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22,700 | It ceased publication in 1973. Notable papers published in Scripta Mathematica included work by Nobelist Percy Williams Bridgman concerning the implications for physics of set-theoretic paradoxes, and Hermann Weyl's obituary of Emmy Noether.Some sources describe Scripta Mathematica as having been assigned ISSN 0036-971... | Scripta Mathematica | 0.809244 |
22,701 | Setting up an RSA system involves choosing large prime numbers p and q, computing n = pq and k = φ(n), and finding two numbers e and d such that ed ≡ 1 (mod k). The numbers n and e (the "encryption key") are released to the public, and d (the "decryption key") is kept private. A message, represented by an integer m, wh... | Totient function | 0.809243 |
22,702 | The application level, just as in any other engineering and economics disciplines, strives to make a practical impact and be a driver for change in the real world.The management scientist's mandate is to use rational, systematic and science-based techniques to inform and improve decisions of all kinds. The techniques o... | Management scientist | 0.809242 |
22,703 | Helping to improve, stabilize or otherwise manage profit margins in enterprises.Management science research can be done on three levels: The fundamental level lies in three mathematical disciplines: probability, optimization, and dynamical systems theory. The modeling level is about building models, analyzing them math... | Management scientist | 0.809242 |
22,704 | In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian category, or in the category of groups), if the rows are short exact sequences, and if g and h ... | Short five lemma | 0.809241 |
22,705 | In 2012 there were about 19,400 biomedical engineers employed in the US, and the field was predicted to grow by 5% (faster than average) from 2012 to 2022. Biomedical engineering has the highest percentage of female engineers compared to other common engineering professions. | Biomedical engineer | 0.809239 |
22,706 | His works have been cited more than 20,000 times cumulatively. P. Hunter Peckham – Donnell Professor of Biomedical Engineering and Orthopaedics at Case Western Reserve University. Pioneer in Functional Electrical Stimulation (FES) Nicholas A. Peppas – Chaired Professor in Engineering, University of Texas at Austin, pio... | Biomedical engineer | 0.809239 |
22,707 | A biomaterial is any matter, surface, or construct that interacts with living systems. As a science, biomaterials is about fifty years old. The study of biomaterials is called biomaterials science or biomaterials engineering. It has experienced steady and strong growth over its history, with many companies investing la... | Biomedical engineer | 0.809239 |
22,708 | J. Thomas Mortimer – Emeritus professor of biomedical engineering at Case Western Reserve University. Pioneer in Functional Electrical Stimulation (FES) Robert M. Nerem – professor emeritus at Georgia Institute of Technology. Pioneer in regenerative tissue, biomechanics, and author of over 300 published works. | Biomedical engineer | 0.809239 |
22,709 | Alfred E. Mann – Physicist, entrepreneur and philanthropist. A pioneer in the field of Biomedical Engineering. | Biomedical engineer | 0.809239 |
22,710 | Julia Tutelman Apter (deceased) - One of the first specialists in neurophysiological research and a founding member of the Biomedical Engineering Society Earl Bakken (deceased) – Invented the first transistorised pacemaker, co-founder of Medtronic. Forrest Bird (deceased) – aviator and pioneer in the invention of mecha... | Biomedical engineer | 0.809239 |
22,711 | A further consequence of excited-state formation may be reaction of the atom or molecule in its excited state, as in photochemistry. | Excited electronic state | 0.809239 |
22,712 | This term has been used to compare a random buy-and-hold strategy to professional traders' records. This same concept has received numerous different names, as the New York Times notes: "In 1957, for example, a statistician named James Hanna called his theorem Bayesian Regret. He had been preceded by David Blackwell, a... | Bayesian regret | 0.809238 |
22,713 | In stochastic game theory, Bayesian regret is the expected difference ("regret") between the utility of a Bayesian strategy and that of the optimal strategy (the one with the highest expected payoff). The term Bayesian refers to Thomas Bayes (1702–1761), who proved a special case of what is now called Bayes' theorem, w... | Bayesian regret | 0.809238 |
22,714 | His method anticipated that of the Dedekind cut in the modern definition of real numbers by Richard Dedekind (1831–1916).In the Posterior Analytics, Aristotle (384–322 BC) laid down the axiomatic method for organizing a field of knowledge logically by means of primitive concepts, axioms, postulates, definitions, and th... | Foundation of mathematics | 0.809237 |
22,715 | While the practice of mathematics had previously developed in other civilizations, special interest in its theoretical and foundational aspects was clearly evident in the work of the Ancient Greeks. Early Greek philosophers disputed as to which is more basic, arithmetic or geometry. Zeno of Elea (490 – c. 430 BC) produ... | Foundation of mathematics | 0.809237 |
22,716 | Stillwell writes on page 120 ... projective geometry is simpler than algebra in a certain sense, because we use only five geometric axioms to derive the nine field axioms. The algebra of throws is commonly seen as a feature of cross-ratios since students ordinarily rely upon numbers without worry about their basis. How... | Foundation of mathematics | 0.809237 |
22,717 | Then he created a means of expressing the familiar numeric properties with his Algebra of Throws. English language versions of this process of deducing the properties of a field can be found in either the book by Oswald Veblen and John Young, Projective Geometry (1938), or more recently in John Stillwell's Four Pillars... | Foundation of mathematics | 0.809237 |
22,718 | One of the traps in a deductive system is circular reasoning, a problem that seemed to befall projective geometry until it was resolved by Karl von Staudt. As explained by Russian historians: In the mid-nineteenth century there was an acrimonious controversy between the proponents of synthetic and analytic methods in p... | Foundation of mathematics | 0.809237 |
22,719 | Subfields within imaging science include: image processing, computer vision, 3D computer graphics, animations, atmospheric optics, astronomical imaging, biological imaging, digital image restoration, digital imaging, color science, digital photography, holography, magnetic resonance imaging, medical imaging, microdensi... | Scientific imaging | 0.809237 |
22,720 | Polyaspartic ester chemistry was first introduced in the early 1990s making it a relatively new technology. The patents were issued to Bayer in Germany and Miles Corporation in the United States. It utilizes the aza-Michael addition reaction. These products are then used in coatings, adhesives, sealants and elastomers. | Polyaspartic esters | 0.809237 |
22,721 | In probability theory, a Markov reward model or Markov reward process is a stochastic process which extends either a Markov chain or continuous-time Markov chain by adding a reward rate to each state. An additional variable records the reward accumulated up to the current time. Features of interest in the model include... | Markov reward model | 0.809237 |
22,722 | The International Conference on Reachability Problems series, previously known as Workshop on Reachability Problems, is an annual academic conference which gathers together researchers from diverse disciplines and backgrounds interested in reachability problems that appear in algebraic structures, computational models,... | Reachability problem | 0.809235 |
22,723 | The Breakthrough Prize in Life Sciences is a scientific award, funded by internet entrepreneurs Mark Zuckerberg and Priscilla Chan of Facebook; Sergey Brin of Google; entrepreneur and venture capitalist Yuri Milner; and Anne Wojcicki, one of the founders of the genetics company 23andMe.The award of $3 million, the larg... | Breakthrough Prize in Life Sciences | 0.809235 |
22,724 | In modular representation theory, while Maschke's theorem does not hold when the characteristic divides the group order, the group algebra may be decomposed as the direct sum of a maximal collection of two-sided ideals known as blocks. When the field F has characteristic 0, or characteristic coprime to the group order,... | Modular representation | 0.809234 |
22,725 | Maik Maurer considers complexity as a reality in engineering. He proposed a methodology for managing complexity in systems engineering: 1. Define the system. 2. | Complexity | 0.809234 |
22,726 | There has also been interest in measuring the complexity of classification problems in supervised machine learning. This can be useful in meta-learning to determine for which data sets filtering (or removing suspected noisy instances from the training set) is the most beneficial and could be expanded to other areas. Fo... | Complexity | 0.809234 |
22,727 | Warren Weaver posited in 1948 two forms of complexity: disorganized complexity, and organized complexity.Phenomena of 'disorganized complexity' are treated using probability theory and statistical mechanics, while 'organized complexity' deals with phenomena that escape such approaches and confront "dealing simultaneous... | Complexity | 0.809234 |
22,728 | In mathematics, Krohn–Rhodes complexity is an important topic in the study of finite semigroups and automata. In Network theory complexity is the product of richness in the connections between components of a system, and defined by a very unequal distribution of certain measures (some elements being highly connected an... | Complexity | 0.809234 |
22,729 | It is possible to treat different kinds of Kolmogorov complexity as particular cases of axiomatically defined generalized Kolmogorov complexity. Instead of proving similar theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily deduce all such results from one corresponding... | Complexity | 0.809234 |
22,730 | Systems theory has long been concerned with the study of complex systems (in recent times, complexity theory and complex systems have also been used as names of the field). These systems are present in the research of a variety disciplines, including biology, economics, social studies and technology. Recently, complexi... | Complexity | 0.809234 |
22,731 | In particle physics, the GIM mechanism (or Glashow–Iliopoulos–Maiani mechanism) is the mechanism through which flavour-changing neutral currents (FCNCs) are suppressed in loop diagrams. It also explains why weak interactions that change strangeness by 2 (ΔS = 2 transitions) are suppressed, while those that change stran... | GIM mechanism | 0.809231 |
22,732 | According to the theorem, it is possible to expand any nonnegative integer power of x + y into a sum of the form where n ≥ 0 {\displaystyle n\geq 0} is an integer and each ( n k ) {\displaystyle {\tbinom {n}{k}}} is a positive integer known as a binomial coefficient. (When an exponent is zero, the corresponding power e... | Binomial theorem | 0.80923 |
22,733 | The binomial theorem is valid more generally for two elements x and y in a ring, or even a semiring, provided that xy = yx. For example, it holds for two n × n matrices, provided that those matrices commute; this is useful in computing powers of a matrix.The binomial theorem can be stated by saying that the polynomial ... | Binomial theorem | 0.80923 |
22,734 | The term topological game was first introduced by Claude Berge, who defined the basic ideas and formalism in analogy with topological groups. A different meaning for topological game, the concept of “topological properties defined by games”, was introduced in the paper of Rastislav Telgársky, and later "spaces defined ... | Topological game | 0.809228 |
22,735 | Some other notable topological games are: the binary game introduced by Ulam — a modification of the Banach–Mazur game; the Banach game — played on a subset of the real line; the Choquet game — related to siftable spaces; the point-open game — in which player I chooses points and player II chooses open neighborhoods of... | Topological game | 0.809228 |
22,736 | There are a large variety of different adversarial attacks that can be used against machine learning systems. Many of these work on both deep learning systems as well as traditional machine learning models such as SVMs and linear regression. A high level sample of these attack types include: Adversarial Examples Trojan... | Adversarial machine learning | 0.809227 |
22,737 | Black box attacks in adversarial machine learning assumes that the adversary can only get outputs for provided inputs and has no knowledge of the model structure or parameters. In this case, the adversarial example is generated either using a model created from scratch, or without any model at all (excluding the abilit... | Adversarial machine learning | 0.809227 |
22,738 | An adversarial example refers to specially crafted input which is designed to look "normal" to humans but causes misclassification to a machine learning model. Often, a form of specially designed "noise" is used to elicit the misclassifications. Below are some current techniques for generating adversarial examples in t... | Adversarial machine learning | 0.809227 |
22,739 | Adversarial machine learning is the study of the attacks on machine learning algorithms, and of the defenses against such attacks. A survey from May 2020 exposes the fact that practitioners report a dire need for better protecting machine learning systems in industrial applications.To understand, note that most machine... | Adversarial machine learning | 0.809227 |
22,740 | In the extreme case, model extraction can lead to model stealing, which corresponds to extracting a sufficient amount of data from the model to enable the complete reconstruction of the model. On the other hand, membership inference is a targeted model extraction attack, which infers the owner of a data point, often by... | Adversarial machine learning | 0.809227 |
22,741 | Model extraction involves an adversary probing a black box machine learning system in order to extract the data it was trained on. This can cause issues when either the training data or the model itself is sensitive and confidential. For example, model extraction could be used to extract a proprietary stock trading mod... | Adversarial machine learning | 0.809227 |
22,742 | In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurable functions and for the F-space of sequences with F–norm ( x n ) ↦ ∑ n 2 − n x n / ( 1 + x n ) . {\textstyle (x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}.} Here we mean by F-norm some real-valued functi... | Absolute-value norm | 0.809226 |
22,743 | In microbiology, the phyllosphere is the total above-ground surface of a plant when viewed as a habitat for microorganisms. The phyllosphere can be further subdivided into the caulosphere (stems), phylloplane (leaves), anthosphere (flowers), and carposphere (fruits). The below-ground microbial habitats (i.e. the thin-v... | Phyllosphere | 0.809223 |
22,744 | In mathematical optimization, the network simplex algorithm is a graph theoretic specialization of the simplex algorithm. The algorithm is usually formulated in terms of a minimum-cost flow problem. The network simplex method works very well in practice, typically 200 to 300 times faster than the simplex method applied... | Network simplex algorithm | 0.809216 |
22,745 | Statistical learning has been shown to play a large role in language acquisition, but social interaction appears to be a necessary component of learning as well. In one study, infants presented with audio or audiovisual recordings of Mandarin speakers failed to distinguish the phonemes of the language. This implies tha... | Probabilistic models of language acquisition | 0.809216 |
22,746 | In this experiment, syllables that went together in words had a much higher transitional probability than did syllables at word boundaries that just happened to be adjacent. Incredibly, infants, after a short two-minute presentation, were able to keep track of these statistics and recognize high probability words. Furt... | Probabilistic models of language acquisition | 0.809216 |
22,747 | The system was refined in collaboration with Berthollet, de Fourcroy and Lavoisier, and promoted by the latter in a textbook that would survive long after his death at the guillotine in 1794. The project was also espoused by Jöns Jakob Berzelius, who adapted the ideas for the German-speaking world. The recommendations ... | Substitutive nomenclature | 0.809215 |
22,748 | These differing aims can have profound effects on valid understanding in chemistry, especially with regard to chemical classes that have achieved mass attention. Examples of the impact of these can be seen in considering the examples of: resveratrol, a single compound clearly defined by this common name, but that can b... | Substitutive nomenclature | 0.809215 |
22,749 | In October 2017, Google released the Google Pixel 2 which featured their Pixel Visual Core (PVC), a fully programmable image, vision and AI processor for mobile devices. The PVC supports TensorFlow for machine learning (and Halide for image processing). | TensorFlow | 0.809213 |
22,750 | In May 2016, Google announced its Tensor processing unit (TPU), an application-specific integrated circuit (ASIC, a hardware chip) built specifically for machine learning and tailored for TensorFlow. A TPU is a programmable AI accelerator designed to provide high throughput of low-precision arithmetic (e.g., 8-bit), an... | TensorFlow | 0.809213 |
22,751 | In July 2018, the Edge TPU was announced. Edge TPU is Google's purpose-built ASIC chip designed to run TensorFlow Lite machine learning (ML) models on small client computing devices such as smartphones known as edge computing. | TensorFlow | 0.809213 |
22,752 | Google JAX is a machine learning framework for transforming numerical functions. It is described as bringing together a modified version of autograd (automatic obtaining of the gradient function through differentiation of a function) and TensorFlow's XLA (Accelerated Linear Algebra). It is designed to follow the struct... | TensorFlow | 0.809213 |
22,753 | TensorFlow.nn is a module for executing primitive neural network operations on models. Some of these operations include variations of convolutions (1/2/3D, Atrous, depthwise), activation functions (Softmax, RELU, GELU, Sigmoid, etc.) and their variations, and other operations (max-pooling, bias-add, etc.). | TensorFlow | 0.809213 |
22,754 | TensorFlow serves as a core platform and library for machine learning. TensorFlow's APIs use Keras to allow users to make their own machine learning models. In addition to building and training their model, TensorFlow can also help load the data to train the model, and deploy it using TensorFlow Serving.TensorFlow prov... | TensorFlow | 0.809213 |
22,755 | TensorFlow offers a set of optimizers for training neural networks, including ADAM, ADAGRAD, and Stochastic Gradient Descent (SGD). When training a model, different optimizers offer different modes of parameter tuning, often affecting a model's convergence and performance. | TensorFlow | 0.809213 |
22,756 | TensorFlow is a free and open-source software library for machine learning and artificial intelligence. It can be used across a range of tasks but has a particular focus on training and inference of deep neural networks.TensorFlow was developed by the Google Brain team for internal Google use in research and production... | TensorFlow | 0.809213 |
22,757 | The name TensorFlow derives from the operations that such neural networks perform on multidimensional data arrays, which are referred to as tensors. During the Google I/O Conference in June 2016, Jeff Dean stated that 1,500 repositories on GitHub mentioned TensorFlow, of which only 5 were from Google.In December 2017, ... | TensorFlow | 0.809213 |
22,758 | TensorFlow also offers a variety of libraries and extensions to advance and extend the models and methods used. For example, TensorFlow Recommenders and TensorFlow Graphics are libraries for their respective functionalities in recommendation systems and graphics, TensorFlow Federated provides a framework for decentrali... | TensorFlow | 0.809213 |
22,759 | TensorFlow also has a library for machine learning in JavaScript. Using the provided JavaScript APIs, TensorFlow.js allows users to use either Tensorflow.js models or converted models from TensorFlow or TFLite, retrain the given models, and run on the web. | TensorFlow | 0.809213 |
22,760 | TensorFlow Extended (abbrev. TFX) provides numerous components to perform all the operations needed for end-to-end production. Components include loading, validating, and transforming data, tuning, training, and evaluating the machine learning model, and pushing the model itself into production. | TensorFlow | 0.809213 |
22,761 | Starting in 2011, Google Brain built DistBelief as a proprietary machine learning system based on deep learning neural networks. Its use grew rapidly across diverse Alphabet companies in both research and commercial applications. Google assigned multiple computer scientists, including Jeff Dean, to simplify and refacto... | TensorFlow | 0.809213 |
22,762 | Numpy is one of the most popular Python data libraries, and TensorFlow offers integration and compatibility with its data structures. Numpy NDarrays, the library's native datatype, are automatically converted to TensorFlow Tensors in TF operations; the same is also true vice versa. This allows for the two libraries to ... | TensorFlow | 0.809213 |
22,763 | In order to assess the performance of machine learning models, TensorFlow gives API access to commonly used metrics. Examples include various accuracy metrics (binary, categorical, sparse categorical) along with other metrics such as Precision, Recall, and Intersection-over-Union (IoU). | TensorFlow | 0.809213 |
22,764 | Examining the properties of this field and its subfields leads to necessary conditions on a number to be constructible, that can be used to show that specific numbers arising in classical geometric construction problems are not constructible. It is convenient to consider, in place of the whole field of constructible nu... | Constructible number | 0.80921 |
22,765 | The definition of algebraically constructible numbers includes the sum, difference, product, and multiplicative inverse of any of these numbers, the same operations that define a field in abstract algebra. Thus, the constructible numbers (defined in any of the above ways) form a field. More specifically, the constructi... | Constructible number | 0.80921 |
22,766 | The starting information for the geometric formulation can be used to define a Cartesian coordinate system in which the point O {\displaystyle O} is associated to the origin having coordinates ( 0 , 0 ) {\displaystyle (0,0)} and in which the point A {\displaystyle A} is associated with the coordinates ( 1 , 0 ) {\displ... | Constructible number | 0.80921 |
22,767 | Because this specific instance of the trisection problem cannot be solved by compass and straightedge, the general problem also cannot be solved. Squaring the circle A square with area π {\displaystyle \pi } , the same area as a unit circle, would have side length π {\displaystyle {\sqrt {\pi }}} , a transcendental num... | Constructible number | 0.80921 |
22,768 | As a cubic polynomial whose only real root is irrational, this polynomial must be irreducible, because if it had a quadratic real root then the quadratic conjugate would provide a second real root. Angle trisection In this problem, from a given angle θ {\displaystyle \theta } , one should construct an angle θ / 3 {\dis... | Constructible number | 0.80921 |
22,769 | )In particular, the algebraic formulation of constructible numbers leads to a proof of the impossibility of the following construction problems: Doubling the cube The problem of doubling the unit square is solved by the construction of another square on the diagonal of the first one, with side length 2 {\displaystyle {... | Constructible number | 0.80921 |
22,770 | According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree n {\displaystyle n} is the characteristic polynomial of some companion matrix of order n {\displaystyle n} .) Theref... | Geometric multiplicity | 0.809209 |
22,771 | Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the determinant, which for an n × n {\displaystyle n\times n} matrix is a sum of n ! {\displaystyle n!} different products.Explicit algebraic formulas for the roots of a polynomial exis... | Geometric multiplicity | 0.809209 |
22,772 | Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Such equations are usually solved by an iteration procedure, called in this case self-consistent field method. In quantum chemistry, one often represents the Hartree–Fock equation in a non-orthogonal basis set. This particular rep... | Geometric multiplicity | 0.809209 |
22,773 | In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenve... | Geometric multiplicity | 0.809209 |
22,774 | The difficulty with generating an all-hexahedral mesh on a complex geometry is that mesh needs to take into consideration the local geometric detail as well as the global connectivity constraint. This is the STC, and it is only present in an all-hexahedral mesh. This is the reason why it is relatively easy to automate ... | Spatial twist continuum | 0.809207 |
22,775 | Life arose on Earth once it had cooled enough for oceans to form. The last universal common ancestor (LUCA) was an organism which had ribosomes and the genetic code; it lived some 4 billion years ago. It gave rise to two main branches of prokaryotic life, the bacteria and the archaea. From among these small-celled, rap... | Eukaryogenesis | 0.809207 |
22,776 | Open source and non-commercial software: The open source CFD software OpenFOAM includes particle methods, including DEM, and solvers that couple CFD-DEM. CFDEMcoupling (DCS Computing GmbH) couples CFD from OpenFOAM with open source DEM software, LIGGGHTS. MFiX(Open Source multiphase flow simulation package). The commer... | CFD-DEM model | 0.809204 |
22,777 | For example, even when viewing a static image, humans perform saccades, rapid changes of the direction of gaze. The image projected onto the retinal photoreceptors changes therefore every few hundred milliseconds (Chapter 1.5 in ) Despite its shortcomings, the concept of a spike-count rate code is widely used not only ... | Neural coding | 0.809202 |
22,778 | A sequence, or 'train', of spikes may contain information based on different coding schemes. In some neurons the strength with which an postsynaptic partner responds may depend solely on the 'firing rate', the average number of spikes per unit time (a 'rate code'). At the other end, a complex 'temporal code' is based o... | Neural coding | 0.809202 |
22,779 | The study of neural coding involves measuring and characterizing how stimulus attributes, such as light or sound intensity, or motor actions, such as the direction of an arm movement, are represented by neuron action potentials or spikes. In order to describe and analyze neuronal firing, statistical methods and methods... | Neural coding | 0.809202 |
22,780 | Continuous attractor neural networks, also known as CANN, are routinely used when studying spatial view cells from an idiothetic stand point. CANNs allow researchers to closely monitor the associated head direction cells and place cells along with the spatial view cells as one close "packet of neural activity". == Refe... | Spatial view cells | 0.809198 |
22,781 | BASCOE - A data assimilation system based on a chemical transport model and created by the Belgian Institute for Space Aeronomy (BIRA-IASB) Boream - Model for the degradation of alpha-pinene BOXMOX - Box model extensions to KPP CMAQ - Community Multiscale Air Quality model DSMACC - Dynamically Simple Model of Atmospher... | Kinetic PreProcessor | 0.809196 |
22,782 | The Kinetic PreProcessor (KPP) is an open-source software tool used in atmospheric chemistry. Taking a set of chemical reactions and their rate coefficients as input, KPP generates Fortran 90, FORTRAN 77, C, or Matlab code of the resulting ordinary differential equations (ODEs). Solving the ODEs allows the temporal int... | Kinetic PreProcessor | 0.809196 |
22,783 | Let V be a vector space over a field F. For instance, suppose V is Rn or Cn, the standard n-dimensional space of column vectors over the real or complex numbers, respectively. In this case, the idea of representation theory is to do abstract algebra concretely by using n × n matrices of real or complex numbers. There a... | Representation Theory | 0.809196 |
22,784 | This means that they have a class of representations that can be understood in the same way as representations of semisimple Lie algebras. Affine Lie algebras are a special case of Kac–Moody algebras, which have particular importance in mathematics and theoretical physics, especially conformal field theory and the theo... | Representation Theory | 0.809196 |
22,785 | There are many classes of infinite-dimensional Lie algebras whose representations have been studied. Among these, an important class are the Kac–Moody algebras. They are named after Victor Kac and Robert Moody, who independently discovered them. These algebras form a generalization of finite-dimensional semisimple Lie ... | Representation Theory | 0.809196 |
22,786 | Hopf algebras provide a way to improve the representation theory of associative algebras, while retaining the representation theory of groups and Lie algebras as special cases. In particular, the tensor product of two representations is a representation, as is the dual vector space. The Hopf algebras associated to grou... | Representation Theory | 0.809196 |
22,787 | Important results in the theory include the Selberg trace formula and the realization by Robert Langlands that the Riemann–Roch theorem could be applied to calculate the dimension of the space of automorphic forms. The subsequent notion of "automorphic representation" has proved of great technical value for dealing wit... | Representation Theory | 0.809196 |
22,788 | In the case of the representation theory of the group SU(2) (or equivalently, of its complexified Lie algebra s l ( 2 ; C ) {\displaystyle \mathrm {sl} (2;\mathbb {C} )} ), the decomposition is easy to work out. The irreducible representations are labeled by a parameter l {\displaystyle l} that is a non-negative intege... | Representation Theory | 0.809196 |
22,789 | Suppose ϕ 1: G → G L ( V 1 ) {\displaystyle \phi _{1}:G\rightarrow \mathrm {GL} (V_{1})} and ϕ 2: G → G L ( V 2 ) {\displaystyle \phi _{2}:G\rightarrow \mathrm {GL} (V_{2})} are representations of a group G {\displaystyle G} . Then we can form a representation ϕ 1 ⊗ ϕ 2 {\displaystyle \phi _{1}\otimes \phi _{2}} of G a... | Representation Theory | 0.809196 |
22,790 | One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the a... | Representation Theory | 0.809196 |
22,791 | The applications of representation theory are diverse. In addition to its impact on algebra, representation theory generalizes Fourier analysis via harmonic analysis, is connected to geometry via invariant theory and the Erlangen program, has an impact in number theory via automorphic forms and the Langlands program.Th... | Representation Theory | 0.809196 |
22,792 | The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices such that the group operation is matrix multiplication.Representation theory is a useful method because it reduces problems in abstract algebra to problems... | Representation Theory | 0.809196 |
22,793 | Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing... | Representation Theory | 0.809196 |
22,794 | Linear algebraic groups (or more generally, affine group schemes) are analogues in algebraic geometry of Lie groups, but over more general fields than just R or C. In particular, over finite fields, they give rise to finite groups of Lie type. Although linear algebraic groups have a classification that is very similar ... | Representation Theory | 0.809196 |
22,795 | In one sense, associative algebra representations generalize both representations of groups and Lie algebras. A representation of a group induces a representation of a corresponding group ring or group algebra, while representations of a Lie algebra correspond bijectively to representations of its universal enveloping ... | Representation Theory | 0.809196 |
22,796 | The method of (hypergraph) containers is a powerful tool that can help characterize the typical structure and/or answer extremal questions about families of discrete objects with a prescribed set of local constraints. Such questions arise naturally in extremal graph theory, additive combinatorics, discrete geometry, co... | Container method | 0.809194 |
22,797 | One of the foundational problems of extremal graph theory, dating to work of Mantel in 1907 and Turán from the 1940s, asks to characterize those graphs that do not contain a copy of some fixed forbidden H as a subgraph. In a different domain, one of the motivating questions in additive combinatorics is understanding ho... | Container method | 0.809194 |
22,798 | The calculus of variations is concerned with the maxima or minima (collectively called extrema) of functionals. A functional maps functions to scalars, so functionals have been described as "functions of functions." Functionals have extrema with respect to the elements y {\displaystyle y} of a given function space defi... | Variational method | 0.809194 |
22,799 | This device proved viable for producing 3D printing filament. The Recyclebot v2.2 is being developed by the Michigan Tech in Open Sustainability Technology Research Group.Many makers or DIY enthusiasts have made various versions of RecycleBots. | Recyclebot | 0.809193 |
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