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# Chapter 20 What’s next?

Having discussed the main aspects of ADAM and how to use it in different circumstances, I want to pause to look back at what we have covered and what is left behind.

The reason why I did not call this monograph “Forecasting with ETS” or “Forecasting with State space models” is because the framework proposed here is not the same as for ETS, and it does not rely on the standard state space model. The combination of ETS, ARIMA and Regression in one unified model has not been discussed in the literature before. This is then extended by introducing a variety of distributions: typically, dynamic models rely on Normality, which is not realistic in real life, but ADAM supports several real-valued and several positive distributions. Furthermore, the model that can be applied to both regular and intermittent demand has been developed only by Svetunkov and Boylan (2019) but has not been published yet (and god knows when it will be). This is included in ADAM. In addition, the model then can be extended with multiple seasonal components, making it applicable to high-frequency data. Moreover, ADAM supports not only the location, but the scale model as well, allowing to model and predict the scale of distribution explicitly (giving it a connection with GARCH models). All of the aspects mentioned above are united in one approach, giving immense flexibility to the analyst.

But what’s next? While we have discussed the important aspects of ADAM, there are still several things left that I did not have time to make work yet.

The first one is the ADAM with Asymmetric Laplace and similar distributions with the non-zero mean (the related to this is the estimation of ADAM with non-standard loss functions, such as pinball). While it is possible to use such distributions, in theory, they do not work as intended in dynamic models, because the latter rely on the assumption that the mean of the error term is zero. They work perfectly in the case of the regression model (e.g. see how the alm() from the greybox works with the Asymmetric Laplace) but fail when a model has MA-related terms. This is because the model becomes more adaptive to the changes, pulls to the centre and cannot maintain the desired quantile. An introduction of such distributions would imply changing the architecture of the state space model.

Second, model combination and selection literature has seen several bright additions to the field, such as a stellar paper by Kourentzes et al. (2019a) on pooling. This is neither implemented in the adam() function nor discussed in the monograph. Yes, one can use ADAM to do pooling, but it would make sense to introduce it as a part of the ADAM approach.

Third, related to the previous point, is the selection and combinations based on cross-validation techniques (and specifically using rolling origin discussed in Section 2.4). The selected and combined models in this case would differ from the AIC-based ones.

Fourth, we have not discussed multiple frequency models in detail that they deserve. For example, we have not mentioned how to diagnose such models when the sample includes thousands of observations. The classical statistical approaches discussed in Section 14 typically fail in this situation, and other tools can be used in this context.

Fifth, adam() has a built-in missing values approach that relies on interpolation and intermittent state space model (from Section 13). While this already works in practice, there are some aspects of this that are worth discussing that have been left outside this monograph.

Sixth, when discussing pure multiplicative ETS models (Chapter 6), we only mentioned that it is possible to apply pure additive ETS to the data in logarithms to achieve similar results to the conventional pure multiplicative model. Akram et al. (2009) have done investigations in this direction, but these models are not yet supported by ADAM.

Finally, while I tried to introduce examples of application of ADAM, case studies for several contexts would be helpful. This would show how ADAM can be used for decisions in inventory management (we have touched the topic in Subsection 18.4.4), scheduling, staff allocation etc.

All of this will hopefully come in the next editions of this monograph.

### References

• Akram, M., Hyndman, R.J., Ord, J.K., 2009. Exponential Smoothing and Non-negative Data. Australian & New Zealand Journal of Statistics. 51, 415–432. https://doi.org/10.1111/j.1467-842X.2009.00555.x
• Kourentzes, N., Barrow, D., Petropoulos, F., 2019a. Another look at forecast selection and combination: Evidence from forecast pooling. International Journal of Production Economics. 209, 226–235. https://doi.org/10.1016/j.ijpe.2018.05.019
• Svetunkov, I., Boylan, J.E., 2019. Multiplicative state-space models for intermittent time series. https://doi.org/10.13140/RG.2.2.35897.06242


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# Multi-particle refinement¶

Now that our particle poses are mostly correct and we have a good estimate of the reconstructed density, there are several parameters that can be tweaked a posteriori to further refine our result. We will perform these optimisations in M.

## Create Population and Species¶

A project in M is referred to as a population. To create a new one press the big + button in the center of the window, and provide a directory. Here, we put it in root/M/1.6Apx/.

We then need to add a data source. Click on Manage data sources to add a local source, and select the *.settings file inside the frames directory. Make sure to Include items outside of filter ranges (since we did not use the filters for this project), give it a name such as HIV and press CREATE. We also only want to Use only first 15 frames/tilts in order to refine only based on the high resolution information.

We can then create a Species by pressing the next big ‘+’, selecting from scratch and filling in the parameters as follows:

Afterwards, select the two last halfmaps generated by the RELION refinements at 1.6Å/px. For the mask, we will generate a binary mask around the central hexamer (160 Å diameter).

Mask one of the halfmaps to include only the central hexamer in Dynamo then use your favourite program and determine an appropriate threshold for binarisation. Use this as the --ini_threshold parameter in the following relion_mask_create command, replacing <latest_halfmap> as appropriate:

relion_mask_create --i <latest_halfmap>.mrc --angpix 1.6 --extend_inimask 5 --o mask_1.6Apx.mrc --ini_threshold 0.05

Provide this mask to M, and then select the final particle positions (e.g: run_it019_data.star) as the input for particle poses. Click FINISH to generate the new species.

## Refinement Parameters¶

To start the refinement, click on the REFINE button and fill in the parameters as follows.

Note

We will do a few iterations of refinements: first some with only spatial parameter optimizations, and then with most parameters. We chose to first optimise a smaller number of parameters to avoid falling into a local minimum, there may be a better way to do this!

First, run four iterations with these parameters:

Afterwards, three iterations as follows:

## End Result!¶

If everything went well, after the second round of refinements you should be able to get to something like this:

Here’s a closer look to the map compared to the available structures:

format: plain


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text:
# What is cross elasticity of demand ? Explain it with examples?

The cross elasticity of demand or cross-price elasticity of demand measures the responsiveness of the demand for a good to a change in the price of another good. It is measured as the percentage change demand for the first good that occurs in response to a percentage change in price of the second good. For example, if, in response to a 10% increase in the price of fuel, the demand of new cars that are fuel inefficient decreased by 20%, the cross elasticity of demand would be: $frac{-20 %}{10 %}=-2$

• 1

it measures the change in demand for main commodity due to change in prices of substitute or complementary commodity

• 2
What are you looking for?


text:
## The effect of extra Perimeters in 3D printing.

After my discussion with Nophead in my previous post I realised that my attempt to compensate for hole shrinking was flawed. It seems that the dominant effect wasn’t what I thought but turned out instead to be the number and order of perimeters I was using in my Slic3r settings.

This has already been researched before, but I decided to investigate it further. This time I used RapCAD to generate the test pieces since its multi-material support made creating the Slic3r print modifiers a doddle.

I decided to use square holes since I can then measure them accurately with a digital vernier calliper.

Because of the way Slic3r handles multi materials, and modifiers I had to export the main() part first, then comment it out and export the mods() as a separate multi material part.

I applied the mods such that the first hole had 1 perimeter, the second 2 perimeters, etc. up to 5 perimeters. Unfortunately Slic3r decided to add extra perimeters internally to the part. This is probably a Slic3r bug but not a problem for this test piece.

I printed the parts out and took 4 measurements of each hole, 2 widthways and 2 lengthways. I then took the average of these four measurements and subtracted them from the specified width of 6mm to get the error. I divided this value by two since this error is doubled up, (there are two sides to the hole).

The blue line in this graph shows the results:

The red line shows the results from a second test piece in which the “Print outer perimeters first” setting was used.

So in summary I don’t think I’ve learned anything new here. Outer perimeters first seems to be the solution as has been suggested before. I was quite surprised to find that the error to perimeters relationship is not linear, but this might be due to errors in measurement, since there is a straight line that goes between all the error bars.

## Polyholes Revisited

Some time ago Nophead did some experiments to investigate why holes printed by reprap machines come out undersized.

I’ve also noticed this in my prints and have in some cases manually just ‘added a bit’ and re-printed the part. This isn’t very methodical so I decided to give Nopheads polyholes compensation a try. However even on paper the hole sizes suggested seem to be lower than some of the values I found by trial and error.

I decided to do a formal test and came up with the following OpenSCAD script. I have a full range of 19 drill bits from 1mm up to 10mm in 0.5mm increments.

The polyholes function is left untouched from Nopheads original and the rest just creates a nice shaped test piece.

Unfortunately the test confirmed what I suspected, and not a single drill bit fit in my test piece. In fact I had to shift all the bits up one space, the 9.5mm bit fitting into the 10mm hole and so on. For the smaller hole sizes even this didn’t work.

I decided to revisit the initial effects that Nophead outlined in his post which were:

• Faceting error
• Segment pausing
• Arc shrinkage
• Corner cutting

The faceting error comes about from CAD programs using the outer circumcircle as the basis for the radius of the polygon.

For a hole you actually need to use the smaller incircle radius. To convert from the incircle radius for a regular polygon with n sides you simply apply the following:

$R_{outer} = R_{inner} / \cos(\frac{\pi}{n})$

This formula can actually be quite useful for calculating the number of sides needed given a radius and tolerance. Simply re-arrange in terms of n to give:

$n = \pi / \cos^{-1}(\frac{r-t}{r})$

For values of r between 1mm and 20mm, and a tolerance of 0.1mm this gives values for n ranging between 7, and 23. This brings me on to segment pausing. I find it hard to believe that current reprap electronics will have significant delay when trying to produce 23 short line segments. Nophead dismissed this since his electronics were very low latency, and so will I.

The next factor is arc shrinkage. I found it quite odd that Nophead acknowledged this factor and referenced Adrian Bower’s calculations which compensate for this, but then dismisses it because the numbers come out too small. It occurred to me that Adrian’s calculations are based on a perfect circle, and so the faceting error calculations have to be used in conjunction with arc compensation.

$R_{outer} = \frac{t + \sqrt{t^2 + 4r^2}}{2} / \cos(\frac{\pi}{n})$

The value for t in this equation needs to be the track width of the extruded path. I wanted to use Slic3r’s formula for this which assumes the area of a track is the same as area of the filament at the nozzle, and that the shape of the track is a rectangle with two semicircles for the sides:

The area at the nozzle is just the area formula for a circle where d is the nozzle diameter:

$A_{nozzle} = \frac{\pi d^2}{4}$

The area of the track is just the sum of a circle area and rectangular area where h is the layer height (which is also the semicircle diameter):

$A_{track} = (x \times h) + \frac{\pi h^2}{4}$

Solving for x when both areas are the same gives:

$x = -\frac{\pi(h^2-d^2)}{4 h}$

The total track width then is x plus the width of the two semicircles:

$w = x + h$

Which gives:

$w = h -\frac{\pi (h^2-d^2)}{4 h}$

Putting this all together I created this new OpenSCAD script:

Here are the results:

Every drill bit fit snugly in the hole it was meant for. Even the small sizes worked surprisingly going down to a 1mm drill bit. But wait… I haven’t accounted for corner cutting. Well this is the aspect where I think I am going to have to disagree with Nophead. Not least because I didn’t account for it in my test above, but also because the assumption is that the filament is being stretched as it goes round corners, and this seems wrong to me. Surely the filament should be being extruded at a rate equal to the motion of the nozzle so that no stretching should occur. In fact in contrary to this Slic3r utilises a faster motion than the extrude rate to deliberately stretch the filament for making bridges, but not during normal perimeter generation. Its odd because Nophead says this is this is the dominant effect on his machines, and is the main justification for using less facets for the holes. Perhaps his extrusion rate is lower and so the filament does get stretched.

For holes below 2mm in diameter Nophead’s formula says you need 4, or 3 facets, which are not really holes at all. I would argue that smaller holes require a higher tolerance to produce accurate results, and from the formula for n above that means you need more facets, not less.

You can really notice the how the difference in facets effects the quality of the holes in the two test pieces:

As a final note I thought its worth mentioning that even a machined hole will be considered an interference fit (tight) if its drilled at the same diameter as the shaft or bolt that its intended for. For this reason a table of clearance holes is used

 Size Pitch (mm) Tapping Diameter (mm) Clearance Holes (mm) Coarse Fine Coarse Fine Close Medium Coarse M1.6 0.35 1.25 1.7 1.8 2.0 M2 0.4 1.6 2.2 2.4 2.6 M2.5 0.45 2.05 2.7 2.9 3.1 M3 0.5 2.5 3.2 3.4 3.6 M4 0.7 3.3 4.3 4.5 4.8 M5 0.8 4.2 5.3 5.5 5.8 M6 1.0 5.0 6.4 6.6 7.0 M8 1.25 1.0 6.8 7.0 8.4 9.0 10.0 M10 1.5 1.25 8.5 8.7 10.5 11.0 12.0 M12 1.75 1.25 10.2 10.8 13 14 15 M16 2.0 1.5 14.0 14.5 17.0 18.0 19.0 M20 2.5 1.5 17.5 18.5 21.0 22.0 24.0 M24 3.0 2.0 21.0 22.0 25.0 26.0 28.0 M30 3.5 2.0 26.5 28.0 31.0 33.0 35.0 M36 4.0 3.0 32.0 33.0 37.0 39.0 42.0

## Virtual Machines and Swap

For a while now I have been using VirtualBox to run various windows guest operating systems on my Linux host. One thing that I have always been wary of when using them is how the disk image gradually grows over time. This is because on the host operating system the disk container uses a sparse file so that the virtual disk can appear to be bigger than it actually is, but it can only do this when the guest operating system is not using the space, and the data is effectively set to zero. When you delete a file on the guest operating system it doesn’t bother to set the file contents back to zero, it just removes the record of where the files data was located.

This is not really a problem and most people know that you can use a tool such as sysinternals SDelete to zero the free space. Then using the command

vboxmanage modifyhd --compact <diskimage.vdi>

the space can be reclaimed.

However in windows there is one file in particular called pagefile.sys (also known as the swap file) that is constantly being written to and then subsequently erased. Its possible that windows does this intelligently and re-uses the same sectors of the virtual disk for the pagefile, but it seems more likely to me that windows doesn’t do this, after all it isn’t aware that its really writing to a virtual disk.

What made sense to me was to create a second virtual disk specifically for the pagefile. I made the second disk image immutable. What this means is that all writes to the disk are stored in a differencing image which is thrown away when the guest machine reboots. Because of this I can ensure that it never gets any bigger. This also has some other advantages.

Firstly I can use the same immutable image for all my Virtual machines. These currently include; Windows XP, Windows 7, Windows 8.1, and Windows 10. Secondly because the image is immutable it allows all guests to use the image while all running at the same time.

Thirdly the fun part, I can host the differencing images to the immutable base image in tmpfs meaning that the guests get the advantage of having very fast i/o for their pagefile.sys. This is the advantage I like the most because it certainly seems to improve performance.

## Binary Universe

I have recently been wondering about probability. What is it? Well obviously I know that its “The chance of something happening” but how does probability manifest itself in our universe. Firstly I have made the assumption that for something to happen, you must have a time component. For example a universe with 0 dimensions, and no time, cannot have probability. A point could exist in such a universe, since a point is as Euclid defined “that which has no part” or in other words a point has no dimensions. But in a 0 dimensional space there is no time and so it cannot appear. However in a 1 dimensional universe where the only dimension is temporal, this gives rise to something being able to happen. Admittedly this kind of universe is still only big enough to contain a point, but it can either be there, or not be there. It is a binary universe, and the only event that can ever happen in that universe is the emergence of a point. You could go so far as to say that the entropy of this simple universe starts low, and then at some point becomes high. The trouble with this model is that its time reversible. If you look at this simple universe with time going backwards the point simply exists to start with, and then disappears. So in that sense the temporal dimension is not temporal at all, since it has no arrow of time.

I have decided to move the RapCAD website to a new hosting solution. There are two reasons for this, firstly I don’t see the need any more to host two separate blogs, and secondly because the new solution is cheaper. As a result of this change the bug tracker and forums have gone, (they were not being used anyway) and git hosting has moved primarily to github.

## How to fix your buttons – Ubuntu

For some reason, with each new major version, Ubuntu keeps changing the location of the minimize maximize and close buttons on the window decorations. Its really annoying because as soon as you get used to one way, they put it back the other way. Once and for all I wanted to fix this problem and have my buttons where they should be, where they have always been (for me this was on the right having come from windows)

gconftool-2 --set /apps/metacity/general/button_layout --type string ":minimize,maximize,close"

Now at least I can get used to them in the right place!

EDIT: For ubuntu 13.04 you need to do this:

gsettings set org.gnome.desktop.wm.preferences button-layout ':minimize,maximize,close'


text:
## mitchelsewbaran 2 years ago Find the value of tan θ for the angle shown.

1. mitchelsewbaran

2. hba

$\tan \theta=P/B$

3. hba

I guess we would be using the Pythagoras first of all.


text:
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# Is 1/s=(s^2)*(t/u)?

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09 Mar 2017, 19:10
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Is $$\frac{1}{s} = s^2*\frac{t}{u}$$?

(1) $$(\frac{1}{s})^3 = \frac{t}{u}$$

(2) $$s^3 = ut$$

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09 Mar 2017, 19:45
D
Statement 1
Sufficient - yes

Statement 2
Sufficient-no is the answer to the question stem

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10 Mar 2017, 02:59
1
St 1: 1/s^3 = t/u
rearranging, 1/s = s^2 x t/u. same expression. will always hold true. ANSWER

St 2: s^3 = ut. therefore 1/s = s^2 x 1/ut. not the sameexpression, but will hold true if t = 1. INSUFFICIENT

Option A
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07 Oct 2017, 09:48
If the statement gives t is not equal to 1, then the answer would be D ?
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07 Oct 2017, 10:39
@s wrote:
If the statement gives t is not equal to 1, then the answer would be D ?

not even then B would be sufficient.

what if t = -1? In this case as well t^2 will be 1. so with statement-2 along with information as t not equal 1, B will not be sufficient.
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31 Dec 2017, 14:36
hazelnut wrote:
Is $$\frac{1}{s} = s^2*\frac{t}{u}$$?

(1) $$(\frac{1}{s})^3 = \frac{t}{u}$$

(2) $$s^3 = ut$$

VERITAS PREP OFFICIAL SOLUTION:

Explanation: To simplify the question, it may be helpful to rewrite the question stem by cross-multiplying both sides of the equation to arrive at $$s^3t = u$$.

Therefore, the question may be rephrased: Is $$s^3t=u$$?
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09 May 2018, 09:57
Bunuel wrote:
hazelnut wrote:
Is $$\frac{1}{s} = s^2*\frac{t}{u}$$?

(1) $$(\frac{1}{s})^3 = \frac{t}{u}$$

(2) $$s^3 = ut$$

VERITAS PREP OFFICIAL SOLUTION:

Explanation: To simplify the question, it may be helpful to rewrite the question stem by cross-multiplying both sides of the equation to arrive at $$s^3t = u$$.

Therefore, the question may be rephrased: Is $$s^3t=u$$?

Isn't statement 1 and statement 2 contradicting each other ?

statement 1 -> $$s ^3$$ =$$\frac {u}{t}$$
statement 2 -> $$s^3$$ = ut

Am i missing anything ?
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01 Jun 2018, 00:40
Bunuel wrote:
hazelnut wrote:
Is $$\frac{1}{s} = s^2*\frac{t}{u}$$?

(1) $$(\frac{1}{s})^3 = \frac{t}{u}$$

(2) $$s^3 = ut$$

VERITAS PREP OFFICIAL SOLUTION:

Explanation: To simplify the question, it may be helpful to rewrite the question stem by cross-multiplying both sides of the equation to arrive at $$s^3t = u$$.

Therefore, the question may be rephrased: Is $$s^3t=u$$?

Bunuel

How is the 2nd statement not sufficient? We are able to answer the question right?Also, as someone mentioned above, the two statements never contradict each other in official questions. Atleast I have not seen any such question.
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01 Jun 2018, 03:58
KS15 wrote:
Bunuel wrote:
hazelnut wrote:
Is $$\frac{1}{s} = s^2*\frac{t}{u}$$?

(1) $$(\frac{1}{s})^3 = \frac{t}{u}$$

(2) $$s^3 = ut$$

VERITAS PREP OFFICIAL SOLUTION:

Explanation: To simplify the question, it may be helpful to rewrite the question stem by cross-multiplying both sides of the equation to arrive at $$s^3t = u$$.

Therefore, the question may be rephrased: Is $$s^3t=u$$?

Bunuel

How is the 2nd statement not sufficient? We are able to answer the question right?Also, as someone mentioned above, the two statements never contradict each other in official questions. Atleast I have not seen any such question.

Hello

Please look at the following solution to this question:
https://gmatclub.com/forum/is-1-s-s-2-t ... l#p1817685

Second statement is not the same as first statement, but if t=1, then answer to the question stem will be YES
but if t is not 1, then answer to the question stem will be NO
So the second statement is not sufficient to answer the question
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### Show Tags

01 Jun 2018, 04:07

Explanation: To simplify the question, it may be helpful to rewrite the question stem by cross-multiplying both sides of the equation to arrive at $$s^3t = u$$.

Therefore, the question may be rephrased: Is $$s^3t=u$$?[/quote]
Bunuel

How is the 2nd statement not sufficient? We are able to answer the question right?Also, as someone mentioned above, the two statements never contradict each other in official questions. Atleast I have not seen any such question.[/quote]

Hello

Please look at the following solution to this question:
https://gmatclub.com/forum/is-1-s-s-2-t ... l#p1817685

Second statement is not the same as first statement, but if t=1, then answer to the question stem will be YES
but if t is not 1, then answer to the question stem will be NO
So the second statement is not sufficient to answer the question[/quote]

Thanks for sharing the link-quick couple of qs
1. In St 2, cant be just take reciprocal on both sides and check?
2. You did not talk about the two statements contradicting each other. Any official question you can share where this is happening?
Is 1/s=(s^2)*(t/u)? &nbs [#permalink] 01 Jun 2018, 04:07
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# Is 1/s=(s^2)*(t/u)?

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text:
# A general relativistic signature in the galaxy bispectrum: the local effects of observing on the lightcone

Research output: Research - peer-reviewArticle

Next-generation galaxy surveys will increasingly rely on the galaxy bispectrum to improve cosmological constraints, especially on primordial non-Gaussianity. A key theoretical requirement that remains to be developed is the analysis of general relativistic effects on the bispectrum, which arise from observing galaxies on the past lightcone, {as well as from relativistic corrections to the dynamics}. {As an initial step towards a fully relativistic analysis of the galaxy bispectrum, we compute for the first time the local relativistic lightcone effects on the bispectrum,} which come from Doppler and gravitational potential contributions. For the galaxy bispectrum, the problem is much more complex than for the power spectrum, since we need the lightcone corrections at second order. Mode-coupling contributions at second order mean that relativistic corrections can be non-negligible at smaller scales than in the case of the power spectrum. In a primordial Gaussian universe, we show that the local lightcone corrections for squeezed shapes at $z\sim1$ mean that the bispectrum can differ from the Newtonian prediction by $\gtrsim 10\%$ when the short modes are $k\lesssim (50\,{\rm Mpc})^{-1}$. These relativistic projection effects, if ignored in the analysis of observations, could be mistaken for primordial non-Gaussianity. For upcoming surveys which probe equality scales and beyond, {all relativistic lightcone effects and relativistic dynamical corrections should be included} for an accurate measurement of primordial non-Gaussianity.
Original language English 034 1-10 11 Journal of Cosmology and Astroparticle Physics 03 10.1088/1475-7516/2017/03/034 Published - 14 Mar 2017

### Documents

• pdf

Rights statement: The final published version of this article by Obinna Umeh et al JCAP03(2017)034 is available online at http://dx.doi.org/10.1088/1475-7516/2016/03/034. © 2016 IOP Publishing Ltd and Sissa Medialab srl.

Accepted author manuscript (Post-print), 394 KB, PDF-document

Due to publisher’s copyright restrictions, this document is not freely available to download from this website until: 14/03/18

ID: 6714876


text:
Rationalize the Denominator with Conjugates

Examples, formula and the Steps!

Examples of rationalizing the denominator

Example 1: Conjugates
(more on rationalizing denominators with conjugates)

Rationalize $$\frac{3}{2 + \sqrt{5}}$$ Step 1

Simplify the square root in the denominator (if possible)
$$\sqrt{5}$$ is already simplified, so nothing to do here
Step 2

Multiply the numerator and denominator by the denominator's conjugate

$$2 \color{red}{-} \sqrt{5}$$
$$\frac{3}{2 + \sqrt{5}} \color{red}{ \frac{2 - \sqrt{5}} {2 - \sqrt{5}}} = \frac{3 (2 \color{red}{-} \sqrt{5} )} { (2\color{red}{+} \sqrt{5} )(2\color{red}{-} \sqrt{5} ) } \\ = \frac{ 6 -3\sqrt{5} } { 4 \color{red}{+} 2\sqrt{5} \color{red}{-} 2\sqrt{5}+ 5 } = \frac{ 6 -3\sqrt{5} } { 4 \cancel { \color{red}{+}2\sqrt{5} \color{red}{-} 2\sqrt{5} } + 5 } \\ = \frac{ 6 -3\sqrt{5} } { 4 + 5 } = \frac{ 6 -3\sqrt{5} } { 9 }$$
Step 3
Simplify
$$\frac{ 6 -3\sqrt{5} } { 9 } = \frac{ \color{red}{3}(2 -1\sqrt{5}) } { \color{red}{3}(3) } \\ \frac{ \color{red}{\cancel{3}}(2 -1\sqrt{5} ) } { \color{red}{\cancel{3} } (3 ) } = \frac{ (2 - \sqrt{5} )} { 3 }$$
Step 1

Identify the conjugate of the denominator
$$\frac{ 14}{ \color{red}{ 2 - \sqrt{7}} }$$

$$2 \color{red}{+} \sqrt{7}$$
Step 2

Multiply the numerator and denominator by the conjugate

$$\frac{ 14}{ \color{red}{2 - \sqrt{7}} } \cdot \frac{2 \color{red}{+} \sqrt{7} }{2 \color{red}{ + } \sqrt{7} }$$
Step 3
$$\frac{ 14}{ 2 - \sqrt{7} } \cdot \frac{2 \color{red}{+} \sqrt{7} }{2 \color{red}{ + } \sqrt{7} } \\ = \frac{ 14(2 \color{red}{ + } \sqrt{7}) }{ (2 - \sqrt{7})(2 \color{red}{ + } \sqrt{7}) } \\ = \frac{ 28 + 14\sqrt{7} }{ 4 \color{red}{- 2\sqrt{7} + 2\sqrt{7}} -7 } \\ = \frac{ 28 + 14\sqrt{7} }{ -3 }$$
Step 1

Identify the conjugate of the denominator
$$\frac{ 9}{ \color{red}{ 3 + \sqrt{5} } }$$

$$3 \color{red}{ -} \sqrt{5}$$
Step 2

Multiply the numerator and denominator by the conjugate

$$\frac{ 9}{ \color{red}{ 3 + \sqrt{5} } } \cdot \frac{ 3 \color{red}{ - } \sqrt{5} }{ 3 \color{red}{ - } \sqrt{5}}$$
Step 3
$$\frac{ 9}{ 3 + \sqrt{5} } \cdot \frac{ 3 \color{red}{ - } \sqrt{5} }{ 3 \color{red}{ - } \sqrt{5}} \\ \frac{ 9 (3 \color{red}{ - } \sqrt{5} ) }{ (3 + \sqrt{5})( 3 \color{red}{ - } \sqrt{5} )} \\ \frac{ 27 - 9\sqrt{5} } { 9 \color{red}{+3 \sqrt{5} -3 \sqrt{5} } - 5 } \\ \\ \frac{ 27 - 9\sqrt{5} } { 4 }$$
Step 1

Identify the conjugate of the denominator
$$\frac{ 3}{ \color{red}{\sqrt{7} +2 }}$$

$$\sqrt{7} \color{red}{-} 2$$
Step 2
Multiply the numerator and denominator by the conjugate
$$\frac{ 3}{ \color{red}{\sqrt{7} +2 }} \cdot \frac{ \sqrt{7} \color{red}{-} 2 }{ \sqrt{7} \color{red}{-} 2 }$$
Step 3
$$\frac{ 3}{ \sqrt{7} +2 } \cdot \frac{ \sqrt{7} \color{red}{-} 2 }{ \sqrt{7} \color{red}{-} 2 } \\ \frac{ 3( \sqrt{7} \color{red}{-} 2) }{ (\sqrt{7} +2)(\sqrt{7} \color{red}{-} 2) } \\ \frac{ 3 \sqrt{7} \color{red}{-} 6 } {7 \color{red}{ +2\sqrt{7}-2\sqrt{7}} -4 } \\ \frac{ 3 \sqrt{7} \color{red}{-} 6 } {3} \\ \frac{ 3 ( \sqrt{7} \color{red}{-} 2) } {3(1)} \\ \frac{ \cancel{ 3} ( \sqrt{7} \color{red}{-} 2) } {\cancel{ 3}(1)} \\ \sqrt{7} \color{red}{-} 2$$
Step 1

Identify the conjugate of the denominator
$$\frac{ 5}{ \color{red}{ 3 - 2\sqrt{6}} }$$

$$3 \color{red}{+} 2\sqrt{6}$$
Step 1

Multiply the numerator and denominator by the conjugate

$$\frac{ 5}{ \color{red}{ 3 - 2\sqrt{6}} } \cdot \frac{ 3 \color{red}{+} 2\sqrt{6} }{ 3 \color{red}{+} 2\sqrt{6} }$$
Step 1
$$\frac{ 5}{3 - 2\sqrt{6} } \cdot \frac{ 3 \color{red}{+} 2\sqrt{6} }{ 3 \color{red}{+} 2\sqrt{6} } \\ \frac{ 5( 3 \color{red}{+} 2\sqrt{6}) }{ (3 - 2\sqrt{6} ) ( 3 \color{red}{+} 2\sqrt{6})} \\ \frac { 15 + 10 \sqrt{6} }{ 9 \color{red}{ -6 \sqrt{6} +6\sqrt{6}} -24 } \\ \frac { 15 + 10 \sqrt{6} }{ -15 } \\ \frac { 15 + 10 \sqrt{6} }{ -15 } \\ \frac { 5(3 + 2 \sqrt{6}) }{ 5(-3) } \\ \frac { \cancel{5}(3 + 2 \sqrt{6}) }{ \cancel{5}(-3) } \\ \frac { 3 + 2 \sqrt{6} }{ -3 }$$
Step 1

Identify the conjugate of the denominator
$$\frac{ 5}{ \color{red}{ 5 + 2\sqrt{7}} }$$

$$5 \color{red}{-} 2\sqrt{ 7 }$$
Step 2

Multiply the numerator and denominator by the conjugate

$$\frac{ 2\sqrt{3}}{ \color{red} {5 + 2\sqrt{7}} } \cdot \frac{5 \color{red}{-} 2\sqrt{ 7 } }{5 \color{red}{-} 2\sqrt{ 7 } }$$
Step 3
$$\frac{ 2\sqrt{3}}{ 5 + 2\sqrt{7} } \cdot \frac{5 \color{red}{-} 2\sqrt{ 7 } }{5 \color{red}{-} 2\sqrt{ 7 } } \\ \frac{ 2\sqrt{3}(5\color{red}{-} 2\sqrt{ 7 } ) }{ (5 + 2\sqrt{7 })( 5 \color{red}{-} 2\sqrt{ 7 } ) } \\ \frac{ 10\sqrt{3} - 4\sqrt{21} }{ 25 \color{red}{ +10\sqrt{ 7} -10 \sqrt{ 7} } + 28 } \\ \frac{ 10\sqrt{3} - 4\sqrt{21} }{ 53}$$

Challenge Problems

Step 1

Identify the conjugate of the denominator
$$\frac{ 7}{ \color{red}{\sqrt{2} + \sqrt{5}}}$$

$$\sqrt{2} \color{red}{-} \sqrt{5}$$
Step 2

Multiply the numerator and denominator by the conjugate

$$\frac{ 7}{ \sqrt{2} + \sqrt{5}} \cdot \frac{ \sqrt{2} \color{red}{-} \sqrt{5} } { \sqrt{2} \color{red}{-} \sqrt{5}}$$
Step 3
$$\frac{ 7}{ \sqrt{2} + \sqrt{5}} \cdot \frac{ \sqrt{2} \color{red}{-} \sqrt{5} } { \sqrt{2} \color{red}{-} \sqrt{5}} \\ \frac{ 7( \sqrt{2} \color{red}{-} \sqrt{5} )} { ( \sqrt{2} + \sqrt{5})( \sqrt{2} \color{red}{-} \sqrt{5})} \\ \frac{ 7 \sqrt{2} - 7\sqrt{5} } { \sqrt{2}\sqrt{2} \color{red}{ - \sqrt{2}\sqrt{5} + \sqrt{2}\sqrt{5}} - \sqrt{5}\sqrt{5} } \\ \frac{ 7 \sqrt{2} - 7\sqrt{5} } { 2 \color{red}{ - \sqrt{10} + \sqrt{10}} - 5 } \\ \frac{ 7 \sqrt{2} - 7\sqrt{5} } { 2 - 5 } \\ \frac{ 7 \sqrt{2} - 7\sqrt{5} } { -3}$$
Step 1

Identify the conjugate of the denominator
$$\frac{ \sqrt{7} -\sqrt{5} }{ \color{red}{ \sqrt{5} + \sqrt{7} }}$$

$$\sqrt{5} \color{red}{-} \sqrt{7}$$
Step 2

Multiply the numerator and denominator by the conjugate

$$\frac{ \sqrt{7} -\sqrt{5} }{ \color{red}{ \sqrt{5} + \sqrt{7} }} \cdot \frac{ \sqrt{5} \color{red}{-} \sqrt{7}} { \sqrt{5} \color{red}{-} \sqrt{7} }$$
Step 3
$$\frac{ \sqrt{7} -\sqrt{5} }{ \sqrt{5} + \sqrt{7} } \cdot \frac{ \sqrt{5} \color{red}{-} \sqrt{7}} { \sqrt{5} \color{red}{-} \sqrt{7} } \\ \frac{ (\sqrt{7} -\sqrt{5})( \sqrt{5} \color{red}{-} \sqrt{7})} { ( \sqrt{5} + \sqrt{7})( \sqrt{5} \color{red}{-} \sqrt{7} )} \\ \frac{ \sqrt{7} \sqrt{5} -\sqrt{7}\sqrt{7} -\sqrt{5}\sqrt{5} + \sqrt{5}\sqrt{7} } { \sqrt{5} \sqrt{5} \color{red}{ -\sqrt{5}\sqrt{7}+\sqrt{5}\sqrt{7} }-\sqrt{7}\sqrt{7}} \\ \frac{ \sqrt{35} -7 -5 + \sqrt{35} } { 5 -7} \\ \frac{ -12 + 2 \sqrt{35} } { -2} \\ \frac{-2( 6 -1 \sqrt{35}) } { -2 (1)} \\ \frac{\cancel{-2}( 6 -1 \sqrt{35}) } { \cancel{-2 }(1)} \\ 6 - \sqrt{35}$$

Rationalize Denominator Widget

Simply type into the app below and edit the expression. The Math Way app will solve it form there. You can visit this calculator on its own page here.


text:
Article Contents
Article Contents

On saddle points of a class of augmented lagrangian functions

• For constrained nonconvex optimization, we first show that under second-order sufficient conditions, a class of augmented Lagrangian functions possess local saddle points, and then prove that global saddle points of these augmented Lagrangian functions exist under certain mild additional conditions.
Mathematics Subject Classification: 90C26, 90C46.

 Citation:


text:
crosscutplot {DOS2} R Documentation

## Scatterplot Associated with the Cross-Cut Test

### Description

A scatterplot of y against x, with the points used by the cross-cut test in black, and the remaining points in grey. See Figure 19.2 in "Design of Observational Studies", second edition.

### Usage

crosscutplot(x, y, ct = 0.25, xlab = "", ylab = "", main = "", ylim = NULL)

### Arguments

 x Variable to be plotted on the horizontal axis. y Variable to be plotted on the vertical axis. ct The quantile that defines the crosscut. By default, the crosscut is at the outer .25 of the data, the lower 25 percent and the upper 75 percent. xlab Label for the x axis. ylab Label for the y axis. main Title of the plot. ylim Limits for the y axis. See example.

A scatter plot.

### Author(s)

Paul R. Rosenbaum

### References

Rosenbaum, P. R. (2016) <doi:10.1111/biom.12373> "The crosscut statistic and its sensitivity to bias in observational studies with ordered doses of treatment". Biometrics, 72(1), 175-183.

Rosenbaum, P. R. (2017) <doi:10.1214/17-STS621> "The general structure of evidence factors in observational studies". Statistical Science 32, 514-530.

### Examples

# Figure 1 in Rosenbaum (2017)
data(periodontal)
attach(periodontal)
m<-matrix(1:2,1,2)
layout(m,widths=c(1,2))
boxplot(pcteither[z==1]-pcteither[z==0],ylab="Smoker-Control Difference",
main="(i)",xlab="Matched Pairs",ylim=c(-100,100))
abline(h=0,lty=2)
crosscutplot(cigsperday[z==1],pcteither[z==1]-pcteither[z==0],
ct=.2,ylab="Smoker-Control Difference",
xlab="Cigarettes per Day",main="(ii)",ylim=c(-100,100))
abline(h=0,lty=2)
detach(periodontal)
layout(1)

[Package DOS2 version 0.5.2 Index]


text:
• ### ARPES study of the epitaxially grown topological crystalline insulator SnTe(111)(1610.07759)

Oct. 25, 2016 cond-mat.mtrl-sci
SnTe is a prototypical topological crystalline insulator, in which the gapless surface state is protected by a crystal symmetry. The hallmark of the topological properties in SnTe is the Dirac cones projected to the surfaces with mirror symmetry, stemming from the band inversion near the L points of its bulk Brillouin zone, which can be measured by angle-resolved photoemission. We have obtained the (111) surface of SnTe film by molecular beam epitaxy on BaF2(111) substrate. Photon-energy-dependence of in situ angle-resolved photoemission, covering multiple Brillouin zones in the direction perpendicular tothe (111) surface, demonstrate the projected Dirac cones at the Gamma_bar and M_bar points of the surface Brillouinzone. In addition, we observe a Dirac-cone-like band structure at the Gamma point of the bulk Brillouin zone,whose Dirac energy is largely different from those at the Gamma_bar and M_bar points.
• ### Ubiquitous strong electron phonon coupling at the interface of FeSe/SrTiO3(1608.01740)

Aug. 5, 2016 cond-mat.supr-con
The high temperature superconductivity in single-unit-cell (1UC) FeSe on SrTiO3 (STO)(001) and the observation of replica bands by angle-resolved photoemission spectroscopy (ARPES) have led to the conjecture that the coupling between FeSe electron and the STO phonon is responsible for the enhancement of Tc over other FeSe-based superconductors1,2. However the recent observation of a similar superconducting gap in FeSe grown on the (110) surface of STO raises the question of whether a similar mechanism applies3,4. Here we report the ARPES study of the electronic structure of FeSe grown on STO(110). Similar to the results in FeSe/STO(001), clear replica bands are observed. We also present a comparative study of STO (001) and STO(110) bare surfaces, where photo doping generates metallic surface states. Similar replica bands separating from the main band by approximately the same energy are observed, indicating this coupling is a generic feature of the STO surfaces and interfaces. Our findings suggest that the large superconducting gaps observed in FeSe films grown on two different STO surface terminations are likely enhanced by a common coupling between FeSe electrons and STO phonons.
• ### Observation of Unusual Topological Surface States in Half-Heusler Compounds LnPtBi (Ln=Lu, Y)(1602.05633)

Topological quantum materials represent a new class of matter with both exotic physical phenomena and novel application potentials. Many Heusler compounds, which exhibit rich emergent properties such as unusual magnetism, superconductivity and heavy fermion behaviour, have been predicted to host non-trivial topological electronic structures. The coexistence of topological order and other unusual properties makes Heusler materials ideal platform to search for new topological quantum phases (such as quantum anomalous Hall insulator and topological superconductor). By carrying out angle-resolved photoemission spectroscopy (ARPES) and ab initio calculations on rare-earth half-Heusler compounds LnPtBi (Ln=Lu, Y), we directly observed the unusual topological surface states on these materials, establishing them as first members with non-trivial topological electronic structure in this class of materials. Moreover, as LnPtBi compounds are non-centrosymmetric superconductors, our discovery further highlights them as promising candidates of topological superconductors.
• ### High Resolution Angle Resolved Photoemission with Tabletop 11eV Laser(1509.01311)

We developed a table-top vacuum ultraviolet (VUV) laser with $113.778$nm wavelength (10.897eV) and demonstrated its viability as a photon source for high resolution angle-resolved photoemission spectroscopy (ARPES). This sub-nanosecond pulsed VUV laser operates at a repetition rate of 10MHz, provides a flux of 2$\times$10$^{12}$ photons/second, and enables photoemission with energy and momentum resolutions better than 2meV and 0.012\AA$^{-1}$, respectively. Space-charge induced energy shifts and spectral broadenings can be reduced below 2meV. The setup reaches electron momenta up to 1.2\AA$^{-1}$, granting full access to the first Brillouin zone of most materials. Control over the linear polarization, repetition rate, and photon flux of the VUV source facilitates ARPES investigations of a broad range of quantum materials, bridging the application gap between contemporary low energy laser-based ARPES and synchrotron-based ARPES. We describe the principles and operational characteristics of this source, and showcase its performance for rare earth metal tritellurides, high temperature cuprate superconductors and iron-based superconductors.
• ### Observation of universal strong orbital-dependent correlation effects in iron chalcogenides(1506.03888)

July 24, 2015 cond-mat.supr-con
Establishing the appropriate theoretical framework for unconventional superconductivity in the iron-based materials requires correct understanding of both the electron correlation strength and the role of Fermi surfaces. This fundamental issue becomes especially relevant with the discovery of the iron chalcogenide (FeCh) superconductors, the only iron-based family in proximity to an insulating phase. Here, we use angle-resolved photoemission spectroscopy (ARPES) to measure three representative FeCh superconductors, FeTe0.56Se0.44, K0.76Fe1.72Se2, and monolayer FeSe film grown on SrTiO3. We show that, these FeChs are all in a strongly correlated regime at low temperatures, with an orbital-selective strong renormalization in the dxy bands despite having drastically different Fermi-surface topologies. Furthermore, raising temperature brings all three compounds from a metallic superconducting state to a phase where the dxy orbital loses all spectral weight while other orbitals remain itinerant. These observations establish that FeChs display universal orbital-selective strong correlation behaviors that are insensitive to the Fermi surface topology, and are close to an orbital-selective Mott phase (OSMP), hence placing strong constraints for theoretical understanding of iron-based superconductors.
• ### Discovery of a Weyl Semimetal in non-Centrosymmetric Compound TaAs(1507.00521)

Three-dimensional (3D) topological Weyl semimetals (TWSs) represent a novel state of quantum matter with unusual electronic structures that resemble both a "3D graphene" and a topological insulator by possessing pairs of Weyl points (through which the electronic bands disperse linearly along all three momentum directions) connected by topological surface states, forming the unique "Fermi-arc" type Fermi-surface (FS). Each Weyl point is chiral and contains half of the degrees of freedom of a Dirac point, and can be viewed as a magnetic monopole in the momentum space. Here, by performing angle-resolved photoemission spectroscopy on non-centrosymmetric compound TaAs, we observed its complete band structures including the unique "Fermi-arc" FS and linear bulk band dispersion across the Weyl points, in excellent agreement with the theoretical calculations. This discovery not only confirms TaAs as the first 3D TWS, but also provides an ideal platform for realizing exotic physical phenomena (e.g. negative magnetoresistance, chiral magnetic effects and quantum anomalous Hall effect) which may also lead to novel future applications.
• ### Extremely large magnetoresistance and ultrahigh mobility in the topological Weyl semimetal NbP(1502.04361)

June 22, 2015 cond-mat.mtrl-sci
Recent experiments have revealed spectacular transport properties of conceptually simple semimetals. For example, normal semimetals (e.g. WTe$_2$) have started a new trend to realize a large magnetoresistance, which is the change of electrical resistance by an external magnetic field. Weyl semimetal (WSM) is a topological semimetal with massless relativistic electrons as the three-dimensional analogue of graphene and promises exotic transport properties and surface states, which are different from those of the famous topological insulators (TIs). In this letter, we choose to utilize NbP in magneto-transport experiments because its band structure is on assembly of a WSM and a normal semimetal. Such a combination in NbP indeed leads to the observation of remarkable transport properties, an extremely large magnetoresistance of 850,000 % at 1.85 K (250 % at room temperature) in a magnetic field of 9 T without any signs of saturation, and ultrahigh carrier mobility of 5$\times$10$^6$ cm$^2$ V$^{-1}$ s$^{-1}$ accompanied by strong Shubnikov-de Hass (SdH) oscillations. NbP presents a unique example to consequent design the functionality of materials by combining the topological and conventional phases.
• ### Experimental Observation of Incoherent-Coherent Crossover and Orbital Dependent Band Renormalization in Iron Chalcogenide Superconductors(1505.03074)

The level of electronic correlation has been one of the key questions in understanding the nature of superconductivity. Among the iron-based superconductors, the iron chalcogenide family exhibits the strongest electron correlations. To gauge the correlation strength, we performed systematic angle-resolved photoemission spectroscopy study on the iron chalcogenide series Fe$_{1+y}$Se$_x$Te$_{1-x}$ (0$<$x$<$0.59), a model system with the simplest structure. Our measurement reveals an incoherent to coherent crossover in the electronic structure as the selenium ratio increases and the system evolves from the weakly localized to more itinerant state. Furthermore, we found that the effective mass of bands dominated by the d$_{xy}$ orbital character significantly decreases with increasing selenium ratio, as compared to the d$_{xz}$/d$_{yz}$ orbital-dominated bands. The orbital dependent change in the correlation level agrees with theoretical calculations on the band structure renormalization, and may help to understand the onset of superconductivity in Fe$_{1+y}$Se$_x$Te$_{1-x}$.
• ### Direct observation of the transition from indirect to direct bandgap in atomically thin epitaxial MoSe2(1401.3386)

Quantum systems in confined geometries are host to novel physical phenomena. Examples include quantum Hall systems in semiconductors and Dirac electrons in graphene. Interest in such systems has also been intensified by the recent discovery of a large enhancement in photoluminescence quantum efficiency and a potential route to valleytronics in atomically thin layers of transition metal dichalcogenides, MX2 (M = Mo, W; X = S, Se, Te), which are closely related to the indirect to direct bandgap transition in monolayers. Here, we report the first direct observation of the transition from indirect to direct bandgap in monolayer samples by using angle resolved photoemission spectroscopy on high-quality thin films of MoSe2 with variable thickness, grown by molecular beam epitaxy. The band structure measured experimentally indicates a stronger tendency of monolayer MoSe2 towards a direct bandgap, as well as a larger gap size, than theoretically predicted. Moreover, our finding of a significant spin-splitting of 180 meV at the valence band maximum of a monolayer MoSe2 film could expand its possible application to spintronic devices.
• ### Measurement of Coherent Polarons in the Strongly Coupled Antiferromagnetically Ordered Iron-Chalcogenide Fe1.02Te using Angle-Resolved Photoemission Spectroscopy(1212.4946)

The nature of metallicity and the level of electronic correlations in the antiferromagnetically ordered parent compounds are two important open issues for the iron-based superconductivity. We perform a temperature-dependent angle-resolved photoemission spectroscopy study of Fe1.02Te, the parent compound for iron chalcogenide superconductors. Deep in the antiferromagnetic state, the spectra exhibit a "peak-dip-hump" line shape associated with two clearly separate branches of dispersion, characteristics of polarons seen in manganites and lightly-doped cuprates. As temperature increases towards the Neel temperature (T_N), we observe a decreasing renormalization of the peak dispersion and a counterintuitive sharpening of the hump linewidth, suggestive of an intimate connection between the weakening electron-phonon (e-ph) coupling and antiferromagnetism. Our finding points to the highly-correlated nature of Fe1.02Te ground state featured by strong interactions among the charge, spin and lattice and a good metallicity plausibly contributed by the coherent polaron motion.
• ### Observation of Temperature-Induced Crossover to an Orbital-Selective Mott Phase in A$_{x}$Fe$_{2-y}$Se$_2$ (A=K, Rb) Superconductors(1208.5192)

Aug. 26, 2012 cond-mat.supr-con
In this work, we study the A$_{x}$Fe$_{2-y}$Se$_2$ (A=K, Rb) superconductors using angle-resolved photoemission spectroscopy. In the low temperature state, we observe an orbital-dependent renormalization for the bands near the Fermi level in which the dxy bands are heavily renormliazed compared to the dxz/dyz bands. Upon increasing temperature to above 150K, the system evolves into a state in which the dxy bands have diminished spectral weight while the dxz/dyz bands remain metallic. Combined with theoretical calculations, our observations can be consistently understood as a temperature induced crossover from a metallic state at low temperature to an orbital-selective Mott phase (OSMP) at high temperatures. Furthermore, the fact that the superconducting state of A$_{x}$Fe$_{2-y}$Se$_2$ is near the boundary of such an OSMP constraints the system to have sufficiently strong on-site Coulomb interactions and Hund's coupling, and hence highlight the non-trivial role of electron correlation in this family of iron superconductors.
• ### Ambipolar Field Effect in Topological Insulator Nanoplates of (BixSb1-x)2Te3(1107.0535)

Topological insulators represent a new state of quantum matter attractive to both fundamental physics and technological applications such as spintronics and quantum information processing. In a topological insulator, the bulk energy gap is traversed by spin-momentum locked surface states forming an odd number of surface bands that possesses unique electronic properties. However, transport measurements have often been dominated by residual bulk carriers from crystal defects or environmental doping which mask the topological surface contribution. Here we demonstrate (BixSb1-x)2Te3 as a tunable topological insulator system to manipulate bulk conductivity by varying the Bi/Sb composition ratio. (BixSb1-x)2Te3 ternary compounds are confirmed as topological insulators for the entire composition range by angle resolved photoemission spectroscopy (ARPES) measurements and ab initio calculations. Additionally, we observe a clear ambipolar gating effect similar to that observed in graphene using nanoplates of (BixSb1-x)2Te3 in field-effect-transistor (FET) devices. The manipulation of carrier type and concentration in topological insulator nanostructures demonstrated in this study paves the way for implementation of topological insulators in nanoelectronics and spintronics.
• ### Observation of Single Dirac Cone Topological Surface State in Compounds TlBiTe2 and TlBiSe2 from a New Topological Insulator Family(1006.3843)

June 19, 2010 cond-mat.mtrl-sci
Angle resolved photoemission spectroscopy (ARPES) studies were performed on two compounds (TlBiTe$_2$ and TlBiSe$_2$) from a recently proposed three dimensional topological insulator family in Thallium-based III-V-VI$_2$ ternary chalcogenides. For both materials, we show that the electronic band structures are in broad agreement with the $ab$ $initio$ calculations; by surveying over the entire surface Brillouin zone (BZ), we demonstrate that there is a single Dirac cone reside at the center of BZ, indicating its topological non-triviality. For TlBiSe$_2$, the observed Dirac point resides at the top of the bulk valance band, making it a large gap ($\geq$200$meV$) topological insulator; while for TlBiTe$_2$, we found there exist a negative indirect gap between the bulk conduction band at $M$ point and the bulk valance band near $\Gamma$, making it a semi-metal at proper doping. Interestingly, the unique band structures of TlBiTe$_2$ we observed further suggest TlBiTe$_2$ may be a candidate for topological superconductors.