Finding the number of minima in sum of square roots of cosines
Suppose you are given a set of functions $f_1, \ldots, f_n$. Every function is defined as follows$$f_i(x) = \sqrt{1+C^2_i-2C_i\cos (x-D_i)}$$where $0<C_i<1$ and $0\leq D_i<2\pi$ are...
View ArticleProve that if $\|f'(x)\| \leq \|x\|$ then $\|f(x)\| \leq \frac{1}{2} \|x\|^2$...
Let $U \subset \mathbb{R}^m$ be open and convex, with $0 \in U$. Suppose that $f : U \rightarrow \mathbb{R}^n$ is $C^1$ with $f(0) = 0$ and $||f'(x)|| \leq ||x||$ for every $x \in U$. I have to show...
View ArticleProve that if $f\in C[a,b]$ satisfies $\int_{[a,b]}|f|d\lambda=0$, then...
ProblemLet $[a,b]$ be a closed and bounded interval, and let $C[a,b]$ be the vector space of all continuous real-valued functions on $[a,b]$. Define the function $\|\cdot\|_1:C[a,b]\to\mathbb{R}$ by...
View ArticleGeneralized chain rule for multivariable functions
Let x(t) = $(x_1(t),\dots,x_m(t))$ and let $f: \mathbb{R}^m \to \mathbb{R}$ be a multivariable function. Then let $\hat f$(t) = f (x(t)).I know that $$\frac{d\hat f(t)}{dt} = \sum_{h=1}^m...
View ArticleSolving the equation of an ellipse using the constant and the foci [closed]
Find the equation of the ellipse having foci$(-8,2) \text{and} (4,2)$ and for the which constant referred in the definition is $18$? I already get the value of $h$ and $c=6$ but I don't how to get the...
View ArticleNowhere analytic smooth solutions to analytic differential equations
There are several parts to this question. Clearly there exist "analytic" differential equations of a certain variety that have nowhere analytic solutions (e.g., $f'(x)=2f(2x)$ has the Fabius function...
View ArticleBorel sets and absolutely continuous functions
Let $A\subset [0,1]$ be a Borel set such that $0<m(A\cap I)<m(I)$ for any subinterval $I$ of $[0,1]$. Let $F(x)=m([0,x]\cap A)$. Show that $F$ is absolutely continuous and strictly increasing on...
View ArticleEstimation of the absolute value of an infinite sum by its coefficient
I got this problem from a complex funtion, but it now has little to do with the complex analysis.$$\Theta(s)=\sum_{n=0}^{\infty}a_{2n}s^{2n}$$I already have:$$c_1^n\frac{(\ln n)^{2n}}{(2n)!}\leq...
View ArticleQuestion regarding set of discontinuities.
Suppose that we have a function $f : [a,b] \to \Bbb{R} $ with countable discontinuities on $[a,b]$. Let call this set $D$Take now the set$\space$$E =$ { $x \in [a,b] : f(x) \gt c $}$\space$ for a...
View ArticleThe boundness of sequence $\{y^{k+\frac{1}{2}}\}$
Let sequence $\{y^{k+\frac{1}{2}}\}$ generated by$$y^{k+\frac{1}{2}}=y^{k}+z^{k},$$where $z^{k}$ not be necessary bounded. If $\{y^{k+\frac{1}{2}}\}$ is bounded, can we deduce$\{y^{k}\}$ is...
View ArticleIs there a function satisfying the following conditions? [closed]
Let $\Omega$ be a bounded smooth open set of $\mathbb R^N$.Is there a function $\psi \in C^2(\overline \Omega)$ satisfying the following conditions:$\frac{\partial \psi}{\partial n} = n \cdot D\psi =...
View ArticleCan the fundamental theorems of real analysis be proven/developed without...
I've been reading about philosophical debates between mathematicians, and some seemed to reject the ideas of real analysis (such as the extreme value theorem) based on a school called "intuitionism",...
View ArticleUsing Peano's axioms to disprove the existence of self-looping tendencies in...
Let me clarify by what I mean by "self-looping". So, we know that Peano's axioms use primitive terms like zero, natural number and the successor operation. Now, I want to prove that the following isn't...
View ArticleIf the sum $\sum_{n=1}^\infty n a_n$ converges for positive $a_n$, what can...
Let $(a_n)$ be a sequence of positive numbers and assume that$$\sum_{n=1}^\infty n a_n < \infty.$$What can we then say about $a_n$? There are a few obvious things; $a_n\to 0$ and the series...
View ArticleLebesgue measure simple property. Seems intuitevely true but can't formalise...
Consider the usual Lebesgue measure in $\mathbb R^n$ and let $f,g \colon \mathbb R^n \to \mathbb R$ be two arbitrary functions that satisfy$$ | \{ x \in \mathbb R^n : g(x) \neq h(x) \} | > 0 \quad...
View ArticleA Ramanujan style series
Recently, someone asked a question involving the expression$$ \sum_{n=0}^{\infty} (-1)^n (4n+1) \left(\frac{(2n-1)!!}{(2n)!!}\right)^3 $$At first glance, I knew that the expression was the value of a...
View ArticleRemark from baby Rudin about continuous function
"We defined the notion of continuity for functions defined on a subset $E$ of a metric space $X$. However, the complement of $E$ in $X$ plays no role whatever in this definition (note that the...
View Articleupper bound for $\int \frac{1}{|x-y|}$
Let $\alpha>0$, determine some upper bound for this integral for $x\in B_{1}$, $$\int_{B_{1}\cap B_{(1+1/\alpha)|x|}}\frac{1}{|x-y|}\,dy.$$My approach: If $x\in B_{1}$ then $|x|\leq 1$ and...
View Articleelementary analysis proof verification
Show that if X, Y and Z are convergent sequences, then the sequence $w:=mid\{x_n,y_n,z_n\}$ is convergent.The attempt:(1) Suppose $x_n$ goes to x, $y_n$ goes to y, and $z_n$ goes to z. WLOG, assume $x...
View ArticleFourier transform $\mathcal{F}$ of smooth functions $f\in C^l$ decays fast.
I want to prove the above statement, which motivates why the Fourier transform on the Schwartz space is automorphic.Let $f\in C^l(\mathbb{R}^d)$ and $\alpha$ a multiindex of at most order $l$. From the...
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