Can a non-trivial continuous function "undo" the discontinuities of another...
Apologies for the unclear title, I have no idea if the property I'm looking for has a better name.I'm wondering if there exists a pair of functions $f, g : \mathbb{R} \rightarrow \mathbb{R}$ such that...
View ArticleAccurate estimate of $\sum_{k=0}^\infty z^{k^2+ck}$
I encountered a problem, where I am interested in determining (or estimating) a series of the form$$S = \sum_{k=0} ^\infty z^{k^2+ck},$$while $z\in (0,1)$ and $c>0$. The most simple estimate I can...
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How to find parameter values that minimises the total deviation of a function...
Suppose I have a continuous differentiable function $f:\mathbb{R}\rightarrow\mathbb{R}^+$ such that $\int_{\mathbb{R}}f(x)dx<\infty$. Let's assume the function to be convex and increasing.Now, for...
View ArticleProve $f$ does not attain its minimum
Let $C[0,1]$ be a family of continuous functions on $[0,1]$ with$||x||_{\infty}=\sup\limits_{t\in [0,1]}|x(t)|$. Define $$A=\{x\in C[0,1]: x(1)=1, -1\leq x(t)\leq 1\}.$$$a)$ Prove...
View Articleim searching for new book in ito calculus.i nedd help:) [closed]
I am a student studying financial mathematics and I am searching for a new book on Ito calculus. Can you assist me?
View ArticleEvaluate the binomial integral
Evaluate the integral $$\int_{-\infty}^{\infty}\binom{n}{x}dx$$This question came in Cambridge Integration Bee and I have no clue what to do in this.I rewrote $\binom{n}{x}$ as...
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Comparing the Kullback-Leibler divergence to the total variation distance on...
I am trying to get a clearer understanding on how the Kullback_Leibler divergence ranks distributions with respect to the total variation in the discrete setting.let $P,Q$ be two probability measures...
View ArticleIs the limit of this sequence of positive numbers equal to zero?
Let $\theta_1, \theta_2 >1$ and $c>0$. Let $\{a_k\}$ be a sequence of positive numbers such that for any $k\in\mathbb N$:$$1. \quad 0 < a_0\le 1;$$$$2.\quad a_{k+1}\le c^k(a_k^{\theta_1}...
View ArticleDoubt regarding Riemann Integrable problems
I was trying to learn Riemann integration. So after that I wanted to practice some questions and got this.Given a function, $f(x) = \sin{x}\ \ \text{if}\ \ x = \frac{1}{n}$ and $f(x) = \cos{x} \ \text{...
View ArticleQuestion in the proof of the Riesz Representation theorem of non-negative...
The problem is from the proof of Theorem 1.5.12 in Leon Simon's book: Geometric Measure TheorySuppose $X$ is a locally compact Hausdorff space, $\mathcal{K}^{+}$ is the set of all non-negative...
View ArticleProve that the function sequence $f_n(x)=n^2\left(\mathrm{e}^{\frac{1}{n...
Prove that the function sequence $f_n(x)=n^2\left(\mathrm{e}^{\frac{1}{n x}}-1\right) \sin \frac{1}{n x}(n=1,2, \cdots)$ convergent uniformly on $[a,+\infty)(a>0)$ .Proof1. For every $x...
View ArticleGive an example of a bounded set $H \subset \mathbb{R}^p$ and a continuous...
In my analysis book there is the following problem:Give an example of a bounded set $H \subset \mathbb{R}^p$ and a continuous,injective function$f : H \to \mathbb{R}^q$ such that $f^{−1}$ is not...
View ArticleStrict proof of infinitesimal equivalency between $\sin{x}$ and $x$
When I was teaching infinitesimal equivalency between $\sin(x)$ and $x$ ($x\rightarrow0$) for Calculus, I realized that it was not very easy to have a pure elementary proof for it without using the...
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$n$-th derivative of $\exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right)$: is...
Let $\lambda$ and $\mu$ be two positive real numbers and let denote $f$ the function defined as:$$\forall x>0,~f(x):= \exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right)$$According to WolframAlpha...
View ArticleSome concerns about using "a subset $X$ of a metric space is closed if and...
I've seen this proof and I have some concerns about using this theoremIn some questions, we know the given set $X$ is closedBased on the information given that $X$ is closed, is that always admissible...
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N-Epsilon proof
I was working on the questions seen below. Namely question 7 and 8. I was able to figure them both out but noticed that in question 7 we do not include the absolute value sign after where it says...
View ArticleHow to construct a nonzero real number between two given nonzero real numbers?
Statement:Let $$X=$$$$\{(a,b) \in \mathbb{R} \setminus \{0\} \times \mathbb{R}\setminus \{0\}:a<b\}$$There exists a function $f:X \rightarrow \mathbb{R} \setminus \{0\}$ such that for all $(a,b) \in...
View ArticleInvertibility of an integral matrix expression
Given $A:\mathbb R^n \rightarrow \mathbb R^{n\times n}$, $x\mapsto A(x)$ invertible for all $x$. In particular, it is known that $$A(x) = \frac{\partial}{\partial x} f(x)$$ with $f:\mathbb R^n...
View ArticleHow to prove...
How to prove $\lim_{n\to\infty}(1+1/n)^n=\lim_{n\to\infty}1+1/1!+1/2!+...+1/n!$ rigorously?I have read many threads regarding the natural constant e, but couldn't find out how to prove this.
View ArticleUpper bound for $f(x) := e^{-\delta x^2 }\frac{\sinh\left(\sqrt{\delta^2 x^4...
Let $\delta >0$ and consider $j\in \mathbb{N}$ such that $\delta 2^j >\frac{1}{2\sqrt{2}}.$ I am looking for an upper bound of$$f(x) := e^{-\delta x^2 }\frac{\sinh\left(\sqrt{\delta^2 x^4...
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