i am looking for a couple ALMOST impossible math problems-Alan B Fabian
Alan B Fabian
I appreciate anything you have to offer.
this is an advanced high school class.
Thank you, Alan B Fabian
Joshua Cranmer
What type of "high school" class? At my high school, courses ranged from
Geometry to Differential Equations.
If it's a multivariable calculus or linear algebra course, you could try
pushing them to prove that the shortest distance between two points is a
straight line.
N. Silver
Google will work for you.
Or you can start here: http://mathforum.org
Or here: http://www.artofproblemsolving.com/
Robert Israel
> Alan B Fabian wrote:
>
> >i am a relatively new teacher and i want something to really push
> > my advanced math class. i have a couple amazingly in-touch
> > students that are ready for some advanced material far beyond
> > the rest. i wanted to see if anyone had some stuff for me to give
> > to them. I appreciate anything you have to offer. this is an
> > advanced high school class.
>
> Google will work for you.
But be careful: if you can google the question, they may be able to
google the answer.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Gerry Myerson
<31448564.1214504950...@nitrogen.mathforum.org>,
Here are some books you might find useful:
Hungarian Problem Book I and
Hungarian Problem Book II. Don't worry, they're in English,
translated by Elvira Rapaport. Originally published by Singer,
I'm pretty sure they've been republished by the MAA.
The William Lowell Putnam Mathematical Competition.
There are three books in this series. The 2nd was something of
a cut'n'paste job, the other two are more substantial volumes.
The Inquisitive Problem Solver, by Vaderlind, Guy, and Larson.
Two books by Peter Winkler: Mathematical Puzzles
and Mathematical Mind-Benders.
Halmos, Problems for Mathematicians Young and Old.
Kornhauser, Velleman, and Wagon, Which Way did the Bicycle Go?
Steven Barr has written some nice books of puzzles,
as has Ross Honsberger. Then there's always Dudeney,
and Sam Loyd.
--
Gerry Myerson (ge...@maths.mq.edi.ai) (i -> u for email)
Dave L. Renfro
Of the several thousand posts I've made to sci.math and
a few other internet-archived groups in the past 9 years,
several hundred of these discuss topics that would likely
be of interest to some of your students. I've included a
few of these posts below, but they only represent a very
quick search of my posts for a handful of topics. Incidentally,
if it's of any interest, for several years in the 1990s I
taught at one of the top few math/science academies in
the U.S. and some of the topics/ideas in my posts have
actually been used in my classes (often as extra credit
assignments or supplementary work for near-olympiad level
to olympiad level students).
"Inspired" mathematical induction examples
http://mathforum.org/kb/thread.jspa?messageID=5965460
http://mathforum.org/kb/message.jspa?messageID=5966861
http://mathforum.org/kb/message.jspa?messageID=5968462
Rationalizing 1 / [ sqrt(2) + sqrt(10) + sqrt(12) + sqrt(56) ]
http://groups.google.com/group/sci.math/msg/a51025968669c23e
http://groups.google.com/group/sci.math/msg/145effd98eac1b1f
Which is larger, e^pi or pi^e?
http://mathforum.org/kb/message.jspa?messageID=6016525
BIG NUMBERS #1, #2, #3
http://groups.google.com/group/sci.math/msg/403051f310ff3dfc
http://groups.google.com/group/sci.math/msg/d12962e3af2c74b7
http://groups.google.com/group/sci.math/msg/4f2ed8e0385b72f2
A "paradox" in differentiating x^2 = x+x+...+x (x-times)
http://mathforum.org/kb/message.jspa?messageID=676045
http://mathforum.org/kb/message.jspa?messageID=5219431
Derivative of rational powers of x using limit definition
http://mathforum.org/kb/message.jspa?messageID=6017409
A lot of examples, remarks, and references for L'Hopital's rule
http://tinyurl.com/2comud
Limits of the form infinity - infinity
http://mathforum.org/kb/message.jspa?messageID=5928321
The graph of x^2 + (x^2 + 1)(y^2 - 10 - x)^2 = 100
http://mathforum.org/kb/message.jspa?messageID=5846292
Implicit differentiation and evolute of a curve
http://mathforum.org/kb/message.jspa?messageID=6128959
Estimates of partial sums of harmonic series
http://mathforum.org/kb/message.jspa?messageID=4725124
Examples of finding ODEs for families of curves
http://groups.google.com/group/sci.math/msg/f751b566953ca0eb
Number of vertices, edges, 2-faces, 3-faces, etc. of an n-cube
http://groups.google.com/group/sci.math/msg/f8853d9754fba265
Various associativity relations (examples, proofs, references)
http://groups.google.com/group/sci.math/msg/f4eece2eec10824c
Dave L. Renfro
Dave L. Renfro
[omitted]
Here's another example that I wrote not very long ago that
would serve for strong students in a trigonometry class:
The approximation sin(x) = x - (1/6)x^3
http://mathforum.org/kb/message.jspa?messageID=6189492
Dave L. Renfro
David Bernier
The following problem is too hard -- I think. But it's interesting ...
From sci.math, Feb./March 2005:
thread: "f is constant?" started by Betelgeuse (also known as Jenny ).
Ref.:
http://groups.google.com/group/sci.math/msg/25121c9fcec86eda
Betelgeuse wrote in part:
<< Let f:(a,b)->R be differentiable in every point.
Assume also that I know that f'(x)=0 for
all x in A, where A is an open dense subset of (a,b).
Can I conclude that f is a constant function? >>
I think the question was not settled by any sci.math post in the "f is
constant?"
thread from around early March 2005 ...
David Bernier
Pubkeybreaker
Can't we construct a counter example??
Let A be the interval (c,d) with a < c < d < b. It is thus
an open dense subset of (a,b).
Now consider a function that is a quarter-circle in (a,c] ,
a horizontal line segment in (c,d) and another quarter circle
from [d,b) such that the line segment is tangent to each of the
quarter circles. Clearly this is differentiable at every point,
and clearly f' is 0 only on the line segment.
Is there something wrong with this construction? It seems too easy.
Dave L. Renfro
> The following problem is too hard -- I think. But it's
> interesting ...
> From sci.math, Feb./March 2005:
> thread: "f is constant?" started by Betelgeuse (also known
> as Jenny ).
> Ref.:
> http://groups.google.com/group/sci.math/msg/25121c9fcec86eda
>
> Betelgeuse wrote in part:
> << Let f:(a,b)->R be differentiable in every point.
> Assume also that I know that f'(x)=0 for
> all x in A, where A is an open dense subset of (a,b).
> Can I conclude that f is a constant function? >>
>
> I think the question was not settled by any sci.math post
> in the "f is constant?"
> thread from around early March 2005 ...
The answer is no. If E is a subset of the open interval that
has an uncountable complement, then there exists a non-constant
differentiable function f on the interval such that f'(x) = 0
at each point of E [*]. Conversely, if f is differentiable
on an open interval and f'(x) = 0 on a co-countable subset
of that interval (i.e. on a set whose complement relative
to the interval is countable), then f is constant. In the
first statement, the zero set of f' can't always be limited
to just the points of E, because the zero set of a derivative
is a Borel set, so not every subset E of the open interval
can be the zero set of a derivative. (This is similar to the
fact that not every set can be the zero set of a continuous
function. Only closed sets can be the zero set of a continuous
function.)
[*] The March 2005 condition is that f'(x) = 0 on a set whose
complement is nowhere dense. But a nowhere dense set can
be uncountable (even have cardinality c; even have positive
Lebesgue measure), so having a nowhere dense complement
does not give us a set that is large enough for us to be
able to interpolate the property of having a zero derivative
on the set to having a zero derivative throughout the interval.
[As an analogy, for continuous functions E being dense
in the open interval is the precise largeness condition
needed in order to be able to interpolate from having zero
value on the set to having a zero value throughout the
interval.]
More precise results are given in the following post,
which also contains a number of references:
http://groups.google.com/group/sci.math/msg/443ba725a7d6722e
As for the
Dave L. Renfro
Dave L. Renfro
> Let A be the interval (c,d) with a < c < d < b.
> It is thus an open dense subset of (a,b).
The interval (c,d) is dense in itself, but
not dense in (a,b). It's a good bet that
"open dense subset of (a,b)" was intended
to mean "open and relatively dense subset
of (a,b)". Indeed, the empty set is a set
that is open and dense (relative to itself).
Dave L. Renfro
David Bernier
Thanks for the info.
I tried to get a copy of the article by Maurey and Tacchi, but haven't
been able
to get a connection to math.jussieu.fr .
http://www.math.jussieu.fr/~maurey/articles/
David Bernier
Dave L. Renfro
> Thanks for the info.
> I tried to get a copy of the article by Maurey and Tacchi,
> but haven't been able to get a connection to math.jussieu.fr .
>
> http://www.math.jussieu.fr/~maurey/articles/
I tried several searches just now and couldn't find a
digital copy. I'm certain a digital copy was once available
because I remember downloading and printing a hard copy
of it a few years ago. I looked through the files I have
on my work computer and couldn't find it. These files include
everything I had on my home & work computers up to the end
of July 2005, when I moved to another state, but not all
the work I've done since then. (I'm not in an academic position
anymore, so virtually all my math research/explorations/dabblings
are done at home, except for some "breaks from work for internet
posting and internet math searches" type of things.) However,
I'm pretty sure that I downloaded and printed the Maurey/Tacchi
paper before July 2005, so I'm pretty sure I don't have a
digital copy on my home computer either. I can't check for
sure, because right now all of my math stuff (including computer)
is: (1) in a large storage unit, (2) is filling up half of
someone's formerly empty car garage, and (3) is filling up the
entire floor space of a bedroom in an apartment I moved into
a week ago (after staying in a hotel for 9 days, which I had to
when my former apartment building was evacuated about 3 weeks ago).
Thus, while at some point I suppose I could snail-mail you a
photocopy of the paper if you're _really_ interested in having
a copy of the paper, it might be a few weeks to a couple of
months before things get sorted out enough for me to lay my
hands on it.
-------------------------------
http://groups.google.com/group/sci.math/msg/6f5944f2dc9dc297 [see end]
http://mathforum.org/kb/thread.jspa?messageID=6267102 [see beginning]
I lived about 100 yards from the lowest part of the "Coralville
Strip",
and my location was flooding several days before the flooding on the
Coralville Strip began. More specifically, I lived on the portion of
4th Avenue (1/3 mile long) that connects between the Coralville Strip
(named because so many restaurants and businesses are sprinkled along
this road) and 5th street. By the way, the Coralville Strip is the
same as "Highway 6".
http://www.google.com/search?q=flooding+coralville-strip
http://www.google.com/search?q=flooding+coralville+4th-avenue
The flood was so bad that pretty much everything in the 500-year
flood plane was flooded. This means the present flood's magnitude
was estimated to be a level that is reached or exceeded about once
every 500 years (or, more precisely, the probability of it occurring
in any specified 12-month period had been estimated to be 0.2%).
-------------------------------
Dave L. Renfro
David Bernier
> David Bernier wrote:
>
>> Thanks for the info.
>> I tried to get a copy of the article by Maurey and Tacchi,
>> but haven't been able to get a connection to math.jussieu.fr .
>>
>> http://www.math.jussieu.fr/~maurey/articles/
>
> I tried several searches just now and couldn't find a
> digital copy. I'm certain a digital copy was once available
> because I remember downloading and printing a hard copy
> of it a few years ago. I looked through the files I have
> on my work computer and couldn't find it. These files include
> of July 2005, when I moved to another state, but not all
> the work I've done since then. (I'm not in an academic position
> anymore, so virtually all my math research/explorations/dabblings
> are done at home, except for some "breaks from work for internet
> posting and internet math searches" type of things.) However,
> I'm pretty sure that I downloaded and printed the Maurey/Tacchi
> paper before July 2005, so I'm pretty sure I don't have a
> digital copy on my home computer either. I can't check for
> sure, because right now all of my math stuff (including computer)
> is: (1) in a large storage unit, (2) is filling up half of
> someone's formerly empty car garage, and (3) is filling up the
> entire floor space of a bedroom in an apartment I moved into
> a week ago (after staying in a hotel for 9 days, which I had to
> when my former apartment building was evacuated about 3 weeks ago).
> Thus, while at some point I suppose I could snail-mail you a
> photocopy of the paper if you're _really_ interested in having
> a copy of the paper, it might be a few weeks to a couple of
> months before things get sorted out enough for me to lay my
> hands on it.
Thanks for the offer. It seems entirely possible that the Web server
at www.math.jussieu.fr is having temporary problems.
If you know of a few books or less ``obscure" journals with relevant
material (e.g. references to the Semenov paper/results, maybe...),
and can mention a few, I'd appreciate that.
I might go to the local university library and have a look.
Empirically, I'd say a counterexample to what Betelgeuse was questioning
(actually Betelgeuse wasn't conjecturing anything ...),
seems harder to find than space-filling curves.
I was wondering if any relevant result is in the fairly well-known book
_Counterexamples in Analysis_ . I don't have that book.
Regards,
David Bernier
P.S. I seem to remember reading that Serge Lang's office was full of
papers,
files and so on.
** Posted from http://www.teranews.com **
Dave L. Renfro
> If you know of a few books or less ``obscure" journals
> with relevant material (e.g. references to the Semenov
> paper/results, maybe...), and can mention a few, I'd
> appreciate that. I might go to the local university
> library and have a look.
>
> Empirically, I'd say a counterexample to what Betelgeuse
> was questioning (actually Betelgeuse wasn't conjecturing
> anything ...), seems harder to find than space-filling curves.
>
> I was wondering if any relevant result is in the fairly
> well-known book _Counterexamples in Analysis_ . I don't
> have that book.
I glanced through the extensive ASCII-written notes I wrote
a few years ago, when I was planning to post a greatly
expanded version of [1] (which I've more or less abandoned,
since it's so tedious to write everything in ASCII format),
and came up with the four references below. The second
reference gives a weaker result for the property you want,
since f' is zero on a measure dense set instead of an open
dense set, but I thought it might be of interest anyway.
The first reference is on the internet (at google-books),
and it appears that an example isn't actually given, but
it does mention that one can be found in a 1950 Trans. AMS
paper by Z. Zahorski (a paper in French that I happen to
have an English translation of, but only in hard-copy form).
However, the expository remarks in the first paper are
definitely relevant to the things you're asking about.
I'm also almost certain that an example, probably examples
having a variety of other properties as well, can be
found in the fourth reference below.
[1] HISTORICAL ESSAY ON CONTINUITY OF DERIVATIVES
http://groups.google.com/group/sci.math/msg/814be41b1ea8c024
You might also want to look through Bruckner/Bruckner/Thomson's
1997 book "Real Analysis", which is freely available on the
internet as a .pdf file at
http://www.math.ucsb.edu/~akemann/Bruckner.pdf
----------------------
Andrew M. Bruckner and John L. Leonard, "On differentiable
functions having an everywhere dense set of intervals of
constancy", Canadian Mathematical Bulletin 8 (1965), 73-76.
[MR 30 #4870; Zbl 144.05103]
http://www.emis.de/cgi-bin/Zarchive?an=0144.05103
http://books.google.com/books?id=b4gHogA6iLQC
Search in this book = "Bruckner" (without quotes)
Choose p. 73.
----------------------
Frank S. Cater, "A derivative often zero and discontinuous",
Real Analysis Exchange 11 (1985-86), 265-270.
[MR 87c:26007; Zbl 646.26011]
http://www.emis.de/cgi-bin/MATH-item?0646.26011
A bounded derivative f' is constructed in [a,b] such that
if G is the set of points where f' is continuous and H is
the set of points where f' = 0, then G has measure zero
and H has positive measure in every subinterval of [a,b].
----------------------
Zofia Denkowska, "An example of a function which is locally
constant in an open dense set, everywhere differentiable but
not constant", Annales Polonici Mathematici 28 (1973), 195-199.
[MR 48 #2320; Zbl 269.26006]
http://www.emis.de/cgi-bin/MATH-item?0269.26006
Zbl review by A. Smajdor: "In the paper is given a construction
of a function which is locally constant in an open dense set
in [0,1] and everywhere differentiable but not constant.
A construction of such a function has been given in several
papers (cf. references of this paper) but the methods used
in this paper are simpler and more elementary."
----------------------
Solomon Marcus, "Sur les dérivées dont les zéros forment un
ensemble frontière partout dense" [On derivatives that are
zero on a set whose boundary is everywhere dense], Rendiconti
del Circolo Matematico di Palermo (2) 12 (1963), 5-40.
[MR 29 #4844; Zbl 124.03202]
http://www.zentralblatt-math.org/zmath/en/search/?an=0124.03202
----------------------
Dave L. Renfro
David Bernier
This seems to be very interesting book, at first glance.
> ----------------------
>
> Andrew M. Bruckner and John L. Leonard, "On differentiable
> functions having an everywhere dense set of intervals of
> constancy", Canadian Mathematical Bulletin 8 (1965), 73-76.
> [MR 30 #4870; Zbl 144.05103]
> http://www.emis.de/cgi-bin/Zarchive?an=0144.05103
>
> http://books.google.com/books?id=b4gHogA6iLQC
> Search in this book = "Bruckner" (without quotes)
> Choose p. 73.
>
> ----------------------
>
> Frank S. Cater, "A derivative often zero and discontinuous",
> Real Analysis Exchange 11 (1985-86), 265-270.
> [MR 87c:26007; Zbl 646.26011]
> http://www.emis.de/cgi-bin/MATH-item?0646.26011
>
> A bounded derivative f' is constructed in [a,b] such that
> if G is the set of points where f' is continuous and H is
> the set of points where f' = 0, then G has measure zero
> and H has positive measure in every subinterval of [a,b].
>
> ----------------------
>
> Zofia Denkowska, "An example of a function which is locally
> constant in an open dense set, everywhere differentiable but
> not constant", Annales Polonici Mathematici 28 (1973), 195-199.
> [MR 48 #2320; Zbl 269.26006]
> http://www.emis.de/cgi-bin/MATH-item?0269.26006
Probably the Zofia Denkowska mentioned on a Web page of the
Institute of Mathematics of the Jagiellonian University, (Warsaw?), Poland:
http://www.im.uj.edu.pl/institute/chairs/krr/
> Zbl review by A. Smajdor: "In the paper is given a construction
> of a function which is locally constant in an open dense set
> in [0,1] and everywhere differentiable but not constant.
> A construction of such a function has been given in several
> papers (cf. references of this paper) but the methods used
> in this paper are simpler and more elementary."
I couldn't learn about the references. Incidentally, I found out about
Minkowski's question-mark function:
http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function
It's strictly increasing, continuous and has a derivative of
zero at every rational number.
I think we can construct a subset of [0, 1] with nowhere-dense complement
of positive Lebesgue measure that might be useful here. (named 'E' below).
A = { x in ]0, 1[ s.t. if a0 +1/(a_1 + 1/(a_2 + 1/(a_3 + ...
.....)))) is the continued fraction
rep. for x, (with a0 = 0), then we have at least one of:
a_1 = 2
a_2 = 4
...
a_k = 2^k ... (some k>=1) }.
Let D = [0, 1] \ interior(A) ; Is D a fat Cantor set? [ perfect set
of positive measure].
E = [0, 1] \ D is open, dense in [0, 1].
Maybe we can try to construct a Lebesgue integrable function g based on
E and D.
If x is in E, then g(x) = 0. To define g on D, we have to be careful,
else the
Lebesgue integral function ( with derivative g) won't be differentiable
everywhere.
Actually, since D is supposed to have Lebesgue measure > 0 , extending g
to that f(a) := int_{0, a} g dmu , mu Lebesgue measure, seems really
hard. The general idea would be to have g get closer to zero as one
approaches E (where g is zero). Since E is dense in [0, 1], there
might have to be fractal like scaling or something in extending g.
> ----------------------
>
> Solomon Marcus, "Sur les dérivées dont les zéros forment un
> ensemble frontière partout dense" [On derivatives that are
> zero on a set whose boundary is everywhere dense], Rendiconti
> del Circolo Matematico di Palermo (2) 12 (1963), 5-40.
> [MR 29 #4844; Zbl 124.03202]
> http://www.zentralblatt-math.org/zmath/en/search/?an=0124.03202
>
> ----------------------
>
> Dave L. Renfro
David Bernier
Bob F
I am enjoying the book "Euler The Master of Us All" by William Dunham
(MAA, 1999) and it has lots of interesting, wonderful things to learn
about that I think would be of interest to you and your students.
There is also a wonderful web site devoted to Euler at
http://www.math.dartmouth.edu/~euler/ and there are several English
translations of his papers.
"Study the masters and not the pupils" wrote Niels Abel...
-Bob
David Bernier
Actually, it seems the boundary of E consists of rationals:
0, 1
1/2 , 1/3
4/5, 5/6, 4/13, 5/16 , + smaller numbers ...
...
An irrational that has infinitely many very good rational approximations
by those
rationals should get a lower value for the 'g' function....
If S denotes the boundary of E, (presumably rationals .... )
and alpha is some irrational,
consider inf_{ y>0 such that {b in S, b = r/s, (r,s)=1, | r/s -
alpha| < 1/(s^{1/y}) } is infinite }.
if the inf_ < 0.01, then |r/s - alpha | < 1/(s^100) for rationals r/s in
S, (r,s)=1, with s as
large as one wants. (infinite set).
So I'd propose
g(alpha) = inf_{ y>0 such that {b in S, b = r/s, (r,s)=1, | r/s -
alpha| < 1/(s^{1/y}) } is infinite }
for irrational alpha, and alpha rational but not in S.
If we integrate g, get f, and look at f(alpha+h)/h for |h|>0 but very tiny,
where alpha irrational as above, I see no reason why this
would be close to g(alpha), as h->0.
I give up..
I.N. Galidakis
[snip]
> I couldn't learn about the references. Incidentally, I found out about
> Minkowski's question-mark function:
> http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function
>
> It's strictly increasing, continuous and has a derivative of
> zero at every rational number.
A totally fascinating function. Many thanks for this reference.
If I may temporarily hijack this thread, the above reference led to the notion
of "absolutely continuous", which I believe I have not encountered before, so I
tried it on a couple of functions, with d(x,y) = |x - y|.
f(x) = x is absolutely continuous in any finite interval I, because we can take
delta = epsilon.
Then I tried f(x) = x^2, which appears to be absolutely continuous on any finite
interval, since we can take delta = epsilon /(n*x_n*y_n), where n is the number
of sub-intervals and [x_n, y_n] is the last sub-interval (to the right).
Looks like f(x) = x^2 is NOT absolutely continuous on [0,oo) (because x_n and
y_n can grow without bounds)
Then I tried it on f(x) = x*sin(1/x), which is one of the examples given on
Wiki:
http://en.wikipedia.org/wiki/Absolutely_continuous
Wiki says that f(x) is NOT absolutely continuous on any sub-interval I
containing the origin.
The easiest way to see that, is of course to note that it is not of bounded
variation, but I wanted to check it directly from the epsilon-delta definition.
So, I choose just ONE sub-interval of I, [x_n, y_n] = [0, 1/(Pi/2 + 2*n*Pi)]
(definition holds for any finite collection of sub-intervals, so n can be 1).
Then,
sum(|x_k - y_k|, k = 1..n) = 1/(Pi/2 + 2*n*Pi), and by increasing n, I can make
this as small as I want.
But then,
sum(|f(x_k) - f(y_k)|, k = 1..n) = |0 - 1/(Pi/2 + 2*n*Pi))*sin(Pi/2 + 2*n*Pi)| =
1/(Pi/2 + 2*n*Pi).
I am having a bout of stupidity here, so can someone show me what I am doing
wrong with this specific case (for n=1 and only this sub-interval)?
Many thanks,
[snip]
--
I.N. Galidakis
Dave L. Renfro
> Incidentally, I found out about Minkowski's question-mark function:
> http://en.wikipedia.org/wiki/Minkowski%27s_question_mark_function
>
> It's strictly increasing, continuous and has a derivative of
> zero at every rational number.
I've got a lot of photocopies of papers on the Minkowski ? function,
but it seems this is one of those topics (among probably many) that
I have quite a bit of literature on but which I have not gotten
around to posting any literature bibliographies on. Indeed, I only
got around to posting a lot of references on the ruler function
about 1.5 years ago.
Differentiability of the Ruler Function
http://groups.google.com/group/sci.math/msg/95b4aabac073ca91
http://groups.google.com/group/sci.math/msg/6be4464698ebe19a
Functions that are increasing and have a zero derivative almost
everywhere are called "singular functions". I believe continuity
might sometimes be assumed in the definition, but I think it's more
convenient to define this without continuity, since we can say
"continuous singular function" when the need arises, whereas
we can't really use the continuity-inclusive term when we want
to consider increasing-with-zero-derivative functions that could
be continuous or discontinuous. (Note that "discontinuous singular
functions" or "possibly discontinuous singular functions" wouldn't
work when the continuity-inclusive term is used.) There is also an
ambiguity in "increasing", but note that if f(x) is non-decreasing
with a zero derivative almost everywhere, then f(x) + x is strictly
increasing and has the same property and, for most purposes, adding
x to a function isn't going to alter the properties one would be
interested in.
I didn't mention anything about continuous singular functions
in my previous posts in this thread, mainly because they aren't
finitely differentiable everywhere. However, if we relax
"differentiable" to include infinite derivatives (of the
bilateral type -- at each point, both the left and right
derivatives are equal as extended real numbers), then these
functions still don't qualify for what the original sub-thread
question was about, since these have zero derivative on a
co-measure-zero set, not a co-nowhere-dense set. Nonetheless,
if the issue is simply to have examples of non-constant
continuous functions that have a zero derivative in many
places and are differentiable (at least, in the extended
real number sense), then continuous singular functions should
be tossed into the pile too.
Below are a couple of my posts that have some connection with
these topics, if you're interested.
derivative of strictly increasing continuous function
http://groups.google.com/group/sci.math/msg/a4b84fcaa9789d2b
ESSAY ON NON-DIFFERENTIABILITY POINTS OF MONOTONE FUNCTIONS
http://groups.google.com/group/sci.math/msg/1bd39d992c91e950
I.N. Galidakis wrote (in another post in this thread) about
absolutely continuous functions. This is a major topic in
Lebesgue integration, so most any book on Lebesgue integration
or on measure theory will have quite a bit of information
and examples. A quick search of my sci.math posts turned up
the following:
Functions that take sets of measure zero to sets of measure zero
http://groups.google.com/group/sci.math/msg/1ecb9de0b2ca7a36
http://groups.google.com/group/sci.math/msg/1e5355704b854ecf
http://groups.google.com/group/sci.math/msg/f380a07221107e1e
Absolutely continuous, xsin(1/x)
http://groups.google.com/group/sci.math/msg/67c09f8ee05142d2
By the way, for those who are interested in this kind of stuff
(meaning you're not interested going through a formal graduate
level textbook on real analysis, but rather you're more specifically
interested in looking at neat counterexamples and at the "theory
of pathology" in real analysis), I strongly recommend getting
the following excellent book:
A.C.M. van Rooij and W.H. Schikhof, "A Second Course on Real
Functions", Cambridge University Press, 1982.
Among textbooks, if that's the route you want to go, the best
by a very long shot for these things (counterexamples and the
"theory of pathology" kinds of things) is the Bruckner, Bruckner,
Thomson text I cited in a previous post, the book that's freely
available in .pdf form on the internet.
Dave L. Renfro
Greg Calvino
Greg Calvino
David Bernier
As you may know, if a function F on [a, b] { or maybe (a, b) } has a
derivative f,
recovering F from f and F(a) or F( (a+b)/2 ) can be hard to do,
if it's not true that f>= 0. Actually, it may be that f not being of
bounded variation may be close to the real problem-area in
using the Lebesgue integral to try to get F from f.
So the "construct the anti-derivative" problem could be of
independent interest. There's a survey paper by P S Bullen
on "Non-absolute integrals" available here:
http://www.emis.de/proceedings/Toronto2000/papers/bullen.pdf
math.jussieu.fr was back, and I had a look at the Maurey and Tacchi paper;
it seems to cover developments in real analysis around 1870 or so,
and the "theoreme des accroissements finis" in particular:
http://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_des_accroissements_finis
So, this should be the Mean value theorem; it's fairly long, probably
mostly historical; I didn't take a careful look, so their paper could have
material helpful to this sub-thread on differentiable functions
with discontinuous derivatives and all that.
Dave L. Renfro
> So the "construct the anti-derivative" problem could
> be of independent interest. There's a survey paper
> by P S Bullen on "Non-absolute integrals" available here:
> http://www.emis.de/proceedings/Toronto2000/papers/bullen.pdf
Related to this, I posted some comments back on May 21
that might be of interest:
------------------------
http://groups.google.com/group/sci.math/msg/bc4d738500d2c961
The problem of finding some kind of descriptive
or topological way to classify derivatives (such as
classifying/describing the inverse images of open
intervals under derivatives) was the driving force
behind much of this work, a problem that I think
originates from William H. Young. Relatively
recently (1990s), some logicians got involved and
showed that, in a certain sense, no relatively
simple characterization is possible [8].
[snip]
[8] http://www.google.com/search?q=complexity-of-antidifferentiation
------------------------
I know almost nothing about the Denjoy integral or about
the omega_1 height Cantor-Bendixson complexity scale methods
that Kechris and his students have used to define hierarchies
of real analysis pathology phenomena (a survey of this is in
Chapter 33 of Kechris' book "Classical Descriptive Set Theory"),
by the way.
Dave L. Renfro
David Bernier
I know basically nothing about "non-absolute integrals", except their
existence and some uses.
I had another look at the article by Maurey and Tacchi. It's almost
a historical investigation of some subtle results on derivatives of
functions and their anti-derivatives for real-valued functions centered
around the early to mid- 1880's in Europe and Britain. It's the kind
of topic that could be suitable for a senior's thesis (the mathematical
results, that is; getting the history right is another story, I'd say).
Scheeffer was very much admired by Cantor, who was 14 years his
senior. He died around the age of 26 from typhus. The results
discussed appear in 3 papers or so, some or all in German in
Acta Mathematica.
Below, I've made a summary of the main results as I see it; of
course, a better place to look is the Maurey-Tacchi paper; however,
as I said, it's quite a historical-like "investigation" of who did what/
wrote what/to whom/when.
Cf.:
http://www.math.jussieu.fr/~maurey/articles/mata.pdf
Proposition IV (page 37)
-----------------------------------
Application: If f: [0, 1] -> R is continuous,
has f'(r) = 0 for all irrationals r in [0,1], with
f'(q) for rational q either existing as a limit (or
un-defined, i.e. limit {delta_x ->0} (delta_f/delta_x)
doesn't exist), then f is a constant function.
[ main fact is that zero derivative except on a countable
set implies constant function, assuming continuity ...]
----
Section 7. Le theoreme ensembliste de Scheeffer (page 42)
Consequence:
<< Il obtiendra ainsi a partir de la fonction singuliere
des exemples de fonctions continues non constantes,
dont la derivee est pourtant nulle
en tout point rationnel. >>
Or, construction of a continuous function, which isn't constant,
but has a derivative of zero at every rational point.
[ seems to be a question-mark-like function ...]
David Bernier
Dave L. Renfro
> Scheeffer was very much admired by Cantor, who was
> 14 years his senior. He died around the age of 26
> from typhus.
I didn't know Scheeffer died so young. I guess that's
why I'm not aware of many papers by him. I have two or
three of his papers from an 1880s (I think) volume of
Acta Mathematica, one of which contains a result I've
posted about before [1], and at least one of these
papers is moderately high on my list of "classical
papers in real analysis and point set theory" to
get translated (by someone else if it's in German,
as I can't read a bit of German except for being
able to recognize a handful of math terms relevant
to my interests) and then edited and LaTeX'ed by me.
Scheeffer also seems to be *essentially* responsible
for proving the sharpness of the result that monotone
functions have countably many discontinuities by showing
any countable set can be the discontinuity set of some
monotone (continuous, in fact) function. See Theorem 1'
and my "HISTORY" comment for Theorem 1' in [2]. [I'm not
sure, but my guess is that Scheeffer probably constructed
a strictly increasing continuous function whose discontinuity
set is the set of rational numbers [3], and the same method
would allow such a construction for any countable set that
is dense in the reals. To extend the result to any countable
set then requires piecing together various constructions
where the set is dense in an interval and where the set
is scattered in an interval, or something or other.] Finally,
the result you mentioned ("zero derivative except on a countable
set implies constant function, assuming continuity") is more
precisely described in [4].
[1] http://groups.google.com/group/sci.math/msg/ce542e3d90896bf1
[2] http://groups.google.com/group/sci.math/msg/69d1332208a81a33
[3] The idea of an arbitrary countable set (i.e. the notion of
countability as a property of a set, especially the only
property a set of reals is required to have) was probably
very incompletely known/used/aware-of at this time, since
only a few people (less than 20 or so, I'd bet) knew, in
the early 1880s, about Cantor's work (and probably only 5
or 6 even mentioned it in print before the mid 1880s).
[4] http://groups.google.com/group/sci.math/msg/443ba725a7d6722e
Dave L. Renfro
David Bernier
> David Bernier wrote (in part):
>
>> Scheeffer was very much admired by Cantor, who was
>> 14 years his senior. He died around the age of 26
>> from typhus.
>
> I didn't know Scheeffer died so young. I guess that's
> why I'm not aware of many papers by him. I have two or
> three of his papers from an 1880s (I think) volume of
> Acta Mathematica, one of which contains a result I've
> posted about before [1], and at least one of these
> papers is moderately high on my list of "classical
> papers in real analysis and point set theory" to
> get translated (by someone else if it's in German,
> as I can't read a bit of German except for being
> able to recognize a handful of math terms relevant
> to my interests) and then edited and LaTeX'ed by me.
I wouldn't hesitate to recommend the article by Maurey
and Tacchi as a reference on Scheefer's work and those
of the same period for singular functions [in an informal sense].
But the exposition has long sentences in French, and in my opinion
it's at a rather high level, interspersed with historical analysis.
There are proofs presented, but mostly I skimmed over them, as
what I wanted was clear statements of Scheefer's results.
One interesting fact is that Scheefer, in a paper, exclaimed that
a theorem of Harnack "was completely false" ...
My German is basic, but I don't feel like I can do a good job of
translating his Acta Mathematica papers from German to English.
So I thank you for bringing to light the Maurey/Tacchi paper.
I see. Some theorems to the effect that
f' = 0 except on exceptional set 'E' + f continuous ==> f constant
used the idea of "derived sets" (topology). Since I only skimmed
trough the proofs in Maurey/Tacchi, I don't know how Scheefer's arguments
went. I think you do a great service by posting about little-known papers.
Getting them translated into English seems quite non-trivial.
David Bernier
> [4] http://groups.google.com/group/sci.math/msg/443ba725a7d6722e
>
> Dave L. Renfro
David Bernier
> David Bernier wrote:
>
>> The following problem is too hard -- I think. But it's
>> interesting ...
>> From sci.math, Feb./March 2005:
>> thread: "f is constant?" started by Betelgeuse (also known
>> as Jenny ).
>> Ref.:
>> http://groups.google.com/group/sci.math/msg/25121c9fcec86eda
>>
>> Betelgeuse wrote in part:
>> << Let f:(a,b)->R be differentiable in every point.
>> Assume also that I know that f'(x)=0 for
>> all x in A, where A is an open dense subset of (a,b).
>> Can I conclude that f is a constant function? >>
>>
>> I think the question was not settled by any sci.math post
>> in the "f is constant?"
>> thread from around early March 2005 ...
>
> The answer is no. If E is a subset of the open interval that
> has an uncountable complement, then there exists a non-constant
> differentiable function f on the interval such that f'(x) = 0
> at each point of E [*]. Conversely, if f is differentiable
What if E is [0, 1] \ C, C being the Cantor set?
In the thread:
``Increasing function whose derivative is densely often equal to 0"
David Ullrich wrote in part in:
http://groups.google.com/group/sci.math/msg/cc23edf4a11502e3:
<< (Note that if the function is supposed to be differentiable
everywhere and have f' = 0 "almost everywhere" in the sense of
Lebesgue measure then there is no such example. But saying f' = 0
on a dense set is a much weaker condition.) >>
I would take a look at the article by Zofia Denkowska,
however it seems to be closed access and the Zbl reviewer
didn't mention other papers with more complicated
examples.
I'm now looking through Bruckner, Bruckner and Thomson's
_Real Analysis_ .
David Bernier
Dave L. Renfro
>> The answer is no. If E is a subset of the open interval that
>> has an uncountable complement, then there exists a non-constant
>> differentiable function f on the interval such that f'(x) = 0
>> at each point of E [*]. Conversely, if f is differentiable
David Bernier wrote (in part):
> What if E is [0, 1] \ C, C being the Cantor set?
Yes, this is included. The complement of E is the Cantor set,
which is uncountable.
> In the thread:
> ``Increasing function whose derivative is densely often equal to 0"
> David Ullrich wrote in part in:
> http://groups.google.com/group/sci.math/msg/cc23edf4a11502e3:
>
> << (Note that if the function is supposed to be differentiable
> everywhere and have f' = 0 "almost everywhere" in the sense of
> Lebesgue measure then there is no such example. But saying f' = 0
> on a dense set is a much weaker condition.) >>
David Ullrich is talking about functions that are also assumed
to be monotone, which imposes a lot more restrictions on things.
Dave L. Renfro
Gerry Myerson
<9de25753-b56b-42b7...@y21g2000hsf.googlegroups.com>,
"Dave L. Renfro" <renf...@cmich.edu> wrote:
> David Bernier wrote (in part):
> > David Ullrich wrote in part in:
I feel like I've stumbled into a David convention.
David Bernier
> David Bernier wrote:
>
>> The following problem is too hard -- I think. But it's
>> interesting ...
>> From sci.math, Feb./March 2005:
>> thread: "f is constant?" started by Betelgeuse (also known
>> as Jenny ).
>> Ref.:
>> http://groups.google.com/group/sci.math/msg/25121c9fcec86eda
>>
>> Betelgeuse wrote in part:
>> << Let f:(a,b)->R be differentiable in every point.
>> Assume also that I know that f'(x)=0 for
>> all x in A, where A is an open dense subset of (a,b).
>> Can I conclude that f is a constant function? >>
>>
>> I think the question was not settled by any sci.math post
>> in the "f is constant?"
>> thread from around early March 2005 ...
>
> The answer is no. If E is a subset of the open interval that
> has an uncountable complement, then there exists a non-constant
> differentiable function f on the interval such that f'(x) = 0
> at each point of E [*]. Conversely, if f is differentiable
What if E is [0, 1] \ C, C being the Cantor set?
In the thread:
``Increasing function whose derivative is densely often equal to 0"
David Ullrich wrote in part in:
http://groups.google.com/group/sci.math/msg/cc23edf4a11502e3:
<< (Note that if the function is supposed to be differentiable
everywhere and have f' = 0 "almost everywhere" in the sense of
Lebesgue measure then there is no such example. But saying f' = 0
on a dense set is a much weaker condition.) >>
I would take a look at the article by Zofia Denkowska,
however it seems to be closed access and the Zbl reviewer
didn't mention other papers with more complicated
examples.
I'm now looking through Bruckner, Bruckner and Thomson's
_Real Analysis_ .
David Bernier
David Bernier
> David Bernier wrote:
>
>> The following problem is too hard -- I think. But it's
>> interesting ...
>> From sci.math, Feb./March 2005:
>> thread: "f is constant?" started by Betelgeuse (also known
>> as Jenny ).
>> Ref.:
>> http://groups.google.com/group/sci.math/msg/25121c9fcec86eda
>>
>> Betelgeuse wrote in part:
>> << Let f:(a,b)->R be differentiable in every point.
>> Assume also that I know that f'(x)=0 for
>> all x in A, where A is an open dense subset of (a,b).
>> Can I conclude that f is a constant function? >>
>>
>> I think the question was not settled by any sci.math post
>> in the "f is constant?"
>> thread from around early March 2005 ...
>
> The answer is no. If E is a subset of the open interval that
> has an uncountable complement, then there exists a non-constant
> differentiable function f on the interval such that f'(x) = 0
> at each point of E [*]. Conversely, if f is differentiable
What if E is [0, 1] \ C, C being the Cantor set?
In the thread:
``Increasing function whose derivative is densely often equal to 0"
David Ullrich wrote in part in:
http://groups.google.com/group/sci.math/msg/cc23edf4a11502e3:
<< (Note that if the function is supposed to be differentiable
everywhere and have f' = 0 "almost everywhere" in the sense of
Lebesgue measure then there is no such example. But saying f' = 0
on a dense set is a much weaker condition.) >>
I would take a look at the article by Zofia Denkowska,
however it seems to be closed access and the Zbl reviewer
didn't mention other papers with more complicated
examples.
I'm now looking through Bruckner, Bruckner and Thomson's
_Real Analysis_ .
David Bernier
David Bernier
I haven't read all the integration theory and
"Real Analysis Exchange"-type papers that
you have.
I saw that you replied to someone (Jennifer)
about Lusin's Theorem, with many references here:
http://groups.google.com/group/sci.math/msg/f04e2f4466d338f1
The list of authors includes Jack B. Brown,
who has a Web page here:
http://www.auburn.edu/~brownj4/
Since he was an active Contributing Editor
to _Real Analysis Exchange_ for 10 years,
1993-2003, it seems good to know for difficult
real analysis questions.
There's also an annual symposium:
http://www.stolaf.edu/people/analysis/
D. Pompeiu's Math. Annalen paper
``Sur les fonctions dérivées" from around 1907
also seems interesting:
http://www.digizeitschriften.de/index.php?id=loader&tx_jkDigiTools_pi1[IDDOC]=361831
David Bernier
Dave L. Renfro
> Since he was an active Contributing Editor
> to _Real Analysis Exchange_ for 10 years,
> 1993-2003, it seems good to know for difficult
> real analysis questions.
>
> There's also an annual symposium:
> http://www.stolaf.edu/people/analysis/
For what it's worth, I know these people fairly well,
at least professionally. Also, my Ph.D. advisor was
one of the founders of this journal and I've given
several talks at these annual symposiums (although
not recently), as well as several talks at a yearly Spring
conference that Brown used to host at Auburn University
(pictures below).
If you're interested in classical papers on "real analysis
pathology" topics, and you can read French, the best items
to look at are the early volumes (1920 until about WW II)
of the journal _Fundamenta_Mathematica_. The complete texts
of these journal volumes are available at:
http://matwbn.icm.edu.pl/spis.php?wyd=1&jez=pl
Some pictures of me (mid to upper 30s then) --->
9th Real Analysis MiniConference (Auburn University, March 1993)
[I'm 2nd from left on the front row.]
http://topo.math.auburn.edu/pub/photos/AubMini/JB92.jpg
10th Real Analysis MiniConference (Auburn University, April 1995)
[I'm 3rd from the right on the front row. Brown and his wife
are to my left, right when looking at the picture.]
http://topo.math.auburn.edu/pub/photos/AubMini/JB94.jpg
11th Real Analysis MiniConference (Auburn University, March 1996)
[I think I'm 2nd from the right at the very back.]
http://topo.math.auburn.edu/pub/photos/AubMini/JB95.jpg
Dave L. Renfro
David Bernier
Thanks. I found this review from 1915 of de la Vallee Poussin's
``Cours d'Analyse":
http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183423393
It was written by M. B. Porter and appeared in the Bulletin
of the AMS. It seems that the reviewer (Porter) wrote
to de la Vallee Poussin because Schoenflies had said that
a theorem in de la Vallee Poussin's Cours, as stated,
was illogical or contradictory.
At the end of the review, an extract of
de la Vallee Poussin's reply is reproduced.
I think the "problem" has to do with
de la Vallee Poussin's usage of "sauf peut-etre",
which I can imagine might confuse a non-native
speaker of French.
The result(theorem) has to do with conditions on the upper right
Dini derivates of continuous functions f and f_1
on an interval ]a, b[ and conditions that imply that
f and f_1 differ by a constant. de la Vallee Poussin's
reply appears on pages 84 and 85.
I'd be interested to know how you would translate the
statement of the theorem(?) in English.
Similar results are discussed around page 29 of
the Maurey & Tacchi paper on Scheeffer's result.
Cf.:
"On est tres pres d'un des enonces de Lebesgue [...] "
on page 29.
Thanks,
David Bernier
Dave L. Renfro
> At the end of the review, an extract of
> de la Vallee Poussin's reply is reproduced.
> I think the "problem" has to do with
> de la Vallee Poussin's usage of "sauf peut-etre",
> which I can imagine might confuse a non-native
> speaker of French.
>
> The result(theorem) has to do with conditions on the
> upper right Dini derivates of continuous functions
> f and f_1 on an interval ]a, b[ and conditions that
> imply that f and f_1 differ by a constant. de la Vallee
> Poussin's reply appears on pages 84 and 85.
>
> I'd be interested to know how you would translate the
> statement of the theorem(?) in English.
My French is very spotty, and I often use google's
translator, a dictionary, and (marking those things
I can't figure out) someone I work with who has a
fair knowledge of French. I have a copy of the review
you mention (140% magnified photocopy from the original
Bull. AMS volume), as well as other reviews of Vallee
Poussin's book, so at some point (because I'm interested
in documenting/translating/writing-about the early
history of descriptive set theory, and Vallee Poussin's
book played an important role due to the later material
on the Borel set and Baire function hierarchies; indeed,
the idea of ambiguous Borel classes began here) I might
have some questions for you when I get around to writing
about Vallee Poussin's book (because I'm sure I'd want to
include the Bull. AMS misreading). I have made a hard copy
of your post and I'll put it with my relevant papers
(when I get to them ... much of my stuff is still in
a state of disorder due to my recent rapid move due to
flooding).
Oh, I see I'm talking about the wrong book. You're talking
about Vallee Poussin's 1915 book, not his 1916 book. The
1916 book is the one that has some historically interesting
material on descriptive set theory. In any event, I also
have reviews of his 1915 book, and I've almost certainly
highlighted the part about Scheeffer's theorem.
A couple of things occur to me looking at the .pdf file
of the Bull. AMS review of the 1915 book:
1. A comment is made that Schoenflies said the theorem is
illogical. While Schoenflies deserves great credit for
the first serious surveys of set theory (which then
meant also results about real analysis and topology),
in 1899 and then again in the middle 1910s decade,
a number of mistatements and misconceptions made by
him have been documented in the literature (quite a
few in the several hundred papers by William H. Young).
[Given how much Schoenflies wrote on the topic, and how
wide ranging his scope, this is almost to be expected.]
2. The corresponding result when infinite derivatives are
considered fails (i.e. "derivative exists" means at each
point the 2-sided derivative exists as a real number,
as +oo, or as -oo). I think H. Hahn was the first to show
this, around 1903 or 1904, by constructing two continuous
functions that are everywhere differentiable in the finite
or infinite sense and which take the same value at each point
not belonging to the Cantor middle-thirds set but different
values at each point belonging to the Cantor middle-thirds
set. Simpler constructions were published in the early 1920s
in the journal Fundamenta Mathematica (available on internet,
via web page I posted URL to earlier in this thread).
Dave L. Renfro
David Bernier
Yes, it sounds interesting.
> of your post and I'll put it with my relevant papers
> (when I get to them ... much of my stuff is still in
> a state of disorder due to my recent rapid move due to
> flooding).
>
> Oh, I see I'm talking about the wrong book. You're talking
> about Vallee Poussin's 1915 book, not his 1916 book. The
> 1916 book is the one that has some historically interesting
> material on descriptive set theory. In any event, I also
> have reviews of his 1915 book, and I've almost certainly
> highlighted the part about Scheeffer's theorem.
>
> A couple of things occur to me looking at the .pdf file
> of the Bull. AMS review of the 1915 book:
>
> 1. A comment is made that Schoenflies said the theorem is
> illogical. While Schoenflies deserves great credit for
> the first serious surveys of set theory (which then
> meant also results about real analysis and topology),
> in 1899 and then again in the middle 1910s decade,
> a number of mistatements and misconceptions made by
> him have been documented in the literature (quite a
> few in the several hundred papers by William H. Young).
> [Given how much Schoenflies wrote on the topic, and how
> wide ranging his scope, this is almost to be expected.]
I thought "sauf sur un ensemble de mesure nulle" would
be as good as "sauf peut-etre...", if used today. Was the empty
set a standard, ordinary set in the common terminology
or thinking of 1915?
>
> 2. The corresponding result when infinite derivatives are
> considered fails (i.e. "derivative exists" means at each
> point the 2-sided derivative exists as a real number,
> as +oo, or as -oo). I think H. Hahn was the first to show
> this, around 1903 or 1904, by constructing two continuous
> functions that are everywhere differentiable in the finite
> or infinite sense and which take the same value at each point
> not belonging to the Cantor middle-thirds set but different
> values at each point belonging to the Cantor middle-thirds
> set. Simpler constructions were published in the early 1920s
> in the journal Fundamenta Mathematica (available on internet,
> via web page I posted URL to earlier in this thread).
I got a digital copy of Vallee Poussin's book from a site in
Michigan. It is listed at the Digital Math. Library here:
http://www.mathematik.uni-bielefeld.de/~rehmann/DML/dml_links_author_A.html
David Bernier
Dave L. Renfro
> I thought "sauf sur un ensemble de mesure nulle" would
> be as good as "sauf peut-etre...", if used today.
> Was the empty set a standard, ordinary set in the
> common terminology or thinking of 1915?
I don't think the empty set was in common use in 1915,
but whether it was in use at all or not, I don't know.
In older literature one often sees the phrase "doesn't
exist" when a set is empty. A good reference for the
history of the empty set (indeed, the only reference
I'm aware of) is the following, which is available
on the internet. (I haven't looked at this paper in
over a year, but I have read it.)
Akihiro Kanamori, "The empty set, the singleton, and the
ordered pair", The Bulletin of Symbolic Logic 9 #3
(September 2003), 273-298.
http://tinyurl.com/5z6esb [.pdf files]
http://tinyurl.com/593x6r
http://tinyurl.com/6gwvef [.ps file]
Dave L. Renfro
Dave L. Renfro
> I got a digital copy of Vallee Poussin's book from
> a site in Michigan.
I intended to mention the following earlier and forgot.
Some very useful historical information, written in
French, can be found published the journal "Cahiers du
séminair d'histoire des mathématiques", of which volumes
1-12 (1980-1991) are on the internet at
http://www.numdam.org/numdam-bin/browse?j=CSHM
Dave L. Renfro
daniel tisdale
Generalization--inscribe curves other than the quarter circle. For which (if any) does the limit exist?
John De Vries Kelowna
Dave L. Renfro
> Some very useful historical information, written in
> French, can be found published the journal "Cahiers du
> séminair d'histoire des mathématiques", of which volumes
> 1-12 (1980-1991) are on the internet at
>
> http://www.numdam.org/numdam-bin/browse?j=CSHM
I suppose this thread is as good as any for two historical
items I recently came across that others might be interested in.
Johannes John Carel Kuiper, "Ideas and Explorations:
Brouwer's Road to Intuitionism", Ph.D. Dissertation
(under Dirk van Dalen), Department of Philosophy,
Universiteit Utrecht, 2004, xviii + 359 pages.
http://www.ozsl.uu.nl/articles/kuiper01.pdf [1.57 MB .pdf file]
The first part of the Acknowledgements (p. v) are interesting.
Chapter 1 gives some useful historical information about
Cantor's work, although I note that neither Joseph Dauben's
book nor Jose Ferreiros Dominguez's book appear in Kuiper's
bibliography at the end of the Dissertation (highly unusual
omissions, given what he discusses and given the other items
in his bibliography). I've hardly glanced at this Dissertation
so far, but one thing did catch my eye. In footnote 7 on p. 4,
Kuiper writes: "In fact there was also Young and Young's
_The_theory_of_sets_of_points_ ([Young and Chisholm Young
1906]), but probably Brouwer was not familiar with this
book; he never mentioned the name Young, neither in his
notebooks, although W.H. Young published in German in the
_Mathematische_Annalen_ in 1905." On p. 298 of the 1972
Chelsea edition, which includes additional notes written/assembled
by the Youngs' for a 2nd edition that never appeared (at least,
not by the Youngs), there is an extract of a 4 April 1913 letter
from Brouwer to Grace C. Young. (See my 2nd paragraph in [1] for
more details about what Brouwer's letter dealt with.) Also, although
I haven't looked to be sure (the papers in question are at
home, where I'm not; plus, it would take me quite a while
to locate them because of my recent rapid move due to my previous
apartment being flooded with 2+ feet of water), I'm pretty
sure that Brouwer mentions the Youngs' book several times in
his 1910s papers on perfect sets, the Cantor-Bendixson theorem,
and G_delta sets (English translations of several of those that
were originally written in Dutch can be found in Brouwer's
Collected Works).
[1] http://groups.google.com/group/sci.math/msg/6098777da158cb95
The other thing I wanted to call attention to is a paper
on infinity in mathematics that appeared just 3 years
before Cantor's first proof that the real numbes are
uncountable and 1-2 years after Cantor's use of, in some
trigonometric series uniqueness papers, countable sets
of real numbers with finite Cantor-Bendixson rank (these
are sets that, for some finite iteration of "limit points
of limit points of limit points of ...", you get the emtpy set).
This paper might be of interest to those who are interested
in how infinity was thought of when Cantor began his work.
This paper is on the internet (see first URLs below).
Abel Transon, "Sur l'emploi de l'infini en Mathématiques",
Comptes Rendus Hebdomadaires des Séances de l'Académie des
Sciences (Paris) 73 #6 (7 August 1871), 367-369.
http://math-doc.ujf-grenoble.fr/RBSM/cr-gallica.html [all C. R.
volumes]
http://gallica.bnf.fr/ark:/12148/bpt6k3030d [C. R. volume #73]
Do a "find word in page" browser search for 'Transon'.
Dave L. Renfro
Dave L. Renfro
> The other thing I wanted to call attention to is a paper
> on infinity in mathematics that appeared just 3 years
> before Cantor's first proof that the real numbes are
> uncountable and 1-2 years after Cantor's use of, in some
Ooops! In the last line, "after" should be "before".
The trig. series stuff I was talking about was done
around 1872-1873.
Dave L. Renfro
Keith Ramsay
|It was written by M. B. Porter and appeared in the Bulletin
|of the AMS. It seems that the reviewer (Porter) wrote
|to de la Vallee Poussin because Schoenflies had said that
|a theorem in de la Vallee Poussin's Cours, as stated,
|was illogical or contradictory.
|
|At the end of the review, an extract of
|de la Vallee Poussin's reply is reproduced.
|I think the "problem" has to do with
|de la Vallee Poussin's usage of "sauf peut-etre",
|which I can imagine might confuse a non-native
|speaker of French.
It strikes me as fairly lame for Schoenflies to
make this sort of mistake. "Peut-etre" is "possibly";
so the term is "except possibly on a set...".
de la Vallee Poussin says that this phrase has a
subjective sense, and the reviewer says he thinks it
can be taken in an objective sense. Schoenflies
objection is that the derivatives would be equal
everywhere, and not everywhere except on a set E1,
if they differ by a constant.
[...]
|I'd be interested to know how you would translate the
|statement of the theorem(?) in English.
My French is not outstanding, but it seems to be this:
If, in an interval (a,b), two functions f_1(x) and
f_2(x) have their right superior derivatives (?):
1) finite in each point except perhaps in a set E_1,
and 2) equal except perhaps in a set of measure zero,
the two functions only differ by a constant unless
E_1 contains a perfect set.
Keith Ramsay
David Bernier
I wonder why there are those "perhaps", "peut-etre", twice.
The Dini derivates, AFAIK, consider
(a) right limits or left limits (but not both) and
(b) limsups or liminfs (but not both)
for a total of four combinations. Extended real values are
ok by some. In de la Vallee Poussin's reply, I think he says
that two +oo right superior Dini derivates aren't necessarily
the same. Also, the perfect set may not be empty,
which was probably the case in the terminology of the day.
From what you wrote, I tend to think that (as a trivial example)
with f_1(x) = 0 and f_2(x) = 1, conditions (1) and (2)
when using "for all" quantifiers produce what we call
vacuously true sentences, that that was Poussin's way
of saying that kind of thing, but Schoenflies decoded
it in a way where "except" implies there is an x1 where
(1) is not true and an x2 where (2) is not true ...
It still seems an overstatement to call the theorem
"illogical" if, as may be the case, Schoenflies
didn't ask Poussin for clarification before calling
it illogical.
David Bernier