Math Is Fun Forum

I've only glanced at the algorithm, but it seems like it will terminate in a finite number of steps whether or not the matrix is positive definite.  Assuming this to be true, the algorithm can be used as a check.

You're assuming it's possible.  Another option is to use the Cholesky decomposition algorithm.  A symmetric matrix is positive definite if and only if it has a Cholesky decomposition, and there exists an algorithm for computing this.  Use the algorithm, and if it blows up somewhere (i.e. division by zero or a certain condition is not met like A^(n) = I), then the matrix must not be positive definite.

Off hand, I know of no easy computable equivalent condition on positive definite matrices.  You could use the criteria and if that fails then check the hard way.

That integral does not exist as a Riemann integral.  We can take the limit as Δx goes to zero by choosing all rationals for Δx, in which case each term in the sequence is 1, or we can take the limit by choosing Δx to be irrationals, in which case each term in the sequence is 0.  This prove the limit does not exist.

Now that is using the calculus definition of the integral, rather than the analysis definition.  The analysis definition will also not exist.

The integral is however zero when you consider it as a Lebesgue integral.  Lebesgue integration works like this: come up with a lower approximation for the integral using functions you can compute the integral of, and then call the highest one of those the value of the integral.  We make these approximations using simple functions which are functions which take on finitely many values.  The integral of a simple function f is

Where a_i is a value the simple function takes, E_i = f^{-1}(a_i), and m(E_i) is the measure of E_i.  This uses a bit of a fancy definition, but in this case you're helped a bit by the theorem that the measure of any countable set is 0, and m(A U B) = m(A) + m(B) when A and B are disjoint.  This means the measure of the rationals in [0,1] is 0, and so your function has integral 0.

Before doing any work on proving the statement, make sure you understand it first.  The statement on the left reads:

If p(x) is true, then at least one of q(x) and r(x) is false.

It should be intuitive that this means at least one of {p(x), q(x), r(x)} is false, which is the statement on the right.

Your first statement is wrong.  When you negate a statement "a or b" using DeMorgan, it becomes "not a and not b".  You negated "a or b" to "not a or b".

It's actually pretty easy to prove the generalization: If f is a continuous map from a compact space X, then f(X) is compact.

calccrypto, you got a half of what he is asking.

There are two parts to making a program look like something "I might download off the internet":

1. Run natively (i.e. an executable)
2. Graphic user interface (GUI)

For the first, it all depends upon which implementation of Python you are using.  If you are using it as a scripting language (e.g. the implementation CPython), then the short answer is: No.   Scripting languages, by definition, require an interpreter to run the code.  This is what I believe you're doing.

But Python is also implemented as a compiled language: you compile code into an executable that you can then run on the machine itself.  Jython is one example of such an implementation.  It actually takes Python code, compiles it into Java byte code, and then you can use that on a machine.  If you go this route, then you can actually use Java GUI libraries to make a GUI.

For the second, you need an external library to interact with the operating system's graphical API.  For Python, check out this site.

It would be much more beneficial if you wrote what you thought was rigorous, and let us critique it.

Is that a mutually exclusive or?

You're having trouble because the proposition is false.  Let f = e^x, then

Which certainly does not fit the conclusion.

If you're not using this for programming, then

works.

And once again you sound like you're talking about in general.  And even if you only meant it referring to this problem (I don't see how...), it's a ridiculous statement to make.

Were you really of the opinion that I thought a brain dead person could program?  Obviously there is less thinking when one uses a brute force method, that's why it's called brute force.  This is the difference we're talking about.