" But climate change is "more like solving a quadratic
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Now consider an alternative approach, which we call Quadratic Voting (QV).
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Also among the materials were math problems involving geometry and quadratic equations.
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Speed matters when the quadratic equation is part of a larger problem.
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Quadratic Capital is based in Greenwich, Connecticut, and is majority owned by women.
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"I have never used the quadratic formula in my personal life," he acknowledged.
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As that number increases linearly, the possibility for interactions increases at a quadratic rate.
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This alternate method for solving quadratic equations uses the fact that parabolas are symmetrical.
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Quadratic Capital Management's Nancy Davis said the next "big short" is betting against volatility.
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But Nancy Davis, CIO of Quadratic Capital, thinks volatility will remain high in the foreseeable future.
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Fitting a quadratic function to this data, I get an angular acceleration of 0.00038 rad/s2.
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After all, it's highly unlikely anyone will seriously wash their hands to, say, the quadratic formula.
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For instance, the seashells above were digitally graphed using a series of exponential and quadratic functions.
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So a pal of his, hearing about his plight, told him about a new idea: quadratic voting.
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That means I can find the vertical acceleration from the quadratic fit of the vertical bullet motion.
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But the world of math goes far beyond chalk-etched numbers and quadratic equations on a blackboard.
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The teenager next door named Lisa, played by Justin, has come to get help with the quadratic equation.
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Here's what I get: The quadratic fit for this data gives a t2 coefficient of –7.00 m/s2.
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His need to understand hhhhwwhy — and mine too — wasn't limited to quadratic equations or the speed of light.
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You might even remember banging your head against the quadratic equation (ax² + bx + c = 0) in algebra class.
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We only hear how loudly someone shouts; quadratic voting allows us to adjust the volume, creating more reliable signals.
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Now if we could only come up with a national holiday for the quadratic formula, we'd probably remember that, too.
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Quadratic increases also "become more and more rapid," but the exponential wins out — strikingly so — once enough time has elapsed.
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The data, they say, closely matches the quadratic formula, which, when plotted on a chart, gives a steady upward swing.
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"It's not ripping because economic data is good," said Nancy Davis, chief investment officer at Quadratic Capital Management in Greenwich, Conn.
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One of her prints (pictured above) -- the Joal print -- was inspired by a class on exponential and quadratic functions, she says.
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But some private equity investments have as much as a 10-year lockup period, said Quadratic Capital Chief Investment Officer Nancy Davis.
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As of now, the feature's still under development but GitHub notes indicate users would be able to graph linear, quadratic, and exponential equations.
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Reciprocity laws are generalizations of the 200-year-old quadratic reciprocity law, a cornerstone of number theory and one of Scholze's personal favorite theorems.
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Concerns over the possibility of recession — something Nancy Davis, the founder of the hedge fund Quadratic Capital, told attendees "is coming" — were not ignored.
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The upshot is that the quadratic formula ensures that, in our town, the collective well-being of the public is maximized through the voting system.
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Quadratic Capital Management is recommending that investors bet against two companies that are likely to be sensitive to a Federal Reserve cutback on stimulative policies.
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Lee about how she repeated the quadratic formula in her lessons so much that she might as well have tattooed it onto her students' foreheads.
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"These advantages that you end up seeing, they're modest; they're not exponential, but they are quadratic," said Nathan Wiebe, a quantum-computing researcher at Microsoft Research.
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Quadratic Capital Management's unusual strategy of investing almost exclusively in options means that it makes money during upheaval because the price of options increases with volatility.
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For the last 50 years, reformers have wanted to teach kids to reason mathematically, to think nimbly about topics like quadratic equations that otherwise come off flat.
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Quadratic voting solves those problems by giving everyone an equal number of credits in a given election, which can be converted into votes of the individual's choosing.
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Comparing the results on each graphic, I started synthesizing these conclusions and translating them into actual mathematical quadratic equations, and later on evolving these into more complex parabolas.
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The quadratic formula expresses a relationship between three known quantities — A, B, and C — and an unknown quantity, X.It is most often written as AX2 + BX + C = 0.
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Jonathan Rochelle, the director of Google's education apps group, said last year at an industry conference that he "cannot answer" why his children should learn the quadratic equation.
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Elementary math facts and the sounds of letters are obvious choices, but any information that is needed with high frequency is a candidate — in algebra, that's the quadratic equation.
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If, while watching the Super Bowl, you had wanted to estimate how far a pass thrown by Patrick Mahomes traveled through the air, you would have been solving a quadratic equation.
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Quadratic Capital Management's Nancy Davis — who correctly predicted the blow-up in the popular wager on low volatility by hedge funds before last week's plunge — thinks the market will remain turbulent.
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To rectify this, Du and his colleagues used a Linear-Quadratic Regulator to automatically adjust the parameters of the controller so it is more in tune with the custom drone it's controlling.
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At risk of giving the quadratic dudes a seizure, I turned to Baseball Reference and selected 22014 pitchers from the past half-century, men who ran the gamut from solid to superb.
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As the 10-year Treasury yield sank below 2% this year, investors went looking for better investment alternatives, according to Nancy Davis, chief investment officer and founder of advisory firm Quadratic Capital.
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Kids who look up the quadratic equation may end up like the child who looked up "meticulous"; they have a definition, but they don't have the background knowledge to use it correctly.
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An opinion article last week about why you still need your brain and cannot outsource knowledge to the web inaccurately described where Jonathan Rochelle, a Google executive, made comments about the quadratic equation.
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Take the humble quadratic (typified by the formula "x to the power 2," or x²) which, for instance, tracks how the distance traveled by an object in free-fall increases with x, the time.
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When a referendum is held, people exchange their voice credits for votes, according to a quadratic formula: one vote costs one credit; two votes cost four credits; three votes cost nine credits; and so on.
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But it is destined to remain so much inert philosophy, no more life-changing than the quadratic equation, until you're able to actually glimpse your little impostor, to fix him in your mental cross hairs.
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SUNDAY REVIEW An opinion article last week about why you still need your brain and cannot outsource knowledge to the web inaccurately described where Jonathan Rochelle, a Google executive, made comments about the quadratic equation.
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She thought back to that one time in 19603th grade math class when she had been too keen to answer the teacher's question and how the boys had laughed when she confused quadratic equations with quadrilaterals.
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Students have always been able to look up the quadratic equation rather than memorize it, but opening a new browser tab takes moments, not the minutes required to locate the right page in the right book.
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If an object has a constant acceleration, then its position should agree with the following kinematic equation: If I fit a quadratic equation to the position data, I can find the coefficient in front of the t22.11 term.
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"The market is just trading off a sci-fi movie where it's a viral pandemic and we're all going to die," said Nancy Davis, founder of Quadratic Capital Management and portfolio manager of the IVOL exchange-traded fund.
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One of the stronger theories was that it was a teaching aid for checking quadratic problems, but new research conducted by UNSW scientists Daniel Mansfield and Norman Wildberger now confirms the markings on the tablet as a trigonometry table.
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That means its motion obeys the following kinematic equation (where y is vertical position, v is velocity, and t is time): Since this is a quadratic equation, a plot of vertical position versus time should have the shape of a parabola.
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If you've ever taken an introductory physics course, you've seen this famous kinematic equation: Fitting a quadratic function to this data shows that the coefficient in front of the t22 term should be equal to the acceleration divided by two.
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Type: Trigonometry Signature move: Law of quadratic reciprocity Pikawack is known to hide in the shadows, seeing by the light of its own generated electricity and waiting to strike unsuspecting victims by rubbing its tiny feet on the carpet and tapping them on the forearm.
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Mark a spot on the midline of a piece of paper, and fold the paper in as many ways as you can that touch its bottom edge to that spot; the folds will inscribe a parabola, as described by quadratic functions such as y=x2.
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Less a full-length sequel to "How Children Succeed" than a short companion, "Helping Children Succeed" argues that skills like emotional regulation and stick-to-it-iveness can't be taught in the same way children are trained to decode phonemes and solve quadratic equations.
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Not everyone has to learn quadratic equations" — which, he points out, most people forget the minute they leave school anyway — "but at some point in our lives, we will all be in stressful situations and we need to be able to keep our cool.
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You noted in the fifth edition that mathematicians have come up with at least 196 different proofs of the "quadratic reciprocity" theorem (concerning which numbers in "clock" arithmetics are perfect squares) and nearly 100 proofs of the fundamental theorem of algebra (concerning solutions to polynomial equations).
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Unlike the asymmetric encryption largely used today, AES is considered quantum-safe, and a long-enough key length should be enough to protect the communication from the super-charged quantum hacking software that will be working very quickly (taking into account the quadratic search speed-up implied in Grover's Algorithm).
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Starting from a certain height y12.53 and an initial velocity v0, we can write the relation between vertical position (y) and time (t) using this famous kinematics equation: Since this depends on both time and time squared, it's a quadratic equation; if we graphed it, it would trace out a parabola.
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As the accompanying image shows, our app uses intuitive geometry rather than mathematical formulas to convey the basic idea behind QV. A bar at the top keeps track of the budget of voice credits, while the circles fill up, and the top bar draws down, at the quadratic rate as the survey respondent exchanges credits for votes.
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Think of what STEM might reasonably be expected to cover: fluid mechanics, C++, the periodic table, PEMDAS, Python, botany, the Krebs cycle, Instagram curation, polymer chemistry, robotics, making an investor deck, formal logic, electrodynamics, the quadratic formula, GIFs, quantum mechanics, JavaScript, civil engineering, machine learning, virology, drones, particle physics, acoustics, the supply chain, astronomy, YouTube memes, natural selection, anatomy, multiplication tables, remote surgery using 5G, and … Everything else.
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Zero, while a square, is not considered to be a quadratic residue. Every finite field of this type has exactly quadratic residues and exactly quadratic non-residues. The quadratic residues form a group under multiplication. The properties of quadratic residues are widely used in number theory.
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The cover topics in elementary number theory, algebraic number theory and analytic number theory, including modular arithmetic, quadratic congruences, quadratic reciprocity and binary quadratic forms.
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A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers.
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In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers. The map d ↦ Q() is a bijection from the set of all square-free integers d ≠ 0,1 to the set of all quadratic fields. If d > 0, the corresponding quadratic field is called a real quadratic field, and for d < 0 an imaginary quadratic field or complex quadratic field, corresponding to whether or not it is a subfield of the field of the real numbers. Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms.
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We can find quadratic residues or verify them using the above formula. To test if 2 is a quadratic residue modulo 17, we calculate 2(17 − 1)/2 = 28 ≡ 1 (mod 17), so it is a quadratic residue. To test if 3 is a quadratic residue modulo 17, we calculate 3(17 − 1)/2 = 38 ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue.
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Quadratic programming (QP) is the process of solving a special type of mathematical optimization problem—specifically, a (linearly constrained) quadratic optimization problem, that is, the problem of optimizing (minimizing or maximizing) a quadratic function of several variables subject to linear constraints on these variables. Quadratic programming is a particular type of nonlinear programming.
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One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations.
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Quadratic Jordan algebras are a generalization of (linear) Jordan algebras introduced by . The fundamental identities of the quadratic representation of a linear Jordan algebra are used as axioms to define a quadratic Jordan algebra over a field of arbitrary characteristic. There is a uniform description of finite-dimensional simple quadratic Jordan algebras, independent of characteristic: in characteristic not equal to 2 the theory of quadratic Jordan algebras reduces to that of linear Jordan algebras.
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Quadratic forms are not to be confused with a quadratic equation which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of homogeneous polynomials.
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No general-purpose sorts run in linear time, but the change from quadratic to sub-quadratic is of great practical importance.
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The electrogyration effect has been revealed for the first time in quartz crystals [2] as an effect quadratic in the external field. Later on, both the linear and quadratic Vlokh O.G., Krushel'nitskaya T.D. (1970). "Axial four-rank tensors and quadratic electrogyration", Kristallografiya 15(3), 587-589 [Vlokh O.G., Krushel'nitskaya T.D. (1970). "Axial four-rank tensors and quadratic electrogyration", Sov.Phys.Crystallogr.
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The eventually periodic nature of the continued fraction is then reflected in the eventually periodic nature of the orbit of a quadratic form under reduction, with reduced quadratic irrationalities (those with a purely periodic continued fraction) corresponding to reduced quadratic forms.
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Since late 1980, there are many algebraic decoding algorithms were developed for correcting errors on quadratic residue codes. These algorithms can achieve the (true) error-correcting capacity ⌊(d − 1)/2⌋ of the quadratic residue codes with the code length up to 113. However, decoding of long binary quadratic residue codes and non-binary quadratic residue codes continue to be a challenge. Currently, decoding quadratic residue codes is still an active research area in the theory of error- correcting code.
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In mathematics, quadratic Jordan algebras are a generalization of Jordan algebras introduced by . The fundamental identities of the quadratic representation of a linear Jordan algebra are used as axioms to define a quadratic Jordan algebra over a field of arbitrary characteristic. There is a uniform description of finite-dimensional simple quadratic Jordan algebras, independent of characteristic. If 2 is invertible in the field of coefficients, the theory of quadratic Jordan algebras reduces to that of linear Jordan algebras.
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In mathematics, Scholz's reciprocity law is a reciprocity law for quadratic residue symbols of real quadratic number fields discovered by and rediscovered by .
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Andrianov was an Invited Speaker at the ICM in 1970 in Nice with talk On the zeta function of the general linear group and in 1983 in Warsaw with talk Integral representation of quadratic forms by quadratic forms: multiplicative properties.Andrianov, A. N. "Integral representations of quadratic forms by quadratic forms: multiplicative properties." In Proc. Intern. Congress of Mathematicians, Warsaw (1983), vol.
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The 2-principalization in unramified quadratic extensions of imaginary quadratic fields with 2-class group of type (2,2) was studied by H. Kisilevsky in 1976. Similar investigations of real quadratic fields were carried out by E. Benjamin and C. Snyder in 1995.
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In mathematics, a quadratic form is a polynomial with terms all of degree two. For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and we speak of a quadratic form over K. If K = ℝ, and the quadratic form never takes zero for a value, then it is a definite quadratic form, otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric, second fundamental form), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form).
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When compared to each other, progressions with a quadratic nonresidue remainder have typically slightly more elements than those with a quadratic residue remainder (Chebyshev's bias).
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Carlyle circle of the quadratic equation x2 − sx + p = 0\. Given the quadratic equation :x2 − sx + p = 0 the circle in the coordinate plane having the line segment joining the points A(0, 1) and B(s, p) as a diameter is called the Carlyle circle of the quadratic equation. E. John Hornsby, Jr.: Geometrical and Graphical Solutions of Quadratic Equations. The College Mathematics Journal, Vol.
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The earliest methods for solving quadratic equations were geometric. Babylonian cuneiform tablets contain problems reducible to solving quadratic equations. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation. The Greek mathematician Euclid (circa 300 BC) used geometric methods to solve quadratic equations in Book 2 of his Elements, an influential mathematical treatise.
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For any bilinear form , there exists an associated quadratic form defined by . When , the quadratic form Q is determined by the symmetric part of the bilinear form B and is independent of the antisymmetric part. In this case there is a one- to-one correspondence between the symmetric part of the bilinear form and the quadratic form, and it makes sense to speak of the symmetric bilinear form associated with a quadratic form. When and , this correspondence between quadratic forms and symmetric bilinear forms breaks down.
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The law of quadratic reciprocity says that if p and q are distinct odd primes, at least one of which is Pythagorean, then p is a quadratic residue mod q if and only if q is a quadratic residue mod p; by contrast, if neither p nor q is Pythagorean, then p is a quadratic residue mod q if and only if q is not a quadratic residue mod p., p. 103. In the finite field Z/p with p a Pythagorean prime, the polynomial equation x2 = −1 has two solutions. This may be expressed by saying that −1 is a quadratic residue mod p.
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In mathematics, Kaplansky's theorem on quadratic forms is a result on simultaneous representation of primes by quadratic forms. It was proved in 2003 by Irving Kaplansky..
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In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is :ax^2+bx+c=0, where a ≠ 0. The quadratic equation on a number x can be solved using the well-known quadratic formula, which can be derived by completing the square. That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated as a decimal fraction only by applying an additional root extraction algorithm.
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Fermat, Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries established theoremsLemmemeyer, Ch. 1 and formed conjecturesLemmermeyer, pp 6-8, p. 16 ff about quadratic residues, but the first systematic treatment is § IV of Gauss's Disquisitiones Arithmeticae (1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that if the context makes it clear, the adjective "quadratic" may be dropped. For a given n a list of the quadratic residues modulo n may be obtained by simply squaring the numbers 0, 1, ..., .
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That can be determined by evaluating the quadratic term of a divided difference formula. If the quadratic term is negligible—meaning that the linear term is sufficiently accurate without adding the quadratic term—then linear interpolation is sufficiently accurate. If the problem is sufficiently important, or if the quadratic term is nearly big enough to matter, then one might want to determine whether the _sum_ of the quadratic and cubic terms is large enough to matter in the problem. Of course, only a divided-difference method can be used for such a determination.
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In number theory, quadratic integers are a generalization of the integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and integers. When algebraic integers are considered, the usual integers are often called rational integers. Common examples of quadratic integers are the square roots of integers, such as , and the complex number , which generates the Gaussian integers.
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As noted in the introduction, inverse quadratic interpolation is used in Brent's method. Inverse quadratic interpolation is also closely related to some other root-finding methods. Using linear interpolation instead of quadratic interpolation gives the secant method. Interpolating f instead of the inverse of f gives Muller's method.
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The picture to the right indicates a quadratic relationship between the mean and the variance. As we saw above, the Gamma variance function is quadratic in the mean.
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Adding an overall parity-check digit to a quadratic residue code gives an extended quadratic residue code. When p\equiv 3 (mod 4) an extended quadratic residue code is self-dual; otherwise it is equivalent but not equal to its dual. By the Gleason–Prange theorem (named for Andrew Gleason and Eugene Prange), the automorphism group of an extended quadratic residue code has a subgroup which is isomorphic to either PSL_2(p) or SL_2(p).
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In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum. These objects are named after Carl Friedrich Gauss, who studied them extensively and applied them to quadratic, cubic, and biquadratic reciprocity laws.
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It may be possible to express a quadratic equation as a product . In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if or . Solving these two linear equations provides the roots of the quadratic.
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The early proofs of quadratic reciprocity are relatively unilluminating. The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomic fields. His proof was cast in modern form by later algebraic number theorists. This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity.
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In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process.
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Let , , be three quadratic spaces over a field k. Assume that : (V_1,q_1)\oplus(V,q) \simeq (V_2,q_2)\oplus(V,q). Then the quadratic spaces and are isometric: : (V_1,q_1)\simeq (V_2,q_2). In other words, the direct summand appearing in both sides of an isomorphism between quadratic spaces may be "cancelled".
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Quadratic forms over the ring of integers are called integral quadratic forms, whereas the corresponding modules are quadratic lattices (sometimes, simply lattices). They play an important role in number theory and topology. An integral quadratic form has integer coefficients, such as ; equivalently, given a lattice Λ in a vector space V (over a field with characteristic 0, such as Q or R), a quadratic form Q is integral with respect to Λ if and only if it is integer-valued on Λ, meaning if . This is the current use of the term; in the past it was sometimes used differently, as detailed below.
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In mathematical finite group theory, a quadratic pair for the odd prime p, introduced by , is a finite group G together with a quadratic module, a faithful representation M on a vector space over the finite field with p elements such that G is generated by elements with minimum polynomial (x − 1)2. Thompson classified the quadratic pairs for p ≥ 5\. classified the quadratic pairs for p = 3\. With a few exceptions, especially for p = 3, groups with a quadratic pair for the prime p tend to be more or less groups of Lie type in characteristic p.
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An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form. More generally, these definitions apply to any vector space over an ordered field..
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The Pythagorean theorem and the law of quadratic reciprocity are contenders for the title of theorem with the greatest number of distinct proofs.See, for example, proofs of quadratic reciprocity for more.
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The proof is straightforward. From the fraction itself, one can construct the quadratic equation with integral coefficients that x must satisfy. Lagrange proved the converse of Euler's theorem: if x is a quadratic irrational, then the regular continued fraction expansion of x is periodic. Given a quadratic irrational x one can construct m different quadratic equations, each with the same discriminant, that relate the successive complete quotients of the regular continued fraction expansion of x to one another.
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There is a connection between the theory of integral binary quadratic forms and the arithmetic of quadratic number fields. A basic property of this connection is that D0 is a fundamental discriminant if, and only if, D0 = 1 or D0 is the discriminant of a quadratic number field. There is exactly one quadratic field for every fundamental discriminant D0 ≠ 1, up to isomorphism. Caution: This is the reason why some authors consider 1 not to be a fundamental discriminant.
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In § VI of the Disquisitiones ArithmeticaeGauss, DA, arts 329-334 Gauss discusses two factoring algorithms that use quadratic residues and the law of quadratic reciprocity. Several modern factorization algorithms (including Dixon's algorithm, the continued fraction method, the quadratic sieve, and the number field sieve) generate small quadratic residues (modulo the number being factorized) in an attempt to find a congruence of squares which will yield a factorization. The number field sieve is the fastest general-purpose factorization algorithm known.
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A special case of the natural exponential families are those with quadratic variance functions. Six NEFs have quadratic variance functions (QVF) in which the variance of the distribution can be written as a quadratic function of the mean. These are called NEF-QVF. The properties of these distributions were first described by Carl Morris.
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A quadratic closure of a field F is a quadratically closed field containing F which embeds in any quadratically closed field containing F. A quadratic closure for any given F may be constructed as a subfield of the algebraic closure Falg of F, as the union of all iterated quadratic extensions of F in Falg.
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Another common example is the non-real cubic root of unity , which generates the Eisenstein integers. Quadratic integers occur in the solutions of many Diophantine equations, such as Pell's equations, and other questions related to integral quadratic forms. The study of rings of quadratic integers is basic for many questions of algebraic number theory.
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In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative ring where 2 is invertible, and if (V_1, q_1) and (V_2,q_2) are two quadratic spaces over R, then their tensor product (V_1 \otimes V_2, q_1 \otimes q_2) is the quadratic space whose underlying R-module is the tensor product V_1 \otimes V_2 of R-modules and whose quadratic form is the quadratic form associated to the tensor product of the bilinear forms associated to q_1 and q_2. In particular, the form q_1 \otimes q_2 satisfies : (q_1\otimes q_2)(v_1 \otimes v_2) = q_1(v_1) q_2(v_2) \quad \forall v_1 \in V_1,\ v_2 \in V_2 (which does uniquely characterize it however). It follows from this that if the quadratic forms are diagonalizable (which is always possible if 2 is invertible in R), i.e., :q_1 \cong \langle a_1, ... , a_n \rangle :q_2 \cong \langle b_1, ... , b_m \rangle then the tensor product has diagonalization :q_1 \otimes q_2 \cong \langle a_1b_1, a_1b_2, ... a_1b_m, a_2b_1, ... , a_2b_m , ... , a_nb_1, ... a_nb_m \rangle.
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Namely, Cl can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.
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If A is a unital associative algebra over K with multiplication × then a quadratic map Q can be defined from A to EndK(A) by Q(a) : b ↦ a × b × a. This defines a quadratic Jordan algebra structure on A. A quadratic Jordan algebra is special if it is isomorphic to a subalgebra of such an algebra, otherwise exceptional.
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Gauss's lemma in number theory gives a condition for an integer to be a quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity. It made its first appearance in Carl Friedrich Gauss's third proof (1808) of quadratic reciprocity and he proved it again in his fifth proof (1818).
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When eh is zero, the quadratic function is a horizontal straight line. When eh is negative, the quadratic function is a parabola opening to the bottom. A quadratic equation with real or complex coefficients has two solutions, called roots. These two solutions may or may not be distinct, and they may or may not be real.
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Completing the square can be used to derive a general formula for solving quadratic equations, called the quadratic formula., Chapter 13 §4.4, p. 291 The mathematical proof will now be briefly summarized.Himonas, Alex.
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A quadratic residue code is a type of cyclic code.
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Euler's criterion is related to the Law of quadratic reciprocity.
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Let be a quadratic space over a field k. Then it admits a Witt decomposition: : (V,q)\simeq (V_0,0)\oplus(V_a, q_a)\oplus (V_h,q_h), where is the radical of q, is an anisotropic quadratic space and is a split quadratic space. Moreover, the anisotropic summand, termed the core form, and the hyperbolic summand in a Witt decomposition of are determined uniquely up to isomorphism. Quadratic forms with the same core form are said to be similar or Witt equivalent.
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The quadratic loss function is also used in linear-quadratic optimal control problems. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions. In the context of stochastic control, the expected value of the quadratic form is used.
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If there is only one solution, one says that it is a double root. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation :ax^2+bx+c=a(x-r)(x-s)=0 where and are the solutions for . Completing the square on a quadratic equation in standard form results in the quadratic formula, which expresses the solutions in terms of , , and .
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Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation. Babylonian mathematicians from circa 400 BC and Chinese mathematicians from circa 200 BC used geometric methods of dissection to solve quadratic equations with positive roots. Rules for quadratic equations were given in The Nine Chapters on the Mathematical Art, a Chinese treatise on mathematics.
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The Bakhshali Manuscript written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic indeterminate equations (originally of type ). Muhammad ibn Musa al-Khwarizmi (Persia, 9th century), inspired by Brahmagupta, developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric proofs in the process.
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In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of . According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "always nonnegative" and "always nonpositive", respectively. In other words, it may take on zero values.
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The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. In 1637 René Descartes published La Géométrie containing special cases of the quadratic formula in the form we know today.
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Q(x) divided by x(x + k) is a quadratic polynomial.
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His solution was largely based on Al-Khwarizmi's work. The writing of the Chinese mathematician Yang Hui (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi. By 1545 Gerolamo Cardano compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by Simon Stevin in 1594. In 1637 René Descartes published La Géométrie containing the quadratic formula in the form we know today.
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Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function . This is the quadratic function whose first and second derivatives are the same as those of at a given point. The formula for the best quadratic approximation to a function around the point is :f(x) \approx f(a) + f'(a)(x-a) + \tfrac12 f(a)(x-a)^2. This quadratic approximation is the second-order Taylor polynomial for the function centered at .
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Quadratic surd: An algebraic number that is the root of a quadratic equation. Such a number can be expressed as the sum of a rational number and the square root of a rational. Constructible number: A number representing a length that can be constructed using a compass and straightedge. These are a subset of the algebraic numbers, and include the quadratic surds.
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Consider the following quadratic polynomial: :x^2 + 10x + 28. This quadratic is not a perfect square, since 28 is not the square of 5: :(x+5)^2 \,=\, x^2 + 10x + 25. However, it is possible to write the original quadratic as the sum of this square and a constant: :x^2 + 10x + 28 \,=\, (x+5)^2 + 3. This is called completing the square.
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If there is a vector v ∈ V such that f(v) = 0, then f is an isotropic quadratic form. If f has the same sign for all vectors, it is a definite quadratic form or an anisotropic quadratic form. There is the closely related notion of a unimodular form and a perfect pairing; these agree over fields but not over general rings.
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1 (1901), pp. 44–63, 213–237. As stated by Kaplansky, "The 11th Problem is simply this: classify quadratic forms over algebraic number fields." This is exactly what Minkowski did for quadratic form with fractional coefficients.
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The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a weight, a value, and an extra profit that can be earned if two items are selected, determine the number of item to include in a collection without exceeding capacity of the knapsack, so as to maximize the overall profit. Usually, quadratic knapsack problems come with a restriction on the number of copies of each kind of item: either 0, or 1. This special type of QKP forms the 0-1 quadratic knapsack problem, which was first discussed by Gallo et al. The 0-1 quadratic knapsack problem is a variation of knapsack problems, combining the features of unbounded knapsack problem, 0-1 knapsack problem and quadratic knapsack problem.
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If V is a linear space with a real quadratic form F:V → R, then { p ∈ V : F(p) = 1 } may be called the unit sphereTakashi Ono (1994) Variations on a Theme of Euler: quadratic forms, elliptic curves, and Hopf maps, chapter 5: Quadratic spherical maps, page 165, Plenum Press, F. Reese Harvey (1990) Spinors and calibrations, "Generalized Spheres", page 42, Academic Press, or unit quasi-sphere of V. For example, the quadratic form x^2 - y^2, when set equal to one, produces the unit hyperbola which plays the role of the "unit circle" in the plane of split-complex numbers. Similarly, quadratic form x2 yields a pair of lines for the unit sphere in the dual number plane.
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Quadratic probing is an open addressing scheme in computer programming for resolving hash collisions in hash tables. Quadratic probing operates by taking the original hash index and adding successive values of an arbitrary quadratic polynomial until an open slot is found. An example sequence using quadratic probing is: H + 1^2 , H + 2^2 , H + 3^2 , H + 4^2 , ... , H + k^2 Quadratic probing can be a more efficient algorithm in an open addressing table, since it better avoids the clustering problem that can occur with linear probing, although it is not immune. It also provides good memory caching because it preserves some locality of reference; however, linear probing has greater locality and, thus, better cache performance.
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A fairly complicated quadratic equation is used to calculate the daily energy.
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One can introduce a vertically inverted (\cap)-quadratic distribution in analogous fashion.
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There are also quadratic reciprocity laws in rings other than the integers.
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He proved that every finite tournament contains an odd number of Hamiltonian paths. He gave several proofs of the theorem on quadratic reciprocity. He proved important results concerning the invariants of the class groups of quadratic number fields. Iyanaga's pamphlet discusses and generalizes one of Rédei's theorems; it gives a "necessary and sufficient condition for the existence of an ideal class (in the restricted sense) of order 4 in a quadratic field k() ..." In several cases, he determined if the ring of integers of the real quadratic field Q() is Euclidean or not.
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A quadratic form (not quadratic equation) is any polynomial in which each term has variables appearing exactly twice. The general form of such an equation is ax2 \+ bxy + cy2. (All coefficients must be whole numbers.) A given quadratic form is said to represent a natural number if substituting specific numbers for the variables gives the number. Gauss and those who followed found that if we change variables in certain ways, the new quadratic form represented the same natural numbers as the old, but in a different, more easily interpreted form.
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The equations of the circle and the other conic sections--ellipses, parabolas, and hyperbolas--are quadratic equations in two variables. Given the cosine or sine of an angle, finding the cosine or sine of the angle that is half as large involves solving a quadratic equation. The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation. Descartes' theorem states that for every four kissing (mutually tangent) circles, their radii satisfy a particular quadratic equation.
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An intuitive way to understand an ε-quadratic form is to think of it as a quadratic refinement of its associated ε-symmetric form. For instance, in defining a Clifford algebra over a general field or ring, one quotients the tensor algebra by relations coming from the symmetric form and the quadratic form: and v^2=Q(v). If 2 is invertible, this second relation follows from the first (as the quadratic form can be recovered from the associated bilinear form), but at 2 this additional refinement is necessary.
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For example, given a framed manifold, one can produce such a refinement. For singly even dimension, the Arf invariant of this skew-quadratic form is the Kervaire invariant. Given an oriented surface Σ embedded in R3, the middle homology group H1(Σ) carries not only a skew-symmetric form (via intersection), but also a skew-quadratic form, which can be seen as a quadratic refinement, via self-linking. The skew- symmetric form is an invariant of the surface Σ, whereas the skew-quadratic form is an invariant of the embedding , e.g.
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If a quadratic effect is expected for a factor, a more complicated experiment should be used, such as a central composite design. Optimization of factors that could have quadratic effects is the primary goal of response surface methodology.
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Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC. Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation only contains powers of that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.
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Quadratic differentials on a Riemann surface X are identified with the tangent space at (X, f) to Teichmüller space., Chapter VI.D The Weil–Petersson metric is the Riemannian metric defined by the L^2 inner product on quadratic differentials.
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The trace also plays a central role in the distribution of quadratic forms.
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Note that the convergence of c_n \,\\!, and therefore also of a_n \,\\!, is quadratic.
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An extension of the above concept of Koszul duality was formulated by Ginzburg and Kapranov who introduced the notion of a quadratic operad and defined the quadratic dual of such an operad.Ginzburg, Victor; Kapranov, Mikhail. Koszul duality for operads. Duke Math.
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The L-groups are defined algebraically in terms of quadratic forms or in terms of chain complexes with quadratic structure. See the main article for more details. Here only the properties of the L-groups described below will be important.
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When minimizing a function in the neighborhood of some reference point , is set to its Hessian matrix and is set to its gradient . A related programming problem, quadratically constrained quadratic programming, can be posed by adding quadratic constraints on the variables.
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On quadratic forms :Chapter 5. Determination of the class number of binary quadratic forms :Supplement I. Some theorems from Gauss's theory of circle division :Supplement II. On the limiting value of an infinite series :Supplement III. A geometric theorem :Supplement IV. Genera of quadratic forms :Supplement V. Power residues for composite moduli :Supplement VI. Primes in arithmetic progressions :Supplement VII. Some theorems from the theory of circle division :Supplement VIII.
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The Casimir operator is a distinguished quadratic element of the center of the universal enveloping algebra of the Lie algebra. In other words, it is a member of the algebra of all differential operators that commutes with all the generators in the Lie algebra. In fact all quadratic elements in the center of the universal enveloping algebra arise this way. However, the center may contain other, non- quadratic, elements.
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This is further generalised to quadratic forms in linear spaces via the inner product. The inertia tensor in mechanics is an example of a quadratic form. It demonstrates a quadratic relation of the moment of inertia to the size (length). There are infinitely many Pythagorean triples, sets of three positive integers such that the sum of the squares of the first two equals the square of the third.
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One may interpret D0 = 1 as the degenerate "quadratic" field Q (the rational numbers).
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In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor product of quadratic forms :\langle\\!\langle a_1, a_2, \ldots , a_n \rangle\\!\rangle \cong \langle 1, -a_1 \rangle \otimes \langle 1, -a_2 \rangle \otimes \cdots \otimes \langle 1, -a_n \rangle, for some nonzero elements a1, ..., an of F.Elman, Karpenko, Merkurjev (2008), section 9.
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Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation . Although the display shows only five significant figures of accuracy, the retrieved value of is 0.732050807569, accurate to twelve significant figures. A quadratic function without real root: .
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In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusual number of proofs. Several hundred proofs of the law of quadratic reciprocity (most of them being variants of previously known proofs) have been published.
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Their efforts resulted in the development of three algorithms: Quadratic Extrapolation, BlockRank and Adaptive PageRank.
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The fastest known algorithm to compute the WF-Semantics in general, is of quadratic complexity.
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This lemma was introduced by Yegor Ivanovich Zolotarev in an 1872 proof of quadratic reciprocity.
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Jacques Helmstetter, Artibano Micali: Quadratic Mappings and Clifford Algebras, Birkhäuser, 2008, Introduction, p. ix ff.
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From the point of view of classification of quadratic forms, anisotropic spaces are the basic building blocks for quadratic spaces of arbitrary dimensions. For a general field F, classification of anisotropic quadratic forms is a nontrivial problem. By contrast, the isotropic forms are usually much easier to handle. By Witt's decomposition theorem, every inner product space over a field is an orthogonal direct sum of a split space and an anisotropic space.
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The quadratic residuosity problem in computational number theory is to decide, given integers a and N, whether a is a quadratic residue modulo N or not. Here N = p_1 p_2 for two unknown primes p_1 and p_2, and a is among the numbers which are not obviously quadratic non-residues (see below). The problem was first described by Gauss in his Disquisitiones Arithmeticae in 1801. This problem is believed to be computationally difficult.
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He used this theory of equivalent quadratic forms to prove number theory results. Lagrange, for example, had shown that any natural number can be expressed as the sum of four squares. Gauss proved this using his theory of equivalence relations by showing that the quadratic w^2 + x^2 + y^2 + z^2 represents all natural numbers. As mentioned earlier, Minkowski created and proved a similar theory for quadratic forms that had fractions as coefficients.
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The quadratic Frobenius test (QFT) is a probabilistic primality test to test whether a number is a probable prime. It is named after Ferdinand Georg Frobenius. The test uses the concepts of quadratic polynomials and the Frobenius automorphism. It should not be confused with the more general Frobenius test using a quadratic polynomial – the QFT restricts the polynomials allowed based on the input, and also has other conditions that must be met.
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Let p be an odd prime. The quadratic excess E(p) is the number of quadratic residues on the range (0,p/2) minus the number in the range (p/2,p) . For p congruent to 1 mod 4, the excess is zero, since −1 is a quadratic residue and the residues are symmetric under r ↔ p−r. For p congruent to 3 mod 4, the excess E is always positive.
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Around the turn of the millennium, he coined, together with his co-authors, the now widely used terms "Standard Quadratic Optimization" and "Copositive Optimization" or "Copositive Programming".I. Bomze, On standard quadratic optimization problems. Journal of Global Optimization 13, 369–387 (1998); I. Bomze, M. Dür, E. de Klerk, A. Quist, C. Roos and T. Terlaky, On copositive programming and standard quadratic optimization problems. Journal of Global Optimization 18, 301–320 (2000).
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This distribution is a useful model for symmetric bimodal processes. Other continuous distributions allow more flexibility, in terms of relaxing the symmetry and the quadratic shape of the density function, which are enforced in the U-quadratic distribution – e.g., beta distribution and gamma distribution.
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Sound diffusers have been based on number-theoretic concepts such as primitive roots and quadratic residues.
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Supplement XI introduces ring theory, and from then on, especially after the 1897 publication of Hilbert's Zahlbericht, the theory of binary quadratic forms lost its preeminent position in algebraic number theory and became overshadowed by the more general theory of algebraic number fields. Even so, work on binary quadratic forms with integer coefficients continues to the present. This includes numerous results about quadratic number fields, which can often be translated into the language of binary quadratic forms, but also includes developments about forms themselves or that originated by thinking about forms, including Shanks's infrastructure, Zagier's reduction algorithm, Conway's topographs, and Bhargava's reinterpretation of composition through Bhargava cubes.
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A further generalization is given by the Veronese variety, when there is more than one input variable. In the theory of quadratic forms, the parabola is the graph of the quadratic form (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form (or scalings), and the hyperbolic paraboloid is the graph of the indefinite quadratic form . Generalizations to more variables yield further such objects. The curves for other values of are traditionally referred to as the higher parabolas and were originally treated implicitly, in the form for and both positive integers, in which form they are seen to be algebraic curves.
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Vieta's approximation and the quadratic formula then start diverging again because the quadratic formula experiences loss of significance error. When c equals four hundred thousand, the difference between Vieta's approximation and the quadratic formula reaches a minimum at approximately b equals ten to the seventh. The curves are both straight to the left of the minimum, indicating a simple monomial power relationship between the difference and b. Likewise, the curves are both approximately straight to the right of the minimum, indicating a power relationship, except that the straight lines have squiggles in them due to the loss of significance errors in the quadratic formula.
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2009 Turkish 10 Lira note In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2. In the special case of the 2-element field F2 the Arf invariant can be described as the element of F2 that occurs most often among the values of the form.
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A quadratic form q over a field F is multiplicative if, for vectors of indeterminates x and y, we can write q(x).q(y) = q(z) for some vector z of rational functions in the x and y over F. Isotropic quadratic forms are multiplicative.
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The polynomial : P(x)=a_0x^4+a_1x^3+a_2x^2+a_1 m x+a_0 m^2 is almost palindromic, as (it is palindromic if ). The change of variables in produces the quadratic equation . Since , the quartic equation may be solved by applying the quadratic formula twice.
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One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension n is equivalent to the standard diagonal form :Q(z) = z_1^2 + z_2^2 + \ldots + z_n^2. Thus, for each dimension n, up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form. We will denote the Clifford algebra on Cn with the standard quadratic form by Cl(C).
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For , if Q has diagonalization diag(a), that is there is a non-zero vector x such that , then is algebra-isomorphic to a K-algebra generated by an element x satisfying , the quadratic algebra . In particular, if (that is, Q is the zero quadratic form) then is algebra-isomorphic to the dual numbers algebra over K. If a is a non-zero square in K, then . Otherwise, is isomorphic to the quadratic field extension K() of K.
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A null cone where q(x,y,z) = x^2 + y^2 - z^2 . In mathematics, given a vector space X with an associated quadratic form q, written , a null vector or isotropic vector is a non-zero element x of X for which . In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.
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Linear basis functions Quadratic basis functions The figures show the linear and the quadratic basis functions for the knots {..., 0, 1, 2, 3, 4, 4.1, 5.1, 6.1, 7.1, ...} One knot span is considerably shorter than the others. On that knot span, the peak in the quadratic basis function is more distinct, reaching almost one. Conversely, the adjoining basis functions fall to zero more quickly. In the geometrical interpretation, this means that the curve approaches the corresponding control point closely.
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These are degrees of freedom that contribute in a quadratic function to the energy of the system.
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Optimisation control schemes include: linear-quadratic regulator design (LQR), model predictive control (MPC) and eigenstructure assignment methods.
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Closing the gap between the known linear lower bounds and quadratic upper bounds remains an open problem.; ; .
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Viswanathan, M. Pan, and C. Chu. FastPlace3.0: A Fast Multilevel Quadratic Placement Algorithm with Placement Congestion Control.
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If has characteristic not equal to 2, then a bilinear form is associated with the quadratic form .
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An extension of the almost ideal demand system is the Quadratic Almost Ideal Demand System (QUAIDS) which was developed by James Banks, Richard Blundell, and Arthur Lewbel.Banks, James, Richard Blundell, and Arthur Lewbel. "Quadratic Engel curves and consumer demand." Review of Economics and statistics 79.4 (1997): 527-539.
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The pair of binary quadratic forms (ax^2+2bxy+cy^2, dx^2+2exy+fy^2) can be represented by a doubly symmetric Bhargava cube as in the figure. The law of composition of cubes is now used to define a composition law on pairs of binary quadratic forms.
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The quadratic Fourier transform extends this further to the group of all linear symplectic transformations in phase space (of which rotations are a subgroup). More specifically, for every member of the metaplectic group (which is a double cover of the symplectic group) there is a corresponding quadratic Fourier transform.
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The appearance of complex values in interpolation methods can be avoided by interpolating the inverse of f, resulting in the inverse quadratic interpolation method. Again, convergence is asymptotically faster than the secant method, but inverse quadratic interpolation often behaves poorly when the iterates are not close to the root.
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An Investigation of Secondary School Algebra Teachers' Mathematical Knowledge for Teaching Algebraic Equation Solving, p. 56 (ProQuest, 2007): "The quadratic formula is the most general method for solving quadratic equations and is derived from another general method: completing the square."Rockswold, Gary. College algebra and trigonometry and precalculus, p.
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Direct methods for solving the standard or generalized eigenvalue problems Ax = \lambda x and Ax = \lambda B x are based on transforming the problem to Schur or Generalized Schur form. However, there is no analogous form for quadratic matrix polynomials. One approach is to transform the quadratic matrix polynomial to a linear matrix pencil ( A-\lambda B), and solve a generalized eigenvalue problem. Once eigenvalues and eigenvectors of the linear problem have been determined, eigenvectors and eigenvalues of the quadratic can be determined.
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Suppose that p and q are rational primes congruent to 1 mod 4 such that the Legendre symbol (p/q) is 1. Then the ideal (p) factorizes in the ring of integers of Q() as (p)=𝖕𝖕' and similarly (q)=𝖖𝖖' in the ring of integers of Q(). Write εp and εq for the fundamental units in these quadratic fields. Then Scholz's reciprocity law says that :[εp/𝖖] = [εq/𝖕] where [] is the quadratic residue symbol in a quadratic number field.
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The complex plane is associated with two distinct quadratic spaces. For a point z = x + iy in the complex plane, the squaring function z2 and the norm-squared x^2 + y^2 are both quadratic forms. The former is frequently neglected in the wake of the latter's use in setting a metric on the complex plane. These distinct faces of the complex plane as a quadratic space arise in the construction of algebras over a field with the Cayley–Dickson process.
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A quadratic polynomial with A ≈ 11.3, currently the highest known value, has been discovered by Jacobson and Williams.
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The proof has since appeared in preprint form.Bhargava, M., & Hanke, J., Universal quadratic forms and the 290-theorem.
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C. Hooley, On the greatest prime factor of a quadratic polynomial, Acta Math., 117 ( 196 7), 281–299.
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The conjugate of a split-octonion x is given by :\bar x = x_0 - x_1\,i - x_2\,j - x_3\,k - x_4\,\ell - x_5\,\ell i - x_6\,\ell j - x_7\,\ell k , just as for the octonions. The quadratic form on x is given by :N(x) = \bar x x = (x_0^2 + x_1^2 + x_2^2 + x_3^2) - (x_4^2 + x_5^2 + x_6^2 + x_7^2) . This quadratic form N(x) is an isotropic quadratic form since there are non-zero split-octonions x with N(x) = 0. With N, the split-octonions form a pseudo-Euclidean space of eight dimensions over R, sometimes written R4,4 to denote the signature of the quadratic form.
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Gauss, BQ § 8 In the vocabulary of group theory, the first set is a subgroup of index 4 (of the multiplicative group Z/pZ×), and the other three are its cosets. The first set is the biquadratic residues, the third set is the quadratic residues that are not quartic residues, and the second and fourth sets are the quadratic nonresidues. Gauss proved that −1 is a biquadratic residue if p ≡ 1 (mod 8) and a quadratic, but not biquadratic, residue, when p ≡ 5 (mod 8).Gauss, BQ § 10 2 is a quadratic residue mod p if and only if p ≡ ±1 (mod 8). Since p is also ≡ 1 (mod 4), this means p ≡ 1 (mod 8).
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There is an alternative solution using algebraic geometry In brief, one interprets the roots as the intersection of two quadratic curves, then finds the three reducible quadratic curves (pairs of lines) that pass through these points (this corresponds to the resolvent cubic, the pairs of lines being the Lagrange resolvents), and then use these linear equations to solve the quadratic. The four roots of the depressed quartic may also be expressed as the coordinates of the intersections of the two quadratic equations and i.e., using the substitution that two quadratics intersect in four points is an instance of Bézout's theorem. Explicitly, the four points are for the four roots of the quartic.
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Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125-146 and 262 of Disquisitiones Arithmeticae in 1801. In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard statement is: . }} This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form x^2\equiv a \bmod p for an odd prime p; that is, to determine the "perfect squares" modulo p.
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Quadratic functions are extensively studied and were characterised in detail, but more general results were derived also for higher-order functions. While quadratic functions can indeed model many problems of practical interest, they are limited by the fact they can represent only binary interactions between variables. The possibility to capture higher-order interactions allows to better capture the nature of the problem and it can provide higher quality results that could be difficult to achieve with quadratic models. For instance in computer vision applications, where each variable represents a pixel or voxel of the image, higher-order interactions can be used to model texture information, that would be difficult to capture using only quadratic functions.
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Bhargava cube with the integers a, b, c, d, e, f, g, h at the corners In mathematics, in number theory, a Bhargava cube (also called Bhargava's cube) is a configuration consisting of eight integers placed at the eight corners of a cube. This configuration was extensively used by Manjul Bhargava, a Canadian-American Fields Medal winning mathematician, to study the composition laws of binary quadratic forms and other such forms. To each pair of opposite faces of a Bhargava cube one can associate an integer binary quadratic form thus getting three binary quadratic forms corresponding to the three pairs of opposite faces of the Bhargava cube. These three quadratic forms all have the same discriminant and Manjul Bhargava proved that their composition in the sense of Gauss is the identity element in the associated group of equivalence classes of primitive binary quadratic forms.
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For example, a quadratic for the numerator and a cubic for the denominator is identified as a quadratic/cubic rational function. The rational function model is a generalization of the polynomial model: rational function models contain polynomial models as a subset (i.e., the case when the denominator is a constant).
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In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras.
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The square root of 2 was the first number proved irrational, and that article contains a number of proofs. The golden ratio is another famous quadratic irrational number. The square roots of all natural numbers which are not perfect squares are irrational and a proof may be found in quadratic irrationals.
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The classification error rates of different types (false positives and false negatives) are integrals of the normal distributions within the quadratic regions defined by this classifier. Since this is mathematically equivalent to integrating a quadratic form of a normal variable, the result is an integral of a generalized-chi-squared variable.
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It is sometimes convenient to reduce a quadratic equation so that its leading coefficient is one. This is done by dividing both sides by a, which is always possible since a is non-zero. This produces the reduced quadratic equation:Alenit͡syn, Aleksandr and Butikov, Evgeniĭ. Concise Handbook of Mathematics and Physics, p.
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It uses Lucas sequences to perform exponentiation in a quadratic field. It is analogous to Pollard's p − 1 algorithm.
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There is semi-conjugacy between the dyadic transformation (the doubling map) and the quadratic polynomial case of c = –2.
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In DAC, pp. 526 - 529, 1997. like Capo. Quadratic placement later outperformed combinatorial solutions in both quality and stability.
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In mathematics, Yamamoto's reciprocity law is a reciprocity law related to class numbers of quadratic number fields, introduced by .
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Strong Frobenius pseudoprimes are also defined. Details on implementation for quadratic polynomials can be found in Crandall and Pomerance.
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The articulation of the quadratic equation can be sung to the tune of various songs as a mnemonic device.
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These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular.
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Higher-order functions can be reduced in polynomial time to a quadratic form that can be optimised with QPBO.
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Investigating pairs of distinct integer-valued quadratic forms that represent every integer the same number of times, Schiemann found that such quadratic forms must be in four or more variables, and the least possible discriminant of a four-variable pair is 1729. \- D1 mentions the Hardy–Ramanujan number. 1729 is the lowest number which can be represented by a Loeschian quadratic form a² + ab + b² in four different ways with a and b positive integers. The integer pairs (a,b) are (25,23), (32,15), (37,8) and (40,3).
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Without supersymmetry, a solution to the hierarchy problem has been proposed using just the Standard Model. The idea can be traced back to the fact that the term in the Higgs field that produces the uncontrolled quadratic correction upon renormalization is the quadratic one. If the Higgs field had no mass term, then no hierarchy problem arises. But by missing a quadratic term in the Higgs field, one must find a way to recover the breaking of electroweak symmetry through a non-null vacuum expectation value.
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Rules for quadratic equations appear in the Chinese The Nine Chapters on the Mathematical Art circa 200 BC. In his work Arithmetica, the Greek mathematician Diophantus (circa 250 AD) solved quadratic equations with a method more recognizably algebraic than the geometric algebra of Euclid. His solution gives only one root, even when both roots are positive. The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD,Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p.
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The quadratic isoperimetric inequality for mapping tori of free-group automorphisms. Memoirs of the American Mathematical Society, to appear. that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality; a proof of algorithmic solvability of the conjugacy problem for free-by-cyclic groups;O.
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In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, who used similar ideas to prove Gauss's conjecture on imaginary quadratic fields of class number one.
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Cocks IBE scheme is an identity based encryption system proposed by Clifford Cocks in 2001.Clifford Cocks, An Identity Based Encryption Scheme Based on Quadratic Residues , Proceedings of the 8th IMA International Conference on Cryptography and Coding, 2001 The security of the scheme is based on the hardness of the quadratic residuosity problem.
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A quadratic Bézier curve segment is defined by two end points and one control point. This circle is eight contiguous segments. The squares are end points and the circles are control points. The outlines of the characters (or glyphs) in TrueType fonts are made of straight line segments and quadratic Bézier curves.
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Her thesis Certain quaternary quadratic forms and diophantine equations by generalized quaternion algebras earned her a doctorate degree in 1927.
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For example, the Aitken method applied to an iterated fixed point is known as Steffensen's method, and produces quadratic convergence.
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The perimeter approaches infinity as n increases. The boundary of the Vicsek fractal is the Type 1 quadratic Koch curve.
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In a pseudo-Euclidean space, the magnitude of a vector is the value of the quadratic form for that vector.
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A simple solution is to use rounded electrodes with a large radius, so that a quadratic pump profile is obtained.
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A major step in the adoption of quadratic voting for political purposes happened in April 2019, when the Democratic caucus of the Colorado House of Representatives used quadratic voting to make budgetary decisions. The experiment in Colorado was generally viewed as successful in that the budget allocations were reported to be reasonable, and the process was smooth. However, it most importantly allowed the caucus to clearly see which bills to fund in a democratic fashion. Several organizations and communities have formed to promote adoption of quadratic voting concurrent with its continued academic research, including Democracy Earth (an online platform for quadratic voting), Collective Decision Engines (an app to facilitate QV adoption), and RadicalxChange (a community dedicated to decentralized forms of society and governance).
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Hilbert's eleventh problem is one of David Hilbert's list of open mathematical problems posed at the Second International Congress of Mathematicians in Paris in 1900. A furthering of the theory of quadratic forms, he stated the problem as follows: :Our present knowledge of the theory of quadratic number fields puts us in a position to attack successfully the theory of quadratic forms with any number of variables and with any algebraic numerical coefficients. This leads in particular to the interesting problem: to solve a given quadratic equation with algebraic numerical coefficients in any number of variables by integral or fractional numbers belonging to the algebraic realm of rationality determined by the coefficients.David Hilbert, Bulletin of the American Mathematical Society, vol. 8, no. 10 (1902), pp. 437-479.
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In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic forms, though it can be generalized to quadratic forms over any algebraic number field. In 0 and 1 dimensions the mass formula is trivial, in 2 dimensions it is essentially equivalent to Dirichlet's class number formulas for imaginary quadratic fields, and in 3 dimensions some partial results were given by Gotthold Eisenstein. The mass formula in higher dimensions was first given by , though his results were forgotten for many years.
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An extremely well-studied formulation in stochastic control is that of linear quadratic Gaussian control. Here the model is linear, the objective function is the expected value of a quadratic form, and the disturbances are purely additive. A basic result for discrete-time centralized systems with only additive uncertainty is the certainty equivalence property: that the optimal control solution in this case is the same as would be obtained in the absence of the additive disturbances. This property is applicable to all centralized systems with linear equations of evolution, quadratic cost function, and noise entering the model only additively; the quadratic assumption allows for the optimal control laws, which follow the certainty-equivalence property, to be linear functions of the observations of the controllers.
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Equivalently, it concerns the density of a different class of graphs, the locally linear graphs in which the neighborhood of every vertex is an induced matching. Neither of these types of graph can have a quadratic number of edges, but constructions are known for graphs of this type with nearly-quadratic numbers of edges.
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The function is a quadratic function. The graph of any quadratic function has the same general shape, which is called a parabola. The location and size of the parabola, and how it opens, depend on the values of , , and . As shown in Figure 1, if , the parabola has a minimum point and opens upward.
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The "3" is the imaginary part of the x-intercept. The real part is the x-coordinate of the vertex. Thus the roots are . The solutions of the quadratic equation :ax^2+bx+c=0 may be deduced from the graph of the quadratic function :y=ax^2+bx+c, which is a parabola.
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Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space.This is true only over a field of characteristic other than 2, but here we consider only ordered fields, which necessarily have characteristic 0. A symmetric bilinear form is also described as definite, semidefinite, etc. according to its associated quadratic form.
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If we use a polynomial fit to remove the quadratic part of the finite difference used in the Secant method, so that it better approximates the derivative, we obtain Steffensen's method, which has quadratic convergence, and whose behavior (both good and bad) is essentially the same as Newton's method but does not require a derivative.
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There he had the opportunity to investigate quadratic and convex programming. This developed into his doctoral dissertation under the guidance of Dantzig and Edmund Eisenberg. Cottle's first research contribution, "Symmetric Dual Quadratic Programs," was published in 1963. This was soon generalized in the joint paper "Symmetric Dual Nonlinear Programs," co-authored with Dantzig and Eisenberg.
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Example 1: Finding primes for which a is a residue Let a = 17. For which primes p is 17 a quadratic residue? We can test prime p's manually given the formula above. In one case, testing p = 3, we have 17(3 − 1)/2 = 171 ≡ 2 ≡ −1 (mod 3), therefore 17 is not a quadratic residue modulo 3. In another case, testing p = 13, we have 17(13 − 1)/2 = 176 ≡ 1 (mod 13), therefore 17 is a quadratic residue modulo 13. As confirmation, note that 17 ≡ 4 (mod 13), and 22 = 4.
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The theory of optimal control is concerned with operating a dynamic system at minimum cost. The case where the system dynamics are described by a set of linear differential equations and the cost is described by a quadratic function is called the LQ problem. One of the main results in the theory is that the solution is provided by the linear–quadratic regulator (LQR), a feedback controller whose equations are given below. The LQR is an important part of the solution to the LQG (linear–quadratic–Gaussian) problem.
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In mathematics, a parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family. For example, for the family of quadratic functions having the general form : y = ax^2 + bx + c\,, the simplest function is : y = x^2. This is therefore the parent function of the family of quadratic equations. For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function by simple translations and stretches parallel to the axes.
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In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface. If F is a Seifert surface of a knot, then the homology group H1(F, Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an imbedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot.
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The S-procedure or S-lemma is a mathematical result that gives conditions under which a particular quadratic inequality is a consequence of another quadratic inequality. The S-procedure was developed independently in a number of different contextsFrank Uhlig, A recurring theorem about pairs of quadratic forms and extensions: a survey, Linear Algebra and its Applications, Volume 25, 1979, pages 219–237.Imre Pólik and Tamás Terlaky, A Survey of the S-Lemma, SIAM Review, Volume 49, 2007, Pages 371–418. and has applications in control theory, linear algebra and mathematical optimization.
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As the linear coefficient increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse. This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see step response).
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Although quadratic TFRs offer perfect temporal and spectral resolutions simultaneously, the quadratic nature of the transforms creates cross-terms, also called "interferences". The cross-terms caused by the bilinear structure of TFDs and TFRs may be useful in some applications such as classification as the cross- terms provide extra detail for the recognition algorithm. However, in some other applications, these cross-terms may plague certain quadratic TFRs and they would need to be reduced. One way to do this is obtained by comparing the signal with a different function.
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In mathematics, a function or sequence is said to exhibit quadratic growth when its values are proportional to the square of the function argument or sequence position. "Quadratic growth" often means more generally "quadratic growth in the limit", as the argument or sequence position goes to infinity – in big Theta notation, f(x) = Θ(x2).. This can be defined both continuously (for a real-valued function of a real variable) or discretely (for a sequence of real numbers, i.e., real-valued function of an integer or natural number variable).
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Among others are: the geometry of numbers, isoperimetric problems, recurrence of random walks, quadratic reciprocity, the central limit theorem, Heisenberg's inequality.
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This relation is indeed an equivalence relation in the set of integer binary quadratic forms and it preserves discriminants and primitivity.
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The signature formula expresses the index of an analytic vector field in terms of the signature of a certain quadratic form.
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For a proof that the square root of any non-square natural number is irrational, see quadratic irrational or infinite descent.
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In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property.
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Nonlinear FM signals are common both in nature and in engineering applications. For example, the sonar system of some bats use hyperbolic FM and quadratic FM signals for echo location. In radar, certain pulse-compression schemes employ linear FM and quadratic signals. The Wigner–Ville distribution has optimal concentration in the time- frequency plane for linear frequency modulated signals.
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Let F be a field of characteristic not 2 and . If we consider the general element of V, then the quadratic forms and are equivalent since there is a linear transformation on V that makes q look like r, and vice versa. Evidently, and are isotropic. This example is called the hyperbolic plane in the theory of quadratic forms.
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If all the hard constraints are linear and some are inequalities, but the objective function is quadratic, the problem is a quadratic programming problem. It is one type of nonlinear programming. It can still be solved in polynomial time by the ellipsoid method if the objective function is convex; otherwise the problem may be NP hard.
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For quadratic polynomials, the above method may be adapted, leading to the so-called ac method of factorization.Stover, Christopher AC Method - Mathworld Consider the quadratic polynomial :ax^2 + bx + c with integer coefficients. If it has a rational root, its denominator must divide evenly. So, it may be written as a possibly reducible fraction \frac ra.
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Differential dynamic programming (DDP) is an optimal control algorithm of the trajectory optimization class. The algorithm was introduced in 1966 by Mayne and subsequently analysed in Jacobson and Mayne's eponymous book. The algorithm uses locally-quadratic models of the dynamics and cost functions, and displays quadratic convergence. It is closely related to Pantoja's step- wise Newton's method.
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Simultaneously generalizing the case of imaginary quadratic fields and cyclotomic fields is the case of a CM field K, i.e. a totally imaginary quadratic extension of a totally real field. In 1974, Harold Stark conjectured that there are finitely many CM fields of class number 1. He showed that there are finitely many of a fixed degree.
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For the credit to Carter and Wegman, see the chapter notes, p. 252. As well as in the hash function, prime numbers are used for the hash table size in quadratic probing based hash tables to ensure that the probe sequence covers the whole table. See "Quadratic probing", p. 382, and exercise C–9.9, p. 415.
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Thus every ε-quadratic form determines an ε-symmetric form. Composing these two maps either way: or yields multiplication by 2, and thus these maps are bijective if 2 is invertible in R, with the inverse given by multiplication with 1/2. An ε-quadratic form is called non-degenerate if the associated ε-symmetric form is non-degenerate.
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In Jigu Suanjin, Wang established and solved 25 cubic equations of the form: x^3+px^2+qx=N, along with 2 quadratic equations and 2 double quadratic equations. Wang's work influence later Chinese mathematicians, but it is said that it was his ideas on cubic equations which influenced the Italian mathematician Fibonacci after transmission via the Islamic world.
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For example, the stiffness matrix when piecewise quadratic finite elements are used will have more degrees of freedom than piecewise linear elements.
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The smallest value of k for which a given graph is k-outerplanar (its outerplanarity index) can be computed in quadratic time.
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The result is similar to the integration by parts theorem for the Riemann–Stieltjes integral but has an additional quadratic variation term.
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D. Shanks observed the infrastructure in real quadratic number fields when he was looking at cycles of reduced binary quadratic forms. Note that there is a close relation between reducing binary quadratic forms and continued fraction expansion; one step in the continued fraction expansion of a certain quadratic irrationality gives a unary operation on the set of reduced forms, which cycles through all reduced forms in one equivalence class. Arranging all these reduced forms in a cycle, Shanks noticed that one can quickly jump to reduced forms further away from the beginning of the circle by composing two such forms and reducing the result. He called this binary operation on the set of reduced forms a giant step, and the operation to go to the next reduced form in the cycle a baby step.
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Quadratic profile For the one-dimensional domain shown in the figure the Φ value at a control volume face is approximated using three-point quadratic function passing through the two bracketing or surrounding nodes and one other node on upstream side. In the figure, in order to calculate the value of the property at the face, we should have three nodes i.e. two bracketing or surrounding nodes and one upstream node. # Φw when uw > 0 and ue > 0 a quadratic fit through WW, W and P is used, # Φe when uw > 0 and ue > 0 a quadratic fit through W, P and E is used, # Φw when uw < 0 and ue < 0 values of W, P and E are used, # Φe when uw < 0 and ue < 0 values of P, E and EE are used.
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Namely, Berkeley relies upon Apollonius's determination of the tangent of the parabola in Berkeley's own determination of the derivative of the quadratic function.
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For polyoxymethylene the yield stress is a linear function of the pressure. However, polypropylene shows a quadratic pressure-dependence of the yield stress.
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The first thing to notice when working within the ring Z of integers is that if the prime number q is ≡ 3 (mod 4) then a residue r is a quadratic residue (mod q) if and only if it is a biquadratic residue (mod q). Indeed, the first supplement of quadratic reciprocity states that −1 is a quadratic nonresidue (mod q), so that for any integer x, one of x and −x is a quadratic residue and the other one is a nonresidue. Thus, if r ≡ a2 (mod q) is a quadratic residue, then if a ≡ b2 is a residue, r ≡ a2 ≡ b4 (mod q) is a biquadratic residue, and if a is a nonresidue, −a is a residue, −a ≡ b2, and again, r ≡ (−a)2 ≡ b4 (mod q) is a biquadratic residue.Gauss, BQ § 3 Therefore, the only interesting case is when the modulus p ≡ 1 (mod 4). Gauss provedGauss, BQ §§ 4–7 that if p ≡ 1 (mod 4) then the nonzero residue classes (mod p) can be divided into four sets, each containing (p−1)/4 numbers.
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Another instance of the separation principle arises in the setting of linear stochastic systems, namely that state estimation (possibly nonlinear) together with an optimal state feedback controller designed to minimize a quadratic cost, is optimal for the stochastic control problem with output measurements. When process and observation noise are Gaussian, the optimal solution separates into a Kalman filter and a linear-quadratic regulator. This is known as linear-quadratic- Gaussian control. More generally, under suitable conditions and when the noise is a martingale (with possible jumps), again a separation principle applies and is known as the separation principle in stochastic control. . . . .
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In mathematics, the universal invariant or u-invariant of a field describes the structure of quadratic forms over the field. The universal invariant u(F ) of a field F is the largest dimension of an anisotropic quadratic space over F, or ∞ if this does not exist. Since formally real fields have anisotropic quadratic forms (sums of squares) in every dimension, the invariant is only of interest for other fields. An equivalent formulation is that u is the smallest number such that every form of dimension greater than u is isotropic, or that every form of dimension at least u is universal.
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The Brāhmasphuṭasiddhānta ("Correctly Established Doctrine of Brahma", abbreviated BSS) is the main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including a good understanding of the role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta's identity, and Brahmagupta’s theorem. The book was written completely in verse and does not contain any kind of mathematical notation. Nevertheless, it contained the first clear description of the quadratic formula (the solution of the quadratic equation).
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Equivalence of a quadratic Bézier curve and a parabolic segment A quadratic Bézier curve is also a segment of a parabola. As a parabola is a conic section, some sources refer to quadratic Béziers as "conic arcs". With reference to the figure on the right, the important features of the parabola can be derived as follows: # Tangents to the parabola at the end-points of the curve (A and B) intersect at its control point (C). # If D is the midpoint of AB, the tangent to the curve which is perpendicular to CD (dashed cyan line) defines its vertex (V).
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In numerical analysis, inverse quadratic interpolation is a root-finding algorithm, meaning that it is an algorithm for solving equations of the form f(x) = 0. The idea is to use quadratic interpolation to approximate the inverse of f. This algorithm is rarely used on its own, but it is important because it forms part of the popular Brent's method.
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The Jenkins–Traub algorithm described earlier works for polynomials with complex coefficients. The same authors also created a three-stage algorithm for polynomials with real coefficients. See Jenkins and Traub A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration.Jenkins, M. A. and Traub, J. F. (1970), A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration, SIAM J. Numer. Anal.
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In mathematics, a Carlyle circle (named for Thomas Carlyle) is a certain circle in a coordinate plane associated with a quadratic equation. The circle has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.
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FortMP is a software package for solving large-scale optimization problems. It solves linear programming problems, quadratic programming problems and mixed integer programming problems (both linear and quadratic). Its robustness has been explored and published in the Mathematical Programming journal. FortMP is available as a standalone executable that accepts input in MPS format and as a library with interfaces in C and Fortran.
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Brahmagupta biography In Europe this problem was studied by Brouncker, Euler and Lagrange. In 1801 Gauss published Disquisitiones Arithmeticae, a major portion of which was devoted to a complete theory of binary quadratic forms over the integers. Since then, the concept has been generalized, and the connections with quadratic number fields, the modular group, and other areas of mathematics have been further elucidated.
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Proving unimodality is often hard. One way consists in using the definition of that property, but it turns out to be suitable for simple functions only. A general method based on derivatives exists, but it does not succeed for every function despite its simplicity. Examples of unimodal functions include quadratic polynomial functions with a negative quadratic coefficient, tent map functions, and more.
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Hardware Sieves: Function and Applications, and other projects In cryptography, he developed in 1994 with Renate Scheidler and Johannes Buchmann a method of public key cryptography based on real quadratic number fields.Buchmann, Williams: Quadratic fields and cryptography. In: Loxton (Hrsg.): Number theory and cryptography. 1989 Williams developed algorithms for calculating invariants of algebraic number fields such as class numbers and regulators.
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In 1967, Samuel F. Gingrich published his idea of comparing basal area per acre, trees per acre, and quadratic mean diameter in one graph. He called this the stocking diagram. These same principles are used to make the stand density management diagram work.Fischer Basal area and density are plotted against one another and quadratic mean diameter lines are plotted through the plot.
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Stifel was the first, who had a standard method to solve quadratic equations. He was able to reduce the different cases known to one case, because he uses both, positive and negative coefficients. He called his method/rule AMASIAS. The letters A, M, A/S, I, A/S each are representing a single operation step when solving a quadratic equation.
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The least quadratic residue mod p is clearly 1. The question of the magnitude of the least quadratic non-residue n(p) is more subtle, but it is always prime. The Pólya–Vinogradov inequality above gives O( log p). The best unconditional estimate is n(p) ≪ pθ for any θ>1/4, obtained by estimates of Burgess on character sums.
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These numbers occur also in other, related areas. In matrix theory, the Radon–Hurwitz number counts the maximum size of a linear subspace of the real n×n matrices, for which each non-zero matrix is a similarity transformation, i.e. a product of an orthogonal matrix and a scalar matrix. In quadratic forms, the Hurwitz problem asks for multiplicative identities between quadratic forms.
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"On quadratic differentials and extremal quasi-conformal mappings." In Proceedings of the International Congress of Mathematicians, vol. 2, p. 223. Canadian Mathematical Congress, 1975.
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A Bézier triangle is a special type of Bézier surface, which is created by (linear, quadratic, cubic or higher degree) interpolation of control points.
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Quadratic reciprocity arises from certain subtle factorization patterns involving perfect square numbers. In this section, we give examples which lead to the general case.
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Francis Buekenhout (born 23 April 1937 in Ixelles near Brussels) is a Belgian mathematician who introduced Buekenhout geometries and the concept of quadratic sets.
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The problem statement resembles that of the assignment problem, except that the cost function is expressed in terms of quadratic inequalities, hence the name.
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The function is given by a quadratic polynomial in three variables :\lambda(x,y,z) \equiv x^2 + y^2 + z^2 - 2xy - 2yz - 2zx.
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139 In particular, over p-adic fields the construction of quaternion algebras can be viewed as the quadratic Hilbert symbol of local class field theory.
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Since the only residue (mod 3) is 1, we see that −3 is a quadratic residue modulo every prime which is a residue modulo 3.
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68-71 (German) DeTemple used in 1989 and 1991 Carlyle circles to devise Compass- and-straightedge constructions for regular polygons, in particular the pentagon, the heptadecagon, the 257-gon and the 65537-gon. Ladislav Beran described in 1999, how the Carlyle circle can be used to construct the complex roots of a normed quadratic function.Ladislav Beran: The Complex Roots of a Quadratic from a Circle.
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The four exponentials conjecture rules out a special case of non-trivial, homogeneous, quadratic relations between logarithms of algebraic numbers. But a conjectural extension of Baker's theorem implies that there should be no non-trivial algebraic relations between logarithms of algebraic numbers at all, homogeneous or not. One case of non-homogeneous quadratic relations is covered by the still open three exponentials conjecture.Waldschmidt, "Variations…" (2005), consequence 1.9.
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Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form : q(x)=a_1 x_1^2 + a_2 x_2^2+ \cdots +a_n x_n^2. Such a diagonal form is often denoted by \langle a_1,\ldots,a_n\rangle. Classification of all quadratic forms up to equivalence can thus be reduced to the case of diagonal forms.
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The unitary group of a quadratic module is a generalisation of the linear algebraic group U just defined, which incorporates as special cases many different classical algebraic groups. The definition goes back to Anthony Bak's thesis.Bak, Anthony (1969), "On modules with quadratic forms", Algebraic K-Theory and its Geometric Applications (editors--Moss R. M. F., Thomas C. B.) Lecture Notes in Mathematics, Vol. 108, pp.
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Quadratic plane vector fields with four limit cycles are known. An example of numerical visualization of four limit cycles in a quadratic plane vector field can be found in . In general, the difficulties in estimating the number of limit cycles by numerical integration are due to the nested limit cycles with very narrow regions of attraction, which are hidden attractors, and semi-stable limit cycles.
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297 (F1 being the first frequency, F2 the second) These are audible most of the time and especially when the level of the original tone is low. Hence they have a greater effect on psychoacoustic tuning curves than quadratic difference tones. Quadratic difference tones are the result of F2 – F1 This happens at relatively high levels hence have a lesser effect on psychoacoustic tuning curves.
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Thus the solutions in the diagram are −AX1/SA and −AX2/SA. Carlyle circle of the quadratic equation x2 − sx + p = 0\. The Carlyle circle, named after Thomas Carlyle, has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the horizontal axis. Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons.
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Today, Diophantine analysis is the area of study where integer (whole-number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: , , and .
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In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irrationals or simply rational numbers. It is named after Harold Davenport and Wolfgang M. Schmidt.
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36 The invariants of a quadratic forms over a local field are precisely the dimension, discriminant and Hasse invariant.Serre (1973) p.39 For quadratic forms over a number field, there is a Hasse invariant ±1 for every finite place. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the signatures coming from real embeddings.
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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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In the figure, Excel is used to find the smallest root of the quadratic equation x2 + bx + c = 0 for c = 4 and c = 4 × 105. The difference between direct evaluation using the quadratic formula and the approximation described above for widely spaced roots is plotted vs. b. Initially the difference between the methods declines because the widely spaced root method becomes more accurate at larger b-values. However, beyond some b-value the difference increases because the quadratic formula (good for smaller b-values) becomes worse due to round-off, while the widely spaced root method (good for large b-values) continues to improve.
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There is a simple reduction from breaking this cryptosystem to the problem of determining whether a random value modulo N with Jacobi symbol +1 is a quadratic residue. If an algorithm A breaks the cryptosystem, then to determine if a given value x is a quadratic residue modulo N, we test A to see if it can break the cryptosystem using (x,N) as a public key. If x is a non- residue, then A should work properly. However, if x is a residue, then every "ciphertext" will simply be a random quadratic residue, so A cannot be correct more than half of the time.
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A closed geodesic on a Riemannian manifold is a closed curve that is also geodesic. One can give an effective description of the set of such curves in an arithmetic surface or three—manifold: they correspond to certain units in certain quadratic extensions of the base field (the description is lengthy and shall not be given in full here). For example, the length of primitive closed geodesics in the modular surface corresponds to the absolute value of units of norm one in real quadratic fields. This description was used by Sarnak to establish a conjecture of Gauss on the mean order of class groups of real quadratic fields.
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Because Lovelock action contains, among others, the quadratic Gauss–Bonnet term (i.e. the four-dimensional Euler characteristic extended to D dimensions), it is usually said that Lovelock theory resembles string-theory-inspired models of gravity. This is because a quadratic term is present in the low energy effective action of heterotic string theory, and it also appears in six- dimensional Calabi–Yau compactifications of M-theory. In the mid 1980s, a decade after Lovelock proposed his generalization of the Einstein tensor, physicists began to discuss the quadratic Gauss–Bonnet term within the context of string theory, with particular attention to its property of being ghost- free in Minkowski space.
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In number theory, Meyer's theorem on quadratic forms states that an indefinite quadratic form Q in five or more variables over the field of rational numbers nontrivially represents zero. In other words, if the equation :Q(x) = 0 has a non-zero real solution, then it has a non-zero rational solution (the converse is obvious). By clearing the denominators, an integral solution x may also be found. Meyer's theorem is usually deduced from the Hasse–Minkowski theorem (which was proved later) and the following statement: : A rational quadratic form in five or more variables represents zero over the field Qp of the p-adic numbers for all p.
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The quadratic Hill yield criterion depends only on the deviatoric stresses and is pressure independent. It predicts the same yield stress in tension and in compression.
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The intractability of the quadratic residuosity problem is the basis for the security of the Blum Blum Shub pseudorandom number generator and the Goldwasser–Micali cryptosystem.
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Nick Bobick (February 1998) "Rotating Objects Using Quaternions", Game Developer (magazine) Quaternions have received another boost from number theory because of their relation to quadratic forms.
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Originally theories which were quadratic in the second derivative of the scalar field were studied, but DHOST theories up to cubic order have now been studied.
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19, no. 1, pp. 153-183, January 2009. The most useful and popular methods form a class referred to as "quadratic" or bilinear time–frequency distributions.
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In the geometry of quadratic forms, an isotropic line or null line is a line for which the quadratic form applied to the displacement vector between any pair of its points is zero. An isotropic line occurs only with an isotropic quadratic form, and never with a definite quadratic form. Using complex geometry, Edmond Laguerre first suggested the existence of two isotropic lines through the point that depend on the imaginary unit :Edmond Laguerre (1870) "Sur l’emploi des imaginaires en la géométrie", Oeuvres de Laguerre 2: 89 : First system: (y - \beta) = (x - \alpha) i, : Second system: (y - \beta) = -i (x - \alpha) . Laguerre then interpreted these lines as geodesics: :An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero.
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ANTIGONE is an evolution of GloMIQO, a global Mixed-Integer Quadratic Programming solver written by Ruth Misener. ANTIGONE extends the functionality of GloMIQO to general MINLP problems.
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In one study of 38 children, only five of the children had an inflection point in their rate of word acquisition as opposed to a quadratic growth.
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While computing the correct positions takes O(n) time for every single element, thus resulting in a quadratic time algorithm, the number of writing operations is minimized.
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As a special case one can define Stiefel–Whitney classes for quadratic forms over fields, the first two cases being the discriminant and the Hasse–Witt invariant .
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The topics covered include fractions, square roots, arithmetic and geometric progressions, solutions of simple equations, simultaneous linear equations, quadratic equations and indeterminate equations of the second degree.
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The subcutaneous pedicle rhomboid flap is a relatively new technique in the treatment of linear, wide or quadratic postburn scar contractures with two or more contracture lines.
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Section IV develops a proof of quadratic reciprocity; Section V, which takes up over half of the book, is a comprehensive analysis of binary and ternary quadratic forms. Section VI includes two different primality tests. Finally, Section VII is an analysis of cyclotomic polynomials, which concludes by giving the criteria that determine which regular polygons are constructible, i.e., can be constructed with a compass and unmarked straightedge alone.
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An example of an ovoid is the elliptic quadric, the set of zeros of the quadratic form ::: x1x2 \+ f(x3, x4), where f is an irreducible quadratic form in two variables over GF(q). [f(x,y) = x2 \+ xy + y2 for example]. If q is an odd power of 2, another type of ovoid is known – the Suzuki–Tits ovoid. Theorem. Let q be a positive integer, at least 2.
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In control theory, optimal projection equations constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller. The linear-quadratic-Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, incomplete state information (i.e. not all the state variables are measured and available for feedback) also disturbed by additive white Gaussian noise and quadratic costs.
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A common instance has F = real numbers in which case and are hyperbolas. In particular, is the unit hyperbola. The notation has been used by Milnor and Husemoller for the hyperbolic plane as the signs of the terms of the bivariate polynomial r are exhibited. The affine hyperbolic plane was described by Emil Artin as a quadratic space with basis satisfying , where the products represent the quadratic form.
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The central nave is separated from the side aisles by archways consisting of large compound piers. The extension of the side aisles by one bay each enables the inclusion of the western tower in the interior architecture. On the eastern side, the nearly-quadratic hall is appended by the polygon-shaped basilican chancel. Attached to the chancel on the north side is the quadratic sacristy, covering four bays.
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The contrasting case of real quadratic fields is very different, and much less is known. That is because what enters the analytic formula for the class number is not h, the class number, on its own — but h log ε, where ε is a fundamental unit. This extra factor is hard to control. It may well be the case that class number 1 for real quadratic fields occurs infinitely often.
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The coefficient of t1t2 … tq in the above expression is q! times λ(v1, …, vq); it follows that λ = 0. Note: φ is independent of a choice of basis; so the above proof shows that ψ is also independent of a basis, the fact not a priori obvious. Example: A bilinear functional gives rise to a quadratic form in a unique way and any quadratic form arises in this way.
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But efficient factoring algorithms had not been studied much at the time, and a lot of progress was made in the following decades. Atkins et al. used the quadratic sieve algorithm invented by Carl Pomerance in 1981. While the asymptotically faster number field sieve had just been invented, it was not clear at the time that it would be better than the quadratic sieve for 129-digit numbers.
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No better worst-case time bound is possible because, for any fixed value of β smaller than one, there exist point sets in general position (small perturbations of a regular polygon) for which the β-skeleton is a dense graph with a quadratic number of edges. In the same quadratic time bound, the entire β-spectrum (the sequence of circle-based β-skeletons formed by varying β) may also be calculated.
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For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed. If one is given a quadratic equation in the form , the sought factorization has the form , and one has to find two numbers and that add up to and whose product is (this is sometimes called "Vieta's rule". and is related to Vieta's formulas). As an example, factors as .
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The more general case where does not equal can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection. Except for special cases such as where or , factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.
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Diamond EDGE 3D 2120 Market interest in the product quickly ended when Microsoft announced the DirectX specifications, based upon triangle polygon rendering. This release by Microsoft of a major industry-backed API that was generally incompatible with NV1 ended Nvidia's hopes of market leadership immediately. While demos of quadratic rendered round spheres looked good, experience had proved working with quadratic texture maps was extremely difficult. Even calculating simple routines was problematic.
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The standard parametrization of the quadratic equation is :ax^2+bx+c=0\ \ . Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as : ax^2 - 2b_1 x + c = 0, where b_1 = -b/2, or : ax^2 + 2b_2 x + c = 0, where b_2 = b/2. These alternative parametrizations result in slightly different forms for the solution, but which are otherwise equivalent to the standard parametrization.
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In the case of Karatsuba multiplication this technique is substantially faster than quadratic multiplication, even for modest-sized inputs. This is especially true when implemented in parallel hardware.
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The discrete logarithm problem, the quadratic residuosity problem, the RSA inversion problem, and the problem of computing the permanent of a matrix are each random self-reducible problems.
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Although the modifications presented by Mehrotra were intended for interior point algorithms for linear programming, the ideas have been extended and successfully applied to quadratic programming as well.
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We can do these calculations faster by using various modular arithmetic and Legendre symbol properties. If we keep calculating the values, we find: :(17/p) = +1 for p = {13, 19, ...} (17 is a quadratic residue modulo these values) :(17/p) = −1 for p = {3, 5, 7, 11, 23, ...} (17 is not a quadratic residue modulo these values). Example 2: Finding residues given a prime modulus p Which numbers are squares modulo 17 (quadratic residues modulo 17)? We can manually calculate it as: : 12 = 1 : 22 = 4 : 32 = 9 : 42 = 16 : 52 = 25 ≡ 8 (mod 17) : 62 = 36 ≡ 2 (mod 17) : 72 = 49 ≡ 15 (mod 17) : 82 = 64 ≡ 13 (mod 17). So the set of the quadratic residues modulo 17 is {1,2,4,8,9,13,15,16}. Note that we did not need to calculate squares for the values 9 through 16, as they are all negatives of the previously squared values (e.g. 9 ≡ −8 (mod 17), so 92 ≡ (−8)2 = 64 ≡ 13 (mod 17)).
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In mathematics, the genus is a classification of quadratic forms and lattices over the ring of integers. An integral quadratic form is a quadratic form on Zn, or equivalently a free Z-module of finite rank. Two such forms are in the same genus if they are equivalent over the local rings Zp for each prime p and also equivalent over R. Equivalent forms are in the same genus, but the converse does not hold. For example, x2 \+ 82y2 and 2x2 \+ 41y2 are in the same genus but not equivalent over Z. Forms in the same genus have equal discriminant and hence there are only finitely many equivalence classes in a genus.
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Excel graph of the difference between two evaluations of the smallest root of a quadratic: direct evaluation using the quadratic formula (accurate at smaller b) and an approximation for widely spaced roots (accurate for larger b). The difference reaches a minimum at the large dots, and round-off causes squiggles in the curves beyond this minimum. The bottom line is that in doing this calculation using Excel, as the roots become farther apart in value, the method of calculation will have to switch from direct evaluation of the quadratic formula to some other method so as to limit round-off error. The point to switch methods varies according to the size of coefficients a and b.
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There is circumstantial evidence of protohistoric knowledge of algebraic identities involving binary quadratic forms. The first problem concerning binary quadratic forms asks for the existence or construction of representations of integers by particular binary quadratic forms. The prime examples are the solution of Pell's equation and the representation of integers as sums of two squares. Pell's equation was already considered by the Indian mathematician Brahmagupta in the 7th century CE. Several centuries later, his ideas were extended to a complete solution of Pell's equation known as the chakravala method, attributed to either of the Indian mathematicians Jayadeva or Bhāskara II. The problem of representing integers by sums of two squares was considered in the 3rd century by Diophantus.
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Hence, in such a case, the harmonics are not so distinguishable in the spectrum. The time frequency approach for machine fault diagnosis can be divided into two broad categories: linear methods and the quadratic methods. The difference is that linear transforms can be inverted to construct the time signal, thus, they are more suitable for signal processing, such as noise reduction and time-varying filtering. Although the quadratic method describes the energy distribution of a signal in the joint time frequency domain, which is useful for analysis, classification, and detection of signal features, phase information is lost in the quadratic time-frequency representation; also, the time histories cannot be reconstructed with this method.
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Solving multivariate quadratic equations (MQ) over a finite set of numbers is an NP-hard problem (in the general case) with several applications in cryptography. The XSL attack requires an efficient algorithm for tackling MQ. In 1999, Kipnis and Shamir showed that a particular public key algorithm, known as the Hidden Field Equations scheme (HFE), could be reduced to an overdetermined system of quadratic equations (more equations than unknowns). One technique for solving such systems is linearization, which involves replacing each quadratic term with an independent variable and solving the resultant linear system using an algorithm such as Gaussian elimination. To succeed, linearization requires enough linearly independent equations (approximately as many as the number of terms).
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The origins of class field theory lie in the quadratic reciprocity law proved by Gauss. The generalization took place as a long-term historical project, involving quadratic forms and their 'genus theory', work of Ernst Kummer and Leopold Kronecker/Kurt Hensel on ideals and completions, the theory of cyclotomic and Kummer extensions. The first two class field theories were very explicit cyclotomic and complex multiplication class field theories. They used additional structures: in the case of the field of rational numbers they use roots of unity, in the case of imaginary quadratic extensions of the field of rational numbers they use elliptic curves with complex multiplication and their points of finite order.
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This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book The Pythagorean Proposition contains 370 proofs.
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Lagrange completed a proof in 1775Nouv. Mém. Acad. Berlin, année 1771, 125; ibid. année 1773, 275; ibid année 1775, 351. based on his general theory of integral quadratic forms.
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In the theory of Lorentzian manifolds, particularly in the context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann.
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Fix an integer d and let D be the discriminant of the imaginary quadratic field Q(√-d). The Zimmert set Z(d) is the set of positive integers n such that 4n2 < -D-3 and n ≠ 2; D is a quadratic non-residue of all odd primes in d; n is odd if D is not congruent to 5 modulo 8. The cardinality of Z(d) may be denoted by z(d).
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For applying the above general construction of finite fields in the case of , one has to find an irreducible polynomial of degree 2. For , this has been done in the preceding section. If is an odd prime, there are always irreducible polynomials of the form , with in . More precisely, the polynomial is irreducible over if and only if is a quadratic non-residue modulo (this is almost the definition of a quadratic non-residue).
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In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing.
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There is thus a group homomorphism whose kernel has two elements denoted , where is the identity element. Thus, the group elements and of are equivalent after the homomorphism to ; that is, for any in . The groups and are all Lie groups, and for fixed they have the same Lie algebra, . If is real, then is a real vector subspace of its complexification , and the quadratic form extends naturally to a quadratic form on .
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The notion of a sheaf and sheafification of a presheaf date to early category theory, and can be seen as the linear form of the calculus of functors. The quadratic form can be seen in the work of André Haefliger on links of spheres in 1965, where he defined a "metastable range" in which the problem is simpler. This was identified as the quadratic approximation to the embeddings functor in Goodwillie and Weiss.
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In mathematics, the Hasse invariant (or Hasse–Witt invariant) of a quadratic form Q over a field K takes values in the Brauer group Br(K). The name "Hasse–Witt" comes from Helmut Hasse and Ernst Witt. The quadratic form Q may be taken as a diagonal form :Σ aixi2. Its invariant is then defined as the product of the classes in the Brauer group of all the quaternion algebras :(ai, aj) for i < j.
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Vanderbei later developed algorithms for quadratic problems, convex, and finally nonlinear optimization problems.Vanderbei, R.J.: LOQO: An interior point code for quadratic programming, Optimization Methods and Software, 12:451–484, 1999.Vanderbei, R.J.; Shanno, D.F.: An Interior-Point Algorithm for Nonconvex Nonlinear Programming, Computational Optimization and Applications, 13:231–252, 1999. Vanderbei is the author of a textbook on linear programmingVanderbei, R.J.: Linear Programming: Foundations and Extensions, Kluwer Academic Publishers, 3rd edition, 2007.
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In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. They were introduced by Gilles Robert in 1973, and were used by John Coates and Andrew Wiles in their work on the Birch and Swinnerton-Dyer conjecture. Elliptic units are an analogue for imaginary quadratic fields of cyclotomic units. They form an example of an Euler system.
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Part 1 covers the theory of general number fields, including ideals, discriminants, differents, units, and ideal classes. Part 2 covers Galois number fields, including in particular Hilbert's theorem 90. Part 3 covers quadratic number fields, including the theory of genera, and class numbers of quadratic fields. Part 4 covers cyclotomic fields, including the Kronecker–Weber theorem (theorem 131), the Hilbert–Speiser theorem (theorem 132), and the Eisenstein reciprocity law for lth power residues (theorem 140) .
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Seeber is known for his study of positive ternary quadratic forms in 1831, which was applauded by Carl Friedrich Gauss (1831) and later simplified by Peter Gustav Lejeune Dirichlet (1847).
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Examples of quadratic residue codes include the (7,4) Hamming code over GF(2), the (23,12) binary Golay code over GF(2) and the (11,6) ternary Golay code over GF(3).
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This forms the basis of several factorization algorithms (such as the quadratic sieve) and can be combined with the Fermat primality test to give the stronger Miller–Rabin primality test.
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By calculus it can be shown that a point load will lead to a linearly varying moment diagram, and a constant distributed load will lead to a quadratic moment diagram.
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If then the biquadratic function : Q(x) = a_4x^4+a_2x^2+a_0\,\\! defines a biquadratic equation, which is easy to solve. Let the auxiliary variable . Then becomes a quadratic in : .
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Some refer to the Pumi as the "Hungarian herding terrier" because it has some terrier-like attributes such as quick movement, alert temperament, and a quadratic, lean and muscular body type.
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Helmut Hasse's book Vorlesungen über Zahlentheorie was published in 1950, and is different from and more elementary than his book Zahlentheorie. It covers elementary number theory, Dirichlet's theorem, and quadratic fields.
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In mathematics, the Fatou conjecture, named after Pierre Fatou, states that a quadratic family of maps from the complex plane to itself is hyperbolic for an open dense set of parameters.
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Vogtmann's early work concerned homological properties of orthogonal groups associated to quadratic forms over various fields.Karen Vogtmann, Spherical posets and homology stability for O_{n,n}. Topology, vol. 20 (1981), no.
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The inner product is more correctly called a scalar product in this context, as the nondegenerate quadratic form in question need not be positive definite (need not be an inner product).
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The Hasse–Minkowski theorem is a fundamental result in number theory which states that two quadratic forms over a number field are equivalent if and only if they are equivalent locally at all places, i.e. equivalent over every completion of the field (which may be real, complex, or p-adic). A related result is that a quadratic space over a number field is isotropic if and only if it is isotropic locally everywhere, or equivalently, that a quadratic form over a number field nontrivially represents zero if and only if this holds for all completions of the field. The theorem was proved in the case of the field of rational numbers by Hermann Minkowski and generalized to number fields by Helmut Hasse.
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An explanation of this is that although the logarithm of the lognormal density function is quadratic in , yielding a "bowed" shape in a log–log plot, if the quadratic term is small relative to the linear term then the result can appear almost linear, and the lognormal behavior is only visible when the quadratic term dominates, which may require significantly more data. Therefore, a log–log plot that is slightly "bowed" downwards can reflect a log-normal distribution – not a power law. In general, many alternative functional forms can appear to follow a power-law form for some extent. Stumpf proposed plotting the empirical cumulative distribution function in the log-log domain and claimed that a candidate power-law should cover at least two orders of magnitude.
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This agreement is because in the classical statistical theory of Ludwig Boltzmann, the heat capacity of solids approaches a maximum of 3R per mole of atoms because full vibrational-mode degrees of freedom amount to 3 degrees of freedom per atom, each corresponding to a quadratic kinetic energy term and a quadratic potential energy term. By the equipartition theorem, the average of each quadratic term is kBT, or RT per mole (see derivation below). Multiplied by 3 degrees of freedom and the two terms per degree of freedom, this amounts to 3R per mole heat capacity. The Dulong-Petit law fails at room temperatures for light atoms bonded strongly to each other, such as in metallic beryllium and in carbon as diamond.
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Goldfeld's research interests include various topics in number theory. In his thesis,Goldfeld, Dorian, Artin's conjecture on the average, Mathematika, 15 1968 he proved a version of Artin's conjecture on primitive roots on the average without the use of the Riemann Hypothesis. In 1976, Goldfeld provided an ingredient for the effective solution of Gauss' class number problem for imaginary quadratic fields.Goldfeld, Dorian, The class number of quadratic fields and the conjectures of Birch and Swinnerton-Dyer. Ann.
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Each factor defines a point at infinity on the curve: if bx − ay is such a factor, then it defines the point at infinity (a, b, 0). Over the reals, pd factors into linear and quadratic factors. The irreducible quadratic factors define non-real points at infinity, and the real points are given by the linear factors. If (a, b, 0) is a point at infinity of the curve, one says that (a, b) is an asymptotic direction.
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Two elements v and w of V are called orthogonal if . The kernel of a bilinear form B consists of the elements that are orthogonal to every element of V. Q is non-singular if the kernel of its associated bilinear form is {0}. If there exists a non-zero v in V such that , the quadratic form Q is isotropic, otherwise it is anisotropic. This terminology also applies to vectors and subspaces of a quadratic space.
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Complementarity problems were originally studied because the Karush–Kuhn–Tucker conditions in linear programming and quadratic programming constitute a linear complementarity problem (LCP) or a mixed complementarity problem (MCP). In 1963 Lemke and Howson showed that, for two person games, computing a Nash equilibrium point is equivalent to an LCP. In 1968 Cottle and Dantzig unified linear and quadratic programming and bimatrix games. Since then the study of complementarity problems and variational inequalities has expanded enormously.
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Onorato Timothy O'Meara (January 29, 1928 – June 17, 2018) was an American mathematician known for his work in number theory, linear groups and quadratic forms. He was provost emeritus and professor emeritus of mathematics at the University of Notre Dame. O’Meara was the author of Symplectic Groups, Introduction to Quadratic Forms, and The Classical Groups and K-Theory (co- author, ). In 2008, the University of Notre Dame Mathematics Library was rededicated and named in O'Meara's honor.
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Sequential quadratic programming (SQP) is an iterative method for constrained nonlinear optimization. SQP methods are used on mathematical problems for which the objective function and the constraints are twice continuously differentiable. SQP methods solve a sequence of optimization subproblems, each of which optimizes a quadratic model of the objective subject to a linearization of the constraints. If the problem is unconstrained, then the method reduces to Newton's method for finding a point where the gradient of the objective vanishes.
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In control theory, the linear–quadratic–Gaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns linear systems driven by additive white Gaussian noise. The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic cost criterion. Output measurements are assumed to be corrupted by Gaussian noise and the initial state, likewise, is assumed to be a Gaussian random vector.
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104 The modular function j(τ) is algebraic on imaginary quadratic numbers τ:Serre (1967) p. 293 these are the only algebraic numbers in the upper half-plane for which j is algebraic. If Λ is a lattice with period ratio τ then we write j(Λ) for j(τ). If further Λ is an ideal a in the ring of integers OK of a quadratic imaginary field K then we write j(a) for the corresponding singular modulus.
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In number theory, the Baker–Heegner–Stark theorem calls this the Stark–Heegner theorem (cognate to Stark–Heegner points as in page xiii of ) but omitting Baker's name is atypical. gratuitously adds Deuring and Siegel in his paper's title. states precisely which quadratic imaginary number fields admit unique factorisation in their ring of integers. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number.
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In statistics, the rational quadratic covariance function is used in spatial statistics, geostatistics, machine learning, image analysis, and other fields where multivariate statistical analysis is conducted on metric spaces. It is commonly used to define the statistical covariance between measurements made at two points that are d units distant from each other. Since the covariance only depends on distances between points, it is stationary. If the distance is Euclidean distance, the rational quadratic covariance function is also isotropic.
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Closer analysis suggests that Kleiber's law does not hold over a wide variety of scales. Metabolic rates for smaller animals (birds under , or insects) typically fit to much better than ; for larger animals, the reverse holds. As a result, log-log plots of metabolic rate versus body mass appear to "curve" upward, and fit better to quadratic models. But note that a quadratic curve has undesirable theoretical implications; see In all cases, local fits exhibit exponents in the range.
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These concepts can even assist with in number- theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence of square roots modulo integer prime numbers. Early attempts to prove Fermat's Last Theorem led to Kummer's introduction of regular primes, integer prime numbers connected with the failure of unique factorization in the cyclotomic integers., Section I.7, p.
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If A is a commutative semigroup, then one has :\forall x, y \isin A \quad (xy)^2 = xy xy = xx yy = x^2 y^2 . In the language of quadratic forms, this equality says that the square function is a "form permitting composition". In fact, the square function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by L. E. Dickson to produce the octonions out of quaternions by doubling.
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Although Cereceda's conjecture itself remains open even for degeneracy , it is known that for any fixed value of the diameter of the space of -colorings is polynomial (with a different polynomial for different values of ). More precisely, the diameter is . When the number of colorings is at least , the diameter is quadratic. A related question concerns the possibility that, for numbers of colors greater than , the diameter of the space of colorings might decrease from quadratic to linear.
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148 He was a Founder of the Quadratic Lodge, Hampton Court; the Æsculapius Lodge, London; the Navy Lodge, London; and the Belgrave Chapter, London.Anonymous (2003), p. 147 He was a Past Master (a former Worshipful Master, the senior officer of a Masonic Lodge) of the Quadratic Lodge; the Æsculapius Lodge; the Prince of Wales Lodge; the Phoenix Lodge, Jamaica; and the Pentangle Lodge, Kent. He was a member of the Orders of Knights Templar and the Knights of Malta.
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Operators are also involved in probability theory, such as expectation, variance, and covariance. Indeed, every covariance is basically a dot product; every variance is a dot product of a vector with itself, and thus is a quadratic norm; every standard deviation is a norm (square root of the quadratic norm); the corresponding cosine to this dot product is the Pearson correlation coefficient; expected value is basically an integral operator (used to measure weighted shapes in the space).
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The fact that the CARS signal is quadratic in the distance makes it quadratic with respect to the concentration and therefore especially sensitive to the majority constituent. The total CARS signal also contains an inherent non-resonant background. This non-resonant signal can be considered as the result of (several) far off-resonance transitions that also add coherently. The resonant amplitude contains a phase shift of π radians over the resonance whereas the non-resonant part does not.
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Over F2, the Arf invariant is 0 if the quadratic form is equivalent to a direct sum of copies of the binary form xy, and it is 1 if the form is a direct sum of x^2+xy+y^2 with a number of copies of xy. William Browder has called the Arf invariant the democratic invariantMartino and Priddy, p.61 because it is the value which is assumed most often by the quadratic form.Browder, Proposition III.
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Schwarzschild, Sitzungsberichten der Kgl. Preuss. Akad. d. Wiss. April 1916, p. 548 were independently able to derive equations for the linear and quadratic Stark effect in hydrogen. Four years later, Hendrik KramersH.
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Arc diagrams were used by to visualize the state diagram of a shift register, and by to show that the crossing number of every graph is at least quadratic in its cutwidth.
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In particular, is correct to 12 decimal places. We see that the number of correct digits after the decimal point increases from 2 (for ) to 5 and 10, illustrating the quadratic convergence.
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Many different methods to derive the quadratic formula are available in the literature. The standard one is a simple application of the completing the square technique., Chapter 13 §4.4, p. 291Li, Xuhui.
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The generalized chi-square distribution is obtained from the quadratic form z′Az where z is a zero-mean Gaussian vector having an arbitrary covariance matrix, and A is an arbitrary matrix.
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A Hebrew treatise on practical geometry and Islamic algebra, the book contains the first known complete solution of the quadratic equation x^2 - ax + b = c, and influenced the work of Leonardo Fibonacci.
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This is equivalent to quadratic reciprocity. He could not prove it, but he did prove the second supplement.Ireland & Rosen, pp. 69-70\. His proof is based on what are now called Gauss sums.
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Several second-order wave properties, i.e. quadratic in the wave amplitude a, can be derived directly from Airy wave theory. They are of importance in many practical applications, e.g. forecasts of wave conditions.
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Let (M, N) be the pair of 2 × 2 matrices associated with a pair of opposite sides of a Bhargava cube; the matrices are formed in such a way that their rows and columns correspond to the edges of the corresponding faces. The integer binary quadratic form associated with this pair of faces is defined as :Q=-\det (Mx+Ny) The quadratic form is also defined as :Q =-\det(Mx-Ny) However, the former definition will be assumed in the sequel.
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Many meshes use linear elements, where the mapping from the abstract to realized element is linear, and mesh edges are straight segments. Higher order polynomial mappings are common, especially quadratic. A primary goal for higher-order elements is to more accurately represent the domain boundary, although they have accuracy benefits in the interior of the mesh as well. One of the motivations for cubical meshes is that linear cubical elements have some of the same numerical advantages as quadratic simplicial elements.
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Then he treats geometric measurements—employing 62,832/20,000 (= 3.1416) for π—and develops properties of similar right-angled triangles and of two intersecting circles. Using the Pythagorean theorem, he obtained one of the two methods for constructing his table of sines. He also realized that second-order sine difference is proportional to sine. Mathematical series, quadratic equations, compound interest (involving a quadratic equation), proportions (ratios), and the solution of various linear equations are among the arithmetic and algebraic topics included.
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More recently, E. Glen Weyl has done research on the mechanism, invented the name, and extensively promoted the mechanism.This paper has been revised several times, but was originally published online in 2012. After circulating working papers on the idea starting in 2012, Weyl worked on QV with Steven Lalley and Eric Posner to further refine the formalism of quadratic voting and its applications. Many experiments at various scales and simulations, and theoretical analyses have been done on quadratic voting since 2012.
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Let be a finite-dimensional real or complex vector space with a nondegenerate quadratic form . The (real or complex) linear maps preserving form the orthogonal group . The identity component of the group is called the special orthogonal group . (For real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, has a unique connected double cover, the spin group .
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In 1937 Theodor Schneider proved the aforementioned result that if is a quadratic irrational number in the upper half plane then is an algebraic integer. In addition he proved that if is an algebraic number but not imaginary quadratic then is transcendental. The function has numerous other transcendental properties. Kurt Mahler conjectured a particular transcendence result that is often referred to as Mahler's conjecture, though it was proved as a corollary of results by Yu. V. Nesterenko and Patrice Phillipon in the 1990s.
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In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graphs allow graph-theoretic tools to be applied to the number theory of quadratic residues, and have interesting properties that make them useful in graph theory more generally. Paley graphs are named after Raymond Paley.
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First assume K is a field of characteristic different from 2. Let E be an elliptic curve over K of the form: : y^2 = x^3 + a_2 x^2 +a_4 x + a_6. \, Given d eq 0 not quadratic residue, the quadratic twist of E is the curve E^d, defined by the equation: : dy^2 = x^3 + a_2 x^2 + a_4 x + a_6. \, or equivalently : y^2 = x^3 + d a_2 x^2 + d^2 a_4 x + d^3 a_6.
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The short-term Fourier transform (STFT) and the Gabor transform are two algorithms commonly used as linear time-frequency methods. If we consider linear time-frequency analysis to be the evolution of the conventional FFT, then quadratic time frequency analysis would be the power spectrum counterpart. Quadratic algorithms include the Gabor spectrogram, Cohen's class and the adaptive spectrogram. The main advantage of time frequency analysis is discovering the patterns of frequency changes, which usually represent the nature of the signal.
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Riccati, Jacopo (1724) "Animadversiones in aequationes differentiales secundi gradus" (Observations regarding differential equations of the second order), Actorum Eruditorum, quae Lipsiae publicantur, Supplementa, 8 : 66-73. Translation of the original Latin into English by Ian Bruce. More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation.
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The simple group SO(q) can always be defined as the maximal smooth connected subgroup of O(q) over k.) When k is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this group SO(n). For a general field k, different quadratic forms of dimension n can yield non- isomorphic simple groups SO(q) over k, although they all have the same base change to the algebraic closure \overline k.
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The conic sections have some very similar properties in the Euclidean plane and the reasons for this become clearer when the conics are viewed from the perspective of a larger geometry. The Euclidean plane may be embedded in the real projective plane and the conics may be considered as objects in this projective geometry. One way to do this is to introduce homogeneous coordinates and define a conic to be the set of points whose coordinates satisfy an irreducible quadratic equation in three variables (or equivalently, the zeros of an irreducible quadratic form). More technically, the set of points that are zeros of a quadratic form (in any number of variables) is called a quadric, and the irreducible quadrics in a two dimensional projective space (that is, having three variables) are traditionally called conics.
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In time-frequency signal processing, a filter bank is a special quadratic time-frequency distribution (TFD) that represents the signal in a joint time-frequency domain. It is related to the Wigner-Ville distribution by a two-dimensional filtering that defines the class of quadratic (or bilinear) time-frequency distributions.B. Boashash, editor, "Time-Frequency Signal Analysis and Processing – A Comprehensive Reference", Elsevier Science, Oxford, 2003; The filter bank and the spectrogram are the two simplest ways of producing a quadratic TFD; they are in essence similar as one (the spectrogram) is obtained by dividing the time-domain in slices and then taking a Fourier transform, while the other (the filter bank) is obtained by dividing the frequency domain in slices forming bandpass filters that are excited by the signal under analysis.
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The use of a quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because of the properties of variances, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target. If the target is t, then a quadratic loss function is :\lambda(x) = C (t-x)^2 \; for some constant C; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. Many common statistics, including t-tests, regression models, design of experiments, and much else, use least squares methods applied using linear regression theory, which is based on the quadratic loss function.
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As with the orthogonal group, the projective orthogonal group can be generalized in two main ways: changing the field or changing the quadratic form. Other than the real numbers, primary interest is in complex numbers or finite fields, while (over the reals) quadratic forms can also be indefinite forms, and are denoted PO(p,q) by their signature. The complex projective orthogonal group, PO(n,C) should not be confused with the projective unitary group, PU(n): PO preserves a symmetric form, while PU preserves a hermitian form – PU is the symmetries of complex projective space (preserving the Fubini–Study metric). In fields of characteristic 2 there are added complications: quadratic forms and symmetric bilinear forms are no longer equivalent, I = -I, and the determinant needs to be replaced by the Dickson invariant.
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McCrimmon was a Sloan Fellow in 1968 and an Invited Speaker of the International Congress of Mathematicians in 1974 in Vancouver.McCrimmon, Kevin. "Quadratic methods in nonassociative algebras." In International Congress of Mathematicians, Vancouver, 1974.
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To construct a regular 257-gon using Carlyle circles, as many as 24 Carlyle circles are to be constructed. One of these is the circle to solve the quadratic equation x2 + x − 64 = 0.
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The author has given a mean quadratic error (RMS) of 0.049 g·cm−3 for 166 checked components. Only for two components (acetonitrile and dibromochloromethane) has an error greater than 0.1 g·cm −3 been found.
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The contents of Professor John Stillwell's 1999 translation of the are as follows :Chapter 1. On the divisibility of numbers :Chapter 2. On the congruence of numbers :Chapter 3. On quadratic residues :Chapter 4.
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The Smith–Minkowski–Siegel mass formula gives the weight or mass of the quadratic forms in a genus, the count of equivalence classes weighted by the reciprocals of the orders of their automorphism groups.
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Leipzig, Germany, 1801. New Haven, CT: Yale University Press, 1965. He further advanced modular arithmetic, greatly simplifying manipulations in number theory. On 8 April he became the first to prove the quadratic reciprocity law.
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Then τ defines a Moufang set structure on J. The Hua maps ha of the Moufang structure are just the quadratic Ua . Note that the link is more natural in terms of J-structures.
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Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59 (1980), no. 2, 189–204. Kobayashi and Ochiai also characterized the situation of as being biholomorphic to a quadratic hypersurface of complex projective space.
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An alternative way of deriving the quadratic formula is via the method of Lagrange resolvents,Clark, A. (1984). Elements of abstract algebra. Courier Corporation. p. 146. which is an early part of Galois theory.
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GORDIAN formulates the wirelength cost as a quadratic function while still spreading cells apart through recursive partitioning. The algorithmH. Eisenmann and F. M. Johannes. Generic Global Placement and Floorplanning. In DAC, pages 269–274, 1998.
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While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the theory of quadratic forms) was an achievement of the twentieth century.
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In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms. Quadratus is Latin for square.
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He received his Ph.D. in Mathematics from MIT for his thesis "The 4-part of the class group of a quadratic field", in 1974. His advisor for both his masters and Ph.D was Harold Stark.
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Using the Chinese remainder theorem these are equivalent to p ≡ 1, 9 (mod 20) or p ≡ 3, 7 (mod 20). The generalization of the rules for −3 and 5 is Gauss's statement of quadratic reciprocity.
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In numerical analysis, Steffensen's method is a root-finding technique named after Johan Frederik Steffensen which is similar to Newton's method. Steffensen's method also achieves quadratic convergence, but without using derivatives as Newton's method does.
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If the restriction of Q to a subspace U of V is identically zero, U is totally singular. The orthogonal group of a non-singular quadratic form Q is the group of the linear automorphisms of V that preserve Q, that is, the group of isometries of into itself. If a quadratic space has a product so that A is an algebra over a field, and satisfies :\forall x, y \isin A \quad Q(x y) = Q(x) Q(y) , then it is a composition algebra.
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The FICO Xpress optimizer is a commercial optimization solver for linear programming (LP), mixed integer linear programming (MILP), convex quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP) and their mixed integer counterparts. Xpress includes a general purpose non-linear solver, Xpress NonLinear, including a successive linear programming algorithm (SLP, first- order method), and Artelys Knitro (second-order methods). Xpress was originally developed by Dash Optimization, and was acquired by FICO in 2008. "Dash Optimization acquired by FICO" Jan 22, 2008.
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Polynomial equations of degree up to four can be solved exactly using algebraic methods, of which the quadratic formula is the simplest example. Polynomial equations with a degree of five or higher require in general numerical methods (see below) or special functions such as Bring radicals, although some specific cases may be solvable algebraically, for example :4x^5 - x^3 - 3 = 0 (by using the rational root theorem), and :x^6 - 5x^3 + 6 = 0 \, , (by using the substitution , which simplifies this to a quadratic equation in ).
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In numerical analysis, Brent's method is a root-finding algorithm combining the bisection method, the secant method and inverse quadratic interpolation. It has the reliability of bisection but it can be as quick as some of the less-reliable methods. The algorithm tries to use the potentially fast- converging secant method or inverse quadratic interpolation if possible, but it falls back to the more robust bisection method if necessary. Brent's method is due to Richard Brent and builds on an earlier algorithm by Theodorus Dekker.
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A geometric construction of the quadratic mean and the Pythagorean means (of two numbers a and b). Harmonic mean denoted by H, geometric by G, arithmetic by A and quadratic mean (also known as root mean square) denoted by Q. Comparison of the arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are asymptotes for the harmonic means. In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM).
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The square function is defined in any field or ring. An element in the image of this function is called a square, and the inverse images of a square are called square roots. The notion of squaring is particularly important in the finite fields Z/pZ formed by the numbers modulo an odd prime number . A non-zero element of this field is called a quadratic residue if it is a square in Z/pZ, and otherwise, it is called a quadratic non-residue.
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Suppose that the function has a zero at , i.e., , and is differentiable in a neighborhood of . If is continuously differentiable and its derivative is nonzero at , then there exists a neighborhood of such that for all starting values in that neighborhood, the sequence will converge to .. If the function is continuously differentiable and its derivative is not 0 at and it has a second derivative at then the convergence is quadratic or faster. If the second derivative is not 0 at then the convergence is merely quadratic.
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A similar but more complicated method works for cubic equations, where one has three resolvents and a quadratic equation (the "resolving polynomial") relating and , which one can solve by the quadratic equation, and similarly for a quartic equation (degree 4), whose resolving polynomial is a cubic, which can in turn be solved. The same method for a quintic equation yields a polynomial of degree 24, which does not simplify the problem, and, in fact, solutions to quintic equations in general cannot be expressed using only roots.
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Translated into modern notation, Euler stated Lemmermeyer, p. 5, Ireland & Rosen, pp. 54, 61 that for distinct odd primes p and q: # If q ≡ 1 (mod 4) then q is a quadratic residue (mod p) if and only if there exists some integer b such that p ≡ b2 (mod q). # If q ≡ 3 (mod 4) then q is a quadratic residue (mod p) if and only if there exists some integer b which is odd and not divisible by q such that p ≡ ±b2 (mod 4q).
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One concern is that PBL may be inappropriate in mathematics, the reason being that mathematics is primarily skill-based at the elementary level. Transforming the curriculum into an over-reaching project or series of projects does not allow for necessary practice of particular mathematical skills. For instance, factoring quadratic expressions in elementary algebra requires extensive repetition . On the other hand, a teacher could integrate a PBL approach into the standard curriculum, helping the students see some broader contexts where abstract quadratic equations may apply.
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In multivariate statistics, if \varepsilon is a vector of n random variables, and \Lambda is an n-dimensional symmetric matrix, then the scalar quantity \varepsilon^T\Lambda\varepsilon is known as a quadratic form in \varepsilon.
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And also Yee and Hadi (2014) show that RCIMs can be used to fit unconstrained quadratic ordination models to species data; this is an example of indirect gradient analysis in ordination (a topic in statistical ecology).
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An important case of the quadratic mapping is c=0. In this case, we get \alpha_1 = 0 and \alpha_2=1. In this case, 0 is a superattractive fixed point, and 1 belongs to the Julia set.
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Of course, this makes no sense if , but since the constant term of is , is a root of if and only if , and in this case the roots of can be found using the quadratic formula.
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The pattern of variation in the spectrum suggest there are regions of enhanced strontium, chromium, iron, titanium, and magnesium on the surface of the star. The averaged quadratic field strength of the surface magnetic field is .
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The ancient Egyptians were the first civilization to develop and solve second-degree (quadratic) equations. This information is found in the Berlin Papyrus fragment. Additionally, the Egyptians solve first-degree algebraic equations found in Rhind Mathematical Papyrus.
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Their operational accuracies may also be reliably estimated from the theory of Minimum-Norm Quadratic Unbiased Estimation (MINQUE) of C. R. Rao (1920- ) and used for controlling the stability of this optimal fast Kalman filtering (Lange, 2015).
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A Clifford algebra is a unital associative algebra that contains and is generated by a vector space V over a field K, where V is equipped with a quadratic form . The Clifford algebra is the "freest" algebra generated by V subject to the conditionMathematicians who work with real Clifford algebras and prefer positive definite quadratic forms (especially those working in index theory) sometimes use a different choice of sign in the fundamental Clifford identity. That is, they take . One must replace Q with −Q in going from one convention to the other.
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A common source of such sensitivity to initial conditions is that the map represents a repeated folding and stretching of the space on which it is defined. In the case of the logistic map, the quadratic difference equation describing it may be thought of as a stretching-and-folding operation on the interval . The following figure illustrates the stretching and folding over a sequence of iterates of the map. Figure (a), left, shows a two-dimensional Poincaré plot of the logistic map's state space for , and clearly shows the quadratic curve of the difference equation ().
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As in the case of translation surfaces there is an analytic interpretation: a half-translation surface can be interpreted as a pair (X, \phi) where X is a Riemann surface and \phi a quadratic differential on X. To pass from the geometric picture to the analytic picture one simply takes the quadratic differential defined locally by (dz)^2 (which is invariant under half-translations), and for the other direction one takes the Riemannian metric induced by \phi, which is smooth and flat outside of the zeros of \phi.
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Algebra began with computations similar to those of arithmetic, with letters standing for numbers. This allowed proofs of properties that are true no matter which numbers are involved. For example, in the quadratic equation :ax^2+bx+c=0, a, b, c can be any numbers whatsoever (except that a cannot be 0), and the quadratic formula can be used to quickly and easily find the values of the unknown quantity x which satisfy the equation. That is to say, to find all the solutions of the equation.
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Gauss also considered a coarser notion of equivalence, with each coarse class called a genus of forms. Each genus is the union of a finite number of equivalence classes of the same discriminant, with the number of classes depending only on the discriminant. In the context of binary quadratic forms, genera can be defined either through congruence classes of numbers represented by forms or by genus characters defined on the set of forms. A third definition is a special case of the genus of a quadratic form in n variables.
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The derivative gives the best possible linear approximation of a function at a given point, but this can be very different from the original function. One way of improving the approximation is to take a quadratic approximation. That is to say, the linearization of a real-valued function at the point is a linear polynomial , and it may be possible to get a better approximation by considering a quadratic polynomial . Still better might be a cubic polynomial , and this idea can be extended to arbitrarily high degree polynomials.
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Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "reals" in many programming languages). In this context, the quadratic formula is not completely stable. This occurs when the roots have different order of magnitude, or, equivalently, when and are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause loss of significance or catastrophic cancellation in the smaller root.
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Vieta's approximation for the smaller root is not accurate for small b but is accurate for large b. The direct evaluation of the smaller root using the quadratic formula is accurate for small b with roots of comparable value, but experiences loss of significance errors for large b and widely spaced roots. When c equals 4, Vieta's approximation starts off poorly at the left, but gets better with larger b, the difference between Vieta's approximation and the quadratic formula reaching a minimum at approximately b equals ten to the fifth.
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The inner product that is defined to define Euclidean spaces is a positive definite bilinear form. If it is replaced by an indefinite quadratic form which is non-degenerate, one gets a pseudo-Euclidean space. A fundamental example of such a space is the Minkowski space, which is the space-time of Einstein's special relativity. It is a four-dimensional space, where the metric is defined by the quadratic form :x^2+y^2+z^2-t^2, where the last coordinate (t) is temporal, and the other three (x, y, z) are spatial.
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If a point of Teichmüller space is represented by a Riemann surface R, then the cotangent space at that point can be identified with the space of quadratic differentials at R. Since the Riemann surface has a natural hyperbolic metric, at least if it has negative Euler characteristic, one can define a Hermitian inner product on the space of quadratic differentials by integrating over the Riemann surface. This induces a Hermitian inner product on the tangent space to each point of Teichmüller space, and hence a Riemannian metric.
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One may want to express the solutions as explicit numbers; for example, the unique solution of is . Unfortunately, this is, in general, impossible for equations of degree greater than one, and, since the ancient times, mathematicians have searched to express the solutions as algebraic expression; for example the golden ratio (1+\sqrt 5)/2 is the unique positive solution of x^2-x-1=0. In the ancient times, they succeeded only for degrees one and two. For quadratic equations, the quadratic formula provides such expressions of the solutions.
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In computational fluid dynamics QUICK, which stands for Quadratic Upstream Interpolation for Convective Kinematics, is a higher-order differencing scheme that considers a three-point upstream weighted quadratic interpolation for the cell face values. In computational fluid dynamics there are many solution methods for solving the steady convection–diffusion equation. Some of the used methods are the central differencing scheme, upwind scheme, hybrid scheme, power law scheme and QUICK scheme. The QUICK scheme was presented by Brian P. Leonard – together with the QUICKEST (QUICK with Estimated Streaming Terms) scheme – in a 1979 paper.
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As the gate–source voltage (VGS) is increased, the drain–source current (IDS) increases exponentially for VGS below threshold, and then at a roughly quadratic rate (IDS ∝ (VGS − VT)2) (where VT is the threshold voltage at which drain current begins) in the "space-charge-limited" region above threshold. A quadratic behavior is not observed in modern devices, for example, at the 65 nm technology node. For low noise at narrow bandwidth, the higher input resistance of the FET is advantageous. FETs are divided into two families: junction FET (JFET) and insulated gate FET (IGFET).
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Chevalley showed that the classification of split reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. For example, every nondegenerate quadratic form q over a field k determines a reductive group SO(q), and every central simple algebra A over k determines a reductive group SL1(A). As a result, the problem of classifying reductive groups over k essentially includes the problem of classifying all quadratic forms over k or all central simple algebras over k.
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The problem of optimizing higher-order pseudo- boolean functions is generally difficult. The process of reducing a high-order function to a quadratic one is known as "quadratization". It is always possible to reduce a higher-order function to a quadratic function which is equivalent with respect to the optimisation, problem known as "higher-order clique reduction" (HOCR), and the result of such reduction can be optimized with QPBO. Generic methods for reduction of arbitrary functions rely on specific substitution rules and in the general case they require the introduction of auxiliary variables.
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In the discrete-time case with uncertainty about the parameter values in the transition matrix (giving the effect of current values of the state variables on their own evolution) and/or the control response matrix of the state equation, but still with a linear state equation and quadratic objective function, a Riccati equation can still be obtained for iterating backward to each period's solution even though certainty equivalence does not apply.ch.13 The discrete-time case of a non-quadratic loss function but only additive disturbances can also be handled, albeit with more complications.
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If is field of complex numbers, the fundamental theorem of algebra implies that all have degree one, and all numerators a_{ij} are constants. When is the field of real numbers, some of the may be quadratic, so, in the partial fraction decomposition, quotients of linear polynomials by powers of quadratic polynomials may also occur. In the preceding theorem, one may replace "distinct irreducible polynomials" by "pairwise coprime polynomials that are coprime with their derivative". For example, the may be the factors of the square-free factorization of .
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If the field K is perfect, then every nonsingular quadratic form over K is uniquely determined (up to equivalence) by its dimension and its Arf invariant. In particular, this holds over the field F2. In this case, the subgroup U above is zero, and hence the Arf invariant is an element of the base field F2; it is either 0 or 1. If the field K of characteristic 2 is not perfect (that is, K is different from its subfield K2 of squares), then the Clifford algebra is another important invariant of a quadratic form.
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By the Seleucid period, the Babylonians had developed a zero symbol as a placeholder for empty positions; however it was only used for intermediate positions. This zero sign does not appear in terminal positions, thus the Babylonians came close but did not develop a true place value system. Other topics covered by Babylonian mathematics include fractions, algebra, quadratic and cubic equations, and the calculation of regular reciprocal pairs. The tablets also include multiplication tables and methods for solving linear, quadratic equations and cubic equations, a remarkable achievement for the time.
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A J-structure has a Peirce decomposition into subspaces determined by idempotent elements.Springer (1973) p.90 Let a be an idempotent of the J-structure (V,j,e), that is, a2 = a. Let Q be the quadratic map.
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A simple oval in the real plane can be constructed by glueing together two suitable halves of different ellipses, such that the result is not a conic. Even in the finite case there exist ovals (see quadratic set).
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In contrast, a priori certified path tracking goes beyond heuristics to provide step size control that guarantees that for every step along the path, the current point is within the domain of quadratic convergence for the current path.
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This can be applied to any quadratic equation. When the x2 has a coefficient other than 1, the first step is to divide out the equation by this coefficient: for an example see the non-monic case below.
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An algorithm is said to be subquadratic time if T(n) = o(n2). For example, simple, comparison-based sorting algorithms are quadratic (e.g. insertion sort), but more advanced algorithms can be found that are subquadratic (e.g. shell sort).
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Tsit Yuen Lam Tsit Yuen Lam (;Faculty Website, retrieved 2014-08-13. born 6 February 1942Curriculum vitae: T. Y. Lam, retrieved 2013-01-12) is a Hong Kong-American mathematician specializing in algebra, especially ring theory and quadratic forms.
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In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
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Quadratic coupling or cubic elastic terms lead to a warping along this "minimum energy path", replacing this infinite manifold by three equivalent potential minima and three equivalent saddle points. In other JT systems, linear coupling results in discrete minima.
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The field K is a pure cubic field if, and only if, d = −3. This is the case for which the quadratic field contained in the Galois closure of K is the cyclotomic field of cube roots of unity.
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While a parabolic arch may resemble a catenary arch, a parabola is a quadratic function while a catenary is the hyperbolic cosine, , a sum of two exponential functions. One parabola is , and hyperbolic cosine is . The curves are unrelated.
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Nvidia did manage to put together limited Direct3D support, but it was slow and buggy (software- based), and no match for the native triangle polygon hardware on the market. Subsequent NV1 quadratic-related development continued internally as the NV2.
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Vitalik Buterin in collaboration with Zoë Hitzig and E. Glen Weyl proposed quadratic funding, a way to allocate the distribution of funds (for example, from a government's budget, a philanthropic source, or collected directly from participants) based on quadratic voting, noting that such a mechanism allows for optimal production of public goods without needing to be determined by a centralized legislature. Weyl argues that this fills a gap with traditional free markets - which encourage the production of goods and services for the benefit of individuals, but fail to create outcomes desirable to society as a whole - while still benefiting from the flexibility and diversity free markets have compared to many government programs. The Gitcoin Grants initiative is an early adopter of quadratic funding. Led by Kevin Owocki, Scott Moore, and Vivek Singh, the initiative has distributed more than $2,000,000 to open-source software development projects as of early 2020.
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Oleg Tomovich Izhboldin (; 1963 - 2000) was a Russian mathematician who was first to provide a non-trivial example of an odd u-invariant field solving a classical Kaplansky's conjecture. Oleg Izhboldin graduated from the 45th Physics-Mathematics School in Saint Petersburg, then from the Faculty of Mathematics and Mechanics of Leningrad State University. He received his Ph.D. from the same University in 1988 and Doktor nauk degree in 2000.. According to Alexander Merkurjev: Oleg found his niche in algebra, namely, the algebraic theory of quadratic forms. ... This needed knowledge in different areas of mathematics was especially important in light of the recently discovered interaction (by Oleg, among others) between the theory of quadratic forms and various branches of mathematics that had seemed absolutely unrelated before... Oleg mastered the algebraic theory of quadratic forms very quickly and became one of the acknowledged experts in that field.
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In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V,Q)A quadratic space is a vector space V together with a quadratic form Q; the Q is dropped from notation when it is clear. on the associated projective space P(V). Explicitly, the projective orthogonal group is the quotient group :PO(V) = O(V)/ZO(V) = O(V)/{±I} where O(V) is the orthogonal group of (V) and ZO(V)={±I} is the subgroup of all orthogonal scalar transformations of V – these consist of the identity and reflection through the origin. These scalars are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" is because the scalar transformations are the center of the orthogonal group.
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The Gaussian periods are related to the Gauss sums G(1,\chi) for which the character χ is trivial on H. Such χ take the same value at all elements a in a fixed coset of H in G. For example, the quadratic character mod p described above takes the value 1 at each quadratic residue, and takes the value -1 at each quadratic non-residue. The Gauss sum G(1,\chi) can thus be written as a linear combination of Gaussian periods (with coefficients χ(a)); the converse is also true, as a consequence of the orthogonality relations for the group (Z/nZ)×. In other words, the Gaussian periods and Gauss sums are each other's Fourier transforms. The Gaussian periods generally lie in smaller fields, since for example when n is a prime p, the values χ(a) are (p − 1)-th roots of unity.
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A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form. Compatibility means that it is invariant under the adjoint representation. Examples of such are semisimple Lie algebras, such as su(n) and sl(n,R).
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Rationalisation can be extended to all algebraic numbers and algebraic functions (as an application of norm forms). For example, to rationalise a cube root, two linear factors involving cube roots of unity should be used, or equivalently a quadratic factor.
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From these texts it is known that ancient Egyptians understood concepts of geometry, such as determining the surface area and volume of three-dimensional shapes useful for architectural engineering, and algebra, such as the false position method and quadratic equations.
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44–45; Singh, pp. 56–58. Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation x2 \+ y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC).
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Divisibility properties of the Bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening in the Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.
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The theory of heights plays a prominent role in the arithmetic of abelian varieties. For instance, the canonical Néron–Tate height is a quadratic form with remarkable properties that appear in the statement of the Birch and Swinnerton-Dyer conjecture.
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We say that a binary quadratic form q(x,y) represents an integer n if it is possible to find integers x and y satisfying the equation n = f(x,y). Such an equation is a representation of n by f.
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A constant error in attitude rate (gyro) results in a quadratic error in velocity and a cubic error growth in position. Positional tracking systems like GPS can be used to continually correct drift errors (an application of the Kalman filter).
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In number theory, a Frobenius pseudoprime is a pseudoprime that passes a specific probable prime test described by Jon Grantham in a 1998 preprint and published in 2000. It has been studied by other authors for the case of quadratic polynomials.
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The Victorians and the Stuart Heritage: Interpretations of a Discordant Past. Cambridge University Press, p. 119 In mathematics, he is known for the Carlyle circle, a method used in quadratic equations and for developing ruler-and-compass constructions of regular polygons.
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Each program involves two robots, Edie and Charon, who work on an assembly line in a high-tech factory. The robots discuss their desire to learn about quadratic equations, and they are subsequently provided with lessons that further their education.
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Sylvester's law of inertia states that two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form.
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Villa Ehinger. The Portico with Balcony und the Main Entrance Villa Ehinger, Main Entrance The villa has a virtually quadratic ground plan. It is a two-storey building. On the western side of the house there is a Portico as entrance.
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The coefficients must be inferred from measured data, such as the Hiroshima Leukemia data. With higher orders being of lesser importance and the total survival fraction being the product of the two functions, the model is aptly called linear-quadratic.
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In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell-Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron and John Tate.
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Biquadratic fields are the simplest examples of abelian extensions of Q that are not cyclic extensions. According to general theory the Dedekind zeta-function of such a field is a product of the Riemann zeta-function and three Dirichlet L-functions. Those L-functions are for the Dirichlet characters which are the Jacobi symbols attached to the three quadratic fields. Therefore taking the product of the Dedekind zeta-functions of the quadratic fields, multiplying them together, and dividing by the square of the Riemann zeta-function, is a recipe for the Dedekind zeta-function of the biquadratic field.
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A field F is called an ordered field if any two elements can be compared, so that and whenever and . For example, the reals form an ordered field, with the usual ordering . The Artin-Schreier theorem states that a field can be ordered if and only if it is a formally real field, which means that any quadratic equation :x_1^2 + x_2^2 + \dots + x_n^2 = 0 only has the solution . The set of all possible orders on a fixed field is isomorphic to the set of ring homomorphisms from the Witt ring of quadratic forms over , to .
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In mathematics, a variable is a symbol which functions as a placeholder for varying expression or quantities, and is often used to represent an arbitrary element of a set. In addition to numbers, variables are commonly used to represent vectors, matrices and functions. Making algebraic computations with variables as if they were explicit numbers allows one to solve a range of problems in a single computation. A typical example is the quadratic formula, which allows one to solve every quadratic equation—by simply substituting the numeric values of the coefficients of the given equation for the variables that represent them.
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In this section, dual quaternions are constructed as the even Clifford algebra of real four-dimensional space with a degenerate quadratic form. Let the vector space V be real four- dimensional space R4, and let the quadratic form Q be a degenerate form derived from the Euclidean metric on R3. For v, w in R4 introduce the degenerate bilinear form : d(v, w) = v_1 w_1 + v_2 w_2 + v_3 w_3 . This degenerate scalar product projects distance measurements in R4 onto the R3 hyperplane. The Clifford product of vectors v and w is given by :v w + w v = -2 \,d(v, w).
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The algebra immediately following the octonions is called the sedenions. It retains an algebraic property called power associativity, meaning that if s is a sedenion, s^n s^m = s^{n + m}, but loses the property of being an alternative algebra and hence cannot be a composition algebra. The Cayley–Dickson construction can be carried on ad infinitum, at each step producing a power-associative algebra whose dimension is double that of the algebra of the preceding step. All the algebras generated in this way over a field are quadratic: that is, each element satisfies a quadratic equation with coefficients from the field.
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Scharlau's research deals with number theory and, in particular, the theory of quadratic forms, about which he wrote a 1985 monograph Quadratic and Hermitian Forms in Springer's series Grundlehren der mathematischen Wissenschaften. Scharlau is also an amateur ornithologist and author of two novels, I megali istoria - die große Geschichte (2nd edition 2001), set on the Greek island of Naxos, and Scharife (2001), set on the island of Zanzibar in the 19th century. He also deals with the history of mathematics and wrote, with Hans Opolka,Hans Opolka (b. 1949) is a German professor of mathematics, specializing in algebra and number theory.
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Advances in Cryptology: Proceedings of CRYPTO 84, Lecture Notes in Computer Science, 7:47--53, 1984 He was however only able to give an instantiation of identity-based signatures. Identity- based encryption remained an open problem for many years. The pairing-based Boneh–Franklin schemeDan Boneh, Matthew K. Franklin, Identity-Based Encryption from the Weil Pairing Advances in Cryptology - Proceedings of CRYPTO 2001 (2001) and Cocks's encryption schemeClifford Cocks, An Identity Based Encryption Scheme Based on Quadratic Residues, Proceedings of the 8th IMA International Conference on Cryptography and Coding, 2001 based on quadratic residues both solved the IBE problem in 2001.
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More precisely, only totally ramified primes have a chance of being Eisenstein primes for the polynomial. (In quadratic fields, ramification is always total, so the distinction is not seen in the quadratic case like above.) In fact, Eisenstein polynomials are directly linked to totally ramified primes, as follows: if a field extension of the rationals is generated by the root of a polynomial that is Eisenstein at then is totally ramified in the extension, and conversely if is totally ramified in a number field then the field is generated by the root of an Eisenstein polynomial at .
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The discriminant of a cubic field K can be written uniquely as df2 where d is a fundamental discriminant. Then, K is cyclic if, and only if, d = 1, in which case the only subfield of K is Q itself. If d ≠ 1, then the Galois closure N of K contains a unique quadratic field k whose discriminant is d (in the case d = 1, the subfield Q is sometimes considered as the "degenerate" quadratic field of discriminant 1). The conductor of N over k is f, and f2 is the relative discriminant of N over K. The discriminant of N is d3f4.
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Alon et al. use the Lovász local lemma to prove that the Thue number of any graph is at most quadratic in its maximum degree; they provide an example showing that for some graphs this quadratic dependence is necessary. In addition they show that the Thue number of a path of four or more vertices is exactly three, and that the Thue number of any cycle is at most four, and that the Thue number of the Petersen graph is exactly five. The known cycles with Thue number four are C5, C7, C9, C10, C14, and C17.
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Brown earned a B.A. at Rice University in 1965.Alumni In The News: Ezra “Bud” Brown Rice Magazine He then studied mathematics at Louisiana State University (LSU), getting an M.S. in 1967 and a Ph.D. in 1969 with the dissertation "Representations of Discriminantal Divisors by Binary Quadratic Forms" under Gordon Pall.Representations of discriminantal divisors by binary quadratic forms Journal of Number Theory, Volume 3, Issue 2, May 1971, pp. 213-225 He joined Virginia Tech in 1969 becoming Assistant Professor (1969-73), Associate Professor (1973-81), Professor (1981-2005), and Alumni Distinguished Professor of Mathematicsand Distinguished Professor of Mathematics (2005-2017).
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Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAST is diagonal, then the number of negative elements in the diagonal of D is always the same, for all such S; and the same goes for the number of positive elements. This property is named after James Joseph Sylvester who published its proof in 1852.
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This article describes periodic points of some complex quadratic maps. A map is a formula for computing a value of a variable based on its own previous value or values; a quadratic map is one that involves the previous value raised to the powers one and two; and a complex map is one in which the variable and the parameters are complex numbers. A periodic point of a map is a value of the variable that occurs repeatedly after intervals of a fixed length. These periodic points play a role in the theories of Fatou and Julia sets.
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Mathematical functions have one or more arguments that are designated in the definition by variables. A function definition can also contain parameters, but unlike variables, parameters are not listed among the arguments that the function takes. When parameters are present, the definition actually defines a whole family of functions, one for every valid set of values of the parameters. For instance, one could define a general quadratic function by declaring :f(x)=ax^2+bx+c; Here, the variable x designates the function's argument, but a, b, and c are parameters that determine which particular quadratic function is being considered.
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If the objective function is concave (maximization problem), or convex (minimization problem) and the constraint set is convex, then the program is called convex and general methods from convex optimization can be used in most cases. If the objective function is quadratic and the constraints are linear, quadratic programming techniques are used. If the objective function is a ratio of a concave and a convex function (in the maximization case) and the constraints are convex, then the problem can be transformed to a convex optimization problem using fractional programming techniques. Several methods are available for solving nonconvex problems.
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TeX provides a different text syntax specifically for mathematical formulas. For example, the quadratic formula (which is the solution of the quadratic equation) appears as: The formula is printed in a way a person would write by hand, or typeset the equation. In a document, entering mathematics mode is done by starting with a $ symbol, then entering a formula in TeX syntax, and closing again with another of the same symbol. Knuth explained in jest that he chose the dollar sign to indicate the beginning and end of mathematical mode in plain TeX because typesetting mathematics was traditionally supposed to be expensive.
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The spin–statistics relation was first formulated in 1939 by Markus Fierz and was rederived in a more systematic way by Wolfgang Pauli. Fierz and Pauli argued their result by enumerating all free field theories subject to the requirement that there be quadratic forms for locally commuting observables including a positive-definite energy density. A more conceptual argument was provided by Julian Schwinger in 1950. Richard Feynman gave a demonstration by demanding unitarity for scattering as an external potential is varied, which when translated to field language is a condition on the quadratic operator that couples to the potential.
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In contrast, this equation has no solution in the finite fields Z/p where p is an odd prime but is not Pythagorean., p. 100. The Paley graph with 13 vertices For every Pythagorean prime p, there exists a Paley graph with p vertices, representing the numbers modulo p, with two numbers adjacent in the graph if and only if their difference is a quadratic residue. This definition produces the same adjacency relation regardless of the order in which the two numbers are subtracted to compute their difference, because of the property of Pythagorean primes that −1 is a quadratic residue..
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The 15 theorem or Conway–Schneeberger Fifteen Theorem, proved by John H. Conway and W. A. Schneeberger in 1993, states that if a positive definite quadratic form with integer matrix represents all positive integers up to 15, then it represents all positive integers. The proof was complicated, and was never published. Manjul Bhargava found a much simpler proof which was published in 2000. Bhargava used the occasion of his receiving the 2005 SASTRA Ramanujan Prize to announce that he and Jonathan P. Hanke had cracked Conway's conjecture that a similar theorem holds for integral quadratic forms, with the constant 15 replaced by 290.
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The research interests of Prof. V. Koshmanenko concern modeling of complex dynamical systems, fractal geometry, functional analysis, operator theory, mathematical physics. He proposed the construction of wave and scattering operators in terms of bilinear functionals, introduced the notion of singular quadratic form and produced the classification of pure singular quadratic forms, developed the self-adjoint extensions approach to the singular perturbation theory in scales of Hilbert spaces, investigated the direct and inverse negative eigenvalues problem under singular perturbations. Volodymyr Koshmanenko developed the original theory of conflict dynamical systems and built a serious new models of complex dynamical systems with repulsive and attractive interaction.
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In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity. There are several different ways to express reciprocity laws. The early reciprocity laws found in the 19th century were usually expressed in terms of a power residue symbol (p/q) generalizing the quadratic reciprocity symbol, that describes when a prime number is an nth power residue modulo another prime, and gave a relation between (p/q) and (q/p). Hilbert reformulated the reciprocity laws as saying that a product over p of Hilbert norm residue symbols (a,b/p), taking values in roots of unity, is equal to 1.
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The fundamental ideas of Fermat's factorization method are the basis of the quadratic sieve and general number field sieve, the best-known algorithms for factoring large semiprimes, which are the "worst-case". The primary improvement that quadratic sieve makes over Fermat's factorization method is that instead of simply finding a square in the sequence of a^2 - n, it finds a subset of elements of this sequence whose product is a square, and it does this in a highly efficient manner. The end result is the same: a difference of square mod n that, if nontrivial, can be used to factor n.
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Carl Friedrich Gauss's Disquisitiones Arithmeticae, first edition Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations—for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to m X^2 + n Y^2)—defining their equivalence relation, showing how to put them in reduced form, etc. Adrien-Marie Legendre (1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions. He gave a full treatment of the equation a x^2 + b y^2 + c z^2 = 0 and worked on quadratic forms along the lines later developed fully by Gauss.
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Emil Artin (1957) Geometric Algebra, page 119 Through the polarization identity the quadratic form is related to a symmetric bilinear form . Two vectors u and v are orthogonal when . In the case of the hyperbolic plane, such u and v are hyperbolic-orthogonal.
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The first three iterations give (approximations given up to and including the first incorrect digit): :3.140\dots :3.14159264\dots :3.1415926535897932382\dots The algorithm has quadratic convergence, which essentially means that the number of correct digits doubles with each iteration of the algorithm.
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Explanation of drag by NASA. As mentioned, the drag equation with a constant drag coefficient gives the force experienced by an object moving through a fluid at relatively large velocity (i.e. high Reynolds number, Re > ~1000). This is also called quadratic drag.
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Numerical ranges and numerical radii are useful in the study of matrix and operator theory. These topics have applications in many subjects in pure and applied mathematics, such as quadratic forms, Banach spaces, dilation theory, control theory, numerical analysis, quantum information science.
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It is so one of the oldest buildings in Eskişehir. The almost-quadratic rectangular- plan mosque is constructed in rubble masonry. An octagonal -diameter dome sits in the middle of the flat wooden roof. Entrance to the mosque is at the northern side.
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The rotation group SO(3) is of rank one, and thus has one Casimir operator. It is three- dimensional, and thus the Casimir operator must have order (3 − 1) = 2 i.e. be quadratic. Of course, this is the Lie algebra of A_1.
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Fukuda and Terlaky. Compare Ziegler. It has been applied to linear- fractional programming, quadratic-programming problems, and linear complementarity problems. Outside of combinatorial optimization, OM theory also appears in convex minimization in Rockafellar's theory of "monotropic programming" and related notions of "fortified descent".
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Weights and measures, length, area, volume, etc. It describes addition, subtraction, multiplication, division, square, square root, cube and cube root. The problems of linear and quadratic equations described here are more complex than in earlier works.M. S. Sriram, Mathematics in India, Lecture 25.
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Amplitude amplification is a technique that allows the amplification of a chosen subspace of a quantum state. Applications of amplitude amplification usually lead to quadratic speedups over the corresponding classical algorithms. It can be considered to be a generalization of Grover's algorithm.
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Quadratics is a six-part Canadian instructional television series produced by TVOntario in 1993. The miniseries is part of the Concepts in Mathematics series. The program uses computer animation to demonstrate quadratic equations and their corresponding functions in the Cartesian coordinate system.
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A quadratic equation, for example, has two solutions. However, in some cases they are both the same number. Thus the multiset of solutions of the equation could be , or it could be . In the latter case it has a solution of multiplicity 2.
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The invariants most often considered are polynomial invariants. These are polynomials constructed from contractions such as traces. Second degree examples are called quadratic invariants, and so forth. Invariants constructed using covariant derivatives up to order n are called n-th order differential invariants.
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The function was standardized in ANSI C (1989). In 1991, Bell Labs employees observed that McMahon's and BSD versions of qsort would consume quadratic time for some simple inputs. Thus Jon Bentley and Douglas McIlroy engineered a new faster and more robust implementation.
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This results in an O(n log n) efficiency. This simple example demonstrates what is capable with an input enhancement technique such as presorting. The algorithm went from quadratic runtime to linearithmic runtime which will result in speed-ups for large inputs.
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In 2009, a gas giant planet was found in orbit around the star. The quadratic drift in the radial velocities did indicate the presence of an additional outer planet in the system, which was confirmed in 2011 as brown dwarf HIP 5158 c.
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In mathematics, the Dirichlet energy is a measure of how variable a function is. More abstractly, it is a quadratic functional on the Sobolev space . The Dirichlet energy is intimately connected to Laplace's equation and is named after the German mathematician Peter Gustav Lejeune Dirichlet.
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Explicitly considers fiber and matrix subcells from periodic repeating unit cell. Assumes 1st-order displacement field in subcells and imposes traction and displacement continuity. It was developed into the High-Fidelity GMC (HFGMC), which uses quadratic approximation for the displacement fields in the subcells.
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Several cryptographic methods rely on its hardness, see Applications. An efficient algorithm for the quadratic residuosity problem immediately implies efficient algorithms for other number theoretic problems, such as deciding whether a composite N of unknown factorization is the product of 2 or 3 primes.
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He developed the theory of dispersion managed interactions of few-cycle pulses in quadratically nonlinear layered media. He has investigated the nonlinear refraction, total internal reflection and scattering of optical beams and pulses in defocusing media with Kerr, cascaded quadratic, photorefractive, and thermal nonlinearities.
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Consider g(x) = ax^2+bx+c for the ring Z/pkZ. Lemma: for k=1 (i.e. Z/pZ) such polynomial defines a permutation only in the case a=0 and b not equal to zero. So the polynomial is not quadratic, but linear.
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The most commonly-used Casimir invariant is the quadratic invariant. It is the simplest to define, and so is given first. However, one may also have Casimir invariants of higher order, which correspond to homogeneous symmetric polynomials of higher order; their definition is given last.
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In mathematics, the Chowla–Selberg formula is the evaluation of a certain product of values of the Gamma function at rational values in terms of values of the Dedekind eta function at imaginary quadratic irrational numbers. The result was essentially found by and rediscovered by .
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Because of its quadratic complexity in time and space, it often cannot be practically applied to large-scale problems and is replaced in favor of less general but computationally more efficient alternatives such as (Gotoh, 1982), (Altschul and Erickson, 1986), and (Myers and Miller, 1988).
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This discussion started from the definition of a parabola as a conic section, but it has now led to a description as a graph of a quadratic function. This shows that these two descriptions are equivalent. They both define curves of exactly the same shape.
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Alice receives (c1, ..., cn). She can recover m using the following procedure: #For each i, using the prime factorization (p, q), Alice determines whether the value ci is a quadratic residue; if so, mi = 0, otherwise mi = 1. Alice outputs the message m = (m1, ..., mn).
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Some locally linear graphs have a number of edges that is near-quadratic. The question of how dense these graphs can be is one of the formulations of the Ruzsa–Szemerédi problem. The densest planar graphs that can be locally linear are also known.
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In mathematics, algebraic L-theory is the K-theory of quadratic forms; the term was coined by C. T. C. Wall, with L being used as the letter after K. Algebraic L-theory, also known as 'hermitian K-theory', is important in surgery theory.
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Similar problems occur even when the root is only "nearly" double. For example, let :f(x) = x^2(x-1000)+1. Then the first few iterations starting at are : = 1 : = … : = … : = … : = … : = … : = … : = … it takes six iterations to reach a point where the convergence appears to be quadratic.
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495--534 In general, if the separation principle applies, then filtering also arises as part of the solution of an optimal control problem. For example, the Kalman filter is the estimation part of the optimal control solution to the linear-quadratic-Gaussian control problem.
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Fast algorithms include the WSPD spanner and the Theta graph which both construct spanners with a linear number of edges in O(n \log n) time. If better weight and vertex degree is required the Greedy spanner can be computed in near quadratic time.
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Born on 30 October 1907 in Huncoat, Lancashire, Davenport was educated at Accrington Grammar School, the University of Manchester (graduating in 1927), and Trinity College, Cambridge. He became a research student of John Edensor Littlewood, working on the question of the distribution of quadratic residues.
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When the values are quadratic in the amplitude (e.g. power), they are first linearised by taking the square root before the logarithm is taken, or equivalently the result is halved. In the International System of Quantities, the neper is defined as .Thor, A. J. (1994).
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Within each of these plots, lidar metrics are calculated by calculating statistics such as mean, standard deviation, skewness, percentiles, quadratic mean, etc. Airborne Lidar Bathymetric Technology-High-resolution multibeam lidar map showing spectacularly faulted and deformed seafloor geology, in shaded relief and coloured by depth.
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Anatoli (or Anatoly) Nikolaievich Andrianov (Анатолий Николаевич Андрианов, born 21 July 1936) is а Russian mathematician. Andrianov received in 1962 his Ph.D. under Yuri Linnik at the Leningrad State University with thesis Investigation of quadratic forms by methods of the theory of correspondences and in 1969 his Russian doctorate of sciences (Doctor Nauk). A Community of Scholars, Institute for Advanced Study, Faculty and Members 1930–1980 He is a professor at the Steklov Institute in Saint Petersburg. His research deals with the multiplicative arithmetic of quadratic forms, zeta functions of automorphic forms, modular forms in several variables (such as Siegel modular forms, Hecke operators, spherical functions, and theta functions).
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Volume 1 on elementary and additive number theory includes the topics such as Dirichlet's theorem, Brun's sieve, binary quadratic forms, Goldbach's conjecture, Waring's problem, and the Hardy–Littlewood work on the singular series. Volume 2 covers topics in analytic number theory, such as estimates for the error in the prime number theorem, and topics in geometric number theory such as estimating numbers of lattice points. Volume 3 covers algebraic number theory, including ideal theory, quadratic number fields, and applications to Fermat's last theorem. Many of the results described by Landau were state of the art at the time but have since been superseded by stronger results.
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In a couple of papers in 1838 and 1839 Peter Gustav Lejeune Dirichlet proved the first class number formula, for quadratic forms (later refined by his student Leopold Kronecker). The formula, which Jacobi called a result "touching the utmost of human acumen", opened the way for similar results regarding more general number fields. Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory. He first used the pigeonhole principle, a basic counting argument, in the proof of a theorem in diophantine approximation, later named after him Dirichlet's approximation theorem.
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In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl2,0(R) and Cl1,1(R), which are both isomorphic to the ring of two-by-two matrices over the real numbers.
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The Witt group of k can be given a commutative ring structure, by using the tensor product of quadratic forms to define the ring product. This is sometimes called the Witt ring W(k), though the term "Witt ring" is often also used for a completely different ring of Witt vectors. To discuss the structure of this ring we assume that k is of characteristic not equal to 2, so that we may identify symmetric bilinear forms and quadratic forms. The kernel of the rank mod 2 homomorphism is a prime ideal, I, of the Witt ringMilnor & Husemoller (1973) p. 66 termed the fundamental ideal.
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If there is no nonlinearity (purple), all the amplitude in a mode will stay in that mode. If a quadratic nonlinearity is introduced in the elastic chain, energy can spread among all the mode, but if you wait long enough (two minutes, in this animation), you will see all the amplitude coming back in the original mode. In the summer of 1953 Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou conducted numerical experiments (i.e. computer simulations) of a vibrating string that included a non-linear term (quadratic in one test, cubic in another, and a piecewise linear approximation to a cubic in a third).
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The most visual way to see this automorphism is to give an interpretation via algebraic geometry over finite fields, as follows. Consider the action of S6 on affine 6-space over the field k with 3 elements. This action preserves several things: the hyperplane H on which the coordinates sum to 0, the line L in H where all coordinates coincide, and the quadratic form q given by the sum of the squares of all 6 coordinates. The restriction of q to H has defect line L, so there is an induced quadratic form Q on the 4-dimensional H/L that one checks is non-degenerate and non-split.
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In their own words, Goldwasser, Micali, and Rackoff say: > Of particular interest is the case where this additional knowledge is > essentially 0 and we show that [it] is possible to interactively prove that > a number is quadratic non residue mod m releasing 0 additional knowledge. > This is surprising as no efficient algorithm for deciding quadratic > residuosity mod m is known when m’s factorization is not given. Moreover, > all known NP proofs for this problem exhibit the prime factorization of m. > This indicates that adding interaction to the proving process, may decrease > the amount of knowledge that must be communicated in order to prove a > theorem.
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The paper also points out that the recursion can accommodate arbitrary gap penalization formulas: > A penalty factor, a number subtracted for every gap made, may be assessed as > a barrier to allowing the gap. The penalty factor could be a function of the > size and/or direction of the gap. [page 444] A better dynamic programming algorithm with quadratic running time for the same problem (no gap penalty) was first introduced by David Sankoff in 1972. Similar quadratic-time algorithms were discovered independently by T. K. Vintsyuk in 1968 for speech processing ("time warping"), and by Robert A. Wagner and Michael J. Fischer in 1974 for string matching.
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The alt=The Ulam spiral Euler noted that the function :n^2 - n + 41 yields prime numbers for 1\le n\le 40, although composite numbers appear among its later values.The sequence of these primes, starting at n=1 rather than n=0, is listed by The search for an explanation for this phenomenon led to the deep algebraic number theory of Heegner numbers and the class number problem. The Hardy-Littlewood conjecture F predicts the density of primes among the values of quadratic polynomials with integer coefficients in terms of the logarithmic integral and the polynomial coefficients. No quadratic polynomial has been proven to take infinitely many prime values.
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An additive view of this group arises from consideration of hyperbolic sectors and their hyperbolic angles. From the point of view of the classical groups, the group of squeeze mappings is , the identity component of the indefinite orthogonal group of 2 × 2 real matrices preserving the quadratic form . This is equivalent to preserving the form via the change of basis :x=u+v,\quad y=u-v\,, and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group (the connected component of the definite orthogonal group) preserving quadratic form as being circular rotations.
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In his old age, he was the first to prove "Fermat's last theorem" for n=5 (completing work by Peter Gustav Lejeune Dirichlet, and crediting both him and Sophie Germain). Carl Friedrich Gauss In his Disquisitiones Arithmeticae (1798), Carl Friedrich Gauss (1777–1855) proved the law of quadratic reciprocity and developed the theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation (congruences) and devoted a section to computational matters, including primality tests. The last section of the Disquisitiones established a link between roots of unity and number theory: > The theory of the division of the circle...which is treated in sec.
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On the other hand, when the diameter is bounded by a polynomial function of , this suggests that the mixing time might also be polynomial. In his 2007 doctoral dissertation, Cereceda investigated this problem, and found that (even for connected components of the space of colors) the diameter can be exponential for -colorings of -degenerate graphs. On the other hand, he proved that the diameter of the color space is at most quadratic (or, in big O notation, ) for colorings that use at least colors. He wrote that "it remains to determine" whether the diameter is polynomial for numbers of colors between these two extremes, or whether it is "perhaps even quadratic".
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The Indian mathematician Brahmagupta, in Brahma-Sphuta- Siddhanta (written c. AD 630), discussed the use of negative numbers to produce the general form quadratic formula that remains in use today. He also found negative solutions of quadratic equations and gave rules regarding operations involving negative numbers and zero, such as "A debt cut off from nothingness becomes a credit; a credit cut off from nothingness becomes a debt." He called positive numbers "fortunes", zero "a cipher", and negative numbers "debts".Colva M. Roney-Dougal, Lecturer in Pure Mathematics at the University of St Andrews, stated this on the BBC Radio 4 programme "In Our Time," on 9 March 2006.
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In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group. In general the map from the Pin group to the orthogonal group is not onto or a universal covering space, but if the quadratic form is definite (and dimension is greater than 2), it is both. The non-trivial element of the kernel is denoted −1, which should not be confused with the orthogonal transform of reflection through the origin, generally denoted −I.
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Quadratic production function Any of these equations can be plotted on a graph. A typical (quadratic) production function is shown in the following diagram under the assumption of a single variable input (or fixed ratios of inputs so they can be treated as a single variable). All points above the production function are unobtainable with current technology, all points below are technically feasible, and all points on the function show the maximum quantity of output obtainable at the specified level of usage of the input. From point A to point C, the firm is experiencing positive but decreasing marginal returns to the variable input.
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This configuration, like Möbius, can also be represented as two tetrahedra, mutually inscribed and circumscribed: in the integer representation the tetrahedra can be 0347 and 1256. However, these two 8_4 configurations are non-isomorphic, since Möbius has four pairs of disjoint planes, while the latter one has no disjoint planes. For a similar reason (and because pairs of planes are degenerate quadratic surfaces), the Möbius configuration is on more quadratic surfaces of three-dimensional space than the latter configuration. The Levi graph of the Möbius configuration has 16 vertices, one for each point or plane of the configuration, with an edge for every incident point-plane pair.
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Quadratic magnetic rotation (also known as QMR or QMR effect) is a type of magneto-optic effect, discovered in the mid 1980s by a team of Ukrainian physicists. QMR, like the Faraday effect, establishes a relationship between the magnetic field and rotation of polarization of the plane of linearly polarized light. In contrast to the Faraday effect, QMR originates from the quadratic proportionality between the angle of the rotation of the plane of polarization and the strength of the magnetic field. Mostly QMR can be observed in the transverse geometry when the vector of the magnetic field strength is perpendicular to the direction of light propagation.
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The ancient Egyptians knew how to solve equations of degree 2 in this manner. The Indian mathematician Brahmagupta (597–668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. In the 9th century Muhammad ibn Musa al-Khwarizmi and other Islamic mathematicians derived the quadratic formula, the general solution of equations of degree 2, and recognized the importance of the discriminant. During the Renaissance in 1545, Gerolamo Cardano published the solution of Scipione del Ferro and Niccolò Fontana Tartaglia to equations of degree 3 and that of Lodovico Ferrari for equations of degree 4.
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Decision-makers are assumed to make their decisions (such as, for example, portfolio allocations) so as to maximize the expected value of the utility function. Notable special cases of HARA utility functions include the quadratic utility function, the exponential utility function, and the isoelastic utility function.
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His name has been given to Kneser graphs which he studied in 1955. He also gave a simplified proof of the Fundamental theorem of algebra. Kneser was an Invited Speaker of the ICM in 1962 at Stockholm. His main publications were on quadratic forms and algebraic groups.
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There is a procedure involving Carlyle circles for the construction of a regular 65537-gon. However there are practical problems for the implementation of the procedure; for example, it requires the construction of the Carlyle circle for the solution of the quadratic equation x2 + x − 214 = 0.
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Typically, the standard graph coloring approaches produce quality code, but have a significant overhead, the used graph coloring algorithm having a quadratic cost. Owing to this feature, linear scan is the approach currently used in several JIT compilers, like the Hotspot compiler, V8 and Jikes RVM.
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The chakravala method () is a cyclic algorithm to solve indeterminate quadratic equations, including Pell's equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) Hoiberg & Ramchandani – Students' Britannica India: Bhaskaracharya II, page 200Kumar, page 23 although some attribute it to Jayadeva (c. 950 ~ 1000 CE).
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Not all spinors are pure. In general pure spinors may be separated from impure spinors via a series of quadratic equations called pure spinor constraints. However, in 6 or less real dimensions all spinors are pure. In 8 dimensions there is, projectively, a single pure spinor constraint.
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Factor bases are used in, for example, Dixon's factorization, the quadratic sieve, and the number field sieve. The difference between these algorithms is essentially the methods used to generate (x, y) candidates. Factor bases are also used in the Index calculus algorithm for computing discrete logarithms.
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The split-octonions, like the octonions, are noncommutative and nonassociative. Also like the octonions, they form a composition algebra since the quadratic form N is multiplicative. That is, :N(xy) = N(x)N(y). The split-octonions satisfy the Moufang identities and so form an alternative algebra.
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Kock, N., & Gaskins, L. (2016). Simpson's paradox, moderation and the emergence of quadratic relationships in path models: An information systems illustration. International Journal of Applied Nonlinear Science, 2(3), 200-234. It is also referred to as Simpson's reversal, Yule–Simpson effect, amalgamation paradox, or reversal paradox.
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This Euclidean norm defines a quadratic form on L, under which the lattice is isomorphic to the E8 lattice. This construction shows that the Coxeter group H_4 embeds as a subgroup of E_8. Indeed, a linear isomorphism that preserves the quaternion norm also preserves the Euclidean norm.
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In 1962 he retired as professor emeritus but continued publishing mathematical papers into the 1970s. In the 1950s Myberg published several fundamental papers on the iteration of rational functions (especially quadratic functions).some of Myrberg's publications online Annales Acad. Sci. Fennicae 1958, 1959, 1963, J. Math.
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Cahit Arf (; 11 October 1910 - 26 December 1997) was a Turkish mathematician. He is known for the Arf invariant of a quadratic form in characteristic 2 (applied in knot theory and surgery theory) in topology, the Hasse-Arf theorem in ramification theory, Arf semigroups, and Arf rings.
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The equation for the drawn line is . The equation for the intersection of the line and circle is then a quadratic equation involving . The two solutions to this equation are and . This allows us to write the latter as rational functions of (solutions are given below).
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One root to this quadratic is , so by Vieta's formulas the other root may be written as follows: . # The first equation shows that is an integer and the second that it is positive. Because as long as . # The base case we arrive at is the case where .
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Grantham's stated goal when developing the algorithm was to provide a test that primes would always pass and composites would pass with a probability of less than 1/7710. The test was later extended by Damgård and Frandsen to a test called extended quadratic Frobenius test (EQFT).
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This solution method is now the most prevalent solution method for NP-hard optimisation problems. Land implemented her linear and integer programming algorithms in Fortran. Later, with Susan Powell, she collected her implementations in a book, Fortran Codes for Mathematical Programming: Linear, Quadratic and Discrete (Wiley, 1973).
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In mathematics, Shimura's reciprocity law, introduced by , describes the action of ideles of imaginary quadratic fields on the values of modular functions at singular moduli. It forms a part of the Kronecker Jugendtraum, explicit class field theory for such fields. There are also higher-dimensional generalizations.
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The inequality can also be used to prove Beck's theorem, that if a finite point set does not have a linear number of collinear points, then it determines a quadratic number of distinct lines.. Similarly, Tamal Dey used it to prove upper bounds on geometric k-sets.
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This example will demonstrate standard quadratic sieve without logarithm optimizations or prime powers. Let the number to be factored N = 15347, therefore the ceiling of the square root of N is 124. Since N is small, the basic polynomial is enough: y(x) = (x + 124)2 − 15347\.
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It is important to review the proof of quadratic convergence of Newton's method before implementing it. Specifically, one should review the assumptions made in the proof. For situations where the method fails to converge, it is because the assumptions made in this proof are not met.
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Such results are significant in geometric analysis, following the original energy quantization result of Yum-Tong Siu and Shing-Tung Yau in their proof of the Frankel conjecture.Siu, Yum Tong; Yau, Shing Tung. Complete Kähler manifolds with nonpositive curvature of faster than quadratic decay. Ann. of Math.
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Al-Khwarizmi's popularizing treatise on algebra (The Compendious Book on Calculation by Completion and Balancing, c. 813–833 CEOaks, J. (2009). Polynomials and equations in Arabic algebra. Archive for History of Exact Sciences, 63(2), 169–203.) presented the first systematic solution of linear and quadratic equations.
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The fact that a triangle with edges 1, \sqrt\varphi and \varphi, forms a right triangle follows directly from rewriting the defining quadratic polynomial for the golden ratio \varphi: :\varphi^2 = \varphi + 1 into the form of the Pythagorean theorem: :(\varphi)^2 = (\sqrt\varphi)^2 + (1)^2.
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Doyle's early work was in the mathematics of robust control, linear-quadratic-Gaussian control robustness, (structured) singular value analysis, H-infinity. He has coauthored books and software toolboxes, a control analysis tool for high performance commercial and military aerospace systems, as well as other industrial systems.
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Although quadratic residues appear to occur in a rather random pattern modulo n, and this has been exploited in such applications as acoustics and cryptography, their distribution also exhibits some striking regularities. Using Dirichlet's theorem on primes in arithmetic progressions, the law of quadratic reciprocity, and the Chinese remainder theorem (CRT) it is easy to see that for any M > 0 there are primes p such that the numbers 1, 2, ..., M are all residues modulo p. > For example, if p ≡ 1 (mod 8), (mod 12), (mod 5) and (mod 28), then by the > law of quadratic reciprocity 2, 3, 5, and 7 will all be residues modulo p, > and thus all numbers 1-10 will be. The CRT says that this is the same as p ≡ > 1 (mod 840), and Dirichlet's theorem says there are an infinite number of > primes of this form. 2521 is the smallest, and indeed 12 ≡ 1, 10462 ≡ 2, > 1232 ≡ 3, 22 ≡ 4, 6432 ≡ 5, 872 ≡ 6, 6682 ≡ 7, 4292 ≡ 8, 32 ≡ 9, and 5292 ≡ > 10 (mod 2521).
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Scuola Norm. Sup. Pisa Cl. Sci. (4) 3 (1976), no. 4 Specifically, he proved an effective lower bound for the class number of an imaginary quadratic field assuming the existence of an elliptic curve whose L-function had a zero of order at least 3 at s=1/2.
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For example, the solutions to the quadratic Diophantine equation x2 \+ y2 = z2 are given by the Pythagorean triples, originally solved by the Babylonians (c. 1800 BC).Aczel, pp. 14–15. Solutions to linear Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c.
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The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012). including zero and negative solutions, to quadratic equations was described by Brahmagupta in his book Brahmasphutasiddhanta, published in 628 AD.Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p.
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Traditionally, ICs have been designed with dedicated point-to-point connections, with one wire dedicated to each signal. This results in a dense network topology. For large designs, in particular, this has several limitations from a physical design viewpoint. It requires power quadratic in the number of interconnections.
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Trisection, like many constructions impossible by ruler and compass, can easily be accomplished by the more powerful operations of paper folding, or origami. Huzita's axioms (types of folding operations) can construct cubic extensions (cube roots) of given lengths, whereas ruler- and-compass can construct only quadratic extensions (square roots).
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The mosque is a single-dome, quadratic-plan building having stone masonry walls. The inside of the dome is decorated with hand-carved figures. Marble columns and capitals support pointed-arches of the narthex in the architectural style of the classical period. The narthex is topped with five domes.
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For an abelian variety, there is no a priori preferred representation, though, as a projective variety. Both halves of the proof have been improved significantly, by subsequent technical advances: in Galois cohomology as applied to descent, and in the study of the best height functions (which are quadratic forms).
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For example, the two roots of a quadratic equation are typically labelled and . In spaceflight, beta angle describes the angle between the orbit plane of a spacecraft or other body and the vector from the sun. β is sometimes used to mean the proton-to-electron mass ratio.
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A mechanical device that computes area integrals is the planimeter, which measures the area of plane figures by tracing them out: this replicates integration in polar coordinates by adding a joint so that the 2-element linkage effects Green's theorem, converting the quadratic polar integral to a linear integral.
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The Ulam spiral arranges the natural numbers in a two-dimensional grid, spiraling in concentric squares surrounding the origin with the prime numbers highlighted. Visually, the primes appear to cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than others.
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Quadratic. 17 rules and 10 examples. Includes a variant of the Chakravala method. Ganita Kaumudi contains many results from continued fractions. In the text Narayana Pandita used the knowledge of simple recurring continued fraction in the solutions of indeterminate equations of the type nx^2+k^2=y^2.
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For this article the bivector will be considered only in real geometric algebras. This in practice is not much of a restriction, as all useful applications are drawn from such algebras. Also unless otherwise stated, all examples have a Euclidean metric and so a positive-definite quadratic form.
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The simplest nontrivial Vassiliev invariant of knots is given by the coefficient of the quadratic term of the Alexander–Conway polynomial. It is an invariant of order two. Modulo two, it is equal to the Arf invariant. Any coefficient of the Kontsevich invariant is a finite type invariant.
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Equations in the book are presently called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the Arithmetica problems lead to quadratic equations. In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes.
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Newton's method is an extremely powerful technique—in general the convergence is quadratic: as the method converges on the root, the difference between the root and the approximation is squared (the number of accurate digits roughly doubles) at each step. However, there are some difficulties with the method.
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Newton's method is only guaranteed to converge if certain conditions are satisfied. If the assumptions made in the proof of quadratic convergence are met, the method will converge. For the following subsections, failure of the method to converge indicates that the assumptions made in the proof were not met.
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TrueType is a font system originally developed by Apple Inc. It was intended to replace Type 1 fonts, which many felt were too expensive. Unlike Type 1 fonts, TrueType glyphs are described with quadratic Bezier curves. It is currently very popular and implementations exist for all major operating systems.
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Any deviation from the above assumptions—a nonlinear state equation, a non-quadratic objective function, noise in the multiplicative parameters of the model, or decentralization of control—causes the certainty equivalence property not to hold. For example, its failure to hold for decentralized control was demonstrated in Witsenhausen's counterexample.
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Hydrological optimization applies mathematical optimization techniques (such as dynamic programming, linear programming, integer programming, or quadratic programming) to water-related problems. These problems may be for surface water, groundwater, or the combination. The work is interdisciplinary, and may be done by hydrologists, civil engineers, environmental engineers, and operations researchers.
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A corrected version of Arf's original statement is that if the degree [K: K2] is at most 2, then every quadratic form over K is completely characterized by its dimension, its Arf invariant and its Clifford algebra.Falko Lorenz and Peter Roquette. Cahit Arf and his invariant. Section 9.
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One of these introduced the Fermi–Ulam model, an extension of Fermi's theory of the acceleration of cosmic rays. Another, with Paul Stein and Mary Tsingou, titled "Quadratic Transformations", was an early investigation of chaos theory and is considered the first published use of the phrase "chaotic behavior".
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Referring to the quadratic form applied to the displacement vector of a point on the quasi-sphere from the centre (i.e. ) as the radial scalar square, in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radial scalar square, those with negative radial scalar square, those with zero radial scalar square. In a space with a positive-definite quadratic form (i.e. a Euclidean space), a quasi-sphere with negative radial scalar square is the empty set, one with zero radial scalar square consists of a single point, one with positive radial scalar square is a standard -sphere, and one with zero curvature is a hyperplane that is partitioned with the -spheres.
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Let these correspond to the points P1, P2, P3, P4. Letting :p1 = ω + ω4, p2 = ω2 + ω3 we have :p1 + p2 = −1, p1p2 = −1\. (These can be quickly shown to be true by direct substitution into the quartic above and noting that ω6 = ω, and ω7 = ω2.) So p1 and p2 are the roots of the quadratic equation :x2 + x − 1 = 0\. The Carlyle circle associated with this quadratic has a diameter with endpoints at (0, 1) and (−1, −1) and center at (−1/2, 0). Carlyle circles are used to construct p1 and p2. From the definitions of p1 and p2 it also follows that :p1 = 2 cos (2/5), p2 = 2 cos (4/5).
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The study of these two hyperboloids is, therefore, in this way connected very simply, through biquaternions, with the study of the sphere; ... In this passage is the operator giving the scalar part of a quaternion, and is the "tensor", now called norm, of a quaternion. A modern view of the unification of the sphere and hyperboloid uses the idea of a conic section as a slice of a quadratic form. Instead of a conical surface, one requires conical hypersurfaces in four-dimensional space with points determined by quadratic forms. First consider the conical hypersurface :P = \lbrace p \ : \ w^2 = x^2 + y^2 + z^2 \rbrace and :H_r = \lbrace p \ :\ w = r \rbrace , which is a hyperplane.
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The study of particular quadratic forms, in particular the question of whether a given integer can be the value of a quadratic form over the integers, dates back many centuries. One such case is Fermat's theorem on sums of two squares, which determines when an integer may be expressed in the form , where x, y are integers. This problem is related to the problem of finding Pythagorean triples, which appeared in the second millennium B.C.Babylonian Pythagoras In 628, the Indian mathematician Brahmagupta wrote Brāhmasphuṭasiddhānta which includes, among many other things, a study of equations of the form . In particular he considered what is now called Pell's equation, , and found a method for its solution.
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In the light of later work on Galois theory, the principles of these proofs have been clarified. It is straightforward to show from analytic geometry that constructible lengths must come from base lengths by the solution of some sequence of quadratic equations.. In terms of field theory, such lengths must be contained in a field extension generated by a tower of quadratic extensions. It follows that a field generated by constructions will always have degree over the base field that is a power of two. In the specific case of a regular n-gon, the question reduces to the question of constructing a length :cos , which is a trigonometric number and hence an algebraic number.
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Under these assumptions an optimal control scheme within the class of linear control laws can be derived by a completion-of-squares argument. This control law which is known as the LQG controller, is unique and it is simply a combination of a Kalman filter (a linear–quadratic state estimator (LQE)) together with a linear–quadratic regulator (LQR). The separation principle states that the state estimator and the state feedback can be designed independently. LQG control applies to both linear time- invariant systems as well as linear time-varying systems, and constitutes a linear dynamic feedback control law that is easily computed and implemented: the LQG controller itself is a dynamic system like the system it controls.
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The minimum number of induced matchings into which the edges of a graph can be partitioned is called its strong chromatic index, by analogy with the chromatic index of the graph, the minimum number of matchings into which its edges can be partitioned. It equals the chromatic number of the square of the line graph. Brooks' theorem, applied to the square of the line graph, shows that the strong chromatic index is at most quadratic in the maximum degree of the given graph, but better constant factors in the quadratic bound can be obtained by other methods. The Ruzsa–Szemerédi problem concerns the edge density of balanced bipartite graphs with linear strong chromatic index.
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Conversely such a decomposition uniquely determines a contact lift of a surface which envelops two one parameter families of spheres; the image of this contact lift is given by the null 2-dimensional subspaces which intersect σ and τ in a pair of null lines. Such a decomposition is equivalently given, up to a sign choice, by a symmetric endomorphism of R4,2 whose square is the identity and whose ±1 eigenspaces are σ and τ. Using the inner product on R4,2, this is determined by a quadratic form on R4,2. To summarize, Dupin cyclides are determined by quadratic forms on R4,2 such that the associated symmetric endomorphism has square equal to the identity and eigenspaces of signature (2,1).
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Frederick Gehring showed in 1977 that U is the interior of the closed subset of Schwarzian derivatives of univalent functions. For a compact Riemann surface S of genus greater than 1, its universal covering space is the unit disc D on which its fundamental group Γ acts by Möbius transformations. The Teichmüller space of S can be identified with the subspace of the universal Teichmüller space invariant under Γ. The holomorphic functions g have the property that :g(z) \, dz^2 is invariant under Γ, so determine quadratic differentials on S. In this way, the Teichmüller space of S is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on S.
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Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue. The first supplementGauss, DA, art 111 to the law of quadratic reciprocity is that if p ≡ 1 (mod 4) then −1 is a quadratic residue modulo p, and if p ≡ 3 (mod 4) then −1 is a nonresidue modulo p. This implies the following: If p ≡ 1 (mod 4) the negative of a residue modulo p is a residue and the negative of a nonresidue is a nonresidue. If p ≡ 3 (mod 4) the negative of a residue modulo p is a nonresidue and the negative of a nonresidue is a residue.
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In mathematics, the quadratic bottleneck assignment problem (QBAP) is one of fundamental combinatorial optimization problems in the branch of optimization or operations research, from the category of the facilities location problems.Assignment Problems, by Rainer Burkard, Mauro Dell'Amico, Silvano Martello, 2009 It is related to the quadratic assignment problem in the same way as the linear bottleneck assignment problem is related to the linear assignment problem, the "sum" is replaced with "max" in the objective function. The problem models the following real-life problem: :There are a set of n facilities and a set of n locations. For each pair of locations, a distance is specified and for each pair of facilities a weight or flow is specified (e.g.
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With Montserrat Alsina, Bayer is the author of the book Quaternion Orders, Quadratic Forms, and Shimura Curves (American Mathematical Society, 2004). As well as quaternion algebras, Eichler orders, quadratic forms, and Shimura curves (the subject of the book), other topics in her research include automorphic forms, diophantine equations, elliptic curves, modular curves, and zeta functions. Beyond number theory, with Jordi Guàrdia and Artur Travesa she is the author of Arrels germàniques de la matemàtica contemporània: amb una antologia de textos matemàtics de 1850 a 1950 (Institut d'Estudis Catalans, 2012), on the history of mathematics in Germany from the mid-19th century to the mid-20th century. In total she is an author or editor of 19 books.
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Up to a high order of approximation, mutual gravitational perturbations between major or minor planets only cause periodic variations in their orbits, that is, parameters oscillate between maximum and minimum values. The tidal effect gives rise to a quadratic term in the equations, which leads to unbounded growth. In the mathematical theories of the planetary orbits that form the basis of ephemerides, quadratic and higher order secular terms do occur, but these are mostly Taylor expansions of very long time periodic terms. The reason that tidal effects are different is that unlike distant gravitational perturbations, friction is an essential part of tidal acceleration, and leads to permanent loss of energy from the dynamic system in the form of heat.
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On a pseudo-Riemannian manifold, -forms can be identified with vector fields; vector fields have additional distinct algebraic structures, which are listed here for context and to avoid confusion. Firstly, each (co)tangent space generates a Clifford algebra, where the product of a (co)vector with itself is given by the value of a quadratic form – in this case, the natural one induced by the metric. This algebra is distinct from the exterior algebra of differential forms, which can be viewed as a Clifford algebra where the quadratic form vanishes (since the exterior product of any vector with itself is zero). Clifford algebras are thus non-anticommutative ("quantum") deformations of the exterior algebra.
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One form of TFR (or TFD) can be formulated by the multiplicative comparison of a signal with itself, expanded in different directions about each point in time. Such representations and formulations are known as quadratic or "bilinear" TFRs or TFDs (QTFRs or QTFDs) because the representation is quadratic in the signal (see Bilinear time–frequency distribution). This formulation was first described by Eugene Wigner in 1932 in the context of quantum mechanics and, later, reformulated as a general TFR by Ville in 1948 to form what is now known as the Wigner–Ville distribution, as it was shown in B. Boashash, "Note on the use of the Wigner distribution for time frequency signal analysis", IEEE Trans. on Acoust. Speech.
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Quickselect is linear-time on average, but it can require quadratic time with poor pivot choices. This is because quickselect is a divide and conquer algorithm, with each step taking O(n) time in the size of the remaining search set. If the search set decreases exponentially quickly in size (by a fixed proportion), this yields a geometric series times the O(n) factor of a single step, and thus linear overall time. However, if the search set decreases slowly in size, such as linearly (by a fixed number of elements, in the worst case only reducing by one element each time), then a linear sum of linear steps yields quadratic overall time (formally, triangular numbers grow quadratically).
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Eventually, Dawe arrived at a culminating sketch. He then adapted his sketch into a carefully plotted matrix that identified the precise curvature and blended color gradient of the final quadratic surface. His final configuration, which sharply fanned off the two dimensional plane, maximized the angles available within the room’s tight width.
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A model which extends the Standard Model should predict one or more of these parameters or shed light on some other issue such as why there are three quark-lepton families or, the most common motivation, the naturalness or hierarchy problem associated with the quadratic divergences appearing in the scalar sector.
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In this case, the extension is a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension is an extension that has a finite degree. Given two extensions and , the extension is finite if and only if both and are finite.
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It is also supported in the AMPL modeling system. The main algorithms implemented in FortMP are the primal and dual simplex algorithms using sparse matrices. These are supplemented for large problems and quadratic programming problems by interior point methods. Mixed integer programming problems are solved using branch and bound algorithm.
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On Gauss's recommendation, Friedrich Bessel was awarded an honorary doctor degree from Göttingen in March 1811.Bessel never had a university education. Around that time, the two men engaged in a correspondence.Helmut Koch, Introduction to Classical Mathematics I: From the Quadratic Reciprocity Law to the Uniformization Theorem, Springer, p. 90.
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The graph shows time (average of 100 instances in ms using a 933 MHz Pentium III) vs.problem size for knapsack problems for a state-of-the-art specialized algorithm. Quadratic fit suggests that empirical algorithmic complexity for instances with 50–10,000 variables is O((log(n))2).Pisinger, D. 2003.
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Nomogram for the law of sines Nomogram for solving the quadratric x^2+px+q=0 Nomogram for solving the cubic x^3+px+q=0 Using a ruler, one can readily read the missing term of the law of sines or the roots of the quadratic and cubic equation.
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Mongol soldiers at the time of Öljeitü, in Jami al-Tawarikh by Rashid-al-Din Hamadani, 1305-1306. Letter of Öljeitu to Philippe le Bel, 1305. In classical Mongolian script, with the Chinese script seal of the Great Khaan in Mongolian Quadratic Script (Dörböljin Bichig). The huge roll measures 302x50 cm.
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The goals are two meters high and three meters wide. They must be securely bolted either to the floor or the wall behind. The goal posts and the crossbar must be made out of the same material (e.g., wood or aluminium) and feature a quadratic cross section with sides of .
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To every quaternion algebra A, one can associate a quadratic form N (called the norm form) on A such that :N(xy) = N(x)N(y) for all x and y in A. It turns out that the possible norm forms for quaternion F-algebras are exactly the Pfister 2-forms.
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Non-local dynamics were developed to improve the scaling to a quadratic scaling (see the Wolff algorithm), beating the critical slowing down. However, it is still an open question whether there is a local dynamics that does not suffer from critical slowing down in spin systems like the Ising model.
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Tonelli; the algorithm requires O(log4n) steps. (in 1891) and Cipolla; the algorithm requires O(log3 n) steps and is also nondetermisitic. found efficient algorithms that work for all prime moduli. Both algorithms require finding a quadratic nonresidue modulo n, and there is no efficient deterministic algorithm known for doing that.
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The theorem allows one to define the conductor of K as the smallest integer n such that K lies inside the field generated by the n-th roots of unity. For example the quadratic fields have as conductor the absolute value of their discriminant, a fact generalised in class field theory.
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If the quadratic form is negative-definite, the second-order conditions for a maximum are met. An important example of such an optimization arises in multiple regression, in which a vector of estimated parameters is sought which minimizes the sum of squared deviations from a perfect fit within the dataset.
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When f is a convex quadratic function with positive-definite Hessian B, one would expect the matrices H_k generated by a quasi-Newton method to converge to the inverse Hessian H = B^{-1}. This is indeed the case for the class of quasi-Newton methods based on least-change updates.
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Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables. Siegel modular forms were first investigated by for the purpose of studying quadratic forms analytically. These primarily arise in various branches of number theory, such as arithmetic geometry and elliptic cohomology.
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Here, also, the "limiting stand" helps somewhat to isolate the variable of interest. For instance, at the upper treeline, where the tree is "cold limited", it's unlikely that nonlinear effects of high temperature ("inverted quadratic") will have numerically significant impact on ring width over the course of a growing season.
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In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers. Specific examples include imaginary quadratic fields, cyclotomic fields, and, more generally, CM fields. Any number field that is Galois over the rationals must be either totally real or totally imaginary.
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Vegeholm Castle () is located in Ängelholm Municipality in Scania, Sweden. The castle is a three-story stone house with a high, split roof that lies around an almost quadratic yard. In two corners there are large, square towers. On both sides of the north facade there are two free laying long buildings.
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If q is an anisotropic quadratic form over a field F, and if q becomes hyperbolic over every extension field E such that q becomes isotropic over E, then q is isomorphic to aφ for some nonzero a in F and some Pfister form φ over F.Elman, Karpenko, Merkurjev (2008), Corollary 23.4.
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59, 297 (1969).Matrix elements of the quadratic stark effect on atoms with hyperfine structure, R. W. Schmieder, Am. J. Phys. 40, 297 (1972). While still an undergraduate, he worked at the CIT synchrotron laboratory, and he participated in the discovery of a new isotope (In106) using the Berkeley 60-inch cyclotron.
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Eva Bayer-Fluckiger (born 25 June 1951) is a Hungarian and Swiss mathematician. She is an Emmy Noether Professor Emeritus at École Polytechnique Fédérale de Lausanne. She has worked on several topics in topology, algebra and number theory, e.g. on the theory of knots, on lattices, on quadratic forms and on Galois cohomology.
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This embeds as a subgroup of , and hence we may realise as a subgroup of . Furthermore, is the complexification of . In the complex case, quadratic forms are determined uniquely up to isomorphism by the dimension of . Concretely, we may assume and :Q(z_1,\ldots, z_n) = z_1^2+ z_2^2+\cdots+z_n^2.
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Historically, and in current teaching, the study of algebra starts with the solving of equations such as the quadratic equation above. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered.
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The NLP level of WORHP is based on SQP, while the quadratic subproblems are solved using an interior point method. This approach was chosen to benefit from the robustness of SQP methods and the reliable runtime complexity of IP methods, since traditional active set methods may be unsuitable for large-scale problems.
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Those that can be recognized now are from the conductus repertory, and are mainly note against note in texture. The notation was in semi-quadratic neumes with pairs of four-line staves. Two songs survive with music intact: Primus parens hominum, a monophonic song, and a two-part work, Sol oritur occasus.
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They are closely related to the Paley construction for constructing Hadamard matrices from quadratic residues . They were introduced as graphs independently by and . Sachs was interested in them for their self- complementarity properties, while Erdős and Rényi studied their symmetries. Paley digraphs are directed analogs of Paley graphs that yield antisymmetric conference matrices.
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Developments since around 1960 have certainly contributed. Before that in his dissertation used Hilbert modular forms to study abelian extensions of real quadratic fields. Complex multiplication of abelian varieties was an area opened up by the work of Shimura and Taniyama. This gives rise to abelian extensions of CM-fields in general.
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In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643.
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Additionally, numeric control over size, position, and other aspects of objects is available in the Geometry panel. Path drawing tools. The program has dedicated tools for drawing quadratic (2nd order) and cubic (3rd order) splines, as well as an Arc tool to draw consecutive arcs in a single Bézier curve. Reusable items.
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The mathematical programming school employed classical gradient-based methods to structural optimization problems. The method of usable feasible directions, Rosen's gradient projection (generalized reduce gradient) method, sequential unconstrained minimization techniques, sequential linear programming and eventually sequential quadratic programming methods were common choices. Schittkowski et al. reviewed the methods current by the early 1990s.
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The Earley parser executes in cubic time in the general case {O}(n^3), where n is the length of the parsed string, quadratic time for unambiguous grammars {O}(n^2), p.145 and linear time for all deterministic context-free grammars. It performs particularly well when the rules are written left-recursively.
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Newton's method assumes the function f to have a continuous derivative. Newton's method may not converge if started too far away from a root. However, when it does converge, it is faster than the bisection method, and is usually quadratic. Newton's method is also important because it readily generalizes to higher-dimensional problems.
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The invariant may be computed for a specific symbol φ taking values ±1 in the group C2.Milnor & Husemoller (1973) p.79 In the context of quadratic forms over a local field, the Hasse invariant may be defined using the Hilbert symbol, the unique symbol taking values in C2.Serre (1973) p.
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A form of degree 1 is a linear form.Linear forms are defined only for finite-dimensional vector space, and have thus to be distinguished from linear functionals, which are defined for every vector space. "Linear functional" is rarely used for finite-dimensional vector spaces. A form of degree 2 is a quadratic form.
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He further contributed significantly to the understanding of perfect numbers, which had fascinated mathematicians since Euclid. Euler made progress toward the prime number theorem and conjectured the law of quadratic reciprocity. The two concepts are regarded as the fundamental theorems of number theory, and his ideas paved the way for Carl Friedrich Gauss.
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A large class of quadrature rules can be derived by constructing interpolating functions that are easy to integrate. Typically these interpolating functions are polynomials. In practice, since polynomials of very high degree tend to oscillate wildly, only polynomials of low degree are used, typically linear and quadratic. Illustration of the rectangle rule.
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Acland, E.L., Colasante, T., & Malti, T. (2019). Respiratory sinus arrhythmia and prosociality in childhood: Evidence for a quadratic effect. Developmental Psychobiology, 00, 1-11. doi:10.1002/dev.21872 Dys, S.P., Peplak, J., Colasante, T., & Malti, T. (2019). Children’s sympathy and sensitivity to excluding economically disadvantaged peers. Developmental Psychology, 55(3), 482–487.
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The Macusani volcanics are located in the Carabaya Province, Puno Department of Peru. The towns of Macusani, Crucero and Ananea lie in the region. The Macusani River flows east of the field, and the high mountain range surrounds the area, forming a quadratic depression. The Macusani volcanics lie in the Cordillera de Carabaya.
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For example, in section V, article 303, Gauss summarized his calculations of class numbers of proper primitive binary quadratic forms, and conjectured that he had found all of them with class numbers 1, 2, and 3. This was later interpreted as the determination of imaginary quadratic number fields with even discriminant and class number 1, 2, and 3, and extended to the case of odd discriminant. Sometimes called the class number problem, this more general question was eventually confirmed in 1986 (the specific question Gauss asked was confirmed by Landau in 1902 for class number one). In section VII, article 358, Gauss proved what can be interpreted as the first nontrivial case of the Riemann hypothesis for curves over finite fields (the Hasse–Weil theorem).
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The systematic use of algebraic manipulations for simplifying expressions (more specifically equations)) may be dated to 9th century, with al-Khwarizmi's book The Compendious Book on Calculation by Completion and Balancing, which is titled with two such types of manipulation. However, even for solving quadratic equations, factoring method was not used before Harriot's work published in 1631, ten years after his death.In , the author notes "In view of the present emphasis given to the solution of quadratic equations by factoring, it is interesting to note that this method was not used until Harriot's work of 1631". In his book Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas, Harriot drew, in a first section, tables for addition, subtraction, multiplication and division of monomials, binomials, and trinomials.
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The CAPM can be derived from the following special cases of the CCAPM: (1) a two-period model with quadratic utility, (2) two-periods, exponential utility, and normally-distributed returns, (3) infinite-periods, quadratic utility, and stochastic independence across time, (4) infinite periods and log utility, and (5) a first-order approximation of a general model with normal distributions. Formally, the CCAPM states that the expected risk premium on a risky asset, defined as the expected return on a risky asset less the risk free return, is proportional to the covariance of its return and consumption in the period of the return. The consumption beta is included, and the expected return is calculated as follows:Romer, David. Advanced Macroeconomics, ch. 7.
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Many areas have been proposed for quadratic voting, including corporate governance in the private sector, allocating budgets, cost-benefit analyses for public goods, more accurate polling and sentiment data, and elections and other democratic decisions. Quadratic voting was conducted in an experiment by the Democratic caucus of the Colorado House of Representatives in April 2019. Lawmakers used it to decide on their legislative priorities for the coming two years, selecting among 107 possible bills. Each member was given 100 virtual tokens that would allow them to put either 9 votes on one bill (as 81 virtual tokens represented 9 votes for one bill) and 3 votes on another bill or 5 votes each (25 virtual tokens) on 4 different bills.
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Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant , the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by due to the convention of the term being rather than ; formally, the discriminant (of the associated quadratic form) is , with the factor of 4 dropped for simplicity. # (elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth.
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During the 600s, negative numbers were in use in India to represent debts. Diophantus' previous reference was discussed more explicitly by Indian mathematician Brahmagupta, in Brāhmasphuṭasiddhānta in 628, who used negative numbers to produce the general form quadratic formula that remains in use today. However, in the 12th century in India, Bhaskara gives negative roots for quadratic equations but says the negative value "is in this case not to be taken, for it is inadequate; people do not approve of negative roots". European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debts (chapter 13 of Liber Abaci, 1202) and later as losses (in ).
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In geometric topology, many properties of manifolds depend only on their dimension mod 4 or mod 8; thus one often studies manifolds of singly even and doubly even dimension (4k+2 and 4k) as classes. For example, doubly even-dimensional manifolds have a symmetric nondegenerate bilinear form on their middle-dimension cohomology group, which thus has an integer-valued signature. Conversely, singly even-dimensional manifolds have a skew-symmetric nondegenerate bilinear form on their middle dimension; if one defines a quadratic refinement of this to a quadratic form (as on a framed manifold), one obtains the Arf invariant as a mod 2 invariant. Odd-dimensional manifolds, by contrast, do not have these invariants, though in algebraic surgery theory one may define more complicated invariants.
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David Pingree / Encyclopaedia Iranica states that he originally hailed from Khuttal or Gilan. He wrote a work on algebra of which only a chapter called "Logical Necessities in Mixed Equations", on the solution of quadratic equations, has survived. He authored a manuscript entitled Logical Necessities in Mixed Equations, which is very similar to al-Khwarzimi's Al-Jabr and was published at around the same time as, or even possibly earlier than, Al-Jabr. The manuscript gives exactly the same geometric demonstration as is found in Al-Jabr, and in one case the same example as found in Al-Jabr, and even goes beyond Al-Jabr by giving a geometric proof that if the discriminant is negative then the quadratic equation has no solution.
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A core member of this class is the Wigner–Ville distribution (WVD), as all other TFDs can be written as a smoothed or convolved versions of the WVD. Another popular member of this class is the spectrogram which is the square of the magnitude of the short- time Fourier transform (STFT). The spectrogram has the advantage of being positive and is easy to interpret, but also has disadvantages, like being irreversible, which means that once the spectrogram of a signal is computed, the original signal can't be extracted from the spectrogram. The theory and methodology for defining a TFD that verifies certain desirable properties is given in the "Theory of Quadratic TFDs".B. Boashash, “Theory of Quadratic TFDs”, Chapter 3, pp.
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For every sub-Riemannian manifold, there exists a Hamiltonian, called the sub-Riemannian Hamiltonian, constructed out of the metric for the manifold. Conversely, every such quadratic Hamiltonian induces a sub- Riemannian manifold. The existence of geodesics of the corresponding Hamilton–Jacobi equations for the sub-Riemannian Hamiltonian is given by the Chow–Rashevskii theorem.
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Jenkins, M. A. (1975), Algorithm 493: Zeros of a Real Polynomial, ACM TOMS, 1, 178–189. The methods have been extensively tested by many people. As predicted they enjoy faster than quadratic convergence for all distributions of zeros. However, there are polynomials which can cause loss of precision as illustrated by the following example.
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A quasi-sphere } in a quadratic space has a counter- sphere }. Furthermore, if and is an isotropic line in through , then , puncturing the union of quasi-sphere and counter-sphere. One example is the unit hyperbola that forms a quasi-sphere of the hyperbolic plane, and its conjugate hyperbola, which is its counter-sphere.
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They are hindered by material defects. Lord Rayleigh investigated this first and quantified the magnetization M as a linear and quadratic term in the field: :M = \chi_0 H + \alpha_R \mu_0 H^2. Here \chi_0 is the initial susceptibility, describing the reversible part of magnetisation reversal. The Rayleigh constant \alpha_R describes the irreversible Barkhausen jumps.
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Contemporary algebraic geometry treats blowing up as an intrinsic operation on an algebraic variety. From this perspective, a blowup is the universal (in the sense of category theory) way to turn a subvariety into a Cartier divisor. A blowup can also be called monoidal transformation, locally quadratic transformation, dilatation, σ-process, or Hopf map.
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Dually, it can be tiled with 56 equilateral triangles, with 24 vertices, each of degree 7, as a quotient of the order-7 triangular tiling. Klein's quartic arises in many fields of mathematics, including representation theory, homology theory, octonion multiplication, Fermat's last theorem, and Stark's theorem on imaginary quadratic number fields of class number 1.
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C. Naylor, R. Donelly, and L. Sha. Non-Linear Optimization System and Method for Wire Length and Delay Optimization for an Automatic Electric Circuit Placer. In US Patent 6301693, 2001. first models wirelength by exponential (nonlinear) functions and density by local piece-wise quadratic functions, in order to achieve better accuracy thus quality improvement.
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The question mark function provides a one-to-one mapping from the non-dyadic rationals to the quadratic irrationals, thus allowing an explicit proof of countability of the latter. These can, in fact, be understood to correspond to the periodic orbits for the dyadic transformation. This can be explicitly demonstrated in just a few steps.
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The inner ward consisted of a palace, a gatehouse, a battlement with bartizan (on the woodcut in the centre of the castle and oversized) and a quadratic keep. Access to the inner ward was protected by a drawbridge, which is not visible on the woodcut. Several investigations, including by Karl Dietel, support this hypothesis, however.
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When b is greater than zero, its vertex is shifted to the left of and below the origin. The vertices of the family of curves created by varying b follow along a parabolic curve. The right plot illustrates varying eh. When eh is positive, the quadratic function is a parabola opening to the top.
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Hallgren showed that a quantum computer can find a product representation, as described above, for the solution to Pell's equation in polynomial time. Hallgren's algorithm, which can be interpreted as an algorithm for finding the group of units of a real quadratic number field, was extended to more general fields by Schmidt and Völlmer.
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Quadratic unconstrained binary optimization (QUBO) is a pattern matching technique, common in machine learning applications. QUBO is an NP hard problem. Examples of problems that can be formulated as QUBO problems are the Maximum cut, Graph coloring and the Partition problem. QUBO problems may sometimes be well-suited to algorithms aided by quantum annealing.
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In 1919 Watson was elected a Fellow of the Royal Society, and in 1946, he received the Sylvester Medal from the Society. He was president of the London Mathematical Society from 1933 to 1935. He is sometimes confused with the mathematician G. L. Watson, who worked on quadratic forms, and G. Watson, a statistician.
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In 1962, Simon entered Harvard with a stipend. He became a Putnam Fellow in 1965 at 19 years old. He received his A.B. in 1966 from Harvard College and his Ph.D. in Physics at Princeton University in 1970, supervised by Arthur Strong Wightman. His dissertation dealt with Quantum mechanics for Hamiltonians defined as quadratic forms.
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Brent's method is a combination of the bisection method, the secant method and inverse quadratic interpolation. At every iteration, Brent's method decides which method out of these three is likely to do best, and proceeds by doing a step according to that method. This gives a robust and fast method, which therefore enjoys considerable popularity.
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The Bunyakovsky conjecture generalizes Dirichlet's theorem to higher-degree polynomials. Whether or not even simple quadratic polynomials such as (known from Landau's fourth problem) attain infinitely many prime values is an important open problem. The Dickson's conjecture generalizes Dirichlet's theorem to more than one polynomial. The Schinzel's hypothesis H generalizes these two conjectures, i.e.
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The Néron–Tate height, or canonical height, is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron, who first defined it as a sum of local heights, and John Tate, who defined it globally in an unpublished work.
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The algebra of invariants of the quadratic form ax2 \+ 2bxy + cy2 is a polynomial algebra in 1 variable generated by the discriminant b2 − ac of degree 2. The algebra of covariants is a polynomial algebra in 2 variables generated by the discriminant together with the form f itself (of degree 1 and order 2).
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On the other hand, convergence (even to a local extremum) is not guaranteed when using this method in isolation. For example, if the three points are collinear, the resulting parabola is degenerate and thus does not provide a new candidate point. Furthermore, if function derivatives are available, Newton's method is applicable and exhibits quadratic convergence.
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In the opposite direction, all these torsion structures occur infinitely often over Q, since the corresponding modular curves are all genus zero curves with a rational point. A complete list of possible torsion groups is also available for elliptic curves over quadratic number fields. There are substantial partial results for quartic and quintic number fields .
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Ernests Fogels (12 October 1910 – 22 February 1985) was a Latvian mathematician who specialized in number theory. Fogels discovered new proofs of the Gauss-Dirichlet formula on the number of classes of positively definite quadratic forms and of the de la Vallée-Poussin formula for the asymptotic location of prime numbers in an arithmetic progression.
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All solution techniques perform transcription, a process by which the trajectory optimization problem (optimizing over functions) is converted into a constrained parameter optimization problem (optimizing over real numbers). Generally, this constrained parameter optimization problem is a non-linear program, although in special cases it can be reduced to a quadratic program or linear program.
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His sister claimed that he could not read proficiently until he was 12. He replied "no, but I knew a lot of quadratic equations!" After leaving Highgate School in 1921, he studied at City and Guilds College (part of Imperial College). He won a Governors' scholarship and joined the second year of the course.
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Like automatic groups, automatic semigroups have word problem solvable in quadratic time. Kambites & Otto (2006) showed that it is undecidable whether an element of an automatic monoid possesses a right inverse. Cain (2006) proved that both cancellativity and left-cancellativity are undecidable for automatic semigroups. On the other hand, right- cancellativity is decidable for automatic semigroups (Silva & Steinberg 2004).
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The above arithmetic can be generalized to calculate second order and higher derivatives of multivariate functions. However, the arithmetic rules quickly grow complicated: complexity is quadratic in the highest derivative degree. Instead, truncated Taylor polynomial algebra can be used. The resulting arithmetic, defined on generalized dual numbers, allows efficient computation using functions as if they were a data type.
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The 2006 contest required entries to count word occurrences, but have vastly different runtimes on different platforms. To accomplish the task, entries used fork implementation errors, optimization problems, endian differences and various API implementation differences. The winner called strlen() in a loop, leading to quadratic complexity which was optimized out by a Linux compiler but not by Windows.
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A space with quadratic form is split (or metabolic) if there is a subspace which is equal to its own orthogonal complement; equivalently, the index of isotropy is equal to half the dimension. The hyperbolic plane is an example, and over a field of characteristic not equal to 2, every split space is a direct sum of hyperbolic planes.
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A. Graham & S. Driver (2007), A Log-Quadratic Relation for Predicting Supermassive Black Hole Masses from the Host Bulge Sérsic Index Sérsic profiles can also be used to describe dark matter halos, where the Sérsic index correlates with halo mass.D. Merritt et al. (2005), A Universal Density Profile for Dark and Luminous Matter? D. Merritt et al.
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Sometimes, some term grouping lets appear a part of a recognizable pattern. It is then useful to add terms for completing the pattern, and subtract them for not changing the value of the expression. A typical use of this is the completing the square method for getting quadratic formula. Another example is the factorization of x^4 + 1.
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There are three windows at each side, and two in the front and backside each, all arched. The two windows on the qibla wall are closed with masonry. It has a covered area of , and can hold up to 500 worshipers. The stand-alone minaret in the form of a quadratic prism is erected on four massive stone columns.
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The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. Each of the algebras Cl(R) and Cl(C) is isomorphic to A or , where A is a full matrix ring with entries from R, C, or H. For a complete classification of these algebras see classification of Clifford algebras.
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Holckenhavn Castle Situated on an almost quadratic castle bank, Holckenhavn is a four-winged complex designed in the Renaissance style and built over the course of three generations. The north and east wings, as well as the gate wing, were completed by 1585. The large bell tower was added somewhat later. The master builder was probably Domenicus Badiaz.
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Recently, the 15 and 290 theorems have completely characterized universal integral quadratic forms: if all coefficients are integers, then it represents all positive integers if and only if it represents all integers up through 290; if it has an integral matrix, it represents all positive integers if and only if it represents all integers up through 15.
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Level-crossing measurements of the lifetimes and hyperfine constants of the 2P3/2 states of the stable alkali atoms, R. W. Schmieder, A. Lurio, W. Happer, and A. Khadjavi, Phys. Rev. A2, 1216 (1970).Quadratic stark effect in the 2P3/2 states of the alkali atoms, R. W. Schmieder, A. Lurio, and W. Happer, Phys. Rev. A3, 1209 (1970).
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In Mauritius, Additional Mathematics, more commonly referred to as Add Maths, is offered in secondary school as an optional subject in the Arts Streams, and a compulsory subject in the Science, Technical and Economics Stream. This subject is included in the University of Cambridge International Examinations, with covered topics including functions, quadratic equations, differentiation and integration (calculus).
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A simple Atsumari puzzle and solution. Atsumari (; translates as "collection", "meeting", or "cluster") is a binary-determination puzzle that was originally developed by Quadratic Games for the iPhone platform. The puzzle is played on a hexagonal grid. A rectangular board shape is standard but variations to the board shape can be part of the puzzle design.
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In economics, the Cobb–Douglas production function is a family of production functions parametrized by the elasticities of output with respect to the various factors of production. In algebra, the quadratic equation, for example, is actually a family of equations parametrized by the coefficients of the variable and of its square and by the constant term.
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Huygens stated what is now known as the second of Newton's laws of motion in a quadratic form.Ernst Mach, The Science of Mechanics (1919), e.g. pp. 143, 172, 187 . In 1659 he derived the now standard formula for the centripetal force, exerted on an object describing a circular motion, for instance by the string to which it is attached.
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Together with J.-P. Serre he is one of the cofounders of the theory of cohomological invariants of linear algebraic groups. He has also made numerous contributions to the theory of torsors, quadratic forms, central simple algebras, Jordan algebras (the Rost-Serre invariant), exceptional groups, and essential dimension. Most of his results are available only on his webpage.
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Throughout the 19th century mathematics became increasingly abstract. Carl Friedrich Gauss (1777–1855) epitomizes this trend. He did revolutionary work on functions of complex variables, in geometry, and on the convergence of series, leaving aside his many contributions to science. He also gave the first satisfactory proofs of the fundamental theorem of algebra and of the quadratic reciprocity law.
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Rosenberg's research was in the area of abstract algebra, including the application of homology to Galois theory and to the theory of quadratic forms. With Gerhard Hochschild and Bertram Kostant, he is one of the namesakes of the Hochschild–Kostant–Rosenberg theorem, which they published in 1962 and which describes the Hochschild homology of some algebras..
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The algorithm can be further simplified for linear feasibility problems, that is for linear systems with nonnegative variables; these problems can be formulated for oriented matroids. The criss-cross algorithm has been adapted for problems that are more complicated than linear programming: There are oriented-matroid variants also for the quadratic-programming problem and for the linear-complementarity problem.
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18, No. 2, March–April 2001 DPM conserves power by shutting down parts of the sensor node which are not currently used or active. A DVS scheme varies the power levels within the sensor node depending on the non-deterministic workload. By varying the voltage along with the frequency, it is possible to obtain quadratic reduction in power consumption.
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For example the shear stress variation in a rotating pipe cannot be predicted with quadratic tensors. Hence, the EASM was extended with a cubic tensor. In order to do not affect the performance in 2D flows, a tensor was chosen that vanish in 2d flows. This offers the concentration of the coefficient determination in 3d flows.
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In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It was introduced by . The Rost invariant is a generalization of the Arason invariant to other algebraic groups.
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These modular forms, for real quadratic fields, were first treated in the 1901 Göttingen University Habilitationssschrift of Otto Blumenthal. There he mentions that David Hilbert had considered them initially in work from 1893-4, which remained unpublished. Blumenthal's work was published in 1903. For this reason Hilbert modular forms are now often called Hilbert-Blumenthal modular forms.
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For imaginary quadratic number fields, the (fundamental) discriminants of class number 1 are: :d=-3,-4,-7,-8,-11,-19,-43,-67,-163. The non-fundamental discriminants of class number 1 are: :d=-12,-16,-27,-28. Thus, the even discriminants of class number 1, fundamental and non-fundamental (Gauss's original question) are: :d=-4,-8,-12,-16,-28.
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The 6-room building with stoa facing the center is designed in quadratic form surrounding a large courtyard. It covers an area of including the courtyard. The building is considered to have served as home of Ahmet Arif. The restoration of the building for this purpose cost 93,000 while another 75,000 were spent for its decoration and furnishing.
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The sieve methods discussed in this article are not closely related to the integer factorization sieve methods such as the quadratic sieve and the general number field sieve. Those factorization methods use the idea of the sieve of Eratosthenes to determine efficiently which members of a list of numbers can be completely factored into small primes.
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Analogous to symmetric polynomials are alternating polynomials: polynomials that, rather than being invariant under permutation of the entries, change according to the sign of the permutation. These are all products of the Vandermonde polynomial and a symmetric polynomial, and form a quadratic extension of the ring of symmetric polynomials: the Vandermonde polynomial is a square root of the discriminant.
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The Douady rabbit, a quadratic filled Julia set, is named after him. Douady taught at the University of Nice and was a Professor at the Paris-Sud 11 University, Orsay. He was a member of Bourbaki. and an invited speaker at the International Congress of Mathematicians in 1966 at Moscow and again in 1986 in Berkeley.
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It maps quadratic irrationals to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the binary expansions of the rationals, given by Arnaud Denjoy in 1938. In addition, it maps rational numbers to dyadic rationals, as can be seen by a recursive definition closely related to the Stern–Brocot tree.
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QRA may be viewed as an extension of combining point forecasts. The well-known ordinary least squares (OLS) averaging uses linear regression to estimate weights of the point forecasts of individual models. Replacing the quadratic loss function with the absolute loss function leads to quantile regression for the median, or in other words, least absolute deviation (LAD) regression.
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In computer graphics, Doo–Sabin subdivision surface is a type of subdivision surface based on a generalization of bi-quadratic uniform B-splines. It was developed in 1978 by Daniel Doo and Malcolm Sabin.D. Doo: A subdivision algorithm for smoothing down irregularly shaped polyhedrons, Proceedings on Interactive Techniques in Computer Aided Design, pp. 157 - 165, 1978 (pdf)D.
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Lagrange extrapolations of the sequence 1,2,3. Extrapolating by 4 leads to a polynomial of minimal degree ( line). A polynomial curve can be created through the entire known data or just near the end (two points for linear extrapolation, three points for quadratic extrapolation, etc.). The resulting curve can then be extended beyond the end of the known data.
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However, one cannot, in every case expect things to be so easy; the quadratic formula's analogue for fourth-order polynomials is very convoluted and no such analogue exists for 5th-or-higher order polynomials. See Galois theory for a theoretical explanation of why this is so, and see numerical analysis for ways to approximate roots of polynomials numerically.
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Nike Dattani is a scientist known for breaking the world-record for largest number factored on a quantum device in 2014. He is also known for co-inventing the Morse/Long-range potential energy function, and for inventing several novel methods for quadratization of high-degree discrete optimization problems into quadratic problems which are much easier to solve.
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Because a2 ≡ (n − a)2 (mod n), the list of squares modulo n is symmetrical around n/2, and the list only needs to go that high. This can be seen in the table below. Thus, the number of quadratic residues modulo n cannot exceed n/2 + 1 (n even) or (n + 1)/2 (n odd).Gauss, DA, art.
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Paley graphs are dense undirected graphs, one for each prime p ≡ 1 (mod 4), that form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley digraphs are directed analogs of Paley graphs, one for each p ≡ 3 (mod 4), that yield antisymmetric conference matrices. The construction of these graphs uses quadratic residues.
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David Hilbert, Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919–1920. But the proof of this number's transcendence was published by Kuzmin in 1930, well within Hilbert's own lifetime. Namely, Kuzmin proved the case where the exponent b is a real quadratic irrational, which was later extended to an arbitrary algebraic irrational b by Gelfond and by Schneider.
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The Ulam spiral of numbers, with black pixels showing prime numbers. This diagram hints at patterns in the distribution of prime numbers. Number theory is concerned with the properties of numbers in general, particularly integers. It has applications to cryptography and cryptanalysis, particularly with regard to modular arithmetic, diophantine equations, linear and quadratic congruences, prime numbers and primality testing.
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Bhāskara II (1114 – c. 1185) was the leading mathematician of the 12th century. In Algebra, he gave the general solution of Pell's equation. He is the author of Lilavati and Vija-Ganita, which contain problems dealing with determinate and indeterminate linear and quadratic equations, and Pythagorean triples and he fails to distinguish between exact and approximate statements.
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This is the basis of the secant method. Three values define a quadratic function, which approximates the graph of the function by a parabola. This is Muller's method. Regula falsi is also an interpolation method, which differs secant method by using, for interpolating by a line, two points that are not necessarily the last two computed points.
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If the given polynomial only has real coefficients, one may wish to avoid computations with complex numbers. To that effect, one has to find quadratic factors for pairs of conjugate complex roots. The application of the multidimensional Newton's method to this task results in Bairstow's method. The real variant of Jenkins–Traub algorithm is an improvement of this method.
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All of these developments with optimal linear decision rules can be thought of today as optimal feedback rules which are computed using dynamic programming methods on linear-quadratic systems that yield Riccati equations, which are used to obtain the key components of the feedback gain matrix. This approach is sometimes called “modern control” to distinguish it from “classical control”.
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The main building dates from 1856-1857 and was designed by Ferdinand Meldahl with inspiration from Renaissance architecture. The building is constructed in red brick and stands on a foundation of boulders. It consists of a two-storey main wing with a short side wing. Towards the courtyard is a centrally placed quadratic tower topped by a dome.
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The first provably-secure probabilistic public-key encryption scheme was proposed by Shafi Goldwasser and Silvio Micali, based on the hardness of the quadratic residuosity problem and had a message expansion factor equal to the public key size. More efficient probabilistic encryption algorithms include Elgamal, Paillier, and various constructions under the random oracle model, including OAEP.
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As a professor, Stern taught Gauss's student Bernhard Riemann. Stern was very helpful to Gotthold Eisenstein in formulating a proof of the quadratic reciprocity theorem. Stern was interested in primes that cannot be expressed as the sum of a prime and twice a square (now known as Stern primes). He is known for formulating Stern's diatomic series.
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The example given below can produce simultaneous lowpass, highpass and bandpass outputs from a single input. This is a second-order (biquad) filter. Its derivation comes from rearranging a high-pass filter's transfer function, which is the ratio of two quadratic functions. The rearrangement reveals that one signal is the sum of integrated copies of another.
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Phil.) at the University of Oxford in 1974. Her dissertation The Enumeration of Perfect Quadratic Forms in Seven Variables concerned number theory and was supervised by Bryan John Birch. She also has a Diploma of Education from Monash University. With Leone Burton and John Mason, Stacey is the author of the book Thinking Mathematically (Addison-Wesley, 1982; 2nd ed.
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The primary motivation for the creation of QV was that an optimal voting mechanism for decisions involving public goods was created in the 60's and 70's by Vickrey, Clarke, and Groves (VCG). Despite high initial excitement about this mechanism, including Vickrey receiving a Nobel Prize for the work, the VCG mechanism was not sufficiently robust and practical to be implemented, and the mechanism found almost no practical adoption. Quadratic voting was intended by its inventors to deliver similarly optimal outcomes as the VCG mechanism while being easier for people to use and understand, and being more robust with respect to collusion and other practical considerations. A mechanism closely resembling quadratic voting was first published in 1977 by Groves and Ledyard, with a similar mechanism being proposed by Hylland and Zeckhauser in 1980.
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The general quintic may be reduced into what is known as the principal quintic form, with the quartic and cubic terms removed: :y^5 + c_2y^2 + c_1y + c_0 = 0 \, If the roots of a general quintic and a principal quintic are related by a quadratic Tschirnhaus transformation :y_k = x_k^2 + \alpha x_k + \beta \, , the coefficients α and β may be determined by using the resultant, or by means of the power sums of the roots and Newton's identities. This leads to a system of equations in α and β consisting of a quadratic and a linear equation, and either of the two sets of solutions may be used to obtain the corresponding three coefficients of the principal quintic form. This form is used by Felix Klein's solution to the quintic.
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The constructive existence proof shows that, in the case of two moduli, the solution may be obtained by the computation of the Bézout coefficients of the moduli, followed by a few multiplications, additions and reductions modulo n_1n_2 (for getting a result in the interval (0, n_1n_2-1)). As the Bézout's coefficients may be computed with the extended Euclidean algorithm, the whole computation, at most, has a quadratic time complexity of O((s_1+s_2)^2), where s_i denotes the number of digits of n_i. For more than two moduli, the method for two moduli allows the replacement of any two congruences by a single congruence modulo the product of the moduli. Iterating this process provides eventually the solution with a complexity, which is quadratic in the number of digits of the product of all moduli.
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Dickson proved many interesting results in number theory, using results of Vinogradov to deduce the ideal Waring theorem in his investigations of additive number theory. He proved the Waring's problem for k\ge 7 under the further condition of :(3^k + 1)/(2^k - 1)\le [1.5^k] + 1 independently of Subbayya Sivasankaranarayana Pillai who proved it for k\ge 6 ahead of him. The three-volume History of the Theory of Numbers (1919–23) is still much consulted today, covering divisibility and primality, Diophantine analysis, and quadratic and higher forms. The work contains little interpretation and makes no attempt to contextualize the results being described, yet it contains essentially every significant number theoretic idea from the dawn of mathematics up to the 1920s except for quadratic reciprocity and higher reciprocity laws.
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Since the order of multiplication does not matter, one can switch and and the values of and will not change: one can say that and are symmetric polynomials in and . In fact, they are the elementary symmetric polynomials – any symmetric polynomial in and can be expressed in terms of and The Galois theory approach to analyzing and solving polynomials is: given the coefficients of a polynomial, which are symmetric functions in the roots, can one "break the symmetry" and recover the roots? Thus solving a polynomial of degree is related to the ways of rearranging ("permuting") terms, which is called the symmetric group on letters, and denoted . For the quadratic polynomial, the only way to rearrange two terms is to swap them ("transpose" them), and thus solving a quadratic polynomial is simple.
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Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ.Math.IHES 69(1989), 119-171; Addendum: ibid, 71(1990); with A.Borel. [7]. Values of isotropic quadratic forms at S-integral points, Compositio Mathematica, 83 (1992), 347-372; with A.Borel. [8]. Unrefined minimal K-types for p-adic groups, Inventiones Math. 116(1994), 393-408; with Allen Moy. [9].
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It can be shown that, provided L is chosen sufficiently large, sλ always converges to a root of P. The algorithm converges for any distribution of roots, but may fail to find all roots of the polynomial. Furthermore, the convergence is slightly faster than the quadratic convergence of Newton–Raphson iteration, however, it uses at least twice as many operations per step.
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Therefore, Apollonius' problem has at most eight independent solutions (Figure 2). One way to avoid this double-counting is to consider only solution circles with non-negative radius. The two roots of any quadratic equation may be of three possible types: two different real numbers, two identical real numbers (i.e., a degenerate double root), or a pair of complex conjugate roots.
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The Hill yield criterion developed by Rodney Hill, is one of several yield criteria for describing anisotropic plastic deformations. The earliest version was a straightforward extension of the von Mises yield criterion and had a quadratic form. This model was later generalized by allowing for an exponent m. Variations of these criteria are in wide use for metals, polymers, and certain composites.
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Another pressure-dependent extension of Hill's quadratic yield criterion which has a form similar to the Bresler Pister yield criterion is the Deshpande, Fleck and Ashby (DFA) yield criterion Deshpande, V. S., Fleck, N. A. and Ashby, M. F. (2001). Effective properties of the octet-truss lattice material. Journal of the Mechanics and Physics of Solids, vol. 49, no. 8, pp. 1747-1769.
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In particular, if there is exactly one non-linear factor, it will be the polynomial left after all linear factors have been factorized out. In the case of a cubic polynomial, if the cubic is factorizable at all, the rational root test gives a complete factorization, either into a linear factor and an irreducible quadratic factor, or into three linear factors.
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This computer-based generation of fractal objects is an endless process. In theory, images can be calculated infinitely but in practice are approximated to a certain level of detail. Mandelbrot used quadratic formulas described by the French mathematician Gaston Julia. The maximum fractal dimension that can be produced varies according to type and is sometimes limited according to the method implemented.
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In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system. The specification of all microstates of a system is a point in the system's phase space. In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer. It is often useful to specify quadratic degrees of freedom.
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In number theory, Shanks is best known for his book Solved and Unsolved Problems in Number Theory. Hugh Williams described it as "a charming, unconventional, provocative, and fascinating book on elementary number theory." It is a wide-ranging book, but most of the topics depend on quadratic residues and Pell's equation. The third edition contains a long essay on "judging conjectures".
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' (German for Lectures on Number Theory) is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Kronecker, Edmund Landau, and Helmut Hasse. They all cover elementary number theory, Dirichlet's theorem, quadratic fields and forms, and sometimes more advanced topics.
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Many of them started as ger-temples. When they needed to be enlarged to accommodate the growing number of worshippers, the Mongolian architects used structures with 6 and 12 angles with pyramidal roofs to approximate to the round shape of a ger. Further enlargement led to a quadratic shape of the temples. The roofs were made in the shape of marquees.
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In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or a quadric (sphere, cylinder, hyperboloid, etc.) lead to quadratic equations that can be easily solved. Intersections between quadrics lead to quartic equations that can be solved algebraically.
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Kaplansky's theorem states that a prime p congruent to 1 modulo 16 is representable by both or none of x2 + 32y2 and x2 + 64y2, whereas a prime p congruent to 9 modulo 16 is representable by exactly one of these quadratic forms. This is remarkable since the primes represented by each of these forms individually are not describable by congruence conditions..
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Clock and shift matrices find routine applications in numerous areas of mathematical physics, providing the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces. The concept of a spinor can further be linked to these algebras. The term Generalized Clifford Algebras can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.
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Another mention of the underlying problem occurred in a 1956 letter written by Kurt Gödel to John von Neumann. Gödel asked whether theorem-proving (now known to be co-NP- complete) could be solved in quadratic or linear time, and pointed out one of the most important consequences—that if so, then the discovery of mathematical proofs could be automated.
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In 10-digit floating-point arithmetic: :(-200 - 200.0000001) / 2 = -200.00000005, :(-200 + 200.0000001) / 2 = 0.00000005. Notice that the solution of greater magnitude is accurate to ten digits, but the first nonzero digit of the solution of lesser magnitude is wrong. Because of the subtraction that occurs in the quadratic equation, it does not constitute a stable algorithm to calculate the two roots.
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Randers Kulturhus actually comprise three adjoined buildings. The oldest is the former main building for Randers Tekniske Skole (Randers Technical School) from 1891, designed by J.P. Jensen Wærum. Another building was added in 1937, designed by I.P. Hjersing. But what now comprise Randers Kulturhus, was founded by the quadratic modernist concrete structure built in 1964-69 and designed by Flemming Lassen.
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Sampled differential dynamic programming (SaDDP) is a Monte Carlo variant of differential dynamic programming. It is based on treating the quadratic cost of differential dynamic programming as the energy of a Boltzmann distribution. This way the quantities of DDP can be matched to the statistics of a multidimensional normal distribution. The statistics can be recomputed from sampled trajectories without differentiation.
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This one-point second-order method is known to show a locally quadratic convergence if the root of the equation is simple. SRA strictly implies this one-point second- order interpolation by a simple rational function. We can notice that even third order method is a variation of Newton's method. We see the Newton's steps are multiplied by some factors.
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Visualization of the expected score under various predictions from some common scoring functions. Dashed black line: forecaster's true belief, red: linear, orange: spherical, purple: quadratic, green: log. In decision theory, a score function, or scoring rule, measures the accuracy of probabilistic predictions. It is applicable to tasks in which predictions must assign probabilities to a set of mutually exclusive outcomes.
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Linear-quadratic regulator rapidly-exploring random tree (LQR-RRT) is a sampling based algorithm for kinodynamic planning. A solver is producing random actions which are forming a funnel in the state space. The generated tree is the action sequence which fulfills the cost function. The restriction is, that a prediction model, based on differential equations, is available to simulate a physical system.
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If the error structure depends on an unknown variable or an unobserved variable the Goldfeld–Quandt test provides little guidance. Also, error variance must be a monotonic function of the specified explanatory variable. For example, when faced with a quadratic function mapping the explanatory variable to error variance the Goldfeld–Quandt test may improperly accept the null hypothesis of homoskedastic errors.
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86 (Infobase Publishing 2006). Later, Persian and Arabic mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ad hoc methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, negative numbers or zero, thus he had to distinguish several types of equations.
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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 algorithm can solve quadratic programming problems and linear complementarity problems, even in the setting of oriented matroids.
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The book was a compilation and extension of known rules for solving quadratic equations and for some other problems, and considered to be the foundation of algebra, establishing it as an independent discipline. The word algebra is derived from the name of one of the basic operations with equations described in this book, following its Latin translation by Robert of Chester.
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In statistics, a central composite design is an experimental design, useful in response surface methodology, for building a second order (quadratic) model for the response variable without needing to use a complete three-level factorial experiment. After the designed experiment is performed, linear regression is used, sometimes iteratively, to obtain results. Coded variables are often used when constructing this design.
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One nave with narrower five sided presbytery and side tower by the southwestern corner is in the ground plan of the church. The quadratic sacristy with entrance hall completes the construction. The ground plan is created by a long rectangle made from five bays. The exterior of the nave is relatively stern, decorated by only ten massive load-bearing pillars.
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Hesse points can be used to solve cubic equations as follows. If A, B, C are three roots of a cubic, then the Hesse points can be found as roots of a quadratic equation. If the Hesse points are then transformed to 0 and ∞ by a fractional linear transformation, the cubic equation is transformed to one of the form x3 = D.
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1999, 219, 16–79. and Kharlampovich O. Kharlampovich, A. Myasnikov, Irreducible affine varieties over a free group. I: Irreducibility of quadratic equations and nullstellensatz, J. Algebra, V. 200, 492–516 (1998), O. Kharlampovich, A. Myasnikov, Irreducible affine varieties over a free group. II: Systems in row-echelon form and description of residually free groups, J. Algebra, V. 200, 517–570 (1998).
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Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations. In the 18th century, Leonhard Euler studied the rotational motion of a rigid body, and discovered the importance of the principal axes. Joseph- Louis Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.
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The quadratic upper bounds are also appreciated by the computer-graphics literature: Ghali notesSh. Ghali. A survey of practical object space visibility algorithms. SIGGRAPH Tutorial Notes, 1(2), 2001. that the algorithms by Devai and McKenna "represent milestones in visibility algorithms", breaking a theoretical barrier from O(n2 log n) to O(n2) for processing a scene of n edges.
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His doctoral thesis introduced sequential quadratic programming, which became a leading iterative method for nonlinear programming. With other mathematical economists at the Stanford Business School, he helped to reformulate the economics of industrial organization and organization theory using non-cooperative game theory. His research on nonlinear pricing has influenced policies for large firms, particularly in the energy industry, especially electricity.
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A QEP can result in part of the dynamic analysis of structures discretized by the finite element method. In this case the quadratic, Q(\lambda) has the form Q(\lambda)=\lambda^2 M + \lambda C + K, where M is the mass matrix, C is the damping matrix and K is the stiffness matrix. Other applications include vibro-acoustics and fluid dynamics.
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Regression analysis, in the context of sensitivity analysis, involves fitting a linear regression to the model response and using standardized regression coefficients as direct measures of sensitivity. The regression is required to be linear with respect to the data (i.e. a hyperplane, hence with no quadratic terms, etc., as regressors) because otherwise it is difficult to interpret the standardised coefficients.
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Although the best model adds a quadratic term to defined International Atomic Time, the team encountered problems with this theory. This then led to non-uniform time in relation to a constant acceleration as the most likely theory.non-uniform time in relation to a constant acceleration is a summarized term derived from the source or sources used for this sub- section.
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In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as . This integral domain is a particular case of a commutative ring of quadratic integers. It does not have a total ordering that respects arithmetic.
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Faugeras, O. D. and Hebert, M., "Segmentation of Range Data into Planar and Quadratic Patches," Proceedings of IEEE conference on Computer Vision and Pattern Recognition , Arlington,VA, pp. 8–13, June 1983.Medioni, G. and Parvin, B., "Segmentation of Range Images into planar Surfaces by Split and Merg", Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, pp. 415–417, 1986.
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Plotting the logarithm of the number of trees per acre against the logarithm of the quadratic mean diameter (or the dbh of the tree of average basal area) of maximally stocked stands generally results in a straight-line relationship.Nyland, Ralph. 2002. Silvicultural Concepts and Applications 2nd edition. In most cases the line is used to define the limit of maximum stocking.
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In mathematics, a Bost–Connes system is a quantum statistical dynamical system related to an algebraic number field, whose partition function is related to the Dedekind zeta function of the number field. introduced Bost–Connes systems by constructing one for the rational numbers. extended the construction to imaginary quadratic fields. Such systems have been studied for their connection with Hilbert's Twelfth Problem.
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The programmer must translate the algorithm into a language that the simulator/computer/computor can effectively execute. Stone gives an example of this: when computing the roots of a quadratic equation the computor must know how to take a square root. If they don't, then the algorithm, to be effective, must provide a set of rules for extracting a square root.Stone 1972:5.
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The signature of a metric tensor is defined as the signature of the corresponding quadratic form. It is the number of positive and zero eigenvalues of any matrix (i.e. in any basis for the underlying vector space) representing the form, counted with their algebraic multiplicities. Usually, is required, which is the same as saying a metric tensor must be nondegenerate, i.e.
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BLAS functionality is categorized into three sets of routines called "levels", which correspond to both the chronological order of definition and publication, as well as the degree of the polynomial in the complexities of algorithms; Level 1 BLAS operations typically take linear time, , Level 2 operations quadratic time and Level 3 operations cubic time. Modern BLAS implementations typically provide all three levels.
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Trellis walls, roof poles and layers of felt were eventually replaced by stone, brick beams and planks. Mongolian artist and art historian N. Chultem identified three styles of traditional Mongolian architecture (Mongolian, Tibetan and Chinese), alone or in combination. Batu- Tsagaan (1654), designed by Zanabazar, was an early quadratic temple. The Dashchoilin Khiid monastery in Ulaanbaatar is an example of yurt-style architecture.
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The 18th century Lavrin Temple in the Erdene Zuu lamasery was built in the Tibetan tradition. The Choijin Lama Süm temple (1904), now a museum, is an example of a temple built in the Chinese tradition. The quadratic Tsogchin Temple, in Ulaanbaatar's Gandan monastery, combines Mongolian and Chinese traditions. The Maitreya Temple (demolished in 1938) was an example of Tibeto-Mongolian architecture.
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Naum Zuselevich Shor () (1 January 1937 – 26 February 2006) was a Soviet and Ukrainian mathematician specializing in optimization. He made significant contributions to nonlinear and stochastic programming, numerical techniques for non-smooth optimization, discrete optimization problems, matrix optimization, dual quadratic bounds in multi-extremal programming problems. Shor became a full member of the National Academy of Science of Ukraine in 1998.
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Helmut Hasse's local–global principle is said to hold for an equation if it can be solved over the rational numbers if and only if it can be solved over the real numbers and over the -adic numbers for every prime . This principle holds, for example, for equations given by quadratic forms, but fails for higher polynomials in several indeterminates.
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The equations 3x + 2y = 6 and 3x + 2y = 12 are parallel and cannot intersect, and is unsolvable. Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common solution. In the above example, a solution exists. However, there are also systems of equations which do not have any solution.
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In the course of this work they found a more elementary proof of Harish-Chandra's fundamental theorem on the local integrability of characters of Lie groups. With H. Donnelly and I. Singer, he extended Hirzebruch's formula (relating the signature defect at cusps of Hilbert modular surfaces to values of L-functions) from real quadratic fields to all totally real fields.
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To indicate this the background color of the text changes to light blue and yellow. If the user specifies text movements should not be detected, its algorithm runs in (m log n) time, which is an improvement from the standard quadratic time often seen in software of this type. m and n refer to the sizes of the original and modified texts.
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The condition for a nonzero number mod p to be a quadratic non-residue is to be an odd power of a primitive root. The lemma therefore comes down to saying that i is odd when j is odd, which is true a fortiori, and j is odd when i is odd, which is true because p − 1 is even (p is odd).
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One of the main applications of the maximum entropy principle is in discrete and continuous density estimation. Similar to support vector machine estimators, the maximum entropy principle may require the solution to a quadratic programming and thus provide a sparse mixture model as the optimal density estimator. One important advantage of the method is able to incorporate prior information in the density estimation.
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It is also a strictly non- palindromic number. 167 is the smallest multi-digit prime such that the product of digits is equal to the number of digits times the sum of the digits, i. e., 1×6×7 = 3×(1+6+7) 167 is the smallest positive integer d such that the imaginary quadratic field Q() has class number = 11.
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Bad character shift function is identical to the one proposed in Boyer–Moore–Horspool algorithm. A modern formulation of a similar pre-check is found in , a linear/quadratic string-matcher, in libc++ and libstdc++. Assuming a well-optimized version of , not skipping characters in the "original comparison" tends to be more efficient as the pattern is likely to be aligned.
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In linear programming, it was the language in which Robert G. Bland formulated his pivoting rule, by which the simplex algorithm avoids cycles. Similarly, it was used by Terlaky and Zhang to prove that their criss-cross algorithms have finite termination for linear programming problems. Similar results were made in convex quadratic programming by Todd and Terlaky.Björner et alia, Chapters 8–9.
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Some alternate models have been proposed which prevent shock formation. One alternative is to modify the "pressure term" in the momentum equation, but it results in a complicated expression for kinetic energy. Another option is to modify the non-linear terms in all equations, which gives a quadratic expression for kinetic energy, avoids shock formation, but conserves only linearized potential vorticity.
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Given any two of these, their intersection has exactly the four points. The reducible quadratics, in turn, may be determined by expressing the quadratic form as a matrix: reducible quadratics correspond to this matrix being singular, which is equivalent to its determinant being zero, and the determinant is a homogeneous degree three polynomial in and and corresponds to the resolvent cubic.
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In mathematics, a function f of n variables :x1, ..., xn leads to a Chisini mean M if for every vector , there exists a unique M such that :f(M,M, ..., M) = f(x1,x2, ..., xn). The arithmetic, harmonic, geometric, generalised, Heronian and quadratic means are all Chisini means, as are their weighted variants. They were introduced by Oscar Chisini in 1929.
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When the function is linear, selection is directional. Directional selection favors one extreme of a trait over another. An individual with the favored extreme value of the trait will survive more than others, causing the mean value of that trait in the population to shift in the next generation. When the relationship is quadratic, selection may be stabilizing or disruptive.
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This quadratic equation has two solutions, − 2 and 0. But if zero is substituted for x into the original equation, the result is the invalid equation 2 = 0\. This counterintuitive result occurs because in the case where x=0, multiplying both sides by x multiplies both sides by zero, and so necessarily produces a true equation just as in the first example.
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Myrelaion in Constantinople. A cross-in-square church is centered around a quadratic naos (the ‘square’) which is divided by four columns or piers into nine bays (divisions of space). The inner five divisions form the shape of a quincunx (the ‘cross’). The central bay is usually larger than the other eight, and is crowned by a dome which rests on the columns.
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2 Kronecker's Theorem, 176–177. This generalized an earlier result of Carl Friedrich Gauss from Disquisitiones Arithmeticae (1801), which classified quadratic forms; Kronecker cited this result of Gauss's. The theorem was stated and proved in the language of groups by Ferdinand Georg Frobenius and Ludwig Stickelberger in 1878.G. Frobenius, L. Stickelberger, Uber Grubben von vertauschbaren Elementen, J. reine u. angew. Math.
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This makes the transcendental numbers uncountable. No rational number is transcendental and all real transcendental numbers are irrational. The irrational numbers contain all the real transcendental numbers and a subset of the algebraic numbers, including the quadratic irrationals and other forms of algebraic irrationals. Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument.
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The spiric sections result from the intersection of a torus with a plane that is parallel to the rotational symmetry axis of the torus. Consequently, spiric sections are fourth-order (quartic) plane curves, whereas the conic sections are second-order (quadratic) plane curves. Spiric sections are a special case of a toric section, and were the first toric sections to be described.
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When used with low-quality hash functions that fail to eliminate nonuniformities in the input distribution, linear probing can be slower than other open-addressing strategies such as double hashing, which probes a sequence of cells whose separation is determined by a second hash function, or quadratic probing, where the size of each step varies depending on its position within the probe sequence..
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Amplitude amplification is a technique in quantum computing which generalizes the idea behind the Grover's search algorithm, and gives rise to a family of quantum algorithms. It was discovered by Gilles Brassard and Peter Høyer in 1997, and independently rediscovered by Lov Grover in 1998. In a quantum computer, amplitude amplification can be used to obtain a quadratic speedup over several classical algorithms.
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151) Privately, Gauss referred to it as the "golden theorem."E.g. in his mathematical diary entry for April 8, 1796 (the date he first proved quadratic reciprocity). See facsimile page from Felix Klein's Development of Mathematics in the 19th century He published six proofs for it, and two more were found in his posthumous papers. There are now over 240 published proofs.
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According to Kopell and Ermentrout (2004), a limitation of the theta lies in its relative difficulty in electrically coupling two theta neurons. It is possible to create large networks of theta neurons – and much research has been done with such networks – but it may be advantageous to use Quadratic Integrate-and-Fire (QIF) neurons, which allow for electrical coupling in a "straightforward way".
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U.S. POWs were then able to communicate securely with one another via the quadratic alphabet code. The tune has been used innumerable times as a coda or ending in musical pieces. It is strongly associated with the stringed instruments of bluegrass music, particularly the 5-string banjo. Earl Scruggs often ended a song with this phrase or a variation of it.
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LINDO (Linear, Interactive, and Discrete Optimizer) is a software package for linear programming, integer programming, nonlinear programming, stochastic programming and global optimization.Linus E. Schrage, Linear, Integer, and Quadratic Programming with Lindo, Scientific Press, 1986, Lindo also creates "What'sBest!" which is an add-in for linear, integer and nonlinear optimization. First released for Lotus 1-2-3 and later also for Microsoft Excel.
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Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov functions for physical systems.
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The (strong) torsion conjecture first posed by has been completely resolved in the case of elliptic curves. Barry Mazur proved uniform boundedness for elliptic curves over the rationals. His techniques were generalized by and , who obtained uniform boundedness for quadratic fields and number fields of degree at most 8 respectively. Finally, Loïc Merel () proved the conjecture for elliptic curves over any number field.
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NPSOL is a software package that performs numerical optimization. It solves nonlinear constrained problems using the sequential quadratic programming algorithm. It was written in Fortran by Philip Gill of UCSD and Walter Murray, Michael Saunders and Margaret Wright of Stanford University. The name derives from a combination of NP for nonlinear programming and SOL, the Systems Optimization Laboratory at Stanford.
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Conceptually, in the Levenberg–Marquardt algorithm, the objective function is iteratively approximated by a quadratic surface, then using a linear solver, the estimate is updated. This alone may not converge nicely if the initial guess is too far from the optimum. For this reason, the algorithm instead restricts each step, preventing it from stepping "too far". It operationalizes "too far" as follows.
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In 2016 the London Mathematical Society gave Wolf their Anne Bennett Prize "in recognition of her outstanding contributions to additive number theory, combinatorics and harmonic analysis and to the mathematical community." The award citation particularly cited her work with Gowers on counting solutions to systems of linear equations over abelian groups, and her work on quadratic analogues of the Goldreich–Levin theorem.
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Artin reformulated the reciprocity laws as a statement that the Artin symbol from ideals (or ideles) to elements of a Galois group is trivial on a certain subgroup. Several more recent generalizations express reciprocity laws using cohomology of groups or representations of adelic groups or algebraic K-groups, and their relationship with the original quadratic reciprocity law can be hard to see.
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The whole of Part 2, running to about 307 pages, constitutes just one chapter numbered as Chapter 3 of the book. Some of the topics discussed in this chapter are linear equations with one unknown and with two unknowns, quadratic equations, linear indeterminate equations, solutions of equations of the form Nx2 \+ 1 = y2, indeterminate equations of higher degrees, and rational triangles.
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While explicit solutions can be found for equations that are quadratic, cubic, and quartic in y, the same is not in general true for quintic and higher degree equations, such as : y^5 + 2y^4 -7y^3 + 3y^2 -6y - x = 0. Nevertheless, one can still refer to the implicit solution y = f(x) involving the multi-valued implicit function f.
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The method has links to the method of multipliers and dual ascent method and multiple generalizations exist. One drawback of the method is that it is only provably convergent if the objective function is strictly convex. In case this can not be ensured, as for linear programs or non-strictly convex quadratic programs, additional methods such as proximal gradient methods have been developed.
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It is possible to recognize read-once functions from their disjunctive normal form expressions in polynomial time., Theorem 10.8, p. 541; ; . It is also possible to find a read-once expression for a positive read- once function, given access to the function only through a "black box" that allows its evaluation at any truth assignment, using only a quadratic number of function evaluations.
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There is a proof reducing its security to the computational difficulty of factoring. When the primes are chosen appropriately, and O(log log M) lower-order bits of each xn are output, then in the limit as M grows large, distinguishing the output bits from random should be at least as difficult as solving the Quadratic residuosity problem modulo M.
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Trends do not have to be linear or log-linear. For example, a variable could have a quadratic trend: :Y_t = a \cdot t + c \cdot t^2 + b + e_t. This can be regressed linearly in the coefficients using t and t2 as regressors; again, if the residuals are shown to be stationary then they are the detrended values of Y_t.
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On the Pell equation :Supplement IX. Convergence and continuity of some infinite series This translation does not include Dedekind's Supplements X and XI in which he begins to develop the theory of ideals. The German titles of supplements X and XI are: :Supplement X: Über die Composition der binären quadratische Formen (On the composition of binary quadratic forms) :Supplement XI: Über die Theorie der ganzen algebraischen Zahlen (On the theory of algebraic integers) Chapters 1 to 4 cover similar ground to Gauss' , and Dedekind added footnotes which specifically cross-reference the relevant sections of the . These chapters can be thought of as a summary of existing knowledge, although Dirichlet simplifies Gauss' presentation, and introduces his own proofs in some places. Chapter 5 contains Dirichlet's derivation of the class number formula for real and imaginary quadratic fields.
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Buff's homepage at Université Toulouse III In 2010 he was an invited speaker at the International Congress of Mathematicians in Hyderabad and gave a talk Quadratic Julia Sets with Positive Area based on joint work with Arnaud Chéritat. In 2006 Buff and Chéritat received the Prix Leconte of the French Academy of Sciences for their collaborative work on Julia sets with positive mass; they proved the existence of quadratic polynomials that have positive Lebesgue measure.Jean-Christophe Yoccoz: Ensembles de Julia de mesure positive et disques de Siegel des polynômes quadratiques, d’après X. Buff et A. Chéritat, Séminaire Bourbaki 966, 2006 In 2008 Buff, Chéritat, and Pascale Roesch received a Young Researchers grantProject number ANR-08-JCJC-0002-01, At the Boundary of Chaos from Agence Nationale de la Recherche. In 2009 Buff became a member of the Institut Universitaire de France.
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Hofreiter went to school in Linz and studied from 1923 in Vienna with Hans Hahn, Wilhelm Wirtinger, Emil Müller at the Technische Universität Wien on descriptive geometry, and Philipp Furtwängler, with whom he obtained his doctorate in 1927 on the reduction theory of quadratic forms (Eine neue Reduktionstheorie für definite quadratische Formen). In 1928 he passed the Lehramtsprüfung examination and completed the probationary year as a teacher in Vienna, but then returned to the university (first as a scientific assistant at the TU Vienna) where in 1929 he was assistant to Furtwängler and then habilitated in 1933. He was even then an excellent teacher, and gave lectures not only in Vienna but also in Graz. His dissertation and habilitation thesis dealt with the reduction theory of quadratic forms, which Gauss, Charles Hermite and Hermann Minkowski had worked on previously.
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In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applied to the displacement vector from a centre point is a constant value, with the inclusion of hyperplanes as a limiting case.
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The circuit path embossed on the PET membrane, acts as the voice coil unit. The diaphragm (now, as a unit) is then housed between 4 stacks of steel pole-plate pieces positioned at 45° within a high-intensity, quadratic, opposing magnetic field. The air motion transformer with its sheet film equally exposed at 180° behaves as a dipole speaker, exciting front and rear sonic waves simultaneously.
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Michael Somos is an American mathematician, who was a visiting scholar in the Georgetown University Mathematics and Statistics department for four years and is a visiting scholar at Catholic University of America. In the late eighties he proposed a conjecture about certain polynomial recurrences, now called Somos sequences, that surprisingly in some cases contain only integers. Somos' quadratic recurrence constant is also named after him.
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Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of Milan to Bologna. Lodovico settled in Bologna, and he began his career as the servant of Gerolamo Cardano. He was extremely bright, so Cardano started teaching him mathematics. Ferrari aided Cardano on his solutions for quadratic equations and cubic equations, and was mainly responsible for the solution of quartic equations that Cardano published.
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The trellis walls, roof poles and layers of felt were replaced by stone, brick, beams and planks, and became permanent. Chultem distinguished three styles in traditional Mongolian architecture: Mongolian, Tibetan and Chinese as well as combinations of the three. Among the first quadratic temples was Batu-Tsagaan (1654) designed by Zanabazar. An example of the ger-style architecture is the lamasery Dashi-Choiling in Ulaanbaatar.
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For binary quadratic forms there is a group structure on the set C equivalence classes of forms with given discriminant. The genera are defined by the generic characters. The principal genus, the genus containing the principal form, is precisely the subgroup C2 and the genera are the cosets of C2: so in this case all genera contain the same number of classes of forms.
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According to an inscription attached at the minaret, the mosque was built by Hajji Hüseyin, son of Hajji Ömer during the reign of Ag Qoyunlu Sultan Kasım Han in 1500. Locally, it is also known as the "Kasım Padishah Mosque". It is owned by the General Directorate of Foundations. The mosque is a single-dome, quadratic-plan building having stone masonry walls alternating with brick.
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He has over 120 publications, including co-authorship with Richard Crandall of Prime numbers: a computational perspective (Springer-Verlag, first edition 2001, second edition 2005). He is the inventor of one of the integer factorization methods, the quadratic sieve algorithm, which was used in 1994 for the factorization of RSA-129. He is also one of the discoverers of the Adleman–Pomerance–Rumely primality test.
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The 2D histogram of SDSS SFGs is shown in logarithmic scale and their best likelihood fit is shown by a black solid line. The subset of 62 GPs are indicated by circles and their best linear fit is shown by a dashed line. For comparison we also show the quadratic fit presented in Amorin et al. 2010 for the full sample of 80 GPs.
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Among his best known mathematical works are "Versal deformations of equivariant vector fields for cases of symmetry of order two and three" (Ph.D. thesis, 1979), "On the number of limit cycles in perturbations of quadratic Hamiltonian systems" (joint with I. D. Iliev), "Some functions that generalize the Krall-Laguerre polynomials" (joint with F. A Grünbaum and L. Haine), and "Perturbations of the spherical pendulum and Abelian integrals".
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His most important work is Talqih al-afkar bi rushum huruf al-ghubar (Fertilization of Thoughts with the Help of Dust Letters (Western Arabic Numerals)). It is a book of two hundred folios about (among other things) the science of calculation and geometry. He also wrote three poems (urzaja), one on algebra, one on irrational quadratic numbers and one on the method of false position.
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Scharlau received his doctorate in 1967 from the University of Bonn. His doctoral thesis Quadratische Formen und Galois-Cohomologie (Quadratic Forms and Galois Cohomology) was supervised by Friedrich Hirzebruch. Scharlau was at the Institute for Advanced Study for the academic year 1969–1970 and in spring 1972. From 1970 he was a professor (most recently Institutsdirektor) at the University of Münster, where he has now retired.
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A hyperbola has two vertices, one on each branch; they are the closest of any two points lying on opposite branches of the hyperbola, and they lie on the principal axis. On a parabola, the sole vertex lies on the axis of symmetry and in a quadratic of the form: :ax^2 + bx + c\,\\! it can be found by completing the square or by differentiation., p. 127.
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The Cocks IBE scheme is based on well-studied assumptions (the quadratic residuosity assumption) but encrypts messages one bit at a time with a high degree of ciphertext expansion. Thus it is highly inefficient and impractical for sending all but the shortest messages, such as a session key for use with a symmetric cipher. A third approach to IBE is through the use of lattices.
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The latter three methods can be found in Ramanujan's theory of elliptic functions to alternative bases. The inversion applied in high-precision calculations of elliptic function periods even as their ratios become unbounded. A related result is the expressibility via quadratic radicals of the values of at the points of the imaginary axis whose magnitudes are powers of 2 (thus permitting compass and straightedge constructions).
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For the regression effective degrees of freedom, appropriate definitions can include the trace of the hat matrix,Trevor Hastie, Robert Tibshirani, Jerome H. Friedman (2009), The elements of statistical learning: data mining, inference, and prediction, 2nd ed., 746 p. , , (eq.(5.16)) tr(H), the trace of the quadratic form of the hat matrix, tr(H'H), the form tr(2H – H H'), or the Satterthwaite approximation, .
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The design of SCIP is based on the notion of constraints. It supports about 20 constraint types for mixed-integer linear programming, mixed-integer nonlinear programming, mixed-integer all- quadratic programming and Pseudo-Boolean Pseudo-Boolean challenge 2009 Feb 11, 2011. optimization. It can also solve Steiner Trees and multi-objective optimization problems.A Generic Approach to Solving the Steiner Tree Problem and Variants Nov 9, 2015.
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The gamma factor, with ℝ as base field, builds split-complex numbers as a composition algebra. Reviewing Albert for Mathematical Reviews, N. H. McCoy wrote that there was an "introduction of some new algebras of order 2e over F generalizing Cayley–Dickson algebras."N.H. McCoy (1942) Review of "Quadratic forms permitting composition" by A.A. Albert, Mathematical Reviews #0006140 Taking and corresponds to the algebra of this article.
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In abstract algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components x and y, and is written , where . The conjugate of z is . Since , the product of a number z with its conjugate is , an isotropic quadratic form, . The collection D of all split complex numbers for forms an algebra over the field of real numbers.
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This is indeed correct, because 7 is not a quadratic residue modulo 11. The above sequence of residues : 7, 3, 10, 6, 2 may also be written : −4, 3, −1, −5, 2. In this form, the integers larger than 11/2 appear as negative numbers. It is also apparent that the absolute values of the residues are a permutation of the residues : 1, 2, 3, 4, 5.
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By modifying the method of de Fraysseix et al., found an embedding of any planar graph into a triangular subset of the grid consisting of 4n2/9 points. A universal point set in the form of a rectangular grid must have size at least n/3 × n/3; ; . A weaker quadratic lower bound on the grid size needed for planar graph drawing was given earlier by .
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Quadratic is a collection of four science fiction works by Olaf Stapledon and Murray Leinster. It was edited by William L. Crawford and published in 1953 by Fantasy Publishing Company, Inc. in an edition of 300 copies. The book is an omnibus of Stapledon's Worlds of Wonder and Leinster's Murder Madness, created by combining unbound sheets from the publisher's previous editions of the two volumes.
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The 2-principalization in unramified quadratic extensions of cyclic cubic fields with 2-class group of type (2,2) was investigated by A. Derhem in 1988. Seven years later, M. Ayadi studied the 3-principalization in unramified cyclic cubic extensions of cyclic cubic fields K\subset\Q(\zeta_f), \zeta_f^f=1, with 3-class group of type (3,3) and conductor f divisible by two or three primes.
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In CRS microscopy, we can regard SRS and CARS as two aspects of the same process. CARS signal is always mixed with non-resonant four-wave mixing background and has a quadratic dependence on concentration of chemicals being imaged. SRS has much smaller background and depends linearly on the concentration of the chemical being imaged. Therefore, SRS is more suitable for quantitative imaging than CARS.
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He also introduced an arithmetical approach to the concept of irrationality, as he attributes the following to irrational magnitudes: The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850 - 930) was the first to accept irrational numbers as solutions to quadratic equations or as coefficients in an equation, often in the form of square roots, cube roots and fourth roots.Jacques Sesiano, "Islamic mathematics", p. 148, in .
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This is a trivial modular square root, because 3^2 ot\geq n and so the modulus is not involved when squaring. The integer b_2 = 15 is also Very Smooth Quadratic Residue modulo n. All prime factors are smaller than 7.37 and the Modular Square Root is x_2 = 20 since 20^2 = 400 \equiv 15 (mod n). This is thus a non-trivial root.
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In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class w_2(M) vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H^2(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952.
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Quadratic configuration interaction (QCI) is an extension of configuration interaction that corrects for size-consistency errors in single and double excitation CI methods (CISD). Size-consistency means that the energy of two non-interacting (i.e. at large distance apart) molecules calculated directly will be the sum of the energies of the two molecules calculated separately. This method called QCISD was developed in the group of John Pople.
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Using bubbles to represent scalar (one- dimensional) values can be misleading. The human visual system naturally experiences a disk's size in terms of its area. However, charting software may request the radius or diameter of the bubble as the third data value (after horizontal and vertical axis data). If so, the apparent size differences among the disks will be non-linear (quadratic) and misleading.
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Linear and quadratic programs can be solved via the primal simplex method, the dual simplex method or the barrier interior point method. All mixed integer programming variants are solved by a combination of the branch and bound method and the cutting-plane method. Infeasible problems can be analyzed via the IIS (irreducible infeasible subset) method. Xpress provides a built-in tuner for automatic tuning of control settings.
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As the son of a Swiss horologist, Guillaume took an interest in marine chronometers. For use as the compensation balance he developed a slight variation of the invar alloy which had a negative quadratic coefficient of expansion. The purpose of doing this was to eliminate the "middle temperature" error of the balance wheel. The Guillaume Balance (a type of balance wheel) in horology is named after him.
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In statistics, an efficient estimator is an estimator that estimates the quantity of interest in some “best possible” manner. The notion of “best possible” relies upon the choice of a particular loss function — the function which quantifies the relative degree of undesirability of estimation errors of different magnitudes. The most common choice of the loss function is quadratic, resulting in the mean squared error criterion of optimality.
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It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters; twisting a curve E by a quartic twist, one obtains precisely four curves: one is isomorphic to E, one is its quadratic twist, and only the other two are really new. Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.
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In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It was later rederived by Jean Ville in 1948 as a quadratic (in signal) representation of the local time- frequency energy of a signal,J. Ville, "Théorie et Applications de la Notion de Signal Analytique", Câbles et Transmission, 2, 61–74 (1948). effectively a spectrogram.
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Needham, Volume 3, 105. However, the first books of Euclid to be translated into Chinese was by the cooperative effort of the Italian Jesuit Matteo Ricci and the Ming official Xu Guangqi in the early 17th century.Needham, Volume 3, 106. Yang's writing represents the first in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi.Needham, Volume 3, 46.
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Hanski proposed a random walk model, modulated by the presumed multiplicative effect of reproduction. Hanski's model predicted that the power law exponent would be constrained to range closely about the value of 2, which seemed inconsistent with many reported values. Anderson et al formulated a simple stochastic birth, death, immigration and emigration model that yielded a quadratic variance function. The Lewontin Cohen growth model.
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The tensor \alpha describes the linear magnetoelectric effect, which corresponds to a polarization induced linearly by a magnetic field, and vice versa. The higher terms with coefficients \beta and \gamma describe quadratic effects. For instance, the tensor \gamma describes a linear magnetoelectric effect which is, in turn, induced by an electric field. The possible terms appearing in the expansion above are constrained by symmetries of the material.
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Let K be a quadratic extension of Q, and let be its ring of integers. By extending to a Z-basis, we see that every order O in K has the form for some positive integer c. The conductor of this order equals the ideal cOK. Indeed, it is clear that cOK is an ideal of OK contained in O, so it is contained in the conductor.
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As the people became more sedentary, the temples evolved into multi-angular and quadratic structures. The roof, supported by pillars and walls, served also as the ceiling. Stupa at Erdene Zuu monastery Zanabazar, the first Bogd Gegeen of the Khalkha Mongols, designed many temples and monasteries in traditional Mongolian style and supervised their construction. He merged Oriental architecture with the designs of Mongolian yurts and marquees.
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This formula can be used to calculate the VaR of the portfolio by ignoring higher order terms. Typically cubic or higher terms are truncated. Quadratic terms, when included, can be expressed in terms of (multi-variate) bond convexity. One can make assumptions about the joint distribution of the interest rates and then calculate VaR by Monte Carlo simulation or, in some special cases (e.g.
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Avoiding the huge data transfer requires a suitable (as stated in Overview) computational problem, whose description is short. Dziembowski et al. achieve this by constructing what they call an (m − δ, ε)-uncomputable hash function, which can be computed in quadratic time using memory of size m, but with memory of size m − δ it can be computed with at most a negligible probability ε.
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Every such prime is the sum of a square and twice a square.Gauss, DA Art. 182 Gauss proved Let q = a2 \+ 2b2 ≡ 1 (mod 8) be a prime number. Then :2 is a biquadratic residue (mod q) if and only if a ≡ ±1 (mod 8), and :2 is a quadratic, but not a biquadratic, residue (mod q) if and only if a ≡ ±3 (mod 8).
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In 2001, Cocks developed one of the first secure identity-based encryption (IBE) schemes, based on assumptions about quadratic residues in composite groups. The Cocks IBE scheme is not widely used in practice due to its high degree of ciphertext expansion. However, it is currently one of the few IBE schemes which do not use bilinear pairings, and rely for security on more well-studied mathematical problems.
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In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by . An example is the universal enveloping algebra of a Lie algebra, with generators a basis of the Lie algebra and relations of the form XY – YX – [X, Y] = 0\.
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Brown and Lohmann introduced a technique to calculate computer generated holograms of 3D objects. Calculation of the light propagation from three-dimensional objects is performed according to the usual parabolic approximation to the Fresnel-Kirchhoff diffraction integral. The wavefront to be reconstructed by the hologram is, therefore, the superposition of the Fourier transforms of each object plane in depth, modified by a quadratic phase factor.
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If only the length of the LCS is required, the matrix can be reduced to a 2\times \min(n,m) matrix with ease, or to a \min(m,n)+1 vector (smarter) as the dynamic programming approach only needs the current and previous columns of the matrix. Hirschberg's algorithm allows the construction of the optimal sequence itself in the same quadratic time and linear space bounds.
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The launch of Excel 2007 increased the maximum row limit to 1,048,576. DiffEngineX's row alignment algorithm runs in m log n time, where m and n refer to the number of rows in the two spreadsheets being compared. Typically longest common subsequence problem algorithms run in quadratic time and as such would be ill-suited to comparing spreadsheets with hundreds of thousands of rows.
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In mathematics, a Picard modular group, studied by , is a group of the form SU(J,L), where L is a 3-dimensional lattice over the ring of integers of an imaginary quadratic field and J is a hermitian form on L of signature (2, 1). Picard modular groups act on the unit sphere in C2 and the quotient is called a Picard modular surface.
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If the shear force is linear over an interval, the moment equation will be quadratic (parabolic). Another note on the shear force diagrams is that they show where external force and moments are applied. With no external forces, the piecewise functions should attach and show no discontinuity. The discontinuities on the graphs are the exact magnitude of either the external force or external moments that are applied.
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Brinkmann, Hyperbolic automorphisms of free groups. Geometric and Functional Analysis, vol. 10 (2000), no. 5, pp. 1071-1089 proving that for an automorphism α of Fn the mapping torus group of α is word-hyperbolic if and only if α has no periodic conjugacy classes; a theorem of Bridson and GrovesMartin R. Bridson and Daniel Groves. The quadratic isoperimetric inequality for mapping tori of free-group automorphisms.
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Memoirs of the American Mathematical Society, to appear. that for every automorphism α of Fn the mapping torus group of α satisfies a quadratic isoperimetric inequality; a proof of algorithmic solvability of the conjugacy problem for free-by-cyclic groups;O. Bogopolski, A. Martino, O. Maslakova, E. Ventura, The conjugacy problem is solvable in free-by-cyclic groups. Bulletin of the London Mathematical Society, vol.
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The BG cryptosystem is semantically secure based on the assumed intractability of integer factorization; specifically, factoring a composite value N = pq where p, q are large primes. BG has multiple advantages over earlier probabilistic encryption schemes such as the Goldwasser–Micali cryptosystem. First, its semantic security reduces solely to integer factorization, without requiring any additional assumptions (e.g., hardness of the quadratic residuosity problem or the RSA problem).
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In mathematical finance, the Cheyette Model is a quasi-Gaussian, quadratic volatility model of interest rates intended to overcome certain limitations of the Heath-Jarrow-Morton framework. By imposing a special time dependent structure on the forward rate volatility function, the Cheyette approach allows for dynamics which are Markovian, in contrast to the general HJM model. This in turn allows the application of standard econometric valuation concepts.
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There is a standard technique (see for example ) for computing the change of variables to normal coordinates , at a point as a formal Taylor series expansion. If the coordinates , at (0,0) are locally orthogonal, write : : where , are quadratic and , cubic homogeneous polynomials in and . If and are fixed, and can be considered as formal power series solutions of the Euler equations: this uniquely determines , , , , and .
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The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra.
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In the reform of provincial administration at the end of the 19th century its status was reduced from Mueang to Tambon (commune). Four monuments are located inside the area of 736,000 m² enclosed by a laterite wall. The south wall winds along the Khwae Noi river course, while the other three sides are quadratic. The main monument is in the center of the area.
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Another technique is solution by substitution.Joseph J. Rotman. (2010). Advanced modern algebra (Vol. 114). American Mathematical Soc. Section 1.1 In this technique, we substitute x = y+m into the quadratic to get: :a(y+m)^2 + b(y+m) + c =0\ \ . Expanding the result and then collecting the powers of y produces: :ay^2 + y(2am + b) + \left(am^2+bm+c\right) = 0\ \ .
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Trivially 1 is a quadratic residue for all primes. The question becomes more interesting for −1. Examining the table, we find −1 in rows 5, 13, 17, 29, 37, and 41 but not in rows 3, 7, 11, 19, 23, 31, 43 or 47. The former set of primes are all congruent to 1 modulo 4, and the latter are congruent to 3 modulo 4.
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The quadratic integrate-and-fire (QIF) model was created by Latham et al. in 2000 to explore the many questions related to networks of neurons with low firing rates. It was unclear to Latham et al. why networks of neurons with "standard" parameters were unable to generate sustained low frequency firing rates, while networks with low firing rates were often seen in biological systems.
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In 1992, he computed all solutions to the inverse Fermat equation. The Cohen–Lenstra heuristics is a set of precise conjectures about the structure of class groups of quadratic fields that are partially named after him. Three of his brothers, Arjen Lenstra, Andries Lenstra, and Jan Karel Lenstra, are also mathematicians. Jan Karel Lenstra is the former director of the Netherlands Centrum Wiskunde & Informatica (CWI).
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He estimated the size of the penguin population in Antarctica, and the effect of repealing the motorcycle helmet law in the United States. In 1966 he was elected as a Fellow of the American Statistical Association.View/Search Fellows of the ASA, accessed 2016-08-20. He is sometimes confused with the mathematician G. L. Watson, who worked on quadratic forms, and G. N. Watson, a mathematical analyst.
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The j-invariant of the Tate curve is given by a power series in q with leading term q−1.Silverman (1994) p.423 Over a p-adic local field, therefore, j is non- integral and the Tate curve has semistable reduction of multiplicative type. Conversely, every semistable elliptic curve over a local field is isomorphic to a Tate curve (up to quadratic twist).
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In mathematics, Kronecker's congruence, introduced by Kronecker, states that : \Phi_p(x,y)\equiv (x-y^p)(x^p-y)\bmod p, where p is a prime and Φp(x,y) is the modular polynomial of order p, given by :\Phi_n(x,j) = \prod_\tau (x-j(\tau)) for j the elliptic modular function and τ running through classes of imaginary quadratic integers of discriminant n.
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In mathematics, the Hurwitz class number H(N), introduced by Adolf Hurwitz, is a modification of the class number of positive definite binary quadratic forms of discriminant –N, where forms are weighted by 2/g for g the order of their automorphism group, and where H(0) = –1/12. showed that the Hurwitz class numbers are coefficients of a mock modular form of weight 3/2.
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Ross found Moore's method exciting, and his pedagogy influenced Ross's own. Ross graduated with a B.S. degree and continued his study as Leonard Eugene Dickson's research assistant. Ross earned a M.S. degree and finished his Ph.D. in number theory at the University of Chicago in 1931 with Dickson as his adviser. Ross's dissertation was entitled "On Representation of Integers by Indefinite Ternary Quadratic Forms".
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The tower was 160 meters (525 feet) high. Built in 1966, it was constructed with metal tubing in a quadratic lattice structure and has been designed to handle vibrations and a wind velocity of up to 90 meters per second. 102 meters up, there was a two-story observation deck. Below this, the tower was painted in white and above it in red and white.
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His handwriting is on screen during a scene at the beginning of the film where Katherine Johnson solves a quadratic equation. He appeared on the interview series In the Know. Horne completed a Mathematical Association of America Maths Fest tour where he discussed the mathematics in Hidden Figures, focussing on the calculations that concerned Glenn's orbit around in 1962. He appeared on NPR's Closer Look.
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FS.DF + FS^2 - ES^2 = 0 . Solving the quadratic for DF, in the limit as ES approaches FS, the smaller root, DF = ES - FS . More simply, as DF approaches zero, in the limit the DF^2 term can be ignored: 2\cdot FS\cdot DF + FS^2 - ES^2 = 0 leading to the same result. Clearly df has the same limit, justifying Newton’s claim.
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Classical models are given by the plane sections of a quadratic surface S in real projective 3-space; if S is a sphere, the geometry is called a Möbius plane. The plane sections of a ruled surface (one-sheeted hyperboloid) yield the classical Minkowski plane, cf. for generalizations. If S is an elliptic cone without its vertex, the geometry is called a Laguerre plane.
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The "Nonlinear Order Statistics Filters" were a special case of linear median, order statistics, homomorphic, a-trimmed median, generalized mean, nonlinear mean and fuzzy nonlinear filters. New versions of polynomial filters, such as quadratic filters, were also studied by Professor Venetsanopoulos. He designed new morphological filters, which lead to various detection and recognition applications. Finally, he conducted extensive research in the area of Adaptive filters.
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The simplest chain-of-state method is the linear synchronous transit (LST) method. It operates by taking interpolated points between the reactant and product geometries and choosing the one with the highest energy for subsequent refinement via a local search. The quadratic synchronous transit (QST) method extends LST by allowing a parabolic reaction path, with optimization of the highest energy point orthogonally to the parabola.
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Numerous technical details have to be taken care of to show that this limit exists and is independent of the particular sequence of partitions. Typically, the left end of the interval is used. Important results of Itô calculus include the integration by parts formula and Itô's lemma, which is a change of variables formula. These differ from the formulas of standard calculus, due to quadratic variation terms.
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About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra: :Historically, the first non- associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras... In 1919 Leonard Dickson advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers. He introduced a new imaginary unit , and for quaternions and writes a Cayley number . Denoting the quaternion conjugate by , the product of two Cayley numbers is :(q + Qe)(r + Re) = (qr - R'Q) + (Rq + Q r')e . The conjugate of a Cayley number is , and the quadratic form is , obtained by multiplying the number by its conjugate.
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In optimal control theory, the evolution of n state variables through time depends at any time on their own values and on the values of k control variables. With linear evolution, matrices of coefficients appear in the state equation (equation of evolution). In some problems the values of the parameters in these matrices are not known with certainty, in which case there are random matrices in the state equation and the problem is known as one of stochastic control. A key result in the case of linear-quadratic control with stochastic matrices is that the certainty equivalence principle does not apply: while in the absence of multiplier uncertainty (that is, with only additive uncertainty) the optimal policy with a quadratic loss function coincides with what would be decided if the uncertainty were ignored, this no longer holds in the presence of random coefficients in the state equation.
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A real quadric surface in the Euclidean space of dimension three is a surface that may be defined as the zeros of a polynomial of degree two in three variables. As for the conic sections there are two discriminants that may be naturally defined. Both are useful for getting information on the nature of a quadric surface. Let P(x,y,z) be a polynomial of degree two in three variables that defines a real quadric surface. The first associated quadratic form, Q_4, depends on four variables, and is obtained by homogenizing ; that is :Q_4(x,y,z,t)=t^2P(x/t,y/t, z/t). Let us denote its discriminant by\Delta_4. The second quadratic form, Q_3, depends on three variables, and consists of the terms of degree two of ; that is :Q_3(x,y,z)=Q_4(x, y,z,0). Let us denote its discriminant by\Delta_3.
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The asymptotic behaviour is very good: generally, the iterates xn converge fast to the root once they get close. However, performance is often quite poor if the initial values are not close to the actual root. For instance, if by any chance two of the function values fn−2, fn−1 and fn coincide, the algorithm fails completely. Thus, inverse quadratic interpolation is seldom used as a stand-alone algorithm.
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Kenthaber The tower is situated at the southern side of the place, where the land walls of the city join the sea walls. The structure consists of a circular tower rising on a quadratic pedestal. The tower's gate at the eastern side leads to a small room, from where a narrow staircase goes up. There are signs of restoration work on the upper part done in the Seljuk and Ottoman eras.
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Biquadratic fields are all obtained by adjoining two square roots. Therefore in explicit terms they have the form :K = Q(,) for rational numbers a and b. There is no loss of generality in taking a and b to be non-zero and square-free integers. According to Galois theory, there must be three quadratic fields contained in K, since the Galois group has three subgroups of index 2.
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Kevin Mor McCrimmon (born September 1941) is an American mathematician, specializing in Jordan algebras. He is known for his introduction of quadratic Jordan algebras in 1966. McCrimmon attended secondary school in Champaign- Urbana, Illinois and then received his bachelor's degree in mathematics in 1960 from Reed College in Portland, Oregon. He received his Ph.D. from Yale University in 1965 with thesis Norms and Noncommutative Jordan Algebras supervised by Nathan Jacobson.
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As observes, it can also be seen as an instance of the Davis–Putnam algorithm for solving satisfiability problems using the principle of resolution. Its correctness follows from the more general correctness of the Davis–Putnam algorithm. Its polynomial time bound follows from the fact that each resolution step increases the number of clauses in the instance, which is upper bounded by a quadratic function of the number of variables..
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Hoffstein graduated with a bachelor's degree in 1974 from Cornell University. He received his Ph.D. in 1978 from Massachusetts Institute of Technology with thesis Class numbers of totally complex quadratic extensions of totally real fields under the supervision of Harold Stark. Hoffstein was a postdoc at the Institute for Advanced Study and then at the University of Cambridge. From 1980 to 1982 he was an assistant professor at Brown University.
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The temple Lavrin (18th century) in the Erdene Zuu lamasery was built in the Tibetan tradition. An example of a temple built in the Chinese tradition is the lamasery Choijing Lamiin Sume (1904), which is a museum today. The quadratic temple Tsogchin in lamasery Gandan in Ulaanbaatar is a combination of the Mongolian and Chinese tradition. The temple of Maitreya (disassembled in 1938) is an example of the Tibeto-Mongolian architecture.
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The action for a classical membrane is simply the surface area of the world sheet. The quantum version is harder to write down, is non-linear and very difficult to solve. Unlike the superstring action which is quadratic, the supermembrane action is quartic which makes it exponentially harder. Adding to this the fact that a membrane can represent many particles at once not much progress has been made on supermembranes.
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The first applications of the FPM focused on adaptive compressible flow problems (Fischer, Onate & Idelsohn, 1995; Oñate, Idelsohn & Zienkiewicz, 1995a; Oñate, Idelsohn, Zienkiewicz & Fisher, 1995b). The effects on the approximation of the local clouds and weighting functions were also analyzed using linear and quadratic polynomial bases (Fischer, 1996). Additional studies in the context of convection-diffusion and incompressible flow problems gave the FPM a more solid base; cf.
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According to the center vortex picture, the string tension should depend on the way the matter fields transform under the center, i.e. their so-called N-ality. This seems to be correct for the large-distance string tension, but at smaller distances the string tension is instead proportional to the quadratic Casimir of the representation -- so-called Casimir scaling. This has been explained by domain formation around center vortices.
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The Ruzzo–Tompa algorithm is a linear-time algorithm for finding all non- overlapping, contiguous, maximal scoring subsequences in a sequence of real numbers. This algorithm is an improvement over previously known quadratic time algorithms. The maximum scoring subsequence from the set produced by the algorithm is also a solution to the maximum subarray problem. The Ruzzo–Tompa algorithm has applications in bioinformatics, web scraping, and information retrieval.
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It is a slowly pulsating B star with a frequency of 0.26877 d−1 and an amplitude of 0.0046 magnitude. The averaged quadratic field strength of the star's longitudinal magnetic field is . The star is around 32 million years old and is spinning rapidly with a projected rotational velocity of 264 km/s. It has an estimated 5.5 times the mass of the Sun and 3.8 times the Sun's radius.
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The Fortran subroutine NLPQLP, a newer version of NLPQL, solves smooth nonlinear programming problems by a sequential quadratic programming (SQP) algorithm. The new version is specifically tuned to run under distributed systems. In case of computational errors, caused for example by inaccurate function or gradient evaluations, a non-monotone line search is activated. The code is easily transformed to C by f2c and is widely used in academia and industry.
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Using properties of this function, he generalized Fermat's little theorem to what is now known as Euler's theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. He proved that the relationship shown between even perfect numbers and Mersenne primes earlier proved by Euclid was one-to- one, a result otherwise known as the Euclid–Euler theorem. Euler also conjectured the law of quadratic reciprocity.
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Applying Lagrange's theorem again, we note that there can be no more than values of that make the first factor zero. But as we noted at the beginning, there are at least distinct quadratic residues (mod ) (besides 0). Therefore, they are precisely the residue classes that make the first factor zero. The other residue classes, the nonresidues, must make the second factor zero, or they would not satisfy Fermat's little theorem.
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The discriminant of the conic section's quadratic equation (or equivalently the determinant of the 2×2 matrix) and the quantity (the trace of the 2×2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes,Pettofrezzo, Anthony, Matrices and Transformations, Dover Publ., 1966, p. 110. as is the determinant of the 3×3 matrix above. The constant term and the sum are invariant under rotation only.
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The elements of the natural basis are multiplicative, namely, Tyw=Ty Tw whenever l(yw)=l(y)+l(w), where l denotes the length function on the Coxeter group W. 3\. Elements of the natural basis are invertible. For example, from the quadratic relation we conclude that T = q Ts \+ (q-1). 4\. Suppose that W is a finite group and the ground ring is the field C of complex numbers.
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The nuclear motion Schrödinger equation can be solved in a space-fixed (laboratory) frame, but then the translational and rotational (external) energies are not accounted for. Only the (internal) atomic vibrations enter the problem. Further, for molecules larger than triatomic ones, it is quite common to introduce the harmonic approximation, which approximates the potential energy surface as a quadratic function of the atomic displacements. This gives the harmonic nuclear motion Hamiltonian.
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Parsippany, NJ: Pearson ducation, Inc. as Dale Seymor Publications. ., and popularized by the Persian mathematician Al-Khwarizmi,Khwarizmi, Abu Jafar Muhammad ibn Musa al-, Oxford Islamic Studies Online when Latin translation of his work on the Indian numerals introduced the decimal positional number system to the Western world. His Compendious Book on Calculation by Completion and Balancing presented the first systematic solution of linear and quadratic equations in Arabic.
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If the discriminant of such a polynomial is negative, then both roots of the quadratic equation have imaginary parts. In particular, if b and c are real numbers and b2 − 4c < 0, all the convergents of this continued fraction "solution" will be real numbers, and they cannot possibly converge to a root of the form u + iv (where v ≠ 0), which does not lie on the real number line.
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The contrast distortions are weighted according to their individual visibilities approximated by the HVS. With these models, an objective function that defines the tone curve can be created and solved using a fast quadratic solver. With the addition of filters, this method can also be extended to videos. The filters ensure that the rapid changing of the tone-curve between frames are not salient in the final output image.
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The MSSM Higgs Mass is a prediction of the Minimal Supersymmetric Standard Model. The mass of the lightest Higgs boson is set by the Higgs quartic coupling. Quartic couplings are not soft supersymmetry-breaking parameters since they lead to a quadratic divergence of the Higgs mass. Furthermore, there are no supersymmetric parameters to make the Higgs mass a free parameter in the MSSM (though not in non-minimal extensions).
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The computation of pair-wise interactions between atoms, which is a prerequisite for the operation of many virtual screening programs, is of O(N^{2}) computational complexity, where N is the number of atoms in the system. Because of the quadratic scaling with respect to the number of atoms, the computing infrastructure may vary from a laptop computer for a ligand- based method to a mainframe for a structure-based method.
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Despite his health, Eisenstein continued writing papers on quadratic partitions of prime numbers and the reciprocity laws. In 1851, at the instigation of Gauss, he was elected to the Academy of Göttingen; one year later, this time at the recommendation of Dirichlet, he was also elected to the Academy of Berlin. He died of tuberculosis at the age of 29. Humboldt, then 83, accompanied his remains to the cemetery.
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Semantically secure encryption algorithms include Goldwasser-Micali, El Gamal and Paillier. These schemes are considered provably secure, as their semantic security can be reduced to solving some hard mathematical problem (e.g., Decisional Diffie-Hellman or the Quadratic Residuosity Problem). Other, semantically insecure algorithms such as RSA, can be made semantically secure (under stronger assumptions) through the use of random encryption padding schemes such as Optimal Asymmetric Encryption Padding (OAEP).
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Neodymium isotopes in the Colorado Front Range and crust – mantle evolution in the Proterozoic. Nature 291, 193–197. The initial 143Nd/144Nd ratios of the samples analyzed are plotted on a ɛNd versus time diagram shown in the figure. DePaolo (1981) fitted a quadratic curve to the Idaho Springs and average ɛNd for the modern oceanic island arc data, thus representing the neodymium isotope evolution of a depleted reservoir.
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Fix a field k of characteristic not two. All vector spaces will be assumed to be finite-dimensional. We say that two spaces equipped with symmetric bilinear forms are equivalent if one can be obtained from the other by adding a metabolic quadratic space, that is, zero or more copies of a hyperbolic plane, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector.Milnor & Husemoller (1973) p.
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Store Dyrehave has an almost quadratic shape. Præstevang, an area on the northwestern side of the forest, is bounded by the town of Hillerød on three sides. The small town of Ny Hammersholt and Hillerød Golf Club are located on the southwest side while the northeastern margin of the forest is bounded by the Istedrødvej motorway. To the southeast is the small village of Kirkelte in Allerød Municipality.
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Saeid Abbasbandy is an Iranian mathematicianSelected works and university professor at Imam Khomeini International University. Abbasbandy was born on March 17, 1967 in Tehran. He finished his high school course in Shariati High school and attended in university entrance exam, then could enter University of Tehran. His paper "Homotopy analysis method for quadratic Riccati differential equation" was singled out by Science Watch as a "Hot Paper in Mathematics" in March 2009.
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For a number field F, the group K2(F) is related to class field theory, the Hilbert symbol, and the solvability of quadratic equations over completions. In contrast, finding the correct definition of the higher K-groups of rings was a difficult achievement of Daniel Quillen, and many of the basic facts about the higher K-groups of algebraic varieties were not known until the work of Robert Thomason.
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The points of the upper half-plane τ which correspond to the period ratios of elliptic curves over the complex numbers with complex multiplication are precisely the imaginary quadratic numbers.Silverman (1986) p. 339 The corresponding modular invariants j(τ) are the singular moduli, coming from an older terminology in which "singular" referred to the property of having non-trivial endomorphisms rather than referring to a singular curve.Silverman (1994) p.
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The term "waveguide" is used to describe horns with low acoustic loading, such as conic, quadratic, oblate spheroidal or elliptic cylindrical horns. These are designed more to control the radiation pattern rather than to gain efficiency via improved acoustic loading. All horns have some pattern control, and all waveguides provide a degree of acoustic loading, so the difference between a waveguide and a horn is a matter of judgement.
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Albrecht Pfister (born July 30, 1934) is a German mathematician specializing in algebra and in particular quadratic forms. Albrecht Pfister 1976 Pfister received his doctoral degree in 1961 at the Ludwig Maximilian University of Munich. The title of his doctoral thesis was Über das Koeffizientenproblem der beschränkten Funktionen von zwei Veränderlichen ("On the coefficient problem of the bounded functions of two variables"). His thesis advisors were Martin Kneser and Karl Stein.
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So the longevity and the size of the reference frame are of quadratic relation in this particular case. In this spin-j system, the degradation is due to the loss of purity of the reference frame state. On the other hand, degradation can also be caused by misalignment of background reference. It has been shown, in such case, the longevity has a linear relation with the size of the reference frame.
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The low-mass membrane sheet is suspended within a quadratic magnetic housing, concentrating an intense field around the diaphragm. When signal current passes through the aluminum strips, the ensuing bellows-like motion of the folded pleats moves air five times faster than a conventional cone driver. This rapid acceleration of air-motion is claimed to provide enhanced sound reproduction, including high dynamic range and over an extremely broad frequency range.
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History of the Theory of Numbers is a three-volume work by L. E. Dickson summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various authors, with little further discussion. The central topic of quadratic reciprocity and higher reciprocity laws is barely mentioned; this was apparently going to be the topic of a fourth volume that was never written .
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These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to than any other fraction with the same or a smaller denominator. Because is known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, cannot have a periodic continued fraction.
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For quadratic polynomials \scriptstyle x^2-Px+Q, every Frobenius (P,Q) pseudoprime is also a Lucas (P,Q) pseudoprime. This immediately follows from condition (1) which defined a Lucas (P,Q) pseudoprime. The converse is not true, making the Frobenius pseudoprimes a subset of the Lucas pseudoprimes for a given (P,Q). The condition on \scriptstyle V_k means it is a Dickson pseudoprime of the second kind.
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Modulo 2, every integer is a quadratic residue. Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues, by Euler's criterion. In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field Z/pZ. (In other words, every congruence class except zero modulo p has a multiplicative inverse.
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This implies that there are more quadratic residues than nonresidues among the numbers 1, 2, ..., (q − 1)/2. > For example, modulo 11 there are four residues less than 6 (namely 1, 3, 4, > and 5), but only one nonresidue (2). An intriguing fact about these two theorems is that all known proofs rely on analysis; no-one has ever published a simple or direct proof of either statement.
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The fact that finding a square root of a number modulo a large composite n is equivalent to factoring (which is widely believed to be a hard problem) has been used for constructing cryptographic schemes such as the Rabin cryptosystem and the oblivious transfer. The quadratic residuosity problem is the basis for the Goldwasser-Micali cryptosystem. The discrete logarithm is a similar problem that is also used in cryptography.
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Finsler's lemma is a mathematical result named after Paul Finsler. It states equivalent ways to express the positive definiteness of a quadratic form Q constrained by a linear form L. Since it is equivalent to another lemmas used in optimization and control theory, such as Yakubovich's S-lemma, Finsler's lemma has been given many proofs and has been widely used, particularly in results related to robust optimization and linear matrix inequalities.
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The roots of a polynomial expression of degree n, or equivalently the solutions of a polynomial equation, can always be written as algebraic expressions if n < 5 (see quadratic formula, cubic function, and quartic equation). Such a solution of an equation is called an algebraic solution. But the Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just for some of them) if n \ge 5\.
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He discovered that the so-called Weil representation, previously introduced in quantum mechanics by Irving Segal and David Shale, gave a contemporary framework for understanding the classical theory of quadratic forms. This was also a beginning of a substantial development by others, connecting representation theory and theta functions. He also wrote several books on the history of Number Theory. Weil was elected Foreign Member of the Royal Society (ForMemRS) in 1966.
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It is selected as the glibc (and the derived newlib; str-two-way.h) and musl algorithm for the memmem and strstr family of substring functions. However, as with most advanced string-search algorithms, there tends to be a break-even point in the size of both the haystack and the needle, before which a naive quadratic (memchr-memcmp) implementation is more efficient. Glibc provides the Breslauer algorithm in both forms.
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In geometry, the Euclidean distance is the square root of a quadratic form. Homogeneous polynomials are ubiquitous in mathematics and physics.Homogeneous polynomials in physics often appear as a consequence of dimensional analysis, where measured quantities must match in real-world problems. They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials.
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This star has a stellar classification of B5/7 III/IV, suggesting it is an evolving star that is entering the giant stage. However, according to Zorec and Royer (2012) it is only 56% of the way through its main sequence lifespan. It is a chemically peculiar magnetic B star, showing an averaged quadratic field strength of . Helium-weak, it displays an underabundance of helium in its spectrum.
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If the data exhibit a trend, the regression model is likely incorrect; for example, the true function may be a quadratic or higher order polynomial. If they are random, or have no trend, but "fan out" - they exhibit a phenomenon called heteroscedasticity. If all of the residuals are equal, or do not fan out, they exhibit homoscedasticity. However, a terminological difference arises in the expression mean squared error (MSE).
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They will also learn how to solve systems of equations, as well as how to simplify exponents, quadratic equations, exponential functions, polynomials, radicals, and rational expressions. Other topics included are probability and statistics. Some schools divided Algebra 1 into a two-year sequence. The students who receive it begin with Algebra 1A, and will cover the rest of the Algebra 1 topics in Algebra 1B in the next school year.
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In many settings, such a linear relationship may not hold. For example, if we are modeling the yield of a chemical synthesis in terms of the temperature at which the synthesis takes place, we may find that the yield improves by increasing amounts for each unit increase in temperature. In this case, we might propose a quadratic model of the form : y = \beta_0 + \beta_1x + \beta_2 x^2 + \varepsilon.
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The ring of invariants of a sum of m linear forms and n quadratic forms is generated by m(m–1)/2 + n(n+1)/2 generators in degree 2, nm (m+1)/2 + n(n–1)(n–2)/6 in degree 3, and m(m+1)n(n –1)/4 in degree 4. For the number of generators of the ring of covariants, change m to m+1.
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Algebraic operations in the solution to the quadratic equation. The radical sign, √ denoting a square root, is equivalent to exponentiation to the power of ½. The ± sign means the equation can be written with either a + or with a – sign. In mathematics, a basic algebraic operation is any one of the common operations of arithmetic, which include addition, subtraction, multiplication, division, raising to an integer power, and taking roots (fractional power).
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It is a quadratic building, that consists of a rectangular prayer room, a smaller room for special religious services, an impressive portal that precedes the entrance and a türbe. The walls within the structure are decorated with various frescoes and murals. Built in 1770, the Dollma Teqe stands within the fortification of Krujë and includes a türbe and hamam. The flat dome rests on a low octagonal tholobate.
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Oppenheim's research focused on the ergodic properties of actions of subgroups of semisimple Lie groups. In 1929, Oppenheim's conjecture was published and presented to The National Academy of Sciences. In 1930, Oppenheim was awarded the PhD at the University of Chicago, after defending his thesis, "Minima of Indefinite Quadratic Quaternary Forms". Oppenheim was awarded a second doctorate, a DSc from the University of Oxford for additional academic work.
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In response surface methodology, the objective is to find the relationship between the input variables and the response variables. The process starts from trying to fit a linear regression model. If the P-value turns out to be low, then a higher degree polynomial regression, which is usually quadratic, will be implemented. The process of finding a good relationship between input and response variables will be done for each simulation test.
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In dimension two, this yields a torus, and taking the connected sum of g tori yields the surface of genus g, whose middle homology has the standard hyperbolic form. With additional structure, this ε-symmetric form can be refined to an ε-quadratic form. For doubly even dimension this is integer valued, while for singly even dimension this is only defined up to parity, and takes values in Z/2.
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This shaped his views on corporate culture and his future role as an employer. Goodnight returned to North Carolina State University after working on the Apollo project. He earned a PhD in statistics with a thesis titled Quadratic unbiased estimation of variance components in linear models with an emphasis on the one-way classification under the supervision of Robert James Monroe, and became a faculty member from 1972 to 1976.
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In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley. The Paley construction uses quadratic residues in a finite field GF(q) where q is a power of an odd prime number. There are two versions of the construction depending on whether q is congruent to 1 or 3 (mod 4).
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The Schnorr–Seysen–Lenstra probabilistic algorithm has been rigorously proven by Lenstra and Pomerance to have expected running time L_n\left[\tfrac12,1+o(1)\right] by replacing the GRH assumption with the use of multipliers. The algorithm uses the class group of positive binary quadratic forms of discriminant Δ denoted by GΔ. GΔ is the set of triples of integers (a, b, c) in which those integers are relative prime.
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One implementation of gridless reflectron utilizes a curved field where the electric potential V(x) along the mirror axis depends non-linearly on distance x to the mirror entrance. Time of flight compensation for ions with different kinetic energy can be obtained by adjusting voltage on the elements producing the electric field inside the mirror, which values follow the equation of an arc of a circle: R2 = V(x)2 \+ kx2, where k and R are some constants. The electric potential in some other implementation of gridless reflectron (a so-called quadratic-field reflectron) is proportional to a square of a distance x to the mirror entrance: V(x)= kx2 thus exhibiting a case of one-dimensional harmonic field. If both the ion source and the detector are placed at the reflectron entrance and if the ions travel in a close proximity of the ion mirror axis, the flight times of ions in the quadratic-field reflectron are almost independent on ion kinetic energy.
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The Rado graph may also be formed by a construction resembling that for Paley graphs, taking as the vertices of a graph all the prime numbers that are congruent to 1 modulo 4, and connecting two vertices by an edge whenever one of the two numbers is a quadratic residue modulo the other. By quadratic reciprocity and the restriction of the vertices to primes congruent to 1 mod 4, this is a symmetric relation, so it defines an undirected graph, which turns out to be isomorphic to the Rado graph. Another construction of the Rado graph shows that it is an infinite circulant graph, with the integers as its vertices and with an edge between each two integers whose distance (the absolute value of their difference) belongs to a particular set S. To construct the Rado graph in this way, S may be chosen randomly, or by choosing the indicator function of S to be the concatenation of all finite binary sequences., Section 1.2.
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Instead of passing through points, a different condition on a curve is being tangent to a given line. Being tangent to five given lines also determines a conic, by projective duality, but from the algebraic point of view tangency to a line is a quadratic constraint, so naive dimension counting yields 25 = 32 conics tangent to five given lines, of which 31 must be ascribed to degenerate conics, as described in fudge factors in enumerative geometry; formalizing this intuition requires significant further development to justify. Another classic problem in enumerative geometry, of similar vintage to conics, is the Problem of Apollonius: a circle that is tangent to three circles in general determines eight circles, as each of these is a quadratic condition and 23 = 8\. As a question in real geometry, a full analysis involves many special cases, and the actual number of circles may be any number between 0 and 8, except for 7\.
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Certain mathematical models suggest that until the early 1970s the world population underwent hyperbolic growth (see, e.g., Introduction to Social Macrodynamics by Andrey Korotayev et al.). It was also shown that until the 1970s the hyperbolic growth of the world population was accompanied by quadratic-hyperbolic growth of the world GDP, and developed a number of mathematical models describing both this phenomenon, and the World System withdrawal from the blow-up regime observed in the recent decades. The hyperbolic growth of the world population and quadratic-hyperbolic growth of the world GDP observed till the 1970s have been correlated by Andrey Korotayev and his colleagues to a non-linear second order positive feedback between the demographic growth and technological development, described by a chain of causation: technological growth leads to more carrying capacity of land for people, which leads to more people, which leads to more inventors, which in turn leads to yet more technological growth, and on and on.
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One can also think of adapting this parametrization during the optimization. Should the objective function be based on a norm other than the Euclidean norm, we have to leave the area of quadratic optimization. As a result, the optimization problem becomes more difficult. In particular, when the L^1 norm is used for quantifying the data misfit the objective function is no longer differentiable: its gradient does not make sense any longer.
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In 1873 he wrote an important paper on binary and ternary quadratic forms which was also translated into French and cited by Henri Poincaré, Émile Picard and Paul Gustav Heinrich Bachmann. Beginning with 1877 he also became concerned with insurance, and participated in the reorganization of the pensions in Bavaria on behalf of the Bavarian government. His application for a promotion to professor ordinarius was declined in 1891. In 1906 he became emeritus.
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The Mincer earnings function is a single-equation model that explains wage income as a function of schooling and experience, named after Jacob Mincer. The equation has been examined on many datasets and Thomas Lemieux argues it is "one of the most widely used models in empirical economics". Typically the logarithm of earnings is modelled as the sum of years of education and a quadratic function of "years of potential experience".Lemieux, Thomas.
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From 1899 he was an adjunct professor at the University of Göttingen, where he taught descriptive geometry and oversaw the collection of mathematical equipment. In 1904 he became a professor at the TH Danzig, where he was rector from 1917 to 1919. He retired in 1936. In his dissertation, he developed a new interpretation of the formulas of spherical trigonometry as a relationship between the invariants of three quadratic forms and their functional determinants.
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Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation and the Siegel mass formula for quadratic forms. He was named as one of the most important mathematicians of the 20th century.Pérez, R. A. (2011) A brief but historic article of Siegel, NAMS 58(4), 558–566.
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In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is not a perfect square, it is a realization of a length incommensurable with the unit, such as , , . For more detail, see Quadratic irrational. Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers.
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One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry.
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But when she had an affair with a townsman after marriage, it caused a scandal and the lover was killed for punishment. The princess was banned from home, for she enjoyed lots of sympathy, so she escaped the death penalty. Instead, she might have been exposed in that Utsuro-bune to leave her to destiny. If this should be correct, the quadratic box may contain the head of the woman's deceased lover.
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In Hong Kong, the syllabus of HKCEE additional mathematics covered three main topics, algebra, calculus and analytic geometry. In algebra, the topics covered include mathematical induction, binomial theorem, quadratic equations, trigonometry, inequalities, 2D-vectors and complex number, whereas in calculus, the topics covered include limit, differentiation and integration. In the HKDSE (i.e. the module 2 of mathematics), some new topics are added: matrix and determinant, and an introduction to the Euler's number.
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This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). Complex conjugates are important for finding roots of polynomials. According to the complex conjugate root theorem, if a complex number is a root to a polynomial in one variable with real coefficients (such as a quadratic or a cubic equation), then so is its conjugate.
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The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved. However, proved that elliptic curves defined over real quadratic fields are modular.
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Flahive graduated from St. Joseph's College in New York in 1969. She completed her Ph.D. at the Ohio State University in 1976. Her dissertation, On The Minima Of Indefinite Binary Quadratic Forms, was supervised by Alan C. Woods, and cites the mentorship of another Ohio State mathematician, Jill Yaqub. She published it under the name Mary Flahive Gbur, and some of her journal papers from this period use the name Mary E. Gbur.
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Suppose that the number of puzzles sent by Bob is m, and it takes both Bob and Alice n steps of computation to solve one puzzle. Then both can deduce a common session key within a time complexity of O(m+n). Eve, in contrast, is required to solve all puzzles, which takes her O(mn) of time. If m ≈ n, the effort for Eve has roughly quadratic complexity compared to Alice and Bob.
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First approaches to optimization using adaptive coordinate system were proposed already in the 1960s (see, e.g., Rosenbrock's method). PRincipal Axis (PRAXIS) algorithm, also referred to as Brent's algorithm, is an derivative-free algorithm which assumes quadratic form of the optimized function and repeatedly updates a set of conjugate search directions. The algorithm, however, is not invariant to scaling of the objective function and may fail under its certain rank- preserving transformations (e.g.
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If the points are random variables, then for a narrow but commonly encountered class of probability density functions, this throw-away pre-processing step will make a convex hull algorithm run in linear expected time, even if the worst-case complexity of the convex hull algorithm is quadratic in n.Luc Devroye and Godfried Toussaint, "A note on linear expected time algorithms for finding convex hulls," Computing, Vol. 26, 1981, pp. 361-366.
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In mathematics, the split-octonions are an 8-dimensional nonassociative algebra over the real numbers. Unlike the standard octonions, they contain non-zero elements which are non-invertible. Also the signatures of their quadratic forms differ: the split-octonions have a split signature (4,4) whereas the octonions have a positive-definite signature (8,0). Up to isomorphism, the octonions and the split-octonions are the only two 8-dimensional composition algebras over the real numbers.
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Therefore, the first condition is \delta_0 S_3 +\delta_1 S_2=0. One has to mod out by the trivial solutions that result from nonlinear field redefinitions in the free action. The deformation procedure may not stop at this order and one may have to add quartic terms S_4 and further corrections \delta_2 to the gauge transformations that are quadratic in the fields and so on. The systematic approach is via BV-BRST techniques.
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The advent of tensor calculus in dynamics goes back to Lagrange, who originated the general treatment of a dynamical system, and to Riemann, who was the first to think about geometry in an arbitrary number of dimensions. He was also influenced by the works of Christoffel and of Lipschitz on the quadratic forms. In fact, it was essentially Christoffel’s idea of covariant differentiation that allowed Ricci-Curbastro to make the greatest progress.
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Finding limit cycles, in general, is a very difficult problem. The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of Hilbert's sixteenth problem. It is unknown, for instance, whether there is any system x'=V(x) in the plane where both components of V are quadratic polynomials of the two variables, such that the system has more than 4 limit cycles.
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A US$100 token prize was awarded by RSA Security for the factorization, which was donated to the Free Software Foundation. The value and factorization are as follows: RSA-129 = 114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541 RSA-129 = 3490529510847650949147849619903898133417764638493387843990820577 × 32769132993266709549961988190834461413177642967992942539798288533 The factorization was found using the Multiple Polynomial Quadratic Sieve algorithm. The factoring challenge included a message encrypted with RSA-129. When decrypted using the factorization the message was revealed to be "The Magic Words are Squeamish Ossifrage".
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The numbers with periodic continued fraction expansion are precisely the irrational solutions of quadratic equations with rational coefficients; rational solutions have finite continued fraction expansions as previously stated. The simplest examples are the golden ratio φ = [1;1,1,1,1,1,...] and = [1;2,2,2,2,...], while = [3;1,2,1,6,1,2,1,6...] and = [6;2,12,2,12,2,12...]. All irrational square roots of integers have a special form for the period; a symmetrical string, like the empty string (for ) or 1,2,1 (for ), followed by the double of the leading integer.
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1QBit has divisions focused on universal quantum computing, advanced AI techniques, cloud based quantum processing, and hardware innovation. 1QBit's 1Qloud platform is focused on optimization including reformulating optimization problems into the quadratic unconstrained binary optimization (QUBO) format necessary to compute with quantum annealing processors and similar devices from organizations such as Fujitsu, D-Wave, Hitachi and NTT, while their QEMIST platform is focused on advanced materials and quantum chemistry research with universal quantum computing processors.
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RSA-100 has 100 decimal digits (330 bits). Its factorization was announced on April 1, 1991 by Arjen K. Lenstra. Reportedly, the factorization took a few days using the multiple- polynomial quadratic sieve algorithm on a MasPar parallel computer. The value and factorization of RSA-100 are as follows: RSA-100 = 1522605027922533360535618378132637429718068114961380688657908494580122963258952897654000350692006139 RSA-100 = 37975227936943673922808872755445627854565536638199 × 40094690950920881030683735292761468389214899724061 It takes four hours to repeat this factorization using the program Msieve on a 2200 MHz Athlon 64 processor.
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D-Wave Two (project code name Vesuvius) is the second commercially available quantum computer, and the successor to the first commercially available quantum computer, D-Wave One. Both computers were developed by Canadian company D-Wave Systems. The computers are not general purpose, but rather are designed for quantum annealing. Specifically, the computers are designed to use quantum annealing to solve a single type of problem known as quadratic unconstrained binary optimization.
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This corresponds to a prior belief in small parameter values (and therefore smooth output functions) in a Bayesian framework. RBF networks have the advantage of avoiding local minima in the same way as multi- layer perceptrons. This is because the only parameters that are adjusted in the learning process are the linear mapping from hidden layer to output layer. Linearity ensures that the error surface is quadratic and therefore has a single easily found minimum.
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In a simplistic one-electron model described below, the total energy of an electron is a negative inverse quadratic function of the principal quantum number n, leading to degenerate energy levels for each n 1.Here we ignore spin. Accounting for s, every orbital (determined by n and ℓ) is degenerate, assuming absence of external magnetic field. In more complex systems—those having forces other than the nucleus–electron Coulomb force—these levels split.
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