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1,900 | If G {\displaystyle G} is a finite group, then for any group element a , {\displaystyle a,} the elements in the conjugacy class of a {\displaystyle a} are in one-to-one correspondence with cosets of the centralizer C G ( a ) . {\displaystyle \operatorname {C} _{G}(a).} This can be seen by observing that any two eleme... | Conjugate subgroup | 0.836865 |
1,901 | The versatility of polymerase chain reaction (PCR) has led to modifications of the basic protocol being used in a large number of variant techniques designed for various purposes. This article summarizes many of the most common variations currently or formerly used in molecular biology laboratories; familiarity with th... | Variants of PCR | 0.836862 |
1,902 | To release the DNA from the cells, the PCR is either started with an extended time at 95 °C (when standard polymerase is used), or with a shortened denaturation step at 100 °C and special chimeric DNA polymerase.The digital polymerase chain reaction simultaneously amplifies thousands of samples, each in a separate drop... | Variants of PCR | 0.836862 |
1,903 | A company might analyze its own potential (productivity, market position) by comparing it to those of the competitors (benchmarking). A market can be analyzed to estimate its potential for a certain product. Processes can be structurally analyzed due to their optimization. | Potential analysis | 0.836838 |
1,904 | For small elements X , Y {\displaystyle X,Y} of the Lie algebra, the structure of the Lie group near the identity element is given by exp ( X ) exp ( Y ) ≈ exp ( X + Y + 1 2 ) . {\displaystyle \exp(X)\exp(Y)\approx \exp(X+Y+{\tfrac {1}{2}}).} Note the factor of 1/2. They also appear in explicit expressions for d... | Structure constants | 0.836821 |
1,905 | The structure constants play a role in Lie algebra representations, and in fact, give exactly the matrix elements of the adjoint representation. The Killing form and the Casimir invariant also have a particularly simple form, when written in terms of the structure constants. The structure constants often make an appear... | Structure constants | 0.836821 |
1,906 | The linear expansion of the Lie bracket of pairs of generators then looks like = ∑ c f a b c T c {\displaystyle =\sum _{c}f_{ab}^{\;\;c}T_{c}} .Again, by linear extension, the structure constants completely determine the Lie brackets of all elements of the Lie algebra. All Lie algebras satisfy the Jacobi identity. | Structure constants | 0.836821 |
1,907 | For a Lie algebra, the basis vectors are termed the generators of the algebra, and the product rather called the Lie bracket (often the Lie bracket is an additional product operation beyond the already existing product, thus necessitating a separate name). For two vectors A {\displaystyle A} and B {\displaystyle B} in ... | Structure constants | 0.836821 |
1,908 | Given a set of basis vectors { e i } {\displaystyle \{\mathbf {e} _{i}\}} for the underlying vector space of the algebra, the product operation is uniquely defined by the products of basis vectors: e i ⋅ e j = c i j {\displaystyle \mathbf {e} _{i}\cdot \mathbf {e} _{j}=\mathbf {c} _{ij}} .The structure constants or str... | Structure constants | 0.836821 |
1,909 | The algebra s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} of the special unitary group SU(2) is three-dimensional, with generators given by the Pauli matrices σ i {\displaystyle \sigma _{i}} . The generators of the group SU(2) satisfy the commutation relations (where ε a b c {\displaystyle \varepsilon ^{abc}} is the Le... | Structure constants | 0.836821 |
1,910 | For a given α {\displaystyle \alpha } , there are as many α i {\displaystyle \alpha _{i}} as there are H i {\displaystyle H_{i}} and so one may define the vector α = α i H i {\displaystyle \alpha =\alpha _{i}H_{i}} , this vector is termed a root of the algebra. The roots of Lie algebras appear in regular structures (fo... | Structure constants | 0.836821 |
1,911 | The dimension r {\displaystyle r} of this subalgebra is called the rank of the algebra. In the adjoint representation, the matrices a d ( H i ) {\displaystyle \mathrm {ad} (H_{i})} are mutually commuting, and can be simultaneously diagonalized. The matrices a d ( H i ) {\displaystyle \mathrm {ad} (H_{i})} have (simulta... | Structure constants | 0.836821 |
1,912 | Given a Lie algebra g {\displaystyle {\mathfrak {g}}} , the Cartan subalgebra h ⊂ g {\displaystyle {\mathfrak {h}}\subset {\mathfrak {g}}} is the maximal Abelian subalgebra. By definition, it consists of those elements that commute with one-another. An orthonormal basis can be freely chosen on h {\displaystyle {\mathfr... | Structure constants | 0.836821 |
1,913 | One conventional approach to providing a basis for a Lie algebra is by means of the so-called "ladder operators" appearing as eigenvectors of the Cartan subalgebra. The construction of this basis, using conventional notation, is quickly sketched here. An alternative construction (the Serre construction) can be found in... | Structure constants | 0.836821 |
1,914 | Primer-BLAST is widely used, and freely accessible from the National Center for Biotechnology Information (NCBI) website. On the other hand, FastPCR, a commercial application, allows simultaneous testing of a single primer or a set of primers designed for multiplex target sequences. | In silico PCR | 0.836798 |
1,915 | In silico PCR refers to computational tools used to calculate theoretical polymerase chain reaction (PCR) results using a given set of primers (probes) to amplify DNA sequences from a sequenced genome or transcriptome.These tools are used to optimize the design of primers for target DNA or cDNA sequences. Primer optimi... | In silico PCR | 0.836798 |
1,916 | These graphs have been used to solve a problem in extremal graph theory, of constructing a graph with a given number of edges and vertices whose largest tree induced as a subgraph is as small as possible.All eigenvalues of the adjacency matrix A of a line graph are at least −2. The reason for this is that A can be writ... | Line graph | 0.836787 |
1,917 | The concept of the line graph of G may naturally be extended to the case where G is a multigraph. In this case, the characterizations of these graphs can be simplified: the characterization in terms of clique partitions no longer needs to prevent two vertices from belonging to the same to cliques, and the characterizat... | Line graph | 0.836787 |
1,918 | However, the algorithm of Degiorgi & Simon (1995) uses only Whitney's isomorphism theorem. It is complicated by the need to recognize deletions that cause the remaining graph to become a line graph, but when specialized to the static recognition problem only insertions need to be performed, and the algorithm performs t... | Line graph | 0.836787 |
1,919 | For instance if edges d and e in the graph G are incident at a vertex v with degree k, then in the line graph L(G) the edge connecting the two vertices d and e can be given weight 1/(k − 1). In this way every edge in G (provided neither end is connected to a vertex of degree 1) will have strength 2 in the line graph L(... | Line graph | 0.836787 |
1,920 | Put another way, the Whitney graph isomorphism theorem guarantees that the line graph almost always encodes the topology of the original graph G faithfully but it does not guarantee that dynamics on these two graphs have a simple relationship. One solution is to construct a weighted line graph, that is, a line graph wi... | Line graph | 0.836787 |
1,921 | The N-terminal "ring" can be from 7 to 9 amino acids long and is formed by an isopeptide bond between the N-terminal amine of the first amino acid of the peptide and the carboxylate side chain of an aspartate or glutamate residue. The C-terminal "tail" ranges from 7 to 15 amino acids in length.The first amino acid of l... | Ribosomally synthesized and post-translationally modified peptide | 0.836777 |
1,922 | Lasso peptides are short peptides containing an N-terminal macrolactam macrocycle "ring" through which a linear C-terminal "tail" is threaded. Because of this threaded-loop topology, these peptides resemble lassos, giving rise to their name. They are a member of a larger class of amino-acid-based lasso structures. Addi... | Ribosomally synthesized and post-translationally modified peptide | 0.836777 |
1,923 | Commonly, the B protein is referred to as the lasso protease, and the C protein is referred to as the lasso cyclase. Some lasso peptide biosynthetic gene clusters also require an additional protein of unknown function for biosynthesis. Additionally, lasso peptide gene clusters usually include an ABC transporter (D prot... | Ribosomally synthesized and post-translationally modified peptide | 0.836777 |
1,924 | Instruments such as the astrolabe, the quadrant, and others were used to measure and accurately record the relative positions and movements of planets and other celestial objects. The sextant and other related instruments were essential for navigation at sea. Most instruments are used within the field of geometry, incl... | Mathematical instruments | 0.836748 |
1,925 | This is a method to find each digit of the square root in a sequence. This method is based on the binomial theorem and basically an inverse algorithm solving ( x + y ) 2 = x 2 + 2 x y + y 2 {\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}} . It is slower than the Babylonian method, but it has several advantages: It can be easi... | Methods of computing square roots | 0.836744 |
1,926 | A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square ... | Methods of computing square roots | 0.836744 |
1,927 | The most common analytical methods are iterative and consist of two steps: finding a suitable starting value, followed by iterative refinement until some termination criterion is met. The starting value can be any number, but fewer iterations will be required the closer it is to the final result. The most familiar such... | Methods of computing square roots | 0.836744 |
1,928 | Methods of computing square roots are numerical analysis algorithms for approximating the principal, or non-negative, square root (usually denoted S {\displaystyle {\sqrt {S}}} , S 2 {\displaystyle {\sqrt{S}}} , or S 1 / 2 {\displaystyle S^{1/2}} ) of a real number. Arithmetically, it means given S {\displaystyle S} , ... | Methods of computing square roots | 0.836744 |
1,929 | Its case study is an investigation into possible gender bias in student admission at the University of California, Berkeley in the 1970s, in which the admission statistics for six separate departments showed a small bias in favor of women in admissions, and yet when grouped together into a single set the same statistic... | Math on Trial | 0.836735 |
1,930 | Noah Giansiracusa complains that the authors sometimes perpetrate the same fallacies or mistaken calculations that they warn of, that their treatment of legal reasoning can be superficial, and that their accounts of some cases appear to exhibit bias by the authors instead of presenting the cases neutrally. Daniel Ullma... | Math on Trial | 0.836735 |
1,931 | Ludwig Paditz write that it "vividly shows how the desire for scientific certainty can lead even well-meaning courts to commit grave injustice". Paul H. Edelman singles out the wide range of times and places of the cases presented as a particular strength of the book.Several reviewers suggest that, beyond a general aud... | Math on Trial | 0.836735 |
1,932 | Surface science is closely related to interface and colloid science. Interfacial chemistry and physics are common subjects for both. The methods are different. In addition, interface and colloid science studies macroscopic phenomena that occur in heterogeneous systems due to peculiarities of interfaces. | Surface Physics | 0.836725 |
1,933 | Surface science is the study of physical and chemical phenomena that occur at the interface of two phases, including solid–liquid interfaces, solid–gas interfaces, solid–vacuum interfaces, and liquid–gas interfaces. It includes the fields of surface chemistry and surface physics. Some related practical applications are... | Surface Physics | 0.836725 |
1,934 | Surface physics can be roughly defined as the study of physical interactions that occur at interfaces. It overlaps with surface chemistry. Some of the topics investigated in surface physics include friction, surface states, surface diffusion, surface reconstruction, surface phonons and plasmons, epitaxy, the emission a... | Surface Physics | 0.836725 |
1,935 | Surface chemistry can be roughly defined as the study of chemical reactions at interfaces. It is closely related to surface engineering, which aims at modifying the chemical composition of a surface by incorporation of selected elements or functional groups that produce various desired effects or improvements in the pr... | Surface Physics | 0.836725 |
1,936 | The book is sectioned into four parts. The first part, Genetics and the Scientific Method briefly review the History of genetics and the various methods used in genetic study. The second part focus on Mendelian inheritance, the third part deals with Molecular genetics and the last section deals with Quantitative geneti... | Principles of genetics | 0.836719 |
1,937 | The Principle of genetics is a genetics textbook authored by D. Peter Snustad & Michael J. Simmons, an emeritus professor of biology, published by John Wiley & Sons, Inc.. The 6th edition of the book was published on 2012. | Principles of genetics | 0.836719 |
1,938 | There are two common ways to describe the steepness of a road or railroad. One is by the angle between 0° and 90° (in degrees), and the other is by the slope in a percentage. See also steep grade railway and rack railway. The formulae for converting a slope given as a percentage into an angle in degrees and vice versa ... | Slope | 0.83671 |
1,939 | The concept of a slope or gradient is also used as a basis for developing other applications in mathematics: Gradient descent, a first-order iterative optimization algorithm for finding the minimum of a function Gradient theorem, theorem that a line integral through a gradient field can be evaluated by evaluating the o... | Slope | 0.83671 |
1,940 | The full solution thus is ( y = x ) ∪ ( n 1 / ( n − 1 ) , n n / ( n − 1 ) ) for n > 0 ; n ≠ 1 {\displaystyle (y=x)\cup \left(n^{1/(n-1)},n^{n/(n-1)}\right){\text{ for }}n>0;n\neq 1} . Based on the above solution, the derivative d y / d x {\displaystyle dy/dx} is 1 for the ( x , y ) {\displaystyle (x,y)} pairs on the li... | Equation xy = yx | 0.836662 |
1,941 | A similar solution was found by Euler.J. van Hengel pointed out that if r , n {\displaystyle r,n} are positive integers with r ≥ 3 {\displaystyle r\geq 3} , then r r + n > ( r + n ) r ; {\displaystyle r^{r+n}>(r+n)^{r};} therefore it is enough to consider possibilities x = 1 {\displaystyle x=1} and x = 2 {\displaystyle... | Equation xy = yx | 0.836662 |
1,942 | In universal algebra, an abstract algebra A is called simple if and only if it has no nontrivial congruence relations, or equivalently, if every homomorphism with domain A is either injective or constant. As congruences on rings are characterized by their ideals, this notion is a straightforward generalization of the n... | Simple algebra (universal algebra) | 0.836662 |
1,943 | Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. The applied methods usually refer to nontrivial mathematical techniques or approaches. Mathematical economics is based on statistics, probability, mathematical programming (as well as other computa... | Applied geometry | 0.836646 |
1,944 | In mathematics, especially in the area of algebra studying the theory of abelian groups, a pure subgroup is a generalization of direct summand. It has found many uses in abelian group theory and related areas. | Pure subgroup | 0.836628 |
1,945 | The first ESA was held in 1993 and contained 35 papers. The intended scope was all research in algorithms, theoretical as well as applied, carried out in the fields of computer science and discrete mathematics. An explicit aim was to intensify the exchange between these two research communities. | Workshop on Algorithms Engineering | 0.836618 |
1,946 | The European Symposium on Algorithms (ESA) is an international conference covering the field of algorithms. It has been held annually since 1993, typically in early Autumn in a different European location each year. Like most theoretical computer science conferences its contributions are strongly peer-reviewed; the art... | Workshop on Algorithms Engineering | 0.836618 |
1,947 | WAOA, the Workshop on Approximation and Online Algorithms, has been part of ALGO since 2003. ATMOS, the Workshop on Algorithmic Approaches for Transportation Modeling, Optimization and Systems, formerly the Workshop on Algorithmic Methods and Models for Optimization of Railways, has been part of ALGO in 2003–2006 and 2... | Workshop on Algorithms Engineering | 0.836618 |
1,948 | Since 2001, ESA is co-located with other algorithms conferences and workshops in a combined meeting called ALGO. This is the largest European event devoted to algorithms, attracting hundreds of researchers. Other events in the ALGO conferences include the following. WABI, the Workshop on Algorithms in Bioinformatics, i... | Workshop on Algorithms Engineering | 0.836618 |
1,949 | In both the UK and the US, professional societies had long existed for civil and mechanical engineers. The Institution of Electrical Engineers (IEE) was founded in the UK in 1871, and the AIEE in the United States in 1884. These societies contributed to the exchange of electrical knowledge and the development of electr... | Power engineering | 0.83659 |
1,950 | Power engineering, also called power systems engineering, is a subfield of electrical engineering that deals with the generation, transmission, distribution, and utilization of electric power, and the electrical apparatus connected to such systems. Although much of the field is concerned with the problems of three-phas... | Power engineering | 0.83659 |
1,951 | The installation powered a 100 horsepower (75 kW) synchronous motor at Telluride, Colorado with the motor being started by a Tesla induction motor. On the other side of the Atlantic, Oskar von Miller built a 20 kV 176 km three-phase transmission line from Lauffen am Neckar to Frankfurt am Main for the Electrical Engine... | Power engineering | 0.83659 |
1,952 | One DNA or RNA molecule differs from another primarily in the sequence of nucleotides. Nucleotide sequences are of great importance in biology since they carry the ultimate instructions that encode all biological molecules, molecular assemblies, subcellular and cellular structures, organs, and organisms, and directly e... | Nucleic acids | 0.836589 |
1,953 | Let us redesign the original library with extensibility in mind using the ideas from the paper Extensibility for the Masses. We use the same implementation as in the first code example but now add a new interface containing the functions over the type as well as a factory for the algebra. Notice that we now generate th... | Expression problem | 0.836587 |
1,954 | There are various solutions to the expression problem. Each solution varies in the amount of code a user must write to implement them, and the language features they require. Multiple dispatch Open classes Coproducts of functors Type classes Tagless-final / Object algebras Polymorphic Variants | Expression problem | 0.836587 |
1,955 | For PLT, the problem had shown up in the construction of DrScheme, now DrRacket, and they solved it via a rediscovery of mixins. To avoid using a programming language problem in a paper about programming languages, Krishnamurthi et al. used an old geometry programming problem to explain their pattern-oriented solution.... | Expression problem | 0.836587 |
1,956 | Most importantly, he discussed situations in which there was more flexibility than Reynolds considered, including internalization and optimization of methods. At ECOOP '98, Shriram Krishnamurthi et al. presented a design pattern solution to the problem of simultaneously extending an expression-oriented programming lang... | Expression problem | 0.836587 |
1,957 | Philip Wadler formulated the challenge and named it "The Expression Problem" in response to a discussion with Rice University's Programming Languages Team (PLT). He also cited three sources that defined the context for his challenge: The problem was first observed by John Reynolds in 1975. Reynolds discussed two forms ... | Expression problem | 0.836587 |
1,958 | As of 2009, the primary of the care physicians did not have adequate training in genetics or genomics. Although medical school curricula typically include medical genetics, fewer than half offer a standalone course, and the emphasis on practical applications is weak.In the United States, Stanford University was the fir... | Education in personalized medicine | 0.836553 |
1,959 | Personalized medicine involves medical treatments based on the characteristics of individual patients, including their medical history, family history, and genetics. Although personal genetic information is becoming increasingly important in healthcare, there is a lack of sufficient education in medical genetics among ... | Education in personalized medicine | 0.836553 |
1,960 | In the differential equations, the nabla symbol, ∇, denotes the three-dimensional gradient operator, del, the ∇⋅ symbol (pronounced "del dot") denotes the divergence operator, the ∇× symbol (pronounced "del cross") denotes the curl operator. | Maxwell equation | 0.836527 |
1,961 | List of numerical computational geometry topics enumerates the topics of computational geometry that deals with geometric objects as continuous entities and applies methods and algorithms of nature characteristic to numerical analysis. This area is also called "machine geometry", computer-aided geometric design, and ge... | List of numerical computational geometry topics | 0.836513 |
1,962 | The College Board suggested as preparation for the test four years of mathematics, including two years of algebra, one year of geometry, and one year of either precalculus or trigonometry.While the precalculus or trigonometry course may have been good preparation for this test, students could have needed to buy extra r... | SAT Subject Test in Mathematics Level 2 | 0.836496 |
1,963 | Compared to Mathematics 1, Mathematics 2 was more advanced. Whereas the Mathematics 1 test covered Algebra II and basic trigonometry, a pre-calculus class was good preparation for Mathematics 2. On January 19, 2021, the College Board discontinued all SAT Subject tests, including the SAT Subject Test in Mathematics Leve... | SAT Subject Test in Mathematics Level 2 | 0.836496 |
1,964 | In the U.S., the SAT Subject Test in Mathematics Level 2 (formerly known as Math II or Math IIChest, the "C" representing chest) was a one-hour multiple choice test. The questions covered a broad range of topics. Approximately 10-14% of questions focused on numbers and operations, 48-52% focused on algebra and function... | SAT Subject Test in Mathematics Level 2 | 0.836496 |
1,965 | MOWChIP-seq is enhanced and low-input ChIP-seq thus it applies to all molecular biology that can be probed using ChIP-seq. This includes analysis of histone modifications, RNA pol II binding, and transcription factor binding. Published MOWChIP-seq results include studies of various histone marks (H3K4me3, H3K27ac, H3K2... | MOWChIP-seq | 0.836481 |
1,966 | MOWChIP-seq (Microfluidic Oscillatory Washing–based Chromatin ImmunoPrecipitation followed by sequencing) is a microfluidic technology used in molecular biology for profiling genome-wide histone modifications and other molecular bindings using as few as 30-100 cells per assay. MOWChIP-seq is a special type of ChIP-seq ... | MOWChIP-seq | 0.836481 |
1,967 | Despite the obvious power of the approach, eDNA metabarcoding is affected by precision and accuracy challenges distributed throughout the workflow in the field, in the laboratory and at the keyboard. As set out in the diagram at the right, following the initial study design (hypothesis/question, targeted taxonomic grou... | DNA metabarcoding | 0.836453 |
1,968 | In microbiology, genes can move freely even between distantly related bacteria, possibly extending to the whole bacterial domain. As a rule of thumb, microbiologists have assumed that kinds of Bacteria or Archaea with 16S ribosomal RNA gene sequences more similar than 97% to each other need to be checked by DNA-DNA hyb... | DNA metabarcoding | 0.836453 |
1,969 | Ecosystem-wide applications of eDNA metabarcoding have the potential to not only describe communities and biodiversity, but also to detect interactions and functional ecology over large spatial scales, though it may be limited by false readings due to contamination or other errors. Altogether, eDNA metabarcoding increa... | DNA metabarcoding | 0.836453 |
1,970 | eDNA production is dependent on biomass, age and feeding activity of the organism as well as physiology, life history, and space use.By 2019 methods in eDNA research had been expanded to be able to assess whole communities from a single sample. This process involves metabarcoding, which can be precisely defined as the ... | DNA metabarcoding | 0.836453 |
1,971 | Since the inception of high‐throughput sequencing (HTS), the use of metabarcoding as a biodiversity detection tool has drawn immense interest. However, there has yet to be clarity regarding what source material is used to conduct metabarcoding analyses (e.g., environmental DNA versus community DNA). Without clarity bet... | DNA metabarcoding | 0.836453 |
1,972 | This requires a complete description of the geometry of the member, its constraints, the loads applied to the member and the properties of the material of which the member is composed. The applied loads may be axial (tensile or compressive), or rotational (strength shear). With a complete description of the loading and... | Mechanical strength | 0.836418 |
1,973 | The field of strength of materials (also called mechanics of materials) typically refers to various methods of calculating the stresses and strains in structural members, such as beams, columns, and shafts. The methods employed to predict the response of a structure under loading and its susceptibility to various failu... | Mechanical strength | 0.836418 |
1,974 | Support vector machines are based upon the idea of maximizing the margin i.e. maximizing the minimum distance from the separating hyperplane to the nearest example. The basic SVM supports only binary classification, but extensions have been proposed to handle the multiclass classification case as well. In these extensi... | Multiclass classification | 0.83641 |
1,975 | SequenceBase is a privately held company, is an international patent sequence information provider with headquarters located in Edison, NJ, USA. SequenceBase develops and markets the SequenceBase Research Portal to the biotechnology, legal, pharmaceutical, scientific, technical and academic bioinformatics communities. ... | SequenceBase | 0.836382 |
1,976 | In mathematics, in algebra, in the realm of group theory, a subgroup H {\displaystyle H} of a finite group G {\displaystyle G} is said to be semipermutable if H {\displaystyle H} commutes with every subgroup K {\displaystyle K} whose order is relatively prime to that of H {\displaystyle H} . Clearly, every permutable s... | Semipermutable subgroup | 0.83638 |
1,977 | In computational geometry, a standard technique to build a structure like a convex hull or Delaunay triangulation is to randomly permute the input points and then insert them one by one into the existing structure. The randomization ensures that the expected number of changes to the structure caused by an insertion is ... | Randomized complexity | 0.83637 |
1,978 | Prior to the popularization of randomized algorithms in computer science, Paul Erdős popularized the use of randomized constructions as a mathematical technique for establishing the existence of mathematical objects. This technique has become known as the probabilistic method. Erdős gave his first application of the pr... | Randomized complexity | 0.83637 |
1,979 | The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb-meter (C⋅m). The debye (D) is another unit of measurement used in atomic physics and chemistry... | Electric dipole | 0.836366 |
1,980 | Some authors may split d in half and use s = d/2 since this quantity is the distance between either charge and the center of the dipole, leading to a factor of two in the definition. A stronger mathematical definition is to use vector algebra, since a quantity with magnitude and direction, like the dipole moment of two... | Electric dipole | 0.836366 |
1,981 | Often in physics the dimensions of a massive object can be ignored and can be treated as a pointlike object, i.e. a point particle. Point particles with electric charge are referred to as point charges. Two point charges, one with charge +q and the other one with charge −q separated by a distance d, constitute an elect... | Electric dipole | 0.836366 |
1,982 | Phase Genomics is an American biotechnology company based in Seattle, Washington. The company develops proximity ligation kits and Hi-C sequencing technology used to analyze chromosomes. Phase Genomics sells proximity ligation kits, scientific services, and computational analyses. | Phase Genomics | 0.836362 |
1,983 | This result was shown by Seinosuke Toda in 1989 and is known as Toda's theorem. This is evidence of how hard it is to solve problems in PP. The class #P is in some sense about as hard, since P#P = PPP and therefore P#P includes PH as well. | PP (complexity class) | 0.83636 |
1,984 | If H is a normal subgroup of G, then the quotient group G/H becomes a topological group when given the quotient topology. It is Hausdorff if and only if H is closed in G. For example, the quotient group R / Z {\displaystyle \mathbb {R} /\mathbb {Z} } is isomorphic to the circle group S1. In any topological group, the i... | Closed subgroup | 0.83636 |
1,985 | If H is a subgroup of G, the set of left cosets G/H with the quotient topology is called a homogeneous space for G. The quotient map q: G → G / H {\displaystyle q:G\to G/H} is always open. For example, for a positive integer n, the sphere Sn is a homogeneous space for the rotation group SO(n+1) in R {\displaystyle \mat... | Closed subgroup | 0.83636 |
1,986 | S {\displaystyle S} is a complete uniform space (under the point-set topology definition of "complete uniform space") when S {\displaystyle S} is endowed with the uniformity induced on it by the canonical uniformity of X {\displaystyle X} ; A subset S {\displaystyle S} is called a sequentially complete subset if every ... | Closed subgroup | 0.83636 |
1,987 | In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other.Topological ... | Closed subgroup | 0.836359 |
1,988 | Examples include all finite-dimensional alternative algebras, and the algebra of real 2-by-2 matrices. Up to isomorphism the only alternative, quadratic real algebras without divisors of zero are the reals, complexes, quaternions, and octonions. The Cayley–Dickson algebras (where K is R), which begin with: C (a commuta... | Quadratic representation | 0.836343 |
1,989 | These include most of the algebras of interest to multilinear algebra, such as the tensor algebra, symmetric algebra, and exterior algebra over a given vector space. Graded algebras can be generalized to filtered algebras. | Quadratic representation | 0.836343 |
1,990 | Power-associative algebras, are those algebras satisfying the power-associative identity. Examples include all associative algebras, all alternative algebras, Jordan algebras over a field other than GF(2) (see previous section), and the sedenions. The hyperbolic quaternion algebra over R, which was an experimental alge... | Quadratic representation | 0.836343 |
1,991 | The most important examples of alternative algebras are the octonions (an algebra over the reals), and generalizations of the octonions over other fields. All associative algebras are alternative. Up to isomorphism, the only finite-dimensional real alternative, division algebras (see below) are the reals, complexes, qu... | Quadratic representation | 0.836343 |
1,992 | In contrast to the Lie algebra case, not every Jordan algebra can be constructed this way. Those that can are called special. Alternative algebras are algebras satisfying the alternative property. | Quadratic representation | 0.836343 |
1,993 | Every associative algebra gives rise to a Lie algebra by using the commutator as Lie bracket. In fact every Lie algebra can either be constructed this way, or is a subalgebra of a Lie algebra so constructed. Every associative algebra over a field of characteristic other than 2 gives rise to a Jordan algebra by defining... | Quadratic representation | 0.836343 |
1,994 | Euclidean space R3 with multiplication given by the vector cross product is an example of an algebra which is anticommutative and not associative. The cross product also satisfies the Jacobi identity. Lie algebras are algebras satisfying anticommutativity and the Jacobi identity. Algebras of vector fields on a differen... | Quadratic representation | 0.836343 |
1,995 | Power associative: the subalgebra generated by any element is associative, i.e., nth power associative for all n ≥ 2. nth power commutative with n ≥ 2: xn−kxk = xkxn−k for all integers k so that 0 < k < n. Third power commutative: x2x = xx2. Fourth power commutative: x3x = xx3 (compare with fourth power associative abo... | Quadratic representation | 0.836343 |
1,996 | Let x, y and z denote arbitrary elements of the algebra A over the field K. Let powers to positive (non-zero) integer be recursively defined by x1 ≝ x and either xn+1 ≝ xnx (right powers) or xn+1 ≝ xxn (left powers) depending on authors. Unital: there exist an element e so that ex = x = xe; in that case we can define x... | Quadratic representation | 0.836343 |
1,997 | They carry two multiplications, turning them into commutative algebras and Lie algebras in different ways. Genetic algebras are non-associative algebras used in mathematical genetics. Triple systems | Quadratic representation | 0.836343 |
1,998 | Hypercomplex algebras are all finite-dimensional unital R-algebras, they thus include Cayley-Dickson algebras and many more. The Poisson algebras are considered in geometric quantization. | Quadratic representation | 0.836343 |
1,999 | The quaternions and octonions are not commutative. Of these algebras, all are associative except for the octonions. Quadratic algebras, which require that xx = re + sx, for some elements r and s in the ground field, and e a unit for the algebra. | Quadratic representation | 0.836343 |
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