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10,800 | At the time it was unclear how many of each work together. In addition, when people spoke of a dimer, it was unclear whether they were referring to αβ or to (αβ)2. Chetverin suggested to call αβ a protomer and (αβ)2 a diprotomer. Protomers usually arrange in cyclic symmetry to form closed point group symmetries. In che... | Protomer | 0.81855 |
10,801 | In structural biology, a protomer is the structural unit of an oligomeric protein. It is the smallest unit composed of at least two different protein chains that form a larger hetero-oligomer by association of two or more copies of this unit. The term was introduced by Chetverin to make nomenclature in the Na/K-ATPase ... | Protomer | 0.81855 |
10,802 | Ultracold atom trapping in optical lattices is an experimental tool commonly used in condensed matter physics, and in atomic, molecular, and optical physics. The method involves using optical lasers to form an interference pattern, which acts as a lattice, in which ions or atoms can be placed at very low temperatures. ... | Condensed Matter | 0.818546 |
10,803 | Theoretical condensed matter physics involves the use of theoretical models to understand properties of states of matter. These include models to study the electronic properties of solids, such as the Drude model, the band structure and the density functional theory. Theoretical models have also been developed to study... | Condensed Matter | 0.818546 |
10,804 | Some states of matter exhibit symmetry breaking, where the relevant laws of physics possess some form of symmetry that is broken. A common example is crystalline solids, which break continuous translational symmetry. Other examples include magnetized ferromagnets, which break rotational symmetry, and more exotic states... | Condensed Matter | 0.818546 |
10,805 | One can also replace presheaves of sets by presheaves of simplicial sets (or of infinity groupoids). Then, in presence of an appropriate homotopic machinery one can develop a notion of derived stack as such a presheaf on the infinity category of derived affine schemes, which is satisfying certain infinite categorical v... | Computational algebraic geometry | 0.818543 |
10,806 | The language of schemes, stacks and generalizations has proved to be a valuable way of dealing with geometric concepts and became cornerstones of modern algebraic geometry. Algebraic stacks can be further generalized and for many practical questions like deformation theory and intersection theory, this is often the mos... | Computational algebraic geometry | 0.818543 |
10,807 | Versions of a tropical geometry, of an absolute geometry over a field of one element and an algebraic analogue of Arakelov's geometry were realized in this setup. Another formal generalization is possible to universal algebraic geometry in which every variety of algebras has its own algebraic geometry. The term variety... | Computational algebraic geometry | 0.818543 |
10,808 | Sheaves can be furthermore generalized to stacks in the sense of Grothendieck, usually with some additional representability conditions leading to Artin stacks and, even finer, Deligne–Mumford stacks, both often called algebraic stacks. Sometimes other algebraic sites replace the category of affine schemes. For example... | Computational algebraic geometry | 0.818543 |
10,809 | The gluing is along Zariski topology; one can glue within the category of locally ringed spaces, but also, using the Yoneda embedding, within the more abstract category of presheaves of sets over the category of affine schemes. The Zariski topology in the set theoretic sense is then replaced by a Grothendieck topology.... | Computational algebraic geometry | 0.818543 |
10,810 | The modern approaches to algebraic geometry redefine and effectively extend the range of basic objects in various levels of generality to schemes, formal schemes, ind-schemes, algebraic spaces, algebraic stacks and so on. The need for this arises already from the useful ideas within theory of varieties, e.g. the formal... | Computational algebraic geometry | 0.818543 |
10,811 | For example, the circle of equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} is a rational curve, as it has the parametric equation x = 2 t 1 + t 2 {\displaystyle x={\frac {2\,t}{1+t^{2}}}} y = 1 − t 2 1 + t 2 , {\displaystyle y={\frac {1-t^{2}}{1+t^{2}}}\,,} which may also be viewed as a rational map from the... | Computational algebraic geometry | 0.818543 |
10,812 | In contrast to the preceding sections, this section concerns only varieties and not algebraic sets. On the other hand, the definitions extend naturally to projective varieties (next section), as an affine variety and its projective completion have the same field of functions. If V is an affine variety, its coordinate r... | Computational algebraic geometry | 0.818543 |
10,813 | Thus its elements are called the irreducible components of the algebraic set. An irreducible algebraic set is also called a variety. It turns out that an algebraic set is a variety if and only if it may be defined as the vanishing set of a prime ideal of the polynomial ring. Some authors do not make a clear distinction... | Computational algebraic geometry | 0.818543 |
10,814 | For various reasons we may not always want to work with the entire ideal corresponding to an algebraic set U. Hilbert's basis theorem implies that ideals in k are always finitely generated. An algebraic set is called irreducible if it cannot be written as the union of two smaller algebraic sets. Any algebraic set is a ... | Computational algebraic geometry | 0.818543 |
10,815 | One of the founding methods of this area is the theory of Gröbner bases, introduced by Bruno Buchberger in 1965. Another founding method, more specially devoted to real algebraic geometry, is the cylindrical algebraic decomposition, introduced by George E. Collins in 1973. See also: derived algebraic geometry. | Computational algebraic geometry | 0.818543 |
10,816 | The prototypical examples are the elliptic curves, which have a rich theory. They were instrumental in the proof of Fermat's Last Theorem and are also used in elliptic-curve cryptography. In parallel with the abstract trend of the algebraic geometry, which is concerned with general statements about varieties, methods f... | Computational algebraic geometry | 0.818543 |
10,817 | One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds. This is obtained by extending the notion of point: In classical algebra... | Computational algebraic geometry | 0.818543 |
10,818 | A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers. It consists mainly of algorithm design and software development for the stud... | Computational algebraic geometry | 0.818543 |
10,819 | The mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more generally to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real algebraic varieties. Diophantine geometry and, more generally, arithmetic geo... | Computational algebraic geometry | 0.818543 |
10,820 | As a study of systems of polynomial equations in several variables, the subject of algebraic geometry begins with finding specific solutions via equation solving, and then proceeds to understand the intrinsic properties of the totality of solutions of a system of equations. This understanding requires both conceptual t... | Computational algebraic geometry | 0.818543 |
10,821 | Basic questions involve the study of points of special interest like singular points, inflection points and points at infinity. More advanced questions involve the topology of the curve and the relationship between curves defined by different equations. Algebraic geometry occupies a central place in modern mathematics ... | Computational algebraic geometry | 0.818543 |
10,822 | These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. | Computational algebraic geometry | 0.818543 |
10,823 | Algebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in alge... | Computational algebraic geometry | 0.818543 |
10,824 | One may cite counting the number of connected components, testing if two points are in the same components or computing a Whitney stratification of a real algebraic set. They have a complexity of d O ( n 2 ) {\displaystyle d^{O(n^{2})}} , but the constant involved by O notation is so high that using them to solve any n... | Computational algebraic geometry | 0.818543 |
10,825 | Such algorithms are rarely implemented because, on most entries Faugère's F4 and F5 algorithms have a better practical efficiency and probably a similar or better complexity (probably because the evaluation of the complexity of Gröbner basis algorithms on a particular class of entries is a difficult task which has been... | Computational algebraic geometry | 0.818543 |
10,826 | The basic general algorithms of computational geometry have a double exponential worst case complexity. More precisely, if d is the maximal degree of the input polynomials and n the number of variables, their complexity is at most d 2 c n {\displaystyle d^{2^{cn}}} for some constant c, and, for some inputs, the complex... | Computational algebraic geometry | 0.818543 |
10,827 | During the same period, Blaise Pascal and Gérard Desargues approached geometry from a different perspective, developing the synthetic notions of projective geometry. Pascal and Desargues also studied curves, but from the purely geometrical point of view: the analog of the Greek ruler and compass construction. Ultimatel... | Computational algebraic geometry | 0.818543 |
10,828 | Such techniques of applying geometrical constructions to algebraic problems were also adopted by a number of Renaissance mathematicians such as Gerolamo Cardano and Niccolò Fontana "Tartaglia" on their studies of the cubic equation. The geometrical approach to construction problems, rather than the algebraic one, was f... | Computational algebraic geometry | 0.818543 |
10,829 | Apollonius in the Conics further developed a method that is so similar to analytic geometry that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years.His application of reference lines, a diameter and a tangent is essentially no different from our modern use of a coordinate frame, ... | Computational algebraic geometry | 0.818543 |
10,830 | Some of the roots of algebraic geometry date back to the work of the Hellenistic Greeks from the 5th century BC. The Delian problem, for instance, was to construct a length x so that the cube of side x contained the same volume as the rectangular box a2b for given sides a and b. Menaechmus (c. 350 BC) considered the pr... | Computational algebraic geometry | 0.818543 |
10,831 | Real algebraic geometry also investigates, more broadly, semi-algebraic sets, which are the solutions of systems of polynomial inequalities. For example, neither branch of the hyperbola of equation x y − 1 = 0 {\displaystyle xy-1=0} is a real algebraic variety. However, the branch in the first quadrant is a semi-algebr... | Computational algebraic geometry | 0.818543 |
10,832 | Real algebraic geometry is the study of real algebraic varieties. The fact that the field of the real numbers is an ordered field cannot be ignored in such a study. For example, the curve of equation x 2 + y 2 − a = 0 {\displaystyle x^{2}+y^{2}-a=0} is a circle if a > 0 {\displaystyle a>0} , but has no real points if a... | Computational algebraic geometry | 0.818543 |
10,833 | Just as continuous functions are the natural maps on topological spaces and smooth functions are the natural maps on differentiable manifolds, there is a natural class of functions on an algebraic set, called regular functions or polynomial functions. A regular function on an algebraic set V contained in An is the rest... | Computational algebraic geometry | 0.818543 |
10,834 | In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that simultaneously satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as... | Computational algebraic geometry | 0.818543 |
10,835 | Given a set S of polynomials, when is S = I(V(S))?The answer to the first question is provided by introducing the Zariski topology, a topology on An whose closed sets are the algebraic sets, and which directly reflects the algebraic structure of k. Then U = V(I(U)) if and only if U is an algebraic set or equivalently a... | Computational algebraic geometry | 0.818543 |
10,836 | The V stands for variety (a specific type of algebraic set to be defined below). Given a subset U of An, can one recover the set of polynomials which generate it? If U is any subset of An, define I(U) to be the set of all polynomials whose vanishing set contains U. The I stands for ideal: if two polynomials f and g bot... | Computational algebraic geometry | 0.818543 |
10,837 | Let S be a set of polynomials in k. The vanishing set of S (or vanishing locus or zero set) is the set V(S) of all points in An where every polynomial in S vanishes. Symbolically, V ( S ) = { ( t 1 , … , t n ) ∣ p ( t 1 , … , t n ) = 0 for all p ∈ S } . {\displaystyle V(S)=\{(t_{1},\dots ,t_{n})\mid p(t_{1},\dots ,t_{n... | Computational algebraic geometry | 0.818543 |
10,838 | First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed. We consider the affine space of dimension n over k, denoted An(k) (or more simply An, when k is clear from the context). Wh... | Computational algebraic geometry | 0.818543 |
10,839 | Later, from about 1960, and largely led by Grothendieck, the idea of schemes was worked out, in conjunction with a very refined apparatus of homological techniques. After a decade of rapid development the field stabilized in the 1970s, and new applications were made, both to number theory and to more classical geometri... | Computational algebraic geometry | 0.818543 |
10,840 | B. L. van der Waerden, Oscar Zariski and André Weil developed a foundation for algebraic geometry based on contemporary commutative algebra, including valuation theory and the theory of ideals. One of the goals was to give a rigorous framework for proving the results of the Italian school of algebraic geometry. In part... | Computational algebraic geometry | 0.818543 |
10,841 | A body of mathematical theory complementary to symbolic methods called numerical algebraic geometry has been developed over the last several decades. The main computational method is homotopy continuation. This supports, for example, a model of floating point computation for solving problems of algebraic geometry. | Computational algebraic geometry | 0.818543 |
10,842 | One may date the origin of computational algebraic geometry to meeting EUROSAM'79 (International Symposium on Symbolic and Algebraic Manipulation) held at Marseille, France, in June 1979. At this meeting, Dennis S. Arnon showed that George E. Collins's Cylindrical algebraic decomposition (CAD) allows the computation of... | Computational algebraic geometry | 0.818543 |
10,843 | In chemistry, IUPAC changed its definition of standard temperature and pressure in 1982: Until 1982, STP was defined as a temperature of 273.15 K (0 °C, 32 °F) and an absolute pressure of exactly 1 atm (101.325 kPa). Since 1982, STP has been defined as a temperature of 273.15 K (0 °C, 32 °F) and an absolute pressure of... | Standard temperature and pressure | 0.81854 |
10,844 | Standard temperature and pressure (STP) are various standard sets of conditions for experimental measurements to be established to allow comparisons to be made between different sets of data. The most used standards are those of the International Union of Pure and Applied Chemistry (IUPAC) and the National Institute of... | Standard temperature and pressure | 0.81854 |
10,845 | In statistics The optimistic knowledge gradient is a approximation policy proposed by Xi Chen, Qihang Lin and Dengyong Zhou in 2013. This policy is created to solve the challenge of computationally intractable of large size of optimal computing budget allocation problem in binary/multi-class crowd labeling where each l... | Optimistic knowledge gradient | 0.818539 |
10,846 | In mathematics and particularly in algebra, a system of equations (either linear or nonlinear) is called consistent if there is at least one set of values for the unknowns that satisfies each equation in the system—that is, when substituted into each of the equations, they make each equation hold true as an identity. I... | Consistent and inconsistent equations | 0.818538 |
10,847 | The Euler equations can be generalized to any simple Lie algebra. The original Euler equations come from fixing the Lie algebra to be s o ( 3 ) {\displaystyle {\mathfrak {so}}(3)} , with generators t 1 , t 2 , t 3 {\displaystyle {t_{1},t_{2},t_{3}}} satisfying the relation = ϵ a b c t k {\displaystyle =\epsilon _{abc}... | Euler's equation of motion | 0.818534 |
10,848 | The study of the rumen required Hungate to develop techniques for culturing anaerobic microbes, and he also pioneered a quantitative approach to the study of microbes and their ecological activities that differentiated the relative contributions of species and catabolic pathways.Progress in microbial ecology has been t... | Microbial Ecology | 0.818534 |
10,849 | Sergei Winogradsky was one of the first researchers to attempt to understand microorganisms outside of the medical context—making him among the first students of microbial ecology and environmental microbiology—discovering chemosynthesis, and developing the Winogradsky column in the process. : 644 Beijerinck and Windog... | Microbial Ecology | 0.818534 |
10,850 | Biotechnology may be used alongside microbial ecology to address a number of environmental and economic challenges. For example, molecular techniques such as community fingerprinting or metagenomics can be used to track changes in microbial communities over time or assess their biodiversity. Managing the carbon cycle t... | Microbial Ecology | 0.818534 |
10,851 | Microbial ecology contributes to the evolution to many different parts of the world. For example, different microbial species evolved CRISPR dynamics and functions, allowing a better understanding of human health. | Microbial Ecology | 0.818534 |
10,852 | Microbial ecology (or environmental microbiology) is the ecology of microorganisms: their relationship with one another and with their environment. It concerns the three major domains of life—Eukaryota, Archaea, and Bacteria—as well as viruses.Microorganisms, by their omnipresence, impact the entire biosphere. Microbia... | Microbial Ecology | 0.818534 |
10,853 | Microbes exist in all areas, including homes, offices, commercial centers, and hospitals. In 2016, the journal Microbiome published a collection of various works studying the microbial ecology of the built environment.A 2006 study of pathogenic bacteria in hospitals found that their ability to survive varied by the typ... | Microbial Ecology | 0.818534 |
10,854 | Bracketing methods determine successively smaller intervals (brackets) that contain a root. When the interval is small enough, then a root has been found. They generally use the intermediate value theorem, which asserts that if a continuous function has values of opposite signs at the end points of an interval, then th... | Root-finding method | 0.818531 |
10,855 | : 19-- It does not require to compute the topological degree - it only requires to compute the signs of function values. The number of required evaluations is at least log 2 ( D / ϵ ) {\displaystyle \log _{2}(D/\epsilon )} , where D is the length of the longest edge of the characteristic polyhedron. : 11, Lemma.4.7 N... | Root-finding method | 0.818531 |
10,856 | The Poincaré–Miranda theorem gives a criterion for the existence of a root in a rectangle, but it is hard to verify, since it requires to evaluate the function on the entire boundary of the triangle. Another criterion is given by a theorem of Kronecker. It says that, if the topological degree of a function f on a recta... | Root-finding method | 0.818531 |
10,857 | The story sparked mainstream media attention, including a mention in The New York Times. The technology podcast This Week in Tech also discussed the incident.According to The Wall Street Journal, as summarized by MediaPost commentator Ross Fadner, "'The Internet is a Series of Tubes!' spawned a new slogan that became a... | Series of tubes | 0.818528 |
10,858 | The European Chapter on Combinatorial Optimization (also, EURO Working Group on Combinatorial Optimization, or EWG ECCO) is a working group whose objective is to promote original research in the field of combinatorial optimization at the European level. | European Chapter on Combinatorial Optimization | 0.818523 |
10,859 | The group is suitable for people who are presently engaged in Combinatorial Optimization, either in theoretical aspects or in business, industry or public administration applications. Currently (2022), the group has over 1,600 members from 75 countries. | European Chapter on Combinatorial Optimization | 0.818523 |
10,860 | The stereoscope is an instrument used to see the 3D overlapping aerial images. Reaching the requirements of the two chosen, overlapping images is simple. The principal points (central point of the image in geometry) of the two photos must be in different locations on the terrain. Another restriction is that the scale o... | Aerial photograph interpretation | 0.81852 |
10,861 | The cement chemist notation is not restricted to cement applications but is in fact a more general notation of oxide chemistry applicable to other domains than cement chemistry sensu stricto. For instance, in ceramics applications, the kaolinite formula can also be written in terms of oxides, thus the corresponding for... | Cement chemist notation | 0.818519 |
10,862 | Viral protein interactomes consist of interactions among viral or phage proteins. They were among the first interactome projects as their genomes are small and all proteins can be analyzed with limited resources. Viral interactomes are connected to their host interactomes, forming virus-host interaction networks. Some ... | Interactome | 0.818518 |
10,863 | In molecular biology, an interactome is the whole set of molecular interactions in a particular cell. The term specifically refers to physical interactions among molecules (such as those among proteins, also known as protein–protein interactions, PPIs; or between small molecules and proteins) but can also describe sets... | Interactome | 0.818518 |
10,864 | An alternative construction of W12 is obtained by use of the 'kitten' of R.T. Curtis, which was intended as a "hand calculator" to write down blocks one at a time. The kitten method is based on completing patterns in a 3x3 grid of numbers, which represent an affine geometry on the vector space F3xF3, an S(2,3,9) system... | Steiner triple system | 0.81851 |
10,865 | Established in 1971, IPL is one of the oldest journals in computer science. In its now over 50-year old history, IPL has published research contributions from leading figures in computer science research, including multiple Turing Award winners: Alan Perlis, Edsger Dijkstra, Donald Knuth, Robert Floyd, Stephen Cook, Ni... | Information Processing Letters | 0.81851 |
10,866 | This naturally covers topics in the broadly understood field of theoretical computer science, including algorithms, formal languages and automata, computational complexity, computational logic, distributed and parallel algorithms, computational geometry, learning theory, computational number theory, computational biolo... | Information Processing Letters | 0.81851 |
10,867 | Information Processing Letters is a peer-reviewed scientific journal in the field of computer science, published by Elsevier. The aim of the journal is to enable fast dissemination of results in the field of information processing in the form of short papers. Submissions are limited to nine double-spaced pages. The sco... | Information Processing Letters | 0.81851 |
10,868 | The Steiner system S(5, 8, 24), also known as the Witt design or Witt geometry, was first described by Carmichael (1931) and rediscovered by Witt (1938). This system is connected with many of the sporadic simple groups and with the exceptional 24-dimensional lattice known as the Leech lattice. The automorphism group of... | Steiner triple system | 0.81851 |
10,869 | The Earth, among other planets, orbits the Sun because the Sun exerts a gravitational pull that acts as a centripetal force, holding the Earth to it, which would otherwise go shooting off into space. If the Sun's pull is considered an action, then Earth simultaneously exerts a reaction as a gravitational pull on the Su... | Reaction (physics) | 0.818504 |
10,870 | One problem frequently observed by physics educators is that students tend to apply Newton's third law to pairs of 'equal and opposite' forces acting on the same object. This is incorrect; the third law refers to forces on two different objects. In contrast, a book lying on a table is subject to a downward gravitationa... | Reaction (physics) | 0.818504 |
10,871 | The forces on ball and player are both explained by their nearness, which results in a pair of contact forces (ultimately due to electric repulsion). That this nearness is caused by a decision of the player has no bearing on the physical analysis. As far as the physics is concerned, the labels 'action' and 'reaction' c... | Reaction (physics) | 0.818504 |
10,872 | Safety shutoff valve Safety shut-off valves are safety valves used to close a line and stop the flow of material.Safety valve A safety valve is a valve mechanism for the automatic release of a gas from a boiler, pressure vessel, or other system when the pressure or temperature exceeds preset limits.Salt bridge A salt b... | Bipolar plate | 0.818501 |
10,873 | e.g. 1 milligram per kilogram. 1 part in 106.Parts per million by volume In atmospheric chemistry and in air pollution regulations, the parts per notation is commonly expressed with a v following, such as ppmv, to indicate parts per million by volume.Pascal The pascal (symbol: Pa) is the SI derived unit of pressure, st... | Bipolar plate | 0.818501 |
10,874 | Palladium Palladium is a rare and lustrous silvery-white metal with the symbol Pd, and its atomic number is 46.Parallel circuit Parallel circuit, if two or more components are connected in parallel they have the same potential difference (voltage) across their ends. The potential differences across the components are t... | Bipolar plate | 0.818501 |
10,875 | For O K {\displaystyle {\mathcal {O}}_{K}} the ring of integers of an algebraic number field K {\displaystyle K} , an element α ∈ O K {\displaystyle \alpha \in {\mathcal {O}}_{K}} is a unit if and only if N K / Q ( α ) = ± 1 {\displaystyle N_{K/\mathbb {Q} }(\alpha )=\pm 1} . For instance N Q ( ζ 3 ) / Q ( ζ 3 ) = 1 {\... | Field norm | 0.818492 |
10,876 | Let K be a field and L a finite extension (and hence an algebraic extension) of K. The field L is then a finite dimensional vector space over K. Multiplication by α, an element of L, m α: L → L {\displaystyle m_{\alpha }\colon L\to L} m α ( x ) = α x {\displaystyle m_{\alpha }(x)=\alpha x} ,is a K-linear transformation... | Field norm | 0.818492 |
10,877 | Reduce became free software in 2008. As of today, the most popular commercial systems are Mathematica and Maple, which are commonly used by research mathematicians, scientists, and engineers. Freely available alternatives include SageMath (which can act as a front-end to several other free and nonfree CAS). Other signi... | Computer Algebra System | 0.818492 |
10,878 | The Texas Instruments company in 1995 released the TI-92 calculator with a CAS based on the software Derive; the TI-Nspire series replaced Derive in 2007. The TI-89 series, first released in 1998, also contains a CAS. Casio released their first CAS calculator with the CFX-9970G and succeeded it with the Algebra FX Seri... | Computer Algebra System | 0.818492 |
10,879 | MATHLAB ("mathematical laboratory") should not be confused with MATLAB ("matrix laboratory"), which is a system for numerical computation built 15 years later at the University of New Mexico. In 1987, Hewlett-Packard introduced the first hand-held calculator CAS with the HP-28 series, and it was possible, for the first... | Computer Algebra System | 0.818492 |
10,880 | Computer algebra systems began to appear in the 1960s and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence. A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martinus Vel... | Computer Algebra System | 0.818492 |
10,881 | For example, the computation of polynomial greatest common divisors is systematically used for the simplification of expressions involving fractions. This large amount of required computer capabilities explains the small number of general-purpose computer algebra systems. Significant systems include Axiom, Maxima, Magm... | Computer Algebra System | 0.818492 |
10,882 | General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as: a user interface allowing a user to enter and display mathematical fo... | Computer Algebra System | 0.818492 |
10,883 | A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th c... | Computer Algebra System | 0.818492 |
10,884 | In petroleum geology, the maturity of a rock is a measure of its state in terms of hydrocarbon generation. Maturity is established using a combination of geochemical and basin modelling techniques. Rocks with high total organic carbon, (termed source rocks), will alter under increasing temperature such that the organic... | Thermal maturity | 0.818487 |
10,885 | Leading examples are the gravitational potential and the electric potential, from which the motion of gravitating or electrically charged bodies may be obtained. Specific forces have associated potentials, including the Coulomb potential, the van der Waals potential, the Lennard-Jones potential and the Yukawa potential... | Potential | 0.818484 |
10,886 | Several languages have a potential mood, a grammatical construction that indicates that something is potential. These include Finnish, Japanese, and Sanskrit.In physics, a potential may refer to the scalar potential or to the vector potential. In either case, it is a field defined in space, from which many important ph... | Potential | 0.818484 |
10,887 | Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple release of energy by objects to the realization of abilities in people. The... | Potential | 0.818484 |
10,888 | Ronald Fisher introduced random effects models to study the correlations of trait values between relatives. In the 1950s, Charles Roy Henderson provided best linear unbiased estimates of fixed effects and best linear unbiased predictions of random effects. Subsequently, mixed modeling has become a major area of statist... | Mixed models | 0.818479 |
10,889 | The standard reduction potentials in acidic aqueous solution for some common iron ions are given below: The red-purple tetrahedral ferrate(VI) anion is such a strong oxidizing agent that it oxidizes nitrogen and ammonia at room temperature, and even water itself in acidic or neutral solutions: 4 FeO2−4 + 10 H2O → 4 Fe3... | Iron compounds | 0.818476 |
10,890 | Due to its electronic structure, iron has a very large coordination and organometallic chemistry. Many coordination compounds of iron are known. A typical six-coordinate anion is hexachloroferrate(III), 3−, found in the mixed salt tetrakis(methylammonium) hexachloroferrate(III) chloride. | Iron compounds | 0.818476 |
10,891 | Iron shows the characteristic chemical properties of the transition metals, namely the ability to form variable oxidation states differing by steps of one and a very large coordination and organometallic chemistry: indeed, it was the discovery of an iron compound, ferrocene, that revolutionalized the latter field in th... | Iron compounds | 0.818476 |
10,892 | Pi bonds are usually weaker than sigma bonds. The C-C double bond, composed of one sigma and one pi bond, has a bond energy less than twice that of a C-C single bond, indicating that the stability added by the pi bond is less than the stability of a sigma bond. From the perspective of quantum mechanics, this bond's wea... | Pi electron | 0.818475 |
10,893 | The usage of the term "mathematical physics" is sometimes idiosyncratic. Certain parts of mathematics that initially arose from the development of physics are not, in fact, considered parts of mathematical physics, while other closely related fields are. For example, ordinary differential equations and symplectic geome... | Mathematical physicist | 0.818474 |
10,894 | The effort to put physical theories on a mathematically rigorous footing not only developed physics but also has influenced developments of some mathematical areas. For example, the development of quantum mechanics and some aspects of functional analysis parallel each other in many ways. The mathematical study of quant... | Mathematical physicist | 0.818474 |
10,895 | The term "mathematical physics" is sometimes used to denote research aimed at studying and solving problems in physics or thought experiments within a mathematically rigorous framework. In this sense, mathematical physics covers a very broad academic realm distinguished only by the blending of some mathematical aspect ... | Mathematical physicist | 0.818474 |
10,896 | The special and general theories of relativity require a rather different type of mathematics. This was group theory, which played an important role in both quantum field theory and differential geometry. This was, however, gradually supplemented by topology and functional analysis in the mathematical description of co... | Mathematical physicist | 0.818474 |
10,897 | Following mathematics: the theory of partial differential equation, variational calculus, Fourier analysis, potential theory, and vector analysis are perhaps most closely associated with mathematical physics. These were developed intensively from the second half of the 18th century (by, for example, D'Alembert, Euler, ... | Mathematical physicist | 0.818474 |
10,898 | (2006), Mathematical Methods for Physics and Engineering (3rd ed. ), Cambridge University Press, ISBN 978-0-521-86153-3 Stakgold, Ivar (2000), Boundary Value Problems of Mathematical Physics, Vol 1-2., Society for Industrial and Applied Mathematics, ISBN 0-89871-456-7 Starkovich, Steven P. (2021), The Structures of Mat... | Mathematical physicist | 0.818474 |
10,899 | (1970), Mathematical Methods of Physics (2nd ed. ), W. A. Benjamin, ISBN 0-8053-7002-1 Menzel, Donald H. (1961), Mathematical Physics, Dover Publications, ISBN 0-486-60056-4 Riley, Ken F.; Hobson, Michael P.; Bence, Stephen J. | Mathematical physicist | 0.818474 |
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