id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
8,400 | In 1938, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model (the constructible universe) which satisfies ZFC and thus showing that ZFC is consistent if ZF itself is consistent. In 1963, Paul Cohen employed the technique of forcing, developed for this purpose... | Axiom of Choice | 0.821443 |
8,401 | In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of excluded middle (unlike in Martin-Löf type theory, where it does not). Thus the axiom of choice is not generally available in constructive set theory. A cause for this difference is that the axiom of choice in ty... | Axiom of Choice | 0.821443 |
8,402 | The status of the axiom of choice varies between different varieties of constructive mathematics. In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is (depending on approach) included as an axiom or provable as a theorem. Errett Bishop argued that the axiom ... | Axiom of Choice | 0.821443 |
8,403 | The debate is interesting enough, however, that it is considered of note when a theorem in ZFC (ZF plus AC) is logically equivalent (with just the ZF axioms) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the ty... | Axiom of Choice | 0.821443 |
8,404 | Without the axiom of choice, one cannot assert that such a function exists for pairs of socks, because left and right socks are (presumably) indistinguishable. Tarski tried to publish his theorem in Comptes Rendus, but Fréchet and Lebesgue refused to present it. | Axiom of Choice | 0.821443 |
8,405 | General topology A uniform space is compact if and only if it is complete and totally bounded. Every Tychonoff space has a Stone–Čech compactification. Mathematical logic Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion. That is, every consistent set of ... | Axiom of Choice | 0.821443 |
8,406 | The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem. On every infinite-dimensional topological vector space there is a discontinuous linear map. | Axiom of Choice | 0.821443 |
8,407 | The additive groups of R and C are isomorphic. Functional analysis The Hahn–Banach theorem in functional analysis, allowing the extension of linear functionals The theorem that every Hilbert space has an orthonormal basis. The Banach–Alaoglu theorem about compactness of sets of functionals. | Axiom of Choice | 0.821443 |
8,408 | Every infinite-dimensional vector field contains an infinite linearly independent subset (this requires dependent choice, but not the full axiom of choice). Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem. The Nielsen–Schreier theorem, that every subgroup of a free group is fre... | Axiom of Choice | 0.821443 |
8,409 | The Banach–Tarski paradox. Algebra Every field has an algebraic closure. Every field extension has a transcendence basis. | Axiom of Choice | 0.821443 |
8,410 | Every infinite game G S {\displaystyle G_{S}} in which S {\displaystyle S} is a Borel subset of Baire space is determined. Measure theory The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable. The Hausdorff paradox. | Axiom of Choice | 0.821443 |
8,411 | For an infinite collection of pairs of socks (assumed to have no distinguishing features), there is no obvious way to make a function that forms a set out of selecting one sock from each pair, without invoking the axiom of choice.Although originally controversial, the axiom of choice is now used without reservation by ... | Axiom of Choice | 0.821443 |
8,412 | In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory equivalent to the statement that a Cartesian product of a collection of non-empty sets is non-empty. Informally put, the axiom of choice says that given any collection of sets, each containing at least one element, it is possible to c... | Axiom of Choice | 0.821443 |
8,413 | In this dollar game, negative integer values (representing debt) are assigned to some of the vertices of the finite graph G. A game is called winnable if there exists a state where all the vertices can be made positive. A a graph-theoretic analogue of Riemann–Roch theorem can be used to characterize if a game is winnab... | Chip-firing game | 0.821439 |
8,414 | Post-pass optimizers usually work on the assembly language or machine code level (in contrast with compilers that optimize intermediate representations of programs). One such example is the Portable C Compiler (pcc) of the 1980s, which had an optional pass that would perform post-optimizations on the generated assembly... | Optimizing compilers | 0.821432 |
8,415 | There can be a wide range of optimizations that a compiler can perform, ranging from the simple and straightforward that take little compilation time to the elaborate and complex that involve considerable amounts of compilation time. Accordingly, compilers often provide options to their control command or procedure to ... | Optimizing compilers | 0.821432 |
8,416 | Although many of these also apply to non-functional languages, they either originate in or are particularly critical in functional languages such as Lisp and ML. Tail-call optimization A function call consumes stack space and involves some overhead related to parameter passing and flushing the instruction cache. Tail-r... | Optimizing compilers | 0.821432 |
8,417 | Additionally, if the comparison number exceeds the size of word of the platform, in standard loop order, multiple instructions would need to be executed in order to evaluate the comparison, which is not the case with loop reversal.Loop unrolling Unrolling duplicates the body of the loop multiple times, in order to decr... | Optimizing compilers | 0.821432 |
8,418 | For correct implementation, this technique must be used with loop inversion, because not all code is safe to be hoisted outside the loop.Loop nest optimization Some pervasive algorithms such as matrix multiplication have very poor cache behavior and excessive memory accesses. Loop nest optimization increases the number... | Optimizing compilers | 0.821432 |
8,419 | When two adjacent loops would iterate the same number of times regardless of whether that number is known at compile time, their bodies can be combined as long as they make no reference to each other's data.Loop inversion This technique changes a standard while loop into a do/while (also known as repeat/until) loop wra... | Optimizing compilers | 0.821432 |
8,420 | Some optimization techniques primarily designed to operate on loops include: Induction variable analysis Roughly, if a variable in a loop is a simple linear function of the index variable, such as j := 4*i + 1, it can be updated appropriately each time the loop variable is changed. This is a strength reduction, and als... | Optimizing compilers | 0.821432 |
8,421 | The term physics envy is used to criticize modern writing and research of academics working in areas such as "softer sciences", liberal arts, business administration education, humanities, and social sciences. The term argues that writing and working practices in these disciplines have overused confusing jargon and com... | Physics envy | 0.821425 |
8,422 | Social science has been accused of inferiority complex, which has been associated with physics envy. For instance, positivist scientists accept a mistaken image of natural science so it can be applied to the social sciences. The phenomenon also exists in business strategy research as demonstrated by historian Alfred Ch... | Physics envy | 0.821425 |
8,423 | Evolutionary biologist Ernst Mayr discusses the issue of the inability to reduce biology to its mathematical basis in his book What Makes Biology Unique?. Noam Chomsky discusses the ability and desirability of reduction to its mathematical basis in his article "Mysteries of Nature: How Deeply Hidden." Chomsky contribut... | Physics envy | 0.821425 |
8,424 | The success of physics in "mathematicizing" itself, particularly since Isaac Newton's Principia Mathematica, is generally considered remarkable and often disproportionate compared to other areas of inquiry. "Physics envy" refers to the envy (perceived or real) of scholars in other disciplines for the mathematical preci... | Physics envy | 0.821425 |
8,425 | 3. Auflage, John Wiley & Sons, New York 2004. ISBN 0-471-19350-X. Bruce Alberts, Alexander Johnson, Peter Walter, Julian Lewis, Martin Raff, Keith Roberts: Molecular Biology of the Cell, 5. Auflage, Taylor & Francis 2007, ISBN 978-0815341062. | Leucines | 0.821423 |
8,426 | Jeremy M. Berg, John L. Tymoczko, Lubert Stryer: Biochemie. 6 Auflage, Spektrum Akademischer Verlag, Heidelberg 2007. ISBN 978-3-8274-1800-5. Donald Voet, Judith G. Voet: Biochemistry. | Leucines | 0.821423 |
8,427 | Some researchers posit environments of water-ammonia mixtures as possible solvents for hypothetical types of biochemistry.Environmental Stability: Where organisms adaptively evolve to the conditions of the environments in which they reside, environmental stability is considered necessary for life to exist. This presupp... | Astrobiology | 0.82142 |
8,428 | However, theoretical astrobiology entertains the potential for other organic molecular bases for life, thus astrobiological research often focuses on identifying environments that have the potential to support life based on the presence of organic compounds. Liquid water: Liquid water is a common molecule that provides... | Astrobiology | 0.82142 |
8,429 | The discovery of microbial life in extreme environments on Earth, such as deep-sea hydrothermal vents, helped to clarify the feasibility of potential life existing in harsh conditions. The development of new techniques for the detection of biosignatures, such as the use of stable isotopes, also played a significant rol... | Astrobiology | 0.82142 |
8,430 | In 1957, the Soviet Union launched Sputnik 1, the first artificial satellite, which marked the beginning of the Space Age. This event led to an increase in the study of the potential for life on other planets, as scientists began to consider the possibilities opened up by the new technology of space exploration. In 195... | Astrobiology | 0.82142 |
8,431 | The term astrobiology was first proposed by the Russian astronomer Gavriil Tikhov in 1953. It is etymologically derived from the Greek ἄστρον, "star"; βίος, "life"; and -λογία, -logia, "study". A close synonym is exobiology from the Greek Έξω, "external"; βίος, "life"; and -λογία, -logia, "study", coined by American mo... | Astrobiology | 0.82142 |
8,432 | Their extreme radiation tolerance and presence of DNA protection proteins may provide answers as to whether life can survive away from the protection of the Earth's atmosphere.Jupiter's moon, Europa, and Saturn's moon, Enceladus, are now considered the most likely locations for extant extraterrestrial life in the Solar... | Astrobiology | 0.82142 |
8,433 | Characterization of these organisms, their environments and their evolutionary pathways, is considered a crucial component to understanding how life might evolve elsewhere in the universe. For example, some organisms able to withstand exposure to the vacuum and radiation of outer space include the lichen fungi Rhizocar... | Astrobiology | 0.82142 |
8,434 | This chemosynthesis revolutionized the study of biology and astrobiology by revealing that life need not be sunlight-dependent; it only requires water and an energy gradient in order to exist. Biologists have found extremophiles that thrive in ice, boiling water, acid, alkali, the water core of nuclear reactors, salt c... | Astrobiology | 0.82142 |
8,435 | Biology cannot state that a process or phenomenon, by being mathematically possible, has to exist forcibly in an extraterrestrial body. Biologists specify what is speculative and what is not. The discovery of extremophiles, organisms able to survive in extreme environments, became a core research element for astrobiolo... | Astrobiology | 0.82142 |
8,436 | The second and third terms, fp, stars with planets and fe, planets with habitable conditions, are being evaluated for the star's neighborhood. Drake originally formulated the equation merely as an agenda for discussion at the Green Bank conference, but some applications of the formula had been taken literally and relat... | Astrobiology | 0.82142 |
8,437 | Most astronomy-related astrobiology research falls into the category of extrasolar planet (exoplanet) detection, the hypothesis being that if life arose on Earth, then it could also arise on other planets with similar characteristics. To that end, a number of instruments designed to detect Earth-sized exoplanets have b... | Astrobiology | 0.82142 |
8,438 | This logic is itself a subset of the 9-valued logic standard called IEEE 1164 and implemented in Very High Speed Integrated Circuit Hardware Description Language, VHDL's std_logic. One should not confuse four-valued mathematical logic (using operators, truth tables, syllogisms, propositional calculus, theorems and so o... | Four-valued logic | 0.821419 |
8,439 | In theoretical computer science and network routing, Suurballe's algorithm is an algorithm for finding two disjoint paths in a nonnegatively-weighted directed graph, so that both paths connect the same pair of vertices and have minimum total length. The algorithm was conceived by John W. Suurballe and published in 1974... | Suurballe's algorithm | 0.821405 |
8,440 | In mathematics, specifically in topology and functional analysis, a subspace S of a uniform space X is said to be sequentially complete or semi-complete if every Cauchy sequence in S converges to an element in S. X is called sequentially complete if it is a sequentially complete subset of itself. | Sequentially complete space | 0.821403 |
8,441 | Edwards proved the Edwards-Erdős bound using probabilistic method; Crowston et al. proved the bound using linear algebra and analysis of pseudo-boolean functions. The proof of Crowston et al. allows us to extend the Edwards-Erdős bound to the Balanced Subgraph Problem (BSP) on signed graphs G = (V, E, s), i.e. graphs w... | Maximum cut | 0.821401 |
8,442 | In statistical physics and disordered systems, the Max Cut problem is equivalent to minimizing the Hamiltonian of a spin glass model, most simply the Ising model. For the Ising model on a graph G and only nearest-neighbor interactions, the Hamiltonian is H = − ∑ i j ∈ E ( G ) J i j s i s j {\displaystyle H=-\sum _{ij\... | Maximum cut | 0.821401 |
8,443 | This is among the oldest unsolved problems in universal algebra. Until it is answered, the theory of finite algebras is incomplete since, given a finite algebra, it is unknown whether there are, a priori, any restrictions on the shape of its congruence lattice. | Finite lattice representation problem | 0.821388 |
8,444 | A lattice is called algebraic if it is complete and compactly generated. In 1963, Grätzer and Schmidt proved that every algebraic lattice is isomorphic to the congruence lattice of some algebra. Thus there is essentially no restriction on the shape of a congruence lattice of an algebra. The finite lattice representatio... | Finite lattice representation problem | 0.821388 |
8,445 | That is, does every finite lattice occur as the congruence lattice of a finite algebra? In 1980, Pálfy and Pudlák proved that this problem is equivalent to the problem of deciding whether every finite lattice occurs as an interval in the subgroup lattice of a finite group. For an overview of the group theoretic approac... | Finite lattice representation problem | 0.821388 |
8,446 | In mathematics, the finite lattice representation problem, or finite congruence lattice problem, asks whether every finite lattice is isomorphic to the congruence lattice of some finite algebra. | Finite lattice representation problem | 0.821388 |
8,447 | Based on the Poisson distribution with mean 50, the probability of observing more than 61 significant tests is less than 0.05, so if more than 61 significant results are observed, it is very likely that some of them correspond to situations where the alternative hypothesis holds. A drawback of this approach is that it ... | Multiple testing correction | 0.821387 |
8,448 | The photon is said to have traveled as a wave if the build up results in the typical interference pattern of waves (see Double-slit experiment § Interference of individual particles for an animation showing the build up). However, if one of the slits is closed or two orthogonal polarizers are placed in front of the sli... | Delayed choice experiment | 0.821386 |
8,449 | "Wheeler's delayed-choice experiment" refers to a series of thought experiments in quantum physics, the first being proposed by him in 1978. Another prominent version was proposed in 1983. All of these experiments try to get at the same fundamental issues in quantum physics. Many of them are discussed in Wheeler's 1978... | Delayed choice experiment | 0.821386 |
8,450 | These moves must only satisfy a basic condition of balance in order for the equilibrium to be properly described, but detailed balance, a stronger condition, is usually imposed when designing new algorithms. An additional advantage is that some systems, such as the Ising model, lack a dynamical description and are only... | Monte Carlo molecular modeling | 0.821384 |
8,451 | Thus, it is the application of the Metropolis Monte Carlo simulation to molecular systems. It is therefore also a particular subset of the more general Monte Carlo method in statistical physics. It employs a Markov chain procedure in order to determine a new state for a system from a previous one. | Monte Carlo molecular modeling | 0.821384 |
8,452 | Major databases include GenBank for DNA sequences and PubMed, a bibliographic database for biomedical literature. Other databases include the NCBI Epigenomics database. All these databases are available online through the Entrez search engine. NCBI was directed by David Lipman, one of the original authors of the BLAST ... | National Center for Biotechnology Information | 0.821374 |
8,453 | The National Center for Biotechnology Information (NCBI) is part of the United States National Library of Medicine (NLM), a branch of the National Institutes of Health (NIH). It is approved and funded by the government of the United States. The NCBI is located in Bethesda, Maryland, and was founded in 1988 through legi... | National Center for Biotechnology Information | 0.821374 |
8,454 | Many cell assays have been developed to assess specific parameters or response of cells (biomarkers, cell physiology). Techniques used to study cells include: reporter assays using i.e. Luciferase, calcium signaling assays using Coelenterazine, CFSE or Calcein Immunostaining of cells on slides by Microscopy (ImmunoHist... | Biological assays | 0.821358 |
8,455 | Archaebacteria are a type of single-cell organism and have a number of characteristics not seen in more "modern" cell types. These characteristics include: Unique cell membrane chemistry Unique gene transcription Capable of methanogenesis Differences in ribosomal RNATypes of Archaebacteria: Thermoproteota: Extremely he... | Marine botany | 0.821353 |
8,456 | Marine ecology and marine botany include: Benthic zone Coral reef Kelp forests Mangroves Phytoplankton Salt marsh Sea grass Seaweed | Marine botany | 0.821353 |
8,457 | Marine botany is the study of flowering vascular plant species and marine algae that live in shallow seawater of the open ocean and the littoral zone, along shorelines of the intertidal zone and coastal wetlands, even in low-salinity brackish water of estuaries. It is a branch of marine biology and botany. | Marine botany | 0.821353 |
8,458 | TORQUE provides enhancements over standard OpenPBS in the following areas: Fault Tolerance Additional failure conditions checked/handled Node health check script support Scheduling Interface Extended query interface providing the scheduler with additional and more accurate information Extended control interface allowin... | TORQUE | 0.821353 |
8,459 | Writing q {\displaystyle q} for the position coordinate and p {\displaystyle p} for the body's momentum, the Hamiltonian is In this example, Hamilton's equations are and Evaluating these partial derivatives, the former equation becomes which reproduces the familiar statement that a body's momentum is the product of its... | Newton's 3rd law | 0.821349 |
8,460 | These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values c... | Newton's 3rd law | 0.821349 |
8,461 | Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result. The expectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence.The Ehrenfest theorem provides a connection between ... | Newton's 3rd law | 0.821349 |
8,462 | Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The c... | Newton's 3rd law | 0.821349 |
8,463 | Research has traditionally focused on certain tasks in machine learning and control. Somewhat recently, interest has turned towards stochastic decoding, which applies stochastic computing to the decoding of error correcting codes. More recently, stochastic circuits have been successfully used in image processing tasks ... | Stochastic computing | 0.821331 |
8,464 | For simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression. Typically this occurs when the mass density is constant, but in some cases the density can vary throughout the object as well. In general, it may not be straightforward to symbolically express t... | List of moments of inertia | 0.821331 |
8,465 | Photosystem – They are functional and structural units of protein complexes involved in photosynthesis that together carry out the primary photochemistry of photosynthesis: the absorption of light and the transfer of energy and electrons. They are found in the thylakoid membranes of plants, algae and cyanobacteria (in ... | Outline of cell biology | 0.82133 |
8,466 | Branch of biology – The study of life and living organisms, including their structure, function, growth, evolution, distribution, and taxonomy. Academic discipline – Focused study in one academic field or profession. A discipline incorporates expertise, people, projects, communities, challenges, studies, inquiry, and r... | Outline of cell biology | 0.82133 |
8,467 | Edmund Beecher Wilson – Known as America's first cellular biologist, discovered the sex chromosome arrangement in humans. Albert Claude – Shared the Nobel Prize in 1974 "for describing the structure and function of organelles in biological cells" Theodor Boveri – In 1888 identified the centrosome and described it as th... | Outline of cell biology | 0.82133 |
8,468 | The following outline is provided as an overview of and topical guide to cell biology: Cell biology – A branch of biology that includes study of cells regarding their physiological properties, structure, and function; the organelles they contain; interactions with their environment; and their life cycle, division, and ... | Outline of cell biology | 0.82133 |
8,469 | Unicellular – Organisms which are composed of only one cell. Multicellular – Organisms consisting of more than one cell and having differentiated cells that perform specialized functions. Tissues – A collection of interconnected cells that perform a similar function within an organism. Cellular differentiation – A conc... | Outline of cell biology | 0.82133 |
8,470 | Developmental biology – The study of the process by which organisms grow and develop, including the genetic control of cell growth, differentiation and "morphogenesis", which is the process that gives rise to tissues, organs and anatomy. Microbiology – The study of microorganisms, which are unicellular or cell-cluster ... | Outline of cell biology | 0.82133 |
8,471 | It deals with the structure and function of cellular components, such as proteins, carbohydrates, lipids, nucleic acids, and other biomolecules. Cytochemistry – The biochemistry of cells, especially that of the macromolecules responsible for cell structure and function. Molecular biology – The study of biology at a mol... | Outline of cell biology | 0.82133 |
8,472 | Cytopathology – A branch of pathology that studies and diagnoses diseases on the cellular level. The most common use of cytopathology is the Pap smear, used to detect cervical cancer at an early treatable stage. Genetics – The science of heredity and variation in living organisms. Biochemistry – The study of the chemic... | Outline of cell biology | 0.82133 |
8,473 | All cells come from preexisting cells and contain the hereditary information necessary for regulating cell functions and for transmitting information to the next generation of cells. Cell biology – (formerly cytology) The study of cells. Cell division – The process of one parent cell separating into two or more daughte... | Outline of cell biology | 0.82133 |
8,474 | Cell biology can be described as all of the following: Branch of science – A systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Branch of natural science – The branch of science concerned with the description, prediction, and understanding ... | Outline of cell biology | 0.82133 |
8,475 | See also Cell biologists belowHistory of cell biology – is intertwined with the history of biochemistry and the history of molecular biology. Other articles pertaining to the history of cell biology include: History of cell theory, embryology and germ theory History of biochemistry, microbiology, and molecular biology ... | Outline of cell biology | 0.82133 |
8,476 | This kind of solubility is of great importance in geology, where it results in formation of metamorphic rocks. In principle, both congruent and incongruent dissolution can lead to the formation of secondary solid phases in equilibrium. So, in the field of Materials Science, the solubility for both cases is described mo... | Solubility rules | 0.821321 |
8,477 | As quoted from: J.A. Dean (ed), Lange's Handbook of Chemistry (15th Edition), McGraw-Hill, 1999; Section 3; Table 3.2 Physical Constants of Inorganic CompoundsUnit is °C | Melting points of the elements (data page) | 0.821321 |
8,478 | Lavrukhina, Avgusta Konstantinovna; Pozdnyakov, Aleksandr Aleksandrovich (1970). Analytical Chemistry of Technetium, Promethium, Astatine, and Francium. Translated by R. Kondor. Ann Arbor–Humphrey Science Publishers. | Melting points of the elements (data page) | 0.821321 |
8,479 | The bcc phase and the P-T and VT phase diagrams below 2 K". Journal of Low Temperature Physics. 23 (1): 63–102. | Melting points of the elements (data page) | 0.821321 |
8,480 | Haynes, William M., ed. (2011). CRC Handbook of Chemistry and Physics (92nd ed.). CRC Press. | Melting points of the elements (data page) | 0.821321 |
8,481 | The paper consists of single correct type questions. There are questions from high school level physics, mathematics and some questions from general astronomy. | National Standard Examination in Astronomy | 0.821315 |
8,482 | The National Standard Examination in Astronomy or NSEA is an examination in astronomy and astrophysics for higher secondary school students, usually conducted in the end of November (last or second last Sunday). Organized by the Indian Association of Physics Teachers in association with Homi Bhabha Centre for Science E... | National Standard Examination in Astronomy | 0.821315 |
8,483 | Surveillance is the act of infection investigation using the CDC definitions. Determining the presence of a hospital acquired infection requires an infection control practitioner (ICP) to review a patient's chart and see if the patient had the signs and symptom of an infection. Surveillance definitions exist for infect... | Infection barrier | 0.821314 |
8,484 | Practitioners can come from several different educational streams: many begin as registered nurses, some as public health inspectors (environmental health officers), some as medical technologists (particularly in clinical microbiology), and some as physicians (typically infectious disease specialists). Specialized trai... | Infection barrier | 0.821314 |
8,485 | Automorphisms may be defined in this way both for directed graphs and for undirected graphs. The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Fru... | Graph automorphism problem | 0.82131 |
8,486 | In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph G = (V, E) is a permutation σ of the vertex set V, such that the pair of vertices (u, v) form an edge i... | Graph automorphism problem | 0.82131 |
8,487 | It is a greedy algorithm in graph theory as in each step it adds the next lowest-weight edge that will not form a cycle to the minimum spanning forest.This algorithm first appeared in Proceedings of the American Mathematical Society, pp. 48–50 in 1956, and was written by Joseph Kruskal. It was rediscovered by Loberman ... | Kruskal's algorithm | 0.821309 |
8,488 | In plane geometry, a triangle ABC contains a triangle having one-seventh of the area of ABC, which is formed as follows: the sides of this triangle lie on cevians p, q, r where p connects A to a point on BC that is one-third the distance from B to C, q connects B to a point on CA that is one-third the distance from C t... | One-seventh area triangle | 0.821302 |
8,489 | In surface chemistry, disjoining pressure (symbol Πd) according to an IUPAC definition arises from an attractive interaction between two surfaces. For two flat and parallel surfaces, the value of the disjoining pressure (i.e., the force per unit area) can be calculated as the derivative of the Gibbs energy of interacti... | Disjoining force | 0.821302 |
8,490 | As a special case of mapping, they can be used for studies of gene duplication or deletion. Researchers use microsatellites in population genetics and in species conservation projects. Plant geneticists have proposed the use of microsatellites for marker assisted selection of desirable traits in plant breeding. | Simple Sequence Repeat | 0.821301 |
8,491 | As the common divisors of two polynomials are not changed if the polynomials are multiplied by invertible constants (in Q), the last nonzero term in a pseudo-remainder sequence is a GCD (in Q) of the input polynomials. Therefore, pseudo-remainder sequences allows computing GCD's in Q without introducing fractions in Q.... | Euclidean algorithm for polynomials | 0.8213 |
8,492 | More precisely, subresultants are defined for polynomials over any commutative ring R, and have the following property. Let φ be a ring homomorphism of R into another commutative ring S. It extends to another homomorphism, denoted also φ between the polynomials rings over R and S. Then, if P and Q are univariate polyno... | Euclidean algorithm for polynomials | 0.8213 |
8,493 | In theoretical computer science, the separating words problem is the problem of finding the smallest deterministic finite automaton that behaves differently on two given strings, meaning that it accepts one of the two strings and rejects the other string. It is an open problem how large such an automaton must be, in th... | Separating words problem | 0.821293 |
8,494 | Although it is true that Hardy preferred pure mathematics, which he often compared to painting and poetry, Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express physical truth in a mathematical framework, whereas pure mathematics expressed truths that wer... | Pure Mathematics | 0.821287 |
8,495 | Nevertheless, almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories. Also, many mathematical theories, which had seemed to be totally pure mathematics, were eventually used in applied areas, mainly physics and computer science. A famous ea... | Pure Mathematics | 0.821287 |
8,496 | Moreover, Hardy briefly admitted that—just as the application of matrix theory and group theory to physics had come unexpectedly—the time may come where some kinds of beautiful, "real" mathematics may be useful as well. Another insightful view is offered by American mathematician Andy Magid: I've always thought that a ... | Pure Mathematics | 0.821287 |
8,497 | Parametric statistics that assume no error often fail on such mixture densities – for example, statistics that assume normality often fail disastrously in the presence of even a few outliers – and instead one uses robust statistics. In meta-analysis of separate studies, study heterogeneity causes distribution of result... | Mixture density | 0.821268 |
8,498 | In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the s... | Mixture density | 0.821268 |
8,499 | Today it is accepted that the neutrinos produced in the Sun are not massless particles as predicted by the Standard Model but rather mixed quantum states made up of defined-mass eigenstates in different (complex) proportions. That allows a neutrino produced as a pure electron neutrino to change during propagation into ... | Solar neutrino problem | 0.821266 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.