Hi guys
I'm reading a book about linear programming and network flows. In chapter 2 when it talks about convex sets and their analysis it talks about extreme points and extreme directions of a convex set. I understand the definitions of extreme points and extreme directions, but I don't know...
Homework Statement
Show that a differentiable function f is convex if and only if the following inequality holds for each fixed point x0 in Rn:
f(x) ≥ f(x0) + ∇tf(x0)(x-x0) for all x in Rn, where ∇tf(x0) is the gradient vector of f at x0.
Homework Equations
The Attempt at a...
This one isn't obvious to me at all. Actually the nature of these operations is quite different for me. One is a binary operation on G×G, the other is a unitary operation on G and so far in the sources I've read, with my limited knowledge, they require both the multiplication and inversion to be...
well, yea. That's what I initially thought too. But actually let's make it a bit more general. Imagine that instead of ℝ we're working in a general field F. F could be any field even with a non-zero characteristic like Zp.
Please check this proof...
Hi guys,
This is a general question that I'm thinking about now. Imagine that I've been given a set which is a group and we have defined a topology on it. how can I show that the group operation is continuous? Actually to begin with, how can I know if the group operation is really continuous...
Another question, is matrix multiplication continuous as well? If yes, how can I prove that? In general, how do we show that a function from G \times G \to G is continuous?
Yup. Sorry for the typo.
But I still need to know how much small it should be! and rearranging the inequality doesn't help, at least I can't see how it helps at this point :(
I hope that I'm not mistaken, but just start with the fact that |sin(x)|\leq|x| for all real numbers.
then prove that: |sin(x)-sin(a)| \leq |x-a|
you can also prove that |cos(x)-cos(a)| \leq |x-a|
That tells you why sin(x) and cos(x) is continuous. This solves the first and second problems...
Homework Statement
Hi guys,
I'm trying to prove that matrix inversion is continuous. In other words, I'm trying to show that in a normed vector space the map \varphi: GL(n,R) \to GL(n,R) defined by \varphi(A) = A^{-1} is continuous.Homework Equations
The norm that we're working in the...
So the title says everything. Let's assume R is a set equipped with vector addition the same way we add real numbers and has a scalar multiplication that the scalars come from the field Q. I believe the dimension of this vector space is infinite, and the reason is we have transcendental numbers...
Actually this is how it's going to get proved, I think you have done the same thing, maybe a little different:
By Wilson's theorem we know that
1 \times 2 \times ... \times \frac{p-1}{2} \times \frac {p+1}{2} ... \times (p-2) \times (p-1) \equiv -1 (mod p)
By what I had shown before we...
well, I give you a hint:
-1 \equiv p-1 (mod p)
-2 \equiv p-2 (mod p)
now, what is \frac{p-1}{2} congruent to?
Also, pay attention to the fact that if p is of the form 4k+1 (i.e p \equiv 1 (mod 4) ), then \frac{p-1}{2} is even.
Another thing that you need to remember is that x^2 \equiv 1...
well, I know that the professor who teaches Manifold I has her own book which is not famous. I have her lecture notes taught in the previous semester class, she first covers some basic definitions like local charts, coordinate functions, local coordinate system, atlases, maximal atlas...