Intereting Posts

$\operatorname{func}(f)\to((x,y)\in f\to f(x)=y) $ and $\operatorname{func}(f)\to((x \in \operatorname{dom}(f) \wedge f(x)=y)\to (x,y)\in f)$
Why are punctured neighborhoods in the definition of the limit of a function?
Surjective Function from a Cantor Set
How do you prove $\def\rank{\operatorname{rank}}\rank(f_3 \circ f_2) + \rank(f_2 \circ f_1) \leq \rank(f_3 \circ f_2 \circ f_1) + \rank(f_2) $?
Why does this pattern of “nasty” integrals stop?
Fun but serious mathematics books to gift advanced undergraduates.
What is… A Parsimonious History?
Calculate limit with summation index in formula
How many ways to arrange people on a bench so that no woman sits next to another woman?
Does $\frac{(30n)!n!}{(15n)!(10n)!(6n)!}$ count something?
Find the Theta class for the recursion $T(n) = T(3n/4) + T(n/6) + 5n$
How to show that $X_t = \frac{1}{\left| B_t -x\right|}\mathbb{1}_{\left\{ B_t \neq x\right\}}$ (“inverse brownian”) is a martingale?
If $f\in L^1(\mathbb{R})$, then $\sum_{n\ge 1}f(x+n)$ Converges for a.e. $x$.
Probability distribution of sign changes in Brownian motion
What does actually probability mean?

Given $A$ and $B$, $n\times n$ complex matrices. If $\langle x,y\rangle =y^{*}x$ for all $x,y\in \mathbb C^{n}$, then the following are equivalent:

(1) $\langle Ax,y\rangle=\langle Bx,y\rangle$, for all $x,y\in \mathbb C^{n}$.

(2) $\langle Ax,x\rangle=\langle Bx,x\rangle$, for all $x,y\in \mathbb C^{n}$.

- Why are every structures I study based on Real number?
- Is $\|x\| = \| \overline{x} \|$ in an inner product space?
- Maximizing and minimizing dot products
- Proving a given formula for projection matrix
- $||u||\leq ||u+av|| \Longrightarrow \langle u,v\rangle=0$
- Isometry group of a norm is always contained in some Isometry group of an inner product?

(1) implies (2) is easy, how to prove (2) implies (1)?

- Show that $T$ is normal
- What does orthogonality mean in function space?
- Orthogonal in inner product space
- Geometrical or Physical significance (interpretation) of the inner-product $\langle A,B \rangle := Trace (AB^t)$ over $M_n(\mathbb R)$
- What is the geometric meaning of the inner product of two functions?
- Scalar Product for Vector Space of Monomial Symmetric Functions
- Inner product and norms for random vectors
- Orthonormal basis
- Derivation of the polarization identities?
- A map which commutes with Hodge dual is conformal?

Hint: Use the algebraic properties of the inner product to expand each side of the following equations:

- $\langle A(x+y),x+y\rangle=\langle B(x+y),x+y\rangle$
- $\langle A(x+iy),x+iy\rangle=\langle B(x+iy),x+iy\rangle$

- Prove that there's no fractions that can't be written in lowest term with Well Ordering Principle
- What is the smallest square into which one can pack a trisected disc?
- What is a real world application of polynomial factoring?
- Neumann series in an incomplete normed algebra
- $\epsilon $ $\delta$ proof of geometric series $x$<1
- Differential forms turn infinitesimal stuff rigorous?
- Show $ \frac{\cosh(a( \pi -x))}{\sinh(a \pi)} = \frac{1}{a \pi} + \frac{2}{\pi} \sum_{n = 1}^{\infty}\frac{a}{a^2+n^2}\cos(nx). $
- Sine of natural numbers
- matrix representations and polynomials
- What are sharp lower and upper bounds of the fast growing hierarachy?
- Evaluate $\sum\limits_{k=1}^{\infty}\frac{(18)^2}{(2k)!}$
- Can this be counted with stars and bars method?
- The least value of the function $f(x)=|x-a|+|x-b|+|x-c|+|x-d|$
- A family having 4 children has 3 girl children. What is the probability that their 4th child is a son?
- Integral of square of Brownian motion with respect to Brownian Motion