# Do mathematicians Switch Fields of Expertise?

However, it is much more common to start in field $X$, get interested in an adjacent field $X'$, then in an adjacent field $X''$, etc.If you define "area of expertise" narrowly, then all mathematicians change areas of expertise.

# Using the state transition matrix to recover the state matrix

I have a state transition matrix $\Phi(t,\tau) = \left(\begin{matrix} e^{-(t-\tau)} & t-\tau \\ 0 & 1+t(t-\tau) \end{matrix}\right)$.I am tasked to find the state matrix $A(t)$ that corresponds to it.

# Proof of the First Sylow Theorem in Herstein's $Abstract$ $Algebra$

I am reading the following proof of Sylow's First Theorem given in Herstein's Abstract Algebra:

Suppose $G$ is a group of order $p^n m$ where $p$ a prime and $p$ does not divide $m$.For the induction step he assumes that the result is true for all groups $H$ such that $|H| < |G|$.

# Group of units of finite type - related to the factorization of ideals

For $p \in P$ and $x \in K^{\times}$ write $v_{p}$ for the exponent of $p$ in the factorization of the $Ax$ into a product of prime ideals.Put $v_{p}(0) = + \infty$.

# 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.

# subadditive function of brownian motion

Let $W_t$ be Brownian, and let $g$ be integrable , and odd, and subadditive
$g(x+y)\le g(x)+g(y)$.How to show that $g(W_t)$ is a supermartingale?

# Corollary of Floquet Theorem

pdf

On page 18, in Corollary 2.24, I do not understand how
the line $y' = P^{-1}(AP - P'y)$ is obtained.

# 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 $n$.

# Roots of $f_n(x) = 1 + (1-x)^2 - (x+3)(1-x)^{n+1}$ in the interval $[0,1]$

Does the polynomial $f_n(x) = 1 + (1-x)^2 - (x+3)(1-x)^{n+1}$ have exactly one root in the interval $[0,1]$ for all non-negative integers $n$?It has at least one root because $f_n(0) = -1$ and $f_n(1) = 1$, but I am not sure if the root is unique.

# 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 $g$.

# Expectation of absorption time for a random walk which remains at n with probability 1/2

A random walk moves from k to k+1 with probability 1/2 and to k-1 with probability 1/2, except when k=n, in which case it remains at n with probability 1/2 and moves to n-1 with probability 1/2.Let T be the first time the path reaches 0.

# covariance of normal distribution

In particular, in your case if you knew the joint density function $f(x,y)$ of $X$ and $Y$, you could compute the covariance.Then $\text{Cov}(X,Y)=E(XY)-E(X)E(Y)=E(X)E(Y)-E(X)E(Y)=0$.

# probability - find the variance of an event X

How would you calculate the variance of $X$?Can we treat this problem as a binomial such that
$p$ = probability of landing on a $6$, $p = \frac{1}{6}$ and
$q$ = probability of not landing on a $6$, $q=\frac{5}{6}$.

# 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.

# Why do we categorize all other (iso.) singularities as “essential”?

When dealing with isolated singularities, we classify each of these points as removable, pole (of order $k$), or essential.Do we not care about essential singularities to classify them further?

# 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.

# Solving quadratic Diophantine equations: $5n^2+2n+1=y^2$ [duplicate]

This question already has an answer here:

How to find integer solutions to $M^2=5N^2+2N+1$?Start by completing the square on the left:
$$5\big( n+\tfrac15)^2 + \tfrac45 = y^2$$
Multiply by 5 to clear denominators:

(5n+1)^2 + 4 = 5y^2.