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SIMULATING THREE-DIMENSIONAL NONTHERMAL HIGH-ENERGY PHOTON EMISSION IN COLLIDING-WIND BINARIES

# Simulating Three-Dimensional Nonthermal High-Energy Photon Emission in Colliding-Wind Binaries

## Abstract

Massive stars in binary systems have long been regarded as potential sources of high-energy  rays. The emission is principally thought to arise in the region where the stellar winds collide and accelerate relativistic particles which subsequently emit  rays. On the basis of a three-dimensional distribution function of high-energy particles in the wind collision region – as obtained by a numerical hydrodynamics & particle transport model – we present the computation of the three-dimensional nonthermal photon emission for a given line of sight. Anisotropic inverse Compton emission is modelled using the target radiation field of both stars. Photons from relativistic bremsstrahlung and neutral pion decay are computed on the basis of local wind plasma densities. We also consider photon photon opacity effects due to the dense radiation fields of the stars. Results are shown for different stellar separations of a given binary system comprising of a B star and a Wolf–Rayet star. The influence of orbital orientation with respect to the line of sight is also studied by using different orbital viewing angles. For the chosen electron-proton injection ratio of 10, we present the ensuing photon emission in terms of two-dimensional projections maps, spectral energy distributions and integrated photon flux values in various energy bands. Here, we find a transition from hadron-dominated to lepton-dominated high-energy emission with increasing stellar separations. In addition, we confirm findings from previous analytic modeling that the spectral energy distribution varies significantly with orbital orientation.

acceleration of particles – binaries: general – gamma rays: stars – hydrodynamics – stars: winds, outflows

## 1 Introduction

The idea that binary systems of massive stars without compact objects provide suitable environments for efficient particle acceleration and subsequent nonthermal emission has been thoroughly discussed in literature (e.g., Pollock, 1987; White & Chen, 1994). Electrons and protons are thought to be accelerated at the shock fronts tracing the region where the stellar winds collide with supersonic velocities. Analytical models have predicted that the ensuing population of high-energy particles is liable to yield considerable emission of nonthermal radio waves, hard X rays and  rays via various emission channels (see e.g.; Eichler & Usov, 1993; Benaglia & Romero, 2003; Reimer et al., 2006).

The observational evidence concerning nonthermal photon emission from these particle-accelerating colliding-wind binaries (CBWs) was recently summarized by De Becker & Raucq (2013) who provide a unified census of these systems. Their list of 43 confirmed or suspected binary systems together with their properties helps to pinpoint some important discrepancies between observations and model predictions, such as Fermi-LAT limits on the gamma-ray flux towards WR 140 or WR 147 (see also Werner et al., 2013), contrasted by the enigmatic properties of the  Carinae binary system. The latter shows a comparably strong high-energy -ray signal that has been found to exhibit variability on orbital time scales and a peculiar two-component spectrum (see Fermi Large Area Telescope observation in Reitberger et al., 2012).

A better understanding of CWB systems can be obtained by dedicated hydrodynamical (HD) simulations. Studies such as presented in Pittard (2009) have explored the highly dynamical nature of the WCR and its strong dependence on stellar and orbital parameters. The complex density, velocity and temperature structure of the colliding winds have further been used to model the thermal radio and X-ray emission in such systems (Pittard, 2010; Pittard & Parkin, 2010). Using magnetohydrodynamic (MHD) simulations Falceta-Gonçalves & Abraham (2012) have recently explored the important role of the magnetic field in the wind collision region (WCR) of these systems and its impact on nonthermal radio emission. Focusing on the properties of a specific binary system, Madura et al. (2013) used three-dimensional (3D) smoothed particle hydrodynamics (SPH) to investigate the turbulent WCR structure in  Carinae.

The present study provides simulations of high-energy nonthermal photon emission of CWBs, based on 3D distributions of high-energy particles. The latter has been obtained by combining a 3D HD description of the WCR with the solution of the transport equation of high-energy particles.

In Reitberger et al. (2014) (paper I) we presented a method to numerically compute the spectral energy distribution (SED) of charged particles within a numerical HD model of the WCR in a binary system of two massive stars. By solving the transport equation including spatial convection, diffusion, diffusive shock acceleration (DSA) and various cooling processes for electrons and protons at every grid point and time step of the HD simulation, we derived the time-dependent 3D spatial distribution of particles at different energies.

In this work, we present the resulting nonthermal photon emission along a given line of sight on the basis of the previously derived electron and proton spectra. The interactions of high-energy particles with the stellar radiation fields and the dense wind material in the WCR give rise to several mechanisms of photon emission, including anisotropic inverse Compton (IC) scattering, relativistic bremsstrahlung and neutral pion decay. We compute the emission throughout the chosen computational domain and present results in terms of two-dimensional (2D) projection maps, spectral energy distributions and total integrated photon flux values.

As not unexpected, the emission is very sensible to the conditions in the WCR, the position of the stars and orbital orientation. We present a case study of three binary systems differing in stellar separation. Non-thermal high-energy photon emission is computed for these systems with respect to different lines of sight.

In Section 2 we provide a detailed description of our calculation of various emission processes, as well as the application of photon photon opacity effects. We investigate the ensuing photon emission in Section 3, first for two stationary stars, then including orbital motion. Sections 4 and 5 provide a summary of our findings as well as an outlook on future developments.

## 2 Non-Thermal High-Energy Photon Emission

In order to estimate the nonthermal high-energy photon emission of a CWB system, we consider the three principal continuum emission processes for energies 10 MeV. These are IC scattering of the photons in the dense stellar radiation field to high energies, relativistic bremsstrahlung by the interaction of the energetic electrons with the ions in the wind, and the decay of neutral pions that are produced in hadronic nucleon nucleon collisions.

The computation of all three processes is based on a 3D distribution of high-energy electrons (IC scattering, bremsstrahlung) and protons (neutral pion decay) which we obtain by the method described in paper I. There, all relevant details concerning the hydrodynamics, the transport equation and the numerical implementation are provided.

In principle, the distribution of high-energy electrons can also be used to compute nonthermal low-energy emission such as synchrotron emission. This is highly relevant in modeling specific binary systems, where such observations of a nonthermal signal at radio wavelengths exists (see e.g; Williams et al., 1994, for the case of WR140). A comparison of measured and modelled synchrotron emission yields constraints concerning the magnetic field in conjunction with the electron injection fraction in the WCR and hence the level of IC emission. As it is not the aim of the present study to investigate a specific binary system with fixed astronomical properties, we restrict ourselves to the high-energy part of the nonthermal emission spectrum considering the three emission channels as detailed below.

1. Anisotropic IC scattering of high-energy electrons on the photons in the radiation field of both stars: This process depends on the local energy density of radiation, the spatial and energetic distribution of electrons and the scattering angle, being the angle between the line of sight to the observer and the direction of the incoming stellar photons. The stars are approximated as monochromatic point sources. The high-energy electrons are considered to have an isotropic distribution function.

2. Relativistic bremsstrahlung of high-energy electrons passing through the wind plasma. This process depends on the number density of the wind plasma and the spatial and energetic distribution of high-energy electrons. We assume a typical interstellar medium (ISM) metallicity of 90% H, 10% He.

3. The decay of neutral pions produced in collisions of high-energy protons with nucleons of the wind plasma. This process depends on the number density of the wind plasma and the spatial and energetic distribution of high-energy protons.

We compute the above nonthermal emission components for each numerical grid cell of the computational domain. For the calculation of IC scattering, additional information about the orientation of the system with respect to the observer (in terms of inclination and argument of periastron of the binary system ) must be provided. Bremsstrahlung and neutral pion decay give isotropically distributed radiation.

In a second step we use the resulting 3D distribution of nonthermal photon emission per energy and volume and the information of direction and distance to the observer, in order to compute

1. 2D projection maps of the nonthermal emission for a given energy interval and line of sight,

2. the SED of the various nonthermal emission components integrated over the whole emission region, and

3. the total integrated flux for various energy bands.

Additionally, we consider effects of photon photon opacity due to the dense radiation fields of the stars in the CWB system. Details are given in Section 2.4. Other possible absorption effects (e.g., via interaction with the ISM) are, presently, not considered since they also depend on the specific system.

In the following, we give a detailed account of how we calculate the individual emission components and derive their 2D projections, SEDs and integrated flux values.

### 2.1 Inverse Compton Scattering

The computation of the emission by IC scattering requires information on the high-energy electrons, as well as on the radiation fields involved in the scattering process. For consistency with paper I, we use the approximation of monochromatic radiation from pointlike stars. The differential target photon field of star can then be expressed as

 dnph,jdEph({r},Eph,μ)=n0,j({r})δ(Eph−ET,j)δ(μ−μsc,j) (1)

where is the energy of a single photon from star with surface temperature . The cosine of the scattering angle is determined via the inner product of the unit vectors pointing from the position of the cell at r toward the observer and toward star . is the local photon number density at the position of the cell r given by

 n0,j({r})=uph,jET,j=L∗,j4πr2jcET,j (2)

with the distance to the star and its stellar luminosity.

The local differential number density of the scattering electrons is directly obtained from the output of the simulations described in paper I. Differential number densities of target photons and high-energy electrons enter into the expression for the IC photon production rate at energy . The complete formula in the context of CWB systems for the case of an isotropic particle distribution is given by Reimer et al. (2006). It involves integration over all target photon energies, all target photon incoming directions and all electron energies. By applying the approximations of monochromacity and pointlike stars, only the integral over the energy of the electrons remains. Specific care has to be taken in determining the lower integration limit which marks the lowest energy at which electrons are liable to scatter photons to an energy with regard to the scattering angle . The upper limit is determined by the maximum energy of the electrons. The integration is performed numerically.

The IC flux from a single cell with volume reaching an observer along the line of sight at distance (neglecting absorption effects) then is

 Extra open brace or missing close brace (3)

where the sum accounts for the contribution of both stars. The integrated IC flux of the total emission region can be determined by a sum over all grid cells and an integration over the photon energy .

### 2.2 Relativistic Bremsstrahlung

According to Blumenthal & Gould (1970), the bremsstrahlung spectrum emitted by a single electron of energy passing through a medium containing various species of ions with number densities is

 d˙NγdEγ(Ee)=c∑iniZ2idσidEγ(Ee) (4)

where the sum is over all elements relevant for the scattering process.

The differential cross-section is greatly simplified if we assume that all contributing species are fully ionized. This is approximately true in the high temperature plasma inside the WCR. With this assumption (the case of “weak shielding”) the differential cross-section can be expressed as

 dσidEγ(Ee)=αr2043E2e−43EeEγ+E2γEγE2eϕweak (5)

with

 ϕweak=⎧⎪⎨⎪⎩4[ln(2Ee(Ee−Eγ)Eγmec2)−12],for Eγ

We further make the assumption of only dealing with hydrogen and helium (90% H, 10% He), allowing to rewrite Equation (4) as

 d˙NγdEγ=c(0.9×12+0.1×22)nHdσdEγ (7)

where the number density of particles in the wind plasma is obtained via from the output of the hydrodynamic simulations described in paper I. Computing the emission for isotropically distributed electrons requires an integration over their energy spectrum.

 d˙nγdEγ=∫dEednedEe(Ee)d˙NγdEγ=1.3cnH∫dEednedEe(Ee)dσdEγ (8)

The integral is solved with the same numerical scheme as discussed in Section 2.1. To obtain the resulting flux emitted from a single computational cell at a distance , we multiply by the cell volume and divide by the respective spherical distance surface.

 dFbremsdEγ=Vcell4πd2ld˙nγdEγ (9)

### 2.3 Neutral Pion Decay

In computing the -ray emission due to the decay of neutral pions, we follow Kelner et al. (2006) who use a -functional approximation for the cross-section with being the total energy of the incident proton (the other one is assumed to be at rest). is the energy of the produced pion. For a distribution of protons the omnidirectional differential neutral pion spectrum can then be expressed as

 dnπ0dEπ(Eπ) =cnHσπpp(mpc2+EπKπ)dnpdEp(mpc2+EπKπ) (10)

where is the mean fraction of the kinetic energy of the protons that is transferred to the secondary neutral pions. According to Aharonian & Atoyan (2000) the parameterization yields results of high accuracy in a broad region from GeV to TeV energies. It can be applied down to the threshold energy of neutral pion production. The total cross-section is approximated (following Kelner et al., 2006) as

 σπpp(Ep)=(34.3+1.88L+0.25L2)[1−(EthEp)4]2×10−31m2 (11)

with and .

Equation (10) allows to directly convert the proton spectra (obtained as described in paper I) to the resulting pion spectra. Via interpolation, can be obtained for any given energy . The omnidirectional differential photon production rate can then be computed via the integral

 d˙ndEγ(Eγ)=2∫∞Eγ+m2πc44EγdEπdnπ0dEπ(Eπ)√E2π−m2πc4 (12)

As in case of the bremsstrahlung component, the flux reaching an observer at a distance then is

 dFπ0decaydEγ=Vcell4πd2ld˙ndEγ. (13)

### 2.4 Photon-Photon Opacity in the Stellar Radiation Fields

Due to the intense stellar radiation field close to the stars, the emitted nonthermal photon flux above  100 GeV may be attenuated significantly via the electron pair production process

 γ+γ′→e++e−. (14)

This attenuation can be expressed as

 dFdEγ({r},Eγ)⟶dFdEγ({r},Eγ)exp(−τγγ({r},Eγ)) (15)

with being the optical depth due to photon-photon pair production. For each position within the emission region, the computation of requires an integration along the path toward the observer. The optical depth is proportional to the cross-section as well as to the differential photon number density of the stellar radiation field. The general expression can be written as

 τγγ({r},Eγ)= ∫dℓ∫dEph∫dμdnphdEph ({x}(ℓ),Eph,μ)σγγ(Eph,Eγ,μ)(1−μ)

where the integrals run along the path , over the energy of the photons in the stellar radiation fields , and over the stellar surface described by the cosine of the angle between the two interacting photons .

Analogous to the computation of the IC emission in Section 2.1, we assume the stars to be pointlike and their radiation field to be monochromatic. Thus, the differential photon number density can be expressed as in Equation (1) and

 τγγ({r},Eγ)= ∫dℓstars∑jn0,j({x}(ℓ)) σγγ(Eph,ET,j,μj({x}% (ℓ)))(1−μj({x}(ℓ))) (17)

where is the angle between the line of sight (LOS) and the direction toward the star j. The integral along path is solved numerically.

According to Gould & Schréder (1967) the cross-section of the process is

 σγγ(ET,Eγ,μ)=12πr20(1−β2)[(3−β4)ln1+β1−β−2β(2−β2)] (18)

with

 β=√1−s−1ands=ETEγ2m2ec4(1−μ) (19)

The threshold for photon photon absorption is at (in the most favourable case of ). For a star of temperature , opacity effects do occur for 40 GeV.

In order to test the validity of the monochromatic approximation, we also computed the absorption assuming a black-body spectrum for both stars. This requires an additional integration over the energies of the stellar photons. We approximate the spectrum by 20 energy bins (equally spaced on a logarithmic scale) in the interval from 0.1 to 100 eV and integrate numerically. A comparison between the two cases is given in Figure 1 where we show the optical depth for a single computational cell as a function of for either the WR star, the B star or both stars (a), as well as the impact of the opacity effects on the SED emitted by a representative fraction of the computational domain (b). The monochromatic approximation is found to overestimate the optical depth near its maximum and underestimates it for lower and higher energies. However, the differences found in the SED are sufficiently small (see Figure 1 b)) to warrant the use of the monochromatic approximation, keeping in mind that it has significantly lower computational cost, as the black-body description of the radiation field (with 20 energy bins) increases the overall computation time by a factor of 20.

## 3 Results

In the following, we will investigate three different binary systems with different stellar separations regarding their wind plasma properties, their distribution of charged particles and their ensuing nonthermal high-energy emission. The latter is explored using 2D projection maps, SEDs and integrated flux values for various energy bands. We also present the results for an additional case including effects of orbital motion.

### 3.1 HD Model Parameters

To allow for quick comparison, we consider three binary systems (Wolf–Rayet (WR) + B star) that differ only in their stellar separation for which we chose 720 (case A), 1440 (case B) and 2880 (case C) R. All stellar and stellar wind parameters (listed in Table 1) are the same as in the parameter studies in Reimer et al. (2006) and in paper I. The important parameter which indicates the point of ram pressure balance between the two stars has a value of 0.1 for the chosen parameters. In a first approach, we neglect orbital motion and study the three systems in a converged state after all traces of the initial conditions have vanished. For all three cases we chose the identical computational domain in the shape of a cube with side length 4000 . The two stars are located on the axis. Figure 2 schematically depicts the computational domain containing the two stars. It also shows the line of centers and various arrows indicating the different viewing angles in respect to which we investigate the nonthermal high-energy emission of the system.

In Section 3.6 we repeat the analysis of case A, now including effects of orbital motion. Again, the two stars are located initially on the axis and move on a Keplerian orbit in the plane.

### 3.2 Properties of the Wind Plasma

As expected, the three different stellar separations show considerable contrasts in terms of properties and structure of the WCR. In Figure 3 we show density, absolute velocity and temperature in the - plane at for cases A, B, and C.

Shape of WCR The thickness of the WCR at the apex widens considerably with increasing stellar separation. Along the line of centers, its width is 100, 170 and 310 for cases A, B, and C, respectively. Analogously, the distance between the WCR and the B star (which is generally closer because of ) increases. The star is located 140, 290 and 600 from the edge, and 190, 380 and 760 from the center of the WCR for cases A, B, and C. Positions and curvatures of the WCR in all models agree well with the analytical approximation by Stevens et al. (1992) in which the distance of the apex of the WCR from the B star is determined by and the curvature of the downstream WCR is approximated by solving the differential equation

 dydx=(√ηd2WR+d2B)y√ηd2WRx+d2B(x−d). (20)

The resulting curves are indicated in the second row of Figure 3. Contrasting the effect of increasing WCR thickness with increasing stellar separation, the opening angle (or curvature) of the collision region significantly decreases. Among the three systems under investigation, case A has the narrowest WCR at the apex, but the widest WCR close to the edge of the computational domain.

Density The maximum number density values within the WCR are 33.1, 7.6 and 1.8 m for cases A, B, and C. The approximate inverse proportionality to is understandable from the fact that, for a strong shock, and . The density contrast of the unshocked winds manifests itself in comparable contrasts within the WCR where the side toward the WR star shows considerable higher density values of the shocked wind plasma.

Temperature The maximum temperature reached at the apex of the WCR shows only little variation. It is 2.14, 2.17 and 2.21 10 K for cases A, B, and C. In contrast to that the temperature gradient is far more affected by the change of stellar separation. The closer the stars, the larger the drop in temperature with increasing distance from the apex of the WCR. The minimum temperature of the shocked wind plasma reached at the edge of the computational domain varies accordingly, with values of 6.9, 20.9, 61.7 K for cases A, B, and C.

Absolute velocity In all three systems, the wind is slowed down to near zero velocity at the apex. We observe a considerable difference concerning the size of the region in which the velocity remains comparatively low. The distance downstream from the apex at which the shocked wind plasma is efficiently re-accelerated is significantly shorter for the case of smallest stellar separation. Another notable difference is the maximum velocity reached within the computational domain in the unshocked wind outside the WCR. For case A, the B wind close to the WCR at the edge of the domain reaches values up to 5100 km s in contrast to merely 4700 and 4300 km s for case B and C. The reason for these velocity values which lie all above the B wind’s terminal velocity of 4000 km s is the radiative wind acceleration mechanism which allows additional acceleration of the unshocked B star’s wind due to the radiation of the WR star. The effect increases with increasing proximity of the WR star, thus it is greatest for case A.

Other properties The local magnetic field strength and the energy density of radiation are two important properties that critically determine the energy distribution of the particles accelerated at the shocks by influencing energy loss by synchrotron emission and IC cooling. In estimating the magnetic field strength in dependence of the distance of the stars we rely on the approximations following Usov & Melrose (1992) (details also in paper I). The respective values at the center of the WCR for cases A, B and C are 0.64 G, 0.16 G, and 0.04 G.
The energy density of radiation is proportional to the luminosity of the stars and the inverse square of their distance. Its values at the center of the WCR are 0.77, 0.19, and 0.05 J m for cases A, B and C, respectively.

### 3.3 High-Energy Particles

Along with the hydrodynamic variables of density, velocity and temperature, we consider 200 advected scalar fields containing the number densities of high-energy electrons and protons at different energies accelerated at the shock fronts of the WCR. A transport equation for both electrons and protons in energy space is solved after every hydrodynamical time step. The injection rate of electrons and protons at the shock front is treated proportional to the number density of particles in the wind. An important free parameter is the electron-proton injection ratio for which we chose a typical value of 10 (for details, see paper I). Figure 4 depicts the spatial distribution of electrons and protons at two different energies for all three cases A, B, and C.

At energies of 10 MeV, electrons as well as protons reach higher densities for smaller stellar separation. This is due to the proportionality of the electron and proton injection rate to the wind density. The most important difference between the three cases lies in the maximum energy attained by electrons. Whereas electrons of 100 GeV are distributed throughout most of the WCR for case C, they are confined to large stellar distances for case B or even vanish completely for case A. This has two reasons. For smaller stellar separations, the winds collide long before reaching their terminal velocity. This produces lower shock velocities . As the diffusive shock acceleration term depends on , the maximum attainable energies are significantly reduced for small stellar separations. In addition, close proximity to the stars leads to much higher energy densities of radiation and magnetic field. Together with higher wind plasma densities in the WCR, this greatly increases IC losses, synchrotron losses and bremsstrahlung losses.

As protons are not affected by these loss terms, energies of 100 GeV are reached for all three cases. However – mainly due to higher downstream velocities – they cannot efficiently propagate into the inner downstream regions of the WCR for low stellar separations. This leads to a notable drop in number density of high-energy protons from the shock fronts toward the inner WCR which is most notable for case A. For case C, low downstream velocities allow the protons to fill the WCR homogeneously.

### 3.4 The 3D Opacity Structure

We illustrate the spatial variation of the optical depth due to photon photon absorption caused by the radiation fields of the two stars in Figure 5 which depicts 3D illustrations of for an incident photon energy of 200 GeV. All four images represent case B for different orientations of the binary system relative to the observer as indicated in the figure. They cover the entire computational domain.

Each point shows the total value of integrated from the point itself along the line of sight toward the observer. Thus, the region that lies precisely behind a star with respect to the line of sight has a very high opacity. Its emitted flux is reduced to very low levels according to .

Photon photon absorption is most effective if the line of centers and the line of sight are aligned such that the opacity contributions of both stars add up. In Figure 5 b) we see that the high- region notably widens at the -position of the B star.

We find that the photon photon absorption in the radiation field of the stars remains inefficient below incident photon energies of 100 GeV. The value of 200 GeV – chosen for the illustration – is close to the energy where absorption is at maximum for the given binary system.

### 3.5 Photon Emission

Applying the formalism described in Section 2, we can now compute 2D projection maps, SEDs and integrated flux values of the nonthermal photon emission of the three discussed binary systems. Here, we start with the former of those.

#### 2D Projection Maps

Figure 6 illustrates how the integrated flux above 100 MeV from the individual nonthermal emission channels appears in the face-on configuration (0,0) for an ideal observatory at 1 kpc distance with infinite angular resolution and without absorption in the ISM. Several features noted previously in Sections 3.2 and 3.3 concerning the wind structure and the high-energy particle distribution have an impact on the resulting photon emission maps. The dominant emission channel shifts from IC-scattering to neutral pion decay as the stellar separation decreases. This is because of the lack of high-energy electrons when the WCR is close to the stars (case A).

Figures 7 to 9 provide further insight by illustrating the 2D projection maps of the individual nonthermal high-energy emission channels for different orientations of the system relative to the observer. The scaling of the colour bars has been kept identical in order to allow quick comparison. The first and second rows show a case where the line of centers and the line of sight are aligned. Note the different sizes of the occultation region due to the stellar disk for 90,0 (smaller WR star in front of B star) and 90,180 (larger B star in front of WR star).

Studying the first and second row clearly reveals the anisotropic character of the IC-component. Both orientations look identical (except for the occultation by the stellar disks) for photons from bremsstrahlung (Figure 8) and neutral pion decay (Figure 9). This is rather different for photons from IC-scattering where the relative angle of stellar positions and line of sight cause notable contrasts between the two orientations due to a different scattering angle (see first and second row in Figure 7).

The general shape and extent of the emission region becomes especially clear in the projection for an observer at 45,45 as it is shown in the third row of Figures 7 to 9.

One of the most striking features is the missing photon emission from IC scattering close to the apex of case A, which is due to the lack of high-energy electrons in that region. Again, the dominance of the IC-component for case C in contrast to the dominance of the pion decay component for case A becomes apparent.

#### Spectral Energy Distribution

We calculate the SED by summation over all emitting grid cells for a fixed energy throughout the computational domain. This is shown for case A, B, and C and various orientations in Figure 10 a), b) and c). Figure 10 d) explores the variation of the IC-component for a single cell at the apex of the WCR for case B as a function of orientation. It also discriminates between the two components of IC-emission, one being due to the scattering of electrons on the radiation field of the B star (red dotted curves), the other due to scattering on the radiation field of the WR star (dashed black). Whereas the first one is maximal at (90,0) and disappears at (90,180), the other behaves in the exact opposite manner. This is due to the dependence on the scattering angle . For the case (90,0) the scattering angle of photons from the B star is 180 . The corresponding IC emission from the apex of the WCR is therefore at its maximum. For the photon field of the WR star the scattering angle is zero and there is no emission from the apex via IC scattering. The spectra of photons by neutral pion decay and bremsstrahlung remain the same, as they do not depend on the scattering angle. This last statement excludes the effects of photon photon opacity which is not taken into account in Figure 10 d). Studying the spectra in Figure 10 (a) to (c) (for case A to C) significant differences between small and large stellar separation become apparent.

Case A shows a neutral pion component which dominates the spectrum at E 100 MeV for all orientations. The lack of high-energy electrons near the apex of the WCR leads to an early cutoff of the IC-component at 10 MeV, followed by a steady decline where high-energy electrons in the wings of the WCR contribute a low number of photons at higher energies. It is interesting to note that amongst the three depicted orientations, the IC flux is highest for (, ). The same can be seen in Figure 7 by comparing the first and second row for case A. Because of the lack of high-energy electrons at the apex, it is the wings of the WCR that mainly contribute to the IC emission at energies 100 MeV. In these regions, the scattering angle for the radiation field of the WR star is much more favourable for (, ) than for (, ), whereas the scattering angle for the B-star is unfavourable for both. This is an effect of the curvature of the WCR and the different distance from the WCR to the stars. Figure 10 a) also shows effects of photon photon opacity in the spectrum of the neutral pion component which completely dominates the emission at high energies. Owing to the denser radiation field of the WR star, opacity effects are highest for the orientation (, ) in which emission from the apex has to pass by close to the star. The higher temperature of the star also causes the onset of photon photon absorption at lower energies than for the orientation (, ) for which high-energy photons from the apex come close to the cooler B star. Due to the gap between the pion-bump and the cutoff of the IC spectrum, the total emitted photon spectrum shows a pronounced dip at 50 MeV.

Case B is already quite different from case A. Due to higher energies reached by the electrons, the cutoff in the IC component occurs at higher photon energies, causing a broad overlap with the neutral pion component. At the same time, lower plasma densities in the WCR lead to a lower flux for the bremsstrahlung and neutral pion component. The maximum energy of photons by bremsstrahlung increases with the maximum electron energy. As some regions in the wings of the WCR now produce sufficiently high electron energies, both, the IC and the bremsstrahlung component reach up to 1 TeV. Effects of photon-photon opacity become more pronounced for these components also. Again, they are largest for the case (, ) where the emission from the WCR has to pass by the luminous WR star.

For case C, a large population of high-energy electrons leads to complete dominance of the IC-component. Whereas lower plasma densities lead to still lower flux values for the neutral pion and the bremsstrahlung component, the latter along with the IC component again reaches energies of up to 1 TeV. As the IC flux also decreases with increasing stellar distance (and thus decreasing energy density of radiation), it shows lower flux values with respect to case B below 1 GeV. For higher energies, flux values unprecedented for lower stellar separations are reached. Effects of photon-photon opacity are visible for all three emission components. For the same reasons as above, they are strongest for (, ). Note that the IC component is now maximal for the case (, ) which can be understood by the increased density of high-energy electrons at the apex where the radiation field of the B star dominates. The latter has most favourable scattering angles for (, ) . Because of the effects of photon photon opacity, the face-on orientation shows the highest resulting IC flux at E100 GeV.

Comparing all three cases, we find that (for an electron-proton injection ratio of ) growing stellar separation leads to a transition from hadron-dominated emission (neutral pion decay) to lepton-dominated emission (IC) at energies E100 MeV. The latter is inhibited for close stellar separations because of a lack of high-energy electrons at the apex of the WCR. As an additional effect, different viewing angles produce significant variation in the IC emission spectra. These dependencies on stellar separation and viewing angle can be quantified using integrated flux values for various energy bands.

#### Integrated Fluxes

By integrating over energy and space, we obtain the total emitted flux, which we computed for three energy intervals. Table 2 provides flux values for three different orientations for case A, B, and C. The main characteristics in this table will be further discussed below.

The flux above 10 GeV As suggested by the spectral shapes discussed in the previous section, the dominant radiation process at E 10 GeV changes with stellar separation. From Table 2 one can assess that, roughly, the total IC and bremsstrahlung fluxes increase by 3 orders of magnitude from case A to C, whereas the neutral pion component decreases on a much smaller scale. Only for case C, the IC-component dominates over the neutral pion component. Studying effects of changing orientation, we find that the variations in the bremsstrahlung and neutral pion component due to changing magnitude of photon photon opacity are very low. For the IC component, its anisotropic nature as well as the larger effect of absorption cause flux variations due to the orientation of the system of a factor of 3 for case A and lower factors for cases B and C.

The flux above 100 MeV Including also lower energies down to 100 MeV, we find that growing stellar separation has the opposite effect than for the previous case. For E10 GeV, higher maximum energies for electrons for large stellar separation led to an increase of the IC and bremsstrahlung photon flux with increasing stellar separation. At lower energies, higher plasma densities in the WCR, along with denser radiation fields for smaller stellar separations, reverse this order and cause IC and bremsstrahlung photon fluxes to decrease with increasing stellar separation. For E100 MeV this transition is only partly realised, as we still find an increase in flux from case A to B for the IC component. However, all components decrease in their average flux from case B to C. There is no such reversal of proportionality for the neutral pion component. We also find that the IC component is now dominant for cases B and C, albeit for some orientations of case B that are least favourable for IC emission, it is just a factor of 1.2 above the neutral pion component. For case A, the latter still dominates all possible orientations. Variations due to changes in the viewing angle cause differences of a factor of 8 in the IC flux component for case A with a smaller contrast for cases B and C.

The flux above 1 keV Including still lower energies, all components now show the identical trend that increasing stellar separation causes decreasing flux values. Furthermore, the IC component dominates for all three cases. The total flux of the neutral pion component is even below the total flux of the bremsstrahlung component. This can be understood in terms of the lack of photons from pion decay below a few MeV due to the shape of the pion bump. Effects of varying orientation are still lower (factor 2) . However the previously observed order of larger fluxes for orientation (90,180) than (90,0) in case A and the opposite trend in case C remains in place.

### 3.6 Effects of Orbital Motion

We finally address the impact of circular orbital motion and the ensuing deformation of the WCR. As the effects increase with orbital velocity, we choose the case of smallest stellar separation (case A) and let it evolve until the WCR adjusts to the new situation and there is no further change in its curvature. We analyse the system after 1.5 orbits when the stars are once more on the axis. For the given stellar masses and a distance of 720 R, a Keplerian orbit has a period of 290 days. The orbital velocity of the stars is 263 km s. With orbital motion, the two arms of the WCR (henceforth forward arm and trailing arm) develop considerable differences.

Figure 11 shows density, temperature and velocity, as well as particle distributions for different energies on the plane (the stars move counter-clockwise). Concerning the hydrodynamic wind variables, the main difference to the stationary case (see Figure 3) lies not in minimal and maximal values but in the now asymmetric shape of the WCR. In paper I, we found that the significant contrasts in the particle distribution between forward and trailing arm are mainly due to the different angle between the stars and the edge of the WCR, resulting in different shock velocities and (as ) in different maximum particle energies. For the present case, the ensuing reduction of the DSA rate in the trailing arm of the shock front toward the B star results in a lack of electrons already at 1 GeV (see center plot of Figure 11). In the forward arm, the opposite effect occurs. As wind velocity components are slightly higher close to the edge of the computational domain in the forward arm, an increased DSA rate can accelerate electrons to energies up to 100 GeV. (Note the thin shaded area at -6 log(MeVm) in the center right plot of Figure 11.) There, electrons reach higher energies than in the case where orbital motion is not taken into account. Although we also find a certain degree of asymmetry for protons, the effects of orbital motion are less severe.

Figure 12 shows the resulting projected photon emission maps for different orientations with respect to the observer. To simplify comparison of the case including orbital motion with the stationary case, we compensate the reversed order of the stars along the axis by a redefinition of the angle : and use the same representation of projection maps as discussed previously for .

For all three components the face-on orientation (first row) shows considerable asymmetry due to the lack of high-energy particles in the trailing arm of the shock front toward the B star. This produces the apparent lack of emission in the upper left section of the plots. In contrast, a wealth of particles in the trailing arm of the shock front toward the WR star produces significant emission that even dominates the entire emission region for the IC component.

In the case of alignment of the line of centers and the line of sight (second and third row), the differences of trailing and forward arm manifest themselves most strikingly for the IC component. The circular feature bordering the low-emission center region (most prominent for 90, 180, third row) is caused by the emission of the forward arm toward the B star touching the edge of the computational domain. If seen from the opposite direction (second row), it is highly suppressed because of an unfavourable scattering angle. The lack of emission next to this feature is due to the low number of high-energy electrons in the forward arm towards the WR star. This becomes also apparent in the corresponding plots for bremsstrahlung and neutral pion decay. For the same reasons as in the stationary case, we find that the configuration (90,180) exhibits considerably higher flux for the IC component than the configuration (90, 0).

A comparison of the spectra with and without orbital motion is shown in Figure 13. We find a decrease in flux for the neutral pion decay and the bremsstrahlung component. Concerning the IC component we find a new spectral feature emerging at 100 MeV that persists for all three depicted orientations. We interpret this as the influence of the forward arm toward the B star. There, a population of electrons reaching up to 100 GeV is still sufficiently close to the stars in order to exhibit favourable conditions for IC emission. To a lesser degree, the same region also influences the bremsstrahlung component which shows a similar feature at 1 GeV. The above shows that orbital motion has considerable impact on the resulting non-thermal high-energy emission.

## 4 Discussion

We have computed different components of nonthermal high-energy photon emission in colliding wind binaries – for the first time on the basis of a 3D distribution of high-energy particles that was obtained by hydrodynamic simulations and the simultaneous solution of the transport equation. Our approach includes the radiative acceleration of the stellar winds, radiative cooling in the hot shocked gas, diffusive shock acceleration at the shock fronts bordering the WCR and various energy loss mechanisms affecting the high-energy particles. The resulting nonthermal high-energy photon emission has been explored in terms of 2D projection maps, SEDs as well as integrated flux predictions of nonthermal photon emission components based on 3D distributions of high-energy particles in colliding wind binary systems.

In computing the anisotropic IC-component of the photon emission we take into account the dependence on the scattering angle, as well as the radiation fields of both stars of the binary system. Bremsstrahlung and neutral pion decay components of the photon flux are computed using local wind plasma densities and the present distribution of electrons and protons. The effect of photon photon opacity in the dense radiation fields of the stars has been taken into account for all three nonthermal photon emission mechanisms.

Studying three cases of varying stellar separation, we find significant differences in terms of total flux value and also in the identity of the dominant emission component. We showed that low stellar separations (720 R for the studied WR-B system, case A) inhibits the acceleration of high-energy electrons at the apex of the WCR due to strong IC and synchrotron losses. This produces an early cutoff in the photon flux ensuing from IC-scattering and bremsstrahlung. Although denser radiation fields and higher plasma densities cause higher photon flux values at lower energies, this early cutoff leads to dominance of the hadronic neutral pion decay component at energies E 100 GeV for the chosen electron-proton injection ratio of . In contrast to that, large stellar separations lead to a dominance of the IC component throughout the studied energy range. As the magnetic and radiation energy density at the WCR are low, the electrons – and also the ensuing photon fluxes – reach higher energies. Both, IC and bremsstrahlung components reach up to 1 TeV if we increase stellar separation by a factor of four (case C). Lower wind plasma densities in the WCR further diminish the significance of the neutral pion decay component. We find that increasing the stellar separation by a factor of four results in an increase of 3 orders of magnitude in the IC and bremsstrahlung components of the photon flux at energies E10 GeV, whereas the neutral pion decay component decreases by about an order of magnitude. For lower energies the importance of high-energy electrons diminishes. Accordingly, the total flux for E1 keV decreases for all three flux components if we increase stellar separation by a factor of four.

The strong dependence on stellar separation suggests that -ray binaries are liable to show a significant variation in their spectra in course of an elliptical orbit with high eccentricity. Spectra can exhibit multiple emission components (as in Figure 10 a)) for low separations (i.e. during periastron), contrasted by longer periods of one-component spectra (as in Figure 10 c)) for larger stellar separations. We note that a situation resembling this trend of two apparently distinct emission components has already been observed for the peculiar case of Carinae (see Abdo et al., 2010; Farnier et al., 2011). However, this can also be understood in terms of -ray absorption due to the presence a hot X-ray gas (see Reitberger et al., 2012). The application of our numerical model to other specific candidates for particle-acceleration CWB systems (see catalogue of De Becker & Raucq, 2013) will provide predictions for the nonthermal high-energy flux that can then be related to detections and upper limits from observations.

Varying the orientation of the binary system by changing the viewing angles and , we find a strong dependence of the IC flux component. Photons from bremsstrahlung and neutral pion decay are influenced only indirectly via the changing conditions for photon photon opacity which affect all components at high energies alike. This effect remains small compared to the variation of the IC component due to its dependence on the scattering angle. Since we consider the radiation field of both stars, the IC component does not vanish as the flux from one population of target photons is maximal when the other is minimal. However, since the two radiation fields differ in intensity depending on distance and identity of the star, there remains a significant level of variation. It is largest for the case of small stellar separation where the B star is very close to the WCR. Here, the IC-flux above 100 MeV varies a factor of 8 between the two extreme cases of (, ) and (, ). Although this dependence is small for the cases we investigated, it becomes significantly stronger if we have a binary system with greater differences concerning the two stellar radiation fields and the ram pressure of the winds. The IC contribution to the -ray flux will differ by several orders of magnitude in course of an orbital cycle if the thermal photon density of one stellar component is dominant and the orbital inclination is close to the edge-on case of .

The effects of photon photon opacity in the radiation field of the star are restricted to incident photons with GeV. Given their low prominence in case of low stellar separation, there is no noticeable impact. However, for larger stellar separation, the absorption leaves a clear signature in the spectral components of IC scattering and bremsstrahlung.

Studying the impact of orbital motion for the case of smaller stellar separation, we find a number of effects that are all significantly less severe than the ones obtained for varying the stellar separation by a factor of two. The distortion of the WCR leads to regions of altered wind velocity normal to the shock and, thus, to notable changes in the DSA acceleration rate. Whereas IC emission is greatly reduced in the trailing arm toward the B star (and also in the forward arm toward the WR star), we find regions of higher maximum energies for electrons (than in the case without orbital motion) in the side of the forward arm toward the B star. In terms of SEDs, this leads to an additional bump in the IC component. The changes due to orbital motion are less pronounced for the bremsstrahlung and neutral pion decay components.

## 5 Outlook

The presented numerical model can be used to study the high-energy nonthermal flux of specific particle-accelerating CWB systems as a function of time. Possible variations in course of the orbit can be quantitatively assessed and favourable observation periods can be predicted. Compared to observational -ray data (e.g. from the Fermi-LAT instrument), our simulation aims for constraining and refinement of poorly known parameters, such as as the injection fraction of electrons and protons at the shock of the wind collision region. Dependencies on other parameters, such as the magnetic field and the diffusion coefficient, can be studied as well.

The consequent next step will be the application of our simulation procedure to archetypal nonthermal binary systems. We will confront hardly understood discrepancies between existing flux predictions from analytical models with the present absence of detections of particular enigmatic systems in the Fermi-LAT data like e.g. WR104 or  Carinae.

This work is supported through the EU FP7 program by Marie Curie IRG grant 248037; K.R. is supported by the Marietta Blau-Stipendium der OeAD-GmbH, financed by the Austrian Ministry of Science BMWF. AR acknowledges financial support through the Austrian Science Fund (FWF) grant P 24926-N27. Additional support was given by the Austrian Ministry of Science BMWF as part of the UniInfrastrukturprogramm of the Research Platform Scientific Computing at the University of Innsbruck and the Austrian Science Fund (FWF) supported Doctorate School DK+ W1227-N16.

### References

1. Abdo, A. A., Ackermann, M., Ajello, M., et al. 2010, ApJ, 723, 649
2. Aharonian, F. A. & Atoyan, A. M. 2000, A&A, 362, 937
3. Benaglia, P. & Romero, G. E. 2003, A&A, 399, 1121
4. Blumenthal, G. R. & Gould, R. J. 1970, Reviews of Modern Physics, 42, 237
5. De Becker, M. & Raucq, F. 2013, ArXiv e-prints
6. Eichler, D. & Usov, V. 1993, ApJ, 402, 271
7. Falceta-Gonçalves, D. & Abraham, Z. 2012, MNRAS, 423, 1562
8. Farnier, C., Walter, R., & Leyder, J.-C. 2011, A&A, 526, A57+
9. Gould, R. J. & Schréder, G. P. 1967, Phys. Rev., 155, 1404
10. Kelner, S. R., Aharonian, F. A., & Bugayov, V. V. 2006, Phys. Rev. D, 74, 034018
11. Madura, T. I., Gull, T. R., Okazaki, A. T., et al. 2013, MNRAS, 436, 3820
12. Pittard, J. M. 2009, MNRAS, 396, 1743
13. Pittard, J. M. 2010, MNRAS, 403, 1633
14. Pittard, J. M. & Parkin, E. R. 2010, MNRAS, 403, 1657
15. Pollock, A. M. T. 1987, A&A, 171, 135
16. Reimer, A. et al. 2006, MNRAS, 644, 1118
17. Reitberger, K., Kissmann, R., Reimer, A., Reimer, O., & Dubus, G. 2014, ApJ, 782, 96
18. Reitberger, K., Reimer, O., Reimer, A., et al. 2012, A&A, 544, A98
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23. Williams, P. M., van der Hucht, K. A., & Spoelstra, T. A. T. 1994, A&A, 291, 805
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4

For both schemes: I am going to lump $B$ and $C$ into a single matrix $B$ for convenience of communicating the idea. Likewise, $y$ and $z$ are now $y$, and $F_y$ and $F_z$ are $F_y$. $Ax+B^Ty=F_x$ or $x = A^{-1}(F_x-B^Ty)$ $Bx=F_y$ or $BA^{-1}(F_x-B^Ty)=F_y$ $BA^{-1}B^Ty=BA^{-1}F_x-F_y$ Note that $BA^{-1}B^T$ is symmetric and invertible, and can be ...

4

One approach to convert this into an ODE is with index reduction methods. These allow you to convert high-index DAEs into low-index DAEs or ODEs. See section VII.2 of "Solving Ordinary Differential Equations II" by Hairer and Wanner. Consider a generic, Hessenberg index-2 DAE \begin{align} y' &= f(y, z) \\ 0 &= g(y) \end{align} ...

3

The smooth solution turned out to have BC's applied in the following way: Walls and inlet: $\frac{\partial p}{\partial n}=0$ Outlet: $p=0$ Actually thought that we need only one value of P to pin, not really outlet P = 0; can take any point, the reason behind is that infinite number of answers are imposed by all Neuman conditions, need one point to fix the ...

3

There are many variations of the idea of Taylor-Hood elements that remain stable. The Taylor-Hood element on quadrilateral and hexahedral elements are generally understood (though historically incorrect) to be the space $Q_2\times Q_1$ for velocity and pressure. But this is not optimal: We can make the pressure space larger and instead use $Q_2 \times P_{-1}... 2 Consider a FV method as an approximation of the integral conservation law. Starting from the one-dimensional, scalar conservation equation $$u_t + f(u,\nabla u)_x =0,$$ with the definitions of the step sizes$\Delta x$and$\Delta t$t_n = n \Delta t,~~~x_i = i \Delta x,~~~x_{i+1/2}=\frac{1}{2}\left(x_i + x_{i+... 2 Disclaimer: I wrote this answer in a rush, may not be up to the standards. Also, I don't have enough information about your problem to give more detailed advice. However, the information presented here should be enough to get you going. I will suggest you to use MINRES with an appropriate preconditioner. For a moment, let's assume that this is not a twofold ... 2 You get a divergence-free$Z_h$if you choose for example$V_h$and$Q_h$such that$\forall v_h\in V_h(div(v_h)\in Q_h)$. (An example for such a pair is the Scott-Vogelius element with$V_h=P^k,Q_h=P^{k-1}_{discont}$) Because if you take$v_h\in Z_h$it fulfills$(div(v_h),q_h)=0 \ \forall q_h\in Q_h $. Now by the special choice of$V_h$and$Q_h$above we ... 1 I don't see what you're not seeing. You've written out a perfectly good non-linear weak form of the PDE in your last two equations. If you invert$M$and apply it, then you have a non-linear equation for$\dot{u}(t)$that you can try to solve. It's a continuous in space and time PDE that you would need to discretize in space to then have only an ODE, with$M\$...

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# Websites

Attached you will find some helpful websites for your child to practice on.

homographs-https://www.quia.com/cb/96929.html

http://www.mathsisfun.com/numbers/math-trainer-multiply.html

http://www.mathplayground.com/multiplication01.html
http://www.superteachertools.us/jeopardyx/jeopardy-review-game.php?gamefile=1765770#.Vtm9IflVgSU

http://pbskids.org/sid/weatherwheel.html
http://www.sheppardsoftware.com/mathgames/fruitshoot/fruitshoot_subtraction.htm

http://mrnussbaum.com/speedmath/

http://www.kidsgeo.com/geography-games/latitude-longitude-map-game.php

http://www.harcourtschool.com/ss1/


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# Greens functions and density of states

1. Feb 22, 2010

### Niles

Hi guys

Here they say that whenever we write the Hamiltonian for a non-interacting system in its eigenbasis, then we have that

$$A(\nu, \omega) = 2\pi \delta(\omega-\varepsilon_\nu).$$

How can this statement be proven? Do you have any hints for this?

Any help will be greatly appreciated.

Sincerely,
Niles.

2. Feb 23, 2010

### Niles

Ok, perhaps I should reformulate my question: I keep reading that the imaginary part of the retarded Greens function is the density of states, and I am quite sure it has got something to do with

$$A(\nu, \omega) = 2\pi \delta(\omega-\varepsilon_\nu).$$

since if we sum over ν we get something proportional to the density of states on the RHS. SO I guess my question is: How do we show that in the eigenbasis of the Hamiltonian for a non-interacting system, the imaginary part of the Green's function is equal to the density of states?

3. Feb 24, 2010

### DrDu

I don't know where the eigenbasis should enter as the Greensfunctions do not depend on a basis. But for a free particle, the greens function is basically $$\sum (\omega - \epsilon_\nu)^{-1}$$. Now using $$1/x=P(1/x)+2\pi i \delta (x)$$ where P stands for taking the principal value, you should be able to derive your formula.

4. Feb 24, 2010

### ZapperZ

Staff Emeritus
You need to consider this as the spectral function of the single-particle propagator. To get the density of states, you sum up over all momentum. So you show that the result of the DOS matches the density of states for a free non-interacting electron gas.

Mattuck actually did this in his book "A Guide to Feynman Diagrams in the Many-Body Problem" for this very case. It is a Dover book, so it's cheap. It is also extremely useful.

Zz.

5. Feb 24, 2010

### Niles

In my book the Greens functions in real-space and momentum-space look different; so how it is that they cannot depend on the chosen basis?

The point is that in the book they only do it in momentum-space, and add as a side-note that "this in fact is true for all non-interacting fermions in any bases you like" - that is the statement I think needs more explanation.


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# Generalized Wick's theorem

1. Jun 12, 2007

### da_willem

I have the following contour integral form of Wick's theorem (C indicating contraction):

$$C[A(z):BC:(w)]=\frac{1}{2 \pi i} \int _w \frac{dx}{x-w} C[A(z)B(x)]C(w) + B(x)C[A(z)C(w)]$$

Does anybody know how to evaluate contractions like C[:AB:(z)C(w)]?

2. May 31, 2009

### Super_Leunam

Yes, it is outlined in Di Francesco's book "Conformal Field Theory" page 189: I'll give you a link to google books since there is a free preview of that chapter :

I imagine you found this in a takehome exercise sheet for a String theory course in the Netherlands (it was a takehome midterm exam at UvA)

i also know this is a very late reply but ... oh well :D

Last edited by a moderator: Apr 24, 2017


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# How to wrap the text within the width of the window in JavaFX?

JavaObject Oriented ProgrammingProgramming

In JavaFX, the text node is represented by the Javafx.scene.text.Text class. To insert/display text in JavaFx window you need to −

• Instantiate the Text class.

• Set the basic properties like position and text string, using the setter methods or, bypassing them as arguments to the constructor.

• Add the created node to the Group object.

If the length of the lines in the text you have passed, longer than the width of the window part of the text will be chopped as shown below −

As a solution you can wrap the text within the width of the window by setting the value to the property wrapping with, using the setWrappingWidth() method.

This method accepts a double value representing the width (in pixels) of the text. If you pass a value that is less than the width of the window, the text will be wrapped within it(the given width).

## Example

import java.io.FileInputStream;
import java.io.FileNotFoundException;
import java.io.InputStream;
import java.util.Scanner;
import javafx.application.Application;
import javafx.scene.Group;
import javafx.scene.Scene;
import javafx.scene.paint.Color;
import javafx.stage.Stage;
import javafx.scene.text.Font;
import javafx.scene.text.FontPosture;
import javafx.scene.text.FontWeight;
import javafx.scene.text.Text;
public class WrappingTheText extends Application {
public void start(Stage stage) throws FileNotFoundException {
//Reading the contents of a text file.
InputStream inputStream = new FileInputStream("D:\sample.txt");
Scanner sc = new Scanner(inputStream);
StringBuffer sb = new StringBuffer();
while(sc.hasNext()) {
sb.append(" "+sc.nextLine()+"\n");
}
//Creating a text object
Text text = new Text(10.0, 25.0, sb.toString());
//Wrapping the text
text.setWrappingWidth(590);
//Setting the stage
Group root = new Group(text);
Scene scene = new Scene(root, 595, 300, Color.BEIGE);
stage.setTitle("Wrapping The Text");
stage.setScene(scene);
stage.show();
}
public static void main(String args[]){
launch(args);
}
}

## sample.txt

Assume following are the contents of the sample.txt file −

JavaFX is a Java library used to build Rich Internet Applications. The applications written
using this library can run consistently across multiple platforms. The applications developed
using JavaFX can run on various devices such as Desktop Computers, Mobile Phones, TVs, Tablets, etc..
To develop GUI Applications using Java programming language, the programmers rely on libraries
such as Advanced Windowing Tool kit and Swing. After the advent of JavaFX, these Java programmers
can now develop GUI applications effectively with rich content.

## Output

Font Name: Brush Script MT

## Output

Updated on 14-Apr-2020 08:40:29


text:
# Broker-dealers, operational risk and CCPs

## The week on Risk.net, April 1–7, 2016

BROKER-DEALERS fear SEC leverage ratio plans

OPERATIONAL RISK out of control at US public-sector pension funds

CCPS are an "enormous risk" to the financial system, the Minnesota Fed hears

COMMENTARY: Centrally planned

Too clever, and possibly Communist: it sounds like an attack on US presidential hopeful Bernie Sanders. In fact, it is criticism of the idea that common equity should be supplemented with a layer of loss-absorbing debt – a mix known as total loss-absorbing capacity (TLAC).

In TLAC theory, a fatter layer of equity makes banks less likely to fail, and the ability to write down – or bail-in – outstanding bonds means spillover losses can be mopped up without taxpayer support.

In practice, bail-in is untested and Adam Posen, a former member of the Bank of England's rate-setting committee, worries it will not work.

"I really don't buy TLAC. I do not buy the idea of conditional capital. We had this wonderful idea ... well, it's all too clever by half," he told a symposium at the Federal Reserve Bank of Minnesota this week.

Posen is not alone. In a Risk.net interview, Thomas Hoenig, vice-chair of the Federal Deposit Insurance Corporation, argued TLAC should not exist – instead, equity should "do its job" – and he attacked one possible outcome of the regime, which is a requirement for some banks to borrow more.

"That's not a market at work – that is a central planner telling everyone how they should operate and I think that's bad," said Hoenig.

Regulation was being second-guessed elsewhere as well, as banks concluded the trading desk structures they drew up for compliance with the Volcker Rule would not be fit for the purpose of calculating regulatory capital.

And mooted new capital requirements for non-bank broker-dealers could cause a kind of financial whiplash in the US Treasury market. Banks have already been hit by new leverage ratio requirements, which gave non-bank broker-dealers an advantage - leading, as Risk.net reported last year, to the insurgents dominating Treasury volumes on at least one major platform. But now the tables could be turned again, with the threat of new leverage limits on non-banks potentially restricting their activities. This might give banks the chance to re-enter the market on better terms or it might further strain liquidity in an already fragile market.

STAT OF THE WEEK

A bank maintaining at least 50,000 reported open positions in interest rate swaps each month would save more than $1 million a year under CME's fee schedule for trade reporting, which includes an annual cap of$250,000 per asset class, analysis conducted by Risk.net reveals.

QUOTE OF THE WEEK

"No single rule served to concentrate risk more than risk-based capital requirements" – Thomas Hoenig, FDIC vice-chair

ALSO THIS WEEK

Buy side targets leverage cap in SEC derivatives plans
AQR says its managed futures fund could suffer larger drawdowns under new rule

Capital sums set to drive FRTB desk decisions
Using Volcker desk structure may hurt model approval chances, banks say

When it comes to correlation, cleaning is a chore that pays
Recent trends in research may help firms obtain reliable correlations from limited data

Basel Committee to amend leverage ratio calculation
CEM to be ditched, but regulators still considering treatment of client margin

ABN Amro cuts rates swaps from €24bn covered bond programme
High cost of downgrade triggers forces Dutch bank to consider alternative hedge

Questions remain after final EU uncleared margin rules
Industry gives a positive welcome, but the treatment of non-netting jurisdictions is still uncertain


text:
# How to change font on definition's optional title, but not on theorem's optional titles?

My preamble is the following:

\newtheorem{teo}{Theorem}[subsection]
\theoremstyle{definition}
\newtheorem{defi}{Definition}[subsection]

The default is set so

\begin{defi}[Some defi]
\end{defi}

\begin{teo}[Some theorem]
\end{teo}

Outputs both titles [Some xxx] between parentheses. I want to change only the definition title to bold and remove its parentheses. I have tried the following:

\makeatletter
\thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2}}%
\thmnote{ {\the\thm@notefont\textbf{#3}}}}
\makeatother

Which correctly changes the title font to bold and removes parentheses, but in all cases. I have also tried,

\makeatletter
\def\th@definition{%
\normalfont % body font
}
\makeatother

Which leaves theorem titles as they are, changes definition title font to bold but keeps parentheses.

Finally I tried,

\makeatletter
\thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2}}%
\thmnote{ {\the\thm@notefont\textbf{#3}}}}
\makeatother

Which doesn't work at all. Any idea on how to proceed?

• Welcome to TeX SX! I suppose your preamble also loads amsthm. You should take a look at the thmtools package, which provides powerful tools to customise the layout of theorem-like structures, and cooperates with amsthm as well as ntheorem. – Bernard Jun 9 '17 at 13:32

The following code uses \pdfstrcmp to condition on whether \thmhead is called as a definition or a theorem. That way you can format things to suit your needs based on the environment:

\documentclass{article}

\usepackage{amsthm}

\newtheorem{teo}{Theorem}[subsection]
\theoremstyle{definition}
\newtheorem{defi}{Definition}[subsection]

\makeatletter
\thmname{#1}\thmnumber{\@ifnotempty{#1}{ }\@upn{#2}}%
\ifnum\pdfstrcmp{#1}{Definition}=0
\thmnote{ {\the\thm@notefont\textbf{#3}}}%
\else
~\thmnote{{\the\thm@notefont (#3)}}%
\fi
}
\makeatother

\begin{document}

\section{A section}

\subsection{A subsection}

\begin{defi}[Some defi]
\end{defi}

\begin{teo}[Some theorem]
\end{teo}

\end{document}


text:
# Math Help - Number formed from different number of digits.

1. ## Number formed from different number of digits.

Q27: The sum of all possible two digit number formed from three different
one digit natural number when divided by the sum of the original three
number is equal to:?
A. 18
B. 22
C. 36
D. None of these
how....................?

Q27: The sum of all possible two digit number formed from three different
one digit natural number when divided by the sum of the original three
number is equal to:?
A. 18
B. 22
C. 36
D. None of these
how....................?

Let $a,b,c$ be three different digits, so the sum of the different 2-digit numbers formed with them is:

$(a\times 10 + b)+(b\times 10 + a)+(a\times 10 + c)+(c\times 10 + a)+(b\times 10 + c)+(c\times 10 + b)=$

$=2(a+b+c)(10+1)=$...so the answer is B.

Tonio


text:
Dean T.

# Express the complex number in polar form

Express the complex number in polar form and use DeMoivreʹs Theorem to find the given power. Write your answer in

29) (- 3 + i)6

Bobosharif S.

Is this (-3+i)6?
Report

04/29/18

Dean T.

Yes.
Report

04/29/18


text:
# Set of numbers from $1-9$ multiplied together to get the smallest possible value

The numbers $$x_1,$$ $$x_2,$$ $$x_3,$$ $$y_1,$$ $$y_2,$$ $$y_3,$$ $$z_1,$$ $$z_2,$$ $$z_3$$ are equal to the numbers $$1,$$ $$2,$$ $$3,$$ $$\dots,$$ $$9$$ in some order. Find the smallest possible value of $$x_1 x_2 x_3 + y_1 y_2 y_3 + z_1 z_2 z_3.$$

I would assume the lowest number, $$1,$$ would have to be multiplied by $$9,$$ the highest. I do not know how to approach this with AM-GM, though.

• You in fact can have a proof by exhaustion... A dirty trick – user12986714 May 10 '20 at 1:22
• Sorry, is there a slick way to do this? I don't really want to brute force, though I can. – Frost Bite May 10 '20 at 1:23
• I don't want to brute force this problem, but rather to do it smartly. – Frost Bite May 10 '20 at 1:30
• $\sqrt[3]{9!}\approx 70.327$ and $3\times\sqrt[3]{9!}\approx 213.98$, so $72+72+70=214$ should be minimal – Daniel Mathias May 10 '20 at 1:34
• $9\cdot8\cdot 1 + 6\cdot 5 \cdot 2 +4\cdot3\cdot 7= 72+60+84=216$ is also pretty close. – mechanodroid May 10 '20 at 1:53

The following Mathematica script confirms that there is only one solution:

prod[p_] := p[[1]] p[[2]] p[[3]] + p[[4]] p[[5]] p[[6]] + p[[7]] p[[8]] p[[9]];
perms = Permutations[{1, 2, 3, 4, 5, 6, 7, 8, 9}];
unique = Select[perms, (#[[1]] < #[[2]] < #[[3]] && #[[4]] < #[[5]] < #[[6]] && #[[7]] < #[[8]] < #[[9]] && #[[1]] < #[[4]] < #[[7]]) &];
products = Map[prod, unique];
min = Min[products];
result = Select[unique, (prod[#] == min) &];
Print[min, " ", result]

The script prints:

214 {{1, 8, 9, 2, 5, 7, 3, 4, 6}}

The next best two solutions are:

215 {{1, 7, 9, 2, 5, 8, 3, 4, 6}}
216 {{1, 8, 9, 2, 5, 6, 3, 4, 7}, {1, 8, 9, 2, 6, 7, 3, 4, 5}}

This isn't as systematic or as linear as I'd like but:

$$\frac {M+N+P}3 \le \sqrt[3]{MNP}=\sqrt[3]{9!}\approx 71.3$$ and equality is closest when the $$w_1w_2w_3$$ clusters are each closet to the geometric mean of $$71.3$$.

So we must have one term contaning the $$9$$. That is, wolog, $$9x_1x_2 \approx 71.3$$ so $$x_1x_2 \approx 7.9$$. Clearly the best option is $$x_1x_2 = 8$$ and $$x_1,x_2 = 2,4$$ or $$1,8$$.

$$1,8$$ has the advantage as it simultaneously finds clusters for the far off values of $$1$$ and $$8$$ as well.

In any event we also have to find, wolog, $$7y_1y_2 \approx 71.3$$ so $$y_1x_2\approx 10.1$$ and the best solution for that is $$7,2,5$$.

So with $$9,1,8$$ and $$7,2,5$$ that leaves $$z_1, z_2, z_3 = 3,4,6$$ and $$3*4*6=72$$ with our clusters being $$70, 72, 72$$ around our geometric mean of $$71.3$$ the only possible smaller sum would be be a tighter cluster and the only tighter cluster that is smaller it $$71,71,71$$, which is not an option.

So we can be assured that is the least.

We were "lucky" though

Still if say we had gotten something pretty close first, say,

$$9*8*1 + 6*5*2 + 7*4*2 = 72 + 60 + 84$$ we could see if we can improve it be attempting to finding a tighter cluster of values closer to $$71.3$$, than $$60$$ and $$84$$ are. The contenders that can formed by trios less than or equal to $$9$$ are:

$$63=7*1*9; 64=8*2*4; 70=7*2*5; 72=9*8*1=9*2*4=6*3*4;80=8*2*5$$ ... and it would take brute force to choose among those. But even among that we want to get as close to $$71.3$$ as possible and we have $$3$$ ways to do $$72$$ and one way to do $$70$$.... well, that screams to uss to look if two of the ways to do $$72$$ are mutually compatible with the way to do $$70$$.


text:
# How do/did you go about learning math? [closed]

I have wanted to improve in math, so I could do computer science, and generally develop math skills. However, the process has been chunky. I can summarize it into these three issues: Understanding, Slow learning, Note taking. But i'd like to hear more about how you handled these things.

Understanding: I'm not good at figuring out problems myself, and things often go over my head. I need to get what i'm thinking about to be able to take my notes, so often i'll write something and not have a clue what it means or what i'm doing. For example, I was learning how to multiply decimals. 3.2 x 3.9. The answer was 12.48, but I kept getting 60. For weeks I didn't know what I was doing wrong. I crossed multiplied like they asked, but I kept getting 45 + 15 = 60. I asked for help online, and realized- I forgot place value. It was quite disappointing to figure out that I couldn't figure something like that out myself, but it wasn't asking about it that made me sad; It was the fact that I was trying to improve my problem solving/intuition skills by figuring out how to solve the problem, and everything went over my head, again. What would you suggest for developing intuition, or being able to figure out what you're doing wrong yourself? Did you struggle with this, and what did you do?

Slow learning: I take more time to learn something than I do to forget it. I often take weeks/months to get a snippet of a topic, and that doesn't usually work when the curriculum needs you to get it in 2 weeks for the test, so I try to cram and-fail. This has more to do with Understanding + Note taking, however, but I wanted to know if there are any slow learners here and what you do to be able to be sharp and progress in math, while still being legible for college snap-snap curriculum pace. What would you/do you do?

Note taking: Memorization is near impossible for me because my notes are horrible. I have to take them by hand, and generally there's a lot of problems with them. My teacher goes quite fast (or, more likely-i'm too slow) so my notes are often scribbly, or don't have text to explain them at all. I'll simply copy what my teacher wrote on the board, yet not getting a single part of the concept or why it works. I won't know where to place certain types of knowledge so the placement of my notes is also difficult, I might write something incoherent in a panic in some random page and not be able to retrieve it, tips and tricks are placed in irrelevant spaces, it's just generally difficult and unusable. While no one writes their notes the same way, my notes generally seem to not work for me. How do you write your notes, what would you say defines terrible notes for you, and what do/did you do to improve and prevent them?

How do you go about with learning how to learn math? What is your process, what do you believe defines a bad math learning process, and how would you improve that process? I'd like to hear what you would do. Thank you for reading, your input is welcome.

## closed as off-topic by José Carlos Santos, John Doe, Lord Shark the Unknown, Mostafa Ayaz, user223391 Mar 16 '18 at 23:22

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Seeking personal advice. Questions about choosing a course, academic program, career path, etc. are off-topic. Such questions should be directed to those employed by the institution in question, or other qualified individuals who know your specific circumstances." – John Doe, Lord Shark the Unknown, Mostafa Ayaz, Community
If this question can be reworded to fit the rules in the help center, please edit the question.

• Khanacademy. ${}{}$ – user535339 Mar 16 '18 at 18:12
• Instead of memorizing try to understand why something is true. Don’t look at the answer of an exercise before really trying to do it by yourself. And when you feel like you understand something (hard, but you finally did it) try to throw away the sheet of paper on which you figured it out and do it all over again on a blank sheet of paper. – John Cataldo Mar 16 '18 at 18:17
• My main concern is the "narrative" you're convinced holds true for you. Everyone learns things differently, in some ways better than others. But you seem to have convinced yourself that "I'm not good at problem solving... I'm a slow learner... I can't take good notes. Are you willing to change that narrative/self description? Have you tried to restrict yourself to at most two pages for writing notes, while specifically trying to follow the lecture, explanations, examples? You're too young to draw the conclusions you have about yourself. Mainly, try to relax about math. You can do this! – Namaste Mar 16 '18 at 18:19
• You should consider joining, or starting, a study group for students taking the same course you are. You'll quickly learn you are not alone. Most campuses of on-campus tutoring for math classes: take advantage of some one-to-one time to hone your skills, and check your progress on assignments. – Namaste Mar 16 '18 at 18:22

My guess is that this question isn't a good fit for this site. You are asking for advice and this advice can be very subjective.

That said, a few general comments and thoughts for you:

1) There is nothing wrong with memorization and I am guessing that you are no worse at this than most. Have you memorized the small multiplication table? Try memorizing some formulas for areas and such.

2) Don't worry about note taking. Yes, in school we focus on having students take notes, but it is often way over emphasized. You need to internalize what you are learning. Having a nice well organized notebook can be helpful, but it means nothing if you can't do much without it.

3) Instead of trying to memorize your way out of everything, try to understand what is really going on. Take your time. It is better to spend a lot of time on one problem/concept and then fully understand this, than memorizing some "method" that you are just going to forget again. If you are having a hard time with multiplying two two digit numbers, then focus on the one digit numbers and get those down to a point where you feel comfortable. Then work you way up this way. This is take time, but you wouldn't have to start over and over again.

4) Don't worry about being a slow learner. I am a slow learner. I have often been in situations where a problem is posed. Everyone else gets the answer quickly and I still don't get it. But when I do get something I typically get it very well. The key thing is to not be satisfied with just knowing the answer to a problem. You have to understand the methods involved.

For example: You are trying to learn how to multiply $3.2 \cdot3.9$. You might have learned to do like $3$ times $3$ plus $3$ and the $9$ and so on... Instead, maybe try the following \begin{align} 3.2\cdot 3.9 &= (3 + 0.2)\cdot(3 + 0.9) \\ &= 3\cdot 3 + 3\cdot 0.9 + 0.2\cdot 3 + 0.3\cdot 0.9 \\ &= \dots \end{align} It is exactly the same as the rule you have learned, but it shows why you are doing it like that. You have to write a lot more, but that is ok.

Or: if you don't like the decimal, then maybe do the following $$3.2\cdot 3.9 = 32\cdot 39 \cdot 10^{-2} = \dots$$

Play around with the multiplication. To to rewrite the expression in various ways and you will see that you always get the same thing.

One last note: I know that many students struggle with the rigour of mathemtics. It is unforgiving. You make a simple mistake and your answer is completely wrong. Don't let this get to you. Keep trying and identify your mistakes and repeat repeat and repeat over and over again until you can do the same problem in your sleep. After you can do $3.2\cdot 3.9$, then do $3.3\cdot 3.9$, then $2.9\cdot 3.8$ and so on. Don't be afraid of repeating!


text:
• ## Bainite

It's not pearlite or martensite. A blog written by Mathew Peet.

## Kinetic energy recovery for buses

Anyone who has watched formula one racing and experience a bus journey will appreciate the similarities, as buses repeatedly accelerate and decelerate for bus stops and junctions in the same way as the formula one car is repeatedly changing acceleration to allow it to be steered around corners.

A kinetic energy recovery system (KERS) usually stores energy during breaking in a device such as a flywheel, battery or capacitor.

Torotrak has recognised this too, and sees a good fit between capability of their kinetic energy recovery system and the service of London buses.

http://www.gizmag.com/torotrak-mechanical-kers-system-for-buses/13023/

I think this presents a good opportunity to remove pollution caused by stresses on engines as they try to move off from a stop. Just look at any place where a bus regularly stops (like rising bollards, junctions) and you will see how the smoke has built up on the road and surrounding street furniture.

Suppliers of kinetic energy recovery systems will emphasise the devices can often be retrofitted onto existing vehicles. However I think they could also enable vehicles to achieve the same accelerations or torques with lower powered engines. Heavy vehicles like a bus could be redesigned with smaller engines, not normally able to accelerate from a stop, by storing power in the KERS system while the bus is not moving, rather than only during braking.

Just to try out latex in wordpress (only included since 2007!)…
$\mathrm{K.E.} = \frac{1}{2} m v^2$

(extensive use of latex in wordpress can be seen here)


text:
# Automatically inserting “Section”, “Subsection”, etc

Is there a package that automatically inserts the ToC level of the reference I inserted?

Section 1.1
\label{a}
Subsection 1.1.1
\label{b}
Subsection 1.1.2
\label{c}

So that the text:

Blah \superref{a} and \superref{b} and \superref{c}

becomes:

Blah Section 1.1 and Subsection 1.1.1 and Subsection 1.1.2

So that it automatically adapts if I change the level of a label?

-

Use the cleveref package and its \cref macro. Note that sectioning levels below \section are referred to as "section" by default, but this can be customized.

EDIT: In case you want the first letter of the reference name capitalized, use \Cref.

\documentclass{book}

\usepackage{cleveref}
\crefname{subsection}{subsection}{subsections}

\begin{document}

\chapter{First}

\section{First-first}\label{a}

\subsection{First-first-first}\label{b}

\subsection{First-first-second}\label{c}

See \cref{a} and \cref{b} and \cref{c}~\dots

\end{document}

-
\Cref is for starting a sentence, akin to \Citet. For consistent capitalization, use the capitalize package option. Use noabbrev to fully spell out the reference names. (See Section 6 of the manual.) –  krlmlr Jun 15 '13 at 13:44
If one writes See \cref{a,b,c}, the output is See section 1.1 and subsections 1.1.1 and 1.1.2. A \cref instruction can take multiple inputs, and cleveref parses, sorts, and (if necessary) compresses the cross-references on its own. –  Mico Mar 1 at 21:24

The hyperref package provides \autoref{<label>} which inserts the name of the reference type as well. These names can also be changed e.g. redefine \sectionautorefname to change the name for sections.

-


text:
zbMATH — the first resource for mathematics

Optimal HIV treatment by maximising immune response. (English) Zbl 1057.92035
Summary: We present an optimal control model of drug treatment of the human immunodeficiency virus (HIV). Our model is based upon ordinary differential equations that describe the interaction between HIV and the specific immune response as measured by levels of natural killer cells. We establish stability results for the model. We approach the treatment problem by posing it as an optimal control problem in which we maximise the benefit based on levels of healthy $$CD4+$$ $$T$$ cells and immune response cells, less the systemic cost of chemotherapy. We completely characterise the optimal control and compute a numerical solution of the optimality system via analytic continuation.

MSC:
 92C50 Medical applications (general) 49N90 Applications of optimal control and differential games 49K15 Optimality conditions for problems involving ordinary differential equations 49J15 Existence theories for optimal control problems involving ordinary differential equations
COLNEW; XPPAUT
Full Text:


text:
### What is the arterial blood gas test?

An arterial blood gas (ABG) test is a blood test that measures:

• The amount of oxygen in your blood. Oxygen is breathed into the lungs and carried to the body’s tissues through the blood.

• The amount of carbon dioxide in your blood. Carbon dioxide is a waste that the body creates. It is carried away from the body’s tissues by the blood to the lungs and breathed out.

• The balance of chemicals called acids and bases in your blood. Your body needs a careful balance of these chemicals to be able to function well. Problems with your kidneys or heart failure, diseases like diabetes, or a severe infection can cause an unhealthy imbalance.

• The amount of bicarbonate in your blood. Bicarbonate is released or absorbed by your kidneys to keep the acid-base balance stable.

It may be done to:

## Subscription Options

### AccessPharmacy Full Site: One-Year Subscription

Connect to the full suite of AccessPharmacy content and resources including 30+ textbooks such as Pharmacotherapy: A Pathophysiologic Approach and Goodman & Gilman's The Pharmacological Basis of Therapeutics, high-quality videos, images, and animations, interactive board review, drug and herb/supplements databases, and more.


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# A burger eating contest had 25 contestants and 450 burgers were consum

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A burger eating contest had 25 contestants and 450 burgers were consum  [#permalink]

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19 Dec 2018, 01:14
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A burger eating contest had 25 contestants and 450 burgers were consumed. The bottom $$\frac{3}{5}$$ of the contestants ate at least 15 burgers each but not more than 20. And the next 9 contestants ate at least 20 burgers each. What is the maximum number of burgers that the winner has consumed to win the contest.

A. 44
B. 45
C. 46
D. 47
E. 48

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Re: A burger eating contest had 25 contestants and 450 burgers were consum  [#permalink]

### Show Tags

19 Dec 2018, 01:33
Bunuel wrote:
A burger eating contest had 25 contestants and 450 burgers were consumed. The bottom $$\frac{3}{5}$$ of the contestants ate at least 15 burgers each but not more than 20. And the next 9 contestants ate at least 20 burgers each. What is the maximum number of burgers that the winner has consumed to win the contest.

A. 44
B. 45
C. 46
D. 47
E. 48

lets take min values for each contestant details given;
so 15 contestants i.e 3/5 ate 15*15=225 burgers
9 ate 9*20 = 180 burgers
so out of 450- (180+225)= 45 is the max burger winner has consumed IMO B
Re: A burger eating contest had 25 contestants and 450 burgers were consum   [#permalink] 19 Dec 2018, 01:33
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text:
# 2 member conversation with each input in left and right boxes headed with extra info

with Expl3, environ and tcolorbox one can make an environment that will put each piece of text ending with a white line in a box. I'd like to modify its code given below so that it also fills the "Title" optional field of a tcolobox box and the upper - lower parts. For example in a cyclic way filling in the same order "Tile", "upper" part, and "lower" part as it finds lines ending with a blank line or \end{mynewenviron}. "mynewenviron" being the name of the environ created for this purpose.

\documentclass{article}
\usepackage[many]{tcolorbox}
\usepackage{xcolor}
\usepackage{varwidth}
\usepackage{environ}
\usepackage{xparse}

\newlength{\bubblesep}
\newlength{\bubblewidth}
\setlength{\bubblesep}{2pt}
\AtBeginDocument{\setlength{\bubblewidth}{.75\textwidth}}

\newcommand{\bubble}[1]{%
\tcbox[
on line,
arc=2.5mm,
rounded corners,
]{\begin{varwidth}{\bubblewidth}#1\end{varwidth}}%
}

\ExplSyntaxOn
\seq_new:N \l__ooker_bubbles_seq
\tl_new:N \l__ooker_bubbles_first_tl
\tl_new:N \l__ooker_bubbles_last_tl

\NewEnviron{rightbubbles}
{
\begin{flushright}
\sffamily
\seq_set_split:NnV \l__ooker_bubbles_seq { \par } \BODY
\int_compare:nTF { \seq_count:N \l__ooker_bubbles_seq < 2 }
{
\bubble{\BODY}\par
}
{
\seq_pop_left:NN \l__ooker_bubbles_seq \l__ooker_bubbles_first_tl
\seq_pop_right:NN \l__ooker_bubbles_seq \l__ooker_bubbles_last_tl
\bubble{\l__ooker_bubbles_first_tl}
\par\nointerlineskip
\seq_map_inline:Nn \l__ooker_bubbles_seq
{
\bubble{##1}
\par\nointerlineskip
}
\bubble{\l__ooker_bubbles_last_tl}
\par
}
\end{flushright}
}
\ExplSyntaxOff

\begin{document}

\begin{rightbubbles}
Right-aligned gray bubbles (103, 184,104) with white text

Bubbles only break after a paragraph (equivalent to an enter press
when chatting). Long message with multiple lines will be kept in one bubble.

Left and right edges are round.

Single
\end{rightbubbles}

\begin{rightbubbles}
Single
\end{rightbubbles}

\end{document}

• First, note that a blank line is equivalent to \par. So, messing around with blank lines could also be achieved by redefining \par, I think. Second, I personally don't think \par should be hijacked. For example, you could instead use the tcbitemize environment by the tcolorbox package, if you don't mind adding \tcbitem before each speaker. Maybe there is a way to define \par as \tcbitem in a custom environment but I couldn't figure it out. – jessexknight Nov 25 '20 at 14:50
• the advantage in the code i inserted in my post is that it allows to do a simple copy and paste into the .tex file and build it. In the worst case the user might need to manually add blank lines where he wants new box creations to be done. Instead of a blank line \par is more complicated as it's a piece of code. The Expl3 syntax should allow a cyclic processing lines in \BODY as here already the environment treats differently the first and last lines. – user1850133 Nov 25 '20 at 16:53
• Yes but if you redefine \par then blank lines will also invoke the new definition. – jessexknight Nov 25 '20 at 19:17


text:
# A counterexample to monotonicity of relative mass in random walks

Electronic Communications in Probability, pp. 82016.

Cited by: 1|Bibtex|Views18|

Abstract:

For a finite undirected graph $G = (V,E)$, let $p_{u,v}(t)$ denote the probability that a continuous-time random walk starting at vertex $u$ is in $v$ at time $t$. In this note we give an example of a Cayley graph $G$ and two vertices $u,v \in G$ for which the function $r_{u,v}(t) = \frac{p_{u,v}(t)}{p_{u,u}(t)} \qquad t \geq 0$ is ...More

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HomeEnglishClass 11MathsChapterBinomial Theorem

In the binomial expansion of (...

In the binomial expansion of (a-b)^(n) , n ge 5, the sum of 5th and 6th terms is zero, then (a)/(b) equals

Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.

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Text Solution

(a) (n-5)/(6)(b) (n-4)/(5)(c) (5)/(n-4)(d) (6)/(n-5)

B

Transcript

TimeTranscript
00:00 - 00:59when is 25 and 60 then option in minus 5 by 6 option B and minus 4 by 5 option C5 and -4 option D 16 - 52 here in the questions are saying that in the Binomial expansion of a minus b power and the sum of m and n is greater than equal to 5 then we need to find a baby's value the term given to us with power and the general form of this is a + b general for comparison b is minus b and n
01:00 - 01:59the question that this we need to find the fifth and sixth term before that will see the formula for general class B power given by a + 1 equal to m c h a power n minus 2 find the 5th term that will put 24 so when we put our equal to four we get it is equal to 4 power n minus into minus the power 4 this is the first expression now we need for that will put equal to
02:00 - 02:59so will get 30 6 is equal to 5 and minus 5 into minus b power 5 that E5 plus 30 x is equal to zero substituting the values near me fees and C4 into minus b power 4 + 36 yrs and C5 a power n minus 5 minus 5 is equal to zero and C4 a power n minus 4 minus b power power power then this becomes plus so this will become the power 4 minus b power 5 when the power is odd then this becomes negative value
03:00 - 03:59555 equal to zero and C4 a power n minus 4 the power 4 is equal to 5 and minus 5 by 4 into B power 4 is equal to hear and see 5 and minus A into B 45 identify and is equal to a for a used this sunao
04:00 - 04:591234 is equal to into but we need the value of a b b sorry arranging a bi bi will be equal to 5 by 4 find the value and PR is equal to any factorial by 125 problems and Ariel B minus 5 into 5 factorial MP4 that is and factorial by -4 factorial in 24 factorial equal to
05:00 - 05:59into 5 factorial in to convert the division into in 2010 - 4 tutorial in 24 and it and -4 hours and minus into and minus into four factorial by and -5 and 24 will be equal to and minus 4 by the correct answer is option be thus we can conclude therefore
06:00 - 06:59option B and minus 4 by 5 thank you


text:
# Problems of the Week

Contribute a problem

Jane, Kate, and Lily are perfect logicians and always tell the truth.

They are each wearing a hat marked with a distinct positive integer. They know that one of the hats has a single-digit integer, $N,$ written on it, and that the other two hats each have a divisor of $N$ written on them. Each person can only see the two numbers on the others’ hats.

• Jane: "I don't know my number."
• Kate: "I don't know my number."
• Lily: "But I know my number!"
• Kate: "Now I know my number, too!"
• Jane: "I still don't know my number."

What is the sum of the numbers written on Kate and Lily’s hats?

RNA molecules are long chains of different repeating units called bases, which can either fold and pair with any other base on the same molecule, or remain unpaired. For short RNA chains, the number of possible arrangements can be quickly enumerated by inspection. Which arrangement the RNA tends to fold into depends on its sequence of bases: given a sequence, every arrangement can be "scored", as some pairs are more likely to form stable bonds than others.

For biological RNAs, the typical length is greater than 100 bases. As with shorter lengths, the arrangement that the RNA folds into depends on its sequence, but the number of possible arrangements to consider is outrageously large, including many common structural motifs shown below.

For an RNA molecule consisting of $N=100$ bases, there are $n$ different arrangements which it can choose from.

How many digits are in $n$ expressed in base $10?$

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A cuboid has base dimensions $30 \times 20$ and a height of $10$ and is lying flat on the $xy$-plane.

It undergoes two rotations—the first is $30^{\circ}$ about the edge measuring $20$, and the second is also $30^{\circ}$ about the edge measuring $30$. Both rotations are such that the center of the cuboid is raised upward.

Find the $z$-coordinate (that is, the altitude) of the highest point of the cuboid after applying the two rotations (to 2 decimal places).

$f(x)=\overbrace{\sin\sin ... \sin}^{70\text{ sines}} (x).$

Calculate the seventh derivative of $f$ evaluated at $x=0.$

Note: Using computer software to answer this question is an even deadlier sin than nesting 70 sines inside each other, so don't do it.

In the common time meter, one bar of music consists of note values equivalent to $4$ quarter notes. Using the note value system, $1$ whole note is equivalent to $2$ half notes, $1$ half note is equivalent to $2$ quarter notes, $1$ quarter note is equivalent to $2$ eighth notes, and $1$ eighth note is equivalent to $2$ sixteenth notes. (For our British friends, a whole note is a semibreve, a half note is a minim, a quarter note is a crotchet, an eighth note is a quaver, and a sixteenth note is a semiquaver.)

An example of a rhythm in one bar of common time meter with $1$ quarter note, $2$ sixteenth notes, $1$ eighth note, and $1$ half note is shown below.

Using only whole notes, half notes, quarter notes, eighth notes, and sixteenth notes (with no rests or dotted modifiers), how many unique rhythms can be made in one bar of common time meter?

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