Voters with a strong preference can vote multiple times on a single issue or candidate by abstaining from other votes, but the cost of doing that goes up quadratically: One credit grants one vote, four credits grants two votes, but 400 credits only grants 20 votes.
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Grover's algorithm runs quadratically faster than the best possible classical algorithm for the same task, a linear search.
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As Burr showed, orientations of graphs whose chromatic number grows quadratically as a function of n are universal for polytrees.
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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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Supersymmetry does not address the cosmological constant problem, since supersymmetry cancels the M4Planck contribution, but not the M2Planck one (quadratically diverging).
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These iterations all converge quadratically; that is, each step roughly doubles the number of correct digits. The golden ratio is therefore relatively easy to compute with arbitrary precision.
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When A_i = 0 for i = 1,\dots,m, the SOCP reduces to a linear program. When c_i = 0 for i = 1,\dots,m, the SOCP is equivalent to a convex quadratically constrained linear program. Convex quadratically constrained quadratic programs can also be formulated as SOCPs by reformulating the objective function as a constraint. Semidefinite programming subsumes SOCPs as the SOCP constraints can be written as linear matrix inequalities (LMI) and can be reformulated as an instance of semidefinite program.
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The fuel saving by doing this is very substantial, as a ship's specific fuel consumption increases quadratically with the speed. A slow-steaming vessel may use less than half the fuel of a ship travelling at its "cruise speed".
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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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More generally, any quadratically closed subfield of or will suffice for this purpose (e.g., algebraic numbers, constructible numbers). However, in the cases where it is a proper subfield (i.e., neither nor ), even finite- dimensional inner product spaces will fail to be metrically complete.
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The possible neuron (node) connections increase quadratically as nodes are added to a network. Computation time depends on the number of nodes and their connections, any increase has drastic consequences for processing time. Assigning specific subtasks to individual modules reduce the number of necessary connections.
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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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Multidimensional versions of this method exist. Halley's method exactly finds the roots of a linear-over-linear Padé approximation to the function, in contrast to Newton's method or the Secant method which approximate the function linearly, or Muller's method which approximates the function quadratically.
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By experiments described in his 1954 paper, Bagnold showed that when a shear flow is applied to the suspension, then the shear and normal stresses in the suspension may vary linearly or quadratically with the shear rate, depending on the strength of viscous effects compared to the particles' inertia. If the shear and normal stresses in the mixture (suspension: mixture of solid and fluid) vary quadratically with the shear rate, the flow is said to satisfy Bagnold’s grain-inertia flow. If this relation is linear, then the motion is said to satisfy Bagnold’s macro-viscous flow. These relationships, particularly the quadratic relationship, are referred to as the Bagnold rheology.
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Address into the truncated 16 MSBs, and that plus 1. Linearly interpolate using the 8 MSBs as weights. (One could instead use 3 LUTs instead and quadratically interpolate). This can result in decreased distortion for the same amount of memory at the cost of some multipliers.
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For the coherent addition to be additive, phase-matching must be fulfilled. For tight focusing conditions this is generally not a restriction. Once phase-matching is fulfilled the signal amplitude grows linearly with distance so that the power grows quadratically. This signal forms a collimated beam that is therefore easily collected.
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In a nonlinear poroelastic system, elastic tubes begin straight. When bent, elastic strain increases proportionally with the distance from the initial position. This induces a bending elastic energy per unit of volume that is quadratically related to the transverse radius. The system will lower this elastic energy by squeezing its cross section.
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39 (2011), no. 3, 259–292. Utilizing an argument of Perelman's, Cao and Detang Zhou showed that complete gradient shrinking Ricci solitons have a Gaussian character, in that for any given point of , the function must grow quadratically with the distance function to . Additionally, the volume of geodesic balls around can grow at most polynomially with their radius.
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Metric and similarity learning naively scale quadratically with the dimension of the input space, as can easily see when the learned metric has a bilinear form f_W(x, z) = x^T W z . Scaling to higher dimensions can be achieved by enforcing a sparseness structure over the matrix model, as done with HDSL, and with COMET.
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If the first order stochastic dominance constraint is employed, the utility function u(x) is nondecreasing; if the second order stochastic dominance constraint is used, u(x) is nondecreasing and concave. A system of linear equations can test whether a given solution if efficient for any such utility function. Third-order stochastic dominance constraints can be dealt with using convex quadratically constrained programming (QCP).
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Quantum computers, which are still in the early phases of research, have potential use in cryptanalysis. For example, Shor's Algorithm could factor large numbers in polynomial time, in effect breaking some commonly used forms of public-key encryption. By using Grover's algorithm on a quantum computer, brute-force key search can be made quadratically faster. However, this could be countered by doubling the key length.
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Scalable manufacturing techniques have yet to be developed. In trilayer graphene, the two stacking configurations exhibit very different electronic properties. The region between them consists of a localized strain soliton where the carbon atoms of one graphene layer shift by the carbon–carbon bond distance. The free-energy difference between the two stacking configurations scales quadratically with electric field, favoring rhombohedral stacking as the electric field increases.
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DE431 was created in 2013 and is intended for analysis of earlier historical observations of the Sun, Moon, and planets. It covers a longer time span than DE430 (13201 BC to AD 17191) agreeing with DE430 within 1 meter over the time period covered by DE430. Position of the Moon is accurate within 20 meters between 1913-2113 and that error grows quadratically outside of that range.
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At the other extreme, bubble sort can be viewed as a hill climbing algorithm (every adjacent element exchange decreases the number of disordered element pairs), yet this approach is far from efficient for even modest N, as the number of exchanges required grows quadratically. Hill climbing is an anytime algorithm: it can return a valid solution even if it's interrupted at any time before it ends.
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Just as the electrostatic force, the frictional force scales quadratically with size F ~ L2. Friction is an ever plaguing problem regardless of the scale of a device. It becomes all the more prominent when a device is scaled down. In the nano scale it can wreak havoc if not accounted for because the parts of a Nano-Electro- Mechanical-Systems (NEMS) device are sometimes only a few atoms thick.
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A sequence profile represents a multiple alignment of homologous sequences and describes what amino acids are likely to occur at each position in related sequences. With this method substitution matrices are unnecessary. In addition, there is no need for transition probabilities as a result of the fact that context information is encoded within the context profiles. This makes computation simpler and allows for runtime to be scaled linearly instead of quadratically.
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With this, for later values of one would set X_0 = Z_{k-1}^{-1} and B = Z_k, and then use Z_k^{-1} = X_n for some small n (perhaps just 1), and similarly for Y_k^{-1}. Convergence is not guaranteed, even for matrices that do have square roots, but if the process converges, the matrix Y_k converges quadratically to a square root 1/2, while Z_k converges to its inverse, −1/2.
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A multi-tape Turing machine is a variant of the Turing machine that utilizes several tapes. Each tape has its own head for reading and writing. Initially, the input appears on tape 1, and the others start out blank. This model intuitively seems much more powerful than the single-tape model, but any multi-tape machine—no matter how many tapes—can be simulated by a single-tape machine using only quadratically more computation time.
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The use of a bipolar NRZ drive voltage instead of a DC drive voltage avoids dielectric charging whereas the electrostatic force exerted on the beam is maintained, because the electrostatic force varies quadratically with the DC drive voltage. Electrostatic biasing implies no current flow, allowing high- resistivity bias lines to be used instead of RF chokes. Fig. 2: Electrostatic biasing of a capacitive fixed-fixed beam RF MEMS switch, switched capacitor or varactor.
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The optical design for CELT is a Ritchey-Chretien two-mirror system, with a segmented mirror mosaic with 1080 segments. This rather naturally provides a large, 20 arcminute field of view with less than 0.5 arcsecond images (100% enclosed energy). This focus is free of coma and only suffers from astigmatism, which grows quadratically with field angle. The primary was planned to be in diameter, and for compactness, the primary f-number will be f/1.5.
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The coherence function provides a quantification of deviations from linearity in the system which lies between the input and output measurement sensors. The bicoherence measures the proportion of the signal energy at any bifrequency that is quadratically phase coupled. It is usually normalized in the range similar to correlation coefficient and classical (second order) coherence. It was also used for depth of anasthesia assessment and widely in plasma physics (nonlinear energy transfer) and also for detection of gravitation waves.
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Since the amplitude at these reflections grows linearly with the number N of scatterers, the observed intensity of these spots should grow quadratically, like N2. In other words, using a crystal concentrates the weak scattering of the individual unit cells into a much more powerful, coherent reflection that can be observed above the noise. This is an example of constructive interference. In a liquid, powder or amorphous sample, molecules within that sample are in random orientations.
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Some complex systems, for example, are also complex networks, which have properties such as phase transitions and power- law degree distributions that readily lend themselves to emergent or chaotic behavior. The fact that the number of edges in a complete graph grows quadratically in the number of vertices sheds additional light on the source of complexity in large networks: as a network grows, the number of relationships between entities quickly dwarfs the number of entities in the network.
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Figure 2 shows the Maxwell–Boltzmann distribution for the speeds of the atoms in four noble gases. In this example, the key point is that the kinetic energy is quadratic in the velocity. The equipartition theorem shows that in thermal equilibrium, any degree of freedom (such as a component of the position or velocity of a particle) which appears only quadratically in the energy has an average energy of kBT and therefore contributes kB to the system's heat capacity. This has many applications.
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Such a technique extends the powerful protocol of quantum illumination to its more natural spectral domain, namely microwave wavelengths. In 2019, a three-dimensional enhancement quantum radar protocol was proposed. It could be understood as a quantum metrology protocol for the localization of a non-cooperative point-like target in three-dimensional space. It employed quantum entanglement to achieve an uncertainty in localization that is quadratically smaller for each spatial direction than what could be achieved by using independent, unentangled photons.
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The C matrix in the naive algorithm grows quadratically with the lengths of the sequences. For two 100-item sequences, a 10,000-item matrix would be needed, and 10,000 comparisons would need to be done. In most real-world cases, especially source code diffs and patches, the beginnings and ends of files rarely change, and almost certainly not both at the same time. If only a few items have changed in the middle of the sequence, the beginning and end can be eliminated.
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He completed his habilitation thesis Über quadratisch konvergente Iterationsverfahren zur Lösung von algebraischen Gleichungen und Eigenwertproblemen ("On quadratically convergent iteration methods for solving algebraic equations and eigenvalue problems") in 1954 at the Technical University of Munich. After teaching as privatdozent at Ludwig- Maximilians-Universität from 1954 to 1958, he became extraordinary professor for applied mathematics at the University of Mainz. Since 1963, he worked as a professor of mathematics and (since 1972) computer science at Technical University of Munich. He retired in 1989.
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This property results in the cipher's security degrading quadratically, and needs to be taken into account when selecting a block size. There is a trade-off though as large block sizes can result in the algorithm becoming inefficient to operate. Earlier block ciphers such as the DES have typically selected a 64-bit block size, while newer designs such as the AES support block sizes of 128 bits or more, with some ciphers supporting a range of different block sizes.
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In the simplest arrangement, all routers within a single AS and participating in BGP routing must be configured in a full mesh: each router must be configured as peer to every other router. This causes scaling problems, since the number of required connections grows quadratically with the number of routers involved. To alleviate the problem, BGP implements two options: route reflectors (RFC 4456) and BGP confederations (RFC 5065). The following discussion of basic UPDATE processing assumes a full iBGP mesh.
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The Aberth method, or Aberth–Ehrlich method, named after Oliver Aberth and Louis W. Ehrlich, is a root-finding algorithm developed in 1967 for simultaneous approximation of all the roots of a univariate polynomial. This method converges cubically, an improvement over the Durand–Kerner method, another algorithm for approximating all roots at once, which converges quadratically. (However, both algorithms converge linearly at multiple zeros.) This method is used in MPSolve, which is the reference software for approximating all roots of a polynomial to an arbitrary precision.
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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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DSSS uses rake receivers to compensate for multipath and is used by CDMA systems. DMT uses interleaving and coding to eliminate ISI and is representative of OFDM systems. The analysis was performed by deriving the MIMO channel matrix models for the three modulation schemes, quantifying the computational complexity and assessing the channel estimation and synchronization challenges for each. The models showed that for a MIMO system using QAM with an equalizer or DSSS with a rake receiver, computational complexity grows quadratically as data rate is increased.
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Lowering speed reduces fuel consumption because the force of drag imparted by a fluid increases quadratically with increase in speed. Thus traveling twice as fast requires four times as much energy and therefore fuel for a given distance. The power needed to overcome drag is the product of the force times speed and thus becomes the cube of the speed at high Reynolds numbers. This is why driving an automobile at requires less than 85% of the power required by the same automobile driving at .
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Molpro was designed and maintained by Wilfried Meyer and Peter Pulay in the late 1960s. At that moment, Pulay developed the first analytical gradient code called Hartree-Fock (HF), and Meyer researched his PNO-CEPA (pseudo-natural orbital coupled-electron pair approximation) methods. In 1980, Werner and Meyer developed a new state-averaged, quadratically convergent (MC-SCF) method, which provided geometry optimization for multireference cases. By the same year, the first internally contracted multireference configuration interaction (IC-MRCI) program was developed by Werner and Reinsch.
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About four years later (1984), Werner and Knowles developed on a new generation program called CASSCF (complete active space SCF). This new CASSCF program combined fast orbital optimization algorithms with determinant-based full CI codes, and additional, more general, unitary group configuration interaction (CI) codes. This resulted in the quadratically convergent MCSCF/CASSCF code called MULTI, which allowed modals to be optimized a weighted energy average of several states, and is capable of treating both completely general configuration expansions. In fact, this method is still available today.
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Solving the general case is an NP-hard problem. To see this, note that the two constraints x1(x1 − 1) ≤ 0 and x1(x1 − 1) ≥ 0 are equivalent to the constraint x1(x1 − 1) = 0, which is in turn equivalent to the constraint x1 ∈ {0, 1}. Hence, any 0–1 integer program (in which all variables have to be either 0 or 1) can be formulated as a quadratically constrained quadratic program. Since 0–1 integer programming is NP-hard in general, QCQP is also NP-hard.
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In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty,+\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as [a,b] for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable.
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The quality of the initial values can have a considerable impact on the success or lack of such of the search algorithm. This is because the fitness function or objective function (in many cases a sum of squared errors (SSE)) can have difficult shapes. In some parts of the search region, the function may increase exponentially, in others quadratically, and there may be regions where the function asymptotes to a plateau. Starting values that fall in an exponential region can lead to algorithm failure because of arithmetic overflow.
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The Klee–Minty cube has also inspired research on average-case complexity. When eligible pivots are made randomly (and not by the rule of steepest descent), Dantzig's simplex algorithm needs on average quadratically many steps (on the order of O(D2). Standard variants of the simplex algorithm takes on average D steps for a cube.More generally, for the simplex algorithm, the expected number of steps is proportional to D for linear-programming problems that are randomly drawn from the Euclidean unit sphere, as proved by Borgwardt and by Smale.
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Cancellation of the Higgs boson quadratic mass renormalization between fermionic top quark loop and scalar top squark Feynman diagrams in a supersymmetric extension of the Standard Model The original motivation for proposing the MSSM was to stabilize the Higgs mass to radiative corrections that are quadratically divergent in the Standard Model (hierarchy problem). In supersymmetric models, scalars are related to fermions and have the same mass. Since fermion masses are logarithmically divergent, scalar masses inherit the same radiative stability. The Higgs vacuum expectation value is related to the negative scalar mass in the Lagrangian.
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The IBM ILOG CPLEX Optimizer solves integer programming problems, very large linear programming problems using either primal or dual variants of the simplex method or the barrier interior point method, convex and non-convex quadratic programming problems, and convex quadratically constrained problems (solved via second- order cone programming, or SOCP). The CPLEX Optimizer has a modeling layer called Concert that provides interfaces to the C++, C#, and Java languages. There is a Python language interface based on the C interface. Additionally, connectors to Microsoft Excel and MATLAB are provided.
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The Grothendieck-Witt ring GW is a related construction generated by isometry classes of nonsingular quadratic spaces with addition given by orthogonal sum and multiplication given by tensor product. Since two spaces that differ by a hyperbolic plane are not identified in GW, the inverse for the addition needs to be introduced formally through the construction that was discovered by Grothendieck (see Grothendieck group). There is a natural homomorphism GW → Z given by dimension: a field is quadratically closed if and only if this is an isomorphism.Lam (2005) p.
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The strong electric field of high intensity light (such as output of a laser) may cause a medium's refractive index to vary as the light passes through it, giving rise to nonlinear optics. If the index varies quadratically with the field (linearly with the intensity), it is called the optical Kerr effect and causes phenomena such as self-focusing and self-phase modulation. If the index varies linearly with the field (a nontrivial linear coefficient is only possible in materials that do not possess inversion symmetry), it is known as the Pockels effect.
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The above two criteria are normally applied iteratively until convergence, defined as the point at which no more rotamers or pairs can be eliminated. Since this is normally a reduction in the sample space by many orders of magnitude, simple enumeration will suffice to determine the minimum within this pared-down set. Given this model, it is clear that the DEE algorithm is guaranteed to find the optimal solution; that is, it is a global optimization process. The single-rotamer search scales quadratically in time with total number of rotamers.
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For a single waveplate changing the wavelength of the light introduces a linear error in the phase. Tilt of the waveplate enters via a factor of 1/cos θ (where θ is the angle of tilt) into the path length and thus only quadratically into the phase. For the extraordinary polarization the tilt also changes the refractive index to the ordinary via a factor of cos θ, so combined with the path length, the phase shift for the extraordinary light due to tilt is zero. A polarization-independent phase shift of zero order needs a plate with thickness of one wavelength.
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For this reason, the PML absorption coefficient σ is typically turned on gradually from zero (e.g. quadratically) over a short distance on the scale of the wavelength of the wave. In general, any absorber, whether PML or not, is reflectionless in the limit where it turns on sufficiently gradually (and the absorbing layer becomes thicker), but in a discretized system the benefit of PML is to reduce the finite-thickness "transition" reflection by many orders of magnitude compared to a simple isotropic absorption coefficient. In certain materials, there are "backward- wave" solutions in which group and phase velocity are opposite to one another.
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However, if the polynomial has a real root, which is larger than the larger real root of its derivative, then Newton's method converges quadratically to this largest root if x_0 is larger that this larger root (there are easy ways for computing an upper bound of the roots, see Properties of polynomial roots). This is the starting point of Horner method for computing the roots. When one root has been found, one may use Euclidean division for removing the factor from the polynomial. Computing a root of the resulting quotient, and repeating the process provides, in principle, a way for computing all roots.
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Given a Blum complexity measure (\varphi, \Phi) and a total computable function f with two parameters, then there exists a total computable predicate g (a boolean valued computable function) so that for every program i for g, there exists a program j for g so that for almost all x :f(x, \Phi_j(x)) \leq \Phi_i(x) \, f is called the speedup function. The fact that it may be as fast-growing as desired (as long as it is computable) means that the phenomenon of always having a program of smaller complexity remains even if by "smaller" we mean "significantly smaller" (for instance, quadratically smaller, exponentially smaller).
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If a perturbation set the system into motion somehow, the object would pick up speed exponentially in time, not quadratically. Standing on the surface of the Moon in 1971, David Scott famously repeated Galileo's experiment by dropping a feather and a hammer from each hand at the same time. In the absence of a substantial atmosphere, the two objects fell and hit the Moon's surface at the same time. The first convincing mathematical theory of gravity – in which two masses are attracted toward each other by a force whose effect decreases according to the inverse square of the distance between them – was Newton's law of universal gravitation.
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The first phase of acceleration is called "boost" and the second phase "sustain". Not all dual-thrust motors are in a tandem arrangement but non-tandem motors function much the same; they just have a different physical layout of fuel. For example, they might burn from the inside to the outside (core burning), rather than from the end in (end burning). The advantage of dual-thrust motors is that, if the fuel were all of the fast-burning kind, the rocket would accelerate up to a higher speed initially but because air resistance increases quadratically with speed, the rocket would slow down very rapidly.
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The stop squark is a key ingredient of a wide range of SUSY models that address the hierarchy problem of the Standard Model (SM) in a natural way. A boson partner to the top quark would stabilize the Higgs boson mass against quadratically divergent quantum corrections, provided its mass is close to the electroweak symmetry breaking energy scale. If this was the case then the stop squark would be accessible at the Large Hadron Collider. In the generic R-parity conserving Minimal Supersymmetric Standard Model (MSSM) the scalar partners of right-handed and left-handed top quarks mix to form two stop mass eigenstates.
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Informally, an algorithm can be said to exhibit a growth rate on the order of a mathematical function if beyond a certain input size n, the function times a positive constant provides an upper bound or limit for the run-time of that algorithm. In other words, for a given input size n greater than some n0 and a constant c, the running time of that algorithm will never be larger than . This concept is frequently expressed using Big O notation. For example, since the run-time of insertion sort grows quadratically as its input size increases, insertion sort can be said to be of order O(n2).
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The zero-phonon line is an optical analogy to the Mössbauer lines, which originate in the recoil-free emission or absorption of gamma rays from the nuclei of atoms bound in a solid matrix. In the case of the optical zero-phonon line, the position of the chromophore is the physical parameter that may be perturbed, whereas in the gamma transition, the momenta of the atoms may be changed. More technically, the key to the analogy is the symmetry between position and momentum in the Hamiltonian of the quantum harmonic oscillator. Both position and momentum contribute in the same way (quadratically) to the total energy.
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The most commonly used fluorophores have excitation spectra in the 400-500 nm range, whereas the laser used to excite the two-photon fluorescence lies in the ~700-1000 nm (infrared) range produced by Ti-sapphire lasers. If the fluorophore absorbs two infrared photons simultaneously, it will absorb enough energy to be raised into the excited state. The fluorophore will then emit a single photon with a wavelength that depends on the type of fluorophore used (typically in the visible spectrum). Because two photons are absorbed during the excitation of the fluorophore, the probability for fluorescent emission from the fluorophores increases quadratically with the excitation intensity.
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There are several major uses of the square function in geometry. The name of the square function shows its importance in the definition of the area: it comes from the fact that the area of a square with sides of length is equal to . The area depends quadratically on the size: the area of a shape times larger is times greater. This holds for areas in three dimensions as well as in the plane: for instance, the surface area of a sphere is proportional to the square of its radius, a fact that is manifested physically by the inverse-square law describing how the strength of physical forces such as gravity varies according to distance.
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Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems,Quantum Algorithm Zoo – Stephen Jordan's Homepage including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Pell's equation. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely. However, quantum computers offer polynomial speedup for some problems. The most well- known example of this is quantum database search, which can be solved by Grover's algorithm using quadratically fewer queries to the database than that are required by classical algorithms.
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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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A standard round-robin tournament is used, in which all teams play each other once. Because the number of total games increases quadratically with respect to the number of teams, scheduling too many teams will result in an unwieldy number of games, particularly when there are a limited number of playing surfaces (championship curling arenas usually only have four or five sheets). Therefore, the number of teams is usually capped at around a dozen; if this is not possible or desirable, teams may be separated into groups playing separate round-robins and either having the top teams combining for the Page playoff or playing separate ones in each group and having the winners play each other after.
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Geomipmapping or geometrical mipmapping is a real-time block-based terrain rendering algorithm developed by W.H. de Boer in 2000 that aims to reduce CPU processing time which is a common bottleneck in level of detail approaches to terrain rendering. Prior to geomipmapping, techniques such as quadtree rendering were used to divide the terrain into square tiles created by binary division with quadratically diminishing size. The subdivision step is typically performed on the CPU which creates a bottleneck as geometry commands are buffered to the GPU. Unlike quadtrees which send 1x1 polygon units to the GPU, to reduce the CPU processing time geomipmapping divides the terrain into grid-based tiles which are themselves regularly subdivided.
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Scaling factors also account for the relative disadvantage of the small cyclist in descending, although this is a result of physics, not physiology. A larger rider will be subject to a greater gravitational force because of their greater body mass. Additionally, as mentioned, the frontal area that creates aerodynamic drag increases only quadratically with the rider's size, and hence the larger rider would be expected to accelerate faster or attain a greater terminal velocity. Although these factors might seem to cancel each other out, the climber still has an advantage on a course with long ascents and long descents: adding several miles per hour on a slow, time-consuming climb is much more valuable than the same increase on a fast and brief descent.
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In practice, this generally happens. Firstly, while the sample variance (using Bessel's correction) is an unbiased estimator of the population variance, its square root, the sample standard deviation, is a biased estimate of the population standard deviation; because the square root is a concave function, the bias is downward, by Jensen's inequality. There is no general formula for an unbiased estimator of the population standard deviation, though there are correction factors for particular distributions, such as the normal; see unbiased estimation of standard deviation for details. An approximation for the exact correction factor for the normal distribution is given by using n − 1.5 in the formula: the bias decays quadratically (rather than linearly, as in the uncorrected form and Bessel's corrected form).
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More sophisticated approaches do not attempt to apply a delta function constraint to the gauge transformation degrees of freedom. Instead of "fixing" the gauge to a particular "constraint surface" in configuration space, one can break the gauge freedom with an additional, non-gauge-invariant term added to the Lagrangian density. In order to reproduce the successes of gauge fixing, this term is chosen to be minimal for the choice of gauge that corresponds to the desired constraint and to depend quadratically on the deviation of the gauge from the constraint surface. By the stationary phase approximation on which the Feynman path integral is based, the dominant contribution to perturbative calculations will come from field configurations in the neighborhood of the constraint surface.
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The surgeon uses a pointed or blade shaped electrode called the "active electrode" to make contact with the tissue and exert a tissue effect...vaporization, and its linear propagation called electrosurgical cutting, or the combination of desiccation and protein coagulation used to seal blood vessels for the purpose of Hemostasis. The electric current oscillates between the active electrode and the dispersive electrode with the entire patient interposed between the two. Since the concentration of the RF current reduces with distance from the active electrode the current density rapidly (quadratically) decreases. Since the rate of tissue heating is proportional to the square of current density, the heating occurs in a very localized region, only near the portion of the electrode, usually the tip, near to or in contact with the target tissue.
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Still other braking methods even transform kinetic energy into different forms, for example by transferring the energy to a rotating flywheel. Brakes are generally applied to rotating axles or wheels, but may also take other forms such as the surface of a moving fluid (flaps deployed into water or air). Some vehicles use a combination of braking mechanisms, such as drag racing cars with both wheel brakes and a parachute, or airplanes with both wheel brakes and drag flaps raised into the air during landing. Since kinetic energy increases quadratically with velocity (K=mv^2/2), an object moving at 10 m/s has 100 times as much energy as one of the same mass moving at 1 m/s, and consequently the theoretical braking distance, when braking at the traction limit, is 100 times as long.
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Valve timing gears on a Ford Taunus V4 engine. The balance shaft runs off the small gear on the left (the large gear is for the camshaft, causing it to rotate at half the speed of the crankshaft). Balance shafts are often used in inline-four engines, to reduce the second-order vibration (a vertical force oscillating at twice the engine RPM) that is inherent in the design of a typical inline-four engine. This vibration is generated because the movement of the connecting rods in an even-firing inline-four engine is not symmetrical throughout the crankshaft rotation; thus during a given period of crankshaft rotation, the descending and ascending pistons are not always completely opposed in their acceleration, giving rise to a net vertical force twice in each revolution (which increases quadratically with RPM).
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The connections transfer forces through load-bearing surface contacts, requiring that the characteristic dimensions of the connections scale with the cross section of the attached strut members, t2, because this dimension determines the maximum stress transferable through the joint. These definitions give a cubic scaling relation between the relative mass contribution of the joints and the strut's thickness-to-length ratio (ρc/ρs ∝ Cc(t/l)3, where Cc is the connection contribution constant determined by the lattice geometry). The struts' relative density contribution scales quadratically with the thickness-to-length ratio of the struts (ρm/ρs ∝ Cm (t/l)2), which agrees with the literature on classical cellular materials. Mechanical properties (such as modulus and strength) scale with overall relative density, which in turn scales primarily with the strut and not the connection, considering only open cell lattices with slender struts [t/l < 0.1 (7)], given that the geometric constants Cc and Cm are of the same order of magnitude [ρ/ρs ∝ Cc (t/l)3 + Cm (t/l)2].
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In engineering and economics, many problems involve multiple objectives which are not describable as the-more-the-better or the-less-the-better; instead, there is an ideal target value for each objective, and the desire is to get as close as possible to the desired value of each objective. For example, energy systems typically have a trade-off between performance and cost or one might want to adjust a rocket's fuel usage and orientation so that it arrives both at a specified place and at a specified time; or one might want to conduct open market operations so that both the inflation rate and the unemployment rate are as close as possible to their desired values. Often such problems are subject to linear equality constraints that prevent all objectives from being simultaneously perfectly met, especially when the number of controllable variables is less than the number of objectives and when the presence of random shocks generates uncertainty. Commonly a multi-objective quadratic objective function is used, with the cost associated with an objective rising quadratically with the distance of the objective from its ideal value.
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