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## April 30, 2009

### mellifluous \muh-LIF-loo-us\,

Flowing as with honey; smooth; flowing sweetly or smoothly; as, a mellifluous voice.

There's no honey in today's word of the day, just oil and water. I am most definitely not the first to photograph the two together, nor are my photos likely the best. But I would bet money that I'm the first to do so on a blog dedicated to dictionary.com's word of the day.

The shortened definition of today's word, as it shows up on my word-of-the-day widget, is flowing sweetly or smoothly, and while nothing in my photo is actually flowing, I thought it might be a fun photo. It's just set up with two lights, one pointed below a glass of water at the white backdrop and the other above where the oil will be. Then just add oil, and snap away.

Focus is typically the toughest part of this type of photo. I was lucky to get what I got, even though it's not necessarily that sharp. Bumping the contrast can sometimes help with that problem, though. And having a real, sturdy tripod might also work for you.

Anyhow, get out there and take some photos for mellifluous and send them along to me in an email so I can include them with mine! Happy Thursday!

Update: Thanks again to Ronny for sending this photo -- it's the first photo he ever sold, on permanent display in the Tally Student Center on N.C. State. I like when you guys do that, so keep sending them over (even retroactively)!

## April 29, 2009

### bumptious \BUMP-shuhs\,

Crudely, presumptuously, or loudly self-assertive.

The photo for today's word of the day, in my very humble opinion, is fantastic. This is a definite response to an earlier photo starring roommate Bryan, which was much more intense and not quite as conducive to blackmail.

Clowns, I think, are loudly self-assertive. I had originally planned to get Bryan in some ridiculous makeup, but the wig and nose were $10 at the Magic Corner (the place for wigs and noses, apparently), and I think they got the job done. The setup, as usual, was not too difficult. Two lights: one in front of the subject and to the right through an umbrella at relatively low power, and the other to his right and behind him at high power facing a white wall. I exposed for Bryan's (hilarious) face and so the light behind him was blown out, giving the side of his wig that effect of being almost feathered out at its end. This was definitely a fun one, and I hope you'll all have something to match it. Send me whatever you've got for bumptious in an email and I'll be sure to include it! Update: Another one from Ronny shows a friend, Clark, a little too close for comfort. Thanks, Ronny! ## April 28, 2009 ### eleemosynary \el-uh-MOS-uh-ner-ee\, adjective: 1. Of or for charity; charitable; as, "an eleemosynary institution." For today's word of the day, I headed down to the Shepher's Table Soup Kitchen on Morgan Street in Raleigh. The homeless problem in our city is pretty bad, and the line outside the door was probably about 50 people long before 11 a.m. And as a woman passing by the soup kitchen mentioned to me (rather bitterly), photographing people in line at a soup kitchen brings up an interesting ethical question: when is it OK to take someone's photo without their permission? The answer to that question is complex. There is, of course, a very easy answer to the question of legality: if you're in public and your subject is in public, you can pretty much just go ahead and shoot it (with a few exceptions). But on which side of the ethical line would you find yourself while shooting a dying man, a cancer patient, or homeless people without their permission? I think the answer is different on a case-by-case basis. For example, I didn't feel like the photo for today's word intruded on someone's privacy - not the sort of privacy you find while in your own home per se, but the "personal bubble," borderline harassment type. In cases where I expect my presence while taking someone's photo will affect their physical or mental being, I tend to talk to them in the first place; permission often makes the photo easier to get to, especially when you want to get close and keep your subject from being visibly awkward (or angry). I wouldn't categorize this blog as journalistic. If it was, though it might not be necessary, I might feel compelled to ask for the names of my subjects. Nor is it commercial. This blog, I think, is some sort of "art," and in that case, I don't feel I've crossed any ethical lines in today's photo. But there are many sides to this kind of conversation. So if you've got any thoughts, please put them into the "thoughts" section at the bottom of this post, and I'd love to discuss. In the meantime, eleemosynary is an interesting word of the day, and I'd love for you to send some photos along to me in an email! ## April 27, 2009 ### fi⋅an⋅cée /fee-ahn-sey; fee-ahn-sey/, noun: a woman engaged to be married; a woman to whom a man is engaged. Doing a little something different today: I got engaged yesterday, so I'm cheating for today's word of the day! This is going to be a little more personal than the other posts, but you'll just have to deal with that. It goes like this: yesterday, I got up at the crack of 3:30 a.m. and got Jamie, put her in the car, and told her we're going on a surprise vacation to the beach. It was pretty believable, I think, because we haven't had the same day off in months, and we had been talking about taking a few days off together. We went down to Wrightsville Beach and made it just before the sun came up. (Fantastic planning, I must say.) We watched the sun rise, and I proposed, and luckily for me, she said yes. Though, if she hadn't said yes, I would have left her here in her pajamas. We stayed over night in a hotel room my sister got us with her points from traveling (thanks Katie!) and we're still just hanging out today, which is why the photo for today's word is so fantastic. There's no color correction, or even cropping, because I just got engaged and I don't care that much about the photo. Hope you're all having a great Monday! I'll be back with a real word of the day tomorrow! ## April 24, 2009 ### gainsay \gayn-SAY; GAYN-say\, transitive verb: 1. To deny or dispute; to declare false or invalid. 2. To oppose; to contradict. Today's word of the day was a relatively easy one. I took it right down the street at the Salvation Army. (I think I've taken enough photos of that place for this blog to start a brochure. If you're reading this, Salvation Army, and you want to pay me, give me a call.) It's a pretty incredible day outside. There aren't any clouds in the blue sky, which can make for a pretty awesome background. As usual, I'll be working this weekend, so I won't be posting anything. But I encourage you all to look back at previous days' words and send any photos to me that might fit. I'll update the posts with your photos, and hopefully I'll find some way to showcase them retroactively. If you've got something good, send it to me in an email! ## April 23, 2009 ### paladin \PAL-uh-din\, noun: 1. A knight-errant; a distinguished champion of a medieval king or prince; as, the paladins of Charlemagne. 2. A champion of a cause. I didn't find too many knights wandering the streets downtown for today's word of the day. Instead, I met one of the nicest people in Raleigh: Miss Carol. Miss Carol apparently runs a hot dog stand on Fayetteville Street. (For those foreigners among you, Fayetteville Street is Raleigh's main strip during the day, with offices and restaurants and stuff. But at night it's fairly dead with the exception of a few little bars.) The side of her cart reads, "The Hot Dog Lady," and if she wasn't the most kind person alive, I might have inserted a dirty joke here. I told Miss Carol that today's word of the day is paladin, and asked her what her cause would be. "Just being the best person I can be," she said. "Helping everyone get through their day." And within the 45 seconds it took to introduce myself and take a photo, she was helping someone on the street get into the bank adjacent to her hot dog stand. That's her in the photo telling him what buttons to press to get in. And when it didn't work, she left her post to go around to the window and find someone to help. At least four people said "Hello, Miss Carol," on their way by. Evidently she's had some effect on the people that pass her cart every morning simply by standing on the same corner, day after day, being sweet. How refreshing; what better cause could there be? Anyway, if you happen to see any knights roaming the streets, or even someone championing a cause, send me your paladin photos in an email so I can include them with Miss Carol. Happy Thursday! ## April 22, 2009 ### invidious \in-VID-ee-uhs\, adjective: 1. Tending to provoke envy, resentment, or ill will. 2. Containing or implying a slight. 3. Envious. Today's word of the day showcases the ridiculously sized governor's mansion on Blount Street in Raleigh. It was extremely windy today, so riding my bike was less fun than it can be. But it's also awesome outside, so I might as well get a little exercise. Not much involved with today's photo. Shot it wider than I probably had to, but I can't stop using the 12-24. I'm obsessed. I think I have a problem. (And judging by Ronny's recent photos, I think he has the same problem.) Today's Earth Day (and, as it so happens, also my birthday), so I'm going to quit writing for now and go out and enjoy the day! I hope you all have an excellent Earth Day, and send me some photos of invidious in an email! ## April 21, 2009 ### peccadillo \peck-uh-DIL-oh\, noun: A slight offense; a petty fault. I got a bit of a new perspective for today's word of the day. I always say (or at least, I should always say) the dumber you look taking the photo, the more likely it is to come out well. In my opinion, that goes for most out-of-studio photos. Look for a different angle: get low or get up on top of something, and you're bound to find a perspective more interesting to viewers than what they can already see at eye level. Well that's the lesson for today, anyway. I stumbled across these windows (with bricks behind them, for some reason) on Salisbury Street in Raleigh. I actually could have sworn I'd seen them somewhere else, like on cell phone camera aficionado Shawn Rocco's blog (which is awesome if you get a chance to look through it), but I couldn't find it anywhere. That said, if you happen to come across them, let me know. It was an awesome day out, which is why I decided to hop on my bike in the first place, so it's nice to have gotten the blue sky in the photo. What you're looking at is just what I saw laying on the ground in a small parking lot. As always, if you've got a photo for peccadillo, then send it to me in an email! ## April 20, 2009 ### stormy petrel \STOR-mee-PET-ruhl\, noun: 1. Any of various small sea birds of the family Hydrobatidae, having dark plumage with paler underparts; also called storm petrel. 2. One who brings discord or strife, or appears at the onset of trouble. Today's word of the day, like so many before it, describes a bird whose name doubles as a way to describe someone. It's a little strange that the guys at Dictionary.com keep picking bird words, but I won't judge. Sadly, today's photo isn't quite what I was going for. I wanted to try my hand at an HDR photo, but since I've got to work on my usual Monday off, I won't have the time today to go gallivanting around taking pictures with a high dynamic range. But in any case, the sky did look pretty awesome today, and the word of the day folks seemed to finally get the word to match the weather. For once. If you've got a picture for stormy petrel, send it to me in an email. And though I may not have been clear before, you can always retroactively send a photo for any other word of the day, and I'll be happy to include it! ## April 17, 2009 ### ephemeral \ih-FEM-er-ul\, adjective: Short-lived; existing or continuing for a short time only. In the true fashion of 12-year-olds stuck in taller-but-just-as-squirrely men's bodies, friend Kevin and I set off a firecracker in my house for today's word of the day. I should probably publicly apologize to Josh, as we set the firecracker off in my "studio," which shares a wall with his room. He may or may not have been sleeping at the time, and if he was, it made for a less than enjoyable awakening. To be sure I would capture the small but extremely loud explosion, I brought the lights down for a 15-second exposure, which is why you can see both the sparks from the fuse as well as the light from the explosion. Kevin was standing by with a glass of water for safety (as if that would have done anything when I lost a finger). As always, I won't be posting anything this weekend. But if you've got any photos for ephemeral, send it to me in an email! Update: Thanks to Bryan for sending this photo of my brother with a sparkler for the word of the day. How exciting! ## April 16, 2009 ### etiolate \EE-tee-uh-layt\, transitive verb: 1. (Botany) To bleach and alter the natural development of (a green plant) by excluding sunlight. Figures that the only word of the day that involves the exclusion of sunlight falls on the only day this week without rain. But no matter, I went for a little bike ride anyway. It took me a while to find an idea I wanted to stick with and actually follow through. What I ended up with was a long exposure (the longest I could get in the shade on a nice day like this was about four seconds), shining a flashlight on the bright leaf you see in the photo. I'm thinking about another project that may be fun with this same idea, but you'll have to wait for that. And by the way, did you notice the fantastic lightbox that Tyler helped me put my photos into? If not, click on the photos to see what I mean. Hopefully some of my many readers with botany backgrounds have some good photos for etiolate. If you do, send them to me in an email! ## April 15, 2009 ### outré \oo-TRAY\, adjective: Unconventional; eccentric; bizarre. Today's word of the day also happens to be Pig Latin for "true" (starting the blog off right with a crappy joke today). It also made me realize just how normal so many people walking around Raleigh are. Unfortunately, I started off the day with one shot and then a dead battery. While I was walking back to my car, however, I happened to pass a man with four-foot dreads and a cross dresser. But these things happen, and I found a harp player instead, playing for a little crowd at the farmer's market on Moore Square today. I also happened to see Ronny out shooting, which is always a lovely experience. He was out shooting video for work, and showed me the harp player. Shorter post today, but hopefully you've got a few photos for outré that you can send my way via email! Update: Thanks to Jessi for sending along the first photo in weeks. It is most definitely bizarre, and quite fantastic. Keep 'em coming! ## April 14, 2009 ### labile \LAY-byl\, adjective: 1. Open to change; apt or likely to change; adaptable. 2. Constantly or readily undergoing chemical, physical, or biological change or breakdown; unstable. I love Mondays and Tuesdays, because since I get those days off from work, I can focus on (extremely more enjoyable) photo projects and my word of the day. For today's, I went downtown looking for a construction site. Today's project is a do-it-yourself tilt/shift lens. Similar, non-do-it-yourself lenses, like the recently released arsenal of Canon tilt/shifts, can cost over$1,000, and I haven't really got the money for anything like that. A tilt/shift lens works like any other lens, but has a very big advantage (and again, for those uninterested in the technical side of these things, go ahead and click here).

The depth of field (or, area of focus) on any regular lens attached to the camera is like a wall. That is to say, the range of focus will be parallel to your camera body. With a regular lens, that depth of field can be manipulated so that the "wall" is thinner or thicker, farther away or closer to the camera. (For more on camera basics, visit the best reference for basic camera functions. Ever.)

With a tilt/shift lens, the lens can move independent from the camera body, allowing that "wall" to move be manipulated in ways a normal lens wouldn't allow. Now the "wall" isn't necessarily parallel to your camera body. It can be at any reasonable diagonal, which can cause severe blurriness in whatever parts of the photo you choose. It works like the "bellows" of a large-format camera.

My do-it-yourself brand of tilt/shift is a little less sharp than what I'm sure the Canon lenses look like. But the concept is there, and it works well. (I used a lot of sources for reference on this, but here's the one that I think helped the most.) The most expensive part of this project was finding a medium-format lens (I sort of regret going with the cheapest I could find: a $70 Bronica from ebay), which focuses farther from the censor than a 35mm lens, allowing me to bring it away from the camera for tilts and shifts. To connect it to the camera, I just used black cloth, a body cap with a hole in it, and a lot of gaffer tape. Hopefully I'll be able to use this lens for more than just today's project. I hope to bring you sharper focus with the next one, but for now, go out and take some labile photos, and send them to me in an email! ## April 13, 2009 ### susurrus \su-SUHR-uhs\, noun: A whispering or rustling sound; a murmur. Sorry about the late post for today's word of the day. I spent a little while today working on a project for the backlight that goes along with the picture. Thanks firstly goes to Josh and Bryan for modeling in this shot. I know you can't see their faces, but that's what makes the photo so good. (Too harsh?) It's easily done with a flash behind them, keeping all the light off their faces. This flash was actually at about half power and the aperture was small, letting only a little light in. The room was lit with a regular overhead bulb, but it has no effect with a fast shutter and a small aperture. On the flash was a homemade "gridspot," made from a stolen sign that you would find on the side of the road (the type that politicians use for their campaigns). You can find the directions here, and they're pretty easily constructed. The light is a reddish purple because of a piece of red saran wrap fashioned over the flash with a rubber band. I could probably afford some gels, but who needs 'em? Hope you liked this (better late than never) word of the day. If you've got any photos for susurrus, send them to me in an email! ## April 10, 2009 ### frisson \free-SOHN\, noun: A moment of intense excitement; a shudder; an emotional thrill. Today's word of the day showcases a former habit of mine that I've kicked. But this photo may be the thing that gets me back off (or is it on?) the wagon. I did manage to win$1 with this lottery ticket (which actually cost $1 in the first place), so at least I don't feel like I got ripped off. I'm sure I'll lose the ticket before I end up back at the gas station to redeem my huge prize. Setup was, as always, pretty simple: two lights, both through umbrellas (as I've found the missing piece to my umbrella adapter) on either side of the lottery ticket, which was just right on top of a sheet I found at the thrift store. They really do have the best props and backgrounds. I've got to work this weekend (just like every weekend), so I won't be posting again until Monday. But hopefully you'll all be out taking photos of frisson (though, if the past is any indicator, you probably won't). If you've got a good one, send it to me in an email! ## April 9, 2009 ### impugn \im-PYOON\, transitive verb: To call in question; to make insinuations against; to oppose or challenge as false; to gainsay. Today's word of the day was a quick one. As some of you may have guessed, I am very important and often have tons of things to do. So thankfully, Josh and Lauren were around to help with today's word. We had to roll Josh out of bed, but it made the whole thing a lot easier since he was already angry. The lighting setup for this shot is not very complicated: two bare flashes on either side of the subject at half power, and the aperture on the camera very small to block out a good bit of the bright sunlight and put the focus on my angry subjects. We were out in the street in front of my house, which was only a little bit terrifying. Update: I reposted this photo with a little texture over it by taking a photo of some dirt/grass/whatever and using layer mode "overlay." It's still not my best shot, but it's at least a little more interesting! Short post today, but if you've got any photos for impugn, send them to me in an email! ## April 8, 2009 ### chimerical \ky-MER-ih-kuhl; -MIR-; kih-\, adjective: 1. Merely imaginary; produced by or as if by a wildly fanciful imagination; fantastic; improbable or unrealistic. 2. Given to or indulging in unrealistic fantasies or fantastic schemes. Today's word of the day just had so many possibilities. Sadly, a small lack of time and slightly larger lack of money kept me from doing anything too elaborate, but those are ideas I'll be saving for another day. Instead, I donned my flight gear and took to a field where I put the camera on self-timer and just took a couple flying laps around. Maybe it was chimerical, or maybe it was the wind, but I could have sworn I was really flying around out there for a little while. Of course, no one was with me to corroborate that story. So if I wasn't, I just looked like an idiot running and jumping around in a field. It's a beautiful day for flying, so get out there and get some chimerical photos, and send them to me in an email! ## April 7, 2009 ### nescience \NESH-uhn(t)s; NESH-ee-uhn(t)s\, noun: Lack of knowledge or awareness; ignorance. Today's word of the day features one of my roommates, Bryan. For those of you who may be confused about the influx of roommates I have with the name Bryan, I've only got two. Perhaps I should give you a little background about my usual models in this post. Today's Bryan (also featured in spoonerism) moved in with us just a few months ago. He's great at Fifa '09 and sports a stylish mustache. Then, there's another Bryan (yes, both spell their names with a "y") who has lived here all along. We were roommates last year; he's a singer and designer and skinny lover (most famous work includes ululate). And then there's Josh, owner and operator of Silver Lining Brewry (based out of my kitchen). We've lived together since freshman year of college. His most important work was in inanition. So for nescience today, I used the old Canon FL 55mm f/1.2 (from 1968-ish) featured in my very first post. It give the photo its signature ring of soft light around whatever it's "focused" on, likely because it's relatively ancient. But it give some great colors that I always enjoy. It's beautiful out, so take some nescience photos, and send them to me in an email! ## April 6, 2009 ### aegis \EE-jis\, noun: 4. A shield or protective armor; -- applied in mythology to the shield of Zeus. Today's word of the day post may turn into a quick rant about how technology ruins my life, but I'll try to keep that to a minimum. First off, I've been in cahoots with the folks from dictionary.com. I was hoping to get a word of the day a few weeks in advance to plan something fantastic for you, my many readers. But instead, they sent the words for today, tomorrow and Wednesday. It's just as well, though; I used that opportunity to shoot in an empty building down the street with my brother. Now for why technology is terrible: after shooting last night, I plugged my card reader into the computer and lost everything. I'm still not sure why it happened, but when I went to retrieve the photos, there was nothing there. In fact, the only reason I have anything for you at all is because I downloaded a recovery software that costs$130 if you want the full-sized image back.

And that's why today's photo is so small (when you click on it). I ended up just taking a screen shot of the tiny photo in the recovery program, which is unfortunate.

And for the photo itself: it's a pretty simple, but very fun way of using light. There's a flash to the right with a softbox pointed at Steven, my brother (thanks for modeling!), and another in the back shot through an umbrella to the wall. The "bars" in the photo are streaks of light made with LED key chains. Helen and I drew them in while Steven hung out looking awesome.

Well that's it for my anti-technology post. Luckily, there wasn't much more than a few photos and the 7th Harry Potter in mp3 on the card. But hopefully it won't happen again.

As always, if you've got a photo for aegis, send it to me in an email!

## April 3, 2009

### longueur \long-GUR\,

noun:
A dull and tedious passage in a book, play, musical composition, or the like.

This word of the day brings me right back to my high school days, when I thought everything I read was longueur. Of course, now I read only the most complicated and rambling pieces of literature.

Bryan insists that his book, One Hundred Years of Solitude, is neither dull nor tedious, but the title implies something different. He was already reading at the Morning Times coffee shop downtown, so there wasn't much setup involved. The windows let in a good bit of natural light, accented by the soft yellow of the restaurant's tungsten bulbs in the ceiling.

Lucky for me, the Times has free Wi-Fi, so I can still hang out a while after the photo.

A few have asked how I shot this: I was standing on a chair opposite from Bryan, holding the camera out over his head. Lots of practice shooting this way has gotten me to the point that I can usually aim pretty well without looking through the viewfinder, though it may still take a few tries. I was not hanging from the ceiling mission impossible style.

This is a tough one, but if you've got any longueur photos, send them to me in an email! As usual, I won't be posting this weekend because of work, so you should have more time to shoot and submit!

## April 2, 2009

### gambol \GAM-buhl\,

intransitive verb:
1. To dance and skip about in play; to frolic.

Too bad this excellent word of the day fell on such a dreary, depressing, rainy day. But it's alright, because many of my friends tend to gambol even on rainy days, so it wasn't hard to convince them to go out frolicking for just a little while.

We went to the second most popular place to photograph Raleigh: Western and Salisbury, just south of downtown (the first most popular being the Boylan Street Bridge). There's a nice little grassy spot where you can frolic if you pull off the road and don't let the authorities get at you.

Today I shot with my favorite, the Tokina 12-24 f/4, but it tended to be just a little too wide that far away from town. But no worries, the girls filled the frame really well with their dancing feet. I'm starting to discover that I like to use people's feet and legs in my photos sometimes, as opposed to their faces, to provide a sort of anonymity and so that the subject isn't just a face, but the entire frame.

Hopefully some of my many readers will brave the rain to go out and do a little gamboling (not gambling), and if you do, send me your photos in an email!

## April 1, 2009

### jape \JAYP\,

noun, verb:
1. A joke or jest.
2. A trick or prank.

The guys at dictionary.com are clearly in an April Fool's Day mood with today's word of the day. I didn't even know it was April 1 until I guessed after seeing their word.

So today, I looked online and found a magic shop down Capital Boulevard in Raleigh. When I got to the little store, there was a pudgy little man in red suspenders and a little yipping white dog, so I knew I was in the right place. I've never been in a magic shop before, but that's how I would have imagined it.

I bought that little flower for \$2 and brought it back home to shoot. For lights, one flash camera left facing subject (myself) in that homemade softbox (only because I lost a piece for my other umbrella adapter) and one flash through an umbrella camera right, pointed where the water would go.

Also, did anyone note that this is the second time in as many days that I've worn a suit in the photo? Classy.

Anyhow, if you've got any April Fool's Day photos that will go with jape, send them to me in an email!


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# some interesting examples and families¶

PHCpack has been tested on many examples of polynomial systems taken from the research literature. The module examples exports some of those examples. Running python examples.py at the command prompt performs a regression test, solving all examples.

Polynomial systems often occur in families and are defined for any number of equations and variables.

## interactive regression testing¶

An interactive use of examples.py at the Python prompt can go as follows:

>>> from phcpy.examples import noon3
>>> f = noon3()
>>> for p in f: print(p)
...
x1*x2^2 + x1*x3^2 - 1.1*x1 + 1;
x2*x1^2 + x2*x3^2 - 1.1*x2 + 1;
x3*x1^2 + x3*x2^2 - 1.1*x3 + 1;

The functions in examples.py returns the polynomials as lists of strings. If we want to solve the system defined by f, we continue the above session as

>>> from phcpy.solver import solve
>>> s = solve(f,silent=True)
>>> len(s)
21
>>> print(s[0])
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
x1 : -1.65123467890611E-01  -7.61734168646636E-01
x2 :  8.98653694263692E-01  -3.48820047576431E-01
x3 :  8.98653694263692E-01  -3.48820047576431E-01
== err :  3.034E-16 = rco :  2.761E-01 = res :  5.974E-16 =

The example session continues in the description of the module solutions.

## the cyclic n-roots problem¶

One such noteworthy family is the cyclic n-roots problem:

>>> from phcpy.families import cyclic
>>> c4 = cyclic(4)
>>> for p in c4: print(p)
...
x0 + x1 + x2 + x3;
x0*x1 + x1*x2 + x2*x3 + x3*x0;
x0*x1*x2 + x1*x2*x3 + x2*x3*x0 + x3*x0*x1;
x0*x1*x2*x3 - 1;

format: plain


text:
# Second order differential equation

hi. I cant figure out this question:

d2y/dx2 - 2 dy/dx - 3y = x

(i) find complementary function
(ii) find particular integral
(iii) using (i) and (ii) find the general solution
(iv) find the solution that satisfies the initial conditions:
y=2/9 at x=0 and dy/dx=-13/3 at x=0

i did:

m^2 - 2m - 3 = 0
(m-3)(m+1)=0
real and distinct solutions thus

y = Ae^(3x) + Be^(-x)

thus dy/dx = 3Ae^3x - Be^-x

d2y/dx2 = 9Ae^3x + Be^-x

now i have no idea how to continue. As i understood what i found above is the complementary function. I think. Thank you

rock.freak667
Homework Helper
y = Ae^(3x) + Be^(-x)

Yes this would be the complementary solution.

What the particular integral for a polynomial?

i think its something to do with the x on the right hand side but i dont know how to do it. Do i integrate that side?

I know when you have a y in the equation you integrate the constants infront of the y. But in this case i dont have this.

HallsofIvy
Try a specific integral of the form $Ax^2+ Bx+ C$. What must A, B, and C equal to satisfy the equation? Recall that the general solution to the entire equation is the general solution to the associated homogeneous equation (the "complementary solution") plus a specific integral.


text:
# 空心线圈传感器的变阻尼标定方法研究 Research on Variable Damping Calibration Method of Air-Core Coil Sensor

DOI: 10.12677/JSTA.2020.84013, PDF, HTML, XML, 下载: 183  浏览: 420

Abstract: The inductive air-core coil sensor is an important time-varying magnetic field measurement tool. Due to the inherent inductance and distributed capacitance of the coil, the output voltage of the coil cannot accurately follow the sudden induced voltage. This phenomenon is called the transition process of the coil. The existing correction methods can only be completed by the laboratory, and it is difficult to calibrate the parameter disturbance caused by the field environment. This paper takes the air-core coil sensor used in transient electromagnetic detection as an example, and proposes a scheme to eliminate the transition process by using the transfer function. By analyzing the equivalent circuit of the receiver coil, a variable damping coil calibration method suitable for solving the parameters of the receiver coil in the field environment is proposed. The device has a simple form and high reliability, which meets the needs of field calibration in the construction environment.

1. 引言

Figure 1. The effect of the transition process of the receiving coil on the signal

2. 空心线圈的过渡过程

Figure 2. Equivalent circuit of receiving coil distributed parameter

$H\left(s\right)=\frac{u\left(s\right)}{\epsilon \left(s\right)}=\frac{1}{{s}^{2}LC+s\left(\frac{L}{{R}_{b}}+RC\right)+\frac{R+{R}_{b}}{{R}_{b}}}$ (1)

$\frac{\epsilon \left(t\right)}{LC}=\frac{{\text{d}}^{2}u\left(t\right)}{\text{d}{t}^{2}}+2\delta \frac{\text{d}u\left(t\right)}{\text{d}t}+{\omega }_{p}^{2}u\left(t\right)$ (2)

$\xi =\frac{{\delta }_{1}}{{\omega }_{p}}=\frac{{R}_{b}RC+L}{2\sqrt{LC{R}_{b}\left(R+{R}_{b}\right)}}$ (3)

${R}_{0}=\frac{L}{RC+2\sqrt{LC}}$ (4)

3. 变阻尼标定法

$\iint v\left(t\right)\text{d}{t}^{2}=\frac{{\tau }_{l}{\tau }_{c}}{\alpha }u\left(t\right)+\left({\tau }_{l}+{\tau }_{c}\right)\int u\left(t\right)\text{d}t+\left(1+\alpha \right)\iint u\left(t\right)\text{d}{t}^{2}$ (5)

0.8倍关系选取三个阻尼电阻值，获得三个参数方程。

$\left\{\begin{array}{l}V\left(t\right)={m}_{1}×{u}_{1}\left(t\right)+{m}_{2}×\int {u}_{1}\left(t\right)\text{d}t+\left(1+\alpha \right)×\iint {u}_{1}\left(t\right)\text{d}{t}^{2}\\ V\left(t\right)={m}_{1}×{u}_{2}\left(t\right)+{m}_{2}×\int {u}_{2}\left(t\right)\text{d}t+\left(1+\alpha \right)×\iint {u}_{2}\left(t\right)\text{d}{t}^{2}\\ V\left(t\right)={m}_{1}×{u}_{3}\left(t\right)+{m}_{2}×\int {u}_{3}\left(t\right)\text{d}t+\left(1+\alpha \right)×\iint {u}_{3}\left(t\right)\text{d}{t}^{2}\end{array}$ (6)

${u}_{2}\left(t\right)$${u}_{3}\left(t\right)$ 可以通过A/D采样直接得到。求解方程组(6)便可以确定参数 ${m}_{1}$${m}_{2}$，进而获取线圈的传递函数 $H\left(s\right)$

$\eta =\mathrm{max}\left\{\begin{array}{l}|\frac{L-{L}_{ct}}{L}|\ast 1000\\ |\frac{C-{C}_{ct}}{C}|\ast 1000\end{array}\begin{array}{c}‰\\ \begin{array}{l}\\ ‰\end{array}\end{array}$ (7)

Figure 3. The measured voltage curve of the variable damping calibration method

4. 讨论

4.1. 采样率对结果的影响

4.2. 阻尼电阻Rb对结果的影响

${R}_{b}$ 按临界值的不同倍数关系选取三个阻尼电阻值，在相同的激励源作用下，得到三组响应信号 ${u}_{1}\left(t\right)$${u}_{2}\left(t\right)$${u}_{3}\left(t\right)$ 组成方程组，考察阻尼电阻的选取对参数求解精度的影响。

Table 1. The influence of sampling rate on parameter solution error rate

Table 2. The relationship between parameter error rate and tolerance of damping resistance

4.3. 积分区间对结果的影响

Table 3. The influence of the integration starting point on the error rate of parameter solution

5. 实验结果

(a) (b) (c)

Figure 4. Comparison of apparent resistivity imaging results before and after the transition process, (a) the schematic diagram of the air-raid shelter distribution; (b) before the transition process; (c) after the transition process

6. 小结

 [1] Smith, R.S. and Balch, S.J. (2000) Robust Estimation of the Band-Limited Inductive-Limit Response from Im-pulse-Response TEM Measurements Taken during the Transmitter Switch-Off and the Transmitter Off-Time. Theory and an Example from Voisey’s Bay, Labrador, Canada. Geophysics, 65, 476-481. https://doi.org/10.1190/1.1444741 [2] Wang, Z., Lin, J., Yu, S.B., Ji, Y.J. and Zhou, G.H. (2006) ATTEM: An Instrument System Using Transient Electromagnetic Pulse for Subsurface Imaging. IEEE Instrumentation and Meas-urement Technology Conference, Sorrento, 2231-2235. [3] Spies, B.R. (1989) Depth of Investigation in Electromag-netic Sounding Methods. Geophysics, 54, 872. https://doi.org/10.1190/1.1442716 [4] 谢伟伦, 黄志威, 刘小沛. 杂散电容对CVT谐波测量影响的处理方法研究[J]. 电力电容器与无功补偿, 2020, 41(2): 59-63. [5] Yu, C.G. and Fu, Z.H. (2014) Transient Process and Op-timal Design of Receiver Coil for Small-Loop Transient Electromagnetics. Geophysical Prospecting, 62, 377-384. https://doi.org/10.1111/1365-2478.12093 [6] Lin, J. (2014) Research and Development on the Air-Core Coil Sensor for Mine Transient Electromagnetic Exploration. Chinese Journal of Geophysics, 59, 721-730. [7] Chu, X.Y. and Zhang, S. (2012) An Optimal Transfer Characteristic of an Air Cored Transient Electromagnetic Sensor. 2012 Inter-national Conference on Industrial Control and Electronics Engineering, Xi’an, 23-25 August 2012, 482-485. https://doi.org/10.1109/ICICEE.2012.134 [8] Hemmati, E. and Shahrtash, S.M. (2013) Digital Compensation of Rogowski Coil’s Output Voltage. IEEE Transactions on Instrumentation and Measurement, 62, 71-82. https://doi.org/10.1109/TIM.2012.2212502 [9] 万云霞, 王言章, 程德福. 感应式磁传感器标定磁场处理方法研究[J]. 传感技术学报, 2013, 26(4): 520-524. [10] Gurleyuk, S.S., Taskin, H. and Sarac, Z. (2012) Measurement of the Parameters and the Resonance Frequency in Semiconductor Controlled Tesla Transformer. International Journal of Electrical Power and Energy Systems, 43, 6-10. https://doi.org/10.1016/j.ijepes.2012.05.026 [11] Kozhevnikov, N.O. (2012) Testing TEM Systems Using a Large Horizontal Loop Conductor. Russian Geology and Geophysics, 53, 1243-1251. https://doi.org/10.1016/j.rgg.2012.09.010 [12] Persova, M.G., Soloveichik, Y.G., Trigubovich, G.M., et al. (2014) Transient Electromagnetic Modelling of an Isolated Wire Loop over a Conductive Medium. Geophysical Prospecting, 62, 1193-1201. https://doi.org/10.1111/1365-2478.12122 [13] Davis, A. and Macnas, J. (2008) Quantifying AEM System Char-acteristics Using a Ground Loop. Geophysics, 4, F179-F188. https://doi.org/10.1190/1.2943189 [14] Wang, H.M., Liu, F., Zhang, H.L. and Zheng, S.X. (2005) Analysis of the Thermal Expansion Effect on Measurement Precision of Rogowski Coils. International Conference on Power Electronics and Drives Systems, Kuala Lumpur, 1658-1661. [15] 林君, 王琳, 王晓光, 等. 矿井瞬变电磁探测中空芯线圈传感器的研制[J]. 地球物理学报, 2016, 59(2): 721-730.


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Let $\triangle ABC$ be an isosceles triangle such that $BC = 30$ and $AB = AC.$ We have that $I$ is the incenter of $\triangle ABC,$ and $IC = 18.$ What is the length of the inradius of the triangle?

michaelcai  Jul 27, 2017
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Let triangle ABC be an isosceles triangle such that BC = 30 and AB = AC.
We have that I is the incenter of triangle ABC, and IC = 18.
What is the length of the inradius of the triangle?

$$\begin{array}{|rcll|} \hline \left( \frac{30}{2} \right)^2 + r^2 &=& 18^2 \\ 15^2 + r^2 &=& 18^2 \\ r^2 &=& 18^2-15^2 \\ r^2 &=& 324-225 \\ r^2 &=& 99 \\ r^2 &=& 9\cdot 11 \\ r &=& 3\cdot \sqrt{11} \\ \hline \end{array}$$

r = 9.95

heureka  Jul 28, 2017

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# Regulating Cryptocurrencies and the Investors’ Search for Clarity

Few regulations around cryptocurrencies exist but institutional investors, already wary of fully entering the market, want countries to provide clear statements laying out how these new assets will be treated. By Emilia David

In the institutional space, regulatory bodies have taken a soft touch when it comes to cryptocurrencies has stunted adoption, but Facebook’s proposed virtual currency, Libra, has helped to put the spotlight back on digital asset regulation.

The US, like most large economies, has been cautious in releasing regulations governing cryptocurrencies and their derivatives. While most countries have not enacted policies around these instruments, institutional investors are still interested in entering the cryptocurrency market and diversifying their investments. Some countries, particularly smaller economies, have managed to come up with a few rules, but there has not been a large take-up in many of the more established economies yet.

Institutional investors, however, have made it clear that their interest is not solely defined by regulation friendly to digital money and are seeking clarity from regulators in defining these assets. Investors see adoption of cryptocurrencies growing if regulators can say with certainty that financial instruments backed by these instruments are legal and fines will not be levied retroactively. Market participants agree there is no single regulatory model from other countries to follow, but that any rule coming out should ensure the assets and the investing process are clearly defined.

Ricky Li, founder of crypto-trading firm Altonomy, says institutional investors prefer clear definitions coming from regulators so risk is mitigated and they are assured they have legal cover.

“I don’t think it matters how lenient or strict regulations are to attract more institutional investors; it really depends on clarifications of definitions,” Li says. “The market will still be there even if it is a strict market. Investors shy away from assets if there is a lot of risk around it, which normally means there is no clear guidance on how they can avoid risk.”

He adds any regulatory gray areas around how an asset is defined is where the potential risk comes in, as investors can take advantage of loopholes or aren’t able to fully manage their risk tolerances.

There is a perception that countries with friendly cryptocurrency regulations attract more investment, but traders point out that it’s not the regulation, itself, that lures digital-currency companies to set up shop, but rather these countries have laid out how they will treat the assets. Companies then feel more comfortable that regulators will not turn around and tell them they should treat these assets differently.

Complicated Markets

In the US, cryptocurrencies fall under three different potential regulatory frameworks: as a commodity and a futures contract, a security through an initial coin offering (ICO) or an exchange-traded fund, or as currency traded for other digital currencies on an exchange.

The US has yet to fully enact rules around cryptocurrencies, which is partially due to the fact that it has somewhat of a complicated regulatory regime. The Commodity Futures Trading Commission (CFTC) focuses on the commodities and futures market while the Securities and Exchange Commission (SEC) oversees the securities market. Each state also has the ability to lay out additional rules, particularly for companies that move money—for example, New York requires a BitLicense for any firm that wants to trade crypto-to-crypto.

Laurian Cristea, general counsel of cryptocurrency derivatives exchange ErisX, says the complicated regulatory environment in the US proves how hard it can be to lay out regulations in the country.

“The regulatory environment in the US is complicated—the states don’t substantively regulate and on the federal side, it’s bifurcated on the part of securities and futures regulators. It’s a testament to the approach of the CFTC that they looked at their core principles and rules and figured out how these can be extended to cryptocurrencies. They gave entities a path to comply with their principles and as a result we’ve seen a lot of new licenses and products announced,” Cristea says.

He adds that “the SEC is taking its time, and currently the market is awaiting the same degree of certainty from SEC’s guidance.”

The CFTC declared in 2014 that it believes some cryptocurrencies are commodities and therefore fall under its jurisdiction. It also clarified that futures contracts based on cryptocurrencies—particularly bitcoin—may be self-certified by exchanges. This allowed the CME Group and the CBOE Futures Exchange to self-certify and start trading bitcoin futures in 2017.

The SEC, on the other hand, has not yet released final guidance on cryptocurrencies in terms of whether they trade as normal securities or something different. The regulator did provide clarity on ICOs and has begun fining ICOs that break its rules. The SEC has not yet released any rules on cryptocurrency exchange-traded funds (ETFs). Its most recent release around cryptocurrencies dealt with the custody and clearing of cryptocurrency securities and other digital tokens, though some market participants believe the statement still did not provide enough clarity on post-trade processes for digital assets.

Cristea points out that institutional investors want much more clarity from regulators because they have a fiduciary duty unlike retail investors, which are the lifeblood of cryptocurrency trading.

“In retail, it’s the individual involved who makes decisions based on their own risk profile, but when it comes to institutional investing, most have a fiduciary duty and institutional investors cannot take undue risks, so there are additional due diligence and [risk] factors involved in transacting in crypto, which is why institutions need clarity and certainty,” says Cristea.

Despite the lack of clarity around cryptocurrencies, some are still founding brokerages, exchanges and trading venues, sometimes in the US. Established firms like Eris Exchange and trueEx incubated trading venues like ErisX and trueDigital. But many of these firms do exercise an abundance of caution by applying for licenses. In many cases, companies decide to move their operations to a country that has more legal cover.

While there is no single model to follow in terms of regulations, market participants do believe countries like the US that are still figuring how best to provide guidance for use of cryptocurrencies can take best practices from first-mover jurisdictions.

Small Size, Big Influence

Much has been said about the more crypto-friendly countries like Malta—where many of the largest cryptocurrency exchanges by volume are registered—which have managed to attract businesses around digital assets.

Two jurisdictions in particular, Malta and the Seychelles, were of interest to market participants. Both are smaller countries and therefore tend to move more quickly around innovations like cryptocurrencies. Asian countries have also traditionally embraced things like crypto, though regulatory actions around it in the region have ranged from easing requirements to outwardly banning the trade of cryptocurrencies, as China did in 2017.

According to a report from crypto data provider Crypto Compare, Malta-registered exchanges traded the largest chunk of cryptocurrencies in the world at $56.1 billion in March 2019, followed by Hong Kong at$53.1 billion and South Korea at $40.2 billion. Other countries have also begun moving forward with cryptocurrency trading by releasing their own rules aimed at providing more transparency and accountability. The Monetary Authority of Singapore (MAS) said in 2017 that it must approve the trade of digital assets by an exchange, and before virtual currencies can be traded, exchanges must verify identities of customers. In April of this year, France announced it has approved its own rules around crypto trading, which came into force in July. Under French rules, crypto companies have to agree to follow capital requirements, identify a legal entity responsible for the issuance of the crypto offering and pay taxes in the country. Another way countries and territories have tested out regulations around cryptocurrencies is to set up regulatory sandboxes. Hong Kong, for example, explored sandboxes that would experiment by allowing crypto exchanges to opt in and follow conceptual rules. Regulators in Hong Kong came out with rules stipulating only professional investors—those with at least HK$8 million in assets—can participate in the market and funds investing in digital currencies must be licensed by Hong Kong’s Securities and Futures Commission (SFC). Trading platforms in Hong Kong must join the sandbox while they are negotiating with the SFC on licensing requirements.

But sandboxes have to provide some legal cover for companies that they can’t be retroactively fined later. And conversely, Altonomy’s Li points out that if a sandbox is too restrictive and unclear, there may not be a lot of take-up from companies.

Li says it’s a good idea to look at the kinds of rules released by other countries and figure out how to implement something similar, or cherry-pick the best ideas for an individual market.

“Singapore and France kind of have a trial period [for rules], while the US is in a wait-and-see pattern. Either way, you do it, just adopt a position. I don’t think the US is lagging, but it is looking around at the other policies being enacted,” he says. “Participants will always adapt to the market, but it’s best to provide best practices. Sometimes companies have an overabundance of caution and get every possible license there is and that can get expensive.”

Of course, regulation is just one of the issues concerning institutional investors when it comes to virtual currencies.

Market participants, though, in search of clarity coming from regulatory bodies, know there’s a reason why these rules are not yet forthcoming. Mike Belshe, CEO of digital wallet and custody provider BitGo, says it’s understandable that some countries have not even come out with definitions for how they will treat these assets because they’re grappling with something fundamentally different from how money and traditional assets move.

“Regulators globally are still trying to understand and grasp what cryptocurrencies are and how they behave,” Belshe says. “They’re used to a world where there are borders and it’s easy to contain assets that trade in their jurisdiction, but cryptocurrencies don’t work that way. Each country has different rules—the US is known for strong regulatory enforcement while some countries, say in Asia, are not.”

But even if investors understand why some countries are taking their time to provide clarity around cryptocurrencies and its derivatives, many market participants still want regulators to offer more information so they can best plan how to go about their business.

Extending Current Rules

In light of the potential to glean best practices from around the globe, some market participants have even started groups that call for the sharing of cryptocurrency policies and creating harmonized standards, as most cryptocurrency trades cross borders.

Simon Taylor, co-founder of Global Digital Finance, a group that is working on harmonized regulations, points out a huge part of promoting cryptocurrencies is to ensure cross-border trades are recognized and treated in accordance with traditional policies. He says he does accept that more countries first have to figure out how they are going to define cryptocurrencies.

There is also a possibility that regulators may not even have to come out with any new definitions at all. Cristea says a possible—and easier—path to clarity is extending existing regulations to cover cryptocurrencies and declare them as assets that function a lot like traditional assets.

“What investors want to hear from regulators is clarity and certainty—to know that what they need to do to comply and the regulators won’t turn around and tell them to materially change or shut down. So regulators either have to make new rules or extend the ones we currently have,” Cristea says. “The regulations are there and I believe that cryptocurrencies are an asset class that fits within existing regulations. It’s really in the interpretation of the regulations in light of the specific characteristics of the asset class that needs to be spelled out.”

Perhaps the greatest worry, though, is that as more time passes, it will lead to more knee-jerk-reaction regulations. The US House of Representatives held a hearing on Facebook’s Libra in July and is considering drafting a law that would regulate digital currencies and limit possible harm to consumers. The idea of the US House of Representatives leading the charge on regulation might not be the most comforting idea for many in the crypto space.

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## Pat Blythe: Cherie Currie….The One and Only Cherry Bomb

Posted in Opinion, Review with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on November 4, 2015 by segarini

Let’s go back to September 25 and a late night visit to Cherry Cola’s Rock ‘n Rolla Cabaret and Lounge. With my sisters in tow, we ducked into the club so I could introduce them to the exquisite Cherish Stevenson, the club’s owner and gracious hostess, and give them a small taste of the nightlife in Toronto. Cherish pulled me aside to let me know Cherie Currie was playing the club and would I be interested in shooting the event. It took me a nanosecond to say yes, I’d love to.


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Home > Standard Error > Margin Of Error And Standard Error

# Margin Of Error And Standard Error

## Contents

Census Bureau. How to Compute the Margin of Error The margin of error can be defined by either of the following equations. The margin of error is the half-width of the associated confidence interval, so for the 95% confidence level, you would have $z_{0.975}=1.96$ resulting in a margin of error $0.0081\cdot1.96=0.0158$. The more people that are sampled, the more confident pollsters can be that the "true" percentage is close to the observed percentage. navigate here

It does not represent other potential sources of error or bias such as a non-representative sample-design, poorly phrased questions, people lying or refusing to respond, the exclusion of people who could The survey results also often provide strong information even when there is not a statistically significant difference. In the Newsweek poll, Kerry's level of support p = 0.47 and n = 1,013. This level is the percentage of polls, if repeated with the same design and procedure, whose margin of error around the reported percentage would include the "true" percentage.

## Margin Of Error Confidence Interval

Check out the grade-increasing book that's recommended reading at Oxford University! Often, however, the distinction is not explicitly made, yet usually is apparent from context. If an approximate confidence interval is used (for example, by assuming the distribution is normal and then modeling the confidence interval accordingly), then the margin of error may only take random A Bayesian interpretation of the standard error is that although we do not know the "true" percentage, it is highly likely to be located within two standard errors of the estimated

Why is JK Rowling considered 'bad at math'? The critical value is either a t-score or a z-score. Otherwise, we use the t statistics, unless the sample size is small and the underlying distribution is not normal. Sample Size Standard Error If an approximate confidence interval is used (for example, by assuming the distribution is normal and then modeling the confidence interval accordingly), then the margin of error may only take random

Linearization and resampling are widely used techniques for data from complex sample designs. The area between each z* value and the negative of that z* value is the confidence percentage (approximately). Here are the steps for calculating the margin of error for a sample mean: Find the population standard deviation and the sample size, n. http://stattrek.com/estimation/margin-of-error.aspx?Tutorial=AP For tolerance in engineering, see Tolerance (engineering).

The true p percent confidence interval is the interval [a, b] that contains p percent of the distribution, and where (100 − p)/2 percent of the distribution lies below a, and Sampling Error Standard Error The confidence interval is a way to show what the uncertainty is with a certain statistic (i.e. and Bradburn N.M. (1982) Asking Questions. This chart can be expanded to other confidence percentages as well.

## Margin Of Error Sampling Error

MathWorld. http://www.dummies.com/education/math/statistics/how-to-calculate-the-margin-of-error-for-a-sample-mean/ Margins of error are commonly used in election polls.A poll might report that a certain candidate is going to win an election with 51 percent of the vote; The confidence level Margin Of Error Confidence Interval For example, if the true value is 50 percentage points, and the statistic has a confidence interval radius of 5 percentage points, then we say the margin of error is 5 Confidence Interval Standard Error The likelihood of a result being "within the margin of error" is itself a probability, commonly 95%, though other values are sometimes used.

References Sudman, Seymour and Bradburn, Norman (1982). http://facetimeforandroidd.com/standard-error/mean-standard-error.php Margin of error applies whenever a population is incompletely sampled. The margin of error is a measure of how close the results are likely to be. The stated confidence level was 95% with a margin of error of +/- 2, which means that the results were calculated to be accurate to within 2 percentages points 95% of Standard Deviation Standard Error

Step 2: Find the Standard Deviation or the Standard Error. As a random variable, it has a (sampling) distribution that can be characterized by mean, variance, distribution function, etc. It asserts a likelihood (not a certainty) that the result from a sample is close to the number one would get if the whole population had been queried. http://facetimeforandroidd.com/standard-error/mean-standard-deviation-and-standard-error-calculator.php WE GENERATED FORM 38 AFTER GENERATED INTERMEDIATE NUMBER?

A random sample of size 7004100000000000000♠10000 will give a margin of error at the 95% confidence level of 0.98/100, or 0.0098—just under1%. How Is Margin Of Error Calculated In Polls For example, a 95% confidence interval with a 4 percent margin of error means that your statistic will be within 4 percentage points of the real population value 95% of the The margin of error can be calculated in two ways, depending on whether you have parameters from a population or statistics from a sample: Margin of error = Critical value x


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# 4.5 – Parallel and Perpendicular Lines

Key Terms

• Perpendicular – Crossing at a right angle.
• The symbol $\perp$ means “is perpendicular to.”
• Slopes of perpendicular lines are negative reciprocals of one another: $\frac{4}{1}$ and $\frac{-1}{4}$
• If you multiply the slopes of two perpendicular lines, the product of their slopes is -1.
• Example: $\frac{4}{1}\cdot\frac{-1}{4}=-1$
• Parallel Lines – Have the same slope, but different y-intercepts

Notes

• Parallel Lines
• If two lines are parallel, their slopes are equal (the same).
• Parallel Examples: The three lines below all have the same slope: 4
• y = 4x + 2
• y = 4x – 3
• y = 4x + 15

• Perpendicular Lines
• If two slopes are perpendicular, they have negative reciprocal slopes.
• They may possibly have different y-intercepts.
• You MUST know the point where the 2 lines intersect.
• Horizontal and Vertical Lines
• If the slope of a line is 0 (horizontal), the slope of perpendicular line is undefined (vertical).
• If the slope of a line is undefined (vertical), the slope of perpendicular line is 0 (horizontal).

• Perpendicular Example: Intersection Point = (-2, -1), Slope = 4, Negative Reciprocal Slope = $\frac{-1}{4}$
• Use Point-Slope Form: $y-y_1=m(x-x_1)$
• Step 1: Substitute the known point and slope: $y-(-1)=4(x-(-2))$
• Step 2: Simplify: $y+1=4(x+2)$
• Step 3: Change to Slope-Intercept Form
• Distribute the 4: $y+1=4x+8$
• Subtract 1 on both sides: $y=4x+7$
• Step 4: Now, substitute the known point and the negative reciprocal slope: $y-(-1)=\frac{-1}{4}(x-(-2))$
• Step 5: Simplify: $y+1=\frac{-1}{4}(x+2)$
• Step 6: Change to Slope-Intercept Form
• Distribute the $\frac{-1}{4}$: $y+1=\frac{-1x}{4}+\frac{-2x}{4}$
• Simplify: $y+1=\frac{-x}{4}+\frac{-x}{2}$

More Examples

• Ex 1. The lines below are parallel. If the slope of the green line is -3, what is the slope of the red line?

Answer: Since parallel lines have the same slope, the red line has a slope of -3 also!

• Ex 2. The lines below are perpendicular. If the slope of the green line is $\frac{3}{2}$, what is the slope of the red line?

Answer: Since perpendicular lines have negative reciprocal slopes, the red line has a slope of $\frac{-2}{3}$.

• Ex 3. Two lines are perpendicular. If one line has a slope of $\frac{1}{21}$, what is the slope of the other line?


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# Dispersion Power Penalty Modeling (Part 3)

#### Deriving Equation 2

Equation 2 is derived from Equation 7 by noting the following items.

• A true normal pulse has infinite length, so we cannot have a high speed data system that sends true normal pulses.
• A common choice is to select a bit time that will contain 95% of the bit energy at the point of transmission.
• If you set the bit time equal to ±2$\sigma_0$ about the signal peak ($4 \cdot \sigma_0$ total), the bit time will contain 95% of the normal pulse's energy.
• This choice is much more conservative than the choice in Equation 1, which can be shown to only have 31% of the pulse energy contained in the bit time.

Given these assumptions, we can state Equation 10 directly.

Eq. 10      $\sigma_0 = \frac{T}{4} = \frac{1}{4 \cdot B}$

We can substitute Equation 10 into Equation 7 to obtain Equation 2.

$P{P_D} = 10 \cdot \log \left( {\sqrt {1 + {{\left( {\frac{{D \cdot L \cdot {\sigma _\lambda }}}{{\frac{1}{{4 \cdot B}}}}} \right)}^2}} } \right)$

$P{P_D} = 5 \cdot \log \left( {1 + {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2} } \right)$

#### Deriving Equation 3

Equation 3 is derived from Equation 7 by noting the following items.

• Select a bit time that will contain 95% of the signal energy within the bit time at the point of reception.
• If you set the bit time equal to $\pm 2 \cdot \sigma_0$ about the signal peak ($4 \cdot \sigma_0$ total), the bit time will contain 95% of the normal pulse's energy.

This derivation is a bit more complicated than the previous derivations because we need to calculate $\sigma_0$ given that the signal $\sigma$ at distance L has been stretched by the factor k, which is computed using the following equation.

$k = \sqrt {1 + {{\left( {\frac{{D \cdot L \cdot {\sigma _\lambda }}}{{{\sigma _0}}}} \right)}^2}}$

Given the factor k, we can compute $\sigma_0$ as follows.

$\sigma\left(\text{Distance = L}\right) = \frac{1}{{4 \cdot B}} \Rightarrow {\sigma _0} = \frac{1}{{4 \cdot B \cdot k}}$

Substituting this equation into Equation 7 gives us the following.

${k^2} = 1 + {\left( {\frac{{D \cdot L \cdot {\sigma _\lambda }}}{{\frac{1}{{4 \cdot B \cdot k}}}}} \right)^2} = 1 + {\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda } \cdot k} \right)^2}$

This equation can be solved for the k to yield

$k = \sqrt {\frac{1}{{1 - {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}}}}$

This equation does not only represent the elongation of the pulse – it also represents the reduction in amplitude of the pulse. Therefore, we simply convert this equation to dB to get Equation 3.
$P{P_D} = 10 \cdot \log \left( {\sqrt {\frac{1}{{1 - {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}}}} } \right) = - 5 \cdot \log \left( {1 - {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right)$

This equation is very commonly used. It is different from the Equations 1 and 2 in that it contains a singularity. This singularity occurs because there are distances beyond which the dispersion is so severe that 95% of the bit energy cannot be contained in a bit time at the receiver. This characteristic has confused many people because they are not used to thinking about ISI at the receiver.

#### Deriving Equation 4

Equation 4 is Equation 3 after applying a Taylor series approximation for the reciprocal of the square root. The approximation is shown below and is only valid for small x.

$\frac{1}{{\sqrt {1 - {x^2}} }} \approx 1 + \frac{1}{2} \cdot {x^2}$

This approximation can used to simplify the $PP_D$ a bit.

$PP_D = 10 \cdot \log \left( {\frac{1}{\sqrt {1 - {\left( 4 \cdot B\cdot D \cdot L \cdot \sigma_\lambda \right)}^2}} }\right) \approx 10 \cdot \log \left( {1 + \frac{1}{2} \cdot {{\left( {4 \cdot B \cdot D \cdot L \cdot {\sigma _\lambda }} \right)}^2}} \right)$

This gives us Equation 4. It is important to remember that this equation is only valid when the dispersion power penalty is small. I do not view this as a serious restriction because I would not deploy a system with a large dispersion power penalty.

### Conclusion

I know this was long, but it will be useful. Here is what was accomplished:

• The commonly used dispersion power penalty formulas were put in one place.
• Their underlying assumptions were examined in detail.
• It was demonstrated that they all come from the same basic equation.

My next blog posts will be shorter and less specialized.

This entry was posted in Fiber Optics and tagged , , . Bookmark the permalink.

### 2 Responses to Dispersion Power Penalty Modeling (Part 3)

1. mruffini says:

Thanks for your post. The definition of a power penalty thought it's usually the increase in power at the receiver to obtain the same BER compared to the case without the impairment.
It seems here that power penalty is only related to the loss of power due to pulse broadening, while the effect of ISI is not taken into account. Is this the reason why usually the maximum dispersion power penalty allowed is 2 dB?

• mathscinotes says:

Hi Marco,

This blog post was outcome of a discussion I had with several optics folks on the various equations used to model dispersion. The models I covered are the most common and they only model pulse broadening. All of the engineers in our little discussion group have made extensive measurements of dispersion and had found that the equations were really only useful at low power penalties. We only use the equations in this blog post to tell us whether dispersion is significant or not. We never allow a PON to be deployed with significant dispersion power penalty. For example, my company has never deployed a PON with more than 1 dB of dispersion power penalty.

At one time, I had a salesman that used one of these equations to recommend that we increase our power penalty budget, reduce our PON split, and run the PONs longer distance. We took a look at this in the lab and found that PONs with a computed power penalty significantly greater than 1 dB turned out to have erratic performance. The spectral width of the laser varies randomly and the worst-case power penalty proved difficult to predict. My customers need rock solid performance and a PON with variable BER characteristics was not acceptable.

I see you work at Trinity College. I was just there with my wife in September. I had a great time and I plan to spend more time in Ireland.

mathscinotes


text:
Prealgebra

# Key Terms

PrealgebraKey Terms
commission
A commission is a percentage of total sales as determined by the rate of commission.
discount
An amount of discount is a percent off the original price, determined by the discount rate.
mark-up
The mark-up is the amount added to the wholesale price, determined by the mark-up rate.
percent
A percent is a ratio whose denominator is $100100$.
percent decrease
The percent decrease is the percent the amount of decrease is of the original amount.
percent increase
The percent increase is the percent the amount of increase is of the original amount.
proportion
A proportion is an equation of the form $ab=cdab=cd$, where $b≠0b≠0$, $d≠0d≠0$.The proportion states two ratios or rates are equal. The proportion is read “$aa$ is to $bb$, as $cc$ is to $dd$”.
sales tax
The sales tax is a percent of the purchase price.
simple interest
If an amount of money, $PP$, the principal, is invested for a period of $tt$ years at an annual interest rate $rr$, the amount of interest, $II$, earned is $I=PrtI=Prt$. Interest earned according to this formula is called simple interest.
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text:
## Sunday, 28 July 2013

### 1 Kaala alavu or Measurement of Time

Our ancestors had observed the periodic occurrences of various seasons. Their agricultural society forced them to try predicting the season cycle. This need forced them to prepare, what is now called, calendar. However, this task was not very easy.

They began the process by recording the periodic movements in the celestial sky. The sun rises from the east and sets in the west. A day or pagal is the period between sunrise and sunset and a night or iravu is the period between sunset and next sunrise. A day and night together makes a naal (usually, in English we refer this as day). Today we know this is due to the rotation of earth on its axis. Indian mathamatician Arybhatta has knowledge of this phenomenon of spinning on its axis. Thus, one complete rotation of earth on its axis makes a naal. Our ancestors measured the rotation of earth on its axis as the duration between successive sunrises. Seen from the north-pole of the earth, the earth spins on its axis from west to east. This is the reason behind sun rising in the east and setting in the west.

One naal (complete day) was subdivided in to 60 division, called nazhigai. There was a reason to choose the 60 subdivision. Recall that we are talking of times when counting was manual. Thus, any subdivision of a day should facilitate usage of fractions of a day in day-to-day life. This is equivalent to ﬁnding the smallest number which has most of the early natural numbers as factors. For instance, $12$ is the ﬁrst number which has factors $1,2,3,4,6$ but not $5$. Thus, if $12$ was chosen as a subdivision of a day one can talk of $1∕6$, $1∕4$, $1∕3$ or $1∕2$ but not $1∕5$ fraction of a day. With this idea, $60$ is the ﬁrst number which has $1,2,3,4,5,6$ has its factors. If, in addition, one demanded the fraction $1∕7$ then we would have to go as big as $420$. So, for all practical reasons, $60$ was good enough and manageable.

Two nazhigai makes one muhurtam. Each nazhigai was, further, subdivided in to 60 parts, called thuli. Each thuli was, further, subdivided in to 60 parts, called nodi. Each nodi was subdivided in to 60 parts, called kuzhi or kuRRuzhi. One kuRRuzhi or kuzhi is the time taken by the krithigai natchatiram (Pleiades stars) to glitter once.

In modern terminology, one day is of $24$ hours, thus one nazhigai is of $24$ minutes and one muhurtam is of $48$ minutes. Similarly, one thuli is of $24$ seconds, one nodi is $0.4$ seconds and one kuRRuzhi is $0.04∕6=0.00666$ seconds. One kannimai, the time taken for normal wink of an eye, is equal to 10 kuRRuzhi. One aNu is $4$ seconds and one kaNam is $4$ minutes. One saamam referred to the modern day one hour which was deﬁned as $2.5$ nazhigai. $4$ saamam (or 4 hours) constituted one siRupozhuthu and $6$ siRupozhuthu made one complete day. They had names for the siRupozhuthu as follows: Assuming $6:00$ am sunrise,

 S.No. Time Slot Name 1 06:00 - 10:00 kaalai 2 10:00 - 14:00 nanpagal 3 14:00 - 18:00 yerpaadu 4 18:00 - 22:00 maalai 5 22:00 - 02:00 yaamam 6 02:00 - 06:00 vaikarai

#### 1.1 Raasi (Zodaic)

The concept of raasi might be a contribution of Sanskrit culture. I feel so because the names of raasi have Sanskrit origin and do not have a tamil equivalent. Our ancestors seem to have known that the earth is spherical in shape. The proof of this is the illustration of earth as a spherical object in Varaha avatar (third) of lord Vishnu. Our ancestors (who believed in geo-centric theory) divided the celestial sphere containing stars and planets in to 12 segments, each division is called raasi or zodiac. They observed the motion of planets against the backdrop of raasis. Today, we know this is the division of the elliptical path of earth around sun. Each raasi identiﬁes a 30 degrees segment of the celestial sphere. Half-a-siRupozhuthu or $2.5$ muhurtam (deﬁned in previous section) is the time taken by earth to spin from one raasi to other. The twelve raasi are:

 No. Raasi Symbol 1 Mesham Ram (Goat) 2 Rishabam Bull 3 Mithunam Woman-Man pair 4 Katakam Crab 5 Simmam Lion 6 Kanni Virgin (in a boat) 7 Thulaam Man holding balance 8 Vrichigam Scorpion 9 Dhanus Man holding Bow 10 Makaram Deer faced Crocodile 11 Kumbham Water Bearer 12 Meenam Two ﬁshes

Note that the raasis have no order (as given above). It is cyclic with no beginning and end. However, later we shall see that the raasi gets an order once we introduce the concept of new year and ﬁrst month of a year. The raasis are spread on the elliptical path within a span of 8-9 degrees (north-south) from the equator of earth.

#### 1.2 Muhurtam, Nazhigai and Laganam (Spin of Earth)

As mentioned above, in Tamil calendar a complete day starts with sunrise and ends with next day sunrise. Imagine this as the earth’s 360 degree spin on its axis, which makes a complete day. Imagine dividing the earth’s surface into 60 parts along longitude direction (north-south), with $6$ degrees between successive lines. The time taken by earth to spin 6 degrees is called a nazhigai, a duration of 24 minutes. 2 nazhigais makes a muhurtam, a spin of 12 degrees by earth on its axis. Thus, a complete day contains 30 muhurtams and each muhurtam is of 48 minutes duration. The names of the 30 muhurtams, in order in which they occur after sunrise till next sunrise, are as follows (assuming sunrise is at 0600 am):

 S.No. Time Slot Name Eﬀect 1 06:00 - 06:48 Rudra bad 2 06:48 - 07:36 Ahi bad 3 07:36 - 08:24 Mitra good 4 08:24 - 09:12 Pitru bad 5 09:12 - 10:00 Vasu good 6 10:00 - 10:48 Vara good 7 10:48 - 11:36 Visvadeva good 8 11:36 - 12:24 Vidhi good (except monday + friday) 9 12:24 - 13:12 Satamukhi good 10 13:12 - 14:00 Puruhuta bad 11 14:00 - 14:48 Vahini bad 12 14:48 - 15:36 Naktancara bad 13 15:36 - 16:24 Varuna good 14 16:24 - 17:12 Aryama good (except sunday) 15 17:12 - 18:00 Bhaga bad 16 18:00 - 18:48 Girisha bad 17 18:48 - 19:36 Ajapad bad 18 19:36 - 20:24 Ahirbudhnya good 19 20:24 - 21:12 Pusa good 20 21:12 - 22:00 Aswini good 21 22:00 - 22:48 Yama bad 22 22:48 - 23:36 Agni bad 23 23:36 - 24:24 Vidhatr good 24 24:24 - 01:12 Canda good 25 01:12 - 02:00 Aditi good 26 02:00 - 02:48 Jiva good 27 02:48 - 03:36 Visnu good 28 03:36 - 04:24 Yumigadyuti good 29 04:24 - 05:12 Brahma very good 30 05:12 - 06:00 Samudram good

Of the above muhurtams, Brahma muhurtam (29) is the most auspicious and considered good for all spiritual purposes, especially meditation. It is considered good to have bath in the Brahma muhurtam. Superstitious? No idea! In fact, all the three sandhi kaalam viz. day meeting night, night meeting day and mid-day are all considered good for spiritual purposes.

Let us explain the concept of lagnam. The division of lagnam is same as raasi. It is the same twelve division of a complete day each of duration two hours. However, the deﬁnition of lagnam is related to an individual time and place of birth. This is to diﬀerentiate between two persons born on the same day but at diﬀerent places (or diﬀerent longitudinal divisions) or two persons born on same day, same place but at diﬀerent time of the day. In modern terms this is to diﬀerentiate people born on the same day but at diﬀerent time zones. Thus, a person born on same day but in diﬀerent longitudinal division is born while Earth is facing diﬀerent raasi. Thus, lagnam of a person is the raasi faced by the part of the Earth surface on which he is born. Fix a point on the surface of earth. Note that the point will face all the 12 segments (raasis) of celestial sky, while the earth spins on its axis. Since the raasis are 30 degree segment, the point will take 2 hours (or 5 nazhigais or 2.5 muhurtam) to move from one raasi to other. A day is divided in to 12 lagnams or ascendant with each lagnam of two hour duration. A person’s lagnam is the lagnam on which he is born on a given day of the given place.

#### 1.3 Nakshathiram (Revolution of Moon)

Our ancestors also divided the celestial sphere in to 27 divisions, called nakshathiram, as a means to describe the motion of moon across the sky. The moon rose in one segment each day taking 27 days to rise again in the same segment, hence the 27 parts of celestial sphere. A raasi, introduced in previous section, approximately covered the region of $2.5$ nakshathiram. Today, we know that moon takes approximately 27 days to revolve around earth, called the sidereal month. The 27 nakshathiram are as follows:

 No. Nakshathiram Constellation 1 Asvini 3 stars (horse face) 2 Barani or Apa Barani 3 stars (pot) 3 Krithigai or Krittika 6 stars (razor) 4 Rohini 5 stars (chariot) 5 Mrigaseersham or Mrigasira 3 stars (deer head) 6 Thiruvaadhirai or Aarudra 1 star (bead) 7 Punarpoosam or Punarvasu 5 stars (bow) 8 Poosam or Pushya 3 stars (ﬂower) 9 Aayilyam or Aslesha 6 stars (serpent) 10 Magam or Magha 5 stars (palanquin) 11 Pooram or Poorva Palguni 4 stars (leg of cot) 12 Uttaram or Uttara Palguni same as above 13 Hastham or Hastha 5 stars (palm) 14 Chithirai or Chitra 1 star (pearl) 15 Svaathi 1 star (sapphire) 16 Visaakam or Visaka 3 stars (potter wheel) 17 Anusham or Anuradha 3 stars (umbrella) 18 Kettai or Jyesta same as above 19 Moolam or Moola 6 stars (crouching lion) 20 Pooraadam or Poorvashada 4 stars (square) 21 Uttaraadam or Uttarashada same as above 22 Thiruvonam or Sravana 3 stars (arrow) 23 Avittam or Sravishta or Dhanista 4 stars (drum) 24 Sadhayam or Sathabishak 100 stars (ﬂower) 25 Poorataadhi or Poorva Broshtapadha 4 stars (leg of cot) 25 Uttarataadhi or Uttara Broshtapadha same as above 27 Revathi 3 stars (ﬁsh)
 No. Raasi Nakshathiram 1 Mesham Asvini Barani Krithigai (1/4) 2 Rishabam Krithigai (3/4) Rohini Mrigaseersham (1/2) 3 Mithunam Mrigaseersham (1/2) Thiruvaadhirai Punarpoosam (3/4) 4 Katakam Punarpoosam (1/4) Poosam Aayilyam 5 Simmam Magam Pooram Uttaram (1/4) 6 Kanni Uttaram Hastham Chithirai (1/2) 7 Thulaam Chithirai (1/2) Svaathi Visaakam (3/4) 8 Vrichigam Visaakam (1/4) Anusham Kettai 9 Dhanus Moolam Pooraadam Uttaraadam (1/4) 10 Makaram Uttaraadam (3/4) Thiruvonam Avittam (1/2) 11 Kumbham Avittam (1/2) Sadhayam Poorataadhi (3/4) 12 Meenam Poorataadhi (1/4) Uttarataadhi Revathi

A person’s janma nakshathiram or birth star is the nakshathiram on which moon rose (resided) on the day of his/her birthday. However, a careful note here is that the janma raasi of a person is not deﬁned as the raasi on which sun rose on his birthday. The janma raasi of a person is the raasi corresponding to the person’s janma nakshathiram. Thus, note that our birth-raasi is not same as the sun-sign (zodiac).

#### 1.4 Lunar and Solar Month (Revolution of Earth)

A thingal or lunar month (also called synodic month) is the period between two amavaasai(new moon) or pournami (full moon). The duration of a lunar month is approximately 29.5 days. The Tamil name for month is thingal and a synonym of Moon is thingal.

A solar month is the duration in which Sun rises with the same raasi as its backdrop. Today, we know that ignoring the spin of earth on its axis, the earth takes 30 days to shift from one raasi to the other. Thus, we have twelve months, each month symbolising the shift of earth on its orbit by 30 degrees. The tamil months are as follows, corresponding to the raasi in which earth (or sun) resides:

 S.No. Month number of days Modern duration 1 Cittirai (Mesham) 31 mid-April–mid-May 2 Vaikaci (Rishabam) 31 mid-May–mid-June 3 Aani (Mithunam) 32 mid-June–mid-July 4 Aadi (Katakam) 31 mid-July–mid-Aug 5 Aavani (Simmam) 31 mid-Aug–mid-Sept 6 Purattaci (Kanni) 31 mid-Sept–mid-Oct 7 Aippaci (Thulaam) 30 mid-Oct–mid-Nov 8 Karttikai (Vrichigam) 29 mid-Nov–mid-Dec 9 Markazhi (Dhanus) 30 mid-Dec–mid-Jan 10 Tai (Makaram) 29 mid-Jan–mid-Feb 11 Maci (Kumbham) 30 mid-Feb–mid-Mar 12 Pankuni (Meenam) 30 mid-Mar–mid-April

The name of tamil month is motivated from the nakshathiram on which moon rises on the pournami (full-moon) day of the tamil month.

The duration for the Sun to return to its rising raasi is one aaNdu or year. The number of days in a aaNdu is, approximately, 365 and 15 nazhigai, 31 vinaadi and 15 Tharparai where 1440 vinaadi makes one nazhigai and 60 Tharparai makes one vinaadi. Today, we know this is precisely the duration in which the Earth goes around Sun once, in its elliptical orbit. This duration is $11$ minutes short of $365\frac{1}{4}$ days. On some years, we have an excess day in Aadi or Aavani month to make 366 days in year to compensate the small error in each year.

The ﬁrst Tamil month starts in mid-April, the new year day. Why mid-April? The reason is to choose the month that has the same length of day and night. With modern precision and metric this may seem wrong! We know today that equal length of day and night happens when the Sun rises exactly in east and sets in west. Today, we know this happens exactly twice a year, one on March 21 and the other on September 23. Both these days have equal length of day and night. We also know the reason behind this phenomenon to be the inclination of the axis of rotation of earth by 23.25 degrees. Thus, in mid-Jul the latitude line 23.25 degrees north of equator, called tropic of Cancer, is where the Sun rises and sets. One ayanam is six months. As seen from Earth, Mid-July to mid-Jan is called dakshinaayanam, when sunrise is towards south-east. In mid-Jan the latitude line 23.25 degrees south of equator, called tropic of Capricon, is where the sun rises and sets. Mid-Jan to mid-July is called uttarayanam, when sunrise direction is towards north-east.

But recall that our ancestors calculated length of day (or night) based on the number of nazhigai. With a total of 60 nazhigai in one full day, if both day and night had 30 nazhigai’s each then they have the same length. This happens in the month of Cittirai and Aippaci. Then, why choose Cittirai as the ﬁrst month? Because the ﬁve months after Cittirai have days longer than night. Day gets the priority! Similarly, the ﬁve months after Aippaci have nights longer than day. There is a tamil poem which describes these aspects. It also gives the formula (vee-doo-poo-mu-di) to compute the number of nazhigai’s by which a day or night is longer in a particular month. Thus, it says: the day is longer in the months of Vaikaci, Aani, Aadi, Aavani and Purattaci by 18, 30, 36, 30 and 18 minutes, respectively; and the night is longer in the months of Karttikai, Markazhi, Tai, Maci and Pankuni by 18,30,36,30 and 18 minutes, respectively.

#### 1.5 Perum-pozhuthu, aaNdu, Yugam etc.

Recall that our ancestors had divided a complete day into six siRupozhuthu. Similarly, they had divided a year (aaNdu) into six perum-pozhuthu or seasons. Two thingal (or months) constituted one perum-pozhuthu or season. The six perum-pozhuthu in a aaNdu are:

 No. Season Description Period 1 ila-venil Light warmth (Vasantam) Mid Apr - Mid Jun 2 mudhu-venil Harsh warmth (Grishmam) Mid Jun - Mid Aug 3 kaar Dark clouds (Rain) Mid Aug - Mid Oct 4 kulir Chill, Cold (Sharadam) Mid Oct - Mid Dec 5 mun-pani Early dew (Hemantam) Mid Dec - Mid Feb 6 pin-pani Late dew (Sishiram) Mid Feb - Mid Apr

One vattam or century is made of 60 years. After every sixty years, the years are repeated, as we do in months and weeks. The 60 years are as follows:

 S.No. Year (Sanskrit) Name Tamil Name Period 1 Prabhava Nantrontral 1987–1988 2 Vibhava Uyarthontral 1988-1989 3 Sukla Velouli 1989-1990 4 Pramodoota Peruvagai 1990-1991 5 Prachorpaththi Makkatchelvam 1991-1992 6 Aangirasa Ayalmuni 1992-1993 7 Srimukha Thirumugam 1993-1994 8 Bhava Thotrram 1994-1995 9 Yuva Ilamai 1995-1996 10 Dhaatu Maazhai 1996-1997 11 Eesvara Eechchuram 1997-1998 12 Bahudhanya Kulavalam 1998-1999 13 Pramathi Munmai 1999-2000 14 Vikrama Nerniral 2000-2001 15 Vishu Vilaipayan 2001-2002 16 Chitrabaanu Oviyakathir 2002-2003 17 Subhaanu Narkathir 2003-2004 18 Dhaarana Thaangkezhil 2004-2005 19 Paarthiba Nilvaraiyan 2005-2006 20 Viya Virimaanbu 2006-2007 21 Sarvajith Mutrarivu 2007-2008 22 Sarvadhari Muzhuniraivu 2008-2009 23 Virodhi Theerpagai 2009-2010 24 Vikruthi Valmattram 2010-2011 25 Kara Seinerththi 2011-2012 26 Nandhana Narkuzhavi 2012-2013 27 Vijaya Uyarvaagai 2013-2014 28 Jaya Vaagai 2014-2015 29 Manmatha Kathanmai 2015-2016 30 Dhunmuki Vemmugam 2016-2017 31 Hevilambi Potradai 2017-2018 32 Vilambi Atti 2018-2019 33 Vikari Ezhilmaral 2019-2020 34 Sarvari Veeriyezhal 2020-2021 35 Plava Keezharai 2021-2022 36 Subakrith Narseikai 2022-2023 37 Sobakrith Mangalam 2023-2024 38 Krodhi Pagaikedu 2024-2025 39 Visuvaasuva Ulaganiraivu 2025-2026 40 Parabhaava Aruttotram 2026-2027 41 Plavanga Nachchupuzhai 2027-2028 42 Keelaka Pinaiviragu 2028-2029 43 Saumya Azhagu 2029-2030 44 Sadharana Pothunilai 2030-2031 45 Virodhikrithu Eegalveeru 2031-2032 46 Paridhaabi Kazhivirakkam 2032-2033 47 Pramaadhisa Natralaimai 2033-2034 48 Aanandha Perumagazhchi 2034-2035 49 Rakshasa Perumaram 2035-2036 50 Nala Thaamarai 2036-2037 51 Pingala Ponmai 2037-2038 52 Kalayukthi Karumaiveechchu 2038-2039 53 Siddharthi Munniyamudithal 2039-2040 54 Raudhri Azhali 2040-2041 55 Dunmathi Kodumathi 2041-2042 56 Dhundubhi Perikai 2042-2043 57 Rudhrodhgaari Odingi 2043-2044 58 Raktakshi Semmai 2044-2045 59 Krodhana Ethiretram 2045-2046 60 Akshaya Valangkalan 2046-2047

This is the reason for celebrating the $60$-th birthday of a person because he is supposed to have lived a century and moves in to the same year he was born. This is very similar to celebrating centenary birthdays.

The time is divided in to four yugams, viz.

1.
Krita/Sathya yugam ($17,28,000=4×4,32,000$ years)
2.
Tretha yugam ($12,96,000=3×4,32,000$ years)
3.
Dvaapara yugam ($8,64,000=2×4,32,000$ years)
4.
Kali yugam ($4,32,000$ years)

The four yugas together comprise one chaturyuga or maha-yuga, which comprises of 4.32 million years (4,320,000 years). Thousand maha-yugas constitute one “kalpam”, i.e., 4,320,000,000 (4.32 billion years). One kalpam is one day of Brahma and equal number of years for night, making a complete day of Brahma, a time period of 8.64 billion years. Manvantaram is one-fourteenth of a kalpam. According to Aryabhata (of 6th century), the kali-yuga started in 23 January, 3102 BCE of Gregorian calendar. This division of yugas is disputed by Swami Sri Yukteswar Giri in his book The Holy Science.

### 2 Panchangam

A day was ascribed 5 attributes called panchangam. The word is a combination of words “pancha” meaning ﬁve and “angam” meaning limbs. Thus, a day is determined by ﬁve factors, viz.,

1.
Tithi (Phase of moon)
2.
Nakshathiram (Home of moon)
3.
Kizhamai (day of week)
4.
Yogam
5.
Karnam (2 Karnam is one tithi)

#### 2.1 Tithi

We have already seen that the period between two amavasai (new moon) is 29.5 days (thingal or synodic month). This period has 15 days of waxing (sukhla paksham) and equal number of waning period (krishna paksham). These paksham is a Sanskrit name. A $15$ day period, in Tamil, is called azhuvam. Thus, the period is divided in to 14 tithis for both waxing and waning period. The zeroth starts at amavasai, followed by 14 thithis (waxing) and ﬁfteenth being pournami (full moon), followed by the same 14 tithis (waning), followed by amavaasai again. The names of the tithi are:

 S. No. Tithi 0 Amavasya, the new moon day 1 Prathamai 2 Dvithiyai 3 Thrithiyai 4 Chathurthi 5 Panchami 6 Sashti 7 Sapthami 8 Ashtami 9 Navami 10 Dasami 11 Ekaadasi 12 Dvaadasi 13 Thrayodasi 14 Chathurdasi 15 Pournami, the full moon day

After Pournami, the tithis once again go on from 1 to 14 to complete one cycle.

#### 2.2 Nakshathiram

We have already elaborated on what nakshathiram is and have already listed them. The Moon rises each day on a diﬀerent nakshtram. Thus, a day is distinguished based on the nakshathiram in which Moon rises that day.

#### 2.3 Kizhamai

Our ancestors were able to distinguish planets, from stars, by their motion in the celestial sphere. They identiﬁed what they called “nava graha” or nine planets. The ﬁrst seven of these planets are: Moon, Sun, Mercury, Venus, Mars, Jupiter, Saturn. In addition, our ancestors imagined two diametrically opposite points on the celestial sphere as Raahu and Kethu, respectively. This was to accommodate the occurrence of eclipses. These imaginary points were the intersection of the orbits of Sun and Moon in the celestial sphere.

They ordered the planets according to their speed in the celestial sphere. We have already seen that the sun takes approximately 30 days to shift from one raasi to other. We shall see the duration taken by every other planet to cover one raasi (and all 12 raasis):

1.
Moon 2.5 days (30 days)
2.
Mercury 30 days (one year)
3.
Venus 30 days (one year)
4.
Sun 30 days (one year)
5.
Mars 45 days (540 days or approx. 1.5 years)
6.
Jupiter one year (12 years)
7.
Rahu 1.5 years (18 years)
8.
Kethu 1.5 years (18 years)
9.
Saturn 2.5 years (30 years)

Thus ordering from slowest to fastest, we have: Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon. The time between sunrise and sunset is divided by 12 and sunset and next sunrise by 12 to get what is called planetary hours or horai. Each horai is ruled by the planets in the above order. The concept of a planet dominating (or ruling) an horai is not comprehensible to me. Suppose the ﬁrst horai after sunrise of a day is ruled by Saturn, we call it Saturday. The next horai is then ruled by Jupiter and so on, so that the 12th horai, at sunset, is ruled by Venus. Continuing this way, we see that the ﬁrst horai (or 25th horai) on the next day is ruled by Sun, called Sunday. In this procedure, the next day is Monday. Proceeding this way, we can identify a day as:

1.
Saturday (Saturn) Sani
2.
Sunday (Sun) Gnayayiru
3.
Monday (Moon) Thingal
4.
Tuesday (Mars) Sevvai
5.
Wednesday (Mercury) Pudhan
6.
Thursday (Jupiter) Viyazhan
7.
Friday (Venus) Velli

This attribute of a day is called kizhamai or week. The above explanation also justiﬁes the reason behind the speciﬁc order of the days of a week. This, probably, also explains why Friday was holiday being the last day of the week which, may be, changed after british politically took over the country.

#### 2.6 Raahu Kaalam and Yama Gandam

The Raahu kaalam and Yama gandam is calculated by dividing the duration between sunrise and sunset by 8, each division called the “octant”. Thus, each octant is approximately of 90 minutes duration (assuming 12 hours between sunrise and sunset). The raahu kaalm is assigned in following order:

 II octant Monday III octant Saturday IV octant Friday V octant Wednesday VI octant Thursday VII octant Tuesday VIII octant Sunday