HDTenEightyP/NetNews huggingface dataset:
From: allenk@heckle.ugcs.caltech.edu (Allen Knutson)
Subject: Re: Square roots of polynomials
Organization: California Institute of Technology, Pasadena
Date: Mon, 27 Jul 1992 21:26:34 GMT

dzk@cs.brown.edu (Danny Keren) writes:

>I am looking for

>1) An algorithm that can decide whether a polynomial in one variable
>   (over the integers or reals) has a square root.

>2) An algorithm to compute such roots.

It seems the best way is to start at the constant term and figure out
inductively what the coefficients of a supposed square root must be. 
This essentially calculates the Taylor series at 0. (Unsurprisingly, 
this method asks you to divide by the constant term, so you have to start 
by dividing by x^2n to get that to be nonzero.) If your polynomial is
of degree 2N, stop at N, and square the attempted root to see if you get 
the given (i.e. if the Taylor series of the square root terminates at
that point).

Here's a more fun method that calculates a little bit more than you want.
(I have only thought about it in the real case.)

Let p be our poly, p' be its derivative, (p,p') be their GCD. The roots
of (p,p') are all the same as those of p, just raised to 1 lower power.
So p/(p,p') has all the same roots as p, but to the first power.
So if p is a square, (p/(p,p')) ^2 divides p. Reduce now to p / (p/(p,p'))^2,
which is of lower degree, and a square iff p is. (In order to finish up
with the actual root, keep a running product of the p/(p,p')'s.)

In particular, this algorithm will compute the largest square dividing p.
								Allen K.
From: rl@socrates.ucsf.edu (Robert Langridge)
Newsgroups: sci.math
Subject: Re: The Value of Science
Date: 22 Jul 92 21:46:23 GMT
Organization: UCSF Computer Graphics Lab

sylviab@sequent.com (Sylvia Berkowitz) writes:

>I'm reading  an English translation of "The Value of Science"
>by Henri Poincare.  The copy I have was published by Dover
>Publications in 1958.  It, however, doesn't give a date for
>the original (presumably French) publication.  

>Two of the chapters in the book are, "The Measure of Time" and
>"The Notion of Space".  My curiosity concerning the original 
>publication date for the book steams from a desire to know 
>which came first, these writings or relativity.  (It's kind
>of amusing actually; I'm trying to establish cause and effect
>on writings concerning simultaneity.) :-)

I recommend "History of the Theories of Aether and Electricity" Vol 2 
"The Modern Theories 1900-1926" by E W Whittaker (Thomas Nelson 1953)
particularly Chapter II, "The Relativity Theory of Poincare and Lorentz".

On page 30 we find:

"Even before the end of the nineteeth century, the failure of so many
promising attempts to measure the velocity of the earth relative to
the aether had suggested to the penetrating and original mind of
Poincare a new possibility.  In his lectures at the Sorbonne in 1899,
after describing the experiments so far made [he said]...
'I regard it as very probable that optical phenomena depend only on the
RELATIVE motions of the material bodies, luminous sources, and optical
apparatus concerned...'  In other words, Poincare believed in 1899 that
ABSOLUTE MOTION IS INDETECTIBLE IN PRINCIPLE..."

Later:

"In a lecture ... on 24 Sep 1904, Poincare gave to a generalized form
of this principle the name, THE PRINCIPLE OF RELATIVITY ...[he declared]...
'From all these results there must arise an entirely new kind of dynamics,
WHICH WILL BE CHARACTERISED ABOVE ALL BY THE RULE, THAT NO VELOCITY CAN
EXCEED THE VELOCITY OF LIGHT"

On page 40:

"[in 1905]...Einstein published a paper which set forth the relativity
theory of Poincare and Lorentz with some amplifications, and which attracted
much attention"

[Whittaker gives the original references.  I have retained his spelling 
and put in upper case the italicised text in the original]

Whittaker's two volumes are fascinating reading.

Bob Langridge				Phone: +1 415 476-2630, -1540, -5128
Computer Graphics Laboratory		               FAX: +1 415 476-0688 
926 Medical Sciences
University of California	                    E-Mail: rl@cgl.ucsf.edu
San Francisco  CA  94143-0446
From: cjh@tinton.ccur.com (Christopher J. Henrich)
Newsgroups: sci.math
Subject: Re: Quadratic Residue Question
Date: 20 Jul 92 19:00:49 GMT
Organization: Concurrent Computer Corp., Tinton Falls, NJ

In article <1992Jul17.210332.2105@cs.umn.edu> pires@cs.umn.edu (Luiz S. Pires) writes:
>
>
>  Given an arbitrary number a and an odd prime p, I am trying to write a 
> subroutine that decides whether a is a quadratic residue mod p.
>
>  I am familiar with a few of the rules available for manipulating the
> "Legendre Symbol", including the law of quadratic reciprocity. However I 
> can't think of an efficient way for handling this problem in the general
> case.
>  For example, if we have, say, a=31123 and p=45667, then I am not clear on
> how to proceed. 

Besides the law of quadratic reciprocity, the key to this is
the rule that

       /  AB  \          / A \      / B \
      |  ----  |   =    | --- |  * | --- | .
       \  P   /          \ P /      \ P /

And also,  if C == D (mod P) then

       / C \         / D \
      | --- |   =   | --- |
       \ P /         \ P /

(*What* a notation! In the rest of this posting I will write
LEG(C, P) etc.)

To take up your example, 

LEG(31123, 45667) = - LEG(45667, 31123) by Q.R.
          
                  = - LEG(14544, 31123) by second rule

                  = - LEG(2**4 * 3**2 * 101, 31123) courtesy of
                                                    "factor"
                  = - LEG(101, 31123) by first rule

                  = - LEG(31123, 101) by Q.R.

                  = - LEG(35, 101) by second rule

                  = - LEG(5, 101) * LEG(7, 101) by first rule

                  = - LEG(101, 5) * LEG(101,7) by Q.R. 

                  = - LEG(1, 5) * LEG(3, 7) by second rule

                  = - LEG(3,7)  = LEG(7,3) = LEG(1,3) = 1.

References on the number theory: I suppose most "first courses" will
show you this; I know that Vinogradov's book does.  Also
refer to Hardy & Wright, which is a real encyclopedia of
basic number theory.  

As a programming project, the difficult part is probably the
factoring.  But you can generalize the Legendre symbol to have
a composite second argument, as long as thetwo arguments are
relatively prime.  I will stick my neck out and affirm that the'
Q.R. formula is still true for the generalization.

Regards,
Chris Henrich
Newsgroups: sci.math
From: luzeaux@bellini.berkeley.edu (Dominique Luzeaux)
Subject: Re: A question from an ignorant philosopher.
Reply-To: luzeaux@bellini.berkeley.edu (Dominique Luzeaux)
Organization: University of California, Berkeley
Date: Wed, 29 Jul 1992 20:28:37 GMT

In article <Bs5LC1.7HE@acsu.buffalo.edu>, v5875bza@ubvmsd.cc.buffalo.edu
(Michael M Gorman Jr) writes:
|> First I'm trying to get clear on what
|> "priority" means, and it has to do with order.  My wife pointed out
|> to me that order is something that mathematicians have defined
|> rigorously, so I consulted some math dictionaries.  Unfortunately,
|> I found two different definitions.

Let R be a binary relation. It is an order (or a partial order) iff:
aRa (reflexive)
aRb and bRa imply a=b (antisymmetric)
aRb and bRc imply aRc (transitive)

The distinction between partial and total order comes from the fact that
in the previous definition nobody claimed that aRb for any a and b.
So a total order is an order for which we have: aRb or bRa, in other words
two elements can always be compared.
A strict order can be defined by adding the further constraint: the elements
have to be distinct.

It is important, when working with ordered structures, to keep in mind that not
every order works like < with numbers. For instance divisibility is an order
and the two numbers 3 and 5 cannot be compared.

This has led to define other structures like the lattice, where any two
elements
have a lower bound and an upper bound (a greatest minorant and a
smallest majorant)
For instance, divisibility yields a lattice in which 3 and 5 have as
lower bound 1
and as upper bound 15.

Remark: for total orders, a set of elements can be ordered into
something called 
a chain, this is why a total order is sometimes called a linear order.

Concerning priority, I guess you could say that if some order < has been
defined,
then some object a has a higher priority than b if a > b, or a < b
(depending on
whether you consider a smaller element to have a higher priority or not). I
would assume priority is a total order; actually, when the term priority
is used in
computer science (e.g. priority requests), this is the case.

	Dominique Luzeaux
From: martin@lyra.cis.umassd.edu (Gary Martin)
Newsgroups: sci.math
Subject: Re: An interesting limit problem.
Date: 29 Jul 92 17:52:03 GMT
	<1992Jul29.000223.27339@massey.ac.nz>
	<1992Jul29.115656.23253@gdr.bath.ac.uk>
Organization: University of Massachusetts Dartmouth

In article <1992Jul29.115656.23253@gdr.bath.ac.uk> mapsj@gdr.bath.ac.uk (Simon Juden) writes:

   ...I seem to have started an interesting discussion. I should make it
   absolutely clear that:

	    My views are my own and do NOT represent those of the University of
	    Bath School of Mathematical Sciences

   That said, I agree that it is more important to understand the meaning of 
   the operations than carry them out like a trained monkey. The point is that
   when faced with a new concept, IMHO the best way to understand it is to 
   work through some easy examples, and then test that understanding with some
   harder ones.  If students get into the habit of thinking about things such
   as limits rather than running to the nearest machine and asking it to do 
   the problem for them they may actually turn into mathematicians. But if all
   that is taught in undergraduate mathematics courses is what buttons to push
   then the people that emerge from those courses will not be mathematicians.
   I don't think this is such an antedeluvian view! The problem that started
   this thread off was a good example of a pretty argument that those who 
   misuse computers would have missed out on.

   Smile! ;-)

   Simon the Slightly Toasted Analyst

And have we even had a proof that the limit is 1 that didn't use a machine?
There was one posting that made the substitution  y=tan(sin x) (or something
similar) and claimed that the result was the reciprocal of the original
expression, but I don't think that claim was correct.  It's been awhile,
so if you've forgotten, the problem was to evaluate:
           sin(tan x) - tan(sin x)
lim  ----------------------------------
x->0 Arcsin(Arctan x) - Arctan(Arcsin x)

I think I know how my students would do this:
      2        2
   sin      sin
   ---  x - --- x
   cos      cos             0
----------------------- =  --- = 0.  (Though they might leave out the
   2   2       2   2        0         equal signs, too.  :( )
Arc sin     Arc sin
------- x - ------- x
     2          2
  cos        cos

--
Gary A. Martin, Assistant Professor of Mathematics, UMass Dartmouth
Martin@cis.umassd.edu
From: ow0a+@andrew.cmu.edu (Oswald Wyler)
Newsgroups: sci.math
Subject: Re; prime quadruplets ... partial apology
Date: 29 Jul 92 07:00:45 GMT
Article-I.D.: andrew.keRf_Ru00iUy02UVMt
Organization: Mathematics, Carnegie Mellon, Pittsburgh, PA

My memory was at fault, the problem whether there are infinitely many
quadruplets 30n+11, 30n+13, 30n+17, 30n+19 of primes, as is the problem
whether there are infinitely many twin primes p, p+2.  However, I cannot
buy the "almost certainly" designation -- as little as I can buy Math
by authority.  The question about quadruplet records still makes sense,
however.
Newsgroups: sci.math
Subject: Re: HELP. TEST FOR EXTREMA IN f(x,y,z)
From: a_rubin@dsg4.dse.beckman.com (Arthur Rubin)
Date: 29 Jul 92 23:04:41 GMT
Keywords: Partial derivatives, Extrema, function of 3 variables.

In <1992Jul29.010542.8650@eng.ufl.edu> armando@synapse.ee.ufl.edu (Armando Barreto) writes:


>Hello,

>Most Calculus books include a theorem like this to test extrema of a function  
>of two variables (This one taken from Calculus ..  by Swokowsky):

>Let f(x,y) be a function of two variables which has continuous second partial  
>derivatives on a rectangular region Q, and let

>		g(x,y) = fxx(x,y)fyy(x,y) - [fxy(x,y)]^2
>		
>for all (x,y) in Q. If (a,b) is in Q and fx(a,b)=0, fy(a,b) =0, then the  
>following statements hold.

>(i) f(a,b) is a local maximum if g(a,b)>0 and fxx(a,b) < 0.
>(ii) f(a,b) is a local maximum if g(a,b)>0 and fxx(a,b) > 0.
>(iii) f(a,b) is not an extremum of f if g(a,b)<0.

><<< NOTATION: fx(x,y) is the first partioal w.r.t. x, fxx(x,y) is the second  
>partial w.r.t. x and fxy(x,y) is the second mixed partial >>> 

>My question is :

>IS THERE SUCH A TEST FOR EXTREMA IN A FUNCTION OF 3 VARIABLES :f(x,y,z) ?

Actually there is; someone else will have to provide a reference, but:

(i) f(a,b) is a local maximum if the Hessian is negative definite
(ii) f(a,b) is a local minimum if the Hessian is positive definite
(iii) f(a,b) is a local minimum if the Hessian is indefinite.

The Hessian, of course is the matrix of 2nd partial derivatives.

A matrix is positive definite 
iff all eigenvalues are positive
iff all diagonal submatrices have positive determinate
iff all upper-left square submatrices have positive determinate

A matrix is negative definite 
iff all eigenvalues are negative
iff all diagonal submatrices have determinate sign (-1)^m
iff all upper-left square submatrices have determinate sign (-1)^m

A matrix is indefinite
iff all there are both positive and negative eigenvalues
iff it is not singular and is neither positive nor negative definite

(All of the equivalences are only for symmetric matrices.)



--
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.
Our news system is unstable; if you want to be sure I see a post, mail it.
From: mattm@lure.latrobe.edu.au
Newsgroups: sci.math
Subject: An interesting limit problem.
Date: 25 Jul 92 11:28:44 GMT
Organization: VAX Cluster, Computer Centre, La Trobe University

A challenge to all mathematicians. A 100 years ago, this would probably have 
been solved fairly simply in a natural way, but can you? I think that this
problem was first posed by the Russian mathematician Arnold. Hope you find this
problem as interesting as I did when I first solved it.

                sin(tan x)    -    tan(sin x)
	lim  ----------------------------------  = ???
	x->0 arcsin(arctan x) - arctan(arcsin x)

Tim    email: mattm@lure.latrobe.edu.au
Newsgroups: sci.math,sci.math.num-analysis,sci.math.symbolic
From: xjzhu@jeeves.waterloo.edu (Xiao Jun Zhu)
Subject: Lattice points in a polytope? (How to find all of them?)
Keywords: polytope, integer points
Organization: University of Waterloo
Date: Thu, 23 Jul 1992 15:05:40 GMT

Hi, there:

   I am in need of an efficient program or algorithm(prefer in C) which can 
find all the integer points in a polytope defined by sets of inequalities.
(Suppose that I know that the number of lattice points is not so huge.)
If you know such a thing exists, please send a message to me. Thanks very
much for your help.

Regards, Xiaojun.
Newsgroups: sci.math
From: prener@watson.ibm.com (Dan Prener)
Subject: Re: Square roots
Date: Wed, 29 Jul 1992 08:22:06 GMT
Disclaimer: This posting represents the poster's views, not necessarily those of IBM
Organization: IBM T.J. Watson Research Center, Hawthorne, New York

In article <Bs4HD3.ID5@usenet.ucs.indiana.edu> hpassaue@silver.ucs.indiana.edu (Billy Passauer) writes:

>I ran across this algorithm for finding the square root of a number a
>long time ago.  I was wondering if someone could tell me where I could
>find more information about it.

>To find the INTEGER portion of a square root:
>  Take the number you want the square root of, and successively
>subtract the odd numbers (starting with 1.)  Each time the result is
>positive, add 1 to a counter.  Repeat with the new number.  When the
>subtraction yields an odd number, the counter contains the answer.
>(integer only)  For instance the square root of 24:

>	              Counter
>	24 - 1 = 23  :   1
>	23 - 3 = 20  :   2
>	20 - 5 = 15  :   3
>	15 - 7 =  8  :   4
>	 8 - 9 = <0 

>Any information about this would be greatly appreciated.

Hint: What is the sum of the first N odd numbers?
--
                                   Dan Prener (prener@watson.ibm.com)
Xref: sparky sci.math:9443 soc.culture.british:10552
From: dlm@sjfc.UUCP (Don Muench)
Newsgroups: sci.math,soc.culture.british
Subject: Re: American mathematical hegemony
Summary: Susan misunderstood
Date: 23 Jul 92 03:44:38 GMT
Organization: St. John Fisher College, Rochester, NY

In article <sooz.711677426@vincent1.iastate.edu>, sooz@iastate.edu (Susan F Bradley) writes:
> 
> N.B. I do not intend for the following to be a direct reply to Mr. Muench and
> hope that he takes no offense by this!
> 
>  Personally, I don't care what a person's accent is and it seems to me that too
> many people get caught up in going on about who's pronunciation is best. 
> 
> Susan Bradley

I believe Ms. Bradley misunderstood what I was saying.  I was not asserting
that one accent or another was "best" or "right".  I was merely saying that
I feel "at home" in a rather special way, when I am talking to a very
friendly kinsman from "back home" who speaks with the rhythm and cadence
of the place where i was brought up.  It isn't a matter of right or wrong
or best.  The other point I was trying to make, in a rather humorous way, is
that I'm proud to be different.

So, Sue, I don't take offense.  However, I do want to say also that I don't
recall that anyone has been saying that this-or-that pronunciation is
superior.  An accent is part of what a person is, and often (usually) "accents"
the person in a wonderfully interesting way.

Don
From: columbus@strident.think.com (Michael Weiss)
Newsgroups: sci.math
Subject: Mathematical reality (was: You know, the integers)
Date: 23 Jul 92 08:59:53
Organization: Thinking Machines Corporation, Cambridge MA, USA
	<1992Jul22.094640.1@amherst.edu>
	<11667.Jul2300.06.3692@virtualnews.nyu.edu>
	<1992Jul23.033720.860@galois.mit.edu>
NNTP-Posting-Host: strident.think.com
In-reply-to: tycchow@riesz.mit.edu's message of Thu, 23 Jul 92 03:37:20 GMT


[discussion by Dan Bernstein and Timothy Y. Chow on whether formalism is
the "correct" philosophy of mathematics or not-- though they didn't put it
that way-- omitted]

Gee, do you suppose that Peano, Russell, Brouwer, Hilbert, Godel, and co.
would have come to an agreement and settled all these questions if only
they'd had email?
Newsgroups: sci.math
From: fc03@ns1.cc.lehigh.edu (Frederick W. Chapman)
Newsgroups: sci.math
Subject: Re: Advice Needed : How to contain a net defamation campaign ?
Date: 24 Jul 92 18:09:38 GMT
Organization: Lehigh University

In article <92206.010542U53644@uicvm.uic.edu>, U53644@uicvm.uic.edu writes:

>As much as I dread bringing this silly flame war over here.....

Evidentally, you do not dread it as much as the rest of us.  If you
feel compelled to argue, then please be so kind as to contain your
arguments to the newsgroup on which this ridiculous mess began.

[decidedly biased point of view deleted]

-- 

o ------------------------------------------------------------------------- o
|  Frederick W. Chapman, User Services, Computing Center, Lehigh University |
|    Campus Phone:  8-3218     Preferred E-mail Address:  fc03@Lehigh.Edu   | 
|  "I do comedy and magic; what you don't find funny -- that's the magic."  |
o ------------------------------------------------------------------------- o
Newsgroups: sci.math
From: chepan@gsusgi1.gsu.edu (Parthasarathy Nambi)
Subject: Need help!
Summary: Need help
Organization: Georgia State University
Date: Thu, 30 Jul 1992 14:19:26 GMT



 Hi,

 I need help in solving the equation:

  


   e^(-x) - x = e^(-y) + y


 For what values of x and y this equation is true?

 P.Nambi

Newsgroups: sci.math
From: dbradley@symcom.math.uiuc.edu (David Bradley)
Subject: Fractional Integral Problem
Organization: Math Dept., University of Illinois at Urbana/Champaign
Date: Thu, 30 Jul 1992 19:59:12 GMT

Can anyone suggest some techniques for dealing with the following
function involving a fractional integral?
                         oo
                        /   -u   b-1       -a
Let F(a,b,z) = GAMMA(a) |  e    u    (1+zu)   du,  Re(a)>0, Re(b)>0,
                        / 
                       0

and complex z not on the negative real axis.  I have reason to believe
the above is symmetric in a and b, but have no rigorous proof. 

It's easy to show that the z-derivatives of S(a,b,z) := F(a,b,z)-F(b,a,z)
all vanish at 0, but it appears F(a,b,z) is not analytic at z=0, so I 
can't equate S(a,b,z) with its Maclaurin series.

The transformations v=zt, z=1/w give

                  oo       y     
            a+b  /   -wy  /       a-1  b-1      -a
F(a,b,z) = w     |  e     |  (y-v)    v    (1+v)   dv dy,
                 /        /     
                0        0

so it would be enough to show the inner integral is symmetric in a and
b for each y>0.  I can do this for 0<y<1 by expanding (1+v)^(-a), so
that the inner integral becomes an infinite sum of Beta functions
multiplied by suitable Gamma functions to make everything symmetric in
a and b.  Unfortunately, this idea doesn't seem to carry over when y>1.
 
Any suggestions would be appreciated.  I suppose I should add that this is
not an assigned problem except insofar as I assigned it to myself.  The
integral arises in a proof for a continued fraction expansion for the
incomplete Gamma function.  Reference: H.S. Wall, Analytic Theory of
Continued Fractions.

-D.M.Bradley
Newsgroups: sci.math
From: palais@binah.cc.brandeis.edu
Subject: Re: HELP. TEST FOR EXTREMA IN f(x,y,z)
Reply-To: palais@binah.cc.brandeis.edu
Organization: Brandeis University
Date: Thu, 30 Jul 1992 14:40:53 GMT

Armando Barre asked                                                     

>My question is :

>IS THERE SUCH A TEST FOR EXTREMA IN A FUNCTION OF 3 VARIABLES :f(x,y,z) ?

Actually there is; someone else will have to provide a reference, but:

(i) f(a,b) is a local maximum if the Hessian is negative definite
(ii) f(a,b) is a local minimum if the Hessian is positive definite
(iii) f(a,b) is a local minimum if the Hessian is indefinite.

Arthur Rubin answered: 

   I think Arthur meant to say saddle point for case (iii) (but one must
assume that the Hessian matrix is non-singular). The same result is true 
in any dimension (and even in Hilbert space). The best reference is to any 
proof of the "Morse-Lemma", which says that after a change of variables the 
function locally is equal to a constant (its value at the critical point) 
plus the quadratic function given bythe Hessian. (See Lang's "Differentiable 
Manifolds" for the Hilbert space proof.)
From: schoudhu@ucbeh.san.uc.edu
Newsgroups: sci.math
Subject: a simple question..
Date: 22 Jul 92 10:32:42 EST
Distribution: world
Organization: Univ. of Cincinnati


Foll. is an easy question that I have no patience to solve and which is needed 
for some work I am doing. Please *mail* me your responses in order to save
bandwidth:


Given expressions for c1(n), c2(n), c3(n),   {for n >=5}  :

    c1(n) = c1(n-1) + c1(n-2) + c1(n-3);
    c2(n) = c2(n-1) + c2(n-2) + c2(n-3);
    c3(n) = c3(n-1) + c3(n-2) + c3(n-3);


with the initial conditions:

     c1(2) = c2(3) = c3(4) = 1;
     c2(2)=c3(2)=c1(3)=c3(3)=c1(4)=c2(4)=0;

what will be the non-recursive expressions for c1(n), c2(n) and c3(n), in terms
of n {for all n >= 2} ?

   --Spandan Choudury (Computer Science Dept.)

Newsgroups: sci.math