Intereting Posts

Help needed with the integral of an infinite series
Proving that $e$ is irrational using these results
Is linear algebra more “fully understood” than other maths disciplines?
Principle of Transfinite Induction
How do you prove that a function $f \in L^1( \bf R)$ and its Fourier transform cannot simultaneously be very small at infinity?
Length of two sides in a quadrilateral with given angles
Holomorphic function $\varphi$ with fixed point $z_0$ such that $\varphi'(z_o)=1$ is linear?
An odd question about induction.
Series RLC Circuit Step Response
Geometric meaning of Cauchy functional equation
A proof about random variables
Evaluate $\int \cos(3x) \sin(2x) \, dx$.
Integrate and measure problem.
Secret Santa Combinatorics with couples
Order of elements modulo p

Let $u\in \mathcal{C}^1[a,b]$ be such that $u(a)=u(b)=0$. Show that

$$\int_a^b u^2(x)dx\leq (b-a)^2\int_a^b (u')^2(x)dx$$

using the Schwarz’s inequality.

- Convolution of compactly supported function with a locally integrable function is continuous?
- Prove $2^n > n^3$
- Integral eigenvectors and eigenvalues
- Show that if the sum of components of one vector adds up to 1 then the sum of the squares of the same vector is at least 1/n
- limit of a recursively defined function
- Why is the rational number system inadequate for analysis?

- Fourier transform of $f(x)=\frac{1}{e^x+e^{-x}+2}$
- If an IVP does not enjoy uniqueness, then there are infinitely many solutions.
- Prove: If the function $f$ is continuous on $$, differentiable on $(a,b)$ and $f'(x) = 0$ on $(a,b)$, then f must be a constant function
- Prove integral is greater than $0$
- Every Cauchy sequence in a metric space is bounded
- Proof that the limit of the square root is the square root of the limit
- complex conjugates of holomorphic functions
- What's $\sum{\frac{x^n}{n^3}}$?
- Doubling measure is absolutely continuous with respect to Lebesgue
- Compact multiplication operators

We have $|u(x)|\leq \int_a^x|u'(t)|dt\leq \sqrt{x-a}\sqrt{\int_a^b|u'(t)|^2}dt$ so

$$u(x)^2\leq (x-a)\int_a^bu'(t)^2dt$$ and integrating

$$\int_a^bu(x)^2dx\leq \frac{(b-a)^2}2\int_a^bu'(t)^2dt.$$

- Monty hall problem with leftmost goat selection.
- Mujica's “Complex analysis in Banach spaces” exercise 1.2.B
- Show that if $ar + bs = 1$ for some $r$ and $s$ then $a$ and $b$ are relatively prime
- Proving that a particular restriction of a projection is a quotient map
- How to explain brackets to young students
- Dyadic rational boundary points of the Mandelbrot set
- $\forall p\geq 3, E:y^2=x^3+x$ satisfies $\#E(\mathbb{F}_p)=0\mod4$
- Gaussian distribution on a $2$-sphere
- Radon–Nikodym derivative and “normal” derivative
- What is the Direction of a Zero (Null) Vector?
- Picking a $\delta$ for a convenient $\varepsilon$?
- The standard role of intuitive numbers in the foundations of mathematics
- Similar matrices and field extensions
- A closed form of the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$
- Relation between a generalization of Weil's abstract varieties and algebraic schemes