Michael Seifert
  • Member for 6 years, 5 months
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  • New London, CT
9 answers
29 votes
6k views
Do $3/8$ (37.5%) of Quadratics Have No $x$-Intercepts?
45 votes

It's helpful to think of each quadratic polynomial as corresponding to a point in "coefficient space", an abstract 3D space whose coordinates are $(a,b,c)$. The question "What fraction of quadratic ...

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7 answers
10 votes
1k views
What's a fair way to share fees in a group road trip with a personal and a rental car?
16 votes

Here's how I would go about dividing the expenses: Tally up the expenses. This should be the total of rental fees, fuel, tolls, and an agreed-upon amount for amortization & insurance for the ...

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4 answers
19 votes
3k views
$f : \mathbb{R}^+ → \mathbb{R}$ with $f(0) = f'(0) = 0$ and $f(x) < x^2$ and $f',f'',f''' > 0$?
16 votes

Your intuition is correct for a slightly different statement: There is no function $f(x)$ on $x \ge 0$ such that $f(0)=0$, $f'(0)=0$, $f(x)&lt;x^2$ for $x&gt;0$ and that the third derivative of $f(...

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2 answers
6 votes
878 views
Clay Institute Navier Stokes Part 2
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16 votes

I think you're misunderstanding the problem. The problem is not to show that functions $\{\mathbf{u}(\mathbf{x},t), P(\mathbf{x},t)\}$ exist which satisfy the Navier-Stokes equations; that much is ...

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6 answers
20 votes
4k views
Is the arc length always irrational between two rational points?
15 votes

Consider the two points $(-\frac12,0)$ and $(\frac12,0)$. For any real value of $y_0$, we can draw a circular arc between these two points which is centered at $(0,y_0)$ and which lies entirely in ...

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3 answers
5 votes
481 views
The functional equation $f(x+x) = f(x)f(x)$
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14 votes

Summary: There is an uncountably infinite family of $\mathcal{C}^\infty$ functions on $(0, +\infty)$ satisfying this equation, and these functions are also continuous at $x = 0$ if we demand $f(0) = 1$...

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3 answers
8 votes
616 views
Unknown syntax with multiplication?
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12 votes

It means "the product over all pairs $i, j$ where $i &lt; j$." For example, if $p(i)$ is defined for $i = 1, 2, 3$, then $$ \prod_{i&lt;j} \;[\,p(j) - p(i)\;] = [ p(2) - p(1) ] [p(3) - p(1)] [p(3) - ...

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2 answers
2 votes
404 views
Is $(p-1)!+p$ a prime for every prime $p$?
12 votes

You just didn't go far enough. While $(p-1)! + p$ is prime for $p = 2, 3, 5, 7, $ and $11$, it is composite for $p = 13, 17, 19, 23, 29, 31, 37, 41, 43,$ and $47$. $p = 53$ is the next prime for ...

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2 answers
7 votes
241 views
Why the substitution is not working even though its bijective?
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9 votes

It sounds like what you did was to write $$ \int_0^\infty \left( \sin(1/x) - \frac{\sin(\pi/x)}{\pi} \right) \,dx = \int_0^\infty \sin(1/x)\, dx - \int_0^\infty \frac{\sin(\pi/x)}{\pi} \,dx $$ You ...

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6 answers
2 votes
60 views
Choose $a, b$ so that $\cos(x) - \frac{1+ax^2}{1+bx^2}$ would be as infinitely small as possible on ${x \to 0}$ using Taylor polynomial
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8 votes

The quick-and-dirty method: \begin{align*} f(x) = \frac{1 + a x^2}{1 + bx^2} &amp;= (1 + a x^2) \left( 1 - b x^2 + b^2 x^4 - b^3 x^6 + \cdots \right) \\&amp;= 1 - (b - a) x^2 + (b^2 - ab) x^4 - (b^3 - ...

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1 answers
6 votes
533 views
Is it possible to convert a linear system in an ODE of higher order
7 votes

This is not a complete answer (yet), and I encourage other folks to build on it: A set of autonomous coupled linear first-order ODEs can always be written in the form $$ y' = A y $$ where $y$ stands ...

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2 answers
4 votes
65 views
How to calculate the dimensions of the required 20 regular hexagons and 12 regular pentagons for a sphere of given diameter (the soccer ball issue)
6 votes

The &quot;soccer ball&quot; polyhedron is formally called a truncated icosahedron. According to that Wikipedia article, if the sides are of length $a$, then the circumscribing sphere has a radius of ...

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2 answers
3 votes
175 views
If $A$, $B$ idempotent and $AB=0$, then $A+B$ idempotent.
6 votes

Counterexample: Let $$ A = \begin{bmatrix} 0 &amp; -1 \\ 0 &amp; 1 \end{bmatrix} \qquad B = \begin{bmatrix} 1 &amp; 0 \\ 0 &amp; 0 \end{bmatrix} $$ Both matrices are idempotent. Then $AB = 0$ but $...

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3 answers
6 votes
187 views
Is this a valid convergence test (for sequences)?
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6 votes

Idea of counterexample: Construct an oscillating function whose "rate of oscillation" decreases as $n$ increases. The idea is to make the differences between successive values smaller due to the ...

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3 answers
6 votes
199 views
Coloring of $\mathbb{R}^3$ into 3 colors
6 votes

You have proven that for any distance $\delta$, we can find two points at that distance whose colors match. What you haven't proven (and what is requested) is that we can always find two points at ...

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1 answers
5 votes
2k views
PDF of uniform distribution over the hypersphere and the hyperball
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6 votes

Maybe I'm missing something, but I'm not sure why it would be any more complicated than $$ \mathcal{P}(\vec{x};\vec{x_0},r) = \frac{1}{V_n} \Theta(r - ||\vec{x} - \vec{x_0}||) $$ for the interior of a ...

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3 answers
3 votes
7k views
Is curl of a given vector always perpendicular to the given vector field?
6 votes

No. As an example, consider the vector field $\vec{F} = y \hat{i} - x \hat{j} + \hat{k}$, whose curl is $\vec{\nabla} \times \vec{F} = 2 \hat{k}$. This is obviously not perpendular to $\vec{F}$ ...

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2 answers
8 votes
2k views
Is this contraction of metric tensor derivatives symmetric?
6 votes

It does appear to be symmetric, though the proof I came up with requires the introduction of a covariant derivative operator. There may be another proof out there that doesn't require quite so much ...

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1 answers
1 votes
43 views
Feynman Lectures p264: How does $r^2 = \rho^2 + a^2$ imply $\rho\, {\rm d}\rho = r\, {\rm d}r$
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5 votes

This is a mathematical technique known as taking the total differential of both sides of the previous equation. On the left-hand side, we have $\mathrm{d}(r^2) = 2 r \, \mathrm{d}r$, and similarly ...

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1 answers
5 votes
107 views
Find the determinant of a particular matrix without a calculator.
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5 votes

HINT: Note that $$ \begin{bmatrix}283 \\ 3136 \\ 6776 \\ 2464\end{bmatrix} - 56 \begin{bmatrix}5 \\ 56 \\ 121 \\ 44\end{bmatrix} = \begin{bmatrix} 3 \\ 0 \\ 0 \\ 0 \end{bmatrix} $$ which allows you ...

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1 answers
1 votes
81 views
Wanted: metric in which the volume of euclidean space is finite
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5 votes

If we express our metric in polar coordinates, it becomes a bit easier to see what to do. Let's suppose that our desired metric is related to the Euclidean metric by a conformal transformation of the ...

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3 answers
5 votes
218 views
A simple integral with one question
5 votes

You can use the negative square root if you want; it all works out the same way in the end. If we denote the positive square root of $x$ by $\sqrt{x}$, we can substitute $t = -\sqrt{x}$ instead. We ...

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2 answers
0 votes
115 views
Asymptotics for ODE with no closed form solution
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5 votes

To do this formally, expand $x(t)$ in a "power series at infinity", multiplied by an unknown power of $t$: $$ x(t) = t^\alpha \sum_{n=0}^\infty \frac{a_n}{t^n}. $$ We can assume WLOG that $a_0 \neq 0$,...

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5 answers
28 votes
4k views
Large powers of sine appear Gaussian -- why?
5 votes

Another observation too long for a comment: It's not actually that good of an approximation if you look at the ratio of $f_k(x) = \sin^k(x)$ to the value of the $g_k(x) = \exp ( - k (x- \pi/2)^2)$ for ...

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1 answers
1 votes
109 views
is $T = \frac{d}{dx}$ continuous on the space of continuously differentiable functions on an interval?
5 votes

No. Consider the sequence of functions defined by $$ f_n(x) = \frac{1}{n} \cos (n^2 x). $$ on $[0, 2\pi]$. We have $ \max |f_n(x)| = \frac{1}{n}$, but $\max|f'_n(x)| = n$. Thus, $f_n \to 0$ as $n \...

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15 answers
13 votes
6k views
Counting the Number of Real Roots of $y^{3}-3y+1$
5 votes

Just for fun: we note that since we have a "depressed cubic" of the form $t^3 + pt + q = 0$ (i.e., no quadratic term), the roots have a nice geometric interpretation: $$ t_k = 2 \sqrt{ -\frac{p}{3}} \...

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1 answers
3 votes
2k views
Alternative to Arnold's mathematical methods
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5 votes

I'm a physics professor, and more mathematically inclined than your average physicist. Arnol'd is a beautiful textbook, but it is most emphatically not the one to learn Lagrangian mechanics from. (I ...

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2 answers
3 votes
2k views
Divergence $0$ everywhere implies Flux $0$?
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5 votes

If the divergence of a vector field is zero ($\nabla \cdot \vec{v} = 0$), then the flux of that vector field through any closed surface is zero. This is a consequence of the divergence theorem: for ...

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1 answers
3 votes
106 views
Integral of a product of Bessel functions of the first kind
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4 votes

The general relationship you need is Eqn. 10.22.63 in the Digital Library of Mathematical Functions: $$ \int_{0}^{\infty}J_{\mu}\left(ax\right)J_{\mu-1}\left(bx\right)\mathrm{d}x=% \begin{cases}b^{\mu-...

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2 answers
4 votes
129 views
contour integration of $\int_0^\infty \frac{\ln(x)}{x^2-1}dx$
Accepted answer
4 votes

If I'm not mistaken, you've done your substitutions incorrectly. For the straight-line segment of the integral in the upper half-plane (call it $\gamma_1$), for example, the integral is $$ I_1 = \...

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