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Bernice Barnett
science forum beginner

Joined: 08 May 2005
Posts: 5

Posted: Wed Apr 12, 2006 12:30 pm    Post subject: Re: A conditional random number generation problem (please help me!)

Giovanni Resta wrote:
 Quote: rjmachado3 wrote: I need to know the formula for the random function that return random numbers in a range of a and b integers [a,b] but that obey on a custom probability (possibly different!) for each integer number on this [a,b] range (of course the sum of all integer number probabilities are = 1!). Finally, what i want is the general function formula that simulate the random behavior (based on a custom probability value for each integer number between the [a,b] range. confuse? i hope not! please help me!!!! what i know so far is that the function formula for generating a "pure" random number between [a,b] range is: rand()*(b-a)+a where rand() return a random number between 0 and 1. I'm not completely sure to have correctly understood you question. Anyway... Here is a very naive apprach that can work, at least if the interval is small. Maybe if the interval is large one can think about something more efficient. I make an example with 4 values, each with a custom prob., that can be easily generalized. Let the integer values be a,b,c,d (they do not need to be consecutive numbers) and let p_a, p_b, p_c, p_d the probability to extract respectively a,b,c, and d. Clearly p_a+p_b+p_c+p_d = 1. First you build these constants: P_a = p_a P_b = p_a+p_b P_c = p_a+p_b+p_c then you extract a random number U between 0 and 1, and if U <= P_a you select a, else if U <= P_b you select b else if U <= P_c you select c else you select d. I hope this example helps you giovanni There is an improvement if the range is large: Make a table of the

distribution (your P_x above). Then generate a uniform random as above.
Finally, do a binary search on the table - complexity O(log(n)) where n
is the number of table entries. I think this is the best you can do for
distributions where you don't have an inverse of the distribution function.

-- Jeff Barnett
Matt Heath
science forum beginner

Joined: 04 Apr 2006
Posts: 3

 Posted: Fri May 05, 2006 2:21 pm    Post subject: Re: Absolutely continuous functions on the circle and disc algebra function You are quite right. A false argument lead me to think it was the same as case with f anti-analytic, which is really what I am trying to solve.
John Baez
science forum Guru Wannabe

Joined: 01 May 2005
Posts: 220

Posted: Tue May 30, 2006 10:06 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 233)

Daniel Moskovich <dmoskovich@gmail.com> wrote:

 Quote: And, R^3 minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)! This is actually incredibly interesting for me- what is a reference for this? (I couldn't find it in either cited paper, and Gannon gives no source).

As usual, I gave all the references I know. I too find this fact
incredibly interesting. I first heard of it from Chris Hillman:

http://www.lns.cornell.edu/spr/2002-04/msg0040885.html

I feel I can *almost* prove it, but not quite.

SL(2,R)/SL(2,Z) is the space of unit-area lattices in the plane.
If we take the hexagonal lattice

* * * *

* * *

* * * *

and gradually rotate it 60 degrees, we get back to the
same lattice. So, we have traced out a certain loop A in
SL(2,R)/SL(2,Z). If we take the square lattice:

* * * *

* * * *

* * * *

and rotate it 90 degrees, we get another loop B.

I believe that these define elements of pi_1(SL(2,R)/SL(2,Z))
satisfying A^3 = B^2. This is the usual presentation for the
fundamental group of the complement of the trefoil knot:

http://en.wikipedia.org/wiki/Trefoil_knot

I think the loop A corresponds to going around a "meridian"
and B corresponds to going around a "longitude" - or maybe
vice versa, since I can never remember the difference between
the "meridian" and the "longitude" of a knot.

But, there should be some more direct way to see what's going on!

Since the Wikipedia article gives an analytic formula for the
trefoil knot, maybe someone come up with an analytic formula
for a diffeomorphism

f: SL(2,R)/SL(2,Z) -> R^3 - {trefoil}

Help, anyone?
Bruce Ikenaga
science forum beginner

Joined: 04 Nov 2005
Posts: 3

Posted: Thu Jun 01, 2006 6:05 am    Post subject: Re: This Week's Finds in Mathematical Physics (Week 233)

On Tue, 30 May 2006 22:06:24 +0000, John Baez wrote:

 Quote: In article <1148262406.418804.233710@i39g2000cwa.googlegroups.com>, Daniel Moskovich wrote: And, R^3 minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)! This is actually incredibly interesting for me- what is a reference for this? (I couldn't find it in either cited paper, and Gannon gives no source). As usual, I gave all the references I know. I too find this fact incredibly interesting. I first heard of it from Chris Hillman: http://www.lns.cornell.edu/spr/2002-04/msg0040885.html I feel I can *almost* prove it, but not quite. SL(2,R)/SL(2,Z) is the space of unit-area lattices in the plane. If we take the hexagonal lattice * * * * * * * * * * * and gradually rotate it 60 degrees, we get back to the same lattice. So, we have traced out a certain loop A in SL(2,R)/SL(2,Z). If we take the square lattice: * * * * * * * * * * * * and rotate it 90 degrees, we get another loop B. I believe that these define elements of pi_1(SL(2,R)/SL(2,Z)) satisfying A^3 = B^2. This is the usual presentation for the fundamental group of the complement of the trefoil knot: http://en.wikipedia.org/wiki/Trefoil_knot I think the loop A corresponds to going around a "meridian" and B corresponds to going around a "longitude" - or maybe vice versa, since I can never remember the difference between the "meridian" and the "longitude" of a knot. But, there should be some more direct way to see what's going on! Since the Wikipedia article gives an analytic formula for the trefoil knot, maybe someone come up with an analytic formula for a diffeomorphism f: SL(2,R)/SL(2,Z) -> R^3 - {trefoil} Help, anyone?

Quillen's proof is on pages 84-85 of Milnor's "Introduction to
Algebraic K-Theory".

Bruce Ikenaga
Joe Christy
science forum beginner

Joined: 01 Jun 2006
Posts: 2

Posted: Thu Jun 01, 2006 6:07 am    Post subject: Re: This Week's Finds in Mathematical Physics (Week 233)

Vis-a-vis John's note of 05/30/2006 03:06 PM:
 Quote: In article <1148262406.418804.233710@i39g2000cwa.googlegroups.com>, Daniel Moskovich wrote: And, R^3 minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)! This is actually incredibly interesting for me- what is a reference for this? (I couldn't find it in either cited paper, and Gannon gives no source). As usual, I gave all the references I know. I too find this fact incredibly interesting. I first heard of it from Chris Hillman: http://www.lns.cornell.edu/spr/2002-04/msg0040885.html ...

I wouldn't be surprised if this was known to Seifert in the 30's,
though I can't lay my hands on Seifert & Threfall at the moment to
check. Likewise for Hirzebruch, Brieskorn, Pham & Milnor in the 60's in
relation to singularities of complex hypersurfaces and exotic spheres.
When I was learning topology in the 80's it was considered a warm up
case of Thurston's Geometrization Program - the trefoil knot complement
has PSL_2(R) geometric structure.

In any case, peruse Milnor's Annals of Math Studies for concrete
references. There is a (typically) elegant proof on p.84 of
"Introduction to Algebraic K-theory" [study 72], which Milnor credits to
to Quillen. It contains the missing piece of John's argument:
introducing the Weierstrass P-function and remarking that the
differential equation that it satisfies gives the diffeomorphism to
S^3-trefoil as the boundary of the pair (discriminant of diff-eq, C^2 =
(P,P')-space).
This point of view grows out of some observations of Zariski, fleshed
out in "Singular Points of Complex Hypersurfaces" [study 61]. The
geometric viewpoint is made explicit in the paper "On the Brieskorn
Manifolds M(p,q,r)" in "Knots, Groups, and 3-manifolds" [study 84].

It is also related to the intermediate case between the classical
Platonic solids and John's favorite Platonic surface - the Klein quartic
http://www.math.ucr.edu/home/baez/klein.html. By way of a hint, look to
relate the trefoil, qua torus knot, the seven-vertex triangulation of
the torus, and the dual hexagonal tiling of a (flat) Clifford torus in S^3.

Joe

--
============================= Joe Christy ==============================
------------------ http://xri.net/=joe.christy ------------------
== If I can save you any time, give it to me, I'll keep it with mine. ==
Joe Christy
science forum beginner

Joined: 01 Jun 2006
Posts: 2

Posted: Thu Jun 01, 2006 3:00 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 233)

Vis-a-vis John's note of 05/30/2006 03:06 PM:
 Quote: In article <1148262406.418804.233710@i39g2000cwa.googlegroups.com>, Daniel Moskovich wrote: And, R^3 minus the trefoil knot is secretly the same as SL(2,R)/SL(2,Z)! This is actually incredibly interesting for me- what is a reference for this? (I couldn't find it in either cited paper, and Gannon gives no source). As usual, I gave all the references I know. I too find this fact incredibly interesting. I first heard of it from Chris Hillman: http://www.lns.cornell.edu/spr/2002-04/msg0040885.html ...

I wouldn't be surprised if this was known to Seifert in the 30's,
though I can't lay my hands on Seifert & Threfall at the moment to
check. Likewise for Hirzebruch, Brieskorn, Pham & Milnor in the 60's in
relation to singularities of complex hypersurfaces and exotic spheres.
When I was learning topology in the 80's it was considered a warm up
case of Thurston's Geometrization Program - the trefoil knot complement
has PSL_2(R) geometric structure.

In any case, peruse Milnor's Annals of Math Studies for concrete
references. There is a (typically) elegant proof on p.84 of
"Introduction to Algebraic K-theory" [study 72], which Milnor credits to
to Quillen. It contains the missing piece of John's argument:
introducing the Weierstrass P-function and remarking that the
differential equation that it satisfies gives the diffeomorphism to
S^3-trefoil as the boundary of the pair (discriminant of diff-eq, C^2 =
(P,P')-space).
This point of view grows out of some observations of Zariski, fleshed
out in "Singular Points of Complex Hypersurfaces" [study 61]. The
geometric viewpoint is made explicit in the paper "On the Brieskorn
Manifolds M(p,q,r)" in "Knots, Groups, and 3-manifolds" [study 84].

It is also related to the intermediate case between the classical
Platonic solids and John's favorite Platonic surface - the Klein quartic
http://www.math.ucr.edu/home/baez/klein.html. By way of a hint, look to
relate the trefoil, qua torus knot, the seven-vertex triangulation of
the torus, and the dual hexagonal tiling of a (flat) Clifford torus in S^3.

Joe

--
============================= Joe Christy ==============================
------------------ http://xri.net/=joe.christy ------------------
== If I can save you any time, give it to me, I'll keep it with mine. ==
Swiatoslaw Gal
science forum beginner

Joined: 02 Jun 2006
Posts: 1

Posted: Fri Jun 02, 2006 12:38 pm    Post subject: Re: This Week's Finds in Mathematical Physics (Week 233)

 Quote: f: SL(2,R)/SL(2,Z) -> R^3 - {trefoil} That would be amazing really- because then we could compose with modular forms and maybe obtain something interesting! I would also definitely like the answer to this.

Not realy.
In fact the isomorphism is a part of the modular theory:

Looking for

f: Gl(2,R)/Sl(2,Z)\to C^2-{x^2=y^3}

(there is an obvious action of R_+ on both sides:
M\to tM (M\in Gl(2,R) , x\to t^6 x, y\to t^4 y,
and that the quotient is what we want).

Gl(2,R)/Sl(2,Z) is a space of lattices in C.
Such a lattice L has classical invariants
g_2(L) = 60 \sum_{z\in L'} z^{-4}, and
g_3(L) = 140 \sum_{z\in L'} z^{-6},
where L'=L-{0}

The modular theory asserts that
1. For every pain (g_2,g_3) there exist a lattice L,
such that g_2(L)=g_2, and g_3(L)=g_3 provided
(g_2/3)^2\neq g_3.
2. Such a lattice is unique.

Best,
S. R. Gal
Philippe Flajolet
science forum beginner

Joined: 18 Jun 2006
Posts: 1

Posted: Sun Jun 18, 2006 10:45 am    Post subject: Re: Refree's query

 Quote: I am considering differential equations of the form: y'*x*(1+y*(d/dz){log(F_{k}(z)}_{z=y})=y

I hope I interpret the statement correctly, though I don't see the exact meaning of "IDENTICALLY complex functions" in the original specification. Also, the z,y notation business seems to me to be a bit obscuring the simplicity of the problem.

Instead of a very specific F_k, subsitute an arbitrary function F(y). Then the differential equation reads

y'*x(1+y*F'(y)/F(y))=y

that is,

y'/y+F'(y)/F(y)=1/x,

which integrates to give

log(y)+log(F(y))=log(x), that is, y*F(y)=x.

Thus, a generalized form of the original equation is plainly solved via inversion of an explicit function, no matter what the structure of F is. This is a type of equation encountered when analysing the Lambert function (y*exp(y)=x) and, in the variant form y=x*G(y), is related to tree enumerations in combinatorial analysis as well as to Lagrange inversion.

On an another register, the particular class of functions F_k that you introduce seem to be composed of some sort of "towers of exponentials": depending on the way you specify F_0, they might be described as the class of functions defined as corresponding to terms expressed using the variable "z" [or better "y"?], the unary construction exp(.), the binary operation "*", and perhaps "+" and/or some other operations. There has been quite a bit of research in mathematical logic for deciding equality or dominance of functions of this sort [eg, exp(2*y)=exp(y)*exp(y), etc]. A starting point to this literature might be

Some Application of Nevanlinna Theory to Mathematical Logic: Identities
Exponential Functions. C. Ward Henson, Lee A. Rubel
Transactions of the American Mathematical Society, Vol. 282, No. 1
(Mar., 1984) , pp. 1-32

and references therein.

Philippe
Anton Deitmar
science forum beginner

Joined: 25 Jul 2005
Posts: 24

 Posted: Thu Jul 20, 2006 2:00 pm    Post subject: Re: A Combinatorics/Graph Theory Question It seems to me that p equals C(n-1,k)+r. The number of elements of Y to which a given x in X is not connected, equals the number of k-element subsets of X which do not contain x, that is C(n-1,k). Hence, to make sure you pick at least r sets which do contain x, you must pick C(n-1,k)+r sets. Did I miss anything? Anton

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