HDTenEightyP/NetNews huggingface dataset:
From: torkel@sics.se (Torkel Franzen)
Subject: Re: Poincare on Set Theory (was Re: Philosophies...)
Organization: Swedish Institute of Computer Science, Kista
	<1992Sep10.205022.15408@CSD-NewsHost.Stanford.EDU>
	<TORKEL.92Sep11100055@isis.sics.se>
	<1992Sep15.002413.23869@unixg.ubc.ca>
Date: Tue, 15 Sep 1992 06:03:52 GMT

In article <1992Sep15.002413.23869@unixg.ubc.ca> ramsay@unixg.ubc.ca
 (Keith Ramsay) writes:

   >It appears that this remark was misquoted by someone and then
   >repeated. Wasn't Poincare responsible for much of the development of
   >point-set topology anyway? This is from memory, so if anyone has the
   >exact reference, please let me/us know.

  I don't believe Poincare worked on point set topology. As for the
"disease" comment, I read about this in "Basic concepts of algebraic
topology" by Croom:

   "Indeed, in an address to the International Mathematical Congress
    of 1908, Poincare referred to point-set theory as a 'disease'
    from which future generations would recover."

  I have no first-hand knowledge of this address. Presumably somebody here
knows bout it?
From: gjm11@cus.cam.ac.uk (G.J. McCaughan)
Newsgroups: sci.math
Subject: Re: nth prime
Date: 10 Sep 92 16:41:33 GMT
Organization: U of Cambridge, England

In article <1992Sep10.190905.10741@waikato.ac.nz> bill@waikato.ac.nz writes:

>Sorry to ask such a FAQ, but what's the quickest algorithm to
>find the nth prime? Is it O(nlogn) or something else?

I think there is no way to get p_n exactly that's much better than
the good old sieve of Eratosthenes. So in sieving out multiples of 2
you do n/2 things, and in [... er, we're suddenly working out the
biggest prime less than n. This is of equal complexity...] total
we do n(1/2+1/3+...+1/p) things, where p is the biggest prime less
than root(n). Now it's well known that the series there comes to
about log log root(n). So the time for "primes up to n" is about
n log log n, and so the time for "nth prime" is about
(n log n) log log (n log n), which is about n log n log log n.
As I say, I don't think you can do much better than this.
-- 
Gareth McCaughan     Dept. of Pure Mathematics & Mathematical Statistics,
gjm11@cus.cam.ac.uk  Cambridge University, England.    [Research student]
From: pvl2@cunixb.cc.columbia.edu (Priscilla V Loanzon)
Newsgroups: sci.math
Subject: Couple of questions
Date: 9 Sep 92 10:24:57 GMT
Organization: Columbia University

Could someone please explain to me a few basic things:

What is the logic used to answer questions of the below type?

1) If the finite group G contains a subgroup of order 7 but no element
(other than the identity) is its own inverse then the order of group G
could be (a)27   (b)28   (c)35   (d)37   (e)42.

It says that the correct anwer is (c). I know why we can eliminate (a)
and (d) but don't know how to proceed further.



Also could someone explain what is elementary Riemann and Lebesgue
integration, and why fuctions can be integrable but now Riemann
itegrable etc? I looked over a couple of books but did not quite get
it.


Thanks
From: bubai@matt.ksu.ksu.edu (P.Chatterjee)
Newsgroups: sci.math
Subject: A Question
Date: 14 Sep 1992 01:20:29 -0500
Organization: Kansas State University
NNTP-Posting-Host: matt.ksu.ksu.edu

I've been stumped by this problem and can't seem to figure out a way to tackle  it.

It goes like this:

Characterize the functions f: X --> Y for which P(X) --> P(Y), F(A)=f(A), is
  
a)injective,

b)surjective, or

c)bijective.

Would appreciate if somebody threw some light on it. Thanks.
Newsgroups: sci.math
From: berling@infko.uni-koblenz.de
Subject: decomposing sets of equations
Organization: University of Koblenz, Germany
Date: Wed, 09 Sep 1992 15:06:15 GMT


Hello!

I have some questions concerning methods for solving large sets 
of mostly nonlinear equations (it's a kind of constraint solving problem).
 
I am seeking for methods to decompose a possible large set of
equations into sets of smaller ones which have to be solved
simultaneously in a certain order. Using a bipartite graph with 
vertices representing the equations and variables 
and undirected arcs representing
the "belongs-to" relationship, one may use a maximum-matching algorithm
to determine a partition into smaller sets. My question is, are there
any other techniques, especially techniques which allow a better 
control of the quality of the generated partition (e.g. number of sets of
more than one equation is minimal)?

Having a set of equations which have to be solved simultaneously, I am
further interested in methods for eliminating as much variables as
possible before solving it using Newton-Raphson iteration. Having for
example a set of 4 equations determining the variables a, b, c, d:

	1. a = f (c, d)
	2. b = g (a, c)
	3. c = h (d)
	4. d = i (b)

the system is reducible to the equation:

	b = g (f (h (i (b)), i (b)), h (i (b)))
or
	d = i (g (f (h (d), d), h (d)))

which may be easier to solve using Newton-Raphson. Knowing wich equation 
has to be solved for which variable, the problem has probably something
to do with finding the feedback vertices in strong components of
directed graphs, a problem known to be NP-complete. Again, my question is
are there any other techniques or heuristics which solve this problem
or special cases perhaps more efficient?

I will thank you for any hints!

+----------------------------+-----------------------------------------+
|                            |                                         |
|  Roland Berling            |  Tel.:   +49 261 9119 436               |
|  Universit"at Koblenz      |  Fax.:   +49 261 9119 499               |
|  Institut f"ur Informatik  |  e-mail: berling@infko.uni-koblenz.de   |
|  Rheinau 3-4               |                                         |
|  D-5400 Koblenz            |                                         |
|  Germany                   |                                         |
|                            |                                         |
+----------------------------+-----------------------------------------+
Newsgroups: sci.math
Subject: Re: Need Help on Matrix Differentiation
From: news@massey.ac.nz (USENET News System)
Date: Tue, 8 Sep 92 20:29:57 GMT
Organization: Massey University

In article <TZq-0-=@engin.umich.edu>, takriti@engin.umich.edu (samer Takriti) writes:
> 
> In article <18h8pnINNdp2@matt.ksu.ksu.edu> bubai@matt.ksu.ksu.edu (P.Chatterjee) writes:
> >Hi,
> >
> >I was just wondering if somebody could help out regarding matrix differentiationThe problem is to minimize:   
> >
> >y= x'Ax + 2x1 + 3x2 - 10
> >
> >where A is a 2x2 matrix :   25  7
> >                             7  13
> >
> >
> >and x is the (x1 x2) column vector and x' denotes the transpose.
> >
> >It's easy if one expands the x'Ax term but I was wondering if there was some way doing it using matrix differentiation.
> >
> >Thanks for any help in this regard.
> >
> 
> Write the problem as:
> y = x'.A.x + c'.x
> (The constant is not important), 
> c = (2, 3)'.
> You need the derivative to be zero, i.e., 
> 2.A.x + c = 0
> x = -0.5 A^(-1) c
> x is the solution for your problem. Make sure that A is
> positive definite, this will guarantee that x is a minimum
> otherwise it may be a maximum.
> -Samer
> 
Sorry, ignore my last posting. A _was_ symmetric.

Unfortunately, my newsreader doesn't have the ability to cancel.

Terry Moore
Newsgroups: sci.math,comp.lang.misc,comp.lang.fortran,comp.std.c,comp.lang.c
From: hybl@umbc4.umbc.edu (Dr. Albert Hybl (UMAB-BIOPHYS))
Subject: Re: Scientists as Programmers (was Re: Small Language Wanted)
Organization: Univ. of MD, Baltimore County
Date: Fri, 11 Sep 1992 17:46:18 GMT

>
>ANSI conforming IMPLEMENTATIONS are allowed to do it because, when the
>standard doesn't define the result, the implementations can do whatever
>they want.
>--  Norman Diamond       diamond@jit081.enet.dec.com

Well the suicide gene has been spliced into ANSI C, has it.  This suggests
that the lawyers and business administration types have legally colluded
(because creating a standard is an allowed violation to the anti-trust laws)
to produce an inadequate language standard thus allowing each vendor to write
compilers that contain proprietary extensions.

Panic,
Albert Hybl (hybl@umbc3.umbc.edu)

Newsgroups: sci.math
From: msb@sq.sq.com (Mark Brader)
Subject: Re: measures of the `size' of infinite sets
Organization: SoftQuad Inc., Toronto, Canada
Date: Tue, 15 Sep 92 04:04:24 GMT

> Here is an exercise (alleged to work, I haven't done it...):
> Take the front page of the New York Times.  Write down in decimal form
> all of the numbers mentioned.  Then about 30% of the numbers will
> begin with the digit '1'. ...
> Or, take an atlas, and look up the lengths of the rivers (in km).
> About 30% of them begin with the digit '1'.  Repeat, using miles.
> Still 30%.

Just a side remark here.  It occurs to me that a larger fraction of
students today will find this "30% effect" surprising or counter-
intuitive than was the case when I was in school.  You see, in those
days, we used *slide rules*...
-- 
Mark Brader             "True excitement lies in doing 'sdb /unix /dev/kmem'"
utzoo!sq!msb, msb@sq.com                                -- Pontus Hedman

This article is in the public domain.
From: bubai@matt.ksu.ksu.edu (P.Chatterjee)
Newsgroups: sci.math
Subject: Set-Theory Doubts: Got some conceptual glitches
Date: 12 Sep 1992 01:28:45 -0500
Organization: Kansas State University
NNTP-Posting-Host: matt.ksu.ksu.edu

Hello there,

I was just wondering if somebody on the net could help clarify a few conceptual glitches I have regarding very elementary concepts of set-theory.

a) According to my textbook, the ORDERED PAIR of two objects x and y is the set <x,y> = {{x}, {x,y}}. It also goes on to state that by this definition, the ordered pair is determined by x and y; and the order, x first and y second, is important unless x=y.

What throws me off is the definition; what is the motivation behind it?                      
b) Let f: X --> Y be an injective mapping. By definition, this means:
  for all x, x' in X: f(x) = f(x') ==> x=x'.

Equivalently, x not equal to x' ==> f(x) not equal to f(x').

My question here is: for the firstmentioned implication why is the CONVERSE not true? (I know it's not but am having trouble finding an intuitive/logical answerfor it).

Would really appreciate if somebody helped clarify these concepts.

Thanks.
Newsgroups: sci.math
From: frank@Cookie.secapl.com (Frank Adams)
Subject: Re: computable reals
Date: Tue, 15 Sep 1992 23:16:44 GMT
Organization: Security APL, Inc.

In article <9209120452.AA03328@ucbvax.Berkeley.EDU> lucas@WATSON.IBM.COM ("Bruce Lucas") writes:
>So, suppose we decided to limit the discussion to the set of "computable"
>reals.  This set could be defined in a number of ways which I suspect
>are all equivalent: for example a real is computable if there is a
>computable function f(n) that produces its nth digit; or, a real is
>computable if there are computable functions f(n) and g(n) that produce
>an approximation f(n)/g(n) to within 1/n.

Actually, these are *not* equivalent.  If you have a number which is very
close to (say) 1/2, you could have f(n) and g(n) which gave you 1/2 for very
large n, but not know whether the first digit is 4 or 5.  You cannot in
general know that for some sufficiently large n you won't suddenly get (say)
either n+1/2n or n-1/2n.

The second definition is the one you want.  Among other problems, the first
depends on the base you choose to represent your numbers.  As others have
noted, these numbers have been studied a fair amount, especially by
intuitionists.
From: lusardi@sybil.cs.Buffalo.EDU (Christopher Lusardi)
Newsgroups: sci.math
Subject: Partial Derivative of LN
Date: 11 Sep 92 17:47:57 GMT
Organization: State University of New York at Buffalo/Comp Sci
Originator: lusardi@sybil.cs.Buffalo.EDU

I would like to review partial derivatives of natural logs. Would some 
people electronically mail me some detailed explanations of examples?

I.e:  D/D@ Ln (function) etc. etc. etc.

					Thank you,

-- 
Christopher M. Lusardi     University At Buffalo  lusardi@sybil.cs.buffalo.edu
From: anders@mentor.cc.purdue.edu (Anders Lund)
Newsgroups: sci.math
Subject: Derivative of x^n with respect to n
Date: 9 Sep 92 19:23:32 GMT
Organization: Purdue University Computing Center


It has been a while since I toyed with calculus, I know there should be 
a straightforward answer, but ......

The problem, if you did not read the Subject line, is 
the deriviative of x^n with respect to n

Any and all answers, flames, threats of war, ... should be emailed to
me.

Thanks!
Anders
Newsgroups: sci.math
From: chalcraft@uk.tele.nokia.fi (Adam Chalcraft)
Subject: Re: what are quadturnians?
Organization: cpd
	<1992Sep8.234757.26306@maths.tcd.ie>
Date: Wed, 9 Sep 1992 07:26:53 GMT

In article <1992Sep8.234757.26306@maths.tcd.ie> tim@maths.tcd.ie (Timothy Murphy) writes:

>   The quaternions are a non-commutative extension of the complex numbers.
>   A quaternion is expressible in the form
>
>	   q = t + xi + yj + zk
>
>   where t, x, y, z are real numbers.
>   Multiplication is defined by the rules:
>
>	   i*i = j*j = k*k = -1,
>	   i*j = k = - j*i,
>	   j*k = i = - k*j,
>	   k*i = j = - i*k.
>
>   Multiplication is non-commutative but associative.

Correct, of course, but the guy wanted to know how to use them for 3-D
co-ordinate transformations.
I love, incidentally, the mis-heard "quadturnians", which is a perfectly
logical reconstruction of a plausible spelling given the intended use.

The extra information is that you represent (x,y,z) in the obvious way by
xi+yj+zk, and then translations correspond to addition. This much is
obvious, and gains you nothing so far. The clever bit is that if

  (x1i+y1j+z1k)*(x2i+y2j+z2k)=-t3+x3i+y3j+z3k

then t3 is (x1,y1,z1).(x2,y2,z2) [dot product] and (x3,y3,z3) is
(x1,y1,z1)^(x2,y2,z2) [cross product] (check sign before use :-).

This still gains you nothing if it is just used as a formula, because you
have just re-written an unilluminating formula in a different notation.
The only place you really gain is when you start using the fact that the
algebra is distributive and associative, so you can combine lots of
transformations into one fairly easily.
--
  ____________________________________________________________________________
 /  _Name: Adam|Names are linguistic constructs expressed in some language._  \
|\_|/|Chalcraft|They correspond to objects in some universe of discourse. |\|_/|
|`   |         |The corresponence between names (in the language) and     |`   |
|`   |Opinions:|objects (in the universe of discourse) is the  relation of|`   |
|`   |     Mine|identifying. A name identifies the object to which it is  |`   |
|`   |_________|bound.__________________________ISO_7498-3_:_1989_(E)_5.1_|`   |
 \__/                                                                      \__/

Newsgroups: sci.math
From: torkel@sics.se (Torkel Franzen)
Subject: Re: ZFC etc. (was Re: Report on Philosophies of Physicists)
Organization: Swedish Institute of Computer Science, Kista
	<1992Sep15.165635.12446@galois.mit.edu>
Date: Tue, 15 Sep 1992 19:07:07 GMT

In article <1992Sep15.165635.12446@galois.mit.edu> jbaez@riesz.mit.edu 
(John C. Baez) writes:

   >The propositional calculus might be inconsistent and we have somehow failed
   >to notice this so far; 

  This is an observation of the same kind as "perhaps the fundamental
theorem of arithmetic is not really true", or "perhaps Euclid's
algorithm sometimes gives the wrong answer", and so on. Not very
exciting stuff.
Newsgroups: sci.math
Subject: Re: nth prime
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 10 Sep 92 23:08:38 GMT

In <1992Sep10.164133.27763@infodev.cam.ac.uk> gjm11@cus.cam.ac.uk (G.J. McCaughan) writes:

>In article <1992Sep10.190905.10741@waikato.ac.nz> bill@waikato.ac.nz writes:

>>Sorry to ask such a FAQ, but what's the quickest algorithm to
>>find the nth prime? Is it O(nlogn) or something else?

>I think there is no way to get p_n exactly that's much better than
>the good old sieve of Eratosthenes. So in sieving out multiples of 2
...

This isn't quite correct.  There is an integral for (the number of primes
<= n) which takes o(n) to evaluate.  (I don't know the exact order.)  I'm
pretty sure that the time to find the first n primes is about  O(n log n),
but you don't have to all n of them in order to find the n_th.
--
Arthur L. Rubin: a_rubin@dsg4.dse.beckman.com (work) Beckman Instruments/Brea
216-5888@mcimail.com 70707.453@compuserve.com arthur@pnet01.cts.com (personal)
My opinions are my own, and do not represent those of my employer.
My interaction with our news system is unstable; please mail anything important.
Xref: sparky sci.math:11110 comp.graphics:9513
From: czako@yoko.rutgers.edu (Gabor M. Czako)
Newsgroups: sci.math,comp.graphics
Subject: Help: inscribed circle of three circles
Date: 8 Sep 92 13:46:27 GMT
Followup-To: sci.math
Organization: Rutgers Univ., New Brunswick, N.J.

In a computer program I am writing I need to find the inscribed
circle(s) of three circles.  In detail, I have three circles with
centers (x_1, y_1), (x_2, y_2), and (x_3, y_3) and radii r_1, r_2,
r_3.  I need to find the inscribed circle(s) with center (x, y) and
radius r which satisfies the following equations:

	(x - x_1)^2 + (y - y_1)^2 = (r + r_1)^2
	(x - x_2)^2 + (y - y_2)^2 = (r + r_2)^2
	(x - x_3)^2 + (y - y_3)^2 = (r + r_3)^2

I have not been able to analytically solve for the 0, 1, or 2
solutions of x, y, and r.  

Any help would be greatly appreciated.

Gabe
Newsgroups: sci.math
From: rscott@csws17.ic.sunysb.edu (Robert Scott)
Subject: Re: Report on Philosophies of Physicists
Organization: State University of New York at Stony Brook
Date: Tue, 15 Sep 1992 19:28:47 GMT

In article <1992Sep15.050118.15796@CSD-NewsHost.Stanford.EDU>
pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:

>For ordinary consumers of topological spaces it's healthiest to go
>right to compact Hausdorff spaces without messing around with
>non-Hausdorff or non-compact counterexamples---not only is CGHaus
>closed but very remarkably it is algebraic, i.e. essentially a variety
>or equational class, just like groups or lattices, only with operations
>at arities of every cardinality.


AREN'T YOU ENGAGING IN SOME CONFUSION HERE BETWEEN PROPERTIES OF THE
CATEGORY OF COMPACT HAUSDORFF SPACES, AND PROPERTIES OF THE CATEGORY OF
COMPACTLY _GENERATED_ HAUSDORFF SPACES?

IT'S THE CATEGORY OF COMPACT HAUSDORFF SPACES THAT'S ALGEBRAIC; THE
FREE COMPACT HAUSDORFF SPACE ON A SET IS JUST THE SPACE OF
ULTRAFILTERS ON IT, ALSO CALLED THE "STONE-CZECH (SP.?)
COMPACTIFICATION" OF THE DISCRETE TOPOLOGY).

THE CATEGORY OF COMPACTLY _GENERATED_ HAUSDORFF SPACES IS CARTESIAN
CLOSED, BUT I'M PRETTY SURE IT COULDN'T POSSIBLY BE ALGEBRAIC.  I THINK
THAT THE CLASS OF CATEGORIES WHICH ARE SIMULTANEOUSLY CARTESIAN CLOSED
AND ALGEBRAIC MUST BE VERY NARROW, THOUGH OFFHAND I CAN'T REMEMBER JUST
HOW NARROW.

ANYWAY, IT SEEMS SOMEWHAT PLAUSIBLE THAT YOU MIGHT BE ABLE TO TEACH
PEOPLE ABOUT COMPACT HAUSDORFF SPACES WITHOUT FIRST TEACHING THEM ABOUT
TOPOLOGICAL SPACES, BUT IT SEEMS MUCH HARDER TO TEACH THEM ABOUT
COMPACTLY GENERATED HAUSDORFF SPACES WITHOUT FIRST TEACHING THEM
ABOUT TOPOLOGICAL SPACES.




-JAMES DOLAN
Xref: sparky sci.math:11232 sci.physics:14430
From: kriman@acsu.buffalo.edu (Alfred M. Kriman)
Newsgroups: sci.math,sci.physics
Subject: Re: What about 3.4 dimension?
Date: 10 Sep 92 21:37:01 GMT
Organization: UB

In article <Sep10.194827.29156@yuma.ACNS.ColoState.EDU>
be231642@longs.LANCE.ColoState.EDU (Bret Egan) asks:
>Is there any literature about the possibilites of dimensions of real numbers?
>
>R^1.2 or R^pi

Yes, literally tons of it.  It arises in at least two ways in physics:
(1) In Renormalization Group (RG) calculations, integrals and sums occur in
which the dimension d is a parameter.  It is common to expand these in Taylor
series in d about the "upper or lower critical dimension." (Which are
integers in every case I know.)  This has even been tried in cases where
d-dependent expressions are not available, with extrapolation schemes that
assume smoothness and use a small number of points in d (small integers).
(2) Hausdorf and other dimensions are defined.  See any book on "fractals"
or by Benoit Mandelbrot.  Although most examples of fractals (noninteger-
dimension geometric objects) are constructed within an embedding space of
integer dimension, this is not required but only convenient.  The general
idea is that the volume V(r) within a ball of radius r varies as r^d in
spaces for which a dimension d can be defined.  Space-filling curves have
Hausdorf dimension equal to the ordinary dimension of the space they fill.
If you imagine space filling curves as being crinkly, then you can imagine
that some curves are of intermediate crinkliness.  These are fractals, and
among these the ones that have a certain degree of homogeneity have a
Hausdorf dimension defined.
   The definition is roughly d = limit as r -> 0 or log(V(r))/log(r).  This
means that the crinkliness must occur at arbitrarily small scales.
   Cases (1) and (2) above are related.  In physical systems near "critical"
transitions described by RG, fractal structures occur.  For example, in 
typical percolation problems, there is a sharp transition as some parameter
(pore size, temperature, whatever the model has) is varied through the
percolation threshold.  Precisely at the percolation threshold, percolation
(diffusion of a mobile species through a random porous medium, roughly)
takes place via a connected region of fractal dimension.

Question:
   Do the fact that fractal structures occur in critical systems, and the
fact that continuous-dimension ideas are useful in RG, have some deep
connection?
Newsgroups: sci.math
Subject: Re: Functional Equation
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 15 Sep 92 21:16:09 GMT
Distribution: sci.math

In <424@nttcvg.ntt.JP> herve@nttcvg.NTT.JP (Herve Delingette) writes:


>I would like to find all the functions f C1 continuous such that :

>f'(x)= [f(x+ x0)-f(x-x0)]/2x0 where f'(x) is the derivative of f
>and where x0 is a real number ..

>I am pretty sure that the only solutions are the linear functions 
>f(x)=ax+b but I am not sure.

>Please answer at herve@nttcvg.ntt.jp

Still looks like homework, but the answer is:

If x0 can vary, then differentiating the equation

2x0 f'(x) = f(x+x0) + f(x-x0) by x0, we get

2 f'(x) = f'(x+x0) + f'(x-x0),

so f' must be linear, and f must be quadradic.  Evaluating, we determine
that f must be linear.

If x0 is fixed, then let c=a+ib be a solution of sinh(c) = c.  (They DO
exist.)  Then 

f(x) = (E^(a x/x0)) cos(b x/x0)

is a solution.
--
Arthur L. Rubin: a_rubin@dsg4.dse.beckman.com (work) Beckman Instruments/Brea
216-5888@mcimail.com 70707.453@compuserve.com arthur@pnet01.cts.com (personal)
My opinions are my own, and do not represent those of my employer.
My interaction with our news system is unstable; please mail anything important.
Newsgroups: sci.math