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Application of weak maximum principle.

You are very close. Since $v$ is subharmonic, from the maximum principle you have that $$ u(x)+ \frac 1{2n} \vert x\vert^2\sup_{\Omega}\vert f\vert =v\leq\sup_\Omega v = \sup_{\partial \Omega} v \leq \...
JackT's user avatar
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Why is the Daniell integral not so popular?

Measure theory in the vein of Lebesgue and Caratheodory relies on an abstract axiomatic system and resembles sprawling computer code more than it does math. Worse yet, it is not clear whether the ...
Hey's user avatar
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1 vote

Tricky Application of Rouche's Theorem

You can try the symmetric version of Rouché': If $f$ and $g$ are analytic in a neighbourhood of $K$ and $|f(z) - g(z)| < |f(z)| + |g(z)|$ for $z \in \partial K$, then $f$ and $g$ have the same ...
Robert Israel's user avatar
2 votes
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Tricky Application of Rouche's Theorem

For “small” $z$ is $e^z \approx 1 + z $ or $e^z - z \approx 1 \ne 0$. That suggests to apply Rouché's theorem to the functions $f(z) = e^z-z$ and $g(z) =1$: For $|z| = 1$ is $$ |f(z)-g(z)| = \left| \...
Martin R's user avatar
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1 vote

$sin(1/x)$: expectation vs infimum

A probability distribution $\mathcal{Q}$ just gives 'weight' to different regions of the real line (hence the names probability mass/density functions). Then $\mathbb{E}_\mathcal{Q}[X]$ is just a ...
SammyH's user avatar
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Prove the following relationship of the number e and then show that it is irrational

appears you need $ \frac{1}{(n+1)!} + \frac{1}{(n+2)!} + ... $ as $$ \frac{1}{n!} \left( \frac{1}{n+1} + \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} +... \right) < \frac{1}{n!} \...
Will Jagy's user avatar
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$sin(1/x)$: expectation vs infimum

Take any probability distribution $X$ on $\mathbb{R}$, let its associated density function be $f_X$. Then, $$\mathbb{E}(sin(1/X))=\displaystyle\int_{-\infty}^{+\infty}sin(1/x) f_X(x) dx$$ So, it ...
Julio Puerta's user avatar
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Trivial cases of a Theorem of Criterion for Multiple Zeros

Your counterexample doesn't work. In fact, we have for example $g(x)=x^2$ that it is common factor of $f(x)=f'(x)=0$ because $x^2*b=f(x)=f'(x)$ with $b=0 \in \mathbb{R}$. What's more, every polynomial ...
Mike_Oxlong's user avatar
2 votes

Is $\int_0^{\pi}\log \sin \theta d\theta$ not well-defined?

As Lebesgue integral it's always well-defined because log is Lebesgue integrable on (0,1). What's not defined is the log function on the complex plane. More precisely, the imaginary part of log will ...
Liding Yao's user avatar
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How to evaluate double integral: $\iint \frac{y}{x} \, dx \, dy$ if it is in the first quadrant and is bounded by: $y=0$, $y=x$, and $x^2 + 4y^2 = 4$

Divide the domain $D$ into two subdomains: $$D_1:0\leq x\leq\frac{2}{\sqrt 5},0\leq y\leq x;\qquad D_2:\frac{2}{\sqrt 5}\leq x\leq2,0\leq y\leq\frac{\sqrt{4-x^2}}{2},$$ then \begin{align*} \iint_{D} \...
Riemann's user avatar
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Existence of $1/(e-1)$ value of a continous and differentiable function $f:[0,1]\to \mathbb R$ given $f(1)=1, f(0)=0$

While the problem was solved above, here is a method that uses a stronger result (Darboux's theorem) but shows how to arrive systematically at the function used in the MVT. Suppose that there doesn't ...
Sarvesh Ravichandran Iyer's user avatar
3 votes
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Existence of $1/(e-1)$ value of a continous and differentiable function $f:[0,1]\to \mathbb R$ given $f(1)=1, f(0)=0$

Let $$F(x)=e^{-x}\left(f(x)+\frac{1}{e-1}\right),\quad x\in[0,1],$$ then $F$ is continous on $[0,1]$ and differentiable on $(0,1)$, $$F(0)=F(1)=\frac{1}{e-1},\qquad F'(x)=e^{-x}\left(f'(x)-f(x)-\frac{...
Riemann's user avatar
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1 vote

Proving a function isn't bounded

If $f: \mathbb{R}\setminus \{0\} \to \mathbb{R}$ was bounded there would exist a constant $M >0$ such that $$ |f(x)| < M, \quad \forall x\ne 0. $$ However, this generates a contradiction since $...
PierreCarre's user avatar
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4 votes
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Proving a function isn't bounded

The formulas that you've written actually prove something stronger, namely that $$ \lim_{x \to 0}\; \biggl\lvert \frac1x \biggr\rvert = \infty. $$ To establish that $f(x) = \frac1x$ is unbounded, we ...
Sammy Black's user avatar
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2 votes

How to evaluate double integral: $\iint \frac{y}{x} \, dx \, dy$ if it is in the first quadrant and is bounded by: $y=0$, $y=x$, and $x^2 + 4y^2 = 4$

Let's take a look at the domain $$\Omega=\{(x,y)\in\mathbb{R}^2\colon 0\leq y\leq x,\, x^2+4y^2\leq4\}$$ and notice that $$x^2+4y^2\leq4\iff \frac{x^2}{2^2}+y^2\leq 1$$ In red we have $\Omega$, in ...
Davide's user avatar
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Assumption of a partial derivative Lars Hörmander does in "The Analysis of Linear Partial Operators I"

But with product and chain rule I get that for the left side $\frac{\partial}{\partial \epsilon}(\epsilon^{-n} \psi (\frac{x}{\epsilon})) = -n\epsilon^{-n-1}\psi(\frac{x}{\epsilon})-\epsilon^{-2}\sum_{...
psl2Z's user avatar
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1 vote
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Proof of the Chain Rule of multivariable functions in Munkres' Analysis on Manifolds

The reason is the metric used here is "sup metric". As a result, we have $|\mathbf h|=\max\{|h_1|,\cdots,|h_m|\}$ and $|Df(\mathbf a)|=\max\{|D_jf_i(\mathbf a)|;i=1,2,\cdots,n \ and\ j=1,2,\...
studyhard's user avatar
  • 151
3 votes
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Differential of the function $\frac{1}{\sqrt{x^2+y^2}}\begin{pmatrix} x \\ y \end{pmatrix} $

Yes, your thinking is correct. They seem to have made numerous errors. We have a map $\Bbb R^2-\{0\}\to\Bbb R^2$, so the derivative at the point $(x,y)$ should be a linear map from $\Bbb R^2$ to $\Bbb ...
Ted Shifrin's user avatar
4 votes

Convergence of $\sum_{n=1}^\infty \frac{\sin\frac{n^2+1}{n+1}}{\sqrt n}$ (again)

The problem can be generalized. Let $p$ and $q$ be polynomials such that $\deg p=\deg q+1.$ Then the series $$\sum {\sin(p(n)/q(n))\over \sqrt{n}}$$ is convergent. Indeed we have $$p(x)=(x+a)q(x)+r(x),...
Ryszard Szwarc's user avatar
7 votes
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Convergence of $\sum_{n=1}^\infty \frac{\sin\frac{n^2+1}{n+1}}{\sqrt n}$ (again)

You can use \begin{align} \left|\sin\left(\frac{n^2 + 1}{n+1}\right) - \sin(n-1)\right| &= \left|\sin\left(n-1+\frac2{n+1}\right) - \sin(n-1)\right|\\ & \le \frac2{n+1} \end{align} this proves ...
Kroki's user avatar
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1 vote
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Convergence of $\sum_{n=1}^\infty \frac{\sin\frac{n^2+1}{n+1}}{\sqrt n}$

Your intuition is correct. Use the angle sum formula: $$ \sin\left(n+\frac{n-1}{n+1}\right) = \sin(n)\cos\left(\frac{n-1}{n+1}\right) + \sin\left(\frac{n-1}{n+1}\right)\cos(n) $$ that way you can ...
Dark Malthorp's user avatar
1 vote

Determining Conditions for the Exchange of Limit and Integral

Point 1. We typically expect the limit and integral to be interchangeable if there is no "redistribution of mass", either because all the mass distributions are confined in a region of ...
Sangchul Lee's user avatar
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Validity of the proof $\mathbb R$ is uncountable

Just gathering up the several apt comments: there is simply no assurance, no general reason, why a countable subset of $(0,1)$ (etc.) should occur as a monotone increasing sequence. The comments give ...
paul garrett's user avatar
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Can Darboux Theorem be stronger by making the derivative at c continuous?

Here is a better counter-example such that only one such $c$ exists, at which $f'$ is discontinuous. Define the triangular function $h_n$ such that $h_n(2^{-n})=1$, and $h_n(2^{-n} - 2^{-2(n+2)}) = ...
Willow Wisp's user avatar
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3 votes
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How to prove $\oint_{A}\mathbf{E}\cdot d\mathbf{A}=\frac{q}{\epsilon_{0}}$ mathematically and rigorously?

You can obtain a rigorous proof by first obtaining the result for $\ A=RS^2 ,$ the surface of a sphere of radius $ R\ $ centred on the origin, and then using Gauss's divergence theorem to extend it to ...
lonza leggiera's user avatar
0 votes

A Challenging Logarithmic Integral $\int_0^1 \frac{\log(x)\log(1-x)\log^2(1+x)}{x}dx$

Okay, I simplified my solution. My original solution was about 3 times longer than this, so it is so lucky for me to find a shortcut like this :) Step 1. Let $I$ be the integral in question: \begin{...
I hate over moderation's user avatar
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Is $\mathbb{R}^3\setminus\{(x,y,z)\in\mathbb{R}^3\mid x\leq 1,y\leq1,-1\leq z\leq 1\}$ a star domain?

Call your set $U$, and suppose $(x,y,z) \in U$ can be used as base point. If $z\neq 0$, take the line that passes through the origin and $(x,y,z)$, that is, $$f(t)=(tx,ty,tz), \quad t\in\mathbb{R}$$ ...
Julio Puerta's user avatar
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Is $\mathbb{R}^3\setminus\{(x,y,z)\in\mathbb{R}^3\mid x\leq 1,y\leq1,-1\leq z\leq 1\}$ a star domain?

A simpler approach might be as follows. Assume for the sake of contradiction that $a\in M$ is a center. Now find a pair of points $(0,0,1+\epsilon)\in M$ and $(0,0,-1-\epsilon)\in M$ such that at ...
RobPratt's user avatar
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Example for proper inclusion in Lorentz space

Try $f=\sum_{j=1}^\infty j^{-q}2^{-j/p}\chi_{(0,2^{-j})}$, or equivalently $f(x)=x^{-1/p}|\log x|^{-q}\chi_{(0,\frac12)}$.
Liding Yao's user avatar
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1 vote

What is the definition of an "edge" and a "degenerate vertex" in a polytope?

I'm a little late but I'm currently studying for an exam in Optimization and we actually have a definition given for degeneracy! It reads as follows: If you have given a matrix $\text{A} \in \mathbb{R}...
moldbellchains's user avatar
0 votes
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Are convergent sequences closed under uniform convergence?

The answer to the second question is no. If $\lim_{k\to\infty}a_{\bullet,k}=A$ then $\lim_{n\to\infty}A_n$ exists and equals $A$.Proof: Let $\epsilon>0$. Let $N$ be such that $\forall n\ge N,\...
Anne Bauval's user avatar
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I want to prove $\lim_{x \to0}\frac{\sqrt{x+1}- \sqrt{1-x}}{x}=1$

To restate the problem to make it easier to reference: $$ \text{I want to prove }\lim_{x \to 0}\frac{\sqrt{x+1}- \sqrt{1-x}}{x}=1 \tag{Eq. 1}$$ The precise definition of the Calculus Limit is given at ...
Stephen Elliott's user avatar
3 votes

I want to prove $\lim_{x \to0}\frac{\sqrt{x+1}- \sqrt{1-x}}{x}=1$

The simplest way is to rationalise the given function with $$\frac{\sqrt{x+1}+\sqrt{1-x}}{\sqrt{x+1}+\sqrt{1-x}}$$
Mahendra Varma's user avatar
4 votes

Prove $\lim_{x\rightarrow +\infty}f(x)=0$ under the conditions $|f'(x)|\leq 1/x$ and $\lim_{R\rightarrow +\infty}\frac{1}{R}\int_0^R|f(x)|dx=0$

Put $\displaystyle F(x)=\int_0^x f(t)dt$. For any $\lambda>1$ and $n\in\mathbb N$ there exists $c_{\lambda,n}\in ]\lambda^n;\lambda^{n+1}[$ such that $\displaystyle \frac{F(\lambda^{n+1})-F(\lambda^...
Fred Bernard's user avatar
1 vote

L'Hopital's rule with dual numbers

Non-standard analysis is, at least in calculus, about replacing limits with algebraic operations. The dual numbers, and in generalization, truncated Taylor series, are one step towards this, doing &...
Lutz Lehmann's user avatar
1 vote

Proving $e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu}+\left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2}$

I will show that any $0 < c < \dfrac13$ will do. Start with $\dfrac{1-x^n}{1-x} =\sum_{k=0}^{n-1} x^k $, so $\dfrac1{1-x} =\sum_{k=0}^{n-1} x^k+\dfrac{x^n}{1-x} $ or, putting $-x$ for $x$, $\...
marty cohen's user avatar
2 votes

Proving $e^{-\mu}\left(\left(\frac{e}{1+\delta}\right)^{(1+\delta)\mu}+\left(\frac{e}{1-\delta}\right)^{(1-\delta)\mu}\right) \le 2e^{-C\mu\delta^2}$

Why not using: $$\frac{e}{1+\delta} \ge 1 \implies \left(\frac{e}{1+\delta}\right)^{\mu (1-\delta)} \le \left(\frac{e}{1+\delta}\right)^{\mu (1+\delta)}$$ and use the inequality from the other ...
Kroki's user avatar
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0 votes
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Limit Interchange for double sequence

Your proof seems correct to me, and here is a proof of your conjecture that we can drop the 3rd hypothesis: Assuming $\lim_{k\to\infty}f_n(k)= \ell_n$, $\lim_{n\to\infty}\ell_n= \ell$, and $f_n\to f$ ...
Anne Bauval's user avatar
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4 votes
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Confusion about Limits (Rationals)

Your argument is correct.If you want a formal proof you can use the definition of continuity in terms of limit of sequences e.g., Define the sequence $x_n=\frac{1}{2^n}$ and $y_n=\frac{\pi}{2^n}$ ...
Marco's user avatar
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3 votes

How to rewrite $\cos{2n\theta}$ as a summation of $\sin\theta$

This appears as an exercise in G. H. Hardy's A Course of Pure Mathematics, 10th edition, on page 397. Let us start with an integral $$J_m=\int_0^x \sin^m t \sin a(x-t) \, dt\tag{1}$$ where $m$ is a ...
Paramanand Singh's user avatar
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0 votes
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Example for a particular function

What about defining $f(\frac{1}{2^{n}}) = \frac{1}{n}$ if $n$ is even and $f(\frac{1}{2^n}) = 0$ if $n$ is odd. Then you can extend $f$ to a continuous function on $(0,1)$ by interpolating linearly. ...
Alex Ortiz's user avatar
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5 votes
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Confusions about the definition of "residue" in Ahlfors' Complex Analysis

If you look back a couple of pages to p.147 you'll find this: The numbers $P_i=\int_{\gamma_i}f\,dz$ depend only on the function, and not on $\gamma$. They are called modules of periodicity of the ...
Gareth McCaughan's user avatar
0 votes

Prove $a^\alpha b^{1-\alpha} \le \alpha a + (1 - \alpha)b, \; a,b > 0,\; 0 < \alpha < 1$

-Let prove that: $ x^{a}y^{\left(1-a\right)} \leq ax+\left(1-a\right)y$ for $0 \leq a \leq 1$ for $0<a<1$ and $x,y>0$. -We have that $ ax+(1-a)y = y(1+ a\frac{(x-y)}{y}) $, now let focus on $(...
OffHakhol's user avatar
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2 votes
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Kernel of an operator defined by a power series

Let $L^\intercal$ be the transpose operator of $L$. Using the integral you can easily prove that $$(L_1 L_2)^\intercal=L_2^\intercal L_1^\intercal.$$ Indeed, \begin{align} \left(L_1L_2\right)^{\...
Kroki's user avatar
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1 vote

Problems regarding the denseness of $a\mathbb{Z}+\mathbb{Z}$

Since $\{ n\alpha \}$ is dense in $[0,1]$, Part $(a)$ is almost obvious. For Part $(b)$, let's assume: $$B_1=[\{m_1\alpha\} -\frac{1}{9^{m_1}}, \{m_1\alpha\} +\frac{1}{9^{m_1}}],$$ where $m_1 \in \...
Reza Rajaei's user avatar
  • 4,773
3 votes

Proving the convergence of a series with very little information

This is essentially very similar to Sangchul's solution, but I think the approach might be easier to understand/motivate. Hints/Observations towards a solution. If you're stuck, explain what you've ...
Calvin Lin's user avatar
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Kernel of an operator defined by a power series

This is more of a comment that wouldn't fit rather than a proper answer, but we can find "simple" expressions of $A_K(y,x)$ and $B_K(y,x)$ relatively to $A_{K^*}$ and $B_{K^*}$ thanks to the ...
Bruno B's user avatar
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5 votes
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Proving the convergence of a series with very little information

Let $(a_n)$ be any sequence as in OP. We begin by establishing key lemmas. Lemma 1. For any $l \geq m$, $$ \sum_{n=l}^{\infty} a_n \leq m a_{\lfloor l/m \rfloor}. $$ Proof of Lemma 1. By the ...
Sangchul Lee's user avatar
2 votes

Is it possible that it exists a $C>0$ that verify for all continuous function on $[a;b]$; $\|f\|_{\infty} \leq C \|f\|_2 $?

As you see based on Matt's example, a continuous function (even a smooth function) can't be controlled in $L^\infty$ by its $L^2$ norm. The example in that answer is a function which has a derivative $...
Alex Ortiz's user avatar
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3 votes
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Is it possible that it exists a $C>0$ that verify for all continuous function on $[a;b]$; $\|f\|_{\infty} \leq C \|f\|_2 $?

The answer is no, there is no such $C$. For example, it is possible to construct a sequence of functions $(f_n)$ which are all continuous and satisfies both $$ \lim_{n \rightarrow \infty} \|f_n\|_2 \...
Matt Werenski's user avatar

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