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Talk:Search and rescue/SAR mission 2008 02

Scene
Jeff, from your description I gather Brandon drove west on the road bisecting Eidsvold Section 6, shown here. Is that correct? --Una Smith 23:23, 30 January 2010 (UTC)
 * Jeff has confirmed this for me, here. --Una Smith 03:37, 31 January 2010 (UTC)
 * I have a photo of this field road. I would be happy to upload it with some help.--Jeff Hasse 20:17, 31 January 2010 (UTC)
 * I don't think I need to see a photo. Nonetheless, see Uploading files.  I usually upload them to Wikimedia Commons (a companion wiki);  photos there can be used on pages here as if the photos were here. --Una Smith 20:59, 31 January 2010 (UTC)

Some questions about the fields:

 * What are the sizes of the fields?
 * They range from 40 acres to 640 acres (1 mile square). 80s and 160s are common.

 * What crops were planted in the fields the prior crop year?
 * Primarily corn and soybeans.

 * Had the fields been planted?
 * About 60% of the fields to the NW of the LKP were planted prior to Brandon's disappearance. (We conducted a survey of the area.)--Jeff Hasse 20:17, 31 January 2010 (UTC)

 * What are the the sizes of the equipment used in the fields?
 * They range from moderate size for the small plots to very large for the big plots.--Jeff Hasse 20:17, 31 January 2010 (UTC)

Laguyon 01:50, 31 January 2010 (UTC)

What type of fences are used around these fields? Are the fence lines clean or do they contain tall grass or brush? Laguyon 18:50, 1 February 2010 (UTC)
 * Most fields do not have any fences around them. However, there are some still in existence (especially between fields with different owners).  These are generally barbed wire fences with steel fence posts mostly overgrown with canary grass and other prarie grasses.

 * The fencelines around the few pastures in the area tend to be a mix of barbed wire and electric fence wire. They tend to be fairly clean.

 * We have also found a number of fencelines in CRP land and along some of the creeks. These also tend to be overgrown with trees, grass, and brush.

Some questions about the river:

 * What types of plants/trees are on the bank? Dense growth may stop someone in the dark.
 * The blog Footprints at the River's Edge has a photo. It looks like the photo was taken in late summer.  It shows a shallow stream bed, mostly dry mud, low banks, and floodplain bosque.  Judging by the topo maps, the floodplain is much narrower west of the car than east of the car.  In May the water would have been high, probably throughout the bosque. --Una Smith
 * This is an accurate description of the area. The banks of the rivers and creeks tend to be wooded, thick with brush, and/or tall Canary grass.  While this has made searching a challenge, most is passable.  The only areas we have found that are nearly impassable are some of the shelter-belts around the farms.  Those are a mix of old trees and very thick undergrowth.--Jeff Hasse 20:17, 31 January 2010 (UTC)

 * Where the dog went into the river, how high was the bank?
 * I don't have an exact location for where he went in the river because they didn't take a GPS waypoint. However, we know to within a couple of hundred feet and the banks all along this section are not high at all (less than a foot at average water levels).  It would be relatively easy to get out of the river if you fell in.--Jeff Hasse 20:17, 31 January 2010 (UTC)

 * How high was the water in the river?
 * Approximately 2-3'. One official has stated that there may be holes as deep as 6-10 feet, but we haven't seen any along this stretch of river.--Jeff Hasse 20:17, 31 January 2010 (UTC)

Laguyon 01:50, 31 January 2010 (UTC)

Cell phone
Some questions about the cell phone:
 * Was Enhanced 911 in use in the area in May 2008?
 * Yes, they were in Phase 2 of developing E911 at the time, but I don't know what their specific capabilities were at the time.--Jeff Hasse 20:17, 31 January 2010 (UTC)

 * Was Brandon's phone GPS capable (E911 "location capable")? For a cell phone used by a 19yo in the USA in 2008, the answer is almost certainly yes.
 * Yes. His phone was a Motorola SLVR L7c.  That model is location capable.  --Una Smith 04:31, 31 January 2010 (UTC)
 * Was his phone subscribed to any location-based services? Did he use it as a navigational aid? --Una Smith 04:31, 31 January 2010 (UTC)
 * No, I don't believe so. The family has never mentioned it and there was a lot of attention paid to the cell phone early in the search.--Jeff Hasse 20:17, 31 January 2010 (UTC)

 * Is it true that cell phone company records were used to find the car?
 * Yes.--Jeff Hasse 20:17, 31 January 2010 (UTC)

 * If true, what information in those records led to the car?
 * The fact that Brandon was connected to the tower just south of Minneota during all of his phone calls that evening.--Jeff Hasse 20:17, 31 January 2010 (UTC)
 * As I hope the discussion here and elsewhere has made clear, there should have been more information than that. --Una Smith 20:48, 31 January 2010 (UTC)
 * Cell phone forensics has been one of the major learning points for me. Thanks.
 * I have been looking into the issue and two new pieces of information have come to light. The family was told that some part of the carrier's system was "down" at the time and they weren't able to access some information that they wanted.  Secondly, the carrier at the time was in transition (Midwest Wireless had been bought out by Alltel which has subsequently been bought out by Verizon) and the tower may have been analogue, not digital, at the time.--Jeff Hasse 20:19, 2 February 2010 (UTC)

These questions are pretty much irrelevant now, but answers may help us all make better use of cell phone records in future searches. If only... It could have gone like this:  Father to Brandon: "Brandon, here is what you do.  Stay at the car.  Hang up and call 911.  Tell 911 you are lost and please locate you.  Stay at the car.  Tell 911 we are driving around looking for you.  Tell 911 my cell phone number.  Stay at the car." --Una Smith 00:22, 31 January 2010 (UTC)
 * I agree. This is one of the lessons of the search.  However, it is clear from his behavoir that Brandon was attempting to avoid a DUI and did not want "official" help.--Jeff Hasse 20:17, 31 January 2010 (UTC)

When the phone company was consulted, did anyone save data logs? Maybe a review of logs, by someone familiar with these types of logs, would reveal some new clues. --Una Smith 00:39, 31 January 2010 (UTC)
 * We do not have access the any cell-phone data logs. I have to make a clear distinction between the Sheriff's investigative role and my own or risk alienating the AHJ I work for.  The lead investigator (one of the Sheriff) did not offer this information and, while I asked about them, I did not push too hard.  From what I was told, the only thing that could be gleaned from the cell-phone was the fact that he was connected to the Minneota tower.--Jeff Hasse 20:17, 31 January 2010 (UTC)
 * The reason I keep on about this is that if the records used on day 1 were saved by someone, somewhere, they might yet exist and be a source of new clues. --Una Smith 20:48, 31 January 2010 (UTC)

When Brandon's parents tried to call him back, they got a synthetic ring sound and then Brandon's voicemail. Did anyone investigate Brandon's cell phone contract to find out if these observations have any meaning? In many cases, they mean nothing, but on some networks that combination would mean the network did contact the phone; if the network failed to make content the caller would hear a recorded "unavailable" message. --Una Smith 00:39, 31 January 2010 (UTC)
 * Many inferences have been made regarding the fact that Brandon's cell phone rang for a time before it went straight to voice mail. Unfortunately, little can be accurately inferred from this. According to one of our members who is an executive at a regional telephone company, the ring that you hear when you call someone has little to do with what is happening at the other end (the recipient's phone). It is an artificial sound that is being generated at a switch in some building, not by the recipient's phone. It is a holdover from the old analog phone days.

 * The way a phone handles voice-mail when it is turned off or disabled is entirely dependant on how this function is programmed at the phone carrier's computer systems. Therefore, there is a great deal of variability regarding how it works between carriers, calling plans, and even models of phones.

 * While there are many possibilities regarding what happened at the end of Brandon's phone call to his father (the phone may have been lost, shorted out in water, battery dead, working but Brandon wasn't able to use it, etc.), the only thing that can accurately be inferred is that Brandon wasn't able to reestablish contact with his father after that point.--Jeff Hasse 20:17, 31 January 2010 (UTC)

 * Given the model of Brandon's phone, almost certainly the ringing his parents heard on their end is not informative. --Una Smith 20:48, 31 January 2010 (UTC)

I have read on the web that Brandon wore his phone on a sterling silver chain. Is this true? Did he clip it to his jeans? Or wear it around his neck and stick the phone in a shirt pocket? --Una Smith 03:40, 31 January 2010 (UTC)
 * That is interesting. I hadn't heard that one before.  No, I believe he kept his phone in his pocket.  The necklace was decorative.--Jeff Hasse 20:17, 31 January 2010 (UTC)
 * I was referring to this from Footprints at the River's Edge: "Glasses, Sterling Silver Chain carrying a Black Motorola SLVR Cell Phone, Wallet/Identification". I may be parsing it incorrectly.  Jeff has not heard the phone was on a chain, and wearing it that way would be remarkable, so Brandon probably carried it in the usual way, loose in a pocket.  That model of phone does not appear to have anywhere to attach a lanyard.  --Una Smith 20:34, 31 January 2010 (UTC)

Cell towers
According to FCC data on this website, there are 3 cell towers in Mineota, and 1 in Porter. Those are the 4 closest towers. The next closest is in Ivanhoe, specifically 5 Mi. S. Of Cr-18 On Cr-5, and there are several more in the vicinity of Ivanhoe, which is marginally higher elevation than the immediate search area. These other towers provide at least a potential for triangulation. --Una Smith 02:18, 1 February 2010 (UTC)

 * The only towers available to Brandon's carrier (alltel) was the tower just south of Minneota (3575 Cr 3 (Minneota 196494523)) and in Canby (.2 mi E Of The Jct Of US Hwy 75 & St. Hwy 68). The cell never connected with the Canby tower.--Jeff Hasse 19:29, 1 February 2010 (UTC)

 * Other carriers' towers in the vicinity and operating in the same radio frequency band would have "heard" Brandon's phone. That is why a cell phone indicating "no service" often can be used to make a 911 call.  --Una Smith 20:39, 1 February 2010 (UTC)

Brandon's Disorientation
There has been considerable speculation as to why Brandon was so disoriented as to his location. We believe that it is relatively easy to explain. His disorientation was likely due to a combination of five factors: 1) fatigue, 2) at least mild intoxication (Remember, his blood alcohol was still rising after he left Canby from that shot of whiskey.), 3) unfamiliar roads (While he had driven Hwy 68 many times, he probably had little reason to drive the gravel roads.), 4) the fact that all the intersections look relatively alike (mostly tilled fields with very few landmarks), and 5) possibly simple distraction. These all could have contributed to him not knowing exactly where he was.

How he came to the conclusion that he was near Lynd is a little more difficult to explain. While we will never know for sure, we believe that when he realized he had travelled down a minimum maintenance road, he conducted what is known as a transderivational search. He searched his memory for other minimum maintenance roads that he knew and came up with the gravel road which lies along a golf course between Marshall and Lynd with which he was apparently familiar.

Direction of travel
Looking at the location of the car on this topo map, my first thought is Brandon had been traveling east on the north edge of Eidsvold Section 6, and turned south onto the dirt road ... which unexpectedly had a 90 degree bend, sending him back west. If he was trying to go southeast, then once he left his car he would have continued southeast ... to the river, but several miles downstream of where the "dog track" scenario would have put him there.
 * Just a clarification. There is no road on the north edge of Eidsvold Section 6 (just another field road that dead-ends at some small ponds).Were these ponds searched?Laguyon 06:35, 4 February 2010 (UTC)  Based on where he came from, where he was going, and where his car was found, I believe there is a high-probability that he did travel south on 290th St, follow the road around Conger's Marsh, turn west on the 90 degree bend, and follow the field road to where his car was found.--Jeff Hasse 20:47, 2 February 2010 (UTC)
 * My Search Area Determination file, associated maps, and IAP describe all of the scenarios the planning team considered. Summary: From the car's position, Brandon could see the glow of lights towards the southeast (Minneota).  From comments he made during his conversation with his father, we believe Brandon mistook Highway 68 for Highway 23.  He stated that he was walking away from Marshall towards Lynd near Highway 23.  That would put him on a path to the northwest.--Jeff Hasse 20:17, 31 January 2010 (UTC)
 * This is assuming Brandon thought Minneota was Marshall and not Lynd. Why?  Clearly he was disoriented, so why assume any orientation? --Una Smith 03:49, 1 February 2010 (UTC)
 * Based on our investigation, I don't believe Brandon was oriented to cardinal direction. He never mentions any direction in any of his phone conversations.  When asked what side of the road he was on, he stated; "On the left side," not a cardinal direction.--Jeff Hasse 20:47, 2 February 2010 (UTC)
 * Not only why assume that, but why did Brandon think he was somewhere near Lynd? In PLSS areas don't many people navigate by the road numbers?  By the signage?  And I see where Brandon's car was found, and east of Lynd, there are similar road numbers.  --Una Smith 04:00, 1 February 2010 (UTC)

 * I am recalling Tomi Finkle's remark that in Minnesota "water" often means specifically a body of water: a lake, not a creek or ditch.  Is that true?  Could Brandon's "water" have been the lake at the NE corner of Section 6?  110 Ave runs N-S except where it swings E around the lake, right on the lakeshore, and makes a dog leg W south of it.  If Brandon drove south around the lake, could he have mistakenly continued straight at the intersection with the dirt road that took him west through Section 6 to Lyon Lincoln Road?  I don't see anything that could be be called a body of water between Marshall and Lynd.  --Una Smith 03:49, 1 February 2010 (UTC)

 * From Google Maps it is clear some terraforming has been going on in that area. It is not possible to drive east on the north edge of Section 6:  Congers Marsh (now a lake) is in the way.  However, my point remains that Brandon may have been trying to go southeast, toward Taunton.  I gather Brandon was driving on a no name road just south of the lake, from a "dog leg" in 110th Ave west to Lyon Lincoln Road.  Fields in the vicinity are 1/4 and 1/8 section.  (A quarter section is a quarter of a square mile and half a mile on a side.)  Many of the fence lines shown on the topo map conform with field boundaries on Google Maps. --Una Smith 05:27, 31 January 2010 (UTC)

Dog on day 2
Jeff Hasse wrote:

Elsewhere, Jeff wrote ''We do have a direction of travel established by a trailing dog on day two of the search. In my opinion, it is a reliable trail.''
 * The dog may have been following Brandon's trail, or not. Pending further discovery, we can only assume.  Consider SAR mission 2008 01, and zoom on the second map there:  that dog may have started on the subject's trail, but then the dog went north.  The subject was found west of PLS, across a river.  The dog went west on a road heading to a bridge across the river, but turned north before the bridge.  --Una Smith 18:49, 31 January 2010 (UTC)
 * We have 573 additional discoveries that suggest this trail is reliable.--Jeff Hasse 20:17, 31 January 2010 (UTC)

 * Returning to Brandon, did this dog handler find any footprints or other sign? Was the handler solo, or was there a navigator / comm / recorder along?  SAR dog teams in New Mexico generally work in pairs, one person to watch the dog like a hawk, the second person to keep track of where they are, take messages, and write down time and location of alerts and changes of direction by the dog, speed and direction of wind at the time of alert, etc. --Una Smith 18:49, 31 January 2010 (UTC)

Other scenarios
Without judging the validity of the above dog's direction of travel, let us consider scenarios in which it was spurious. --Una Smith 18:49, 31 January 2010 (UTC)

Hole in the ground
On MSAR-Riders Tomi Finkle has asked about concealed hand-dug wells. Others, on other forums, have asked the same question. Not just wells, but underground cisterns too. I agree that the possibility of such "manholes" deserves consideration. Such wells are notorious in some SAR communities; some others have similar problems with small mines. Comparing the old topo maps on the MN DNR site to new satellite images on Google Maps, I see numerous abandoned homesteads within 2 miles of Brandon's car. --Una Smith 05:40, 31 January 2010 (UTC)

 * Here is a 1998 Usenet thread by Jorene Downs about a search for a 2 year old child fallen in a well. --Una Smith 05:56, 31 January 2010 (UTC)

Lynd
Re why Brandon thought he was near Lynd, on Help Find the Missing Jeff wrote:

and:

On Google Maps I don't see a golf course within 2 miles of Lynd. I do see one 6 miles away, in Marshall. According to the CNN article he gave them an exact location and they drove directly there. Where exactly did Brandon ask his parents to pick him up? --Una Smith 04:42, 1 February 2010 (UTC)
 * He told them to meet them along the road that "goes between Savannah Golf Course and Lynd"--Jeff Hasse 00:23, 2 February 2010 (UTC)

Google Maps has a proximity search tool. Plugging in "golf loc: Lynd, MN", I get a display of golf courses in the area. The second closest is the one I mentioned above, in Marshall. The closest is Savannah Oaks Golf Course, except that I don't see it on the satellite image so I think the marker is in the wrong place. Is this the golf course in question? Where exactly is it? --Una Smith 04:56, 1 February 2010 (UTC)
 * Yes, I had trouble with it at first too. The search engines don't seem to know it exists (it is relatively new).  This is the correct golf course.  I sent you a map of it since I can't upload here for some reason.--Jeff Hasse 00:23, 2 February 2010 (UTC)

Timeline
Jeff has added the detailed timeline to SAR mission 2008 02 (diff). I think it is too detailed for there. How about we move it to a subpage, linked from the Situation section? Or put it in that section, but collapse it (if Wikiversity supports collapsed text boxes)? --Una Smith 18:08, 1 February 2010 (UTC)

I like it because it is the most accurate timeline available. But I am biased. ;) I wouldn't have any trouble with moving the two paragraphs regarding why Brandon was disoriented to this or a separate page, since it is more speculative.--Jeff Hasse 00:23, 2 February 2010 (UTC)

Maps
We really need some topo maps on the page. Do you have any that we can use? USGS topos are public domain, so can be used. The topo linked above looks like USGS work but I am uncertain if I can take it from that website. --Una Smith 03:54, 31 January 2010 (UTC)
 * I have many maps that I developed that I would be happy to share. I just need help in uploading them.--Jeff Hasse 20:17, 31 January 2010 (UTC)
 * Jeff, you could e-mail them to me and I could upload them, but it is much simpler for you to upload them yourself. On the left edge of the window in which you view this, there is a "toolbox" pane, and in the toolbox is a link "Upload file".  It is pretty simple.  Have the files you want to upload on the computer from which you are reading this.  --Una Smith 21:04, 31 January 2010 (UTC)
 * The Census Tiger Map Browser tool is not working correctly in that it cannot locate place names, but I have worked around that and produced a couple of maps now on SAR mission 2008 02. Also, MN DNR has cleared us to use maps from their website. --Una Smith 18:20, 1 February 2010 (UTC)

 * DNR maps
 * 1:100000 topo map centered on location of the car
 * 1:24000 topo map centered ditto
 * topo map symbols
 * Tiger Map Server Browser

Planning Scenarios Discussion

 * This is the process we used to evaluate the various scenarios regarding what happened to Brandon before and after the cell phone went dead. Feel free to add to the arguments.

Initial Direction of Travel Scenarios (During the cell-phone conversation):

 * 1. Brandon traveled due east across the field road (395th St) to 110th Ave around Congers Marsh, then east on Lyon-Yellow Medicine Rd.
 * Arguments for: Brandon may have been backtracking his route.  (It is unknown in what direction he thought the Lyndwood Tavern laid.)
 * Arguments against: Brandon thought he was between Marshall and Lynd.  There is no evidence that he was backtracking towards Marshall.

 * 2. Brandon traveled east by first going south on Lyon-Lincoln County Rd., then east on 390th St.
 * Arguments for:
 * Taunton was the closest town to the LKP.
 * During the cell phone call, he stayed on the Minneota Tower.
 * Arguments against:
 * Taunton was nearly invisible (no glow of lights).

 * 3. Brandon traveled south on the Lyon-Lincoln Road.
 * Arguments for: None known
 * Arguments against: He would have reached the highway within the 47-minute conversation.  If so, he would likely have mentioned it to his parents.  No mention was made.

 * 4. Brandon traveled westerly by first going south on Lyon-Lincoln County Rd, west on 390th St., then either south on 270th St to Highway 68 or north on 270th St.
 * If Brandon intended to go west from his LKP, he only had two choices if he wanted to walk along a road. 1) To go south along the road he was on and turn west at the nearest intersection or 2) Go north and turn west.  Either way, he would likely turn west at the nearest opportunity.
 * Arguments for:
 * Brandon’s vehicle was heading west when it went into the ditch. Accordingly, he likely thought his destination was to the west or northwest.
 * The brightest glow in the sky was Minneota, which was east-southeast of Brandon’s LKP. He would likely have thought that this was Marshall and headed away from it towards where he thought Lynd was located.
 * At one point in the conversation, when asked on which side of the road his car was located, he replied; “On the left side.” This indicates that he was facing south at that time, which may suggest an intended direction of travel.
 * Later, he mentioned that he could clearly see the moon (suggesting that it was within his field of vision). At the time he was walking, the moon was due west of his position near or at the horizon.  While certainly a “soft” clue, this suggests that he may have been heading westerly at the time he made the statement.
 * Brandon stated that he believed he was on the road between Savannah Hills Golf Course and Lynd. He also believed that he is just off Highway 23.  If one compares the relative positions of Marshall, the Lyndwood Tavern, and Highway 23 with the positions of Minneota and Highway 68, a westerly direction of travel makes sense.
 * The trail established by trailing dog on Day 2 of the search.
 * Arguments against:
 * Porter is farther away than Taunton, (but Brandon probably didn’t know that).

 * 5. Brandon traveled westerly by first going north on Lyon-Lincoln County Rd, then west on 400th St.
 * Arguments for:
 * Brandon’s vehicle was heading west when it went into the ditch. Accordingly, he likely thought his destination was to the west.
 * The brightest glow in the sky was Minneota, which was east-south-east of Brandon’s LKP. He would likely have thought that this was Marshall and headed away from it towards where he thought Lynd was located.
 * Later, he mentioned that he could clearly see the moon (suggesting that it was within his field of vision). At the time he was walking, the moon was due west of his position near or at the horizon.  While certainly a “soft” clue, this suggests that he may have been heading westerly at the time he made the statement.
 * Brandon stated that he believed he was on the road between Savannah Hills Golf Course and Lynd. He also believed that he is just off Highway 23.  If one compares the relative positions of Marshall, the Lyndwood Tavern, and Highway 23 with the positions of Minneota and Highway 68, a westerly direction of travel makes sense.
 * Arguments against:
 * At one point in the conversation, when asked on which side of the road his car was located, he replied; “On the left side.” This indicates that he was facing south at that time, which may suggest an intended direction of travel (towards the south).
 * Porter is farther away than Taunton, (but Brandon probably didn’t know that).

 * 6. Brandon traveled westerly by first going north on Lyon-Lincoln County Rd, then west on 170th Ave (or 180th, 190th, etc.).
 * Arguments for: He may have decided to turn west at any intersection.
 * Arguments against: If he wanted to head west, it is more likely that he turned west at the nearest opportunity.  The phone conversation suggests that he was single-minded in his determination to go to the tavern.

 * 7. Brandon traveled north on the Lyon-Lincoln County Rd.
 * Arguments for:
 * There was a direct line of travel north.
 * It is unknown in what direction he thought the Lyndwood Tavern laid.
 * Arguments against:
 * At some point in the conversation, he left the road and went cross-country, so it is clear that Brandon believed that the tavern was not in a direct path along the road.

 * 8. Brandon traveled northeasterly.
 * Arguments for: None known
 * Arguments against: None known

Scenarios After Phone Went Dead:
 * 1. Brandon was killed at the point the phone went dead (or became immobilized by injury and later perished in the same area).
 * Notes: Brandon was traveling cross-country at the time.  During the 47-minute conversation, Brandon likely traveled between ¾ to 2.5 miles.

 * 1.1 Brandon drowned in one of the many waterways in the area.
 * Possible waterways include:
 * The Yellow Medicine River to the south and west.
 * The Mud Creek to the north.
 * Any of the many drainage ditches in the area.
 * Arguments for:
 * During the phone conversation, Brandon noted hearing moving water nearby.
 * Drowning would certainly account for Brandon’s disappearance.
 * Water searches are difficult to search to a 100% POD, therefore it is possible that he is still missing in the ¾ - 2.5 mile radius from the LKP.
 * Arguments against:
 * While it is clear that Brandon’s phone was rendered inoperable when he uttered the expletive, it is dangerous to assume that Brandon was unable to continue walking after that point.
 * The area within the ¾ - 2.5 mile radius from the LKP has been searched extensively without success (especially the Yellow Medicine River), suggesting that Brandon is located elsewhere.

 * 1.2 Brandon stumbled, injured himself (thereby becoming immobilized), and later succumbed to the cold weather.
 * Arguments for:
 * During the night in question, it is nearly certain that an immobile subject would have become hypothermic (especially with the light clothing that Brandon was wearing).
 * Arguments against:
 * A subject located on the ground within the ¾ - 2.5-mile radius from the LKP would likely have been found by now.

 * 2. Brandon slipped and his phone was somehow rendered inoperable, but he was able to continue walking.
 * 2.1 Brandon became wet from falling, wet grass, and/or sweating from the exertion of walking and became hypothermic over time, eventually succumbing to it.
 * Arguments for:
 * One cannot assume that Brandon stopped moving at the point the cell phone went dead. Cell phones are relatively fragile devices.
 * The area within the ¾ - 2.5 mile radius from the LKP has been searched extensively without success, suggesting that Brandon is located elsewhere. Per Robert Koester in Lost Person Behavior, “Hill, in an examination of suspended searches, found that in almost every case the search planners failed to search outside the median.”  In this case, the median distance is 1.3 miles (which has been searched extensively).  Statistically, there is a 50% chance that Brandon will be located beyond this distance.
 * Arguments against: None known

 * 2.2 Brandon eventually fell into some sort of well, cistern, or tank and perished there.
 * Arguments for:
 * This scenario would certainly account for why no one has spotted his body.
 * Arguments against:
 * Modern well casings for residential use are generally 4-6” in diameter. Small enough for a puppy to fall into, but not a human.  Municipal and commercial casing are often in the range of 16-20” in diameter, which is large enough to fall into.  However, there are none in the area that aren’t enclosed in buildings.  The only well that is large enough for an adult male to fall into that would have been accessible to Brandon is a hand-dug brick or timber lined well.  Any in the area would be over 100 years old and most likely be collapsed.  An intact pioneer well is an exceptionally rare thing in this area (if they are in the area at all).  The likelihood that he found the one intact pioneer well in the entire region is extremely low.
 * For many years, Minnesota law has required that old wells and cisterns be filled in and/or capped.
 * A more likely possibility is falling into a cistern. Many, if not most, old farms in the area had one.  Windmills would pump water into a cistern where it is more readily accessible.  They are often covered with a wooden covering which can rot away.  While all the teams have been looking for them, none have been found.
 * There just aren’t a lot of abandoned farms within the search area. We have found at least three old farmsteads on the map that have been completely wiped off the land.  The farmland in this area is so valuable that every footing, cistern, and piece of concrete has been removed.
 * All of the search teams that I have worked with on this search are aware of old cisterns and the like. The three or four abandoned farms we have found have been searched thoroughly.

 * 3. Brandon was picked up by someone and taken outside of the search area.
 * 3.1 Brandon was later harmed by whoever picked him up.
 * Arguments for: None known
 * Arguments against: The likelihood of meeting a predator in this rural area on a weeknight is quite remote.

 * 3.2 Brandon obtained a ride from someone who was able to transport him to safety and Brandon is alive and well and living elsewhere.
 * Arguments for: None known.
 * Arguments against:
 * Brandon did not have any means of supporting himself.


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Hi Jeff. Have you watchlisted the case page (clicked the "watch" tab at the top of the page)? There are some questions for you on the discussion page. --Una Smith 03:45, 31 January 2010 (UTC)

Hello Jeff, what was the Yellow Medicine River levels like that Thursday in may 2008?

regards. --[User:Directangle] 8:35, 4 April 2017 (UTC)

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text:
User:The Evil IP address/monobook.css

importStylesheetURI('http://meta.wikimedia.org/w/index.php?title=User:The_Evil_IP_address/global.css&action=raw&ctype=text/css');

title: User:The Evil IP address/monobook.css


text:
User talk:Hannu.linturi

 Hello Hannu.linturi, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:Hannu.linturi


text:
User talk:Kumarp

 Hello Kumarp, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:Kumarp


text:
User talk:KaitlinA

 Hello KaitlinA, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:KaitlinA


text:
User talk:LawrenceF

 Hello LawrenceF, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:LawrenceF


text:
User talk:Mary horgan

 Hello Mary horgan, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:Mary horgan


text:
User talk:Mirarahikainen

 Hello Mirarahikainen, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:Mirarahikainen


text:
User talk:Yvonnecleary

 Hello Yvonnecleary, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:Yvonnecleary


text:
User talk:DanteLearning

 Hello DanteLearning, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:DanteLearning


text:
User talk:Wikiblogger~enwikiversity

 Hello Wikiblogger, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Wikiblogger. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Wikiblogger~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:47, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:25, 19 April 2015 (UTC)

title: User talk:Wikiblogger~enwikiversity


text:
User talk:Aoifelen

 Hello Aoifelen, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:Aoifelen


text:
User talk:Nleahy~enwikiversity

 Hello Nleahy, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Nleahy. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Nleahy~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:40, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:17, 19 April 2015 (UTC)

title: User talk:Nleahy~enwikiversity


text:
User talk:CianUraguay

 Hello CianUraguay, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:18, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:18, 31 January 2010 (UTC)

title: User talk:CianUraguay


text:
User talk:Rick Charles

 Hello Rick Charles, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 31 January 2010 (UTC)

title: User talk:Rick Charles


text:
User talk:Sign Tarrant

 Hello Sign Tarrant, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 31 January 2010 (UTC)

title: User talk:Sign Tarrant


text:
User talk:Michaelnewham

 Hello Michaelnewham, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 31 January 2010 (UTC)

title: User talk:Michaelnewham


text:
User talk:Maedhbhs~enwikiversity

 Hello Maedhbhs, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 31 January 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Maedhbhs. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Maedhbhs~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:37, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:15, 19 April 2015 (UTC)

title: User talk:Maedhbhs~enwikiversity


text:
User talk:EGM6341.s10.Team1.Kumanchik

 Hello EGM6341.s10.Team1.Kumanchik, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 31 January 2010 (UTC)

title: User talk:EGM6341.s10.Team1.Kumanchik


text:
User talk:Smoothy~enwikiversity

 Hello Smoothy, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 31 January 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Smoothy. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Smoothy~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:43, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:22, 19 April 2015 (UTC)

title: User talk:Smoothy~enwikiversity


text:
User talk:Kram~enwikiversity

 Hello Kram, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random project.
 * Browse Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! If you're a twitter user, please follow http://twitter.com/Wikiversity. --Mu301Bot 08:19, 31 January 2010 (UTC)

Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 31 January 2010 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Kram. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Kram~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

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Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:41, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:13, 19 April 2015 (UTC)

title: User talk:Kram~enwikiversity


text:
User talk:Robrichard

 Hello Robrichard, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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Feel free to leave a message on my talk page if you have any questions or need help. --mikeu talk 08:19, 31 January 2010 (UTC)

title: User talk:Robrichard


text:
User:Ali k

Looking to learn a good level of French.

title: User:Ali k


title: File:Eas4200c.f08.1.PDF


title: File:Eas4200c.f08.2.PDF


title: File:Eas4200c.f08.3.PDF


title: File:Eas4200c.f08.4.PDF


title: File:Eas4200c.f08.5.PDF


title: File:Eml4500.f08.2.PDF


title: File:Eml4500.f08.1.PDF


title: File:Eml4500.f08.3.PDF


title: File:Eml4500.f08.4.PDF


text:
Category:Eas4200c.f08

Eas4200c.f08

title: Category:Eas4200c.f08


title: File:Eml4500.f08.5.PDF


title: File:Eml4500.f06.1.PDF


title: File:Eml4500.f06.2.PDF


title: File:Eml4500.f06.4.PDF


title: File:Eml4500.f06.3.PDF


title: File:Eml4500.f06.5.PDF


text:
Java Programming/Swing

About Swing Swing library is an official Java GUI toolkit released by Sun Microsystems. Swing is a lightweight component and does not use the native API's of the OS. The main characteristics of the Swing toolkit:
 * platform independent
 * customizable
 * extensible
 * configurable
 * lightweight

Swing is no longer the latest GUI toolkit for use with Java, being replaced by JavaFX. However, it will remain included in Java for the forseeable future. For more information, see the JavaFX Frequently Asked Questions.

Packages

 * javax.swing
 * javax.swing.border
 * javax.swing.colorchooser
 * javax.swing.event
 * javax.swing.filechooser
 * javax.swing.plaf
 * javax.swing.plaf.basic
 * javax.swing.plaf.metal
 * javax.swing.plaf.multi
 * javax.swing.plaf.synth
 * javax.swing.table
 * javax.swing.text
 * javax.swing.text.html
 * javax.swing.text.html.parser
 * javax.swing.text.rtf
 * javax.swing.tree
 * javax.swing.undo

Java Swing first programs
A basic test application for Swing is as follows: import javax.swing.JFrame;

public class Test extends JFrame { public Test { setSize(300, 200); setTitle("Test"); setDefaultCloseOperation(EXIT_ON_CLOSE); } public static void main(String[] args) { Test test = new Test; test.setVisible(true); } }

The constructor contains three function calls. The first two set the size and title of the window. The third statement is used to terminate the application once the window is closed; by default, the program will keep running even though the main function finishes. This is a change from non-graphical Java applications.

title: Java Programming/Swing


text:
User:Egm6341.s10.team3.ynahn81/HW2-3

= (3) Newton-Cotes Method for $$ f(x)=\frac{e^{x}-1}{x} $$ = Ref: Lecture notes [[media:Egm6341.s10.mtg9.pdf|p.9-2]]

Problem Statement
Consider $$ f(x)=\frac{e^{x}-1}{x} $$ on $$ x\in[0,1], \quad x_{0}=a=0, \quad x_{n}=b=1 $$.

(i) Construct $$ f_{n}(x)= \sum_{i=0}^{n}l_{i,n}(x)f(x_{i}) $$ for n = 1, 2, 4, 8, 16.

(ii) Plot $$\displaystyle f(x) $$ and $$\displaystyle f_{n}(x) $$ for n = 1, 2, 4, 8, 16.

(iii) Compute $$ I_{n}=\int_{b}^{b}f_{n}(x)dx $$ for n = 1, 2, 4, 8 and compare to $$\displaystyle I $$.

(iv) For n = 4, plot $$\displaystyle l_{0}, l_{1}, l_{2} $$.

(v) Why don't we have to take a look at $$\displaystyle l_{3}, l_{4} $$ for n = 4?

Solution
Recall the formula
 * {| style="width:100%" border="0" align="left"

l_{i,n}(x) = \Pi_{j=0,i\neq j}^{n}\frac{x-x_{j}}{x_{i}-x_{j}} $$ $$ from [[media:Egm6341.s10.mtg7.pdf|p.7-3]] of lecture note.
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 1)
 * }
 * }

(i)

When n = 1 $$\displaystyle \rightarrow x_{0} = 0, x_{1} = 1 $$.

 * {| style="width:100%" border="0" align="left"

f_{1}=\sum_{i=0}^{1}l_{i,1}(x)f(x_{i})=l_{0,1}(x)f(x_{0})+l_{1,1}(x)f(x_{1}) $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 2)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

l_{0,1}=\Pi_{j=0,j\neq 0}^{1}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{x-1}{0-1}, \quad f(x_{0})=f(0)=1, $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{1,1}=\Pi_{j=0,j\neq 1}^{1}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{x-0}{1-0}, \quad f(x_{1})=f(1)=e-1. $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

When n = 2 $$ \rightarrow x_{0} = 0, x_{1} = \frac{1}{2}, x_{2} = 1 $$.

 * {| style="width:100%" border="0" align="left"

f_{2}=\sum_{i=0}^{2}l_{i,2}(x)f(x_{i})=l_{0,2}(x)f(x_{0})+l_{1,2}(x)f(x_{1})+l_{2,2}(x)f(x_{2}) $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 4)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

l_{0,2}=\Pi_{j=0,j\neq 0}^{2}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{(x-\frac{1}{2})(x-1)}{(0-\frac{1}{2}(0-1)}, \quad f(x_{0})=f(0)=1, $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{1,2}=\Pi_{j=0,j\neq 1}^{2}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{(x-0)(x-1)}{(\frac{1}{2}-0)(\frac{1}{2}-1)}, \quad f(x_{1})=f(\frac{1}{2})=2\left(e^{\frac{1}{2}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{2,2}=\Pi_{j=0,j\neq 2}^{2}\frac{x-x_{j}}{x_{2}-x_{j}}=\frac{(x-0)(x-\frac{1}{2})}{(1-0)(1-\frac{1}{2})}, \quad f(x_{2})=f(1)=e-1. $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

When n = 4 $$ \rightarrow x_{0} = 0, x_{1} = \frac{1}{4}, x_{2} = \frac{2}{4}, x_{3} = \frac{3}{4}, x_{4} = 1 $$.

 * {| style="width:100%" border="0" align="left"

f_{4}=\sum_{i=0}^{4}l_{i,4}(x)f(x_{i})=l_{0,4}(x)f(x_{0})+l_{1,4}(x)f(x_{1})+l_{2,4}(x)f(x_{2})+l_{3,4}(x)f(x_{3})+l_{4,4}(x)f(x_{4}) $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 5)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

l_{0,4}=\Pi_{j=0,j\neq 0}^{4}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{(x-\frac{1}{4})(x-\frac{2}{4})(x-\frac{3}{4})(x-1)}{(0-\frac{1}{4})(0-\frac{2}{4})(0-\frac{3}{4})(0-1)}, \quad f(x_{0})=f(0)=1, $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{1,4}=\Pi_{j=0,j\neq 1}^{4}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{(x-0)(x-\frac{2}{4})(x-\frac{3}{4})(x-1)}{(\frac{1}{4}-0)(\frac{1}{4}-\frac{2}{4})(\frac{1}{4}-\frac{3}{4})(\frac{1}{4}-1)}, \quad f(x_{1})=f(\frac{1}{4})=4\left(e^{\frac{1}{4}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{2,4}=\Pi_{j=0,j\neq 2}^{4}\frac{x-x_{j}}{x_{2}-x_{j}}=\frac{(x-0)(x-\frac{1}{4})(x-\frac{3}{4})(x-1)}{(\frac{2}{4}-0)(\frac{2}{4}-\frac{1}{4})(\frac{2}{4}-\frac{3}{4})(\frac{2}{4}-1)}, \quad f(x_{2})=f(\frac{2}{4})=2\left(e^{\frac{2}{4}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{3,4}=\Pi_{j=0,j\neq 3}^{4}\frac{x-x_{j}}{x_{3}-x_{j}}=\frac{(x-0)(x-\frac{1}{4})(x-\frac{2}{4})(x-1)}{(\frac{3}{4}-0)(\frac{3}{4}-\frac{1}{4})(\frac{3}{4}-\frac{2}{4})(\frac{3}{4}-1)}, \quad f(x_{3})=f(\frac{3}{4})=\frac{4}{3}\left(e^{\frac{3}{4}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{4,4}=\Pi_{j=0,j\neq 4}^{4}\frac{x-x_{j}}{x_{4}-x_{j}}=\frac{(x-0)(x-\frac{1}{4})(x-\frac{2}{4})(x-\frac{3}{4})}{(1-0)(1-\frac{1}{4})(1-\frac{2}{4})(1-\frac{3}{4})}, \quad f(x_{4})=f(1)=e-1, $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

When n = 8 $$ \rightarrow x_{0} = 0, x_{1} = \frac{1}{8}, x_{2} = \frac{2}{8}, x_{3} = \frac{3}{8}, x_{4} = \frac{4}{8}, x_{5} = \frac{5}{8}, x_{6} = \frac{6}{8}, x_{7} = \frac{7}{8}, x_{8} = 1 $$.

 * {| style="width:100%" border="0" align="left"

f_{8}=\sum_{i=0}^{8}l_{i,8}(x)f(x_{i})=l_{0,8}(x)f(x_{0})+l_{1,8}(x)f(x_{1})+l_{2,8}(x)f(x_{2})+l_{3,8}(x)f(x_{3})+l_{4,8}(x)f(x_{4})+l_{5,8}(x)f(x_{5})+l_{6,8}(x)f(x_{6})+l_{7,8}(x)f(x_{7})+l_{8,8}(x)f(x_{8}) $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 6)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

l_{0,8}=\Pi_{j=0,j\neq 0}^{8}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(0-\frac{1}{8})(0-\frac{2}{8})(0-\frac{3}{8})(0-\frac{4}{8})(0-\frac{5}{8})(0-\frac{6}{8})(0-\frac{7}{8})(0-1)}, \quad f(x_{0})=f(0)=1, $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{1,8}=\Pi_{j=0,j\neq 1}^{8}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{(x-0)(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{1}{8}-0)(\frac{1}{8}-\frac{2}{8})(\frac{1}{8}-\frac{3}{8})(\frac{1}{8}-\frac{4}{8})(\frac{1}{8}-\frac{5}{8})(\frac{1}{8}-\frac{6}{8})(\frac{1}{8}-\frac{7}{8})(\frac{1}{8}-1)}, \quad f(x_{1})=f(\frac{1}{8})=8\left(e^{\frac{1}{8}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{2,8}=\Pi_{j=0,j\neq 2}^{8}\frac{x-x_{j}}{x_{2}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{2}{8}-0)(\frac{2}{8}-\frac{1}{8})(\frac{2}{8}-\frac{3}{8})(\frac{2}{8}-\frac{4}{8})(\frac{2}{8}-\frac{5}{8})(\frac{2}{8}-\frac{6}{8})(\frac{2}{8}-\frac{7}{8})(\frac{2}{8}-1)}, \quad f(x_{2})=f(\frac{2}{8})=\frac{8}{2}\left(e^{\frac{2}{8}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{3,8}=\Pi_{j=0,j\neq 3}^{8}\frac{x-x_{j}}{x_{3}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{3}{8}-0)(\frac{3}{8}-\frac{1}{8})(\frac{3}{8}-\frac{2}{8})(\frac{3}{8}-\frac{4}{8})(\frac{3}{8}-\frac{5}{8})(\frac{3}{8}-\frac{6}{8})(\frac{3}{8}-\frac{7}{8})(\frac{3}{8}-1)}, \quad f(x_{3})=f(\frac{3}{8})=\frac{8}{3}\left(e^{\frac{3}{8}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{4,8}=\Pi_{j=0,j\neq 4}^{8}\frac{x-x_{j}}{x_{4}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{4}{8}-0)(\frac{4}{8}-\frac{1}{8})(\frac{4}{8}-\frac{2}{8})(\frac{4}{8}-\frac{3}{8})(\frac{4}{8}-\frac{5}{8})(\frac{4}{8}-\frac{6}{8})(\frac{4}{8}-\frac{7}{8})(\frac{4}{8}-1)}, \quad f(x_{4})=f(\frac{4}{8})=\frac{8}{4}\left(e^{\frac{4}{8}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{5,8}=\Pi_{j=0,j\neq 5}^{8}\frac{x-x_{j}}{x_{5}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{6}{8})(x-\frac{7}{8})(x-1)}{(\frac{5}{8}-0)(\frac{5}{8}-\frac{1}{8})(\frac{5}{8}-\frac{2}{8})(\frac{5}{8}-\frac{3}{8})(\frac{5}{8}-\frac{4}{8})(\frac{5}{8}-\frac{6}{8})(\frac{5}{8}-\frac{7}{8})(\frac{5}{8}-1)}, \quad f(x_{5})=f(\frac{5}{8})=\frac{8}{5}\left(e^{\frac{5}{8}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{6,8}=\Pi_{j=0,j\neq 6}^{8}\frac{x-x_{j}}{x_{6}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{7}{8})(x-1)}{(\frac{6}{8}-0)(\frac{6}{8}-\frac{1}{8})(\frac{6}{8}-\frac{2}{8})(\frac{6}{8}-\frac{3}{8})(\frac{6}{8}-\frac{4}{8})(\frac{6}{8}-\frac{5}{8})(\frac{6}{8}-\frac{7}{8})(\frac{6}{8}-1)}, \quad f(x_{6})=f(\frac{6}{8})=\frac{8}{6}\left(e^{\frac{6}{8}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{7,8}=\Pi_{j=0,j\neq 7}^{8}\frac{x-x_{j}}{x_{7}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-1)}{(\frac{7}{8}-0)(\frac{7}{8}-\frac{1}{8})(\frac{7}{8}-\frac{2}{8})(\frac{7}{8}-\frac{3}{8})(\frac{7}{8}-\frac{4}{8})(\frac{7}{8}-\frac{5}{8})(\frac{7}{8}-\frac{6}{8})(\frac{7}{8}-1)}, \quad f(x_{7})=f(\frac{7}{8})=\frac{8}{7}\left(e^{\frac{7}{8}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{8,8}=\Pi_{j=0,j\neq 8}^{8}\frac{x-x_{j}}{x_{8}-x_{j}}=\frac{(x-0)(x-\frac{1}{8})(x-\frac{2}{8})(x-\frac{3}{8})(x-\frac{4}{8})(x-\frac{5}{8})(x-\frac{6}{8})(x-\frac{7}{8})}{(1-0)(1-\frac{1}{8})(1-\frac{2}{8})(1-\frac{3}{8})(1-\frac{4}{8})(1-\frac{5}{8})(1-\frac{6}{8})(1-\frac{7}{8})}, \quad f(x_{8})=f(1)=e-1. $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

When n = 16 $$ \rightarrow x_{0} = 0, x_{1} = \frac{1}{16}, x_{2} = \frac{2}{16}, x_{3} = \frac{3}{16}, x_{4} = \frac{4}{16}, x_{5} = \frac{5}{16}, x_{6} = \frac{6}{16}, x_{7} = \frac{7}{16}, x_{8} = \frac{8}{16}, x_{9} = \frac{9}{16}, x_{10} = \frac{10}{16}, x_{11} = \frac{11}{16}, x_{12} = \frac{12}{16}, x_{13} = \frac{13}{16}, x_{14} = \frac{14}{16}, x_{15} = \frac{15}{16}, x_{16} = 1 $$.

 * {| style="width:100%" border="0" align="left"

f_{16}=\sum_{i=0}^{16}l_{i,16}(x)f(x_{i})=l_{0,16}(x)f(x_{0})+l_{1,16}(x)f(x_{1})+l_{2,16}(x)f(x_{2})+l_{3,16}(x)f(x_{3})+l_{4,16}(x)f(x_{4})+l_{5,16}(x)f(x_{5})+l_{6,16}(x)f(x_{6})+l_{7,16}(x)f(x_{7})+l_{8,16}(x)f(x_{8})+l_{9,16}(x)f(x_{9})+l_{10,16}(x)f(x_{10})+l_{11,16}(x)f(x_{11})+l_{12,16}(x)f(x_{12})+l_{13,16}(x)f(x_{13})+l_{14,16}(x)f(x_{14})+l_{15,16}(x)f(x_{15})+l_{16,16}(x)f(x_{16}) $$ $$ where
 * $$\displaystyle
 * $$\displaystyle
 * $$\displaystyle (Eq. 7)
 * }
 * }
 * {| style="width:100%" border="0" align="left"

l_{0,16}=\Pi_{j=0,j\neq 0}^{16}\frac{x-x_{j}}{x_{0}-x_{j}}=\frac{(x-\frac{1}{16})(x-\frac{2}{16})(x-\frac{3}{16})(x-\frac{4}{16})(x-\frac{5}{16})(x-\frac{6}{16})(x-\frac{7}{16})(x-\frac{8}{16})(x-\frac{9}{16})(x-\frac{10}{16})(x-\frac{11}{16})(x-\frac{12}{16})(x-\frac{13}{16})(x-\frac{14}{16})(x-\frac{15}{16})(x-1)}{(0-\frac{1}{16})(0-\frac{2}{16})(0-\frac{3}{16})(0-\frac{4}{16})(0-\frac{5}{16})(0-\frac{6}{16})(0-\frac{7}{16})(0-\frac{8}{16})(0-\frac{9}{16})(0-\frac{10}{16})(0-\frac{11}{16})(0-\frac{12}{16})(0-\frac{13}{16})(0-\frac{14}{16})(0-\frac{15}{16})(0-1)}, \quad f(x_{0})=f(0)=1, $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * {| style="width:100%" border="0" align="left"
 * {| style="width:100%" border="0" align="left"

l_{1,16}=\Pi_{j=0,j\neq 1}^{16}\frac{x-x_{j}}{x_{1}-x_{j}}=\frac{(x-0)(x-\frac{2}{16})(x-\frac{3}{16})(x-\frac{4}{16})(x-\frac{5}{16})(x-\frac{6}{16})(x-\frac{7}{16})(x-\frac{8}{16})(x-\frac{9}{16})(x-\frac{10}{16})(x-\frac{11}{16})(x-\frac{12}{16})(x-\frac{13}{16})(x-\frac{14}{16})(x-\frac{15}{16})(x-1)}{(\frac{1}{16}-0)(\frac{1}{16}-\frac{2}{16})(\frac{1}{16}-\frac{3}{16})(\frac{1}{16}-\frac{4}{16})(\frac{1}{16}-\frac{5}{16})(\frac{1}{16}-\frac{6}{16})(\frac{1}{16}-\frac{7}{16})(\frac{1}{16}-\frac{8}{16})(\frac{1}{16}-\frac{9}{16})(\frac{1}{16}-\frac{10}{16})(\frac{1}{16}-\frac{11}{16})(\frac{1}{16}-\frac{12}{16})(\frac{1}{16}-\frac{13}{16})(\frac{1}{16}-\frac{14}{16})(\frac{1}{16}-\frac{15}{16})(\frac{1}{16}-1)}, \quad f(x_{1})=f(\frac{1}{16})=16\left(e^{\frac{1}{16}}-1\right), $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

$$\cdot$$ $$\cdot$$ $$\cdot$$ $$\cdot$$ $$\cdot$$ $$\cdot$$

 * {| style="width:100%" border="0" align="left"

l_{16,16}=\Pi_{j=0,j\neq 16}^{16}\frac{x-x_{j}}{x_{16}-x_{j}}=\frac{(x-0)(x-\frac{1}{16})(x-\frac{2}{16})(x-\frac{3}{16})(x-\frac{4}{16})(x-\frac{5}{16})(x-\frac{6}{16})(x-\frac{7}{16})(x-\frac{8}{16})(x-\frac{9}{16})(x-\frac{10}{16})(x-\frac{11}{16})(x-\frac{12}{16})(x-\frac{13}{16})(x-\frac{14}{16})(x-\frac{15}{16})}{(1-0)(1-\frac{1}{16})(1-\frac{2}{16})(1-\frac{3}{16})(1-\frac{4}{16})(1-\frac{5}{16})(1-\frac{6}{16})(1-\frac{7}{16})(1-\frac{8}{16})(1-\frac{9}{16})(1-\frac{10}{16})(1-\frac{11}{16})(1-\frac{12}{16})(1-\frac{13}{16})(1-\frac{14}{16})(1-\frac{15}{16})}, \quad f(x_{16})=f(1)=e-1, $$
 * $$\displaystyle
 * $$\displaystyle
 * }
 * }

(ii)

 Matlab code of this plots is provided in the end of this document

In order to compare the plots of different functions, it is definitely efficient to plot all functions in the same figure. In this case, however, the plots of functions $$\displaystyle f_{n}(x) $$ for n = 2, 4, 8, 16 almost overlap with $$\displaystyle f(x) $$ for $$\displaystyle x \in [0, 1] $$, and thus we cannot distinguish a specific plot from the others. Thus, we use separate figures for each plot of functions.

Plot $$\displaystyle f(x) = \frac{e^{x}-1}{x} $$

Plot $$\displaystyle f_{1}(x)$$

Plot $$\displaystyle f_{2}(x)$$

Plot $$\displaystyle f_{4}(x)$$

Plot $$\displaystyle f_{8}(x)$$

Plot $$\displaystyle f_{16}(x)$$

(iii)

 Matlab code of this calculations is provided in the end of this document

(iv)

 Matlab code of this plots is provided in the end of this document

Plots of $$\displaystyle l_{0}, l_{1}, l_{2} $$ when n = 4.

(v)

 Matlab code of this plots is provided in the end of this document

As the plots of $$\displaystyle l_{i} $$ for all $$\displaystyle i $$ in the figure below, the plots of $$\displaystyle l_{3} $$ and $$\displaystyle l_{4} $$ are the mirror images of $$\displaystyle l_{1} $$ and $$\displaystyle l_{0} $$ respectively with respect to $$\displaystyle x = x_{2}=0.5 $$. Therefore, we don't need to take a look at the plots of $$\displaystyle l_{3} $$ and $$\displaystyle l_{4} $$ if we already get the plots of $$\displaystyle l_{0} $$ and $$\displaystyle l_{1} $$.

Matlab code for (ii), (iii), (iv) and (v)

For this problem, one global code is much more efficient than the several short codes for each parts of problem. Therefore, we are providing a comprehensive code below.

title: User:Egm6341.s10.team3.ynahn81/HW2-3


text:
User:RjorjorjoHW2

EGM6341 HW 2

=Problem 9-2 (2)=

Statement
$$f\left( x \right)=\!\frac{e^x-1}{x}\,\,on\,\,[\!\,0,1\,]\!$$

$$x_0=\!a=\!0,\,x_n=\!b=\,1$$

Consider $$n=\!1,2,4,8,16$$

Constr. $$f_n\left( x \right)=\,\sum_{i=0}^{n}l_{i,n}\left ( x \right)f\left ( x_i \right)$$

Plot $$f.\!\,\, f_n, n=\!1,\,2,\,4,\,8,\,16$$

Comp. $$ I_n=\!\int_{a}^{b}f_n\left ( x \right)dx,\,n=\!1,2,4,8\,$$ and compare to $$ I\!$$

For $$ n=\!4\,$$, plot $$\,l_0,\,l_1,\,l_2 $$ why not $$\,l_3,\,l_4,\,$$ also $$2\,$$

Solution

 * {| style="width:100%" border="1" align="left"

 * Ref: Lecture Notes [[media:Egm6341.s10.mtg7.pdf|P7-3]]
 * }
 * }

MATLAB
Plot $$f.\!\,\, f_n, n=\!1,\,2,\,4,\,8,\,16:\!$$

For $$ n=\!4\,$$, plot $$\,l_0,\,l_1,\,l_2 $$ why not $$\,l_3,\,l_4,\,$$

For $$ n=\!2\,$$, plot $$\,l_0,\,l_1,\,l_2 \,$$

=Problem 12-2=

Statement
Why :$$E_{n}^{n+1}\left ( x \right )=F^{n+1}\left ( x \right)-0$$

Solution

 * {| style="width:100%" border="1" align="left"

 * Ref: Lecture Notes [[media:Egm6341.s10.mtg6.pdf|P6-4]], [[media:Egm6341.s10.mtg7.pdf|P7-3]]
 * }
 * }

From [[media:Egm6341.s10.mtg6.pdf|P6-4]]

\begin{align} \\E_{n} &= I-I_{n}\; \\ &= \int_{0}^{1}f\left ( x \right )dx-\int_{0}^{1}f_{n}\left ( x \right )dx\,. \end{align} $$

the first derivative of $$E_{n}\,$$ is

 * $$E_n^{1}=\int_{0}^{1}f^{1}\left ( x \right )dx-\int_{0}^{1}f_{n}^{1}\left ( x \right )dx\,$$

the second derivative of $$E_{n}\,$$ is

 * $$E_n^{2}=\int_{0}^{1}f^{2}\left ( x \right )dx-\int_{0}^{1}f_{n}^{2}\left ( x \right )dx\,$$

 * {| style=text-align:center

 * width=2% | $$.\,$$
 * width=2% | $$.\,$$
 * width=2% | $$.\,$$
 * }
 * width=2% | $$.\,$$
 * }
 * }
 * }

 * $$E_n^{n}=\int_{0}^{1}f^{n}\left ( x \right )dx-\int_{0}^{1}f_{n}^{n}\left ( x \right )dx\,$$

 * $$E_n^{n+1}=\int_{0}^{1}f^{n+1}\left ( x \right )dx-\int_{0}^{1}f_{n}^{n+1}\left ( x \right )dx\,$$

From [[media:Egm6341.s10.mtg7.pdf|P7-3]]
 * $$f_{n}\left ( x \right )=\sum _{i=0}^{n}l_{i,n}\left ( x \right ) f\left ( x_i \right)\,$$

 * (i)where $$l_{i,n}\left ( x \right)=l_{i}\left ( x \right)=\prod_{j=0,i \neq j}^{n}\frac{x-x_{j}}{x_{0}-x_{j}}\,$$

 * when $$i=0\,$$
 * {| style=text-align:center

\begin{align} \\n=1, \rightarrow & l_i\left ( x \right)=\frac{x-x_{1}}{x_{0}-x_{1}} \rightarrow l_{0}=l\left ( x^{1}\right)\; \\n=2, \rightarrow & l_i\left ( x \right)=\frac{x-x_{1}}{x_{0}-x_{1}}\frac{x-x_{2}}{x_{0}-x_{2}} \rightarrow l_{0}=l\left ( x^{2}\right)\; \\.\, \\.\, \\.\, \\n=n, \rightarrow & l_i\left ( x \right)=\frac{x-x_{1}}{x_{0}-x_{1}}\frac{x-x_{2}}{x_{0}-x_{2}}...\frac{x-x_{n-1}}{x_{0}-x_{n-1}}\frac{x-x_{n}}{x_{0}-x_{n}} \rightarrow l_{0}=l\left ( x^{n}\right)\, \end{align} $$
 * }
 * }

 * (ii)$$f\left( x_{i} \right)\,$$ is constant, so $$f_n\left( x \right)\,$$ is $$x^{n}\,$$'s function

 * $$f_n^{n+1}\left ( x \right)=0\,$$

 * $$E_n^{n+1}=\int_{0}^{1}f^{n+1}\left ( x \right )dx-\int_{0}^{1}f_{n}^{n+1}\left ( x \right )dx\,$$

 * $$\rightarrow E_n^{n+1}=\int_{0}^{1}f^{n+1}\left ( x \right )dx-\int_{0}^{1} 0 dx\,$$

 * $$\rightarrow E_n^{n+1}=\!F^{n+1}\left ( x \right )dx- 0 \,$$

 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\ E_n^{n+1}=\!F^{n+1}\left ( x \right )dx- 0 \, $$
 * $$\displaystyle\!$$
 * }
 * }

=Problem 15-1=

Statement
Prove why $$\alpha^{1}\left ( t \right )=F\left ( -t \right)+F\left ( t \right)\,$$

Given
$$\alpha\left ( t \right )=\int_{-t}^{k}f\left ( x\left ( t \right) \right)dt+\int_{k}^{t}f\left ( x\left ( t \right) \right)dt, k\in\![-t,t]\, $$

Solution

 * {| style="width:100%" border="1" align="left"

 * Ref: Lecture Notes [[media:Egm6341.s10.mtg14.pdf|P14-2]]
 * }
 * }

(i)$$\int_{k}^{t}F\left ( x\left ( t \right) \right)dt=\mathbb{F}\!\left ( x\left ( t \right) \right)-\mathbb{F}\!\left ( x\left ( k \right) \right)\, $$

 * $$\frac{d}{dt}(\int_{k}^{t}F\left ( x\left ( t \right) \right)dt)=\frac{d}{dt}(\mathbb{F}\!\left ( x\left ( t \right) \right))-\frac{d}{dt}(\mathbb{F}\!\left ( x\left ( k \right) \right))\, $$

 * since$$k=constant\,\rightarrow\mathbb{F}\!\left ( x\left ( k \right) \right)\, $$ is constant

\begin{align} \\\frac{d}{dt}(\mathbb{F}\!\left ( x\left ( t \right) \right)dt)-\frac{d}{dt}(constant) &=F\left ( t \right)-0\; \\ &= F\left ( t \right)\, \end{align} $$

(ii)$$\int_{-t}^{k}F\left ( x\left ( t \right) \right)dt=\mathbb{F}\!\left ( x\left ( k \right) \right)-\mathbb{F}\!\left ( x\left ( -t \right) \right)\, $$

 * $$\frac{d}{dt}(\int_{-t}^{k}F\left ( x\left ( t \right) \right)dt)=\frac{d}{dt}(\mathbb{F}\!\left ( x\left ( k \right) \right))+\frac{d}{dt}(\mathbb{F}\!\left ( x\left ( -t \right) \right))\, $$
 * P.S. Chain rule :$$\frac{d}{dt}(\int_{-t}^{k}F\left ( x\left ( t \right) \right)dt)=\frac{d}{dt}(\mathbb{F}\!\left ( x\left ( k \right) \right))-(\frac{d(-t)}{dt})\frac{d}{dt}(\mathbb{F}\!\left ( x\left ( -t \right) \right))\, $$

 * since$$k=constant\,\rightarrow\mathbb{F}\!\left ( x\left ( k \right) \right)\, $$ is constant

\begin{align} \\\frac{d}{dt}(constant)+\frac{d}{dt}(\mathbb{F}\!\left ( x\left ( -t \right) \right)dt) &=0+F\left ( t \right)+0\; \\ &=F\left ( -t \right)\, \end{align} $$

From (i) and (ii)

\begin{align} \\\alpha^{1}\left ( t \right ) &= \frac{d}{dt}(\int_{-t}^{k}F\left ( x\left ( t \right) \right)dt)+\frac{d}{dt}(\int_{k}^{t}F\left ( x\left ( t \right) \right)dt)\; \\ &= F\left ( -t \right)+F\left ( t \right)\, \end{align} $$

 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

\ alpha^{1}\left ( t \right )=\! F\left ( -t \right)+F\left ( t \right) \, $$
 * $$\displaystyle\!$$
 * }
 * }

=Problem 15-2=

Statement
Prove why $$G^{(2)}\left ( 0 \right )=0\,$$

Given
$$G^{(1)}\left ( 0 \right )=0\,$$ and $$G^{(1)}\left ( \xi_1 \right )=0\,$$ Apply Rolle's theorem
 * $$ \exists\!\,\,\,\,\xi_2\,\,\,\,\in\!\,\,\,\,[\,0,\xi_1\,]:\,\,\,\,\!G^{(2)}\left ( \xi_2 \right)=0\,$$

Solution

 * {| style="width:100%" border="1" align="left"

 * Ref: Lecture Notes [[media:Egm6341.s10.mtg15.pdf|P15-1]],[[media:Egm6341.s10.mtg14.pdf|P14-2]]
 * }
 * }

From [[media:Egm6341.s10.mtg14.pdf|P14-2]]

$$G\left( t \right):=\!\,e\left( t \right)-\!t^{5}G\left( 1 \right)\,$$

Do [[media:Egm6341.s10.mtg15.pdf|P15-1]] again to find out $$G^{(2)}\left ( 0 \right)=\!0$$

$$G^{(1)}\left( t \right)=\!\,e^{(1)}\left( t \right)-\!5t^{4}e\,$$

$$G^{(2)}\left( t \right)=\!\,e^{(2)}\left( t \right)-\!20t^{3}e\,$$

We need $$ e^{(2)}\left ( t \right)$$

 * $$e\left( t \right):=\!\alpha\left ( t \right)-\!\alpha_2\left ( t \right)\,$$

 * $$\alpha\left ( t \right)=\!\int_{-t}^{k}F\left ( -t \right)+\!\int_{k}^{t}F\left ( t \right)\,\,\,\,k\in\!\,\,[\,-t,t\,]\,$$

 * $$\alpha_2\left ( t \right)=\!\frac{t}{3}[\!\,F\left ( -t \right)+\!4F\left ( 0 \right)+\!F\left ( t \right)]\!\,$$

$$\rightarrow e^{(2)}\left( t \right):=\!\alpha^{(2)}\left ( t \right)-\!\alpha_2^{(2)}\left ( t \right)\,$$

We need $$\alpha^{(2)}\left ( t \right)\,\,\,$$ and $$\,\,\, \alpha_2^{(2)}\left ( t \right)\,$$

 * $$\alpha^{(1)}\left ( t \right)=\!F\left ( -t \right)+\!F\left ( t \right)\,$$

 * $$\alpha_2^{(1)}\left ( t \right)=\!\frac{1}{3}[\!\,F\left ( -t \right)+\!4F\left ( 0 \right)+\!F\left ( t \right)]\!+\!\frac{t}{3}[\!\,-F^{(1)}\left ( -t \right)+\! F^{(1)}\left ( t \right)]\!\,$$

$$\alpha^{(2)}\left ( t \right)=\!-\!F^{(1)}\left ( -t \right)+\!F^{(1)}\left ( t \right)\,$$

$$\alpha_2^{(2)}\left ( t \right)=\!\frac{1}{3}[\!\,-\!F^{(1)}\left ( -t \right)+\!\,F^{(1)}\left ( t \right)]\!+\!\,\frac{1}{3}[\!\,-F^{(1)}\left ( -t \right)+\! F^{(1)}\left ( t \right)]\!+\!\frac{t}{3}[\!\,+\!F^{(2)}\left ( -t \right)+\! F^{(2)}\left ( t \right)]\!\,$$

Finally

\begin{align} \\G^{(2)}\left( t \right) &= e^{(2)}\left( t \right)-\!20t^{3}e\; \\ &= \alpha^{(2)}\left ( t \right)-\!\alpha_2^{(2)}\left ( t \right)-\!20t^{3}e\; \\ &= -\!F^{(1)}\left ( -t \right)+\!F^{(1)}\left ( t \right)-\!\frac{1}{3}[\!\,-\!F^{(1)}\left ( -t \right)+\!F^{(1)}\left ( t \right)]\!-\!\frac{1}{3}[\!\,-F^{(1)}\left ( -t \right)+\! F^{(1)}\left ( t \right)]\!-\!\frac{t}{3}[\!\,+\!F^{(2)}\left ( -t \right)+\! F^{(2)}\left ( t \right)]\!-\!20t^{3}e\,. \end{align} $$

When $$t=\!0$$

\begin{align} \\G^{(2)}\left( 0 \right) &= e^{(2)}\left( 0 \right)-\!0\; \\ &= \alpha^{(2)}\left ( 0 \right)-\!\alpha_2^{(2)}\left ( 0 \right)-\!0\; \\ &= -\!F^{(1)}\left ( 0 \right)+\!F^{(1)}\left ( 0 \right)-\!\frac{1}{3}[\!\,-\!F^{(1)}\left ( 0 \right)+\!F^{(1)}\left ( 0 \right)]\!-\!\frac{1}{3}[\!\,-F^{(1)}\left ( 0 \right)+\! F^{(1)}\left ( 0 \right)]\!-\!0-\!0\; \\ &= 0\, \end{align} $$

 * {| style="width:100%" border="0" align="left"

 * style="width:50%; padding:10px; border:2px solid #8888aa" |
 * $$ \displaystyle
 * $$ \displaystyle

G^{(2)}\left ( 0 \right)\,=\!\,0 \, $$
 * $$\displaystyle\!$$
 * }
 * }

title: User:RjorjorjoHW2


text:
Votian/Nouns and Their Derivation

== Nouns  and their  derivation  ==

adjective-to-noun

 *  * -ut
 * It is declined like the XV Declension: -t stems.
 * Pronunciation varies (  -uz ,  -üz  ) according to vowel harmony e.g.  лad́d́uz  ("width " <  лad́d́a  "wide"),  лaiskuz  ("lazyness " <  лaiska  "lazy"),  iлozuz  ("beauty" <  iлoza  "pretty"),  terveüz  ("health " <  terve  "healthy"),  nōruz  (" youth " <  nōri  "young"),  nōre̮puz  (" being younger " <  nōre̮pi  "younger"),  pēnüz  ("being small " <  pēni  "small").
 * Should not be confused with  * -u-ks .

verb-to-noun & noun-to-noun

 *  * -ks
 * It is declined like the XI Declension: -ks stems.
 * Pronunciation is (  -z  ) in the end of a word e.g.  jaлgaz  ("runner (of a sleigh) " <  jaлka  "foot"),  arjaz  ("bristle " <  arja  "brush"),  teräz  ("steel" <  terä  "blade").
 * Compear with  * -u-ks .

 *  * -u-ks
 * It is declined like the XI Declension: -ks stems.
 * Pronunciation varies (  -uz ,  -üz  ) according to vowel harmony e.g.  kagлuz  ("collar " <  kagлa  "neck"),  se̮rmuz  (" ring " <  se̮rmi  "finger"),  arvatuz  (" riddle " <  arvat-a  "to guess"),  pelehüz  ("ghost " ~  peleht-ellä  "to frighten").
 * The stem is usually in the weak grade if there is a such one available e.g.  kannuz  ("spur " <  kanta  "heel"),  rinnuz  ("horse-collar's strap" <  rinta  "chest"),  jätüz  ("remnant " <  jättǟ  "to remain").
 * Should not be confused with  * -ut .

 *  * -(u-)m
 * It is declined like the XIII Declension: -m stems.
 * Any final -m went through a protofinnic sound change *  m > n . The resulting final -n is usually pronounced as a prolongation of the preceding vowel e.g.  ve̮tī : ve̮ttime̮D  (" key : keys" <  ve̮ttā  "to take"),  vȫtī : vȫttimeD  (" ridge band : (plural)" <  vȫttǟ  "to gird"),  viē̮  : vite̮me̮D  (" scutching-blade: (plural)").
 * This suffix takes a vowel stem. Therefore the stem vary according to  consonant gradation  e.g.  ve̮d́d́ē̮  : ve̮ite̮me̮D  (" ointment : (plural)" <  ve̮itā  "to salve"),  jaлgū : jaлkume̮D  ("stirrup " <  jaлka  "foot").
 * Compear with  * -ma .

title: Votian/Nouns and Their Derivation


title: File:TauntonMN-2.gif


text:
Talk:Graphic Design/2D Design

I'd like to vote for this being primarily called graphic design. I've never heard of 2D design. Anyone else feel this way?Wmcleod 02:40, 1 February 2010 (UTC)

title: Talk:Graphic Design/2D Design


text: Category:UCNISS

title: Category:UCNISS


text:
User:Wolf-tracker

About myself
Hello, I'm Wolf-tracker or wolf for short.

I live in Brooklyn, Ny.

Things I like:

title: User:Wolf-tracker