If $f(x,y)=e^{-xy}\sin x\sin y$, $\int_0^\infty\int_0^\infty f(x,y)dxdy$...
Let $f(x,y)=e^{-xy}\sin x\sin y$ for $(x,y)\in\mathbb{R_{\geq 0}^2}$ and $f(x,y)=0$ wherever else.I want to prove that$$\int_{\mathbb{R}}\int_{\mathbb{R}} f(x,y)dxdy$$exists but$$\iint_{\mathbb{R}^2}...
View ArticleShow that $\sum_{j \in J} \sum_{i\in I_j}f(i) = \sum_{i\in I}f(i)\cdot...
Let I be a finite indexing set and $f :I \rightarrow \mathbb{R}$ be a function.We know that if $(I_j)_{j \in J}$ is a partition of I, then $\sum_{j \in J} \sum_{i\in I_j}f(i) = \sum_{i \in I}f(i)$I'm...
View ArticleWhy would you need to define a series as an ordered pair?
I've seen a series defined formally as an ordered pair of sequences $(a_n,s_n)$, such that $s_0=a_0$ and $s_{n}=s_{n-1}+a_n$. I was wondering why would you define it this way, like what is the...
View ArticleFind the value of integration: $\lim_{n\to\infty}...
$$\lim_{n\to\infty}\int_0^{\infty}{\frac{x}{1+x^{n}}\,\mathrm dx}$$Consider the integral$$\int_{0}^{\infty} \frac{x}{1+x^n} \,\mathrm dx$$This is an improper integral, but we can solve it using...
View ArticleFirst learn definition of limit of a sequence or a function?
I am having a problem in understanding the $\epsilon$ - $\delta$ definition of limit of a function. Actually, I understand what the definition says perfectly, that, no matter how close we get to $L$ if...
View ArticleInduction proof for floor function composition
Given the following function:$$P(x)=\left\lfloor\frac{x}{2}\right\rfloor$$I want to prove the following simplification:$$\underbrace{P(P(\cdots P(x)\cdots))}_{l \text{...
View ArticleProve that $g(x)=0$ for all $x$ in $\mathbb{R}$
Suppose that the function $g:\mathbb{R}\rightarrow\mathbb{R}$ is continuous and that $g(x)=0$ if $x$ is rational. Prove that $g(x)=0$ for all $x$ in $\mathbb{R}$.Proof:Let $\{x_n\}$ be a sequence of...
View ArticleUnderstanding some concepts regarding metric spaces.
I already know the base definition and 4 axioms. The things I don't know are completeness, compactness, and connectivity.Also, in the example:$\gamma: [0, 1]\to D$, $\gamma(0) = z_1, \gamma(1) = z_2$...
View ArticleUnderstanding Folland's definition of two complex measures being mutually...
In Folland's "Real Analysis", two complex measures $\nu=\nu_r + i\nu_i$ and $\mu=\mu_r + i\mu_i$ are said to be mutually singular (in symbols: $\nu\perp \mu$) if "$\nu_a \perp \mu_b$ for $a,b = r,i$"....
View Articleshow that if set B is symmetric, that is B=-B, then $\bigcup_{n=1}^\infty...
I’ve been trying to proof the following:If the set $B\subset R$ is symmetric, that is B=-BLet $B_n=(-n,n)$ with $n \in \mathbb{R^+}$$\bigcup_{n=1}^\infty B_n$ is also symmetricI’ve tried to work with...
View ArticleRiemann integrability of the composition of an integrable multivariate...
$f(\boldsymbol{x})$ is Riemann integrable on a volume-measurable bounded closed domain $\Omega\subset\mathbb{R}^n$, and $\boldsymbol{u}(\boldsymbol{x})$ is a $C^1$ homeomorphism mapping from a...
View Articlepositive eigenfunctions of elliptic PDE
I was wondering the following: Suppose one has a divergence free vector field $u$, and is interested in solutions to the elliptic PDE$$ \lambda f+u \cdot \nabla f=\Delta f$$where $Re(\lambda)>0$. Is...
View ArticleWhy $f(x)=x^2$ and $f(x)=|x|^2$ are not equal [closed]
Describe by mapping is $f(x)=|x|^2$ and $f(x)=x^2$ are equal.
View ArticleExtreme points of the unit ball in $L_p$
Definition of extreme point: A point $f$ in $L$ with $\|f\| = 1$ is called an extreme point of the unit ball if $$f = (1 - \alpha)f_1 + \alpha f_2, \, 0 < \alpha < 1, \, \|f_1\|, \|f_2\| \leq 1...
View ArticleUnder what conditions on $f(x)$ is $g(x)=(x+a)f(x)$ unimodal?
$f(x)$ is assumed positive and decreasing. What additional conditions are necessary for $f(x)$ to satisfy so that $g(x)=(x+a)f(x)$ is unimodal?($a$ is a constant)
View ArticleBartle Elements of Integration Exercise 7W
I have solved problem $7W$ in a particular case when $\varphi$ is uniformly continuous and I wonder if there are any counterexamples when uniform continuity isn't satisfied.7.W. If $f_n$ converges to...
View ArticleWhat is the topological classification of spaces with non-constant cauchy...
To be clear, I'm not asking about completeness of the space, just if there is some classification that can be placed on spaces where there exists a Cauchy sequence that is (eventually) non-constant....
View ArticleIf the sum of the partial distances is convergent, prove the sequence is Cauchy.
I am trying to prove that, given a sequence $(x_n)$ such that$$ \sum_{n=1}^{\infty} d(x_n, x_{n+1}) < \infty$$then the sequence is a Cauchy sequence. So far this is what I have:Let $S_n =...
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Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition...
Why the derivatives $f^{(n)}(x)$ of Flat functions grows so fast? (intuition behind)In this other question I did about Bump functions, other user told in an answer that these kind of functions "tends...
View ArticleNumber of unit vectors in $\mathbb{R}^n$ with sparsity s within an epsilon net
It is well-known that the size of an $\varepsilon$-net in $\mathbb{R}^n$ is upper bounded by $(1+2/\varepsilon)^n$. What if I only wanted to counted the number of elements within this net which is...
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