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January 2

Sum of differences

Is there an algorithm for finding the maximum sum of absolute differences of N integers (1..N or 0..N-1 : it doesn't matter which) when arranged in a circle.

EG N=4 we can have 1 2 3 4 (1) with absolute differences 1 1 1 3 so a sum of 6. Or we can have 1 4 2 3 (1) -> 3 2 1 2 with sum 8. Or we can have 1 2 4 3 (1) -> 1 2 1 2 again sum 6.

I want to work out the "biggest" circle for N=26 without having to try every permutation.

-- SGBailey (talk) 13:10, 2 January 2020 (UTC)[reply]

This looks like what you need: Geeks for Geeks: Maximize sum of consecutive differences in a circular array. --Canley (talk) 13:36, 2 January 2020 (UTC)[reply]

Thanks. -- SGBailey (talk) 20:23, 2 January 2020 (UTC)[reply]

@SGBailey: for the particular case of N consecutive integers, see this OEIS entry. --JBL (talk) 21:15, 4 January 2020 (UTC)[reply]

I think there is an OEIS sequence for the original problem (as a function of N), because I think I remember working on it. Bubba73 ^{You talkin' to me?} 21:35, 9 January 2020 (UTC)[reply]

Entia multiplicata praeter necessitatem

It appears to be not uncommon in mathematics that, when a topic is new, a multitude of terms are introduced which the student all has to learn. This occurs several times in The Whetstone of Witte, e.g. in the names for multiples, the table of proportion, and the signs and names for powers (of which we only kept the names “square” and “cube”). (For these names he lists an alternative, more systematic nomenclature, which he discourages because he feels the names of his first list “do expresse the qualitie of the nomber, better then these later names doe.” However, he lets his “Scholar” thank the Master “double, sith you are contente to teache me double names”!)

Similarly, before Adam Riese introduces multiplication and division, he inserts a chapter each for duplication and “mediation”.

The same occurs in other areas, such as chemical nomenclature, but due to their direct connection with physical nature they are not as clearly a figment of the human mind as in mathematics.

Do we have an article on this phenomenon? What can we learn from it for contemporary areas of mathematics? Will they, too, get simpler and more consistent over time, and can we foresee in which direction? ◄ Sebastian 16:40, 2 January 2020 (UTC)[reply]

Concepts, even apparently fundamental ones, evolve over time. I don't think this phenomenon is limited to mathematics, though you may have a point that the changes are more apparently mental than physical in that case. I don't think the evolution necessarily goes in the direction of simplification either since since concepts often become more formal, abstract and complex as they are generalized to cover different situations. For example, with multiplication the idea of a product has been extended to apply in abstract algebra to such objects as rings and fields. Then there's the Cartesian product which has been generalized to a meaning in category theory which is virtually unrecognizable when compared with the original idea. So I think your question is more about philosophy than mathematics, and more philosophy proper than the philosophy of mathematics which tends to be concerned with finding fundamental principles upon which the rest of mathematics can be built. Probably the article which comes closest to what you're getting at is Conceptual history, though it's described as branch of semantics. As for the other questions, see the guidelines above about asking for predictions. In other words if I knew the future then I'd be busy getting rich on Wall Street, not answering questions on the internet. --RDBury (talk) 08:04, 4 January 2020 (UTC)[reply]

Including/excluding the first day

Apparently indicating a range of time/duration (e.g. a month) paradoxically produces two different results, depending on whether the initial day is included or excluded. E.g. 30 days actually yields either 29 or 30 depending on whether the first day is included. While normally the first day is excluded for such purposes, for example the 30-day transit pass in Poland somewhat misleadingly includes the first day, even if both cases involve 30 days. Apparently both cannot be true simultaneously (by that inclusive count common parlance deadlines like "after two days" would include the initial day as well). Is there a name for this oddity? 212.180.235.46 (talk) 17:11, 2 January 2020 (UTC)[reply]

It's some version of the fencepost error. (In real life, I call this the "posts-versus-rails" problem, which I find more descriptive than "fencepost", but I don't know if this is common.) --JBL (talk) 04:25, 3 January 2020 (UTC)[reply]

Separately, your statement of reality seems garbled. Here is an accurate version: the interval of time that starts on the day of the month numbered a and ends on the day of the same month numbered b contains

b - a + 1

days. Thus, in a string of d consecutive days, the difference between the number of the first day and the number of the last day in the string is

d - 1

. The phrase "after two days" is ambiguous, but for unrelated reasons. --JBL (talk) 16:58, 3 January 2020 (UTC)[reply]

I'm not sure it's for unrelated reasons. Or rather, you may have an example ambiguity in mind that's unrelated, but there's also an ambiguity that is related.

An example appropriate to the current season is the circumcision of Christ, which traditionally is dated "eight days after" his birth. But his traditional birth date is December 25, and eight days after that (meaning 8 · 24 hours) is January 2. But his circumcision is traditionally dated January 1. I'm pretty sure the discrepancy is precisely the fencepost thing. --Trovatore (talk) 22:58, 3 January 2020 (UTC)[reply]

I wrote a thing, but I'm not totally committed to it, so here it is small. Depending on where you are, some rail passes are "30 day passes" in the sense that they expire after 11:59 PM and are valid on 30 days, starting with the first day on which they are used. If you use such a pass on the first day of March, it will correctly expire at the end of the day on the 30th, and not be valid at all on the 31st. In some other places, a "30 day pass" may actually be valid for 720 hours beginning with first use, and if you first use such a pass on the first day of March (at any time other than 12:00 AM exactly) the pass will be valid for some portion of time on the 31st. However, I do not think this is the same kind of confusion as not understanding what is meant by "after eight days", whose genuine ambiguity is more akin to expecting a 30-day rail pass to be valid both at 8 AM on March 1st and at 5 PM on March 31st. --JBL (talk) 00:04, 4 January 2020 (UTC)[reply]

In French two weeks is commonly referred to as a quinzaine (fifteenth) and a week a huitaine (eighth). Christian mythology has Jesus crucified on a Friday and resurrected on the following Sunday, the "third day". 2A01:E34:EF5E:4640:41ED:BCB4:C707:D0ED (talk) 22:32, 3 January 2020 (UTC)[reply]

In English, "see you in a week" does not unambiguously indicate the next day on which we will see each other; in French, if you say "see you in a huitaine", am I right that this ambiguity is absent? --JBL (talk) 00:04, 4 January 2020 (UTC)[reply]

the same ambiguity exists. The phrase would typically be followed by naming the day. — Preceding unsigned comment added by 2a01:e34:ef5e:4640:41ed:bcb4:c707:d0ed (talk)

Interesting! --JBL (talk) 15:23, 4 January 2020 (UTC)[reply]

"In a week" is not ambiguous. If today is Wednesday, it means on Wednesday next week. In informal use it may be used imprecisely to mean "in about a week", but that's not the same as claiming that it can have another specific meaning. --142.112.159.101 (talk) 03:35, 5 January 2020 (UTC)[reply]

That seems definite, but it's not. If it's Wednesday early morning, 'in a week' may well be also the next Wednesday late evening. So a week in this case may be actually almost 8 times 24 hours. --CiaPan (talk) 10:10, 6 January 2020 (UTC)[reply]