mm-183
recently sent a paper on Quintic and Riccati
differentialequations, to the Journal of mathematics of Kyoto
University.The quintic itself (multi variable) has much
simpler form but theRiccati solution by Maple has difficult
hypergeometric functions.Maple could convert the
hypergeometric functions, but the solution isvery
equations has sent two of my paperson New methods of Riccati
differential equations I and II for theirreferee
problem with constraintsHow good is Matlab at solving linear
optimization problems withconstraints(we have 227,760 equality
Optimization problem with constraints>How good is Matlab at
solving linear optimization problems with>constraints(we have
227,760 equality and inequality contraints, all>linear)?
constraintsalewando_tego_nie@oddpost_tego_tez_nie.com says...How good is 
Matlab at solving linear optimization problems
with>constraints(we have 227,760 equality and inequality
a problem of this size and structure.-- Philip A. VitonOhio
use mathcad. I installed version 2001i this weekend.
Afterinstallition I noticed that c-dilla had also been
introduced to my registry.Google c-dilla to find out what this
program does. Anyhow I have spent 2days rebuilding my hard
drive because of the damage that Mathcad did.Steven O.
in> I am considering purchasing Mathcad 2001 (not the latest
release,> Version 11, but the one before that), and I have
posted a few> questions regarding that purchase. I have one
more question...I wanted to know if Mathcad 2001 can provide
closed form solutions for> at least some partial differential
equations. I posted this question> several places including
Mathsofts own collaboratory (their> user-helping-user Web
site). One or two people responded in the> 
affirmative.However,
Mathsofts own people claim it does not. I know that> 
sometimes
tech support people are not always 100% accurate even about>
their own products (especially one that they no longer support
or> sell). And Im just cynical enough to think that they 
might
even> deliberately mislead me, in an effort to get me to buy
the latest> version of the product. (I will not buy MC 11,
because it requires> Product Activation, and only allows
installation on two computers. I> find that entire technology
unacceptable in principle, and in practise> I have three
computers -- Im certainly not buying a third copy of the>
software for the third computer...)Anyway, the question is,
can someone confirm that MC 2001 (not 2001i,> but just 2001)
does or does not provide closed-form solutions to PDEs?> (I
understand it wont solve all such equations, but it would
help if> it at least has the ability to solve some such
advance for all replies.> Steve O.Standard Anti§ame
Disclaimer: Please dont §ame me. I may actually*be* an 
idiot,
released and we will be offering FREE theStandard Edition with
the purchase of the book:Ferreira, C., Gene Expression
Programming: Mathematical Modeling by anArtificial
Intelligence, 2002, 272 pp.For further information please go
to:http://www.gene-expression-programming.com/gep/Books/
index.aspGeneral features of APS 3.0 Standard Edition:o
Translates the evolved models into 8 different languages (C,
C#, C++,Java, JavaScript, Visual Basic, VB.NET, and Fortran).o
Draws the parse trees of the evolved models.o Translates the
evolved models virtually into any programming languagethrough
User Defined Grammars.o A total of 70 different built-in
mathematical functions and comparisonrules plus Dynamic UDFs
and Static UDFs for modeling.o A total of 11 built-in fitness
functions for Function Finding.o A total of 10 built-in 
fitness
functions for Classification.o A total of 11 built-in 
fitness
functions for Times Series Prediction.o User Defined Fitness
Functions for all problem categories.o Implements a new
algorithm for handling random numerical constants.o Data
screening engine for preprocessing.o Time series
transformation engine.o Evolution from seed models.o Change
seed utilities.o Saves all the best-of-generation models of a
run.o Plots the evolutionary dynamics of the run.o Complexity
increase engine.o Supports Databases and Text Files both for
loading input data andscoring.o Recursive testing and
prediction for Time Series.o Implements an extensive package
of statistical indexes for modelevaluation.o and much more.For
more information go to:http://www.gepsoft.com/gepsoftEnjoy and
spread the word!All the best,Candida
Ferreira------------------------------------------------------
-----Candida Ferreira, Ph.D.Chief Scientist, Gepsoft73 Elmtree
DriveBristol BS13 8NA, UKph: +44 (0) 117 330
9272http://www.gepsoft.com/gepsofthttp://
www.gene-expression-programming.com/author.asp----------------
support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
I have such a hard timewith it. I attempted it, but Im all
over the place. I was hoping,maybe I could get a little
direction?Questionshow that u(r)~exp -
[(((2mE/(h(bar))^2)^(1/2))r] is a form of thesolution to the
radial differential equation for the hydrogen atomwhen r is
<jlwenthe@uiuc.edu> schrieb im Newsbeitrag> Ok, 
schrodingers
equation is everywhere, and I have such a hard time> with it.
I attempted it, but Im all over the place. I was hoping,>
maybe I could get a little direction?Question> show that
u(r)~exp - [(((2mE/(h(bar))^2)^(1/2))r] is a form of the>
solution to the radial differential equation for the hydrogen
atom> when r is large.If r is large the schr.9adinger equation
Schrodingers equation> Ok, schrodingers 
equation is
everywhere, and I have such a hard time> with it. I attempted
it, but Im all over the place. I was hoping,> maybe I could
get a little direction?> Question> show that u(r)~exp -
[(((2mE/(h(bar))^2)^(1/2))r] is a form of the> solution to the
radial differential equation for the hydrogen atom> when r is
large.Substitute into the equation and see what you get. Try a
solutionof the form u(r) = A*exp[-b*r] where b is a positive
constant tobe determined, and where A is an overall
normalization (that dropsout, of course).-- Julian V.
NobleProfessor Emeritus of
^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ God
is not willing to do everything and thereby take away our free
will and that share of glory that rightfully belongs to us. --
challenge, risingIve been fortunate enough to discover some
challenging mathematics,which is also easy to elaborate on,
and has been made easier for me byan example from a professor
who happens to disagree with my position.That professor put
forward the quadratic example:(5a_1(x) + 7)(5a_2(x) + 7) =
7(25x^2 + 30x + 2) where his as are roots of a^2 - (x - 1)a 
+
7(x^2 + x). (reference info at bottom)The mathematics is
immediately challenging as here you have apolynomial
multiplied by 7, so it hardly is amazing to considerdividing
that 7 off, and at first blush, it makes sense to divide itoff
so as to get(5a_1(x)/7 + 1)(5a_2(x) + 7) = 25x^2 + 30x +
2where I arbitrarily picked the first to divide through, as 
you
couldjust as easily have(5a_1(x) + 7)(5a_2(x)/7 + 1) = 25x^2 +
30x + 2.The mathematics here has just become challenging, as
if you try to dothat in the ring of algebraic integers, you
find that the result isoutside that ring, like try 
x=1.Theres
something going on here! But what?Part of the joy of
mathematical research Id think is finding 
outexactly whats
going on, where now a constant like 7 has inexplicablybecome
such a major issue. Its just a number! Numbers like 7
areunderstood! Whats happening?Some have reacted to the
challenge by turning to a dogmaticinterpretation of
mathematics. For instance that professor hasdecided to
conclude that whatever it takes you must be able to stay inthe
ring of algebraic integers.Why not rise to the challenge of
asking, what if?After all, theres nothing in the 
definition of
algebraic integers tohandle this situation. Their definition 
is
that they are roots ofmonic polynomials with integer
coefficients. Theres nothing thereabout(5a_1(x) 
+ 7)(5a_2(x) +
7) = 7(25x^2 + 30x + 2) where the as are roots of a^2 - (x 
-
1)a + 7(x^2 + x). Now given two algebraic integers you know
that multiplying themtogether or adding them will give you an
algebraic integer. Its alsotrue that you can factor any 
given
algebraic integer an infinitenumber of ways into algebraic
integer factors.But none of that handles the current
situation.Just like that, the mathematical world shifts, as
that 7 on the end,which defies just dividing off without
contention, unlike so manyother constant factors of
polynomials, changes the mathematical scene.The challenge to
you is to accept that change, or quietly hold on tothe belief
that nothing has happened! You can try to go back to
yourresearch, back to your life, back to comfort where
constants behave!Or like the professor who came up with that
example, you can fight,fight, fight 
for dogma, to see more in
past mathematics than is thereto try and claim that there IS a
way to stay in the ring of algebraicintegers after dividing off
7.The choice is yours: rise to the challenge, or not.James
Harris Decker Quadratic Source
Information---------------------Recently Rick Decker, a
professor at Hamilton College, apparentlytrying to refute my
research came up with a quadratic example, which Ilike because
its a quadratic, and easier to manipulate than thecubics 
Ive
used before.If you wish to see his original post here are some
headers which alsoshow that he posts from Hamilton
===
College:Subject: Re: Mathematical consistency, courageDecker
put forward the quadratic(5a_1(x) + 7)(5a_2(x) + 7) = 7(25x^2
+ 30x + 2) where his as are roots of a^2 - (x - 1)a + 7(x^2 
+
have a hydraulic system I want to model. I have it running in
an excelspread sheet now but it is very messy and very slow. I
wish to convert themodel to Mathcad but I need to solve a
system of linear and non-lineardifferential equations AND have
limits. For instance the pressures can gobelow 0 and the
position cant go beyond the end of the cylinders. I
havewritten this in C++ using Runge Kutta but I get bogged
down in the graphicsas I am not a windows programmer.I could
roll my own own Runge Kutta algorithm with limits but this
would notbe as easy a doing it in C and buying a graphic
library.Any hints as to the best tools and
Quadratic, algebraic integers, done> Turns out theres a
rather direct approach to showing a problem with> the old
concepts about the ring of algebraic integers.> Ive 
abandoned
this entire approach, as it seems that you *can*indeed, despite
what I thought find algebraic integer factors in thering of
algebraic integers for the example I gave.James
fit my experimental data to my model. The values of
allparameters must be greater than zero. However, when I use
the mrqminroutine from numerical recipes, the fitted values 
are
below zero.Is there a way to apply constraints to the
parameters supplied tomrqmin function? This issue has been
addressed here before but wereunsatisfactory.Please help if
am trying to fit my experimental data to my model. The values
of all> parameters must be greater than zero. However, when I
use the mrqmin> routine from numerical recipes, the fitted
values are below zero.Is there a way to apply constraints to
the parameters supplied to> mrqmin function? This issue has
been addressed here before but were> unsatisfactory.Please
minimization , you could replace each of yourparameters with
its square.That is, if your present parameters are say a, b,
c, then wherever theseappear in your function, put a^2, b^2,
Millerhttp://users.bigpond.net.au/amillerRetired
fit my experimental data to my model. The values of all>
parameters must be greater than zero. However, when I use the
mrqmin> routine from numerical recipes, the fitted values are
below zero.Is there a way to apply constraints to the
parameters supplied to> mrqmin function? This issue has been
addressed here before but were> unsatisfactory.Please help if
, you could replace each of your> parameters with its square.>
That is, if your present parameters are say a, b, c, then
wherever these> appear in your function, put a^2, b^2, c^2.>
I simplifiedthe problem to avoid confusing the readers. I am
fitting my data to abiexponential function and the range of
amplitudes and exponents haveto be in the range [0 500].
Unconstrained mrqmin() from NumericalRecipes always gives me
negative values for the best fit. However,when I use
constrained minimization in Matlab using lsqcurvefitfunction, 
I
get the correct results.I must use a constrained minimization.
seem to be getting better-than-expected help from all you
people. SoI will eliminate the simplification and explain the
entire problem.Heres the rig:I have an array of data points
(x) from my experiment. I would like tofit a bi-exponential
equation to the data points.Equation: y = A*exp(t/TA) +
B*exp(t/TB)Independent params are A, TA, B, TBt is an array
for the times for which I have the data points.x is array of
experimental data pointsA & B are in range [0 2000]TA & TB are
in range [0 500]Objective: Obtain values for A, TA, B, TB such
that (y - x) is theminimum. In other words, the best fit.Like 
I
said, Matlab constrained non-linear minimization
function,lsqcurvefit, gives me the correct results. mrqmin()
gives me incorrectresults since atleast one the parameters is
negative, therebyrendering the entire solution meaningless.Any
fit my experimental data to my model. The values of all>
parameters must be greater than zero. However, when I use the
mrqmin> routine from numerical recipes, the fitted values are
below zero.Is there a way to apply constraints to the
parameters supplied to> mrqmin function? This issue has been
addressed here before but were> unsatisfactory.Please help if
, you could replace each of your> parameters with its square.>
That is, if your present parameters are say a, b, c, then
wherever these> appear in your function, put a^2, b^2, c^2.>
case. I simplified> the problem to avoid confusing the 
readers.
I am fitting my data to a> biexponential function and the 
range
of amplitudes and exponents have> to be in the range [0 500].
Unconstrained mrqmin() from Numerical> Recipes always gives me
negative values for the best fit. However,> when I use
constrained minimization in Matlab using lsqcurvefit> 
function,
I get the correct results.> I must use a constrained
athttp://plato.asu.edu/guide.htmlIn the least squares section
you find several codes. I am sure you candeal with 
Fortran.Hans
unfortunately it will not work in my case. I simplified >the
problem to avoid confusing the readers. I am fitting my data 
to
a >biexponential function and the range of amplitudes and
exponents have >to be in the range [0 500]. Unconstrained
mrqmin() from Numerical >Recipes always gives me negative
values for the best fit. However, >when I use constrained
minimization in Matlab using lsqcurvefit >function, I get the
correct results. >I must use a constrained minimization. Any
solutions?http://plato.la.asu.edu/nlolsq.html -> gaussfit ->
elsunc -> enlsip ->/port/n2fb (allows bounds, the large
am trying to fit my experimental data to my model. The values 
of
all> parameters must be greater than zero. However, when I use
the mrqmin> routine from numerical recipes, the fitted values
are below zero.Is there a way to apply constraints to the
parameters supplied to> mrqmin function? This issue has been
addressed here before but were> unsatisfactory.Please help if
, you could replace each of your> parameters with its square.>
That is, if your present parameters are say a, b, c, then
wherever these> appear in your function, put a^2, b^2, c^2.>
better than y =exp(x), which will not create multiple
you suggest a book that is not too heavy on the math? I am
mostlylooking for implementation examples. I have looked at
enoughderivations and would rather look at real world
examples.Also Im still looking for comments on the approach 
I
describe. Itseems fairly straightforward and I had hoped
someone would comment onthe filter model.TIA> 1. You should
pick up a book on Inertial Navigation. These typically > will
have discussions and derivations on Kalman Filter
inplementations.> 2. In general, the position mesurements
are, in fact, treated as > measurement data for the filter.
However, accelerometer and vehicle > orientation (both from an
IMU device) are usually treated as inputs to a > Kalman Filter
using as its model the error states. That is, the error >
model is derived from the Inertial Navigation mechanization
equations > and the effects of erroneous sensor data, etc.
These error equations > would then contain states to estimate
sensor biases, etc. as well as the > position and
orientatation error states. Once the error states are >
estimated they are applied to the navigator in order to
of product of bessel functionsX-PAA-AntiVirus:
PassedX-PAA-AntiVirus-Message: Scanned by
http://www.pandasoftware.com/PAAHi all,I need to calculate
this integralint_{0}^{infty} du frac{BesselJ(k,u)
BesselJ(l,u)}{u^2 + a^2}for different orders k,l (can go up to
5 or 10 or so) and different valuesfor the parameter a (which
is real and positive).Is there a simplification formula at
hand? Or some transformation (it looksquite like a Hankel
transform, not?). For the numerical quadrature, I could cut
the interval at a large enoughvalue. The problem is in the
oscillation (I cant integrate between zerosbecause for each
case I would need to recalculate the zeros (unless someonehas
a huge table at hand ;-) ). all tips
bessel functions> Hi all,> I need to calculate this
integral> int_{0}^{infty} du frac{BesselJ(k,u)
BesselJ(l,u)}{u^2 + a^2}> for different orders k,l (can go
up to 5 or 10 or so) and different values> for the parameter a
(which is real and positive).> Is there a simplification
formula at hand? Or some transformation (it looks> quite like
a Hankel transform, not?).> For the numerical quadrature, I
could cut the interval at a large enough> value. The problem
is in the oscillation (I cant integrate between zeros>
because for each case I would need to recalculate the zeros
(unless someone> has a huge table at hand ;-) ).> all tips
appreciated,> gertMy advice would be to use the integral
representations of theBessel functions to get finite integrals
of range (0, 2pi).-- Julian V. NobleProfessor Emeritus of
^^^^^^^^^^^^^^^^^^http://galileo.phys.virginia.edu/~jvn/ God
is not willing to do everything and thereby take away our free
will and that share of glory that rightfully belongs to us. --
been released and we will be offering FREE theStandard Edition
with the purchase of the book:Ferreira, C., Gene Expression
Programming: Mathematical Modeling by anArtificial
Intelligence, 2002, 272 pp.For further information please go
to:http://www.gene-expression-programming.com/gep/Books/
index.aspGeneral features of APS 3.0 Standard Edition:o
Translates the evolved models into 8 different languages (C,
C#, C++,Java, JavaScript, Visual Basic, VB.NET, and Fortran).o
Draws the parse trees of the evolved models.o Translates the
evolved models virtually into any programming languagethrough
User Defined Grammars.o A total of 70 different built-in
mathematical functions and comparisonrules plus Dynamic UDFs
and Static UDFs for modeling.o A total of 11 built-in fitness
functions for Function Finding.o A total of 10 built-in 
fitness
functions for Classification.o A total of 11 built-in 
fitness
functions for Times Series Prediction.o User Defined Fitness
Functions for all problem categories.o Implements a new
algorithm for handling random numerical constants.o Data
screening engine for preprocessing.o Time series
transformation engine.o Evolution from seed models.o Change
seed utilities.o Saves all the best-of-generation models of a
run.o Plots the evolutionary dynamics of the run.o Complexity
increase engine.o Supports Databases and Text Files both for
loading input data andscoring.o Recursive testing and
prediction for Time Series.o Implements an extensive package
of statistical indexes for modelevaluation.o and much more.For
more information go to:http://www.gepsoft.com/gepsoftEnjoy and
spread the word!All the best,Candida
Ferreira------------------------------------------------------
-----Candida Ferreira, Ph.D.Chief Scientist, Gepsoft73 Elmtree
DriveBristol BS13 8NA, UKph: +44 (0) 117 330
9272http://www.gepsoft.com/gepsofthttp://
www.gene-expression-programming.com/author.asp----------------
algoritm wanted...X-Beer: Yes pleaseX-Woobinda: Oh yesX-Kebab:
 
available SVD algorithm either written in C orwhich can easily
be called from C?I have been using the svd algorithm from
Numerical Recipes in C but now, asI need to distribute some
code, I need to replace it with a free alternative.Is the best
way to use one of the subroutines from LAPACK? If this is
the case, can anyone point me in the direction of some simple
examples where people have called these subroutines from C? I
have never called Fortranfrom within C so Im 
finding my feet
here a bit!Are there any other alternatives which I could
a freely available SVD algorithm either written in C or >which
can easily be called from C? >I have been using the svd
algorithm from Numerical Recipes in C but now, as >I need to
distribute some code, I need to replace it with a free
>alternative. >Is the best way to use one of the subroutines
from LAPACK? If this is the >case, can anyone point me in the
direction of some simple examples where >people have called
these subroutines from C? I have never called Fortran >from
within C so Im finding my feet here a bit! >Are 
there any
>Lazclapack in http://www.netlib.orgor c/meschach in the same
algorithm either written in C or> which can easily be called
from C?I have been using the svd algorithm from Numerical
Recipes in C but now, as> I need to distribute some code, I
need to replace it with a free> alternative.Is the best way to
use one of the subroutines from LAPACK? If this is the> case,
can anyone point me in the direction of some simple examples
where> people have called these subroutines from C? I have
never called Fortran> from within C so Im 
finding my feet here
a bit!Are there any other alternatives which I could
to C from original codes inHandbook for Automatic Computation
are available athttp://www.crbond.com/linear.htm--There are
two things you must never attempt to prove: the unprovable --
and theobvious.--Democracy: The triumph of popularity over
DON T KNOW HOW TO SOLVE THIS!1.DETERMINE DOMAIN OF THE
FUNCTIONf(X)=X3/X2+2X+1 FIND EXTREME FUNCTIONS f AND DRAW ITS
GRAPH2.BY NEWTONS METHOD SOLVE EQUATION:X3-X-2=0DRAW A
understand this two,the procedure and all,please help me, i
mustunderstand this.> CAN ANYONE PLEASE HELP ME?????I WILL
DOMAIN OF THE FUNCTION>>f(X)=X3/X2+2X+1>> FIND EXTREME
FUNCTIONS f AND DRAW ITS GRAPH>>2.BY NEWTONS METHOD SOLVE
ANYONE PLEASE HELP ME?????>CAN ANYONE PLEASE HELP ME?????I
YOU know the answer, not if WE do.Most of the people here are
willing to help you learn and understandwhat you should learn
and understand, if you give hints on why and where youhave
trouble in learning and understanding, but not to pass the
Process Control with PCA (T2 and SPE charts)The group that I
work for (see below) has done a lot of research in thisarea. I
would say though that your best bet for reading about T2 and
SPE ingeneral is the book by Jackson (very expensive book
though!). As fordetermining the variable(s) at fault, the
paper by Miller, on contributionplots looks at this problem. A
tutorial paper on how to actually use andinterpret all these
plots is given by Kourti and MacGregor. The paper byNomikos
and MacGregor gives an alternative method in the appendix
tocalculate the SPE charts limit.These books and papers
should be in most university libraries. Finding
thecontributing variables to SPE (or even T2) violations is
the real purpose ofapplying these statistical methods to 
ones
data: if it is data from aproduction process, then these would
be the variables to target in order toimprove the process. One
very simple way to compute the Q-charts limitsfrom your 
given
data is to take the Q-values that you have computed, and
findthe point which splits between 95% and the remaining 5% of
your values.This point is your 95% confidence limit, based on
the distribution of yourdata.Jackson, J.E. A Users Guide to
Principal Components, John Wiley, New York,1991.P Miller, R.E.
Swanson and C.F. Heckler, Contribution plots: a missing linkin
multivariate quality control, Applied Math. and Comp. Science,
vol 8(4),p 775-792, 1998. [There is a version available from
Paige Millers website?]T. Kourti and J.F. MacGregor, 
Process
analysis, monitoring and diagnosis,using multivariate
projection methods, Chemometrics and IntelligentLaboratory
Systems, vol 28, p 3-21, 1995.P. Nomikos and J.F. MacGregor,
Multivariate SPC Charts for Monitoring BatchProcesses,
Dunn______________________________________Kevin Dunn: Research
EngineerMcMaster Advanced Control Consortium (MACC)Dept. of
Chemical Engineering,McMaster University,Hamilton, ON, L8S
4L7CanadaWebsite:
http://macc.mcmaster.ca/______________________________________
> Hi... Ive been looking all over the Net for equations
regarding Hotellings> T2 and Squared Prediction Error (SPE)
calculations for a multivariate SPC> chart Im trying to put
together:> While I did (finally) find out how to 
calculate T2
and SPE, I still havent> found out:> 1) How to determine
the control limit for an SPE (Q) graph?> 2) How to properly
drill-down from an SPE graph to find out which variable> is
response. I apologize for my imprecise language. Ivaguely
remember from school something about being able to form
aone-to-one relationship between the integers and rationals.
Thatwould mean that the density of these number sets would be
the same (Idont know if density is the right word). It 
seems
that this wouldalso be the case for the computable numbers but
I am not sure aboutthat. I forgot the essence of the proof for
this, but I remember thatthe reals do not have this one-to-one
relationship and are thereforemore dense than the rationals.
This is a strange concept for anon-mathematician since all of
these sets are infinite. The number ofrationals between 0 and 
1
is infinite and the same goes for reals, butthere are still 
more
reals than rationals in this interval. I took abunch of math
classes in college, but dont remember much all theseyears
later. I never took number theory which a lot of people told
meI missed out on a very important subject for understanding
othersubjects. I just had this stuck and my head and was
numbers consist of the following:> 1) rationals> 2) numbers
that can be represented by finite algorithms (example: e> and
pi)> 3) numbers that are neither rational nor representable by
a finite> algorithm> The last item would include numbers like
this:> 0.123245223643792134....(infinite sequence of digits
that are> impossible to represent with any algorithm.)> Why
is there not a well known set of numbers that does not include
#3?> Theres more than one :-). For instance, the so-called
algebraic numbers> (numbers that are roots of polynomials with
integer coefficients)> lie between #1 and #2. Your #2 is 
itself
a well known set, generally> called the computable numbers.>
It would be more dense than the rationals but less than than
the> reals.> That sounds plausible, but actually (for most
meanings of dense)> its wrong, because in a very strong 
sense
almost all numbers> are in set 3.> What is the purpose of
describing numbers like #3?> I dont understand that
your response. I apologize for my imprecise language. I>
vaguely remember from school something about being able to
form a> one-to-one relationship between the integers and
rationals. That> would mean that the density of these number
sets would be the same (I> dont know if density is the 
right
word). I think the word you are looking for is cardinality.
The rationals andthe reals are both dense (density does not
come in degrees; a set may bedense or not dense). The
cardinality of a set is a way of specifying itssize, i.e. the
number of elements it has. The cardinality of therationals is
aleph_0. The cardinality of the reals is 2^aleph_0, alsocalled
c for the cardinality of the continuum. It is known 
byCantors
theorem that x < 2^x for every cardinal number x, and
thereforealeph_0 < c.>It seems that this would> also be the
case for the computable numbers but I am not sure about> that.
The computable numbers are countable. They have the same
cardinality asthe rationals.>I forgot the essence of the proof
for this, but I remember that> the reals do not have this
one-to-one relationship and are therefore> more dense than the
rationals. This is a strange concept for a> non-mathematician
since all of these sets are infinite. The number of> rationals
between 0 and 1 is infinite and the same goes for reals, but>
there are still more reals than rationals in this interval.
Correct. That means there is no bijection (no mapping that is
1-1 andonto) between the rationals and the reals.>I took a>
bunch of math classes in college, but dont remember much 
all
these> years later. I never took number theory which a lot of
people told me> I missed out on a very important subject for
understanding other> subjects. I just had this stuck and my
doesnt really have much to do with cardinality; 
thats
aconcept from set theory.-- Dave SeamanJudge Yohns mistakes
revealed in Mumia Abu-Jamal
ruling.<http://www.commoncouragepress.com/index.cfm?action=
the Lagrange interpolation formula in Abramowitz & Stegun,>
Handbook of Mathematical Functions. It is just a simple way to
write a> polynomial of degree N that goes through N+1 points of
a function.For example, suppose you know f(x) at the points x0,
x1 and x2. Call> these values f0, f1 and f2. Then you write> f0
* (x-x1)*(x-x2) f1 * (x-x0)*(x-x2) f2 * (x-x0)*(x-x1)> f(x) ~
------------------ + ------------------ + ------------------>
(x0-x1)*(x0-x2) (x1-x0)*(x1-x2) (x2-x0)*(x2-x1)You see this is
quadratic in x. Higher-order polynomials are essentially> the
same.>This formula directly gives the value of f at any x.
What Im actually> after is the kernel. Ie. | g(x) 0 <= |x| 
<=
1> k(x) = | h(x) 1 <= |x| <= 2> | 0 |x| >= 2>What is g and
h?f(x) = 1 - 1/2x - x^2 + 1/2x^3, 0<=x<1 1 - 11/6x + x^2 -
1/6x^3, 1<=x<2 0 , 2<=x f(-x) , otherwiseI assume one can
easily convert this into the form using absolute valuesthat
you want. I havent gone through it in detail, but I would
think thenthat in your notation it would simply be:g(x) = 1 -
1/2|x| - |x|^2 + 1/2|x|^3 andh(x) = 1 - 11/6|x| + |x|^2 -
f(x) = f0 * (x-x1) + f1 * (x-x0)> ----------- -----------> -1
+1Let d(n) = {1, n=0; 0, otherwise} be the discrete unit
impulse function and fi = d(xi). For -1 < x <= 0, setting x0 =
-1, x1 = 0 yields fa(x) = 1*(x+1) - 0*(x-0) = 1+x. Likewise, 0
< x <= 1 corresponds to fb(x) = 1-x. Theres your kernel.It
works the same way for higher odd-degree interpolating
polynomials, choosing the xi such that x ranges between the
two central locations when finding each kernel piece. For
instance, the cubic Lagrange interpolator isf(x) =
sum{j=0..3}[fj * product{k=0..3, k=/=j} (x-xk)/(xj-xk)].The
kernel is zero when |x| >= 2. For -2 < x <= -1, let xi = i-3
and again fi = d(xi), i=0..3. Only one term in f(x) is 
nonzero,
and we get fa(x) = (x-x0)*(x-x1)*(x-x2)/(x3-x0)/(x3-x1)/(x3-x2)
= (x+3)*(x+2)*(x+1)/3/2/1 = x^3/6 + x^2 + 11/6*x + 1.When the
interpolating polynomial has even degree, the kernel pieces
are spliced together halfway between integer locations, i.e. x
ranges over the middle one of the xi +-1/2. The process is the
same apart from that, but the resulting kernels are
discontinuous. The quadratic kernel has support |x| <= 3/2.You
can find some more kernels in Olli Niemitalos 
paper on
oversampled
interpolation:http://www.biochem.oulu.fi/~oniemita/dsp/
deip.pdfOh, and try
kernelAnd one that doesnt appear on the first 
two pages:
Survey: Interpolation Methods in Medical Image
Processinghttp://bigwww.ep§.ch/publications/meijering0301.
pdfMartin-- 2 + 2 = 5, for sufficiently large values of
solve this equation?> y-a y^4 + b = 0I would 
like to know if
there is a solution, or how to solve it> numerically.boundary
conditions y(x=0)= T> y(x=0)= c> How can I solve that in
mathematica?> As mentioned elsewhere, you can use a program
for first order systems.Another method, taking advantage of 
the
circumstances, namely that theequation is autonomous of second
order:First, constant solutions would satisfy -a*y^4+b=0 ;
check if the (at mosttwo) real solutions happen to fit into 
the
initial conditions, that is, if -a*T^4+b=0 and c=0.It would be
a rare coincidence, but one has to be insured against
losingthe singular solutions.If the constants do not fit, you
are free to multiply by y, and obtain aso-called 
first
integral (or law of conservation): (1/2)*(y)^2 - 
(1/5)*a*y^5
+ b*y = Kwith K a constant obtained from initial conditions: K
= (1/2)*c^2 - (1/5)*a*T^5+b*TThen the resulting first order 
ODE
has no elementary solution unless a=0(the easy case). If a is
not zero, you have y = (+ or -) sqrt(2(K + (2/5)*a*y^5 -
2*b*y)with initial condition y(0)=c. And you use your
favorite ODE solver forthat. (The (+ or -) is determined by
y(0), of course.)Danger: The reduced equation can run into 
a
singularity that the secondorder ODE may not have had. I am
convinced that a change of variable wouldbe possible that
bypasses the singularity.To conclude, it may be safer to use
the 1st order system, and use theconservation law to monitor,
skeleton tracing and loopextraction? Here are the details:What
I have:I have a binary image that contains a skeleton of a
contour. Thisskeleton can be thought of as a collection of
links connecting featurepoints of three types: end points, T-
and X-intersections. Thisskeleton may contain loops ofvarious
degrees of complexity, e.g. a simple loop where a link
startsand ends at the same FP or a complex loop where several
FPs areconnected into a single loop.What I need:I need to
trace this contour, i.e. to turn a 2d image into a sequenceof
skeleton pixels going from one FP to another in a certain
order. Ialso need to detect loops. This implies that when
traced, all linksbelonging to the same loop should be listed
successively without anygaps.Any help will be greatly
numbers by two objects> oo and -oo and to define some limited
arithmetic for them, as -oo < x < oo for any real number x> x
+ oo = oo for any real number x> x + -oo = x - oo = -oo for
any real number x> x * oo = oo for any positive real number x>
x * oo = -oo for any negative real number x> oo + oo = oo> -oo
+ -oo = -ooBut caution: you cannot define oo - oobecause any
definition of this term would lead to a 
contradiction]Id be
interested to see what, in your opinion, such a contradiction>
elaborating.> When you said any definition of this term would
lead to a contradiction,> I must now suppose that you intended
us to infer from context that you> meant any definition within
the two-point extension of the reals... I> agree that, in that
system, there is no element to which we may suitably> equate oo
- oo.> But if we extend the system further, perhaps by
adjoining merely one other> improper element, then surely
defining oo - oo need not lead to a> contradiction. Consider,
for example, standard §oating-point arithmetic,> in which oo -
oo is defined as NaN.Of course. NaN! is the 
copros way of
saying the result of thisoperation cant be 
defined in a
reasonable way. If you map everyundefined operation to some
exceptional object which is no longerallowed to take part in
operations then you are stuffing a structurewith dead objects.
But this doesnt take away undefinedness, 
itmerely renames it.
utilises Internet technology and expert system totrack all the
professional activity of the racing experts when they
much.If a is an r-cycle, prove that a is an odd permutation if
have the same parityHello everyone, I need some help in
proving this problem. Your helpwould greatly be appreciated.
The problem is as follows: If a, g E Sn, prove that the
permutations a and gag have thesame parity.I know that the
parity of the permutations can either be both even orboth
odd.Also I know that g*g=e=g*g.Again thank 
you for your
this problem. Your help| would greatly be appreciated. The
problem is as follows:|| If a, g E Sn, prove that the
permutations a and gag have the| same parity.|| I know that
the parity of the permutations can either be both even or|
both odd.| Also I know that g*g=e=g*g.|Any 
2-cycle in g if
you write it out containing only 2-cycles, is one in galso.
Therefore, in gag the extra 2-cycles occor two times
each,therefore conserving the parity.(You might want to check
teaching Maths in Victoria?Hi All,(This has also been posted
in aus.education - forgot to cross post :( )I am considering a
career move to teaching, I wish to teach maths athigh school
level and would welcome some advice as to the best way
toproceed. I have a degree in Electronics Engineering with
honours and Iwould have to do a diploma of education. Would
the maths containedwithin the engineering course be sufficient
to teach and to whatlevel?Any advice most appreciated.Iain