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# Deming and Passing Bablok Regression in R

## Regression Methods

In this post we will be discussing how to perform Passing Bablok and Deming regression in R. Those who work in Clinical Chemistry know that these two approaches are required by the journals in the field. The idiosyncratic affection for these two forms of regression appears to be historical but this is something unlikely to change in my lifetime–hence the need to cover it here.

Along the way, we shall touch on the ways in which Deming and Passing Bablok differ from ordinary least squares (OLS) and from one another.

## Creating some random data

Let's start by making some heteroscedastic random data that we can use for regression. We will use the command set.seed() to begin with because by this means, the reader can generate the same random data as the post. This function takes any number you wish as its argument, but if you set the same seed, you will get the same random numbers. We will generate 100 random $$x$$ values in the uniform distribution and then an accompanying 100 random $$y$$ values with proportional bias, constant bias and random noise that increases with $$x$$. I have added a bit of non–linearity because we do see this a fair bit in our work.

The constants I chose are arbitrary. I chose them to produce something resembling a comparison of, say, two automated immunoassays.

Let's quickly produce a scatter plot to see what our data looks like:

## Residuals in OLS

OLS regression minimizes the sum of squared residuals. In the case of OLS, the residual of a point is defined as the vertical distance from that point to the regression line. The regression line is chosen so that the sum of the squares of the residuals in minimal.

OLS regression assumes that there is no error in the $$x$$–axis values and that there is no heteroscedasticity, that is, the scatter of $$y$$ is constant. Neither of these assumptions is true in the case of bioanaytical method comparisons. In contrast, for calibration curves in mass–spectrometry, a linear response is plotted as a function of pre–defined calibrator concentration. This means that the $$x$$–axis has very little error and so OLS regression is an appropriate choice (though I doubt that the assumption about homoscedasticity is generally met).

OLS is part of R's base package. We can find the OLS regression line using lm() and we will store the results in the variable lin.reg.

Just to demonstrate the point about residuals graphically, the following shows them in vertical red lines.

## Deming Regression

Deming regression differs from OLS regression in that it does not make the assumption that the $$x$$ values are free of error. It (more or less) defines the residual as the perpendicular distance from a point to its fitted value on the regression line.

Deming regression does not come as part of R's base package but can be performed using the MethComp and mcr packages. In this case, we will use the latter. If not already installed, you must install the mcr package with install.packages("mcr").

Then to perform Deming regression, we will load the mcr library and execute the following using the mcreg() command, storing the output in the variable dem.reg.

By performing the str() command on dem.reg, we can see that the regression parameters are stored in the slot @para. Because the authors have used an S4 object as the output of their function, we don't address output as we would in lists (with a \$), but rather with an @.

The intercept and slope are stored in demreg@para[1] and dem.reg@para[2] respectively. Therefore, we can add the regression line as follows:

To emphasize how the residuals are different from OLS we can plot them as before:

We present the figure above for instructional purposes only. The usual way to present a residuals plot is to show the same picture rotated until the line is horizontal–this is a slight simplification but is essentially what is happening:

### Ratio of Variances

It is important to mention that if one knows that the $$x$$–axis method is subject to a different amount of random analytical variability than the $$y$$–axis method, one should provide the ratio of the variances of the two methods to mcreg(). In general, this requires us to have “CV” data from precision studies already available. Another approach is to perform every analysis in duplicate by both methods and use the data to estimate this ratio.

If the methods happen to have similar CVs throughout the analytical range, the default value of 1 is assumed. But suppose that the ratio of the CVs of the $$x$$ axis method to the $$y$$–axis method was 1.2, we could provide this in the regression call by setting the error.ratio parameter. The resulting regression parameters will be slightly different.

### Weighting

In the case of heteroscedastic data, it would be customary to weight the regression which in the case of the mcr package is weighted as $$1/x^2$$. This means that having 0's in your $$x$$–data will cause the calculation to “crump”. In any case, if we wanted weighted regression parameters we would make the call:

And plotting both on the same figure:

## Passing Bablok

Passing Bablok regression is not performed by the minimization of residuals. Rather, all possible pairs of $$x$$–$$y$$ points are determined and slopes are calculated using each pair of points. Work–arounds are undertaken for pairs of points that generate infinite slopes and other peculiarities. In any case, the median of the $$\frac{N(N-1)}{2!}$$ possible slopes becomes the final slope estimate and the corresponding intercept can be calculated. With regards to weighted Passing Bablok regression, I’d like to acknowledge commenter glen_b for bringing to my attention that there is a paradigm for calculating the weighted median of pairwise slopes. See the comment section for a discussion.

Passing Bablok regression takes a lot of computational time as the number of points grows, so expect some delays on data sets larger than $$N=100$$ if you are using an ordinary computer. To get the Passing Bablok regression equation, we just change the method.reg parameter:

and the procedures to plot this regression are identical. The mcreg() function does have an option for Passing Bablok regression on large data sets. See the instructions by typing help("mcreg") in the R terminal.

## Outlier Effects

As a consequence of the means by which the slope is determined, the Passing Bablok method is relatively resistant to the effect of outlier(s) as compared to OLS and Deming. To demonstrate this, we can add on outlier to some data scattered about the line $$y=x$$ and show how all three methods are affected.

Because of this outlier, the OLS slope drops to 0.84, the Deming slope to 0.91, while the Passing Bablok is much better off at 0.99.

## Generating a Pretty Plot

The code authors of the mcr package have created a feature such that if you put the regression model inside the plot function, you can quickly generate a figure for yourself that has all the required information on it. For example,

But this method of out–of–the–box figure is not very customizable and you may want it to appear differently for your publication. Never fear. There is a solution. The MCResult.plot() function offers complete customization of the figure so that you can show it exactly as you wish for your publication.

In this example, I have created semi–transparent “darkorchid4” (hex = #68228B) points and a semi–transparent blue (hex = #0000FF) confidence band of the regression. Maybe darkorchid would not be my first choice for a publication after all, but it demonstrates the customization. Additionally, I have suppressed my least favourite features of the default plot method. Specifically, the sub="" term removes the sentence at the bottom margin and the add.grid = FALSE prevents the grid from being plotted. Enter help(MCResult.plot) for the complete low–down on customization.

## Conclusion

We have seen how to perform Deming and Passing Bablok regression in the R programming language and have touched on how the methods differ “under the hood”. We have used the mcr to perform the regressions and have shown how you can beautify your plot.

The reader should have a look at the rlm() function in the MASS package and the rq() function in the quantreg package to see other robust (outlier–resistant) regression approaches. A good tutorial can be found here

I hope that makes it easy for you.

-Dan

May all your paths (and regressions) be straight:


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# If $a,b > 1$ and $r>2$ does $ax^2+by^2=z^r$ have any rational solutions?

I have been trying to solve the following equation for months without much success. It has been so far a very frustrating endeavor.Please help.

Consider the diophantine equation: $x^2+y^2=z^r$ where $\gcd(x,y,z)=1$ and $r>2$.
Assuming that there exists a non-trivial triplet: $(x_0,y_0,z_0)$ satisfying: $$x_0^2+y_0^2=z_0^r$$ How do I find the parametrization of:$$ax^2+by^2=z^r$$?

#### Solutions Collecting From Web of "If $a,b > 1$ and $r>2$ does $ax^2+by^2=z^r$ have any rational solutions?"

It’s quite easy to parameterize,

$$\color{red}ax^2+\color{red}by^2 =z^k$$

for odd $k$. Assume,

$$x^2+by^2 = (p^2+bq^2)^k$$

$$(x+y\sqrt{-b})(x-y\sqrt{-b}) = (p+q\sqrt{-b})^k(p-q\sqrt{-b})^k$$

Equate factors and solve for $x,y$. Hence,

$$x =\frac{\alpha+\beta}{2},\quad y = \frac{\alpha-\beta}{2\sqrt{-b}},\quad\text{where}\quad \alpha = (p+q\sqrt{-b})^k,\quad \beta = (p-q\sqrt{-b})^k$$

For example, let $k = 3$. Then,

$$(p (p^2 – 3 b q^2))^2 + b(3 p^2q – b q^3)^2 = (p^2+b q^2)^3$$

Let $p = \sqrt{a}p$. Then,

$$\color{red}a(ap^2 – 3 b q^2)^2 + \color{red}b(3 ap^2q – b q^3)^2 = (ap^2+b q^2)^3\tag{k=3}$$

for free variables $p,q$. Let $k = 5$,

$\color{red}a(a^2p^4 – 10 a b p^2q^2 + 5 b^2 q^4)^2 + \color{red}b(5 a^2p^4q – 10 a bp^2 q^3 + b^2 q^5)^2 = (ap^2+b q^2)^5\tag{k=5}$

and so on.

format: plain


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# Posts Tagged place constraints

## Recent Postings from place constraints

### X-ray Transients: Hyper- or Hypo-Luminous?

The disk instability picture gives a plausible explanation for the behavior of soft X-ray transient systems if self-irradiation of the disk is included. We show that there is a simple relation between the peak luminosity (at the start of an outburst) and the decay timescale. We use this relation to place constraints on systems assumed to undergo disk instabilities. The observable X-ray populations of elliptical galaxies must largely consist of long-lived transients, as deduced on different grounds by Piro and Bildsten (2002). The strongly-varying X-ray source HLX-1 in the galaxy ESO 243-49 can be modeled as a disk instability of a highly super-Eddington stellar-mass binary similar to SS433. A fit to the disk instability picture is not possible for an intermediate-mass black hole model for HLX-1. Other, recently identified, super-Eddington ULXs might be subject to disk instability.

### Constrains on Dark Matter sterile neutrino resonant production in the light of Planck

Recently, few independent detections of a weak X-ray emission line at an energy of ~3.5 keV seen toward a number of astrophysical sites have been reported. If this signal will be confirmed to be the signature of decaying DM sterile neutrino with a mass of ~7.1 keV, then the cosmological observables should be consistent with its properties. In this paper we place constraints on the sterile neutrino resonant production parameters and asymmetry lepton number by using most of the present cosmological measurements. We compute the radiation and matter perturbations including the full resonance sweep solution for active – sterile neutrino flavor conversion and place constraints on the cosmological parameters and sterile neutrino properties. We find the sterile neutrino upper limits for mass and mixing angle of 7.86 keV (equivalent to 2.54 keV thermal mass) and 9.41 x 10^{-9} (at 95% CL) respectively, for a lepton number per flavor of 0.0042, that is significantly higher than that inferred in Abazajian 2014 from the linear large scale structure constraints. This reflects the sensitivity of the high precision CMB anisotropies to the helium abundance yield which in turn is set by the $\nu_e$ lepton number and non-thermal spectrum. Other cosmological parameters are in agreement with the predictions of the minimal extension of the base \LambdaCDM model except for the active neutrino total mass upper limit that is decreased to 0.21 eV (95% CL).

### Constrains on Dark Matter sterile neutrino resonant production in the light of Planck [Cross-Listing]

Recently, few independent detections of a weak X-ray emission line at an energy of ~3.5 keV seen toward a number of astrophysical sites have been reported. If this signal will be confirmed to be the signature of decaying DM sterile neutrino with a mass of ~7.1 keV, then the cosmological observables should be consistent with its properties. In this paper we place constraints on the sterile neutrino resonant production parameters and asymmetry lepton number by using most of the present cosmological measurements. We compute the radiation and matter perturbations including the full resonance sweep solution for active – sterile neutrino flavor conversion and place constraints on the cosmological parameters and sterile neutrino properties. We find the sterile neutrino upper limits for mass and mixing angle of 7.86 keV (equivalent to 2.54 keV thermal mass) and 9.41 x 10^{-9} (at 95% CL) respectively, for a lepton number per flavor of 0.0042, that is significantly higher than that inferred in Abazajian 2014 from the linear large scale structure constraints. This reflects the sensitivity of the high precision CMB anisotropies to the helium abundance yield which in turn is set by the $\nu_e$ lepton number and non-thermal spectrum. Other cosmological parameters are in agreement with the predictions of the minimal extension of the base \LambdaCDM model except for the active neutrino total mass upper limit that is decreased to 0.21 eV (95% CL).

### Constrains on Dark Matter sterile neutrino resonant production in the light of Planck [Replacement]

Few independent detections of a weak X-ray emission line at an energy of ~3.5 keV seen toward a number of astrophysical sites have been reported. If this signal will be confirmed to be the signature of decaying DM sterile neutrino with a mass of ~7.1 keV, then the cosmological observables should be consistent with its properties. We compute the radiation and matter perturbations including the full resonance sweep solution for active – sterile neutrino flavor conversion and place constraints on the cosmological parameters and sterile neutrino properties by using most of the present cosmological measurements. We find the sterile neutrino upper limits for mass and mixing angle of 7.86 keV (equivalent to 2.54 keV thermal mass) and 9.41 x 10^{-9} (at 95% CL) respectively, for a lepton number per flavor of 0.0042, that is significantly higher than that inferred in Abazajian (2014) from the linear large scale structure constraints. This reflects the sensitivity of the high precision CMB anisotropies to the helium abundance yield which in turn is set by the electron neutrino lepton number and the non-thermal active neutrino spectra. Other cosmological parameters are in agreement with the predictions of the minimal extension of the base LambdaCDM model except for the active neutrino total mass uper limit that is decreased to 0.21 eV (95% CL).

### Constrains on Dark Matter sterile neutrino resonant production in the light of Planck [Replacement]

Few independent detections of a weak X-ray emission line at an energy of ~3.5 keV seen toward a number of astrophysical sites have been reported. If this signal will be confirmed to be the signature of decaying DM sterile neutrino with a mass of ~7.1 keV, then the cosmological observables should be consistent with its properties. We compute the radiation and matter perturbations including the full resonance sweep solution for active – sterile neutrino flavor conversion and place constraints on the cosmological parameters and sterile neutrino properties by using most of the present cosmological measurements. We find the sterile neutrino upper limits for mass and mixing angle of 7.86 keV (equivalent to 2.54 keV thermal mass) and 9.41 x 10^{-9} (at 95% CL) respectively, for a lepton number per flavor of 0.0042, that is significantly higher than that inferred in Abazajian (2014) from the linear large scale structure constraints. This reflects the sensitivity of the high precision CMB anisotropies to the helium abundance yield which in turn is set by the electron neutrino lepton number and the non-thermal active neutrino spectra. Other cosmological parameters are in agreement with the predictions of the minimal extension of the base LambdaCDM model except for the active neutrino total mass uper limit that is decreased to 0.21 eV (95% CL).

### Constrains on Dark Matter sterile neutrino resonant production in the light of Planck [Replacement]

Few independent detections of a weak X-ray emission line at an energy of ~3.5 keV seen toward a number of astrophysical sites have been reported. If this signal will be confirmed to be the signature of decaying DM sterile neutrino with a mass of ~7.1 keV, then the cosmological observables should be consistent with its properties. We compute the radiation and matter perturbations including the full resonance sweep solution for active – sterile neutrino flavor conversion and place constraints on the cosmological parameters and sterile neutrino properties by using most of the present cosmological measurements. We find the sterile neutrino upper limits for mass and mixing angle of 7.86 keV (equivalent to 2.54 keV thermal mass) and 9.41 x 10^{-9} (at 95% CL) respectively, for a lepton number per flavor of 0.0042, that is significantly higher than that inferred in Abazajian (2014) from the linear large scale structure constraints. This reflects the sensitivity of the high precision CMB anisotropies to the helium abundance yield which in turn is set by the electron neutrino lepton number and the non-thermal active neutrino spectra. Other cosmological parameters are in agreement with the predictions of the minimal extension of the base LambdaCDM model except for the active neutrino total mass uper limit that is decreased to 0.21 eV (95% CL).

### Constrains on Dark Matter sterile neutrino resonant production in the light of Planck [Replacement]

Few independent detections of a weak X-ray emission line at an energy of ~3.5 keV seen toward a number of astrophysical sites have been reported. If this signal will be confirmed to be the signature of decaying DM sterile neutrino with a mass of ~7.1 keV, then the cosmological observables should be consistent with its properties. We compute the radiation and matter perturbations including the full resonance sweep solution for active – sterile neutrino flavor conversion and place constraints on the cosmological parameters and sterile neutrino properties by using most of the present cosmological measurements. We find the sterile neutrino upper limits for mass and mixing angle of 7.86 keV (equivalent to 2.54 keV thermal mass) and 9.41 x 10^{-9} (at 95% CL) respectively, for a lepton number per flavor of 0.0042, that is significantly higher than that inferred in Abazajian (2014) from the linear large scale structure constraints. This reflects the sensitivity of the high precision CMB anisotropies to the helium abundance yield which in turn is set by the electron neutrino lepton number and the non-thermal active neutrino spectra. Other cosmological parameters are in agreement with the predictions of the minimal extension of the base LambdaCDM model except for the active neutrino total mass uper limit that is decreased to 0.21 eV (95% CL).

### Chemical Enrichment RGS cluster sample (CHEERS): Constraints on turbulence

Feedback from AGN, galactic mergers, and sloshing are thought to give rise to turbulence, which may prevent cooling in clusters. We aim to measure the turbulence in clusters of galaxies and compare the measurements to some of their structural and evolutionary properties. It is possible to measure the turbulence of the hot gas in clusters by estimating the velocity widths of their X-ray emission lines. The RGS Spectrometers aboard XMM-Newton are currently the only instruments provided with sufficient effective area and spectral resolution in this energy domain. We benefited from excellent 1.6Ms new data provided by the CHEERS project. The new observations improve the quality of the archival data and allow us to place constraints for some clusters, which were not accessible in previous work. One-half of the sample shows upper limits on turbulence less than 500km/s. For several sources, our data are consistent with relatively strong turbulence with upper limits on the velocity widths that are larger than 1000km/s. The NGC507 group of galaxies shows transonic velocities, which are most likely associated with the merging phenomena and bulk motions occurring in this object. Where both low- and high-ionization emission lines have good enough statistics, we find larger upper limits for the hot gas, which is partly due to the different spatial extents of the hot and cool gas phases. Our upper limits are larger than the Mach numbers required to balance cooling, suggesting that dissipation of turbulence may prevent cooling, although other heating processes could be dominant. The systematics associated with the spatial profile of the source continuum make this technique very challenging, though still powerful, for current instruments. The ASTRO-H and Athena missions will revolutionize the velocity estimates and discriminate between different spatial regions and temperature phases.

### Constraints on Galactic Wino Densities from Gamma Ray Lines [Cross-Listing]

We systematically compute the annihilation rate for neutral winos into the final state gamma + X, including all leading radiative corrections. This includes both the Sommerfeld enhancement (in the decoupling limit for the Higgsino) and the resummation of the leading electroweak double logarithms. Adopting an analysis of the HESS experiment, we place constraints on the mass as a function of the wino fraction of the dark matter and the shape of the dark matter profile. We also determine how much coring is needed in the dark matter halo to make the wino a viable candidate as a function of its mass. Additionally, as part of our effective field theory formalism, we show that in the pure-Standard Model sector of our theory, emissions of soft Higgses are power-suppressed and that collinear Higgs emission does not contribute to leading double logs.

### Constraints on Galactic Wino Densities from Gamma Ray Lines

We systematically compute the annihilation rate for neutral winos into the final state gamma + X, including all leading radiative corrections. This includes both the Sommerfeld enhancement (in the decoupling limit for the Higgsino) and the resummation of the leading electroweak double logarithms. Adopting an analysis of the HESS experiment, we place constraints on the mass as a function of the wino fraction of the dark matter and the shape of the dark matter profile. We also determine how much coring is needed in the dark matter halo to make the wino a viable candidate as a function of its mass. Additionally, as part of our effective field theory formalism, we show that in the pure-Standard Model sector of our theory, emissions of soft Higgses are power-suppressed and that collinear Higgs emission does not contribute to leading double logs.

### Homogeneity and isotropy in the 2MASS Photometric Redshift catalogue

Using the 2MASS Photometric Redshift catalogue we perform a number of statistical tests aimed at detecting possible departures from statistical homogeneity and isotropy in the large-scale structure of the Universe. Making use of the angular homogeneity index, an observable proposed in a previous publication, as well as studying the scaling of the angular clustering and number counts with magnitude limit, we place constraints on the fractal nature of the galaxy distribution. We find that the statistical properties of our sample are in excellent agreement with the standard cosmological model, and that it reaches the homogeneous regime significantly faster than fractal models with dimensions D<2.75. As part of our search for systematic effects, we also study the presence of hemispherical asymmetries in our data, finding no significant deviation beyond those allowed by the concordance model.

### Homogeneity and isotropy in the 2MASS Photometric Redshift catalogue [Replacement]

Using the 2MASS Photometric Redshift catalogue we perform a number of statistical tests aimed at detecting possible departures from statistical homogeneity and isotropy in the large-scale structure of the Universe. Making use of the angular homogeneity index, an observable proposed in a previous publication, as well as studying the scaling of the angular clustering and number counts with magnitude limit, we place constraints on the fractal nature of the galaxy distribution. We find that the statistical properties of our sample are in excellent agreement with the standard cosmological model, and that it reaches the homogeneous regime significantly faster than a class of fractal models with dimensions $D<2.75$. As part of our search for systematic effects, we also study the presence of hemispherical asymmetries in our data, finding no significant deviation beyond those allowed by the concordance model.

### Two- and Many-Body Decaying Dark Matter and Supernovae Type Ia [Cross-Listing]

We present a decaying dark matter scenario where the daughter products are a single massless relativistic particle and a single, massive but possibly relativistic particle. We calculate the velocity distribution of the massive daughter particle and its associated equation of state and derive its dynamical evolution in an expanding universe. In addition, we present a model of decaying dark matter where there are many massless relativistic daughter particles together with a massive particle at rest. We place constraints on these two models using supernovae type Ia observations. We find that for a daughter relativistic fraction of 1% and higher, lifetimes of at least less than 10 Gyrs are excluded, with larger relativistic fractions constraining longer lifetimes.

### Dark matter with two- and many-body decays and supernovae type Ia [Replacement]

We present a decaying dark matter scenario where the daughter products are a single massless relativistic particle and a single, massive but possibly relativistic particle. We calculate the velocity distribution of the massive daughter particle and its associated equation of state and derive its dynamical evolution in an expanding universe. In addition, we present a model of decaying dark matter where there are many massless relativistic daughter particles together with a massive particle at rest. We place constraints on these two models using supernovae type Ia observations. We find that for a daughter relativistic fraction of 1% and higher, lifetimes of at least less than 10 Gyrs are excluded, with larger relativistic fractions constraining longer lifetimes.

### Dark matter with two- and many-body decays and supernovae type Ia [Replacement]

We present a decaying dark matter scenario where the daughter products are a single massless relativistic particle and a single, massive but possibly relativistic particle. We calculate the velocity distribution of the massive daughter particle and its associated equation of state and derive its dynamical evolution in an expanding universe. In addition, we present a model of decaying dark matter where there are many massless relativistic daughter particles together with a massive particle at rest. We place constraints on these two models using supernovae type Ia observations. We find that for a daughter relativistic fraction of 1% and higher, lifetimes of at least less than 10 Gyrs are excluded, with larger relativistic fractions constraining longer lifetimes.

### Two- and Many-Body Decaying Dark Matter and Supernovae Type Ia

We present a decaying dark matter scenario where the daughter products are a single massless relativistic particle and a single, massive but possibly relativistic particle. We calculate the velocity distribution of the massive daughter particle and its associated equation of state and derive its dynamical evolution in an expanding universe. In addition, we present a model of decaying dark matter where there are many massless relativistic daughter particles together with a massive particle at rest. We place constraints on these two models using supernovae type Ia observations. We find that for a daughter relativistic fraction of 1% and higher, lifetimes of at least less than 10 Gyrs are excluded, with larger relativistic fractions constraining longer lifetimes.

### Observed parity-odd CMB temperature bispectrum [Replacement]

Parity-odd non-Gaussianities create a variety of temperature bispectra in the cosmic microwave background (CMB), defined in the domain: $\ell_1 + \ell_2 + \ell_3 = {\rm odd}$. These models are yet unconstrained in the literature, that so far focused exclusively on the more common parity-even scenarios. In this work, we provide the first experimental constraints on parity-odd bispectrum signals in WMAP 9-year temperature data, using a separable modal parity-odd estimator. Comparing theoretical bispectrum templates to the observed bispectrum, we place constraints on the so-called nonlineality parameters of parity-odd tensor non-Gaussianities predicted by several Early Universe models. Our technique also generates a model-independent, smoothed reconstruction of the bispectrum of the data for parity-odd configurations.

### Observed parity-odd CMB temperature bispectrum

Parity-odd non-Gaussianities create a variety of temperature bispectra in the cosmic microwave background (CMB), defined in the domain: $\ell_1 + \ell_2 + \ell_3 = {\rm odd}$. These models are yet unconstrained in the literature, that so far focused exclusively on the more common parity-even scenarios. In this work, we provide the first experimental constraints on parity-odd bispectrum signals in WMAP 9-year temperature data, using a separable modal parity-odd estimator. Comparing theoretical bispectrum templates to the observed bispectrum, we place constraints on the so-called nonlineality parameters of parity-odd tensor non-Gaussianities predicted by several Early Universe models. Our technique also generates a model-independent, smoothed reconstruction of the bispectrum of the data for parity-odd configurations.

### Observed parity-odd CMB temperature bispectrum [Cross-Listing]

Parity-odd non-Gaussianities create a variety of temperature bispectra in the cosmic microwave background (CMB), defined in the domain: $\ell_1 + \ell_2 + \ell_3 = {\rm odd}$. These models are yet unconstrained in the literature, that so far focused exclusively on the more common parity-even scenarios. In this work, we provide the first experimental constraints on parity-odd bispectrum signals in WMAP 9-year temperature data, using a separable modal parity-odd estimator. Comparing theoretical bispectrum templates to the observed bispectrum, we place constraints on the so-called nonlineality parameters of parity-odd tensor non-Gaussianities predicted by several Early Universe models. Our technique also generates a model-independent, smoothed reconstruction of the bispectrum of the data for parity-odd configurations.

### Observed parity-odd CMB temperature bispectrum [Cross-Listing]

Parity-odd non-Gaussianities create a variety of temperature bispectra in the cosmic microwave background (CMB), defined in the domain: $\ell_1 + \ell_2 + \ell_3 = {\rm odd}$. These models are yet unconstrained in the literature, that so far focused exclusively on the more common parity-even scenarios. In this work, we provide the first experimental constraints on parity-odd bispectrum signals in WMAP 9-year temperature data, using a separable modal parity-odd estimator. Comparing theoretical bispectrum templates to the observed bispectrum, we place constraints on the so-called nonlineality parameters of parity-odd tensor non-Gaussianities predicted by several Early Universe models. Our technique also generates a model-independent, smoothed reconstruction of the bispectrum of the data for parity-odd configurations.

### Observed parity-odd CMB temperature bispectrum [Replacement]

Parity-odd non-Gaussianities create a variety of temperature bispectra in the cosmic microwave background (CMB), defined in the domain: $\ell_1 + \ell_2 + \ell_3 = {\rm odd}$. These models are yet unconstrained in the literature, that so far focused exclusively on the more common parity-even scenarios. In this work, we provide the first experimental constraints on parity-odd bispectrum signals in WMAP 9-year temperature data, using a separable modal parity-odd estimator. Comparing theoretical bispectrum templates to the observed bispectrum, we place constraints on the so-called nonlineality parameters of parity-odd tensor non-Gaussianities predicted by several Early Universe models. Our technique also generates a model-independent, smoothed reconstruction of the bispectrum of the data for parity-odd configurations.

### Observed parity-odd CMB temperature bispectrum [Replacement]

Parity-odd non-Gaussianities create a variety of temperature bispectra in the cosmic microwave background (CMB), defined in the domain: $\ell_1 + \ell_2 + \ell_3 = {\rm odd}$. These models are yet unconstrained in the literature, that so far focused exclusively on the more common parity-even scenarios. In this work, we provide the first experimental constraints on parity-odd bispectrum signals in WMAP 9-year temperature data, using a separable modal parity-odd estimator. Comparing theoretical bispectrum templates to the observed bispectrum, we place constraints on the so-called nonlineality parameters of parity-odd tensor non-Gaussianities predicted by several Early Universe models. Our technique also generates a model-independent, smoothed reconstruction of the bispectrum of the data for parity-odd configurations.

### Observed parity-odd CMB temperature bispectrum [Replacement]

Parity-odd non-Gaussianities create a variety of temperature bispectra in the cosmic microwave background (CMB), defined in the domain: $\ell_1 + \ell_2 + \ell_3 = {\rm odd}$. These models are yet unconstrained in the literature, that so far focused exclusively on the more common parity-even scenarios. In this work, we provide the first experimental constraints on parity-odd bispectrum signals in WMAP 9-year temperature data, using a separable modal parity-odd estimator. Comparing theoretical bispectrum templates to the observed bispectrum, we place constraints on the so-called nonlineality parameters of parity-odd tensor non-Gaussianities predicted by several Early Universe models. Our technique also generates a model-independent, smoothed reconstruction of the bispectrum of the data for parity-odd configurations.

### Short-period $g$-mode pulsations in low-mass white dwarfs triggered by H shell burning

The detection of pulsations in white dwarfs with low mass offers the possibility of probing their internal structure through asteroseismology and place constraints on the binary evolutionary processes involved in their formation. In this paper we assess the impact of stable H burning on the pulsational stability properties of low-mass He-core white dwarf models resulting from binary star evolutionary calculations. We found that, apart from a dense spectrum of unstable radial modes and nonradial $g$- and $p$-modes driven by the $\kappa$-mechanism due to the partial ionization of H in the stellar envelope, some unstable $g$-modes with short pulsation periods are powered also by H burning via the $\varepsilon$-mechanism of mode driving. This is the first time that $\varepsilon$-destabilized modes are found in models representative of cool white dwarf stars. The short periods recently detected in the pulsating low-mass white dwarf SDSS J111215.82+111745.0 could constitute the first evidence of the existence of stable H burning in these stars, in particular in the so-called extremely low-mass white dwarfs.

### Multi-redshift limits on the 21cm power spectrum from PAPER [Replacement]

The epoch of reionization power spectrum is expected to evolve strongly with redshift, and it is this variation with cosmic history that will allow us to begin to place constraints on the physics of reionization. The primary obstacle to the measurement of the EoR power spectrum is bright foreground emission. We present an analysis of observations from the Donald C. Backer Precision Array for Probing the Epoch of Reionization (PAPER) telescope which place new limits on the HI power spectrum over the redshift range of $7.5<z<10.5$, extending previously published single redshift results to cover the full range accessible to the instrument. To suppress foregrounds, we use filtering techniques that take advantage of the large instrumental bandwidth to isolate and suppress foreground leakage into the interesting regions of $k$-space. Our 500 hour integration is the longest such yet recorded and demonstrates this method to a dynamic range of $10^4$. Power spectra at different points across the redshift range reveal the variable efficacy of the foreground isolation. Noise limited measurements of $\Delta^2$ at $k=$0.2hMpc$^{-1}$ and z$=7.55$ reach as low as (48mK)$^2$ ($1\sigma$). We demonstrate that the size of the error bars in our power spectrum measurement as generated by a bootstrap method is consistent with the fluctuations due to thermal noise. Relative to this thermal noise, most spectra exhibit an excess of power at a few sigma. The likely sources of this excess include residual foreground leakage, particularly at the highest redshift, and unflagged RFI. We conclude by discussing data reduction improvements that promise to remove much of this excess.

### Multi-redshift limits on the 21cm power spectrum from PAPER

The epoch of reionization power spectrum is expected to evolve strongly with redshift, and it is this variation with cosmic history that will allow us to begin to place constraints on the physics of reionization. The primary obstacle to the measurement of the EoR power spectrum is bright foreground emission. We present an analysis of observations from the Donald C. Backer Precision Array for Probing the Epoch of Reionization (PAPER) telescope which place new limits on the HI power spectrum over the redshift range of $7.5<z<10.5$, extending previously published single redshift results to cover the full range accessible to the instrument. To suppress foregrounds, we use filtering techniques that take advantage of the large instrumental bandwidth to isolate and suppress foreground leakage into the interesting regions of $k$-space. Our 500 hour integration is the longest such yet recorded and demonstrates this method to a dynamic range of $10^4$. Power spectra at different points across the redshift range reveal the variable efficacy of the foreground isolation. Noise limited measurements of $\Delta^2$ at $k=$0.2hMpc$^{-1}$ and z$=7.55$ reach as low as (48mK)$^2$ ($1\sigma$). We demonstrate that the size of the error bars in our power spectrum measurement as generated by a bootstrap method is consistent with the fluctuations due to thermal noise. Relative to this thermal noise, most spectra exhibit an excess of power at a few sigma. The likely sources of this excess include residual foreground leakage, particularly at the highest redshift, and unflagged RFI. We conclude by discussing data reduction improvements that promise to remove much of this excess.

### The Chandra Planetary Nebula Survey (ChanPlaNS). II. X-ray Emission from Compact Planetary Nebulae [Replacement]

We present results from the most recent set of observations obtained as part of the Chandra X-ray observatory Planetary Nebula Survey (ChanPlaNS), the first comprehensive X-ray survey of planetary nebulae (PNe) in the solar neighborhood (i.e., within ~1.5 kpc of the Sun). The survey is designed to place constraints on the frequency of appearance and range of X-ray spectral characteristics of X-ray-emitting PN central stars and the evolutionary timescales of wind-shock-heated bubbles within PNe. ChanPlaNS began with a combined Cycle 12 and archive Chandra survey of 35 PNe. ChanPlaNS continued via a Chandra Cycle 14 Large Program which targeted all (24) remaining known compact (R_neb <~ 0.4 pc), young PNe that lie within ~1.5 kpc. Results from these Cycle 14 observations include first-time X-ray detections of hot bubbles within NGC 1501, 3918, 6153, and 6369, and point sources in HbDs 1, NGC 6337, and Sp 1. The addition of the Cycle 14 results brings the overall ChanPlaNS diffuse X-ray detection rate to ~27% and the point source detection rate to ~36%. It has become clearer that diffuse X-ray emission is associated with young (<~5×10^3 yr), and likewise compact (R_neb<~0.15 pc), PNe with closed structures and high central electron densities (n_e>~1000 cm^-3), and rarely associated with PNe that show H_2 emission and/or pronounced butterfly structures. Hb 5 is one such exception of a PN with a butterfly structure that hosts diffuse X-ray emission. Additionally, of the five new diffuse X-ray detections, two host [WR]-type CSPNe, NGC 1501 and NGC 6369, supporting the hypothesis that PNe with central stars of [WR]-type are likely to display diffuse X-ray emission.

### The Chandra Planetary Nebula Survey (ChanPlaNS). II. X-ray Emission from Compact Planetary Nebulae

We present results from the most recent set of observations obtained as part of the Chandra X-ray observatory Planetary Nebula Survey (ChanPlaNS), the first comprehensive X-ray survey of planetary nebulae (PNe) in the solar neighborhood (i.e., within ~1.5 kpc of the Sun). The survey is designed to place constraints on the frequency of appearance and range of X-ray spectral characteristics of X-ray-emitting PN central stars and the evolutionary timescales of wind-shock-heated bubbles within PNe. ChanPlaNS began with a combined Cycle 12 and archive Chandra survey of 35 PNe. ChanPlaNS continued via a Chandra Cycle 14 Large Program which targeted all (24) remaining known compact (R_neb <~ 0.4 pc), young PNe that lie within ~1.5 kpc. Results from these Cycle 14 observations include first-time X-ray detections of hot bubbles within NGC 1501, 3918, 6153, and 6369, and point sources in HbDs 1, NGC 6337, and Sp 1. The addition of the Cycle 14 results brings the overall ChanPlaNS diffuse X-ray detection rate to ~27% and the point source detection rate to ~36%. It has become clearer that diffuse X-ray emission is associated with young (<~5×10^3 yr), and likewise compact (R_neb<~0.15 pc), PNe with closed structures and high central electron densities (n_e>~1000 cm^-3), and rarely associated with PNe that show H_2 emission and/or pronounced butterfly structures. Hb 5 is one such exception of a PN with a butterfly structure that hosts diffuse X-ray emission. Additionally, of the five new diffuse X-ray detections, two host [WR]-type CSPNe, NGC 1501 and NGC 6369, supporting the hypothesis that PNe with central stars of [WR]-type are likely to display diffuse X-ray emission.

### KIC 10526294: a slowly rotating B star with rotationally split quasi-equally spaced gravity modes

Massive stars are important for the chemical enrichment of the universe. Since internal mixing processes influence their life, it is of high importance to place constraints on the corresponding physical parameters, such as core overshooting and the internal rotation profile, to calibrate their stellar structure and evolution models. Although asteroseismology was shown to be able to deliver the most precise constraints so far, the number of detailed seismic studies delivering quantitative results is limited. Our goal is to extend this limited sample with an in-depth case study and provide a well constrained set of asteroseismic parameters, contributing to the ongoing mapping efforts of the instability strips of the beta Cep and SPB stars. We derived fundamental parameters from high-resolution spectra using spectral synthesis techniques. We used custom masks to obtain optimal light curves from the original pixel level data from the Kepler satellite. We used standard time-series analysis tools to construct a set of significant pulsation modes which provide the basis for the seismic analysis carried out afterwards. We find that KIC 10526294 is a cool SPB star, one of the slowest rotators ever found. Despite this fact, the length of Kepler observations is sufficient to resolve narrow rotationally split multiplets for each of its nineteen quasi-equally spaced dipole modes. The number of detected consecutive (in radial order) dipole modes in this series is higher than ever before. The observed amount of splitting shows an increasing trend towards longer periods, which – largely independent of the seismically calibrated stellar models – points towards a non-rigid internal rotation profile. From the average splitting we deduce a rotation period of 188 d. From seismic modelling we find that the star is young with a central hydrogen mass fraction X_c>0.64; it has a core overshooting alpha_ov<=0.15.

### KIC 10526294: a slowly rotating B star with rotationally split, quasi-equally spaced gravity modes [Replacement]

Massive stars are important for the chemical enrichment of the universe. Since internal mixing processes influence their lives, it is very important to place constraints on the corresponding physical parameters, such as core overshooting and the internal rotation profile, so as to calibrate their stellar structure and evolution models. Although asteroseismology has been shown to be able to deliver the most precise constraints so far, the number of detailed seismic studies delivering quantitative results is limited. Our goal is to extend this limited sample with an in-depth case study and provide a well-constrained set of asteroseismic parameters, contributing to the ongoing mapping efforts of the instability strips of the beta Cep and SPB stars. We derived fundamental parameters from high-resolution spectra using spectral synthesis techniques. We used custom masks to obtain optimal light curves from the original pixel level data from the Kepler satellite. We used standard time-series analysis tools to construct a set of significant pulsation modes that provide the basis for the seismic analysis carried out afterwards. We find that KIC 10526294 is a cool SPB star, one of the slowest rotators ever found. Despite this, the length of Kepler observations is sufficient to resolve narrow rotationally split multiplets for each of its 19 quasi-equally spaced dipole modes. The number of detected consecutive (in radial order) dipole modes in this series is higher than ever before. The observed amount of splitting shows an increasing trend towards longer periods, which – largely independent of the seismically calibrated stellar models – points towards a non-rigid internal rotation profile. From the average splitting we deduce a rotation period of ~188 d. From seismic modelling, we find that the star is young with a central hydrogen mass fraction X_c>0.64; it has a core overshooting alpha_ov<=0.15.

### The Effect of Anisotropic Viscosity on Cold Fronts in Galaxy Clusters

Cold fronts–contact discontinuities in the intracluster medium (ICM) of galaxy clusters–should be disrupted by Kelvin-Helmholtz (K-H) instabilities due to the associated shear velocity. However, many observed cold fronts appear stable. This opens the possibility to place constraints on microphysical mechanisms that stabilize them, such as the ICM viscosity and/or magnetic fields. We performed exploratory high-resolution simulations of cold fronts arising from subsonic gas sloshing in cluster cores using the grid-based Athena MHD code, comparing the effects of isotropic Spitzer and anisotropic Braginskii viscosity (expected in a magnetized plasma). Magnetized simulations with full Braginskii viscosity or isotropic Spitzer viscosity reduced by a factor f ~ 0.1 are both in qualitative agreement with observations in terms of suppressing K-H instabilities. The RMS velocity and turbulence within the sloshing region is only modestly reduced by Braginskii viscosity. We also performed unmagnetized simulations with and without viscosity and find that magnetic fields have a substantial effect on the appearance of the cold fronts, even if the initial field is weak and the viscosity is the same. This suggests that determining the dominant suppression mechanism of a given cold front from X-ray observations (e.g. viscosity or magnetic fields) by comparison with simulations is not straightforward. Finally, we performed simulations including anisotropic thermal conduction, and find that including Braginskii viscosity in these simulations does not significant affect the evolution of cold fronts; they are rapidly smeared out by thermal conduction, as in the inviscid case.

### The Effect of Anisotropic Viscosity on Cold Fronts in Galaxy Clusters [Replacement]

Cold fronts–contact discontinuities in the intracluster medium (ICM) of galaxy clusters–should be disrupted by Kelvin-Helmholtz (K-H) instabilities due to the associated shear velocity. However, many observed cold fronts appear stable. This opens the possibility to place constraints on microphysical mechanisms that stabilize them, such as the ICM viscosity and/or magnetic fields. We performed exploratory high-resolution simulations of cold fronts arising from subsonic gas sloshing in cluster cores using the grid-based Athena MHD code, comparing the effects of isotropic Spitzer and anisotropic Braginskii viscosity (expected in a magnetized plasma). Magnetized simulations with full Braginskii viscosity or isotropic Spitzer viscosity reduced by a factor f ~ 0.1 are both in qualitative agreement with observations in terms of suppressing K-H instabilities. The RMS velocity of turbulence within the sloshing region is only modestly reduced by Braginskii viscosity. We also performed unmagnetized simulations with and without viscosity and find that magnetic fields have a substantial effect on the appearance of the cold fronts, even if the initial field is weak and the viscosity is the same. This suggests that determining the dominant suppression mechanism of a given cold front from X-ray observations (e.g. viscosity or magnetic fields) by comparison with simulations is not straightforward. Finally, we performed simulations including anisotropic thermal conduction, and find that including Braginskii viscosity in these simulations does not significantly affect the evolution of cold fronts; they are rapidly smeared out by thermal conduction, as in the inviscid case.

### The Effect of Anisotropic Viscosity on Cold Fronts in Galaxy Clusters [Replacement]

Cold fronts — contact discontinuities in the intracluster medium (ICM) of galaxy clusters — should be disrupted by Kelvin-Helmholtz (K-H) instabilities due to the associated shear velocity. However, many observed cold fronts appear stable. This opens the possibility to place constraints on microphysical mechanisms that stabilize them, such as the ICM viscosity and/or magnetic fields. We performed exploratory high-resolution simulations of cold fronts arising from subsonic gas sloshing in cluster cores using the grid-based Athena MHD code, comparing the effects of isotropic Spitzer and anisotropic Braginskii viscosity (expected in a magnetized plasma). Magnetized simulations with full Braginskii viscosity or isotropic Spitzer viscosity reduced by a factor f ~ 0.1 are both in qualitative agreement with observations in terms of suppressing K-H instabilities. The RMS velocity of turbulence within the sloshing region is only modestly reduced by Braginskii viscosity. We also performed unmagnetized simulations with and without viscosity and find that magnetic fields have a substantial effect on the appearance of the cold fronts, even if the initial field is weak and the viscosity is the same. This suggests that determining the dominant suppression mechanism of a given cold front from X-ray observations (e.g. viscosity or magnetic fields) by comparison with simulations is not straightforward. Finally, we performed simulations including anisotropic thermal conduction, and find that including Braginskii viscosity in these simulations does not significant affect the evolution of cold fronts; they are rapidly smeared out by thermal conduction, as in the inviscid case.

### Herschel/PACS Observations of the Host Galaxy of GRB 031203

We present Herschel/PACS observations of the nearby (z=0.1055) dwarf galaxy that has hosted the long gamma ray burst (LGRB) 031203. Using the PACS data we have been able to place constraints on the dust temperature, dust mass, total infrared luminosity and infrared-derived star-formation rate (SFR) for this object. We find that the GRB host galaxy (GRBH) 031203 has a total infrared luminosity of 3×10^10 L_sun placing it in the regime of the IR-luminous galaxy population. Its dust temperature and specific SFR are comparable to that of many high-redshift (z=0.3-2.5) infrared (IR)-detected GRB hosts (T_dust>40K ; sSFR>10 Gyr^-1), however its dust-to-stellar mass ratio is lower than what is commonly seen in IR-luminous galaxies. Our results suggest that GRBH 031203 is undergoing a strong starburst episode and its dust properties are different to those of local dwarf galaxies within the same metallicity and stellar mass range. Furthermore, our measurements place it in a distinct class to the well studied nearby host of GRB 980425 (z=0.0085), confirming the notion that GRB host galaxies can span a large range in properties even at similar cosmological epochs, making LGRBs an ideal tool in selecting samples of star-forming galaxies up to high redshift.

### Herschel/PACS Observations of the Host Galaxy of GRB 031203 [Replacement]

We present Herschel/PACS observations of the nearby (z=0.1055) dwarf galaxy that has hosted the long gamma ray burst (LGRB) 031203. Using the PACS data we have been able to place constraints on the dust temperature, dust mass, total infrared luminosity and infrared-derived star-formation rate (SFR) for this object. We find that the GRB host galaxy (GRBH) 031203 has a total infrared luminosity of 3×10^10 L_sun placing it in the regime of the IR-luminous galaxy population. Its dust temperature and specific SFR are comparable to that of many high-redshift (z=0.3-2.5) infrared (IR)-detected GRB hosts (T_dust>40K ; sSFR>10 Gyr^-1), however its dust-to-stellar mass ratio is lower than what is commonly seen in IR-luminous galaxies. Our results suggest that GRBH 031203 is undergoing a strong starburst episode and its dust properties are different to those of local dwarf galaxies within the same metallicity and stellar mass range. Furthermore, our measurements place it in a distinct class to the well studied nearby host of GRB 980425 (z=0.0085), confirming the notion that GRB host galaxies can span a large range in properties even at similar cosmological epochs, making LGRBs an ideal tool in selecting samples of star-forming galaxies up to high redshift.

### Constraining primordial vector mode from B-mode polarization [Replacement]

The B-mode polarization spectrum of the Cosmic Microwave Background (CMB) may be the smoking gun of not only the primordial tensor mode but also of the primordial vector mode. If there exist nonzero vector-mode metric perturbations in the early Universe, they are known to be supported by anisotropic stress fluctuations of free-streaming particles such as neutrinos, and to create characteristic signatures on both the CMB temperature, E-mode, and B-mode polarization anisotropies. We place constraints on the properties of the primordial vector mode characterized by the vector-to-scalar ratio $r_{v}$ and the spectral index $n_{v}$ of the vector-shear power spectrum, from the {\it Planck} and BICEP2 B-mode data. We find that, for scale-invariant initial spectra, the $\Lambda$CDM model including the vector mode fits the data better than the model including the tensor mode. The difference in $\chi^{2}$ between the vector and tensor models is $\Delta\chi^{2} = 3.294$, because, on large scales the vector mode generates smaller temperature fluctuations than the tensor mode, which is preferred for the data. In contrast, the tensor mode can fit the data set equally well if we allow a significantly blue-tilted spectrum. We find that the best-fitting tensor mode has a large blue tilt and leads to an indistinct reionization bump on larger angular scales. The slightly red-tilted vector mode supported by the current data set can also create ${\cal O}(10^{-22})$-Gauss magnetic fields at cosmological recombination. Our constraints should motivate research that considers models of the early Universe that involve the vector mode.

### Constraining primordial vector mode from B-mode polarization [Replacement]

The B-mode polarization spectrum of the Cosmic Microwave Background (CMB) may be the smoking gun of not only the primordial tensor mode but also of the primordial vector mode. If there exist nonzero vector-mode metric perturbations in the early Universe, they are known to be supported by anisotropic stress fluctuations of free-streaming particles such as neutrinos, and to create characteristic signatures on both the CMB temperature, E-mode, and B-mode polarization anisotropies. We place constraints on the properties of the primordial vector mode characterized by the vector-to-scalar ratio $r_{v}$ and the spectral index $n_{v}$ of the vector-shear power spectrum, from the {\it Planck} and BICEP2 B-mode data. We find that, for scale-invariant initial spectra, the $\Lambda$CDM model including the vector mode fits the data better than the model including the tensor mode. The difference in $\chi^{2}$ between the vector and tensor models is $\Delta\chi^{2} = 3.294$, because, on large scales the vector mode generates smaller temperature fluctuations than the tensor mode, which is preferred for the data. In contrast, the tensor mode can fit the data set equally well if we allow a significantly blue-tilted spectrum. We find that the best-fitting tensor mode has a large blue tilt and leads to an indistinct reionization bump on larger angular scales. The slightly red-tilted vector mode supported by the current data set can also create ${\cal O}(10^{-22})$-Gauss magnetic fields at cosmological recombination. Our constraints should motivate research that considers models of the early Universe that involve the vector mode.

### Constraining primordial vector mode from B-mode polarization [Replacement]

The B-mode polarization spectrum of the Cosmic Microwave Background (CMB) may be the smoking gun of not only the primordial tensor mode but also of the primordial vector mode. If there exist nonzero vector-mode metric perturbations in the early Universe, they are known to be supported by anisotropic stress fluctuations of free-streaming particles such as neutrinos, and to create characteristic signatures on both the CMB temperature, E-mode, and B-mode polarization anisotropies. We place constraints on the properties of the primordial vector mode characterized by the vector-to-scalar ratio $r_{v}$ and the spectral index $n_{v}$ of the vector-shear power spectrum, from the {\it Planck} and BICEP2 B-mode data. We find that, for scale-invariant initial spectra, the $\Lambda$CDM model including the vector mode fits the data better than the model including the tensor mode. The difference in $\chi^{2}$ between the vector and tensor models is $\Delta\chi^{2} = 3.294$, because, on large scales the vector mode generates smaller temperature fluctuations than the tensor mode, which is preferred for the data. In contrast, the tensor mode can fit the data set equally well if we allow a significantly blue-tilted spectrum. We find that the best-fitting tensor mode has a large blue tilt and leads to an indistinct reionization bump on larger angular scales. The slightly red-tilted vector mode supported by the current data set can also create ${\cal O}(10^{-22})$-Gauss magnetic fields at cosmological recombination. Our constraints should motivate research that considers models of the early Universe that involve the vector mode.

### The Carnegie Supernova Project: Intrinsic Colors of Type Ia Supernovae

We present an updated analysis of the intrinsic colors of SNe Ia using the latest data release of the Carnegie Supernova Project. We introduce a new light-curve parameter very similar to stretch that is better suited for fast-declining events, and find that these peculiar types can be seen as extensions to the population of "normal" SNe Ia. With a larger number of objects, an updated fit to the Lira relation is presented along with evidence for a dependence on the late-time slope of the B-V color-curves with stretch and color. Using the full wavelength range from u to H band, we place constraints on the reddening law for the sample as a whole and also for individual events/hosts based solely on the observed colors. The photometric data continue to favor low values of Rv, though with large variations from event to event, indicating an intrinsic distribution. We confirm the findings of other groups that there appears to be a correlation between the derived reddening law, Rv, and the color excess, E(B-V), such that larger E(B-V) tends to favor lower Rv. The intrinsic u-band colors show a relatively large scatter that cannot be explained by variations in Rv or by the Goobar (2008) power-law for circumstellar dust, but rather is correlated with spectroscopic features of the supernova and is therefore likely due to metallicity effects.

### Parametric models of the periodogram

The maximum likelihood estimator is used to determine fit parameters for various parametric models of the Fourier periodogram followed by the selection of the best fit model amongst competing models using the Akaike information criteria. This analysis, when applied to light curves of active galactic nuclei can be used to infer the presence of quasi-periodicity and break or knee frequencies. The extracted information can be used to place constraints on the mass, spin and other properties of the putative central black hole and the region surrounding it through theoretical models involving disk and jet physics.

### A Three-Year Multi-Wavelength Study of the Very High Energy Gamma-ray Blazar 1ES 0229+200

The high-frequency-peaked BL Lacertae object 1ES 0229+200 is a relatively distant (z = 0.1396), hard-spectrum (Gamma ~ 2.5), very-high-energy-emitting (E > 100 GeV) gamma-ray blazar. Very-high-energy measurements of this active galactic nucleus have been used to place constraints on the intensity of the extragalactic background light and the intergalactic magnetic field. A multi-wavelength study of this object centered around very-high-energy observations by VERITAS is presented. This study obtained, over a period of three years, an 11.7 standard deviation detection and an average integral flux F(E>300 GeV) = (23.3 +- 2.8_stat +- 5.8_sys) x 10^-9 photons m^-2 s^-1, or 1.7% of the Crab Nebula’s flux (assuming the Crab Nebula spectrum measured by H.E.S.S). Supporting observations from Swift and RXTE are analyzed. The Swift observations are combined with previously published Fermi observations and the very-high-energy measurements to produce an overall spectral energy distribution which is then modeled assuming one-zone synchrotron-self-Compton emission. The chi^2 probability of the TeV flux being constant is 1.6%. This, when considered in combination with measured variability in the X-ray band, and the demonstrated variability of many TeV blazars, suggests that the use of blazars such as 1ES 0229+200 for intergalactic magnetic field studies may not be straightforward and challenges models that attribute hard TeV spectra to secondary gamma-ray production along the line of sight.

### Cosmological constraints from large-scale structure growth rate measurements [Replacement]

We compile a list of $14$ independent measurements of large-scale structure growth rate between redshifts $0.067 \leq z \leq 0.8$ and use this to place constraints on model parameters of constant and time-evolving general-relativistic dark energy cosmologies. With the assumption that gravity is well-modeled by general relativity, we discover that growth-rate data provide restrictive cosmological parameter constraints. In combination with type Ia supernova apparent magnitude versus redshift data and Hubble parameter measurements, the growth rate data are consistent with the standard spatially-flat $\Lambda$CDM model, as well as with mildly evolving dark energy density cosmological models.

### Large-scale structure growth rate measurement cosmological constraints

We compile a list of $14$ independent measurements of large-scale structure growth rate between redshifts $0.067 \leq z \leq 0.8$ and use this to place constraints on model parameters of constant and time-evolving general-relativistic dark energy cosmologies. With the assumption that gravity is well-modeled by general relativity, we discover that growth-rate data provide restrictive cosmological parameter constraints. In combination with type Ia supernova apparent magnitude versus redshift data and Hubble parameter measurements, the growth rate data are consistent with the standard spatially-flat $\Lambda$CDM model, as well as with mildly evolving dark energy density cosmological models.

### Interstellar Absorption Lines in the Direction of the Cataclysmic Variable SS Cygni

We present an analysis of interstellar absorption lines in high-resolution optical echelle spectra of SS Cyg obtained during an outburst in 2013 June and in archival Hubble Space Telescope and Far Ultraviolet Spectroscopic Explorer data. The Ca II K and Na I D lines toward SS Cyg are compared with those toward nearby B and A stars in an effort to place constraints on the distance to SS Cyg. We find that the distance constraints are not very robust from this method due to the rather slow increase in neutral gas column density with distance and the scatter in the column densities from one sight line to another. However, the optical absorption-line measurements allow us to derive a precise estimate for the line-of-sight reddening of E(B-V) = 0.020+/-0.005 mag. Furthermore, our analysis of the absorption lines of O I, Si II, P II, and Fe II seen in the UV spectra yields an estimate of the H I column density and depletion strength in this direction.

### The Kappa Andromedae System: New Constraints on the Companion Mass, System Age & Further Multiplicity

Kappa Andromedae is a B9IVn star at 52 pc for which a faint substellar companion separated by 55 AU was recently announced. In this work, we present the first spectrum of the companion, "kappa And B," using the Project 1640 high-contrast imaging platform. Comparison of our low-resolution YJH-band spectra to empirical brown dwarf spectra suggests an early-L spectral type. Fitting synthetic spectra from PHOENIX model atmospheres to our observed spectrum allows us to constrain the effective temperature to ~2000K, as well as place constraints on the companion surface gravity. Further, we use previously reported log(g) and effective temperature measurements of the host star to argue that the kappa And system has an isochronal age of 220 +/- 100 Myr, older than the 30 Myr age reported previously. This interpretation of an older age is corroborated by the photometric properties of kappa And B, which appear to be marginally inconsistent with other 10-100 Myr low-gravity L-dwarfs for the spectral type range we derive. In addition, we use Keck aperture masking interferometry combined with published radial velocity measurements to rule out the existence of any tight stellar companions to kappa And A that might be responsible for the system’s overluminosity. Further, we show that luminosity enhancements due to a nearly "pole-on" viewing angle coupled with extremely rapid rotation is unlikely. Kappa And A is thus consistent with its slightly evolved luminosity class (IV) and we propose here that kappa And, with a revised age of 220 +/- 100 Myr, is an interloper to the 30 Myr Columba association with which it was previously associated. The photometric and spectroscopic evidence for kappa And B combined with our re-assesment of the system age implies a substellar companion mass of 50^{+16}_{-13} Jupiter Masses, consistent with a brown dwarf rather than a planetary mass companion.

### Occultation of the T Tauri Star RW Aurigae A by its Tidally Disrupted Disk

RW Aur A is a classical T Tauri star, believed to have undergone a reconfiguration of its circumstellar environment as a consequence of a recent fly-by of its stellar companion, RW Aur B. This interaction stripped away part of the circumstellar disk of RW Aur A, leaving a tidally disrupted arm and a short truncated circumstellar disk. We present photometric observations of the RW Aur system from the Kilodegree Extremely Little Telescope (KELT) survey showing a long and deep dimming that occurred from September 2010 until March 2011. The dimming has a depth of ~2 magnitudes, a duration of ~180 days and was confirmed by archival observations from American Association of Variable Star Observers (AAVSO). We suggest that this event is the result of a portion of the tidally disrupted disk occulting RW Aur A, specifically a fragment of the tidally disrupted arm. The calculated transverse linear velocity of the occulter is in excellent agreement with the measured relative radial velocity of the tidally disrupted arm. Using simple kinematic and geometric arguments, we show that the occulter cannot be a feature of the RW Aur A circumstellar disk, and we consider and discount other hypotheses. We also place constraints on the thickness and semi-major axis of the portion of the arm that occulted the star.

### Placing Limits On The Transit Timing Variations Of Circumbinary Exoplanets

We present an efficient analytical method to predict the maximum transit timing variations of a circumbinary exoplanet, given some basic parameters of the host binary. We derive an analytical model giving limits on the potential location of transits for coplanar planets orbiting eclipsing binaries, then test it against numerical N-body simulations of a distribution of binaries and planets. We also show the application of the analytic model to Kepler-16b, -34b and -35b. The resulting method is fast, efficient and is accurate to approximately 1% in predicting limits on possible times of transits over a three-year observing campaign. The model can easily be used to, for example, place constraints on transit timing while performing circumbinary planet searches on large datasets. It is adaptable to use in situations where some or many of the planet and binary parameters are unknown.

### Constraining Superluminal Electron and Neutrino Velocities using the 2010 Crab Nebula Flare and the IceCube PeV Neutrino Events [Replacement]

The observation of two PeV-scale neutrino events reported by Ice Cube can, in principle, allows one to place constraints on Lorentz invariance violation (LIV) in the neutrino sector. After first arguing that at least one of the IceCube events was of extragalactic origin, I derive an upper limit for {\it the difference} between putative superluminal neutrino and electron velocities of $\le \sim 5.6 \times 10^{-19}$ in units where $c = 1$, confirming that the observed PeV neutrinos could have reached Earth from extragalactic sources. I further derive a new constraint on the superluminal electron velocity, obtained from the observation of synchrotron radiation in the Crab Nebula flare of September, 2010. The inference that the $>$ 1 GeV $\gamma$-rays from synchrotron emission in the flare were produced by electrons of energy up to $\sim 5.1$ PeV indicates the non-occurrence of vacuum \’{C}erenkov radiation by these electrons. This implies a new, strong constraint on superluminal electron velocities $\delta_e \le \sim 5 \times 10^{-21}$. It immediately follows that one then obtains an upper limit on the superluminal neutrino velocity {\it alone} of $\delta_{\nu} \le \sim 5.6 \times 10^{-19}$, many orders of magnitude better than the time-of-flight constraint from the SN1987A neutrino burst. However, if the electrons are {\it subluminal} the constraint on $|\delta_e| \le \sim 8 \times 10^{-17}$, obtained from the Crab Nebula $\gamma$-ray spectrum, places a weaker constraint on superluminal neutrino velocity of $\delta_{\nu} \le \sim 8 \times 10^{-17}$.

### Constraining Superluminal Electron and Neutrino Velocities using the 2010 Crab Nebula Flare and the IceCube PeV Neutrino Events [Replacement]

The observation of two PeV-scale neutrino events reported by Ice Cube can, in principle, allows one to place constraints on Lorentz invariance violation (LIV) in the neutrino sector. After first arguing that at least one of the IceCube events was of extragalactic origin, I derive an upper limit for {\it the difference} between putative superluminal neutrino and electron velocities of $\le \sim 5.6 \times 10^{-19}$ in units where $c = 1$, confirming that the observed PeV neutrinos could have reached Earth from extragalactic sources. I further derive a new constraint on the superluminal electron velocity, obtained from the observation of synchrotron radiation in the Crab Nebula flare of September, 2010. The inference that the $>$ 1 GeV $\gamma$-rays from synchrotron emission in the flare were produced by electrons of energy up to $\sim 5.1$ PeV indicates the non-occurrence of vacuum \’{C}erenkov radiation by these electrons. This implies a new, strong constraint on superluminal electron velocities $\delta_e \le \sim 5 \times 10^{-21}$. It immediately follows that one then obtains an upper limit on the superluminal neutrino velocity {\it alone} of $\delta_{\nu} \le \sim 5.6 \times 10^{-19}$, many orders of magnitude better than the time-of-flight constraint from the SN1987A neutrino burst. However, if the electrons are {\it subluminal} the constraint on $|\delta_e| \le \sim 8 \times 10^{-17}$, obtained from the Crab Nebula $\gamma$-ray spectrum, places a weaker constraint on superluminal neutrino velocity of $\delta_{\nu} \le \sim 8 \times 10^{-17}$.

### Constraining Superluminal Electron and Neutrino Velocities using the 2010 Crab Nebula Flare and the IceCube PeV Neutrino Events [Replacement]

The observation of two PeV-scale neutrino events reported by Ice Cube can, in principle, allows one to place constraints on Lorentz invariance violation (LIV) in the neutrino sector. After first arguing that at least one of the IceCube events was of extragalactic origin, I derive an upper limit for {\it the difference} between putative superluminal neutrino and electron velocities of $\le \sim 5.6 \times 10^{-19}$ in units where $c = 1$, confirming that the observed PeV neutrinos could have reached Earth from extragalactic sources. I further derive a new constraint on the superluminal electron velocity, obtained from the observation of synchrotron radiation in the Crab Nebula flare of September, 2010. The inference that the $>$ 1 GeV $\gamma$-rays from synchrotron emission in the flare were produced by electrons of energy up to $\sim 5.1$ PeV indicates the non-occurrence of vacuum \’{C}erenkov radiation by these electrons. This implies a new, strong constraint on superluminal electron velocities $\delta_e \le \sim 5 \times 10^{-21}$. It immediately follows that one then obtains an upper limit on the superluminal neutrino velocity {\it alone} of $\delta_{\nu} \le \sim 5.6 \times 10^{-19}$, many orders of magnitude better than the time-of-flight constraint from the SN1987A neutrino burst. However, if the electrons are {\it subluminal} the constraint on $|\delta_e| \le \sim 8 \times 10^{-17}$, obtained from the Crab Nebula $\gamma$-ray spectrum, places a weaker constraint on superluminal neutrino velocity of $\delta_{\nu} \le \sim 8 \times 10^{-17}$.


text:
# Dijkstra Monads in Monadic Computation

Abstract : The Dijkstra monad has been introduced recently for capturing weakest precondition computations within the context of program verification, supported by a theorem prover. Here we give a more general description of such Dijkstra monads in a categorical setting. We first elaborate the recently developed view on program semantics in terms of a triangle of computations, state transformers, and predicate transformers. Instantiations of this triangle for different monads $T$ show how to define the Dijkstra monad associated with $T$, via the logic involved. Technically, we provide a morphism of monads from the state monad transformation applied to $T$, to the Dijkstra monad associated with $T$. This monad map is precisely the weakest precondition map in the triangle, given in categorical terms by substitution.
Type de document :
Communication dans un congrès
Marcello M. Bonsangue. 12th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2014, Grenoble, France. Lecture Notes in Computer Science, LNCS-8446, pp.135-150, 2014, Coalgebraic Methods in Computer Science. 〈10.1007/978-3-662-44124-4_8〉
Domaine :
Liste complète des métadonnées

Littérature citée [29 références]

https://hal.inria.fr/hal-01408757
Contributeur : Hal Ifip <>
Soumis le : lundi 5 décembre 2016 - 13:24:57
Dernière modification le : lundi 5 décembre 2016 - 14:32:53
Document(s) archivé(s) le : mardi 21 mars 2017 - 10:42:43

### Fichier

328263_1_En_8_Chapter.pdf
Fichiers produits par l'(les) auteur(s)

### Licence

Distributed under a Creative Commons Paternité 4.0 International License

### Citation

Bart Jacobs. Dijkstra Monads in Monadic Computation. Marcello M. Bonsangue. 12th International Workshop on Coalgebraic Methods in Computer Science (CMCS), Apr 2014, Grenoble, France. Lecture Notes in Computer Science, LNCS-8446, pp.135-150, 2014, Coalgebraic Methods in Computer Science. 〈10.1007/978-3-662-44124-4_8〉. 〈hal-01408757〉

### Métriques

Consultations de la notice

## 29

Téléchargements de fichiers


text:
A brainfuck interpreter in python 3

I've been writing a brainfuck interpreter in python 3. It's nothing serious, but I'm trying to make it as good as possible (considering program structure, user experience and etc) for my own learning purpose. Any suggestion is welcomed.

"""A brainfuck interpreter written in Python 3.5."""

import argparse

__all__ = ['Brainfuck']

class Brainfuck:
"""Brainfuck interpreter main class."""

def __init__(self, src, debug=False):
"""
Parameters
----------
src: string
Brainfuck code to be evaluated.
debug: bool (optional, default: False)
Pause at '#' and print some status information.

Public attributes
-----------------
output: string
Output of the brainfuck script.
"""
self._cells = [0]
self._cell_ptr = 0

self._src = self._clean(src, debug)
self._src_ptr = 0

self._open_bracket_indexes = []
self._close_bracket_indexes = []
self._pair_brackets()

self.output = ''

self._evaluate()

@property
def _cell_value(self):
return self._cells[self._cell_ptr]

@_cell_value.setter
def _cell_value(self, value):
self._cells[self._cell_ptr] += value

if self._cells[self._cell_ptr] > 255:
self._cells[self._cell_ptr] = 0

elif self._cells[self._cell_ptr] < 0:
self._cells[self._cell_ptr] = 255

@property
def _command(self):
return self._src[self._src_ptr]

@staticmethod
def _clean(src, debug):
commands = '+-<>[],.#' if debug else '+-<>[],.'
return ''.join(c for c in src if c in commands)

def _pair_brackets(self):
"""
_open_bracket_indexes[i] is paired with _close_bracket_indexes[i].
"""
stack = []

for index, command in enumerate(self._src):
if command == '[':
stack.append(index)

elif command == ']':
self._open_bracket_indexes.append(stack.pop())
self._close_bracket_indexes.append(index)

def _evaluate(self):
while self._src_ptr <= len(self._src)-1:
command = self._command

if command == '+':
self._cell_value = 1

elif command == '-':
self._cell_value = -1

elif command == '<':
self._cell_ptr -= 1

elif command == '>':
self._cell_ptr += 1

if self._cell_ptr > len(self._cells)-1:
self._cells.append(0)

elif command == '[' and self._cell_value == 0:
self._to_paired_close_bracket()

elif command == ']' and self._cell_value != 0:
self._to_paired_open_bracket()

elif command == ',':
self._cell_value = ord(input('> ')[0])

elif command == '.':
new_char = chr(self._cell_value)

self.output += new_char
print(new_char, end='')

elif command == '#':
self._print_status()

self._src_ptr += 1

def _to_paired_open_bracket(self):
index = self._close_bracket_indexes.index(self._src_ptr)
self._src_ptr = self._open_bracket_indexes[index]

def _to_paired_close_bracket(self):
index = self._open_bracket_indexes.index(self._src_ptr)
self._src_ptr = self._close_bracket_indexes[index]

def _print_status(self):
cells_string = self._cells_string()
cell_ptr_string = self._cell_ptr_string(cells_string)

print('\n---------')
print('CELLS:    {}'.format(cells_string))
print('POINTER:  {}'.format(cell_ptr_string))
print('POSITION: {}'.format(self._cell_ptr))
print('---------')

input('[DEBUGGING] \'enter\' to continue')

def _cells_string(self):
"""
Transform cells array into a string.

Example:
array: [0, 87, 100, 33, 10]
string: |0| 87| 100| 33| 10|
"""
return (str(self._cells).replace('[', '|')
.replace(',', '|')
.replace(']', '|'))

def _cell_ptr_string(self, cells_string):
"""
Example:
_cell_ptr: 3
cells_string:    |0| 87| 100| 33| 10|
cell_ptr_string:               ^
"""
index = self._cell_ptr_index_on_cells_string(cells_string)
return ' ' * (index-1) + '^'

def _cell_ptr_index_on_cells_string(self, cells_string):
"""Calculate the index of '^' in cell_ptr_string."""
n = self._cell_ptr + 2
index = cells_string.find('|')

while n > 1 and index >= 0:
index = cells_string.find('|', index+1)
n -= 1

return index

def _parser():
p = argparse.ArgumentParser()

p.add_argument('file', help='brainfuck file to be evaluated')
action='store_true',
help='pause at \'#\' and print some status information')

return p.parse_args()

def _main():
args = _parser()

with open(args.file) as file:
Brainfuck(src, args.debug)

if __name__ == '__main__':
_main()

• Does this code actually work? it looks like command = self._command is missing something. Is it supposed to be command = self._command[self._src_ptr]? – skrrgwasme Jul 11 '16 at 20:13
• It works. self._command is a property, which returns self._src[self._src_ptr]. – Oliver Jul 11 '16 at 20:51
• Looking at it now, maybe it doesn't need to be a property, because I only call it once.. Right? – Oliver Jul 11 '16 at 20:56
• I think making it a property is unnecessary. The point of properties is to hide complexity from code that uses your classes, and allow them to treat some complex value retrieval/calculation as a simple attribute. Since it's an internal object member, there's nothing to hide, so it introduces complexity without much benefit.. My opinion would be that it's unnecessarily complex, and should be a simple list index. – skrrgwasme Jul 11 '16 at 21:02
• Thanks for the explanation. I'm new to this site, should I edit the code above right away? – Oliver Jul 11 '16 at 21:13

Your implementation of the , command, using input('> '), is non-standard. Standard Brainfuck shouldn't print a prompt, nor should it wait for Return to be pressed. Rather, it should just read one byte from sys.stdin.

The condition

while self._src_ptr <= len(self._src)-1:

… would be more idiomatically written as

while self._src_ptr < len(self._src):

Similarly simplify if self._cell_ptr > len(self._cells)-1.

What would you guess that

self._cell_value = 1

does? I find it deceptive that it increments the cell value by 1 (with wraparound) rather than setting the value. I recommend writing it as self._cell_value += 1. Keep the wraparound logic in the setter, if you want.

Looks like fairly reasonable code to me.

In _clean(), rather than commands = '+-<>[],.#' if debug else '+-<>[],.', consider setting the value and then conditionally appending '#' if we're debugging. It's just easier to read, to avoid needing to visually diff them.

In _evaluate(), the giant if is perfectly nice, but there is an opportunity to dispatch using a dict that maps from character to code, if you like. And this expression:

    while self._src_ptr <= len(self._src)-1:

is less natural than a for loop over the "good part" of _src, perhaps using a slice.

EDIT: The expression is odd enough that I just mentally flagged it as "odd" and moved on, without being curious enough to consider its details. Thank you, 200_success, for pointing out that <= is just trouble there. I saw the - 1 and thought that for command in self._src[:-1]: was what was intended, but no, it's actually saying for command in self._src:, just not in the obvious way. There should be one - and preferably only one -obvious way to do it.

• Could you be more specific about what you mean by 'a for loop over the "good part" of _src'? – 200_success Jan 17 '18 at 3:57


text:
# Thread: Need help with Percentage Word Problems.

1. ## Need help with Percentage Word Problems.

There are two word problems that are giving me alot of trouble for an assignment that I have to do.

A 50% saline solution mixed together with a 30% saline solution to get 30L of a 46% concentrated saline solution. How many litres of each are needed to be mixed?

You invest a total of $1800 into two diffrent investments. One investment returned 9% and the other returned 6%. If the interest on the 9% was$102 more of a return than the return of the 6% investment then how much of your $1800 was in each seperate account? I don't know how to set up any of these problems. If I could have help with the set up I could do it. These are the only problems on the sheet giving me problems. Thanks for your assistance. 2. Originally Posted by (?)G You invest a total of$1800 into two diffrent investments. One investment returned 9% and the other returned 6%. If the interest on the 9% was $102 more of a return than the return of the 6% investment then how much of your$1800 was in each seperate account?
Let A be the amount of the first investment

Let B be the amount of the second investment

Originally Posted by (?)G

You invest a total of $1800 into two diffrent investments. $A+B = 1800$ Originally Posted by (?)G If the interest on the 9% was$102 more of a return than the return of the 6% investment then how much of your $1800 was in each seperate account? $1.09A = 102 +1.06B$ Can you solve this system? $A+B = 1800$ $1.09A = 102 +1.06B$ 3. Thanks, I knew the A+B=$1800 part, but the other part really threw me, with the one returned 9 percent was $102 more of a return than the 6 percent investment. So You have the 1.09, The 1 represents 100 percent, plus an extra 9 percent on top of one and 6 percent on top of the other? What about the saline solution one, I can't even make heads or tails of even the first equation. Here is my work just in case it can help somebody else out: A= -B+ 1800 1.09( -B + 1800) =$102 + 1.06B "Equal Quantities Method"

1860 = 2.15B

1860/2.15 = 2.15B/2.15B

(($865.12 = B)) 1.09A = 102 + 1.06 (865.12) 1.09A = 102 + 917.03 1.09A/1.09A = 1019.03/1.09 (($934.89= A))

When you add A and B up it is $1800.01, With rounding off certain numbers it is .01 over the$1800, but it's around the same area.

4. Originally Posted by (?)G

A 50% saline solution mixed together with a 30% saline solution to get 30L of a 46% concentrated saline solution. How many litres of each are needed to be mixed?
I'm thinking to tackle this one similar to that of the second question

A and B are the volumes of the 2 mixtures.

$50\%\times A + 30\%\times B = 46\%\times (A+B)$

$A+B= 30$

can you finish it?

5. Originally Posted by pickslides
I'm thinking to tackle this one similar to that of the second question

A and B are the volumes of the 2 mixtures.

$50\%\times A + 30\%\times B = 46\%\times (A+B)$

$A+B= 30$

can you finish it?
Was I right for the last one, by the way?

6. Looks good.