text:
Annuity notation in LaTeX

How do you write this in LaTeX:

6 Answers

In the thread Actuarial & Annuity notation in LaTeX from text.comp.tex, Donald Arseneau provided the following \annu command:

\def\annu#1{_{%
\vbox{\hrule height .2pt
\kern 1pt
\hbox{$\scriptstyle {#1}\kern 1pt$}%
}\kern-.05pt
\vrule width .2pt
}}

with the additional comment:

Tweak ".2pt" and "1pt" for appearance.

• But you cant do it on this site? Aug 6 '11 at 6:56
• @ annuitt Can't do what on this site? You now have two solutions to your problem on this site. (Or do you mean why LaTeX markup isn't converted on this site? If that's your question, it's because this site is mainly about the markup language (TeX/LaTeX) and having automatic conversion would defeat the purpose.) See this question Can we turn off math-tex for reasons. Aug 6 '11 at 16:47

I had the same problem with the actuarial symbol and the subscript/superscript, sooo I made a package to make my life easier and help other.

Plus, I’ve add some shortcut to save time.

The project page and the CTAN.

All you need is the actuarialsymbol package.

At the beginning of the code you have to write

\usepackage{actuarialsymbol}

For the sub/superscript

\actsymb['subscripLeft']['superscriptL']{<middle>}{'subscriptR'}{'superscriptR'}

Example of output:

Example of shortcut for actuarial symbol :

• Welcome to TeX.SE, and thanks for providing this announcement about the newly available package. It would he helpful is you also posted a compilable example -- borrowing code from some of the earlier answers would be entirely acceptable -- that shows the output produced by your package.
– Mico
Apr 26 '17 at 20:33

I suggest to google for "actuarial" symbols and, of course, "LaTeX". You will find lots of examples, e.g. Annuity or Angle Operation Symbol in LaTeX.

I solved that problem some years ago, if I only remembered the context. . .

EDIT

Some years later: There is a package for the actuarial angle, now in version 2: https://ctan.org/pkg/actuarialangle

• +1 for the lifecon package mentioned there. Aug 5 '11 at 23:15
• @Keks: But how do you do it on sites like this? Aug 6 '11 at 6:56
• How do you do what? On which sites? Sorry, I do not understand the question. Aug 6 '11 at 15:06
• @annuitt Ah, now I understand, you'd like to write it as html here! Well, sorry, no idea! Apr 13 '17 at 14:30
• There is now a better option with actuarialsymbol. Dec 18 '19 at 23:13

You need the actuarialange package.

At the beginning of the code you have to write

\usepackage{actuarialangle}

Then in math mode, to make the subscript “n” with the actuarial angle, you have to use one of these three commands:

\actuarialangle{n}
\angl{n}
\angln

Then for your task the command is:

$a_{\actuarialangle{n}}$

Or if you want to write also the interest rate “i” the command is

$a_{\actuarialangle{n}i}$

Here's an example: in the tabular the construction appears at natural size, but of course the commands should be used as subscripts like in the display.

\documentclass{article}
\usepackage{actuarialangle}

\begin{document}

\begin{tabular}{ll}
\verb|\actuarialangle{n}| & $\actuarialangle{n}$ \\
\verb|\angl{n}| & $\angl{n}$ \\
\verb|\angln| & $\angln$
\end{tabular}

$a_{\actuarialangle{n}} \qquad a_{\actuarialangle{n}i}$

\end{document}

• Welcome to TeX.SX! I reformatted the post to conform to our usual standards. Apr 1 '17 at 20:08

This symbol is in Unicode as U+20E7 (x⃧ ) and in unicode-math as \annuity.

• @AndréC Thanks for catching the typo. Dec 13 '19 at 15:24

\documentclass{article}
\usepackage{amsmath,amssymb,actuarialangle}
%amssymb is needed for the urcorner symbol OR actuarialangle for the taller version

%following two lines are only for this demo
\usepackage[]{graphicx}
\usepackage[colorlinks=true]{hyperref}

\begin{document}
\centering
\includegraphics[scale=0.4]{Actuarial_notation}\\
Actuarial notation from \url{https://en.wikipedia.org/wiki/Actuarial_notation}\\
\vspace{0.4 cm}

%This is the full urcorner template to be reduced as one desires (see a typical example below)
$m\mkern-8mu{\raisebox{0.8em}2} \vert \bar{\raisebox{0.5em}{\large{A}}}{\raisebox{0.8em}1}$
$\mkern-12mu{\raisebox{-0.0em}{\textit{x:n}}\mkern-8mu\big\urcorner}$

\vspace{0.4cm}

or for actuarialangle version
\vspace{0.4cm}

%This is the full Actuarialangle template to be reduced as one desires (see a typical example below)
$m\mkern-8mu{\raisebox{0.8em}2} \vert{\bar{\raisebox{0.5em}{\large{A}}}{\raisebox{0.8em}1}\mkern-10mu{x\mkern-4mu:\mkern-4mu\actuarialangle{n}}}$

\vspace{0.4cm}
or for a n other example
\vspace{0.4cm}

% bar A changed to ddot a, and several ''terms'' removed
$\ddot{\raisebox{0.4em}{{a}}}$$\raisebox{-0.0em}{\textit{x:n}}\mkern-8mu\big\urcorner$\\
$\ddot{a}_{x:\actuarialangle{n}}$

\vspace{0.4cm}
These are here as simple snippets to adjust rather than have to start from fresh
\end{document}


text:
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## Formlabs Form 1 Teardown

I got a Form 1 3D printer!

It’s the first 3D printer I’ve purchased (technically, I “backed” it). I managed to acquire a pre-release beta unit, as I’m affiliated with Neoteny Labs, one of Formlabs’ investors.

What’s the first thing I do with any shiny new gadget? That’s right, I take it apart!

Unboxing

The complete Form 1 kit consists of three boxes: the printer (shown above), bottles of liquid photocurable resin, and a cleaning station. Many optical stereolithography printers require a cleaning step to wash off the uncured resin; for the Form 1 this means rinsing the part in isopropyl alcohol (the same stuff you use to disinfect cuts, but more concentrated) for a few minutes.

The packing method, at least for the beta, is styrofoam-free. It uses a pair of thin plastic sheets suspended on cardboard frames to hold the printer in a cradle. For a box that went as checked luggage from Boston to Singapore through three flights, it held up remarkably well.

The printer box also contains the power supply, cables, and the requisite quickstart guides.

The serial number scheme, at least for the beta units, is “AdjectiveAnimalname”, so mine is simply “ChiefCat” — hurray for easy-to-read serial numbers!

Undressing
You need exactly one tool to disassemble the printer: a 2.5mm hex key (or, if you’re a fan of Torx, a T-10 bit). I really appreciate the balance of good design with practicality: it looks good, and you don’t need whacky pentalobe screwdrivers to service it. It does take a bit of elbow grease to undo the screws; but a grippy Torx driver makes short work of the problem without a single stripped screw.

Two screws release the orange-colored light shield from its hinge, and four screws release the top of the base unit from the frame. The orange-colored shield protects your eyes from stray blue laser light, while also protecting the photosensitive resin bath on the inside from stray ambient light.

Six more screws (two of which are on the bottom) release the mid-frame from the base. The mid-frame is screwed to the front and back panels, but I didn’t need to remove the panels, as there is enough flexability in the sheet aluminum to slide the mid-frame off. The wiring to the front panel is tidily secured and is exactly the right length to allow the frame to sit next to the base without having to undo any cables — super-convenient for service and debug.

The inside of the front panel reveals the adapter for the OLED display, as well as the very handsome back-lit power button.

The Mechanism
One of the great things about a stereolithography 3D printer is that it uses a laser to cure a resin, instead of melting and extruding a plastic. As a result, the operation is whisper-quiet, and there is no odor during operation — you can put the printer in a bedroom and still sleep at night. To borrow the words of Obi-Wan Kenobi: “This is the 3D printer of a Jedi Knight. Not as clumsy or as random as an FDM, but an elegant tool for a more civilized age.”

I count four motors in the Form 1: two stepper motors with lead screws, and two galvonometers (which I refer to as simply “galvos”). Let’s tour the printer’s mechanisms, motor by motor.

A large stepper motor with a lead screw running up the spine of the device pulls the build platform upwards. This causes the printed object to appear to “grow” out of the resin bath.

The build platform is attached to the leadscrew with an anti-backlash nut. The spring in the picture above applies a static force to the leadscrew-nut interface, reducing the play between the two and improving accuracy.

Every time a thin layer of resin is cured, a second motor performs a “peel” operation by tilting one edge of the resin tank downwards. The peel operation separates the newly formed layer from the bottom of the resin tank, allowing the build platform to move up freely. The leadscrew pictured above is responsible for this action, and it is driven by the smaller of two motors inside the Form 1. The white bearing is fixed to the frame, and the pin connecting the leadscrew to the tiltable resin tank is visible at the top.

Ah…and now we peer into the heart of the Form 1! Here are the two remaining motors of the Form 1, the galvos. Galvos are very precise rotational actuators, with a built-in optical feedback loop (take a look inside one). The galvos each have a tiny octagonal mirror attached to their shafts, and are mounted at right angles to each other. One galvo steers the laser beam, entering from the left, along the X axis; then the light bounces to a second galvo that steers along the Y axis. The two working together can thus trace the 2-D vector pattern used to cure a single layer of resin. Below the galvos and laser is a mirror that reflects the light to another, larger mirror that finally shines the beam upwards into the resin bath.

Above is another view of the laser. According to the safety warning bundled with the printer, it is a 405nm, 120mW violet laser diode (take a look inside one).

In order to achieve high performance, the galvo has a sturdy bank of drivers, mounted to the bottom aluminum surface of the case for heat sinking.

A slightly different camera angle shows the power input stage of the Form 1.

The main board wraps around the spine of the Form 1 and on the left side are the brains. An STM32 32-bit ARM Cortex-M3 microcontroller coordinates all of the activities inside the Form 1, and it appears that firmware for the device is stored on an internal, captive SD card.

Above are some photos that give the whole context of the electronics behind the Form 1.

A hall effect sensor inside the base makes sure you can’t turn the device on when the protective light shield is lifted. But, you can defeat the sensor by putting a magnet on it, as I did above to check I hadn’t damaged anything before putting the whole thing back together again.

Re-assembly is a breeze — within a couple minutes I have the whole printer put back together and ready to do my first test print.

My First Print
What’s the first thing I’m going to print? I’m going to download and print a stand, designed by another user (IHeartEngineering) and shared for free, for my Geiger counter!

The Form 1 comes with a print management software package called “Preform”. It’s intuitive and easy to use; it allows you to place multiple objects on the build platform, duplicate, scale, and rotate. There’s an auto-orient feature that suggests an optimal orientation for supports based upon a DFM analysis. The screencap above is the result of the software picking the build orientation and populating the build supports.

I click “Form”, and about 10 hours later the print is done.

The next step is to remove the build platform from the printer and bring it to the cleaning station, one of the accessories shipped with the Form 1. Cleaning is necessary to remove uncured liquid resin residue, and it’s recommended to handle uncleaned parts using gloves to avoid getting the goopy resin on your fingers.

The cleaning station is thoughtfully laid out, with an area designed to hold the build platform. The kit includes some of the requisite tools to peel the part off of the platform, rinse it, and break off the supports. I supplement the kit with a pair of diagonal cutters for removing the supports, although you can also simply snap the supports off by pulling on them.

Two minutes of active shaking in an isopropyl bath, ten minutes of soaking, and the part is ready to be handled.

I use a pair of diagonal cutters to remove all of the supports in about five minutes.

This particular part features large, robust geometries: it’s designed to be printable on an FDM, so it doesn’t show off the high-resolution capability of the Form 1.

Small bumps are left on the area where the supports contact the part. These can be sanded off, if necessary.

The print is a success! It fits my Geiger counter like a glove.

And now for the shameless plug: if you made it all the way through this tear-down, and you’re a top-notch EE, you might want to consider applying for Formlabs’ Electrical Engineering Lead position by dropping them an email with your resume and cover letter.

Tags: , , ,

### 47 Responses to “Formlabs Form 1 Teardown”

1. sk says:

The build quality looks stunning.
You mentioned a while back that you had received a laser cutter/engraver. what ever happened to that? I was considering buying that brand.

• sk says:

It was a Full Spectrum. would you recommend it?

• bunnie says:

It’s been good for me so far. Good value for the money, I’d say.

• sk says:

they offer a multitude of upgrade options for that unit. Any suggestions on what upgrades are necessary?

• Paco belmar says:

10 hours? For print this piece? Lot of time, no?

• bunnie says:

• bunnie says:

hmm….

I got the air compressor, and air purifier so I could use it indoors. But I think the accessories will depend upon what you want to do with it, and how much you want to build/set up your own venting system.

2. […] Formlabs Form 1 Teardown […]

3. db says:

Very nice tear down, the galvo looks very like the one I got for my scanner…
http://exploreideasdaily.wordpress.com/2013/03/18/first-steps-of-the-scanner-building/
Interesting, I can’t wait to have my form 1.
Cheers

4. […] teardown of the new FormLabs Form 1 3d printer (with […]

5. Aven says:

Thanks for publishing the interesting and informative, exclusive teardown bunnie.

I had a question about how the peel/tilt mechanism worked, because with my understanding there was the risk of part location variance between each cycle. Of course the printed object is not pushed up, it’s pulled out — a brief segment of the printer’s operation tucked in a clip on youtube cleared that up for me. Just thought I would leave a note in case anyone else had the same idea.

Very nice, clean design of the printer — my feelings are part jealousy, and part desire to find the performance limits.

6. Justin Tan says:

What resolution did you print that at?

• bunnie says:

50 microns resolution. The software has options ranging from 25 microns to 100 microns resolution, if I’m understanding this pull-down menu correctly.

7. […] One of the investor in FormLabs (Bunnie Huang) has provided a detailed breakdown of the machine which you can check out here. […]

8. You’re the best, Bunny. Keep on keeping on.

9. So the scanning uses galvos (galvAnometers)… That’s clever! This is the fastest method.

10. Dennis Chan says:

Ten hours to print. Seems rather long. Is it typical?

• bunnie says:

It doesn’t seem to be. The print time is a function of height, resin volume cured, and resolution. This example piece is a solid resin block between the supports and the base, so that took a bit longer. I’m printing a similarly sized mathematical knot (so it’s a lattice that doesn’t consume as much resin) at 100 micron resolution and it’s running in 3.5 hours.

11. JSNP says:

Hi Bunnie!

I was wondering if you, or any other owner of the form 1 printer could print a small model for me, and how much would it cost you to do it?
(I understand it takes time and resin so I would gladly pay for it)

I would like to see what it can do before I buy one myself.

My email is joe[dot]stupensky[at]gmail[dot]com

• JSNP says:

I guess you’re too busy to answer or simply not interested
:/

I wish there was a way to try and print something before spending $3500… Oh well… • bunnie says: Hi sorry — I had for some reason missed this comment as I was scrolling through quickly. I…don’t think it would be very cost efficient for me to print you something and then mail it to you from Singapore. Your best bet is to drop a line on the community forum: http://support.formlabs.com/categories/20100046-Community-Forums 12. You mentioned “Two minutes of active shaking in an isopropyl bath” … I’m assuming an ultrasonic cleaner would be a good idea, yes? • bunnie says: Hm, I hadn’t considered that. That’s a good idea. I have an ultrasonic cleaner, I’ll give it a try. 13. Mathilde says: How much does the material cost? What would be your pricing estimation for the example you show? Interesting article showing the product in great details. Wish I had one already. Thanks! • bunnie says: They just posted the price for resin on their home page – listed at$150 for 1 liter, although Kickstarter backers are getting a deal for up to 3 bottles at $129/liter. The resin volume for the print was 74ml so this was about$11. However, as I get to use the printer more, I realize this piece is pretty resin-inefficient, it has large regions of mostly resin which could have been hollowed out, and saved a good deal of print time and resin.

14. […] did a great teardown of the Formlabs Form1. Can’t wait to hear more about the low cost SLA […]

15. Peter Dilworth says:

Hi Bunny, It’s Pete (dino robot guy from the old AI lab) I seem to have lost your contact, I have a few questions I would like to ask you, would you mind if you have a chance, shooting me a quick email so I can converse?

thanks!!
-pete

• Peter Dilworth says:

my email is: freakazoidal AT gmail dot com

16. Luisz says:

10 hours! For God sake! Now i know why the guys at Formlabs never told us anything about actual printing time. The resine finish needs some painting. It looks awful at natural. In one of the pictures i can see a V shaped path. Does it exist on the cad model too or is it an artifact of the printing process? Thanks for your detailed review!

• bunnie says:

The V-shaped artifact does not seem to be in the CAD. It looks like a hairline surface artifact. I took the photo with a specular reflection to highlight the layering artifacts that arise from the printing process, without the light shining at the right angle, it’s difficult to see.

I printed at a 50 micron resolution, and the design is a big block of resin so it takes longer to print.

I’m running a similarly sized piece at a lower resolution (100 microns) which doesn’t have large solid resin features, and it is taking 3.5 hours.

17. mianchen says:

very cool. i want one!

18. Thanks for the post. You are the Fucking boss!

19. Alex says:

Why didn’t you align it the way the holder is supposed to stand when done (approx. 75 degrees to the right)??? You could have saved the support material?

20. dale says:

All those preset potentiometers look scary!…i.e. looks like a lot of scope for things to go out of whack or is this just for prototype?

Dale

21. Leodegar Berger says:

I want to produce small car models with the form 1 ( z scale) May I send you a stl file and you can print a model for me? I pay for it but I want to see what the printer is able to do.

LEO

22. benjamin says:

hi do you know any reseller in Singapore? or do you mind share with me more information:)

23. Anderson Ta says:

Any chance you can put up the part number for that motor used for the tilt action?

24. Mike says:

Excellent site and forum. Wondering who will hack to highest resolution, I did not think it had even 50 micron resolution ( ie is the 50 you state XY or Z resolution ?) Mike

25. Leonard says:

Hi, I received my form 1 yesterday and my first print came out shifted in the x direction.. and the result is a bit distorted as well… do you think it might be a loose galvo? is it difficult to tear it down to check the galvo? Thanks a lot！

26. Max says:

Leonard,

27. Schorhr says:

Nice write up, thank you!
I wish I could afford one right now, would be so much better for my models…

28. You’re pretty courageous taking the Form 1 3D Printer apart, risking to damage it, when hundreds of people are waiting months and months for theirs to be delivered :-)

Thanks for the cool insight into the system though, appreciate it a lot!

Etienne

29. Hiren says:

Hi bunnie
Can you tell me what is the XY resolution of form1 and it is suitable for jewellery or not???

30. I pay a quick visit each day a few sites and information sites to read articles, however this webpage provides quality based posts.

31. Jun Lee says:

Hello?
can you sell your form 1 ? I can pay \$6,000 ?


text:
TL;DR

On with TASK #2 from The Weekly Challenge #127. Enjoy!

# The challenge

You are given a list of intervals.

Write a script to find out if the current interval conflicts with any of the previous intervals.

Example

Input: @Intervals = [ (1,4), (3,5), (6,8), (12, 13), (3,20) ]
Output: [ (3,5), (3,20) ]

- The 1st interval (1,4) do not have any previous intervals to compare with, so skip it.
- The 2nd interval (3,5) does conflict with previous interval (1,4).
- The 3rd interval (6,8) do not conflicts with any of the previous intervals (1,4) and (3,5), so skip it.
- The 4th interval (12,13) again do not conflicts with any of the previous intervals (1,4), (3,5) and (6,8), so skip it.
- The 5th interval (3,20) conflicts with the first interval (1,4).

Input: @Intervals = [ (3,4), (5,7), (6,9), (10, 12), (13,15) ]
Output: [ (6,9) ]

- The 1st interval (3,4) do not have any previous intervals to compare with, so skip it.
- The 2nd interval (5,7) do not conflicts with the previous interval (3,4), so skip it.
- The 3rd interval (6,9) does conflict with one of the previous intervals (5,7).
- The 4th interval (10,12) do not conflicts with any of the previous intervals (3,4), (5,7) and (6,9), so skip it.
- The 5th interval (13,15) do not conflicts with any of the previous intervals (3,4), (5,7), (6,9) and (10,12), so skip it.

# The questions

Well… this challenge seems to go all by induction from the examples. Which means… let’s guess a rule, and stick to that.

In this case, I’ll assume that two conflicting intervals share a sub-interval or an endpoint. Hence, (1,4) and (3,5) conflict because they share the sub-interval (3,4). By this definition, anyway, also (1,3) and (3,5) conflict, because they share one endpoint.

I’ll also assume that it’s OK to print conflicting intervals as we like it and not necessarily in the order they appear in the input sequence of intervals. This is because it’s not written and… my algorithm does not necessarily preserve the input order 🙄

# The solution

Although the challenge talks about “looking backwards” for conflicting predecessors, I opted for a look-ahead approach where I take the interval at the beginning and see if it generates any conflict ahead.

The first interval is always safe, because it has no conflicting predecessors. For each successor that the first interval conflicts with, we eliminate that interval from the list of candidates (because we know it has a conflict) and merge it with the initial one, going on to find out additional conflicting intervals up to the end.

At the end of one such pass, the first element can be safely ignored, all conflicting intervals associated to that element and other conflicting with it have been eliminated from the input list, and we can restart the process with the intervals we’re left with. When we exhaust this list… we’re done.

To be fair, I started with Perl:

sub conflict_intervals (@intervals) {
my @conflicting; # keep the answer
while (@intervals) {

# if there is a "first" one, it's safe because it has not been
# eliminated by its predecessors. Its endpoints will be used to
# possibly eliminate successors, we keep them in two convenience
# variables.
my ($X,$Y) = shift(@intervals)->@*;

# we filter the remaining intervals ditching all those that
# conflict with ($X,$Y) or whatever it becomes on the way. In
# particular, at every conflict we expand ($X,$Y) to also
# include the conflicting item, because we will ditch it from
# the candidate "clean" intervals and put it in @conflicting.
@intervals = grep {
my ($A,$B) = $_->@*; # this is a general check to see if the two intervals are # disjoint. It assumes that touching intervals are conflicting. ($A - $Y) * ($B - $X) > 0 or do { push @conflicting,$_;
$X =$A if $X >$A;  # "eat" the ($A,$B) interval in ($X,$Y)
$Y =$B if $Y <$B;
0; # this interval conflicted and does not get passed along
}
} @intervals;
}
return @conflicting;
}

The Raku translation required very little changes, admittedly:

sub conflict-intervals (@intervals) {
return gather {
while @intervals {

# if there is a "first" one, it's safe because it has not been
# eliminated by its predecessors. Its endpoints will be used to
# possibly eliminate successors, we keep them in two convenience
# variables.
my ($X,$Y) = @intervals.shift.Slip;

# we filter the remaining intervals ditching all those that
# conflict with ($X,$Y) or whatever it becomes on the way. In
# particular, at every conflict we expand ($X,$Y) to also
# include the conflicting item, because we will ditch it from
# the candidate "clean" intervals and put it in @conflicting.
@intervals = @intervals.grep: -> $interval { my ($A, $B) = |$interval;

# this is a general check to see if the two intervals are
# disjoint. It assumes that touching intervals are conflicting.
($A -$Y) * ($B -$X) > 0 or do {
take $interval;$X = $A if$X > $A; # "eat" ($A, $B) in ($X, $Y)$Y = $B if$Y < \$B;
0; # this interval conflicted and does not get passed along
}
}
}
}
}

I’m stubborn as a mule and I insist on using gather/take despite the performance tax. I stand by it, though: it’s such a useful construct that I’ll not be hindered by its slowness. Until I can, anyway. Besides… it might become fast one day, who knows?

I hope you found this interesting! Stay safe and… have -Ofun!


text:
# pymrio.parse_exiobase2¶

pymrio.parse_exiobase2(path, charact=True, popvector='exio2')

Parse the exiobase 2.2.2 source files for the IOSystem

The function parse product by product and industry by industry source file in the coefficient form (A and S).

Filenames are hardcoded in the parser - for any other function the code has to be adopted. Check git comments to find older verions.

Parameters: path (string or pathlib.Path) – Path to the EXIOBASE source files charact (string or boolean, optional) – Filename with path to the characterisation matrices for the extensions (xls). This is provided together with the EXIOBASE system and given as a xls file. The four sheets Q_factorinputs, Q_emission, Q_materials and Q_resources are read and used to generate one new extensions with the impacts. If set to True, the characterisation file found in path is used ( can be in the zip or extracted). If a string, it is assumed that it points to valid characterisation file. If False or None, no characterisation file will be used. popvector (string or pd.DataFrame, optional) – The population vector for the countries. This can be given as pd.DataFrame(index = population, columns = countrynames) or, (default) will be taken from the pymrio module. If popvector = None no population data will be passed to the IOSystem. A IOSystem with the parsed exiobase 2 data IOSystem ParserError – If the exiobase source files are not complete in the given path


text:
# area of trapezoid base on radius of excircles [closed]

We have a trapezoid and we know radiuses of its escribed circles. Now we want to calculate area of this trapezoid with radiuses. Radiuses of two circles that are besides the legs of trapezoid (and between parallel sides) is the same.(r1=r2) And height of trapezoid equal with diameter of these circles.(h=2r1=2r2) Now we have height and one more thing that we need is lengths of the parallel sides. And I don't have any idea how find those.

• I don't know what a trapezoid with radius is. Can you give a picture example, or describe it better? – Calvin Khor Jan 9 at 9:16

Excircles radii are not enough to compute the area of the trapezoid, as you can see from the animation below: excircles have always radii $$1$$, $$1$$, $$1$$ and $$3$$, but the area of the trapezoid varies.


text:
Open access peer-reviewed chapter

# Augmented Reality and Virtual Reality: Initial Successes in Diagnostic Radiology

By David B. Douglas, Demetri Venets, Cliff Wilke, David Gibson, Lance Liotta, Emanuel Petricoin, Buddy Beck and Robert Douglas

Submitted: October 9th 2017Reviewed: January 23rd 2018Published: March 8th 2018

DOI: 10.5772/intechopen.74317

## Abstract

In this chapter, we will review the applications of augmented reality (AR) and virtual reality (VR) in diagnostic radiology. We begin the chapter by discussing state of the art medical imaging techniques to include scanner hardware, spatial resolution, and presentation methods to the radiologist or other medical professionals. We continue by discussing the methodology of a technique called Depth-3-Dimensional (D3D) imaging, which transforms cross-sectional medical imaging datasets into a left and right eye images that that can be viewed with state of the art AR or VR headsets. We include results of the D3D processed AR/VR imaging and conclude with a discussion of the path forward.

### Keywords

• augmented reality
• virtual reality
• D3D
• depth perception

## 1. Introduction

The purpose of this chapter is to review the applications of augmented reality (AR) and virtual reality (VR) in diagnostic radiology. This introduction section will provide a brief general summary of AR technologies, VR technologies, AR/VR applications in medicine and AR/VR applications diagnostic radiology. The remainder of the paper will discuss state-of-the-art medical imaging systems, current methods of viewing medical images, the imaging processing techniques for generating an AR/VR image, AR/VR imaging results and path forward.

### 1.1. Introduction to AR/VR

AR technologies can be classified into AR systems or mixed reality (MR) systems. In both the AR and MR systems, the user wears a head mounted display (HMD), which provides the simultaneous display of a virtual image and the scene of the real-world surroundings [1]. The differences are that in AR, the virtual image is transparent like a hologram and in MR the virtual image appears solid. Examples of AR systems include the Meta and DAQRI systems. An example of an MR system is the Microsoft HoloLens. In AR systems, the user can view the virtual image and interact with the real-world scene.

VR technologies may be classified as fully immersive, semi-immersive or non-immersive [2]. In fully immersive VR systems such as the Oculus Rift and the HTC Vive, the HMD displays a virtual image and the real-world surroundings are completely occluded from the user’s field of view [3]. In semi-immersive VR systems such as the Samsung Gear VR, the HMD displays the virtual image, but the real-world scene is only partially occluded from the user’s field of view [3]. In VR systems, the user can maneuver through the virtual world by head movements via HMD tracking or walking via external camera tracking systems. Additional ways that a user can interact with the virtual environment include voice gestures or through handheld devices with haptic feedback. The HMDs for the AR and VR systems display a unique image to each eye; therefore, stereoscopic imaging and depth perceptions is achieved. Both AR and VR are rapidly growing fields. Worldwide revenues for AR/VR were $5 billion in 2016, but are expected to increase to$162 billion by 2020 [3].

### 1.2. Applications of AR/VR in medicine

In 2015, the United States spent 17% of its gross domestic product on healthcare. 32% of the healthcare spending was during hospital stays [4]. Nearly half of hospital costs are for surgical care [5]. Therefore, approximately 3% of the gross domestic product includes costs related to surgery. There is a drive to improve efficiency in the operation room to both improve patient care and drive down costs. AR/VR holds promise in accomplishing these goals through improving pre-operative planning and enhancing intra-operative surgical procedures [6].

### 1.3. Applications of AR/VR in diagnostic radiology

In the United States, the total cost of diagnostic imaging has been estimated to be \$100 billion in 2006 alone [7]. Utilization rates of diagnostic imaging are on the rise [8, 9, 10]. Currently, there is no United States Food and Drug Administration (FDA) approved AR/VR system used in diagnostic radiology [11]. AR/VR provides enhanced viewing including depth perception and improved human machine interface (HMI) [12, 13]. AR/VR HMDs provide unique images to each eye yielding depth perception. AR/VR systems leverage advanced gaming controllers and joysticks to improve HMI. Because of these features of AR/VR and others discussed later in this chapter, we believe there will be increasing applications of AR/VR in diagnostic radiology in the future.

## 2. Current state-of-the-art diagnostic medical imaging

Diagnostic radiology plays a major role in medicine as it provides precise anatomic and physiologic information to physicians enabling diagnosis of complex disease and monitoring response to treatment. This section will be organized into three subsections including medical imaging equipment, conventional imaging techniques and the advanced 3D rendering methods.

### 2.1. Medical imaging equipment

The field of diagnostic radiology includes a wide variety of imaging equipment including systems that generate inherently 2D images (e.g., chest radiograph) as well as systems that generate volumetric medical imaging datasets (e.g., computed tomography (CT), magnetic resonance imaging (MRI), and positron emission tomography (PET)). In this section, we will focus on the latter.

#### 2.1.1. Computed tomography (CT)

In order to perform a CT scan (also known as a CAT scan), the patient is placed in the horizontal position on the CT scanner table. See Figure 1. The table is then translated through a donut shaped device containing both an X-ray tube and X-ray detector. Multiple projection images are acquired as the X-ray tube and detector assembly rotate around the patient.

Image reconstruction algorithms, such as filtered back projection, are performed in order to generate cross-sectional images in the axial plane (x-y plane). Since sequential, contiguous axial images can be obtained, coronal plane (x-z) and sagittal plane (y-z) plane images can be reconstructed. Data is stored in digital imaging and communication in medicine (DICOM) files with a typical matrix of a CT scan is 512 × 512 pixels. A pixel in the axial plane is a 2D object with a discrete length in the x-direction and discrete length in the y-direction. A voxel is a 3D object created with a pixel by adding a third dimension to create a volume.

Each pixel has an associated gray-scale value called a Hounsfield Unit (HU), which is a function of the density and composition of the tissue. As a reference, water has a HU of 0. Soft tissues (e.g., brain, kidney, muscle, etc.) are slightly denser than water and have a HU of approximately 30–40. Compact bone can have a density of 400. Fat is slightly less dense than water and has a HU of approximately −100. Air is significantly less dense than water and has a HU of −1000.

A process called “windowing and leveling” is performed by the radiologist to set the window “level” and window “width.” The window “level” refers to the HU number where mid-gray can represent. The window “width” is the range of shades of gray such that any value more extreme than the range is white (if more dense) or black (if less dense). See Figure 2.

Modern CT scans can be perform a total body head-to-toe scan in less than 15 seconds with spatial resolution of less than 1 mm. Thus, CT scans can be invaluable in the setting of trauma since it enables the radiologist to diagnose critical injuries anywhere in the body, such as traumatic brain injury [14]. This provides critical information to the neurosurgeon who can then perform life-saving interventions.

#### 2.1.2. Magnetic resonance imaging (MRI)

In order to perform an MRI scan, the patient is placed in the horizontal position on the MRI scanner table within a large cylindrical shaped device. A large magnetic field is directed through the long axis of the cylinder. Transmit coils direct a radiofrequency (RF) pulse into the patient’s body and receive coils process the returning electromagnetic signal from the body to create an image. Similar to CT, contiguous planar images can be stacked and axial, sagittal and coronal reformats can be reconstructed. The imaging data for a MRI scan is similar to that of a CT scan in matrix size and the fact that each pixel has a comparable gray scale.

Unlike CT scanners, MRI scanners do not employ ionizing radiation and are therefore ideal for young children or pregnant women who are more vulnerable to radiation. Furthermore, MRI scans have the ability to perform exceptional contrast resolution between tissues of similar density and can diagnose certain types of traumatic brain injury that cannot be seen on CT scans [15]. Modern MRI scans require significantly more time to perform than CT scans, nearly one-hour of image acquisition time for a MRI scan of the brain.

#### 2.1.3. Positron emission tomography (PET) scanner

In order to perform a PET scan, a radiopharmaceutical (e.g., fludeoxyglucose F-18) is administered to the patient. Then, the patient is placed in the horizontal position on the PET scanner table within a donut shaped device. As the radiopharmaceutical decays, photons are emitted from within the patient and are received by the PET detector crystals. As with CT and MRI, axial images can be stacked and sagittal and coronal images can be reconstructed.

A typical matrix of a PET scan is 128 × 128, which is smaller than that of CT or MRI. However, it is similar in that each pixel has an associated numerical value associated with it indicating gray scale. In the case of PET, the gray scale corresponds to the amount of radioactivity emitted from that location. Since the radiopharmaceuticals can target certain structures in the body, it is possible for improved diagnosis of conditions like Alzheimer’s disease compared with MRI [16].

### 2.2. Current methods of viewing medical imaging

In the previous section, we discussed three types of medical imaging scanners that generate volumetric data. In this section, we will review the current methods of viewing the volumetric data.

#### 2.2.1. Conventional viewing of the volumetric data

The conventional viewing method for reviewing volumetric datasets is a slice-by-slice viewing method for axial, sagittal and coronal imaging planes or on occasion oblique reformats. It is estimated that most radiologists spend more than 95% of their total time on cross-sectional imaging datasets using this conventional slice-by-slice approach [17]. Certain anatomical structures are easier to visualize on particular imaging planes. As an example, some radiologists have a preference for viewing the midline structures in the brain on the sagittal images, as shown in Figure 3.

Occasionally, the radiologist needs to view an abnormality in another plane, other than the axial, sagittal, coronal imaging planes. In these instances, oblique plane reformats can be used with the images still viewed in a conventional slice-by-slice approach. See Figure 4. Curved planar reformats can also be performed and have been shown to be beneficial [18].

A standard viewing method includes a flat screen, high-resolution diagnostic imaging monitor with keyboard and mouse, as shown in Figure 5. Typically, the radiologist will use the wheel on the back of the mouse to scroll through the stacks of images. Note that the radiologist also uses a microphone to dictate the radiology reports.

#### 2.2.2. Challenges in conventional viewing of the volumetric data

First, is the challenge of information overload. The dramatic improvements in spatial resolution (commonly smaller than 1 mm) of CT and MRI coupled with large portions of the body imaged generate immense datasets and the radiologists face the challenge of information overload. As an example, a CT scan of the chest with an axial matrix of 512 pixels (x-direction) by 512 pixels (y-direction) would have 262,144 pixels on a single slice. Thin-cut imaging of the chest provides 500 axial slices, each containing the 262,144 pixels, or roughly 131 million pixels in the data set.

Second, is the challenge of detecting small lesions. One example of this is the challenge is the identification a small pulmonary nodule, which is a topic of great concern for radiologists and a top cause of litigation [19, 20]. Identifying a tumor at an small size and corresponding early stage is important in order to improve patient survival and reduce cost of treatment. A very deliberate slice-by-slice method takes considerable time.

Third, is the challenge of mentally building a 3D image from reviewing slices [21]. Depending on the clinical scenario and body part imaged, the radiologist can be tasked with following certain twisting and turning structures through the body such as following a blood vessel or loops of intestines.

#### 2.2.3. Advanced (non-AR/VR) viewing methods of the volumetric data

Most radiologists spend a small fraction (<5%) of their total time in interpreting their imaging scans with advanced viewing methods [17]. Such non-AR/VR techniques include surface rendering and volume rendering.

#### 2.2.3.1. Surface rendering

The first 3D rendering technique to display the human body’s anatomy was surface rendering (also known as shaded surface display). Through segmentation techniques such as thresholding to display only a prescribed set of pixels, apparent surfaces are displayed within the body. A virtual light source is used to provide surface shading. In surface rendering, only a single surface is used. An advantage of only displaying a single surface is the fact that surface rendering techniques are typically not limited by overlapping tissues within the human body. However, there are a few limitations.

One limitation of only displaying a single surface is the fact that thresholding is used for one tissue type at a time and it can be difficult to understand the anatomic relationship of multiple different organ systems when only a single organ system is displayed. Another limitation is the fact that many organs are of similar density to their surroundings and it can be difficult to segment these structures out. Finally, since surface rendering images have been displayed on flat screen images true depth perception is not achieved. See Figure 6.

#### 2.2.3.2. Volume rendering

The technique of volume rendering has been researched for many years by the computer graphics industry and has recently been applied to diagnostic radiology [22]. In volume rendering, a transfer function is applied to assign a color and opacity to each intensity value. As an example, voxels that correspond to the density of blood vessel are colored red and voxels that correspond to the density of bone are colored white. This has significantly helped radiologists visualize complex 3D structures [23]. See Figure 7.

One of the key limitations of volume rendering is the overlapping structures [24, 25]. This limitation is significantly worse in settings such as viewing of the vasculature of the brain. See Figure 8.

## 3. Review of AR/VR in diagnostic radiology

In the first sections of this chapter, we have reviewed the medical imaging equipment, conventional slice-by-slice techniques and the advanced 3D rendering methods including surface rendering and volume rendering. We will now review AR/VR in diagnostic radiology.

The previously discussed limitation of overlapping structures can be minimized through a process called depth-3-dimensional (D3D) imaging by providing unique images to each eye and display on AR/VR HDUs. D3D transforms cross-sectional imaging datasets and displays them on AR/VR head display units (HDUs). In doing so, an overall immersive viewing experience is created. In AR/VR radiology, the radiologist will wear a head display unit, which can be either a VR, AR or MR system. The basic concept is outlined in Figure 9.

This section of the paper will be organized into three sections. First, we will discuss the process for generating an AR/VR image from a cross-sectional medical imaging dataset. Second, we will include results. Third, we will discuss the path forward and future opportunities of AR/VR in diagnostic radiology.

### 3.1. Image processing for AR/VR

To optimize visualization, the D3D software suite must provide the capability for visualizing medical imagery rendered in a true 3 dimensional representation [26, 27]. In order to accomplish this, input imagery is converted from 2 dimensional images into a 3 dimensional voxel space, segmented into distinct tissue types; and then filtered. Finally, rendering is then performed wherein the rendering engine computes a left and right view. This allows the operator to visualize the data using the same stereopsis that our eyes and brains have spent a lifetime interpreting and processing.

#### 3.1.1. 3D segmentation

During the 3D segmentation process, each image pixel in the input imagery is treated as a voxel, a three dimensional entity with length, width, and height. Each voxel is read into a large 3D array. The 3D array is analyzed for similarity to its neighbors, and for context clues that imply similarity to historical training data.

Mean and variance statistics largely drive the similarity analysis. Every voxel is compared to all the neighboring voxels in the 3x3 (or 5x5 or larger) nearest neighbor region. The most similar voxels are assigned the same class designation. The 3x3 variance is used to determine the width of the local intensity distribution. The width of the local intensity distribution in turn determines the threshold applied for declaring similarity between voxels. Simple local features on the neighborhood are computed around each voxel to describe the local average intensity or texture surrounding a voxel, which provides a classifier algorithm with measurements that are used to distinguish one class from another, where classes correspond to the different tissue types.

Historical data from numerous similar imaging exams will be collected and “ground truthed” so that a neural network or deep learning classifier can be used to improve classification accuracy. Techniques will be used to first classify the input imagery into categories such as skeletal, fat, normal breast tissue, breast cancer, etc. [26, 27]. The user will need to input the tissue-type selection until an intelligent system will automatically determine the anatomy of the input imagery. User input of the tissue-type selection will be used as historical data to drive the classifier.

#### 3.1.2. 3D filtering algorithm

The goal of the 3D filtering algorithm is to make the most understandable representation of the imagery to the radiologist [26, 27]. In order to accomplish this goal, it utilizes the image tissue-type from the 3D segmentation algorithm to determine which tissue types should be given highest priorities.

Current graphic hardware have limitations on how many voxels that can be displayed. Therefore, it is necessary to allocate percentages to the various tissue types based on priority. Those tissue voxels that must be seen from a clinically importance perspective (e.g., tumor) are given the highest percentage. Other tissues that are not as clinically important (e.g., subcutaneous fat) are sparsely sampled. There are two benefits of this process. First of all, this allows the operator to understand the high priority tissue (e.g., tumor) in proper context (e.g., tumor is touching the spine). It can be difficult or impossible to understand exact position of the tumor when all the surrounding tissue has been removed. Second of all, this lower priority tissue needs to be somewhat more transparent so the higher priority tissue can be seen through the lower priority tissue. However, transparency allocated may not necessarily correspond to tissue density. Additionally, thinning the tissue between the viewing perspectives and the area of interest makes it much easier to see the area of interest accurately.

User selected filtering is employed to enable the operator to slice away tissue sections to facilitate viewing of areas of interest. Additional implementation of various geometric volumes and surfaces to temporarily remove sections of the voxel cloud may be beneficial. The operator can also enable or disable display of specific tissue types.

#### 3.1.3. 3D rendering engine

The goal of the 3D rendering is to replicate, insofar as possible, what a radiologist would see if looking with his/her own two eyes through the skin and into the body were possible. A slightly different image would be seen from each eye from its particular vantage point. This difference allows one to see depth. The basic concept is illustrated in Figure 10. The geometry behind the approach is illustrated in Figures 11 and 12.

The Left Eye Viewing Perspective (LEVP) would be at an observer/ radiologist selected point to inspect the volume. The Right Eye Viewing Perspective (REVP) would be the inter-ocular distance to the right of the LEVP and similar math would be applied to generate the slightly different image presented to the right eye display. The rendering engine generates the left and right eye views to provide a true 3D data visualization to be displayed in the AR/VR HDUs [26, 27]. This component relies heavily on the Graphics Processing Units (GPU) built into the graphics card. The rendering engine supports a convergence depth adjustment for fine-tuning the operator’s focal point as shown in Figure 13.

The interocular distance can of course be altered to change stereopsis. The rendering engine allows the operator to comfortably move the viewing position, zoom, and rotate about the pitch, roll or yaw axes. The angular field of view can be changed. The volume of interest (VOI) can be rotated. The viewing perspectives can also be rotated. See Figure 14. And to improve visibility, the operator can select any number of color palettes for different tissue classes. The rendering engine also incorporates a graphical overlay for image markup.

### 3.2. AR/VR control systems used in diagnostic radiology

The field of AR/VR is evolving with new and changing HDUs and gaming control systems coming on the market. In order to illustrate what we have used in our prior research, we illustrate example control systems shown and how various control buttons correspond with basic maneuvers and advanced maneuvers. See Figures 1517.

### 3.3. Initial results of AR/VR in radiology

As previously discussed, there have been numerous applications of AR/VR in surgery to include pre-operative planning, education, and intra-operative assistance [6]. However, AR/VR is not yet FDA approved in diagnostic radiology, but is being actively researched by DXC Technologies/D3D Technologies with an initial focus in breast cancer imaging [11, 12, 13].

Breast Cancer is one of the leading causes of death in women [28, 29]. Breast calcifications are extremely common and are present in up to 86% of mammograms [30]. The calcifications are classified by distribution with a linear and branching pattern suspicious for ductal carcinoma in situ (DCIS) [31, 32]. Standard mammographic views may not reveal the true linear and branching pattern due to suboptimal view point and lack of depth perception. Therefore, the D3D AR imaging system was tested on a simulated set of microcalcifications. The radiologist who rated the AR system found that when the microcalcifications appeared as a cluster when viewed from a single perspective, but with rotation and the AR HDU appeared as a linear and branching pattern. See Figure 18.

In addition to microcalcifications, breast cancer can also present as a mass on imaging. Characterization of the shape and margins is important in determining whether the breast mass is malignant or benign. Dedicated breast CT provides high spatial resolution of breast masses [33]. Recent data has shown the importance of characterizing tumor morphology [34]. Therefore, the D3D AR system was also tested in viewing of a known breast cancer [13]. Malignant features of spiculations were noted to be more conspicuous on the D3D system than the native CT. See Figure 19.

### 3.4. Future work of AR/VR in radiology

Both the fields of AR/VR and diagnostic radiology are large and rapidly growing. One of the most common reasons for performing diagnostic imaging is for cancer. Early determination of whether a particular therapy regimen is working would be extremely helpful to improve survival and would save costs. We outline a flow chart below as a recommended process for introducing AR/VR evaluating a cancer at multiple time points. See Figure 20.

It is foreseeable that AR/VR will one day play a major role in diagnostic radiology. Computer aided diagnosis (CAD) will help to identify the abnormalities. The role of the radiologist will include assessing an abnormality in great detail to appreciate subtle changes to diagnose accurately and assess treatment response. Features of AR/VR including depth perception, head tracking, improved GUIs create an overall immersive environment allowing for new opportunities in diagnostic radiology. Radiologists will interact with medical images in ways never before including voice commands, gestures or through handheld devices with haptic feedback.

## 4. Conclusions

Both diagnostic radiology and AR/VR are large and rapidly growing fields. In this chapter, we reviewed diagnostic medical imaging equipment, data storage, conventional slice-by-slice analysis and advanced 3D rendering techniques including surface rendering and volume rendering. We then introduced D3D processing of images, so volumetric medical imaging can be displayed in AR/VR HDUs. We showed a variety of GUIs including controllers and joysticks with a variety of functions achieved by various functions. Imaging cases were illustrated with a specific focus on breast cancer. We concluded with a discussion on future techniques in comparing how a tumor changes appearance over multiple time points.

## Conflict of interest

All authors either have direct financial interest or are employees of D3D Technologies or DXC Technologies.

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## More

© 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## How to cite and reference

### Cite this chapter Copy to clipboard

David B. Douglas, Demetri Venets, Cliff Wilke, David Gibson, Lance Liotta, Emanuel Petricoin, Buddy Beck and Robert Douglas (March 8th 2018). Augmented Reality and Virtual Reality: Initial Successes in Diagnostic Radiology, State of the Art Virtual Reality and Augmented Reality Knowhow, Nawaz Mohamudally, IntechOpen, DOI: 10.5772/intechopen.74317. Available from:

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Working paper
Chaudru de Raynal P., Menozzi S. math. arxive. Cornell University, 2019
We establish well-posedness results for multidimensional non degenerate α-stable driven SDEs with time inhomogeneous singular drifts in Lr−B−1+γp,q with γ<1 and α in (1,2], where Lr and B−1+γp,q stand for Lebesgue and Besov spaces respectively. Precisely, we first prove the well-posedness of the corresponding martingale problem and then give a precise meaning to the dynamics of the SDE. Our results rely on the smoothing properties of the underlying PDE, which is investigated by combining a perturbative approach with duality results between Besov spaces.
Working paper
Gladkov N., Kolesnikov A., Zimin A. math. arxive. Cornell University, 2018
The multistochastic (n,k)-Monge--Kantorovich problem on a product space ∏ni=1Xi is an extension of the classical Monge--Kantorovich problem. This problem is considered on the space of measures with fixed projections onto Xi1×…×Xik for all k-tuples {i1,…,ik}⊂{1,…,n} for a given 1≤k<n. In our paper we study well-posedness of the primal and the corresponding dual problem. Our central result describes a solution π to the following important model case: n=3,k=2,Xi=[0,1], the cost function c(x,y,z)=xyz, and the corresponding two--dimensional projections are Lebesgue measures on [0,1]2. We prove, in particular, that the mapping (x,y)→x⊕y, where ⊕ is the bitwise addition (xor- or Nim-addition) on [0,1]≅Z∞2, is the corresponding optimal transportation. In particular, the support of π is the Sierpiński tetrahedron. In addition, we describe a solution to the corresponding dual problem.
Working paper
Aleksei Golota. math. arxive. Cornell University, 2017
A celebrated conjecture of Kobayashi and Lang says that the canonical line bundle K_X of a Kobayashi hyperbolic compact complex manifold X is ample. In this note we prove that K_X is ample if X is projective and satisfies a stronger condition of nondegenerate negative total k-jet curvature. We use positivity of direct image sheaves and decomposition of jets in order to produce pluridifferentials on X.
Working paper
Angella D., Tomassini A., Verbitsky M. math. arxive. Cornell University, 2016
We study cohomological properties of complex manifolds. In particular, we provide an upper bound for the Bott-Chern cohomology in terms of Betti numbers for compact complex surfaces, according to the dichotomy b1 even or odd. In higher dimension, a similar result is obtained at degree 1 under additional metric conditions (see Theorem 2.4).
Working paper
Loginov K. math. arxive. Cornell University, 2018
We consider threefold del Pezzo fibrations over a curve germ whose central fiber is non-rational. Under the additional assumption that the singularities of the total space are at worst ordinary double points, we apply a suitable base change and show that there is a 1-to-1 correpspondence between such fibrations and certain non-singular del Pezzo fibrations equipped with a cyclic group action.
Working paper
Sergey Slavnov. math. arxive. Cornell University, 2016
We study the question when a $*$-autonomous Mix-category has a representation as a $*$-autonomous Mix-subcategory of a compact one. We define certain partial trace-like  operation on morphisms of a Mix-category, which we call a mixed trace, and show that any structure preserving embedding of a Mix-category into a compact one induces a mixed trace on the former. We also show that, conversely, if a Mix-category ${\bf K}$ has a mixed trace, then we can construct a compact category and structure preserving embedding of ${\bf K}$ into it, which induces the same mixed trace.   Finally, we find a specific condition expressed in terms of interaction of Mix- and coevaluation maps on a Mix-category ${\bf K}$, which is necessary and sufficient for a structure preserving embedding of ${\bf K}$ into a compact one to exist. When this condition is satisfied, we construct a free'' or minimal'' mixed trace on ${\bf K}$ directly from the Mix-category structure, which gives us also a free'' compactification of ${\bf K}$.   K.
Working paper
Zlotnik A., Čiegis R. math. arxive. Cornell University, 2021. No. ArXiv: 2101.10575v2[math.NA].
We consider an initial-boundary value problem for the $n$-dimensional wave equation with the variable sound speed, $n\geq 1$. We construct three-level implicit in time compact in space (three-point in each space direction) 4th order finite-difference schemes on the uniform rectangular meshes including their one-parameter (for $n=2$) and three-parameter (for $n=3$) families. They are closely connected to some methods and schemes constructed recently by several authors. In a unified manner, we prove the conditional stability of schemes in the strong and weak energy norms together with the 4th order error estimate under natural conditions on the time step. We also give an example of extending a compact scheme for non-uniform in space and time rectangular meshes. We suggest simple effective iterative methods based on FFT to implement the schemes whose convergence rate, under the stability condition, is fast and independent on both the meshes and variable sound speed. A new effective initial guess to start iterations is given too. We also present promising results of numerical experiments.
Working paper
Manita L., Ronzhnina M. math. arxive. Cornell University, 2019. No. 1909.04708.
In this paper we study an optimal control problem that is affine in two-dimensional bounded control. The problem is related to the stabilization of an inverted spherical pendulum in the vicinity of the upper unstable equilibrium. We find solutions stabilizing the pendulum in a finite time, wherein the corresponding optimal controls perform an infinite number of rotations along the circle S1.
Working paper
Shchur L. math. arxive. Cornell University, 2018. No. 1808.09251.
We review recent advances in the analysis of the Wang--Landau algorithm, which is designed for the direct Monte Carlo estimation of the density of states (DOS). In the case of a discrete energy spectrum, we present an approach based on introducing the transition matrix in the energy space (TMES). The TMES fully describes a random walk in the energy space biased with the Wang-Landau probability. Properties of the TMES can explain some features of the Wang-Landau algorithm, for example, the flatness of the histogram. We show that the Wang--Landau probability with the true DOS generates a Markov process in the energy space and the inverse spectral gap of the TMES can estimate the mixing time of this Markov process. We argue that an efficient implementation of the Wang-Landau algorithm consists of two simulation stages: the original Wang-Landau procedure for the first stage and a 1/t modification for the second stage. The mixing time determines the characteristic time for convergence to the true DOS in the second simulation stage. The parameter of the convergence of the estimated DOS to the true DOS is the difference of the largest TMES eigenvalue from unity. The characteristic time of the first stage is the tunneling time, i.e., the time needed for the system to visit all energy levels.
Working paper
Теплицкая Я. И. math. arxive. Cornell University, 2019
We study the properties of sets Σ which are the solutions of the maximal distance minimizer problem, id est of sets having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ R 2 satisfying the inequality maxy∈M dist (y, Σ) ≤ r for a given compact set M ⊂ R 2 and some given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M, where M is the set of customers of the pipeline. In this work it is proved that each maximal distance minimizer is a union of finite number of curves, having one-sided tangent lines at each point. Moreover the angle between these lines at each point of a maximal distance minimizer is greater or equal to 2π/3. It shows that a maximal distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact M, which differs it from a solution of the Steiner problem (an example of a Steiner tree with an infinite number of branching points can be find in [10]). Also we classify the behavior of a minimizer in a neighbourhood of any point of Σ. In fact, all the results are proved for more general class of local minimizer, i. e. sets which are optimal in any neighbourhood of its arbitrary point.
Working paper
Kuznetsov A., Smirnov M. math. arxive. Cornell University, 2018
We define and discuss some general properties of residual categories of Lefschetz decompositions in triangulated categories. In the case of the derived category of coherent sheaves on the Grassmannian G(k,n) we conjecture that the residual category associated with Fonarev's Lefschetz exceptional collection is generated by a completely orthogonal exceptional collection. We prove this conjecture for k=p, a prime number, modulo completeness of Fonarev's collection (and for p=3 we check this completeness).
Working paper
Ostrik V., Etingof P. math. arxive. Cornell University, 2019
We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normnalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group S_n+p in characteristic p, where 0 ≤ n ≤ p−1, and of the Deligne category Rep^ab S_t, where t∈ℕ. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of 𝔰𝔩_2. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of GL(n), SL(n) and PGL(n) in characteristic 2. In the appendix, we classify categorifications of the Grothendieck ring of representations of SO(3) and its truncations.
Working paper
Ostrik V., Etingof P. math. arxive. Cornell University, 2018
We develop the theory of semisimplifications of tensor categories defined by Barrett and Westbury. In particular, we compute the semisimplification of the category of representations of a finite group in characteristic p in terms of representations of the normnalizer of its Sylow p-subgroup. This allows us to compute the semisimplification of the representation category of the symmetric group Sn+p in characteristic p, where 0≤n≤p−1, and of the Deligne category Rep^ab S_t, where t∈ℕ. We also compute the semisimplification of the category of representations of the Kac-De Concini quantum group of the Borel subalgebra of 𝔰𝔩2. We also study tensor functors between Verlinde categories of semisimple algebraic groups arising from the semisimplification construction, and objects of finite type in categories of modular representations of finite groups (i.e., objects generating a fusion category in the semisimplification). Finally, we determine the semisimplifications of the tilting categories of GL(n), SL(n) and PGL(n) in characteristic 2. In the appendix, we classify categorifications of the Grothendieck ring of representations of SO(3) and its truncations.
Working paper
Loginov K. math. arxive. Cornell University, 2019
Consider a family of Fano varieties π:X⟶B∋o over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber F=π^{−1}(o) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension ≤dim F. Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension 3 we reproduce the classification of the semistable degenerations of del Pezzo surfaces obtained by Fujita. We also show that the maximal degeneration is unique and has trivial monodromy in dimension ≤3.
Working paper
Netay I. V., Glutsyuk A. math. arxive. Cornell University, 2019
The paper deals with a three-parameter family of special dou- ble confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are l, λ, μ ∈ R. Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a poly- nomial solution. They have shown that for μ ≠ 0 this happens ex- actly when l ∈ N and the parameters (λ, μ) lie on an algebraic curve Γl ⊂ C2(λ,μ) called the l-spectral curve and defined as zero locus of de- terminant of a remarkable three-diagonal l × l-matrix. They studied the real part of the spectral curve and obtained important results with applications to model of Josephson junction, which is a family of dy- namical systems on 2-torus depending on real parameters (B,A;ω); the parameter ω, called the frequency, is fixed. One of main problems on the above-mentioned model is to study the geometry of boundaries of its phase-lock areas in R2(B,A) and their evolution, as ω decreases to 0. An approach to this problem suggested in the present paper is to study the complexified boundaries. We prove irreducibility of the complex spectral curve Γl for every l ∈ N. We also calculate its genus for l ⩽ 20 and present a conjecture on general genus formula. We apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. The family of real boundaries taken for all ω > 0 yields a countable union of two-dimensional analytic surfaces in R3 . We show that, unexpectedly, its complexification is(B,A,ω−1) a complex analytic subset consisting of just four two-dimensional irreducible components, and we describe them. This is done by using the representation of some special points of the boundaries (the so-called generalized simple intersections) as points of the real spectral curves and the above irreducibility result. We also prove that the spectral curve has no real ovals. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as ω decreases, and a partial positive result towards its confirmation.
Working paper
Bogomolov F. A., Pirutka A., Böhning C. math. arxive. Cornell University, 2018
We study when kernels of inflation maps associated to extraspecial p-groups in stable group cohomology are generated by their degree two components. This turns out to be true if the prime is large enough compared to the rank of the elementary abelian quotient, but false in general.
Working paper
Anton Fonarev. math. arxive. Cornell University, 2018
We construct a mininal Lefschetz decomposition of the bounded derived category of coherent sheaves on the isotropic Grassmannian $\IGr(3,7)$. Moreover, we show that $\IGr(3, 7)$ admits a full exceptional collection consisting of equivariant vector bundles.
Working paper
Amerik E., Guseva L. math. arxive. Cornell University, 2016
Let X be an irreducible holomorphic symplectic fourfold and D a smooth hypersurface in X. It follows from a result by Amerik and Campana that the characteristic foliation (that is the foliation given by the kernel of the restriction of the symplectic form to D) is not algebraic unless D is uniruled. Suppose now that the Zariski closure of its general leaf is a surface. We prove that X has a lagrangian fibration and D is the inverse image of a curve on its base.


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# Tag Archives: Math

## Any questions about this video?

As part of preparing for our impending move from Indy to Grand Rapids, my family and I have made a couple of visits to the area. These by necessity combine business with pleasure, since our three kids (ages 2, 5, and 7) don’t handle extended amounts of business well. On the last visit, we spent some time at the Grand Rapids Childrens Museum, the second floor of which is full of stuff that could occupy children — and mathematicians — for hours. This “exhibit” was, for me, one of the most evocative. Have a look:

I asked this on Twitter a few days ago, but I’ll repost it here: In the spirit of Dan Meyer’s Any Questions? meme, what questions come to mind as you watch this? Particularly math, physics, etc. questions.

One other thing — just after I wrapped up the video on this, someone put one of the little discs rolling on the turntable and it did about a dozen graceful, perfect three-point hypocycloids before falling off the table.

3 Comments

Filed under Geometry, Math, Problem Solving

## The “golden moment”

We’re in final exams week right now, and last night students in the MATLAB course took their exam. It included some essay questions asking for their favorite elements of the course and things that might be improved in the course. I loved what one of my students had to say about the assignment in the course he found to be the most interesting, so I’ve gotten permission from him to share it. The lab problem he’s referring to was to write a MATLAB program to implement the bisection method for polynomials.

It is really hard to decide which project I found most interesting; there are quite a few of them. If I had to choose just one though, I would probably have to say the lab set for April 6. I was having a really hard time getting the program to work, I spent a while tweaking it this way and that way. But when you’re making a program that does not work yet, there is this sort of golden moment, a moment when you realize what the missing piece is. I remember that moment on my April 6 lab set. After I realized what it was, I could not type it in fast enough I was so excited just to watch the program work. After hitting the play button, that .3 seconds it takes for MATLAB to process the program felt like forever. I actually was devastated that I got an error, and thought I had done it all wrong once again, but then I remembered I had entered the error command so it would display an error. I actually started laughing out loud in the lab, quite obnoxiously actually.

Yes!  As somebody once said, true learning consists in the debugging process. And that’s where the fun in learning happens to lie, too. Let’s give students as many shots as possible to experience this process themselves.

2 Comments

## Understanding “understanding”

This past Saturday, I was grading a batch of tests that weren’t looking so great at the time, and I tweeted:

I do ask these two questions a lot in my classes, and despite what I tweeted, I will probably continue to do so. Sometimes when I do this, I get questions, and sometimes only silence. When it’s silence, I am often skeptical, but I am willing to let students have their end of the responsibility of seeking help when they need it and handling the consequences if they don’t.

But in many cases, such as with this particular test, the absence of questions leads to unresolved issues with learning, which compound themselves when a new topic is connected to the old one, compounded further when the next topic is reached, and so on. Unresolved questions are like an invasive species entering an ecosystem. Pretty soon, it becomes impossible even to ask or answer questions about the material in any meaningful way because the entire “ecosystem” of a student’s conceptual framework for a subject is infected with unresolved questions.

Asking if students understand something or if they have questions is, I am realizing, a poor way to combat this invasion. It’s not the students’ fault — though persistence in asking questions is a virtue more students could benefit from. The problem is that students, and teachers too, don’t really know what it means to “understand” something. We tend to base it on emotions — “I understand the Chain Rule” comes to mean “I have a feeling of understanding when I look at the Chain Rule” — rather than on objective measures. This explains the common student refrain of “It made sense when you did it in class, but when I tried it I didn’t know where to start“. Of course not! When you see an expert do a calculation, it feels good, but that feeling does not impart any kind of neural pathway towards your being able to do the same thing.

So what I mean by my tweet is that instead of asking “Do you understand?” or “Do you have any questions?” I am going to try in the future to give students something to do that will let me gauge their real understanding of a topic in an objective way. This could be a clicker question that hits at a main concept, or a quick and simple problem asking them to perform a calculation (or both). If a student can do the task correctly, they’re good for now on the material. If not, then they aren’t, and there is a question. Don’t leave it up to students to self-identify, and don’t leave it up to me to read students’ minds. Let the students do something simple, something appropriate for the moment, and see what the data say instead.

This may have the wonderful side effect of teaching some metacognition as well — to train students how to tell when they do or do not know something.

4 Comments

Filed under Education, Teaching

## Technology making a distinction but not a difference?

This article is the second one that I’ve done for Education Debate at Online Schools. It first appeared there on Tuesday this week, and now that it’s fermented a little I’m crossposting it here.

The University of South Florida‘s mathematics department has begun a pilot project to redesign its lower-level mathematics courses, like College Algebra, around a large-scale infusion of technology. This “new way of teaching college math” (to use the article’s language) involves clickers, lecture capture, software-based practice tools, and online homework systems. It’s an ambitious attempt to “teach [students] how to teach themselves”, in the words of professor and project participant Fran Hopf.

It’s a pilot project, so it remains to be seen if this approach makes a difference in improving the pass rates for students in lower-level math courses like College Algebra, which have been at around 60 percent. It’s a good idea. But there’s something unsettling about the description of the algebra class from the article:

Hopf stands in front of an auditorium full of students. Several straggle in 10 to 15 minutes late.

She asks a question involving an equation with x’s, h’s and k’s.

Silence. A few murmurs. After a while, a small voice answers from the back.

“What was that?” Hopf asks. “I think I heard the answer.”

Every now and then, Hopf asks the students to answer with their “clickers,” devices they can use to log responses to multiple-choice questions. A bar graph projected onto a screen at the front of the room shows most students are keeping up, though not all.

[…]

As Hopf walks up and down the aisles, she jots equations on a hand-held digital pad that projects whatever she writes on the screen. It allows her to keep an eye on students and talk to them face-to-face throughout the lesson.

Students start drifting out of the 75-minute class about 15 minutes before it ends. But afterward, Hopf is exuberant that a few students were bold enough to raise their hands and call out answers.

To be fair: This is a very tough audience, and the profs involved have their work cut out for them. The USF faculty are trying with the best of intentions to teach students something that almost assuredly none of them really want to learn, and this is exceedingly hard and often unrewarding work. I used to teach remedial algebra (well short of “college algebra”) at a two-year institution, and I know what this is like. I also know that the technology being employed here can, if used properly, make a real difference.

But if there’s one main criticism to make here, it’s that underneath the technology, what I’m seeing — at least in the snapshot in the article — is a class that is really not that different than that of ten or twenty years ago. Sure, there’s technology present, but all it seems to be doing is supporting the kinds of pedagogy that were already being employed before the technology, and yielded 60% pass rates. The professor is using handheld sketching devices — to write on the board, in a 250-student, 75-minute long lecture. The professor is using clickers to get student responses — but also still casting questions out to the crowd and receiving the de rigeur painful silence following the questions, and the clickers are not being used in support of learner-centered pedagogies like peer instruction. The students have the lectures on video — but they also still have to attend the lectures, and class time is still significantly instructor-centered. (Although apparently there’s no penalty for arriving 15 minutes late and leaving 15 minutes early. That behavior in particular should tell USF something about what really needs to change here.)

What USF seems not to have fully apprehended is that something about their remedial math system is fundamentally broken, and technology is neither the culprit nor the panacea. Moving from an instructor-centered model of learning without technology to an instructor-centered model of learning with technology is not going to solve this problem. USF should instead be using this technology to create disruptive change in how it delivers these courses by refocusing to a student-centered model of learning. There are baby steps here — the inclusion of self-paced lab activities is promising — but having 75-minute lectures (on college algebra, no less) with 225 students signals a reluctance to change that USF’s students cannot afford to keep.

## An M-file to generate easy-to-row-reduce matrices

In my Linear Algebra class we use a lot of MATLAB — including on our timed tests and all throughout our class meetings. I want to stress to students that using professional-grade technological tools is an essential part of learning a subject whose real-life applications closely involve the use of those tools. However, there are a few essential calculations in linear algebra, the understanding of which benefits from doing by hand. One of those calculations is row-reduction. Nobody does this by hand; but doing it by hand is useful for understanding elementary row operations and for getting a feel for the numerical processes that are going on under the hood. And it helps with understanding later concepts, notably that of the LU factorization of a matrix.

I have students take a mastery exam where they have to reduce a 3×5 or 4×6 matrix to reduced echelon form by hand. They are not allowed any technology on that exam. I’ve learned that making up good matrices for this exam is surprisingly tricky. My first attempt at writing the exam resulted in a nice-looking matrix whose reduced echelon form had mind-bendingly big fractions in it. I want the exam to be about row reduction and not fraction arithmetic, so I  sat down this morning and wrote this MATLAB function called easyRR.m which automatically spits out $m \times n$ random integer matrices whose row-reduction process might involve fractions but which aren’t horrendous:

%% Function to create an mxn matrix that is easy to row-reduce by hand.
% Basic idea: Construct this matrix by building an LU factorization for it
% where both L and U have small integer values.
% R. Talbert, Feb 15, 2011

function A = easyRR(m,n)

%% Create the L in the LU factorization. This matrix encodes the elementary
%% row operations needed to get A to echelon form.

% Start with a random integer square matrix:
L = randi([-10, 10], [m,m]);

% Replace diagonal elements with 1's:
for i=1:m
L(i,i) = 1;
end

% Zero out all entries above the diagonal:
L = tril(L);

%% Now create the U in the LU factorization, using smaller integers so that
%% the back substitution phase isn't too bad.

% This creates an mxn random integer matrix and zeros out all entries below
% the diagonal.
U = triu(randi([-5,5], [m,n]));

%% The easy-to-reduce matrix is the product of L and U.
A = L*U;

Here’s a screenshot:

The fractions involved here have denominators no larger than 25, which is way more doable for students than what I had been having them work with (sorry, guys).

And, if you happen to have the Symbolic Toolbox for MATLAB, you can add the line latex(sym(A)) to the end and the function will spit out the $\LaTeX$ code for that matrix, for easy copy/paste into the exam.

Anyway, I thought this was useful and so I’m giving it away!

2 Comments

Filed under LaTeX, Linear algebra, Math, MATLAB, Teaching

## Another thought from Papert

Image via Wikipedia

Like I said yesterday, I’m reading through Seymour Papert’s Mindstorms: Children, Computers, and Powerful Ideas right now. It is full of potent ideas about education that are reverberating in my brain as I read it. Here’s another quote from the chapter titled “Mathophobia: The Fear of Learning”:

Our children grow up in a culture permeated with the idea that there are “smart people” and “dumb people.” The social construction of the individual is as a bundle of aptitudes. There are people who are “good at math” and people who “can’t do math.” Everything is set up for children to attribute their first unsuccessful or unpleasant learning experiences to their own disabilities. As a result, children perceive failure as relegating them either to the group of “dumb people” or, more often, to a group of people “dumb at x” (where, as we have pointed, x often equals mathematics). Within this framework children will define themselves in terms of their limitations, and this definition will be consolidated and reinforced throughout their lives. Only rarely does some exceptional event lead people to reorganize their intellectual self-image in such a way as to open up new perspectives on what is learnable.

Haven’t all of us who teach seen this among the people in our classes? The culture in which our students grow up unnaturally, and incorrectly, breaks people into “good at math” or “bad at math”, and students who don’t have consistent, lifelong success will put themselves in the second camp, never to break out unless some “exceptional event” takes place. Surely each person has real limitations — I, for example, will never be on the roster of an NFL team, no matter how much I believe in myself — but when you see what students are capable of doing when put into a rich intellectual environment that provides them with challenges and support to meet them, you can’t help but wonder how many of those “limitations” are self-inflicted and therefore illusory.

It seems to me that we teachers are in the business of crafting and delivering “exceptional events” in Papert’s sense.

## Bound for New Orleans

Happy New Year, everyone. The blogging was light due to a nice holiday break with the family. Now we’re all back home… and I’m taking off again. This time, I’m headed to the Joint Mathematics Meetings in New Orleans from January 5 through January 8. I tend to do more with my Twitter account during conferences than I do with the blog, but hopefully I can give you some reporting along with some of the processing I usually do following good conference talks (and even some of the bad ones).

I’m giving two talks while in New Orleans:

• On Thursday at 3:55, I’m speaking on “A Brief Fly-Through of Cryptology for First-Semester Students using Active Learning and Common Technology” in the MAA Session on Cryptology for Undergraduates. That’s in the Great Ballroom E, 5th Floor Sheraton in case you’re there and want to come. This talk is about a 5-day minicourse I do as a guest lecturer in our Introduction to the Mathematical Sciences activity course for freshmen.
• On Friday at 11:20, I’m giving a talk called “Inverting the Linear Algebra Classroom” in the MAA Session on Innovative and Effective Ways to Teach Linear Algebra. Thats in Rhythms I, 2nd floor Sheraton. This talk is an outgrowth of this blog post I did back in the spring following the first non-MATLAB attempt at the inverted classroom approach I did and will touch on the inverted classroom model in general and how it can play out in Linear Algebra in particular.

Both sessions I’m speaking in are loaded with what look to be excellent talks, so I’m excited about participating. I’d be remiss if I didn’t mention that Gil Strang and David Lay are two of the organizers of the linear algebra setting, which is like a council of the linear algebra gods.

I’ll give Casting Out Nines readers a sneak peek at my two talks by telling you I’ve set up a web site that has the Prezis for both talks along with links to the materials I mention in the talks. And if you’re there in New Orleans, come by my talks if you have the slots free or just give me a ring on my Twitter and I’d love to meet up with you.

Comments Off on Bound for New Orleans


text:
Arctan derivatives problem

1. Feb 11, 2006

teng125

Find the derivatives of the following arctan {[(1+x^2)^1/2 ] - 1} / x with respect to arctan x.may i know how to do this??
how to start and the steps??

2. Feb 11, 2006

arildno

Let your independent variable be given as $u=arctan(x)$ that is,
$x(u)=tan(u)$

You have been given the function of x,
$$F(x)=\frac{arctan(\sqrt{1+x^{2}}-1)}{x}$$
Let $h(u)=F(x(u))[/tex] You are asked to find [itex]\frac{dh}{du}$

Note that it is easy to give your final answer in terms of "x" rather than "u", since we have $$\frac{dx}{du}=\frac{1}{\cos^{2}u}=1+\tan^{2}u=1+x^{2}$$

Last edited: Feb 11, 2006
3. Feb 11, 2006

HallsofIvy

Staff Emeritus
You have $\frac{arctan(u)}{x}$. Since that is a quotient of two functions use the "quotient" rule:
$$\frac{d\left(\frac{f(u)}{x}}= \frac{\frac{df}{du}\frac{du}{dx}x- f(u)}{x}$$
where $u= \sqrt{1+x^2}-1$ and f(u)= arctan u.
The $\frac{df}{du}\frac{du}{dx} in that is the "chain" rule. 4. Feb 11, 2006 teng125 sotty,for threat 2 i don't really understand.can u plsexplain more clearly?? 5. Feb 11, 2006 arildno HallsofIvy's "u" is not the same as my "u". He is presenting a technique to evaluate the term [itex]\frac{dF}{dx}$ in my terms.

We have: $\frac{d}{dx}(\sqrt{1+x^{2}}-1)=\frac{x}{\sqrt{1+x^{2}}}$
Thus, we get:
$$\frac{dF}{dx}=\frac{\sqrt{1+x^{2}}}{1+(\sqrt{1+x^{2}}-1)^{2}}-\frac{F(x)}{x}$$
Multiply this with $1+x^{2}$ to get $\frac{dh}{du}$

6. Feb 11, 2006

teng125

can u pls show me the whole steps ??thanx

7. Feb 11, 2006

arildno

Darn, I made a mistake!
the correct expression for the derivative with respect to u=arctan(x), should be:
$$\frac{dh}{du}=\frac{\sqrt{1+x^{2}}}{1+(\sqrt{1+x^{2}}-1)^{2}}-F(x)\frac{1+x^{2}}{x}$$

8. Feb 11, 2006

Calin

$\\int_1^4 \\sqrt{t}(ln(t))dt$

9. Feb 11, 2006

VietDao29

Unfortunately, I don't think we are going to provide you a step by step solution. However, we may help you.
Just tell us where in the post that you don't really understand and we may clarify it for you.

10. Feb 12, 2006

arildno

It might be helpful to remember the following rule:

Suppose you are to differentiate a function F(x) with respect to a (invertible) function f(x).
Then we have, in general:
$$\frac{dF}{df}=\frac{\frac{dF}{dx}}{\frac{df}{dx}}$$


text:
$c=\lambda v$

dstemp
Posts: 20
Joined: Fri Sep 29, 2017 7:07 am

I was attempting to fill out the table, but I was unsure of what type of radiation would be involved with reading (x-ray, suntan, and microwave are all a little more straightforward). Would reading simply be in the "visible light" category?

Minie 1G
Posts: 63
Joined: Fri Sep 29, 2017 7:04 am

Yes, I think you're right. I mean, would you want to (or could you) read under either UV or infrared light rays? They aren't even visible to the human eye.

Chem_Mod
Posts: 17860
Joined: Thu Aug 04, 2011 1:53 pm
Has upvoted: 406 times

The visible spectrum is called as such because the only light that we can see with our eyes falls within this very narrow band of wavelengths (700nm-400nm). Any other light, such as UV and microwave and x-rays can't be seen by the human eye. So if you were reading something with your eyes that would naturally involve light only in the visible band.

dstemp
Posts: 20
Joined: Fri Sep 29, 2017 7:07 am