id int32 0 100k | text stringlengths 21 3.54k | source stringlengths 1 124 | similarity float32 0.78 0.88 |
|---|---|---|---|
13,500 | real line. Its algebraic closure, the (algebraic?) | Space (mathematics) | 0.816113 |
13,501 | In particular, when it is finite-dimensional, over real numbers, and endowed with inner product, it becomes Euclidean space; now geometric. The (algebraic?) field of real numbers is the same as the (geometric?) | Space (mathematics) | 0.816113 |
13,502 | Clearly, groups are algebraic, while Euclidean spaces are geometric. Modules over rings are as algebraic as groups. In particular, when the ring appears to be a field, the module appears to be a linear space; is it algebraic or geometric? | Space (mathematics) | 0.816113 |
13,503 | However, it was never used actively in mathematical practice (not even in the mathematical treatises written by Bourbaki himself). Here are the last phrases from a review by Robert Reed of a book by Leo Corry: Corry does not seem to feel that any formal definition of structure could do justice to the use of the concept... | Space (mathematics) | 0.816113 |
13,504 | According to Kevin Carlson, Neither of these words have a single mathematical definition. The English words can be used in essentially all the same situations, but you often think of a "space" as more geometric and a "structure" as more algebraic. So you could think of "structures" as places we do algebra, and "spaces... | Space (mathematics) | 0.816113 |
13,505 | It is known to be true for p > λ {\displaystyle p>\lambda } when G {\displaystyle G} is an abelian group, and this is known as the First Multiplier Theorem. A more general known result, the Second Multiplier Theorem, says that if D {\displaystyle D} is a ( v , k , λ ) {\displaystyle (v,k,\lambda )} -difference set in a... | Difference sets | 0.816112 |
13,506 | It is sharply transitive on both points and blocks.In particular, if λ = 1 {\displaystyle \lambda =1} , then the difference set gives rise to a projective plane. An example of a (7,3,1) difference set in the group Z / 7 Z {\displaystyle \mathbb {Z} /7\mathbb {Z} } is the subset { 1 , 2 , 4 } {\displaystyle \{1,2,4\}} .... | Difference sets | 0.816112 |
13,507 | Research on intelligent code completion began in 1957, with spelling checkers for bitmap images of cursive writing and special applications to find records in databases despite incorrect entries. In 1961, Les Earnest, who headed the research on this budding technology, saw it necessary to include the first spell checke... | Intelligent code completion | 0.816111 |
13,508 | As of Visual Studio 2005, IntelliSense is now activated by default when the user begins to type, instead of requiring marker characters (though this behavior can be turned off). The IDE has the capability of inferring a greater amount of context based on what the developer is typing, to the point that basic language co... | Intelligent code completion | 0.81611 |
13,509 | The derivation of a useful equation for modelling the effect of eddy currents in a material starts with the differential, magnetostatic form of Ampère's Law, providing an expression for the magnetizing field H surrounding a current density J: Taking the curl on both sides of this equation and then using a common vector... | Eddy current | 0.816103 |
13,510 | The mathematics of dealing with the situation where M(x) varies with x is difficult, except in special cases that usually do not correspond to physical reality. A full mathematical treatment appears in the 1965 monograph by Fröman and Fröman. Their ideas have not been incorporated into physics textbooks, but their corr... | Electron tunneling | 0.816102 |
13,511 | The time-independent Schrödinger equation for one particle in one dimension can be written as or where ℏ {\displaystyle \hbar } is the reduced Planck's constant, m is the particle mass, x represents distance measured in the direction of motion of the particle, Ψ is the Schrödinger wave function, V is the potential ener... | Electron tunneling | 0.816102 |
13,512 | Cold emission of electrons is relevant to semiconductors and superconductor physics. It is similar to thermionic emission, where electrons randomly jump from the surface of a metal to follow a voltage bias because they statistically end up with more energy than the barrier, through random collisions with other particle... | Electron tunneling | 0.816102 |
13,513 | Radioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form a stable product. This is done via the tunnelling of a particle out of the nucleus (an electron tunneling into the nucleus is electron capture). This was the first application of quantum tunnelling. Radioac... | Electron tunneling | 0.816102 |
13,514 | For a mutation to occur, the proton must have tunnelled into the shallower well. The proton's movement from its regular position is called a tautomeric transition. If DNA replication takes place in this state, the base pairing rule for DNA may be jeopardised, causing a mutation. Per-Olov Lowdin was the first to develop... | Electron tunneling | 0.816102 |
13,515 | Quantum tunnelling is among the central non-trivial quantum effects in quantum biology. Here it is important both as electron tunnelling and proton tunnelling. Electron tunnelling is a key factor in many biochemical redox reactions (photosynthesis, cellular respiration) as well as enzymatic catalysis. Proton tunnelling... | Electron tunneling | 0.816102 |
13,516 | Construct a polar coordinate grid on the complex plane with spacing derived from the degree of the polynomial being factored Use the FFT to evaluate the polynomial at each node along the concentric circles of the grid. Search over each 3x3 set of values for relative minima. If the center value is less than the edge val... | Lindsey–Fox algorithm | 0.816101 |
13,517 | This rule is based on the “Minimum Modulus Theorem” which states that if a relative minimum of the absolute value of an analytic function over an open region exists, the minimum must be a zero of the function. Finally, this set of prospective zeros is passed to the second stage. | Lindsey–Fox algorithm | 0.816101 |
13,518 | To counteract this problem, classical control theory uses the Laplace transform to change an Ordinary Differential Equation (ODE) in the time domain into a regular algebraic polynomial in the frequency domain. Once a given system has been converted into the frequency domain it can be manipulated with greater ease. Mode... | Classical control theory | 0.8161 |
13,519 | A Physical system can be modeled in the "time domain", where the response of a given system is a function of the various inputs, the previous system values, and time. As time progresses, the state of the system and its response change. However, time-domain models for systems are frequently modeled using high-order diff... | Classical control theory | 0.8161 |
13,520 | In the 1930s-1950s, the first protein structures were solved by protein crystallography. These early structures suggested that a fixed three-dimensional structure might be generally required to mediate biological functions of proteins. These publications solidified the central dogma of molecular biology in that the ami... | Intrinsically disordered protein | 0.816099 |
13,521 | Façade (interactive story) was released in 2005 and used interactive multiple way dialogs and AI as the main aspect of game. Games have provided an environment for developing artificial intelligence with potential applications beyond gameplay. Examples include Watson, a Jeopardy!-playing computer; and the RoboCup tourn... | Game artificial intelligence | 0.816097 |
13,522 | It serves to improve the game-player experience rather than machine learning or decision making. During the golden age of arcade video games the idea of AI opponents was largely popularized in the form of graduated difficulty levels, distinct movement patterns, and in-game events dependent on the player's input. Modern... | Game artificial intelligence | 0.816097 |
13,523 | In video games, artificial intelligence (AI) is used to generate responsive, adaptive or intelligent behaviors primarily in non-player characters (NPCs) similar to human-like intelligence. Artificial intelligence has been an integral part of video games since their inception in the 1950s. AI in video games is a distinc... | Game artificial intelligence | 0.816097 |
13,524 | Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication i... | Nonnegative integers | 0.816095 |
13,525 | The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties: Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers. Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c an... | Nonnegative integers | 0.816095 |
13,526 | Sociophysics uses tools and concepts from physics and physical chemistry to describe some aspects of social and political behavior. It attempts to explain why and how humans behave much like atoms, at least in some aspects of their collective lives. The law of mass action (generalized if it is necessary) is the main to... | Mass action kinetics | 0.816094 |
13,527 | Computational lattice problems have many applications in computer science. For example, the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) has been used in the cryptanalysis of many public-key encryption schemes, and many lattice-based cryptographic schemes are known to be secure under the assumption th... | Lattice model (mathematics) | 0.81607 |
13,528 | A simple example of a lattice in R n {\displaystyle \mathbb {R} ^{n}} is the subgroup Z n {\displaystyle \mathbb {Z} ^{n}} . More complicated examples include the E8 lattice, which is a lattice in R 8 {\displaystyle \mathbb {R} ^{8}} , and the Leech lattice in R 24 {\displaystyle \mathbb {R} ^{24}} . The period lattice... | Lattice model (mathematics) | 0.81607 |
13,529 | The distance from the axis may be called the radial distance or radius, while the angular coordinate is sometimes referred to as the angular position or as the azimuth. The radius and the azimuth are together called the polar coordinates, as they correspond to a two-dimensional polar coordinate system in the plane thro... | Cylindrical coordinate system | 0.816064 |
13,530 | Zero to the power of zero, denoted by 00, is a mathematical expression that is either defined as 1 or left undefined, depending on context. In algebra and combinatorics, one typically defines 00 = 1. In mathematical analysis, the expression is sometimes left undefined. Computer programming languages and software also h... | Zero to the power of zero | 0.816053 |
13,531 | That is, r0 = 1 for each real number r, including 0. The same argument applies with R replaced by any ring.Defining 00 = 1 is necessary for many polynomial identities. For example, the binomial theorem (1 + x)n = Σnk=0 (nk) xk holds for x = 0 only if 00 = 1.Similarly, rings of power series require x0 to be defined as 1... | Zero to the power of zero | 0.816053 |
13,532 | Also, polynomials can be evaluated by specializing x to a real number. More precisely, for any given real number r, there is a unique unital R-algebra homomorphism evr: R → R such that evr(x) = r. Because evr is unital, evr(x0) = 1. | Zero to the power of zero | 0.816053 |
13,533 | Euler, when setting 00 = 1, mentioned that consequently the values of the function 0x take a "huge jump", from ∞ for x < 0, to 1 at x = 0, to 0 for x > 0. In 1814, Pfaff used a squeeze theorem argument to prove that xx → 1 as x → 0+.On the other hand, in 1821 Cauchy explained why the limit of xy as positive numbers x a... | Zero to the power of zero | 0.816053 |
13,534 | Some authors define 00 as 1 because it simplifies many theorem statements. According to Benson (1999), "The choice whether to define 00 is based on convenience, not on correctness. If we refrain from defining 00, then certain assertions become unnecessarily awkward. ... The consensus is to use the definition 00 = 1, al... | Zero to the power of zero | 0.816053 |
13,535 | Limits involving algebraic operations can often be evaluated by replacing subexpressions by their limits; if the resulting expression does not determine the original limit, the expression is known as an indeterminate form. The expression 00 is an indeterminate form: Given real-valued functions f(t) and g(t) approaching... | Zero to the power of zero | 0.816053 |
13,536 | The external body forces will appear as the independent ("right-hand side") term in the differential equations, while the concentrated forces appear as boundary conditions. An external (applied) surface force, such as ambient pressure or friction, can be incorporated as an imposed value of the stress tensor across that... | Stress analysis | 0.816052 |
13,537 | Stress analysis is simplified when the physical dimensions and the distribution of loads allow the structure to be treated as one- or two-dimensional. In the analysis of a bridge, its three dimensional structure may be idealized as a single planar structure, if all forces are acting in the plane of the trusses of the b... | Stress analysis | 0.816052 |
13,538 | The basic stress analysis problem can be formulated by Euler's equations of motion for continuous bodies (which are consequences of Newton's laws for conservation of linear momentum and angular momentum) and the Euler-Cauchy stress principle, together with the appropriate constitutive equations. These laws yield a syst... | Stress analysis | 0.816051 |
13,539 | Canadian Bioinformatics Workshops (CBW) are a series of advanced training workshops in bioinformatics, founded in 1999 in response to an identified need for a skilled bioinformatics workforce in Canada. | Canadian Bioinformatics Workshops | 0.816048 |
13,540 | The Canadian Bioinformatics Workshops series began offering one and two week short courses in bioinformatics, genomics and proteomics in 1999, in response to an identified need for a skilled bioinformatics workforce in Canada. In partnership with the Canadian Genetics Diseases Network and Human Resources Development Ca... | Canadian Bioinformatics Workshops | 0.816048 |
13,541 | The Canadian Bioinformatics Workshops began offering the 2-day advanced topic workshops in 2008. All workshop material is licensed under a Creative Commons-Share Alike 2.5 license and is available on the Bioinformatics.ca website. The CBW is sponsored by the Canadian Institute of Health Research and the Ontario Institu... | Canadian Bioinformatics Workshops | 0.816048 |
13,542 | In 2007, the Canadian Bioinformatics Workshops moved to Toronto, where it is now hosted by the Ontario Institute for Cancer Research. A new format and series of workshops were designed in the fall of 2007. It was recognized that with the introduction of new technologies and scientific approaches to research, having the... | Canadian Bioinformatics Workshops | 0.816048 |
13,543 | Molecular Ecology Resources is a monthly scientific journal covering development of tools and techniques to address questions in ecology, evolution, behavior, and conservation. It is the companion journal of Molecular Ecology and is published by Wiley-Blackwell. From 2001 to 2007, it was published under the name Molecu... | Molecular Ecology Resources | 0.816043 |
13,544 | The first writer to advocate a lectin-free diet was Peter J. D'Adamo, a naturopathic physician best known for promoting the Blood type diet. He argued that lectins may damage a person's blood type by interfering with digestion, food metabolism, hormones, insulin production—and so should be avoided. D'Adamo provided no ... | Lectin | 0.816043 |
13,545 | A lectin from Carex is used to identify R antigen.In neuroscience, the anterograde labeling method is used to trace the path of efferent axons with PHA-L, a lectin from the kidney bean.A lectin (BanLec) from bananas inhibits HIV-1 in vitro. Achylectins, isolated from Tachypleus tridentatus, show specific agglutinating ... | Lectin | 0.816043 |
13,546 | Long before a deeper understanding of their numerous biological functions, the plant lectins, also known as phytohemagglutinins, were noted for their particularly high specificity for foreign glycoconjugates (e.g., those of fungi and animals) and used in biomedicine for blood cell testing and in biochemistry for fracti... | Lectin | 0.816043 |
13,547 | Completion (ring theory) Formal power series Localization of a ring Local ring Regular local ring Localization of a module Valuation (mathematics) Discrete valuation Discrete valuation ring I-adic topology Weierstrass preparation theorem | List of commutative algebra topics | 0.816035 |
13,548 | Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ... | List of commutative algebra topics | 0.816035 |
13,549 | Noetherian ring Hilbert's basis theorem Artinian ring Ascending chain condition (ACC) and descending chain condition (DCC) | List of commutative algebra topics | 0.816035 |
13,550 | Height (ring theory) Depth (ring theory) Hilbert polynomial Regular local ring Discrete valuation ring Global dimension Regular sequence (algebra) Krull dimension Krull's principal ideal theorem | List of commutative algebra topics | 0.816035 |
13,551 | Flat module Flat map Flat map (ring theory) Projective module Injective module Cohen-Macaulay ring Gorenstein ring Complete intersection ring Koszul complex Hilbert's syzygy theorem Quillen–Suslin theorem | List of commutative algebra topics | 0.816035 |
13,552 | Field (mathematics) Algebraic number field Polynomial ring Integral domain Boolean algebra (structure) Principal ideal domain Euclidean domain Unique factorization domain Dedekind domain Nilpotent elements and reduced rings Dual numbers Tensor product of fields Tensor product of R-algebras | List of commutative algebra topics | 0.816035 |
13,553 | Commutative ring Module (mathematics) Ring ideal, maximal ideal, prime ideal Ring homomorphism Ring monomorphism Ring epimorphism Ring isomorphism Zero divisor Chinese remainder theorem | List of commutative algebra topics | 0.816035 |
13,554 | Algebraic number theory Algebraic geometry Ring theory Field theory (mathematics) Differential algebra Homological algebra | List of commutative algebra topics | 0.816035 |
13,555 | Primary ideal Primary decomposition and the Lasker–Noether theorem Noether normalization lemma Going up and going down | List of commutative algebra topics | 0.816035 |
13,556 | The civilian (commercial and general) drone market is dominated by Chinese companies. Chinese manufacturer DJI alone had 74% of the civil market share in 2018, with no other company accounting for more than 5%. Following increased scrutiny of its activities, the US Interior Department grounded its fleet of DJI drones i... | Unmanned Aerial Vehicles | 0.816035 |
13,557 | The completion is the inverse limit of the factor rings, R ^ I = lim ← ( R / I n ) {\displaystyle {\widehat {R}}_{I}=\varprojlim (R/I^{n})} pronounced "R I hat". The kernel of the canonical map π from the ring to its completion is the intersection of the powers of I. Thus π is injective if and only if this intersecti... | Completion (algebra) | 0.816033 |
13,558 | In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R. The case of a maximal ideal I = m {\displaystyle I={\mathfrak {m}}} is especially important, for example the distinguished maximal ideal of a valuation ri... | Completion (algebra) | 0.816033 |
13,559 | In particular, taking quotients of rings commutes with completion, meaning that for any quotient R-algebra R / I {\displaystyle R/I} , there is an isomorphism R / I ^ ≅ R ^ / I ^ . {\displaystyle {\widehat {R/I}}\cong {\widehat {R}}/{\widehat {I}}.} Cohen structure theorem (equicharacteristic case). Let R be a complete... | Completion (algebra) | 0.816033 |
13,560 | American soil scientist Hans Jenny published in 1941 a state equation for the factors influencing soil formation: S = f(cl, o, r, p, t, …) S soil formation cl (sometimes c) climate o organisms (soil microbiology, soil mesofauna, soil biology) r relief p parent material t timeThis is often remembered with the mnemonic C... | Soil creation | 0.816032 |
13,561 | Soil formation, also known as pedogenesis, is the process of soil genesis as regulated by the effects of place, environment, and history. Biogeochemical processes act to both create and destroy order (anisotropy) within soils. These alterations lead to the development of layers, termed soil horizons, distinguished by d... | Soil creation | 0.816032 |
13,562 | Optimally, printing of calibrators and high and low controls directly on the same chip will then provide for unmatched ability to quantitatively measure each protein over time and between experiments. A problem that is encountered with tissue microarrays is antigen retrieval and the inherent subjectivity of immunohisto... | Reverse phase protein lysate microarray | 0.816031 |
13,563 | In addition, since RPMA can utilize whole-cell or undissected or microdissected cell lysates, it can provide direct quantifiable information concerning post translationally modified proteins that are not accessible with other high-throughput techniques. Thus, RPMA provides high-dimensional proteomic data in a high thro... | Reverse phase protein lysate microarray | 0.816031 |
13,564 | RPMA was first introduced in 2001 in a paper by Lance Liotta and Emanuel Petricoin who invented the technology. The authors used the technique to successfully analyze the state of pro-survival checkpoint protein at the microscopic transition stage using laser capture microdissection of histologically normal prostate ep... | Reverse phase protein lysate microarray | 0.816031 |
13,565 | Symmetric and antimetric networks are often also topologically symmetric and antimetric, respectively. The physical arrangement of their components and values are symmetric or antimetric as in the ladder example above. However, it is not a necessary condition for electrical antimetry. For example, if the example networ... | Symmetrical electrical network | 0.816027 |
13,566 | The rated capacity of a battery is usually expressed as the product of 20 hours multiplied by the current that a new battery can consistently supply for 20 hours at 68 °F (20 °C), while remaining above a specified terminal voltage per cell. For example, a battery rated at 100 A·h can deliver 5 A over a 20-hour period a... | Electrical batteries | 0.81602 |
13,567 | The more electrode material contained in the cell the greater its capacity. A small cell has less capacity than a larger cell with the same chemistry, although they develop the same open-circuit voltage. Capacity is measured in units such as amp-hour (A·h). | Electrical batteries | 0.81602 |
13,568 | A battery's characteristics may vary over load cycle, over charge cycle, and over lifetime due to many factors including internal chemistry, current drain, and temperature. At low temperatures, a battery cannot deliver as much power. As such, in cold climates, some car owners install battery warmers, which are small el... | Electrical batteries | 0.81602 |
13,569 | In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal methods of mathematics.Formal verification can be helpful in proving the correctnes... | Program verification | 0.816011 |
13,570 | Much of number theory can be viewed as the study of rational points of algebraic varieties, a convenient setting being smooth projective varieties. For smooth projective curves, the behavior of rational points depends strongly on the genus of the curve. | Rational points | 0.816005 |
13,571 | Not much is known about that case. The strongest known result on the Bombieri–Lang conjecture is Faltings's theorem on subvarieties of abelian varieties (generalizing the case of curves). Namely, if X is a subvariety of an abelian variety A over a number field k, then all k-rational points of X are contained in a finit... | Rational points | 0.816005 |
13,572 | In higher dimensions, one unifying goal is the Bombieri–Lang conjecture that, for any variety X of general type over a number field k, the set of k-rational points of X is not Zariski dense in X. (That is, the k-rational points are contained in a finite union of lower-dimensional subvarieties of X.) In dimension 1, thi... | Rational points | 0.816005 |
13,573 | Teachers utilize SSI to foster understanding of science content and consequences involved in everyday scientific issues. For example, in a study of ecology, an elementary class might consider whether pesticides confer more benefit or harm to our ecosystem. This type of analysis would require students to research the in... | Socio-scientific issues | 0.816002 |
13,574 | The points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends itself to intuitive visualizations; for example, the set of all elements that are members of both sets S and T, denoted S ∩ T and read "the intersection of S and T", is... | Venn diagrams | 0.815997 |
13,575 | A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diag... | Venn diagrams | 0.815997 |
13,576 | (Compare this with the definition of the dual Boolean algebra.) Hence, with this notation, an algebra A has signature ⟨·, +, ', 0, 1, ∀⟩, with ⟨A, ·, +, ', 0, 1⟩ a Boolean algebra, as before. Moreover, ∀ satisfies the following dualized version of the above identities: ∀1 = 1 ∀x ≤ x ∀(xy) = ∀x∀y ∀x + ∀y = ∀(x + ∀y).∀x ... | Monadic Boolean algebra | 0.815997 |
13,577 | In abstract algebra, a monadic Boolean algebra is an algebraic structure A with signature ⟨·, +, ', 0, 1, ∃⟩ of type ⟨2,2,1,0,0,1⟩,where ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra. The monadic/unary operator ∃ denotes the existential quantifier, which satisfies the identities (using the received prefix notation for ∃): ∃0... | Monadic Boolean algebra | 0.815997 |
13,578 | Monadic Boolean algebras also have an important connection to modal logic. The modal logic S5, viewed as a theory in S4, is a model of monadic Boolean algebras in the same way that S4 is a model of interior algebra. Likewise, monadic Boolean algebras supply the algebraic semantics for S5. Hence S5-algebra is a synonym ... | Monadic Boolean algebra | 0.815997 |
13,579 | They are to monadic predicate logic what Boolean algebras are to propositional logic, and what polyadic algebras are to first-order logic. Paul Halmos discovered monadic Boolean algebras while working on polyadic algebras; Halmos (1962) reprints the relevant papers. Halmos and Givant (1998) includes an undergraduate tr... | Monadic Boolean algebra | 0.815997 |
13,580 | Hence monadic Boolean algebras are the semisimple interior/closure algebras such that: The universal (dually, existential) quantifier interprets the interior (closure) operator; All open (or closed) elements are also clopen.A more concise axiomatization of monadic Boolean algebra is (1) and (2) above, plus ∀(x∨∀y) = ∀x... | Monadic Boolean algebra | 0.815997 |
13,581 | Monadic Boolean algebras have an important connection to topology. If ∀ is interpreted as the interior operator of topology, (1)–(3) above plus the axiom ∀(∀x) = ∀x make up the axioms for an interior algebra. But ∀(∀x) = ∀x can be proved from (1)–(4). Moreover, an alternative axiomatization of monadic Boolean algebras ... | Monadic Boolean algebra | 0.815997 |
13,582 | In 1858, Dedekind proposed a definition of the real numbers in terms of Dedekind cuts of rational numbers, a definition still employed in contemporary texts.Georg Cantor developed the fundamental concepts of infinite set theory. His early results developed the theory of cardinality and proved that the reals and the nat... | Formal logical systems | 0.815994 |
13,583 | Hilbert developed a complete set of axioms for geometry, building on previous work by Pasch. The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and the real line. This would prove to be a major area of research in the first ... | Formal logical systems | 0.815994 |
13,584 | "Mathematical logic has been successfully applied not only to mathematics and its foundations (G. Frege, B. Russell, D. Hilbert, P. Bernays, H. Scholz, R. Carnap, S. Lesniewski, T. Skolem), but also to physics (R. Carnap, A. Dittrich, B. Russell, C. E. Shannon, A. N. Whitehead, H. Reichenbach, P. Fevrier), to biology (... | Formal logical systems | 0.815994 |
13,585 | Theories of logic were developed in many cultures in history, including China, India, Greece and the Islamic world. Greek methods, particularly Aristotelian logic (or term logic) as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrys... | Formal logical systems | 0.815994 |
13,586 | In the mid-19th century, flaws in Euclid's axioms for geometry became known. In addition to the independence of the parallel postulate, established by Nikolai Lobachevsky in 1826, mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these is the th... | Formal logical systems | 0.815994 |
13,587 | Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties. Dedekind proposed a different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work, however, proved theorems inaccessible in Peano's system, including... | Formal logical systems | 0.815994 |
13,588 | Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for fundamental areas of mathematics such as arithmetic, analysis, and geometry. In logic, the term arithmetic refers to the theory of the natural numbers. Giuseppe Peano published a set of axioms for arithme... | Formal logical systems | 0.815994 |
13,589 | J.D. Sneed, The Logical Structure of Mathematical Physics. | Formal logical systems | 0.815994 |
13,590 | An important subfield of recursion theory studies algorithmic unsolvability; a decision problem or function problem is algorithmically unsolvable if there is no possible computable algorithm that returns the correct answer for all legal inputs to the problem. The first results about unsolvability, obtained independentl... | Formal logical systems | 0.815994 |
13,591 | Hilbert, however, did not acknowledge the importance of the incompleteness theorem for some time.Gödel's theorem shows that a consistency proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibil... | Formal logical systems | 0.815994 |
13,592 | In 1931, Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which proved the incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Gödel's incompleteness theorem, establishes severe limitation... | Formal logical systems | 0.815994 |
13,593 | Gödel used the completeness theorem to prove the compactness theorem, demonstrating the finitary nature of first-order logical consequence. These results helped establish first-order logic as the dominant logic used by mathematicians. | Formal logical systems | 0.815994 |
13,594 | Leopold Löwenheim and Thoralf Skolem obtained the Löwenheim–Skolem theorem, which says that first-order logic cannot control the cardinalities of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable mod... | Formal logical systems | 0.815994 |
13,595 | During the time period studied, the US did not have many robots in the economy which restricts the impact of automation. However, automation is expected to triple (conservative estimate) or quadruple (a generous estimate) leading these numbers to become substantially higher.Based on a formula by Gilles Saint-Paul, an e... | Automation | 0.815991 |
13,596 | For example, while a study by the Pew Research Center indicated that 72% of Americans are worried about increasing automation in the workplace, 80% of Swedes see automation and artificial intelligence (AI) as a good thing, due to the country's still-powerful unions and a more robust national safety net.In the U.S., 47%... | Automation | 0.815991 |
13,597 | manufacturing productivity growth fell from 5.2%/yr 1919–29 to 2.76%/yr 1929–41. Alexander Field notes that spending on non-medical instruments increased significantly from 1929 to 1933 and remained strong thereafter.The First and Second World Wars saw major advancements in the field of mass communication and signal pr... | Automation | 0.815991 |
13,598 | In physics, the term dielectric strength has the following meanings: for a pure electrically insulating material, the maximum electric field that the material can withstand under ideal conditions without undergoing electrical breakdown and becoming electrically conductive (i.e. without failure of its insulating propert... | Dielectric strength | 0.815983 |
13,599 | At the end of 1990's Kerner introduced a new traffic phase, called synchronized flow whose basic feature leads to the nucleation nature of the F → S transition at a highway bottleneck. Therefore, Kerner's synchronized flow traffic phase can be used synonymously with the term three-phase traffic theory. In 1998 Kerner f... | Kerner’s breakdown minimization principle | 0.815983 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.