Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Is there a function that gives the same result for a number and its reciprocal? Is there a (non-piecewise, non-trivial) function where $f(x) = f(\frac{1}{x})$?
Why?
It would be nice to compare ratios without worrying about the ordering of numerator and denominator. For example, I might want to know whether the "magnitu... | For lack of anything worse than this, $f(x)=(x - \frac{1}{x})^2$.
| {
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Solve Algebraical.ly $0.5=\dfrac{365!}{365^{n}(365-n)!} $ How does one go about solving this equation? Not sure how to approach this as no factorials will cancel out. Im sorry I meant $\dfrac{365!}{365^{n}(365-n)!}=0.5$.
| We usually solve this equation numerically:
$$a_n=\frac{365!}{365^n(365-n)!}$$
Hence $a_1=1$ and $$a_{n+1}=a_n.\frac{365-n}{365}$$
If you want to solve $a_n=p$, just do a little program that computes $a_n$ from $a_1$ by multiplying at each step by $\frac{365-n}{365}$ until you find $p$.
Here $$a_{23}=0.4927027656$$
| {
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How to solve this sum limit? $\lim_{n \to \infty } \left( \frac{1}{\sqrt{n^2+1}}+\cdots+\frac{1}{\sqrt{n^2+n}} \right)$ How do I solve this limit?
$$\lim_{n \to \infty } \left( \frac{1}{\sqrt{n^2+1}}+\cdots+\frac{1}{\sqrt{n^2+n}} \right)$$
Thanks for the help!
| Extract n from the radical. So, you have to sum n terms which are close to 1 and you divide the sum by n. Are you able to continue with this ?
| {
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Number of $2n$-letter words using double $n$-letter alphabet, without consecutive identical letters How many words with $2n$ letters can be created if I have an alphabet with $n$ letters and each of the letters have to occur exactly twice in the word, but no two consecutive letters are equal?
Thanks!
| There is no simple closed formula for this (I think no known closed formula at all), but one can give a formula as a sum by using inclusion-exclusion.
First consider such orderings where the pairs of identical letters are distinguishable. Then $\displaystyle \sum_{k=0}^n (-1)^k2^k(2n-k)!\binom{n}{k}$ gives the number ... | {
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Basis of Partial Fractions I need some guidance for the following:
For the polynomial q(x) = (x − 1)(x − 2)(x − 3)(x − 4), with degree 4,
describe the basis for P3 that partial fractions asserts we have, and demonstrate that the
collection of polynomials are linearly independent; that is, that
a1p1(x) + a2p2(x) + a3p3(... | Start with $p(x)=a_0+a_1x+a_2x^2+a_3x^3$ and use the partial fractions setup
$$\frac{p(x)}{q(x)}=\frac{b_1}{x-1}+\frac{b_2}{x-2}+\frac{b_3}{x-3}+\frac{b_4}{x-4}.$$
Now as usual multiply both sides by $q(x)=(x-1)(x-2)(x-3)(x-4)$ and you'll get each $b_k$ multiplied by the product of three of the four factors $x-i$ on th... | {
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Axiom of separation The Axiom of separation states that, if A is a set then $\{a \in A ;\Phi(a)\}$ is a set.
Given a set $B \subseteq A$, Suppose I define $B=\{ a \in A ; a\notin B \}$.
This, of course leads to a contradiction. Because we define $B$ by elements not from $B$. My queation is: what part of the axioms sais... | There is nothing at all to stop you defining a set $\Sigma$ such that $x \in \Sigma$ iff $x \in A \land x \notin B$, so $\Sigma = \{x \in A \mid x \notin B\}$.
But what you've shown is that $\Sigma \neq B$!
No problem so far.
What you can't do is then go on (having a knock-down argument to show that $\Sigma \neq B$) ... | {
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I do not know how to start this problem-help needed There are 6 people who are holding hands, such that each person is holding hands with exactly 2 other people. How many ways are there for them to do that?
My friend challenged me to this problem and i dont know where to start...
Thanks for any help...=)
| Assuming we don't care which hand is being used to do the holding then with three people there is only one way. You add a fourth person and he/she can go into 1 of 3 positions making 3.
Another (fifth) person and they have 4 positions to choose from so $4 \times 3 = 12$.
Another (sixth) person and they have 5 position... | {
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$A = \left\{ (1,x) \in \mathbb{R}^2 : x \in [2,4] \right\} \subseteq \mathbb{R}^2$ is bounded and closed but not compact? Is it true that set $A = \left\{ (1,x) \in \mathbb{R}^2 : x \in [2,4] \right\} \subseteq \mathbb{R}^2$ is bounded and closed but is not compact. We consider space $(\mathbb{R}^2, d_C)$ where $$d_C(x... | Yes, that is right. Note that each point $a\in A$ is isolated, because if you choose $\epsilon<d_E(a,0)$, then this ball does not contain any other point $b\in A$, as the distance $d_C(a,b)$ would be larger than $d_E(a,0)$. That means that $A$ is discrete.
$A$ is bounded since the distance from each point in $A$ to $0... | {
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Good book for algebra after Herstein? What is a good book to read after herstein's topics in algebra?
I've read in reviews somewhere that it's a bit shallow...
The main interests are algebraic and differential geometry. I prefer books with challenging excersices.
Something that crossed my mind: prehaps
it's preferable... | One of my favourite texts for mid-level algebra is Dummit and Foote's $\textit{Abstract Algebra}$. Another good text is Eisenbud's $\textit{Commutative Algebra (With a View Towards Algebraic Geometry)}$. This book is more of a graduate level text book.
| {
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On the definition of the direct sum in vector spaces We say that if $V_1 , V_2, \ldots, V_n$ are vector subspaces, the sum is direct if and only if the morphism $u$ from $V_1 \times \cdots \times V_n$ to $V_1 + \cdots + V_n$ which maps $(x_1, \ldots, x_n)$ to $x_1 + \cdots + x_n$ is an isomorphism.
Looking at the defin... | No. It is possible that $V_1$ and $V_2$ are subspaces of some vectorspace $W$ such that the subspace $V_1 + V_2$ of $W$ is isomorphic to $V_1 \times V_2$, but the canonical map $V_1 \times V_2 \to V_1 + V_2$ is not an isomorphism.
Example. Take $W$ the vectorspace of infinite sequences of real numbers (with or without ... | {
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Integrate: $ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $ without using complex analysis methods Can this integral be solved without using any complex analysis methods: $$ \int_0^\infty \frac{\log(x)}{(1+x^2)^2} \, dx $$
Thanks.
| A really easy method using the substitution $x=\tan t$ offered by Jack D'Aurizio but followed by elementary integration by parts without using Fourier series.
$$\begin{align}
&\int_0^\frac{\pi}{2}\cos^2x\ln\tan x dx=\frac12 \int_0^\frac\pi2 \cos2x\ln\tan xdx\\
=&\frac14\sin2x\ln\tan x\bigg|^\frac{\pi}{2}_0-\frac14\int_... | {
"language": "en",
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Concrete Mathematics - Towers of Hanoi Recurrence Relation I've decided to dive in Concrete Mathematics despite only doing a couple of years of undergraduate maths many years ago. I'm looking to work through all the material whilst plugging gaps in my knowledge no matter how large they are.
However, solving recurrence ... | You’re just missing a little algebra. You have $U_n=T_n+1$ for all $n\ge 0$, so $U_{n-1}=T_{n-1}+1$, and therefore $2T_{n-1}+2=2(T_{n-1}+1)=2U_{n-1}$. Combine this with $T_n+1=2T_{n-1}+2$, and $U_n=T_n+1$, and you get $U_n=2U_{n-1}$, with $U_0=1$.
Now notice that $U_n$ is just doubling each time $n$ is increased by $1$... | {
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The Probability of Rolling the Same Number with 10 Dice That Have 19 Sides I like dice and I want to know what the probability is for rolling the same number with 10 dice that have 19 sides. Also, do these dice exist or not?
| You are rolling $10$ dice each with $19$ sides.
The first die can land on anything, but then all of the rest have to land on that same number.
The chance that each die lands on this number is $\frac{1}{19}$ since they have 19 sides. And since our rolls are independent, this means we have a probability of
$$\frac{1}{1... | {
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When proving a statement by induction, how do we know which case is the valid 'base'? For example proving 2^n < n!, 4 is the 'base' that works for this exercise, then starting from there we prove p + 1 considering p has to be at least 4 and we have our result. However, I believe determining the first valid value withou... | The base case is usually so trivial that it is either obvious or can be immediately calculated without difficulty. However, in a case such as yours where the first values do not work, one would usually proceed to show that there are a finite number of cases where the relation is not true, and then proceed to perform i... | {
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Why, while checking consistency in $3\times3$ matrix with unknowns, I check only last row? I would like to know whether my thinking is right. So, having 3 linear equations,
$$
\begin{align}
x_1 + x_2 + 2x_3 & = b_1 \\
x_1 + x_3 & = b_2 \\
2x_1 + x_2 + 3x_3 & = b_3
\end{align}
$$
I build $3\times 3$ matrix
\begin{bmatri... | The second row corresponds to the equation $1x_2+1x_3=b_1-b_2$. This can be solved for any values of $b_1, b_2$. For example, we can take $x_2=b_1-b_2$, and $x_3=0$.
The third row corresponds to the equation $0x_1+0x_2+0x_3=b_3-b_2-b_1$. This cannot be solved if $b_3-b_2-b_1\neq 0$. No matter what $x_1,x_2,x_3$ ar... | {
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Is statistical dependence transitive? Take any three random variables $X_1$, $X_2$, and $X_3$.
Is it possible for $X_1$ and $X_2$ to be dependent, $X_2$ and $X_3$ to be dependent, but $X_1$ and $X_3$ to be independent?
Is it possible for $X_1$ and $X_2$ to be independent, $X_2$ and $X_3$ to be independent, but $X_1$ a... | *
*Let $X,Y$ be independent real-valued variables each with the standard normal distribution $\mathcal N(0,1)$
We have then: $$EX = EY = 0, EX^2 = EY^2 = 1.$$
Consider random variables $U = X +Y$, $V = X - Y$. We have $E(U * V) = 0 .$
Since $U$ and $V$ are also normally distributed, this implies that $U$ and $V$ a... | {
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Is the number 100k+11 ever a square? Is the number 100k+11 ever a square?
Obviously all such numbers end in 11. Can a square end with an 11?
| Ends by 11 means, $n² \mod 100 = 11$ which also implies end by one $n² \mod 10 = 1$
this is true if and only if $(n \mod 100)^2 \mod 100 = 11$ and $(n \mod 10)² \mod 10 = 1$.
Ergo you juste have to check squares of 1, 11, 21, 31, 41, 51, 61, 71, 81 and 91 and see none ends with 11.
| {
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How to determine the dual space of se(2)? In an article there are the following sentences:
The euclidean group $SE(2)=\left\{\left[\begin{array}{cc}1 & 0\\v & R\end{array}\right]:v\in \mathbf{R}^{2\times1}\text{ and }R\in SO(2)\right\}$ is a real three dimensional conected matrix Lie group and its associated Lie alegbr... | On Mathoverflow.net, I found a description of the duality pairing between a Lie Algebra and its dual, namely $\langle X, \alpha \rangle = trace(X\alpha)$ where $X$ is in the algebra and $\alpha$ is in the dual.
But before going into that, let's talk about dual spaces a little. The dual of a vector space $V$ is the set... | {
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Distribution of function of two random variables Let $X$ be the number on a die roll, between 1 and 6. Let $Y$ be a random number which is uniformly distributed on $[0,1]$, independent of $X$. Let $Z = 10X + 10Y$.
What is the distribution of $Z$?
| Hint:
$$
F_Z(x) = P(Z < x) = P(X + Y < x/10)
$$
Work it out by cases from here based on the potential values of $x$. For instance, if $x < 10$ then $x/10 < 1$, so $F_Z(x) = 0$ as $X \geq 1$. Another sample case: if $10 \leq x < 20$, then the value of $X$ in the right hand expression must be $1$ (in order for $X + Y < x... | {
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CDF of a sum of continuous and discrete dependent random variables Let $\psi_1$ be a Normal random variable with mean $\mu_1$ and standard deviation $\sigma_1$. Let $\xi$ be defined as
$$
\xi=c\,\mathbb{1}_{\left\{\psi_2+\psi_1\leq 0\right\}},
$$
where $\mathbb{1}$ is the indicator function, and $\psi_2$ a Normal rand... | Recall that $\{ \psi_2 + \xi \leqslant \alpha \} = \{\psi_2 + c [ \psi_2+\psi_1 \leqslant 0] \leqslant \alpha \} = \{\psi_1 \leqslant -\psi_2, \psi_2 + c \leqslant \alpha \} \lor \{\psi_1 > -\psi_2, \psi_2 \leqslant \alpha \} $
and the latter two events are disjoint. Hence
$$ \begin{eqnarray}
F(\alpha) &=& \Pr\left(\... | {
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For fractional ideal, why $AB=R$ implies $B=A^{-1}$? Let $A,B$ be two fractional ideals of $R$ (an integral domain). Could anyone tell me why $AB=R$ implies $B=A^{-1}$?
| The inverse of an ideal $I\subseteq R$ is defined as $$I^{-1}:=\{x\in k(R) : xI\subseteq R\}$$ Then if $R$ is a Dedekind domain we have $II^{-1}=R$. Now in a Dedekind domain the set of fractional ideals forms a group under the following operation: $$(r^{-1}I)\cdot (s^{-1} J):=(rs)^{-1}IJ$$ where $r,s\in R$ and $I,J$ ar... | {
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open and closed and bounded interval people
interval $[0,1]$, is closed interval
can you say such interval $[0,1]$ is bounded or not??
if it is how to show this interval is bounded ?(proof?)
or interval is not enough to say this is bounded or not
is only apply to the function $f()$?
thank you
| $[0,1]$ is bounded in $\mathbb R$ because there is a point $0$ in $\mathbb R$ from which the distance to any point in this interval is bounded by $1$.
| {
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Irreducibility of a particular polynomial I've got this problem for my homework: find out whether the polynomial
$$f(x)=x(x-1)(x-2)(x-3)(x-4) - a$$
is irreducible over the rationals, where $a$ is integer which is congruent to $3$ modulo $5$.
It is easy to verify that $f(x)$ has no integer zeros (and no rational zeros ... | Observation: In ${\mathbb F}_5$ we have $x(x-1)(x-2)(x-3)(x-4) = x^5 - x$.
This follows from the fact that over a finite field $F$ we have $x^{|F|} - x = \prod_{g\in F} (x-g)$ which you can easily check by noticing that all elements are roots and that a polynomial of degree $|F|$ can't have any more roots.
Claim: More ... | {
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$\varepsilon$-$\delta$ proof of $\sqrt{x+1}$ need to prove that $ \lim_{x\rightarrow 0 } \sqrt{1+x} = 1 $
proof of that is:
need to find a delta such that $ 0 < |x-1| < \delta \Rightarrow 1-\epsilon < \sqrt{x+1} < \epsilon + 1 $ if we choose $ \delta = (\epsilon + 1)^2 -2 $ and consider $ |x-1| < \delta = (\epsilon + 1... | So it remains to show that $\sqrt{4-(\epsilon + 1)^2} -1 > -\epsilon $.
Note that for the $\epsilon-\delta$ proof is enough to show that $|x-1|\lt\delta\Rightarrow\ldots$ only for small epsilons. In your case we may assume therefore that $\epsilon<\frac12$. In that case we have that indeed $\sqrt{4-(\epsilon + 1)^2} -1... | {
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Of Face and Circuit Rank The circuit rank of a graph $G$ is given by
$$r = m - n + c,$$
where $m$ is the number of edges in $G$, $n$ is the number of vertices, and $c$ is the number of connected components.
Doesn't Euler's formula say the same?
$$
\#\text{faces} = \#\text{edges}-\#\text{vertices} +\#\text{component... | We have
$$
\#\text{circuit rank} = \#\text{edges} - \#\text{vertices} + \#\text{components}
$$
and, for a planar graph (or a graph on the surface of a sphere),
$$
\#\text{vertices} - \#\text{edges} + \#\text{faces} = \#\text{components} + 1 \text{.}
$$
This leads to
$$
\#\text{circuit rank} = \#\text{faces} - 1 \text{... | {
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Prove $13|19^n-6^n$ by congruences I am trying to prove $13|19^n-6^n$. With induction its not so bad but by congruences its quite difficult to know how to get started.
Any hints?
| Because $x^n - y^n$ is divisible by $x-y$ as
$$x^n - y^n = (x-y)\sum_{i=0}^{n-1}x^iy^{n-1-i}$$
Substitute $x=19$ and $y = 6$.
$$\begin{align*}
x^n-y^n =& x^n\left[1-\left(\frac yx\right)^n\right]\\
=& x^n \left[1+\left(\frac yx\right)+\left(\frac yx\right)^2+\cdots+\left(\frac yx\right)^{n-1}\right]\left[1-\left(\frac... | {
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Negative Base to non-integer power I'm looking to consistently solve the m^n case, including conditions where m is negative and n is non-integer. I'd like to, additionally, catch the error when it isn't possible.
Some examples to think about.
(-.5)^(.2) which is, effectively, the fifth root of (-1/2) which has a soluti... | The underlying issue here is that (assuming you want to stay within the real numbers) when $c<0$, the function $c^x$ is undefined for most values of $x$. Specifically, it's undefined unless $x$ is a rational number whose denominator is odd. There is no continuous/differentiable function underlying the places where it i... | {
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An inequality related to the number of binary strings with no fixed substring Let $f \in \{0,1\}^k$ and let $S_n(f)$ be the number of strings from $\{0,1\}^n$ that do not contain $f$ as a substring. As an interesting example $S_n(11) = f_{n+2}$ where $f_n$ is the $n$'th Fibonacci number.
I would like to show that if $... | Let $Q_n(f)$ be the number of bit strings of length $n$ that contain $f$ as a substring. Also let $B_n=2^n$ be the number of bit strings at all of length $n$. Then clearly
$$
B_n=Q_n(f)+S_n(f)
$$
which just states that bit strings of length $n$ either does or does not contain $f$ as a substring. Furthermore, it is easy... | {
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} |
Show that the multiplicative group $\mathbb{Z}_{10}^{\times}$ is isomorphic to the additive group $\mathbb{Z}_4$. Show that the multiplicative group $\mathbb{Z}_{10}^{\times}$ is isomorphic to the additive group $\mathbb{Z}_4$.
I'm completely lost with this one.
| write down all the elements of $\mathbb{Z}^{\times}_{10}$ explicitly. any find a generator by its Cayley Table or by explicit calculation. Now map this generator to $1$ in $\mathbb{Z}_4$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integral equations that can be solved elementary
Solve the following integral equations:
$$
\int_0^xu(y)\, dy=\frac{1}{3}xu(x) \tag 1 \label 1
$$
and
$$
\int_0^xe^{-x}u(y)\, dy=e^{-x}+x-1. \tag 2 \label 2
$$
Concerning $\eqref 1$, I read that it can be solved by differentiation. Differentiation on both sides gi... | Hint:
$$\int_0^xe^{-x}u(y)\, dy=e^{-x}+x-1$$ So $$ e^{-x}\int_0^xu(y)\, dy=e^{-x}+x-1$$ and therefore, $$ \ \int_0^xu(y)\, dy=1+\frac{x-1}{e^{-x}} $$
| {
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"timestamp": "2023-03-29T00:00:00",
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If $x^3+\frac{1}{x^3}=18\sqrt{3}$ then to prove $x=\sqrt{3}+\sqrt{2}$
If $x^3+\frac{1}{x^3}=18\sqrt{3}$ then we have to prove $x=\sqrt{3}+\sqrt{2}$
The question would have been simple if it asked us to prove the other way round.
We can multiply by $x^3$ and solve the quadratic to get $x^3$ but that would be unnecessa... | $$t+\frac1t=18\sqrt3\iff t^2-(2\cdot9\sqrt3)t+1=0\iff t_{1,2}=\frac{9\sqrt3\pm\sqrt{(9\sqrt3)^2-1\cdot1}}1=$$
$$=9\sqrt3\pm\sqrt{81\cdot3-1}\quad=\quad9\sqrt3\pm\sqrt{243-1}\quad=9\sqrt3\pm\sqrt{242}\quad=\quad9\sqrt3\pm\sqrt{2\cdot121}=$$
$$=9\sqrt3\pm\sqrt{2\cdot11^2}\quad=\quad9\sqrt3\pm11\sqrt2\quad\iff\quad x_{1,2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The milk sharing problem I found a book with math quizzes. It was my father's when he was young. I encountered a problem with the following quiz. I solved it, but I wonder, is there a faster way to do it? If so, how can I compute the time (polynomial time) that is needed to solve it? Can we build an algorithm?
The pro... | I've found a better way, in 10 steps. I tried to improve it but it seems to me that it's the option with less steps. Here it goes:
A B C
(10, 0, 0)-(3, 7, 0)-(3, 4, 3)-(6, 4, 0)-(6, 1, 3)-(9, 1, 0)-(9, 0, 1)-(2, 7, 1)-(2, 5, 3)-(5, 5, 0)
In any case, i'm going to draw a tree (this may sound a bit childish) in order to... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 6,
"answer_id": 4
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How to evaluate $\lim_{n\to\infty}\{(n+1)^{1/3}-n^{1/3}\}?$ How to calculate $$\lim_{n\to\infty}\{(n+1)^{1/3}-n^{1/3}\}?$$
Of course it's a sequence of positive reals but I can't proceed any further.
| Hint: multiply by $\frac{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3}}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3}}$ to get
$$
(n+1)^{1/3}-n^{1/3}=\frac{(n+1)-n}{(n+1)^{2/3}+(n+1)^{1/3}n^{1/3}+n^{2/3}}
$$
Alternatively, use the definition of the derivative:
$$
\begin{align}
\lim_{n\to\infty}\frac{(n+1)^{1/3}-n^{1/3}}{(n+1)-n}
&=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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${1\over{n+1}} {2n \choose n} = {2n \choose n} - {2n \choose n-1}$ I need to prove that $${1\over{n+1}} {2n \choose n} = {2n \choose n} - {2n \choose n-1}$$
I started by writing out all the terms using the formula ${n!\over{k!(n-k)!}}$ but I can't make the two sides equal.
Thanks for any help.
| That should work. Note that $$ \begin {align*} \dbinom {2n}{n-1} &= \dfrac {(2n)!}{(n-1)! (n+1)!} \\&= \dfrac {n}{n+2} \cdot \dfrac {(2n!)}{n!n!} \\&= \dfrac {n}{n+1} \cdot \dbinom {2n}{n}, \end {align*} $$ so we have: $ \dbinom {2n}{n-1} = \left( 1 - \dfrac {1}{n+1} \right) \cdot \dbinom {2n}{n}, $ which is equivalent... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
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Finding distance between two parallel 3D lines I can handle non-parallel lines and the minimum distance between them (by using the projection of the line and the normal vector to both direction vectors in the line), however, in parallel lines, I'm not sure on how to start. I was thinking of finding a normal vector to o... | Let $P$ be a variable point of $L_1$ and $P_0$ a fixed point of $L_2$. Try to minimize $$\left|\frac{\mathbf{a}\times\vec{PP_0}}{|\mathbf{a}|}\right|$$ where $\mathbf{a}$ is leading vector (for example) for $L_1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
} |
Square root of nilpotent matrix How could I show that $\forall n \ge 2$ if $A^n=0$ and $A^{n-1} \ne 0$ then $A$ has no square root? That is there is no $B$ such that $B^2=A$. Both matrices are $n \times n$.
Thank you.
| I found this post while trying to write solutions for my class's homework assignment. I had asked them precisely this question and couldn't figure it out myself. One of my students actually found a very simple solution, so I wanted to share it here:
Suppose that $N = A^2$. Since $N^n = 0$, we have $A^{2n} = 0$. Thus $A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/583442",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Proof that $\sum_{k=2}^{\infty} \frac{H_k}{k(k-1)} $ where $H_n$ is the sequence of harmonic numbers converges? How to prove that $$\displaystyle \sum_{k=2}^{\infty} \dfrac{H_k}{k(k-1)} $$ where $H_n$ is the sequence of harmonic numbers converges and that $\dfrac{H_n}{n(n-1)}\to 0 \ $
I have already proven by inductio... | $$\sum_{k=2}^{\infty} \frac{H_k}{k(k-1)}=\sum_{k=1}^{\infty} \frac{H_{k+1}}{k(k+1)}=-\int_0^1 \ln(1-x)\sum_{k=1}^\infty \frac{x^k}{k}dx$$
$$=\int_0^1\ln^2(1-x)dx=\int_0^1 \ln^2xdx=2$$
Note that we use $\int_0^1 x^k \ln(1-x)dx=-\frac{H_{k+1}}{k+1}$
| {
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"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Understanding induction proof with inequalities I'm having a hard time proving inequalities with induction proofs. Is there a pattern involved in proving inequalities when it comes to induction? For example:
Prove ( for any integer $n>4$ ):
$$2^n > n^2 \\ $$
Well, the skipping ahead to the iduction portion, here's wha... | Working off Eric's answer/approach:
Let P(n) be the statement that $2^n \gt n^2$.
Basis step: (n=5) $2^5\gt 5^2 $ which is true.
Inductive step: We assume the inductive hypothesis that $P(k)$ is true for an arbitrary integer $k\ge5$.
$$ 2^{k} \gt k^2 \text{ (IH)}$$
Our goal is to show that P(k+1) is true. Let's multipl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ratio of balls in a box A box contains some identical tennis balls. The ratio of the total volume of the tennis
balls to the volume of empty space surrounding them in the box is $1:k$, where $k$ is an
integer greater than one.
A prime number of balls is removed from the box. The ratio of the total volume of the
remaini... | The ratio $1:k$ means in particular that if $\alpha$ is the total volume of the balls originally in the box, then $k\alpha$ is the total volume of the empty space surrounding them in the box. In general, if $V_t$ is the total volume of the tennis balls and $V_b$ is the total volume of the box, then the total volume of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/583712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
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De Moivre's formula to solve $\cos$ equation Use De Moivre’s formula to show that
$$
\cos\left(x\right) + \cos\left(3x\right) = 2\cos\left(x\right)\cos\left(2x\right)
$$
$$
\mbox{Show also that}\quad
\cos^{5}\left(x\right) = \frac{1}{16}\,\cos\left(5x\right) + \frac{5}{16}\,\cos\left(3x\right) + \frac{5}{8}\,\cos\left(... | HINT:
$$ (\cos x + i \sin x)^n = \cos nx + i \sin nx $$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is a divisibility language context free? I am working to see if this language would be context free:
L = { 0n1k : n/k is an integer }
After playing with it for a little while, I believe that the language is not context free. Now I am looking to use the pumping lemma to prove that it is not, but am struggling a bit to... | Let $p$ be the pumping length and pick $k>p$.
As $0^k1^k\in L$, we can write $0^k1^k=xyz$ with $|xy|\le p$, $|y|\ge 1$, and $xy^iz\in L$ for all $i\ge 0$. We conclude that $x=0^r, y=0^s$ wirh $r\ge0$, $s\ge 1$. Then $xy^0z=0^{k-s}1^k\notin L$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/583876",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $1<\frac1{e^2(e-1)}\int_e^{e^2}\frac{x}{\ln x}dxProve the following inequalities:
a) $1.43 < \int_0^1e^{x^2}dx < \frac{1+e}2$
b) $2e <\int_0^1 e^{x^2}dx+\int_0^1e^{2-x^2}dx<1+e^2$
c) $1<\frac1{e^2(e-1)}\int_e^{e^2}\frac{x}{\ln x}dx<e/2$
Source: http://www.sms.edu.pk/downloads/prceedingsfordae1/1_intinequal.pdf
W... | for c use mean value theorem for integral calculus.Ie. take $f(x)=\frac{x}{ln x}$ in the interval $[e,e^2]$.
since maximum value of the function in the given interval is $e$ and $\frac{e^2}{2}$ at $x=e$ and $x=e^2$ using maxima minima test.Hence applying meanvalue theorem we get
$$ e<\frac{\int_e^{e^2} \frac{x}{ln... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Solve $z^4 + 4 = 0$ I'm trying to solve it by using its polar form, but then I get
$$
\begin{align*}
z^4 &= (\rho e^{i\phi})^4 = \rho^4 e^{4i\phi}\\
&= -4 = -4 e^{0i}\\
\end{align*}
$$
From the definition of equality of complex numbers, $\rho^4 = -4$ and $4\phi = 0 + 2\pi k$ for some $k \in \mathbb{Z}$.
This would... | Something that we all never learned in high school, but should have, is the amazing factorization $X^4+4=(X^2+2X+2)(X^2-2X+2)$. With this, the total factorization is easy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/584059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
} |
How to find the limit of $\frac{\ln(n+1)}{\sqrt{n}}$ as $n\to\infty$? I'm working on finding whether sequences converge or diverge. If it converges, I need to find where it converges to.
From my understanding, to find whether a sequence converges, I simply have to find the limit of the function.
I'm having trouble gett... | You need to use L'Hopital's rule, twice. What this rule states is that basically you if you are trying to take the limit of something that looks like $\frac{f(x)}{g(x)}$, you can keep on taking the derivative of the numerator and denominator until you get a simple form where the limit is obvious. Note: L'Hopitals rule ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Exact probability of random graph being connected The problem: I'm trying to find the probability of a random undirected graph being connected. I'm using the model $G(n,p)$, where there are at most $n(n-1) \over 2$ edges (no self-loops or duplicate edges) and each edge has a probability $p$ of existing. I found a simpl... | Here is one way to view the formula. First, note that it suffices to show that
$$
\sum_{i=1}^n {n-1 \choose i-1} f(i) (1-p)^{i(n-i)} = 1,
$$
since ${n-1 \choose n-1}f(n)(1-p)^{n(n-n)}=f(n).$ Let $v_1, v_2, \ldots, v_n$ be $n$ vertices. Trivially,
$$
1= \sum_{i=1}^n P(v_1 \text{ is in a component of size }i).
$$
Now let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/584228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-{1\over 6}+\dots-{1\over 2012}+{1\over 2013}$ The sum $1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-{1\over 6}+\dots-{1\over 2012}+{1\over 2013}$ is equal,
a) ${1\over 1006}+{1\over 1007}+{1\over 1008}+\dots+{1\over 2013}$
b) ${1\over 1007}+{1\over 1008}+{1\over 1009}+\do... | $$\sum_{1\le r\le 2n+1}(-1)^{r-1}\frac1r$$
$$=\sum_{1\le r\le 2n+1}\frac1r-2\left(\sum_{1\le r\le n-1}\frac1{2r}\right)$$
$$=\sum_{1\le r\le 2n+1}\frac1r-\left(\sum_{1\le r\le n-1}\frac1{r}\right)$$
$$=\sum_{n\le r\le 2n+1}\frac1r$$
Here $2n+1=2013\implies n=?$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/584323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Let $L = \mathbb F_2[X]/\langle X^4 + X + 1 \rangle$ is a field. Show $L^* = L / \{0\} = \langle X \rangle$ is cyclic. Let $L = \mathbb F_2[X]/\langle X^4 + X + 1 \rangle$ is a field. Show $L^* = L / \{0\} = \langle X \rangle$ is cyclic.
I've proven that $X^4 + X + 1$ is irreducible, so $L$ is a field. I also know that... | since the group is of order 15 any element must have order 1,3,5 or 15.
in any field the equation $x^k = 1 $ can have at most k roots, so the number of elements with order less than $15$ is at most $1 + 3 + 5 = 9$
hence there must be an element of order 15
i will add, after the query from OP, that a slightly more soph... | {
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"timestamp": "2023-03-29T00:00:00",
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Is this function injective / surjective? A question regarding set theory.
Let $g\colon P(\mathbb R)\to P(\mathbb R)$, $g(X)=(X \cap \mathbb N^c)\cup(\mathbb N \cap X^c)$
that is, the symmetric difference between $X$ and the natural numbers.
We are asked to show if this function is injective, surjective, or both.
I trie... | We can view subsets of $X$ as functions $X \to \{0,1\}$: 0 if the point is not in the set, It is commonly very useful to rewrite questions about subsets as questions about functions.
What form does $g$ take if we do this rewrite?
Terminology wise, if $S \subseteq X$, then $\chi_S$ is the function described above:
$$\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/584601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Are convex function from a convex, bounded and closed set in $\mathbb{R}^n$ continuous? If I have a convex function $f:A\to \mathbb{R}$, where $A$ is a convex, bounded and closed set in $\mathbb{R}^n$, for example $A:=\{x\in\mathbb{R}^n:\|x\|\le 1\}$ the unit ball. Does this imply that $f$ is continuous? I've searched ... | The continuity of convex functions defined on topological vectors spaces is rather well understood. For functions defined on a finite dimensional Banach space, i.e., $\mathbb{R}^n$, the classical monograph Convex Analysis by R.T. Rockafellar is a good place to check. Let me first point out a simple trick used ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/584680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Commutation of exponentials of matrices Given two $n \times n$ real matrices $A$ and $B$, prove that the following are equivalent:
(i) $\left[A,B\right]=0$
(ii) $\left[A,{\rm e}^{tB}\right] = 0,\quad$ $\forall\ t\ \in\ \mathbb{R}$
(iii) $\left[{\rm e}^{sA},{\rm e}^{tB}\right] = 0;\quad$ $\forall\ s,t\ \in\ \mathbb{R}$ ... | The easiest way that I see is to do it in two steps, proving $(iii) \Rightarrow (ii) \Rightarrow (i)$.
To prove $(iii) \Rightarrow (ii)$, differentiate $f(s) = [e^{sA},e^{tB}] \equiv 0$. The proof of $(ii) \Rightarrow (i)$ is quite similar.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/584754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Entropy of a distribution over strings Suppose for some parameter $d$, we choose a string from the Hamming cube ($\{0,1\}^d$) by setting each bit to be $0$ with probability $p$ and $1$ with probability $1-p$. What is the entropy of this distribution on the Hamming cube? Clearly, if $p=\frac{1}{2}$, then the entropy wou... | In the Hamming $d$-cube there are $\binom{d}k$ points with $k$ zeroes and $d-k$ ones, so the entropy is
$$-\sum_{k=0}^d\binom{d}kp^k(1-p)^k\lg\left(p^k(1-p)^{d-k}\right)\;,$$
or
$$-\sum_{k=0}^d\binom{d}kp^k(1-p)^{d-k}\Big(k\lg p+(d-k)\lg(1-p)\Big)\;.$$
Now
$$\begin{align*}
\sum_{k=0}^d\binom{d}kkp^k(1-p)^{d-k}&=d\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/584821",
"timestamp": "2023-03-29T00:00:00",
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Show that if $n$ is not divisible by $2$ or by $3$, then $n^2-1$ is divisible by $24$.
Show that if $n$ is not divisible by $2$ or by $3$, then $n^2-1$ is divisible by $24$.
I thought I would do the following ... As $n$ is not divisible by $2$ and $3$ then $$n=2k+1\;\text{or}\\n=3k+1\;\text{or}\\n=3k+2\;\;\;\;$$for s... | $n^2-1=(n-1)(n+1)$
$n$ is not even so $n-1$ and $n+1$ are even.
Also $n=4t+1$ or $4t+3$, this means at least one of $n-1$ or $n+1$ is divisible by 4.
$n$ is not $3k$ so at least one of $n-1$ or $n+1$ must be divisible by 3.
So $n^2-1$ has factors of 4, 2(distinct from the 4) and 3 so $24|n^2-1$
Edit: I updated my post... | {
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How to notice that $3^2 + (6t)^2 + (6t^2)^2$ is a binomial expansion. The other day during a seminar, in a calculation, a fellow student encountered this expression:
$$\sqrt{3^2 + (6t)^2 + (6t^2)^2}$$
He, without much thinking, immediately wrote down:
$$(6t^2+3)$$
What bothers me, is that I didn't see that. Although I ... | $\bigcirc^2+\triangle+\square^2$ is a complete square if $\left|\triangle\right|=2\cdot\bigcirc\cdot\triangle$.
For example, $49x^2-42x+9$ is a complete square since the only candidate for $\triangle$ in $42x$, and indeed $42x=2\cdot7x\cdot3$. In our special case we have three candidates for $\triangle$; your fellow s... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving $\lim_{n \to\infty} \frac{1}{n^p}=0$ for $p > 0$? I'm trying to prove 3.20a) from baby Rudin. We are dealing with sequences of real numbers.
Theorem.
$$\lim_{n \to {\infty}} \frac{1}{n^p} = 0; \hspace{30 pt}\mbox {$p > 0$}$$
Proof. Let $\epsilon > 0$. Because of the Archimedan property of real numbers, there e... | Another approach: you can show $x^p \rightarrow \infty $ as $x\rightarrow \infty$. Rewrite as $$\frac {x^{p+1}}{x}$$ , which is an indeterminate $\infty/\infty$ , and use L'Hopital, to get $$\frac {(p+1)x^{p}}{1} $$. Since $p$ is fixed and $p+1>1$, you can show this goes to $\infty$ , and then $1/x^p\rightarrow 0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/585148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Showing that $E[X|XI would like to show that:
$\hspace{2mm} E[X|X<x] \hspace{2mm} \leq \hspace{2mm} E[X] \hspace{2mm} $ for any $x$
X is a continuous R.V. and admits a pdf. I'm guessing this isn't too hard but I can't come up with a rigorous proof. Thanks so much.
| Hint: $$E[X] = E[X|X<x]P(X<x) + E[X|X\geq x]P(X\geq x)$$
Also:
$$E(X|X<x)< x \leq E(X|X\geq x)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $\frac{y^2-x^2}{x+y+1}=\pm1$ is a hyperbola? How to prove $\frac{y^2-x^2}{x+y+1}=\pm1$ is a hyperbola, knowing the canonical form is $\frac{y^2}{a^2}-\frac{x^2}{b^2}=\pm1$ where $a$ and $b$ are constants? Thanks !
| Let
$$
\frac{y^2-x^2}{x+y+1}=1\\
\Rightarrow y^2-x^2=x+y+1\\
\Rightarrow y^2-x^2-x-y=1
$$
Complete the squares for x and y . You will get rectangular hyperbola. Similar will be the case if
$$
\frac{y^2-x^2}{x+y+1}=-1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/585301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Fourier series formula with finite sums
Let $f\in C(\mathbb{R}/2\pi\mathbb{Z})$, meaning that $f$ is continuous with period $2\pi$. Let $x_N(j)=2\pi j/N$. Define $$c_N(n)=\dfrac1N\sum_{j=1}^Nf(x_N(j))e^{-ix_N(j)n}.$$ Show that for any integer $M$, $$f(x_N(j))=\sum_{n=-M}^{N-M-1}c_N(n)e^{ix_N(j)n}.$$
This looks like t... | You are asked to prove the formula for the Discrete Fourier Transform. The continuous periodic function is largely irrelevant, since we only deal with its values on the uniform grid $x_j$. Presumably there is some way to get DFT from continuous Fourier transform, but I would not bother: the DFT is simpler, since we wor... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof that the limit of a sequence is equal to the limit of its partial sums divided by n Let $\{ x_n \}_n$ be a sequence of real numbers. Suppose $ \lim_{n \to \infty}x_n=a.$
Show that
$$\lim_{n \to \infty} \frac{x_1+x_2+...+x_n}{n}=a$$
As it is my first proof I'm not really sure whether I am allowed to do the followi... | The proof is quite wrong and almost nonsensical (Sorry!).
1) You cannot assume $\lim_{n \to \infty} S_n$ exists.
2) Since $n$ is the variable which tends to infinity, you cannot repeat $n$ times like that and get it out of the limit.
3) The statement $na = \lim_{n \to \infty} S_n$ is nonsensical.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integration of parabola I have this homework question I am working on:
The base of a sand pile covers the region in the xy-plane that is bounded by the parabola $x^2 +y = 6$ and the line $y = x$: The height of the sand above the point $(x;y)$ is $x^2$: Express the volume of sand as (i) a double integral, (ii) a triple ... |
Now I am REALLY confused what the question means about the x2 being the height. What point are they talking about?
Means exactly that, the height of the surface on the $z$ axis is given by $z=x^2$, you can also look at it like a function on the $xy$ plane given by $z=f(x,y) = x^2$.
Also, if it is a volume then does... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Positivity of the Coulomb energy in 2d Let $$D(f,g):=\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{1}{|x-y|}\overline{f(x)}g(y)~dxdy$$
with $f,g$ real valued and sufficiently integrable be the usual Coulomb energy. Under the assumption $D(|f|,|f|)<\infty$ it can be seen that $D(f,f)\geq 0$ (see for example Lieb-Loss, Anal... | This is true when the support of $f$ is contained in the unit disc. If the support is contained in a disc $|z|<R$, then $(f,f)$ is bounded from below by a constant that depends
on $R$. This minor nuisance makes the logarithmic potential somewhat different from the Newtonian
potential, however most statements of potenti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/585619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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If a function is smooth is 1 over the function also smooth If $f(x):\mathbb{R}\rightarrow\mathbb{C}$ is $C^\infty$-smooth. Is $1/f(x)$ also $C^\infty$-smooth? $f(x)\neq0$
| If $f$ is differentiable and non-zero at some point $a$, then $1/f$ is differentiable at $a$, and $(1/f)'=-f'/f^2$. This is a "base case" of an induction argument for the following statement:
$1/f$ is $n$ times differentiable, and $(1/f)^{(n)}$ equals a polynomial in the functions $f,f',f'',\ldots,f^{(n-1)}$ divided... | {
"language": "en",
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if the matrix such $B-A,A$ is Positive-semidefinite,then $\sqrt{B}-\sqrt{A}$ is Positive-semidefinite Question:
let the matrix $A,B$ such $B-A,A$ is Positive-semidefinite
show that:
$\sqrt{B}-\sqrt{A}$ is Positive-semidefinite
maybe The general is true?
question 2:
(2)$\sqrt[k]{B}-\sqrt[k]{A}$ is Positive-semidefini... | Another proof (short and simple) from "Linear Algebra and Linear Models" by R. B. Bapat.
Lemma Let $A$ and $B$ be $n\times n$ symmetric matrices such that $A$ is positive definite and $AB+BA$ is positive semidefinite, then Y is positive semidefinite.
Proof of $B\geq A \implies B^{\frac{1}{2}}\geq A^{\frac{1}{2}}$
Fir... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Fourier transform supported on compact set Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Show that $$\hat{f}(y)=1_{[-\pi,\pi]}(y)\sum_{n=-\infty}^\infty f(n)e^{-iny}$$ in the sense of $L^2(\mathbb{R})$-norm convergence.
I know that $f$ must be continuous and going to $0$ at $\pm\infty$... | *
*Expand $\hat f$ into a Fourier series on $[-\pi,\pi]$, that is $\hat f(y)=\sum_{n=-\infty}^\infty c_n e^{-iny}$. (I put $-$ in the exponential to get closer to the desired form; this does not change anything since $n$ runs over all integers anyway.)
*Write $c_n $ as an integral of $\hat f(y)e^{iny}$ over $[-\p... | {
"language": "en",
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derivative of an integral from 0 to x when x is negative? Given a function $$F(x) = \int_0^x \frac{t + 8}{t^3 - 9}dt,$$ is $F'(x)$ different when $x<0$, when $x=0$ and when $x>0$?
When $x<0$, is $$F'(x) = - \frac{x + 8}{x^3 - 9}$$ ... since you can't evaluate an integral going from a smaller number to a bigger number?... | No, we have
$$F'(x)=\frac{x+8}{x^3-9}$$
for all $x<\sqrt[3]{9}$. The limitation is due to the fact that the integral is meaningful only when the interval doesn't contain $\sqrt[3]{9}$ and so we must consider only the interval $(-\infty,\sqrt[3]{9})$ that contains $0$.
If $b<a$, one sets, by definition,
$$
\int_a^b f(t)... | {
"language": "en",
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Is a decimal with a predictable pattern a rational number? I'm starting as a private Math tutor for a high school kid; in one of his Math Laboratories (that came with an answer sheet) I was stumped by an answer I encountered in the True or False section (I'm certain it should've been a False):
The number 4.21211211121... | A real number is rational if and only if its decimal expansion terminates or eventually repeats.
Lemma: Every prime $p \neq 2, 5$ divides a repunit.
Proof of Lemma:
Fix a prime $p \neq 2,5$. Let $\textbf{A}$ be the set of repunits, so
$$\textbf{A} = \left\{\displaystyle\sum\limits_{k=1}^{n} 10^{k-1} \, \mid \, n \in \... | {
"language": "en",
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What is the difference between an indefinite integral and an antiderivative? I thought these were different words for the same thing, but it seems I am wrong. Help.
| An anti-derivative of a function $f$ is a function $F$ such that $F'=f$.
The indefinte integral $\int f(x)\,\mathrm dx$ of $f$ (that is, a function $F$ such that $\int_a^bf(x)\,\mathrm dx=F(b)-F(a)$ for all $a<b$) is an antiderivative if $f$ is continuous, but need not be an antiderivative in the general case.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 3
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Zagier's proof of the prime number theorem. In Zagier's paper, "Newman's Short Proof of the Prime Number Theorem", (link below) his theorem ${\bf (V) }$ states that,
$$ \int_{1}^{\infty} \frac{\vartheta(x) - x}{x^2} dx \text{ is a convergent integral.} $$
Note: $\vartheta(x) = \sum_{p \le x} \log(p)$, where $p$ is a pr... | In step $\mathbf{III}$, it was shown that $\vartheta(x) \leqslant C\cdot x$ for some constant $C$. That is enough to ensure
$$\lim_{x\to\infty} x^{-s}\vartheta(x) = 0$$
for $\Re s > 1$.
| {
"language": "en",
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"source": "stackexchange",
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RSA encryption/decryption scheme I've been having trouble with RSA encryption and decryption schemes (and mods as well) so I would appreciate some help on this question: Find an $e$ and $d$ pair with $e < 6$ for the integer $n = 91$ so that $n,e,d$ are the ingredients of an RSA encryption/decryption scheme. Use it to e... | We are given $N$ and that will give us the prime factors $p$ and $q$ as:
$$N = 91 = p \times q = 7 \times 13$$
We need the Euler Totient Function of the modulus, hence we get:
$$\varphi(N) = \varphi(91) = (p-1)(q-1) = 6 \times 12 = 72$$
Now, we choose an encryption exponent $1 \lt e \lt \varphi(N) = 72$. We were told ... | {
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Find all primes $p$ such that $14$ is a quadratic residue modulo $p$. I want to find all primes $p$ for which $14$ is a quadratic residue modulo $p$. I referred to an example that was already posted for finding all odd primes $p$ for which $15$ is a quadratic residue modulo $p$, but I am getting stuck.
This is what I h... | Quadratic reciprocity modulo $2$ works slightly differently. In fact, it holds that
$$\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}.$$
Thus, you have:
$$\left(\frac{14}{p}\right) = (-1)^{(p^2-1)/8} \cdot (-1)^{(p-1)/2} \cdot \left(\frac{p}{7}\right).$$
This means that you need to look at the form of $p$ modulo $8$ (firs... | {
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Prove that Baire space $\omega^\omega$ is completely metrizable? When I tried to prove that Baire space $\omega^\omega$ is completely metrizable, I defined a metric $d$ on $\omega^\omega$ as: If $g,h \in \omega^\omega$ then let $d(g,h)=1/(n+1)$ where $n$ is the smallest element in $\omega$ so that $g(n) \ne h(n)$ is su... | The point here is that two functions are close iff they agree on an initial segment, that is, $d(f,g)\le 1/(n+1)$ iff $f(0)=g(0),f(1)=g(1),\dots,f(n-1)=g(n-1)$. Now, if $(f_n)_n$ is a Cauchy sequence, then, for each $n$, there is $N_n$ such that for all $m,k>N_n$ we have $d(f_m,f_k)\le1/(n+1)$. That is, all functions $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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linear algebra foundation of Riemann integrals Let $V$ be the vector space of real functions $f\colon [a,b]\to \mathbb R$ and let $X$ be the set of characteristic (indicatrix) functions of subintervals: $X=\{\mathbb 1_I\colon I\subset [a,b] $ interval $\}$. We define $T\colon X \to \mathbb R$ as $T(\mathbb 1_I) = |I|$ ... | I think the only thing that needs to be proved for $T$ to be extendable to a linear map on the vector space generated by $X$ is the following. Assume $f=\mathbb{1}_{[c,d)}$ and $g = \mathbb{1}_{[d,e)}$ with $f,g \in X$. Then $f+g$ is also in $X.$ (I am assuming that half-open intervals are used here.) It is then needed... | {
"language": "en",
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Categorization of PBE refinements into forward/backward looking? I have recently come across the term forward / backward looking refinement of a Perfect Bayesian Equilibrium. I am, however, unsure about the meaning of this term, and unable to find any information about this. Does anyone know the difference between the ... | The usual examples are backward induction and forward induction. Somewhat surprisingly, backward induction is forward looking and forward induction is backwards looking.
In backward induction you start at the end of the game- which lies in the future. So it is forward looking. Forward induction, which has many differen... | {
"language": "en",
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Why we need $\sigma$ finite measure to define $L^{p}$ space? I am curious why we need $\sigma$ finite measure to define $L^{p}$ space. More generally, Why we need $\sigma$ finite measure instead of just finite measure?
| You can define $L^p$ on any measure space you like. If it just those measurable functions for which $|f|^p$ is integrable. And then you modulo by functions zero a.e.
But if you don't impose some hypothesis like $\sigma$ finite, many results about $L^p(\mathbb R)$ don't always generalize very well to arbitrary measure... | {
"language": "en",
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Holomorphic problem I have a function $f(z)$ holomorphic in $\mathbb{C}\setminus\mathbb{R}^-$. I have these information:
*
*$f(x+i\epsilon) = f(x-i\epsilon)$ on $\mathbb{R}^+$ (the $\epsilon$ is indented as a shorthand for a limit);
*$f(x+i\epsilon) = - f(x-i\epsilon)$ on $\mathbb{R}^-$;
*$f(z)=\sqrt{z} + O\left(\... | Let $\sqrt{z}$ denote the square root which branch cut on the negative real axis. Is there a reason $f(z)=\sqrt{z}(1+\frac{1}{z})$ does not satisfy your conditions?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How do you prove set with modulo? Given any prime $p$. Prove that $(p-1)! \equiv -1 \pmod p$.
How to prove this?
| This is known as Wilson's theorem: (not completely, but Wilson's theorem is an if and only if while this is only an if)
https://en.wikipedia.org/wiki/Wilson%27s_theorem
The idea is that (p-1)! is the product of an element of each residue class $\bmod p$, also since all numbers less than p are relatively prime to p each... | {
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does this series converge? $\sum_{n=1}^\infty{\left( \sqrt[3]{n+1} - \sqrt[3]{n-1} \right)^\alpha} $
show the the following series converge\diverge
$\sum_{n=1}^\infty{\left( \sqrt[3]{n+1} - \sqrt[3]{n-1} \right)^\alpha} $
all the test i tried failed (root test, ratio test,direct comparison)
please dont use integrals ... | Ratio test is inconclusive, but using Raabe's test we can see that the series converges when $\alpha>\frac{3}{2}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving a limit of a sequence I have to prove that for a > 1, $$\lim_{ n\rightarrow \infty }{ { \left (\frac { 2^{n^a} }{ n! } \right ) } } = \infty$$
I've tried to apply L'Hôpital's rule and d'Alembert's ratio test, but without any success...
Any help would be greatly appreciated!
| Using that $(n-k)/n<1$ for all $k=1,\ldots,n-1$, we have (for $n$ big enough such that $n^{a-1}\log 2\geq \log n$)
$$
\frac{2^{n^a}}{n!}\geq\frac{n^n}{n!}=\frac1{1 \;\frac{n-1}n\;\cdots\;\frac1n}\geq\frac1{\frac1n}=n\to\infty
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/587236",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is this formula for $\zeta(2n+1)$ correct or am I making a mistake somewhere? I am calculating $\zeta(3)$ from this formula:
$$\zeta(2n+1)=\frac{1}{(2n)!}\int_0^{\infty} \frac{t^{2n}}{e^t -1}dt$$
From Grapher.app, I get $\int_0^{\infty} \frac{x^{2}}{e^x -1}dx = .4318$ approximately which, when multiplied by $\frac{1}{2... | Computing the integral with Wolfram Alpha on a truncated interval (I chose $[0, 10000]$) gives $2.40411$, which is just about the correct answer. It would appear to be an issue in the application you're using.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How did they get this answer for critical value I know how to get a critical value but I am not sure now how to do it when they added "when testing the claim that p = 1/2"....
how did they get the answer for critical value ?
| It would be good if we could see the entire question, but if it is the Sign-Test, then it make sense to use p = 1/2 because it is either "yes" (assigned a plus) or a "no" (assigned a minus, hence the name of the test) But again, more information about the question would be useful
| {
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What is the purpose of the first test in an inductive proof? Learning about proof by induction, I take it that the first step is always something like "test if the proposition holds for $n = \textrm{[the minimum value]}$"
Like this:
Prove that $1+2+3+...+n = \frac{n(n + 1)}{2}$ for all $n \ge 1$.
Test it for $n = 1$... | Imagine a pond with an infinite linear progression of lily pads. You have a frog who, if he hops on one pad, he is guaranteed to hop on the next one. If he hops on the first pad, he'll visit them all. But if he never makes the first lilypad, all bets are off.
| {
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"source": "stackexchange",
"question_score": "22",
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Can an algorithm be part of a proof? I am an undergraduate student. I have read several papers in graph theory and found something may be strange: an algorithm is part of a proof. In the paper, except the last two sentences, other sentences describe a labeling algorithm.
Can an algorithm be part of a proof? I do not un... | If you ask me, I would say that everything can be part of a proof as long as you bring a convincing reasonable argument. When I listen to my colleagues topologist, they make a few drawings (generally sketches of knots) on the blackboard and they claim it's a proof. Thus, a drawing or an algorithm, there is no big diffe... | {
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Equality of discriminants of integral bases (statement in Ireland and Rosen, A Classical Introduction to Modern Number Theory) I'm doing independent study and need assistance.
This is taken from Ireland and Rosen's A Classical Introduction to Modern Number Theory,
Chapter 12.
Let F/Q be an algebraic number field, D th... | Just to put this in an answer.
Let $\{\omega_1,\ldots,\omega_n\}$ and $\{\omega_1',\ldots,\omega_n'\}$ be two integral bases for $D$. By definition, this means that $D$ is a free $\mathbb{Z}$-module, and that these are two bases for $D$. Thus, by definition there must exists some $M\in\text{GL}_n(\mathbb{Z})$ such that... | {
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Checking on some convergent series I need some verification on the following 2 problems I attemped:
I have to show that the following series
is convergent: $$1-\frac{1}{3 \cdot 4}+\frac{1}{ 5 \cdot 4^2 }-\frac{1}{7 \cdot 4^3}+ \ldots$$ .
My Attempt: I notice that the general term is given by $$\,\,a_n=(-1)^{n}{1 ... | Hint: For all sufficiently large $n$ (in fact, $n \ge 1$ suffices for this), we have $\ln{n} \le \sqrt{n}$; thus
$$\sum\limits_{n = 2}^{\infty} \frac{\ln n}{n^2} \le \sum\limits_{n = 2}^{\infty} \frac{1}{n^{3/2}}$$
which is a $p$-series.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Integral through Fourier Transform and Parseval's Identity $$
\int_{-\infty}^{\infty}{\rm sinc}^{4}\left(\pi t\right)\,{\rm d}t\,.
$$
Can you help me evaluate this integral with the help of Fourier Transform and Parseval Identity. I could not see how it is implemented. Thank you..
| There are two correct answers to this question, depending on how you understand sinc. My guess is that your convention is $\operatorname{sinc}x = \frac{\sin x}{x}$.
I don't know your FT convention, but I will use $\hat f(\xi)=\int f(x)e^{-2\pi i \xi x}\,dx$. Then
$$\hat \chi_{[-a,a]}(\xi)= \int_{-a}^a e^{-2\pi i \xi x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/587866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Boundedness of functional In the setting of $2\pi$-periodic $C^1$ functions (whose Fourier series converge to themselves), and given a linear functional $D:C^1_{\text{per}}\to\mathbb R$ satisfying $\sup_{n}|D(e^{inx})|<\infty$ I would like to show that $D$ is continuous (or equivalently, bounded).
Attempt
The supremum ... | This is not true. The functions $\{e^{inx}\}$ do not span $C^1_{\text{per}}$; indeed, no countable set can. So using the axiom of choice, one can show the existence of a linear functional that vanishes on all the $e^{inx}$ but is not continuous.
You can't tell whether a linear functional is continuous by looking at a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/588013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$T(v_1), \ldots,T(v_k)$ are independent if and only if $\operatorname{span}(v_1,\ldots,v_k)\cap \ker(T)=\{0\}$ I need help with this:
If $T:V\rightarrow W$ is a linear transformation and $\{v_1,v_2,\ldots,v_k\}$ is a linearly independent set in $V$, prove that $T(v_1), T(v_2),\ldots,T(v_k)$ are independent in $W$ if ... | Let $v=c_1v_1+\cdots+c_nv_n\in\mathrm{ker}~T$, noting that also $v\in\mathrm{span}~\{v_1,\ldots,v_n\}$, then $$T(v)=T(c_1v_1+\cdots+c_nv_n)=c_1T(v_1)+\cdots+c_nT(v_n)=0$$
implies $c_1=\cdots=c_n=0$ since $T(v_1),\ldots,T(v_n)$ are linear independent. It follows that $\mathrm{span}~\{v_1,\ldots,v_n\}\bigcap\mathrm{ker}~... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/588100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to plot circle with Maxima? I cannot get equitation plotted
Expression:
f1: (x^2) + (y^2) = 9;
I try this command:
wxplot2d(f1, [x, -5, 5], [y, -5, 5]);
And it gives:
plot2d: expression evaluates to non-numeric value everywhere in plotting range.
plot2d: nothing to plot.
What is correct way to plot such expressi... | There are 4 methods to draw circle with radius = 3 and centered at the origin :
*
*load(draw); draw2d(polar(3,theta,0,2*%pi));
*load(draw); draw2d(ellipse (0, 0, 3, 3, 0,360)
*plot2d ([parametric, 3*cos(t), 3*sin(t), [t,-%pi,%pi],[nticks,80]], [x, -4, 4])$
*load(implicit_plot);
z:x+y*%i;
implicit_plot (abs(z) = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/588180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Study convergence of the series of the $\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin{(\ln{n})}}{n^a}$ Question:
Study convergence of the series
$$\sum_{n=1}^{\infty}(-1)^{n-1}\dfrac{\sin{(\ln{n})}}{n^a},a\in R$$
My try: Thank you DonAntonio help,
(1):$a>1$,
since
$$|\sin{(\ln{n})}|\le1$$
so
$$\left|\dfrac{\sin{(\ln{n})}... | Use absolute value
$$\left|(-1)^{n-1}\frac{\sin\log n}{n^a}\right|\le\frac1{n^a}$$
and thus your series converges absolutely for $\;a>1\;$ .
It's clear that for $\;a\le 0\;$ the series diverges (why?), so we're left only with the case $\;0<a\le1\;$...and perhaps later I'll think of something.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/588259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 1
} |
Find Laurent series Let $$f(z):=\frac{e^{\frac{1}{z}}}{z^2+1}$$ and let $$\sum_{k}a_kz^k$$ with k in Z the Laurent series of $f(z)$ for $0<|z|<1$. I have to find a formula for $a_k$. I've tried a lot, but I'm stuck. Can somebody help me? I want to do it with ordening of absolute series.
| We have $$e^\frac{1}{z}=1+\frac{1}{z}+\frac{1}{2!} \frac{1}{z^2}+ \dots$$ as well as the geometric expansion$$\frac{1}{z^2+1}=1-z^2+z^4 -+ \dots $$
valid in the annulus $0<|z|<1$. Multiply the two and collect coefficients from terms of the same degree.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/588336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
NormalDistribution: Problem Question:Cans of regular Coke are labeled as containing $12 \mbox{ oz}$.
Statistics students weighted the content of 6 randomly chosen cans, and found the mean weight to be $12.11$.
Assume that cans of Coke are filled so that the actual amounts are normally distributed with a mean of $12... | You need to find $z$ that corresponds to the sample mean from this exercise. So first a comment on your notation - it should be:
$$z=\frac{\bar{x}-\mu_{\bar{x}}}{\sigma_{\bar{x}}}$$
Of course $\mu_{\bar{x}}=\mu$. But $\sigma_{\bar{x}}\neq\sigma$, as you have computed with. Look up the standard deviation of the sample m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/588438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Law of large numbers for Brownian Motion Let $\{B_t: 0 \leq t < \infty\}$ be standard Brownian motion and let $T_n$ be an increasing sequence of finite stopping times converging to infinity a.s. Does the following property hold?
$$\lim_{n \to \infty}\frac{B_{T_n}}{T_n} = 0$$ a.s.
| This is an almost sure property hence the result you are asking to check is equivalent to the following.
Let $f:\mathbb R_+\to\mathbb R$ denote a function such that $\lim\limits_{t\to+\infty}f(t)=0$ and $(t_n)$ a sequence of nonnegative real numbers such that $\lim\limits_{n\to\infty}t_n=+\infty$. Then $\lim\limits_{n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/588499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $\{u,v\}$ is an orthonormal set in an inner product space, then find $\lVert 6u-8v\rVert$ If $\{u,v\}$ is an orthonormal set in an inner product space, then find $\|6u-8v\|$.
That's pretty much it. I'm trying to study for a quiz and can't figure it out. There are no examples in my book to help me out, just a questio... | Compute the square of the norm and use the definition of the inner product as being bilinear:
$\|6u-8v\|^2 = \langle 6u-8v, 6u-8v \rangle = 36\langle u,u \rangle - 48\langle u, v \rangle - 48 \langle v,u \rangle + 64 \langle v,v \rangle$.
Now use that $u$ and $v$ are orthonormal and the fact that $\langle w,w \rangle$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/588700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Relationship between primitive roots and quadratic residues I understand that if $g$ is a primitive root modulo an odd prime $p$, then Euler's Criterion tells us that $g$ cannot be a quadratic residue.
My question is, does this result generalize to prime powers? That is, if $g$ is a primitive root modulo $p^m$ for an o... | The order of a quadratic residue modulo $n$ divides $\varphi(n)/2$. A primitive root has order $\varphi(n)$. Hence a primitive root is always a quadratic nonresidue.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/588774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Show two groups are isomorphic I need to show two groups are isomorphic. Since I know they are of the same order, would finding an element that generates the other elements, in both groups, suffice to show that they are isomorphic?
| Yes. If you can find an element $x$ which generates the finite group $G$ then it is cyclic. If $G=\langle x\rangle$ and $G'=\langle x'\rangle$ are both cyclic, and $|G|=|G'|$ then $G$ is isomorphic to $G'$ by an isomorphism $\phi\colon G\to G'$ which is defined by $\phi(x^k)=x'^k$. Show that this map is an isomorphism.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/588837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If $a=a'$ mod $n$ and $b=b'$ mod $n$ does $ab=a'b'$ mod $n$? So far this is my working out:
$n|a-a'$ and $n|b-b'$
So $n|(a-a')(b-b')$
Expanding: $n|ab-a'b-ab'+a'b'$
I'm not sure what to do next. I need to show that $n|ab-a'b'$
| We can write $a' = a + n k$ for some $k$, and likewise $b' = b + n m$; then
$$a'b' = (a + nk)(b + nm) = ab + n(\text{stuff}) \equiv ab \pmod{n}$$
upon distribution the multiplication.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/588925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Showing there is some point interor to the unit disk, where V=(0,0) using covering maps Let $B$ be the closed unit disk in $\mathbb{R}^2$
and suppose that $V = (p(x, y), q(x, y))$
is a vector field (p,q are continuous functions) defined on B. The boundary
of $B$ is the unit circle $S^1$
. Show that if at every point of... | The map $V/|V|$ had better not be a covering map because the disk is two-dimensional and the circle is one-dimensional.
Here's a guide to a solution using a little bit of homotopy theory. First, observe that the map $V/|V|$ restricted to the boundary circle of $B$ is a degree-one self-map of the circle. Now answer thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/589009",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The contraction mapping theorem Let $f : \mathbb{R} \to \mathbb{R} : f(x) = 1 + x + \mathbb{e}^{-x}$ and let $ U [1,\infty)$. Firstly i need to show that $f$ maps $U$ into itself. But im can only see how $f$ maps $U$ to $[2 +1/\mathbb{e},\infty) $ as if we set $x=1$ then $f(1) = 2 + \mathbb{e}^{-1}$.
As $x \to \infty$ ... | When someone says that $f$ maps $U$ into $U$, they don't mean that every point in $U$ must be $f(x)$ for some $x \in U$. They merely mean that $f(x) \in U$ for every $x \in U$. And since $[2+1/e,\infty) \subset U$, you have solved that part of the problem.
And yes, the mean value theorem would be an excellent way to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/589103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Cyclic Groups - Existence of the identity
How is it claimed that the identity of a cyclic group $G$ with generator 'a' can be written in the form of $e = a^m$ where $m$ is a positive integer.
A proof for a subgroup of $G$ being in itself cyclic claims that any element in the subgroup can be written in the the form $... | First I assume that we're talking about a finite cyclic group. The group $(\mathbb Z, +)$ is an example of what some people would call an infinite cyclic group and it is certainly not true that $n\cdot 1 = 0$ for some $n > 0$ (note everything is translated into additive notation in that example, but I switch back to m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/589175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Computing the area and length of a curve
Using the Riemannian hyperbolic metric $$g = \frac{4}{(1-(u^2+v^2))^2}\pmatrix{ 1 & 0 \\ 0 & 1 \\}$$ on the disk $D_p = \{(u,v)\ | \ u^2 +v^2 \le p^2\}$ compute the area of $D_p$ and the length of the curve $\partial D_p$.
| This is just the hyperbolic metric
$$ds={2|dw|\over 1-|w|^2}, \qquad w:=u+iv,\tag{1}$$
on the unit disk in the $(u+iv)$-plane. Therefore the length of $\partial D_p$ computes to
$$L(\partial D_p)=\int\nolimits_{\partial D_p} ds={2\over 1-p^2}\int\nolimits_{\partial D_p} |dw|={4\pi p\over1-p^2}\ .$$
For any conformal me... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/589295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Residues at singularities I have the following question: Show that the integral
$$\int_{-\infty}^{+\infty}\frac{\cos\pi x}{2x-1}dx = -\frac\pi2$$
Clearly there is a singularity at $z=1/2$ but I think this is a removable singularity so it has $0$ residue. Is this right or have I missed another singularity? If I am righ... | You are right, the function
$$f(z) = \frac{\cos \pi z}{2z-1}$$
is entire. However, to evaluate the integral, one considers a different function,
$$g(z) = \frac{e^{i\pi z}}{2z-1},$$
which has a pole in $z = \frac12$. We then have
$$\int_{-\infty}^\infty f(x)\,dx = \operatorname{Re} \int_{-\infty}^\infty g(x)\,dx.$$
The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/589409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
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