Wikipedia:Reference desk/Archives/Mathematics/2015 June 16

= June 16 =

v
help — Preceding unsigned comment added by 184.191.143.218 (talk) 16:16, 16 June 2015 (UTC)


 * You're going to need to give us an actual question. We don't even know if that v for velocity, a vector, or something else. StuRat (talk) 16:25, 16 June 2015 (UTC)


 * While there's no more info I'd suggest to consult the v page and the V (disambiguation). :) CiaPan (talk) 17:01, 16 June 2015 (UTC)
 * I'd like to think he's asking about VVVVVV. This being WP:RD/math, he must want to know about Reflection symmetry. -- Meni Rosenfeld (talk) 17:07, 16 June 2015 (UTC)
 * If you're drawing Trogdor the Burninator, make sure the Vs are consummate. -- Kinu  t/c 18:41, 16 June 2015 (UTC)


 * Perhaps ∠V<>Λ?  → Michael J Ⓣ Ⓒ Ⓜ 05:44, 20 June 2015 (UTC)

What are the odds of summer starting on a Sunday?
A little less likely than 1 in 7 I imagine with leap years throwing off the days every four years.

Baseballfan (talk) 18:07, 16 June 2015 (UTC)


 * I fail to see why it is less than 1 in 7. What day of the week is the solstice more likely to fall on?  Leap year doesn't do anything to the days of the week or to the Earth's orbit.  I think it will average out to 1 in 7 over a sufficient period of time.  Robert McClenon (talk) 18:35, 16 June 2015 (UTC)


 * If leap year throws it off a day wouldn't that make it slightly less? And the solstice may change slightly by a day or two. So, over time, maybe 1 in 7.0000233 (for example). Anyway way to mathematically figure it out? Baseballfan (talk) 21:56, 16 June 2015 (UTC)


 * Leap year doesn't throw it off a day. Leap year doesn't change the day of the week, only the numbering of the days using the Gregorian calendar.  If it were to change it, what day are you saying would be more likely for the solstice?  Leap year doesn't throw the day of the week off.  It only changes the numbering of the days using the Gregorian calendar.  The date of the solstice can also be specified using a calendar that doesn't have Gregorian leap years, such as the Jewish calendar, which has the same days of the week (after all, the week of seven days is of Jewish origin) and has 29-day and 30-day months and a leap month in 7 out of 19 years.  The solstice will still fall on the same day of the week using the Jewish calendar as the Gregorian calendar.  Robert McClenon (talk) 22:35, 16 June 2015 (UTC)


 * My friend brought it up, hence my original question that he got me wondering (I said 1 in 7). I'll take your word :P Baseballfan (talk) 22:46, 16 June 2015 (UTC)


 * There is some interaction between leap years and days of the week. For example, the 13th of a month is just a slight bit more likely (a 1 in 6.9767 chance) to be a Friday than any other day (only a 1 in 7.0175 chance for Thursday or Saturday).  This works because the 146,097 days in the 400 year cycle of leap years is divisible by 7, the number of days in the week, so the same pattern plays out every cycle.  See the bottom of Friday the 13th. -- ToE 02:49, 17 June 2015 (UTC)


 * But, if you look at it over an even longer time period, eventually additional leap days (and/or negative leap days) not in the 400 year cycle would be needed, shifting all the days and odds over, and this would all eventually even back out to 1/7th for each day. StuRat (talk) 03:43, 17 June 2015 (UTC)


 * Maybe more to the point, the solstice is not on a fixed day of the month. In fact it will be on different days in different time zones.  It's a particular point in the Earth's orbit, and where it falls modulo the seven-day week really has nothing to do with twenty-ninths of February.
 * Certainly, the Earth's orbit changes slightly over time, and the rotational speed of the Earth changes slightly over time, so you can't quite argue from regularity that solstices have to be exactly equally distributed among the days of the week. But it should be pretty bloody close. --Trovatore (talk) 04:18, 17 June 2015 (UTC)


 * The law of large numbers comes into play here. Specifically, as t -> ∞, P -> 1/7. StuRat (talk) 01:59, 20 June 2015 (UTC)


 * As mentioned above, leap years change the matching of day of week to day of month, but they do not change the matching between day of the week (there are always 7 days per week) and physical reality (which cares nothing about our calendars).
 * But more importantly, even if it did affect it somehow... Why do you think it would make Sunday less likely? What is special about Sunday? And if you think it makes any weekday less likely, then, where does all this missing likelihood go? The probabilities for each day must sum to 1 (unless you actually want to introduce weeks with 8 days...) -- Meni Rosenfeld (talk) 15:53, 17 June 2015 (UTC)


 * As mentioned above in Friday the 13th, there is a very slight bias as to the occurrence of particular days of the week and particular dates of the month in the Gregorian calendar. The Gregorian calendar exactly repeats itself every 400 years.  If the tropical year was exactly equal to 365.2425 days, then one could use that to determine what the frequency of the solstice occurring on a particular day of the week was.  However, although the Gregorian calendar was very well designed to keep the occurrence of the solstices consistent with the calendar, it isn't exact, because the tropical year is closer to 365.24218 days.  Also, as noted, the solstice doesn't even occur on the same day of the month everywhere in the world.  You could count the frequency with which the solstice occurs on any particular day of the week in UTC in a particular group of four centuries, such as 1600 to 2000 or 2000 to 2400, and the pattern will not be the same in another group of four centuries.  As mentioned above, the probabilities for the seven days must add to 1.  (The world's Jews, Christians, and Muslims will never allow you to introduce weeks of any number of days other than seven.  To many people, the length of the week really is sacred.)  Robert McClenon (talk) 16:37, 17 June 2015 (UTC)


 * (EC) There certainly shouldn't be any correlation over a long enough period of time, but what led me to thoughts of the excessive Friday-ness of 13ths is that the Gregorian calender was established to better track the tropical year (though as mentioned here and shown in the table at the bottom of Tropical year, it does a considerably worse job tracking the northern solstice than it does the southern solstice or either equinox), and that if the northern solstice dates were largely repeated in successive 400 year cycles, the fact that 400 years have a whole number of weeks could yield a temporary resonance which leads to an imbalance in the distribution of northern solstice days over any continuous 400 day period taken out of a somewhat larger range of years.


 * I don't know if that is what is really going on here, particularly as the time of northern solstice is slipping by about 8 hours every 400 year cycle, but using the Table of dates/times (UTC) of the northern solstice from 1600-2400 (linked to from Summer solstice), we find that only 113 of those 801 northern solstices -- 1 in 7.0885 -- fall on a Sunday. Even if you stack the cards in Sunday's favor by lopping off the three non-Sunday northern solstices at the start and the two at the end, you still only have 113 Sundays of 796 northern solstices for 1 in 7.0442.


 * Testing my hypothesis, of the 401 continuous 400-year periods in the table, every single one has only 56 or 57 Sunday northern solstices, for 1 in 7.1429 or 1 in 7.0175, with an average of 56.55 Sunday northern solstices per 400 year period, for 1 in 7.0736.


 * I too question Baseballfan's logic, but their example of "maybe 1 in 7.0000233" understates the actual ~ 1% temporary deficit of Sunday northern solstices (when viewed in the time frame of the 400 year Gregorian cycle) which we are currently experiencing. -- ToE 17:27, 17 June 2015 (UTC)


 * There is a plan to eliminate all these temporary deficits and put the calendar on an accurate footing.  See immediately above Talk:Tropical year for details. 5.150.92.20 (talk) 19:39, 17 June 2015 (UTC)


 * 113 sunday solistices in 801 years is only 1.4 less than would be expected if it was an even distribution, so it's hardly a big effect, and might just be coincidence. I'm not sure how valid checking every 400 year period is - if the "missing" sunday is around the middle of the table (i.e. around the year 2000), then it's going to show up in most of those periods - and looking at the table, 2003-2004 skips Sat-Mon.  Performing the same for 2000-2800 might well give a different result (since the solstice of 2400 falls on the 20th June, which changes where the skip in the day of the week occurs compared to the leap year cycle) MChesterMC (talk) 08:33, 18 June 2015 (UTC)


 * It's true the summer solstice skipped a Sunday in 2003/4.  That will happen again in 2020/1, 2065/6, 2082/3, 2099/2100 etc.   It seems to happen every 17 or 45 years.   What might be interesting to investigate is if there is any period less than 400 years in which the interval between summer solstices is an exact number of weeks.   Since the summer solstice year (like all equinox and solstice years) is constantly changing it would be easier to use the mean summer solstice year (the mean tropical year) in the investigation.   The mean summer solstice in 1983 fell on a Wednesday at 9.15 AM and each year it falls 1d 5h 48m 45s later in the week.


 * In the Jewish calendar, which has been running for 4,000 years, the tekufah (equinox) of Samuel returns to the same day and hour every 28 years and there may well be some imbalance within that period.  However, there is another equinox (that of Adda) which does not follow the 28 year solar cycle.   That said, the Passover can never commence on a Monday, Wednesday or Friday. 5.150.92.20 (talk) 09:46, 18 June 2015 (UTC)


 * In Reference desk/Archives/Mathematics/2015 February 10 there was a claim (obviously wrong) that a month could have five Fridays, Saturdays and Sundays in it only once in 823 years. 5.150.92.20 (talk) 09:53, 18 June 2015 (UTC)