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8,300 | According to classical mechanics, a diatomic molecule like nitrogen should have more degrees of internal freedom, corresponding to vibration of the two atoms that stretch and compress the bond between them. For thermodynamic purposes, each direction in which an atom can independently vibrate relative to the rest of the... | Molar heat capacity | 0.821543 |
8,301 | Thus, in order to achieve the same increase in temperature, more heat energy will have to be provided to a mol of that substance than to a mol of a monatomic gas. Substances with high atomic count per molecule, like octane, can therefore have a very large heat capacity per mole, and yet a relatively small specific heat... | Molar heat capacity | 0.821543 |
8,302 | In the classical view, each of these modes should store an equal share of the heat energy. However, according to quantum mechanics, the energy difference between the allowed (quantized) rotation states is inversely proportional to the moment of inertia about the corresponding axis of rotation. Because the moment of ine... | Molar heat capacity | 0.821543 |
8,303 | Quantum mechanics also explains why the specific heat of monatomic gases is well predicted by the ideal gas theory with the assumption that each molecule is a point mass that has only the f = 3 translational degrees of freedom. According to classical mechanics, since atoms have non-zero size, they should also have thre... | Molar heat capacity | 0.821543 |
8,304 | 454 Life Sciences was a biotechnology company based in Branford, Connecticut that specialized in high-throughput DNA sequencing. It was acquired by Roche in 2007 and shut down by Roche in 2013 when its technology became noncompetitive, although production continued until mid-2016. | 454 Life Sciences | 0.821521 |
8,305 | Mathematics and the Search for Knowledge is a book by Morris Kline on the developing mathematics ideas, which are partially overlap with his previous book Mathematics: The Loss of Certainty, as a source of human knowledge about the physical world, starting from astronomical theories of Ancient Greek to the modern theor... | Mathematics and the Search for Knowledge | 0.82152 |
8,306 | Some relational database management systems (RDBMS) can enforce referential integrity, normally either by deleting the foreign key rows as well to maintain integrity, or by returning an error and not performing the delete. Which method is used may be determined by a referential integrity constraint defined in a data di... | Declarative Referential Integrity | 0.821519 |
8,307 | An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). This σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets. On the Euclidean space R... | Sigma algebra | 0.821517 |
8,308 | Suppose ( Ω , Σ , P ) {\displaystyle (\Omega ,\Sigma ,\mathbb {P} )} is a probability space and R T {\displaystyle \mathbb {R} ^{\mathbb {T} }} is the set of real-valued functions on T . {\displaystyle \mathbb {T} .} If Y: Ω → X ⊆ R T {\displaystyle \textstyle Y:\Omega \to X\subseteq \mathbb {R} ^{\mathbb {T} }} is mea... | Sigma algebra | 0.821517 |
8,309 | Let ( X 1 , Σ 1 ) {\displaystyle \left(X_{1},\Sigma _{1}\right)} and ( X 2 , Σ 2 ) {\displaystyle \left(X_{2},\Sigma _{2}\right)} be two measurable spaces. The σ-algebra for the corresponding product space X 1 × X 2 {\displaystyle X_{1}\times X_{2}} is called the product σ-algebra and is defined by Observe that { B 1 ×... | Sigma algebra | 0.821517 |
8,310 | The Sylvester–Gallai theorem, in the form that if a finite set of points in the plane has the property that every line through two of the points contains a third point from the set, then the given points must all lie on a single line. Tarski's plank problem, in the form that whenever two infinite strips together cover ... | Combinatorial Geometry in the Plane | 0.821515 |
8,311 | Paradoxes of set decomposition into smaller sets, related to the Banach–Tarski paradox. Radon's theorem that every four points in the plane can be partitioned into two subsets with intersecting convex hulls. Sperner's lemma on colorings of triangulations. | Combinatorial Geometry in the Plane | 0.821515 |
8,312 | Jung's theorem, that (for sets in the plane) the radius of the smallest enclosing circle is at most 1 / 3 {\displaystyle 1/{\sqrt {3}}} times the diameter. This bound is tight for the equilateral triangle. | Combinatorial Geometry in the Plane | 0.821515 |
8,313 | A Helly-like property of visibility related to the art gallery theorem: if every three points of a polygon are visible from some common point within the polygon, then there is a point from which the entire polygon is visible. In this case the polygon must be a star-shaped polygon. The impossibility of covering a closed... | Combinatorial Geometry in the Plane | 0.821515 |
8,314 | The first half of the book provides the statements of nearly 100 propositions in the discrete geometry of the Euclidean plane, and the second half sketches their proofs. Klee's added chapter, lying between the two halves, provides another 10 propositions, including some generalizations to higher dimensions, and the boo... | Combinatorial Geometry in the Plane | 0.821515 |
8,315 | Combinatorial Geometry in the Plane is a book in discrete geometry. It was translated from a German-language book, Kombinatorische Geometrie in der Ebene, which its authors Hugo Hadwiger and Hans Debrunner published through the University of Geneva in 1960, expanding a 1955 survey paper that Hadwiger had published in L... | Combinatorial Geometry in the Plane | 0.821515 |
8,316 | The book is written at a level appropriate for undergraduate students in mathematics, and assumes a background knowledge in real analysis and undergraduate-level geometry. One goal of the book is to expose students at this level to research-level problems in mathematics whose statement is readily accessible. == Referen... | Combinatorial Geometry in the Plane | 0.821515 |
8,317 | The journal was founded in 1979 as a supplement to the first journal in the field of polymer science, the Journal für Makromolekulare Chemie (Journal for Macromolecular Chemistry) as a forum for the rapid publication of the newest and most exciting developments in the field of polymer science. According to the Journal ... | Macromolecular Rapid Communications | 0.821514 |
8,318 | Macromolecular Rapid Communications is a biweekly peer-reviewed scientific journal covering polymer science. It publishes Communications, Feature Articles and Reviews on general polymer science, from chemistry and physics of polymers to polymers in materials science and life sciences. | Macromolecular Rapid Communications | 0.821514 |
8,319 | Z represents: the unit prefix zetta (1021) the atomic number or proton number of an element in chemistry a standardized normal random variable in probability theory and statistics Partition function in meteorology, the radar reflectivity factor Electrical impedance Z boson Compressibility factor Z {\displaystyle \mathb... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,320 | F represents the digit "15" in hexadecimal and other positional numeral systems with a radix of 16 or greater the unit farad of electrical capacity the Helmholtz free energy of a closed thermodynamic system of constant pressure and temperature together with a degree symbol (°) represents the Fahrenheit measurement of t... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,321 | D represents the digit "13" in hexadecimal and other positional numeral systems with a radix of 14 or greater diffusion coefficient or diffusivity in dimensions of the differential operator in Euler's calculus notation dissociation energy Dimension Deuterium Electric displacement D meson Density d represents the diffe... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,322 | C represents: the third point of a triangle the digit "12" in hexadecimal and other positional numeral systems with a radix of 13 or greater the unit coulomb of electrical charge capacitance in electrical theory with indices denoting the number of combinations, a binomial coefficient together with a degree symbol (°), ... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,323 | B represents: the digit "11" in hexadecimal and other positional numeral systems with a radix of 12 or greater the second point of a triangle a ball (also denoted by ℬ ( B {\displaystyle {\mathcal {B}}} ) or B {\displaystyle \mathbb {B} } ) a basis of a vector space or of a filter (both also denoted by ℬ ( B {\displays... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,324 | T represents: the top element of a lattice a tree (a special kind of graph) temperature in physics equations: 4 the unit tesla of magnetic flux density the unit prefix tera (1012) the stress–energy tensor tension in physics an arbitrary monad the time it takes for one oscillation kinetic energy Torque A spectral type T... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,325 | P represents: the pressure in physics equations: 4 the unit prefix peta (1015) probability in statistics and statistical mechanics: 35 an arbitrary point in geometry power, measured in watts Active power in electrical engineering weight measured in newtons Legendre polynomial Phosphorus Polarization P {\displaystyle \m... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,326 | N represents the unit newton of force the nine-point center of a triangle Bessel function of the second kind (uncommon) Nitrogen Normal distribution Normal vector N represents the neutron number The number of particles of a thermodynamical system: 57 NA represents the Avogadro constant N {\displaystyle \mathbb {N} } re... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,327 | K represents: the temperature unit kelvin the functors of K-theory an unspecified (real) constant a field in algebra the area of a polygon kinetic energy Kaon Potassium Sectional curvature A spectral type k represents the unit prefix kilo- (103) the Boltzmann constant, often represented as kB to avoid confusion the ang... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,328 | J represents: the unit joule of energy the current density in electromagnetism denoted J {\displaystyle {\boldsymbol {J}}} the radiosity in thermal mechanics the moment of inertia Total angular momentum quantum number Bessel function of the first kind Impulse J represents: the scheme of a diagram in category theory j r... | Latin letters used in mathematics, science, and engineering | 0.821498 |
8,329 | Examples of super-recursive algorithms include (Burgin 2005: 132): limiting recursive functions and limiting partial recursive functions (E.M. Gold 1965) trial and error predicates (Hilary Putnam 1965) inductive inference machines (Carl Smith) inductive Turing machines, which perform computations similar to computation... | Super-recursive algorithm | 0.821496 |
8,330 | Simple inductive Turing machines and general Turing machines are related to limiting partial recursive functions and trial-and-error predicates as Turing machines are related to partial recursive functions and lambda calculus. The non-halting computations of inductive Turing machines should not be confused with infinit... | Super-recursive algorithm | 0.821496 |
8,331 | Cryogenics The branches of engineering that involve the study of very low temperatures (ultra low temperature i.e. below 123 K), how to produce them, and how materials behave at those temperatures.Cryobiology The branch of biology involving the study of the effects of low temperatures on organisms (most often for the p... | Low-Temperature Phenomena | 0.821492 |
8,332 | In physics, cryogenics is the production and behaviour of materials at very low temperatures. The 13th IIR International Congress of Refrigeration (held in Washington DC in 1971) endorsed a universal definition of "cryogenics" and "cryogenic" by accepting a threshold of 120 K (or –153 °C) to distinguish these terms fro... | Low-Temperature Phenomena | 0.821492 |
8,333 | In particle, atomic and condensed matter physics, a Yukawa potential (also called a screened Coulomb potential) is a potential named after the Japanese physicist Hideki Yukawa. The potential is of the form: V Yukawa ( r ) = − g 2 e − α m r r , {\displaystyle V_{\text{Yukawa}}(r)=-g^{2}{\frac {e^{-\alpha mr}}{r}},} wher... | Yukawa potential | 0.821489 |
8,334 | Olbers's paradox, also known as the dark night sky paradox, is an argument in astrophysics and physical cosmology that says that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. In the hypothetical case that the universe is static, homogeneous at a large scale, and... | Dark night sky paradox | 0.821487 |
8,335 | List of algebras List of axioms List of equations List of mathematical functions List of types of functions List of mathematical jargon List of mathematical abbreviations List of mathematical proofs List of long mathematical proofs List of mathematical symbols List of mathematical symbols by subject List of rules of in... | Outline of mathematics | 0.821486 |
8,336 | Geometry Algebraic geometryList of algebraic geometry topics List of algebrasTrigonometry Differential geometry Topology Fractal geometry | Outline of mathematics | 0.821486 |
8,337 | Combinatorics (outline) Cryptography Graph theory Number theory | Outline of mathematics | 0.821486 |
8,338 | Calculus Vector calculus Differential equations Dynamical systems Chaos theory Analysis | Outline of mathematics | 0.821486 |
8,339 | Algebra Abstract algebra Linear algebra (Outline) Number theory Order theory Function | Outline of mathematics | 0.821486 |
8,340 | without borrowing, Long division, Short division, Modulo operation, Chunking (division), Multiplication and repeated addition, Euclidean division, Division by zeroPlus and minus signs, Multiplication sign, Division sign, Equals sign Equality (mathematics), Inequality (mathematics), Logical equivalence RatioVariable (ma... | Outline of mathematics | 0.821486 |
8,341 | Peano axioms also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. Floating-point arithmetic is arithmetic using formulaic representation of real numbers as an approximation to support a trade-off between r... | Outline of mathematics | 0.821486 |
8,342 | Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. Arithmetic — (from the Greek ἀριθμός arithmos, 'number' and τική , tiké , 'art') is a branch of mathematics that consists of the study of numbers and the properties of the traditional mathematical ... | Outline of mathematics | 0.821486 |
8,343 | Mathematical object — an abstract concept in mathematics; an object is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs. Each branch of mathematics has its own objects. Mathematical structure — a set endowed with some additional features on the... | Outline of mathematics | 0.821486 |
8,344 | Mathematical chemistry Mathematical physics Analytical mechanics Mathematical fluid dynamics Numerical analysis Control theory Dynamical systems Mathematical optimization Operations research Probability Statistics Game theory Engineering mathematics Mathematical economics Financial mathematics Information theory Crypto... | Outline of mathematics | 0.821486 |
8,345 | Mathematical Reviews – journal and online database published by the American Mathematical Society (AMS) that contains brief synopses (and occasionally evaluations) of many articles in mathematics, statistics and theoretical computer science. Zentralblatt MATH – service providing reviews and abstracts for articles in pu... | Outline of mathematics | 0.821486 |
8,346 | In physics, the elastic potential energy gained by a wire during elongation with a tensile (stretching) or compressive (contractile) force is called strain energy. For linearly elastic materials, strain energy is: U = 1 2 V σ ε = 1 2 V E ε 2 = 1 2 V E σ 2 {\displaystyle U={\frac {1}{2}}V\sigma \varepsilon ={\frac {1}{2... | Strain energy | 0.821482 |
8,347 | Given an integer α ≥ 1, a tactical configuration satisfying: For every anti-flag (B, m) there are α flags (A, l) such that B I l and A I m,is called a partial geometry. If there are s + 1 points on a line and t + 1 lines through a point, the notation for a partial geometry is pg(s, t, α). If α = 1 these partial geometr... | Incidence geometry | 0.821475 |
8,348 | A question raised by J.J. Sylvester in 1893 and finally settled by Tibor Gallai concerned incidences of a finite set of points in the Euclidean plane. Theorem (Sylvester-Gallai): A finite set of points in the Euclidean plane is either collinear or there exists a line incident with exactly two of the points. A line cont... | Incidence geometry | 0.821475 |
8,349 | A bound on the number of flags determined by a finite set of points and the lines they determine is given by: Theorem (Szemerédi–Trotter): given n points and m lines in the plane, the number of flags (incident point-line pairs) is: O ( n 2 3 m 2 3 + n + m ) , {\displaystyle O\left(n^{\frac {2}{3}}m^{\frac {2}{3}}+n+m\r... | Incidence geometry | 0.821475 |
8,350 | In graph theory they are called hypergraphs, and in combinatorial design theory they are called block designs. Besides the difference in terminology, each area approaches the subject differently and is interested in questions about these objects relevant to that discipline. Using geometric language, as is done in incid... | Incidence geometry | 0.821475 |
8,351 | Such fundamental results remain valid when additional concepts are added to form a richer geometry. It sometimes happens that authors blur the distinction between a study and the objects of that study, so it is not surprising to find that some authors refer to incidence structures as incidence geometries.Incidence stru... | Incidence geometry | 0.821475 |
8,352 | In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An incidence structure is what is obtained when all other concepts are removed and all... | Incidence geometry | 0.821475 |
8,353 | The symbolic axioms below are from Boolos (1998: 196), and govern how sets behave and interact. As with Z, the background logic for GST is first order logic with identity. Indeed, GST is the fragment of Z obtained by omitting the axioms Union, Power Set, Elementary Sets (essentially Pairing) and Infinity and then takin... | General set theory | 0.821473 |
8,354 | Given any model M of ZFC, the collection of hereditarily finite sets in M will satisfy the GST axioms. Therefore, GST cannot prove the existence of even a countable infinite set, that is, of a set whose cardinality is ℵ0. Even if GST did afford a countably infinite set, GST could not prove the existence of a set whose ... | General set theory | 0.821473 |
8,355 | General set theory (GST) is George Boolos's (1998) name for a fragment of the axiomatic set theory Z. GST is sufficient for all mathematics not requiring infinite sets, and is the weakest known set theory whose theorems include the Peano axioms. | General set theory | 0.821473 |
8,356 | While ST is a small fragment of the well-known canonical set theories ZFC and NBG, ST interprets Robinson arithmetic (Q), so that ST inherits the nontrivial metamathematics of Q. For example, ST is essentially undecidable because Q is, and every consistent theory whose theorems include the ST axioms is also essentially... | General set theory | 0.821473 |
8,357 | With its simple axioms, GST is also immune to the three great antinomies of naïve set theory: Russell's, Burali-Forti's, and Cantor's. GST is Interpretable in relation algebra because no part of any GST axiom lies in the scope of more than three quantifiers. This is the necessary and sufficient condition given in Tarsk... | General set theory | 0.821473 |
8,358 | Reverse power feeding (RPF) is a technology being standardized by ETSI and the Broadband Forum in 2020. It allows sending power from the customer premises to a distribution point (DP), in order to power the distribution point unit (DPU). The DPU includes typically a DSL access multiplexer (DSLAM) that can support up to... | Reverse Power Feeding | 0.821468 |
8,359 | In model theory, a branch of mathematical logic, the notion of an existentially closed model (or existentially complete model) of a theory generalizes the notions of algebraically closed fields (for the theory of fields), real closed fields (for the theory of ordered fields), existentially closed groups (for the theory... | Existentially closed model | 0.821463 |
8,360 | Absolute geometry is an incomplete axiomatic system, in the sense that one can add extra independent axioms without making the axiom system inconsistent. One can extend absolute geometry by adding different axioms about parallel lines and get incompatible but consistent axiom systems, giving rise to Euclidean or hyperb... | Neutral geometry | 0.821461 |
8,361 | A plane that satisfies Hilbert's Incidence, Betweenness and Congruence axioms is called a Hilbert plane. Hilbert planes are models of absolute geometry. | Neutral geometry | 0.821461 |
8,362 | In Euclid's Elements, the first 28 Propositions and Proposition 31 avoid using the parallel postulate, and therefore are valid in absolute geometry. One can also prove in absolute geometry the exterior angle theorem (an exterior angle of a triangle is larger than either of the remote angles), as well as the Saccheri–Le... | Neutral geometry | 0.821461 |
8,363 | Earth science (also known as geoscience), is an all-embracing term for the sciences related to the planet Earth, including geology, geography, geophysics, geochemistry, climatology, glaciology, hydrology, meteorology, and oceanography. Although mining and precious stones have been human interests throughout the history... | Natural sciences | 0.821461 |
8,364 | Materials science is a relatively new, interdisciplinary field that deals with the study of matter and its properties; as well as the discovery and design of new materials. Originally developed through the field of metallurgy, the study of the properties of materials and solids has now expanded into all materials. The ... | Natural sciences | 0.821461 |
8,365 | The formulation of theories about the governing laws of the universe has been central to the study of physics from very early on, with philosophy gradually yielding to systematic, quantitative experimental testing and observation as the source of verification. Key historical developments in physics include Isaac Newton... | Natural sciences | 0.821461 |
8,366 | Physics embodies the study of the fundamental constituents of the universe, the forces and interactions they exert on one another, and the results produced by these interactions. In general, physics is regarded as the fundamental science, because all other natural sciences use and obey the field's principles and laws. ... | Natural sciences | 0.821461 |
8,367 | The basis of materials science involves studying the structure of materials, and relating them to their properties. Once a materials scientist knows about this structure-property correlation, they can then go on to study the relative performance of a material in a certain application. The major determinants of the stru... | Natural sciences | 0.821461 |
8,368 | The Sprague–Grundy theorem implies that a heap of size n is equivalent to a nim heap of a given size, usually noted G(n). The analysis of an octal game then consists in finding the sequence of the nim-values for heaps of increasing size. This sequence G(0), G(1), G(2) ... is usually called the nim-sequence of the game.... | Octal game | 0.82146 |
8,369 | One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory. This states that given a finite Galois extension K / k {\displaystyle K/k} , there is a bijection between the set of subfields k ⊂ E ⊂ K {\displaystyle k\subset E\subset K} and the subgroups H ⊂ G . {\display... | Galois groups | 0.82146 |
8,370 | The initial object in the category of bounded lattices is a Boolean domain. In computer science, a Boolean variable is a variable that takes values in some Boolean domain. Some programming languages feature reserved words or symbols for the elements of the Boolean domain, for example false and true. However, many progr... | Boolean domain | 0.821456 |
8,371 | The Boolean domain {0, 1} can be replaced by the unit interval , in which case rather than only taking values 0 or 1, any value between and including 0 and 1 can be assumed. Algebraically, negation (NOT) is replaced with 1 − x , {\displaystyle 1-x,} conjunction (AND) is replaced with multiplication ( x y {\displaystyle... | Boolean domain | 0.821456 |
8,372 | In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include false and true. In logic, mathematics and theoretical computer science, a Boolean domain is usually written as {0, 1}, or B . {\displaystyle \mathbb {B} .} The algebraic structure that natural... | Boolean domain | 0.821456 |
8,373 | In mathematics, in the realm of group theory, a quotientable automorphism of a group is an automorphism that takes every normal subgroup to within itself. As a result, it gives a corresponding automorphism for every quotient group. All family automorphisms are quotientable, and particularly, all class automorphisms and... | Quotientable automorphism | 0.821452 |
8,374 | Suppose S is a compact connected oriented Riemann surface and M is a non-empty finite set of points in S that contains at least one point from each boundary component of S (the boundary of S is not assumed to be either empty or non-empty). The pair (S, M) is often referred to as a bordered surface with marked points. I... | Cluster algebra | 0.821444 |
8,375 | Simple examples are given by the algebras of homogeneous functions on the Grassmannians. The Plücker coordinates provide some of the distinguished elements. For the Grassmannian of planes in C n {\displaystyle \mathbb {C} ^{n}} , the situation is even more simple. In that case, the Plücker coordinates provide all the d... | Cluster algebra | 0.821444 |
8,376 | If {x} is the cluster of a seed of rank 1, then the only mutation takes this to {2x−1}. So a cluster algebra of rank 1 is just a ring k of Laurent polynomials, and it has just two clusters, {x} and {2x−1}. In particular it is of finite type and is associated with the Dynkin diagram A1. | Cluster algebra | 0.821444 |
8,377 | Cluster algebras are a class of commutative rings introduced by Fomin and Zelevinsky (2002, 2003, 2007). A cluster algebra of rank n is an integral domain A, together with some subsets of size n called clusters whose union generates the algebra A and which satisfy various conditions. | Cluster algebra | 0.821444 |
8,378 | So this cluster algebra has exactly 5 clusters, and in particular is of finite type. It is associated with the Dynkin diagram A2. There are similar examples with b12 = 1, –b21 = 2 or 3, where the analogous sequence of cluster variables repeats with period 6 or 8. These are also of finite type, and are associated with t... | Cluster algebra | 0.821444 |
8,379 | The cluster algebra also comes with the extra structure of the seeds of this graph. A cluster algebra is said to be of finite type if it has only a finite number of seeds. Fomin & Zelevinsky (2003) showed that the cluster algebras of finite type can be classified in terms of the Dynkin diagrams of finite-dimensional si... | Cluster algebra | 0.821444 |
8,380 | The inverse of a mutation is also a mutation, i.e. if A is a mutation of B then B is a mutation of A. A cluster algebra is constructed from an initial seed as follows. If we repeatedly mutate the seed in all possible ways, we get a finite or infinite graph of seeds, where two seeds are joined by an edge if one can be o... | Cluster algebra | 0.821444 |
8,381 | Suppose that F is an integral domain, such as the field Q(x1,...,xn) of rational functions in n variables over the rational numbers Q. A cluster of rank n consists of a set of n elements {x, y, ...} of F, usually assumed to be an algebraically independent set of generators of a field extension F. A seed consists of a c... | Cluster algebra | 0.821444 |
8,382 | Furthermore, various DNA repair functions as well as regulatory functions are based on base pair complementarity. In biotechnology, the principle of base pair complementarity allows the generation of DNA hybrids between RNA and DNA, and opens the door to modern tools such as cDNA libraries. While most complementarity i... | Complementary base sequence | 0.821444 |
8,383 | In molecular biology, complementarity describes a relationship between two structures each following the lock-and-key principle. In nature complementarity is the base principle of DNA replication and transcription as it is a property shared between two DNA or RNA sequences, such that when they are aligned antiparallel ... | Complementary base sequence | 0.821444 |
8,384 | Given an ordinal parameter α ≥ ω+2 — for every set S with rank less than α, S is well-orderable. Given an ordinal parameter α ≥ 1 — for every set S with Hartogs number less than ωα, S is well-orderable. As the ordinal parameter is increased, these approximate the full axiom of choice more and more closely. Other choice... | Axiom of Choice | 0.821443 |
8,385 | When one attempts to solve problems in this class, it makes no difference whether ZF or ZFC is employed if the only question is the existence of a proof. It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF. The axiom of choice is not the only significant statement which is independ... | Axiom of Choice | 0.821443 |
8,386 | Examples of category-theoretic statements which require choice include: Every small category has a skeleton. If two small categories are weakly equivalent, then they are equivalent. Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint (the Freyd... | Axiom of Choice | 0.821443 |
8,387 | Equivalently, a partition of a set S cannot be strictly larger than S. If WPP holds, this already implies the existence of a non-measurable set. Each of the previous three statements is implied by the preceding one, but it is unknown if any of these implications can be reversed. There is no infinite decreasing sequence... | Axiom of Choice | 0.821443 |
8,388 | Set theory Partition principle: if there is a surjection from A to B, there is an injection from B to A. Equivalently, every partition P of a set S is less than or equal to S in size. Converse Schröder–Bernstein theorem: if two sets have surjections to each other, they are equinumerous. Weak partition principle: if the... | Axiom of Choice | 0.821443 |
8,389 | There is a free complete boolean algebra on countably many generators. There is a set that cannot be linearly ordered. | Axiom of Choice | 0.821443 |
8,390 | There is a field with no algebraic closure. In all models of ZF¬C there is a vector space with no basis. There is a vector space with two bases of different cardinalities. | Axiom of Choice | 0.821443 |
8,391 | Tychonoff's theorem: The Cartesian product of any family of compact topological spaces is compact. In the product topology, the closure of a product of subsets is equal to the product of the closures. Mathematical logic If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is include... | Axiom of Choice | 0.821443 |
8,392 | Every set is a projective object in the category Set of sets. Functional analysis The closed unit ball of the dual of a normed vector space over the reals has an extreme point. Point-set topology The Cartesian product of any family of connected topological spaces is connected. | Axiom of Choice | 0.821443 |
8,393 | Krull's theorem: Every unital ring (other than the trivial ring) contains a maximal ideal. Equivalently, in any nontrivial unital ring, every ideal can be extended to a maximal ideal. For every non-empty set S there is a binary operation defined on S that gives it a group structure. | Axiom of Choice | 0.821443 |
8,394 | Abstract algebra Every vector space has a basis (i.e., a linearly independent spanning subset). In other words, vector spaces are equivalent to free modules. | Axiom of Choice | 0.821443 |
8,395 | (The reason for the term "colloquially" is that the sum or product of a "sequence" of cardinals cannot itself be defined without some aspect of the axiom of choice.) Well-ordering theorem: Every set can be well-ordered. Consequently, every cardinal has an initial ordinal. | Axiom of Choice | 0.821443 |
8,396 | The Cartesian product of any family of nonempty sets is nonempty. In other words, every family of nonempty sets has a choice function (i.e. a function which maps each of the nonempty sets to one of its elements). König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a se... | Axiom of Choice | 0.821443 |
8,397 | There are important statements that, assuming the axioms of ZF but neither AC nor ¬AC, are equivalent to the axiom of choice. The most important among them are Zorn's lemma and the well-ordering theorem. In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theor... | Axiom of Choice | 0.821443 |
8,398 | The usual statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function. However, that particular case is a theorem of the Zermelo–Fraenkel set theory without the axiom of choice (ZF);... | Axiom of Choice | 0.821443 |
8,399 | Many theorems which are provable using choice are of an elegant general character: the cardinalities of any two sets are comparable, every nontrivial ring with unity has a maximal ideal, every vector space has a basis, every connected graph has a spanning tree, and every product of compact spaces is compact, among many... | Axiom of Choice | 0.821443 |
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