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# That’s *Mister* IRIG to you, sir…

I’m a bit of a time-nut…  well, not quite as adamant as some of those on that list  ( Allen deviations?   Seriously?   Who would talk about Allen that way?!? ), but I am crazy about keeping time, accurate time (well to within the millisecond or so, anyway), and distributing time.

# IRIG-B Decoder Box

Since my first encounter with IRIG-B in, oh, was that 1988 (?), doing an IRIG-B decoder box at Vansco, on contract for Manitoba Hydro, I’ve been fascinated by the IRIG-B time code.  Here’s the official specification, and an unofficial site by someone trying to sell you something.  Truth be told, the original MC68xx (might have been MC68701 because it had both accumulators and more than one TCAP pin) IRIG-B decoder firmware was written by Filipe Fernandes…   then I took it from there, as Filipe got gobbled up by all the other stuff going on at Vansco 🙂  That decoder box took in modulated IRIG-B only, put out a contact for when it was valid or invalid, and had an RS-232 interface for another device to tell it the IRIG-B time.  What fun!

# Extensions

Then came the IEEE 1344 extensions, which defined a use for the extra IRIG-B undefined bits, giving more information that originally intended.  The original IRIG-B decoder box did not support these extensions, but our next product did.

# APT Power Technologies had IRIG-B

Anyways, while tangled up in all the other matters during development of the APT relay, like running the APT division, hiring staff, arranging trade shows, and designing the entire data acquisition system, I designed the hardware and ported & upgraded the firmware for the on-board MC68711 IRIG-B decoder.  Oh yes, and designed the time coordination system between the multiple processors in the system, which one had the master time and when, etc.  Very interesting.  During that time, we added unmodulated IRIG-B input as well.  The unit cycles between searching for valid modulated IRIG-B and unmodulated IRIG-B.

# Getting IRIG-B for Testing

The only way that we had to generate IRIG-B was with an Arbiter 1084 satellite clock.  It would put out a modulated IRIG-B stream, and an unmodulated IRIG-B stream (TTL level), and a 1 pulse-per-second (generally referred to as 1 PPS or just “PPS”) (also TTL level).  There was also a serial interface – reminiscent of what we did in the original IRIG-B decoder box – but we rarely used it.  It could change settings on the IRIG-B interface signal, but we generally had models with the front panel buttons and display, so we changed parameters using that.

Satellite clocks were expensive back then.  Well, industrial grade ones still are – but we needed IRIG-B signals for testing – in product development, but also in manufacturing test.  So, we had one central satellite clock, and distributed IRIG-B through the building using CAT5 twisted pair, with specially marked RJ45 drops at developer’s desks, in production, in customer support, etc.

Experimenting with various formats was a pain.  We had to get access to the satellite clock (it was in the locked server room), change the settings, then get other users’ acceptance of the change…  oops, did I get that out of sequence?  Ha ha, I did get that out of sequence several times.  It turned out that most locations only had a single drop, so if I changed from unmodulated to modulated when users weren’t expecting it, ugh it caused us some heartache.  To the point where production got two separate direct wire drops.  Then, we had loading issues, so production got its own clock.  It didn’t have to be accurate, just functional, so I’m not sure that it was even always satellite locked.

## Rudimentary Test Recordings

Sometimes, we recorded IRIG-B signals and just played them back.  After all, modulated IRIG-B was developed in the 50s (or was it 60s?) and was intended to be recorded on multi-track audio recorders and strip chart recorders for the telemetry of various military tests (think nuclear weapons testing).  I experimented with cassette tape recorderings in the early days, then with MP3 players like my little Samsung YP-U3 that we used on our trip to England & Scotland in 2005.  It had mixed results.  Modulated worked quite well, but unmodulated was unreliable – since it’s not really audio, and requires DC offset to be maintained, or restored, or something.

## Enter tg

Around 2005, I found out that the NTP project had a program called tg in its “utils” subdirectory.   tg stands for “Timecode Generator”, and it was designed to put out modulated simulated IRIG-B or WWV through the audio output port of a Sun workstation computer (I suspect that this was the original target for the NTP executable).  I could be wrong, maybe it was a different *NIX system, but anyway, it didn’t work on the LINUX that I was running at the time, probably Mandrake LINUX.  So, I hacked it to use OSS to make it run on my computer.  What fun!

tg isn’t good enough for really accurate time sync!  It’s just good enough to test the decoder, test edge cases, etc.

### Deano went a-hacking

Of course, I couldn’t leave well enough alone, so I Weiten-ified it, as Michael Miller used to say.  I added a slew of options and tweaks.  I added optional unmodulated IRIG-B output, tweaked its time code generation, added IEEE 1344 extension support, added the ability to start at an arbitrary specified time & date, added the ability to insert and remove leap seconds on demand, and much more.  I found that the audio card output sample rate wasn’t quite good enough over long term testing to keep accurate time – the output sample rate of 8,000 samples per second and the sinewaves were precisely, you know, 8 samples or 80 samples (IRIG or WWV) per cycle, so if the output sample rate were off by 0.1%, the frequency was off by 0.1%, and this would accumulate to eventually cause time drift.

So, I arranged to insert or remove a single cycle (1 mSec or 10 mSec) at what I decided was an innocuous part of the time code – at least for my decoders, heh heh.  The program compared the time to the LINUX real-time clock (which often is sync’d to global standard using NTP, how cool is that), and if the amount became detectable, it would slip in an extra cycle, or remove an extra cycle per second, to get back in sync.

I also added options to create file output instead of audio, so could be played back later (through audio port, as before, or through an Arbitrary Waveform Generator).

### The Fork Less Travelled

So anyways, when I was done, I was quite proud of my work, so I sent it back to the NTP project – specifically to Dr. David Mills, chief NTP guy , and author of the book on network time sync – and I’m sure he was absolutely horrified at what I had done to the program.  In retrospect, I didn’t read nor respect the code formatting guidelines, I hacked and slashed the original audio driver code, yikes.  Dr. Mills was very polite, but wouldn’t accept my changes to tg on top of tg itself, but instead called it tg2.  If you download NTP today, you will see util/tg2 in there, waiting to be hacked some more 🙂

Oh, and I have hacked it some more.    Source code in tarball with RCS tracking available online.  I am trying to be more respectful of code formatting guidelines, and maintaining backward compatibility these days.

### tg2‘s Greatest Hits?

Around 2006, I re-mastered a Knoppix CD with tg2 built in, so I could demo all its features on a live CD.  At the time, Jason Fuith and I were operating our own business called Elecsys Solutions (and starving, at least financially, while doing it), and we wanted to maybe sell it to folks like Krish Narendra, who was now in charge of Product Development at ERLPhase, and still faced all the problems of satellite clocks, IRIG-B distribution etc.

The idea that we had wasn’t necessarily to sell the code, but to sell support of it – maybe install it on custom embedded hardware, or put in specific customization, etc.

### tg2 + Knoppix = NAN

Although interesting, it didn’t sweep Krish off his feet, and even at a modest cost, he could not justify the expenditure.  Well, I gave him the CD anyway, and I don’t think it went any further.

After Krish left ERLPhase, I worked out of his office for a while in 2018, and I stumbled across that CD, heh heh.  I gave it to Mark Poole, as he is now ERLPhase’s IRIG-B guy – and he was quite interested.  Mark and I had a long chat, and my next “hobby” project was born!  More on that later.

# Meanwhile, Back doing Real Work

I tinkered with tg2 from time to time, per above, but didn’t do much with it.  In the meantime, Elecsys was purchased by Norscan Instruments and I went to work there as Product Development Manager for 3-1/2 years.  They didn’t have any specific interest in IRIG-B, NTP, or tg2 – they needed to make some money to pay our wages, imagine that!

Then back to ERLPhase for a bit, a great year at GE Multilin in Markham, back in Winnipeg for a short time, and off to Phoenix to work for Alstom Grid Digital Instrument Transformers, or Alstom Grid DIT (now GE Grid Solutions DIT).

## Optical IRIG

Now, at DIT, they used satellite clocks all right – Arbiter 1084s, in fact, among others (like Reason, now GE, ones).  There was one important difference though – DIT used the now-standard orange 62.5/125 multimode fibre with 820 nm pulsed light (using the Avago/Broadcom HFPL1414TZ transmitter).  On the Arbiter 1084, this option is called “option 20”, and the single ST output can be configured to output IRIG-B unmodulated (of course 🙂 ), IRIG-B modified manchester coding, or 1 PPS.  DIT used the 1 PPS, because its primary use was to synchronize sampling / output of its product – what’s now known in the industry as a “primary converter merging unit” – so that it can be synchronized in time to other merging units in the system.

We did switch our satellite clocks back and forth between 1 PPS and IRIG-B unmodulated, when we were synchronizing with Reason (now GE) merging units.  I was intrigued.

# Dean Scores!  ( an Arbiter 1084C satellite clock )

I always wanted a satellite clock, but, as I said before, they are expensive.  I started trolling for one in late 2015.  They are still expensive – generally selling for US700 used, and up to US2100 if new in box.  I just happened to find one selling at BMI Surplus for something like US250, talked them down to US236 inc shipping and tax, and it was mine, yay!  As a bonus, it was the Arbiter 1084C model, which has the big seven segment LED display on the front, way cool 🙂

It didn’t come with option 20, so I (carefully!) disassembled one of DIT’s 1084 clocks, made a parts list and took pictures, ordered what I needed, and installed them in my clock.  Now I have option 20, heh heh heh.

This came in handy for testing of the Reason merging unit with RogoFlex at PowerTech in Vancouver in August 2017, when I took my own 1084 clock to sync the systems together.

## What to Do with a Satellite Clock?

My satellite clock didn’t come with a GPS antenna, so I got one through Amazon, of course, put that up with the many other antennas above my house, and got the clock working.

I had a lot of fun with the Arbiter 1084C and my oscilloscope – marvelling at the time code, watching the different signals, cross referencing the different formats (yes, I am that weird).

The clock was affected by a rollover bug.  I ordered new EPROMs for US$60 from Arbiter. Easy peasy, replaced those, just like in my early days. ## GPS Module Troubles At one point, the clock stopped working properly. Sad day! The GPS module had died. It seems that it’s writing something to the flash on the GPS module constantly, and it just wore out. The module, an old Motorola OnCore unit, is of course obsolete. There are non-drop-in replacements, but they aren’t binary compatible. Arbiter will upgrade your unit – as I recall, it’s about US$300 – more than I paid for the satellite clock in the first place!  Now that was a very sad day.

I bought a SMT soldering/desoldering station, flash programmer and some spare flash parts, pulled the old flash off the board and rolled it out to disk… programmed the new part, but could not get it installed back on the board properly.  Seems as though I damaged the board removing the parts, so sad.

Fortunately, I found a (less) used GPS module of the exact same model, at some place in the far east.  I received it, plugged it in, and was back in business.

Oh, and the backup battery had of course died – being nickel-cadmium and probably 25 years old!  But, it had also corroded its connections badly.  I repaired the board and replaced the battery, and all is good again.   The battery is actually optional, probably should have just left it off.

## Now What to do with a Satellite Clock?

The satellite clock sat running on my shelf for a year or so, before I got to thinking that I should use the IRIG-B signal output.  At first, I wanted to time sync a local computer and create my own (effectively) stratum 1 NTP time reference.  I don’t recall, I might have even had it working briefly, but what do I need a stratum 1 NTP time reference for?  That’s kind of a dead-end project, at least for me.

## If You Can’t Generate ‘Em, Decode ‘Em?

Back to that conversation with Mark Poole, he mused that it would be nice to have a tool that could display the details of IRIG-B signals, so Customer Service, Applications and Product Development personnel, could trouble shoot issues, including various clocks that apparently give wrong or conflicting information, get the parity wrong, use reversed time zone-to-UTC offset etc.

So I started playing with NTP’s refclock_irig, not to sync my computer’s clock (in fact, I neutered it, so it would not), but to fully decode and display the signal.  Along the way, I added support for unmodulated IRIG-B as well – that was a challenge, since an audio input port is used to read the signal, and it does not maintain DC levels.  In fact, I did a lot of work on DC restoration à la NTSC TV signal recovery, back in the day.  That was fraught with peril, eventually causing a numerical overflow or a drifting DC baseline, so I eventually settled on a much simpler “look for fast shifts” solution, and that worked well.  I made the printout of the time code optional, using “fudge” tweak configuration bits already in the code, to dump output to the existing log, then tail -f in a separate console to display them.  I also generated some (huge) CSVs from time to time, both raw analog data (do you see what I see?) and various decode internals (do you decode what I decode?).  A lot of work to keep putting debug in, take it out, etc.  There are only 4 fudge tweak bits, and 3 of them are already in use… I pulled one back, so I could use 2, but still, too much hassle to tweak it inside the NTP framework.

## Stand-Alone Decoder Program

Eventually, I undertook an effort to break my refclock_irig code away from NTP, creating read_irig.  This program stands alone, and supports too many options:

Read and decode modulated/unmodulated IRIG-B/modulated IRIG-E from audio signal, v0.43, 2019-05-08 dmw

RCS Info:
Header: /home/dmw/src/ntp/refclock_irig/RCS/read_irig.c,v 1.57 2019/05/08 06:07:07 dmw Exp

Options: -b                             Disable automatic input gain control
-c <csv_output_filename>       Put out CSV file of bunch of internal values for portion of time
-e <lines_between_headers>     Repeat full header every this many lines (default 20)
-f <format_type>               Force only format:
0 = Any (default)
1 = Normal Modulated
2 = Normal Unmodulated
3 = Inverted Unmodulated
4 = Normal or Inverted Unmodulated
-g <initial_gain>              Initial gain setting (default 50), from 10 to 100
-i <input_file>                Input file instead of audio input
-j <csv_output_delay_samples>  Delay in samples for "-c" CSV output after startup (default 8000)
-m                             Enable IRIG-B modulated bandpass filter
-n                             Non-inverting input stream (normally input is inverted)
-o <raw_signal_csv_out>        Put out CSV file of just raw input for as long as program runs
-r <bitmap_of_outputs>         Bitmap of outputs:
bit 0 = decoded data
bit 1 = raw data
bit 2 = do not repeat full header while in low verbosity
bit 3 = 8 lines x 16 chars/line flat panel display output (disables all others)
(default value shows raw and decoded data, and repeats full header periodically)
-s <seconds_to_run>            Run this many seconds then exit (default forever), must be greater than 7
-t <timeout_seconds>           Change timeout on successful decode, must be greater than 4.0 sec
-v                             Increase verbosity of output
-x                             Use 2nd stream (right? left?) instead of 1st
-y                             Invert raw_signal_csv_out from "-o" option
-z <threshold_amount>          Unmodulated edge detect threshold amount (default +/-0.20) in +/- amount of max

Notes: 1. Starting with "  Normal Unmodulated" format, default time out after about 64,000 samples
(about 8.0 sec) without successful decode, then will cycle through "  Normal   Modulated",
"  Normal Unmodulated", and "Inverted Unmodulated", with about 8.0 sec timeout on each format.
2. Setting timeout on successful decode to less than about 8 sec may produce intermittent lock or
failure to lock on modulated IRIG carrier.
3. It can take a few seconds to lock onto a signal, so output may not be <seconds_to_run> frames.
4. Modulated bandpass filter for IRIG-B tends to "shmoosh" the input signal and make it difficult
to decode, and most IRIG-B signals are very clean, so it's not generally necessary.
5. Default outputs include both raw and decoded data.  If both are turned off using "-r" option,
decoded data is turned on alone.

This software licenced under the GPL, derived from refclock_irig, changes made 2018, 2019 by Dean Weiten
Contact: Dean Weiten, Winnipeg, MB, Canada, ph (204)-888-1334, E-mail dmw@weiten.com

The output, I think, is way cool:

#
#--------------------------------------------------------------------------------------------------||------------------------------------------------------------|
#         Inverted Unmodulated IRIG-B Raw (inverted audio) (Gain 100 giving -12.9 dB)              || Inverted Unmodulated IRIG-B (inv   , gain 100 -> -12.9 dB) |
# StrtBinSecs (SBS) |    Control Bits   |   Year  |    Day of Year    |  Hours  | Minutes |Seconds ||Day|   Date and Time    |SBS hex|Leap| DST |Offset|Qual|Par |
# ........ .........|......... .........|.... ....|       .. .... ....|  .. ....| ... ....|........||...|.... .. .. .. .. .. | . ....|....|.....|......|  ..|....|
#                   |                   |         |                   |         |         |        ||   |                    |       |    |     |      |    |    |
.001010000.000011001.000100000.010111000.000101001.000000001.001100000.000100001.001000011.00000101. 130 2019-05-10 11:23:05   0_A019 !lsp  DST  UTC-05   00 1 ok
.001010000.000011010.000100000.010111000.000101001.000000001.001100000.000100001.001000011.00000110. 130 2019-05-10 11:23:06   0_A01A !lsp  DST  UTC-05   00 1 ok
.001010000.000011011.000000000.010111000.000101001.000000001.001100000.000100001.001000011.00000111. 130 2019-05-10 11:23:07   0_A01B !lsp  DST  UTC-05   00 0 ok
.001010000.000011100.000000000.010111000.000101001.000000001.001100000.000100001.001000011.00001000. 130 2019-05-10 11:23:08   0_A01C !lsp  DST  UTC-05   00 0 ok
...

The raw and inverted decode sections can be enabled independently, and the program can auto-detect (slowly, mind you) modulated IRIG, unmodulated IRIG, and inverted unmodulated IRIG.

I’m still a sad case… I can sit and watch the codes go by, constantly.  It turns in a separate TTY on my computer, in the background, all the time.  Heh heh heh.

# What Now?

In keeping with How Could They Possibly Improve Upon That?, you have to wonder where I could go from there.  Stay tuned! 🙂


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# CONGEST model

## Distributed Weighted Min-Cut in Nearly-Optimal Time

Minimum-weight cut (min-cut) is a basic measure of a network's connectivity strength. While the min-cut can be computed efficiently in the sequential setting [Karger STOC'96], there was no efficient way for a distributed network to compute its own min-cut without limiting the input structure or dropping the output quality: In the standard CONGEST model, existing algorithms with nearly-optimal time (e.g. [Ghaffari, Kuhn, DISC'13; Nanongkai, Su, DISC'14]) can guarantee a solution that is $(1+\epsilon)$-approximation at best while the exact \$\tilde O(n^{0.

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# Is there difference between existence of integral and integrability

For Lebesgue integral of a measurable function, its existence and integrability are different, as the latter requires the integral not only to exist but also to be finite.

1. I wonder if there is similar difference between existence of Riemann integral and Riemann integrability of a function over an interval? Or there is no difference, i.e., both require the Riemann integral to exist and to be finite? My question arose when I tried to understand a comment by Shai Covo following this reply

Consider X uniform(0,1), with distribution function F and probability distribution μ. Then, $\int {f(x)dF(x)} = \int_0^1 {f(x)dx}$ is an ordinary Riemann integral, whereas $\int {f(x)d\mu (x)} = \int_0^1 {f(x)dx}$ is Lebesgue integral. The former is not defined if f is (for example) unbounded.

2. Is the answer for Riemann integral same for Riemann-Stieltjes Integral?

3. Is the answer for Lebesgue integral same for Lebesgue-Stieltjes Integral? I guess yes, because the latter is actually defined in terms of the former?

Thanks and regards!

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In my experience, to say something is X integrable is identical to saying it has a finite integral, of type X. –  Jon Beardsley Apr 11 '11 at 2:38

My understanding is that when we say an integral (any type) exists, it means the function is integrable (finite integral) or the integral is infinite.

However, Riemann integral is only defined for bounded functions defined on a bounded interval $[a,b]$. (At least that's the case I've seen in all the text books.) So when the integral exists, it must be finite. One way to define Riemann integral is through upper and lower Riemann integrals. The upper Riemann integral is defined as $$U(f)=\inf\left\{\sum_i \left(\sup_{x_{i-1}\le x \le x_i}f(x)\right)\cdot (x_i-x_{i-1}):\text{ all partitions } (x_0,x_1,\cdots,x_n)\right\},$$ and the lower Riemann integral is defined to be $$L(f)=\sup\left\{\sum_i \left(\inf_{x_{i-1}\le x \le x_i}f(x)\right)\cdot (x_i-x_{i-1}):\text{ all partitions } (x_0,x_1,\cdots,x_n)\right\}.$$ The function $f$ is called Riemann integrable if $U(f)=L(f)$, and the common value is called the Riemann integral. Clearly, since we only consider bounded functions $f$: $|f|\le M$, we always have $$-M\le L(f)\le U(f)\le M,$$ and the Riemann integral is always finite if $L(f)=U(f)$.

The case of Riemann-Stieltjes integral should be similar.

If you take the definitions from Wikipedia, there's no infinite Riemann(-Stieltjes) integral because the definition requires the existence of a real number $A$ which is the limit of Riemann sum.

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Thanks! Is the answer for Riemann integral same for Riemann-Stieltjes Integral? Is the answer for Lebesgue integral same for Lebesgue-Stieltjes Integral? I guess yes, because the latter is actually defined in terms of the former? –  Tim Apr 11 '11 at 3:37
The reason why RI is defined only for bounded functions is because it can be proven that unbounded functions on compact intervals are not RI..... To see this, pick any partition, then at least on one of the intervals of this partition the function is unbounded, which means that we can make the Riemann sum as high as we want by picking the right sample points in this partition. –  N. S. Apr 11 '11 at 3:42
(1) Yes. Riemann or Riemann-Stieltjes integrals, existence = finite (=integrable). (2) Lebesgue or Lebesgue-Stieltjes integrals can exist with infinite value. The reason is exactly as you said. –  GWu Apr 11 '11 at 3:43
@user9176 This is a good point! That's what I suspected but couldn't prove it off the top of my head, so I didn't mention it. –  GWu Apr 11 '11 at 3:43
I added a short argument which explains why does that happen. –  N. S. Apr 11 '11 at 3:46

As described below, this question is very closely related to the question Meaning of non-existence of expectation?.

There is a difference between existence of integral and integrability in Lebesgue (but not Riemann) integration. A measurable function $f$ (taking values in $[-\infty,\infty]$, on some measure space with measure $\mu$) is said to be Lebesgue integrable if $\int {|f|d\mu } < \infty$. As usual, let $f^+$ and $f^-$ be the (nonnegative) functions defined by $f^ + (x) = \max \{ f(x),0\}$ and $f^ - (x) = -\min \{ f(x),0\}$, so that $|f|=f^+ + f^-$. Then, $f$ is integrable if and only if $\int {f^ + d\mu } < \infty$ and $\int {f^ - d\mu } < \infty$; in this case, $$\int {fd\mu } : = \int {f^ + d\mu } - \int {f^ - d\mu }$$ (in particular, the integral of an integrable function is finite). However, $f$ need not be integrable in order that the integral $\int {fd\mu }$ be defined. Indeed, the definition of $\int {fd\mu }$ for nonnegative measurable $f$ allows it to take the value $\infty$.

With the convention that $\infty - c = \infty$ and $c - \infty = -\infty$, for $c$ finite, the above definition of $\int {fd\mu }$ holds whenever $\int {f^ + d\mu } \leq \infty$ and $\int {f^ - d\mu } < \infty$ or $\int {f^ + d\mu } < \infty$ and $\int {f^ - d\mu } \leq \infty$. The integral $\int {fd\mu }$ does not exist if and only if $\int {f^ + d\mu } = \infty$ and $\int {f^ - d\mu } = \infty$ (since $\infty - \infty$ is not defined).

Examples: $\int_{(0,1]} {x^{ - 1} dx} = \infty$ (where $dx$ stands for Lebesgue measure), so the integral exists but $x^{-1}$ is not integrable on $(0,1]$. On the other hand, the integral $\int_{(0,1]} {\frac{{\sin (1/x)}}{x}dx}$ does not exist, since $\int_{(0,1]} {\big[\frac{{\sin (1/x)}}{x}\big]^ + }dx = \infty$ and $\int_{(0,1]} {\big[\frac{{\sin (1/x)}}{x}\big]^ - }dx = \infty$ (in particular, $\frac{{\sin (1/x)}}{x}$ is not integrable on $(0,1]$).

Relation to expectations of random variables.

Let $(\Omega,\mathcal{F},P)$ be a probability space, that is, a measure space with $P(\Omega)=1$. A random variable is a measurable function $X:\Omega \to \mathbb{R}$. So, $X(\omega)$ ($\omega \in \Omega$) and $P$ play the same role as $f(x)$ and $\mu$ above, respectively. The expectation of $X$ is defined by ${\rm E}(X) = \int_\Omega {XdP}$, which is just a special case of $\int {fd\mu }$ above. Accordingly, $X$ is said to be integrable if ${\rm E}|X|:=\int_\Omega {|X|dP} < \infty$, and has expectation if ${\rm E}(X^+) := \int_\Omega {X^ + dP} \le \infty$ and ${\rm E}(X^-) := \int_\Omega {X^ - dP} < \infty$ or ${\rm E}(X^+) < \infty$ and ${\rm E}(X^-) \leq \infty$ (in either case, ${\rm E}(X)= {\rm E}(X^+) - {\rm E}(X^-)$); $X$ does not admit an expectation if and only if ${\rm E}(X^+) = \infty$ and ${\rm E}(X^-) = \infty$.

It is instructive to consider the above examples in the setting of expectations. Let the probability space $(\Omega,\mathcal{F},P)$ be defined by $\Omega = (0,1]$, $\mathcal{F} = \mathcal{B}((0,1])$, and $P$ Lebesgue measure on $(0,1]$ (thus $dP(\omega)=d\omega$). Note that the random variable $X$ defined by $X(\omega)=\omega$ is a uniform$(0,1]$ random variable. Define $X_1$ by $X_1 = 1/X$, that is $X_1 (\omega) = 1/\omega$. Then, ${\rm E}(X_1) = \int_{(0,1]} {\frac{1}{\omega }d\omega } = \infty$; so, $X_1$ has (infinite) expectation but is not integrable. On the other hand, the random variable $X_2$ defined by $X_2 = \frac{{\sin (1/X)}}{X}$, that is $X_2 (\omega) = \frac{{\sin (1/\omega)}}{\omega}$, does not admit an expectation, since the corresponding integral, $\int_{(0,1]} {\frac{{\sin (1/\omega )}}{\omega }d\omega }$, does not exist (as noted above).

Finally, it is interesting to consider the above examples with connection to the strong law of large numbers (SLLN). Let $X_1^1,X_2^1,\ldots$ be a sequence of i.i.d. random variables distributed as $X_1$, and let $S_n^1 = X_1^1 + \cdots + X_n^1$. Then, almost surely, $n^{-1}S_n^1 \to {\rm E}(X_1) = \infty$ (this follows from the standard SLLN by using the monotone convergence theorem). So, the existence of ${\rm E}(X_1) := \int_{(0,1]} {\frac{1}{\omega }d\omega } = \infty$ agrees with SLLN. As for the second example, let $X_1^2,X_2^2,\ldots$ be a sequence of i.i.d. random variables distributed as $X_2$, and let $S_n^2 = X_1^2 + \cdots + X_n^2$. Since $X_2$ does not admit an expectation, with $X_2 ^+$ and $X_2^-$ behaving the same, one may expect that, almost surely, $\lim \sup n^{ - 1} S_n^2 = \infty$ and $\lim \inf n^{ - 1} S_n^2 = -\infty$; see Theorem 1 in the paper The strong law of large numbers when the mean is undefined, by K. Bruce Erickson. Here, it is important to note that $\frac{{\sin (1/x)}}{x}$ is (improperly) Riemann integrable on $(0,1]$; so SLLN may justify the Lebesgue non-integrability of this function.

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# Definition of Enthalpy

Enthalpy is a specific combination of values for the purpose of more simply showing relationships between those properties and another property graphically. There are two types: enthalpy (H) and specific enthalpy (h). Enthalpy combines internal energy (U) by adding the product of pressure (P) and volume (V); that is, . Specific enthalpy uses specific internal energy (u) and specific volume (v), yielding . Enthalpy is measured in kilojoules (kJ), whereas specific enthalpy is measured in kilojoules per kilogram (kJ/kg). An example of a specific type of enthalpy is hfg, or the enthalpy of vaporization, which takes a liquid that is saturated and calculates the energy required to convert one mass unit of it into vapor form.

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text:
What is the probability that I have a allele if my mom/cousin have it?

Say my mom has a gene (it's one of those regular gene, ignore X and Y chromosome)

The gene exist in p% of the population.

Things get complicated due to that p. If p is 100%, for example, then of course the probability that I have the gene to is 100%. If p is 1% then the probability that I got the gene will be 50%.

How to formalize this mathematically?

My gut feeling is it'll just be linear. if p is 1% then my chance will be 50.5% because that means my dad have a 1% chance to have that gamete.

Then i want to extend this to cousins.

-
What is gammete? How does it work between generations? –  Patrick Li Nov 8 '12 at 4:32
a gene? Did I spell that correctly? –  Jim Thio Nov 8 '12 at 4:34

1 Answer

First by 'gamete', you mean allele correct? A gamete is when the sperm and egg have already fused together so there is no more probability. While an allele is a specific gene which can vary (as in your case). So I will assume that you are talking about an allele/gene.

In your case, the formula, that you are talking about is the Hardy-Weinberg formula.

Basically $$(p^2) + (2pq) + (q^2) = 1$$ where p and q are the percentage of two version of the alleles.

Say you have a gene with 2 possible alleles 'A' and 'a' which are present in p% and q% of the the population (so p would be 1% and q can be 99% in your case).

The percentage of the three possible genotypic frequencies in the offspring become:

$f(AA) = p^2$

$f(Aa) = 2pq$

$f(aa) = q^2$

To find the chance that you will get an allele A, you just need to add up $f(AA) + f(Aa)$ (because both of them have a gene/allele 'A' which is the gene that you want to know about)

If your case, if p = 100% = 1, this means $q=0$, then the probability that you will be an allele A would be 100% (because $f(AA) + f(Aa) = 1^2 + 2*1*0=1+0=1$).

In your second case, If p = 1%, this means q = 99% (because all the other eggs would have the other gene/allele). This means $f(AA) + f(Aa) = 0.01^2 + 2*0.01*0.99=0.0001+0.0198=0.0199$ or 1.99%.

-
And what about if my mom have that A allele? You haven't taken that into account. Say 50% of my Mom's allele is A. –  Jim Thio Nov 8 '12 at 4:46
Then p = 0.5 (and 0.5). So $f(AA)+F(Aa) = 0.5^2+2*0.5*0.5 = 0.75$ or 75%. It doesn't matter whether you are looking at the gene from mom or dad side since the frequency of the allele is known at a population level so its assumed that both the egg and sperm have the same chance of having that specific gene/allele. –  BYS2 Nov 8 '12 at 4:48


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# Number of galaxies per redshift

Does exist something like a distribution of number of galaxies per redshift interval? I'm studying galaxy formation and evolution, and I'm curious to know how many todays' galaxies (with redshift <0.01 I would say) we see.

Yes, for a given galaxy type…

The term "a distribution of number of galaxies per redshift interval" does make sense, but usally we restrict ourselves to a certain type of galaxies, e.g. Lyman $$\alpha$$ emitters, Lyman break galaxies, sub-mm galaxies, etc. The reason is that galaxies are detected in a number of different ways, and no single selection technique captures all types of galaxies.

…and comoving volume…

Because the Universe is expanding with time, the "true" (physical) number density of galaxies — and everything else (except dark energy) — decreases with time. To be able to compare densities at different times, it is useful to measure them in comoving coordinates, which is the coordinates that expand along with the Universe. In these coordinates, densities only change if some physical process happen. For instance, the physical energy density of photons decreases both because the volume of the Universe increases and because the photons redshift. In comoving coordinates it also decreases, but only due to the redshifting since the volume change is factored out.

…and luminosity.

Moreover, since galaxies have no well-defined lower size/mass/luminosity threshold, we have to define a threshold above which we count them. For this reason, rather than talking about "the number density of galaxies per redshift", we usually use the very useful observable luminosity function (LF) which measures the number of (a certain type of) galaxies per luminosity bin, and per comoving volume, at a given redshift.

LFs have been probed out to very high redshifts, at the times when galaxies were just beginning to form at $$\sim$$half a gigayear after Big Bang (e.g. Schenker et al. 2013).

You end your question by asking about the number of galaxies at $$z\le0.01$$, which would be considered very low redshift in the context of galaxies. You can find the answer by considering all kinds of selection criteria, checking for overlaps between surveys (most local galaxies will be detected by multiple techniques), and integrating over volume the LFs down to your chosen threshold (e.g. Small Magellanic Cloud-sized). Doable, but not trivial. An alternative is to integrate numerically calculated dark matter halo mass functions, using e.g. this online tool, and assuming the every halo hosts a galaxy. I get a number density of $$n\sim0.4$$ halos per $$h^{-3}\,\mathrm{Mpc}^{3}$$, or $$1.2\,\mathrm{Mpc}^{-1}$$. The comoving volume out to $$z=0.01$$ is $$V=3.3\times10^5$$ Mpc, so the total number of (super-SMC-sized) galaxies is $$N = n V \simeq 3.8\times10^5$$.


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# How to write in LazyTeX instead of strict LaTeX

TL;DR Use Pandoc, write your document in a combination of YAML, Markdown and, when you need it, inline LaTeX. Read Pandoc’s README.

TeX is awesome. LaTeX was made to make it easier to use TeX and produce high-quality documents.

Still, there are two downsides with using LaTeX:

1. the source of your document is a bit cryptic for people who aren’t used to source code
2. TeX was designed for paper as the output and thus comes with its limitations.

Today, most LaTeX documents end up as PDF and/or printed on paper (which is kind of the same). This is nice, but PDF and paper are not mediums that enable others to co-edit the text (unless they can work with the LaTeX source, but most people won’t learn that, see point 1 above).

This is especially sad because LaTeX is not only able to produce awesome typesets, it’s also able to produce part of the content of the document, thanks to a myriad of packages that you can use in LaTeX (for instance, varioref).

× For example, the varioref package is the best program I’ve seen to make automated references to another point in a document. Using varioref, LaTeX is able to print something like: “see section 3, on the facing page” automatically (or “… on the next page” if the document isn’t supposed to be printed twosides aka recto verso).

That’s great, but that only makes sense for documents in pages, like paper or PDF. It does not make sense for a document in HTML that will be a single “web page” (of course, we could also emulate pages in HTML but, seriously, why are people doing that?) although it still makes sense to be able to refer to another part of the doc (like I did above.)

So what I’m doing these days is mostly LazyTeX.

## What’s LazyTeX?

LazyTeX is a way to use TeX that is lazy, but has the potential to overcome the two donwsides of using strict LaTeX that I just described.

Mainly, LazyTeX is just a funny name I have given to the combination of Markdown, YALM and inline LaTeX, that can be used through Pandoc in order to produce beautiful LaTeX PDF.

The upside to doing this, is that the source is way more legible for people like LibreOffice or Microsoft Office users, and the output will not necessarily be PDF but, in some cases, could as well be HTML or plain text, or something else.

Why’s this lazy? There are two reasons to this:

First, the markdown syntax is lazier than the latex syntax. For instance, a list in markdown is as simple as writing:

This is some text.

- This is the first item of a list
- This is the second item of a list

This is some text.

whereas a list in LaTeX cannot really be more simple than:

This is some text.

\begin{itemize}
\item This is the first item of a list
\item This is the second item of a list
\end{itemize}

This is some text.

You get the idea. Sometimes, even documents that I only need as PDF, for which I could use plain, strict LaTeX – I today prefer to write them in a combination of YAML, Markdown and inline LaTeX – that means, what I call from now on LazyTeX.

However, lazy also has a downside. Mainly, if I mike a mistake in the source file, there are more risks of producing a PDF with the mistake showing in plain sight, rather than having a compilation error.

When I do a mistake in a LaTeX file, usually the compilation will give me an error and not produce the result. Thus, the error flags me that I need to fix something.

However, when I do a mistake in a LazyTeX file (for instance, misplacing a list inside a list because of wrong indention, or misplacing an asterisk that’s supposed to make something bold) – in such cases, the LazyTeX file might compile correctly and will just print the mistake. So I may need to review the PDF more thoroughly, which can be cumbersome for long documents. So, in some cases, maybe LazyTeX should be avoided and strict LaTeX prefered.

## How does it work?

Pandoc is what makes this possible. Pandoc has its own Markdown variant, which enables Markdown to be a bigger subset of HTML than the “vanilla” Markdown is. But Pandoc also has some neat tricks that makes Markdown an interesting source for LaTeX. For instance, the pandoc-citeproc program that’s shipped with Pandoc enables you to use the bibliography engines of LaTeX.

Pandoc also parses YAML data, which you can then use to generate parts of your LaTeX document, especially the preamble.

Pandoc also allows you to have inline LaTeX, meaning you can write some LaTeX inside your markdown and Pandoc will work it out. (Although this has some limitations).

Obviously, one of the biggest upside of Pandoc, is the ability to convert documents from one format to another. Here’s the insane diagram of possibilities:

## What’s not working?

The problem is that Pandoc’s LaTeX “reader” isn’t a full LaTeX parser (yet). So the markdown+inlineLaTeX combination may cause issues for non-LaTeX outputs.

So be careful with some commands. See the sample LazyTeX doc and the resulting PDF.

My solution right now, is to add another layer of complexity, to make things worse: I use custom directives in Emacs’ Pandoc-mode.


text:
# Subtracting Fractions

• April 29th 2006, 12:36 PM
stomper12000
Subtracting Fractions
Rosa has 6 3/5 yards of material. Her new skirt will take 4 2/3 yards. How much material will she have left after the skirt is made?

Is this solved by using division or subtraction. It looks like subtraction to me but I can't fix this. Thanks for any assistance.
• April 29th 2006, 09:12 PM
earboth
Quote:

Originally Posted by stomper12000
Rosa has 6 3/5 yards of material. Her new skirt will take 4 2/3 yards. How much material will she have left after the skirt is made?
Is this solved by using division or subtraction.
It looks like subtraction to me ...

Hello,

you are right.

You have to calculate 6 3/5 yd - 4 2/3 yd. Before you can do so, transform the mixed numbers into fractions:

6 3/5 yd = 33/5 yd
4 2/3 yd = 14/3 yd

Then calculate:
$\frac{33}{5} yd-\frac{14}{3} yd$

You need a common denominator, which is here: 15. So you'll get:

$\frac{3 \cdot 33}{3 \cdot 5}yd-\frac{5 \cdot 14}{5 \cdot 3}yd=\frac{99-70}{15}yd$

The result is: $\frac{29}{15} yd = 1 \frac{14}{15} yd$

Greetings

EB
• May 30th 2006, 12:32 AM
mathmaster
Quote:

Originally Posted by stomper12000
Rosa has 6 3/5 yards of material. Her new skirt will take 4 2/3 yards. How much material will she have left after the skirt is made?

Is this solved by using division or subtraction. It looks like subtraction to me but I can't fix this. Thanks for any assistance.

:confused:
easier terms

6 3/5 = 33/5
4 2/3 = 14/3

therefore

33/5 = 99/15
14/3 = 70/15

99-70 = 29

29/15 which is 1 14/15 yard of material is left
:D
:cool:
:rolleyes:
• June 18th 2006, 11:36 PM
moshiman
basically, it means that before doing anything else, you have to make the fractions you're using have the same denominator. :o


text:
# 3.1 Vector fields

 Page 1 / 1
What is a Vector Field?

## Vector fields

A vector field can be considered a map of vectors over some space. . For example if one were to show wind vectors on a weather map; that would be avector field. The electric field surrounding a charge is a vector field (were is the potential around the charge is a scalar field).

The flux of a vector field through a closed surface is the average outward normal component of the vector times the area of thesurfaceflux = (average normal component ) $\cdot$ (surface area)

Three vector fields are shown below. Which represents the electric field eminating from a positive point charge in the middle? (Note that vectors ofsimilar magnitude are colored similarly in these plots)

How we are making nano material?
what is a peer
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
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it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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how did you get the value of 2000N.What calculations are needed to arrive at it
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text:
Now we return to the original graph coloring problem: coloring maps. As indicated in section 1.1, the map coloring problem can be turned into a graph coloring problem. Figure 5.10.1 shows the example from section 1.1.

Figure 5.10.1. A map and its corresponding graph.

Graphs formed from maps in this way have an important property: they are planar.

Definition 5.10.1 A graph $G$ is planar if it can be represented by a drawing in the plane so that no edges cross.

Note that this definition only requires that some representation of the graph has no crossing edges. Figure 5.10.2 shows two representations of $K_4$; since in the second no edges cross, $K_4$ is planar.

Figure 5.10.2. $K_4$ drawn in two ways; the second shows that it is planar.

The number of colors needed to properly color any map is now the number of colors needed to color any planar graph. This problem was first posed in the nineteenth century, and it was quickly conjectured that in all cases four colors suffice. This was finally proved in 1976 (see figure 5.10.3) with the aid of a computer. In 1879, Alfred Kempe gave a proof that was widely known, but was incorrect, though it was not until 1890 that this was noticed by Percy Heawood, who modified the proof to show that five colors suffice to color any planar graph. We will prove this Five Color Theorem, but first we need some other results. We assume all graphs are simple.

Theorem 5.10.2 (Euler's Formula) Suppose $G$ is a connected planar graph, drawn so that no edges cross, with $n$ vertices and $m$ edges, and that the graph divides the plane into $r$ regions. Then $$r=m-n+2.$$

Proof.
The proof is by induction on the number of edges. The base case is $m=n-1$, the minimum number of edges in a connected graph on $n$ vertices. In this case $G$ is a tree, and contains no cycles, so the number of regions is 1, and indeed $1=(n-1)-n+2$.

Now suppose $G$ has more than $n-1$ edges, so it has a cycle. Remove one edge from a cycle forming $G'$, which is connected and has $r-1$ regions, $n$ vertices, and $m-1$ edges. By the induction hypothesis $r-1=(m-1)-n+2$, which becomes $r=m-n+2$ when we add 1 to each side.

$\square$

Lemma 5.10.3 Suppose $G$ is a simple connected planar graph, drawn so that no edges cross, with $n\ge3$ vertices and $m$ edges, and that the graph divides the plane into $r$ regions. Then $m\le 3n-6$.

Proof.
Let $f_i$ be the number of edges that adjoin region number $i$; if the same region is on both sides of an edge, that edge is counted twice. We call the edges adjoining a region the boundary edges of the region. Since $G$ is simple and $n\ge3$, every region is bounded by at least 3 edges. Then $\sum_{i=1}^r f_i=2m$, since each edge is counted twice, once for the region on each side of the edge. From $r=m-n+2$ we get $3r=3m-3n+6$, and because $f_i\ge 3$, $3r\le \sum_{i=1}^r f_i=2m$, so $3m-3n+6\le 2m$, or $m\le 3n-6$ as desired.

$\square$

Theorem 5.10.4 $K_5$ is not planar.

Proof.
$K_5$ has 5 vertices and 10 edges, and $10\not\le 3\cdot 5-6$, so by the lemma, $K_5$ is not planar.

$\square$

Lemma 5.10.5 If $G$ is planar then $G$ has a vertex of degree at most 5.

Proof.
We may assume that $G$ is connected (if not, work with a connected component of $G$). Suppose that $\d(v_i)>5$ for all $v_i$. Then $2m=\sum_{i=1}^n \d(v_i)\ge 6n$. By lemma 5.10.3, $3n-6\ge m$ so $6n-12\ge 2m$. Thus $6n\le 2m \le 6n-12$, a contradiction.

$\square$

Theorem 5.10.6 (Five Color Theorem) Every planar graph can be colored with 5 colors.

Proof.
The proof is by induction on the number of vertices $n$; when $n\le 5$ this is trivial.

Now suppose $G$ is planar on more than 5 vertices; by lemma 5.10.5 some vertex $v$ has degree at most 5. By the induction hypothesis, $G-v$ can be colored with 5 colors. Color the vertices of $G$, other than $v$, as they are colored in a 5-coloring of $G-v$. If $\d(v)\le 4$, then $v$ can be colored with one of the 5 colors to give a proper coloring of $G$ with 5 colors. So we now suppose $\d(v)=5$. If the five neighbors of $v$ are colored with four or fewer of the colors, then again $v$ can be colored to give a proper coloring of $G$ with 5 colors.

Now we suppose that all five neighbors of $v$ have a different color, as indicated in figure 5.10.4.

Figure 5.10.4. Five neighbors of $v$ colored with 5 colors: $v_1$ is red, $v_2$ is purple, $v_3$ is green, $v_4$ is blue, $v_5$ is orange.

Suppose that in $G$ there is a path from $v_1$ to $v_3$, and that the vertices along this path are alternately colored red and green; call such a path a red-green alternating path. Then together with $v$, this path makes a cycle with $v_2$ on the inside and $v_4$ on the outside, or vice versa. This means there cannot be a purple-blue alternating path from $v_2$ to $v_4$. Supposing that $v_2$ is inside the cycle, we change the colors of all vertices inside the cycle colored purple to blue, and all blue vertices are recolored purple. This is still a proper coloring of all vertices of $G$ except $v$, and now no neighbor of $v$ is purple, so by coloring $v$ purple we obtain a proper coloring of $G$.

If there is no red-green alternating path from $v_1$ to $v_3$, then we recolor vertices as follows: Change the color of $v_1$ to green. Change all green neighbors of $v_1$ to red. Continue to change the colors of vertices from red to green or green to red until there are no conflicts, that is, until a new proper coloring is obtained. Because there is no red-green alternating path from $v_1$ to $v_3$, the color of $v_3$ will not change. Now no neighbor of $v$ is colored red, so by coloring $v$ red we obtain a proper coloring of $G$.

$\square$

## Exercises 5.10

Ex 5.10.1 Show $K_{3,3}$ is not planar. (Prove a lemma like lemma 5.10.3 for bipartite graphs, then do something like the proof of theorem 5.10.4.) What is the chromatic number of $K_{3,3}$?


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# If (3 - 2x)^(1/2) = 1, then what is the value of (3 – 2x) + (3 – 2x)^2

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If (3 - 2x)^(1/2) = 1, then what is the value of (3 – 2x) + (3 – 2x)^2  [#permalink]

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If $$\sqrt{3 - 2x} = 1$$, then what is the value of $$(3 – 2x) + (3 – 2x)^2$$ ?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

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Re: If (3 - 2x)^(1/2) = 1, then what is the value of (3 – 2x) + (3 – 2x)^2  [#permalink]

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11 Apr 2019, 01:01
Bunuel wrote:
If $$\sqrt{3 - 2x} = 1$$, then what is the value of $$(3 – 2x) + (3 – 2x)^2$$ ?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

$$(3 – 2x) + (3 – 2x)^2$$
3-2x= $$\sqrt{3 - 2x} * [m]\sqrt{3 - 2x} so [m](3 – 2x) + (3 – 2x)^2$$ = $$\sqrt{3 - 2x} * [m]\sqrt{3 - 2x} + ( [m]\sqrt{3 - 2x} * [m]\sqrt{3 - 2x} )^2 given [m]\sqrt{3 - 2x} = 1$$
so 1+1 = 2
IMO C
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If (3 - 2x)^(1/2) = 1, then what is the value of (3 – 2x) + (3 – 2x)^2  [#permalink]

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11 Apr 2019, 11:20
Archit3110 wrote:
Bunuel wrote:
If $$\sqrt{3 - 2x} = 1$$, then what is the value of $$(3 – 2x) + (3 – 2x)^2$$ ?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

$$(3 – 2x) + (3 – 2x)^2$$
3-2x= $$\sqrt{3 - 2x} * [m]\sqrt{3 - 2x} so [m](3 – 2x) + (3 – 2x)^2$$ = $$\sqrt{3 - 2x} * [m]\sqrt{3 - 2x} + ( [m]\sqrt{3 - 2x} * [m]\sqrt{3 - 2x} )^2 given [m]\sqrt{3 - 2x} = 1$$
so 1+1 = 2
IMO C

Why not square both sides of the first equation to eliminate the square root? and then solve.

I got 0 as my answer
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Re: If (3 - 2x)^(1/2) = 1, then what is the value of (3 – 2x) + (3 – 2x)^2  [#permalink]

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11 Apr 2019, 11:33
thealpine ; squaring both side will give +/- values and this question we can solve without squaring as well √x*√x=x so ; 3-2x = (√3-2x) * (√3-2x)

thealpine wrote:
Archit3110 wrote:
Bunuel wrote:
If $$\sqrt{3 - 2x} = 1$$, then what is the value of $$(3 – 2x) + (3 – 2x)^2$$ ?

(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

$$(3 – 2x) + (3 – 2x)^2$$
3-2x= $$\sqrt{3 - 2x} * [m]\sqrt{3 - 2x} so [m](3 – 2x) + (3 – 2x)^2$$ = $$\sqrt{3 - 2x} * [m]\sqrt{3 - 2x} + ( [m]\sqrt{3 - 2x} * [m]\sqrt{3 - 2x} )^2 given [m]\sqrt{3 - 2x} = 1$$
so 1+1 = 2
IMO C

Why not square both sides of the first equation to eliminate the square root? and then solve.

I got 0 as my answer

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Re: If (3 - 2x)^(1/2) = 1, then what is the value of (3 – 2x) + (3 – 2x)^2  [#permalink]

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11 Apr 2019, 11:42
Archit3110 wrote:
thealpine ; squaring both side will give +/- values and this question we can solve without squaring as well √x*√x=x so ; 3-2x = (√3-2x) * (√3-2x)

I heard this on a teaching platform, that whenever the GMAT gives your the square root on the prompt, you can/it is okay to only consider positive values. However, if the square root is something you come across while solving for a problem then you have to consider both positive and negative values.

Heard this from MikeMcGarry

Thoughts on this?
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If (3 - 2x)^(1/2) = 1, then what is the value of (3 – 2x) + (3 – 2x)^2  [#permalink]

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16 Apr 2019, 09:54
thealpine wrote:

I heard this on a teaching platform, that whenever the GMAT gives your the square root on the prompt, you can/it is okay to only consider positive values. However, if the square root is something you come across while solving for a problem then you have to consider both positive and negative values.

Heard this from MikeMcGarry

Thoughts on this?

Indeed, it is correct to consider only positive values for the square roots on the prompt on GMAT; otherwise, you would be handling with complex roots.

It is important to bear in mind that while a "regular square root" and a "square root of some variable" may seem to be the same case, there is a difference.
Numbers have a real and an imaginary part - which is often ignored by us - but in this specific case the imaginary part plays an important role.

While the "regular square root" has an $$i^0$$ form, the variable solving find positive numbers from both $$i^0$$ and a combination of $$i^2$$ forms - which will multiply as a $$i^4$$ -, as the imaginary part follows the following power pattern:

$$i^0 = 1$$
$$i^1 = i$$
$$i^2 = – 1$$
$$i^3 = i^2 * i = (–1) * i = –i$$
$$i^4 = i^2 * i^2 = (–1) * (– 1) = 1$$
...

Having that in mind, as you may have realized, we can be sure that $$√1 = 1$$, since it is in the $$√(1*i^0)$$ form. On the hand, there is no certain when it comes to $$x^2=1$$ as it could be both:
$$(1*i^0)^2 = (1*1)^2 = 1^2 = 1$$
or
$$(1*i^2)^2 = (1*-1)^2 = (-1)^2 = 1$$

Hope it helps!

As for the problem, I would just consider that 1 to any power equals 1. Hence, the given expression would be the same of $$√1$$ and we could simply do
$$(3–2x)+(3–2x)^2 = 1 + 1^2 = 2$$
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If (3 - 2x)^(1/2) = 1, then what is the value of (3 – 2x) + (3 – 2x)^2   [#permalink] 16 Apr 2019, 09:54
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# If (3 - 2x)^(1/2) = 1, then what is the value of (3 – 2x) + (3 – 2x)^2

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# Quark Matter 2014 - XXIV International Conference on Ultrarelativistic Nucleus-Nucleus Collisions

19-24 May 2014
Europe/Zurich timezone

## Measurement of transverse momentum jet spectra in Pb-Pb collsions at $\sqrt{\rm s_{NN}}$ = 2.76 TeV using calorimeter triggers with the ALICE detector

20 May 2014, 16:30
2h

Board: E-27
Poster Jets

### Speaker

Rosi Jan Reed (Yale University (US))

### Description

In relativistic heavy-ion collisions a hot, dense medium of strongly interacting matter called the Quark-Gluon Plasma (QGP) is formed. It has been determined that jets of hadrons, created from the fragmentation of hard-scattered partons, are suppressed relative to measurements from proton-proton collisions. This jet quenching effect in both strong and weak coupling calculations explicitly depends on the path-length of the parton through the medium. Jets can be reconstructed in the ALICE detector from the tracking systems alone, creating charged jets, or from the combination of the tracking systems and the electromagnetic calorimeter (EMCal). The EMCal was used to trigger on events with a jet within its acceptance, which allows for a measurement of the jet spectra versus centrality and event plane by increasing the available statistics. The centrality and event plane of these events are reconstructed with the VZERO detectors to avoid autocorrelations with the triggered jets. Triggers introduce biases to both the fragmentation of the jets, and to the background. Centrality and event plane dependence of the measured spectra and the background using the EMCal trigger will be shown as well as the reconstruction and correction procedures with the trigger biases.
On behalf of collaboration: ALICE

### Primary author

Rosi Jan Reed (Yale University (US))

 Slides


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# Current conservation for SU (N)

Tian
Homework Statement:
In the "An introduction to Quantum Field Thoery" of Peskin and Schroeder, the equation(15.51) of the chapter 15.3 gives the classical equation of motion, so from this equation to derive the current conservation.
Relevant Equations:
the classical equation of motion for SU(N), please see my picture
Here is my solution

## Answers and Replies

Science Advisor
$$D^{\mu}F_{\mu\nu} = - j_{\nu},$$ Differentiate this covariantly and anti-symmetrized to obtain $$\frac{1}{2}[D^{\mu}, D^{\nu}]F_{\mu\nu} = D^{\nu}j_{\nu}. \ \ \ \ (1)$$ Now, from the definition of the covariant derivative in the adjoint representation (acting on any matrix-valued field) $$D^{\mu}M \equiv \partial^{\mu}M + [A^{\mu} , M],$$ you can show that $$[D^{\mu} , D^{\nu}]M = [F^{\mu\nu} ,M]$$ Thus, for $M = F_{\mu\nu}$, eq(1) becomes $$D^{\nu}j_{\nu} = \frac{1}{2}[F^{\mu\nu} , F_{\mu\nu}] = 0.$$

Tian
Thank you veery much . It should be done in the adjoint repesentation.


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× This section is under development!

# Classical vs High-dimensional

L

et $n$ be # of observations, $p$ be # of variables. The classical regime allows $n$ to diverge, but assumes $p$ fixed. In contrast, the high-dimensional regime permits both $n$ and $p$ to diverge, $p/n \to \gamma > 0$. Many of the classical results break down in that case. Here I consider eigenvalues and eigenvectors of a high-dimensional covariance matrix. This has immediate implications for covariance estimation, but also for all the statistical tools based on covariance estimates: PCA, GLS, GMM, classification, portfolio optimization, etc.

Consider a simple case   $X_i \overset{iid}{\sim} \mathcal{N}_p(\mathbf{0}, \Sigma),\quad i=1,\ldots, n.$ How to estimate $\Sigma$?

Some notation:

Sample covariance estimator   $S = \frac{1}{n}\sum_i^n X_iX_i’ = \frac{1}{n} X’X.$ Eigendecompositions   $\Sigma = ULU’ = \sum_j^p \ell_j \mathrm{u}_j \mathrm{u}_j’, \quad S = V\Lambda V’ = \sum_j^p \lambda_j \mathrm{v}_j \mathrm{v}_j’.$ Eigenvalues distinct, sorted in decreasing order. Eigenvectors chosen with the first element positive.

# Clasical Regime

In a classical regime, $S$ is a very good estimator (Anderson 1963, Van der Vaart 2000): Unbiased   $\E(S) = \Sigma.$
Consistent   $S \as \Sigma$ as $n\to\infty.$
Asymptotically normal eigenvalues   $\sqrt{n}(\lambda_i-\ell_i) \overset{d}{\to} \mathcal{N}(0,2\ell_i^2), \quad j=1,\ldots,p.$
Is invertible.

It gets trickier in high dimensions It is especially interesting what happens to eigenvalues and eigenvectors in high dimensions. There are three key features: eigenvalue spreading, eigenvalue bias and eigenvectors inconsistency.

# High-dimensional Regime

In high dimensions, sample eigenvalues $\lambda_j$ are more spread out than their population counterparts $\ell_j.$ In fact, the higher the dimension, the more is the spreading.
Consider the case when $\Sigma = I_p,$ i.e. $\ell_1 = \ldots = \ell_p = 1,$ and $p/n \to \gamma \le 1.$
Empirical d’n of eigenvalues of sample covariance   $F_p(x) := \frac{1}{p} \# \{ \lambda_j\le x \}$
Ukranian mathematicians Marchenko & Pastur (MP) showed that this empirical d’n converges $F_p(x) \to F(x),$ with the limit pdf given by:
$f^{MP}(x) = \frac{\sqrt{(\lambda_+-x)(x-\lambda_-)}}{2\pi x \gamma}, \quad \lambda_+ = (1+\sqrt{\gamma})^2, \quad \lambda_- = (1-\sqrt{\gamma})^2.$
$\begin{split} F(x) = & \frac{1}{2} + \frac{1}{2\pi \gamma} \Big[\sqrt{(\lambda_+-x)(x-\lambda_-)} \\ & + (1+\gamma)\arcsin(\frac{x-1-\gamma}{2\sqrt{\gamma}}) + (1-\gamma)\arcsin(\frac{(1-\gamma)^2-(1+\gamma)x}{2x\sqrt{\gamma}})\Big]. \end{split}$


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Is there need to patrol for deleted questions?

In a recent discussion Mariano Suárez-Alvarez told us, that if we are aware of users, who delete their questions after receiving an answer, we should bring them to the attention of mods. (IIRC the OP can only delete a question if no of the answers have been upvoted.)

In an older discussion about the same problem Willie Wong said that:

While 10K users can see all deleted posts, the posts are only presented in a nice summary form in the 10K tools panel if they are "deleted by committee". Deleted by owner posts don't show up there. This makes it hard for users to "patrol" for such behaviour. Also, one can always use the free-form flag field.

and

nothing easy that I know of (and there's also a difference between what diamond mods and 10K can see/search). Hence my previous comment.

The fact that self-deleted questions are not listed anywhere was mentioned in this discussion, too.

So I want to ask you this:

• Should we try to find self-deleted questions which could be a sign of cheating? Does this problem appear often enough to warrant additional work for users/moderators?
• If yes, how this could be done?

I was not able to come up with anything more intelligent than downloading links in the form https://math.stackexchange.com/questions/XXXXXX using wget and then checking which of them could not have been downloaded. I've done a test where I found out that in 1000 posts I've tried I have found 14 deleted questions. (The reasons why I was doing this experiment were twofold: I wanted to see whether this method is feasible. And I also wanted to have some data about the proportion of deleted questions among all questions.)

Maybe someone who is more familiar with SE data explorer can come up with a better method how to find such questions.

Feature request mentioned in Jonas Meyer's answer would be another possibility.

EDIT: Since this post was made, there were some changes which now make possible for users to find their own answers which have been deleted (at least in some cases).

Namely:

For details see here: Is there any way to see my deleted questions or answers?

• MathOverflow users used to do it with help of the administrator (which we would not have here). Apparently they ran out of steam about a year ago. – user53153 Feb 21 '13 at 14:04
• +1 for your effort... – draks ... Feb 21 '13 at 14:23
• I tried tinkering with the data explorer, and I have not found any way to locate deleted posts. There is a "Post History" table that records deletions, edits, and other actions, and I can find posts that have been deleted, but only those that have been subsequently undeleted again. – MJD Feb 22 '13 at 5:20
• @MJD A few queries (although they show somewhat incomplete information about the post) can be found here: Are there some tools to follow deletions on this site? (And, of course, further additions and improvements are welcome, it is a CW answer.) – Martin Sleziak May 10 '18 at 4:33
• math.meta.stackexchange.com/users/recently-deleted-answers/… with the users id instead of mine. – Roddy MacPhee Sep 4 at 11:25
• @RoddyMacPhee Yes, that's the first bullet point in the part after "EDIT". However, this is only way how a user can get to their own posts that were deleted. – Martin Sleziak Sep 4 at 14:34

Changes on the SE part do not look likely. One thing an ordinary user can do is to develop a habit to save his/her answers with a service such as Evernote. (Chrome extension makes it straightforward to clip the entire page.) If it later turns out that an answer has disappeared, a search through the notes will recover the answer and provide a point of reference for the moderators, who can undelete the question.

Of course, a simple notification ("your post ... was deleted") would be nice, and if it came with a copy of the deleted post (by email) that would be even better.

Added: I found and adapted a query which retrieves the last 50000 posts ordered by id. (Apparently, 50000 rows is the most DataExplorer can return). Downloaded the data and ran a script to generate the list of the missing ids. There were 7192 of them, which gives the proportion $7192/57192\approx 14\%$ of deleted posts. Despite the apparently large number of processed posts, they only cover the period back to mid-December 2012. Note that posts include both answers and questions, and presumably most of the deleted posts are answers. There seems to be no way to tell which ones are questions (let alone questions with substantive answers) without navigating to each by the URL, if anyone is so inclined.

• By extension, it is easy to get the id numbers of all non-deleted posts by using "between 50000k and 50000(k+1)". Then take the complement of this set. Still no way to separate answers from questions, though... – user53153 Feb 27 '13 at 6:50
• Thanks for providing the list with links. I went through some questions from your list. The list seems to be deleted by now, but it should be easy to generate a similar list. Perhaps if I have some spare time in a future, I'll check a few of such questions again. – Martin Sleziak Mar 10 '13 at 11:17

Personally I wouldn't want the burden the community by requesting that we hunt down cheaters. I think it is enough that people know that this sort of stuff happens and that they can do something (i.e. flag) if they see it happening.

In general, if you spot a contribution to the site that you think is unnecessarily deleted, you can flag for moderator attention and we will do our best to sort it out.

Lastly, I am more in favour of preventative measures (which to me feel more "positive"):

2. Vote early, vote often. By voting up answers to a question, you are affirming that both the question and its answer has some value on the site, which will help prevent their being removed.
• Concerning 2: I find it appropriate to quickly upvote any answer that has no votes at all if, upon a quick scan, it looks like a serious attempt at the given question. I will not necessarily check the correctness at this stage. This is by analogy with an established process (at some better journals): quick pre-refereeing message to journal's editor merely confirms that the paper is something worth looking at, not that it works. If the question was already voted on, then that stage is over, and I have to actually read it to decide how to vote. – user53153 Feb 21 '13 at 14:23
• @5pm If your prior comment proposes that one should employ lower upvoting standards on unvoted answers in order to help prevent question deletion, then I don't think that is a good idea. This could cause problems much worse than the problem of question deletion, e.g. incorrect positive valuation of wrong or misguided answers, and. consequently, possible encouragement of further such problematic posts by the same author. – Math Gems Feb 21 '13 at 15:00
• @5pm Further, some students post answers which they are not sure of, hoping to receive feedback on their understanding, correctness of proof, etc. Lowering the standard of that feedback due to orthogonal (lower-value?) concerns is not fair to those seeking thoughtful, correct feedback on their answers. – Math Gems Feb 21 '13 at 15:11
• @MathGems The upvote arrow says: "This answer is useful". An answer does not have to be entirely correct in order to be useful. – user53153 Feb 21 '13 at 15:59
• @5pm: Some people feel that only if an answer surpasses their own it merits an upvote, otherwise it merits a downvote. Some people don't even vote on threads they don't answer, because they don't visit threads they don't intend to answer. – Asaf Karagila Feb 21 '13 at 16:26
• @5pm Eh? If you believe that incorrect or misguided answers are "useful" then it seems we have a very different conception of the meaning of "useful". I think that, generally, one should exercise more care in voting - not less care. Doing so would help alleviate some of the problems, e.g. that good or later answers often receive fewer votes than poorer but quicker answers. – Math Gems Feb 21 '13 at 16:27
• @AsafKaragila This entire chain of comments can be summarized as "people do things"... Willie may want to clean it up, as it seems my comment sidetracked the discussion of deleted answers... And yes, Math Gems, incorrect answers can be useful to me. Your idea of usefulness may be different. – user53153 Feb 21 '13 at 16:29
• @5pm I think the comments are quite useful in that they help to show they different ways that various users think about votes - so help to promote better understanding of each other. Further, they are quite on topic, having to do with how one should follow the "vote early, vote often" SE advice, i.e. whether or not one should compromise voting standards to do so. – Math Gems Feb 21 '13 at 17:09

There is no reasonable way to find self-deleted questions. They don't appear in the Data Explorer, and crawling the site like you did could cause SE to block you from accessing the site, if you are not careful.

But such automatic measures are not necessary anyway in my opinion, there is always one user that likely notices such behaviour, and that is the user that answered the question. If that user notices such behaviour, he can flag for moderator attention and the moderators can easily see if it happened more often for that user.

An old feature request of mine on Meta.SO might also be of interest in this situation: Preventing misuse of question self-deletion

• SE could block you from accessing the site, if you are not careful. I've tried to be careful as far as this is concerned. I waited 5 seconds between any two downloads, I hope that it is sufficient. Anyway I agree with what you wrote here: My main point is that the moment you're scripting the SE interface to achieve your goal, it indicates that the tools that are available are insufficient. I'd rather convince SE to improve those than hack around these deficiencies. – Martin Sleziak Feb 21 '13 at 14:34
• I don't have a high expectation that the average MSE user playing the role of the answerer would simultaneously: realize we want things to be flagged; be motivated enough to do it; know how to go about doing so. Also, those who answers in comments are much less likely to notice this happening. – user14972 Feb 21 '13 at 15:37
• "there is always one user that likely notices such behaviour, and that is the user that answered the question" How will they notice? I don't know how I would notice if one of 361 questions I answered so far got self-deleted (in many of these questions the answer has 0 score). – user53153 Feb 23 '13 at 3:58
• @MartinSleziak Good luck convincing SE... The 3.5-year-old request Show all of my question/answers to me even if they are deleted has the score of $\mathbf{354}$ and status-declined. Atwood responded to it with "Will not get implemented as long as I am still alive to prevent it from happening." -- and got downvoted to $\mathbf{-107}$ as of now. – user53153 Feb 24 '13 at 0:51
• @5pm I don't know how I would notice if one of 361 questions I answered so far got self-deleted. I know of two possibilities where it is possible to notice that a question has been removed. One case is when the question is in your favorites. Another possibility is to save your reputation breakdown to a text file from time to time. If you then compare two versions, you can see which answers lost rep. (But this only work if your answer has been upvoted. IIRC robjohn mentioned in chat that he saves this file.) – Martin Sleziak Mar 7 '13 at 6:53
• @Martin Thanks but unfortunately neither would help me notice. I use favorites to collect questions I intend/hope to answer, and consequently remove them from favorites once they are answered. And since the unupvoted answers are the ones easiest to remove, reputation count does not help as much. // It's no big deal, of course: answers are not academic papers. – user53153 Mar 7 '13 at 12:59
• @5pm Of course, you are right about upvoted answers. Such questions cannot be deleted by OP. I knew that - I'm not sure how I have overlooked this problem with my suggestion. – Martin Sleziak Mar 7 '13 at 13:07

I will add that in SEDE there is the table PostsWithDeleted which contains data with deleted posts. This gives possibility to find links to deleted questions and if the users with 10k+ reputation can also view them.

For example, it is possible to find all recently deleted question, such query is listed in this answer: Are there some tools to follow deletions on this site?

In particular, we could also try to look for question which could have been deleted by the OP after receiving an answer: query1, query2.

These queries search for questions which have only one answer (a question with two answers cannot be deleted by the OP) and where the answer does not have positive score (if the answer has an upvote, the OP can no longer self-delete).

Still there are som issues with these queries:

• There will be some false positives - since we can use the query to check whether the question satisfies the criteria for the self-deletion to be allowed, but it's still possible that it was deleted in some other way.
• Some questions can be missed by the first query - since we count all deleted answers; if some of the answers were deleted by the answerer and only one remained, the OP is still allowed to delete. (In the second query I only take into account the answers which have the same DeletionDate as the question.)
• The data in SEDE are only updated once a week - in this way we do not find the most recent deletions.


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# incenter of a triangle problems

Posted on Posted in Uncategorized

Word problems on sets and venn diagrams. Only in the equilateral triangle, the incenter, centroid and orthocenter lie at the same point. OTHER TOPICS One of the problems gives a triangle and asks you to construct the incenter, or as it is put, "the intersection of angle bisectors." is represented by 2c, and. It's been noted above that the incenter is the intersection of the three angle bisectors. Incenter-Incircle. the missing component in this study is a . The incenter is the center of the incircle for a polygon or insphere for a polyhedron (when they exist). to support its claims, the company issues advertisements claiming that 8 out of 10 people (chosen randomly from across the country) who tried their product reported improved memory. The point of intersection of angle bisectors of a triangle is called the incenter of the triangle. The second equality follows from the law of sines. Incenter of a Triangle Exploration (pg 42) If you draw the angle bisector for each of the three angles of a triangle, the three lines all meet at one point. Incenter- Imagine that there are three busy roads that form a triangle. Find ,nLADC. The incenter is always located within the triangle. In this video you will learn the basic properties of triangles containing Centroid, Orthocenter, Circumcenter, and Incenter. of the Incenter of a Triangle. It is also the interior point for which distances to the sides of the triangle are equal. Answers and Explanations. Let ABC be a triangle whose vertices are (x 1, y 1), (x 2, y 2) and (x 3, y 3). Percent of a number word problems. Problem. Added 5 minutes 54 seconds ago|1/22/2021 7:06:36 AM Problem 1 (USAMO 1988). Point I is the incenter of triangle CEN. The circumcenter is the intersection of which 3 lines in a triangle… Incenter: Where a triangle’s three angle bisectors intersect (an angle bisector is a ray that cuts an angle in half); the incenter is the center of a circle inscribed in (drawn inside) the triangle. Incenter of Triangles Students should drag the vertices of the triangle to form different triangles (acute, obtuse, and right). The center of the triangle's incircle is known as incenter and it is also the point where the angle bisectors intersect. Show that its circumcenter coincides with the circumcenter of 4ABC. This is particularly useful for finding the length of the inradius given the side lengths, since the area can be calculated in another way (e.g. Word problems on ages. You want to open a store that is equidistant from each road to get as many customers as possible. Theorems and Problems about the Incenter of a triangle Read more: Incenter of a triangle, Collection of Geometry Problems Level: High School, SAT Prep, College geometry. A right triangle has one $\text{90^\circ }$ angle, which is often marked with the symbol shown in the triangle below. The incenter is one of the triangle's points of concurrency formed by the intersection of the triangle's 3 angle bisectors. Let , , for convenience.. 26 degrees. The incenter of a triangle is the point 2. Definition. 27. $\begingroup$ @MathTise The first equality is a property of bisectors in any triangle. No comments: Post a Comment. Triangle has , , , and .Let , , and be the orthocenter, incenter, and circumcenter of , respectively.Assume that the area of pentagon is the maximum possible. The perpendicular bisectors of a triangle are concurrent. AD and CD are angle bisectors of AABC and ,nLABC = 1000. Given two integers r and R representing the length of Inradius and Circumradius respectively, the task is to calculate the distance d between Incenter and Circumcenter.. Inradius The inradius( r ) of a regular triangle( ABC ) is the radius of the incircle (having center as l), which is the largest circle that will fit inside the triangle. Circumcenter And Incenter - Displaying top 8 worksheets found for this concept.. The incenter point always lies inside for right, acute, obtuse or any triangle types. It is also the center of the triangle's incircle. The incenter of a triangle is the intersection point of the angle bisectors. Grade: High School This applet allows for the discovery of the incenter and incircle of a triangle. Challenge Quizzes Triangle Centers: Level 2 Challenges Triangle Centers: Level 3 Challenges Triangle Centers: Level 4 Challenges Triangles - Circumcenter . LT 14: I can apply the properties of the circumcenter and incenter of a triangle in real world applications and math problems. Labels: incenter, incircle, triangle. Construct two angle bisectors. TRIANGLE: Centers: Incenter Incenter is the center of the inscribed circle (incircle) of the triangle, it is the point of intersection of the angle bisectors of the triangle. Creating my incenter for point J. Medial Triangle Attempt The perpendicular bisectors of A XYZ intersect at point W, WT = 12, and Theorem for the Incenter. Log in for more information. The incenter can be constructed as the intersection of angle bisectors. Then, as , it follows that and consequently pentagon is cyclic. The corresponding radius of the incircle or insphere is known as the inradius.. The internal bisectors of the three vertical angle of a triangle are concurrent. Improve your math knowledge with free questions in "Construct the circumcenter or incenter of a triangle" and thousands of other math skills. The altitudes of a triangle are concurrent. Formula: Coordinates of the incenter = ( (ax a + bx b + cx c )/P , (ay a + by b + cy c )/P ) Where P = (a+b+c), a,b,c = Triangle side Length Consider the triangle whose vertices are the circumcenters of 4IAB, 4IBC, 4ICA. The centroid is _____ in the triangle. Then you can apply these properties when solving many algebraic problems dealing with these triangle … Ratio and proportion word problems. The incenter is the position where angle bisectors converge in a triangle. Their common point is the ____. Finding the incenter would help you find this point because the incenter is equidistant from all sides of a triangle. The incenter is deonoted by I. Remark Suppose r is the distance from the incenter to a side of a triangle. It's well-known that , , and (verifiable by angle chasing). This point of concurrency is called the incenter of the triangle. If. If. a triangle ; meet at a point that is equally distant from the three side ; of the triangle. If your answer is yes, that means the manufacturer of clock has used concept of incenter to make sure center of clock coincides exactly with the incenter of the triangle inside which the clock is inscribed. If the coordinates of all the vertices of a triangle are given, then the coordinates of incircle are given by, (a + b + c a x 1 + b x 2 + c x 3 , a + b + c a y 1 + b y 2 + c y 3 ) where CA) 800 900 (E) 1400 1000 28. The area of the triangle is equal to s r sr s r.. Word problems on constant speed. Pythagorean theorem word problems. What is ?. Also draw a circle with center at the incenter and notice that you can make an inscribed circle (the circle touches all three sides). a. centroid b. incenter c. orthocenter d. circumcenter 19. Problem 2 (CGMO 2012). The incenter of a triangle is the intersection point of the _____ bisectors. Orthocenter. The incenter is the center of the incircle. Time and work word problems. Use this online incenter triangle calculator to find the triangle incenter point and radius based on the X, Y … A bisector of a triangle converges at a point called triangle incenter that is equally distant from the triangle sides. The radius of incircle is given by the formula r=At/s where At = area of the triangle and s = ½ (a + b + c). is represented by 2b + c, find the value of b. Recall that the incenter of a triangle is the point where the triangle's three angle bisectors intersect. Remember that each side of the triangle is tangent to the circle, so if you draw a radius from the center of the circle to the point where the circle touches the edge of the triangle, the radius will form a right angle with the edge of the triangle. The formula first requires you calculate the three side lengths of the triangle. all the angle bisector of traingle always lies inside the triangle, and their point of concurrency that is in center also lies inside the traingle hence option A is answer. Where all three lines intersect is the center of a triangle's "incircle", called the "incenter": Try this: find the incenter of a triangle using a compass and straightedge at: Inscribe a Circle in a Triangle. Triangle ABC has incenter I. For a triangle with semiperimeter (half the perimeter) s s s and inradius r r r,. Word problems on average speed Word problems on sum of the angles of a triangle is 180 degree. a. always b. sometimes 1. It is also call the incenter of the triangle. Their common point is the ____. Posted by Antonio Gutierrez at 1:14 PM. Draw a line segment (called the "altitude") at right angles to a side that goes to the opposite corner. An energy drink company claims that its product increases students' memory levels. These three angle bisectors are always concurrent and always meet in the triangle's interior (unlike the orthocenter which may or may not intersect in the interior). 23. It is the center of the circle that can be inscribed in the triangle, making the incenter equidistant from the three sides of the triangle. 2. a. centroid b. incenter c. orthocenter d. circumcenter 20. $\endgroup$ – Lozenges Jun 28 '18 at 14:28 $\begingroup$ Please explain how B1-A1 and B1-C1 are perpendicular and then ∡A1-B1-C1=90∘, if B1-A1 bisects ∡B-B1-C and B1-C1 bisects ∡A-B1-B? Circumcenter of a right triangle is the only center point that lies on the edge of a triangle. The point where they intersect is the incenter. How to Find the Coordinates of the Incenter of a Triangle. Centroid Circumcenter Incenter Orthocenter properties example question. Incenter of a triangle - formula A point where the internal angle bisectors of a triangle intersect is called the incenter of the triangle. Incenter of a Triangle . 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Article Contents
Article Contents

# Bifurcation from infinity with applications to reaction-diffusion systems

• * Corresponding author: Kousuke Kuto

The second author was supported in part by the Ministry of Science and Technology of Taiwan, grant 105-2115-M-007-009-MY3. The third author was partially supported by JSPS KAKENHI Grant-in-Aid Grant Numbers 15K04948 and 19K03581. The fourth author would like to thank the Mathematics Division of NCTS (Taipei Office) for the warm hospitality and the support of the fourth author's visit to Taiwan. The fourth author was partially supported by JSPS KAKENHI Grant Numbers JP26287024, JP15K04963, JP16K13778 and JP16KT0022

• The bifurcation method is one of powerful tools to study the existence of a continuous branch of solutions. However without further analysis, the local theory only ensures the existence of solutions within a small neighborhood of bifurcation point. In this paper we extend the theory of bifurcation from infinity, initiated by Rabinowitz [11] and Stuart [13], to find solutions of elliptic partial differential equations with large amplitude. For the applications to the reaction-diffusion systems, we are able to relax the conditions to obtain the bifurcation from infinity for the following nonlinear terms; (ⅰ) nonlinear terms satisfying conditions similar to [11] (all directions), (ⅱ) nonlinear terms satisfying similar conditions only on the strip domain along the direction determined by the eigenfunction, (ⅲ) $p$-homogeneous nonlinear terms with degenerate conditions.

Mathematics Subject Classification: Primary: 35B32, 35B36; Secondary: 35B44.

 Citation:

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