Intereting Posts

Isomorphic representations on exterior powers
Do groups, rings and fields have practical applications in CS? If so, what are some?
How to prove consistency of Natural Deduction systems
Derivation of binomial coefficient in binomial theorem.
Calculate $\int_0^\infty {\frac{x}{{\left( {x + 1} \right)\sqrt {4{x^4} + 8{x^3} + 12{x^2} + 8x + 1} }}dx}$
Most even numbers is a sum $a+b+c+d$ where $a^2+b^2+c^2=d^2$
How to find $\int_{0}^{1}\dfrac{\ln^2{x}\ln^2{(1-x)}}{2-x}dx$
How to find $\int\frac{\sin x}{x}dx$
Is the set of all conformal structures on $\mathbb{R}^n$ a manifold? Does it have a name?
Limit of Integral of Difference Quotients of Measurable/Bounded $f$ Being $0$ Implies $f$ is Constant
$f\colon\mathbb R^n \rightarrow \mathbb R$ be a linear map with $f(0,0,0,\ldots,0)=0.$
Finding the basis of a null space
How many surjective functions are there from $A=${$1,2,3,4,5$} to $B=${$1,2,3$}?
Show that $\int_0^\infty \frac{x\log(1+x^2)}{e^{2\pi x}+1}dx=\frac{19}{24} – \frac{23}{24}\log 2 – \frac12\log A$
Does convergence in $L^{p}$ implies convergence almost everywhere?

What’s wrong with this argument?

Let $f_n$ be a sequence of functions such that $f_n \to f$ in $L^2(\Omega)$. This means $$\lVert f_n – f \rVert_{L^2(\Omega)} \to 0,$$ i.e.,

$$\int_\Omega(f_n – f)^2 \to 0.$$

Since the integrand is positive, this must mean that $f_n \to f$ a.e.

Why is this not true? Apparently this only true for a subsequence $f_n$ (and in all $L^p$ spaces).

- Classifying the compact subsets of $L^p$
- Accumulation points of $\{ \sqrt{n} - \sqrt{m}: m,n \in \mathbb{N} \}$
- Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$
- Evaluate $ \int_0^\pi \left( \frac{2 + 2\cos (x) - \cos((k-1)x) - 2\cos (kx) - \cos((k+1)x)}{1-\cos (2x)}\right) \mathrm{d}x $
- Estimating rate of blow up of an ODE
- Is there a difference between allowing only countable unions/intersections, and allowing arbitrary (possibly uncountable) unions/intersections?

- An exercise on liminf and limsup
- A derivation of the Euler-Maclaurin formula?
- Compact form of the series $\sum\limits_{n=-\infty}^{\infty} {1\over (x-n)^2}$
- Integration of $\ln $ around a keyhole contour
- Find polynomials such that $(x-16)p(2x)=16(x-1)p(x)$
- Alternating roots of $f(x) = \exp(x) \sin(x) -1$ and $\exp(x)\cos(x) +1$
- the elements of Cantor's discontinuum
- Adjoint of an operator on $L^2$
- Why call this a spectral projection?
- A simple way to evaluate $\int_{-a}^a \frac{x^2}{x^4+1} \, \mathrm dx$?

Consider the following sequence of characteristic functions $f_n \colon [0,1] \to R$ defined as follows:

$f_1 = \chi[0, 1/2]$

$f_2 = \chi[1/2, 1]$

$f_3 = \chi[0, 1/3]$

$f_4 = \chi[1/3, 2/3]$

$f_5 = \chi[2/3, 1]$

$f_6 = \chi[0, 1/4]$

$f_7 = \chi[1/4, 2/4]$

and so on.

Then $f_n \to 0$ in $L^2$, but $f_n$ does not converge pointwise.

- Combining rotation quaternions
- Generators for $S_n$
- About finding the function such that $f(xy)=f(x)f(y)-f(x+y)+1$
- Calculate Point Coordinates
- Degeneracy in Simplex Algorithm
- Fiber bundles with same total spaces, but different base spaces
- When does a SES of vector bundles split?
- What is the inverse cycle of permutation ?
- Is $AA^T$ a positive-definite symmetric matrix?
- Evaluate limit $\lim_{x \rightarrow 0}\left (\frac 1x- \frac 1{\sin x} \right )$
- Show using the definition, that f is differentiable at $x_0+i0$
- Find the upper bound of the derivative of an analytic function
- Famous puzzle: Girl/Boy proportion problem (Sum of infinite series)
- Closed form for definite integral involving Erf and Gaussian?
- There is a $3\times 3 $ orthogonal matrix with all non zero entries.?