How would you find the maximum of this function?...
Like the title says, I am trying to find the maximum of this function, which seems to be near 0.787. This is to "normalize" the function, so that the maximum is 1, so the x value is what I am looking...
View ArticleWhy do sequences need to converge with respect to a norm?
Suppose there is a sequence $x_n$ in $\mathbb{R}^n$, which converges to $x$, with respect to $\lVert\cdot\rVert_2$, say. Then we can write down vectors $x_1, x_2, x_3, ...$, and so on, and, with each...
View Articlewhy the partial derivatives are not continuous?
Here is the function I am trying to prove that its partial derivative is not continuous:$$f(x,y) = \begin{cases} \frac{xy}{x^2 + y^2} & if (x,y)\neq (0,0) \\ 0 & if (x,y) = (0,0) \end{cases}...
View ArticleIs the metric exterior condition necessary to ensure that all Borel sets are...
In Stein's Real Analysis, Chapter 6, Theorem 1.2, it is stated that all Borel sets are measurable for the metric exterior measure. I have a question about the converse: Is the metric exterior condition...
View ArticleFind the limit for this sequence
Let $a \in \mathbb R$ and $x_0=a$$$ x_{n+1}=3 -\frac{2}{x_n} $$Find the limit for different values of $a$.I see that if there is limit then it will be $1$ or $2$Also, if $a=1$ then$(x_n)=(1,1,1,...)...
View ArticleFinding $f$ orthogonal to set.
Let $A \subsetneq \mathbb{Z}$. Consider the following set of functions $$M = \{e^{2 \pi in x}\}_{n \in A} \cup \{xe^{2 \pi i n x}\}_{n \in A^c}$$ belonging to $L^2[0, 1]$. Does there exists a function...
View ArticleProve that sequence is monotonic
Given a sequence $x_{0}=1$, $x_{n+1}=\frac{x_{n}}{2}+\frac{1}{x_{n}}$.I want to show if the sequence is monotonically decreasing. I tried induction as follows:For $n \geq 1$, $x_{n+1}<...
View ArticleWhat is the sum of $\sum _{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}$
I have the following series:$$\sum _{n=1}^\infty \frac{(-1)^{n-1}}{(2n-1)3^{n-1}}$$I already know that it converges but I'm trying to get what does it sum up to. I've tried to find an alternative way...
View ArticleTrace of a function and $L^\infty$ norm
Let $\Omega$ be an open bounded domain in $\mathbb R^n$ and let $C= \lbrace(x, y): x\in\Omega, y\in [0, +\infty)\rbrace$.Let $$X(C)=\left\lbrace u\in L^2(C): u|_{\partial\Omega\times [0, +\infty)}=0 \...
View ArticleDense Sequences
I'm trying to generalize the following result:There exists a sequence whose set of subsequential limits is equal to $\mathbb{R}$.Using the following lemma,Given a sequence $(p_n)$ with range $E$, $E'$...
View ArticleConvolution of $\mathcal{C}^\infty$ is analytic
Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$. Is the convolution (assumed to be well defined) defined as:$$(f*g)(x) =...
View ArticleWhy isn't continuity necessary for existance of partial derivatives?
Consider the graph of a function.$$f(x,y) =\begin{cases}\frac{xy}{x^2+2y^2}, & \text{(x,y) $\neq$ (0,0)} \\0, & \text{elsewhere}\end{cases}$$It is discontinuous at $(0,0)$. As on path $y = mx$,...
View ArticleCompleteness of $sin(kx)_{k=0}^{\infty}$ in $L_1[1, 4]$
I am reading a function analysis book and having trouble proving the following task: is ${\sin(kx)}_{k=0}^{\infty}$ complete in $L_1[1, 4]$?I understand that completeness in $C[1, 4]$ implies...
View ArticleLet $f\colon (a,b)\to \mathbb{R}$ be a nondecreasing and continuous function....
Let $f\colon (a,b)\to \mathbb{R}$ be a nondecreasing and continuous function. If $E=\{x\in (a,b)\;|\; \exists f'(x), f'(x)=0\}$, then $$\lambda(f(E))=0.$$$\lambda$ denote the Lebesgue measure.I'm...
View ArticleIs there a $\sigma$-algebra on $\mathbb{R}$ strictly between the Borel and...
So, after proving that $\mathfrak{B}(\mathbb{R})\subset \mathfrak{L}(\mathbb{R})$, I asked myself, and now asking you, is there a set $\mathfrak{S}(\mathbb{R})$, which...
View ArticleThe variation of a complex measure is finitely additive.
I need to prove the followingThe variation $|\mu|$ of a complex measure $\mu$ is finitely additive.The following is my attempt. Basically, my idea is to prove that $|\mu|(B_1\bigcup...
View ArticleEquivalent definitions of affine function: Dose$f(\gamma x...
For a function $f\colon\mathbb{R}^n\to\mathbb{R}$ the following two are equivalent\begin{align*}&\text{(i) there exist $a\in\mathbb{R}^n$, $b\in\mathbb{R}$ so that...
View ArticleOn the axioms of the real number system as stated in Apostol’s textbook
The typical axiom system for the real numbers states that the real numbers satisfy the axioms of an algebraic field plus a few others.In the mathematical analysis textbook by Apostol, the axioms are...
View ArticleProof of second partials test for saddle points - why does Hessian need to be...
Suppose $f:U \subseteq \mathbb{R}^n \to \mathbb{R}$ is $C^2$ on $U$ and $\nabla f(x)=0$ at $x \in U$. If $Hf(x)$ is indefinite and nonsingular—that is, if it has both positive and negative eigenvalues...
View ArticleTo determine the number of roots for all antiderivative of a cubic polynomial
Let $f(x)$ be a cubic polynomial with real coefficients. Suppose that $f(x)$ has exactly one real root which is simple. Which of the following statements holds for all antiderivative $F(x)$ of $f(x)$...
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