condition to apply integration by parts?
I have two real functions $f, g$ defined on $[a,b]$. Both functions are differentiable on the open interval $(a,b)$.Is this enough to apply integration by parts$$\int_{a}^b f(x) g(x) dx = f(b) G(b) -...
View ArticleShow $f$ is integrable if and only if...
This is a measure theory qualifying exam question: Suppose $f$ is a bounded nonnegative function on $(X,\mu)$ with $\mu(X)=\infty$. Show that $f$ is integrable if and only...
View ArticleDoes vanishing of wronskian of solutions at point $\implies$ solutions are...
Let $u$ and $v$ be two solutions of $y''+P(x)y'+Q(x)y=0$,Let $W(u,v)$ denote the wronskian of $u$ and $v$ then $W(u,v)$ vanishes at a point $x_0\in[a,b]\implies u$ and $v$ are linearly...
View ArticleApproximating $\log x$ by a sum of power functions $a x^b$
Let's approximate $\log x$ on the interval $(0,1)$ by a power function $a x^b$ to minimize the integral of the squared difference$$\delta_0(a,b)=\int_0^1\left(\log x-a x^b\right)^2dx.\tag1$$It's easy...
View ArticleHow to tell what is actually the derivative of a piecewise function
For example, if I have$f(x)= \begin{cases} x & \text{if } x\neq0, \\ 0 & \text{if } x = 0. \end{cases}$and I take the derivative:$f'(x)= \begin{cases} 1 & \text{if } x\neq0, \\ 0 &...
View ArticleMaximum value of $f(x, y, z) = yz + xz + xy − 2xyz$ for $x, y, z \ge 1$
I know I should try to explore a limit such as $\lim_{(x,y,z)\rightarrow(\infty,a, b)} f(x,y,z)$ where $a,b$ are some constants. Is it true that I can directly replace $y$ and $z$ in the limit by way...
View ArticleUnderstanding ordered fields and the subset $P \subseteq \mathbb{F}$ of...
I'm following Real Analysis: A Long-Form Textbook (Jay Cummings) and there is a part about defining the positive set $P \subseteq \mathbb{F}$.The following definitions are given:An ordered field is a...
View ArticleHow to use local approximation spaces to build a global space via the...
I'm working to understand how the partition of unity method is used to build a global PUM space from local approximation spaces. Can someone please explain the mechanics of gluing the local spaces...
View ArticleGeneralized Sobolev-Morrey Embedding Theorem on Unit Sphere
By generalized Sobolev-Morrey embedding theorem, for U an open subset of $\mathbb{R}^n$ with $C^1$ boundary and $k>\frac{n}{p}$, $W^{k,p}(U)$ can be embedded in $C^{k-[\frac{n}{p}]-1,\gamma}(U)$,...
View ArticleTrying to solve an exercise on double integration
I have a problem with the following integral:$$\iint_\Omega y^2 dxdy,$$where$$\Omega=\{(x,y) \in \mathbb{R}^2 \mid x^2+y^2 \le 2, \quad and \quad 0 \le x^3 \le y\}$$MY ATTEMPTMaking a picture of our...
View Articlefind function of $f$ such that the following holds
finding a function $f$ such that the following holds:$f\in lip_0^{\alpha}[0,1] $ where $\alpha =\frac {1}{2}$ and base point is zero, such that $\tilde{f}=\frac {f(x)-f(y)}{ \vert x-y \vert^{1/2}...
View ArticleMarginally continuous measures
Consider a continuous density f on $\mathbb{R}^{2}$, and suppose that $\mu$ is the corresponding Lebesgue-Stieltjes measure on the product space...
View ArticleProof of the fact that an open ball is indeed an open set.
The definition given for an open set is:A set $U\subset X$ of a metric space $(X,d)$ is open if $\forall p\in U$ there exists an $\varepsilon>0$ such that all points $q$ with $d(q,p)<...
View ArticleProof verification: If $(b_n)$ is a bounded (real) sequence with exactly one...
My proof is the following: As $(b_n)$ is bounded, there is a $C>0$ such that $|b_n|\leq C$ for all $n\in \mathbb N$. Let $\epsilon > 0$. We aim to show: for nearly all $n\in \mathbb N$ we have...
View ArticleCalculate: $ \int_{\Omega}\frac{x^{-\frac{3}{4}}}{4...
Calculate an integral:$$ \int_{\Omega} \frac{x^{-\frac{3}{4}}}{4 \sqrt{1 - x^{-\frac{1}{2}}}} \ d \lambda_2,$$where $\Omega$ is an area bounded by a curve: $r = \cos^3 \phi$, $\phi \in \left[...
View ArticleLimit of the convolution of f and the approximation of unity
I have a question regarding the proof of this theorem (from the Textbook Real Analysis by E. M. Stein, R. Shakarchi)Theorem 2.1If ${Kδ}_{δ}>0$ is an approximation to the identity and $f$...
View Articleexample to The sum of mutually singular measures [closed]
let $\mu$ and $\nu_1$ ,$\nu_2$ are measure on a $\sigma$_ algebra $m$, and $\mu$ is positive.If $\nu_1+\nu_2\perp\mu$, so find the counterexample to $\nu_1\perp\mu$ and $\nu_2\perp\mu$.I have tried...
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Is the solution to...
Is the solution to $\ddot{\theta}+0.021\,\text{sgn}(\dot{\theta})\sqrt{|\dot{\theta}|}+0.02\sin(\theta)=0,\,\,\theta(0)=\frac{\pi}{2},\,\dot{\theta}(0) = 0 \quad\text{(Eq. 1)}$ of finite duration?I...
View ArticleProve the subspace of $L^p([a,b])$ determined by the step functions on...
I am reading the proof of the following proposition and got a bit confused about its idea behind the proof:Proposition$\quad$Suppose that $[a,b]$ is a closed bounded interval and that $p$ satisfies...
View ArticleRecommendations for a second look at core subjects
I'm starting a master's program in mathematics in September and I would like to do some review over summer, before I start, so I am looking for book recommendations. My background is in engineering...
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