} is the inverse of the o... | Formation matrix | 0.818929 |
10,442 | {\displaystyle b\leq a.} This is the order-theoretic dual to the notion of cofinal subset. Cofinal (respectively coinitial) subsets are precisely the dense sets with respect to the right (respectively left) order topology. | Final function | 0.818929 |
10,443 | From 2005 until 2008, the third track was dedicated to "Security and Cryptography Foundations", and in 2009, it is devoted to the topic "Foundations of Networked Computation: Models, Algorithms and Information Management". Track C was dropped from the 2020 conference, with submissions from these areas invited to submit... | International Colloquium on Automata, Languages and Programming | 0.818928 |
10,444 | The second ICALP was held in 1974, and since 1976 ICALP has been an annual event, nowadays usually taking place in July. Since 1999, the conference was thematically split into two tracks on "Algorithms, Complexity and Games" (Track A) and "Automata, Logic, Semantics, and Theory of Programming" (Track B), corresponding ... | International Colloquium on Automata, Languages and Programming | 0.818928 |
10,445 | ICALP, the International Colloquium on Automata, Languages, and Programming is an academic conference organized annually by the European Association for Theoretical Computer Science and held in different locations around Europe. Like most theoretical computer science conferences its contributions are strongly peer-revi... | International Colloquium on Automata, Languages and Programming | 0.818928 |
10,446 | The Gödel Prize, a prize for outstanding papers in theoretical computer science and awarded jointly by the EATCS and the ACM SIGACT, is presented every second year at ICALP. Presentation of the prize, which is awarded annually, alternates with the conference STOC (ACM Symposium on Theory of Computing). | International Colloquium on Automata, Languages and Programming | 0.818928 |
10,447 | Nature Biotechnology is a monthly peer-reviewed scientific journal published by Nature Portfolio. The editor-in-chief is Barbara Cheifet who heads an in-house team of editors. The focus of the journal is biotechnology including research results and the commercial business sector of this field. Coverage includes the rel... | Nature Biotechnology | 0.818922 |
10,448 | This journal is indexed in BIOBASE BIOSIS Chemical Abstracts Service CSA Illumina CAB Abstracts EMBASE Scopus Current Contents Science Citation Index Medline (PubMed)According to the Journal Citation Reports, the journal has a 2021 impact factor of 68.164, ranking it 2nd out of 158 journals in the category "Biotechnolo... | Nature Biotechnology | 0.818922 |
10,449 | A paper by Martin Grötschel, László Lovász, and Alexander Schrijver on the ellipsoid method, awarded the 1982 Fulkerson Prize.M. Grötschel, L. Lovász, A. Schrujver: The ellipsoid method and its consequences in combinatorial optimization, Combinatorica, 1(1981), 169–197.József Beck's paper on the discrepancy of hypergra... | Combinatorica | 0.818917 |
10,450 | Combinatorica is an international journal of mathematics, publishing papers in the fields of combinatorics and computer science. It started in 1981, with László Babai and László Lovász as the editors-in-chief with Paul Erdős as honorary editor-in-chief. The current editors-in-chief are Imre Bárány and József Solymosi. ... | Combinatorica | 0.818917 |
10,451 | This problem was studied in detail by Joseph Louis Lagrange, who laid out most of the methods for solving it. The simplest approach is to define new generalized coordinates that are unconstrained; this approach eliminates the algebraic equations and reduces the problem once again to solving an ordinary differential equ... | Constraint algorithm (mechanics) | 0.818915 |
10,452 | The motion of a set of N particles can be described by a set of second-order ordinary differential equations, Newton's second law, which can be written in matrix form M ⋅ d 2 q d t 2 = f = − ∂ V ∂ q {\displaystyle \mathbf {M} \cdot {\frac {d^{2}\mathbf {q} }{dt^{2}}}=\mathbf {f} =-{\frac {\partial V}{\partial \mathbf {... | Constraint algorithm (mechanics) | 0.818915 |
10,453 | The SHAKE algorithm was first developed for satisfying a bond geometry constraint during molecular dynamics simulations. The method was then generalised to handle any holonomic constraint, such as those required to maintain constant bond angles, or molecular rigidity.In SHAKE algorithm, the system of non-linear constra... | Constraint algorithm (mechanics) | 0.818915 |
10,454 | One issue with SHAKE is that the number of iterations required to reach a certain level of convergence does rise as molecular geometry becomes more complex. To reach 64 bit computer accuracy (a relative tolerance of ≈ 10 − 16 {\displaystyle \approx 10^{-16}} ) in a typical molecular dynamics simulation at a temperature... | Constraint algorithm (mechanics) | 0.818915 |
10,455 | In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, including possibly their inverse functions (e.g., arcsin, lo... | Elementary functions | 0.818911 |
10,456 | Engineering is the discipline and profession that applies scientific theories, mathematical methods, and empirical evidence to design, create, and analyze technological solutions cognizant of safety, human factors, physical laws, regulations, practicality, and cost. In the contemporary era, engineering is generally con... | Engineering disciplines | 0.818905 |
10,457 | Electrical engineering comprises the study and application of electricity, electronics and electromagnetism. | Engineering disciplines | 0.818905 |
10,458 | Civil engineering comprises the design, construction, and maintenance of the physical and natural built environments. | Engineering disciplines | 0.818905 |
10,459 | Chemical engineering is the application of chemical, physical and biological sciences to the process of converting raw materials or chemicals into more useful or valuable forms. | Engineering disciplines | 0.818905 |
10,460 | Mechanical engineering comprises the design and analysis of heat and mechanical power for the operation of machines and mechanical systems. | Engineering disciplines | 0.818905 |
10,461 | Newton Game Dynamics is an open-source physics engine for realistically simulating rigid bodies in games and other real-time applications. Its solver is deterministic and not based on traditional LCP or iterative methods. Newton Game Dynamics is actively developed by Julio Jerez. Currently a new version which will take... | Newton Game Dynamics | 0.818905 |
10,462 | Størmer's theorem applies Pell equations to find pairs of consecutive smooth numbers, positive integers whose prime factors are all smaller than a given value. As part of this theory, Størmer also investigated divisibility relations among solutions to Pell's equation; in particular, he showed that each solution other t... | Pell's equation | 0.818904 |
10,463 | Pell's equation is closely related to the theory of algebraic numbers, as the formula x 2 − n y 2 = ( x + y n ) ( x − y n ) {\displaystyle x^{2}-ny^{2}=(x+y{\sqrt {n}})(x-y{\sqrt {n}})} is the norm for the ring Z {\displaystyle \mathbb {Z} } and for the closely related quadratic field Q ( n ) {\displaystyle \mathbb {Q... | Pell's equation | 0.818904 |
10,464 | In a letter to Kenelm Digby, Bernard Frénicle de Bessy said that Fermat found the smallest solution for N up to 150 and challenged John Wallis to solve the cases N = 151 or 313. Both Wallis and William Brouncker gave solutions to these problems, though Wallis suggests in a letter that the solution was due to Brouncker.... | Pell's equation | 0.818904 |
10,465 | The disjunction property is satisfied by a theory if, whenever a sentence A ∨ B is a theorem, then either A is a theorem, or B is a theorem. The existence property or witness property is satisfied by a theory if, whenever a sentence (∃x)A(x) is a theorem, where A(x) has no other free variables, then there is some term ... | Disjunction and existence properties | 0.8189 |
10,466 | A practical unit to be used by electricians and engineers, the ampere, was then defined as equal to one tenth of the electromagnetic unit of current. In another system, the "rationalized metre–kilogram–second (rmks) system" (or alternatively the "metre–kilogram–second–ampere (mksa) system"), km is written as μ0/2π, whe... | Permeability constant | 0.818897 |
10,467 | Suppose that a mathematician is studying geometry and shapes, and she wishes to prove certain theorems about them. She conjectures that "All rectangles are squares", and she is interested in knowing whether this statement is true or false. In this case, she can either attempt to prove the truth of the statement using d... | Counterexample | 0.818894 |
10,468 | A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization “students are lazy”, and both a counter... | Counterexample | 0.818894 |
10,469 | In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures to produce provable theorems. It is sometimes said that mathem... | Counterexample | 0.818894 |
10,470 | One of the most ambitious programs to tackle this problem was Doug Lenat's Cyc project. Cyc established its own Frame language and had large numbers of analysts document various areas of common-sense reasoning in that language. The knowledge recorded in Cyc included common-sense models of time, causality, physics, inte... | Knowledge Representation | 0.818893 |
10,471 | One of the first realizations learned from trying to make software that can function with human natural language was that humans regularly draw on an extensive foundation of knowledge about the real world that we simply take for granted but that is not at all obvious to an artificial agent. Basic principles of common-s... | Knowledge Representation | 0.818893 |
10,472 | Although message passing originated in the object-oriented community rather than AI it was quickly embraced by AI researchers as well in environments such as KEE and in the operating systems for Lisp machines from Symbolics, Xerox, and Texas Instruments.The integration of frames, rules, and object-oriented programming ... | Knowledge Representation | 0.818893 |
10,473 | Knowledge representation and reasoning (KRR, KR&R, KR²) is the field of artificial intelligence (AI) dedicated to representing information about the world in a form that a computer system can use to solve complex tasks such as diagnosing a medical condition or having a dialog in a natural language. Knowledge representa... | Knowledge Representation | 0.818893 |
10,474 | The transformer model has been implemented in standard deep learning frameworks such as TensorFlow and PyTorch. Transformers is a library produced by Hugging Face that supplies transformer-based architectures and pretrained models. | Transformer (machine learning model) | 0.818876 |
10,475 | The transformer has had great success in natural language processing (NLP), for example the tasks of machine translation and time series prediction. Many large language models such as GPT-2, GPT-3, GPT-4, BERT, XLNet, RoBERTa and ChatGPT demonstrate the ability of transformers to perform a wide variety of such NLP-rela... | Transformer (machine learning model) | 0.818876 |
10,476 | In 2016, Google Translate gradually replaced the older statistical machine translation approach with the newer neural-networks-based approach that included a seq2seq model combined by LSTM and the "additive" kind of attention mechanism. They achieved a higher level of performance than the statistical approach, which to... | Transformer (machine learning model) | 0.818876 |
10,477 | An advantage of the fast linear transformers is that their computational complexity grows linearly with sequence length, while modern transformers scale quadratically.In 1993, the IBM alignment models were used for statistical machine translation.In 1997, a precursor of large language model, using recurrent neural netw... | Transformer (machine learning model) | 0.818876 |
10,478 | In 1990, Elman network, using a recurrent network, encoded each word in a training set as a vector, called static "word embedding", and the whole vocabulary as a vector database, allowing it to perform such tasks as sequence-prediction that are beyond the power of a simple multilayer perceptron. Shortcoming of the stat... | Transformer (machine learning model) | 0.818876 |
10,479 | Each encoder consists of two major components: a self-attention mechanism and a feed-forward neural network. The self-attention mechanism accepts input encodings from the previous encoder and weights their relevance to each other to generate output encodings. The feed-forward neural network further processes each outpu... | Transformer (machine learning model) | 0.818876 |
10,480 | This sum of encoded positions, when fed into the attention mechanism, would create attention weights on its neighbors, much like what happens in a convolutional neural network language model. In the author's words, "we hypothesized it would allow the model to easily learn to attend by relative position". In typical imp... | Transformer (machine learning model) | 0.818876 |
10,481 | For example, some attention heads can attend mostly to the next word, while others mainly attend from verbs to their direct objects. The computations for each attention head can be performed in parallel, which allows for fast processing. The outputs for the attention layer are concatenated to pass into the feed-forward... | Transformer (machine learning model) | 0.818876 |
10,482 | This is a somewhat explicit form of the previous definition: Chosen a homogeneous basis v i ∈ V {\displaystyle v_{i}\in V} (i.e. coaction/graduation v i ↦ g i ⊗ v i {\displaystyle v_{i}\mapsto g_{i}\otimes v_{i}} ) one may define skew derivations ∂ i {\displaystyle \partial _{i}} , using the universal property of the t... | Nichols algebra | 0.818871 |
10,483 | diagonals in squares, in the super Lie algebra odd resp. even roots). It visibly has 2 {\displaystyle 2} different types of Weyl chambers (equilateral triangles resp. | Nichols algebra | 0.818871 |
10,484 | The smallest crystallographic hyperplane arrangement, Weyl groupoid, generalized root system, which is not of ordinary Lie type, is as follows. It appears for a diagonal Nichols algebra, even a super Lie algebra. The hyperplane arrangement can be constructed from a cuboctahedron (a platonic solid): It has 7 {\displayst... | Nichols algebra | 0.818871 |
10,485 | The existence of a root system also in the nonabelian case implies rather immediately the following very strong implications: Immediate consequences are implied for rank 2 Nichols algebras B ( O ⊕ O ) {\displaystyle {\mathfrak {B}}\left({\mathcal {O}}_{}\oplus {\mathcal {O}}_{}\right)} which g, h discommuting; then: ... | Nichols algebra | 0.818871 |
10,486 | Especially for irreducible V there are no submodules; however one may use the more abstract notion of subrack only reflecting the braiding of two contained elements. In several papers, Nicolas Andruskiewitsch et al. gave negative criteria excluding groups at all from possessing (indecomposable) Nichols algebras. Their ... | Nichols algebra | 0.818871 |
10,487 | Let V ∈ H H Y D {\displaystyle V\in {}_{H}^{H}{\mathcal {YD}}} . There exists a largest ideal I ⊂ T V {\displaystyle {\mathfrak {I}}\subset TV} with the following properties: I ⊂ ⨁ n = 2 ∞ T n V , {\displaystyle {\mathfrak {I}}\subset \bigoplus _{n=2}^{\infty }T^{n}V,} Δ ( I ) ⊂ I ⊗ T V + T V ⊗ I {\displaystyle \Delta ... | Nichols algebra | 0.818871 |
10,488 | We give examples of finite-dimensional Nichols algebras. Over characteristic p, this effect already may appear in the non-braided situation, namely the truncated universal envelopings of p-restricted Lie algebras. In characteristic zero and with a braiding coming from an abelian group, this seems to be a similarly freq... | Nichols algebra | 0.818871 |
10,489 | There are more diagonal Nichols algebras than Lie algebras in Heckenbergers list, and the root system theory is systematic, but more complicated (see below). In particular is contains also the classification of Super-Lie-Algebras (example below) as well as certain Lie algebras and Super-Lie-Algebras that only appear in... | Nichols algebra | 0.818871 |
10,490 | The unique Hopf pairing V ⊗ T V ∗ → k {\displaystyle V\otimes TV^{*}\to k} factorizes to a nondegenerate Hopf pairing between B ( V ) ⊗ B ( V ∗ ) → k {\displaystyle {\mathfrak {B}}(V)\otimes {\mathfrak {B}}(V^{*})\to k} and this fact characterizes the Nichols algebra uniquely. This theoretically very helpful characteri... | Nichols algebra | 0.818871 |
10,491 | See also this List of finite-dimensional Nichols algebras. The finite-dimensional theory is greatly governed by a theory of root systems and Dynkin diagrams, strikingly similar to those of semisimple Lie algebras. A comprehensive introduction is found in the lecture of Heckenberger. | Nichols algebra | 0.818871 |
10,492 | In algebra, the Nichols algebra of a braided vector space (with the braiding often induced by a finite group) is a braided Hopf algebra which is denoted by B ( V ) {\displaystyle {\mathfrak {B}}(V)} and named after the mathematician Warren Nichols. It takes the role of quantum Borel part of a pointed Hopf algebra such ... | Nichols algebra | 0.818871 |
10,493 | Consider a Yetter–Drinfeld module V in the Yetter–Drinfeld category H H Y D {\displaystyle {}_{H}^{H}{\mathcal {YD}}} . This is especially a braided vectorspace, see Braided monoidal category. The tensor algebra T V {\displaystyle TV} of a Yetter–Drinfeld module V ∈ H H Y D {\displaystyle V\in {}_{H}^{H}{\mathcal {YD}}... | Nichols algebra | 0.818871 |
10,494 | Both negation techniques above have been very fruitful to negate (indecomposable) finite-dimensional Nichols algebras: for Alternating groups A n ≥ 5 {\displaystyle \mathbb {A} _{n\geq 5}} for Symmetric groups S n ≥ 6 {\displaystyle \mathbb {S} _{n\geq 6}} except a short list of examples some group of Lie type (sources... | Nichols algebra | 0.818871 |
10,495 | As the abelian subracks use the structural classification of Heckenberger for Nichols algebras over abelian groups (see above) one can also consider nonabelian subracks. If such a subrack decomposes into several pieces (because now less element are present to conjugate), then the above results on root systems apply. A ... | Nichols algebra | 0.818871 |
10,496 | As a first example, consider the 1-dimensional Yetter–Drinfeld module V ± = k x {\displaystyle V_{\pm }=kx} over the Group Hopf algebra H = k with the Cyclic group multiplicatively denoted (as usual in algebra) and generated by some g. Take as H-coaction (resp. Z/2Z-graduation) on V ± {\displaystyle V_{\pm }}: x ↦ g ⊗ ... | Nichols algebra | 0.818871 |
10,497 | A very remarkable feature is that for every Nichols algebra (under sufficient finiteness conditions) there exists a generalized root system with a set of roots Φ {\displaystyle \Phi } , which controls the Nichols algebra. This has been discovered in for diagonal Nichols algebras in terms of the bicharacter q i j {\disp... | Nichols algebra | 0.818871 |
10,498 | The generators and relations of a Nichols algebra are not readily available from the root system. Rather, one has to perform tedious work with the Lynond words. This has been completely done in | Nichols algebra | 0.818871 |
10,499 | The Nichols algebras of finite dimension over abelian groups in k = C were classified by Istvan Heckenberger in the years 2004–2005 by classifying arithmetic root systems and generalized Dynkin diagrams; where already Kharchenko had proven them to possess a Poincaré–Birkhoff–Witt basis of iterated (braided) commutators... | Nichols algebra | 0.818871 |
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