Choosing the Permutation Which Maximizes a Weighted Sum
ProveConsider ordered real numbers $x_1 \le x_2 \le \dots \le x_n$ and $y_1 \le y_2 \le \dots \le y_n$. Let $ \sigma : \{ 1,2,\dots,n\} \rightarrow \{ 1,2,\dots,n\}$ be a permutation on the integers...
View ArticleWhat is the boundary $ \partial ([a,b] \times [a,b])$ and how can I write it...
What is the boundary of $ \partial ([a,b] \times [a,b])$?I know that $ \partial ([a,b] \times [a,b])=\partial [a,b]^2=\overline{[a,b]^2}\setminus ([a,b]^2)^\circ=[a,b]^2\setminus (a,b)^2$Visually the...
View ArticleConstructing the interval [0, 1) via inverse powers of 2
If $x$ is rational and in the interval ${[0,1)}$, is it always possible to find constants $a_1, a_2, ..., a_n\in\{-1, 0, 1\}$ such that for some integer $n\geq{1}$, $x = a_1\cdot2^{-1} +...
View ArticleIf f is integrable, is it finite almost everywhere?
If $\int_\Omega f d\mu<\infty$, and $f$ is non-negative, can we conclude that $f$ is finite a.e. on $\Omega$?Is being finite a.e. the same as having a finite essential supremum, i.e. there exists an...
View ArticleIs there a function whose maximizers remain the same after any affine...
Let $f: \mathbb{R_+}^n\to \mathbb{R_+}$ be a function that is strictly increasing in each of its arguments. Let $M_f$ be the set of its maximizers on some fixed compact subset $D\subseteq...
View ArticleReference Request: Hausdorff–Young inequality for the inverse Fourier seires
Let $ \hat f : \mathbb Z^d \to \mathbb C $ denote a function in $\ell^p(\mathbb Z^d)$ where $p \in [1,2]$.Let $f : \mathbb T^d = (\mathbb R / 2\pi \mathbb Z)^d \to \mathbb C$ denote the inverse Fourier...
View ArticleIntegration of the product of a compact supported convolution
I know that in general case we have $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(s)g(t-s) ds dt = \left( \int_{-\infty}^{+\infty} f(t) dt \right) \left( \int_{-\infty}^{+\infty}g(t) dt \right)...
View ArticleThe function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ defined by...
Let the function $d \colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ be defined by$$d\big( (x, y) \big) := \lvert x-y \rvert \qquad \mbox{ for all } (x, y) \in \mathbb{R}^2. $$Let $\mathbb{R}$ and...
View ArticleIf $f$ and $\frac{d^2}{dx^2}$ are continuous, then $\frac{d}{dx}f$ is...
Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a function such that both $f$ and $\frac{\partial^2 f}{\partial x^2}$ are continuous on $\mathbb{R}^2$. Is then $\frac{\partial f}{\partial x}$ also...
View ArticleTaylor's Theorem for functions on the Rational Numbers
I have been looking for different proofs of the Taylor's Theorem with Peano form of the remainder. But all proofs I have found use some form of the Mean Value Theorem or L'Hôpital's Rule (which also...
View ArticleFekete's lemma with an extra constant in the index
Say I have an "almost" subadditive sequence $(a_n)_{n\in\mathbb{N}}$ in the sense that$$a_m+a_n\geq a_{m+n+k}$$for some fixed $k\in\mathbb{N}$. Then do I have that the limit...
View ArticleSmallest value of $k$ for which a function approaches $0$ as $x$ goes to...
I was playing around with factorials on desmos and trying to find some inequality between $x!$ and $\sqrt[x]{x}!$. After a bit I formulated the following question:What is the smallest value of $k$ such...
View ArticleEvaluate $\lim_{n \rightarrow \infty} \int_{-n}^n f(1+\frac{x}{n^2}) g(x) dx$
I want to evaluate $\lim_{n \rightarrow \infty} \int_{-n}^n f(1+\frac{x}{n^2}) g(x) dx$, where $g: \mathbb{R} \rightarrow \mathbb{R}$ is (Lebesgue)-integrable, and $f:\mathbb{R} \rightarrow \mathbb{R}$...
View ArticleIf $f$ and $\frac{\partial^2 f}{\partial x^2}$ are continuous, then...
Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a function such that both $f$ and $\frac{\partial^2 f}{\partial x^2}$ are continuous on $\mathbb{R}^2$. Is then $\frac{\partial f}{\partial x}$ also...
View Articlefractional power function inequality
I am facing an issue to prove if its true the following inequality and any help is greatly appreciate. I used this inequality $x^\alpha \le \alpha x+1-\alpha$ but still I could not get the result.Let...
View ArticleSubset of index that minimizes a sum of real values
Given a series of real numbers $c_1, \ldots, c_n$ with $ n \in \mathbb{N} $, is there an algorithm or method to find the subset of indices such that the absolute value of the sum of the values within...
View ArticleDiophantine approximation and asymptotic for $\dfrac{1}{\sin(n\pi\sqrt{3})}$
I have this exercise and i proved the lemma but i couldn't use it to prove the asymptotic formula i couldn't plug the sin function into the inequality because it change variations maybe some choice of...
View ArticleNumber of times a continuous function changes sign in an interval
This is problem 7.23 in Apostol, T., Mathematical Analysis, 2nd edition, Pearson, 1974.Suppose $f$ is continuous on $[0,a]$. Let $f_0(x)=f(x)$ and$$...
View ArticleDerivative of singular integrand
I am trying to differentiate this integral with respect to $x$:$$T(x,t) = {1\over\sqrt\pi} \int_0^t {g(s) \over \sqrt{t-s} } e^{-{x^2\over 4(t-s)}} ds$$According to this paper the derivative with...
View ArticleAsymptotic expression around $x=\infty$ of Infinite sum of the exponential...
I ran across the following conjecture, which I checked numerically and seemed to check out. For $x\to \infty$ the sum$$\Theta(x)=\sum_{i=1,3,5..} \text{Ei}{\left(-\frac{i^2\pi^3}{4 x^2}\right)},$$has...
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