Culture & Recreation

Calculating the Logarithm of a Non-Diagonalizable Matrix

The only non-diagonalizable example we covered in class were of the form

$\lambda I + N$, where $N^r = 0$ for some positive integer $r$, then we used the formula

$\log(\lambda I + N) = \log(\lambda I) + \sum\limits_{n=1}^{r-1}\frac{(-N)^{n}}{j\lambda^{j}}$.The Jordan normal form of a complex matrix can be written as a "block diagonal" matrix, where each block on the diagonal is of the form $\lambda I + N$ in your form.

Non-linear recurrence problem

How to convert $p_n$ to an expression in terms of $n$ if $3p_{n-1}^2 - p_{n-2}=p_{n}$ and $p_0=5, p_1=7$?As said by others, there is no closed form expression of $p_n$ for general $n$.

Little 'o' / Big 'O' Definitions

in "Manifolds, Tensor Analysis and Applications" define a "little o" function as any continuous function $f:E\rightarrow F$ such that
\lim_{x\rightarrow 0}\frac{f(x^k)}{|x|^k} = 0

Is there any hope of reconciling these definitions?The objective of the $o$ and $O$ notations is to compare growth (asymptotic behavior) of 2 functions $f$ and $g$.

How do we check Randomness? [duplicate]

So to test if a string is random, we only have to brute-force check the length of the shortest Turing-program that outputs the first n bits correctly.If the length eventually becomes proportional to n, then we can be fairly certain we have a random sequence, but to be 100% we have to check the whole (infinite) string.

Difference between Bellman and Pontryagin dynamic optimization?

Can someone please explain the difference between dynamic optimization via the Bellman equation and dynamic optimization via Pontryagin's maximization principle?Thanks

The Bellman principle poses an optimization problem using a nonlinear 1st order partial differential equation - the object being optimized is a function.

A Hunt for a Mathematical Machine That Gives Points

The central question is :

Is there any method for Producing the global Points on the curve (any cubic curve, or at least a Degree-2 curve ) , if we have local Part with us?So let us call the set of points over local field as $C(\mathbb{F_P})$ for each prime $P$ , and let us call that set as local set.

Distribution of the second largest event

Consider M events that are all independent and poisson distributed in occurence with individual frequencies $\{\lambda_{k}\}_{k=1}^{M}$.Once they occur, they occur with a certain severity, event $k$ has severity distribution $F_{k}(T)$.

Stiefel Manifold

Am I right that Stiefel manifold $V_k(\mathbb{R}^n)$ (set of all orthonormal k-frames in $\mathbb{R}^n$) homeomorphic to $O(n)/O(n-k)$?This is most easily done by considering it as a subset of the $n \times k$ matrices, namely those for which $A^\top A = I_k$ is the $k \times k$ identity matrix.