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# Continuous and analytical diffusion signal modelling with MAPMRI

We show how to model the diffusion signal as a linear combination of continuous functions from the MAPMRI basis [Ozarslan2013]. This continuous representation allows for the computation of many properties of both the signal and diffusion propagator.

We show how to estimate the analytical Orientation Distribution Function (ODF) and a variety of scalar indices. These include rotationally invariant quantities such as the Mean Squared Displacement (MSD), Q-space Inverse Variance (QIV), Return-To-Origin Probability (RTOP) and Non-Gaussianity (NG). Interestingly, the MAP-MRI basis also allows for the computation of directional indices, such as the Return To the Axis Probability (RTAP), the Return To the Plane Probability (RTPP), and the parallel and perpendicular Non-Gaussianity.

The estimation of these properties from noisy DWIs requires that the fitting of the MAPMRI basis is regularized. We will show that this can be done using both constraining the diffusion propagator to positive values [Ozarslan2013] and analytic Laplacian Regularization (MAPL) [Fick2016a].

First import the necessary modules:

from dipy.reconst import mapmri
from dipy.viz import window, actor
from dipy.data import get_fnames, get_sphere
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.axes_grid1 import make_axes_locatable

MAPMRI requires multi-shell data, to properly fit the radial part of the basis. to False to only download eddy-current/motion corrected data. The total size of the downloaded data is 187.66 MBytes, however you only need to fetch it once.

fraw, fbval, fbvec, t1_fname = get_fnames('cfin_multib')

data contains the voxel data and gtab contains a GradientTable object (gradient information e.g. b-values). For example, to show the b-values it is possible to write:

print(gtab.bvals)

For the values of the q-space indices to make sense it is necessary to explicitly state the big_delta and small_delta parameters in the gradient table.

data, affine = load_nifti(fraw)

big_delta = 0.0365  # seconds
small_delta = 0.0157  # seconds
big_delta=big_delta,
small_delta=small_delta)

data_small = data[40:65, 50:51]

print('data.shape (%d, %d, %d, %d)' % data.shape)

The MAPMRI Model can now be instantiated. The radial_order determines the expansion order of the basis, i.e., how many basis functions are used to approximate the signal.

First, we must decide to use the anisotropic or isotropic MAPMRI basis. As was shown in [Fick2016a], the isotropic basis is best used for tractography purposes, as the anisotropic basis has a bias towards smaller crossing angles in the ODF. For signal fitting and estimation of scalar quantities the anisotropic basis is suggested. The choice can be made by setting anisotropic_scaling=True or anisotropic_scaling=False.

First, we must select the method of regularization and/or constraining the basis fitting.

• laplacian_regularization=True makes it use Laplacian regularization [Fick2016a]. This method essentially reduces spurious oscillations in the reconstruction by minimizing the Laplacian of the fitted signal. Several options can be given to select the regularization weight:

• regularization_weighting=GCV uses generalized cross-validation [Craven1978] to find an optimal weight.

• regularization_weighting=0.2 for example would omit the GCV and just set it to 0.2 (found to be reasonable in HCP data [Fick2016a]).

• regularization_weighting=np.array(weights) would make the GCV use a custom range to find an optimal weight.

• positivity_constraint=True makes it use the positivity constraint on the diffusion propagator [Ozarslan2013]. This method constrains the final solution of the diffusion propagator to be positive at a set of discrete points, since negative values should not exist.

• The pos_grid and pos_radius affect the location and number of constraint points in the diffusion propagator.

Both methods do a good job of producing viable solutions to the signal fitting in practice. The difference is that the Laplacian regularization imposes smoothness over the entire signal, including the extrapolation beyond the measured signal. In practice this results in, but does not guarantee positive solutions of the diffusion propagator. The positivity constraint guarantees a positive solution in a set of discrete points, which in general results in smooth solutions, but does not guarantee it.

A suggested strategy is to use a low Laplacian weight together with the positivity constraint. In this way both desired properties are guaranteed in the final solution.

We use package CVXPY (http://www.cvxpy.org/) to solve convex optimization problems when “positivity_constraint=True”, so we need to first install CVXPY.

For now we will generate the anisotropic models for all combinations.

radial_order = 6
laplacian_regularization=True,
laplacian_weighting=.2)

map_model_positivity_aniso = mapmri.MapmriModel(gtab,
laplacian_regularization=False,
positivity_constraint=True)

laplacian_regularization=True,
laplacian_weighting=.05,
positivity_constraint=True)

Note that when we use only Laplacian regularization, the GCV option may select very low regularization weights in very anisotropic white matter such as the corpus callosum, resulting in corrupted scalar indices. In this example we therefore choose a preset value of 0.2, which has shown to be quite robust and also faster in practice [Fick2016a].

We can then fit the MAPMRI model to the data:

mapfit_laplacian_aniso = map_model_laplacian_aniso.fit(data_small)
mapfit_positivity_aniso = map_model_positivity_aniso.fit(data_small)
mapfit_both_aniso = map_model_both_aniso.fit(data_small)

From the fitted models we will first illustrate the estimation of q-space indices. For completeness, we will compare the estimation using only Laplacian regularization, positivity constraint or both. We first show the RTOP [Ozarslan2013].

# generating RTOP plots
fig = plt.figure(figsize=(10, 5))
ax1 = fig.add_subplot(1, 3, 1, title=r'RTOP - Laplacian')
ax1.set_axis_off()
rtop_laplacian = np.array(mapfit_laplacian_aniso.rtop()[:, 0, :].T,
dtype=float)
ind = ax1.imshow(rtop_laplacian, interpolation='nearest',
origin='lower', cmap=plt.cm.gray)

ax2 = fig.add_subplot(1, 3, 2, title=r'RTOP - Positivity')
ax2.set_axis_off()
rtop_positivity = np.array(mapfit_positivity_aniso.rtop()[:, 0, :].T,
dtype=float)
ind = ax2.imshow(rtop_positivity, interpolation='nearest',
origin='lower', cmap=plt.cm.gray)

ax3 = fig.add_subplot(1, 3, 3, title=r'RTOP - Both')
ax3.set_axis_off()
rtop_both = np.array(mapfit_both_aniso.rtop()[:, 0, :].T, dtype=float)
ind = ax3.imshow(rtop_both, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
divider = make_axes_locatable(ax3)
plt.colorbar(ind, cax=cax)

plt.savefig('MAPMRI_maps_regularization.png')

It can be seen that all maps appear quite smooth and similar. Though, it is possible to see some subtle differences near the corpus callosum. The similarity and differences in reconstruction can be further illustrated by visualizing the analytic norm of the Laplacian of the fitted signal.

fig = plt.figure(figsize=(10, 5))
ax1 = fig.add_subplot(1, 3, 1, title=r'Laplacian norm - Laplacian')
ax1.set_axis_off()
laplacian_norm_laplacian = np.array(mapfit_laplacian_aniso.norm_of_laplacian_signal()[:, 0, :].T,
dtype=float)
ind = ax1.imshow(laplacian_norm_laplacian, interpolation='nearest',
origin='lower', cmap=plt.cm.gray, vmin=0, vmax=3)

ax2 = fig.add_subplot(1, 3, 2, title=r'Laplacian norm - Positivity')
ax2.set_axis_off()
laplacian_norm_positivity = np.array(mapfit_positivity_aniso.norm_of_laplacian_signal()[:, 0, :].T,
dtype=float)
ind = ax2.imshow(laplacian_norm_positivity, interpolation='nearest',
origin='lower', cmap=plt.cm.gray, vmin=0, vmax=3)

ax3 = fig.add_subplot(1, 3, 3, title=r'Laplacian norm - Both')
ax3.set_axis_off()
laplacian_norm_both = np.array(mapfit_both_aniso.norm_of_laplacian_signal()[:, 0, :].T,
dtype=float)
ind = ax3.imshow(laplacian_norm_both, interpolation='nearest', origin='lower',
cmap=plt.cm.gray, vmin=0, vmax=3)
divider = make_axes_locatable(ax3)
plt.colorbar(ind, cax=cax)
plt.savefig('MAPMRI_norm_laplacian.png')

A high Laplacian norm indicates that the gradient in the three-dimensional signal reconstruction changes a lot - something that may indicate spurious oscillations. In the Laplacian reconstruction (left) we see that there are some isolated voxels that have a higher norm than the rest. In the positivity constraint reconstruction the norm is already smoother. When both methods are used together the overall norm gets smoother still, since both smoothness of the signal and positivity of the propagator are imposed.

From now on we just use the combined approach, show all maps we can generate and explain their significance.

fig = plt.figure(figsize=(15, 6))
ax1 = fig.add_subplot(1, 5, 1, title=r'MSD')
ax1.set_axis_off()
msd = np.array(mapfit_both_aniso.msd()[:, 0, :].T, dtype=float)
ind = ax1.imshow(msd, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)

ax2 = fig.add_subplot(1, 5, 2, title=r'QIV')
ax2.set_axis_off()
qiv = np.array(mapfit_both_aniso.qiv()[:, 0, :].T, dtype=float)
ind = ax2.imshow(qiv, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)

ax3 = fig.add_subplot(1, 5, 3, title=r'RTOP')
ax3.set_axis_off()
rtop = np.array((mapfit_both_aniso.rtop()[:, 0, :]).T, dtype=float)
ind = ax3.imshow(rtop, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)

ax4 = fig.add_subplot(1, 5, 4, title=r'RTAP')
ax4.set_axis_off()
rtap = np.array((mapfit_both_aniso.rtap()[:, 0, :]).T, dtype=float)
ind = ax4.imshow(rtap, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)

ax5 = fig.add_subplot(1, 5, 5, title=r'RTPP')
ax5.set_axis_off()
rtpp = np.array(mapfit_both_aniso.rtpp()[:, 0, :].T, dtype=float)
ind = ax5.imshow(rtpp, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
plt.savefig('MAPMRI_maps.png')

From left to right:

• Mean Squared Displacement (MSD) is a measure for how far protons are able to diffuse. a decrease in MSD indicates protons are hindered/restricted more, as can be seen by the high MSD in the CSF, but low in the white matter.

• Q-space Inverse Variance (QIV) is a measure of variance in the signal, which is said to have higher contrast to white matter than the MSD [Hosseinbor2013]. We also showed that QIV has high sensitivity to tissue composition change in a simulation study [Fick2016b].

• Return to origin probability (RTOP) quantifies the probability that a proton will be at the same position at the first and second diffusion gradient pulse. A higher RTOP indicates that the volume a spin is inside of is smaller, meaning more overall restriction. This is illustrated by the low values in CSF and high values in white matter.

• Return to axis probability (RTAP) is a directional index that quantifies the probability that a proton will be along the axis of the main eigenvector of a diffusion tensor during both diffusion gradient pulses. RTAP has been related to the apparent axon diameter [Ozarslan2013] [Fick2016a] under several strong assumptions on the tissue composition and acquisition protocol.

• Return to plane probability (RTPP) is a directional index that quantifies the probability that a proton will be on the plane perpendicular to the main eigenvector of a diffusion tensor during both gradient pulses. RTPP is related to the length of a pore [Ozarslan2013] but in practice should be similar to that of Gaussian diffusion.

It is also possible to estimate the amount of non-Gaussian diffusion in every voxel [Ozarslan2013]. This quantity is estimated through the ratio between Gaussian volume (MAPMRI’s first basis function) and the non-Gaussian volume (all other basis functions) of the fitted signal. For this value to be physically meaningful we must use a b-value threshold in the MAPMRI model. This threshold makes the scale estimation in MAPMRI only use samples that realistically describe Gaussian diffusion, i.e., at low b-values.

map_model_both_ng = mapmri.MapmriModel(gtab, radial_order=radial_order,
laplacian_regularization=True,
laplacian_weighting=.05,
positivity_constraint=True,
bval_threshold=2000)

mapfit_both_ng = map_model_both_ng.fit(data_small)

fig = plt.figure(figsize=(10, 6))
ax1 = fig.add_subplot(1, 3, 1, title=r'NG')
ax1.set_axis_off()
ng = np.array(mapfit_both_ng.ng()[:, 0, :].T, dtype=float)
ind = ax1.imshow(ng, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
divider = make_axes_locatable(ax1)
plt.colorbar(ind, cax=cax)

ax2 = fig.add_subplot(1, 3, 2, title=r'NG perpendicular')
ax2.set_axis_off()
ng_perpendicular = np.array(mapfit_both_ng.ng_perpendicular()[:, 0, :].T,
dtype=float)
ind = ax2.imshow(ng_perpendicular, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
divider = make_axes_locatable(ax2)
plt.colorbar(ind, cax=cax)

ax3 = fig.add_subplot(1, 3, 3, title=r'NG parallel')
ax3.set_axis_off()
ng_parallel = np.array(mapfit_both_ng.ng_parallel()[:, 0, :].T, dtype=float)
ind = ax3.imshow(ng_parallel, interpolation='nearest', origin='lower',
cmap=plt.cm.gray)
divider = make_axes_locatable(ax3)
plt.colorbar(ind, cax=cax)
plt.savefig('MAPMRI_ng.png')

On the left we see the overall NG and on the right the directional perpendicular NG and parallel NG. The NG ranges from 1 (completely non-Gaussian) to 0 (completely Gaussian). The overall NG of a voxel is always higher or equal than each of its components. It can be seen that NG has low values in the CSF and higher in the white matter.

Increases or decreases in these values do not point to a specific microstructural change, but can indicate clues as to what is happening, similar to Fractional Anisotropy. An initial simulation study that quantifies the added value of q-space indices over DTI indices is given in [Fick2016b].

The MAPMRI framework also allows for the estimation of Orientation Distribution Functions (ODFs). We recommend to use the isotropic implementation for this purpose, as the anisotropic implementation has a bias towards smaller crossing angles.

For the isotropic basis we recommend to use a radial_order of 8, as the basis needs more generic and needs more basis functions to approximate the signal.

radial_order = 8
laplacian_regularization=True,
laplacian_weighting=.1,
positivity_constraint=True,
anisotropic_scaling=False)

mapfit_both_iso = map_model_both_iso.fit(data_small)

sphere = get_sphere('repulsion724')

Compute the ODFs.

The radial order s can be increased to sharpen the results, but it might also make the ODFs noisier. Always check the results visually.

odf = mapfit_both_iso.odf(sphere, s=2)
print('odf.shape (%d, %d, %d, %d)' % odf.shape)

Display the ODFs.

# Enables/disables interactive visualization
interactive = False

scene = window.Scene()
sfu = actor.odf_slicer(odf, sphere=sphere, colormap='plasma', scale=0.5)
sfu.display(y=0)
sfu.RotateX(-90)
window.record(scene, out_path='odfs.png', size=(600, 600))
if interactive:
window.show(scene)

Orientation distribution functions (ODFs).

## References

Ozarslan2013(1,2,3,4,5,6,7)

Ozarslan E. et al., “Mean apparent propagator (MAP) MRI: A novel diffusion imaging method for mapping tissue microstructure”, NeuroImage, 2013.

Fick2016a(1,2,3,4,5,6)

Fick, Rutger HJ, et al. “MAPL: Tissue microstructure estimation using Laplacian-regularized MAP-MRI and its application to HCP data.” NeuroImage (2016).

Craven1978

Craven et al. “Smoothing Noisy Data with Spline Functions.” NUMER MATH 31.4 (1978): 377-403.

Hosseinbor2013

Hosseinbor et al. “Bessel fourier orientation reconstruction (bfor): an analytical diffusion propagator reconstruction for hybrid diffusion imaging and computation of q-space indices. NeuroImage 64, 650-670.

Fick2016b(1,2)

Fick et al. “A sensitivity analysis of Q-space indices with respect to changes in axonal diameter, dispersion and tissue composition. ISBI 2016.

Example source code

You can download the full source code of this example. This same script is also included in the dipy source distribution under the doc/examples/ directory.


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### Introduction

The RcppArrayFire package provides an interface from R to and from the ArrayFire library, an open source library that can make use of GPUs and other hardware accelerators via CUDA or OpenCL.

The official R bindings expose ArrayFire data structures as S4 objects in R, which would require a large amount of code to support all the methods defined in ArrayFire’s C/C++ API. RcppArrayFire instead, which is derived from RcppFire by Kazuki Fukui, follows the lead of packages like RcppArmadillo or RcppEigen to provide seamless communication between R and ArrayFire at the C++ level.

### Installation

Please note that RcppArrayFire is developed and tested on Linux systems. There is preliminary support for Mac OS X.

In order to use RcppArrayFire you will need the ArrayFire library and header files. While ArrayFire has been packaged for Debian, I currently prefer using upstream’s binary installer or building from source.

RcppArrayFire is not on CRAN, but you can install the current version via drat:

If you have installed ArrayFire in a non-standard directory, you have to use the configure argument --with-arrayfire, e.g.:

### A first example

Let’s look at the classical example of calculating $\pi$ via simulation. The basic idea is to generate a large number of random points within the unit square. An approximation for $\pi$ can then be calculated from the ratio of points within the unit circle to the total number of points. A vectorized implementation in R might look like this:

pi ~=  3.13999

user  system elapsed
0.047   0.008   0.055

A simple way to use C++ code in R is to use the inline package or cppFunction() from Rcpp, which are both possible with RcppArrayFire. An implementation in C++ using ArrayFire might look like this:

pi ~=  3.1451

   user  system elapsed
0.001   0.000   0.001

Several things are worth noting:

1. The syntax is almost identical. Besides the need for using types and a different function name when generating random numbers, the argument f32 to randu as well as the float type catches the eye. These instruct ArrayFire to use single precision floats, since not all devices support double precision floating point numbers. If you want and can to use double precision, you have to specify f64 and double.

2. The results are not the same since ArrayFire uses a different random number generator.

3. The speed-up can be quite impressive. However, the first invocation of a function is often not as fast as expected due to the just-in-time compilation used by ArrayFire. This can be circumvented by using a warm-up call with (normally) fewer computations.

### Arrays as parameters

Up to now we have only considered simple types like double or int as function parameters and return values. However, we can also use arrays. Consider the case of an European put option that was recently handled with R, Rcpp and RcppArmadillo. The Armadillo based function from this post reads:

This function can be applied to a range of spot prices:

        [,1]
[1,] 5.52021
[2,] 4.58142
[3,] 3.68485
[4,] 2.85517
[5,] 2.11883
[6,] 1.49793

Porting this code to RcppArrayFire is straight forward:

Compared with the implementations in R, Rcpp and RcppArmadillo the syntax is again almost the same. One exception is that ArrayFire does not contain a function for the cumulative normal distribution function. However, the closely related error function is available.

Since an object of type af::array can contain different data types, the templated wrapper class RcppArrayFire::typed_array<> is used to indicate the desired data type when converting from R to C++. Again single precision floats are used with ArrayFire, which leads to differences of the order $10^{-6}$ compared to the results from R, Rcpp and RcppArmadillo:

[1] 5.52021 4.58142 3.68485 2.85517 2.11883 1.49793

### Performance

The reason to use hardware accelerators is of course the quest for increased performance. How does ArrayFire fare in this respect? Using the same benchmark as in the R, Rcpp and RcppArmadillo comparison:

  test replications elapsed relative
2   AF          100   0.489    1.000
1 Arma          100   1.928    3.943

Here a Nvidia GeForce GT 1030 is used together with ArrayFire’s CUDA backend. With a build-in Intel HD Graphics 520 using the OpenCL backend the ArrayFire solution is about 6 times faster. Even without a high performance GPU the performance boost from using ArrayFire can be quite impressive. However, the results change dramatically, if fewer options are evaluated:

  test replications elapsed relative
1 Arma         1000   0.007    1.000
2   AF         1000   0.088   12.571

But is it realistic to process $10^6$ options at once? Probably not in the way used in the benchmark where only the spot price is allowed to vary. However, one can alter the function to process not only arrays of spot prices but also arrays of strikes, risk free rates etc.:

Note that ArrayFire does not recycle elements if arrays with non-matching dimensions are combined. In this particular case this means that all arrays must have the same length. One can ensure that by using a data frame for the values:

        s  k    r    y t sigma           p
1 87.4192 50 0.01 0.02 1  0.05 7.45937e-29
2 48.7060 50 0.01 0.02 1  0.05 2.09402e+00
3 67.2626 50 0.01 0.02 1  0.05 2.34672e-09
4 72.6573 50 0.01 0.02 1  0.05 6.84430e-14
5 68.0854 50 0.01 0.02 1  0.05 5.26394e-10
6 57.8775 50 0.01 0.02 1  0.05 2.57756e-03

### Conclusion

The ArrayFire library provides a convenient way to use hardware accelerators without the need to write low-level OpenCL or CUDA code. The C++ syntax is actually quite similar to properly vectorized R code. The RcppArrayFire package makes this available to useRs. However, one still has to be careful: using hardware accelerators is not a “silver bullet” due to the inherent memory transfer overhead.

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This is not a post about Donald Trump or the election, except tangentially. I have thoughts on Trump - so, so many thoughts - but most of them have been said, meaning I will provide little value added.

Garett Jones did a podcast with The Economics Detective recently on the costs of ethnic diversity. It is particularly worth listening to given that racial identity has re-emerged as a salient element of politics. A quick summary - and the link above includes a nice write-up of relevant sources - would be that diversity within workplaces does not appear to improve outcomes (however those outcomes are measured).

At the same time, there is a parallel literature, touched on in the podcast, about ethnic diversity (or fractionalization, as it is termed in that literature) and economic growth. But one has to be careful drawing a bright line between the two literatures. It does not follow that the results for workplace diversity imply the results regarding economic growth. And this is because the growth results, to the extent that you believe they are robust, all operate through political systems.

So here let me walk through some of the core empirical relationships that have been found regarding ethnic fractionalization and economic growth, and then talk about why you need to take care with over-interpreting them. This is not a thorough literature review, and I realize there are other papers in the same vein. What I’m after is characterizing the essential results.

### Baseline empirics

Easterly and Levine did one of the first studies of fractionalization and its relationship to economic growth, with a focus on Africa. To measure fractionalization they used a simple index that has become the de facto norm for measuring this kind of thing going forward.

where $s_i$ is the share of group i in the population of a given area. You can interpret the fractionalization measure as the probability that two random people in an area belong to different groups. If there is only one group, then the summation term is 1, so Frac is zero. If every single person is in a unique group, so there are N groups, then the summation term approaches 1/N, and Frac approaches 1. EL constructed indices like this for a set of countries using data that originally came from Soviet ethnographers. As I’ll mention in more detail below, their index thus depends on choices made by those ethnographers about what constitutes and ethnicity.

EL then looked at basic growth regressions. That is, they looked at the relationship between the average growth rate of GDP over a long period of time and how that correlated with their measure of fractionalization. In doing this, they controlled for initial GDP per capita and several other characteristics of countries. What they found was that higher fractionalization was associated with lower growth when only basic controls were included. To take the extreme, going from 0 fractionalization to total fractionalization was associated with a growth rate 2 percentage points lower.

As a quick aside, note that this doesn’t mean a permanently lower growth rate. EL are working with a standard growth regression, controlling for initital income. What they are finding is that places with high fractionalization had lower balanced growth paths. In their empirical setting, every country has the same long run growth rate, but highly fractionalized ones will be poorer than homogenous ones. This shows up as a lower growth rate for fractionalized places because they are closer to their balanced growth paths to begin with.

EL go on to explore the idea that the reason for fractionlization’s effect on balanced growth paths is due to the interference in public goods provision. Their hypothesis is that fractionalization implies conflicts between groups, which implies an inability or unwillingness to get broad public support for things like education or infrastructure, and this makes them poorer in the long run. And what EL find is that once you control for public goods provision (crudely, using things like the number of telephones per capita) the effect of fractionalization is no longer significant or as large. For EL, this shows that the impact of fractionalization works through the provision of these public goods.

Stronger support for that idea comes from a different Easterly paper. He, with Alesina and Baqir, published Public Goods and Ethnic Divisions in 1999. This looked directly at US cities, counties, and metro areas, and the share of government expenditures at those levels being spent on various public goods. They found that higher fractionalization was associated with a lower share of spending on roads (about minus 5%), welfare (minus 5%), and education (minus 10-17%). On the other hand, there was a higher share of spending on police (plus 2-4%) and health services (13-25%). The increased police and health share is not always enough to explain the lower shares in all the other areas of spending, implying that places with high fractionalization have a larger share of local government expenditures on “everything else”. That “everything else” includes debt payments and other poorly classified spending. The authors speculate it could be related to patronage, but there is no way to look at that directly. In terms of actual levels of taxes and spending, there found conflicting results. At the city level, fractionalization is associated with both higher taxes and spending per capita, but at the metro area and county levels fractionalization was associated with lower taxes but higher spending per capita. Either way, fractionalization was generally associated with higher spending per capita. And no, that wasn’t simply due to city/metro/county sizes, as this held while controlling for the total population of the city or metro area.

Back to the country level, where Alesina, Devleeschauwer, Easterly, Kurlat, and Wacziarg extended increased the amount of data available on a cross-country basis, including ethnic, linguistic, and religious measures of fractionalization. Similar to Easterly and Levine, they find that fractionalization implies a lower steady state growth path. Also similar, this negative effect dissipates as you control for crude measures of public goods spending and government effectiveness. The authors appeal to a similar argument to before: fractionalization makes consensus hard to build, and so public goods do not get provided at a level more homogenous societies can manage. A few correlates of ethnic fractionalization include lower literacy, lower levels of education attainment, and lower indices of democracy and political rights.

One interesting note to their paper is that religious fractionalization has almost the exact opposite effects of ethnic and linguistic fractionalization. More religiously heterogenous countries tend to have higher growth paths, more education, and better measure of political rights.

### Problems?

Start with a basic measurement issue. The fractionalization index is sensitive to the number of groups you divide everyone up into. That is, if your country has two ethnic groups, then the maximum value of the fractionalization index is 0.5. If your country has 10 ethnic groups, the maximum index is 0.9. That seems to make sense. More groups, more possible fractionalization.

Except that the definition of what counts as a group is not a fixed characteristic. If I give people in the US the options of white, black, Asian, Native American, or Other, then I am allowing for 5 groups, and max fractionalization of 0.8. We could double the number of groups to 10 by allowing each racial group to report as either Hispanic or non-Hispanic, and now the max index is 0.9. Or we could sub-divide white into “German-American”, “Irish-American”, and so on, and get ourselves up to like 40 or 50 groups if we really wanted to, and this would automatically drive up the fractionalization index to one. Or we could say f\$&# it, and stop tracking racial identity completely, and get a fractionalzation index of zero by definition.

The point is that the degree of fractionalization is not immutably fixed by arbitrary racial or ethnic groups designed by the Census Bureau. James Fearon has an excellent paper addressing the arbitrary nature of building an index of ethnic fractionalization. There is no right fractionalization index for any country. The number of ethnic groups tracked in a census or survey may not map to any meaningful political groups. The Hutus and Tutsis in Rwanda were are included in a single ethnic group in the original Soviet data, for example.

Going back to the empirical work, this raises some questions on how to interpret the results. When Alesina, et al, collected their updated measures of fractionalization, they noted that ethnic fractionalization in much of the Americas is based on racial definitions (black versus white), while in Europe and Sub-Saharan Africa ethnic fractionalization is based primarily on languages (think of Switzerland). So the ethnic fractionalization measures are not comparable across countries. What does it mean to compare fractionalization in Belgium (French versus Flemish speakers) with fractionalization in Bolivia (Mestizos versus Aymara versus Quechua)?

Alesina et al mention that they go out of their way to find the most disaggregated data on ethnic groups possible. That is, they make sure they maximize the number of groups in each country, thus maximizing the measure of fractionalization. This has the virtue of being a consistent strategy, but suffers because it is subject to variation in the level of disaggregation tracked by country, which is an inherently political decision.

You could easily turn all of the empirical results I talked about around in the following way. Places with relatively dysfunctional political systems benefitted from setting ethnic groups against each other, preventing effective opposition. They thus focused on ethnic or linguistic differences, and made sure to track these carefully when they did censuses. It need not even be nefarious. Former colonies who were left with dysfunctional governments may also have inherited a tradition of close ethnic group tracking from their colonizers. Either way, places with dysfunctional governments would to have mechanically high fractionalization indices along ethnic lines. And their dysfunction also leads to poor economic outcomes.

And don’t forget those results on religious fractionalization, which was associated with better political and economic outcomes. The effect of diversity can go either way, depending on how you define it. If you started measuring fractionalization along as many dimensions as possible - ethnicity, religion, language, home state, favorite Beatle, height, extroversion/introversion - you’d eventually come across some of them that were positively related, and some negatively, to things like public goods spending or growth, just by chance.

The overarching point here is that even taking the studies on workplace diversity seriously, this does not imply that poor political outcomes and/or economic growth are a necessary outcome of ethnic fractionalization. At the political level, ethnic fractionalization and its salience to public policy is a choice variable.


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## Ordinal Logit Regression and Proportional Odds Assumption

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Posts: 84

# Ordinal Logit Regression and Proportional Odds Assumption

In ordered logit models,  the test for proportional odds tests whether our one-equation model is valid. If we were to reject the null hypothesis, we would conclude that ordered logit coefficients are not equal across the levels of the outcome and we would fit a less restrictive model (i.e., multinomial logit model). If we fail to reject the null hypothesis, we conclude that the assumption holds.

I noticed that If I include the option NOINT in the model statement in Proc logistics, this test is not performed by SAS. Why is that? Does it mean that with no intercept in the model, the assumption of equality of the logit coefficients across the levels of outcomes in no longer necessary?

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05-02-2013 03:54 PM
Posts: 2,655

## Re: Ordinal Logit Regression and Proportional Odds Assumption

Look at the model that is fit WITH intercepts.  There is a common part (which assumes proportionality) and then there are the intercepts, which are the classes into which you are categorizing (as nearly as possible).  WITHOUT intercepts, there is only the common (linear in the predictors) part.  What I think you are seeing is an interaction between one of the predictors and the outcome variable that is not proportional.

Steve Denham

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Posts: 2,655

## Re: Ordinal Logit Regression and Proportional Odds Assumption

I think it means something more like "Without intercepts, and for the cumulative (ordered) logit link, all categories are assumed to be identical."  Thus no test.  The intercepts drive the classification at first is how I think of it.

Steve Denham

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Posts: 84

## Re: Ordinal Logit Regression and Proportional Odds Assumption

Then the question will be " How can I test my assumption about the similarity off the categories? " Or in other words, if I use the intercepts and the assumption is not rejected, then the results should be similar to the ones with no intercept?

The reason that I ask is that the results with and without the intercept are very different.

the assumption is also rejected when I use intercept.

Solution
05-02-2013 03:54 PM
Posts: 2,655

## Re: Ordinal Logit Regression and Proportional Odds Assumption

Look at the model that is fit WITH intercepts.  There is a common part (which assumes proportionality) and then there are the intercepts, which are the classes into which you are categorizing (as nearly as possible).  WITHOUT intercepts, there is only the common (linear in the predictors) part.  What I think you are seeing is an interaction between one of the predictors and the outcome variable that is not proportional.

Steve Denham

Frequent Contributor
Posts: 84

## Re: Ordinal Logit Regression and  Proportional Odds Assumption

Thanks for clarification steve

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Posts: 1

## Re: Ordinal Logit Regression and  Proportional Odds Assumption

Hi.  According to the documentation for Proc Logisti, NOINT sets only the FIRST intercept to zero, not all.  There is an example but I'm not sure of the utility.

My recollection from theory is that generally such a restricted model does not lead to a valid likelihood function, and likelihood ratio tests are distorted.  That may be why SAS suppressed the test.

I would think that if the null is rejected with intercept(s), restricting the model can't improve the situation.

☑ This topic is SOLVED.


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# Four particles A, B, C and D with masses $m_{A}=m, m_{B}=2m, m_{C}=3m\; and\; m_{D}=4m$ are the corners of a square. They have accelerations of equal magnitude with directions as shown. The acceleration of the centre of mass of the particles is : Option 1) $\frac{a}{5}(\widehat{i}-\widehat{j})$   Option 2) $a(\widehat{i}+\widehat{j})$ Option 3) $Zero$   Option 4) $\frac{a}{5}(\widehat{i}+\widehat{j})$

$\vec{a}_{cm}=\frac{ \sum m\vec{a}}{ \sum m}=\frac{m(-a\hat{i}\: )+(2m)(a\hat{j}\: )+(3m)(a\hat{i}\: )+4m(-a\hat{j}\: )}{m+2m+3m+4m}$

$\vec{a}_{cm}=\frac{2(\; \hat{i}-\hat{j}\: )a}{10}=\frac{1}{5}(\: \hat{i}-\hat{j}\: )a$

$\vec{a}_{cm}=\frac{a}{5}(\: \hat{i}-\hat{j}\; )$

Option 1)

$\frac{a}{5}(\widehat{i}-\widehat{j})$

Option 2)

$a(\widehat{i}+\widehat{j})$

Option 3)

$Zero$

Option 4)

$\frac{a}{5}(\widehat{i}+\widehat{j})$

Exams
Articles
Questions


text:
I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in examples of poor writing that introduce logical errors into the proof as well as those that just make the proof hard to read.

I am aware that there is already a question on this site asking for examples of bad proofs. I find the answers there kind of unsatisfactory (one answer simply says to look at student assignments and the other just consists of a broken link), but perhaps it's because there aren't many great examples of poorly written proofs on the internet. Also, that question is only focused on proofs that are wrong or "could be simplified" whereas I am also interested in more general problems with the way proofs are written.

In case there are no good examples online of poorly written proofs, I would like to try to generate some myself. Ideally, for each general type of mistake that undergraduates are prone to make when writing proofs, I want to write a bad proof that isolates and exemplifies that mistake. So I am interested in compiling a "taxonomy" of ways that proof writing by students can go wrong.

Here's my current taxonomy:

1. Backwards proofs: Proofs that start from the desired conclusion and work backwards until they arrive at the premises. Not only can this be confusing to read, it is easy to introduce errors when some step of the proof is not reversible.
2. The "obvious" worst case: It is popular to warn students against "proof by example" but I have found that students are usually already aware that an example is not a proof... unless of course they think that the example they chose is obviously the worst case scenario without bothering to justify it.
3. The tangle: The proof that keeps going and going, introducing a complicated series of manipulations, definitions, hand-waving, etc. until somehow arriving at the right conclusion. Often some step in the middle is wrong (or possibly nonsensical) but it's hard to find because it's buried under a pile of more-or-less-correct but irrelevant details. A special favorite of students taking exams.
4. The mysterious stranger: Some object is used without ever being introduced or defined. Often there is a reasonable way to fill in the definition of the object but the proof becomes a guessing game of what the student meant the object to be. Sometimes this is caused by a student trying to use a variable outside of its proper scope. In that case it can also lead to the student giving the same name to two distinct objects.
5. Inappropriate types: The student claims that some ideal is an element of the ring. Or tries to divide by a set of real numbers. Etc. The point is that they are trying to do something with an object which is of the wrong type to do it with. Or sometimes they are trying to perform a typecast that cannot automatically be done in an unambiguous way like if they try to apply a function of type $$\mathbb{R} \to \mathbb{R}$$ to a complex number. Often in these cases the student does have in mind a way to do the typecast (although sometimes they have just misunderstood a definition or don't even really grok that mathematical objects have types) but problems occur when there are multiple mutually incompatible ways to do it.
6. The incantation: You've just learned a big, important sounding theorem in class so you should probably use it on the homework, right? You don't quite understand it or know when it applies, but it seems like you should be using it so you just go ahead and invoke it anyways. It's pretty common for students to invoke big theorems without checking that the hypotheses of the theorem hold and sometimes in situations where it doesn't even make sense to invoke the theorem.
7. Implication confusion: The student is supposed to prove $$X \implies Y$$ but instead proves $$\lnot X \implies \lnot Y$$ or $$Y \implies X$$ or tries (and typically fails) to prove $$X\land Y$$. Or the student is told to assume that $$X \implies Y$$ and then concludes that $$Y \implies X$$. Most common when the statements of $$X$$ and $$Y$$ are a bit complicated or involve implications as well.

Which leads me to my question.

Question: What other common pitfalls of student proof writing should be added to this taxonomy? What are some poorly written proofs that exemplify one of these pitfalls (either one that I have already listed or a new one)?

Feel free to argue with my current list. Also, if anyone wants to add an answer to the old question on "bad proofs" I'd be very happy.

EDIT: After discussing this with one of my friends, I've added two more entries to the taxonomy.

• Two examples I can think of that might be slightly outside your scope (because even strong mathematicians can do this from time to time) are not considering all possible cases when dividing the proof into separate cases, and in overlooking sometimes problematic exceptional situations (what if the variable is zero, what if the set is the empty set, etc.). – Dave L Renfro Feb 17 '20 at 19:33
• @amWhy The fallacies you link to are rhetorical fallacies, which is explicitly not what I'm asking about. I suppose it's possible that an undergraduate student could include an instance of the ad hominem fallacy in a proof they write, but I have yet to see it. – Patrick Lutz Feb 17 '20 at 22:01
• There is far more overlap than you acknowledge (circular reasoning, argument by authority, etc.) – amWhy Feb 17 '20 at 22:03
• You do need to understand that at the root of many students' logical fallacies are rhetorically/cognitively/psychologically based. Certainly not all. But we can see this comes into play often in combinatorics and probability and stats. Also, wrt the material conditional $p\to q$ which is true except in the case that p is true and q is false, students often expect $p$ to be relevant to $q$, or for $q$ to be dependent on $p$. So some have a hard time accepting the truth of, e.g., "If pigs can fly, then purple unicorns can swim", and even, "If pigs can fly, then $2+2 = 4$." – amWhy Feb 17 '20 at 22:35
• I love this. I'm a TA for vector calc and I most often see backwards proofs and "the tangle". Beautiful example for explaining backwards proofs is one I saw last year: The question asks you to prove that if $r(t)$ is a parametric curve and $r(t)$ is always orthogonal to $r'(t)$, then $r(t)$ traces out a circle. Many students started with $r(t) = \langle R \cos(t), R \sin(t) \rangle$ and then proved $r'(t) \cdot r(t) = 0$ to conclude that $r(t)$ and $r'(t)$ are orthogonal. – Dark Malthorp Feb 25 at 16:39

Perhaps related to "The Tangle" is what I call "Wishful Thinking". This most often happens when the student has a correct algebraic expression/equality and knows the correct final expression/equality, but makes great (or incorrect) leaps to get from one to the other, e.g. in proofs by induction.

There is also the "Using What You Are Proving" (maybe similar to your "Backwards Proof," but more subtle). Example: in proving two elements $$a$$ and $$b$$ commute, they use $$(ab)^2 = a^2b^2$$. Or in an onto proof where they assume a general element of the range has the form of an image.

• Those are both great! I think "Wishful Thinking" is different from "The Tangle" since in the tangle, the errors are usually not caused by trying to get to any particular goal; it's more like lots of unnecessary complexity is introduced which often leads to errors, which often in turn leads to a false proof. And I also think "Using What You Are Proving" is different enough from "Backwards Proof" to constitute its own entry. – Patrick Lutz Feb 17 '20 at 18:50
• Also, it occurs to me that "The 'Obvious' Worst Case," "The Mysterious Stranger" and "The Incantation" are all at least sometimes caused by cases of "Wishful Thinking." – Patrick Lutz Feb 17 '20 at 18:52
• I suppose "using what you are proving" can be better described as "circular reasoning"? – GoodDeeds Feb 17 '20 at 18:57
• @GoodDeeds It's true that "Using What You Are Proving" is an instance of "Circular Reasoning," but what is interesting to me about it is the circumstances that cause students to make that mistake. Most students can recognize bald instances of circular reasoning and know it is wrong. But they still end up engaging in circular reasoning anyway and I am interested in how that occurs. – Patrick Lutz Feb 17 '20 at 19:32

The Q&A so far has gone into a lot of relatively sophisticated stuff. The students I teach are mostly engineering majors at a community college, so their mistakes are a lot simpler. They have generally been exposed to proof writing in high school geometry. However, my experience is that this does not translate well into a more general skill. Many of them learned "two-column proofs," which are pretty artificial and constrained.

Often they will "prove" something by giving an example, e.g., they'll prove that the dot product is commutative by making up a couple of vectors and doing the computations. This also comes up in physics labs, where they claim that their lab results prove Newton's second law. They don't want to believe that a counterexample can disprove a general statement, but one piece of evidence can't prove it.

They write something that they claim is a proof, but they never make use of one of the assumptions, and this assumption is necessary.

In my experience students in elementary classes like first-semester calculus need to get some experience writing really simple, easy proofs so that they can get past these initial stages. Unfortunately, I don't think most instructors assign any such work in these courses. I would imagine that this is a big reason for the meltdowns I hear about when math majors start their upper-division coursework.

Here's an example of what I would like them to be able to do. Suppose they know enough about the dot product to know how to compute it from cartesian components. Then if I ask them to prove that the dot product is commutative, ideally they would be able to write something like this: "The dot product can be expressed in terms of the products of the two vectors' components. Since the components are real numbers, and multiplication of real numbers is commutative, it follows that the dot product is commutative."

But even this extremely simple-looking argument contains some pretty sophisticated mental activity. Students are trying to relate everything to their previous education, and their previous education contains a couple of possible models of this. One model is concrete computation involving numbers, so they produce an example involving numbers and call that a proof. Another model is two-column proofs in high school geometry, but that model doesn't carry over here, because it doesn't help them to focus on the essentials, ignore inessential elements, and write a sentence meant for a human to read.

• Thanks for adding your perspective. At my current university, lower division math students are not really expected to be able to read or write proofs so the issue of having trouble producing correct proofs doesn't come up directly (though it comes up indirectly when they employ faulty reasoning in the course of solving a problem). I agree with you that it's a shame that there is often no opportunity for students to really understand how to write a rigorous proof in a environment which is slow-paced enough that they can just focus on mastering that skill. – Patrick Lutz May 14 at 3:46
• Perhaps it would even be beneficial for lower-division, computation-heavy courses if students had already been through some course on logical reasoning and proof writing. I feel that in such classes (i.e. computation-focused ones) the instructors are often math professors who attempt to use mathematical language to teach the students and often give at least sketches of proofs. But since the students have not been formally taught how to understand mathematical language or how to read proofs, they have trouble following this kind of instruction. – Patrick Lutz May 14 at 3:48
• At my current university, lower division math students are not really expected to be able to read or write proofs Here in the US, students are generally expected to write proofs in high school geometry. I've edited my answer to discuss this a little more. – Ben Crowell May 14 at 15:25
• Perhaps it would even be beneficial for lower-division, computation-heavy courses if students had already been through some course on logical reasoning and proof writing. I don't think this is realistic. Nearly all students in these courses are either engineering majors or life science majors. The engineering majors in particular have a very heavy load of required courses. You can't just add another semester-length course to their requirements. – Ben Crowell May 14 at 15:26
• I agree with you that it's a shame that there is often no opportunity for students to really understand how to write a rigorous proof in a environment which is slow-paced enough that they can just focus on mastering that skill. I actually don't think I agree with you here. There is no need for them to be "rigorous." They just need to get experience writing correct arguments that are understandable to another human. I don't think it's a good idea to focus on just the skill of proof-writing. Writing simple, easy proofs should just be a normal part of every class. – Ben Crowell May 14 at 15:30

My friend, who is currently teaching an abstract algebra course, recently provided me with an example of an incorrect proof that I think showcases a few types of faulty reasoning that are common among students. I'm sharing it here with his permission.

Proposition: Suppose $$R$$ is a ring and $$I$$ is a prime ideal in $$R$$. For all ideals $$J$$ and $$K$$ in $$R$$, if $$JK \subseteq I$$ then either $$J \subseteq I$$ or $$K \subseteq I$$.

Proof. Suppose $$JK \subseteq I$$. Then $$\sum_{n, m} a_nb_m \in I$$. Since $$I$$ is an ideal, it is closed under addition so each $$a_nb_m$$ is in $$I$$. Since $$I$$ is prime, either $$a_n$$ or $$b_m$$ is in $$I$$. Therefore either $$J \subseteq I$$ or $$K \subseteq I$$.

There are a few obvious problems with this proof.

• First, it is not clear what $$a_n$$ and $$b_m$$ are. Presumably the $$a_n$$'s are some finite set of elements in $$J$$ and the $$b_m$$'s are some finite set of elements in $$K$$, but this should be stated explicitly. This is an instance of The Mysterious Stranger.
• Next, saying that "since $$I$$ is closed under addition, each $$a_nb_m$$ is in $$I$$" is not correct. It is mixing up "closed under addition" with its converse, so this is an instance of Implication Confusion.
• The conclusion of the proof is more or less a non-sequitor. All that the proof has really established is that either $$J$$ or $$K$$ contains some element which is in $$I$$, which is not enough to show that either $$J$$ or $$K$$ is entirely contained in $$I$$. I would say this is an example of Wishful Thinking.
• Finally, there is an element of The Tangle to the whole proof. It is not really clear what the strategy of the entire proof is, why the sum $$\sum_{n, m} a_nb_m$$ is being introduced or what the goal of each step is. It's likely that such a proof is mostly the result of pattern-matching the appearance of proofs given in class in the hopes that this will lead to an approximation of a correct proof (and thus partial credit).
• I'm accustomed to tangles being much longer than this. – Andreas Blass May 13 at 2:30
• @AndreasBlass Yeah, I agree. That's why I just said "there is an element of the tangle." But it would be nice to have better examples. – Patrick Lutz May 13 at 2:34
• I think this useful and highlights what I came to say: there should be a category of "The Totally Incoherent" which is likely a synthetic gumbo of all the various errors. – Daniel R. Collins May 15 at 4:28

Three categories of error I often encounter are "Incorrect negation", "trouble with quantifiers" and "Confusing general parameters with specific objects". Everybody knows that students have trouble dealing with negating quantifiers in the context of, for example, an $$\epsilon-\delta$$ proof in an Analysis course. We also see these issues arise in Linear Algebra, where student work like the following is rampant:

Let $$\vec{v}$$ be any nonzero vector; prove that $$\{\vec{v}, \vec{0}\}$$ is linearly dependent.

Solution. Let $$a\vec{v} + b\vec{0} = \vec{0}$$. If $$a = b = 0$$ then this is a trivial relation; therefore the vectors are linearly dependent.

What's happening in this muddle is something like this: Students know that "linearly independent" means "there is no nontrivial linear relation on the vectors"; they try to negate it, but get tangled up in the triple negative ("it is not true that there is no nontrivial..."), and end up thinking that "linearly dependent" means "there is a trivial linear relation on the vectors". The use of the letters $$a$$ and $$b$$, which seem like general parameters in the first sentence, but are subsequently revealed in the very next sentence to have been very specific constants all along, helps to obscure the obvious error.

In fact on our most recent final exam we included, as a true/false question,the statement

"If $$S = \{v_1 \, \dots, v_m \}$$ is a set of vectors, and $$a_1v_1 + \cdots a_nv_n = 0$$ is a trivial relation, then $$S$$ is linearly independent."

A distressingly large fraction of our students marked this as true. They seem to have trouble understanding that "There are no nontrivial relations..." means something different from "There is a trivial relation..."

• Thanks for the nice answer and for the example. I think that "incorrect negation" and "trouble with quantifiers" feel related to "implication confusion." In all three cases, the problem is that students don't understand (or at least don't have an operational understanding of) how to manipulate formulas of propositional or predicate logic. – Patrick Lutz May 14 at 3:30
• I like "confusing general parameters with specific objects." It feels like it is similar in nature to "inappropriate types" and "the mysterious stranger." In all three cases, the error seems to arise from misunderstanding how mathematicians talk about mathematical objects. I think what sometimes causes it is that students are trying to imitate the language that they saw instructors using in class without completely understanding the underlying logic (e.g. they see instructors sometimes talk about "setting a and b equal to 0" and so they think it is something that is valid in any situation). – Patrick Lutz May 14 at 3:32
• For the true/false question that you mentioned, I would guess that part of the reason students got it wrong is that they didn't parse it correctly. Some of them may have read it as "If $S = \{v_1, \ldots, v_m\}$ is a set of vectors and $a_1v_1 + \ldots + a_nv_n = 0$ is a trivial relation for any $a_1, \ldots, a_n$ for which the relation holds then $S$ is linearly independent." This is adding a quantifier that is not in the original statement, but mathematicians have fairly particular conventions about when there are implicit quantifiers which I don't think all students are aware of. – Patrick Lutz May 14 at 3:37


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# zbMATH — the first resource for mathematics

Quantized enveloping algebras associated with simple Lie superalgebras and their universal $$R$$-matrices. (English) Zbl 0821.17005
For any simple Lie superalgebra $$\mathfrak g$$ a quantized superalgebra $$U_{\hbar}({\mathfrak g})$$ is explicitly defined in terms of generators and relations. It is proved that $$U_{\hbar}({\mathfrak g})$$ is a topologically free superalgebra. The relations include the known $$(q)$$-Serre relations and new ones of higher order. The new relations were not classified before even in the classical case.
An analog of the Poincaré-Birkhoff-Witt theorem is proved. The twisted product $$U_{\hbar}({\mathfrak g})^ b$$ of $$U_{\hbar}({\mathfrak g})$$ with the group algebra of $$Z_ 2$$ is introduced. $$U_{\hbar}({\mathfrak g})^ b$$ is a usual Hopf algebra which originates from the quantum double of the corresponding Borel subalgebra. The universal $$R$$-matrix for $$U_{\hbar}({\mathfrak g})^ b$$ is written down explicitly. It corresponds by restriction to the universal $$R$$-matrix for $$U_ q({\mathfrak g})$$ by Khoroshkin-Tolstoy.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 17A70 Superalgebras
Full Text:
##### References:
 [1] Abe., E., Hopf algebras, Cambridge Univ. Press, Cambridge-New York, 1980. · Zbl 0476.16008 [2] Baxter, R.J., Exactly solved models in statistical mechanics, Academic Press, London, 1982. · Zbl 0538.60093 [3] Corwin, L., Ne’eman, Y. and Sternberg, S., Graded Lie algebras in mathematics and physics, Rev. Mod. Phys., 47 (1975), 573-604. · Zbl 0557.17004 [4] Drinfeld, V.G., Quantum groups, Proc. Int. Congr. Math., Berkeley, 1 (1986); Amer. Math. Soc. (1988), 798-820. [5] Jimbo, M., A ^-difference analogue of U(&) and the Yang-Baxter equation, Lett. Math. Phys., 10 (1985), 63-69. · Zbl 0587.17004 [6] Kac, V.G., Infinite dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990. · Zbl 0716.17022 [7] Kac, V.G., Lie superalgebras, Adv. Math., 26 (1977), 8-96. · Zbl 0366.17012 [8] Kirillov, A.N. and Reshetikhin, N.Yu., #-Weyl group and a multiplicative formula for universal .R-matrix, Comm. Math. Phys., 134 (1990), 421-431. · Zbl 0723.17014 [9] Khoroshkin, S.M. and Tolstoy, V.N., Universal jR-matrix for quantized (super)algebras, Comm. Math. Phys,, 141 (1991), 599-617. · Zbl 0744.17015 [10] Levendorskii., S. and Soibelman, Ya., Some applications of quantum Weyl group, J. Geom. Phys., 7 (1990), 241-254. [11] Lusztig, G., Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc., 3 (1990), 257-296. · Zbl 0695.16006 [12] , Quantum groups at roots of 1, Geom. Dedicata, 35 (1990), 89-144. [13] Matsumura, H., Commutative Algebra, Second ed. London: Benjamin, 1980. [14] Okado, M. and Yamane, H., ^-matrices with gauge parameters and multi-parameter quantized enveloping algebras, Special Functions (Proceedings of Hayashibara Forum 1990 held in Fujisaki Institute, Okayama, Japan, August 16-20, 1990) ed. by M. Kashiwara and T. Miwa, Springer-Verlag, Tokyo, New York, Berlin, 1991, 289-293. · Zbl 0774.17022 [15] Perk, J.H.H. and Schultz, C.L., Families of commuting transfer matrices in g-state vertex models, Non-linear Integrable system - Classical theory and quantum theory (Proceedings of RIMS symposium, Kyoto, Japan, 13-16 May 1981) ed. by M. Jimbo and T. Miwa, World Scientific, Singapore, 1983, 326-343. [16] Rosso, M., An analogue of P.B.W. Theorem and the Universal ^-matrix for Uhsl(N+ 1), Comm. Math. Phys., 124 (1989), 307-318. · Zbl 0694.17006 [17] Scheunert, M., The theory of Lie superalgebras, Lecture Note in Math. 716, Springer, Berlin, 1979. · Zbl 0407.17001 [18] Takeuchi, M., Finite-dimensional representations of the quantum Lorents group, Comm. Math. Phys., 144 (1992), 557-580. · Zbl 0747.17022 [19] Tanisaki, T., Harish-Chandra isomorphism for quantum algebras, Comm. Math. Phys., 127 (1990), 555-571. · Zbl 0701.17007 [20] , Killing forms, Harish-Chandra isomorphisms, and universal /^-matrices for quantum algebras, Infinite Analysis (Proceedings of RIMS symposium, Kyoto, Japan, June-August 1991) ed. by A. Tsuchiya, T. Eguchi and M. Jimbo, World scientific, Singapore, 1992, 941-961. [21] Turaev, T.G., The Yang-Baxter equation and invariants of links, Invent. Math., 92 (1988), 527-553. · Zbl 0648.57003 [22] Yamane, H., Universal /^-matrices for quantum groups associated to simple Lie superalgebras, Proc. Japan Acad., Ser. A, No.4 (1991), 108-112. Note added in proof: In this paper, for the datum ((E,H,p),D), we first defined the Hopf superalgebra Uh=Uh(Tl,p)=Uh((£>,Yl,p),D) in an abstract manner in §2. Later, in Theorem 2.10.1, we showed that, if ({$$,Yl,p),D$$ corresponds to a symmetrizable Kac-Moody Lie algebra G, then our Uh coincides with the Drinfeld-type quantized enveloping algebra C/h(G) topologically defined over C[[A]]. Namely, in this case, the defining relations satisfied by the Chevalley generators of Uh are the g-Serre relations. After this paper has been submitted, the author learned that the same definition and result for the Jimbo-type quantized enveloping algebra Uq(G) defined over C(g) are given in the recent book of Prof. G jL,usztig (Introduction to Quantum Groups, Birkhauser, Boston, 1993).}
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
RESEARCH OF SINGLE ROOM DECENTRALIZED HEAT RECOVERY UNIT

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VOLUME 12 , ISSUE 4 (December 2019) > List of articles

### RESEARCH OF SINGLE ROOM DECENTRALIZED HEAT RECOVERY UNIT

Citation Information : Architecture, Civil Engineering, Environment. Volume 12, Issue 4, Pages 109-114, DOI: https://doi.org/10.21307/ACEE-2019-056

License : (CC-BY-NC-ND 4.0)

Published Online: 07-January-2020

### ARTICLE

#### ABSTRACT

Mechanical ventilation systems with heat recovery units are one of the key elements of low energy residential buildings and it has gained increasing interest during the last years. This research presents experimental investigation of the work of the compact recuperator, as a part of the decentralized, single pipe ventilation system, in real conditions. An experimental study was carried out to obtain data for tested unit efficiency calculation, for supply and exhaust mode. The scope of measurement encompassed the air temperature and airflow rate. Tested recuperator fulfilled its role of ventilation with heat recovery, however, some deficiencies were indicated. As a result of measurements it has been noticed that it is possible to improve the device in the future. This possibility of improvement of this device was suggested in the conclusions.

## 1. INTRODUCTION

New devices emerging on the market such as small ventilation equipment contribute to the need for research on their compliance with manufacturers’ data. They are new, poorly known and still not very popular. However, they have great potential to ventilate single rooms. Moreover, they are compact and inexpensive. One of them is single room decentralized heat recovery unit, commonly called compact recuperator, which can be installed in almost every building in the external partition, away from sources of pollution. It is characterized by a very simple construction. Its task, in addition to providing fresh air, is heat recovery using a small accumulated heat exchanger. Together with the accumulated heat exchanger, according to the manufacturers, it is able to obtain up to 90% of heat recovery from exhaust air. It is an energy-efficient way of ventilation of single and residential rooms in the temperature range of -20 to 50°C.

The compact recuperator is a part of a decentralized, single pipe ventilation system. Typical decentralized ventilation (DV) is a type of traditional mechanical ventilation that is implemented in a similar method to fan-assisted natural ventilation. However, fresh outdoor air is supplied and distributed into the room by passing through a compact decentralized air-handling unit rather than being utilized directly [1, 2]. Compared to centralized ventilation, DV has many advantages. DV minimizes duct space, can simplify individual zoning control in spaces and is less influenced by outdoor environmental condition [2]. Furthermore, the use of decentralized installation results in reduced operating costs. However, there are also some disadvantages: possibility of location in buildings and compliance with standards [3] in terms of air inlets and outlets location. This is the reason that sometimes they technically cannot be used [4].

Nowadays, the world attaches great significance to the saving of energy. Energy-efficient construction focuses mainly on the continuous reduction of thermal losses [5]. The airtight construction required by the current state of the art of the building shell increases the need for fresh ventilation air. According to standards [3], heat recovery is obligatory in installations with an air volume above 500 m3/h. Most often this is accomplished by recirculating air or just by the use of equipment for the recovery of heat and moisture. In small detached houses and in residential buildings, heat recovery devices such as recuperators are still seen as expensive and luxurious. However, in the future, in the light of new requirements, they will become a standard. Therefore, a compact recuperator is a noteworthy alternative, in particular for modernized and emerging residential and commercial buildings. The need for using such devices was also discussed by Amanowicz & Szczechowiak [6]. Considering the need to introduce solutions using heat recovery, also with a small supply air stream, in order to obtain a low primary energy ratio, the use of such solutions should soon be common in Poland.

The one-pipe ventilation system is relatively unknown and poorly supported by research. Most of the information comes from catalogs and brochures from manufacturers. In the literature, experimental and theoretical study of the heat exchanger with a periodic change in the flow direction for room ventilation [7], and stochastic analysis of compact plate-fin recuperators for microturbines [8] can be found. Research relating directly single room ventilation units with recuperative or regenerative heat recovery was conducted by Manz et al. [9]. More research results can be found directly on DV in the field of its energy analysis, for example [2, 10].

The aim of this study was experimental investigation of the work of the compact recuperator in real conditions. The device’s efficiency was determined based on measurements results and compared with manufacture’s data.

## 2. OVERVIEW OF THE TESTED RECUPERATOR

The compact recuperator offered by Alnor was tested (Fig. 1). A unit named HRU-WALL-100-25 with the amount of air flow of 25 m3/h was purchased in December 2018 for the price of PLN 1160. It is the smallest of the two types of units offered by the manufacturer. The exact dimensions of this device were given in Fig. 1.

##### Figure 1.

HRU-WALL-100-25 compact recuperator and its dimensions in millimeters [11]

As stated by the manufacturer, HRU-WALL is a single room decentralized heat recovery unit for concealed installation. The unit comes with a ceramic heat exchanger which boasts a maximum heat recovery ratio of 90% (the nominal heat recovery ratio is 74.3% as per standards [12]). The energy-efficient EC fan changes the running direction every 70 seconds to alternate between air supply and air exhaust. Low energy consumption and extremely low operating noise make this heat recovery unit a recommended solution for non-stop operation. The heat recovery unit has three speed levels to choose from, depending on the size of the area and indoor demands. This device operates in the outdoor temperature rangeing from -20 to 50°C. It weighs 2.4 kg and belongs to the “A” energy class [9].

## 3. MEASUREMENTS

Measurements of air temperature and airflow rate were carried out for the tested recuperator. The first tests took place in winter, at a low outdoor temperature in order to measure the air temperature in the individual parts of the device during its operation in heat recovery mode. Fig. 2 shows the test stand. It consisted of the recuperator mounted in the plate with thermal insulation placed in the window frame together with the switch; five thermocouples connected to the thermometer, which recorded the measurement data; and the micromanometer for measuring the pressure difference between the room and the external environment.

##### Figure 2.

Air temperature test stand

Prepared test stand enabled change of the pressure in the room by installing two small fans in the door with adjustable performance.

Mentioned thermocouples were arranged as shown in Fig. 3: in the outdoor environment; in the air intake; between the heat exchanger and the fan; behind the fan (air inlet to the room); and in the room.

##### Figure 3.

Location of the thermocouples; 1) in the air intake, 2) in the outdoor environment, 3) between the heat exchanger and the fan, 4) behind the fan (air inlet to the room), 5) in the room

Temperature measurements in the indicated 5 measurement points were taken for three fan speeds in two different conditions: (1) in real conditions with natural air flow; (2) for balanced pressure indoors and outdoors. The measured temperatures were used to calculate the actual efficiency of the tested recuperator.

Measurement of air velocity distribution flowing through the device was also carried out in order to determine the air volume flow rate. The test was made using a thermoanemometer. The use of other measurement methods was in this case impossible. Prepared test stand (Fig. 4) consisted of tested recuperator together with a switch, a cardboard box with an air intake, a thermoanemometer installed in the channel, a laptop with a software for recording measurements and a micromanometer. Small fans were also attached to the box. Their speed was regulated using the power supply with adjustable voltage.

##### Figure 4.

Scheme of air velocity distribution test stand

Open-box reproduced conditions without differential pressure on both sides of the device; closed and sealed – overpressure conditions on the outside. Mounted adjustable fans enabled the change of pressure inside, measured by the micromanometer.

The air velocity was measured in 6 points at two axes of the device channel, in accordance with the standard [13]. All fan speeds were considered, for supply and exhaust.

For correct and accurate placement of the thermoanemometer in the tube, the tube halves printed in a 3D printer were used (Fig. 5).

##### Figure 5.

Tube halves printed in a 3D printer

The measured air velocity values were averaged and the air flow volume rate for each fan speed for supply and exhaust was calculated. It was based on a product of averaged velocity and area of channel.

## 4. RESULTS AND DISCUSSION

The graph shown in Fig. 6 shows the distribution of measured air temperatures for the second fan speed when pressures in the room and outside were balanced. The indoor and outdoor temperatures were almost constant during all measurements. The remaining temperatures show 70 seconds of air supply and exhaust operation in heat recovery mode. It clearly shows five work cycles of the HRU-WALL-100–25 compact recuperator. There was also a noticeable difference between the indoor temperature and the temperature measured between the exchanger and the fan. Theoretically, the difference should be very small, because there were no elements that could decrease or increase the air temperature. This may be due to the fact that in addition to the heat exchanger, heat accumulation was also carried out on other elements of the device such as fan elements.

##### Figure 6.

Sample graph of air temperatures for the second fan speed at balanced pressures in the room and outside

In order to obtain the efficiency of the device, the averaged air temperature in measured 5 points for 3 fan speeds was calculated, for the time step of 5 s. Table 1 and Table 2 show the calculated efficiency of the unit separately for supply and exhaust air. The exhaust efficiency means the amount of energy kept by the device in relation to the total energy-related to the temperature difference between the indoor and outdoor air, whereas supply efficiency, the amount of energy supplied to the air by the device in relation to the total energy necessary to warm the air to room temperature.

##### Table 1.

Calculated supply efficiency for the tested working conditions

##### Table 2.

Calculated exhaust efficiency for the tested working conditions

The efficiency was calculated from the ratio of heat supplied/received from the heat exchanger (Σq1) to the sum of that energy and energy needed for the air to reach room temperature (Σq2):

##### (1)
$η=Σq1Σq1+Σq2,%$

The values of air flow volume rate used for calculations (converted into mass flow rate) were measured values (Table 35). It has been assumed that the average air flow volume rate is identical to the calculated average value from the measurements, and the ratio of the amount of supply air to exhaust air was adopted so that the amounts of energy delivered to the device and consumed were the same. The calculations were made in Microsoft Excel with the use of the Solver tool.

##### Table 3.

Calculated averaged air flow volume rate for all fan speeds based on measurements for 0 Pa

##### Table 4.

Calculated averaged air flow volume rate for all fan speeds based on measurements for 4 Pa

##### Table 5.

Calculated averaged air flow volume rate for all fan speeds based on measurements for 7 Pa

Air flow volume rate, according to manufacturer data, should be as follows: 25 m3/h for the third fan speed, 18 m3/h for the second fan speed and 10 m3/h for the first fan speed. Thus, the test confirmed the accuracy of the data provided by the manufacturer. Results differed by a maximum of 5 m3/h for atmospheric pressure. Moreover, the device was characterized by unevenness of the air flow in relation to the exhaust and higher efficiency of exhaust than supply mode. Therefore, the air velocity profiles for supply and exhaust mode were different, as shown in Fig. 7.

##### Figure 7.

Air velocity profile for supply mode (left) and exhaust mode (right) for third fan speed and 0 Pa of pressure difference

On the air velocity profile for exhaust mode, the air was blown towards the channel walls. At the walls, the air velocity value was the highest. A small amount of air flowed through the channel. The centrifugal force produced by the fan threw air at the sides of the channel. Air velocity distribution was more even for the air supply, however, slightly more air flowed over the upper part of the channel, probably due to the fact that the intake has an inlet at the bottom. Air velocity values were lower than in the case of exhaust mode.

## 5. CONCLUSIONS

1. The test allowed to calculate the actual efficiency of the device and showed some of its deficiencies.

2. Efficiency given by manufacturers did not differ significantly from those measured and calculated. However, it was higher for operation in the exhaust air mode. This is due to the fact that the exhaust air flows had lower values, and thus the air flowed with lower air velocity. As a result, exhaust air better contact with the surface of the heat exchanger. Moreover, air temperature for exhaust mode changes more than in supply mode, because the air volume is lower, however, the same amount of energy is exchanged.

3. Air flow measurements showed uneven flow efficiency for air supply in relation to exhaust. In real working conditions, the recuperator supplies more air than it removes.

4. In general, the equipment was rated very well. It fulfilled its role of ventilation with heat recovery. In the room in which it was temporarily installed, the improvement in air quality was observed.

5. This device is new and will certainly undergo many modifications to improve its work. One suggestion could be the use of flow guides to more accurately guide the airflow path so that it evenly covers the entire heat exchanger. The use of this solution would probably improve the heat recovery performance of the device.

6. The correctness of operation and the possibility of improving this device will be the subject of many subsequent research works due to the curiosity of engineers.

## References

1. Kim, M.K. (2015). Introduction of decentralized ventilation systems in buildings. Proceedings of the 1st International Conference on Sustainable Buildings and Structures, Suzhou, China, 29–31.
2. Kim, M.K., & Baldini, L. (2016). Energy analysis of a decentralized ventilation system compared with centralized ventilation systems in European climates: Based on review of analyses. Energy and Buildings, 111, 424–433.
[CROSSREF]
3. Regulation of the Minister of Infrastructure of April 12 2002 on technical conditions, which should correspond to the buildings and their location (consolidated text, Journal of Laws of 2019, item 1065, with further amendments).
4. Amanowicz, Ł., Ratajczak, K., Szczechowiak, E. (2019). Testing of single-duct ventilation units for the assessment of mixing of air in inlet and outlet. Ciepłownictwo, Ogrzewnictwo, Wentylacja, 50(6), 231–238 (in Polish).
5. Adamski, B. (2011). Heat recovery in ventilation and air conditioning systems, Inżynier budownictwa, 9 (in Polish).
6. Amanowicz, Ł., & Szczechowiak, E. (2017). Principles for the designing ventilations systems for energy-efficient buildings. Ciepłownictwo, Ogrzewnictwo, Wentylacja, 48(2), 72–78 (in Polish).
7. Nizovtsev, M.I., Borodulin, V.Yu, Letushko, V.N., & Zakharov, A.A. (2016). Analysis of the efficiency of air-to-air heat exchanger with a periodic change in the flow direction. Applied Thermal Engineering, 93, 113–121.
[CROSSREF]
8. Giugno, A., Cuneo, A., & Traverso, A. (2019). Analysis of uncertainties in compact plate-fin recuperators for microturbines. Applied Thermal Engineering, 105, 1243–1251.
[CROSSREF]
9. Manz, H., Huber, H., Schälin, A., Weber, A., Ferrazzini, M., & Studer, M. (2000). Performance of single room ventilation units with recuperative or regenerative heat recovery. Energy and Buildings, 31, 37–47.
[CROSSREF]
10. Kim, M.K., Leibundgut, H., & Choi, J-H. (2014). Energy and exergy analyses of advanced decentralized ventilation system compared with centralized cooling and air ventilation systems in the hot and humid climate. Energy and Buildings, 79, 212–222.
[CROSSREF]
11. www.ventilation-alnor.co.uk (last accessed 24 July 2019)
12. PN-EN 13141-8:2011. Ventilation for buildings – Performance testing of components/products for residential ventilation – Part 8: Performance testing of un-ducted mechanical supply and exhaust ventilation units (including heat recovery) for mechanical ventilation systems intended for a single room.
13. PN-EN 12599:2013-04. Ventilation for buildings – Test procedures and measurement methods to hand over air conditioning and ventilation systems.

### FIGURES & TABLES

Figure 1.

HRU-WALL-100-25 compact recuperator and its dimensions in millimeters [11]

Figure 2.

Air temperature test stand

Figure 3.

Location of the thermocouples; 1) in the air intake, 2) in the outdoor environment, 3) between the heat exchanger and the fan, 4) behind the fan (air inlet to the room), 5) in the room

Figure 4.

Scheme of air velocity distribution test stand

Figure 5.

Tube halves printed in a 3D printer

Figure 6.

Sample graph of air temperatures for the second fan speed at balanced pressures in the room and outside

Figure 7.

Air velocity profile for supply mode (left) and exhaust mode (right) for third fan speed and 0 Pa of pressure difference

### REFERENCES

1. Kim, M.K. (2015). Introduction of decentralized ventilation systems in buildings. Proceedings of the 1st International Conference on Sustainable Buildings and Structures, Suzhou, China, 29–31.
2. Kim, M.K., & Baldini, L. (2016). Energy analysis of a decentralized ventilation system compared with centralized ventilation systems in European climates: Based on review of analyses. Energy and Buildings, 111, 424–433.
[CROSSREF]
3. Regulation of the Minister of Infrastructure of April 12 2002 on technical conditions, which should correspond to the buildings and their location (consolidated text, Journal of Laws of 2019, item 1065, with further amendments).
4. Amanowicz, Ł., Ratajczak, K., Szczechowiak, E. (2019). Testing of single-duct ventilation units for the assessment of mixing of air in inlet and outlet. Ciepłownictwo, Ogrzewnictwo, Wentylacja, 50(6), 231–238 (in Polish).
5. Adamski, B. (2011). Heat recovery in ventilation and air conditioning systems, Inżynier budownictwa, 9 (in Polish).
6. Amanowicz, Ł., & Szczechowiak, E. (2017). Principles for the designing ventilations systems for energy-efficient buildings. Ciepłownictwo, Ogrzewnictwo, Wentylacja, 48(2), 72–78 (in Polish).
7. Nizovtsev, M.I., Borodulin, V.Yu, Letushko, V.N., & Zakharov, A.A. (2016). Analysis of the efficiency of air-to-air heat exchanger with a periodic change in the flow direction. Applied Thermal Engineering, 93, 113–121.
[CROSSREF]
8. Giugno, A., Cuneo, A., & Traverso, A. (2019). Analysis of uncertainties in compact plate-fin recuperators for microturbines. Applied Thermal Engineering, 105, 1243–1251.
[CROSSREF]
9. Manz, H., Huber, H., Schälin, A., Weber, A., Ferrazzini, M., & Studer, M. (2000). Performance of single room ventilation units with recuperative or regenerative heat recovery. Energy and Buildings, 31, 37–47.
[CROSSREF]
10. Kim, M.K., Leibundgut, H., & Choi, J-H. (2014). Energy and exergy analyses of advanced decentralized ventilation system compared with centralized cooling and air ventilation systems in the hot and humid climate. Energy and Buildings, 79, 212–222.
[CROSSREF]
11. www.ventilation-alnor.co.uk (last accessed 24 July 2019)
12. PN-EN 13141-8:2011. Ventilation for buildings – Performance testing of components/products for residential ventilation – Part 8: Performance testing of un-ducted mechanical supply and exhaust ventilation units (including heat recovery) for mechanical ventilation systems intended for a single room.
13. PN-EN 12599:2013-04. Ventilation for buildings – Test procedures and measurement methods to hand over air conditioning and ventilation systems.


text:
The relative spacing between the badge icon, and the badge-count gives a visual impression that the badges are associated to numbers on the left, which is the wrong number ... let me explain by pictures:

For example, in the above, the badges for janos read like "6 silver, 67 bronze" and in Loki's the bronze {} icon is closer to the 43 than the 144, so it looks like 2-silver, and 43 bronze.

This applies for questions/answer taglines too:

See the user profile pages: https://codereview.stackexchange.com/users?tab=Reputation&filter=all

The space to the left of the icon should be increased, and to the right it should be reduced.

• It would be even clearer if the braces went around the number. – 200_success Nov 26 '15 at 1:34
• we gonna investigate this :) – Paweł Nov 26 '15 at 7:54
• @200_success there's no chance for that, sory :c – Paweł Nov 26 '15 at 7:54