HDTenEightyP/NetNews huggingface dataset:
From: M.Titahmboh@lut.ac.uk (Mana TitahMboh)
Subject: Re: Difference between "show" and "prove"
Organization: Loughborough University
Date: Wed, 20 Jan 93 21:33:55 GMT

In article <Bz7KyK.297@ulowell.ulowell.edu>, cdeloge@cs.ulowell.edu (Carolyn
Deloge) writes:
|> Hi, I am a CS graduate student and I have a question for the math people.
|> 
|> What is the difference between "show" and "prove"?
|> 
|> I had a homework problem that asked to "show" the solution.  I answered by
|> giving an example and some reasoning.  The teaching assistant marked it 
|> incorrect, saying that I needed to "prove" it inductively.
|> 
|> Can someone give me his/her definitions on these two words?  I would really
|> appreciate it.  Thanks.
|> 
|> CD

Sounds like "show that 'pigs can fly' for all n such that n is in the set of
positive integers".  Though this does not say "prove", it is not sufficient to 
"show" it for a few integers. The reason for this is that "pigs could fly " for
a lot positive integers, instances of which you would use by your reasoning, but
fail for a few "integers" as well. In this case you certainly could not say that
the original ascertion was true.
The important bit here is the "for all integers". 

Hope this helps. 

|> 
|> P.S. Yes I did look in the dictionary, but I want to hear it from the math
|> experts.

Mana
Newsgroups: sci.math,news.answers
From: pmontgom@math.orst.edu (Peter Montgomery)
Newsgroups: sci.math
Subject: Re: Matrix Problem
Date: 28 Jan 1993 19:15:34 GMT
Organization: Oregon State University Math Department
NNTP-Posting-Host: ilanga.math.orst.edu

In article <C1Knzo.54s@ecf.toronto.edu> rairan@ecf.toronto.edu (RAI Ranjan) 
writes:
>I remember a result from first year linear algebra that said that the
>eigenvectors of AB are the same as the eigenvectors of BA where A and B
>are nxn real matrices.  I don't know how to prove this.  Any pointers or
>references to the proof?

        AB and BA have the same eigenvalues, but not the same eigenvectors.

        Consider two matrices A and B with symbolic entries.
The characteristic polynomials det(lambda*I - AB) and
det(lambda*I - BA) are polynomials in these entries and lambda.
We claim that these are the same polynomial.

        Since A has symbolic entries, it is invertible in
the field generated by its coefficients.  Then

det(lambda*I - AB) = det(lambda*A*A^(-1) - AB)
                   = det(A) * det(lambda*A^(-1) - B)
                   = det(lambda*A^(-1) - B) * det(A)
                   = det(lambda*A^(-1)*A - BA)
                   = det(lambda*I - BA)

in the field of rational functions generated by the symbolic coefficients
of A and B.  Since both det(lambda*I - AB) and det(lambda*I - BA) are
polynomials, they must also agree as polynomials.

        If two polynomials are the same, then they generate the same
polynomial function.  Upon substituting numerical values into the coefficients
of A and B, the two characteristic polynomials will agree.
Hence they have the same roots, which are the eigenvalues of AB and BA.

        The reader may feel uncomfortable if his numerical A is
singular (i.e. non-invertible).  As an analogy, consider the derivation

                    x^6 - 1   (x^3 - 1)(x^3 + 1)
    x^4 + x^2 + 1 = ------- = ------------------ = (x^2 + x + 1)(x^2 - x + 1)
                    x^2 - 1     (x - 1)(x + 1)

of the polynomial identity x^4 + x^2 + 1 = (x^2 + x + 1)(x^2 - x + 1).
The left and right sides agree as rational functions (by the derivation)
and hence must be the same polynomial.  Therefore the two agree
at all numerical values of x, including x = +- 1, even though we
divided by x - 1 and x + 1 during the proof.

        EXERCISE:  Find two 2 x 2 matrices A, B such that AB and BA 
are not similar (i.e., there is no 2 x 2 invertible matrix C such that
(AB)C = C(BA).
-- 
        Peter L. Montgomery              Internet: pmontgom@math.orst.edu
        Dept. of Mathematics, Oregon State Univ, Corvallis, OR 97331-4605 USA
My visiting professorship ends in June.  Interested in program optimization,
compilers, computer arithmetic, number theory.  17 yrs industrial experience.
Newsgroups: sci.math
From: krech@jordan.geom.umn.edu (Daniel Krech)
Subject: sci.math.we_need_subgroups!
Organization: University of Minnesota
Date: Tue, 26 Jan 1993 06:25:56 GMT


Question.  What would be wrong with dividing sci.math up into the  
following groups: algebra, geometry, number theory, calculus, topology,  
history, puzzle, marketplace (for those debatably inappropriate for-sale  
ads which you could then choose to read and/or flame).

Why not even sub-sub groups for sci.math.[subject].homework. For those who  
like to read other peoples homework it will be all in one place for them,  
and for those that do not - well, they do not have to read  
sci.math.algebra.homework now do they! Do you really think that people are  
going to post a homework problem to sci.math for a algebra class when  
there is a subgroup sci.math.algebra? I do not see many people posting  
questions about cartoons, geography, football, abortion, etc. I think this  
is because people (even people who currently waste there time reading the  
collage of posts in sci.math just to whine about having to read cross post  
if we split it up) are smart enough to realize the difference between  
alt.cartoons.roadrunny and sci.math.

krech@geom.umn.edu

PS: For those of you who have plenty of time to read all of sci.math and  
think the rest of us should not get a chance to read the parts we would  
like, just so you will not have to NOT READ one maybe two cross post a day  
-- I would like to thank you all.
From: giesl@kirmes.intellektik.informatik.th-darmstadt.de (Juergen Giesl)
Newsgroups: sci.math
Subject: Algorithm for Polynomial Inequalities?
Date: 27 Jan 1993 15:55:02 GMT
Organization: Technical University Darmstadt
Distribution: world
NNTP-Posting-Host: kirmes.inferenzsysteme.informatik.th-darmstadt.de
Keywords: Polynomial Inequalities


A POLYNOMIAL INEQUATION over the natural numbers
is a formula of the following type:

FOR ALL natural numbers n1,..,nk: p(n1,...,nk) > 0,

where p is a multivariate polynomial of any degree 
with variable coefficients p0,...,pj. 


A mapping from {po,...,pj} to the natural numbers is a 
SOLUTION to the above inequation, if substituting each pi 
by the number it is mapped to makes the inequation true. 


A polynomial equation and its solutions are defined analogously. 


A solution to a set of polynomial inequations and equations 
is a mapping from all occurring variable coefficients to the
natural numbers, which solves each inequation and each equation
in the set.

  _______________________________________________________
 | I am searching for an algorithm working as follows:   |
 | Given a set of polynomial inequations and equations   | 
 | over natural numbers,                                 |
 | it returns "solvable" or "don't know".                |
 | If it returns "solvable" then there exists a solution |
 | to the set of inequations and equations.              |
  -------------------------------------------------------
 

Due to Lankford the problem of deciding inequalities between
n polynomials over the natural numbers is undecidable. (Although
it is decidable over the real numbers).


So if anybody knows where to find an algorithm which is able to find
a solution for a reasonably big class of these problems, please
tell me.


Thank you,

Juergen Giesl.




Newsgroups: sci.math
From: ags@seaman.cc.purdue.edu (Dave Seaman)
Subject: Re: Why Algebra...Important
Organization: Purdue University
Date: Mon, 25 Jan 1993 21:39:12 GMT

In article <1993Jan25.164019.19356@Princeton.EDU> tao@mace.Princeton.EDU  
(Terry Tao) writes:
> Claim: \exists Cup = {(x, y, z) = d((x, y), Cup_0) = r_cup, z_0 < z <  
z_0 +
> h_cup} \cup {(x, y, z_0): d((x, y), Cup_0) < r_cup}:d(Cup, me) <  
\epsilon_0 \and
> \exists Coffee = \{ f(x, y): d((x, y),  Cup_0) < r_cup} s.t. Coffee \in  
\overbar
> {Cup} \and T_coffee > 90 F \and t > t_0 => Hap_me(t) > Hap_me(t_0).

I had heard that a mathematician is someone who converts coffee into  
theorems, but I had not yet seen a demonstration.

--
Dave Seaman
ags@seaman.cc.purdue.edu
Newsgroups: sci.math
From: rivin@SAIL.Stanford.EDU (Igor Rivin)
Subject: Re: pyramid volume
Organization: Computer Science Department,  Stanford University.
Date: Thu, 21 Jan 1993 22:09:08 GMT

In article <1993Jan21.205125.15427@sophia.smith.edu> orourke@sophia.smith.edu (Joseph O'Rourke) writes:
>In article <1993Jan21.185357.28141@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
>>:In article <1993Jan21.140402.25519@mr.med.ge.com> carl@crazyman.med.ge.com (Carl Crawford) writes:
>>:>
>>:>how do show that the volume of a pyramid is 
>>:>	1/3 * area of base * altitude 
>> 
>>A hint: Use a little vector analysis.  Consider 3 non-colinear
>>vectors A,B,C  in R^3. Then consider A * (B x C).
>
>	Why don't you complete the thought?   It may be that you assume
>knowledge more "complicated" than the intended goal...


I assume that the intended goal is an argument that Archimedes would
buy, so that volume can be defined as (say) the total amount of water
displaced by the figure. This pretty much limits us to cutting the
pyramid into pieces of known volume. Now, any pyramid can be cut into
tetrahedra with the same vertex, so it is enough to compute the volume
of a general tetrahedron. On the other hand, a general tetrahedron can
be represented as the union of six right angled tetrahedra with the
same vertex (some of which may be negatively oriented, but this is
very reasonable from the bathtub perspective, if you imagine ladling
water out with European milk containers). In turn, the volume of
a right-angled tetrahedron  is computed in a previous message by
O'Rourke. This method can also simply adapted to show that a
parallelopiped of "determinant" 1 has volume 1, which in turn allows
the indiscriminant use of affine transformations. This train of
thought then also leads to the hairy world of scissors-congruence,
etc. 




Newsgroups: sci.math
From: A8131DAL@vm.univie.ac.at (Peter Schmitt)
Subject: CONFERENCE: Discrete Geometry (Austria)
Organization: University of Vienna
Date: Wed, 27 Jan 1993 16:15:17 GMT

-------------- Announcement ----------------------

May 2-8, 1993.

Austrian Conference on
Discrete Geometry and related topics

The conference will take place in a hotel near Neuhofen, a small village about
100~km west of Vienna (near Amstetten), situated in a pleasant landscape near
some of the most prominent sights of Austria.  The price of full board will be
approximately AS~4500.-- (about $440) (in a single room),
or AS~3250.-- (about $300) per person in a double room.

Information: Peter Schmitt, Institute of Mathematics, University of Vienna,
             Strudlhofgasse 4, A-1090 Wien, Austria
     e-mail: schmitt@awirap.bitnet
        (or) a8131dal@awiuni11.edvz.univie.ac.at
------------------------------------------------------------------------------
From: fc03@ns1.cc.lehigh.edu (Frederick W. Chapman)
Newsgroups: sci.math
Subject: Re: Hare Hare?  (Re: Geometry = the mind of God.)
Date: 21 Jan 93 03:58:12 GMT
Organization: Lehigh University

In article <Vm8LXB1w165w@kalki33.lakes.trenton.sc.us>,
system@kalki33.lakes.trenton.sc.us (Kalki Dasa) writes:
>djdaneh@pbhyc.PacBell.COM (Dan'l DanehyOakes) writes:
>
>> With all the goodwill in the world and no desire to put down anyone else's
>> religion, I've found you can get remarkable results by chanting:
>> 
>> 	Larry Moe Larry Moe Moe Moe Larry Larry
>> 	Larry Curly Larry Curly Curly Curly Larry Larry
>
>What is this nonsense doing in sci.math?
>
>Sincerely,
>Kalki dasa
>
>
>       ---------------------------------------------------------
>      |                Don't forget to chant:                   |
>      |                                                         |
>      |  Hare Krishna Hare Krishna, Krishna Krishna Hare Hare   |
>      |       Hare Rama Hare Rama, Rama Rama Hare Hare          |
>      |                                                         |
>      |    Kalki's Infoline BBS Aiken, South Carolina, USA      |
>      |           (kalki@kalki33.lakes.trenton.sc.us)           |
>       ---------------------------------------------------------


Actually, I and several people I know found the Larry-Moe-Curly chant
hilariously funny (no disrespect intended to anyone's religion).  A
good laugh is always welcome on sci.math; it is a refreshing break
from some of the other things that go on here.

Fred Chapman
-- 

o ------------------------------------------------------------------------- o
|  Frederick W. Chapman, User Services, Computing Center, Lehigh University |
|    Campus Phone:  8-3218     Preferred E-mail Address:  fc03@Lehigh.Edu   | 
o ------------------------------------------------------------------------- o
|             Ecstasy is transitory, but a theorem is forever!              |
o ------------------------------------------------------------------------- o
From: mrw@ukc.ac.uk (M.R.Watkins)
Newsgroups: sci.math
Subject: Smooth manifolds and function extensions
Date: 26 Jan 93 10:45:25 GMT
Organization: Computing Lab, University of Kent at Canterbury, UK.

Given a manifold M with a smooth (C-infinity) structure, it is possible to
define E(M), the linear space of all smooth functions on M.  Now is it
possible to reconstruct the smooth structure (that is the atlas on M) from
the knowledge of E(M) alone?

I thought about something like defining a collection of charts on M of the
form (U,F), where F:U -> |E^{m} has component functions F_i:U -> |R which
are restrictions of E(M) functions to U.  As far as I can see, this collection
will contain the original atlas (provided coordinate functions have smooth
extensions), but I've been unable to show that it doesn't
also contain other noncompatible charts.
From: kwong@mary.cs.fredonia.edu (Harris Kwong)
Newsgroups: sci.math
Subject: Re: Binomial coefficient modulo prime power-- looking for a reference.
Date: 21 Jan 93 15:28:02 GMT
Organization: Math / CS, State Univ. of N. Y. College at Fredonia
X-Newsreader: Tin 1.1 PL3

tsai@cs.uchicago.edu (Shi-Chun Tsai) writes:
: Define an integer function f(i) = i \choose k  mod p^a, where
: i ranges over non-negative integers, k and a are any positive integers,
: and p is a prime number.  Then it is not hard to prove that f is a
: periodic function with period p^{\lfloor log_p k \rfloor + a}.
: 
: Actually, it can be generalized for modulo any composite number by using
: the Chinese Remainder Theorem.  I believe this has been proved for a while,
: but I couldn't find a reference for it.  Could anyone show me a pointer?
: Thanks in advance.
: 
: Shi-Chun

It is indeed quite well-known.  Zabek [5] was probably the first one who
published the result.  Applying Vandermonde's convolution, Trench [4]
obtained identical period.  Fray [1] extended the result to q-binomial
coefficients.  Recently, Kwong [2,3] obtained the same results by
studying generating functions. 

1. R.D. Fray, Congruence properties of ordinary and q-binomial
   coefficients, Duke Math. J. 34 (1967), 467--480.

2. Y.H.H. Kwong, Minimum periods of binomial coefficients modulo M,
   Fibonacci Quarterly 27 (1989), 348--351.

3. Y.H.H. Kwong, Periodicities of a class of infinite integer sequences
   modulo M, J. Number Theory 31 (1989), 64--79.

4. W.F. Trench, On the periodicities of certain sequence of residues,
   Amer. Math. Monthly 67 (1960), 652--656.

5. S. Zabek, Sur la p\'eriodicit\'e modulo m des suites de nombres
   {n\choose k}, Ann. Univ. Mariae Curie-Sklodowska A10 (1956), 37--47.

--
Harris Kwong

Dept. of Math. & Comp. Sci.
SUNY College at Fredonia
Fredonia,  NY   14063
Email: kwong@mary.cs.fredonia.edu
Newsgroups: sci.math
From: roth@csoadm.enet.dec.com (Lee Roth)
Subject: Best cut algorithim?
Organization: Digital Equipment Corporation
Date: Wed, 27 Jan 1993 16:57:48 GMT


Any idea where to start looking for an algorithim to help me figure out
'best cuts' for a sheet of plywood?

For a given size of stock (say, a 4' x 8' sheet) I need to be able to
calculate the points for doing cuts to produce the target size (say, somthing
like 7.5" x 22.5") with the least amount of waste.

Any suggestions where to start? I could code up some iterative guessing
contraption but would like to apply real, working math principles instead.

Thanks-

Lee (good in algebra, weak in calculus but willing to brush up)


-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
Lee Roth  N8JQY                         Digital Equipment Corporation
roth@csoadm.enet.dec.com                
-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
From: zeleny@husc10.harvard.edu (Mikhail Zeleny)
Newsgroups: sci.math
Subject: Re: Non-Standard Analysis and philosophy
Date: 22 Jan 93 00:34:36 GMT
Article-I.D.: husc3.1993Jan21.193437.19639
Organization: The Phallogocentric Cabal

In article <1993Jan21.180359.21766@ulrik.uio.no> 
solan@smaug.uio.no (Svein Olav G. Nyberg) writes:

>Under which philosophies of mathematics is non-standard
>analysis possible, and to what extent?

Precisely under the non-Archimedean ones, and all the way.

>Thanks in advance,
>                Solan

cordially,
mikhail zeleny@husc.harvard.edu
"Le cul des femmes est monotone comme l'esprit des hommes."
Newsgroups: sci.math,sci.math.symbolic
From: rivin@SAIL.Stanford.EDU (Igor Rivin)
Subject: Re: Which CAS have BA's in math?
Organization: Computer Science Department,  Stanford University.
Distribution: sci
Date: Fri, 22 Jan 1993 03:30:03 GMT


In article <scott.727648403@trimtab> bobk@decide.com writes:
>[Posted for friend.]
>
>ilan@leland.Stanford.EDU (ilan vardi) writes:
>
>>In article <1jhjkfINN8m9@function.mps.ohio-state.edu> edgar@math.ohio-state.edu (Gerald Edgar) writes:
>>>
>>>I doubt there is a computer algebra system that can get a degree
>>>in mathematics: they don't write proofs (yet).  However, I tried the
>
>>They write papers though. Shalosh B. Ekhad, Doron's Zeilberger's computer,
>>has written a number of papers (more than I have, I think).
>
>Actually, one system written in Mathematica can do proofs (printed in
>TeX of course).  This system created by a couple of professors from
>Carnegie-Mellon can do the proof of the Wierstass continuous-nowhere
>differentiable function. All of the theorems from Chapter 2 of
>Ramanujan's collected works, properties of stereographic projections
>and on and on. The professors are Ed Clark and Xudong Zhao and they
>have an article in the current issue of the Mathematical Journal. I
>recommend it highly. A BA in analysis is not as far away as some might
>think.
>

(a) Minor point:  Zhao is not a professor, being a graduate student of
Professor Clark.

(b) Major point: Their work is very interesting, but if you did your
exams with as much help as their system (or any theorem proving
system, for that matter) gets, you would get kicked out of school
pronto for plagiarism. On the other hand, maybe their system could
become a US senator (viz. Ted Kennedy). Theorem proving systems now
and for the forseeable future would be better described as proof
verifying systems.






Newsgroups: sci.math
From: rivin@SAIL.Stanford.EDU (Igor Rivin)
Subject: Re: Matrices (problem)
Keywords: Rank
Organization: Computer Science Department,  Stanford University.
Date: Mon, 25 Jan 1993 15:42:08 GMT

In article <Jan.24.22.14.31.1993.13572@inferno.rutgers.edu> gore@inferno.rutgers.edu (Bittu) writes:
>A simple question about the rank of a special matrix (this is not a
>homework problem; I already know a solution, want something better).
>In fact, this was probably on the Math Olympiad a few years ago.
>
>Let A be a matrix with the following properties:
>
>1. A is 2n+1 x 2n+1 for a positive integer n.
>
>2. A has 0's on the diagonal.
>
>3. The remaining entries are either 1's or -1's.
>
>4. Every row has exactly n 1's and n -1's.
>
>We want to show that the rank of this matrix is 2n (over complex
>numbers, say). Clearly the rank is <= 2n (all columns sum to the zero
>vector). 

It is not clear that the below is any more elegant than the GF[2]
solution, but here goes (m=2n+1):

(a) The question is equivalent to showing that the null-space of A is
generated by the vector (1, ... ,1)

(b) Which is equivalent to showing that there is no null-vector v of
A, such that the sum of elements of v adds up to 0.

(c) suppose there is such a vector V -- it is easy to see that it must
also be a null vector of the matrix B which has 0s on the main
diagonal, and 1s elsewhere (do it row by row).

(d) B = -I(m) + J(m), where J has all ones. J(m) vanishes on the
orthogonal complement of the vector w=(1, ... ,1), and multiplies w by
m. hence the eigenvalues of B are (n-1, -1, ..., -1), hence it is
non-singular, hence V is the 0 vector, hence w is the only null-vector
of A, hence A has rank m-1 = 2n.
From: ad99s461@sycom.mi.org (Alex Deburie)
Newsgroups: sci.math
Subject: Want to learn about FFT's ... where?
Date: 26 Jan 93 02:14:42 GMT
Organization: Michigan Information eXchange
X-BBS-Software: EXCELSIOR! V0.976

I'd like to learn about fast fourier transforms, but I don't know exactly 
where to look.
From: ara@zurich.ai.mit.edu (Allan Adler)
Newsgroups: sci.math
Subject: Re: Compact quotients of symmetric spaces
Date: 28 Jan 93 17:41:24 GMT
Organization: M.I.T. Artificial Intelligence Lab.



I wrote my thesis on anti-holomorphic involutions of fibre systems of
abelian varieties over compact quotients of Hermitian symmetric
spaces. It appeared in the Transactions of the AMS around 1978, but I don't
remember exactly. There are relations between these and anti-holomorphic
involutions of the base. The antiholomorphic involutions of Hermitian
symmetric spaces were studied by Harris Jaffee in his thesis. Anti-holomorphic
involutions of compact quoetients of products of the upper half plane
were studied by Steve Kudla in his thesis. We three all wrote our
theses with Prof. Michi Kuga, recently deceased, at SUNY Stony Brook around
1974. Professor Shimura did earlier work and so did one of his friends
or students, whose name I unfortunately forget at the monent, but which
is cited in my thesis. 

Allan Adler
ara@altdorf.ai.mit.edu
Newsgroups: sci.math
From: hdev@dutiai.tudelft.nl (Hans de Vreught)
Subject: Re: Strange Question
Keywords: dictionary 
Organization: Delft University of Technology
Date: Thu, 21 Jan 1993 09:22:47 GMT

mrw9e@kelvin.seas.Virginia.EDU (Michael Robert Williams) writes:

>Is there such a thing as a self-contained dictionary? Meaning, is
>every word defined completely in terms of other words that are 
>contained in that dictionary?

Mathematically speaking this means that you are using circular definitions, I
presume? This isn't a big problem for humans however.

Yes, there do exist such dictionaries. I've one at my work. Each word in the
dictionary is explained into terms of 2000 frequently used words which are
also listed in the dictionary.

These kind of dictionaries are usely intended for foreign people. They are
very handy.
--
Hans de Vreught                          | John von Neumann:
hdev@dutiba.twi.tudelft.nl               | Young man, in mathematics
Delft University of Technology (TWI-ThI) | you don't understand things,
The Netherlands                          | you just get used to them.