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1
Appel, H., E. K. U. Gross, and K. Burke. "Double-pole approximation in time-dependent density functional theory." International Journal of Quantum Chemistry 106, no. 14 (2006): 2840–47. http://dx.doi.org/10.1002/qua.20964.
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Moale, Ionela, and Veronika Pillwein. "A note on uniform approximation of functions having a double pole." LMS Journal of Computation and Mathematics 17, no. 1 (2014): 233–44. http://dx.doi.org/10.1112/s1461157013000387.
Повний текст джерелаАнотація:
AbstractWe consider the classical problem of finding the best uniform approximation by polynomials of$1/(x-a)^2,$where$a>1$is given, on the interval$[-\! 1,1]$. First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.
3
Bredenstein, A., S. Dittmaier, and M. Roth. "Four-fermion production at $\gamma\gamma$ colliders: 2. Radiative corrections in double-pole approximation." European Physical Journal C 44, no. 1 (September 2005): 27–49. http://dx.doi.org/10.1140/epjc/s2005-02343-5.
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Denner, A., S. Dittmaier, M. Roth, and D. Wackeroth. "Electroweak radiative corrections to e+e−→WW→4 fermions in double-pole approximation — the RacoonWW approach." Nuclear Physics B 587, no. 1-3 (October 2000): 67–117. http://dx.doi.org/10.1016/s0550-3213(00)00511-3.
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Liu, Quan, Xiang Mu, Wei Huang, Qiming Fu та Yonggang Zhang. "A Sarsa(λ) Algorithm Based on Double-Layer Fuzzy Reasoning". Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/561026.
Повний текст джерелаАнотація:
Solving reinforcement learning problems in continuous space with function approximation is currently a research hotspot of machine learning. When dealing with the continuous space problems, the classicQ-iteration algorithms based on lookup table or function approximation converge slowly and are difficult to derive a continuous policy. To overcome the above weaknesses, we propose an algorithm named DFR-Sarsa(λ) based on double-layer fuzzy reasoning and prove its convergence. In this algorithm, the first reasoning layer uses fuzzy sets of state to compute continuous actions; the second reasoning layer uses fuzzy sets of action to compute the components ofQ-value. Then, these two fuzzy layers are combined to compute theQ-value function of continuous action space. Besides, this algorithm utilizes the membership degrees of activation rules in the two fuzzy reasoning layers to update the eligibility traces. Applying DFR-Sarsa(λ) to the Mountain Car and Cart-pole Balancing problems, experimental results show that the algorithm not only can be used to get a continuous action policy, but also has a better convergence performance.
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Li, Xiao-ming, Xiu-yu Zhang, Hong Cao, Zhong-wei Lin, Yu-guang Niu, and Jian-guo Wang. "Neural Adaptive Decentralized Coordinated Control with Fault-Tolerant Capability for DFIGs under Stochastic Disturbances." Mathematical Problems in Engineering 2017 (2017): 1–16. http://dx.doi.org/10.1155/2017/6271341.
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At present, most methodologies proposed to control over double fed induction generators (DFIGs) are based on single machine model, where the interactions from network have been neglected. Considering this, this paper proposes a decentralized coordinated control of DFIG based on the neural interaction measurement observer. An artificial neural network is employed to approximate the nonlinear model of DFIG, and the approximation error due to neural approximation has been considered. A robust stabilization technique is also proposed to override the effect of approximation error. A H2 controller and a H∞ controller are employed to achieve specified engineering purposes, respectively. Then, the controller design is formulated as a mixed H2/H∞ optimization with constrains of regional pole placement and proportional plus integral (PI) structure, which can be solved easily by using linear matrix inequality (LMI) technology. The results of simulations are presented and discussed, which show the capabilities of DFIG with the proposed control strategy to fault-tolerant control of the maximum power point tracking (MPPT) under slight sensor faults, low voltage ride-through (LVRT), and its contribution to power system transient stability support.
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Freese, Michael, and Stefan Kulig. "Influence of constructional turbo-generator end region design on end winding inductances." Archives of Electrical Engineering 61, no. 2 (June 1, 2012): 199–210. http://dx.doi.org/10.2478/v10171-012-0017-z.
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Influence of constructional turbo-generator end region design on end winding inductancesThe paper introduces a comprehensive investigation in end winding inductances of large two-pole turbo-generators. With the aid of an analytic-numeric approach, where Neumann's formula is applied, the influence of geometric characteristics of double-layer stator end windings with involute shape is analysed. This parameter study results in approximation formulas for the stator self and mutual inductances at strand level as well as for the common used end winding leakage inductance. In order to consider field affecting components as pressure plate, flux shield, rotor shaft and rotor retaining ring, finite elements models for two machines (250 MVA and 1150 MVA) are created and computed. The results are integrated in the developed approximation formulas. Finally the simulation results of machine 1 are compared to the data of two different measurements. All approaches introduced in this paper show good correlation. The high speed of the analytic-numeric calculation is combined with the accuracy and opportunity to consider field affecting components within the extensive finite element computation successfully.
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Ohta, Koji, and Hatsuo Ishida. "Comparison among Several Numerical Integration Methods for Kramers-Kronig Transformation." Applied Spectroscopy 42, no. 6 (August 1988): 952–57. http://dx.doi.org/10.1366/0003702884430380.
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Several numerical integration methods are compared in order to search out the most effective method for the Kramers-Kronig transformation, using the analytical formula of the Kramers-Kronig transformation of a Lorentzian function as a reference. The methods to be compared involve the use of (1) Maclaurin's formula, (2) trapezium formula, (3) Simpson's formula, and (4) successive double Fourier transform methods. It is found that Maclaurin's formula, in which no special approximation is necessary for the pole part of the integration, gives the most accurate results, and also that its computation time is short. Successive Fourier transform is less accurate than the other methods, but it takes the least time when used without zero-filling. These results have important relevance for programs used to obtain optical constant spectra and to analyze spectral data.
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Baglio, Julien, and Duc Ninh Le. "Polarization Observables in WZ Production at the 13 TeV LHC: Inclusive Case." Communications in Physics 30, no. 1 (March 5, 2020): 35. http://dx.doi.org/10.15625/0868-3166/30/1/14461.
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We present a study of the polarization observables of the $W$ and $Z$ bosons in the process \(p p \to W^\pm Z\to e^\pm \nu_e \mu^+\mu^-\) at the 13 TeV Large Hadron Collider. The calculation is performed at next-to-leading order, including the full QCD corrections as well as the electroweak corrections, the latter being calculated in the double-pole approximation. The results are presented in the helicity coordinate system adopted by ATLAS and for different inclusive cuts on the di-muon invariant mass. We define left-right charge asymmetries related to the polarization fractions between the \(W^+ Z\) and \(W^- Z\) channels and we find that these asymmetries are large and sensitive to higher-order effects. Similar findings are also presented for charge asymmetries related to a P-even angular coefficient.
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Osipov, Dmitry, Aleksandr Gusev, Vitaly Shumikhin, and Steffen Paul. "Noise shaping in SAR ADC." Facta universitatis - series: Electronics and Energetics 33, no. 1 (2020): 15–26. http://dx.doi.org/10.2298/fuee2001015o.
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The successive approximation register (SAR) analog-to-digital converter (ADC) is currently the most popular type of ADC architecture, owing to its power efficiency. They are also used in multichannel systems, where power efficiency is of high importance because of the large number of simultaneously working channels. However, the SAR ADC architecture is not the most area efficient. In SAR ADCs, the binary weighted capacitive digital-to-analog converter (DAC) is used, which means that one additional bit of resolution costs double the increase of area. Oversampling and noise shaping are methods that allow an increase in resolution without an increase of area. In this paper we present the new SAR ADC architectures with a noise shaping. A first-order noise transfer function (NTF) with zero located nearly at one can be achieved. We propose two modifications of the architecture: with zero-only NTF and with the NTF with additional pole. The additional pole theoretically increases the efficiency of noise shaping to further 3 dB. The architectures were applied to the design of SAR ADCs in a 65 nm complementary metal-oxide semiconductor (CMOS) with OSR equal to 10. A 6-bit capacitive DAC was used. The proposed architectures provide nearly 4 additional bits in ENOB. The equalent input bandwitdth is equal to 200 kHz with the sampling rate equal to 4 MS/s.
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WOOD, TOBY S., and MICHAEL E. McINTYRE. "Polar confinement of the Sun's interior magnetic field by laminar magnetostrophic flow." Journal of Fluid Mechanics 677 (April 19, 2011): 445–82. http://dx.doi.org/10.1017/jfm.2011.93.
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The global-scale interior magnetic field Bi needed to account for the Sun's observed differential rotation can be effective only if confined below the convection zone in all latitudes including, most critically, the polar caps. Axisymmetric solutions are obtained to the nonlinear magnetohydrodynamic equations showing that such polar confinement can be brought about by a very weak downwelling flow U ~ 10−5cms−1 over each pole. Such downwelling is consistent with the helioseismic evidence. All three components of the magnetic field B decay exponentially with altitude across a thin, laminar ‘magnetic confinement layer’ located at the bottom of the tachocline and permeated by spiralling field lines. With realistic parameter values, the thickness of the confinement layer ~10−3 of the Sun's radius, the thickness scale being the magnetic advection–diffusion scale δ = η/U where the magnetic (ohmic) diffusivity η ≈ 4.1 × 102cm2s−1. Alongside baroclinic effects and stable thermal stratification, the solutions take into account the stable compositional stratification of the helium settling layer, if present as in today's Sun, and the small diffusivity of helium through hydrogen, χ ≈ 0.9 × 101cm2s−1. The small value of χ relative to η produces a double boundary-layer structure in which a ‘helium sublayer‘ of smaller vertical scale (χ/η)1/2δ is sandwiched between the top of the helium settling layer and the rest of the confinement layer. Solutions are obtained using both semi-analytical and purely numerical, finite-difference techniques. The confinement-layer flows are magnetostrophic to excellent approximation. More precisely, the principal force balances are between Lorentz, Coriolis, pressure-gradient and buoyancy forces, with relative accelerations negligible to excellent approximation. Viscous forces are also negligible, even in the helium sublayer where shears are greatest. This is despite the kinematic viscosity being somewhat greater than χ. We discuss how the confinement layers s at each pole might fit into a global dynamical picture of the solar tachocline. That picture, in turn, suggests a new insight into the early Sun and into the longstanding enigma of solar lithium depletion.
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Brion, CE, and KH Tan. "Satellite Intensities in Photoelectron Spectroscopy and Electron Momentum Spectroscopy-Synchrotron Radiation Studies of Argon 3s and Xenon 5s Photoionisation (hv - 60?130 eV)." Australian Journal of Physics 39, no. 5 (1986): 565. http://dx.doi.org/10.1071/ph860565.
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Binding energy spectra for the main line, satellite lines and associated double ionization continuum for argon 3s and xenon 5s photoionisation have been measured in the photon energy range 60-130 eV using monochromatised synchrotron radiation and magic angle photoelectron spectroscopy. In particular the ratio of satellite to main line intensities has been investigated and compared with the results of earlier experiments using X-ray and ultraviolet photoelectron spectroscopy as well as with recent measurements by electron momentum [i.e. binary (e,2e)] spectroscopy. The results of the presently reported photoelectron measurements show that the satellite intensity relative to that of the main ns -1 line increases with increasing photon energy and approaches the value given by electron momentum spectroscopy. These findings are contrary to predictions based on recent calculations of photoelectron intensities. The present work lends support to the direct interpretation of satellite intensities (pole strengths or spectroscopic factors) in binary� (e, 2e) spectroscopy using the plane wave impulse approximation. The need for improved calculations of photoionisation intensities as well as further photoelectron measurements in the X-ray region is stressed.
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Acho, Thomas M., and Dominic P. Clemence. "The parameter dependent Sturm-Liouville eigenproblem with an interior simple or double pole." ANZIAM Journal 43, no. 4 (April 2002): 479–91. http://dx.doi.org/10.1017/s1446181100012098.
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AbstractBoundary value problems where resonance phenomena are studied are most often transformable to parameter dependent Sturm-Liouville (SL) eigenproblems with interior singularities. The parameter dependent Sturm-Liouville eigenproblem with interior poles is examined. Asymptotic approximations to the solutions are obtained using an extended Langer's method to take care of the resulting complex eigenvalues and eigenfunctions.
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DUNSTER, T. M. "UNIFORM ASYMPTOTIC APPROXIMATIONS FOR THE WHITTAKER FUNCTIONS Mκ,iμ(z) AND Wκ,iμ(z)". Analysis and Applications 01, № 02 (квітень 2003): 199–212. http://dx.doi.org/10.1142/s0219530503000119.
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Uniform asymptotic approximations are obtained for the Whittaker's confluent hypergeometric functions Mκ, iμ(z) and Wκ, iμ(z), where κ, μ and z are real. Three cases are considered, and when taken together, result in approximations which are valid for κ → ∞ uniformly for 0 ≤ μ < ∞, 0 < z < ∞, and also for μ → ∞ uniformly for 0 ≤ κ < ∞, 0 < z < ∞. The results are obtained by an application of general asymptotic theories for differential equations either having a coalescing turning point and double pole with complex exponent, or a fixed simple turning point. The resulting approximations achieve a uniform reduction of free variables from three to two, and involve either modified Bessel functions or Airy functions. Explicit error bounds are available for all the approximations.
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Dunster, T. M. "Uniform asymptotic expansions for associated Legendre functions of large order." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 133, no. 4 (August 2003): 807–27. http://dx.doi.org/10.1017/s0308210500002687.
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Uniform asymptotic expansions are obtained for the associated Legendre functions and , and the Ferrers functions and , as the order μ → ∞. The approximations are uniformly valid for 0 ≤ ν + ½ ≤ μ(1 − δ), where δ ∈ (0, 1) is fixed, x ∈ (−1, 1) in the real-variable case and Re z ≥ 0 in the complex-variable case. Explicit error bounds are available for all approximations. In the complex-variable case, expansions are obtained by an application of two existing general asymptotic theories to the associated Legendre differential equation: the first case (in which ν is fixed) applies to regions containing an isolated simple pole; and the second case (in which 0 ≤ ν + ½ ≤ μ(1 − δ)) applies to regions containing a coalescing turning point and double pole. In both cases, the expansions involve modified Bessel functions. In the real-variable case (in which 0 ≤ ν + ½ ≤ μ(1 − δ)), asymptotic expansions of Liouville–Green type are obtained, which involve elementary functions.
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Meisel, David D., Kenneth F. Kinsey, and Charles H. Recchia. "Microcomputers in an Introductory College Astronomy Laboratory: A Software Development Project." International Astronomical Union Colloquium 105 (1990): 175–76. http://dx.doi.org/10.1017/s0252921100086656.
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We have developed software for the Apple IIe series of microcomputers for use in labs in an introductory astronomy course. This software emphasizes a toolkit approach to data analysis; it has been class tested with over 170 students and was a resounding success as a replacement for previously used graphical approximations. A unique feature of this software is the incorporation of image-processing techniques into a course designed for non-science majors.The five software packages are:(a)Datasheet - A six-column spreadsheet with columnwise operations, statistical functions, and double-high-resolution graphics.(b)Image-Processor Program - Allows 37 × 27 pixel × 8 bit video captured images to be manipulated using standard image-processing techniques such as low pass/high pass filtering and histogram equalization.(c)Picture-Processor Program - Allows 256 × 192 bilevel pictures to be manipulated and measured with functions that include calipers, odometer, planimeter, and protractor.(d)Orrery Program - Simulates planet configurations along the ecliptic. A movable cursor allows selection of specific configurations. Since both relative times and angular positions are given, students can deduce the scale of the solar system using simple trigonometry.(e)Plot Program - Allows orbital positions as observed from above the pole to be plotted on the screen. By entering trial values of elliptical orbit parameters, students obtain and the program plots the best fitting ellipse to the data. The sum of the squares of the residuals in the radial coordinate is given after each trial so that students can discover convergence more easily than by simple visual examination of a plot comparing the trial theoretical points with the raw data points.
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Huba, Mikulas, Pavol Bistak, and Damir Vrancic. "Series PIDA Controller Design for IPDT Processes." Applied Sciences 13, no. 4 (February 4, 2023): 2040. http://dx.doi.org/10.3390/app13042040.
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This paper discusses optimal design of the series proportional–integral–derivative–accelerative (PIDA) controller for integral-plus-dead-time (IPDT) plants. The article starts with the design of disturbance reconstruction and compensation based on proportional-derivative-accelerative (PDA) stabilizing controllers. It shows that by introducing positive feedback by a low-pass filter from the (limited) output of the stabilizing PDA controller, one gets disturbance observer (DOB) for the reconstruction and compensation of input disturbances. Thereby, the DOB functionality is based on evaluating steady-state controller output. This DOB interpretation is in full agreement with the results of the analysis of the optimal setting of the stabilizing PDA controller and of its expanded PIDA version with positive feedback from the controller output. By using the multiple real dominant pole (MRDP) method, it confirms that the low-pass filter time constant in positive feedback must be much longer than the dominant time constant of the stabilized loop. This paper also shows that the constrained PIDA controller with the MRDP setting leads to transient responses with input and output overshoots. Experimentally, such a constrained series PIDA controller can be shown as equivalent to a constrained MRDP tuned parallel PIDA controller in anti-windup connection using conditional integration. Next, the article explores the possibility of removing overshoots of the output and input of the process achieved for MRDP tuning by interchanging the parameters of the controller transfer function, which was proven as very effective in the case of the series PID controller. It shows that such a modification of the controller can only be implemented approximately, when the factorization of the controller numerator, which gives complex conjugate zeros, will be replaced by a double real zero. Neglecting the imaginary part and specifying the feedback time constant with a smaller approximative time constant results in the removal of overshoots, but the resulting dynamics will not be faster than for the previously mentioned solutions. A significant improvement in the closed-loop performance can finally be achieved by the optimal setting of the constrained series PIDA controller calculated using the performance portrait method. This article also points out the terminologically incorrect designation of the proposed structure as series PIDA controller, because it does not contain any explicit integral action. Instead, it proposes a more thorough revision of the interpretation of controllers based on automatic reset from the controller output, which do not contain any integrator, but at the same time represent the core of the most used industrial automation. In the end, constrained structures using automatic reset of the stabilizing controller output can ensure a higher performance of transient responses than the usually preferred solutions based on parallel controllers with integral action that, in order to respect the control signal limitation, must be supplemented with anti-windup circuitry. The excellent properties of the constrained series PIDA controller are demonstrated by an example of controlling a thermal process and proven by the circle criterion of absolute stability.
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Xu, Jian, Ning Guo, Hongyu Li, and Youzhi Tu. "On the double-pole solutions of the complex short-pulse equation." Modern Physics Letters B, December 7, 2020, 2150129. http://dx.doi.org/10.1142/s0217984921501293.
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In non-linear optics, it is well known that the non-linear Schrödinger (NLS) equation was always used to model the slowly varying wave trains. However, when the width of optical pulses is in the order of femtosecond ([Formula: see text] s), the NLS equation becomes less accurate. Schäfer and Wayne proposed the so-called short pulse (SP) equation which provided an increasingly better approximation to the corresponding solution of the Maxwell equations. Note that the one-soliton solution (loop soliton) to the SP equation has no physical interpretation as it is a real-valued function. Recently, an improvement for the SP equation, the so-called complex short pulse (CSP) equation, was proposed in Ref. 9. In contrast with the real-valued function in SP equation, [Formula: see text] is a complex-valued function. Since the complex-valued function can contain the information of both amplitude and phase, it is more appropriate for the description of the optical waves. In this paper, the new types of solutions — double-pole solutions — which correspond to double-pole of the reflection coefficient are obtained explicitly, for the CSP equation with the negative order Wadati–Konno–Ichikawa (WKI) type Lax pair by Riemann–Hilbert problem method. Furthermore, we find that the double-pole solutions can be viewed as some proper limits of the soliton solutions with two simple poles.
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Poncelet, Rene, and Andrei Popescu. "NNLO QCD study of polarised W+W− production at the LHC." Journal of High Energy Physics 2021, no. 7 (July 2021). http://dx.doi.org/10.1007/jhep07(2021)023.
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Abstract Longitudinal polarisation of the weak bosons is a direct consequence of Electroweak symmetry breaking mechanism providing an insight into its nature, and is instrumental in searches for physics beyond the Standard Model. We perform a polarisation study of the diboson production in the pp →$$ {\mathrm{e}}^{+}{v}_{\mathrm{e}}{\mu}^{-}{\overline{v}}_{\mu } $$ e + v e μ − v ¯ μ process at NNLO QCD in the fiducial setup inspired by experimental measurements at ATLAS. This is the first polarisation study at NNLO. We employ the double-pole approximation framework for the polarised calculation, and investigate NNLO effects arising in differential distributions.
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Billoni, M., S. Dittmaier, B. Jäger, and C. Speckner. "Next-to-leading order electroweak corrections to pp → W+W− → 4 leptons at the LHC in double-pole approximation." Journal of High Energy Physics 2013, no. 12 (December 2013). http://dx.doi.org/10.1007/jhep12(2013)043.
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Le, Duc Ninh, Julien Baglio, and Thi Nhung Dao. "Doubly-polarized WZ hadronic production at NLO QCD+EW: calculation method and further results." European Physical Journal C 82, no. 12 (December 7, 2022). http://dx.doi.org/10.1140/epjc/s10052-022-11032-2.
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AbstractThe doubly-polarized production of $$W^\pm Z$$ W ± Z pairs at the Large Hadron Collider (LHC) is presented at next-to-leading order (NLO) accuracy both for the electroweak (EW) and QCD corrections, including a detailed description of the calculational method using the double-pole approximation. Numerical results at the 13 TeV LHC are presented in particular for the $$W^- Z$$ W - Z case in the $$e^-\bar{\nu }_e \mu ^+\mu ^-$$ e - ν ¯ e μ + μ - channel using ATLAS fiducial cuts and for polarized distributions defined in the WZ center-of-mass system. The NLO EW corrections relative to the NLO QCD predictions are found to be smaller than 5% in most kinematic distributions, but can reach the level of 10% in some distributions such as lepton transverse momentum distributions or rapidity separation between the electron and the Z boson. EW corrections are not uniform for different polarizations. A comparison between the new ATLAS measurement of polarization fractions to our theoretical prediction is presented.
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Baker, Gregory, Ching Chang, Stefan G. Llewellyn Smith, and D. I. Pullin. "Long-wavelength equations of motion for thin double vorticity layers." Journal of Fluid Mechanics 942 (May 16, 2022). http://dx.doi.org/10.1017/jfm.2022.342.
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We consider the time evolution in two spatial dimensions of a double vorticity layer consisting of two contiguous, infinite material fluid strips, each with uniform but generally differing vorticity, embedded in an otherwise infinite, irrotational, inviscid incompressible fluid. The potential application is to the wake dynamics formed by two boundary layers separating from a splitter plate. A thin-layer approximation is constructed where each layer thickness, measured normal to the common centre curve, is small in comparison with the local radius of curvature of the centre curve. The three-curve equations of contour dynamics that fully describe the double-layer dynamics are expanded in the small thickness parameter. At leading order, closed nonlinear initial-value evolution equations are obtained that describe the motion of the centre curve together with the time and spatial variation of each layer thickness. In the special case where the layer vorticities are equal, these equations reduce to the single-layer equation of Moore (Stud. Appl. Math., vol. 58, 1978, pp. 119–140). Analysis of the linear stability of the first-order equations to small-amplitude perturbations shows Kelvin–Helmholtz instability when the far-field fluid velocities on either side of the double layer are unequal. Equal velocities define a circulation-free double vorticity layer, for which solution of the initial-value problem using the Laplace transform reveals a double pole in transform space leading to linear algebraic growth in general, but there is a class of interesting initial conditions with no linear growth. This is shown to agree with the long-wavelength limit of the full linearized, three-curve stability equations.
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Denner, Ansgar, and Giovanni Pelliccioli. "NLO QCD corrections to off-shell $$ \mathrm{t}\overline{\mathrm{t}}{\mathrm{W}}^{\pm } $$ production at the LHC." Journal of High Energy Physics 2020, no. 11 (November 2020). http://dx.doi.org/10.1007/jhep11(2020)069.
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Abstract We present results of a computation of NLO QCD corrections to the production of an off-shell top-antitop pair in association with an off-shell W+ boson in proton-proton collisions. As the calculation is based on the full matrix elements for the process $$ \mathrm{pp}\to {\mathrm{e}}^{+}{v}_{\mathrm{e}}{\mu}^{-}{\overline{v}}_{\mu }{\tau}^{+}{v}_{\tau}\mathrm{b}\overline{\mathrm{b}} $$ pp → e + v e μ − v ¯ μ τ + v τ b b ¯ , all off-shell, spin-correlation, and interference effects are included. The NLO QCD corrections are about 20% for the integrated cross-section. Using a dynamical scale, the corrections to most distributions are at the same level, while some distributions show much larger K-factors in suppressed regions of phase space. We have performed a second calculation based on a double-pole approximation. While the corresponding results agree with the full calculation within few per cent for integrated cross-sections, the discrepancy can reach 10% and more in regions of phase space that are not dominated by top-antitop production. As a consequence, on-shell calculations should only be trusted to this level of accuracy.