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PAPER-TopDiffXS8TeV.bib \addbibresourceATLAS.bib \AtlasTitleMeasurement of the differential cross-section of highly boosted top quarks as a function of their transverse momentum in = 8 \TeV proton–proton collisions using the ATLAS detector \AtlasRefCodeTOPQ-2014-15 \PreprintIdNumberCERN-PH-EP-2015-237 \AtlasJournalPhys. Rev. D. \AtlasAbstractThe differential cross-section for pair production of top quarks with high transverse momentum is measured in 20.3 fb of proton–proton collisions at a center-of-mass energy of 8 \TeV. The measurement is performed for events in the lepton+jets channel. The cross-section is reported as a function of the hadronically decaying top quark transverse momentum for values above 300 \GeV. The hadronically decaying top quark is reconstructed as an anti- jet with radius parameter and identified with jet substructure techniques. The observed yield is corrected for detector effects to obtain a cross-section at particle level in a fiducial region close to the event selection. A parton-level cross-section extrapolated to the full phase space is also reported for top quarks with transverse momentum above 300 \GeV. The predictions of a majority of next-to-leading-order and leading-order matrix-element Monte Carlo generators are found to agree with the measured cross-sections. \pdfstringdefDisableCommands\pdfstringdefDisableCommands

## 1 Introduction

The large number of top–antitop quark () pairs produced at the LHC provide a unique opportunity to improve our understanding of production and test the Standard Model (SM) at the \TeV scale. New phenomena beyond the Standard Model may distort the top quark transverse momentum () spectrum, in particular at high (see, e.g., Refs. [Atwood:1994vm, Englert:2012by]), and could thus be revealed by a precise measurement. Moreover, due to their high cross-section at the LHC and rich experimental signature, events constitute a dominant background to a wide range of searches for new massive particles. A better understanding of the production of high-momentum top quarks, including a more precise determination of the parton distribution functions (PDF) of the proton, would be of great benefit to the broader LHC program.

The initial measurements of production at the LHC have focused on a determination of the inclusive production cross-section. Now that the experimental uncertainties on these measurements (see, e.g., Refs. [Aad:2014kva, Chatrchyan:2012bra, Chatrchyan:2013faa]) are comparable to or lower than the uncertainties on the next-to-next-to-leading-order plus next-to-next-to-leading-logarithmic order (NNLO+NLLL) theory prediction [Cacciari:2011hy, Beneke:2011mq, Baernreuther:2012ws, Czakon:2012zr, Czakon:2012pz, Czakon:2013goa], the interest in differential top quark cross-section measurements has gained traction. Measurements of the differential cross-section as a function of the kinematics of the top quark, or the top–antitop quark pair, have been performed by the ATLAS [Aad:2012hg, Aad:2014zka, Aad:2015eia] and CMS collaborations [Chatrchyan:2012saa, Khachatryan:2015oqa], where the highest measured top quark range is 350-800 \GeV [Aad:2014zka].

In this paper a measurement using techniques specifically designed to deal with the collimated decay topology of highly boosted top quarks is presented. In particular, the hadronic top quark decay is reconstructed as a single large-radius (large-) jet. The selection and reconstruction are based on an algorithm developed [ATL-PHYS-PUB-2010-008] and used in resonance searches [Aad:2013nca, Aad:2012raa, Aad:2015fna, Chatrchyan:2012ku] that increases the selection efficiency at high top quark and extends the kinematic reach into the \TeV range. This analysis utilizes the lepton+jets channel where one boson decays hadronically and the other leptonically to an electron or a muon, assuming each top quark decays to a boson and a -quark. The cross-section is measured as a function of the hadronically decaying top quark . A particle-level cross-section is measured in a kinematic region close to the detector-level selection, referred to in the following as fiducial region. A parton-level differential cross-section is also reported as a function of the hadronically decaying top quark , by further extrapolating to the full kinematic phase space except for a lower limit on top quark of 300 \gev. The measured cross-sections are compared to the predictions of several MC generators and PDF sets.

The object definition, event selection, and background determination used in this analysis follow closely the ones used in the search for resonances [Aad:2015fna]. More details of these aspects of the measurement can be found in the corresponding reference.

## 2 The ATLAS detector

ATLAS is a multipurpose detector [Aad:2008zzm] that provides nearly full solid angle111ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the center of the detector and the -axis along the beam pipe. The -axis points from the IP to the center of the LHC ring, and the -axis points upward. Cylindrical coordinates (,) are used in the transverse plane, being the azimuthal angle around the beam pipe. The pseudorapidity is defined in terms of the polar angle as . coverage around the interaction point. Charged-particle trajectories are reconstructed by the inner detector, which covers pseudorapidity and is composed of a silicon pixel detector, a silicon microstrip detector, and a transition radiation tracker (TRT). The inner detector is surrounded by a solenoid that provides a 2 T magnetic field. Sampling calorimeters with several different designs span the pseudorapidity range up to . High-granularity liquid-argon (LAr) electromagnetic (EM) calorimeters are used up to . Hadronic calorimetry based on scintillator-tile active material covers while LAr technology is utilized for hadronic calorimetry from to . The calorimeters are surrounded by a muon spectrometer. A magnetic field in the spectrometer is provided by air-core toroid magnets. Three layers of precision gas chambers track muons up to = 2.7 and muon trigger chambers cover .

## 3 Data and Monte Carlo samples

The cross-section is measured using data from the 2012 LHC run at = 8 \TeV, which corresponds to an integrated luminosity of

## 4 Object definition and event selection

Jets are reconstructed using the anti- algorithm [Cacciari:2008gp] implemented in the FastJet package [Cacciari:2011ma] with radius parameter or , respectively called small- and large- jets in the following, using as input calibrated topological clusters [topocluster, Aad:2011he, CONF-2015-017]. These clusters are assumed to be massless when computing the jet four-vectors and substructure variables. Large- jets containing hadronically decaying top quarks are selected by applying jet substructure requirements, which exploit the fact that they contain several high- objects and have a high mass, unlike most jets originating from the fragmentation of other quarks or gluons. The trimming algorithm [Krohn:2009th] with parameters and is applied to large- jets to mitigate the impact of initial-state radiation, underlying-event activity, and pileup. A correction for the number of additional interactions per bunch crossing is applied to small- jets [Cacciari:2008gn, Cacciari:2007fd, Ellis:1993tq, Catani:1993hr]. The of small- jets and large- trimmed jets and the large- jet mass, obtained from the four-momentum sum of all jet constituents, are calibrated using energy- and -dependent correction factors. After this calibration, the and mass of the jets in simulated events correspond on average to the ones of the corresponding particle-level jets, which are built from the stable particles produced by the MC event generator [Aad:2014bia, Aad:2013gja]. Differences between the small- jet response in data and MC simulation are evaluated from control samples and corresponding corrections are applied to data. Small- jets are required to be in the fiducial region and must have \GeV. The jet vertex fraction (JVF) is a measure of the fraction of the jet’s track momenta that originate from the primary vertex. It is computed as the summed of all tracks matched to the jet and the primary vertex, divided by the summed of all tracks matched to the jet. Small- jets with \GeV and are rejected when , to reduce the contribution of jets generated by pileup interactions.222The jet is retained if no tracks are assigned to the jet. Trimmed large- jets are considered for the analysis if and  \GeV. More details on the reconstruction and performance of highly boosted top quarks in ATLAS can be found in Ref. [Aad:2013gja, ATLAS-CONF-2015-036].

Small- jets containing a -hadron are tagged using a neural-network-based algorithm (MV1) [CON-2014-046] that combines information from the track impact parameters, secondary vertex location, and decay topology inside the jets. The operating point corresponds to an overall 70% -tagging efficiency in events, and to a probability to mistag light-flavor jets of approximately 1%.

Electron candidates are reconstructed as charged-particle tracks in the inner detector associated with energy deposits in the EM calorimeter. They must satisfy identification criteria based on the shower shape in the EM calorimeter, on track quality, and on the transition radiation observed in the TRT detector [Aad:2014fxa]. Electrons are required to be in the pseudorapidity region , excluding the transition region between the barrel and the endcap calorimeters (). The EM clusters must have a transverse energy \GeV. The associated track must have a longitudinal impact parameter  2 mm with respect to the primary vertex, which is the vertex with the highest of the associated tracks in the event.

Muon candidates are defined by matching track segments in the muon spectrometer with tracks in the inner detector. The track is determined through a global fit of the track that takes into account the energy loss in the calorimeters [Aad:2014rra]. The track is required to have a longitudinal impact parameter 2 mm, and a transverse impact parameter significance , indicating the track is consistent with originating from the hard-scattering vertex. Muons are required to have \GeV and be in the fiducial region .

Lepton candidates are required to be isolated to suppress background leptons originating from jets. The variable “mini-isolation” [Rehermann:2010vq] is used. It is defined as , where is the lepton transverse momentum and the sum is over all good-quality tracks (excluding the lepton track) that have \GeV and a distance from the lepton . The parameter is set to 10 \GeV and the isolation requirement is applied for both the electrons and muons. An isolation cone that decreases in size with increasing improves the selection efficiency of the decay of high- top quarks.

Since leptons deposit energy in the calorimeters, an overlap removal procedure is applied in order to avoid double counting of leptons and small- jets. In order to improve the reconstruction efficiency in the highly boosted topology, the same overlap removal procedure as used in Ref. [Aad:2015fna] has been adopted. First, jets close to electrons, with are corrected by subtracting the electron four-vector from the jet four-vector and the JVF is recalculated after removing the electron track. The new -subtracted jet is retained if it satisfies the jet selection criteria listed above, otherwise it is rejected. After this procedure, electrons that lie within from a small- jet are removed and their four-momentum added back to that of the jet. The muon–jet overlap removal procedure removes muons that fall inside a cone of size around a small- jet axis.

The missing transverse momentum is the magnitude of the vector sum of the transverse energy of all calorimeter cells [Aad:2012re]. Their energy is corrected on the basis of the associated physics object. The contribution of muons is added using their transverse momentum obtained from the tracking system and the muon spectrometer.

The event selection proceeds as follows. Each event must have a reconstructed primary vertex with five or more associated tracks with  \GeV. The events are required to contain exactly one reconstructed lepton candidate with . The transverse mass of the lepton and is defined as , where is the azimuthal angle between the lepton and . Events are retained if \GeV and to suppress QCD multijet events.

The selection exploits the fact that the highly boosted top quark decay products tend to be collimated. Therefore events are selected by requiring the presence of at least one small- jet close to the lepton () and the existence of a reconstructed large- trimmed jet with mass  \GeV. To improve the rejection of background jets, originating from light quarks or gluons, a cut on the splitting scale [Ellis:1993tq, Catani:1993hr] of the large- jets is made. The splitting scale is calculated by reclustering the large- jet with the -clustering algorithm, and taking the distance between the two subjets of the final clustering step to be , where and are the transverse momenta of the two subjets and is the distance between them. It is expected to have large values for jets containing two hard subjets, as expected in the decay of massive objects. Events are selected if the large- jet has  \GeV. The large- jet must be well separated from the lepton () and from the small- jet associated with the lepton (). The leading- trimmed large- jet satisfying these requirements is referred to as the top-jet candidate. Finally, at least one of the two top quark candidates must be -tagged. This implies that either the highest- small- jet close to the lepton () or at least one small- jet close to the large- jet () is -tagged333The reconstruction of a large- jet does not prevent the reconstruction of small- jets overlapping with it..

The event selection is summarized in Table 1. After these requirements the data sample contains 4145 and 3603 events in the electron channel and muon channel, respectively, of which are expected to be semileptonic events.

## 5 Background estimation

After the event selection the background is composed primarily, in order of importance, of +jets,  dilepton, single top, and QCD multijet events. The +jets background is obtained from MC simulation with normalization and heavy-flavor content adjusted in data control regions. The dilepton background is determined as a fraction of the full sample predicted by MC simulation. QCD multijet events are estimated with a fully data-driven method. Single top production as well as minor backgrounds (+jets and diboson) are determined from MC simulation normalized to the best available theoretical calculation of their cross-sections.

The +jets background estimate uses as a starting point the Alpgen+Pythia samples normalized to the inclusive boson NNLO cross-section. The normalization and heavy-flavor fraction of the +jets background have large theoretical uncertainties, and are then determined from data. The overall +jets normalization is obtained by exploiting the expected charge asymmetry in the production of and bosons at a collider [Aad:2012hg, ATLAS:2012an]. This asymmetry is predicted precisely by theory, and other processes in the sample are symmetric in charge except for a small contamination from single top and events, which is corrected by MC simulation. The total number of +jets events in the sample can thus be estimated with the following equation:

 NW++NW−=(rMC+1rMC−1)(D+−D−), (1)

where is the ratio of the number of events with positive leptons to the number with negative leptons in the MC simulation, and and are the number of events with positive and negative leptons in the data, respectively. The signal sample has too few events to apply Eq. (1) directly. Instead a sample enhanced in +jets events is obtained by removing the -tagging, , jet mass, and requirements. The heavy-flavor fraction scale factors correct for potential mismodeling in the generator of the fractions of production associated with different flavor components (, , ). They are estimated in a sample with the same lepton and selections as the signal selection, but with only two small- jets and no -tagging requirements. The -jet multiplicity, in conjunction with knowledge of the -tagging and mistag efficiency, is used to extract the heavy-flavor fraction in this sample. A common scale factor is used for the and components. This information is extrapolated to the signal region using the MC simulation, assuming constant relative rates for the signal and control regions. The overall normalization and heavy-flavor scale factors are extracted iteratively because the various flavor components have different charge asymmetries. After correction the +jets events are expected to make up approximately 5% of the total events in the signal region.

QCD multijet events can mimic the lepton+jets signature. This background is estimated directly from data by using the matrix-method technique [ATLAS-CONF-2014-058]. A sample enhanced in fake leptons, i.e., nonprompt leptons or jets misidentified as prompt leptons, is obtained by loosening the lepton identification requirements. The number of events with fake leptons in the signal region can be predicted as: {linenomath}

 Nmultijet=(ϵ−1)fϵ−fNT+ϵfϵ−fNL,

where and are the efficiencies for leptons that passed the loose selections to also pass the tight (signal) selections, for real and fake leptons respectively, is the number of events with a tight lepton, and is the number of events with a loose lepton that failed the tight cuts. The efficiency is measured using data in fake-lepton-enhanced control regions and is extracted from MC simulation and validated in data. QCD multijet events contribute to the total event yield at approximately the percent level.

Top quark pair events with both the top and antitop quarks decaying leptonically (including decays to ) can sometimes pass the event selection, contributing approximately 5% of the total event yield, and are treated as background in the analysis. The fraction of dileptonic events in each bin is estimated using the same MC sample used to model the signal.

## 6 Systematic uncertainties

Systematic uncertainties, which arise from object reconstruction and calibration, MC generator modeling, and background estimation, are described below. The propagation of systematic uncertainties through the unfolding procedure is described in Sec. 8.4.

### 6.1 Detector modeling

The uncertainty on the large- jet energy scale (JES), jet mass scale (JMS), and splitting scale is obtained using two different data-driven methods. For \GeV for JES, and for all for the JMS and splitting scale, the ratio of the large- jets kinematic variables reconstructed from the calorimeter clusters to those from inner-detector tracks is compared between data and MC simulation. For \GeV for JES, the of large- jets are compared to the well-calibrated of photons in a large sample of photon+jets events. An additional MC-based uncertainty, referred to as large- JES topology uncertainty, is included to reflect the fact that the jets in these calibration samples have a different response (gluon or light-quark jets) than those in events (top-jets). The full difference between the response to these two types of jets is conservatively assigned as the corresponding systematic uncertainty. The uncertainty on the large- jet energy resolution (JER) is determined by smearing the jet energy such that the resolution is degraded by 20% [ATLAS:2012am, Aad:2012ag] and evaluating the effect on the final result. The same smearing procedure is applied to determine the uncertainty due to the large- jet mass resolution (JMR). The uncertainties on the large- jets JES are the dominant contribution to the total uncertainty of this measurement, in particular the topology and photon+jet calibration uncertainties.

The small- jet energy scale uncertainty is derived using a combination of simulations, test beam data, and in situ measurements [Aad:2014bia, Aad:2011he, Aad:2012vm]. Additional contributions from the jet flavor composition, calorimeter response to different jet flavors, and pileup are taken into account. Uncertainties in the jet energy resolution are obtained with an in situ measurement of the jet balance in dijet events [JER].

The efficiency to tag -jets and mistag light jets is corrected in Monte Carlo events by applying -tagging scale factors, extracted in and dijet samples, that compensate for the residual difference between data and simulation. The associated systematic uncertainty is computed by varying the scale factors within their uncertainty [ATLAS-CONF-2014-004, CON-2012-043, CONF-2012-040]. The -jet calibration is performed for jets with up to 300 \GeV; for larger transverse momenta an additional MC-based extrapolation uncertainty is applied, which ranges from approximately 10% to 30%, increasing with -jet from 300 \GeV to 1200 \GeV.

The lepton reconstruction efficiency in simulation is corrected by scale factors derived from measurements of these efficiencies in data using enriched control regions. The lepton trigger and reconstruction efficiency scale factors, energy scale, and energy resolution are varied within their uncertainties [Aad:2011mk, Aad:2014rra].

The uncertainty associated with is calculated by propagating the energy scale and resolution systematic uncertainties on all physics objects to the calculation. Additional uncertainties arising from energy deposits not associated with any reconstructed objects are also included [Aad:2012re].

The uncertainty on the integrated luminosity is 2.8% and is derived following a methodology similar to that defined in Ref. [Aad:2013ucp].

### 6.2 Signal and background modeling

The parton shower and hadronization uncertainty is computed by comparing the results obtained with Powheg+Pythia (without electroweak corrections applied) and Powheg+Herwig. The generator uncertainty is evaluated by taking the difference between the results obtained with Powheg+Herwig and MC@NLO+Herwig. Both uncertainties are symmetrized. The procedure to compute the PDF uncertainty on the signal is based on the PDF4LHC recommendations [Botje:2011sn] using the MC@NLO+Herwig sample with three different PDF sets (CT10 [Lai:2010vv], MSTW [Martin:2009iq] and NNPDF [Ball:2012cx]). An intra-PDF uncertainty is obtained for each PDF set by following its respective prescription while an inter-PDF uncertainty is computed as the envelope of the three intra-PDF uncertainties. The modeling of ISR and FSR is evaluated separately using dedicated AcerMC + Pythia samples with variation of the Pythia parameters for QCD radiation.

The +jets shape uncertainty is extracted by varying the renormalization and matching scales in Alpgen. The +jets MC statistical uncertainty is also computed and its contribution to the cross-section uncertainty increases with the top-jet candidate from approximately 1% to 6%. A new set of +jets normalization and heavy-flavor scale factors is extracted for each variation of the most important detector modeling uncertainties, allowing their correlated effect on the +jets background, signal and background, and other MC-based background processes to be assessed.

The uncertainty on the fake-lepton background is determined by varying the definition of loose leptons, changing the selection used to form the fake-enhanced control region, and propagating the statistical uncertainty of parameterizations of the efficiency and the fake rate.

The single-top background is assigned an uncertainty associated with the theory calculations used for its normalization [Kidonakis:2011wy, Kidonakis:2010ux, Kidonakis:2010tc]. A generator uncertainty is included for the channel, which provides the largest single-top contribution, by taking the difference between the yields predicted by Powheg and MC@NLO. An uncertainty on the interference between the and processes is also included. A conservative uncertainty of 50% is applied to the normalization of the subdominant +jets and diboson backgrounds.

## 7 Data and MC comparison at detector level

Table 2 gives the number of observed and expected events for each process, where the systematic uncertainties on the background estimates, objects’ energy scales and reconstruction efficiencies, and MC statistics are taken into account. The prediction is generally found to overestimate the data by approximately one standard deviation.

Agreement of the data with the prediction is further tested by studying the distributions of several variables of interest in Fig. LABEL:fig:reco_all. The systematic uncertainties on the objects’ energy scales and reconstruction efficiencies, on the background estimates, luminosity and MC statistics are shown. While the prediction generally overestimates the data, as already seen in Table 2, the simulation reproduces the observed shapes in most cases. Exceptions include the tails of some kinematic variables such as the top-jet candidate . The distribution of the top-jet candidate constitutes the input to the unfolding procedure and is studied in more detail in the following sections.

## 8 Differential cross-section determination

Differential cross-sections are measured as a function of the \pt of the top-jet candidate at particle level and the of the top quark at parton level. The electron and muon channels are first combined into a +jets sample at the detector level. The detector-level spectrum is corrected for detector inefficiencies and finite resolution to obtain particle- and parton-level differential cross-sections. The particle-level measurement is performed in a specific fiducial region of phase space close to the event selection. The systematic and statistical uncertainties are propagated through the unfolding procedure. Finally a covariance matrix is computed to perform a quantitative comparison of the measured cross-sections with MC predictions.

### 8.1 Combination of decay channels

The +jets and +jets selections are combined into a +jets sample at the detector level. The combined +jets signal and background samples take into account the efficiencies of the two selections. This procedure is well motivated given that the relative yields of the two channels agree well between data and MC simulation, as shown in Table 2. The combination method is cross-checked by performing the unfolding in each channel individually to the +jets phase space described in Sec. 8.2 and comparing these alternative cross-section estimates with the one based on the combined data. The final results are found to be consistent.

### 8.2 Particle- and parton-levels fiducial region definitions

Particle-level corrections to the data are derived from leptons and jets in simulated \ttbar events that are constructed using stable particles, with a mean lifetime greater than seconds, which result directly from the hard-scattering interaction or from subsequent decays of particles with a shorter lifetime.

All leptons (, , ,, ) not from hadron decays are considered as prompt isolated leptons. The leptons from decays are accepted only if the parent is not a hadron decay product itself. The four-momenta of photons within a cone of = 0.1 around the electron or muon direction are added to those of the leptons (dressed leptons). Both the small- and large- jets are reconstructed using all stable particles except for the selected dressed leptons. The trimming procedure applied to detector-level jets is also applied to particle-level jets. A small- jet with \pt  25 \GeV and is considered to be “-tagged” if there exists at least one -hadron with \GeV clustered in the jet.444The -hadrons are not stable and do not contribute to the total four-vector of the jet, only their decay products do. However, they are clustered with their energy set to a negligible value to check that they match the jet geometrically [Cacciari:2008gn].

The missing transverse momentum is the magnitude of the vector sum of the momenta of neutrinos not resulting from hadron decays.

To minimize the theoretical input to the measurement, the fiducial region is chosen to follow the detector-level event selections closely, including the kinematic requirements on the objects and the requirements on the event topology. In contrast to the detector-level selection, no overlap removal procedure is applied to the leptons and jets, and no isolation requirement is imposed on the leptons. Using the particle-level objects defined above, the fiducial region is defined by requiring:

• Exactly one lepton (electron or muon) with 25 \GeV, 2.5.

• \Etmiss

20 \GeV and \Etmiss+\mtw 60 \GeV.

• At least one small- jet with 25 \GeV, 2.5, and a distance 1.5 from the lepton. If there is more than one such jet, the one with the largest \pt is considered to be the leptonic -jet candidate (the -jet associated to the leptonic top quark decay).

• At least one trimmed large- jet with 300 \GeV, mass 100 \GeV, 40 \GeV, and 2, well separated from both the lepton ( 2.3) and the leptonic -jet candidate ( 1.5). The jet mass is reconstructed from the four-vector sum of the particles constituting the jet. If more than one large- jet satisfies these criteria, the one with largest \pt is chosen. The jet passing this selection is referred to as the particle-level top-jet candidate.

• At least one -tagged small- jet such that and/or the leptonic -jet candidate is -tagged.

The particle-level event selection is summarized in Table 1. Fiducial particle-level corrections are determined by using only simulated \ttbar events in which exactly one of the bosons, resulting from the decay of the \ttbar pair, decays to an electron or a muon either directly or through a lepton decay. All other \ttbar events are not used. The cross-section is then determined as a function of the particle-level top-jet candidate transverse momentum, \ptptcl.

For the parton level, the top quark that decays to a hadronically decaying boson is considered just before the decay and after QCD radiation, selecting events in which the momentum of such a top quark, \ptptn, is larger than 300 \GeV. Parton-level corrections are determined by using only simulated \ttbar events in which exactly one of the boson decays to an electron or a muon or a lepton (including hadronic decays). The correction to the full parton-level phase space defined above is obtained by accounting for the branching ratio of \ttbar pairs to the \ljetschannel.

### 8.3 Unfolding to particle and parton levels

The procedure to unfold the distribution of \ptreco, the \pt of the detector-level leading- trimmed large- jet, to obtain the differential cross-section as a function of \ptptcl is composed of several steps, outlined in:

 \dsigma\ttbard\ptptcl(\ptptcli) = NiptclΔ\ptptcli\lum (2) = 1Δ\ptptcli\lumfiptcl!reco⋅∑jM−1ijfjreco!ptclf\ttbar,ℓ+jets(Njreco−Njreco,bgnd),

where is the number of observed events in bin of \ptreco with the detector-level selection applied, is the total number of events in bin of \ptptcl that meet the fiducial region selection, is the size of bin of \ptptcl, and is the integrated luminosity of the data sample. The corrections that are applied to \ptreco are all extracted from the nominal Powheg+Pythia sample.

First, the non- background contamination, , is subtracted from the observed number of events in each \ptreco bin. The contribution from non-\lplus \ttbar events is taken into account by the multiplicative correction , which represents the fraction of +jets events extracted from the nominal Powheg+Pythia sample.

In a second step the correction factor , also referred to as acceptance correction, corrects the \ptreco spectrum for the events that pass the detector-level selection but fail the particle-level selection. For each \ptreco bin , is defined as the ratio of the number of events that meet both the detector-level and particle-level selections to the number of events that satisfy the detector-level selection. The distribution of the acceptance correction is shown in Fig. LABEL:fig:fakecorr(a) for various MC generators.

The third step corrects for detector resolution effects. A migration matrix is constructed to correlate the \ptreco-binned distribution to the \ptptcldistribution. The matrix represents the probability for an event with \ptptcl in bin to have a \ptreco in bin . This matrix is shown in Fig. LABEL:fig:migration(a). It shows that approximately 50% to 85% of events have values of \ptptcl and of \ptreco that fall in the same bin.

The inversion of the migration matrix to correct \ptreco to the particle level is carried out by an unfolding scheme based on Tikhonov regularization which is implemented through the singular value decomposition (SVD) of the matrix [svd]. This scheme is chosen to reduce sizable statistical fluctuations that are introduced by instabilities in the inversion procedure. The unfolding regularization parameter, which characterizes the size of the expansion of the solution to the inversion problem, is optimized according to the procedure described in Ref. [svd]. In parallel the bin size for the \ptptcl (and \ptreco) distribution is optimized such that systematic uncertainties are larger than statistical uncertainties in each bin, and such that the width of each bin corresponds to at least one and a half times the expected resolution in that bin. The former requirement is introduced to minimize statistical fluctuations when estimating systematic uncertainties. The typical expected fractional resolution for \ptreco in \ttbar simulated events ranges from 7% to 3% for \ptreco values between 250 \GeV and 1.2 \TeV. Finally, the optimization requires the unfolding to be unbiased, i.e., that a given input \ptptcl spectrum is recovered on average by the unfolding procedure. After rounding to the nearest 50 \GeV, this procedure results in bin widths of 50 \GeV between 300 \GeV and 550 \GeV, 100 \GeV between 550 \GeV and 750 \GeV, while the last bin spans 750 \GeV to 1200 \GeV. Just one event with reconstructed = 1535 \GeV falls outside this region in the \muplus sample, and none in the \eplus sample.

The fourth step is to apply a bin-by-bin correction factor , also referred to as efficiency correction, which restores the contribution of \ttbar events that fulfill the particle-level selection but not the detector-level selection. This factor is defined as the ratio of the number of events that satisfy both the particle-level and detector-level selections to the number that meet the selection at particle level only. The distribution of the efficiency correction is shown in Fig. LABEL:fig:eff(a).

The ability of the full correction procedure to recover a distribution that is significantly different from the nominal sample is tested. Simulated \ttbar events are reweighted such that the \ptreco distribution matches the data. The corresponding \ptptcl spectrum of the distorted \ptreco input spectrum is recovered with subpercent accuracy after unfolding.

The differential cross-section as a function of \ptptn is then derived according to:

 \dsigma\ttbard\ptptn(\ptptnk) = NkpartonBΔ\ptptnk\lum (3) = 1BΔ\ptptnk\lumfkparton!ptcl⋅∑j^M−1jkfjptcl!partonNjptcl.

Similarly to Eq. (2), is the total number of events in bin of \ptptcl that enter the particle-level fiducial region described in Sec. 8.2, is the number of events in bin of \ptptn in the full phase space, is the size of bin of the parton-level \ptptn (and of \ptptcl), is the total integrated luminosity of the data sample, and =0.438 [BRRef] is the branching ratio for \ttbar events with exactly one of the bosons, from the decay of the \ttbar pair, decaying to an electron or a muon or a lepton.

The corrections that are applied to the \ptptcl variable are derived following steps similar to the ones described to derive . They are also extracted from the nominal Powheg+Pythia sample. First, the factor corrects the \ptptcl spectrum for the events that pass the particle-level selection but fail the parton-level selection, shown in Fig. LABEL:fig:fakecorr(b). Effects relating \ptptn to \ptptclare corrected with the same matrix unfolding procedure used for detector effects. This migration matrix is shown in Fig. LABEL:fig:migration(b). A final correction factor is applied in bins of \ptptn to correct the result from the particle level to the partonic phase space, shown in Fig. LABEL:fig:eff(b).

To test the two-step derivation, the cross-section is also obtained by directly correcting the reconstructed distribution to parton level in a single step. The results are found to be consistent.

### 8.4 Propagation of statistical and systematic uncertainties

The propagation of statistical and systematic uncertainties is performed in the same way for both the particle-level and parton-level results. The impact of the data statistical uncertainty is evaluated by performing 1000 pseudoexperiments in which independent Poisson fluctuations in each \ptreco bin are assumed. The statistical uncertainty due to the limited size of the signal and background MC samples used to correct the data are estimated by performing 1000 pseudoexperiments using the bootstrap method [bootstrap], which builds 1000 statistically connected (co-varied) replicas of individual simulated signal or background spectra and derives the associated corrections.

For each systematic uncertainty arising from detector modeling, background modeling, and the electroweak correction factor, a varied \ptreco distribution is obtained and unfolded using corrections extracted from the nominal signal and background samples. The correlation between each systematic uncertainty’s effect on the signal and background spectra is taken into account. For the \ttbar generator, parton shower, and ISR/FSR uncertainties, a systematic uncertainty variation is defined as the difference between the generated and unfolded cross-section of a given generator, with unfolding corrections extracted with an alternative generator (or alternative generator setting). The PDF uncertainty is computed by unfolding the nominal sample with correction factors extracted by reweighting the nominal sample at the hard-process level for each variation of the PDF.

Figure  LABEL:fig:syst shows the effect of the statistical and systematic uncertainties on and . The total uncertainty generally increases with the measured and ranges from 13% to 29% for the particle-level cross-section, and from 15% to 41% for the parton-level cross-section. The dominant uncertainty for the particle-level cross-section is the large- jet energy scale, in particular its components due to the topology uncertainty at low and the uncertainty from balance in photon+jet events at high . The experimental uncertainties have a comparable size at parton level. However, the reported parton-level cross-section has significantly larger systematic uncertainties than the particle-level cross-section since it is affected by larger modeling uncertainties. The parton shower or generator uncertainties are dominant for nearly all bins of the parton-level cross-section, which illustrates the benefit of defining a particle-level cross-section in a fiducial region closely following the detector-level selection. A detailed breakdown of the systematic uncertainties is provided in Appendix A.

A covariance matrix including the effect of all uncertainties is calculated at particle level to make quantitative comparisons with theoretical predictions. This covariance matrix is obtained by summing two covariance matrices. The first covariance matrix incorporates uncertainties from detector and background modeling by performing 250,000 pseudoexperiments. In each pseudoexperiment, the data \ptreco distribution is varied following a Poisson distribution. Gaussian-distributed shifts are coherently added for each systematic uncertainty effect by scaling each Poisson-fluctuated bin with the relative variation from the associated systematic uncertainty effect. Differential cross-sections are obtained by unfolding each varied \ptreco distribution with the nominal corrections, and the results are used to compute a covariance matrix.

The second covariance matrix is obtained by summing four separate covariance matrices corresponding to the effects of \ttbar generator, parton shower, ISR/FSR, and PDF uncertainties. The standard deviations of the covariance matrices are derived by scaling the measured cross-section with the appropriate relative systematic uncertainty. The bin-to-bin correlation value is set to unity for the generator, parton shower, and ISR/FSR matrices, while it is set to 0.5 for the PDF matrix. This value is motivated by the fraction of the bins in which a single PDF set dominates in the determination of the envelopes used for their respective estimates. The procedure for these signal modeling uncertainties is needed because these effects cannot be represented by a variation at the detector level, and so cannot be included in the pseudoexperiment formalism used to build the first covariance matrix.

The correlation matrix derived from the particle-level covariance matrix is shown in Table 3. Agreement between the measured differential cross-sections and various predictions is quantified by calculating values employing the covariance matrix and by inferring corresponding -values. The are evaluated using: {linenomath}

 χ2=VT⋅Cov−1⋅V,

where is the vector of differences between measured differential cross-section values and predictions, and is the inverse of the covariance matrix.

## 9 Results and interpretation

The unfolding procedure is applied to the observed top-jet candidate distribution. The cross-sections are provided in Table 4 and Fig. 1 for the particle-level cross-section, and in Table 5 and Fig. 2 for the parton-level cross-section. The higher efficiency of reconstruction techniques for highly boosted top quarks allows measurement of the top quark spectrum up to 1200 \GeV. The differential cross-section is measured over two orders of magnitude. The measured differential cross-sections are compared to the predictions from Alpgen+Herwig, MC@NLO+Herwig, Powheg+Herwig, and Powheg+Pythia samples normalized to the NNLO+NNLL inclusive cross-section. The electroweak corrections are not applied to the Powheg+Pythia prediction in these figures in order to compare it on an equal footing with the other generators. All generators produce a top quark spectrum that is harder than the one observed, with a difference that generally increases with . The MC prediction to data ratio is approximately the same at both the particle and parton levels for Powheg+Pythia, which was used to extract the unfolding corrections. However, it changes significantly when going from particle level to parton level for the other MC generators, in particular for Powheg+Herwig, and Alpgen+Herwig, due to the different parton-level corrections in these MC generators. The level of agreement is better at parton level than at particle level because the parton level is affected by larger systematic uncertainties.

The and -values that quantify the level of agreement between the particle-level predictions and data are listed in Table 6. Within uncertainties, the differences are not significant for Powheg+Pythia, Powheg+Herwig and MC@NLO+Herwig, for which -values of 0.11 (for Powheg+Pythia without electroweak corrections), 0.41, and 0.14 are obtained, respectively. Only the prediction of Alpgen+Herwig is significantly disfavored by the data at the particle level with a -value of .

The measured differential cross-sections are compared in Fig. LABEL:fig:ewkcorr to the predictions of Powheg+Pythia with and without the electroweak corrections applied. The electroweak corrections lead to a slightly softer spectrum, increasing the particle-level -value from 0.11 to 0.28 without and with the corrections, respectively. The measured differential cross-sections are also compared in Fig. LABEL:fig:hdamp_herapdf to Powheg+Pythia predictions using either the HERAPDF [Aaron:2009aa] or CT10 PDF sets, and two different values of the Powheg parameter, the nominal value and one with , which increases the amount of hard radiation and yields a lower -value of 0.05. Better agreement with data is obtained when using the HERAPDF set instead of CT10, which reduces the difference between data and MC simulation by up to about 20%. The Powheg+Pythia prediction that provides the best description of the data is the one that simultaneously employs the HERAPDF set and , corresponding to a -value of 0.31 at particle level.

The measured parton-level cross-section is compared to the prediction of the parton-level NLO MCFM generator [Campbell:2010ff], which is interfaced with Applgrid [Carli:2010rw] to convolve the perturbative coefficients with the strong coupling and the PDF. The inclusive cross-section computed by MCFM is used to normalize the prediction and no electroweak corrections are applied. Several PDF sets are compared: CT10, MSTW, NNPDF, and HERAPDF. The renormalization scale and factorization scale are dynamic: , where is the average of the two top quarks in the event. The uncertainties on the prediction include the PDF uncertainties estimated according to the prescription of each set and variations of the strong coupling constant, , and . The predictions are compared to the measured parton-level cross-section in Fig. 3. All predictions are in good agreement with the measured cross-section within the quoted uncertainties, which are dominated by systematic uncertainties correlated between bins.

## 10 Conclusions

The differential production cross-section in \rts = 8 \TeV  collisions has been measured as a function of the hadronically decaying top quark in a high- regime, using a dataset corresponding to an integrated luminosity of 20.3

## Acknowledgments

We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.

We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWFW and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF, DNSRC and Lundbeck Foundation, Denmark; IN2P3-CNRS, CEA-DSM/IRFU, France; GNSF, Georgia; BMBF, HGF, and MPG, Germany; GSRT, Greece; RGC, Hong Kong SAR, China; ISF, I-CORE and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; RCN, Norway; MNiSW and NCN, Poland; FCT, Portugal; MNE/IFA, Romania; MES of Russia and NRC KI, Russian Federation; JINR; MESTD, Serbia; MSSR, Slovakia; ARRS and MIZŠ, Slovenia; DST/NRF, South Africa; MINECO, Spain; SRC and Wallenberg Foundation, Sweden; SERI, SNSF and Cantons of Bern and Geneva, Switzerland; MOST, Taiwan; TAEK, Turkey; STFC, United Kingdom; DOE and NSF, United States of America. In addition, individual groups and members have received support from BCKDF, the Canada Council, CANARIE, CRC, Compute Canada, FQRNT, and the Ontario Innovation Trust, Canada; EPLANET, ERC, FP7, Horizon 2020 and Marie Sk0̆142odowska-Curie Actions, European Union; Investissements d’Avenir Labex and Idex, ANR, Region Auvergne and Fondation Partager le Savoir, France; DFG and AvH Foundation, Germany; Herakleitos, Thales and Aristeia programmes co-financed by EU-ESF and the Greek NSRF; BSF, GIF and Minerva, Israel; BRF, Norway; the Royal Society and Leverhulme Trust, United Kingdom.

The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide.

## Appendix A Detailed tables of systematic uncertainties

Tables 7 and 8 report the detailed breakdown of the systematic uncertainties as a percentage of the measured differential cross sections.

\printbibliography

The ATLAS Collaboration

G. Aad, B. Abbott, J. Abdallah, O. Abdinov, R. Aben, M. Abolins, O.S. AbouZeid, H. Abramowicz, H. Abreu, R. Abreu, Y. Abulaiti, B.S. Acharya, L. Adamczyk, D.L. Adams, J. Adelman, S. Adomeit, T. Adye, A.A. Affolder, T. Agatonovic-Jovin, J. Agricola, J.A. Aguilar-Saavedra, S.P. Ahlen, F. Ahmadov, G. Aielli, H. Akerstedt, T.P.A. Åkesson, A.V. Akimov, G.L. Alberghi, J. Albert, S. Albrand, M.J. Alconada Verzini, M. Aleksa, I.N. Aleksandrov, C. Alexa, G. Alexander, T. Alexopoulos, M. Alhroob, G. Alimonti, L. Alio, J. Alison, S.P. Alkire, B.M.M. Allbrooke, P.P. Allport, A. Aloisio, A. Alonso, F. Alonso, C. Alpigiani, A. Altheimer, B. Alvarez Gonzalez, D. Álvarez Piqueras, M.G. Alviggi, B.T. Amadio, K. Amako, Y. Amaral Coutinho, C. Amelung, D. Amidei, S.P. Amor Dos Santos, A. Amorim, S. Amoroso, N. Amram, G. Amundsen, C. Anastopoulos, L.S. Ancu, N. Andari, T. Andeen, C.F. Anders, G. Anders, J.K. Anders, K.J. Anderson, A. Andreazza, V. Andrei, S. Angelidakis, I. Angelozzi, P. Anger, A. Angerami, F. Anghinolfi, A.V. Anisenkov, N. Anjos, A. Annovi, M. Antonelli, A. Antonov, J. Antos, F. Anulli, M. Aoki, L. Aperio Bella, G. Arabidze, Y. Arai, J.P. Araque, A.T.H. Arce, F.A. Arduh, J-F. Arguin, S. Argyropoulos, M. Arik, A.J. Armbruster, O. Arnaez, H. Arnold, M. Arratia, O. Arslan, A. Artamonov, G. Artoni, S. Artz, S. Asai, N. Asbah, A. Ashkenazi, B. Åsman, L. Asquith, K. Assamagan, R. Astalos, M. Atkinson, N.B. Atlay, K. Augsten, M. Aurousseau, G. Avolio, B. Axen, M.K. Ayoub, G. Azuelos, M.A. Baak, A.E. Baas, M.J. Baca, C. Bacci, H. Bachacou, K. Bachas, M. Backes, M. Backhaus, P. Bagiacchi, P. Bagnaia, Y. Bai, T. Bain, J.T. Baines, O.K. Baker, E.M. Baldin, P. Balek, T. Balestri, F. Balli, W.K. Balunas, E. Banas, Sw. Banerjee, A.A.E. Bannoura, L. Barak, E.L. Barberio, D. Barberis, M. Barbero, T. Barillari, M. Barisonzi, T. Barklow, N. Barlow, S.L. Barnes, B.M. Barnett, R.M. Barnett, Z. Barnovska, A. Baroncelli, G. Barone, A.J. Barr, F. Barreiro, J. Barreiro Guimarães da Costa, R. Bartoldus, A.E. Barton, P. Bartos, A. Basalaev, A. Bassalat, A. Basye, R.L. Bates


text:
# A Julia Example: Finding Pi¶

This notebook develops a complete simulation to solve an interesting, if somewhat artificial problem. The problem can be stated as "Imagine we don't know the value of the constant, $\pi$. Can we write a simple piece of code, based on some mathematic rules, to estimate a value for this constant?"

The mathematical rules we will be using are derived from the the ratio of the area of a cirlce, with a radius $r$ to the area of a square with a sides of length $2r$ (See the description of the finding_pi function below). Image a square circumscribing a circle like so.

Since we can exactly determine the value of $\pi$ if we know the ratio of these areas, we should be able to estimate the value of pi by estimating the ratio of these areas. In this example, we are going to generate a large number of points randomly distributed within a square like the one above, and determine what proportion of points are inside the circle. This should work as an estimate of the ratio of area of the circle to area of the square, and allow us to estimate $\pi$.

The code we write to do this will be divided into three modules. The first develops the library of functions needed to support this simulation. The second tests these functions, and the third writes the code to run a trial of this simulation, and to run a batch of trials to test the properties of this simulation.

Finaly, we'll plot the results.

## FindingPi Module¶

The following text describes the pieces of code in this module, with explanations of the Julia concepts they illustrate. Consider glancing through the code before reading this (much longer) description.

This is where most of our "library" code will go. We will write these functions and types as we need them, and will make sure they are testable. Actual simulation logic will live in another module.

In Julia, Modules are the primary method of creating namespaces. In this module, we do not import any external names, but we do use the export keyword to export some. The types and methods that we export will be made public so that we can use them in other modules we write, through the using keyword.

### The Point Type¶

Stores a point in cartesian space. We will create instances of this type when we are generating the points for our simulation. This user defined type is a Composite Type.

### The is_inside Methods¶

This function takes a point, and a radius of a circle, and returns a boolean, whether the point is inside the circle or not.

The pythagorean formula states that $a^2 + b^2 = c^2$. We'll use this to create a test of whether a point is inside our circle. If we let the x-coordinate of our point be our $a$ and the y-coordinate of our point be $b$, it makes sense that the point is inside the circle, if $x^2 + y^2 <= r^2$

We define three versions of this function. When multiple versions of a function with the same name are defined for different types of arguments, they're known as Methods. When a function is called, the number and types of arguments passed in are used dispatch to the correct method. Since all the arguments are used to determine which of the available methods should be used, this process is known as Multiple Dispatch, as opposed to the single dispatch that most OO languages use.

It is also worth noting that the final two functions could be replaced with a single definition: is_inside(point, radius=1), using default arguments. These forms are semantically identical (See: functions with optional arguments).

### The find_pi function¶

This function takes the number of points inside and outside the circle, and returns an estimate of pi.

The area of a circle with radius $r$, called $area_c$, can be given $area_c = \pi r^2$ and the area of a square circumbscribing the circle, called $area_s$ can be given $area_s = (2r)^2$. If we take the proportion of points inside the circle to outside the circle, to be the same as the proportion, $\frac{area_c}{area_s}$, then

$\frac{inside}{outside} = \frac{\pi r^2}{(2r)^2} = \frac{\pi}{4}$

and then our estimate of pi can be expressed as

$\pi = 4 \frac{inside}{outside}$

Which is what this function returns

### The random_point function¶

This function generates a random x-coordinate and random y-coordinate, each between -1 and 1. These are used to construct a Point object, which is returned.

### The generate_points function¶

Returns n points created with the random_point function.

In [66]:
module FindingPi

export Point, is_inside, find_pi, random_point, generate_points

type Point
x::Real
y::Real
end

# r^2 = x^2 + y^2
is_inside(point) = is_inside(point, 1)

# area_circle = pi*r^2
# area_square = (2r)^2
#  => c / s = pi / 4
find_pi(inside, total) = 4 * inside / total

function random_point()
x = (rand() * 2) - 1
y = (rand() * 2) - 1
Point(x, y)
end

generate_points(n) = [random_point() for i=1:n]

end

Warning: replacing module FindingPi

## The TestingPi Module¶

Here we right some basic tests to make sure that our functions are working as expected. We will only be testing our library code, not our actual simulation code.

In order to test code contained in another module, we first have to be sure those functions are available in the current namespace. There are two keywords that can be used to bring external functions into the current namespace, import and using which have a number of differences. Here, we use using FindingPi because it will bring into scope all the names that were export'd in the FindingPi module.

We also include using Base.Test, which brings in the unit testing functionality that is included with Julia. Specifically, we'll use the @test macro from Base.Test to test our code and provide feedback if something fails. This macro, denoted with the @ sign, is essentially a glorified assert statement. It will pass without output if it's single argument evaluates to true otherwise it will fail with a full traceback.

In [68]:
module TestingPi

using FindingPi
using Base.Test

# is_inside
@test is_inside(0, 0, 1)
@test !is_inside(1, 1, 1)
@test is_inside(1/sqrt(2), 1/sqrt(2), 1)
@test is_inside(Point(0,0))

# find_pi
@test find_pi(100, 100) == 4

# generate_points
@test typeof(random_point()) == Point
@test length(generate_points(4)) == 4

# @test false

end

Warning: replacing module TestingPi

## FindingPiSim Module¶

This module contains two functions. The first runs an individual trial, creating an estimation of pi. The second collects a series of these trials together, and returns an array of the results.

### The trial function¶

In this function, we used the components we developed in the FindingPi module to estimate pi. As described in the introduction, this involves generating a set of randomly distributed points, assessing how many of them lie within the circle, and using that proportion to estimate $\pi$.

This function takes a single argument, n, which controls how many points are generated. This comes with an assumption that as more points are generated, the estimate of pi should become more accurate. This assumption makes intuitive sense, but we can test the assumption by observing the results of the trial function at a variety of values of n.

### The batch function¶

The batch function runs a number of trials, collecting the results for observation. Here, we are moving beyond simply estimating pi, and starting to look at the properties of our estimation techniques.

The batch function takes a range (or other iterable) and a number of iterations. For each value in the iterable, it will run the trial function iterations times. Taking both a range, and number of iterations allows us to observe the behavior of our estimate at specified values of n, as well as show a broad picture of the estimates at those levels of n by controlling the number of iterations.

The output is an array with dimensions $length(range) \cdot iterations \times 2$, where the first column is the number of points, n, used to estimate $\pi$, and the second column is the estimate of $\pi$ for that trial. This array will allow us to easily plot the results of many trials of our estimate of $\pi$.

### A note on performance¶

For people who are used to working with languages such as R, Matlab, or Python/Numpy/Scipy, it might be counterintuitive to write nested for loops, as seen in the batch function below. In those languages, to get decent performance, all expensive computation needs to either be vectorized, or off-loaded to a more performant language.

Benchmarks show that Julia is very fast. While artificial benchmarks are a poor judge of the quality of a language, and many of the contenders in those benchmarks could be improved through optimization, the real benefit of Julia being fast, is that you can write simple, readable code and expect it to be performant.

This performance comes from an LLVM-based just-in-time (JIT) compiler. For the most part, the programmer can simply write code and expect it to be optimized automatically at run time. If the programmer needs more performance, the guide for writing Fast Numeric Computation in Julia is short, straigtforward, and easy to internalize.

In [67]:
module FindingPiSim

using FindingPi

function trial(n)
points = generate_points(n)
points_inside = map(is_inside, points)
inside_count = sum(points_inside)
pi_estimate = find_pi(inside_count, n)
end

function batch(range, iterations)
results = Array(Real, (length(range) * iterations, 2))
i = 1
for n = range
for j = 1:iterations
results[i, :] = [n, trial(n)]
i += 1
end
end
results
end

end

Warning: replacing module FindingPiSim

## Analysis and Plotting with DataFrames and Gadfly¶

At this point we can look at the results of our estimations. In the following cells we'll use two external packages, Gadfly a grammar-of-graphics based plotting library, and DataFrames, a package providing an interface similar to R's dataframes. Before we can use these external packages, we have to fetch and install them. Installing a package is a simple as calling Pkg.add(), passing in the name of the package. Note that packages can be found by searching for them at http://pkg.julialang.org/.

To plot the results of the batch trials we'll be creating a data frame, using the DataFrame constructor. This will be passed in to the plot function from gadfly, with a Geom.point geometry.

Along the way, we'll also print out some timing statistics, showing how quickly we're dealing with the large number of randomly distrubuted points we're using to estimate $\pi$.

This is the end of our analysis for this workbook, but this should give a solid grounding in how to use Julia, and let you jump off to lot of different and interesting topics.

In [38]:
FindingPiSim.trial(1000000)

Out[38]:
3.140244
In [27]:
Pkg.add("Gadfly")

INFO: Nothing to be done
INFO: No packages to install, update or remove
INFO: Package database updated

In [95]:
using Gadfly
using DataFrames

range = 100:10:1000
iterations = 5

total_points = sum(range) * iterations
println("$total_points points") t1 = time() res = FindingPiSim.batch(range, iterations) t2 = time() println("total time:$(t2 - t1)")
println("time per point: \$((t2 - t1) / total_points)")

df = DataFrame(n=res[:, 1], pi_est=res[:, 2])
df[:error] = pi - df[:pi_est]

# A simple plot:
plot(df, x="n", y="pi_est", color="error", Geom.point)

# A more complex plot, with a theme, and a layer to mark the true
# value of pi:
#
# t = Theme(
#     default_point_size=Compose.Measure(0.7),
#     highlight_width=Compose.Measure(0.1)
# )
#
# plot(
#     layer(x->3.1415, range[1], range[end]),
#     layer(df, x="n", y="pi_est", color="error", Geom.point, t)
# )

250250 points
total time: 0.16734004020690918
time per point: 6.686914693582784e-7

Out[95]:

format: plain


text:
# How do you find the arc length of the curve y=lnx over the interval [1,2]?

##### 2 Answers
Jun 28, 2018

Apply the arc length formula.

#### Explanation:

$y = \ln x$

$y ' = \frac{1}{x}$

Arc length is given by:

$L = {\int}_{1}^{2} \sqrt{1 + \frac{1}{x} ^ 2} \mathrm{dx}$

Rearrange:

$L = {\int}_{1}^{2} \frac{\sqrt{{x}^{2} + 1}}{x} \mathrm{dx}$

Multiply numerator and denominator by $\sqrt{{x}^{2} + 1}$:

$L = {\int}_{1}^{2} \frac{{x}^{2} + 1}{x \sqrt{{x}^{2} + 1}} \mathrm{dx}$

Integration is distributive:

$L = {\int}_{1}^{2} \frac{x}{\sqrt{{x}^{2} + 1}} \mathrm{dx} + {\int}_{1}^{2} \frac{1}{x \sqrt{{x}^{2} + 1}} \mathrm{dx}$

Apply the substitution $x = \tan \theta$:

$L = {\left[\sqrt{{x}^{2} + 1}\right]}_{1}^{2} + \int \sec \frac{\theta}{\tan} \theta d \theta$

Simplify:

$L = \sqrt{5} - \sqrt{2} + \int \csc \theta d \theta$

Integrate directly:

$L = \sqrt{5} - \sqrt{2} - \left[\ln | \csc \theta + \cot \theta |\right]$

Rewrite in terms of $\tan \theta$ and $\sec \theta$:

$L = \sqrt{5} - \sqrt{2} - \left[\ln | \frac{1 + \sec \theta}{\tan} \theta |\right]$

Reverse the substitution:

$L = \sqrt{5} - \sqrt{2} - {\left[\ln \left(\frac{1 + \sqrt{{x}^{2} + 1}}{x}\right)\right]}_{1}^{2}$

Insert the limits of integration:

$L = \sqrt{5} - \sqrt{2} - \ln \left(\frac{1 + \sqrt{5}}{2 \left(1 + \sqrt{2}\right)}\right)$

Rearrange for clarity:

$L = \sqrt{5} - \sqrt{2} + \ln 2 - \ln \left(\frac{1 + \sqrt{5}}{1 + \sqrt{2}}\right)$

Jun 28, 2018

Length = $\sqrt{5} + \frac{1}{2} \ln \left(\frac{\sqrt{5} - 1}{\sqrt{5} + 1}\right) - \sqrt{2} - \frac{1}{2} \ln \left(\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right) \approx 1.222$

#### Explanation:

The arc length of a function $y = f \left(x\right)$ over the interval $\left[a , b\right]$ is given by $L = {\int}_{a}^{b} \sqrt{1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}} \mathrm{dx}$.
$y = \ln x \to \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{x}$
$\therefore L = {\int}_{1}^{2} \sqrt{1 + {\left(\frac{1}{x}\right)}^{2}} \mathrm{dx}$
$L = {\int}_{1}^{2} \sqrt{1 + \frac{1}{x} ^ 2} \mathrm{dx}$
I will use this solution $\int \sqrt{1 + \frac{1}{x} ^ 2} \mathrm{dx} = \sqrt{{x}^{2} + 1} + \frac{1}{2} \ln | \frac{\sqrt{{x}^{2} + 1} - 1}{\sqrt{{x}^{2} + 1} + 1} | + C$ courtesy of mason m (https://socratic.org/questions/how-do-you-evaluate-the-integral-int-sqrt-1-1-x-2) to find that
L=sqrt(x^2+1)+1/2ln|(sqrt(x^2+1)-1)/(sqrt(x^2+1)+1)|]_1^2
$\therefore L = \sqrt{5} + \frac{1}{2} \ln \left(\frac{\sqrt{5} - 1}{\sqrt{5} + 1}\right) - \sqrt{2} - \frac{1}{2} \ln \left(\frac{\sqrt{2} - 1}{\sqrt{2} + 1}\right) \approx 1.222$


text:
Assignment 1: DrawSVG

# Assignment 1 Instructions

Project repository: https://github.com/cmu462/DrawSVG

Assignment 1 is due Sept 24th at 10:00pm. Late days may be used on this assignment (see Course Info).

This assignment is to be completed individually. You may discuss the material in general terms with your peers, but the code you submit must be entirely your own.

NOTE: Some of the concepts involved in this assignment will be discussed later in lecture. However, you should not wait until those lectures to get started on the assignment! Almost everything you need to know is detailed in the assignment writeup below. Also, simultaneously working on the assignment and going to lecture will enrich your experience on both sides, i.e., lectures will help to clarify things you're implementing, and implementing will help to clarify ideas discussed in lecture.

Remember that making your solution public now or in the future is a violation of the academic integrity policy. We are using Github to distribute code, but do not work in a public fork!

Your code must run on the GHC 5xxxx cluster machines as we will grade on those machines. Do not wait until the submission deadline to test your code on the cluster machines. Keep in mind that there is no perfect way to run on arbitrary platforms. If you experience trouble building on your computer, while the staff may be able to help, but the GHC 5xxx machines will always work and we recommend you work on them if any troubles arise.

Each task will be graded on the basis of correctness. You are not expected to completely reproduce the reference solution pixel-by-pixel as slight differences in implementation strategy or even the order of floating point arithmetic will causes differences, but your solution should not be very far off. (While we provide a diff view in the application to help you debug, we will mostly be using it as a reference when we grade your code.) If you have any questions about whether your implementation is "sufficiently correct", just ask.

The assignment consists of a total of 100 pts. The point breakdown is as follows:

• Task 1 (Hardware Renderer): 5
• Task 2 (Drawing Lines): 5
• Task 3 (Drawing Triangles): 15
• Task 6 (Scaled Images): 15
• Task 7 (Trilinear Filtering): 15
• Task 8 (Alpha Compositing): 5
• Task 9 (Draw Something!): 5
• Documentation: 5
• Writeup: 5

The project description includes some additional challenges marked as extra credit. The point value next to these problems indicates a typical number of points that would be earned for successful completion, though we may award more or fewer points for these problems based on your particular level of effort. In general, we encourage you to push the envelope and explore extensions that make your program more powerful or more efficient, and we may even award additional points to you for doing so. If you do anything extra, please explain your work in the writeup! However, no matter how much extra work you do, your overall score on this assignment is capped at 110 points.

### Documentation

Clear, well-written documentation is a critical part of software development. Since you (the students) are the ones who will most benefit from good documentation---or will suffer through bad documentation---we are "crowd sourcing" improvements to the course material. What did you find confusing---and then finally figure out?

As 5% of your assignment grade, you will need to submit suggested edits to the assignment documentation. These could be:

• Suggested changes or additions to the assignment writeup
• Suggested changes or additions to assignment Wiki (for A2, A3, and A4)

Do not wait until the due date to submit these suggested edits. They will be most helpful to your classmates (and to us!) if they are submitted ahead of time, so that we can immediately incorporate any good suggestions into the actual course material. To submit your edits, you must therefore:

• Go on Piazza
• Find the thread with the appropriate label: a1wiki, a2wiki, a3wiki or a4wiki
• Make an anonymous (not private) post with your suggested edit

The TAs will monitor this label, and immediately update the docs with any/all useful suggestions. This Piazza post will also be recorded as part of your assignment grade. Note that we do not have to accept your edit in order for you to receive full credit on the assignment. However, we will grade your edits based on whether they appear to be a "good faith effort" to make a useful comment. In other words, did you really think about what is clear/unclear and provide a useful edit? Or did you just write something totally random at the very last minute? :-) Especially useful edits (e.g., those that provide nice insights, or point out serious bugs/errors) may receive extra credit.

### Writeup

You will submit a short document explaining how you believe your implementation performs on each of the tasks. The project directory contains a template in writeup.txt. For each task, if you believe that you have correctly implemented the requested functionality, simply leave the text "Fully implemented." If you were not able to achieve the requested functionality, write a short explanation of what your submission does and does not do correctly, and (optionally) briefly explain your strategy and how you got stuck. If your explanation indicates significant understanding of the problem, we may award partial credit even though the code does not work.

If there are any parts of the documentation that seems confusing, make sure to also include your thoughts/fixes in the writeup.

Failure to submit this writeup will incur a 10 pt penalty on the assignment.

### Code Environment

This codebase should compile on Linux, macOS, and Windows on a typical environment. The build instructions given on the project page will walk you through installing dependencies and building the code. If you have difficulties running the code on your local machine, the GHC 5xxx cluster machines have all the packages required to build the project.

### Handin Instructions

As with the previous assignments, we will be submitting on Autolab. You should create a tar archive of your entire src subdirectory along with the writeup (writeup.txt) and Task 9 submission (task9.svg).

$pwd > (...)/DrawSVG$ tar cvzf handin.tgz src/ writeup.txt task9.svg
> a src
> a src/cmake
> a src/CMakeLists.txt
(...)
> a writeup.txt


text:
In a recent contribution, the authors show that the uncertainty in heat transfer results obtained using the Monte Carlo ray-trace (MCRT) method is related to the median of the radiation distribution factor probability density function (PDF). The value of this discovery would be significantly enhanced if the median could be known a priori without first computing the distribution factors. This would allow the user to determine the number of rays required to achieve the desired accuracy of a subsequent heat transfer analysis. The current contribution presents a correlation for the median of the distribution factor PDF as a function of emissivity and the number of surface elements defining an enclosure. The correlation involves a single parameter whose value is unique for a given enclosure geometry. We find that the radiation behavior of a given enclosure can be classified on a scale ranging from reflection-dominated to geometry-dominated. The correlation is shown to work well for reflection-dominated enclosures but less well for geometry-dominated enclosures.

## References

1.
Mahan
,
J. R.
,
2002
,
Radiation Heat Transfer: A Statistical Approach
,
Wiley
,
New York
.
2.
Mahan
,
J. R.
,
2019
,
The Monte Carlo Ray-Trace Method in Radiation Heat Transfer and Applied Optics
,
Wiley-ASME Press Series
,
New York
.
3.
,
M.
,
Robert Mahan
,
J.
, and
Priestley
,
K. J.
,
2019
, “
,”
ASME J. Heat Transfer
,
141
(
3
), p.
032701
.
4.
Sánchez
,
M. C.
, and
Mahan
,
J. R.
,
2010
, “
Predicting Uncertainty and Confidence Intervals in Thermal Radiative Modeling Using the Monte Carlo Ray-Trace Method
,”
Int. J. Numer. Methods Heat Fluid Flow
,
20
(
1
), pp.
66
83
.
5.
,
M.
,
2018
, “
An Engine for Computing Radiation Distribution Factors for Two-Dimensional Enclosures
,”
Zenodo, Geneva, Switzerland
(
epub
).
6.
,
M.
, and
Robert Mahan
,
J.
,
2018
, “
Verification of Two-Dimensional Monte Carlo Ray-Trace Methodology in Radiation Heat Transfer Analysis
,” Preprint
arXiv:1810.05204
. https://arxiv.org/abs/1810.05204


text:
# What determines a horizontal asymptote?

The line $y = b$ is called a horizontal asymptote of $f$ if at least one of the following limits holds:
${\lim}_{x \to - \infty} f \left(x\right) = b$ or ${\lim}_{x \to \infty} f \left(x\right) = b$.


text:
triangularMatrix-class {Matrix} R Documentation

## Virtual Class of Triangular Matrices in Package Matrix

### Description

The virtual class of triangular matrices,"triangularMatrix", the package Matrix contains square (nrow == ncol) numeric and logical, dense and sparse matrices, e.g., see the examples. A main use of the virtual class is in methods (and C functions) that can deal with all triangular matrices.

### Slots

uplo:

String (of class "character"). Must be either "U", for upper triangular, and "L", for lower triangular.

diag:

String (of class "character"). Must be either "U", for unit triangular (diagonal is all ones), or "N" for non-unit. The diagonal elements are not accessed internally when diag is "U". For denseMatrix classes, they need to be allocated though, such that the length of the x slot does not depend on diag.

Dim, Dimnames:

The dimension (a length-2 "integer") and corresponding names (or NULL), inherited from the Matrix, see there.

### Extends

Class "Matrix", directly.

### Methods

There's a C function triangularMatrix_validity() called by the internal validity checking functions.

Currently, Schur, isSymmetric and as() (i.e. coerce) have methods with triangularMatrix in their signature.

isTriangular() for testing any matrix for triangularity; classes symmetricMatrix, and, e.g., dtrMatrix for numeric dense matrices, or ltCMatrix for a logical sparse matrix subclass of "triangularMatrix".

### Examples

showClass("triangularMatrix")

## The names of direct subclasses:
scl <- getClass("triangularMatrix")@subclasses
directly <- sapply(lapply(scl, slot, "by"), length) == 0
names(scl)[directly]

(m <- matrix(c(5,1,0,3), 2))
as(m, "triangularMatrix")

[Package Matrix version 1.5-3 Index]


text:
How do you simplify -4sqrt(7/8)?

Apr 2, 2018

The expression simplifies to $- \sqrt{14}$.

Explanation:

$\textcolor{w h i t e}{=} - 4 \sqrt{\frac{7}{8}}$

$= - 4 \cdot \sqrt{\frac{7}{8}}$

$= - 4 \cdot \frac{\sqrt{7}}{\sqrt{8}}$

$= - 4 \cdot \frac{\sqrt{7}}{\sqrt{4 \cdot 2}}$

$= - 4 \cdot \frac{\sqrt{7}}{\sqrt{4} \cdot \sqrt{2}}$

$= - 4 \cdot \frac{\sqrt{7}}{2 \cdot \sqrt{2}}$

$= - 4 \cdot \frac{\sqrt{7}}{2 \cdot \sqrt{2}} \textcolor{red}{\cdot \frac{\sqrt{2}}{\sqrt{2}}}$

$= - 4 \cdot \frac{\sqrt{7} \cdot \sqrt{2}}{2 \cdot \sqrt{2} \cdot \sqrt{2}}$

$= - 4 \cdot \frac{\sqrt{7} \cdot \sqrt{2}}{2 \cdot {\left(\sqrt{2}\right)}^{2}}$

$= - 4 \cdot \frac{\sqrt{7} \cdot \sqrt{2}}{2 \cdot 2}$

$= - 4 \cdot \frac{\sqrt{7} \cdot \sqrt{2}}{4}$

$= - \textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} \cdot \frac{\sqrt{7} \cdot \sqrt{2}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}}}$

$= - \sqrt{7} \cdot \sqrt{2}$

$= - \sqrt{14}$

We can verify this answer using a calculator:

Hope this helped!


text:
# 14.14: Channels of Business Communication

### Learning Outcomes

• Differentiate between face-to-face, written, oral, web-based, and other typical channels of business communication.
• Explain the importance of tailoring the message to the audience.

Business communication is held to a higher standard than everyday communication. The consequences of misunderstandings are usually higher and the chances to recognize and correct a mistake are lower. The barriers to communication and technique for improving communication are the same regardless of where the conversation takes place.

There’s a well-known expression that goes “It’s not what you say, it’s how you say it.” It’s really both.

A communication channel is the medium, mean, manner or method through which a message is sent to its intended receiver. The basic channels are written (hard copy print or digital formats), oral or spoken, and electronic and multimedia. Within those channels, business communications can be formal, informal, or unofficial. Finally, communications can be rich or lean.

Channel richness refers to the amount and immediacy of information that can be transmitted. Face-to-face communication is very high in richness because it allows information to be transmitted with immediate feedback. A tweet is very low in richness because it allows only 140 characters to be transmitted with no feedback. On the other hand, face-to-face communication is limited to one person communicating with a few other people in close proximity, whereas a tweet can go out to thousands of followers around the world. The following diagram shows the richness of different types of communication.

Different types of communication media have varying channel richness.

### Oral Communications

Oral channels depend on the spoken word. They are the richest mediums and include face-to-face, in-person presentations, mobile phone conferences, group presentations, telephone, video meetings, conferences, speeches, and lectures. These channels deliver low-distortion messages because body language and voice intonation also provide meaning for the receiver. They allow for immediate feedback of the communication to the sender. They are also the most labor-intensive channels in terms of the number of people involved in the transaction. Oral channels are generally used in organizations when there is a high likelihood of the message creating anxiety, confusion, or an emotional response in the audience. For instance, a senior manager should address rumors about layoffs or downsizing in face-to-face meetings with management staff. This allows the receiver (audience) to get immediate clarification and explanations, even if the explanation is a simple but direct: “At this time, I just don’t know.”

Oral communications are also useful when the organization wants to introduce a key official or change a long-established policy, followed up with a written detailed explanation. Senior managers with high credibility usually deliver complex or disturbing messages. For example, a senior manager will usually announce plans to downsize in person so that everyone gets the same message at the same time. This will often include a schedule so people know when to expect more details.

### Written Communications

Written communications include e-mails, texts, memos, letters, documents, reports, newsletters, spreadsheets, etc. (Even though e-mails are electronic, they are basically digital versions of written memos.) They are among the leaner business communications. With written communications, the writer must provide enough context that the words can be interpreted easily. The receiver should be alert for ambiguity and ask for clarification if needed. An e-mail sender cannot take receipt for granted. Most people receive too much e-mail and sort and filter it quickly, sometimes incorrectly.

Written messages are effective in transmitting large messages. Humans are limited in the amount of data they can absorb at one time. Written information can be studied over time. Reports can include supporting data and detailed explanations when it is important to persuade the receiver about a course of action. Written communications can be carefully crafted to say exactly what the sender means. Formal business communications, such as job offer letters, contracts and budgets, proposals and quotes, should always be written.

### Electronic (Multimedia) Communications

Television broadcasts, web-based communications such as social media, interactive blogs, public and intranet company web pages, Facebook, and Twitter belong in this growing category of communication channels. Electronic communications allow messages to be sent instantaneously and globally. People can talk face-to-face across enormous distances. Marketing and advertising can be targeted to many different types of customers, and business units can easily communicate in real time. This is especially important when customers must be advised of product recalls or security issues.

Although extremely effective, the widespread utilization of electronic communications for business purposes can also be risky. In recent years, the private communications and customer files of many large corporations have been hacked and their data stolen. In 2016, New Jersey Horizon Blue Cross Blue Shield was fined $1.1 million for failing to safeguard the personal information of medical patients. The company stored unencrypted sensitive data including birth dates and Social Security numbers on laptops that were stolen out of their main offices. Major Types of Business Communication Channels TYPE OF CHANNEL ADVANTAGES DISADVANTAGES Oral communications Build relationships and trust; accelerate decision making due to immediate feedback Spontaneous nature may lead to unwise statements; people are unable to refer to the communication once it is said unless a record is made. Written communications Message can be revised for exactness; can be archived for reference; can be studied. Appropriate for legal and formal business functions. Message is static; sender does not receive immediate feedback. Hard for the sender to gauge if the receiver has understood. Multimedia Instant, global, and adaptable to multiple targets. Technical difficulties and hack attacks threaten the security of organizations and their customers/clients. ### Which Channel Is Best? Quite simply, the best channel is the one that most effectively delivers the message so that it is understood as the sender intended. Nuanced or emotionally charged messages require a rich medium; simple, routine messages don’t need the personal touch. If you want to advise your department that at 2 p.m. you want to have a five-minute stand-up meeting in the hallway outside of your office to congratulate them on meeting a goal, then send a quick e-mail. You really don’t want people to reply with questions. E-mail is a lean medium but works very well when the content of the message is neither complex nor emotionally charged. On the other hand, a telephone call is a more appropriate channel to apologize for having to cancel a lunch date. The speaker can hear the sincerity in your voice and can express their disappointment or offer to reschedule. A good rule of thumb is the more emotional the context of the message, the richer the medium should be to deliver the message. But remember—even face-to-face business meetings can be followed up with a written note to ensure that both parties are truly on the same page. If a meeting results in assignments or agreements, then a written note will be useful documentation for future reference. ### Practice Question https://assessments.lumenlearning.co...essments/12315 ## Tailoring a Message to an Audience Steve Jobs, one of the founders of Apple, was a great communicator who used his passion, focus, and storytelling ability to make his presentations memorable. Have you ever had to converse with a young child and really needed him to listen to you? You probably knelt down so that you were at eye level. You may have even put your hand gently on his arm to focus his attention on you. Speaking clearly and slowly in words that you knew he understood, you made exaggerated facial gestures of fear or surprise or happiness to emphasize your points. In brief, you tailored your message to your audience so that it would be effective. Although the channel of communication you select to transmit your message is important, so also is knowing the intended audience. Effective communicators seem to instinctively adjust their styles to their audience, but it is a skill that can also be learned. The way the information is conveyed should complement the audience, whether it is one person or a roomful of people. Three factors are involved in tailoring a message: the sender, the message itself, and the audience. ### The Sender For the majority of everyday business communications, a manager needs only to focus on presenting a professional image. On different occasions, she may want to present herself as a serious and responsible leader, a technical expert, or informally casual. But when uncertain what a situation calls for, always err on the side of professionalism. Studies have shown that an audience reacts initially to the demeanor of the speaker and not to what is being said. If you appear nervous, your audience will pick up on that and become nervous as well. ### The Message The first step in tailoring a message, of course, is to choose the right communication channel. But even then, there are questions to ask about the message itself that will influence how it is presented. Some of these questions are listed below: • Which channel suits the content best? A critical or urgent message may demand the fastest channel, whether that is in-person, text-message, or e-mail. A scripted presentation requires planning and preparation. Businesses can use journalists or Twitter for brief announcements. If you want to demonstrate how your organization does something, clarify values, or construct common meanings for people across an organization, you might tell a memorable story in either an oral or written presentation. Decide which channel best suits the content. • What do you want to achieve by sending the message? Is it intended to persuade or to inform? Are you just reassuring your boss that everything is on track? A face-to-face communication is much more effective than an e-mail or report in trying to convince someone to change a belief or a behavior. • Does the message require interaction from the audience or is it more of an “information dump”? Don’t try to deliver a rationale for a$1 million investment in a phone call, especially if you need supporting evidence. Allow enough time to answer all questions.
• Will visual aids help the message or will they just distract from it? Arguments that a new computer software program will increase your department’s productivity will need to be backed by solid evidence presented in charts or graphs. But excessive or unfocused visuals may irritate senior-level managers who want just the concise summation and the triple bottom line (economic, social and environmental impacts).
• Do you have to establish your credibility? Does the audience know you? Are you speaking to people who know more than you about the topic? The answers to these questions will influence how you design your message.
• Most importantly, before deciding on a communication, ask yourself why the audience should care about it. When you know, then tailor the communication so that it answers that question. The story of United Airlines Flight 4311 is example of a CEO who failed to identify his primary audience and ask himself what they cared about before he spoke. In June of 2017, a seated passenger was informed by the flight attendants that his seat was needed and that he would have to leave the plane. He refused to give up his seat and was eventually dragged, literally kicking and screaming, off the plane. The video immediately went viral. In an effort to prevent a public relations nightmare, the CEO initially blamed the passenger and defended the actions of the United Airlines employees and security guards. When public outrage grew and customers threatened a boycott, the CEO finally realized that the first communication should have focused not on the employees and the company but on the passenger, who was, after all, a paying customer.

### The Audience

You can’t tailor your communication to the sender if you don’t know your audience. Here are some questions to consider when getting to know your audience.

• How big is your audience? Is it one person or a large group? One individual means getting “up close and personal” as opposed to a conference room. If a large group, is it diverse or fairly homogenous? Do you need language translators or signers for the hearing impaired?
• What is the status of your audience? Is the audience mostly senior-level managers, peers, or subordinates? Your body language, presentation, and formality will vary depending upon the status of the group. Although subordinates may be more easily put at ease with an informal, relaxed approach, high-level managers usually want to get straight to the point and move on to the next thing.
• Can you establish empathy with the audience? The more the audience trusts you, the easier it will be to retain their attention. Notice the audience—are people shifting around, avoiding eye contact, yawning, or are they nodding in agreement and focused? You may have to adapt your message to meet the mood of the audience.

### Practice Question

https://assessments.lumenlearning.co...essments/12316