# All Questions

478 views

### Proof that every natural number is the sum of 9 cubes of natural numbers

What types of proof are there of this result and where can I read about it? I think that the Hardy-Littlewood circle method can prove that every number is the sum of something like $100000$ cubes, ...
203 views

### Morrey space and Campanato space.

I'd like to know a lot about Morrey space and Campanato spaces. For example, I'd like to know how can I see the details presents here. I'd like some reference about this. I thank you very much.
80 views

### Explain this statement $\bar 0 \in \partial f(x^*)$ where $\partial f(x^*)$ is subgradient

I haven't understood this theorem "$x^*$ is global minimum iff $\bar 0\in \partial f(x^*)$". What does it mean? Visually? P.s. Studying Nonlinear-optimization -course, 2.3139.
127 views

### Mertens' asymptotic formula for $\prod \left(1-p^{-1}\right)$ without constant

I've heard that there is an easy way to derive the asymptotic $$\prod_{p\le x} \left(1-\frac{1}{p}\right) \sim \frac{c}{\log(x)}$$ if one isn't interested in deriving $c=e^{-\gamma}$. I don't see how ...
58 views

### How to find the derivative with respect to the transformed co-ordinates.

I am stuck with something very simple , would be glad to get help . Suppose if i have a transformation matrix J , how do i find the derivative with respect to new co-ordinates , and derivative of ...
64 views

### Optimized Algorithm for Distance Matrix Solution

I've been looking for an optimized algorithm for solving a distance matrix (a hollow, skew symmetric matrix), but I haven't been able to find anything but papers discussing repopulating sparse ...
349 views

136 views

### Solving the initial value problem of a differential equation

Let $x′′- q(t) x = 0$, $0\le t \lt\infty$ , $x(0)=1$, $x'(0)=1$, where $q(x)$ is monotonically increasing continuous function, then what will be the solution?
417 views

### Proof of closure axiom necessary or is it implied by other axioms?

Sometimes I see books saying that you need to prove the closure axiom to decide if a set is a group. Other times I see books saying you only need to prove the identity, inverse and associativity ...
124 views

### Lie Groups question from Brian Hall's Lie Groups, Lie Algebras and their representations.

In page 60 of Hall's textbook, ex. 8 assignment (c), he asks me to prove that if $A$ is a unipotent matrix then $\exp(\log A))=A$. In the hint he gives to show that for $A(t)=I+t(A-I)$ we get ...
1k views

### Why are continuous functions not dense in $L^\infty$?

Why are the continuous functions not dense in $L^\infty$? I mean both concretely (i.e. a counter example) and intuitively why is this the case.
709 views

### General method for determining stability of equilibrium points

Given a system of ODEs, $\mathbf{x}' = A\mathbf{x}$, one way to determine the stability of an equilibrium point is to look at the eigenvalues of the Jacobian matrix. However, there are cases in ...
304 views

### Book on matrix computation

I'm taking a machine learning course and it involves a lot of matrix computation like compute the derivatives of a matrix with respect to a vector term. In my linear algebra course these material is ...
81 views

### What is the $P( |X-10| > 2)$ of a normal distribution when mean is 10, and standard deviation is 6?

I couldn't figure out this question: What is the $P( |X-10| > 2)$ of a normal distribution when mean is 10, and standard deviation is 6?
192 views

### Basic introduction to algebraic topology using simplicial complexes

What is a basic, graduate-level introduction to algebraic topology? I think Hatcher is a great book, but I want to learn the subject from the point of view of simplicial complexes. Primarily, I want ...
277 views

### For two uncorrelated random variables $X,Y$, why does $\rho(X+Y,2X+2Y)=4?$

Given two uncorrelated random variables $X,Y$ with the same variance $\sigma^2$ I need to compute $\rho= \frac{COV(X,Y)}{\sigma(X)\sigma(Y)}$ between $X+Y$ and $2X+2Y$. I know it should be a ...
58 views

### Write expectation

How can I write the folowing expectation $E[f(X_t,X_s)]$ by means of a Lebesgue integral and the density of $X_t$? where $f$ is a "nice" function and $X_t$ is a process without undependent increments! ...
Definition: $n$ is a psuedo-square if the legendre symbol $(\frac{n}{p}) = 0$ or $= 1$ for $p = 3, 5, 7, 11$. I want to find the probability of $n$ being a square or psuedo-square. I know that any ...
### geometric charaterization of complex interpolation spaces $(H,Y)_\theta$ where $H$ is a Hilbert space?
Let $C$ be the class of Banach spaces $X$ such that there exists $0<\theta<1$, a Hilbert space $H$ and a Banach space $Y$ such that $$X=(H,Y)_\theta$$ (complex interpolation of Calderon). ...