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About the definition of functional derivative and the $L^2$ inner product
There is something I do not understand well about the definition of the functional derivative. In the wikipedia page http://en.wikipedia.org/wiki/Functional_derivative it says:1) This definition...
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Proof of logarithmic equality with floor and ceiling
I have to prove the following identity:$$\left\lfloor \log _2(n)\right\rfloor\equiv\left\lceil \log _2(n+1)-1\right\rceil$$Each side is defined for all $n>0\colon n\in\mathbb{Z}$. So, when plotting...
View ArticleEquivalence proof between sum and integral
Given the following sum:$$\sum_{i=1}^{i=n} \lfloor\log_2(i)\rfloor$$I want to find a closed form for it, or at least the term of greatest asymptotic growth of its outcome. However, to simplify such...
View ArticleImproper integral as Primitive
Let $f\in C(a,b],$$\lim_{x\to a}f(x) = \infty$ and $\int_a^bf(x)dx$ exists (finite) as Improper integral.Can the expression $\int_a^xf(t)dt$ be considered as a Primitive of the function $f(x)$ on a...
View ArticleLet $f\colon (a,b)\to \mathbb{R}$ be nondecreasing and continuous. If...
I need help to understand the proof below of the following theorem.Let $f\colon (a,b)\to \mathbb{R}$ be an arbitrary function. If $E=\{x\in (a,b)\mid f'(x)\text{ exists and }f'(x)=0\}$, then...
View Articlea confusion in checking whether this improper converges?
I have an improper integral as follows:$$\int_1^{\infty}\dfrac{e^{-2y}-e^{-y}}{y}dy$$I want to check whether this integral converges by reasoning as...
View ArticleMinimum of convex functions.
Assume that $ f:\mathbb{R}^n\to[0,+\infty) $ is a convex function and $ f(0)=0 $. It is obvious that $ 0 $ is the minimal point of $ f $. I want to ask if there exists $ \delta>0 $ such thta $...
View ArticlePlease clarify whether the restrictions of $\lambda^{u}$ on the isotropy...
Suppose $\{\lambda^{u}\}_{u\in G^{0}}$ be a haar system of a locally compact groupoid. As we all know $\lambda^{u}$ has support in $G^{u}$.whether the restriction of $\lambda^{u}$ to isotropy subgroup...
View Articleboth iterated integrals exist and are equal, but the double integral does not
Consider the function $ f(x, y) = e^{ - x \cdot y } \sin x \sin y $for $ x \geq 0 $ and $ y \geq 0 $, and $ 0 $ in other cases.Prove that both iterated integrals $$ \int_{\mathbb{R}} \left[...
View ArticleStep 6 Real construction from Rational Baby Rudin
Rudin's Books establishesMultiplication is a little more bothersome than addition in the present context , since products of negative rationals are positive. For this reason we confine ourselves first...
View ArticleQuery regarding Dedekind cuts and multiplicative axioms
I am currently studying Dedekind cuts and I am going through the proof of the 5 multiplication field axioms. The first 4 are trivial (if x $\in$ F and y $\in$ F then xy $\in$ F, xy = yx, (xy)z = x(yz),...
View ArticleRelation properties [duplicate]
I'm studying the properties of relations. I'm struggling in getting the point for the following exercise:Let $R$ be the relation "is strictly higher than" applied to the set of all mountain peaks. The...
View ArticleFourier series of auto-correlation function
Suppose that $f$ is integrable, has period $ T $, and Fourier series$$f(t) \sim \sum_{n=-\infty}^{\infty} c_n e^{\frac{2\pi int}{T}}.$$Determine the Fourier series of the so-called...
View ArticleSup norm can always be made smaller without altering the value of the...
Set $\alpha>0$ and let $V := \{f\in C^0([0,1]) : f(0) = 0\}$ be equipped with the sup norm $\Vert\cdot \Vert_\infty$. I'm trying to show that for every $f\in V$ with $\int_{0}^{1} f(x) \, dx =...
View ArticleA Jacobian determinant from stereographic projection
Let $n \geq 3$. The stereographic projection from $\mathbb{S}^n \backslash\{N\}$ to $\mathbb{R}^n$ is the inverse of$$F: \mathbb{R}^n \rightarrow \mathbb{S}^n \backslash\{N\}, \quad x...
View ArticleVerification of a proof of Dynkin's $\lambda$-$\pi$ system
In our real analysis course, we are asked to prove Dynkin's $\pi$-$\lambda$ theorem. After searching on the Internet for a while, the proofs found are quite...verbose to me. Therefore, I write a...
View ArticleShow that $f(b) - f(a) \geq \sum_{n=1}^{\infty}(f(b_n)-f(a_n))$
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be an increasing function and suppose that:$A = \bigcup_{n=1}^\infty(a_n,b_n]\ \ (\text{ disjoint union }) \ \ (a_n \leq b_n; ,a_n ,b_n \in [-\infty,...
View ArticleIf $y_n>0$, $\lim (y_1+\cdots + y_n)=\infty$ and $\lim \frac{x_n}{y_n}=a$,...
Problem. Let $y_n>0$ for all $n\in \mathbb{N}$, with $\lim (y_1+\cdots + y_n)=\infty$. Prove that if $\lim \frac{x_n}{y_n}=a$ then $\lim \frac{x_1+\cdots+x_n}{y_1+\cdots+y_n}=a$.My approach. Let...
View ArticleQuestion Regarding Lebesgue Integrable
I am reading Royden's Real Analysis(5th edition), and confusing about the definition of Lebesgue Integrable.Here is the definition in chapter 4.2: A bounded function defined on a measurable set is said...
View ArticleProve that $\operatorname{cond}(f ) = \|A^{−1}\|\|A\|.$
Question:I am trying to prove that the condition number of a function $ f(x) = Ax $ is given by:$$\text{cond}(f) = \|A^{-1}\|\|A\|$$The definition of the condition number of a function ( f ) is:$$...
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