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18,500 | In mathematics, a primary cyclic group is a group that is both a cyclic group and a p-primary group for some prime number p. That is, it is a cyclic group of order pm, Cpm, for some prime number p, and natural number m. Every finite abelian group G may be written as a finite direct sum of primary cyclic groups, as stat... | Primary cyclic group | 0.812132 |
18,501 | Automatic control manages a device without the need of human inputs for correction, such as cruise control for regulating a car's speed. Control systems engineering activities are multi-disciplinary in nature. They focus on the implementation of control systems, mainly derived by mathematical modeling. Because instrume... | Instrumentation and control engineering | 0.812131 |
18,502 | Meanwhile, control engineering, also called control systems engineering, is the engineering discipline that applies control theory to design systems with desired behaviors. Control engineers are responsible for the research, design, and development of control devices and systems, typically in manufacturing facilities a... | Instrumentation and control engineering | 0.812131 |
18,503 | This algebra is the derived Lie algebra of b {\displaystyle {\mathfrak {b}}} , the Lie algebra of all upper triangular matrices; in symbols, n = . {\displaystyle {\mathfrak {n}}=.} In addition, n {\displaystyle {\mathfrak {n}}} is the Lie algebra of the Lie group of unitriangular matrices. In fact, by Engel's theorem,... | Simultaneously triangularizable | 0.812131 |
18,504 | The set of unitriangular matrices forms a Lie group. The set of strictly upper (or lower) triangular matrices forms a nilpotent Lie algebra, denoted n . {\displaystyle {\mathfrak {n}}.} | Simultaneously triangularizable | 0.812131 |
18,505 | Upper triangularity is preserved by many operations: The sum of two upper triangular matrices is upper triangular. The product of two upper triangular matrices is upper triangular. The inverse of an upper triangular matrix, if it exists, is upper triangular. The product of an upper triangular matrix and a scalar is upp... | Simultaneously triangularizable | 0.812131 |
18,506 | If all of the entries on the main diagonal of a (upper or lower) triangular matrix are also 0, the matrix is called strictly (upper or lower) triangular. All finite strictly triangular matrices are nilpotent of index at most n as a consequence of the Cayley-Hamilton theorem. | Simultaneously triangularizable | 0.812131 |
18,507 | High demand for engineering professionals is found in fields associated with process automation. Specializations include industrial instrumentation, system dynamics, process control, and control systems. Additionally, technological knowledge, particularly in computer systems, is essential to the job of an instrumentati... | Instrumentation and control engineering | 0.812131 |
18,508 | Many universities teach instrumentation and control engineering as an academic courses at the graduate and postgraduate levels. It is possible to approach this field coming from many standard engineering backgrounds, being the most common among them Electrical and Mechanical Engineering, since these branches cover stro... | Instrumentation and control engineering | 0.812131 |
18,509 | All these results hold if upper triangular is replaced by lower triangular throughout; in particular the lower triangular matrices also form a Lie algebra. However, operations mixing upper and lower triangular matrices do not in general produce triangular matrices. For instance, the sum of an upper and a lower triangul... | Simultaneously triangularizable | 0.812131 |
18,510 | Additionally, this also shows that the upper triangular matrices can be viewed as a Lie subalgebra of the Lie algebra of square matrices of a fixed size, where the Lie bracket given by the commutator ab − ba. The Lie algebra of all upper triangular matrices is a solvable Lie algebra. It is often referred to as a Borel... | Simultaneously triangularizable | 0.812131 |
18,511 | A more precise statement is given by the Jordan normal form theorem, which states that in this situation, A is similar to an upper triangular matrix of a very particular form. The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.In the case of ... | Simultaneously triangularizable | 0.812131 |
18,512 | Any complex square matrix is triangularizable. In fact, a matrix A over a field containing all of the eigenvalues of A (for example, any matrix over an algebraically closed field) is similar to a triangular matrix. This can be proven by using induction on the fact that A has an eigenvector, by taking the quotient space... | Simultaneously triangularizable | 0.812131 |
18,513 | This construction achieves many of the desired properties of F1-geometry: Spec F1 consists of a single point, so behaves similarly to the spectrum of a field in conventional geometry, and the category of affine monoid schemes is dual to the category of multiplicative monoids, mirroring the duality of affine schemes and... | Field with one element | 0.812131 |
18,514 | A Tits–Weyl model of an algebraic group G {\displaystyle {\mathcal {G}}} is a blue scheme G with a group operation that is a morphism in the Tits category, whose base extension is G {\displaystyle {\mathcal {G}}} and whose Weyl extension is isomorphic to the Weyl group of G . {\displaystyle {\mathcal {G}}.} F1-geometry... | Field with one element | 0.812131 |
18,515 | Using this, they managed to provide a notion of several number-theoretic constructions over F1 such as motives and field extensions, as well as constructing Chevalley groups over F12. Along with Matilde Marcolli, Connes and Consani have also connected F1 with noncommutative geometry. It has also been suggested to have ... | Field with one element | 0.812131 |
18,516 | Nikolai Durov constructed F1 as a commutative algebraic monad. Borger used descent to construct it from the finite fields and the integers.Alain Connes and Caterina Consani developed both Soulé and Deitmar's notions by "gluing" the category of multiplicative monoids and the category of rings to create a new category M ... | Field with one element | 0.812131 |
18,517 | He suggested that zeta functions of varieties over F1 would have very simple descriptions, and he proposed a relation between the K-theory of F1 and the homotopy groups of spheres. This inspired several people to attempt to construct explicit theories of F1-geometry. The first published definition of a variety over F1 ... | Field with one element | 0.812131 |
18,518 | The extensions of F1 later on were denoted as Fq with q = 1n. Together with Mikhail Kapranov, Smirnov went on to explore how algebraic and number-theoretic constructions in prime characteristic might look in "characteristic one", culminating in an unpublished work released in 1995. In 1993, Yuri Manin gave a series of ... | Field with one element | 0.812131 |
18,519 | By 1991, Smirnov had taken some steps towards algebraic geometry over F1, introducing extensions of F1 and using them to handle the projective line P1 over F1. Algebraic numbers were treated as maps to this P1, and conjectural approximations to the Riemann–Hurwitz formula for these maps were suggested. These approximat... | Field with one element | 0.812131 |
18,520 | The analogous objects in the theory of buildings are called apartments. Apartments play such a constituent role in the theory of buildings that Tits conjectured the existence of a theory of projective geometry in which the degenerate geometries would have equal standing with the classical ones. This geometry would take... | Field with one element | 0.812131 |
18,521 | In 1957, Jacques Tits introduced the theory of buildings, which relate algebraic groups to abstract simplicial complexes. One of the assumptions is a non-triviality condition: If the building is an n-dimensional abstract simplicial complex, and if k < n, then every k-simplex of the building must be contained in at leas... | Field with one element | 0.812131 |
18,522 | Geometry over a field with one element is also motivated by Arakelov geometry, where Diophantine equations are studied using tools from complex geometry. The theory involves complicated comparisons between finite fields and the complex numbers. Here the existence of F1 is useful for technical reasons. | Field with one element | 0.812131 |
18,523 | In some descriptions of F1-geometry the converse is also true, in the sense that the extension of scalars of F1-varieties to Z are toric. Whilst other approaches to F1-geometry admit wider classes of examples, toric varieties appear to lie at the very heart of the theory. The zeta function of PN(F1) should be ζ(s) = s(... | Field with one element | 0.812131 |
18,524 | More generally, anything defined purely in terms of diagrams of algebraic objects should have an F1-analog in the category of sets. Group actions on sets are projective representations of G over F1, and in this way, G is the group Hopf algebra F1. Toric varieties determine F1-varieties. | Field with one element | 0.812131 |
18,525 | The affine scheme Spec Z is a curve over F1. Groups are Hopf algebras over F1. | Field with one element | 0.812131 |
18,526 | Finite sets are both affine spaces and projective spaces over F1. Pointed sets are vector spaces over F1. The finite fields Fq are quantum deformations of F1, where q is the deformation. Weyl groups are simple algebraic groups over F1: Given a Dynkin diagram for a semisimple algebraic group, its Weyl group is the semis... | Field with one element | 0.812131 |
18,527 | Deitmar's construction of monoid schemes has been called "the very core of F1-geometry", as most other theories of F1-geometry contain descriptions of monoid schemes. Morally, it mimicks the theory of schemes developed in the 1950s and 1960s by replacing commutative rings with monoids. The effect of this is to "forget"... | Field with one element | 0.812131 |
18,528 | One motivation for F1 comes from algebraic number theory. Weil's proof of the Riemann hypothesis for curves over finite fields starts with a curve C over a finite field k, which comes equipped with a function field F, which is a field extension of k. Each such function field gives rise to a Hasse–Weil zeta function ζF,... | Field with one element | 0.812131 |
18,529 | From this point of view the finite field Fq is an algebra over F1n, of dimension d = (q − 1)/n for any n that is a factor of q − 1 (for example n = q − 1 or n = 1). This corresponds to the fact that the group of units of a finite field Fq (which are the q − 1 non-zero elements) is a cyclic group of order q − 1, on whic... | Field with one element | 0.812131 |
18,530 | One of the defining features of theories of F1 is that these new foundations allow more objects than classical abstract algebra does, one of which behaves like a field of characteristic one. The possibility of studying the mathematics of F1 was originally suggested in 1956 by Jacques Tits, published in Tits 1957, on th... | Field with one element | 0.812131 |
18,531 | Most proposed theories of F1 replace abstract algebra entirely. Mathematical objects such as vector spaces and polynomial rings can be carried over into these new theories by mimicking their abstract properties. This allows the development of commutative algebra and algebraic geometry on new foundations. | Field with one element | 0.812131 |
18,532 | In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name "field with one element" and the notation F1 are only suggestive, as ther... | Field with one element | 0.812131 |
18,533 | This construction is reversible, and the incidence matrix of a symmetric 2-design with these parameters can be used to form an Hadamard matrix of order 4a. When a = 2 we obtain the, by now familiar, Fano plane as an Hadamard 2-design. A pairwise balanced design (or PBD) is a set X together with a family of subsets of X... | Combinatorial design | 0.812123 |
18,534 | An example of a (7,3,1) difference set in the group Z / 7 Z {\displaystyle \mathbb {Z} /7\mathbb {Z} } (an abelian group written additively) is the subset {1,2,4}. The development of this difference set gives the Fano plane. Since every difference set gives an SBIBD, the parameter set must satisfy the Bruck–Ryser–Chowl... | Combinatorial design | 0.812123 |
18,535 | A solution of the famous 15 schoolgirl problem is a resolution of a BIBD with v = 15, k = 3 and λ = 1. A Latin rectangle is an r × n matrix that has the numbers 1, 2, 3, ..., n as its entries (or any other set of n distinct symbols) with no number occurring more than once in any row or column where r ≤ n. An n × n Lati... | Combinatorial design | 0.812123 |
18,536 | In physics, the proton-to-electron mass ratio (symbol μ or β) is the rest mass of the proton (a baryon found in atoms) divided by that of the electron (a lepton found in atoms), a dimensionless quantity, namely: μ = mp/me = 1836.15267343(11).The number in parentheses is the measurement uncertainty on the last two digit... | Proton–electron mass ratio | 0.812122 |
18,537 | Recombination and generation are always happening in semiconductors, both optically and thermally. As predicted by thermodynamics, a material at thermal equilibrium will have generation and recombination rates that are balanced so that the net charge carrier density remains constant. The resulting probability of occupa... | Radiative recombination | 0.812122 |
18,538 | Theorem (hashing bound). There exists a stabilizer quantum error-correcting code that achieves the hashing limit R = 1 − H ( p ) {\displaystyle R=1-H\left(\mathbf {p} \right)} for a Pauli channel of the following form:where p = ( p I , p X , p Y , p Z ) {\displaystyle \mathbf {p} =\left(p_{I},p_{X},p_{Y},p_{Z}\right)} ... | Hashing bound | 0.812121 |
18,539 | The LSD theorem states that the coherent information of a quantum channel is an achievable rate for reliable quantum communication. For a Pauli channel, the coherent information has a simple form and the proof that it is achievable is particularly simple as well. We prove the theorem for this special case by exploiting... | Hashing bound | 0.812121 |
18,540 | In the theory of quantum communication, the quantum capacity is the highest rate at which quantum information can be communicated over many independent uses of a noisy quantum channel from a sender to a receiver. It is also equal to the highest rate at which entanglement can be generated over the channel, and forward c... | Hashing bound | 0.812121 |
18,541 | Many different models have been proposed throughout the history of quantum mechanics, but the most prominent system of nomenclature spawned from the Hund-Mulliken molecular orbital theory of Friedrich Hund, Robert S. Mulliken, and contributions from Schrödinger, Slater and John Lennard-Jones. This system of nomenclatur... | Quantum number | 0.812121 |
18,542 | The second-order logic without these restrictions is sometimes called full second-order logic to distinguish it from the monadic version. Monadic second-order logic is particularly used in the context of Courcelle's theorem, an algorithmic meta-theorem in graph theory. The MSO theory of the complete infinite binary tre... | Monadic second order | 0.812117 |
18,543 | The compactness theorem and the Löwenheim–Skolem theorem do not hold for full models of second-order logic. They do hold however for Henkin models. : xi | Monadic second order | 0.812117 |
18,544 | It is a corollary of Gödel's incompleteness theorem that there is no deductive system (that is, no notion of provability) for second-order formulas that simultaneously satisfies these three desired attributes: (Soundness) Every provable second-order sentence is universally valid, i.e., true in all domains under standar... | Monadic second order | 0.812117 |
18,545 | Four-, five-, six-, seven- and eight-dimensional magic hypercubes of order three have been constructed by J. R. Hendricks. Marian Trenkler proved the following theorem: A p-dimensional magic hypercube of order n exists if and only if p > 1 and n is different from 2 or p = 1. A construction of a magic hypercube follows ... | Nasik magic hypercube | 0.812115 |
18,546 | Painsky, Rosset and Feder (2016, 2017) further studied this problem in the context of independent component analysis over finite alphabet sizes. Through a series of theorems they show that the factorial coding problem can be accurately solved with a branch and bound search tree algorithm, or tightly approximated with a... | Factorial code | 0.812114 |
18,547 | But each detector uses a machine learning algorithm to become as unpredictable as possible. The global optimum of this objective function corresponds to a factorial code represented in a distributed fashion across the outputs of the feature detectors. | Factorial code | 0.812114 |
18,548 | The cleanroom software engineering process is a software development process intended to produce software with a certifiable level of reliability. The central principles are software development based on formal methods, incremental implementation under statistical quality control, and statistically sound testing. | Cleanroom Software Engineering | 0.812113 |
18,549 | The basic principles of the cleanroom process are Software development based on formal methods Software tool support based on some mathematical formalism includes model checking, process algebras, and Petri nets. The Box Structure Method might be one such means of specifying and designing a software product. Verificati... | Cleanroom Software Engineering | 0.812113 |
18,550 | The Serre–Tate theorem asserts, roughly speaking, that the deformations of abelian scheme A is controlled by deformations of the p-divisible group A {\displaystyle A} consisting of its p-power torsion points. | Deformation (mathematics) | 0.812113 |
18,551 | Deformation theory was famously applied in birational geometry by Shigefumi Mori to study the existence of rational curves on varieties. For a Fano variety of positive dimension Mori showed that there is a rational curve passing through every point. The method of the proof later became known as Mori's bend-and-break. T... | Deformation (mathematics) | 0.812113 |
18,552 | Another method for formalizing deformation theory is using functors on the category Art k {\displaystyle {\text{Art}}_{k}} of local Artin algebras over a field. A pre-deformation functor is defined as a functor F: Art k → Sets {\displaystyle F:{\text{Art}}_{k}\to {\text{Sets}}} such that F ( k ) {\displaystyle F(k)} is... | Deformation (mathematics) | 0.812113 |
18,553 | Palamodov, V. P., III. Deformations of complex spaces. Complex Variables IV (very down to earth intro) Course Notes on Deformation Theory (Artin) Studying Deformation Theory of Schemes Sernesi, Eduardo, Deformations of Algebraic Schemes Hartshorne, Robin, Deformation Theory Notes from Hartshorne's Course on Deformation... | Deformation (mathematics) | 0.812113 |
18,554 | Mazur, Barry (2004), "Perturbations, Deformations, and Variations (and "Near-Misses" in Geometry, Physics, and Number Theory" (PDF), Bulletin of the American Mathematical Society, 41 (3): 307–336, doi:10.1090/S0273-0979-04-01024-9, MR 2058289 Anel, M., Why deformations are cohomological (PDF) | Deformation (mathematics) | 0.812113 |
18,555 | Strip Algebra is a set of elements and operators for the description of carbon nanotube structures, considered as a subgroup of polyhedra, and more precisely, of polyhedra with vertices formed by three edges. This restriction is imposed on the polyhedra because carbon nanotubes are formed of sp2 carbon atoms. Strip Alg... | Strip algebra | 0.812111 |
18,556 | Strip Algebra has been applied to the construction of nanotube heterojunctions, and was first implemented in the CoNTub v1.0 software, which makes it possible to find the precise position of all the carbon rings needed to produce a heterojunction with arbitrary indices and chirality from two nanotubes. == References == | Strip algebra | 0.812111 |
18,557 | Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background. Paid circulation in 2008 was 9,500 and total circulation was 10,000.Mathematics Magazine is a c... | Mathematics Magazine | 0.812109 |
18,558 | Bland's rule selects an entering variables by comparing values of reduced costs, using the real-number ordering of the eligible pivots. Unlike Bland's rule, the criss-cross algorithm is "purely combinatorial", selecting an entering variable and a leaving variable by considering only the signs of coefficients rather tha... | Criss-cross algorithm | 0.812105 |
18,559 | The criss-cross algorithm is often studied using the theory of oriented matroids (OMs), which is a combinatorial abstraction of linear-optimization theory. Indeed, Bland's pivoting rule was based on his previous papers on oriented-matroid theory. However, Bland's rule exhibits cycling on some oriented-matroid linear-pr... | Criss-cross algorithm | 0.812105 |
18,560 | The criss-cross algorithm is not a simplex-like algorithm, because it need not maintain feasibility. The criss-cross algorithm does not have polynomial time-complexity, however. Researchers have extended the criss-cross algorithm for many optimization-problems, including linear-fractional programming. The criss-cross a... | Criss-cross algorithm | 0.812105 |
18,561 | In mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear inequality constraints and nonlinear objective functions; there are criss-cross algorithms for linear-fractional programming... | Criss-cross algorithm | 0.812105 |
18,562 | A number of dualities relate the above theories. The A-model and B-model on two mirror manifolds are related by mirror symmetry, which has been described as a T-duality on a three-torus. The A-model and B-model on the same manifold are conjectured to be related by S-duality, which implies the existence of several new b... | Topological A-model | 0.812104 |
18,563 | The 2-dimensional worldsheet theory is an N = (2,2) supersymmetric sigma model, the (2,2) supersymmetry means that the fermionic generators of the supersymmetry algebra, called supercharges, may be assembled into a single Dirac spinor, which consists of two Majorana–Weyl spinors of each chirality. This sigma model is t... | Topological A-model | 0.812104 |
18,564 | In the paper Quantum Calabi–Yau and Classical Crystals, Andrei Okounkov, Nicolai Reshetikhin and Cumrun Vafa conjectured that the quantum A-model is dual to a classical melting crystal at a temperature equal to the inverse of the string coupling constant. This conjecture was interpreted in Quantum Foam and Topological ... | Topological A-model | 0.812104 |
18,565 | Since skeletal structures were introduced in the latter half of the 19th century, their appearance has undergone considerable evolution. The graphical conventions in use today date to the 1980s. Thanks to the adoption of the ChemDraw software package as a de facto industry standard (by American Chemical Society, Royal ... | Bond-line formula | 0.812101 |
18,566 | Sulfonate esters are often leaving groups in nucleophilic substitution reactions. See the articles on sulfonyl and sulfonate groups for further information. Bs for the brosyl (p-bromobenzenesulfonyl) group; OBs is the brosylate group Ms for the mesyl (methanesulfonyl) group; OMs is the mesylate group Ns for the nosyl (... | Bond-line formula | 0.812101 |
18,567 | In robust statistics, the convex hull provides one of the key components of a bagplot, a method for visualizing the spread of two-dimensional sample points. The contours of Tukey depth form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested fam... | Convex hull | 0.812099 |
18,568 | In quantum physics, the state space of any quantum system — the set of all ways the system can be prepared — is a convex hull whose extreme points are positive-semidefinite operators known as pure states and whose interior points are called mixed states. The Schrödinger–HJW theorem proves that any mixed state can in fa... | Convex hull | 0.812099 |
18,569 | In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a graph. It is one of several graph polynomials studied in algebraic graph theory. | Matching polynomial | 0.812098 |
18,570 | The Hosoya index of a graph G, its number of matchings, is used in chemoinformatics as a structural descriptor of a molecular graph. It may be evaluated as mG(1) (Gutman 1991). The third type of matching polynomial was introduced by Farrell (1980) as a version of the "acyclic polynomial" used in chemistry. | Matching polynomial | 0.812098 |
18,571 | Magma computer algebra system SageMath Number Theory Library PARI/GP Fast Library for Number Theory | Algorithmic number theory | 0.812098 |
18,572 | In mathematics and computer science, computational number theory, also known as algorithmic number theory, is the study of computational methods for investigating and solving problems in number theory and arithmetic geometry, including algorithms for primality testing and integer factorization, finding solutions to dio... | Algorithmic number theory | 0.812098 |
18,573 | In number theory, Brun's theorem states that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a finite value known as Brun's constant, usually denoted by B2 (sequence A065421 in the OEIS). Brun's theorem was proved by Viggo Brun in 1919, and it has historical importa... | Brun's constant | 0.81209 |
18,574 | In Euclidean geometry, the distance from a point to a line is the shortest distance from a given point to any point on an infinite straight line. It is the perpendicular distance of the point to the line, the length of the line segment which joins the point to nearest point on the line. The algebraic expression for cal... | Distance from a point to a line | 0.812089 |
18,575 | {\displaystyle y=m{\frac {(x_{0}+my_{0}-mk)}{m^{2}+1}}+k.} Using the equation for finding the distance between 2 points, d = ( X 2 − X 1 ) 2 + ( Y 2 − Y 1 ) 2 {\displaystyle d={\sqrt {(X_{2}-X_{1})^{2}+(Y_{2}-Y_{1})^{2}}}} , we can deduce that the formula to find the shortest distance between a line and a point is the ... | Distance from a point to a line | 0.812089 |
18,576 | In theoretical computer science and formal language theory, a formal language is empty if its set of valid sentences is the empty set. The emptiness problem is the question of determining whether a language is empty given some representation of it, such as a finite-state automaton. For an automaton having n {\displayst... | Emptiness problem | 0.812083 |
18,577 | In the theory of stochastic processes in probability theory and statistics, a nuisance variable is a random variable that is fundamental to the probabilistic model, but that is of no particular interest in itself or is no longer of any interest: one such usage arises for the Chapman–Kolmogorov equation. For example, a ... | Nuisance variable | 0.812083 |
18,578 | Whether the estimator is unbiased in this case depends upon the sample selection process; see the inspection paradox. There are a variety of functions that are used to calculate statistics. Some include: Sample mean, sample median, and sample mode Sample variance and sample standard deviation Sample quantiles besides t... | Sample statistic | 0.812077 |
18,579 | Some examples of statistics are: "In a recent survey of Americans, 52% of Republicans say global warming is happening. "In this case, "52%" is a statistic, namely the percentage of Republicans in the survey sample who believe in global warming. The population is the set of all Republicans in the United States, and the ... | Sample statistic | 0.812077 |
18,580 | Important potential properties of statistics include completeness, consistency, sufficiency, unbiasedness, minimum mean square error, low variance, robustness, and computational convenience. | Sample statistic | 0.812077 |
18,581 | Drawing from this observation, Girard described directly this operator from the proof and has given a formula, the so-called execution formula, encoding the process of cut elimination at the level of operators. One of the first significant applications of GoI was a better analysis of Lamping's algorithm for optimal red... | Geometry of interaction | 0.812077 |
18,582 | The Geometry of Interaction (GoI) was introduced by Jean-Yves Girard shortly after his work on linear logic. In linear logic, proofs can be seen as various kinds of networks as opposed to the flat tree structures of sequent calculus. To distinguish the real proof nets from all the possible networks, Girard devised a cr... | Geometry of interaction | 0.812077 |
18,583 | Once the periodontal pockets exceed 6 mm in depth, the effectiveness of deposit removal begins to decrease, and the likelihood of complete healing after one procedure begins to decline as well. The more severe the infection prior to intervention, the greater the effort required to arrest its progress and return the pat... | Scaling and root planing | 0.812076 |
18,584 | A scaling and root planing procedure is to be considered effective if the patient is subsequently able to maintain their periodontal health without further bone or attachment loss and if it prevents recurrent infection with periodontal pathogens.The long term effectiveness of scaling and root planing depends upon a num... | Scaling and root planing | 0.812076 |
18,585 | Several systematic reviews have been made of the effectiveness of scaling and root planing as evidence-based dentistry. A Cochrane review by Worthington et al. in 2013 considered only scaling and polishing of the teeth, but not root planing. After examining 88 papers they found only three studies that met all their req... | Scaling and root planing | 0.812076 |
18,586 | Since it is of the utmost importance to remove the entirety of the deposit in each periodontal pocket, attention to detail during this procedure is crucial. Therefore, depending on the depth of the pocket and amount of calculus deposit versus soft biofilm deposit, hand instruments may be used to complete the fine hand ... | Scaling and root planing | 0.812076 |
18,587 | Ultrasonic scalers vibrate at a high frequency to help with removing stain, plaque and calculus. In addition, ultrasonic scalers create tiny air bubbles through a process known as cavitation. | Scaling and root planing | 0.812076 |
18,588 | Polishing does not remove calculus, but only some plaque and stains, and should therefore be done only in conjunction with scaling. Often, an electric device, known as an ultrasonic scaler, sonic scaler, or power scaler may be used during scaling and root planing. | Scaling and root planing | 0.812076 |
18,589 | The objective for periodontal scaling and root planing is to remove dental plaque and calculus (tartar), which house bacteria that release toxins which cause inflammation to the gum tissue and surrounding bone. Planing often removes some of the cementum or dentine from the tooth.Removal of adherent plaque and calculus ... | Scaling and root planing | 0.812076 |
18,590 | Once the bacteria and calculus are removed from the periodontal pocket, the tissue can begin to heal. The inflammation dissipates as the infection declines, allowing the swelling to decrease which results in the gums once again forming an effective seal between the root of the tooth and the outside environment. However... | Scaling and root planing | 0.812076 |
18,591 | Brushing and flossing are effective only at removing the soft materia alba and biofilm in supragingival areas, and in pockets up to 3 mm deep. Even the best brushing and flossing is ineffective at cleaning pockets of greater depths, and are never effective in removing calculus. Therefore, in order to remove the causati... | Scaling and root planing | 0.812076 |
18,592 | These processes will persist, causing greater damage, until the infectious bacterial agents (plaque) and local irritating factors (calculus) are removed. In order to effectively remove these at this stage in the disease process, brushing and flossing are no longer sufficient. This is due to several factors, the most im... | Scaling and root planing | 0.812076 |
18,593 | This results in an ulceration in the lining of the tissue, which begins to break down the attachment of the gum to the tooth. Gingival attachment begins to loosen further as the bacterial plaque continues to invade the space created by the swelling it causes. This plaque eventually transforms into calculus, and the pro... | Scaling and root planing | 0.812076 |
18,594 | At a depth of 4 mm or greater, the vertical space between the tooth and surrounding gum becomes known as a periodontal pocket. Because tooth brush and floss cannot reach the bottom of a gum pocket 4–5 mm deep, bacteria stagnate in these sites and have the opportunity to proliferate into periodontal disease-causing colo... | Scaling and root planing | 0.812076 |
18,595 | Scaling and root planing, also known as conventional periodontal therapy, non-surgical periodontal therapy or deep cleaning, is a procedure involving removal of dental plaque and calculus (scaling or debridement) and then smoothing, or planing, of the (exposed) surfaces of the roots, removing cementum or dentine that i... | Scaling and root planing | 0.812076 |
18,596 | Through this absorption of calcium and phosphorus from the saliva, oral biofilm is transformed from the soft, easily removable form into a hard substance known as calculus. Commonly known as 'tartar', calculus provides a base for new layers of plaque biofilm to settle on and builds up over time. Calculus cannot be remo... | Scaling and root planing | 0.812076 |
18,597 | and using all things "air and space" as a hook for youth to embark on greater experimentation, studies, and careers in the region's burgeoning space-related industries. One of AYVF's best-known programs, "STEAM@TheMall", served over 200,000 its first two years at the most popular shopping malls and provided free weeken... | STEAM Education | 0.812074 |
18,598 | The artists collaborate with the teachers to integrate math and science into the arts. American Lisa La Bonte, CEO of the Arab Youth Venture Foundation based in the United Arab Emirates, uses the STEAM acronym, but her work does not include arts integration. Starting in 2007, La Bonte created and ran high profile free ... | STEAM Education | 0.812074 |
18,599 | An elevator out in deep space, far from any planet, could mimic a gravitational field to its occupants if it could be accelerated continuously "upward". Whether the acceleration is from motion or from gravity makes no difference in the laws of physics. One can also understand it in terms of the equivalence of so-called... | Paradox of radiation of charged particles in a gravitational field | 0.812074 |
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