All preceding applications, and many others, show that the resultant is a fundamental tool in computer algebra. In fact most computer algebra systems include an efficient implementation of the computation of resultants.
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Resultants were introduced for solving systems of polynomial equations and provide the oldest proof that there exist algorithms for solving such systems. These are primarily intended for systems of two equations in two unknowns, but also allow solving general systems.
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In 1683, Seki pushed ahead with elimination theory, based on resultants, in the Kaifukudai no Hō (解伏題之法). To express the resultant, he developed the notion of the determinant.Eves, Howard. (1990). An Introduction to the History of Mathematics, p. 405.
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Patients are seen with a cyanotic discoloration of the shoulder skin and neck and face, jugular distention, bulging of the eyeballs, and swelling of the tongue and lips. The latter two are resultants of edema, caused by excessive blood accumulating the veins of the head and neck and venous stasis.
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J. Numerical Methods in Engineering, 62, 1606–1635, (2005). # Rashed, Y. F., A boundary/domain element method for analysis of building raft foundations, Engineering Analysis with Boundary Elements, 29, 859-877, (2005). # Rashed, Y. F., A relative quantity integral equation formulation for evaluation of boundary stress resultants in shear deformable plate bending problems, Eng. Analysis with Boundary Elements 32, 152–161, (2008).
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Queer Failure is a concept in queer theory that also calls gender into question, because it examines queer art and the bodies of LGBTQ+ people through the lens of what a parental figure may identify as "failure" on the part of their character. Instead of recognizing these instances as moral or psychological failures, this concept frames them as the resultants of a conflict between a person's sexuality and their gender.
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Uwe Storch Uwe Storch (born 12 July 1940, Leopoldshall- Lanzarote, 17 September 2017) was a German mathematician. His field of research was commutative algebra and analytic and algebraic geometry, in particular derivations, divisor class group, resultants. Storch studied mathematics, physics and mathematical logic in Münster and in Heidelberg. He got his PhD 1966 under the supervision of Heinrich Behnke with a thesis on almost (or Q) factorial rings.
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During 19th century, this has been extended to linear Diophantine equations and abelian group with Hermite normal form and Smith normal form. Before 20th century, different types of eliminants were introduced, including resultants, and various kinds of discriminants. In general, these eliminants are also invariant, and are also fundamental in invariant theory. All these concepts are effective, in the sense that their definition include a method of computation.
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However, when the coefficients are integers, rational numbers or polynomials, these arithmetic operations imply a number of GCD computations of coefficients which is of the same order and make the algorithm inefficient. The subresultant pseudo- remainder sequences were introduced to solve this problem and avoid any fraction and any GCD computation of coefficients. A more efficient algorithm is obtained by using the good behavior of the resultant under a ring homomorphism on the coefficients: to compute a resultant of two polynomials with integer coefficients, one computes their resultants modulo sufficiently many prime numbers and then reconstructs the result with the Chinese remainder theorem. The use of fast multiplication of integers and polynomials allows algorithms for resultants and greatest common divisors that have a better time complexity, which is of the order of the complexity of the multiplication, multiplied by the logarithm of the size of the input (\log(s(d+e)), where is an upper bound of the number of digits of the input polynomials).
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Around 1890, David Hilbert introduced non-effective methods, and this was seen as a revolution, which led most algebraic-geometers of the first half of 20th century to try to "eliminate elimination". Nevertheless Hilbert's Nullstellensatz, may be considered to belong to elimination theory, as it asserts that a system of polynomial equations does not have any solution if and only one may eliminate all unknowns for getting 1. Elimination theory culminated with the work of Kronecker, and, finally, F.S. Macaulay, who introduced multivariate resultants and U-resultants, providing complete elimination methods for systems of polynomial equations, which have been described in chapter Elimination theory of the first editions (1930) of van der Waerden's Moderne Algebra. After that, elimination theory has been considered as old fashioned, removed from next editions of Moderne Algebra, and generally ignored, until the introduction of computers, and more specifically of computer algebra, which set the problem of designing elimination algorithms that are sufficiently efficient for being implemented.
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The appearance of the headmaster made the pupils calm down. :c. The breaking of the dam let the water flow from the storage lake. :d. The abating of the wind let the sailboat slow down. In this series of scenarios, various kinds of causation are described. Furthermore, a basic relationship between the concepts of ‘causing something to happen’ and ‘letting something happen’ emerges, definable in terms of the balance between the force entities and the resultants of the interaction.
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The global warming of over 2 °C would begin to seriously threaten global living. In 2010, UAE examined with the support of the Stockholm Environment Institute's US Center the effects of increasing carbon dioxide emissions and its impact on the weather. The report investigates the effects of climate change on the economy, the infrastructure, the health of citizens and the entire ecosystem. It resultants with a dramatical impact of rising sea levels by affecting 6 percent of its coastal urbanization by the end of the century.
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"Examen du rendement des services d'infrastructure – Sommaire des resultants[sic]", Gatineau, Quebec, p. 9, accessed August 2009 Canada's International Development Research Centre has had at least eighty technical projects in Mali since 1971, including support for community telecentres in Bamako and Timbuktu between 1998 and 2009, to enhance public access to computers and the Internet.International Development Research Centre. "Projects in Mali", web page accessed August 2009. Canada has been active in the development of Mali's energy infrastructure since the mid-1970s, with C$23m.
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In the same period began the algebraization of the algebraic geometry through commutative algebra. The prominent results in this direction are Hilbert's basis theorem and Hilbert's Nullstellensatz, which are the basis of the connexion between algebraic geometry and commutative algebra, and Macaulay's multivariate resultant, which is the basis of elimination theory. Probably because of the size of the computation which is implied by multivariate resultants, elimination theory was forgotten during the middle of the 20th century until it was renewed by singularity theory and computational algebraic geometry.
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A plate is a structural element which is characterized by two key properties. Firstly, its geometric configuration is a three-dimensional solid whose thickness is very small when compared with other dimensions. Secondly, the effects of the loads that are expected to be applied on it only generate stresses whose resultants are, in practical terms, exclusively normal to the element's thickness. Thin plates are initially flat structural members bounded by two parallel planes, called faces, and a cylindrical surface, called an edge or boundary.
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With Israel Gelfand and Andrei Zelevinsky, Kapranov investigated generalized Euler integrals, A-hypergeometric functions, A-discriminants, and hyperdeterminants, and authored Discriminants, Resultants, and Multidimensional Determinants in 1994. According to Gelfand, Kapranov, and Zelevinsky: In 1995 Kapranov provided a framework for a Langlands program for higher-dimensional schemes, and with, Victor Ginzburg and Eric Vasserot, extended the "Geometric Langlands Conjecture" from algebraic curves to algebraic surfaces. In 1998 Kapranov was an Invited Speaker with talk Operads and Algebraic Geometry at the International Congress of Mathematicians in Berlin.
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A shell is a type of structural element which is characterized by its geometry, being a three-dimensional solid whose thickness is very small when compared with other dimensions, and in structural terms, by the stress resultants calculated in the middle plane displaying components which are both coplanar and normal to the surface. Essentially, a shell can be derived from a plate by two means: by initially forming the middle surface as a singly or doubly curved surface, and by applying loads which are coplanar to a plate's plane which generate significant stresses.
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Different choices of α give different pseudo-remainder sequences, which are described in the next subsections. As the common divisors of two polynomials are not changed if the polynomials are multiplied by invertible constants (in Q), the last nonzero term in a pseudo-remainder sequence is a GCD (in Q[X]) of the input polynomials. Therefore, pseudo-remainder sequences allows computing GCD's in Q[X] without introducing fractions in Q. In some contexts, it is essential to control the sign of the leading coefficient of the pseudo- remainder. This is typically the case when computing resultants and subresultants, or for using Sturm's theorem.
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That is, with a 2' stop drawn, pressing middle C sounds the G that is the 12th diatonic note above. Mutations usually sound at pitches in the harmonic series of the unison pitch. In some large organs, non-harmonic mutations are occasionally used, sounding pitches from the harmonic series of one or two octaves below unison pitch. Such mutations that sound at the fifth above (or fourth below) the fundamental can create the impression of a stop an octave (or two) lower than the fundamental, especially when low frequencies are involved; these are often called resultants.
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In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to solve systems of polynomial equations. The classical elimination theory culminated with the work of Macaulay on multivariate resultants, and its description in chapter Elimination theory of the first editions (1930) of van der Waerden's Moderne Algebra. After that, elimination theory was ignored by most algebraic geometers for almost thirty years, until the introduction of new methods for solving polynomial equations, such as Gröbner bases, which were needed for computer algebra.
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The general system of three equations may be solved by the method of resultants. When multiplied out, all three equations have on the left-hand side, and rs2 on the right-hand side. Subtracting one equation from another eliminates these quadratic terms; the remaining linear terms may be re-arranged to yield formulae for the coordinates xs and ys : x_s = M + N r_s : y_s = P + Q r_s where M, N, P and Q are known functions of the given circles and the choice of signs. Substitution of these formulae into one of the initial three equations gives a quadratic equation for rs, which can be solved by the quadratic formula.
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Macaulay's resultant, named after Francis Sowerby Macaulay, also called the multivariate resultant, or the multipolynomial resultant,, Chapter 3. Resultants is a generalization of the homogeneous resultant to homogeneous polynomials in indeterminates. Macaulay's resultant is a polynomial in the coefficients of these homogeneous polynomials that vanishes if and only if the polynomials have a common non-zero solution in an algebraically closed field containing the coefficients, or, equivalently, if the hyper surfaces defined by the polynomials have a common zero in the dimensional projective space. The multivariate resultant is, with Gröbner bases, one of the main tools of effective elimination theory (elimination theory on computers).
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A cantilever Timoshenko beam under a point load at the free end For a cantilever beam, one boundary is clamped while the other is free. Let us use a right handed coordinate system where the x direction is positive towards right and the z direction is positive upward. Following normal convention, we assume that positive forces act in the positive directions of the x and z axes and positive moments act in the clockwise direction. We also assume that the sign convention of the stress resultants (M_{xx} and Q_x) is such that positive bending moments compress the material at the bottom of the beam (lower z coordinates) and positive shear forces rotate the beam in a counterclockwise direction.
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As the computation of a resultant may be reduced to computing determinants and polynomial greatest common divisors, there are algorithms for computing resultants in a finite number of steps. However, the generic resultant is a polynomial of very high degree (exponential in ) depending on a huge number of indeterminates. It follows that, except for very small and very small degrees of input polynomials, the generic resultant is, in practice, impossible to compute, even with modern computers. Moreover, the number of monomials of the generic resultant is so high, that, if it would be computable, the result could not be stored on available memory devices, even for rather small values of and of the degrees of the input polynomials.
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Isolines of frequency are drawn on maps showing the frequency of a particular phenomenon (for example, annual number of days with a thunderstorm or snow cover). Isochrones are drawn on maps showing the dates of onset of a given phenomenon (for example, the first frost and appearance or disappearance of the snow cover) or the date of a particular value of a meteorological element in the course of a year (for example, passing of the mean daily air temperature through zero). Isolines of the mean numerical value of wind velocity or isotachs are drawn on wind maps (charts); the wind resultants and directions of prevailing winds are indicated by arrows of different length or arrows with different plumes; lines of flow are often drawn. Maps of the zonal and meridional components of wind are frequently compiled for the free atmosphere.
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