[off topic] notation

hv at crypt.org hv at crypt.org
Tue Apr 10 16:52:33 CEST 2007

Your u(x) is often written as x^+ and is called the positive part of x.  Similarly, x^- = max(-x, 0) is the negative part of x, and x = x^+ - x^-.


-----Original Message-----
From: hv at zen.crypt.org [mailto:hv at zen.crypt.org] On Behalf Of hv at crypt.org
Sent: Tuesday, April 10, 2007 10:53 AM
To: seqfan at ext.jussieu.fr
Cc: hv at crypt.org
Subject: [off topic] notation

I'm working on an enumeration sequence, and I find that in my calculations I'm regularly needing recourse to two functions that I don't really know how to write, which also makes it harder to think about them.

The first is a truncation function u(x):
  u(x) = max(x, 0)
    = (x + |x|)/2

The second is a similar v(x), with an exception at zero:
  v(x) = u(x) + \delta_x [Kronecker delta]
    = u(x) + c(x = 0)
    = { 1    (x = 0)
      { u(x) (x != 0)

Is there standard notation and/or naming for either of these functions, particularly such as would allow me to search for them and learn how to do useful maths with them?

In case it helps, the domain of 'x' I'm interested in is the integers in both cases.


I was wondering about those; there are many more duplicates that fit into
your three categories than duplicates that have the same definition. One
4 times. Would it be best to submit those to the list? Here are the top 10
duplicates, the number of times they occur regardless of definition,
comments on any obvious groupings, and the other A-numbers that match:

A016176	23 (Inverse of nth cyclotomic polynomial)
A014922 A014988 A015076 A015120 A015142 A015186 A015208 A015252 A015406
A015450 A015516 A015538 A015648 A015670 A015736 A015802 A015846 A015912
A015978 A016000 A016110 A016132

A016258	18 (Inverse of nth cyclotomic polynomial)
A015088 A015166 A015270 A015322 A015348 A015400 A015426 A015478 A015660
A015712 A015790 A015816 A015946 A015972 A016050 A016128 A016180

A016221	10 (Inverse of nth cyclotomic polynomial)
A015157 A015213 A015325 A015493 A015661 A015717 A015885 A015997 A016053

A016236	10 (Inverse of nth cyclotomic polynomial)
A015420 A015522 A015658 A015726 A015760 A015828 A015862 A015930 A016168

A043869	9 (Number n such that the number of runs in Base 4, various
A043593 A043827 A043830 A043834 A043839 A043845 A043852 A043860

A043818	9 (Number n such that the number of runs in Base 3, various
A043583 A043773 A043779 A043783 A043788 A043794 A043801 A043809

A043870	9 (Number n such that the number of runs in Base 4, various
A043594 A043825 A043831 A043835 A043840 A043846 A043853 A043861

A043819	8 (Number n such that the number of runs in Base 3, various
A043776 A043784 A043789 A043584 A043795 A043802 A043810

A016156	8 (Inverse of nth cyclotomic polynomial)
A015586 A015700 A015852 A015928 A015966 A016042 A016080

A043871	8 (Number n such that the number of runs in Base 4, various
A043595 A043828 A043836 A043841 A043847 A043854 A043862

Most of the duplicates are like these. There's another group that I'm
tracking where the sequences might be the same, but I either don't know
enough about the math involved to tell or it's probable that they differ at
higher terms. I can pass those along to the list as I find them if that's
the appropriate thing to do.

-----Original Message-----

The ones where the definition is not the same are also of interest.
There are three possibilities:

1) Although the definitions are different, the actual sequences (out to
infinity) are the same.  In this case, these are duplicates, and should
be merged.
2) The sequences differ at some term beyond what is included in the
database.  In this case, at least one - preferably both - should have a
comment indicating where they differ.
3) We can't tell if the sequences are different or not.  In this case,
they should reference each other with some sort of comment indicating
that the equality of the sequences is unknown.

Franklin T. Adams-Watters

-----Original Message-----

decided in the down-time to find duplicate sequences in the OEIS. I
have a text file of about 1500 sequences that match lexically (same
values, same ordering) that I'm using, then manually comparing the
happy to share this file with anyone who'd like to participate, email
me off-list and I'll send it along (approx. 280k text file). Anyways,
here are a few that I've found this evening by working from the last
line of the file upwards. If it's preferable, I'll send future
duplicates to Neil directly or a designated editor/editors to keep
traffic down on the email list.

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