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13,700 | None of these elements has ever been collected in a macroscopic sample. Transactinide elements are all named after nuclear physicists and chemists or important locations involved in the synthesis of the elements. Chemistry Nobel Prize winner Glenn T. Seaborg, who first proposed the actinide concept which led to the acc... | Period 7 element | 0.815878 |
13,701 | (?) Prediction (*) Exception to the Madelung rule. In many periodic tables, the f-block is erroneously shifted one element to the right, so that lanthanum and actinium become d-block elements, and Ce–Lu and Th–Lr form the f-block tearing the d-block into two very uneven portions. This is a holdover from early erroneous... | Period 7 element | 0.815878 |
13,702 | The Numberphile YouTube channel was started on 15 September 2011. Most videos consist of Haran interviewing an expert on a number, mathematical theorem or other mathematical concept. The expert usually draws out their explanation on a large piece of brown paper and attempts to make the concepts understandable to the av... | Numberphile | 0.815866 |
13,703 | Numberphile is an educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its scope, featuring videos on more advanced mathematical concepts such as Fermat's La... | Numberphile | 0.815866 |
13,704 | A stochastic partial differential equation (SPDE) is an equation that generalizes SDEs to include space-time noise processes, with applications in quantum field theory and statistical mechanics. An ultrametric pseudo-differential equation is an equation which contains p-adic numbers in an ultrametric space. Mathematica... | Differential equations of mathematical physics | 0.815861 |
13,705 | A delay differential equation (DDE) is an equation for a function of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times. Integral equations may be viewed as the analog to differential equations where instead ... | Differential equations of mathematical physics | 0.815861 |
13,706 | The classification of parallelohedra into five types was first made by Russian crystallographer Evgraf Fedorov, as chapter 13 of a Russian-language book first published in 1885, whose title has been translated into English as An Introduction to the Theory of Figures. Some of the mathematics in this book is faulty; for ... | Parallelohedron | 0.815859 |
13,707 | In geometry, a parallelohedron is a polyhedron that can be translated without rotations in 3-dimensional Euclidean space to fill space with a honeycomb in which all copies of the polyhedron meet face-to-face. There are five types of parallelohedron, first identified by Evgraf Fedorov in 1885 in his studies of crystallo... | Parallelohedron | 0.815859 |
13,708 | These derived formulas are called theorems and may be interpreted to be true propositions. A constructed sequence of such formulas is known as a derivation or proof and the last formula of the sequence is the theorem. The derivation may be interpreted as proof of the proposition represented by the theorem. | Truth-functional propositional logic | 0.815859 |
13,709 | We now prove the same theorem A → A {\displaystyle A\to A} in the axiomatic system by Jan Łukasiewicz described above, which is an example of a Hilbert-style deductive system for the classical propositional calculus. The axioms are: (A1) ( p → ( q → p ) ) {\displaystyle (p\to (q\to p))} (A2) ( ( p → ( q → r ) ) → ( ( p... | Truth-functional propositional logic | 0.815859 |
13,710 | The propositional calculus then defines an argument to be a list of propositions. A valid argument is a list of propositions, the last of which follows from—or is implied by—the rest. All other arguments are invalid. The simplest valid argument is modus ponens, one instance of which is the following list of proposition... | Truth-functional propositional logic | 0.815859 |
13,711 | The roots of universal logic as general theory of logical systems, may go as far back as some work of Alfred Tarski in the early twentieth century, but the modern notion was first presented in the 1990s by Swiss logician Jean-Yves Béziau. The term 'universal logic' has also been separately used by logicians such as Ric... | Universal logic | 0.815851 |
13,712 | Originally the expression Universal logic was coined by analogy with the expression Universal algebra. The first idea was to develop Universal logic as a field of logic that studies the features common to all logical systems, aiming to be to logic what Universal algebra is to algebra. A number of approaches to universa... | Universal logic | 0.815851 |
13,713 | In a crystal structure the coordination geometry of an atom is the geometrical pattern of coordinating atoms where the definition of coordinating atoms depends on the bonding model used. For example, in the rock salt ionic structure each sodium atom has six near neighbour chloride ions in an octahedral geometry and eac... | Coordination geometry | 0.815851 |
13,714 | IUPAC have introduced the polyhedral symbol as part of their IUPAC nomenclature of inorganic chemistry 2005 recommendations to describe the geometry around an atom in a compound. IUCr have proposed a symbol which is shown as a superscript in square brackets in the chemical formula. For example, CaF2 would be CaF2, wher... | Coordination geometry | 0.815851 |
13,715 | The geometrical pattern can be described as a polyhedron where the vertices of the polyhedron are the centres of the coordinating atoms in the ligands.The coordination preference of a metal often varies with its oxidation state. The number of coordination bonds (coordination number) can vary from two in K as high as 20... | Coordination geometry | 0.815851 |
13,716 | The coordination geometry of an atom is the geometrical pattern defined by the atoms around the central atom. The term is commonly applied in the field of inorganic chemistry, where diverse structures are observed. The coodination geometry depends on the number, not the type, of ligands bonded to the metal centre as we... | Coordination geometry | 0.815851 |
13,717 | The polar radius has now been measured at just over 6357 km. This was an error only 0.44% less than the modern value. This was another example of advances in astronomy and its tools making possible advances in cartography. | Picard's arc measurement | 0.815849 |
13,718 | The energy of a metallic bond is thus mostly a function of the number of electrons which surround the metallic atom, as exemplified by the embedded atom model. This typically results in metals assuming relatively simple, close-packed crystal structures, such as FCC, BCC, and HCP. Given high enough cooling rates and app... | Sea of electrons | 0.815846 |
13,719 | If an element partakes, its peaks tend to be skewed. Some intermetallic materials, e.g., do exhibit metal clusters reminiscent of molecules; and these compounds are more a topic of chemistry than of metallurgy. The formation of the clusters could be seen as a way to 'condense out' (localize) the electron-deficient bond... | Sea of electrons | 0.815846 |
13,720 | Being a function of random variables, the sample variance is itself a random variable, and it is natural to study its distribution. In the case that Yi are independent observations from a normal distribution, Cochran's theorem shows that S2 follows a scaled chi-squared distribution (see also: asymptotic properties and ... | Sample variance | 0.815842 |
13,721 | To see how, consider that a theoretical probability distribution can be used as a generator of hypothetical observations. If an infinite number of observations are generated using a distribution, then the sample variance calculated from that infinite set will match the value calculated using the distribution's equation... | Sample variance | 0.815842 |
13,722 | In probability theory and statistics, variance is the squared deviation from the mean of a random variable. The variance is also often defined as the square of the standard deviation. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is t... | Sample variance | 0.815842 |
13,723 | Equilibrium segregation is associated with the lattice disorder at interfaces, where there are sites of energy different from those within the lattice at which the solute atoms can deposit themselves. The equilibrium segregation is so termed because the solute atoms segregate themselves to the interface or surface in a... | Segregation (materials science) | 0.815841 |
13,724 | Because of the favorable segregation of some impurities to the surface of the material, a very small concentration of impurity in the bulk of the sample can lead to a very significant coverage of the impurity on a cleaved surface of the sample. In applications where an ultra-pure surface is needed (for example, in some... | Segregation (materials science) | 0.815841 |
13,725 | : 4 The physics concept of force makes quantitative the everyday idea of a push or a pull. Forces in Newtonian mechanics are often due to strings and ropes, friction, muscle effort, gravity, and so forth. Like displacement, velocity, and acceleration, force is a vector quantity. | Newton's laws of motion | 0.815839 |
13,726 | Numerically, a vector can be represented as a list; for example, a body's velocity vector might be v → = ( 3 m / s , 4 m / s ) {\displaystyle {\vec {v}}=(\mathrm {3~m/s} ,\mathrm {4~m/s} )} , indicating that it is moving at 3 metres per second along a horizontal axis and 4 metres per second along the vertical axis. The... | Newton's laws of motion | 0.815839 |
13,727 | Position, when thought of as a displacement from an origin point, is a vector: a quantity with both magnitude and direction. : 1 Velocity and acceleration are vector quantities as well. The mathematical tools of vector algebra provide the means to describe motion in two, three or more dimensions. | Newton's laws of motion | 0.815839 |
13,728 | If the body's location as a function of time is s ( t ) {\displaystyle s(t)} , then its average velocity over the time interval from t 0 {\displaystyle t_{0}} to t 1 {\displaystyle t_{1}} is Here, the Greek letter Δ {\displaystyle \Delta } (delta) is used, per tradition, to mean "change in". A positive average velocity... | Newton's laws of motion | 0.815839 |
13,729 | A laboratory technician is a person who works in a laboratory performing analytical or experimental procedures, maintaining laboratory equipment. According to the Oxford English Dictionary, the first use of the term laboratory technician was in 1896:It is now totally abandoned by the advanced laboratory technicians.Th... | Laboratory technician | 0.815838 |
13,730 | arXiv Bibliography Network Project CiteSeerX DBLP LEABib Networked Computer Science Technical Reference Library | Collection of Computer Science Bibliographies | 0.815838 |
13,731 | As the Collection of Computer Science Bibliographies consists of many subcollections there is a substantial overlap (roughly 1/3). At the end of 2008 there were more than 4.2 million records which represent about 2.8 million unique (in terms of normalized title and authors' last names) bibliographic entries. The number... | Collection of Computer Science Bibliographies | 0.815838 |
13,732 | The Collection of Computer Science Bibliographies (1993–2023) was one of the oldest (if not the oldest) bibliography collections freely accessible on the Internet. As of July 2023 it ceased operations. It is a collection of bibliographies of scientific literature in computer science and (computational) mathematics from... | Collection of Computer Science Bibliographies | 0.815838 |
13,733 | The standard softmax function is often used in the final layer of a neural network-based classifier. Such networks are commonly trained under a log loss (or cross-entropy) regime, giving a non-linear variant of multinomial logistic regression. Since the function maps a vector and a specific index i {\displaystyle i} to... | Softmax activation function | 0.815836 |
13,734 | The softmax function, also known as softargmax: 184 or normalized exponential function,: 198 converts a vector of K real numbers into a probability distribution of K possible outcomes. It is a generalization of the logistic function to multiple dimensions, and used in multinomial logistic regression. The softmax functi... | Softmax activation function | 0.815836 |
13,735 | The softmax function is used in various multiclass classification methods, such as multinomial logistic regression (also known as softmax regression): 206–209 , multiclass linear discriminant analysis, naive Bayes classifiers, and artificial neural networks. Specifically, in multinomial logistic regression and linear d... | Softmax activation function | 0.815836 |
13,736 | In neural network applications, the number K of possible outcomes is often large, e.g. in case of neural language models that predict the most likely outcome out of a vocabulary which might contain millions of possible words. This can make the calculations for the softmax layer (i.e. the matrix multiplications to deter... | Softmax activation function | 0.815836 |
13,737 | In the case of dimension one, varieties are classified by only the topological genus, but, in dimension two, one needs to distinguish the arithmetic genus p a {\displaystyle p_{a}} and the geometric genus p g {\displaystyle p_{g}} because one cannot distinguish birationally only the topological genus. Then, irregularit... | Algebraic surface | 0.815835 |
13,738 | The birational geometry of algebraic surfaces is rich, because of blowing up (also known as a monoidal transformation), under which a point is replaced by the curve of all limiting tangent directions coming into it (a projective line). Certain curves may also be blown down, but there is a restriction (self-intersection... | Algebraic surface | 0.815835 |
13,739 | There are essential three Hodge number invariants of a surface. Of those, h1,0 was classically called the irregularity and denoted by q; and h2,0 was called the geometric genus pg. The third, h1,1, is not a birational invariant, because blowing up can add whole curves, with classes in H1,1. It is known that Hodge cycle... | Algebraic surface | 0.815835 |
13,740 | The Hodge index theorem is used in Deligne's proof of the Weil conjecture. Basic results on algebraic surfaces include the Hodge index theorem, and the division into five groups of birational equivalence classes called the classification of algebraic surfaces. The general type class, of Kodaira dimension 2, is very lar... | Algebraic surface | 0.815835 |
13,741 | For an ample line bundle H on S, the definition { H } ⊥ := { D ∈ N u m ( S ) | D ⋅ H = 0 } . {\displaystyle \{H\}^{\perp }:=\{D\in Num(S)|D\cdot H=0\}.} is used in the surface version of the Hodge index theorem: for D ∈ { { H } ⊥ | D ≠ 0 } , D ⋅ D < 0 {\displaystyle D\in \{\{H\}^{\perp }|D\neq 0\},D\cdot D<0} , i.e. th... | Algebraic surface | 0.815835 |
13,742 | The Nakai criterion says that: A Divisor D on a surface S is ample if and only if D2 > 0 and for all irreducible curve C on S D•C > 0.Ample divisors have a nice property such as it is the pullback of some hyperplane bundle of projective space, whose properties are very well known. Let D ( S ) {\displaystyle {\mathcal {... | Algebraic surface | 0.815835 |
13,743 | One of the fundamental theorems for the birational geometry of surfaces is Castelnuovo's theorem. This states that any birational map between algebraic surfaces is given by a finite sequence of blowups and blowdowns. | Algebraic surface | 0.815835 |
13,744 | In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex manifold, when it is non-singular) and so of dimension four as a smooth manifold. The theory of algebraic surfaces is muc... | Algebraic surface | 0.815835 |
13,745 | In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient map V → Q {\displaystyle V\to Q} , the difference dim V − dim Q is the relative dimension; this equals the dimension of the kernel. In fiber bundles, the relative dimension... | Relative dimension | 0.815834 |
13,746 | The T1 method. is an efficient computational approach developed for calculating accurate heats of formation of uncharged, closed-shell molecules comprising H, C, N, O, F, Si, P, S, Cl and Br, within experimental error. It is practical for molecules up to molecular weight ~ 500 a.m.u. T1 method as incorporated in Sparta... | Quantum chemistry composite methods | 0.815832 |
13,747 | In practice, for systems consisting of more than ~9 non-hydrogen atoms (with C1 symmetry), even the computationally more economical W1 theory becomes prohibitively expensive with current mainstream server hardware. In an attempt to extend the applicability of the Wn ab initio thermochemistry methods, explicitly correla... | Quantum chemistry composite methods | 0.815832 |
13,748 | RI-MP2/6-311+G(2d,p) single point energy with dual basis set. An empirical correction using atom counts, Mulliken bond orders, HF/6-31G* and RI-MP2 energies as variables.T1 follows the G3(MP2) recipe, however, by substituting an HF/6-31G* for the MP2/6-31G* geometry, eliminating both the HF/6-31G* frequency and QCISD(T... | Quantum chemistry composite methods | 0.815832 |
13,749 | This is the G2MP2 method. Sometimes the geometry is obtained using a density functional theory method such as B3LYP and sometimes the QCISD(T) method in step 2 is replaced by the coupled cluster method CCSD(T). The G2(+) variant, where the "+" symbol refers to added diffuse functions, better describes anions than conve... | Quantum chemistry composite methods | 0.815832 |
13,750 | A Hartree–Fock geometry optimization with the 6-31G(d) basis set used to give a geometry for: A frequency calculation with the 6-31G(d) basis set to obtain the zero-point vibrational energy (ZPVE)The various energy changes are assumed to be additive so the combined energy is given by: EQCISD(T) from 2 + + + The secon... | Quantum chemistry composite methods | 0.815832 |
13,751 | The G2 uses seven calculations: the molecular geometry is obtained by a MP2 optimization using the 6-31G(d) basis set and all electrons included in the perturbation. This geometry is used for all subsequent calculations. The highest level of theory is a quadratic configuration interaction calculation with single and do... | Quantum chemistry composite methods | 0.815832 |
13,752 | In comparison, the Gaussian-n methods perform their approximation using additive corrections. Similar to the modified G2(+) method, CBS-QB3 has been modified by the inclusion of diffuse functions in the geometry optimization step to give CBS-QB3(+). The CBS family of methods is available via keywords in the Gaussian 09... | Quantum chemistry composite methods | 0.815832 |
13,753 | Pascal's theorem (also known as the Hexagrammum Mysticum Theorem) states that if an arbitrary six points are chosen on a conic section (i.e., ellipse, parabola or hyperbola) and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon (extended if necessary) meet in ... | Collinearity (geometry) | 0.81583 |
13,754 | Pantelides algorithm in mathematics is a systematic method for reducing high-index systems of differential-algebraic equations to lower index. This is accomplished by selectively adding differentiated forms of the equations already present in the system. It is possible for the algorithm to fail in some instances. Pante... | Pantelides algorithm | 0.815827 |
13,755 | Modeling is an integral part of many technical fields, including engineering, economics, and software engineering. In this context, a model is a formal representation of an organizational system, such as a business model or a formal description of software in UML. | SEQUAL framework | 0.815826 |
13,756 | Risk assessments can be done in individual cases, including in patient and physician interactions. In the narrow sense chemical risk assessment is the assessment of a health risk in response to environmental exposures. The ways statistics are expressed and communicated to an individual, both through words and numbers i... | Risk assessments | 0.815824 |
13,757 | The Staudinger ligation of azides and phosphine has been used extensively in field of chemical biology. Because it is able to form a stable amide bond in living cells and animals, it has been applied to modification of cell membrane, in vivo imaging, and other bioconjugation studies. Contrasting with the classic Staudi... | Bioconjugation | 0.815823 |
13,758 | One strategy is to first install a unique functional group onto a protein, and then a bioorthogonal reaction is used to couple a biomolecule with this unique functional group. The bioorthogonal reactions targeting non-native functional groups are widely used in bioconjugation chemistry. Some important reactions are mod... | Bioconjugation | 0.815823 |
13,759 | Recent advances in the understanding of biomolecules enabled their application to numerous fields like medicine and materials. Synthetically modified biomolecules can have diverse functionalities, such as tracking cellular events, revealing enzyme function, determining protein biodistribution, imaging specific biomarke... | Bioconjugation | 0.815823 |
13,760 | Mean-field game theory is the study of strategic decision making by small interacting agents in very large populations. It lies at the intersection of game theory with stochastic analysis and control theory. The use of the term "mean field" is inspired by mean-field theory in physics, which considers the behavior of sy... | Mean-field game theory | 0.815819 |
13,761 | When the special case of the electromagnetic self-energy or self-force of charged particles is discussed, also in modern texts some sort of "effective" electromagnetic mass is sometimes introduced – not as an explanation of mass per se, but in addition to the ordinary mass of bodies. Many different reformulations of th... | Electromagnetic mass | 0.815808 |
13,762 | (Poincaré added that one should not be too surprised by these assumptions, since they are only mathematical fictions.) In this way, the motion of the COM-frame, including matter, fictitious em-fluid, and fictitious non-em-fluid, at least theoretically remains uniform. However, since only matter and electromagnetic ener... | Electromagnetic mass | 0.815808 |
13,763 | The concepts of longitudinal and transverse mass (equivalent to those of Lorentz) were also used by Einstein in his first papers on relativity. However, in special relativity they apply to the entire mass of matter, not only to the electromagnetic part. Later it was shown by physicists like Richard Chace Tolman that ex... | Electromagnetic mass | 0.815808 |
13,764 | Unsupervised learning is a paradigm in machine learning where, in contrast to supervised learning and semi-supervised learning, algorithms learn patterns exclusively from unlabeled data. | Unsupervised classification | 0.815805 |
13,765 | For instance, neurons change between deterministic (Hopfield) and stochastic (Boltzmann) to allow robust output, weights are removed within a layer (RBM) to hasten learning, or connections are allowed to become asymmetric (Helmholtz). Of the networks bearing people's names, only Hopfield worked directly with neural net... | Unsupervised classification | 0.815805 |
13,766 | Latent variable models are statistical models where in addition to the observed variables, a set of latent variables also exists which is not observed. A highly practical example of latent variable models in machine learning is the topic modeling which is a statistical model for generating the words (observed variables... | Unsupervised classification | 0.815805 |
13,767 | The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal... | Mathematical optimization | 0.815802 |
13,768 | Quasi-Newton methods: Iterative methods for medium-large problems (e.g. N<1000). Simultaneous perturbation stochastic approximation (SPSA) method for stochastic optimization; uses random (efficient) gradient approximation. | Mathematical optimization | 0.815802 |
13,769 | Bundle method of descent: An iterative method for small–medium-sized problems with locally Lipschitz functions, particularly for convex minimization problems (similar to conjugate gradient methods). Ellipsoid method: An iterative method for small problems with quasiconvex objective functions and of great theoretical in... | Mathematical optimization | 0.815802 |
13,770 | Interior point methods: This is a large class of methods for constrained optimization, some of which use only (sub)gradient information and others of which require the evaluation of Hessians. Methods that evaluate gradients, or approximate gradients in some way (or even subgradients): Coordinate descent methods: Algori... | Mathematical optimization | 0.815802 |
13,771 | The extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view. | Mathematical optimization | 0.815802 |
13,772 | Some common applications of optimization techniques in electrical engineering include active filter design, stray field reduction in superconducting magnetic energy storage systems, space mapping design of microwave structures, handset antennas, electromagnetics-based design. Electromagnetically validated design optimi... | Mathematical optimization | 0.815802 |
13,773 | Optimization problems are often expressed with special notation. Here are some examples: | Mathematical optimization | 0.815802 |
13,774 | Nonlinear optimization methods are widely used in conformational analysis. | Mathematical optimization | 0.815802 |
13,775 | A ruler or rule is a tool used in, for example, geometry, technical drawing, engineering, and carpentry, to measure lengths or distances or to draw straight lines. Strictly speaking, the ruler is the instrument used to rule straight lines and the calibrated instrument used for determining length is called a measure, ho... | Numerical measure | 0.8158 |
13,776 | Since accurate measurement is essential in many fields, and since all measurements are necessarily approximations, a great deal of effort must be taken to make measurements as accurate as possible. For example, consider the problem of measuring the time it takes an object to fall a distance of one metre (about 39 in). ... | Numerical measure | 0.8158 |
13,777 | Mathematical Programming is a peer-reviewed scientific journal that was established in 1971 and is published by Springer Science+Business Media. It is the official journal of the Mathematical Optimization Society and consists of two series: A and B. The "A" series contains general publications, the "B" series focuses o... | Mathematical Programming | 0.815799 |
13,778 | The case that a polyphase matrix allows reconstruction of a processed signal from the filtered data, is called perfect reconstruction property. Mathematically this is equivalent to invertibility. According to the theorem of invertibility of a matrix over a ring, the polyphase matrix is invertible if and only if the det... | Polyphase matrix | 0.815799 |
13,779 | The Denjoy–Riesz theorem gives general conditions under which a point set can be covered by the homeomorphic image of a curve. This is true, in particular, for every compact totally disconnected subset of the Euclidean plane. However, it may be necessary for such an arc to have infinite length, failing to meet the cond... | Analyst's Traveling Salesman Theorem | 0.815798 |
13,780 | The analyst's traveling salesman problem is an analog of the traveling salesman problem in combinatorial optimization. In its simplest and original form, it asks which plane sets are subsets of rectifiable curves of finite length. Whereas the original traveling salesman problem asks for the shortest way to visit every ... | Analyst's Traveling Salesman Theorem | 0.815798 |
13,781 | The Traveling Salesman Theorem was shown to hold in general Euclidean spaces by Kate Okikiolu, that is, the same theorem above holds for sets E ⊆ R d {\displaystyle E\subseteq \mathbb {R} ^{d}} , d > 1, where Δ is now the collection of dyadic cubes in R d {\displaystyle \mathbb {R} ^{d}} defined in a similar way as dya... | Analyst's Traveling Salesman Theorem | 0.815798 |
13,782 | Let Δ denote the collection of dyadic squares, that is, Δ = { × : i , j , k ∈ Z } , {\displaystyle \Delta =\{\times :i,j,k\in \mathbb {Z} \},} where Z {\displaystyle \mathbb {Z} } denotes the set of integers. For a set E ⊆ R 2 {\displaystyle E\subseteq \mathbb {R} ^{2}} , define β ( E ) = diam E + ∑ Q ∈ Δ β E ( 3 Q ) ... | Analyst's Traveling Salesman Theorem | 0.815798 |
13,783 | In computational complexity theory, a branch of computer science, Schaefer's dichotomy theorem, proved by Thomas Jerome Schaefer, states necessary and sufficient conditions under which a finite set S of relations over the Boolean domain yields polynomial-time or NP-complete problems when the relations of S are used to ... | Schaefer's dichotomy theorem | 0.815794 |
13,784 | An example of a Mal'tsev operation is the Minority operation given in the modern, algebraic formulation of Schaefer's dichotomy theorem above. Thus, when Γ has the Minority operation as a polymorphism, it is not only possible to decide CSP(Γ) in polynomial time, but to compute #CSP(Γ) in polynomial time. There are a to... | Schaefer's dichotomy theorem | 0.815794 |
13,785 | For Schaefer's dichotomy theorem, the most important concept in universal algebra is that of a polymorphism. An operation f: D m → D {\displaystyle f:D^{m}\to D} is a polymorphism of a relation R ⊆ D k {\displaystyle R\subseteq D^{k}} if, for any choice of m tuples ( t 11 , … , t 1 k ) , … , ( t m 1 , … , t m k ) {\dis... | Schaefer's dichotomy theorem | 0.815794 |
13,786 | A modern, streamlined presentation of Schaefer's theorem is given in an expository paper by Hubie Chen. In modern terms, the problem SAT(S) is viewed as a constraint satisfaction problem over the Boolean domain. In this area, it is standard to denote the set of relations by Γ and the decision problem defined by Γ as CS... | Schaefer's dichotomy theorem | 0.815794 |
13,787 | The analysis was later fine-tuned: CSP(Γ) is either solvable in co-NLOGTIME, L-complete, NL-complete, ⊕L-complete, P-complete or NP-complete and given Γ, one can decide in polynomial time which of these cases holds.Schaefer's dichotomy theorem was recently generalized to a larger class of relations. | Schaefer's dichotomy theorem | 0.815794 |
13,788 | {\displaystyle {\tbinom {n}{k}}.} Like the binomial distribution that involves binomial coefficients, there is a negative binomial distribution in which the multiset coefficients occur. Multiset coefficients should not be confused with the unrelated multinomial coefficients that occur in the multinomial theorem. | Multiset coefficient | 0.815794 |
13,789 | Thus the multiset of solutions of the equation could be {3, 5}, or it could be {4, 4}. In the latter case it has a solution of multiplicity 2. More generally, the fundamental theorem of algebra asserts that the complex solutions of a polynomial equation of degree d always form a multiset of cardinality d. A special cas... | Multiset coefficient | 0.815794 |
13,790 | One of the simplest and most natural examples is the multiset of prime factors of a natural number n. Here the underlying set of elements is the set of prime factors of n. For example, the number 120 has the prime factorization which gives the multiset {2, 2, 2, 3, 5}. A related example is the multiset of solutions of ... | Multiset coefficient | 0.815794 |
13,791 | We can use Taylor's theorem to rewrite the second factor for small h {\displaystyle h} (subscripts of u denote partial derivatives): ( x j + 1 + x j − 1 − 2 x j h 2 ) = u ( x + h , t ) + u ( x − h , t ) − 2 u ( x , t ) h 2 = u x x ( x , t ) + ( h 2 12 ) u x x x x ( x , t ) + O ( h 4 ) . {\displaystyle {\begin{aligned}\... | FPUT problem | 0.815789 |
13,792 | Although the recurrence is easily observed, it eventually became apparent that over much, much longer time periods, the system does eventually thermalize. Multiple competing theories have been proposed to explain the behavior of the system, and it remains a topic of active research. The original intent was to find a ph... | FPUT problem | 0.815789 |
13,793 | In physics, the Fermi–Pasta–Ulam–Tsingou (FPUT) problem or formerly the Fermi–Pasta–Ulam problem was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactly periodic behavior – called Fermi–Pasta–Ulam–Tsingou recurrence (or Fermi–Pasta–Ulam recurrence) – instead of th... | FPUT problem | 0.815789 |
13,794 | : {\displaystyle Y(J_{n_{1}+1}^{a_{1}}J_{n_{2}+1}^{a_{2}}...J_{n_{k}+1}^{a_{k}}1,z)=:{\frac {\partial ^{n_{1}}}{\partial z^{n_{1}}}}{\frac {J^{a_{1}}(z)}{n_{1}! }}{\frac {\partial ^{n_{2}}}{\partial z^{n_{2}}}}{\frac {J^{a_{2}}(z)}{n_{2}! }}\cdots {\frac {\partial ^{n_{k}}}{\partial z^{n_{k}}}}{\frac {J^{a_{k}}(z)}{n_{... | Vertex algebra | 0.815788 |
13,795 | Reconstruction: Let V {\displaystyle V} be a vertex algebra, and let J a {\displaystyle J_{a}} be a set of vectors, with corresponding fields J a ( z ) ∈ E n d ( V ) ] {\displaystyle J^{a}(z)\in \mathrm {End} (V)]} . If V {\displaystyle V} is spanned by monomials in the positive weight coefficients of the fields (i.e.... | Vertex algebra | 0.815788 |
13,796 | Conversely, any commutative ring V {\displaystyle V} with derivation T {\displaystyle T} has a canonical vertex algebra structure, where we set Y ( u , z ) v = u − 1 v z 0 = u v {\displaystyle Y(u,z)v=u_{-1}vz^{0}=uv} , so that Y {\displaystyle Y} restricts to a map Y: V → End ( V ) {\displaystyle Y:V\rightarrow \ope... | Vertex algebra | 0.815788 |
13,797 | A vertex algebra V {\displaystyle V} is commutative if all vertex operators Y ( u , z ) {\displaystyle Y(u,z)} commute with each other. This is equivalent to the property that all products Y ( u , z ) v {\displaystyle Y(u,z)v} lie in V ] {\displaystyle V]} , or that Y ( u , z ) ∈ End ] {\displaystyle Y(u,z)\in \ope... | Vertex algebra | 0.815788 |
13,798 | }},...\right):} if we understand that each term in the expansion of f is normal ordered. The rank n free boson is given by taking an n-fold tensor product of the rank 1 free boson. For any vector b in n-dimensional space, one has a field b(z) whose coefficients are elements of the rank n Heisenberg algebra, whose commu... | Vertex algebra | 0.815788 |
13,799 | The Fock space V0 can be made into a vertex algebra by the following definition of the state-operator map on a basis b j 1 b j 2 . . . | Vertex algebra | 0.815788 |
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