instruction stringlengths 12 30k |
|---|
A divisor $d$ of $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}}$ is _unitary_ if and only if $d = p_{1}^{\varepsilon_{1}} \cdots p_{n}^{\varepsilon_{n}}$, where each exponent $\varepsilon_{i}$ is either $0$ or $r_{i}$. Let $D_{k} = ${ $d$ } be the subset of unitary divisors of an integer $k > 1$ satisfying $\omega(d) = \omega... |
Putting the bound on N at 9 seems quite non-random. The continued fraction method for solving x^2 - dy^2 = 1 also works on x^2 - dy^2 = N as long as |N| < sqrt(d), which corresponds exactly to your bound on N when d = 94. Therefore your question almost sounds like it was set by someone who knew continued fractions. How... |
How compute the total number of common divisors between two given pairs of integers ?
For example,if we consider `(12,24)` the answer is **6** i.e `{1,12,2,6,3,4}`.
EDIT : I got an answer [here][1].The solution boils down to finding the number of divisor of the GCD of the two numbers.
[1]: http://books.google... |
Here is a hint for you:
Take $v = (1, 1, 1, \dots, 1)'$, the vector with only ones. By definition of your Matrix, you have $Xv = v$. Thus, $v$ is an eigenvector with eigenvalue 1.
Now change the first basis vector of your vector space to $v$ and keep the other unchanged. In this new basis, what will $X$ looks lik... |
Interesting problem... I tried solving it through isosceles triangles, and I've proved it doesn't work on equilateral triangles. Can anyone give me some hints towards a geometric proof? |
Can two angle bisectors of a triangle be perpendicular? |
What about the ring *R* with ground-set {x<sub>0</sub>, x<sub>1</sub>, x<sub>2</sub>, ..., x<sub>p−1</sub>}, where
> x<sub>a</sub> + x<sub>b</sub> = x<sub>a+b</sub>  (where addition of subscripts is modulo p)<br>
> x<sub>a</sub> ⋅ x<sub>b</sub> = x<sub>0</sub>
for all a,b ∈ {0, 1, ..... |
Question: Find the distinct number of permutations of four letters of the word MATHEMATICS.
The answer is 11P4/(2!)^3 = 990
Is this correct? What if I only want 1 letter? Then the answer would be 8, which I got by manually listing down all possibilities. But what it the general formula for this sort of proble... |
Go on a hunt for all the ways in which *R* can *fail* to be isomorphic to ℤ/pℤ, and demonstrate the failure of any of them to hold.
Consider the subset *Z* ⊆ *R* of elements such that left-multiplication by z ∈ *Z* **does** map everything to zero. We may easily show that *Z*... |
How to (efficiently) compute the sum of the digits of evey number lying between two given numbers ? |
Your question can also be rephrased as "**How many words can be formed by taking four different letters of the word MATHEMATICS ?**"
First you need to find the number of distinct letters in the given word, here we have 8 (M,A,T,H,E,I,C,S),now you are suppose to arrange four digits among them, this can be done in P(8... |
Your question can also be rephrased as "**How many words can be formed by taking four different letters of the word MATHEMATICS ?**"
First you need to find the number of distinct letters in the given word, here you have 8 (M,A,T,H,E,I,C,S), now you are suppose to arrange four digits among them, this can be done in P... |
Fair warning: I am not a math expert (that's why I'm here).
I would like to be able to calculate the probability of rolling a certain side on a die with *n* sides where any number of those sides has an unfair advantage over the others.
For example: On a 6-sided die, we could assume that the 1 - 6 sides are longer... |
For a ring $R$, an $R$-right-module $M$, an $R$-left-module $N$, and an abelian group $P$, one can use the universal property of the tensor product to construct maps
$$
M\otimes_R N\to P.
$$
It concrete cases, it is often easy to see that the constructed map is surjective by just writing down pre-images.
It seem... |
What is the probability to get $n$ times the same element $k$ choosing randomly from a set $A$ knowing the cardinality $|A| = c$?
Thank you in advance,
rubik |
It depends: is $k$ fixed in advance? if so then the probability is given by $\frac{1}{c^n}$. If you just ask for the probability ot have $n$ times the same element, then it is $\frac{1}{c^{n-1}}$ (since the first time you can have any object). By the way, I supposed you are doing exactly $n$ tries. Of course, the proba... |
Question: Find the distinct number of permutations of four letters of the word MATHEMATICS.
The answer is 11P4/(2!)^3 = 990
Is this correct? This question along with the answer is from a module from my course, so this is all the details that I am able to provide.
My understanding of this question is how m... |
It depends: is $k$ fixed in advance? if so then the probability is given by $\frac{1}{c^n}$. If you just ask for the probability to have $n$ times the same element, then it is $\frac{1}{c^{n-1}}$ (since the first time you can have any object). By the way, I supposed you are doing exactly $n$ tries. Of course, the proba... |
After the edit by OP,here is a similar [problem and solution][1].
[1]: http://sg.answers.yahoo.com/question/index?qid=20100302091836AA9Cf9K |
If $R$ is a ring of order $p$, then it injects into the ring $\text{End}(\mathbb{Z}/p\mathbb{Z})$ of endomorphisms of $\mathbb{Z}/p\mathbb{Z}$. How many elements does this ring have? |
Is the subset consisting of all integrable (or square integrable) smooth functions of the set of all integrable (or square integrable) functions dense under the usual Euclidean or integral of absolute difference metric?
By smooth I mean derivatives of all orders exist. |
Note: This answer presumes that the question is "How many four-letter words can be formed using the letters of the word 'MATHEMATICS'." So MAME should be counted as a word.
The provided answer, ${}_{11}P_4 / (2!)^3$, is not correct.
Let's temporarily change the question to "How many distinct rearrangements are t... |
If $R$ is a ring of order $p$, then it injects into the ring $\text{End}(\mathbb{Z}/p\mathbb{Z})$ of endomorphisms of the abelian group $\mathbb{Z}/p\mathbb{Z}$. How many elements does this ring have? |
Note: This answer presumes that the question is "How many four-letter words can be formed using the letters of the word 'MATHEMATICS'." So MAME should be counted as a word.
The provided answer, ${}_{11}P_4 / (2!)^3$, is not correct.
Let's temporarily change the question to "How many distinct rearrangements are t... |
what is the number of permutations of a set say {1, 2, 3, 4, 5} that keep at least one integer fixed ?
|
The 2nd part of the "Fundamental Theorem of Calculus" has never seemed as earth shaking or as fundamental as the first to me. Why is it "fundamental" -- I mean, the *mean value theorem*, and the *intermediate value theorem*s are both pretty exciting by comparison. And after the joyful union of integration and the deriv... |
what is the number of permutations of a set say {1, 2, 3, 4, 5} that keep at least one integer fixed ?
|
The 2nd part of the "Fundamental Theorem of Calculus" has never seemed as earth shaking or as **fundamental** as the first to me. Why is it "fundamental" -- I mean, the *mean value theorem*, and the *intermediate value theorems* are both pretty exciting by comparison. And after the joyful union of integration and the d... |
I'm a senior undergrad at a top-ish(say, top 15) math school. I'm a solid, not stellar, student. This year I'm taking the qualifying exam grad courses in algebra and analysis and have been taken aback by the "pressure cooker" atmosphere among grad students here. That is, even moreso than in the undergraduate program.
... |
Does it ever make sense NOT to go to the most prestigious graduate school you can get into? |
This is the part of the fundamental theorem that allows you to _compute_ integrals; then you can compute areas, and with more theory even volumes, surfaces and so on. Exciting enough? |
Please let me summarize the method by which L. Euler solved the Basel Problem and how he found the exact value of $\zeta(2n)$ up to $n=13$. Euler used the infinite product $$ \displaystyle f(x) = \frac{sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\pi^2}\Big) , $$ <a href="http://en.wikipedia.org/wiki/Newto... |
Why is the following evaluation of Apery's Constant wrong and do you have suggestions on how, if at all, this method could be improved? |
Please let me summarize the method by which L. Euler solved the Basel Problem and how he found the exact value of $\zeta(2n)$ up to $n=13$. Euler used the infinite product $$ \displaystyle f(x) = \frac{sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\pi^2}\Big) , $$ <a href="http://en.wikipedia.org/wiki/Newto... |
How can I compute the number of common divisors of two naturals ?
For example,if we consider `(12,24)` the answer is **6** i.e `{1,12,2,6,3,4}`.
EDIT : I got an answer [here][1].The solution boils down to finding the number of divisor of the GCD of the two numbers.
[1]: http://books.google.co.in/books?id=di0W... |
Please let me summarize the method by which L. Euler solved the Basel Problem and how he found the exact value of $\zeta(2n)$ up to $n=13$. Euler used the infinite product $$ \displaystyle f(x) = \frac{sin(x)}{x} = \prod_{n=1}^{\infty} \Big(1-\frac{x^2}{n^2\pi^2}\Big) , $$ <a href="http://en.wikipedia.org/wiki/Newto... |
$f(x) = \frac{1}{\cos x}$
$f'(x) = \frac{\sin(x)}{\cos^2(x)}$
$f''(x) = \frac{2\sin^2(x)+\cos^2(x)}{\cos^3(x)}$
$f^{(3)}(x) = \frac{6\sin^3(x)+5\cos^2(x)\sin(x)}{cos^4(x)}$
$\vdots$
$f^{(n)}(x) = \frac{ ?}{cos^{n+1}(x)}$
Some of these are easy: http://darkwing.uoregon.edu/~jcomes/251exn.pdf
Others ar... |
I think that there are several important points.
First, you will learn just as much from your fellow
graduate students, especially in the initial years, than
you will from your professors, and far more from your
fellow students if there is a good atmosphere for it. Thus,
I suggest that you try to land at a schoo... |
Suppose $X$ is a space and $A_1\subseteq A_2\subseteq A_3\subseteq ...\subset X$ is a sequence of subspaces each of which is closed in $X$ and such that $X\cong \varinjlim_{n}A_n$ (i.e. $U$ is open in $X$ if and only if $U\cap A_n$ is open in $A_n$ for each $n$). This topology on $X$ has many names (direct limit, induc... |
There are many good comments above, I'd like to add a few other points.
Prestige is a fairly abstract, borderline meaningless word. It's meant to be an amalgamam of other, *actually important* ideas. And those concepts are the things one really ought to pay attention to.
For example, when you go to grad school... |
There are many good comments above, I'd like to add a few other points.
Prestige is a fairly abstract, borderline meaningless word. It's meant to be an amalgamam of other, *actually important* ideas. And those concepts are the things one really ought to pay attention to.
For example, when you go to grad school... |
How many Natural (Integers) Numbers could you count?
There are infinitely many, yet you can count them.
It's called [Countable Set][1].
How many Real Numbers are there?
Infinitely as well (Since at least every Natural Number is a Real Number).
Yet you won't be able to count them (Intuitively, Let's say you name ... |
Maybe you would like an answer from someone who is there.
I went to undergrad at a "Top 100" school. I did extremely well there, but utterly failed to get into the level of grad schools I was aiming at, and am now a grad student at a "Top 50" school.
I can only answer based on what I've seen at these schools, s... |
A divisor $d$ of $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}}$ is _unitary_ if and only if $d = p_{1}^{\varepsilon_{1}} \cdots p_{n}^{\varepsilon_{n}}$, where each exponent $\varepsilon_{i}$ is either $0$ or $r_{i}$. Let $D_{k} = ${ $d$ } be the subset of unitary divisors of an integer $k > 1$ satisfying $\omega(d) = \omega... |
Lets say I have a 2 x 2 Matrix (Actually [Structure Tensor][1] of a Discrete Image - I):
$$ \begin{bmatrix}
\frac{{\partial}^{2} I}{\partial {x}^{2}} & \frac{{\partial}^{2} I}{\partial x \partial y} \\
\frac{{\partial}^{2} I}{\partial y \partial x} & \frac{{\partial I}^{2}}{\partial {y}^{2}}
\end{bmatrix}$$
... |
Lets say I have a 2 x 2 Matrix (Actually [Structure Tensor][1] of a Discrete Image - I):
$$ \begin{bmatrix}
\frac{{\partial}^{2} I}{\partial {x}^{2}} & \frac{{\partial}^{2} I}{\partial x \partial y} \\\
\frac{{\partial}^{2} I}{\partial y \partial x} & \frac{{\partial I}^{2}}{\partial {y}^{2}}
\end{bmatrix}$$... |
Lets say I have a 2 x 2 Matrix (Actually [Structure Tensor][1] of a Discrete Image - I):
$$ \begin{bmatrix}
\frac{{\partial}^{2} I}{\partial {x}^{2}} & \frac{{\partial}^{2} I}{\partial x \partial y} \\\
\frac{{\partial}^{2} I}{\partial y \partial x} & \frac{{\partial I}^{2}}{\partial {y}^{2}}
\end{bmatrix}$$... |
Consider a situation with a bag with infinity number of balls. Each ball is of some color. Number of colors is finite but it is not known. Balls are drawn from the bag one by one and checked for the color.
We want to stop drawing balls when there is small probability that we will find color that is not already drawn.
... |
I happened to have gone to a toppish school (look up my profile if you really care which one). Even though I didn't realize it at the time, there are distinct advantages from the mathematical environment there. But I also think you shouldn't base your decision on graduate schools purely on artifical rankings of prestig... |
Say $k$ has the prime factorization $p_1^{r_1} p_2^{r_2} \cdots p_n^{r_n}$. Then let $H(k)$ be the "hyperbolicity" of this integer, $H(k) = \sum_{i=1}^n p_i^{-r_i}$. Hyperbolic integers are those integers $k$ that have $H(k) < 1$.
Now, $H(k)$ for some "random" integer $k$ looks like a sum of independent random varia... |
A divisor $d$ of $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}}$ is _unitary_ if and only if $d = p_{1}^{\varepsilon_{1}} \cdots p_{n}^{\varepsilon_{n}}$, where each exponent $\varepsilon_{i}$ is either $0$ or $r_{i}$. Let $D_{k} = ${ $d$ } be the subset of unitary divisors of an integer $k > 1$ satisfying $\omega(d) = \omega... |
There are many good comments above, I'd like to add a few other points.
Prestige is a fairly abstract, borderline meaningless word. It's meant to be an amalgamam of other, *actually important* ideas. And those concepts are the things one really ought to pay attention to.
For example, when you go to grad school... |
A closed curve on a surface can be represented by the boundary it intersects. Suppose the original surface is created by glueing a to A, b to B etc. Pick an orientation, each time it intersects a boundary, add the boundary to the representation. For example in the image, it's
![alt text][1]
The black dot is the star... |
I know how to generate solutions to $x^2-Dy^2=\pm 1$ from the fundamental solution to $x^2-Dy^2=\pm 4$. But how do we know that **all** the solutions are generated from this fundamental solution? |
Generalized Pell's Equation: Why is 4 special? |
A divisor $d$ of $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}}$ is _unitary_ if and only if $d = p_{1}^{\varepsilon_{1}} \cdots p_{n}^{\varepsilon_{n}}$, where each exponent $\varepsilon_{i}$ is either $0$ or $r_{i}$. Let $D_{k} = ${ $d$ } be the subset of unitary divisors of an integer $k > 1$ satisfying $\omega(d) = \omega... |
A divisor $d$ of $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}}$ is _unitary_ if and only if $d = p_{1}^{\varepsilon_{1}} \cdots p_{n}^{\varepsilon_{n}}$, where each exponent $\varepsilon_{i}$ is either $0$ or $r_{i}$. Let $D_{k} = ${ $d$ } be the subset of unitary divisors of an integer $k > 1$ satisfying $\omega(d) = \omega... |
I just read "Generalized Feedback Shift Register Pseudorandom Number Algorithm" by T. G. Lewis and W. H. Payne.
I think that paper settles the question I was raising (going to the source, right?). In essence, the question is
"What is the correct procedure to use the Generalized Feedback Shift Register Algorithm (GF... |
I am trying to self-read through Ian Stewart's Galois theory and am stuck at this paragraph. I quote - "Lagrange observed that all methods for solving polynomial equations by radicals involve constructing rational functions of the roots that take a small number of values when the roots $\alpha_j$ are permuted. Prominen... |
I just read "Generalized Feedback Shift Register Pseudorandom Number Algorithm" by T. G. Lewis and W. H. Payne.
I think that paper settles the question I was raising (going to the source, right?). In essence, the question is
"What is the correct procedure to use the Generalized Feedback Shift Register Algorithm (GF... |
I just read "Generalized Feedback Shift Register Pseudorandom Number Algorithm" by T. G. Lewis and W. H. Payne.
I think that paper settles the question I was raising (going to the source, right?). In essence, the question is
"What is the correct procedure to use the Generalized Feedback Shift Register Algorithm (GF... |
I am taking my first real-analysis course this year and have a midterm on friday. I have NO idea what is going on in this class.
So if anyone could clarify what the following means.
The book starts off with talking about archimedes and his principal which states:
> The value of an infinite series, if it
> exis... |
I am taking my first real-analysis course this year and have a midterm on friday. I have NO idea what is going on in this class.
So if anyone could clarify what the following means.
The book starts off with talking about Archimedes and his principle which states:
> The value of an infinite series, if it
> exis... |
I am taking my first real-analysis course this year and have a midterm on friday. I have NO idea what is going on in this class.
So if anyone could clarify what the following means.
The book starts off with talking about archimedes and his principal which states:
> The value of an infinite series, if it
> exis... |
I encountered a problem today to prove that cos (n!) does not have a limit. I have no idea how to do it formally. Could someone help? The simpler the proof (by that i mean less complex theorems are used) the better. Thanks |
Is there a limit of cos (n!)? |
I am taking my first real-analysis course this year and have a midterm on friday. I have NO idea what is going on in this class.
So if anyone could clarify what the following means.
The book starts off with talking about Archimedes and his principle which states:
> The value of an infinite series, if it
> exis... |
I encountered a problem today to prove that (X_n) with X_n = cos(n!) does not have a limit (when n approaches infinity). I have no idea how to do it formally. Could someone help? The simpler the proof (by that i mean less complex theorems are used) the better. Thanks |
Let $A$ be a stochastic matrix. Thus $A$ has nonnegative entries, and the sum of the elements in each row is 1. This implies that the vector $\begin{bmatrix} 1 & 1 & \cdots & 1 \end{bmatrix}^T$ is an eigenvector corresponding to the eigenvalue 1. Is it true that there exists a vector $b$ such that
$$(A - I)x \geq... |
No solutions to a matrix inequality? |
f(x) = 1/cos x then f^(n)(x) is ...? |
I'm reading paper at [here](http://www.emis.de/journals/EJC/Volume_10/PDF/v10i1r8.pdf) . In figure 2, there is that the best known packing has side length about 3.8772 and tilt angle about 40.182 degrees. How can I verify those values, as another source (Which Way did the Bicycle Go page 105) says that the side length ... |
An algorithm to determine whether a given element of the fundamental group has a simple representative on surfaces with boundary was described by Birman and Series in their paper
[An algorithm for simple curves on surfaces][1]. *J. London Math. Soc.* (2) 29 (1984), no. 2, 331–342
From the MathSciNet review:
>T... |
What is the smallest rectangle which contains 11 non-overlapping (except boundary) unit squares?
I'm reading paper at [here](http://www.emis.de/journals/EJC/Volume_10/PDF/v10i1r8.pdf) . In figure 2, there has been said that the best known packing has side length about 3.8772 and tilt angle about 40.182 degrees. How ... |
As a two-by-two matrix, applying Jacobi in fact gives the answer at once!
It is known that given a two-by-two matrix $\mathbf{A}$, one can construct an orthogonal matrix $\mathbf{V}=\bigl(\begin{smallmatrix}c&s\\\\-s&c\end{smallmatrix}\bigr)$ such that $\mathbf{V}^T\mathbf{A}\mathbf{V}$ is diagonal, where the two nu... |
I need to figure out the Probability that Y > 23 for a chi-square distribution with degrees of freedom = 7.
The book says to do this with some applet. I don't know what it's talking about.
Can I do this on my Ti-89 (preferable) or on some website somewhere? |
chi-square distribution with v = 7 degrees of freedom Ti-89? |
A divisor $d$ of $k = p_{1}^{r_{1}} \cdots p_{n}^{r_{n}}$ is _unitary_ if and only if $d = p_{1}^{\varepsilon_{1}} \cdots p_{n}^{\varepsilon_{n}}$, where each exponent $\varepsilon_{i}$ is either $0$ or $r_{i}$. Let $D_{k} = ${ $d$ } be the subset of unitary divisors of an integer $k > 1$ satisfying $\omega(d) = \omega... |
"The interval between $L$ and $M$" means all the numbers that are between $L$ and $M$; we call $L$ and $M$ the "endpoints" of the interval. If you imagine the real numbers as the points on the real number line, with $L$ and $M$ located somewhere, the "interval" would be the segment of the line that is between the point... |
I need to figure out the Probability that Y > 23 for a chi-square distribution with degrees of freedom = 7.
The book says to do this with some applet. I don't know what it's talking about.
Can I do this on my Ti-89 (preferable) or on some website somewhere?
**EDIT/UPDATE:**
I found a program on my ti-89 for... |
What is the smallest rectangle which contains 11 non-overlapping (except boundary) unit squares?
This question is open but I would like to know a method to verify the best known answer at the moment.
I'm reading paper at [here](http://www.emis.de/journals/EJC/Volume_10/PDF/v10i1r8.pdf) . In figure 2, there has be... |
An algorithm to determine whether a given element of the fundamental group has a simple representative on surfaces with boundary was described by Birman and Series in their paper
[An algorithm for simple curves on surfaces][1]. *J. London Math. Soc.* (2) 29 (1984), no. 2, 331–342
From the MathSciNet review:
>T... |
If you assume there are N colored balls, with equal frequency p=1/N, then this should be pretty do-able. Of course, you're desired confidence interval will affect the answer.
Without working out all the details, consider that you have drawn the point where you see a few colors 2 or 3 times. By considering how many... |
If your manifold is closed you must be using some definition that ensures it's compact? So if $f : M \to \mathbb R^3$ is an embedding, isn't the image simultanously compact and open? |
Can a derivative operation commute over an integral operation irrespective of the properties of the function under the integral ? |
What is the smallest square which contains 11 non-overlapping (except boundary) unit squares?
This question is open but I would like to know a method to verify the best known answer at the moment.
I'm reading paper at [here](http://www.emis.de/journals/EJC/Volume_10/PDF/v10i1r8.pdf) . In figure 2, there has been ... |
Not in general. I recommend Gelbaum and Olmsted's *[Counterexamples in Analysis][1]*, which is where I turned to find a counterexample to your question. Namely, example 15 on page 123 is titled
> A function $f$ for which $d/dx\int_a^b f(x,y)dy\neq\int_a^b[\partial/\partial x f(x,y)]dy$, although each integral is p... |
If your manifold is closed you must be using some definition that ensures it's compact? So if $f : M \to \mathbb R^3$ is an embedding, isn't the image simultanously compact and open?
edit in response to your comment: if $f : M \to \mathbb R^3$ is an embedding, let $B \subset M$ be an open subset of $M$ which is h... |
Not in general. I recommend Gelbaum and Olmsted's *[Counterexamples in Analysis][1]*, which is where I turned to find a counterexample to your question. Namely, example 15 on page 123 is titled
> A function $f$ for which $d/dx\int_a^b f(x,y)dy\neq\int_a^b[\partial/\partial x f(x,y)]dy$, although each integral is p... |
How this operation is called? |
Most of what I've read defines the support of a distribution topologically if it defines support precisely; e.g the smallest closed subset that has full measure (i.e. the intersection of all closed subsets of full measure). I've seen allusions to purely measure-theoretic definitions, and I think I even came across one ... |
$f(x) = \frac1{\cos x}$ then $f^{(n)}(x)$ is ...? |
The Riemann Zeta Function is defined as $ \displaystyle \zeta(s) = \sum\limits_{n=1}^{\infty} \frac{1}{n^s}$. It is not absolutely convergent or conditionally convergent for $\text{Re}(s) \leq 1$. Using analytic continuation, one can derive the fact that $\displaystyle \zeta(-s) = -\frac{B_{s+1}}{s+1}$. Can one obtain ... |
The Riemann Zeta Function is defined as $ \displaystyle \zeta(s) = \sum\limits_{n=1}^{\infty} \frac{1}{n^s}$. It is not absolutely convergent or conditionally convergent for $\text{Re}(s) \leq 1$. Using analytic continuation, one can derive the fact that $\displaystyle \zeta(-s) = -\frac{B_{s+1}}{s+1}$ there $B_{s+1}$ ... |
What a spectacularly rubbish way to define convergence. Here's how to think of it in a simple, yet rigorous way.
For example, suppose we have a sequence $S_{n} = \frac{1}{2^n}$. We know that $S_{0} = 1, S_{1} = \frac{1}{2}, S_{2} = \frac{1}{4}$, and so on and so forth, just as you would have defined it in Calculus. ... |
What a spectacularly rubbish way to define convergence. Here's how to think of it in a simple, yet rigorous way.
For example, suppose we have a sequence $S_{n} = \frac{1}{2^n}$. We know that $S_{0} = 1, S_{1} = \frac{1}{2}, S_{2} = \frac{1}{4}$, and so on and so forth, just as you would have defined it in Calculus. ... |
I presume that it is because one needs to take a limit to define the derivative, but is there another reason? |
Why are holomorphic functions defined in reference to an open set? |
What are some algorithms used for finding the shortest path between 2 points on a (riemannian) manifold (the manifold may have a smooth boundary)?
So far I've had 3 ideas, none of which seem that good:
1) (Rubber band principle) Find any path between the two points, then draw the path taut like a stretched rubber... |
How to prove this trignometrical Identities ? |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.