HDTenEightyP/NetNews huggingface dataset:
From: israel@unixg.ubc.ca (Robert B. Israel)
Subject: Re: Question on Connectedness
Keywords: Topology
Organization: University of British Columbia, Vancouver, B.C., Canada
Date: Wed, 15 Apr 1992 09:39:40 GMT

In article <36937@gremlin.nrtc.northrop.com> jbarnett@nrtc.northrop.com (Jeff Barnett) writes:
>Is the following true in the topology of Euclidean n space:
>
>P is a collection of non-empty (pairwise) disjoint closed sets.  P is
>a partition of S.  Claim -- if S is connected, then the cardinality
>of P is either 1 or uncountable.
>
>Jeff

The answer is "yes" in dimension 1, otherwise no.

For a counterexample if n > 1, take a countable collection of disjoint
curves (each homeomorphic to a closed interval [0,1]), with P_j 
constructed so its distance from P_i is less than 1/j for  1 <= i < j.

If n = 1, note that S is an interval.  You can assume P = { P_j } is
countable and get a contradiction by constructing a pair of sequences
{ a_j } and { b_j } so that
(1) a_j < a_{j+1} < b_{j+1} < b_j  for all j
(2) a_j and b_j are in different members of P
(3) P_j is disjoint from (a_j, b_j).
The nonempty intersection of the intervals [a_j, b_j] contains a point
of S that can't be in any P_j.

Robert Israel                         israel@math.ubc.ca
Department of Mathematics           or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada
From: dwilson@cvbnet.prime.com (David Wilson x5694 4-1600)
Newsgroups: sci.math
Subject: polyomino classification
Date: 9 Apr 92 14:45:03 GMT
Organization: Computervision, A Prime Computer Company, Bedford, MA, USA
Originator: dwilson@wrench



    I was wondering about the ability to tile certain figures with
    polyominoes.  The figures I am interested in, with cartesian
    descriptions, are:

    rectangle (r)		[0, a] X [0, b]
    halfstrip (hs)		R+ X [0, b]
    strip (s)			R X [0, b]
    angled strip (as)		([0, a] X R+) U (R+ X [0, b])
    quadrant (q)		R+ X R+
    halfplane (hp)		R X R+
    three-quarter plane (tp)	(tp)(R X R+) U (R+ X R)
    plane (p)			R X R

    It is easy to show that the tileability of some of the above
    figures implies the tileablity of others.  The main implications I
    have found are summarized in the following table:

	Table:  a tileable -> b tileable

        a\b	r   hs  s   as  q   hp  tp  p

	r	T   T   T   T	T   T   T   T
	hs	?   T   T   T	T   T   T   T
	s	F   F   T   F	F   T   ?   T
	as	F   F   ?   T	T   T   T   T
	q	F   F   ?   ?	T   T   T   T
	hp	F   F   ?   F	F   T   ?   T
	tp	F   F   ?   ?	?   ?   T   ?
	p	F   F   F   F	F   F   F   T

    These imply that any polyomino must tile precisely one of the
    following sets of figures:

	 1.  empty set
	 2.  { p }
	 3.  { tp }
	 4.  { tp, p }
	 5.  { hp, p }
	 6.  { hp, tp, p }
	 7.  { q, hp, tp, p }
	 8.  { as, q, hp, tp, p }
	 9.  { s, hp, p }
	10.  { s, hp, tp, p }
	11.  { s, q, hp, tp, p }
	12.  { s, as, q, hp, tp, p }
	13.  { hs, s, as, q, hp, tp, p }
	14.  { r, hs, s, as, q, hp, tp, p }

    In the following table, I have listed some polyominoes with their
    classifications.  To read a polyomino description, type it in,
    replacing horizontal bar with carriage return and period with
    space.  The asterisks then form picture of the polyomino.
    
	Polyomino		Set
	
	|.*|****|.*|.*|		1
	|.*|***|.*|		2
	|**|.**|.*|		9 or 10
	|**|.**|		12
	|*|			14

    I have been unable to find polyominoes in sets other than these
    five, or to fill in the ? in the above table.  There may be general
    implications I have not worked out (e.g., does a polyomino tiling
    the half strip necessarily tile a rectangle?).  It would make an
    interesting study.


-- 
David W. Wilson (dwilson@cvbnet.prime.com)
J.H.Whitney (was Prime Computer (was Computervision Corp.)), Bedford, MA
Disclaimer: "Truth is just truth...You can't have opinions about truth."
- Peter Schikele, introduction to P.D.Q. Bach's oratorio "The Seasonings."
From: bs@faron.mitre.org (Robert D. Silverman)
Newsgroups: sci.math
Subject: Re: factoid about new prime
Date: 9 Apr 92 11:25:56 GMT
Article-I.D.: linus.1992Apr9.112556.20469
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA

In article <1992Apr9.020906.20650@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
:In article <SMITH.92Apr7102215@gramian.harvard.edu> smith@gramian.harvard.edu (Steven Smith) writes:
::In article <1992Apr1.201324.25572@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
::
::> In article <3987@crl.LABS.TEK.COM> scott@ferrari.LABS.TEK.COM (Scott Huddleston) writes:
::> :It was mentioned on CNN that the new prime number discovered
::> :recently is four times bigger then the previous record.
::> :
::>
::> This is a (slight) exaggeration. 756839/216091 = 3.5024 times as big.
::
::Pardon me, but
::
::                756839
::               2       - 1                                          162781
::               -----------  ~=  2.33397019895780128135136676956 x 10
::                216091
::               2       - 1
::
::which is larger than four.  It appears that CNN was conservative in
::their report.
: 
:Pardon me, but in Complexity theory and in number theory the SIZE of a number
:is its number of digits or bits.
: 
:SIZE(N) = O(log N).
:
:And even your arithmetic is wrong. The above ratio is around 2^540748
:
:I'd like to know where 162781 came from.
 
Oops! Slight case of cognitive dissonance. I didn't notice that he
changed from base 2 to base 10 in his estimate of the quotient.
I (wrongly and implicitly) assumed binary arithmetic throughout and
didn't even LOOK at his change of base.

However, it is still true that SIZE(N) = O(log N), not O(N).

--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
Xref: sparky sci.math:5953 comp.os.msdos.apps:2819 comp.sys.ibm.pc.misc:7439
From: hank@ducvax.auburn.edu
Newsgroups: sci.math,comp.os.msdos.apps,comp.sys.ibm.pc.misc
Subject: Re: Looking for a function plotter?
Date: 11 Apr 92 14:27:49 GMT

In article <1992Apr10.119867.6MLD8@undr.org>, seanp@undr.org (Sean Petty) 
writes:
>         I am in need of a very simple function plotter that will allow
>         me to print what I plot to the screen.  What I am looking for in
>         the program includes the following:
> 
>         It should be able to plot very simple functions like 'sin X +2',
>         etc.
> 
>         I would like the X axis to show radian measures (and be
>         configurable), such as .2 pi, .4 pi, .6 pi, etc.
> 
>         It should be able to have multiple graphs on one screen, and
>         have 3 different line types (solid, dotted, dashed).

I have just started testing GNUPlot. I don't usually try to make
recommendations on software that I've barely used, but the GNU-project software
(this is lots of folks) has produced excellent software in the past. It is a
programming-language-type implementation, so it may not be as simple as you
desire. Others have mentioned it for its ability to prodeuce output for the TeX
environment.

Look on simtel20.army.mil in pd1:<msdos.plot>.

--darrel hankerson hank@ducvax.auburn.edu
Newsgroups: sci.math
From: bs@faron.mitre.org (Robert D. Silverman)
Subject: Re: Mersenne again
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Distribution: na
Date: Tue, 14 Apr 1992 23:03:29 GMT

In article <22708@dog.ee.lbl.gov> sichase@csa2.lbl.gov writes:
>This week's Science reported the new Mersenne prime and added that there 
>is an expectation that at least one lower one has so far been missed.
>Now I know from reading previous posts that certain intervals of primes
>have not been thouroughly explored.  But what is the specific conjecture on 
>the density of Mersenne primes that leads this Science article to claim 
>that there is likely a missing one?
 
While a proof is lacking, there are strong heuristical arguments that
suggest the expected number of Mersenne primes between 2^n-1 and 2^{2n}-1
is exp(gamma), where gamma is Euler constant. The previous one was
at 216091. One expects therefore to find another one before 750K.

Bob
--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
From: dy3@cunixf.cc.columbia.edu (Deane Yang)
Newsgroups: sci.math
Subject: Re: On knowing who's who (was: Fermat's Last Theorem)
Date: 14 Apr 92 02:11:55 GMT
Reply-To: dy3@cunixf.cc.columbia.edu (Deane Yang)
Organization: Columbia University


>In article <29E4E96C.13732@ics.uci.edu>, jeff@madeleine.ics.uci.edu (Jeff Erickson) writes:
>> a_rubin@dsg4.dse.beckman.com (Arthur Rubin) writes:
>> 
>> | Erdos is probably the best known active mathematician, possibly the
>				     ^^^^^^
>> | best known living mathematician.
>> Jeff Erickson 
>
>  And what do you think of Elie Cartan, Jean Dieudonne, Schwartz
>(yes, I'm french) ?
>
>Jacques Blanc-Talon

Um, although Elie Cartan is one of the great mathematicians of
the 20th century, he's not very active anymore (he's dead).
You probably mean his son Henri Cartan, who is also an important
mathematician. However, Cartan, Dieudonne, and Schwartz are all
fairly old, and, I believe, not very active anymore.

The best known active French mathematicians that come to my admittedly
limited mind are Jacques Tits, Alain Connes, Jean-Pierre Serre
(I'm basically trying to think of who I know in the College de France).

There's also Pierre Deligne, but he's Belgian.

Deane Yang
Polytechnic University
From: ender2@husc10.harvard.edu (Matt Ender)
Newsgroups: sci.math
Subject: Re: A simple question about R2 - Q2 (irrational p
Date: 13 Apr 92 04:05:57 GMT
Article-I.D.: husc3.1992Apr13.000558.11027
Organization: Harvard University Science Center

In article <1992Apr11.175123.360@math.okstate.edu> gindrup@math.okstate.edu (Eric `'d'kidd' G..) writes:
>In article <1992Apr7.233720.12411@newshost.anu.edu.au> gerry@macadam.mpce.mq.edu.au (Gerry Myerson) writes:
>>In article <1992Apr7.174510@masg1.epfl.ch>, baumgart@masg1.epfl.ch (Marcel Baumgartner) writes:
>>> 
>>> Keywords: In R2 - Q2, does there exist an oblique straight line ?
>>
>>Sure; x+y=pi. 
>
>Can you PROVE that the sum of any two irrational numbers is irrational?
>From my humble memories comes a rumbling that that is still an open idea...
>
>-- Eric Gindrup ! gindrup@hardy.math.okstate.edu

No, but I can show a counterexample:  pi and -pi.

The line in question is correct; the questions of (sum of irrats ->
irrat) and (R2-Q2 containing oblique line) are different...

-- Matt
Xref: sparky sci.math:6117 sci.physics:6811 sci.optics:507
From: draper@sscvx1.ssc.gov
Newsgroups: sci.math,sci.physics,sci.optics
Subject: Re: Help with reflection equasions PLEASE!
Date: 16 Apr 92 16:30:41 GMT
Article-I.D.: sscvx1.1992Apr16.103041.1
Organization: Superconducting Super Collider Laboratory

In article <92106.054206SIMMONS@UCF1VM.BITNET>, Tad Simmons <SIMMONS@UCF1VM.BITNET> writes:
> Hi all, I need some help.
> 
> For fun (heh!) I am writing a program to simulate beams of light intersecting
> prisms and mirrors.  Unfortunately, I haven't had to use higher math much
> lately, so I'm having trouble with the mirrors (so far, I haven't even TOUCHED
> the refraction function yet).
> 
> Problem:
>   Given a direction vector for the ray (NOT a unit vector) and a vector normal
>    to the mirror (again, not a unit vector) compute the new direction vector.
>   Assume: The mirror normal vector is the relevent one.
>           Ignore wavelength, phase, etc.  Just angle of incidence equals
>           angle of reflection.  (I might add that later, but I REALLY doubt it)
>   Preferences: Integer math, but I'm not fussy.
> 

The answer is easy, if you at least know how to do some vector operations. In a
nutshell, only the component of the ray vector along the mirror normal changes,
and it only changes sign. To do this, first find the UNIT mirror normal vector:
	~N = N/|N| where |N|=sqrt(Nx**2+Ny**2+Nz**2)
Then, for a ray vector A, the component along the mirror normal is the dot
product of A and ~N, multiplied by the normal unit vector.
	A(N) = (A . ~N) ~N
The new ray vector has a component
	A'(N) = - A(N)
and the other components remain unchanged
	A'(perp to N) = A(perp to N) = A - A(N)
So that the total new ray vector is
	A' = A'(perp to N) + A'(N) = A - 2*A(N)
{Here, all dot products and sums are vector sums. To write code for this, do it
component by component. If you don't know what a dot-product between two
vectors is, I suggest you brush up on this before attempting to do more.}

Paul Draper
University of Texas at Arlington
draper@sscvx1.ssc.gov
From: mab@wdl39.wdl.loral.com (Mark A Biggar)
Newsgroups: sci.math
Subject: Re: Does Goedel's work apply to number theory?
Date: 9 Apr 92 22:41:06 GMT
Article-I.D.: wdl.1992Apr9.224106.4243
Distribution: usa
Organization: Loral Western Development Labs

In article <1992Apr9.213653.965@aero.org> sinclair@aero.org (William S. Sinclair) writes:
>A colleague where I work insists that Goedel's work shows that there
>can be unprovable statements in the field of number theory. Even if they
>exist, are the any KNOWN unprovable statements in this field? Has there 
>been much conjecuture about the prvability of conjectures such as Fermat's
>theorem, or Goldbach's conjecture?

I believe that the following problem has been shown to be undecidable in first
order logic + peano axioms (which should be sufficient for number threory):

Call a finite set in positive integers LARGE if the smallest member
of the set is greater then the number of elements in the set.

Take the set of numbers 1..N and partition them as follows: randomly
select a pair out of the set and color (label) one of the pair BLUE
and the other RED.  Continue selecting pairs without replacement and coloring
until the set is empty (if N is odd randomly color the left over element).

Now for any given k, show that there exists a sufficiently large N such that
it is guarenteed that the will exist a LARGE subset of the RED numbers
containing k elements.

It is possible (using higher order logic) to determine the function
N=f(k) and determines the smallest N for a given k, but this function grows
so fast (it's faster the ack(N,N)) that it is impossible to prove that it
the the solution to the problem when limited to the above axiom system.

--
Mark Biggar
mab@wdl1.wdl.loral.com
From: dwilson@cvbnet.prime.com (David Wilson x5694 4-1600)
Newsgroups: sci.math
Subject: Tiling problem
Date: 8 Apr 92 12:48:55 GMT
Organization: Computervision, A Prime Computer Company, Bedford, MA, USA
Originator: dwilson@wrench



    Will the pentomino

        **
         **
         *

    tile the plane minus a quadrant?

-- 
David W. Wilson (dwilson@cvbnet.prime.com)
J.H.Whitney (was Prime Computer (was Computervision Corp.)), Bedford, MA
Disclaimer: "Truth is just truth...You can't have opinions about truth."
- Peter Schikele, introduction to P.D.Q. Bach's oratorio "The Seasonings."
Newsgroups: sci.math
From: sibley@math.psu.edu (David Sibley)
Subject: Re: fun with primes
Reply-To: sibley@math.psu.edu
Organization: Dept. of Mathematics, Penn State University
Date: Mon, 6 Apr 1992 16:18:21 GMT

In article 16293@uniwa.uwa.oz.au, hartley@maths.uwa.oz.au (Mike Hartley) writes:
>Or are there an infinite number of primes
>even in finite integral domains?

A finite integral domain is a field, so the question does not arise.
All the elements except 0 are units (invertible).


>Is there an infinite integral domain
>with a finite (but nonzero) set of primes? Why or why not?

You can get any finite number of primes you want as follows. Let S be
any set (finite or not) of ordinary integer primes and consider the set
A of rational numbers whose denominators, when in lowest terms, are not
divisible by any prime in S. (Here consider that an integer has
denominator 1, of course.) It is easy to check that A is a subring of
the rationals, and therefore a domain.  The only primes in A are the
ones in S.

David Sibley
sibley@math.psu.edu

Newsgroups: sci.math
From: bs@gauss.mitre.org (Robert D. Silverman)
Subject: Re: Best Method(complexity-wise) for Solving System over GF(2)?
Organization: Research Computer Facility, MITRE Corporation, Bedford, MA
Distribution: na
Date: Thu, 9 Apr 1992 20:29:36 GMT

In article <1992Apr9.181436.29316@uwm.edu> leranth@miller.cs.uwm.edu ( ULYSSES ) writes:
:
:
:    I am seeking information that will help me solve a problem.
:
:    Question:   Is there a better method [complexity-wise O[f(n)] ]
:
:		than  G.E. for solving a  linear system over GF(2)  ?
 
Yes. There are several. For example, the "4 Russian" method [see Aho, Hopcroft
& Ullman's book on algorithms] works in time O(N^3/log N) for large, sparse
systems. Wiedemann's method and the Lanczos algorithms work in time O(N^2 log N)
There is also the conjugate gradient method.

See, for example, "Solving sparse linear systems over finite fields" by
Wiedemann in IEEE Trans. Info. Theory. vol 32 (1986)

Andrew Odlyzko has also written a paper comparing these methods. I don't
have a reference, but he can be reached at amo@att.research.com

However, don't overlook GE as a practical method. I routinely solve systems
that are 30,000 x 30,000 with G.E. on a Sparc.

--
Bob Silverman
These are my opinions and not MITRE's.
Mitre Corporation, Bedford, MA 01730
"You can lead a horse's ass to knowledge, but you can't make him think"
Newsgroups: sci.math
From: cyeomans@ms.uky.edu (Charles Yeomans)
Subject: Re: Fermat's Last Theorem
    <3660@tuegate.tue.nl> <1992Apr9.021905.20917@linus.mitre.org>
Date: Thu, 9 Apr 1992 14:40:09 GMT
Organization: University Of Kentucky, Dept. of Math Sciences
X-Bytes: 

In article <1992Apr9.021905.20917@linus.mitre.org> bs@gauss.mitre.org (Robert D. Silverman) writes:
>In article <3660@tuegate.tue.nl> wsadjw@urc.tue.nl writes:
>:In article <STROMME.92Apr6130830@jeeves.math.utah.edu> stromme@math.utah.edu (Stein Stromme) writes:
>
[stuff omitted]
>
>I am just surprised that someone with a presumed interest in math would
>not know one of the most outstanding and prolific mathematicians of
>the 20th century. If I were Erdos, I'd be insulted.
>
>Paul Erdos has been one of the movers and shakers of the 20th century.
>The fact that he is relatively unknown is just another example of the
>piss-poor attitude the American public has towards math and mathematicians.
>
>I'm sure the original poster knows who Michael Jordan is, whose only
>claim to fame is his ability to jump high and stuff a ball through a hoop.
>The contributions of Paul Erdos will have long lasting value to the
>knowledge base of the human race and his work has provided stimulous for
>many others. What long lasting value Michael Jordan's contributions have
>would be a matter of serious debate.
>
>And I did answer the question in a half joking manner. The way the question
>was posed: "who is this Erdos dude" invites such because calling someone a
>'dude' without knowing that person is in itself somewhat flip and insulting.
>--
The fact that anyone knows who Erdos is is amazing.  I don't
think that Erdos has been any more influential than, say, Richard
Swan, but I bet that a majority of the posters to this net don't
know who he is, either.

As for appending "dude" to Erdos' name, the annoyance expressed 
over that reminds me of how I used to get my ass beat in junior
high school for not wishing to automatically say "sir" to 
(male) teachers.  I am willing to make some allowance for the
linguistic customs of other people.

CHarles Yeomans
cyeomans@ms.uky.edu
yeomans@austin.onu.edu
Newsgroups: sci.math
From: graham@sees.bangor.ac.uk (The Land of Confusion)
Subject: Pseudospheres ? 
Organization: University of Wales
Date: Wed, 15 Apr 1992 12:17:05 GMT

I posted a query a while back about pseudospheres. Either it got lost
or nobody felt like replying. So if there are any generous souls out
there that feel like shedding some light on this, it'd be appreciated.
Otherwise feel free to again ignore this request:

Can anybody give a brief description of pseudospheres, (generating)
tractrices, and the connection with non-Euclidean geometry?

--
graham a stephen    (e-mail: graham@sees.bangor.ac.uk)           \/
                                                                 /\
Subliminal Message:      -->                       <--           \/
From: carabalo@phoenix.Princeton.EDU (David G. Caraballo)
Newsgroups: sci.math
Subject: Dirichlet's Problem and the Riemann Mapping Theorem
Date: 6 Apr 92 13:13:30 GMT
Article-I.D.: Princeto.1992Apr6.131330.29664
Organization: Princeton University
Originator: news@ernie.Princeton.EDU

Ahlfors mentions (_Complex Analysis_, p. 251) that, using subharmonic 
functions, one can prove that the Dirichlet Problem "can be solved for any
region whose complement is such that no component reduces to a point."
He attributes this method to O. Perron, and goes on to make the following
claim:

"From this proposition an independent proof of the Riemann Mapping Theorem
 can easily be derived."

I may be missing something, but I have been unable to construct a proof of
the Riemann Mapping Theorem (in the form given below) using this proposition.
If anyone sees a way of doing so (Ahlfors is usually correct when he says
"easily", so I am sure I am missing something obvious.), please let me know.
This is not a homework problem.

Riemann Mapping Theorem:  Let G be any simply connected region (except the
plane), and let w be a point of G.  Then there exists a unique analytic
function f on G, such that f is a bijection between G and the disk |z|<1,
subject to the conditions f(w) = 0, and f'(w) > 0.

David G. Caraballo
From: fenthe@drycas.club.cc.cmu.edu
Newsgroups: sci.math
Subject: groups/fields query
Date: 17 Apr 92 19:26:42 GMT
Organization: Carnegie Mellon Computer Club


 I've got a couple of problems; firstly, is there a field with four elements?
One and two element fields are trivial, and three elements is just
the integers mod 3, but four elements? I can't seem to find anything
that'll work - integers mod 4 fails as 4's not prime, and I'm stumped
to find something along alternative lines that works as well.

 Also, can anyone point me at a proof for;
If R is a ring, and for all x in R, x^3=x, then R is commutative.

 For x^2=x, it's fairly simple (at least, compared to how far I've
got with x^3..) 


 Thanks.
From: loader@vax.oxford.ac.uk (R Loader)
Newsgroups: sci.math
Subject: Re: Fermat's Last Theorem
Date: 11 Apr 92 11:34:51 GMT
Organization: Oxford University VAXcluster

Edited flames . . .
In article <1992Apr9.133316.24615@linus.mitre.org>, bs@gauss.mitre.org (Robert D. Silverman) writes:
> In article <sundquis.702591357@s6.math.umn.edu> sundquis@s6.math.umn.edu (Thomas Sundquist) writes:
> :>   >>No, Erdos only offers money for his own problems.
> :>   >Pardon my ignorance. Who is this Erdos dude?
> :
> :>   You're joking right???  April 1 is tomorrow, not today, right???
> :>   This must be a joke. How can anyone who reads or posts to this newsgroup
> :>   not know who Paul Erdos is???
> :
> :>Come on, Bob, as a reader of this group you must have noticed that its
> :>contributors range from the professional to the not-so-professional.
> :>Why not just answer the question? After all, the poster says "Pardon
> :>my ignorance", and then he does not deserve to be run over like that.
> :>Stein 
> :
> :My thoughts exactly.	Tom
> 
> 
> Huh?  I was simply expressing my surprise that someone who claims an
> interest in math did not know who one of the greatest mathematicians of
> the 20th century is!!! How is that "running over someone"???
Could someone please answer the original question???
  I have heard of Erdos, but know next to nothing about what fields he
contributed to, what his major results are etc etc. Perhaps a sensible end to
this thread of flames would be for someone to post such info.
  Or perhaps the flamers were tring to conceal that they are just as
ignorant as me :-)
--
Ralphy, Oxford
Xref: sparky sci.math:6116 sci.logic:883