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19,500 | A logical framework is based on a general treatment of syntax, rules and proofs by means of a dependently typed lambda calculus. Syntax is treated in a style similar to, but more general than Per Martin-Löf's system of arities. To describe a logical framework, one must provide the following: A characterization of the c... | Edinburgh Logical Framework | 0.811464 |
19,501 | To implement this, Gray code can be used to efficiently step through multiples of a given polynomial. This is completely analogous to the sieving step in other sieving algorithms such as the Number Field Sieve or the index calculus algorithm. Instead of numbers one sieves through functions in F p {\displaystyle \mathb... | Function field sieve | 0.81146 |
19,502 | It is not surprising that there exist two similar algorithms, one with number fields and the other one with function fields. In fact there is an extensive analogy between these two kinds of global fields. The index calculus algorithm is much easier to state than the Function Field Sieve and the Number Field Sieve since... | Function field sieve | 0.81146 |
19,503 | There are two other well known algorithms that solve the discrete logarithm problem in sub-exponential time: the index calculus algorithm and a version of the Number Field Sieve. In their easiest forms both solve the DLP in a finite field of prime order but they can be expanded to solve the DLP in F p n {\displaystyle ... | Function field sieve | 0.81146 |
19,504 | Let C ( x , y ) {\displaystyle C(x,y)} be a polynomial defining an algebraic curve over a finite field F p {\displaystyle \mathbb {F} _{p}} . A function field may be viewed as the field of fractions of the affine coordinate ring F p / ( C ( x , y ) ) {\displaystyle \mathbb {F} _{p}/(C(x,y))} , where ( C ( x , y ) ) {\... | Function field sieve | 0.81146 |
19,505 | Previous work includes the work of D. Coppersmith about the DLP in fields of characteristic two. The discrete logarithm problem in a finite field consists of solving the equation a x = b {\displaystyle a^{x}=b} for a , b ∈ F p n {\displaystyle a,b\in \mathbb {F} _{p^{n}}} , p {\displaystyle p} a prime number and n {\di... | Function field sieve | 0.81146 |
19,506 | It is necessary to use established weather stations that have a long unbroken records so that annual means and other statistics can be determined. Ablation (Ba) is determined from temperature observed at weather stations near the glacier. Daily maximum and minimum temperatures are converted to glacier ablation using tw... | Surface mass balance | 0.811457 |
19,507 | The distribution of seasonal accumulation and ablation rates are measured in-situ. Traditional field methods are combined with remote sensing techniques to track changes in mass, geometry and the flow behaviour of the two glaciers. These investigations contribute to the Swiss Glacier Monitoring Network and the Internat... | Surface mass balance | 0.811457 |
19,508 | Relational databases have often replaced legacy hierarchical databases and network databases, because RDBMS were easier to implement and administer. Nonetheless, relational stored data received continued, unsuccessful challenges by object database management systems in the 1980s and 1990s, (which were introduced in an ... | Database constraints | 0.811454 |
19,509 | Connolly and Begg define database management system (DBMS) as a "software system that enables users to define, create, maintain and control access to the database". RDBMS is an extension of that acronym that is sometimes used when the underlying database is relational. An alternative definition for a relational databas... | Database constraints | 0.811454 |
19,510 | In order for a database management system (DBMS) to operate efficiently and accurately, it must use ACID transactions. | Database constraints | 0.811454 |
19,511 | PTR's which satisfy additional algebraic conditions are given other names. These names are not uniformly applied in the literature. The following listing of names and properties is taken from Dembowski (1968, p. 129). | Planar ternary ring | 0.811453 |
19,512 | A coding interview, technical interview, programming interview or Microsoft interview is a technical problem-based job interview technique to assess applicants for a computer programming or software development position. Modern coding interview techniques were pioneered by Microsoft during the 1990s and adopted by othe... | Coding interview | 0.811441 |
19,513 | The nuclear chemistry associated with the nuclear fuel cycle can be divided into two main areas, one area is concerned with operation under the intended conditions while the other area is concerned with maloperation conditions where some alteration from the normal operating conditions has occurred or (more rarely) an a... | Nuclear Chemistry | 0.811438 |
19,514 | He also discovered the phenomena of radioactive recoil and nuclear isomerism, and pioneered rubidium–strontium dating. In 1938, Hahn, Lise Meitner and Fritz Strassmann discovered nuclear fission, for which Hahn received the 1944 Nobel Prize for Chemistry. Nuclear fission was the basis for nuclear reactors and nuclear w... | Nuclear Chemistry | 0.811438 |
19,515 | In 1936 Cornell University Press published a book in English (and later in Russian) titled Applied Radiochemistry, which contained the lectures given by Hahn when he was a visiting professor at Cornell University in Ithaca, New York, in 1933. This important publication had a major influence on almost all nuclear chemis... | Nuclear Chemistry | 0.811438 |
19,516 | In addition, they bombarded aluminium and magnesium with neutrons to make new radioisotopes. In the early 1920s Otto Hahn created a new line of research. Using the "emanation method", which he had recently developed, and the "emanation ability", he founded what became known as "applied radiochemistry" for the researchi... | Nuclear Chemistry | 0.811438 |
19,517 | This is the UNiversal EXtraction process which was developed in Russia and the Czech Republic, it is a process designed to remove all of the most troublesome (Sr, Cs and minor actinides) radioisotopes from the raffinates left after the extraction of uranium and plutonium from used nuclear fuel. The chemistry is based u... | Nuclear Chemistry | 0.811438 |
19,518 | Radiochemistry is the chemistry of radioactive materials, in which radioactive isotopes of elements are used to study the properties and chemical reactions of non-radioactive isotopes (often within radiochemistry the absence of radioactivity leads to a substance being described as being inactive as the isotopes are sta... | Nuclear Chemistry | 0.811438 |
19,519 | Radiation chemistry is the study of the chemical effects of radiation on matter; this is very different from radiochemistry as no radioactivity needs to be present in the material which is being chemically changed by the radiation. An example is the conversion of water into hydrogen gas and hydrogen peroxide. Prior to ... | Nuclear Chemistry | 0.811438 |
19,520 | As a result, nuclear chemistry greatly assists the understanding of medical treatments (such as cancer radiotherapy) and has enabled these treatments to improve. It includes the study of the production and use of radioactive sources for a range of processes. These include radiotherapy in medical applications; the use o... | Nuclear Chemistry | 0.811438 |
19,521 | Nuclear chemistry is the sub-field of chemistry dealing with radioactivity, nuclear processes, and transformations in the nuclei of atoms, such as nuclear transmutation and nuclear properties. It is the chemistry of radioactive elements such as the actinides, radium and radon together with the chemistry associated with... | Nuclear Chemistry | 0.811438 |
19,522 | Cosmogenic isotopes are formed by the interaction of cosmic rays with the nucleus of an atom. These can be used for dating purposes and for use as natural tracers. In addition, by careful measurement of some ratios of stable isotopes it is possible to obtain new insights into the origin of bullets, ages of ice samples,... | Nuclear Chemistry | 0.811438 |
19,523 | Another important area of nuclear chemistry is the study of how fission products interact with surfaces; this is thought to control the rate of release and migration of fission products both from waste containers under normal conditions and from power reactors under accident conditions. Like chromate and molybdate, the... | Nuclear Chemistry | 0.811438 |
19,524 | Some methods first developed within nuclear chemistry and physics have become so widely used within chemistry and other physical sciences that they may be best thought of as separate from normal nuclear chemistry. For example, the isotope effect is used so extensively to investigate chemical mechanisms and the use of c... | Nuclear Chemistry | 0.811438 |
19,525 | Radiochemistry, radiation chemistry and nuclear chemical engineering play a very important role for uranium and thorium fuel precursors synthesis, starting from ores of these elements, fuel fabrication, coolant chemistry, fuel reprocessing, radioactive waste treatment and storage, monitoring of radioactive elements rel... | Nuclear Chemistry | 0.811438 |
19,526 | Nuclear spectroscopy are methods that use the nucleus to obtain information of the local structure in matter. Important methods are NMR (see below), Mössbauer spectroscopy and Perturbed angular correlation. These methods use the interaction of the hyperfine field with the nucleus' spin. The field can be magnetic or/and... | Nuclear Chemistry | 0.811438 |
19,527 | It includes the study of the chemical effects resulting from the absorption of radiation within living animals, plants, and other materials. The radiation chemistry controls much of radiation biology as radiation has an effect on living things at the molecular scale. To explain it another way, the radiation alters the ... | Nuclear Chemistry | 0.811438 |
19,528 | The Poincaré conjecture, proved in 2002/2003 by Grigori Perelman, is a prominent application of this idea. The influence is not unidirectional, though. For example, algebraic topology makes use of Eilenberg–MacLane spaces which are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on ... | Symmetry point group | 0.811438 |
19,529 | Algebraic topology is another domain which prominently associates groups to the objects the theory is interested in. There, groups are used to describe certain invariants of topological spaces. They are called "invariants" because they are defined in such a way that they do not change if the space is subjected to some ... | Symmetry point group | 0.811438 |
19,530 | An important elaboration of the concept of a group occurs if G is endowed with additional structure, notably, of a topological space, differentiable manifold, or algebraic variety. If the group operations m (multiplication) and i (inversion), m: G × G → G , ( g , h ) ↦ g h , i: G → G , g ↦ g − 1 , {\displaystyle m:G\ti... | Symmetry point group | 0.811438 |
19,531 | For example, if G is finite, it is known that V above decomposes into irreducible parts (see Maschke's theorem). These parts, in turn, are much more easily manageable than the whole V (via Schur's lemma). Given a group G, representation theory then asks what representations of G exist. | Symmetry point group | 0.811438 |
19,532 | The theory of transformation groups forms a bridge connecting group theory with differential geometry. A long line of research, originating with Lie and Klein, considers group actions on manifolds by homeomorphisms or diffeomorphisms. The groups themselves may be discrete or continuous. | Symmetry point group | 0.811438 |
19,533 | Aside from the motivation of energy-efficient computation, reversible logic gates offered practical improvements of bit-manipulation transforms in cryptography and computer graphics. Since the 1980s, reversible circuits have attracted interest as components of quantum algorithms, and more recently in photonic and nano-... | Reversible computation | 0.811437 |
19,534 | There are two major, closely related types of reversibility that are of particular interest for this purpose: physical reversibility and logical reversibility.A process is said to be physically reversible if it results in no increase in physical entropy; it is isentropic. There is a style of circuit design ideally exhi... | Reversible computation | 0.811437 |
19,535 | Reversible computing is any model of computation where the computational process, to some extent, is time-reversible. In a model of computation that uses deterministic transitions from one state of the abstract machine to another, a necessary condition for reversibility is that the relation of the mapping from states t... | Reversible computation | 0.811437 |
19,536 | Landauer's principle (and indeed, the second law of thermodynamics) can also be understood to be a direct logical consequence of the underlying reversibility of physics, as is reflected in the general Hamiltonian formulation of mechanics, and in the unitary time-evolution operator of quantum mechanics more specifically... | Reversible computation | 0.811437 |
19,537 | }}\quad &q(a,b)=q(a,c)\Leftrightarrow q(a,bc)=1,\\{\text{3. }}\quad &q(a,b)=q(c,d)\Rightarrow \exists x\mid q(a,b)=q(a,x),q(c,d)=q(c,x)\end{aligned}}.} Every field F gives rise to a Q-structure by taking G to be F∗/F∗2, Q the set of Brauer classes of quaternion algebras in the Brauer group of F with the split quaternio... | Quaternionic structure | 0.811435 |
19,538 | In mathematics, a quaternionic structure or Q-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field. A quaternionic structure is a triple (G, Q, q) where G is an elementary abelian group of exponent 2 with a distinguished element −1, Q is a pointed set with distinguished eleme... | Quaternionic structure | 0.811435 |
19,539 | The book also explores the conflicts between the academic knowledge of biologists and the local and craft knowledge of fishers. While the main focus of the book—the establishment of the structural components of Canadian fisheries biology—ends around 1940, its importance comes in part from its explication of the histori... | A Science on the Scales | 0.811434 |
19,540 | A Science on the Scales: The Rise of Canadian Atlantic Fisheries Biology, 1898-1939 is a 2011 book by Jennifer Hubbard. The book provides an analysis of Canadian fisheries history with the tools of the professional historian, when most earlier works on the topic came from fisheries scientists themselves.The book traces... | A Science on the Scales | 0.811434 |
19,541 | As in the previous example, the lower triangular matrix has a characteristic polynomial that is the product of its diagonal elements, The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots. The sum of the algebr... | Characteristic value | 0.811434 |
19,542 | The matrix A {\displaystyle A} is invertible if and only if every eigenvalue is nonzero. If A {\displaystyle A} is invertible, then the eigenvalues of A − 1 {\displaystyle A^{-1}} are 1 λ 1 , … , 1 λ n {\textstyle {\frac {1}{\lambda _{1}}},\ldots ,{\frac {1}{\lambda _{n}}}} and each eigenvalue's geometric multiplicity ... | Characteristic value | 0.811434 |
19,543 | Let A {\displaystyle A} be an arbitrary n × n {\displaystyle n\times n} matrix of complex numbers with eigenvalues λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} . Each eigenvalue appears μ A ( λ i ) {\displaystyle \mu _{A}(\lambda _{i})} times in this list, where μ A ( λ i ) {\displaystyle \mu _{A}(\l... | Characteristic value | 0.811434 |
19,544 | Phylogenetic nomenclature is a method of nomenclature for taxa in biology that uses phylogenetic definitions for taxon names as explained below. This contrasts with the traditional approach, in which taxon names are defined by a type, which can be a specimen or a taxon of lower rank, and a description in words. Phyloge... | Phylogenetic classification | 0.811433 |
19,545 | The Biotechnology Journal is a peer-reviewed scientific journal covering all aspects of biotechnology. | Biotechnology Journal | 0.811428 |
19,546 | In radiometry, radiant exitance or radiant emittance is the radiant flux emitted by a surface per unit area, whereas spectral exitance or spectral emittance is the radiant exitance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. This... | Photon exitance | 0.811425 |
19,547 | For centuries, mathematicians and philosophers wrestled with paradoxes involving division by zero or sums of infinitely many numbers. These questions arise in the study of motion and area. The ancient Greek philosopher Zeno of Elea gave several famous examples of such paradoxes. Calculus provides tools, especially the ... | Differential and integral calculus | 0.811424 |
19,548 | : 341–453 Applications of integral calculus include computations involving area, volume, arc length, center of mass, work, and pressure. : 685–700 More advanced applications include power series and Fourier series. Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. | Differential and integral calculus | 0.811424 |
19,549 | While many of the ideas of calculus had been developed earlier in Greece, China, India, Iraq, Persia, and Japan, the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on the work of earlier mathematicians to introduce its basic principles. The Hungarian polymath John von Neumann wr... | Differential and integral calculus | 0.811424 |
19,550 | Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'. There is evidence of an early f... | Differential and integral calculus | 0.811424 |
19,551 | In formal terms, the derivative is a linear operator which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function i... | Differential and integral calculus | 0.811424 |
19,552 | Differential calculus is the study of the definition, properties, and applications of the derivative of a function. The process of finding the derivative is called differentiation. Given a function and a point in the domain, the derivative at that point is a way of encoding the small-scale behavior of the function near... | Differential and integral calculus | 0.811424 |
19,553 | Physics makes particular use of calculus; all concepts in classical mechanics and electromagnetism are related through calculus. The mass of an object of known density, the moment of inertia of objects, and the potential energies due to gravitational and electromagnetic forces can all be found by the use of calculus. A... | Differential and integral calculus | 0.811424 |
19,554 | Calculus is also used to find approximate solutions to equations; in practice it is the standard way to solve differential equations and do root finding in most applications. Examples are methods such as Newton's method, fixed point iteration, and linear approximation. For instance, spacecraft use a variation of the Eu... | Differential and integral calculus | 0.811424 |
19,555 | Or, it can be used in probability theory to determine the expectation value of a continuous random variable given a probability density function. : 37 In analytic geometry, the study of graphs of functions, calculus is used to find high points and low points (maxima and minima), slope, concavity and inflection points. | Differential and integral calculus | 0.811424 |
19,556 | Calculus is used in every branch of the physical sciences,: 1 actuarial science, computer science, statistics, engineering, economics, business, medicine, demography, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired. It allows one to go from (non-constant) rates of... | Differential and integral calculus | 0.811424 |
19,557 | {\displaystyle {\frac {d}{dx}}(x^{2})=2x.} In this usage, the dx in the denominator is read as "with respect to x". : 79 Another example of correct notation could be: g ( t ) = t 2 + 2 t + 4 d d t g ( t ) = 2 t + 2 {\displaystyle {\begin{aligned}g(t)&=t^{2}+2t+4\\{d \over dt}g(t)&=2t+2\end{aligned}}} Even when calculus... | Differential and integral calculus | 0.811424 |
19,558 | Modern calculus was developed in 17th-century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (independently of each other, first publishing around the same time) but elements of it appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. | Differential and integral calculus | 0.811424 |
19,559 | Newton called his calculus "the science of fluxions", a term that endured in English schools into the 19th century. : 100 The first complete treatise on calculus to be written in English and use the Leibniz notation was not published until 1815. Since the time of Leibniz and Newton, many mathematicians have contributed... | Differential and integral calculus | 0.811424 |
19,560 | Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. : 51–52 The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second a... | Differential and integral calculus | 0.811424 |
19,561 | He is now regarded as an independent inventor of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the product rule and chain rule, in their differential and integral forms... | Differential and integral calculus | 0.811424 |
19,562 | He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. | Differential and integral calculus | 0.811424 |
19,563 | In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating flui... | Differential and integral calculus | 0.811424 |
19,564 | The formal study of calculus brought together Cavalieri's infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term. T... | Differential and integral calculus | 0.811424 |
19,565 | Infinitesimals get replaced by sequences of smaller and smaller numbers, and the infinitely small behavior of a function is found by taking the limiting behavior for these sequences. Limits were thought to provide a more rigorous foundation for calculus, and for this reason they became the standard approach during the ... | Differential and integral calculus | 0.811424 |
19,566 | Limits describe the behavior of a function at a certain input in terms of its values at nearby inputs. They capture small-scale behavior using the intrinsic structure of the real number system (as a metric space with the least-upper-bound property). In this treatment, calculus is a collection of techniques for manipula... | Differential and integral calculus | 0.811424 |
19,567 | From this point of view, calculus is a collection of techniques for manipulating infinitesimals. The symbols d x {\displaystyle dx} and d y {\displaystyle dy} were taken to be infinitesimal, and the derivative d y / d x {\displaystyle dy/dx} was their ratio.The infinitesimal approach fell out of favor in the 19th centu... | Differential and integral calculus | 0.811424 |
19,568 | Calculus is usually developed by working with very small quantities. Historically, the first method of doing so was by infinitesimals. These are objects which can be treated like real numbers but which are, in some sense, "infinitely small". For example, an infinitesimal number could be greater than 0, but less than an... | Differential and integral calculus | 0.811424 |
19,569 | In the realm of medicine, calculus can be used to find the optimal branching angle of a blood vessel so as to maximize flow. Calculus can be applied to understand how quickly a drug is eliminated from a body or how quickly a cancerous tumour grows.In economics, calculus allows for the determination of maximal profit by... | Differential and integral calculus | 0.811424 |
19,570 | : 814 In biology, population dynamics starts with reproduction and death rates to model population changes. : 631 Green's theorem, which gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C, is applied in an instrument known as a planim... | Differential and integral calculus | 0.811424 |
19,571 | Alternatively, Newton's second law can be expressed by saying that the net force is equal to the object's mass times its acceleration, which is the time derivative of velocity and thus the second time derivative of spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its path... | Differential and integral calculus | 0.811424 |
19,572 | Johannes Kepler's work Stereometrica Doliorum formed the basis of integral calculus. Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.A significant work was a treatise, the origin being Kepler's methods, written by Bonaventura Cavalie... | Differential and integral calculus | 0.811424 |
19,573 | In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for c... | Differential and integral calculus | 0.811424 |
19,574 | History of engineering Greatest Engineering Achievements of the 20th Century History of chemical engineering History of electrical engineering History of mechanical engineering History of software engineering History of structural engineering Roman engineering Roman military engineering | Outline of engineering | 0.811423 |
19,575 | Engineering education Bachelor of Engineering Bachelor of Applied Science Bachelor of Technology Bachelor of Biomedical Engineering Bachelor of Computer Science Bachelor of Electrical Engineering Bachelor of Software Engineering Master of Engineering Master of Science in Engineering Master of Technology Diplôme d'Ingén... | Outline of engineering | 0.811423 |
19,576 | Index of aerospace engineering articles Index of electrical engineering articles Index of genetic engineering articles Index of mechanical engineering articles Index of software engineering articles | Outline of engineering | 0.811423 |
19,577 | Design (outline) Drawings Computer-aided design (CAD) Drafting Engineering design process Earthworks Ecological engineering methods Engineering, procurement and construction Engineering economics Cost Manufacturing cost Value-driven design Engineering overhead Engineering society Environmental engineering science Explo... | Outline of engineering | 0.811423 |
19,578 | altered to be more like those of Earth. Climate engineering (geoengineering) Software engineering Computer-aided engineering Knowledge engineering Language engineering Release engineering Teletraffic engineering Usability engineering Sustainable engineering Systems engineering – analysis, design, and control of giganti... | Outline of engineering | 0.811423 |
19,579 | Applied engineering – application of management, design, and technical skills for the design and integration of systems, the execution of new product designs, the improvement of manufacturing processes, and the management and direction of physical and/or technical functions of a firm or organization. Packaging engineer... | Outline of engineering | 0.811423 |
19,580 | In mathematics, and more specifically in abstract algebra, a pseudo-ring is one of the following variants of a ring: A rng, i.e., a structure satisfying all the axioms of a ring except for the existence of a multiplicative identity. A set R with two binary operations + and ⋅ such that (R, +) is an abelian group with id... | Pseudo-ring | 0.811421 |
19,581 | In software engineering, architecture tradeoff analysis method (ATAM) is a risk-mitigation process used early in the software development life cycle. ATAM was developed by the Software Engineering Institute at the Carnegie Mellon University. Its purpose is to help choose a suitable architecture for a software system by... | Architecture tradeoff analysis method | 0.811419 |
19,582 | Let X be a set and consider some equivalence relation on it. Let Q be the set of all equivalence classes in X. Then the map q: X → Q {\displaystyle q:X\to Q} that sends an element x to the equivalence class to which x belongs is a quotient. In the above example, Q is a subset of the power set H of X. In algebraic geome... | Quotient by an equivalence relation | 0.811418 |
19,583 | In trigonometry, for all n ∈ Z {\displaystyle n\in \mathbb {Z} } , the functions tan θ {\displaystyle \tan \theta } and sec θ {\displaystyle \sec \theta } are undefined for all θ = π ( n − 1 2 ) {\textstyle \theta =\pi \left(n-{\frac {1}{2}}\right)} , while the functions cot θ {\displaystyle \cot \theta } and csc... | Undefined (mathematics) | 0.811414 |
19,584 | The expression n 0 , n ≠ 0 {\displaystyle {\frac {n}{0}},n\neq 0} is undefined in arithmetic, as explained in division by zero (the 0 0 {\displaystyle {\frac {0}{0}}} expression is used in calculus to represent an indeterminate form). Mathematicians have different opinions as to whether 00 should be defined to equal 1,... | Undefined (mathematics) | 0.811414 |
19,585 | In topology, a topological space may be defined as a set of points endowed with certain properties, but in the general setting, the nature of these "points" is left entirely undefined. Likewise, in category theory, a category consists of "objects" and "arrows", which are again primitive, undefined terms. This allows su... | Undefined (mathematics) | 0.811414 |
19,586 | As these terms are not defined in terms of other concepts, they may be referred to as "undefined terms". A function is said to be "undefined" at points outside of its domain – for example, the real-valued function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} is undefined for negative x {\displaystyle x} (i.e., it assig... | Undefined (mathematics) | 0.811414 |
19,587 | In mathematics, the term undefined is often used to refer to an expression which is not assigned an interpretation or a value (such as an indeterminate form, which has the possibility of assuming different values). The term can take on several different meanings depending on the context. For example: In various branche... | Undefined (mathematics) | 0.811414 |
19,588 | Most compound flexure designs are composed of 3 fundamental types of flexure: Pin flexure- a thin bar or cylinder of material, constrains 3 degrees of freedom when geometry matches a notch cutout. Blade flexure- thin sheet of material, constrains 3 degrees of freedom. Notch flexure- thin cutout on both sides of a thick... | Flexure | 0.811413 |
19,589 | Jason Roberts and Derek So from the University of Ottawa Heart Institute validated a pharmacogenomics approach in patients undergoing percutaneous coronary intervention for acute coronary syndrome or stable coronary artery disease. This study is the first in medicine to incorporate point-of-care testing with genetics i... | Point-of-care genetic testing | 0.811413 |
19,590 | Ecology and evolution provide the explanatory context for biogeographical studies. Biogeographical patterns result from ecological processes that influence range distributions, such as migration and dispersal. and from historical processes that split populations or species into different areas. | Ecological science | 0.811411 |
19,591 | The largest scale of ecological organization is the biosphere: the total sum of ecosystems on the planet. Ecological relationships regulate the flux of energy, nutrients, and climate all the way up to the planetary scale. For example, the dynamic history of the planetary atmosphere's CO2 and O2 composition has been aff... | Ecological science | 0.811411 |
19,592 | ": 8 Ecological studies are necessarily holistic as opposed to reductionistic. Holism has three scientific meanings or uses that identify with ecology: 1) the mechanistic complexity of ecosystems, 2) the practical description of patterns in quantitative reductionist terms where correlations may be identified but nothin... | Ecological science | 0.811411 |
19,593 | Ecology is as much a biological science as it is a human science. Human ecology is an interdisciplinary investigation into the ecology of our species. "Human ecology may be defined: (1) from a bioecological standpoint as the study of man as the ecological dominant in plant and animal communities and systems; (2) from a... | Ecological science | 0.811411 |
19,594 | Other researchers have recently categorized other biomes, such as the human and oceanic microbiomes. To a microbe, the human body is a habitat and a landscape. Microbiomes were discovered largely through advances in molecular genetics, which have revealed a hidden richness of microbial diversity on the planet. The ocea... | Ecological science | 0.811411 |
19,595 | "Elaborate sexual displays and posturing are encountered in the behavioural ecology of animals. The birds-of-paradise, for example, sing and display elaborate ornaments during courtship. These displays serve a dual purpose of signalling healthy or well-adapted individuals and desirable genes. The displays are driven by... | Ecological science | 0.811411 |
19,596 | To be adapted to their environment and face predatory threats, organisms must balance their energy budgets as they invest in different aspects of their life history, such as growth, feeding, mating, socializing, or modifying their habitat. Hypotheses posited in behavioural ecology are generally based on adaptive princi... | Ecological science | 0.811411 |
19,597 | Predator-prey interactions are an introductory concept into food-web studies as well as behavioural ecology. Prey species can exhibit different kinds of behavioural adaptations to predators, such as avoid, flee, or defend. Many prey species are faced with multiple predators that differ in the degree of danger posed. | Ecological science | 0.811411 |
19,598 | This could include investigations of motile sperm of plants, mobile phytoplankton, zooplankton swimming toward the female egg, the cultivation of fungi by weevils, the mating dance of a salamander, or social gatherings of amoeba.Adaptation is the central unifying concept in behavioural ecology. Behaviours can be record... | Ecological science | 0.811411 |
19,599 | All organisms can exhibit behaviours. Even plants express complex behaviour, including memory and communication. Behavioural ecology is the study of an organism's behaviour in its environment and its ecological and evolutionary implications. Ethology is the study of observable movement or behaviour in animals. | Ecological science | 0.811411 |
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