Applying a Wirtinger-like inequality to get a bound for a differential equation.
Here I am considering the following problem.I have some $f:[0,\infty) \to \mathbb R$ smooth with the property that $f''≥-Cf$ for some $C > 0$.We have that $f(0)=0$ but $f'(0)>0$.Can I find an...
View Articledefinite integral of square root with negative squared trigonometric identity...
How do we integrate $\int_0^{\pi/3}\sqrt{1+(-2sin(x))^2}dx$.
View ArticleDoubt in Apostol's Book Theorem 1.6
I have some doubt about the following theorem from Apostol's Mathematical Analysis [2nd Edition] that I found when I was trying to redo it from my understanding of it. I have taken the following...
View ArticleShow right eigenvectors are parallel to a vector that relates to the eigenvalues
I am given a system of first-order linear hyperbolic PDE's that can be written as $n_{\theta} + A\eta_{\xi} = 0, n = (\eta \,\,\,u)^T.$ Thus, A is diagonalizable, $A' = R^{-1}AR$. Multiplying by R from...
View Article$y' = \frac{\sin y + y}{(x^2 + 1)^{-2}e^{\left(\frac{x^3}{3} + x\right)}}$,...
Given the Cauchy problem $y' = \frac{\sin y + y}{(x^2 + 1)^{-2}e^{\left(\frac{x^3}{3} + x\right)}}$, $y(x_0) = y_0$, where $x_0 \in \mathbb{R}^+$$\cup$ {$0$}.a) Does $f(x,y)$ in the vicinity of...
View ArticleSolve the equation: $y''-y'x^{-1}+2yx^{-2}=(\ln x+1)\cos(\ln x)+\ln x\sin(\ln...
Solve the equation: $y''-y'x^{-1}+2yx^{-2}=(\ln x+1)\cos(\ln x)+\ln x\sin(\ln x)$. Firstly, I noticed that it is an Euler-Cauchy equation, so I multiplied it by $x^2$. Then I tackled the homogeneous...
View ArticleProperty of coordinate-wise convex function
A function $f \colon \mathbb{R}^2 \rightarrow \mathbb{R}$ iscoordinate-wise convex if $f_x \colon \mathbb{R} \rightarrow> \mathbb{R}, t \rightarrow f(x, t)$ and $f_y \colon \mathbb{R}>...
View ArticleIntroduce $y(x)=u(x)z(x)$ into the equation $y''-2xy'-2y=0$ so that there...
Introduce $y(x)=u(x)z(x)$ into the equation $y''-2xy'-2y=0$ so that there won't be a term with $z'$ in the new equation. Find all solutions of such equations and also explore the possibility when...
View ArticleA claim for a Schauder basis on Hilbert spaces
Let $H$ be a separable Hilbert space and assume that $\{e_k\}_{k=1}^{\infty} \subset H$ is a Schauder basis for $H$ such that given any $v\in H$ there holds:$$ v = \sum_{k=1}^{\infty} \langle v,e_k...
View ArticleInner regularity of a Radon measure induced from a nonstandard metric on...
Problem: Define the distance function between $(x_1, y_1)$ and $(x_2, y_2)$ for two points (where $x_i, y_i$ are real numbers) in the plane to be$$|y_1 - y_2| \text{ if } x_1=x_2; \quad 1 + |y_1- y_2|...
View ArticleJustification for differentiation under integral
Given a function I seek to find its derivative $$f(x) = \int_{\frac{1}{x}}^{\frac{e^x}{x}} \frac{\cos(xt)}{t} \, dt, \quad (x>0)$$My question is regarding the justification of the differentiation...
View ArticleThe Medians of Lipschitz Functions on $(X,d,\mu)$ (Existence and Uniqueness)
Let $\varphi:(X,d,\mu)\to \Bbb R$ be a Lipschitz function, where $\mu$ is a probability measure on the metric space $(X,d)$. The median $m_\varphi$ of $\varphi$ is defined as the real number such...
View Articleexistence of function in R2 with some property
I would like to know if there exists a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ satisfying :property 1 : $f$ is boundedproperty 2 : $f$ is $1$-Lipshitzproperty 3 : $|f(a)-f(b)|\geq C...
View ArticleWhat does it mean that the distributional extension of ordinary functions is...
Let $\Omega \subset \mathbb{R}^n$ and $K \subset \Omega$ compact. Let $\mathscr{D}_K$ denote the space of all $f \in C^\infty(\mathbb{R}^n)$ whose support lies in $K$ and $\mathscr{D}(\Omega)$ the...
View ArticleTheorem implicit function
I have the following excercise:$f,g: \mathbb R^3 \to \mathbb R$ with:$f(x,y,z)=z^2-2y-xz$$g(x,y,z)=zy+x^2$a) Show that in an open environment of $(1,1,-1)$ there exist functions $h_1(z)$ and $h_2(z)$...
View ArticlePrenex Normal Form of a Simple Proposition Reads Strangely.
I was trying to convert a simple proposition concerning the real numbers to Prenex Normal Form but arrived at a logical statement that didn't appear equivalent to what I started with.The proposition I...
View ArticleProving the right eigenvectors show in the direction of the characteristic lines
Given a system of two linear homogeneous first-order hyperbolic PDE's, $n_{\theta} + An_{\xi} = 0,$ I need to show the right eigenvectors of $A$ show in the direction of the characteristic lines. The...
View ArticleReimann series theorem [closed]
As we know the sum of conditionally convergwnt series changes on rearranging the terms so my question is what all can i do with the series to find the sum which will not change the sum
View ArticleEstimate of supremum of derivative by supremum of function and second derivative
For a homework question I have to solve the following problem:Let $f \in C^2([0,1])$. The task is to show that there exists $M>0$ (independent of $f$) such that$$||f'||_\infty \le M ||f||_\infty...
View ArticleIs it true that |X| vanishes at infinity, then X is integrable?
Suppose $X$ is a random variable on $(\Omega,\mathscr{F},\mathbb{P})$. If there exists $M>0$, such that for all $\lambda>0$, we have $\mathbb{P}[|X|>\lambda]\le\frac{M}{\lambda}$, then is it...
View Article