JavaMan
  • Member for 10 years, 11 months
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Limit involving exponential functions
2 votes

Hint: Writing the numerator of the fraction in the limit as $$ p \left( e^{- \frac{y}{1+p}} + e^{- \frac{y}{1-p}} \right) + e^{- \frac{y}{1+p}} - e^{- \frac{y}{1-p}} $$ should help.

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1 not equal to 0 stated as a condition, why?
Accepted answer
3 votes

The case $1 = 0$ occurs when you are talking about the trivial ring $R = \{0\}$. It turns out $R$ is the trivial ring if and only if $1 = 0$, so stating $1 \neq 0$ is arguably the easiest way to say $...

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Supplemental number theory text to Montgomery and Vaughan
3 votes

I would also add to Eric's answer that Tenenbaum's "Introduction to Analytic and Probabilistic Number" is a great resource if one is limited to the number of books you can carry.

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calculus- limit includes polynomial
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0 votes

Write $P(x) = \displaystyle\sum_{n=0}^k a_n x^n$, and note $|P(x)| \leq k|a_k|x^k$ for all $x \geq 1$. Hint: $$\begin{align} |P(x)| \leq C_k x^k &\Longrightarrow \log P(x) \leq \log (C_k x^k) ...

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some short proofs on group theory
1 votes

Q1. For the first question, since $G$ is a group, each element $a_i$ has an inverse $a_i^{-1}$. Note that inverses are unique. Let $\{b_i\}_{i=1}^k$ denote the subset of elements of order $2$ (I.e., ...

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Proof of $\lim_{n\to \infty} \sqrt[n]{n}=1$
Accepted answer
31 votes

Since $x \mapsto \log x$ is a continuous function, and since continuous functions respect limits: $$ \lim_{n \to \infty} f(g(n)) = f\left( \lim_{n \to \infty} g(n) \right), $$ for continuous ...

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Plots & complex numbers
3 votes

To graph these equations, I would just write out $z = x + iy$, and then use a standard graphing device, recognizing that we can identify $\mathbb{C}$ with the plane $\mathbb{R}^2$. For instance, for ...

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Geometric mean never exceeds arithmetic mean
Accepted answer
8 votes

Hint: Using induction: Show the base case $n = 2$. Show that if the statement is true for $2^n$, then it is true for $2^{n+1}$. Show that if the statement is true for $n$, then it is true for $n - ...

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Calculating the upper Minkowski dimension of the set $\{0,1,\frac{1}{2}, \frac{1}{3}, \ldots \}$
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4 votes

In practice, I prefer the (equivalent) following definition of upper Minkowski dimension: $$\overline{\dim}(E) = \limsup_{\delta \to 0} \frac{\log N(E,\delta)}{\log(1/\delta)}. $$ $N(E, \delta)$ is ...

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Which one result in mathematics has surprised you the most?
9 votes

One of the most surprising results I have ever seen is the Universality Theorem of Voronin which states that any nonvanishing analytic function can be well -approximated by $\zeta(s)$ somewhere in the ...

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Basis functions in Fourier Series
2 votes

As Jonas points out, a basis function is simply a function in the basis of a function space. Most likely, you are (whether it explicitly says so or not) working in the function space $L^2[0,1]$ (or ...

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How can I compute this limit
Accepted answer
1 votes

I hope I'm not misunderstanding the question, but here's what I think (I apologize ahead of time as it is late!) Write $\eta(t) = \displaystyle\mathop{\Delta}_{t} \left[\frac{\phi(t) e^{-itx}}{it} \...

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Implies sign in math
9 votes

The equal sign $"="$ should be used when you have two quantities which are equal. For example: $$x^2 + 3x - 4 = (x+4)(x-1).$$ The $"\Rightarrow"$ should be used when relating ...

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Show that $a^n \mid b^n$ implies $a \mid b$
6 votes

Hint: Write $b^n = q_1^{n \alpha_1} \dots q_k^{n \alpha_k}$, where $\alpha_i > 0$. If $a^n | b^n$, then $a^n = q_1^{n \beta_1} \dots q_k^{n \beta_k}$, where $0 \leq \beta_k \leq \alpha_k$.

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Formula for Geometric Progression
2 votes

HINT: $$\prod_{i=2}^n \left( 1 - \frac{1}{i^2}\right) = \prod_{i=2}^n \left(\frac{i^2 - 1}{i^2} \right)=\prod_{i=2}^n \frac{(i-1)(i+1)}{i^2} = \frac{\displaystyle\prod_{i=2}^n (i-1) \prod_{i=2}^n (i+1)...

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Is it standard to say $-i \log(-1)$ is $\pi$?
7 votes

The (principal value) of the complex logarithm is defined as $\log z = \ln |z| + i Arg(z)$. Therefore, $$\log(-1) = \ln|-1| + i Arg(-1) = 0 + i \pi.$$ and then, one simply gets $$ -i \log(-1) = -...

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Help with implicit differentation
Accepted answer
1 votes

Jason. You should not be solving for $y$, you should be solving for $\frac{dy}{dx}$. There was an error in your calculation, however, when you applied the chain rule to find $\frac{d}{dx} (\sin (\pi ...

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Erroneous numerical approximations of $\zeta\left(\frac{1}{2}\right)$?
17 votes

To supplement Derek's answer, the Riemann zeta function $\zeta(s) = \sum_{n} n^{-s}$ was originally only considered for $s = \sigma + i t$, where $\sigma > 1$. Note that when $\sigma > 1$, the ...

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Big $O$ vs Big $\Theta$
Accepted answer
2 votes

If $g(x)$ and $f(x)$ tends to $\infty$, then there is a value $x_0$ such that for $x > x_0$, $g(x)$ and $f(x)$ are strictly positive. Therefore, if $-C \leq f(x) - g(x) \leq C$, then for $x > ...

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Another question in calculus
Accepted answer
4 votes

Add xy to both sides (To get $y'' = xy$), and start taking derivatives. A pattern emerges. Edit: You should get: $y^{(3)} = y + xy'$ $y^{(4)} = 2y' + xy''$ $y^{(5)} = 3y'' ...

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Examples of non symmetric distances
4 votes

This may or may not be what you're looking for, but I thought the idea was fun, so I'll share: This will be a non-symmetric metric on $\mathbb{Z}^+$: Suppose that one is only allowed to move left ...

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How to pick a thesis advisor?
8 votes

This is an interesting take: http://blogs.ams.org/mathgradblog/2010/11/17/the-minimalist-approach-to-an-advisor/ ${}{}{}{}{}$

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Is the complex exponential function injective, surjective and/or bijective - and why?
4 votes

Richard Palais has a wonderful program for visualizing some complex maps called 3D-XplorMath which can be downloaded for free at http://3d-xplormath.org/. It also has some nice visual for 3d surfaces ...

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Where is $1 gone!
2 votes

The $\$27$ includes the $\$25$ they paid for food, plus the $\$2$ to the waiter. They pocketed the other three dollars.

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Problems that are largely believed to be true, but are unresolved
6 votes

The Kakeya conjecture. It states that a set in $\mathbb{R}^d$ containing a line in every direction has Hausdorff dimension $d$. It is solved for $d = 2$ and open for $d \geq 3$. The finite field ...

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