Note: All integrals are taken with respect to Lebesgue measure. The symbol $\def\avint{\mathop{\rlap{\raise.15em{\scriptstyle -}}\kern-.2em\int}\nolimits} \avint$ denotes the average integral.
We say ...
The Pareto principle says that the top 20% of wealthy people people hold over 80% of the wealth. Suppose we had a non-negative function on $\mathbb R^n$ that satisfied this principle on every open ...
In an recent test I was asked to evaluate the integral
$$ \int_0^1 \frac{\sqrt[3]{x^2(1-x)}}{(1+x)^3} \text{d}x$$
in 8 minutes, but I didn't have a clue what to do with it.
After the test, I tried the ...
Say real-valued function $f(x,y)$ is everywhere differentiable, consider the function $g(y)=\underset{x}{\max}f(x,y)$, I think this funcion is also everywhere differentiable, but how to prove it? And ...
I need counter for this statement :
If the series $ \sum a_n$ is divergent then the series $b_n$= $\sum \text{min}(a_n,\frac{1}{n})$ is also divergent.
I closest I reached to a counter is ,
Define ...
I want to prove that two polynomial functions that are equal over a specific interval, $(a, b) \in \mathbb R$ (closed interval with more than one point if that condition is necessary) are equal over ...
Let $\sum a_n$ be a conditionally convergent sum of real numbers, and $\epsilon_n$ a sequence of independent identically distributed Bernoulli random variables with $\epsilon_n = 1$ or $-1$ with ...
Let $a_n$ be a sequence of positive real numbers with $\sum a_n < \infty$. What are the necessary and sufficient conditions for the following to hold?
For any $S \in \mathbb R$ with $-\sum a_n ...
I stumbled upon these 4 limit/integral identities involving Euler's constant aka gamma (~0.5772). They appear to be valid based on inspection but I have no idea how to prove them. In addition, I have ...
In Stein's Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, he defines tempered distribution ($\mathscr S'$) as continuous linear functionals from the Schwartz class. ...
It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
We say that $g(x)\sim h(x)$ when $x\to x_0$ if
$$
\lim_{x\to x_0}\frac{g(x)}{h(x)}=1,
$$
so, for example, $x\sim\sin(x)$ when $x\to0$.
I would like to find a function $f$ such that $f(x)\sim ...
Let $\mathcal{A}$ be some uncountable index set and $X$ be some separable reflexive Banach space.
For each $\alpha \in \mathcal{A}$, let
\begin{equation}
\{ x_n^{\alpha} \}_{n=1}^\infty
\end{equation}
...
I was motivated for this question while seeking for a new sorting algorithm.
Suppose a continuous function $f : [a, b] \to \mathbb{R}$ is given. I wanted to define the sorted version $g$ of $f$, which ...
Let $U\subseteq \mathbb{R}^n$ be an open convex set. Suppose that, for any direction $\nu$ (i.e. every unit vector $\nu\in\mathbb{R}^n$) there exists $x_\nu\in U$ such that
$$
\nu\cdot x_\nu > 0.
...
Define the set
$$
A = \{ x \in \mathbb{R} \, \vert \, \forall M \in \mathbb{R}: \exists p, q \in \mathbb{Z}: p> M \text{ and }0 < \vert x-\frac{q}{p} \vert < \frac{1}{p^{3}} \}.
$$
Is this ...
How would I go about solving $$\sum_{n=1}^{\infty} \frac{(n-1)x^{n-1}}{(n-1)!}$$ So far I have tried to of course consider the exponential power series, but I seem to get negative factorials.
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
Proofs that the Intermediate Value Theorem (IVF) implies the Least Upper Bound Property for an ordered field usually use a continuous function that is not uniformly continuous like here ...