Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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143 views

Convergence check (No steps/solutions/proofs please)

I just wish to check that I have got these right. Please just indicate whether the ans are right -- please don't show steps (I wish to figure those out myself). Given $\phi_n (x)=n^k(1-x)x^n$ where ...
5
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183 views

Intuitive test of convergence

Are there any intuitive tests that might help one decide whether a sequence of functions converges / converges uniformly? For example, an intuitive test I have recently realized for uniform continuity ...
5
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208 views

How should I understand $u_{\infty}$ in this theorem?

I learned the extension of Green's formula to unbounded domains in Kress's Linear Integral Equations (p.71): Theorem 6.10 Assume that $D$ is a bounded domain of class $C^1$ with a connected boundary ...
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191 views

A relation between permanents and determinants

I have skimmed this video that I found on mathoverflow: http://tube.sfu-kras.ru/video/407?playlist=397 At about 15:05 the lecturer wrote down an equality $\sum F(m_1, \ldots, m_m)z^{m_1}\ldots ...
4
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53 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
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126 views

How do find the numerical average of $x^x$ from $(-4,-2)$?

I wanted to find the approximate average of all real points in $(x)^{x}$ from $[-4,-2]$. This means I am ignoring all real inputs that give a complex output and need average to be a real number. To ...
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47 views

Looking for a function that satisfies some kind of mean value property

Given $a<b\in (0,1)$ and $\delta<1/2$, I need to find an integrable function $\gamma :(a-\delta,b+\delta)\to [0,1]$ such that $$\frac{1}{2\delta}\int_{x-\delta}^{x+\delta}\gamma(y)\; ...
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26 views

Function $f:\mathbb{R}^k \rightarrow \mathbb{H}$ is differentiable if partial derivatives are continuous.

Let $f: \mathbb{R}^k \rightarrow \mathbb{H}$ be a function from the euclidean space to a Hilbert space. Is it true that if the partial derivatives are continuous, then $f$ is differentiable? (the ...
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32 views

Help to understand a Lemma about 'supremum of a family of measures'

I read the following Lemma from a paper but I can't understand the proof. Please help! Lemma: Let $\mu$ be a positive measure defined on the family of open subsets of $\Omega$, which is ...
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25 views

Literature on the convergence of $x_{n+1} = f(x_n)$ in general

When faced with a recurrence of the form $x_{n+1} = f(x_n)$, my toolbelt for proving convergence is very limited: if $f$ isn't $k$-lipschitzian with $k<1$, and/or if I can't find some complete set ...
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61 views

If $m(A)=0$ and $f:[0,1]\rightarrow [0,1]$ has bounded derivative then $m(f(A))=0$

Let $A\subset [0,1]$. Show that if $m(A)=0$ and $f:[0,1]\rightarrow [0,1]$ has bounded derivative then $m(f(A))=0$. ($m$ denotes Lebesgue measure on $[0,1]$) If $m(A)=0$ then for any ...
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38 views

Direct sum of $\mathbb{R}(B)$-modules consisting of all cross sections.

Let $B$ be a Tychonoff space and let $\mathbb{R}(B)$ denote the ring of continuous real valued functions on $B$. For any vector bundle $\xi$ over $B$ let $S(\xi)$ denote the $\mathbb{R}(B)$-module ...
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50 views

“Increasingify” a function

Let $f : [a,b] \rightarrow \mathbb{R}$ be a $C^1$ function such that $f$ is monotonic on each $[t_k, t_{k+1}]$, with $a = t_0 < t_1 < ... < t_N = b$. Let g be the increasing-ified version of ...
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85 views

Minimal conditions to show $\sum \rho_{ij} \Psi_{ij} s_i s_j < \sum s_i s_j $

Consider a sequence of real number $\{s_i\}_{i\leq n}$. Now consider the real numbers $F$, $G$ and $\alpha$ defined below $$F= \sqrt{ \left( \sum ~\rho_{ij} ~\Psi_{ij}~ s_i ~s_j \right)^+}, $$ $$G = ...
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39 views

The proof of Hahn Decomposition Theorem in Folland (real analysis)

On p.86, the following is Folland's proof of Hahn: Before the red line, it is not hard to understand and I think the proof is enough for showing $N$ is negative. Since it proves that: $N$ ...
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64 views

Rigorous proof of this limit

I have shown that the function $$f(x):=\int_{[-\pi,\pi]^n} \frac{e^{-i\langle k,x \rangle}}{1-\frac{1}{n} \sum_{i=1}^n \cos(k_i)}dk$$ exists everywhere for $n \ge 3$. Now, I want to show that ...
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60 views

Show that if $\int_I f=0$ for all interval then $f=0$

Let $f$ integrable on $\mathbb R$ and continuous. I have to show that if for all interval $I\subset \mathbb R$, $$\int_If=0$$ then $f=0$. My attempts Suppose $f\neq 0$. Then, there is a $c\in\mathbb ...
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151 views

Find the complex (or real) roots of $e^{\frac{3 x}{2}}+2 \cos \left(\frac{\sqrt{3} x}{2}\right)$

Define for natural $n\geq 2$ $$G(x,n)= \sum _{k=0}^\infty \frac{x^{k n}}{(k n)!}= \frac{\sum _{k=0}^{n-1} e^{x e^{\frac{2 i \pi k}{n}}}}{n}= G(x e^{\frac{2 i \pi}{n}},n)= \prod_{m=1}^\infty ...
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50 views

Why do these Integration-by-Parts Evaluation Terms Vanish?

The Associated Legendre operator is $$ L_mf = -\frac{d}{dx}\left((1-x^{2})\frac{df}{dx}\right)+\frac{m^{2}}{1-x^{2}}f, $$ where $m$ is a positive integer. For the purposes here, define ...
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60 views

Limit of $\int_E f(nx) dx$ for a $1$-periodic function $f$ on $[0,2\pi]$

Let $E$ be a measurable subset of $[0, 2\pi]$. Assume that $f \in C(\mathbb R)$ is $1$-periodic, i.e. $f(x + 1) = f(x)$. Compute $$\lim_{n\to\infty} \int_{E} f(nx) dx$$. Since $f$ is continuous on ...
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63 views

Codimensions of $\mathbb{Q}$-subspaces of $\mathbb{R}$

Under the Axiom of Choice, we can pick a Hamel basis $H$ for $\mathbb{R}$ as a vector space over $\mathbb{Q}$. Adjoining all but some elements of $H$ to $\mathbb{Q}$ shows that for any cardinality ...
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162 views

trace norm and tensor product

Let $(M_n (\mathbb{C}), n\|.\|)$ , $(M_n (\mathbb{C}), n\|.\|)$ and $(M_{nm} (\mathbb{C}), nm\|.\|)$ be three Banach algebras. where $$\|A\| = \mathrm{tr}\sqrt{(A^* A)}. $$ What is the norm of $\phi$ ...
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104 views

Reference request: are levelsets/levelspaces (as defined here) used anywhere in mathematics?

Thinking about unimodal distributions in multiple dimensions, I came up with the following. Given a function $f : Y \leftarrow X$ where $X$ is a topological space, define that a levelset of $f$ is a ...
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41 views

Show that $(3x_{n}+4y_{n})$ is also Cauchy sequence.

Show that if $(x_{n})$ and $(y_{n})$ are Cauchy sequences in $X$, then the sequence $(3x_{n}+4y_{n})$ is also Cauchy sequence using the definition of a Cauchy sequence. Attempt Let $\epsilon > 0$ ...
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67 views

Is there any flaw in my proof of Darboux's Theorem?

Darboux's Theorem. Let $I$ be an open interval, and let $f : I \to R$ be a differentiable function. If $a$ and $b$ are points of $I$ with $a < b$ and if $y$ lies between $f' (a)$ and $f' (b)$, ...
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109 views

Minimizing the Frobenius norm of a matrix involving the Hadamard product, $\|X(A\odot Y)-S\|_F$

Let $S\in\mathbb{R}^{L\times N}$ and $A\in\mathbb{R}^{M\times N}$ be known and arbitrary. I'd like to solve the following system: \begin{align} \min_{X\in\mathbb{R}^{L\times M},Y\in\mathbb{R}^{M\times ...
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49 views

Computing the Fourier transform of the distribution $\|x\|^{-\alpha}$.

Question: Suppose we are given the tempered distribution $\|x\|^{-\alpha}$. We want to compute the Fourier transform $\mathcal{F}[\|x\|^{-\alpha}](\xi)$. What techniques are available for ...
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47 views

Prove that $a_n$ $\rightarrow$ L $\implies$ |$a_n$| $\rightarrow$ |L|

The book I am using for my Advance Calculus course is Introduction to Analysis by Arthur Mattuck. Prove that $a_n$ $\rightarrow$ L $\implies$ |$a_n$| $\rightarrow$ |L|. We are needed to make a cases ...
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79 views

A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
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74 views

Assumption in PDE theory

I have an exercise in PDE theory. Let $w \in C^2(U)\cap C(\overline{U})$ where $U$ is open, bounded and connected and $c \in C(\overline{U},\mathbb{R})$ with $c(x) \le 0$ everywhere. Moreover, ...
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73 views

Definition of the Limit of a Function for the Extended Reals

Definition 4.33 of Rudin's Principles of Real Analysis: Let $f$ be a real function defined on $E \subset R$. We say that $f(t) \rightarrow A$ as $t \rightarrow x$ where $A$ and $x$ are in the ...
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83 views

Is there a name for this type of integral $\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$?

Given a polynomial of arbitrary degree, $P(x)$, on $[a,b]$ is there a name for this type of integral: $$\int_a^b \frac{P(x)}{\sqrt{1-P(x)^2}}dx$$
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108 views

Alternative Proof of Heine-Borel Theorem

This is regarding the proof of Heine Borel Theorem for closed intervals on real line as given in Hardy's Pure Mathematics. Heine-Borel Theorem: Let $[a, b]$ be a closed interval with $a < b$ and ...
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72 views

Integral of a function with an exponentiated inner product

Let $f:\Bbb R^n\to \Bbb R^n$ be a continuous function such that $\int_{\Bbb R^n}|f(x)|dx\lt\infty$. Let $A$ be a real $n\times n$ invertible matrix and for $x,y\in\Bbb R^n$, let $\langle x,y\rangle$ ...
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62 views

Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$

Assume $\Omega$ is open bounded domain in $\mathbb R^n$ Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$ with inner product ...
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50 views

Convergence and integral for nets.

We say that the non-empty set $S$ with partial order $\leq$ is directed set if for any $s,t\in S$ we have a $u\in S$ such that $s\leq u, t\leq u$. A net is a function from directed $S$ into the any ...
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51 views

Integrate complex function over $\mathbb{C}^2$

I have a question in mind and I would appreciate your help. Usually in complex analysis we consider integrals of the form $\int_\gamma f(z) dz$ where $\gamma $ is a contour and ...
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55 views

Exercise from Pugh's Real Analysis Regarding Zero Derivative on Open Subsets of $\mathbb{R}^m$

Assume that $U$ is a connected open subset of $\mathbb{R}^n$ and $f:U\to\mathbb{R}^m$ is differentiable everywhere on $U$. If $(Df)_p=0$ for all $p\in U$, show that $f$ is constant. I immediately ...
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108 views

uniform convergence in recursive function

Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to \mathbb{R}$ an increasing function ...
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47 views

Computing MMSE and conditional expectation

Suppose we have three independent, zero mean, finite variance random variables $V,W,Z$ and where $W,Z$ are Gaussian random variables. These random variables form a new random variable $Y$ ...
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90 views

I need your suggestions: Does this series converge or diverge?

I have this series given by $$\sum\limits_{n=1}^\infty \frac{1}{n^3 \sin^2n}.$$ I know that the terms are all nonnegative. Can I get a subsequence of $\frac{1}{n^3 \sin^2n}$ such that its sum diverges ...
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69 views

Prove that $\int_{a}^{b}f(x)dx$ $\leq$ $\int_{a}^{b} g(x)dx$.

Suppose $f,g$ are integrable on $[a,b]$ and $f(x)\leq g(x)$ for all $x\in [a,b]$. Prove that $\int_{a}^{b}f(x)dx$ $\leq$ $\int_{a}^{b} g(x)dx$. Okay so I believe I can use either the lower or the ...
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57 views

What is the Dedekind cut for 2?

I was watching a lecture on defining the reals. The professor named a set $\beta$ such that $\beta =\big\{ r\in Q:r<2 \big\}$ and he said that this is not a cut because it does not satisfy our ...
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45 views

real analysis question on equicontinuity

Can anyone verify my proof of the following problem found in Rudin's Principles of Mathematical Analysis chapter 7 exercise 15. Suppose $f$ is a real continuous function on $\mathbb{R}$, ...
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64 views

Sequences of rapidly decaying analytic functions

Note that we have $$\frac{1}{x}\gg\frac{1}{x^2}\gg\frac{1}{x^3}\gg\cdots\gg e^{-x}.$$ I was wondering if a generalization is possible. Namely, if $f_1\gg f_2\gg f_3\gg\cdots$ are analytic from ...
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48 views

Prove that the Weierstrass-type function is nowhere differentiable

Given $0<\alpha\leq1$. Show that the function $$f(x)=\sum_{j=1}^\infty 2^{-j\alpha}\sin(2^jx)$$ is nowhere differentiable. I have solved the case $x=0$. Taking $t_l=2^{-l-1}\pi$, then ...
4
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49 views

How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
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165 views

Convex set of derivatives implies mean value theorem

Let U$ \subset$ $R^{^{n}}\ $be open, $f:U\rightarrow R^{m}$ differentiable on U, and segment $[a,b]\subset U$. Assume that the set of derivatives $\{ f'(x)\in L(R^{^{n}},R^{^{m}}):x\in [a,b] \}$ ...
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104 views

Prob. 10, Sec. 3.2 in Erwine Kreyszig's “Introductory functional analysis with applications”

Here is Prob. 10 in the Problems after Sec. 3.2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: ... Let $T \colon X \to X$ be a bounded linear operator on a complex ...
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64 views

Conditions for integrability of $\int_0^1 \frac{f(x)}{f(x)+f(1-x)} \ dx$

Earlier today there was a question asking, for any $f$ from $[0,1] \rightarrow \Bbb R,$ how to evaluate the following integral, $$ \int_0^1 \frac{f(x)}{f(x)+f(1-x)} \ dx. $$ Link here. The question ...