mm-481
===
Subject:
: Re: Integral of a Vector over Distance> Mathematical
necessity? For the integral of the dot product? There> are
integrals of dot products, and integrals of cross products.
If> you can specify a product K(u,v) of two vectors u and v,
as long as> it has reasonable properties (bilinearity), then
you can calculate> the integral> integral(K(u,dv))> along the
curve traced out by v. So far, you have seemingly pointed> to
the dot product as one case; another poster has pointed out
the> cross product can be integrated.Just out of curiosity,
what kind of a space might the kernel function k(x,y) = h(x:y)
= h(x) + h(y) - h(x,y)where h(x:y) is the mutual algorithmic
entropy and h(x), h(x,y) arealgorithmic information content
(entropy) and joint algorithmicentropy respectively, indicate
through an approximation of thedot-product?===
===
Subject:
: Re:
Periodic function>A function like cos x^2 produces a cosine
wave that compresses itself in>both directions away from x=0.
What function would produce regularly>periodic compressions
and rarefactions?> Try cos(2*pi*cos(kx)). Choose k to your
liking.Maybe that's what Tim wants, but I'd be surprised.
Different choices ofk do not, in a sense, change things. The
graphs are all similar. Forexample, regardless of k, there are
always 4 extrema per cycle.Let me suggest instead considering a
family of functions like sin(p*x + q*sin(r*x))To see two
examples, graph sin(5*x + 3*sin(x)) and sin(16*x +
5*sin(2*x)).One might also read about frequency
modulation.
===
Subject:
: Re: Periodic function>A function like
cos x^2 produces a cosine wave that compresses itself in>both
directions away from x=0. What function would produce
regularly>periodic compressions and rarefactions?> Try
cos(2*pi*cos(kx)). Choose k to your liking.>Maybe that's what
Tim wants, but I'd be surprised. Different choices of>k do
not, in a sense, change things. The graphs are all similar.
For>example, regardless of k, there are always 4 extrema per
cycle.>Let me suggest instead considering a family of
functions like> sin(p*x + q*sin(r*x))>To see two examples,
graph sin(5*x + 3*sin(x)) and sin(16*x + 5*sin(2*x)).>One
might also read about frequency modulation.Here's another that
has infinite periodic compression, undefined at apoint or two
but, hey, who's quibbling:plot(cos(2*Pi*sec(x/5)),x=
-20..20);also try x = 8..9===
===
Subject:
: Re: differential
equations teaching> Spend 3 hours doing basic algebra to clean
up the mess>Skill in integration is a necessary
pre-requisite.Course starts>usually asking students to
eliminate arbitrary constant/s to set up>differential
equation,to appreciate the two way process.For example,y>= C
e^(-x^2/2) sets up dy/dx= -x*y , that integrates back to
given>relation by separable variables. etc..But the overall
emphasis on explicitly solving differential equationsis a
waste of time - that's the Calculus-approach:
Mindlessequation-crunching. Most equations are *not*
integrable. Its betterto teach a qualitative approach: What
are the steady states of theequation? What are the limit sets?
Then teach some numerics: Eulermethod, Runge-Kutte, advanced
methods.===>
===
Subject:
: Re: Conway's comment on large
numbers>Message-id: <jYidncRoruhwfRjdRVn-hg@comcast.com> I was
reading a paper by Conway and Doyle that proves that division
by> three is possible (without the axiom of choice).>
http://math.dartmouth.edu/~doyle/docs/three/three.pdf> In the
last paragraph the authors explain their doubts about the
ZF>axioms, saying,> Indeed, we're somewhat doubtful whether
large natural numbers (like> 80^5000, or even 2^200) exist in
any very real sense...> I don't understand this at all. What
do very large numbers have to do> with the consistency of
ZFC?> Nothing, directly.> Is this a joke?> Well, Conway has
a sense of humor, and this is humorous but it need> not be
taken as a joke; the somewhat serious interpretation is as a>
question about the provenance of numbers that are larger than
the> the number of distinct things in the observable physical
universe --> are they really ontologically different from
actual infinities?>There was a thread about computability
where the question>of theoretical vs actual computers came
up.>A theoretical computer can perform an unbounded, but
finite,>number of operations. I pointed out that finite was
pretty>small for an actual computer (where actual means
conforms>to the laws of physics as we know them).>Assume a
computer can perform an operation in 3 x 10^-44 sec.>(Planck's
time - theoretically, the shortest time an action can occur
in).>Let this computer count for a billion years (10^9).>(The
universe is estimated to be about 13 billion years old.>In 3
billion years, the Milky Way will crash into
Andromeda.)>Google says 1 year = 31 556 926 seconds (3 x
10^7).>So, in a billion years, the computer will count to>3 x
10^44 x 10^9 x 3 x 10^7 = approximately 10^61.Somewhat
doubtful.>Russell>- 2 many 2 countMensanatorAce of
Clubs===
===
Subject:
: Re: um......a little problem..0^0=1>
Integer-power definitions are based on generating the powers
through> some type of multiplicative operation, either
directly or through a> counting process (e.g., mappings form a
set of one cardinalit to a set> of another). Defniitons of
negative exponent powers as multiplicative> inverses is
included in this Such definitions render 0^0 as, in effect,>
an empty product that equals 1 naturally.> 0^0 = 1 makes the
most sense to me.> We may also define a rational-power
operation. If p and q are> relatively prime, then a^(p/q) is
the principal root (which needs to> be secified and may depend
on whether you're working with the real or> the complex domain)
of x^q = a^p. In this algebraic equaiton the powers> are
evaluated via the integer-power defnition. If a = 0, p = 1,
and q => 1 (or -1), then this process still gives 0^0 or
0^(0/1) = 1.> Please pardon me if I'm missing something
obvious, but how does that> get you 0^0=1? I can see how you
get 8^(2/3)=4 from x^3=8^2 , e.g. But> how does> x^1 = 0^0>
give you x=1? > Apply the integer-power definition to the
right-hand side, yielding> x^1 = 0^0 = 1> from which we get x
= 1^(1/1) = 1.But you already have 0^0=1 before you extract x
from x^1=1. !!! Sowhat's the point? I guess it's to show that
the integer definition isconsistent with the rationals
definition and produces no new values.OK.===
===
Subject:
: Re: Mass
versus Weight> The ratio of the gravitational weight [w], or
heavieness of any> object; body, or mass of matter, divided by
the acceleration [g] at> which it will free fall is its mass:
For any given body: Its mass m => w/g = f/a; anywhere;
anytime! ===
===
Subject:
: schools for topologyCan anyone tell me
which schools in southern california is good
intopology/algebraic topology?===
===
Subject:
: Re: Cantor:
ignorant, harmless fool or intentional liar?> Sagan's makes
his contempt for transfinite arithmetic clear in >
The_Demon_Haunted_World, where, in the chapter titled The Fine
Art of > Balony Detection, he mentions that transfinite
arithmetic is suited > to those with a deep devotion and
longing for life after death. The actual quote is Those with a
deep longing for life after death might, it seems, devote
themselves to cosmology, quantum gravity, elementary but it's
clear to anyone who knows anything about Sagan that he's not
expressing contempt for any of these subjects.On the other
hand, Sagan in this chapter talks about the fallacy of
argument from authority, which in view of your slavish quotes
from Wittgenstein, you might want to brush up on (I'd include
your quotes from Hilbert and Sagan, but you don't seem to
understand what they are saying.)===
===
Subject:
: Re: the MVT -- -
-- applications> By using the MVT with f(x)=x^.5 on [25,26],
show that> 26^.5 - 25^.5 < 1> f'(c) = 26^.5 - 25^.5> and f'(x)
= 1/(2(x^.5)), so f'(c) = 1/(2(x^.5)).> but how do I prove the
answer?Last equation is absurd25 < c < 261/2sqr 26 < f'(c) <
1/2sqr 25as f' is decreasing===
===
Subject:
: Re: the MVT -- - --
applicationsQ1.Pick x so that 0 < x < pi/2. (This eliminates
the need for tan(pi/2).)The MVT in this instance states (tan x
- tan 0)/(x - 0) = (sec^2)(c)where c satisfies 0 < c < x. Now
(sec^2)(c) >= 1 . . . .Hope this
helps.________________________________________________________
__> Hey! I'm learning about the MVT, and am stuggling with
being able toapply it to solve these problems.> Question 1:>
Prove that tan x > x for 0 < x < pi/2.> I think there must be
an f'(c) = (tan(pi/2) - tan 0)/(pi/2)> however, since
tan(pi/2) is undefined i am having difficulty trying toanswer
the question.> Question 2:> By using the MVT with f(x)=x^.5 on
[25,26], show that> 26^.5 - 25^.5 < 1> f'(c) = 26^.5 - 25^.5>
and f'(x) = 1/(2(x^.5)), so f'(c) = 1/(2(x^.5)).> but how do I
prove the answer?===
===
Subject:
: Re: Geometric interpretation of
the trace of a matrix?> If you start with unitary eigenvectors
of the transformation, as sides> of a paralelepiped, the trace
is certainly what I said in the last> post, i.e., the sum of
the edges of the image paralelepiped divided by>
2^{n-1}.Almost correct. You have to be careful with the signs
of the eigenvalues.A modification like: if the transformation
reverses top and bottom(or left and right etc), then the
length of the corresponding edgehas to be counted with a minus
sign.When your transformation has complex eigenvalues, you are
more or lessout of luck (ok, you may be able to do something
with rotations)If you make the further assumption that the
transformation is symmetric,then you can choose the
parallelpiped to be rectangular. All the eigenvaluesof a
symmetric transformation are real, so in that case a geometric
interpretationis possible.Jyrki Lahtonen, Turku,
Finland===
===
Subject:
: Complexity in advanced mathematicsI am
looking for resources for problems of considerable difficutly
inthe following subjects.Functionaly AnalysisHarmonic
AnalysisLinear Algebraalso what books do you recommend for
developing ability in proofwriting.===
===
Subject:
: Fields of study
under mathematicsIs there a website resource that lists areas
of mathematics and thecomponents of each field of study,I have
already checked out mathematical atlas and that at times can
beambigous, what I am looking for is a catalog style
heading/descriptionresource.thank you very much.===
===
Subject:
:
central limit theorem, equivalent statementsCLT and related
topics):Let {X_n,i} be a triangular array of random variables
(independentwithin each row). Suppose (*) given epsilon > 0,
sum(over i) E( X_n,i I(|X_n,i| <= epsilon) )---> m(**) given
epsilon > 0, sum(over i) Var(X_n,i I(|X_n,i| <= epsilon) )--->
s^2 finite(***) max(over i) |X_n,i| ---> 0 in probability,where
I(A) is the indicator function of the set A.Then sum(over i)
X_n,i ---> N(m,s^2) in distribution. I'm going tocall this
result R1.The problem is that I haven't been able to convince
myself that thisfollows from (or really is an equivalent
formulation of) any of theversion of the CLT for triangular
independent arrays (probablyLindeberg's) that I'm familiar
with.I'm guessing there's no loss of generality if we set m=0,
s^2=1. ThenI'm acquainted with the following:Let {X_n,i} be a
triangular array of random variables (independentwithin each
row). Suppose(1) E(X_n,i) = 0(2) Sum(i=1,...,k_n) E((X_n,i)^2)
= 1(3) Given epsilon > 0, Sum (i=1,...,k_n) E( (X_n,i)^2
I(|X_n,i| epsilon) ) ---> 0.Then Z_n = Sum(i=1,...,k_n) X_n,i
---> N(0,1) in distribution. I'mgoing to call this result
R2.Can someone help show me how R1 is equivalent to one of the
morepopular formulations of the CLT for triangular arrays (say
R2)? I'vetried doing various manipulations of the hypotheses
of R1 (i.e.flipping around the indicator functions to
I(|X_n,i| > epsilon) butI still can't put it in just the right
form. please help.....!===
===
Subject:
: Re: Yes, these are homework
questions! > I was going to ask my teacher, but ran out of
time... and as for working> these out myself, well there's
only so may times I can shake my fist at> them, I'm lost.>
Anyone care to help me out, please?> Express a in terms of b
and c> a=(b-a)/c> -- Don't have a clue where to begin.>
Variable a represents a tablevalue.> First determine the
weight of each variable.> b>c and b for b=(a*c)+a> c<b and a
for c=(b-a)/a> for c=1 b has to be as big as twice to a:>
(c*a)+a=b> or coming back to a=(b-a)/c:> (c*a)=b-a> In any
table c will be the multiplier to tablevalue a where b will
adept by> +a upgoing the table (from c=1 b=a+a, c=2 b=a+a+a
etc.).> Given a factor, find k> x^3-x^2+kx+n=0> k = (x^2-x)> I
don't see the point in n.I don't understand how you came about
k = (x^2-x), I got k = (-x^2+x).(removing n from the equation
because you saw no point in it)1. x^3-x^2+kx = 02. x(x^2-x+k)
= 03. x(x^2-x+k)/x = 0/x4. x^2-x+k = 0 -x^2 + x^25. k =
(-x^2+x)===
===
Subject:
: Re: Yes, these are homework questions!>
because it's so obvious now... can anyone recommend a book on
algebra - forNotice how you are dealing mostly with linear
things. Even yourpolynomial problem, could be split into
linear factors over anappropriate splitting field. So first of
all you need to get a bookon linear algebra. Since you are
dealing with polynomials, it isessential that you also get a
book on galois theory, which is thestudy of polynomials. Once
you finish those, you are ready to go alittle deeper, and you
should seek a book on abstract algebra.If these books seem a
little bit over your head (I'm not sure how goodyou are), you
might need to step back and start with I.N. Herstein'sTopics
in Algebra, which is written for folks like you gettingstarted
in math.Good luck in all your math endeavors!Your dear
friend,Bruce B.===
===
Subject:
: Re: Method of undetermined
coefficients>Yes, this is an assignment so I am not asking
anyone to solve it for>me. Here is my attempt at a solution
and I was just wondering if>somebody could let me know if I am
doing this correctly. If not,>perhaps a few pointers in the
right direction?>y-y'-6y=-8x>(i) find Yc:> characteristic
equation: r^2-r-6=0> r=-2,3> general solution to Yc is
Yc=C1e^-2t + C2e^3tThis should be Yc=C1e^-2x + C2e^3x. If ' is
differentiation w.r.t. tyou get different particular solutions,
since -8x is a constant then.It is essential, that you
understand that.....>(iii) general solution to y-y'-6y=-8x is
Y=Yc+Yp> Y=C1e^-2t+C2e^3t+4/3x+2/9There was a sign error in
your calculation. It should
be:Y=C1e^-2x+C2e^3x+4/3x+2/9.===
===
Subject:
: Re: Method of
undetermined coefficients>(iii) general solution to
y-y'-6y=-8x is Y=Yc+Yp> Y=C1e^-2t+C2e^3t+4/3x+2/9>There was a
sign error in your calculation. It should
be:>Y=C1e^-2x+C2e^3x+4/3x+2/9. ^^^^Oops!-2/9, of
course.===
===
Subject:
: Re: Method of undetermined
coefficientsThere is an algebra error in going from:>
0-A-6(Ax+B)=-8xto> 0-A-6Ax+6B=-8x .See
it?________________________________> Yes, this is an
assignment so I am not asking anyone to solve it for> me. Here
is my attempt at a solution and I was just wondering if>
somebody could let me know if I am doing this correctly. If
not,> perhaps a few pointers in the right direction?>
y-y'-6y=-8x> (i) find Yc:> characteristic equation: r^2-r-6=0>
r=-2,3> general solution to Yc is Yc=C1e^-2t + C2e^3t> (ii)
find Yp:> Yp=Ax+B> Yp'=A> Yp=0> 0-A-6(Ax+B)=-8x>
0-A-6Ax+6B=-8x> x(-6A)+(-A+6B)=-8x> solve the system of
equations:> -6A=-8x> -A+6B=0> A=4/3> B=2/9> general solution
to Yp is Yp=4/3x+2/9> (iii) general solution to y-y'-6y=-8x is
Y=Yc+Yp> Y=C1e^-2t+C2e^3t+4/3x+2/9===
===
Subject:
: Re: Method of
undetermined coefficients> Yes, this is an assignment so I am
not asking anyone to solve it for> me. Here is my attempt at a
solution and I was just wondering if> somebody could let me
know if I am doing this correctly. If not,> perhaps a few
pointers in the right direction?> y-y'-6y=-8x> (i) find Yc:>
characteristic equation: r^2-r-6=0> r=-2,3> general solution
to Yc is Yc=C1e^-2t + C2e^3t> (ii) find Yp:> Yp=Ax+B> Yp'=A>
Yp=0> 0-A-6(Ax+B)=-8x> 0-A-6Ax+6B=-8x-6B> x(-6A)+(-A+6B)=-8x>
solve the system of equations:> -6A=-8x> -A+6B=0> A=4/3>
B=2/9> general solution to Yp is Yp=4/3x+2/9> (iii) general
solution to y-y'-6y=-8x is Y=Yc+Yp>
Y=C1e^-2t+C2e^3t+4/3x+2/9===
===
Subject:
: Re: Help Needed
Understanding Article> As I said, I don't see anything in this
paper that proves this> theorem, so I would appreciate it if
anyone who is somehow convinced> of this theorem from reading
this paper would tell me how they came to> that conclusion.I
am convinced that there is no computable procedure for
determiningif any arbitrary program written in some general
purpose language canhalt. I am convinced of this because (1)
It is central to the theory of computability, from the little
I understand about computability. (2) I vaguely recall being
taught this in some class I took on computability 20 years
ago. (3) I vaguely remember the proof - really, proof outline
or rigorous argument - having something like the
self-reflexivity of the proof outline David Ulrich, I think it
was, presented recently on this newsgroup.> So far, nobody has
come forth with that argument. > Yet there are still people
who maintain that Boyer and Moore do> somehow prove it.
Without having some sort of logical explanation to> back up
that assertion, it seems pretty disingenuous to take that>
position.I see the title and abstract of this paper. I know
Boyer and Moorehave an absolutely stellar reputation in the
area of systems forformal proofs. There's lots of technical
details in the paper,which I have not read. So I am more than
willing to accept thattheir system has verified it.I don't
maintain that the paper must present a proof of
theunsolvability of the halting problem. I think the purpose
of thepaper is to describe how they used their system to get
the systemto generate or check a proof.By the way, I know that
some formal verification systems usedecision procedures in
restricted domains. The result of adecision procedure, as I
understand it, is not a formal proofshowing the steps in
deriving a theorem from an axiom. I do notknow if the
particular use of the Boyer-Moore theorem proverdescribed in
that paper can be criticized along these grounds.Notice I do
not assert that one who does not know the theoryof
computability should be convinced of the unsolvability ofthe
halting problem on the grounds I give above.Try
http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/
Bukharin.html To solve Linear Programs: .../LPSolver.htmlr c A
game: .../Keynes.html v s a Whether strength of body or of
mind, or wisdom, or i m p virtue, are found in proportion to
the power or wealth e a e of a man is a question fit perhaps
to be discussed by n e . slaves in the hearing of their
masters, but highly @ r c m unbecoming to reasonable and free
men in search of d o the truth. -- Rousseau===
===
Subject:
: Re:
Logistic Curve Goodness of fit> I am analysing some data and
have a simple linear equation and a> logistic/sigmoid
equation. To measure the Goodness of Fit for the> linear
equation I am using simple X:Y R^2 but am unsure of the best>
way to compare this goodness of fit to that of the logistic
equation.> Is a direct comparison of the Sum of Squares
accurate or is there a> better way.I guess it is not a bad
start.I can't help feeling that a real expert would come up
with a proper answer,but as none have come forward I will
offer my two penn'oth.As the parameter count is the same in
both models what comes to mind is thatif you use max log
likelihood to fit the parameters in both models you couldsay
which one more probably fitted the data.The other suggestion
is that for linear regression it is easy to get anestimate of
the error in the parameters. I expect there is a way of
directlyestimating the same for the logistic equation, if not
you can (with a bit ofwork) obtain estimates using max log
likelihood.If the relative error on both the parameters for
one model is smaller thanthe other then presumably this is
good news.> Any advice would be appreciated.>
Chris.===
===
Subject:
: Re: Mathematical Language Equations> Here is
a language equation puzzle where the solutions are
mathematical.> [So I will cross-post this to sci.math. ]> 1)
160 T of 187.this one is a bit hard to fathom. is there an
equation here?> 2) S of D of 70 is 144. Sum Divisors> 3) The
sixth H N is 49/20. Harmonic Number > 4) 12th F N is 12th
positive S. Fibonacci Number Square> 5) F of N L of 10000 is
9. Floor Natural Logarithm> 6) 8 C 3 is 56. Choose> 7) 10th T
N is 10th F N. Triangular Number Fibonacci Number> 8) S of 1st
5 positive S is 10th F N too. Sum Squares Fibonacci Number> 9)
(m+2) C 3 P (m+2) C 2 is S of 1st m positive S. Choose Plus
Choose Sum SquaresS's aren't always equal in an equation:
S=Sum or Square.> 10) For m >= 2, S of T of m is m T P m D B
2.???> 11) The N of P L T 50 is 15. Number Primes Less Than>
12) The P of 1st 4 H N is 275/48. Product Harmonic Numbers>
13) D I B 0 and 1 of -ln(1-x)/x is P S D B 6.Definite Integral
Between Pi Squared Divided By> 14) L, as m A I, of S of 1st m
positive C D B m R B 4 is 1/4. Limit Approaches Infinity Sum
Cubes Divided By Raised By> 15) (2m) F D B m F S is A to 4 R B
m D B S R of P.Factorial Divided By Factorial Squared
Asymptotic Raised By Divided By Square Root Pi