instruction stringlengths 12 30k |
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What is wrong with this?
![alt text][1]
**[SOURCE][2]**
[1]: https://i.stack.imgur.com/GU8wd.jpg
[2]: http://www.bestwtf.com/2010/11/explaining-why-pi-is-4.html |
Is value of $\pi$ = 4 ? |
I'm interested in the potential of such a technique. I got the idea from Moron's answer to <a href="http://math.stackexchange.com/questions/9402/calculating-an-integral-without-using-complex-analysis">this question</a>, which uses the technique of differentiation under the integral.
Now, I'd like to consider this i... |
Will moving differentiation from inside, to outside an integral, change the result? |
I'm interested in the potential of such a technique. I got the idea from Moron's answer to <a href="http://math.stackexchange.com/questions/9402/9409#9409">this question</a>, which uses the technique of differentiation under the integral.
Now, I'd like to consider this integral:
$$\int_{-\pi}^\pi \cos{(y(1-e^{i\... |
Edit:
---------------------
-----------------------------------
So most people have said that this is not possible, but I have come across a formulae for subdividing an icosahedron (which I dont have to hand, but goes something like this)
V = 10 * u² + 2
E = 30 * u²
F = 20 * u²
Where U = the frequency o... |
this step in the proof is confusing me:
$$\sum_1^\infty {\frac{4^{n}}{3^{n-1}}}\qquad \longrightarrow \qquad\sum_1^\infty 4\left(\frac43\right)^{n-1}$$
please explain how/why this happened?
cheers,
gregg
|
Edit:
---------------------
-----------------------------------
The answers I seem to be getting here are that this is not possible, but I have come across a formulae for subdividing an icosahedron (which I dont have to hand, but goes something like this)
V = 10 * u² + 2
E = 30 * u²
F = 20 * u²
Where U ... |
I run a website that generates a specialist form of weather data called [degree days][1]. Degree days are effectively a measure of outside air temperatures that is known to correlate with building energy consumption. Without going into unnecessary detail, temperature is a function of time, and degree days are essenti... |
I run a website that generates [degree days][1], a specialist form of weather data used for calculations relating to building energy consumption. Without going into unnecessary detail, temperature is a function of time, and degree days are essentially the integral of that function.
At the moment our site calculates... |
if i have matrices A and B then wat is x to minimize ||A-xB||_F ? |
If I have matrices $\mathbf A$ and $\mathbf B$ then what is $\mathbf x$ to minimize $\|\mathbf A-\mathbf x\mathbf B\|_F$? |
If $\gcd(a,b) = 1$, $x \equiv 0 \pmod{a}$, $x \equiv 0 \pmod{b}$ then prove that $x \equiv 0 \pmod{ab}$.
My attempt at answering the question:
\begin{align*}
x &\equiv 0 \pmod{a}\\\
&\Longrightarrow x\text{ is divisible by $a$}\\\
&\Longrightarrow x = ma\text{ for some integer $m$}\\\
\ \\\
x &\equiv 0 \pmod... |
I'm trying to figure out whether my hardware function for mapping operating system pages to DDR memory controllers is [injective][1]:
$$F(x) = a x \pmod{b},\text{ where $x$ in $[0,b)$, $a$ and $b$ are co-prime}$$
My hunch is that it is injective and I think it has to do with the [Chinese Remainder Theorem][2], th... |
When $a$ and $b$ are co-prime, is $F(x) = a x \mod b$, $x$ in $[0,b)$ , $a$ and $b$ co-prime, an injective function? |
What is an R-trivial language? What is an R-trivial monoid? |
formal languages: what does R-trivial mean? |
For which values of $\alpha, \beta$, $\int_{0}^{1}\frac{\ln^\alpha(1+x)}{x^\beta(1-x)^\alpha}$ exists? |
What is wrong with this? Fundamentally, that you have jumped in without a definition of the length of a arc. |
The problem:
***Three poles standing at the points A,B and C subtend angles $\alpha$, $\beta$ and $\gamma$ respectively,at the circumcentre of the triangle ABC.If the angles of this poles are in arithmetic progression; then show that $\cot \alpha$, $\cot \beta$ and $\cot \gamma $ are in harmonic progression.***
N... |
The problem:
***Three poles standing at the points A,B and C subtend angles $\alpha$, $\beta$ and $\gamma$ respectively,at the circumcentre of the triangle ABC.If the angles of this poles are in arithmetic progression; then show that $\cot \alpha$, $\cot \beta$ and $\cot \gamma $ are in harmonic progression.***
N... |
One can write out an integral whose solution gives the number of solutions to the NP-Complete Number Partition Problem and I'm wondering if anyone has an suggestions or ideas on who to solve or approximate this integral analytically or numerically.
Given $a_0, a_1, \dots, a_{n-1} \in \mathbb{Z}$, the Number Partitio... |
Let
$R(t) = u + ct - \sum_{k=1}^{X(t)}Z_{k}, t\geq 0 $,
be a risk process, where $u> 0$ is the initial capital of the insurance company and $c> 0$ is a premium rate. We know that the number of claims received by the insurance company is in accordance with a homogenous Poisson Process $X(t), t\geq 0$, with parame... |
What is an R-trivial language? What is an R-trivial monoid?
Context: Formal languages. Afaik, R-trivial languages is a subset of the starfree languages.
I mostly have background in formal languages and automata theory but not so much with the syntactic monoid characterization. So it would be nice to give a basic ... |
what is the product of delta function with itself ? what is the dot product with itself ? |
what is product of delta function with itself ? |
There are a number of ways of dealing with free/bound variables. The simplest (and least user friendly) is [de Bruijn Indices][1]. In this system, a bound variable is named based on how "far" it is from the binder. For example:
$\lambda x. \lambda y.~ x~y$
is
$\lambda . \lambda .~ 1~0$
in de Bruijn notation... |
where can I find a proof that truncated octahedron tiles Euclidean 3D space? |
Truncated octahedron tiles 3D space. Proof? |
Any idea how to proof that when 3D space is tiled with truncated octahedrons, all vertices can be colored black and white such that no 2 vertices sharing the same color are adjacent? |
Truncated octahedron is bipartite. Proof? |
Is every CW complex homotopic to a Delta-Complex? |
Let $\{B(t), t \in \mathbb{R} \}$ be a two sided brownian motion be defined as
$$
B(t) = \begin{cases} B_1(t),\quad t >0 \\
0, \quad t = 0 \\
B_2(-t), \quad t < 0 \end{cases}
$$
where $B_1$ and $B_2$ are independent standard Brownian motions on $\mathbb{R}^+$.
Fix $x_0 > 0$ and let $x_k = B(x_{k-1})$ for $k=1,... |
How do you find out how much a put contract would cost?
Example:
> The premium for a 99.00 Eurodollar futures put option is 0.55 and the
> spot price is 98.51
So I have the intrinsic value which is `$1225`, but I am not sure how to compute the time value and how much that put contract would cost. Still ... |
Any idea how to proof that when 3D space is tiled with truncated octahedra, all vertices can be colored black and white such that no 2 vertices sharing the same color are adjacent? |
<a href="http://books.google.com/books?id=ZlY7gZacogsC&pg=PA172">Exercise 7.11 of Kurzweil–Stellmacher</a> asks me to:
> Prove that the centralizer of an involution is nonsolvable inside any perfect group with generalized quaternion Sylow 2-subgroups of order at least 16.
I would like to oblige, but I'm having t... |
A direct computation shows that
$$F(x+)=F(x)-\frac{1}{2n^2}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^2}=F(x)-\frac{\pi^2}{16n^2},$$
$$F(x-)=F(x)+\frac{1}{2n^2}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^2}=F(x)+\frac{\pi^2}{16n^2},$$
when $x=m/2n$, with $m$ and $2n$ being relatively prime. (The infinite sum is easily derived fro... |
I am trying to see why $\frac{\sin(nx)}{1+\vert x\vert^2}$ has no convergent subsequence in $L^1(\mathbb{R})$. This is an "optional" homework problem from my disorganized analysis professor which is worth no points, and which I wouldn't do anyway, because I'm terrible at analysis and have no clue how to deal with this... |
<a href="http://books.google.com/books?id=ZlY7gZacogsC&pg=PA172">Exercise 7.11 of Kurzweil–Stellmacher</a> asks me to:
> Prove that the centralizer of an involution is nonsolvable inside any perfect group with generalized quaternion Sylow 2-subgroups of order at least 16.
I would like to oblige, but I'm having t... |
The problem:
***Three poles standing at the points A,B and C subtend angles $\alpha$, $\beta$ and $\gamma$ respectively,at the circumcentre of the triangle ABC.If the heights of this poles are in arithmetic progression; then show that $\cot \alpha$, $\cot \beta$ and $\cot \gamma $ are in harmonic progression.***
... |
The problem:
***Three poles standing at the points A,B and C subtend angles $\alpha$, $\beta$ and $\gamma$ respectively,at the circumcentre of the triangle ABC.If the heights of these poles are in arithmetic progression; then show that $\cot \alpha$, $\cot \beta$ and $\cot \gamma $ are in harmonic progression.***
... |
I'm studying statistics, and would like to better understand the Central Limit Theorem. The proof I found on Wikipedia requires some previous knowledge I do not currently possess.
Is there a quick intuitive explanation you can give as to why this theorem is correct? |
For which values of $\alpha, \beta$, $\int_0^1 \frac{\ln^\alpha(1+x)}{x^\beta(1-x)^\alpha}\mathrm dx$ exists? |
Here is a hint for the first part: you can make min$(a_n,b_n)=0$ for all $n$ (it doesn't get more convergent than that), while the two series $\sum a_n$ and $\sum b_n$ are very far from converging. Once you solve this one, let me know and I will give you a hint for the challenging part. It will build on ideas from the ... |
I am wondering how much a smooth function may be non-analytic, because in proofs, whilst there non-analytic smooth functions, it would suffice if a smooth function were analytic on only a "small set". More exactly:
Let $U \in \mathbb R^n$ be open, $C^\infty = C^\infty(U,\mathbb R)$. A smooth function in $C^\infty$ i... |
Is Robinson Arithmetic complete in the sense of Gödels completeness theorem? And is Robinson Arithmetic incomplete in the sense of Gödels first incompleteness theorem?
If RA is both it would be a good example to explain the different notions of "completeness" used in the two theorems mentioned. |
Is Robinson Arithmetic complete and not-complete? |
If I have 16 different symbols, howmany of these symbols can I combine to create no more than 16.777.216 unique combiations?
Eg. I have all symbols of the Hexadecimal language (16).
Howmany combinations can I create with X amount of symbols to not have more than 16.777.216 unique combinations maximum?
The posi... |
If I have 16 different symbols, howmany of these symbols can I combine to create no more than 16.777.216 unique combiations? |
Let $R$ a root system and $\Delta$ be a simple system of roots of a Lie algebra $\mathfrak g$, $\Delta'\subset \Delta$ and $R(\Delta')=R\cap \mathbb Z(\Delta')$. Define
$$p(\Delta')=\mathfrak h \bigoplus_{\alpha \in R(\Delta')} \mathfrak g_{\alpha} \bigoplus_{\alpha \in R^+ \setminus R^+(\Delta')}\mathfrak g_{\alpha... |
Let $R$ a root system and $\Delta$ be a simple system of roots of a Lie algebra $\mathfrak g$, $\Delta'\subset \Delta$ and $R(\Delta')=R\cap \mathbb Z(\Delta')$. Define
$$p(\Delta')=\mathfrak h \bigoplus_{\alpha \in R(\Delta')} \mathfrak g_{\alpha} \bigoplus_{\alpha \in R^+ \setminus R^+(\Delta')}\mathfrak g_{\alpha... |
When you multiply two power series' that have difference intervals of convergence, what is the product's interval of convergence?
Ex. f(x) converges for |x|<6, g(x) converges for |x|<4. What does f(x)g(x) converge for? |
The 4-associahedron is the 4-dimensional version of Mac Lane's pentagon diagram. If you look at [Trimble's notes on tetracategories](http://math.ucr.edu/home/baez/trimble/tetracategories.html), you can see the obvious difficulty in working with such a diagram (using it to verify things like coherence etc). Are there ... |
Practical approaches to working with nonplanar commutative diagrams? |
Consider a parallelogram which is neither a rhombus nor a rectangle. It is well-known that such shapes do not have an axis of symmetry.
> Is there a simple proof for this?
I prefer a proof that I can give to a kid (~12 year old), but more involved proofs will do as well. |
Are there any advanced topics--preferably at the research frontier--which can nevertheless be explained accurately, if not efficiently, using very down-to-earth ideas which would be accessible to most college students, not necessarily even math majors?
As a prototypical example of what I mean, I offer the book *QED... |
For any fixed $x_0$, there is positive probability that the sequence starting from $x_0$ fails to converge. To see why, take, say, $x_0 = 15$. There is positive probability that $50 < B_t < 60$ for $10 < t < 15$ and $10 < B_t < 20$ for $50 < t < 60$. On this event the sequence oscillates between the intervals $(10,... |
It is proved in *Advanced Calculus* by Angus Taylor, § 20.8, that
$$\log n!=\log \left( \left( \frac{n}{e}\right) ^{n}\sqrt{2\pi n}\right)
+r_{n},$$
where
$$r_{n}=\sum_{k=1}^{\infty }S_{k}$$
with
$$S_{k}=\sum_{p=n+1}^{\infty }\frac{k}{2(k+1)(k+2)p^{k+1}}.$$
This formula for $r_{n}$ provides a method ... |
For any fixed $x_0$, there is positive probability that the sequence starting from $x_0$ fails to converge. To see why, take, say, $x_0 = 15$. There is positive probability that $50 < B_t < 60$ for $10 < t < 15$ and $10 < B_t < 20$ for $50 < t < 60$. On this event the sequence oscillates between the intervals $(10,... |
I can think of a couple of books that may be close to what you claim to be looking for. One is [Fearless Symmetry](http://books.google.co.cr/books?id=h7oUr88S7JQC&printsec=frontcover&dq=fearless+symmetry&hl=es&ei=_4H6TLKHFoL88AaU7sWCCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCUQ6AEwAA#v=onepage&q&f=false) by Avner... |
<a href="http://books.google.com/books?id=ZlY7gZacogsC&pg=PA172">Exercise 7.11 of Kurzweil–Stellmacher</a> asks me to:
> Prove that the centralizer of an involution is nonsolvable inside any perfect group with generalized quaternion Sylow 2-subgroups of order at least 16.
I would like to oblige, but I'm having t... |
The issue is complicated by the fact that many (most?) "mainstream mathematicians" are not really keeping track of whether they are using some form of AC in their arguments. Moreover, they often do not write their proofs in such a way which makes clear either whether they are using AC or whether this use of AC is real... |
What order is the field: $\mathbb{F}_p(\zeta_{p-1}^{1/n})$?
$\mathbb{F}_p^{\times}$ is size $p-1$ and cyclic (with a generator we shall call $\zeta_{p-1}$). Naively, it seems that by taking the $n^{th}$ root of $\zeta_{p-1}$ we get a group of units of size $C_{n(p-1)}$. This, of course, doesn't make sense. What shou... |
So I read the answers in <a href="http://math.stackexchange.com/questions/713/how-to-visualize-a-rank-2-tensor">this</a> question, and don't feel that much closer to an answer about how tensors as multi-linear maps and tensors as "multi-dimensional" matrices are truly related. For instance, it seems that a (1,1)-tenso... |
The issue is complicated by the fact that many (most?) "mainstream mathematicians" are not really keeping track of whether they are using some form of AC in their arguments. Moreover, they often do not write their proofs in such a way which makes clear either whether they are using AC or whether this use of AC is real... |
What order is the field: $\mathbb{F}_p(\zeta_{p-1}^{1/n})$?
$\mathbb{F}_p^{\times}$ is size $p-1$ and cyclic (with a generator we shall call $\zeta_{p-1}$). Naively, it seems that by taking the $n^{th}$ root of $\zeta_{p-1}$ we get a group of units of size $C_{n(p-1)}$. This, of course, doesn't make sense. What shou... |
So, I guess divide it into six sections representing the difference between the numbers, so the first is 0->n, the second through fifth are 1->n and the sixth is 0->n again, and that becomes a generating function of ((1+x+x^2...x^n)^2)((x+^2+...x^n)^4). Does this make sense, and what coefficient represents what value o... |
Let $f_{k,n}$ be the number of ways to select $k$ integers in $\{ 1, 2, ... n \}$ such that no two are consecutive. There are two ways to do this: either the largest one is not $n$, and then you choose $k$ integers in $\{ 1, 2, ... n-1 \}$, or the largest one is $n$, and then you choose $k-1$ integers in $\{ 1, 2, ...... |
1. Show that $\Gamma(y) = \int_0^{\infty}{e^{-x}x^{y-1}\,dx}$ is finite for $y>0$ both as an improper Riemann integral and as a Lebesgue integral.
2. Show $\Gamma'(y) = \int_0^{\infty}{e^{-x}x^{y-1}\ln{x}\,dx}$ for $y>0$.
For one: I've tried simply integrating it as an improper Riemann integral, but you always ... |
So, I guess divide it into six sections representing the difference between the numbers, so the first is $0\rightarrow n$, the second through fifth are $1\rightarrow n$ and the sixth is $0\rightarrow n$ again, and that becomes a generating function of $((1+x+x^2+\cdots+x^n)^2)((x^2+\cdots+x^n)^4)$. Does this make sense... |
Let $\{B(t), t \in \mathbb{R} \}$ be a two sided brownian motion defined as
$$
B(t) = \begin{cases} B_1(t),\quad t >0 \\
0, \quad t = 0 \\
B_2(-t), \quad t < 0 \end{cases}
$$
where $B_1$ and $B_2$ are independent standard Brownian motions on $\mathbb{R}^+$.
Fix $x_0 > 0$ and let $x_k = B(x_{k-1})$ for $k=1,2,\... |
Let $R$ a root system and $\Delta$ be a simple system of roots of a Lie algebra $\mathfrak g$, $\Delta'\subset \Delta$ and $R(\Delta')=R\cap \mathbb Z(\Delta')$. Define
$$p(\Delta')=\mathfrak h \bigoplus_{\alpha \in R(\Delta')} \mathfrak g_{\alpha} \bigoplus_{\alpha \in R^+ \setminus R^+(\Delta')}\mathfrak g_{\alpha... |
Perhaps this will be convincing?
1) The axis of symmetry intersects one of the sides at point which is not the vertex.
In this case, by considering the perpendicular to the axis at the point of intersection of the axis and the side, we see that the axis has to be perpendicular to the side and intersects it at the... |
Are there problems that can't be expressed as languages? |
Let T1 and T2 be two theories having the same set of symbols. Assume that any interpretation of T1 is a model of T1 if and only if it is not a model of T2. Then:
T1 and T2 are finitely axiomatizable.
(i.e. there are finite sets of sentences A1 and A2 such that, for any sentence S:
T1 proves S if and only if A1... |
I have to calculate the surface area of the solid of revolution which is produced from rotating $f: (-1,1) \rightarrow \mathbb{R}, \; f(x) = 1-x^2$ by the x-axis. I do know there is a formula:
$S=2 \pi \cdot \int\limits_{a}^b f(x) \cdot \sqrt{1+f'(x)^2} dx$
Which will work very well. However, I am not very comfo... |
Here is a heuristic to suggest that it will be difficult to define the square of the delta function.
The Fourier transform has the property that it takes the convolution of two functions to the product of their Fourier transforms, and vice versa, ie, it takes the product of two functions to their convolution.
R... |
I have to calculate the surface area of the solid of revolution which is produced from rotating $f: (-1,1) \rightarrow \mathbb{R}$, $f(x) = 1-x^2$ about the $x$-axis. I do know there is a formula:
$$S=2 \pi \int_{a}^b f(x) \sqrt{1+f'(x)^2}\, dx$$
Which will work very well. However, I am not very comfortable with th... |
I'm not too sure, but if I recall correctly, one way you can make this a little easier is to note that (1-x^2)sqrt(1+4x^2)dx is even f(-x) = f(x) and it's range is from -a to a.
So you can do 2*Integral f(x) dx from 0 to a.
btw: Can anyone show me a link on how to type out formulas so that they appear in mathem... |
I'm not too sure, but if I recall correctly, one way you can make this a little easier is to note that $g(x)=(1-x^2)\sqrt{1+4x^2}$ is even $g(-x) = g(x)$ and its range is from $-a$ to $a$.
So you can do $2\int_0^a g(x)\, dx$
btw: Can anyone show me a link on how to type out formulas so that they appear in mathem... |
You did misinterpret Pappus' Theorem. You used the geometric centroid of the *region* between $1-x^2$ and the $x$-axis, whereas Pappus' Theorem wants you to use the geometric centroid of the *curve* $1-x^2$. The geometric centroid of the latter is not $\frac{2}{5}$ but
$$\frac{\int_{-1}^1 (1-x^2) \sqrt{1+4x^2} dx... |
You did misinterpret Pappus' Theorem. You used the geometric centroid of the *region* between $1-x^2$ and the $x$-axis, whereas Pappus' Theorem wants you to use the geometric centroid of the *curve* $1-x^2$. The geometric centroid of the latter is not $\frac{2}{5}$ but (by definition)
$$\frac{\int_{-1}^1 (1-x^2) \... |
X|N,P ~ Binomial(N, P)
N ~ Poisson(11)
P ~ Beta(2,3)
What is the moment generating function for X? |
I'm going over a chapter by chapter review for my calculus final and discovered this problem:
$$y=\int_{\sqrt{x}}^{x^3}\sqrt{t}\sin{t} dt$$
They split it up so that it became:
$$-\int_1^{\sqrt{x}}\sqrt{t}\sin{t} dt + \int_1^{x^3}\sqrt{t}\sin{t} dt $$
Why 1? How was that determined?
|
It doesn't have to be $1$; all it needs to be is in the domain of the function.
What is going on is that you have the Fundamental Theorem, which tells you that the function
$$F(x) = \int_a^x f(t)\,dt$$
has $F'(x) = f(x)$. But this requires (i) the lower limit to be constant; and (ii) the upper limit to be the vari... |
Let $A$ be the infinitesimal generator of a $C_0$-semigroup $(S(t))_{t \geq 0}$. Now, for every $x_0 \in X$ the map $t \mapsto S(t) x_0$ is a mild solution of
\begin{equation}\label{eq:1}
\dot{x} = Ax, \quad x(0) = x_0. \qquad \text{(*)}
\end{equation}
Now, a continuous function $x: [0, \infty) \to X$ is called... |
I'm going over a chapter by chapter review for my calculus final and discovered this problem:
$$y=\int_{\sqrt{x}}^{x^3}\sqrt{t}\sin{t}\;\mathrm dt$$
They split it up so that it became:
$$-\int_1^{\sqrt{x}}\sqrt{t}\sin{t}\;\mathrm dt + \int_1^{x^3}\sqrt{t}\sin{t}\;\mathrm dt $$
Why 1? How was that determined... |
I have to calculate the surface area of the solid of revolution which is produced from rotating $f: (-1,1) \rightarrow \mathbb{R}$, $f(x) = 1-x^2$ about the $x$-axis. I do know there is a formula:
$$S=2 \pi \int_{a}^b f(x) \sqrt{1+f'(x)^2}\, \mathrm dx$$
Which will work very well. However, I am not very comfortable... |
It doesn't have to be $1$; all it needs to be is in the domain of the function.
What is going on is that you have the Fundamental Theorem of Calculus, which tells you that if $f(x)$ is continuous on an interval containing $a$, then the function $F(x)$ defined on that interval by:
$$F(x) = \int_a^x f(t)\,dt$$
is di... |
The following is a screenshot of Mathematica notebook: ![http://static.itmages.ru/i/10/1205/h_1291516944_15480acb6f.png][1]
(full image: http://static.itmages.ru/i/10/1205/h_1291516944_15480acb6f.png)
In theory the blue and the red graphs should be identical: the Gamma function. But in reality we have the blue pl... |
Is it problem of Mathematica or my own? |
In [this question](http://math.stackexchange.com/questions/13050/eee79-and-ultrafinitism/), I needed to assume in my answer that $e^{e^{e^{79}}}$ is not an integer. Is there some standard result in number theory that applies to situations like this? |
In some math job advertisments, it shows "the deadeline is...", but to surprised some schools review applications before that, does that mean we submit application as earlier as possible could be in a better position? (because I normally do that when date is close to the deadline). (I just want to make sure I understan... |
Is it true that $$ \max_{0\le x\le \pi/N}\left|\int_x^{\pi/N} \frac{\sin (N+1/2)t}{\sin t/2} dt \right|=O\left(\frac{1}{N}\right) ?$$
This question arises from my attempt to give a rigorous proof of Gibbs phenomenon. In fact, I only need that the limit is 0 as $N\to\infty$. |
The following is a screenshot of Mathematica notebook: ![Mathematica session][1]
(full image: http://static.itmages.ru/i/10/1205/h_1291516944_15480acb6f.png)
In theory the blue and the red graphs should be identical: the Gamma function. But in reality we have the blue plot representing the absolute value of the G... |
Is $$\max_{0\le x\le \pi/N}\left|\int_x^{\pi/N} \frac{\sin (N+1/2)t}{\sin t/2}\mathrm dt \right|=O\left(\frac{1}{N}\right)$$ true?
This question arises from my attempt to give a rigorous proof of Gibbs phenomenon. In fact, I only need that the limit is 0 as $N\to\infty$. |
I attended a panel talk about this recently and the uniform opinion of the panelists was that one should submit job applications as soon as possible regardless of the deadline. Some schools start reviewing applications much earlier than the deadline (this includes my university). Of course, as long as you send in your ... |
Let $x_N$ be the number in $0<x<\pi/N$ such that $\frac{\sin [(N+1/2)t]}{\sin (t/2)}=1$. Is $$\left|\int_{x_N}^{\pi/N} \frac{\sin [(N+1/2)t]}{\sin (t/2)}\mathrm dt \right|=O\left(\frac{1}{N}\right)$$ true?
This question arises from my attempt to give a rigorous proof of Gibbs phenomenon. In fact, I only need that the ... |
Let $$D_N(x)=\frac{\sin [(N+(1/2))t]}{\sin (t/2)}$$ be the Dirichlet kernel. Let $x_N$ be the number in $0<x<\pi/N$ such that $D_N(x)=1$. Is $$|\int_{x_N}^{\pi/N} D_N(t) dt |=O\left(\frac{1}{N}\right)$$ true?
This question arises from my attempt to give a rigorous proof of Gibbs phenomenon. In fact, I only need that t... |
Let $$D_N(x)=\frac{\sin [(N+(1/2))t]}{\sin (t/2)}$$ be the Dirichlet kernel. Let $x(N)$ be the number in $0<x<\pi/N$ such that $D_N(x)=1$. Is $$|\int_{x(N)}^{\pi/N} D_N(t) dt |=O\left(\frac{1}{N}\right)$$ true?
This question arises from my attempt to give a rigorous proof of Gibbs phenomenon. In fact, I only need that... |
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