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1,800 | Around 1945, Alfred Tarski asked whether the free groups on two or more generators have the same first-order theory, and whether this theory is decidable. Sela (2006) answered the first question by showing that any two nonabelian free groups have the same first-order theory, and Kharlampovich & Myasnikov (2006) answere... | Free subgroup | 0.837294 |
1,801 | Free groups first arose in the study of hyperbolic geometry, as examples of Fuchsian groups (discrete groups acting by isometries on the hyperbolic plane). In an 1882 paper, Walther von Dyck pointed out that these groups have the simplest possible presentations. The algebraic study of free groups was initiated by Jakob... | Free subgroup | 0.837294 |
1,802 | Higgins below, is kind of extracted from an approach using covering spaces. It allows more powerful results, for example on Grushko's theorem, and a normal form for the fundamental groupoid of a graph of groups. In this approach there is considerable use of free groupoids on a directed graph. Grushko's theorem has the ... | Free subgroup | 0.837294 |
1,803 | If S has more than one element, then F(S) is not abelian, and in fact the center of F(S) is trivial (that is, consists only of the identity element). Two free groups F(S) and F(T) are isomorphic if and only if S and T have the same cardinality. This cardinality is called the rank of the free group F. Thus for every car... | Free subgroup | 0.837294 |
1,804 | By contrast, a rotation in a plane spanned by a space-like dimension and a time-like dimension is a hyperbolic rotation, and if this plane contains the time axis of the reference frame, is called a "Lorentz boost". These transformations demonstrate the pseudo-Euclidean nature of the Minkowski space. Hyperbolic rotation... | Rotation (mathematics) | 0.83728 |
1,805 | But in mechanics and, more generally, in physics, this concept is frequently understood as a coordinate transformation (importantly, a transformation of an orthonormal basis), because for any motion of a body there is an inverse transformation which if applied to the frame of reference results in the body being at the ... | Rotation (mathematics) | 0.83728 |
1,806 | Rotation in mathematics is a concept originating in geometry. Any rotation is a motion of a certain space that preserves at least one point. It can describe, for example, the motion of a rigid body around a fixed point. Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude ... | Rotation (mathematics) | 0.83728 |
1,807 | Burnside's theorem states that if G is a finite group of order paqb where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. | Solvable subgroup | 0.83727 |
1,808 | Solvability is closed under a number of operations. If G is solvable, and H is a subgroup of G, then H is solvable. If G is solvable, and there is a homomorphism from G onto H, then H is solvable; equivalently (by the first isomorphism theorem), if G is solvable, and N is a normal subgroup of G, then G/N is solvable. T... | Solvable subgroup | 0.837269 |
1,809 | The following sections lay out examples of some of the types of algebraic equations that may be encountered. | Elementary algebra | 0.837256 |
1,810 | It is typically taught to secondary school students and at introductory college level in the United states, and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broade... | Elementary algebra | 0.837256 |
1,811 | Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).This use of variables entails use of algebraic notat... | Elementary algebra | 0.837256 |
1,812 | Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as ( a + b ) = ( b + a ) {\displaystyle (a+b)=(b+a)} . | Elementary algebra | 0.837256 |
1,813 | A more general (algebraic) description may state that the number of seconds, s = 60 × m {\displaystyle s=60\times m} , where m is the number of minutes. Variables allow one to describe mathematical relationships between quantities that may vary. For example, the relationship between the circumference, c, and diameter, ... | Elementary algebra | 0.837256 |
1,814 | For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as C = P + 20 {\displaystyle C=P+20} . Variables allow one to describe general problems, without specifying the values of the quantities that are invol... | Elementary algebra | 0.837256 |
1,815 | Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons. Variables may represent numbers whose values are not yet known. | Elementary algebra | 0.837256 |
1,816 | An exponential equation is one which has the form a x = b {\displaystyle a^{x}=b} for a > 0 {\displaystyle a>0} , which has solution X = log a b = ln b ln a {\displaystyle X=\log _{a}b={\frac {\ln b}{\ln a}}} when b > 0 {\displaystyle b>0} . Elementary algebraic techniques are used to rewrite a given equation in ... | Elementary algebra | 0.837256 |
1,817 | and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example, 3 × x 2 {\displaystyle 3\times x^{2}} is written as 3 x 2 {\displaystyle 3x^{2}} , and 2 × x × y {\displaystyle 2\t... | Elementary algebra | 0.837256 |
1,818 | Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. a , b , c {\displaystyle a,b,c} ) are typically used to represent constants, and those toward the end of the alphabet (e.g. x , y {\displaystyle x,y} and z) are used to represent variables. They are usually printed ... | Elementary algebra | 0.837256 |
1,819 | Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression 3 x 2 − 2 x y + c {\displaystyle 3x^{2}-2xy+c} has the following components: A coefficient is a numerical value, or letter repr... | Elementary algebra | 0.837256 |
1,820 | Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example, Added terms are simplified using coefficients. For example, x + x + x {\displaystyle x+x+x} can be simplified as 3 x {\displays... | Elementary algebra | 0.837256 |
1,821 | Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. As an illustration of this, while exponents are usually formatted using superscripts, e.g., x 2 {\displaystyle x^{2}} , in plain text, and... | Elementary algebra | 0.837256 |
1,822 | Sperner's lemma Brouwer fixed-point theorem Kakutani fixed-point theorem | List of PPAD-complete problems | 0.837255 |
1,823 | Paul R. Thagard proposed another set of principles to try to overcome these difficulties, and argued that it is important for society to find a way of doing so. According to Thagard's method, a theory is not scientific if it satisfies two conditions: The theory has been less progressive than alternative theories over a... | Demarcation Problem | 0.837242 |
1,824 | From 1934 to 1936, NEC engineer Akira Nakashima, Claude Shannon and Victor Shestakov introduced switching circuit theory in a series of papers showing that two-valued Boolean algebra, which they discovered independently, can describe the operation of switching circuits. Using this property of electrical switches to imp... | Discrete logic | 0.83722 |
1,825 | Many important complexity classes can be defined by bounding the time or space used by the algorithm. Some important complexity classes of decision problems defined in this manner are the following: The logarithmic-space classes (necessarily) do not take into account the space needed to represent the problem. It turns ... | Computational complexity theory | 0.837203 |
1,826 | The process is most productive when the absorption of quanta is equal in both PSII and PSI, assuring that input energy from the antenna complex is divided between the PSI and PSII systems, which in turn powers the photochemistry.Robert Hill thought that a complex of reactions consisted of an intermediate to cytochrome ... | Photosynthesis | 0.837195 |
1,827 | Cornelis Van Niel made key discoveries explaining the chemistry of photosynthesis. By studying purple sulfur bacteria and green bacteria he was the first to demonstrate that photosynthesis is a light-dependent redox reaction, in which hydrogen reduces (donates its atoms as electrons and protons to) carbon dioxide. Robe... | Photosynthesis | 0.837195 |
1,828 | In the late 1940s at the University of California, Berkeley, the details of photosynthetic carbon metabolism were sorted out by the chemists Melvin Calvin, Andrew Benson, James Bassham and a score of students and researchers utilizing the carbon-14 isotope and paper chromatography techniques. The pathway of CO2 fixatio... | Photosynthesis | 0.837195 |
1,829 | This table organizes the elements by increasing atomic number into rows ("periods") in which the columns ("groups") share recurring ("periodic") physical and chemical properties. The periodic table summarizes various properties of the elements, allowing chemists to derive relationships between them and to make predicti... | Chemical elements | 0.837187 |
1,830 | While there might seem to be an obvious difference between the class of problems that are efficiently solvable and the class of problems whose solutions are merely efficiently checkable, P and NP are actually at the center of one of the most famous unsolved problems in computer science: the P versus NP problem. While i... | Complexity classes | 0.837168 |
1,831 | A number of complexity classes are defined using interactive proof systems. Interactive proofs generalize the proofs definition of the complexity class NP and yield insights into cryptography, approximation algorithms, and formal verification. Interactive proof systems are abstract machines that model computation as th... | Complexity classes | 0.837168 |
1,832 | A counting problem asks not only whether a solution exists (as with a decision problem), but asks how many solutions exist. For example, the decision problem CYCLE {\displaystyle {\texttt {CYCLE}}} asks whether a particular graph G {\displaystyle G} has a simple cycle (the answer is a simple yes/no); the corresponding ... | Complexity classes | 0.837168 |
1,833 | The study of the relationships between complexity classes is a major area of research in theoretical computer science. There are often general hierarchies of complexity classes; for example, it is known that a number of fundamental time and space complexity classes relate to each other in the following way: NL⊆P⊆NP⊆PSP... | Complexity classes | 0.837168 |
1,834 | The complexity classes EXPSPACE and NEXPSPACE are the space analogues to EXPTIME and NEXPTIME. That is, EXPSPACE is the class of problems solvable in exponential space by a deterministic Turing machine and NEXPSPACE is the class of problems solvable in exponential space by a nondeterministic Turing machine. Or more for... | Complexity classes | 0.837168 |
1,835 | Important corollaries of Savitch's theorem are that PSPACE = NPSPACE (since the square of a polynomial is still a polynomial) and EXPSPACE = NEXPSPACE (since the square of an exponential is still an exponential). These relationships answer fundamental questions about the power of nondeterminism compared to determinism.... | Complexity classes | 0.837168 |
1,836 | Savitch's theorem establishes the relationship between deterministic and nondetermistic space resources. It shows that if a nondeterministic Turing machine can solve a problem using f ( n ) {\displaystyle f(n)} space, then a deterministic Turing machine can solve the same problem in f ( n ) 2 {\displaystyle f(n)^{2}} s... | Complexity classes | 0.837168 |
1,837 | quotient group Given a group G {\displaystyle G} and a normal subgroup N {\displaystyle N} of G {\displaystyle G} , the quotient group is the set G {\displaystyle G} / N {\displaystyle N} of left cosets { a N: a ∈ G } {\displaystyle \{aN:a\in G\}} together with the operation a N ∗ b N = a b N . {\displaystyle aN*bN=abN... | Glossary of group theory | 0.83716 |
1,838 | permutation group A permutation group is a group whose elements are permutations of a given set M (the bijective functions from set M to itself) and whose group operation is the composition of those permutations. The group consisting of all permutations of a set M is the symmetric group of M. p-group If p is a prime nu... | Glossary of group theory | 0.83716 |
1,839 | Both subgroups and normal subgroups of a given group form a complete lattice under inclusion of subsets; this property and some related results are described by the lattice theorem. Group homomorphism. These are functions f: ( G , ∗ ) → ( H , × ) {\displaystyle f\colon (G,*)\to (H,\times )} that have the special proper... | Glossary of group theory | 0.83716 |
1,840 | Systems biology is the computational and mathematical analysis and modeling of complex biological systems. It is a biology-based interdisciplinary field of study that focuses on complex interactions within biological systems, using a holistic approach (holism instead of the more traditional reductionism) to biological ... | Molecular physiology | 0.837153 |
1,841 | Since the objective is a model of the interactions in a system, the experimental techniques that most suit systems biology are those that are system-wide and attempt to be as complete as possible. Therefore, transcriptomics, metabolomics, proteomics and high-throughput techniques are used to collect quantitative data f... | Molecular physiology | 0.837153 |
1,842 | Systems biology can be considered from a number of different aspects. As a field of study, particularly, the study of the interactions between the components of biological systems, and how these interactions give rise to the function and behavior of that system (for example, the enzymes and metabolites in a metabolic p... | Molecular physiology | 0.837153 |
1,843 | Researchers begin by choosing a biological pathway and diagramming all of the protein interactions. After determining all of the interactions of the proteins, mass action kinetics is utilized to describe the speed of the reactions in the system. Mass action kinetics will provide differential equations to model the biol... | Molecular physiology | 0.837153 |
1,844 | Deep learning Deep belief networks Deep Boltzmann machines Deep Convolutional neural networks Deep Recurrent neural networks Hierarchical temporal memory Generative Adversarial Network Style transfer Transformer Stacked Auto-Encoders | Machine learning algorithms | 0.837143 |
1,845 | Bayesian statistics Bayesian knowledge base Naive Bayes Gaussian Naive Bayes Multinomial Naive Bayes Averaged One-Dependence Estimators (AODE) Bayesian Belief Network (BBN) Bayesian Network (BN) | Machine learning algorithms | 0.837143 |
1,846 | Artificial Intelligence and Security (AISec) (co-located workshop with CCS) Conference on Neural Information Processing Systems (NIPS) ECML PKDD International Conference on Machine Learning (ICML) ML4ALL (Machine Learning For All) | Machine learning algorithms | 0.837143 |
1,847 | Applications of machine learning Bioinformatics Biomedical informatics Computer vision Customer relationship management – Data mining Earth sciences Email filtering Inverted pendulum – balance and equilibrium system. Natural language processing (NLP) Named Entity Recognition Automatic summarization Automatic taxonomy c... | Machine learning algorithms | 0.837143 |
1,848 | Amazon Machine Learning Microsoft Azure Machine Learning Studio DistBelief – replaced by TensorFlow | Machine learning algorithms | 0.837143 |
1,849 | Machine Learning Journal of Machine Learning Research (JMLR) Neural Computation | Machine learning algorithms | 0.837143 |
1,850 | Computational learning theory – studying the design and analysis of machine learning algorithms. Grammar induction Meta-learning | Machine learning algorithms | 0.837143 |
1,851 | Almeida–Pineda recurrent backpropagation ALOPEX Backpropagation Bootstrap aggregating CN2 algorithm Constructing skill trees Dehaene–Changeux model Diffusion map Dominance-based rough set approach Dynamic time warping Error-driven learning Evolutionary multimodal optimization Expectation–maximization algorithm FastICA ... | Machine learning algorithms | 0.837143 |
1,852 | Alberto Broggi Andrei Knyazev Andrew McCallum Andrew Ng Anuraag Jain Armin B. Cremers Ayanna Howard Barney Pell Ben Goertzel Ben Taskar Bernhard Schölkopf Brian D. Ripley Christopher G. Atkeson Corinna Cortes Demis Hassabis Douglas Lenat Eric Xing Ernst Dickmanns Geoffrey Hinton – co-inventor of the backpropagation and... | Machine learning algorithms | 0.837143 |
1,853 | In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution to an optimization problem. The main characteristics of problems that lie in PLS are that the cost of a solution can be calculated in polynomial time and the neighborh... | PLS (complexity) | 0.837122 |
1,854 | A local search problem L {\displaystyle L} is PLS-complete, if L {\displaystyle L} is in PLS every problem in PLS can be PLS-reduced to L {\displaystyle L} The optimization version of the circuit problem under the Flip neighborhood structure has been shown to be a first PLS-complete problem. | PLS (complexity) | 0.837122 |
1,855 | Two linear systems using the same set of variables are equivalent if each of the equations in the second system can be derived algebraically from the equations in the first system, and vice versa. Two systems are equivalent if either both are inconsistent or each equation of each of them is a linear combination of the ... | Linear equation system | 0.837116 |
1,856 | Putting it another way, according to the Rouché–Capelli theorem, any system of equations (overdetermined or otherwise) is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If, on the other hand, the ranks of these two matrices are equal, the system must have at least o... | Linear equation system | 0.837116 |
1,857 | The first science with a theoretical foundation based in mathematics was mechanics; the underlying principles of mechanics were first delineated by Isaac Newton in his 1687 book Philosophiæ Naturalis Principia Mathematica. One of the earliest works to define applied mechanics as its own discipline was the three volume ... | Applied Mechanics | 0.837083 |
1,858 | It has numerous applications in a wide variety of fields and disciplines, including but not limited to structural engineering, astronomy, oceanography, meteorology, hydraulics, mechanical engineering, aerospace engineering, nanotechnology, structural design, earthquake engineering, fluid dynamics, planetary sciences, a... | Applied Mechanics | 0.837083 |
1,859 | System theory and design Optimal control system System and control applications Robotics Manufacturing | Applied Mechanics | 0.837083 |
1,860 | Electrical Engineering Civil engineering Mechanical Engineering Nuclear engineering Architectural engineering Chemical engineering Petroleum engineering | Applied Mechanics | 0.837083 |
1,861 | In aerospace engineering, applied mechanics is used in aerodynamics, aerospace structural mechanics and propulsion, aircraft design and flight mechanics. In materials engineering, applied mechanics’ concepts are used in thermoelasticity, elasticity theory, fracture and failure mechanisms, structural design optimisation... | Applied Mechanics | 0.837083 |
1,862 | Engineering problems are generally tackled with applied mechanics through the application of theories of classical mechanics and fluid mechanics. Because applied mechanics can be applied in engineering disciplines like civil engineering, mechanical engineering, aerospace engineering, materials engineering, and biomedic... | Applied Mechanics | 0.837083 |
1,863 | All of the Nano Quest missions are related to nanotechnology and its applications: Individual Atom Manipulation - up to 40 points Smell (pizza molecules) - up to 15 points each Stain-Resistant Fabric - up to 30 points Dirt (particles) - up to 5 points each Atomic Force Microscopy - 40 points Self-Assembly - 30 points S... | Nano Quest | 0.837057 |
1,864 | Teams were tasked with identifying an application of nanotechnology, see where current research stands and creatively improve upon it. Then teams shared their project with the community and with judges at competition. | Nano Quest | 0.837057 |
1,865 | Nano Quest was the name of the 2006–07 challenge theme of FIRST Lego League. It revolved around the emerging field of nanotechnology. | Nano Quest | 0.837057 |
1,866 | Transcriptomics studies generate large amounts of data that have potential applications far beyond the original aims of an experiment. As such, raw or processed data may be deposited in public databases to ensure their utility for the broader scientific community. For example, as of 2018, the Gene Expression Omnibus co... | Transcriptomics technique | 0.837044 |
1,867 | Analysis of over 1000 isolates of Plasmodium falciparum, a virulent parasite responsible for malaria in humans, identified that upregulation of the unfolded protein response and slower progression through the early stages of the asexual intraerythrocytic developmental cycle were associated with artemisinin resistance i... | Transcriptomics technique | 0.837044 |
1,868 | Aging-related preventive interventions are not possible without personal aging speed measurement. The most up to date and complex way to measure aging rate is by using varying biomarkers of human aging is based on the utilization of deep neural networks which may be trained on any type of omics biological data to predi... | Transcriptomics technique | 0.837044 |
1,869 | Legend: NCBI SRA – National center for biotechnology information sequence read archive. Currently RNA-Seq relies on copying RNA molecules into cDNA molecules prior to sequencing; therefore, the subsequent platforms are the same for transcriptomic and genomic data. Consequently, the development of DNA sequencing technol... | Transcriptomics technique | 0.837044 |
1,870 | Secondary structures in the DNA can result in folding or knotting of DNA template or primers, leading to decreased product yield or failure of the reaction. Hairpins, which consist of internal folds caused by base-pairing between nucleotides in inverted repeats within single-stranded DNA, are common secondary structure... | PCR optimization | 0.837031 |
1,871 | The PCR method is extremely sensitive, requiring only a few DNA molecules in a single reaction for amplification across several orders of magnitude. Therefore, adequate measures to avoid contamination from any DNA present in the lab environment (bacteria, viruses, or human sources) are required. Because products from p... | PCR optimization | 0.837031 |
1,872 | Several highly regarded tactile tests have been developed as rigorous replacements to the 2PD test. In psychophysics research laboratories, a favored test of tactile spatial acuity has for many years been the grating orientation task (GOT). In the GOT, the participant attempts to discern the orientation of parallel gro... | Two-point discrimination | 0.837005 |
1,873 | In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of a Lie group G, then H is an embedded Lie group with the smooth structure (and hence the group topology) agreeing with the embedding. One of severa... | Closed subgroup theorem | 0.836987 |
1,874 | A subgroup is locally closed if every point has a neighborhood in U ⊂G such that H ∩ U is closed in U. If H = AB = {ab | a ∈ A, b ∈ B}, where A is a compact group and B is a closed set, then H is closed. If h ⊂ g is a Lie subalgebra such that for no X ∈ g \ h, ∈ h, then Γ(h), the group generated by eh, is closed in G.... | Closed subgroup theorem | 0.836987 |
1,875 | Organic compounds form the basis of all earthly life and constitute the majority of known chemicals. The bonding patterns of carbon, with its valence of four—formal single, double, and triple bonds, plus structures with delocalized electrons—make the array of organic compounds structurally diverse, and their range of a... | Organic Chemistry | 0.836944 |
1,876 | The range of chemicals studied in organic chemistry includes hydrocarbons (compounds containing only carbon and hydrogen) as well as compounds based on carbon, but also containing other elements, especially oxygen, nitrogen, sulfur, phosphorus (included in many biochemicals) and the halogens. Organometallic chemistry i... | Organic Chemistry | 0.836944 |
1,877 | Organic chemistry is a subdiscipline within chemistry involving the scientific study of the structure, properties, and reactions of organic compounds and organic materials, i.e., matter in its various forms that contain carbon atoms. Study of structure determines their structural formula. Study of properties includes p... | Organic Chemistry | 0.836944 |
1,878 | For this relatively simple organism, the study of protein complexes is now genome wide and the elucidation of most of its protein complexes is ongoing. In 2021, researchers used deep learning software RoseTTAFold along with AlphaFold to solve the structures of 712 eukaryote complexes. They compared 6000 yeast proteins ... | Protein complex | 0.836903 |
1,879 | Despite its interesting complexity theoretic status, parity game solving can be seen as the algorithmic backend to problems in automated verification and controller synthesis. The model-checking problem for the modal μ-calculus for instance is known to be equivalent to parity game solving. Also, decision problems like ... | Parity game | 0.836886 |
1,880 | This explains the word "parity" in the title. Parity games lie in the third level of the Borel hierarchy, and are consequently determined.Games related to parity games were implicitly used in Rabin's proof of decidability of the monadic second-order theory of n successors (S2S for n = 2), where determinacy of such game... | Parity game | 0.836886 |
1,881 | In empirical risk minimization, the supervised learning algorithm seeks the function g {\displaystyle g} that minimizes R ( g ) {\displaystyle R(g)} . Hence, a supervised learning algorithm can be constructed by applying an optimization algorithm to find g {\displaystyle g} . When g {\displaystyle g} is a conditional p... | Supervised Machine Learning | 0.836883 |
1,882 | Other norms include the L 1 {\displaystyle L_{1}} norm, ∑ j | β j | {\displaystyle \sum _{j}|\beta _{j}|} , and the L 0 {\displaystyle L_{0}} "norm", which is the number of non-zero β j {\displaystyle \beta _{j}} s. The penalty will be denoted by C ( g ) {\displaystyle C(g)} . The supervised learning optimization probl... | Supervised Machine Learning | 0.836883 |
1,883 | Structural risk minimization seeks to prevent overfitting by incorporating a regularization penalty into the optimization. The regularization penalty can be viewed as implementing a form of Occam's razor that prefers simpler functions over more complex ones. A wide variety of penalties have been employed that correspon... | Supervised Machine Learning | 0.836883 |
1,884 | The hypothesis of the inverse function theorem is satisfied with Φ analytic, and thus there are open sets U1 ⊂ g, V1 ⊂ GL(n, R) with 0 ∈ U1 and I ∈ V1 such that Φ is a real-analytic bijection from U1 to V1 with analytic inverse. It remains to show that U1 and V1 contain open sets U and V such that the conclusion of the... | Closed-subgroup theorem | 0.836871 |
1,885 | We begin by establishing the key lemma stated in the "overview" section above. Endow g with an inner product (e.g., the Hilbert–Schmidt inner product), and let h be the Lie algebra of H defined as h = {X ∈ Mn(R) = g | etX ∈ H ∀t ∈ R}. Let s = {S ∈ g | (S, T) = 0 ∀T ∈ h}, the orthogonal complement of h. Then g decompose... | Closed-subgroup theorem | 0.836871 |
1,886 | The proof is given for matrix groups with G = GL(n, R) for concreteness and relative simplicity, since matrices and their exponential mapping are easier concepts than in the general case. Historically, this case was proven first, by John von Neumann in 1929, and inspired Cartan to prove the full closed subgroup theorem... | Closed-subgroup theorem | 0.836871 |
1,887 | Next, construct coordinate charts on H. First define φ1: e(U) ⊂ G → g, g ↦ log(g). This is an analytic bijection with analytic inverse. Furthermore, if h ∈ H, then φ1(h) ∈ h. By fixing a basis for g = h ⊕ s and identifying g with R n {\displaystyle \mathbb {R} ^{n}} , then in these coordinates φ1(h) = (x1(h), …, xm(h),... | Closed-subgroup theorem | 0.836871 |
1,888 | For j ≥ i, the image in H of Bj under Φ form a neighborhood basis at I. This is, by the way it is constructed, a neighborhood basis both in the group topology and the relative topology. Since multiplication in G is analytic, the left and right translates of this neighborhood basis by a group element g ∈ G gives a neigh... | Closed-subgroup theorem | 0.836871 |
1,889 | While the map exp: h → (H, τr) is an analytic bijection, its inverse is not continuous. That is, if U ⊂ h corresponds to a small open interval −ε < θ < ε, there is no open V ⊂ (H, τr) with log(V) ⊂ U due to the appearance of the sets V. However, with the group topology τg, (H, τg) is a Lie group. With this topology the... | Closed-subgroup theorem | 0.836871 |
1,890 | This means that H is not locally path connected. In the group topology, the small open sets are single line segments on the surface of the torus and H is locally path connected. The example shows that for some groups H one can find points in an arbitrarily small neighborhood U in the relative topology τr of the identit... | Closed-subgroup theorem | 0.836871 |
1,891 | For an example of a subgroup that is not an embedded Lie subgroup, consider the torus and an "irrational winding of the torus". and its subgroup with a irrational. Then H is dense in G and hence not closed. In the relative topology, a small open subset of H is composed of infinitely many almost parallel line segments o... | Closed-subgroup theorem | 0.836871 |
1,892 | Every closed subgroup yields a homogeneous space. In a similar way, the closed subgroup theorem simplifies the hypothesis in the following theorem. If X is a set with transitive group action and the isotropy group or stabilizer of a point x ∈ X is a closed Lie subgroup, then X has a unique smooth manifold structure suc... | Closed-subgroup theorem | 0.836871 |
1,893 | It is important to note, however, that the topology on H {\displaystyle H} coming from these coordinates is not the subset topology. That it so say, the identity component of H {\displaystyle H} is an immersed submanifold of G {\displaystyle G} but not an embedded submanifold. In particular, the lemma stated above does... | Closed-subgroup theorem | 0.836871 |
1,894 | Since h {\displaystyle {\mathfrak {h}}} is just a subspace of g {\displaystyle {\mathfrak {g}}} , this means that h ⊂ g {\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}} is just like R k ⊂ R n {\displaystyle \mathbb {R} ^{k}\subset \mathbb {R} ^{n}} , with k = dim ( h ) {\displaystyle k=\dim({\mathfrak {h}})} an... | Closed-subgroup theorem | 0.836871 |
1,895 | The challenge is that H {\displaystyle H} is not assumed to have any smoothness and therefore it is not clear how one may define its tangent space. To proceed, define the "Lie algebra" h {\displaystyle {\mathfrak {h}}} of H {\displaystyle H} by the formula It is not difficult to show that h {\displaystyle {\mathfrak {h... | Closed-subgroup theorem | 0.836871 |
1,896 | Let G {\displaystyle G} be a Lie group with Lie algebra g {\displaystyle {\mathfrak {g}}} . Now let H {\displaystyle H} be an arbitrary closed subgroup of G {\displaystyle G} . It is necessary to show that H {\displaystyle H} is a smooth embedded submanifold of G {\displaystyle G} . The first step is to identify someth... | Closed-subgroup theorem | 0.836871 |
1,897 | Because of the conclusion of the theorem, some authors chose to define linear Lie groups or matrix Lie groups as closed subgroups of GL(n, R) or GL(n, C). In this setting, one proves that every element of the group sufficiently close to the identity is the exponential of an element of the Lie algebra. (The proof is pra... | Closed-subgroup theorem | 0.836871 |
1,898 | An embedded Lie subgroup H ⊂ G is closed so a subgroup is an embedded Lie subgroup if and only if it is closed. Equivalently, H is an embedded Lie subgroup if and only if its group topology equals its relative topology. | Closed-subgroup theorem | 0.836871 |
1,899 | A frequently used theorem is that, given any subset S ⊆ G , {\displaystyle S\subseteq G,} the index of N ( S ) {\displaystyle \operatorname {N} (S)} (the normalizer of S {\displaystyle S} ) in G {\displaystyle G} equals the order of Cl ( S ) {\displaystyle \operatorname {Cl} (S)}: This follows since, if g , h ∈ G ,... | Conjugate subgroup | 0.836865 |
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