Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Proof of parallel lines
The quadrilateral ABCD is inscribed in circle W. F is the intersection point of AC and BD. BA and CD meet at E. Let the projection of F on AB and CD be G and H, respectively. Let M and N be the midpoints of BC and EF, respectively. If the circumcircle of triangle MNH only meets segment CF at Q... | Possibly the last steps of a proof
This is no full proof, just some observations which might get you started, but which just as well might be leading in the completely wrong direction.
You could start from the end, i.e. with the last step of your proof, and work backwards.
$P$ is the midpoint of $FB$ and $Q$ is the mid... | {
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"url": "https://math.stackexchange.com/questions/431742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Implication with a there exists quantifier When I negate
$ \forall x \in \mathbb R, T(x) \Rightarrow G(x) $
I get $ \exists x \in \mathbb R, T(x) \wedge \neg G(x) $
and NOT
$ \exists x \in \mathbb R, T(x) \Rightarrow \neg G(x) $
right?
What would it mean if I said $ \exists x \in \mathbb R, T(x) \Rightarrow \neg G(... | You're correct that the negation of $\forall x (T(x) \rightarrow G(x))$ is $\exists x (T(x) \wedge \neg G(x))$.
The short answer is that $\exists x (\varphi(x) \rightarrow \psi(x))$ doesn't really have a good English translation. You could try turning this into a disjunction, so that $\exists x (\varphi(x) \rightarrow ... | {
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"uniquely written" definition I'm having troubles with this definition:
My problem is with the uniquely part, for example the zero element:
$0=0+0$,
but $0=0+0+0$
or $0=0+0+0+0+0+0$.
Another example, if $m \in \sum_{i=1}^{10} G_i$ and $m=g_1+g_2$, with $g_1\in G_1$ and $g_2\in G_2$,
we have: $m=g_1+g_2$ or $m=g_1+g... | Well notice what the definition says. It says that for each $m \in M$, you need to be able to write $m= \sum\limits_{\lambda \in \Lambda} g_{\lambda}$ where this sum is over all $\lambda$. So for $0$, the only possibility is a sum of $0$ $\lambda$-many times.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/431931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Proving the normed linear space, $V, ||a-b||$ is a metric space (Symmetry) The following theorem is given in Metric Spaces by O'Searcoid
Theorem: Suppose $V$ is a normed linear space. Then the function $d$ defined on $V \times V$ by $(a,b) \to ||a-b||$ is a metric on $V$
Three conditions of a metric are fairly straight... | $$\|a-b\|=\|(-1)(b-a)\|=|-1|\cdot\|b-a\|=\|b-a\|$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/431999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How can I calculate a $4\times 4$ rotation matrix to match a 4d direction vector? I have two 4d vectors, and need to calculate a $4\times 4$ rotation matrix to point from one to the other.
edit - I'm getting an idea of how to do it conceptually: find the plane in which the vectors lie, calculate the angle between the v... | Consider the plane $P\subset \mathbf{R}^4$ containing the two vectors given to you. Calculate the inner product and get the angle between them. Call the angle $x$.
Now there is a 2-d rotation $A$ by angle $x$ inside $P$. And consider identity $I$ on the orthogonal complement of $P$ in $\mathbf{R}^4$.
Now $A\oplus... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Math Parlor Trick
A magician asks a person in the audience to think of a number $\overline {abc}$. He then asks them to sum up $\overline{acb}, \overline{bac}, \overline{bca}, \overline{cab}, \overline{cba}$ and reveal the result. Suppose it is $3194$. What was the original number?
The obvious approach was modular ar... | Let $S$ be the sum,
$$S \text{ mod} 10=A$$
$$S \text{ mod} 100=B$$
$$A=2b+2a+c$$
$$\frac{B-A}{10}=(2c+2a+b)$$
$$\frac{S-B}{100}=(a+2b+2c)$$
$$\text{Now just solve the system of equations for $a$ $b$ and $c$}$$
$$\text{ The original number will be a+10b+100c}$$
$$\text{ Now memorize this formula and do the addition in y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Two questions on topology and continous functions I have two questions:
1.) I have been thinking a while about the fact, that in general the union of closed sets will not be closed, but I could not find a counterexample, does anybody of you have one available?
2.) The other one is, that I thought that one could possibl... | (1) Take the closed sets
$$\left\{\;C_n:=\left[0\,,\,1-\frac1n\right]\;\right\}_{n\in\Bbb N}\implies \bigcup_{n\in\Bbb N}C_n=[0,1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/432244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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Infimum and supremum of the empty set Let $E$ be an empty set. Then, $\sup(E) = -\infty$ and $\inf(E)=+\infty$. I thought it is only meaningful to talk about $\inf(E)$ and $\sup(E)$ if $E$ is non-empty and bounded?
Thank you.
| There might be different preferences to how one should define this. I am not sure that I understand exactly what you are asking, but maybe the following can be helpful.
If we consider subsets of the real numbers, then it is customary to define the infimum of the empty set as being $\infty$. This makes sense since the i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
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Bott periodicity and homotopy groups of spheres I studied Bott periodicity theorem for unitary group $U(n)$ and ortogonl group O$(n)$ using Milnor's book "Morse Theory". Is there a method, using this theorem, to calculate $\pi_{k}(S^{n})$? (For example $U(1) \simeq S^1$, so $\pi_1(S^1)\simeq \mathbb{Z}$).
| In general, no. However there is a strong connection between Bott Periodicity and the stable homotopy groups of spheres. It turns out that
$\pi_{n+k}(S^{n})$ is independent of $n$ for all sufficiently large $n$ (specifically $n \geq k+2$). We call the groups
$\pi_{k}^{S} = \lim \pi_{n+k}(S^{n})$
the stable homotopy gr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Integral $\int_0^\frac{\pi}{2} \sin^7x \cos^5x\, dx $ im asked to find the limited integral here but unfortunately im floundering can someone please point me in the right direction?
$$\int_0^\frac{\pi}{2} \sin^7x \cos^5x\, dx $$
step 1 brake up sin and cos so that i can use substitution
$$\int_0^\frac{\pi}{2} \sin^7(x)... | $$
\begin{aligned}
\int_0^{\frac{\pi}{2}} \sin ^7 x \cos ^5 x & \stackrel{x\mapsto\frac{\pi}{2}-x}{=} \int_0^{\frac{\pi}{2}} \cos ^5 x \sin ^7 x d x \\
&=\frac{1}{2} \int_0^{\frac{\pi}{2}} \sin ^5 x\cos ^5 x\left(\sin ^2 x+\cos ^2 x\right) d x \\
&=\frac{1}{64} \int_0^{\frac{\pi}{2}} \sin ^5(2 x)d x \\
&=\frac{1}{128} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 5
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Finding the rotation angles of an plane I have the following question:
First I have a global coordinate system x y and z.
Second I have a plane defined somewhere in this coordinate system, I also have the normal vector of this plane.
Assuming that the plane originally was lying in the xy plane and the normal vector of ... | First off, there is no simple rule (at least as far as I know) to represent a rotation as a cascade of single axis rotations. One mechanism is to compute a rotation matrix and then compute the single axis rotation angles. I should also mention that there is not a unique solution to the problem you are posing, since the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432502",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Triangle integral with vertices
Evaluate $$I=\iint\limits_R \sin \left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)\, dA,$$ where $R$ is the triangle with vertices $(0,0),(2,0)$ and $(1,1)$.
Hint: use $u=\dfrac{x+y}{2},v=\dfrac{x-y}{2}$.
Can anyone help me with this question I am very lost. Please help
I know y... | $\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)=\frac{1}{2}\left(\sin x+\sin y\right)$
The line joining $(0,0)$ and $(2,0)$ has an equation $y=0$ and $0\leq x\leq 2$
The second line: $y=-x+2$
The third line: $y=x$
The integral becomes:
$$I=\frac{1}{2}\left(\int\limits_{0}^{1}\int\limits_{0}^{x}+\int\limit... | {
"language": "en",
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Favourite applications of the Nakayama Lemma Inspired by a recent question on the nilradical of an absolutely flat ring, what are some of your favourite applications of the Nakayama Lemma? It would be good if you outlined a proof for the result too. I am also interested to see the Nakayama Lemma prove some facts in Alg... | You might be interested in this.
It contains some applications of Nakayama's Lemma in Commutative Algebra and Algebraic Geometry.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/432659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 5,
"answer_id": 4
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If $\lim_{x \rightarrow \infty} f(x)$ is finite, is it true that $ \lim_{x \rightarrow \infty} f'(x) = 0$? Does finite $\lim_{x \rightarrow \infty} f(x)$ imply that $\lim_{x \rightarrow \infty} f'(x) = 0$? If not, could you provide a counterexample?
It's obvious for constant function. But what about others?
| Simple counterexample: $f(x) = \frac{\sin x^2}{x}$.
UPDATE: It may seem that such an answer is an unexplicable lucky guess, but it is not. I strongly suggest looking at Brian M. Scott's answer to see why. His answer reveals exactly the reasoning that should first happen in one's head. I started thinking along the same ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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"answer_id": 2
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Determining whether a coin is fair I have a dataset where an ostensibly 50% process has been tested 118 times and has come up positive 84 times.
My actual question:
*
*IF a process has a 50% chance of testing positive and
*IF you then run it 118 times
*What is the probability that you get AT LEAST 84 successes?
... | Of course, 118 is in the "small numbers regime", where one
can easily (use a computer to) calculate the probability exactly.
By wolframalapha,
the probability that you get at least 84 successes $\;\;=\;\; \frac{\displaystyle\sum_{s=84}^{118}\:\binom{118}s}{2^{118}}$
$=\;\; \frac{392493659183064677180203372911}{16615... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Norms in extended fields let's have some notation to start with:
$K$ is a number field and $L$ is an extension of $K$. Let $\mathfrak{p}$ be a prime ideal in $K$ and let its norm with respect to $K$ be denoted as $N_{\mathbb{Q}}^K(\mathfrak{p})$.
My question is this: If $|L:K|=n$, what is $N_{\mathbb{Q}}^L(\mathfrak{p... | The queston is simple, if one notices the equality:
Let $p\in \mathfrak p\cap \mathbb Z$ be a prime integer in $\mathfrak p$. And let $f$ be the inertia degree of $\mathfrak p$. Then $N^K_{\mathbb Q}\mathfrak p=p^f$.
Now the result follows from the multiplicativity of the inertia degree $f$.
P.S. The above equali... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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How do we take second order of total differential? This is the total differential
$$df=dx\frac {\partial f}{\partial x}+dy\frac {\partial f}{\partial y}.$$
How do we take higher orders of total differential, $d^2 f=$?
Suppose I have $f(x,y)$ and I want the second order total differential $d^2f$?
| I will assume that you are referring to the Fréchet derivative. If $U\subseteq\mathbb{R}^n$ is an open set and we have functions $\omega_{j_1,\dots,j_p}:U\to\mathbb{R}$, then $$
D\left(\sum_{j_1,\dots,j_p} \omega_{j_1,\dots,j_p} dx_{j_1}\otimes\cdots\otimes dx_{j_p}\right) = \sum_{j_1,\dots,j_p}\sum_{j=1}^n \frac{\part... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/432955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Odds of being correct X times in a row Is there a simple way to know what the chances are of being correct for a given number of opportunities?
To keep this simple: I am either right or wrong with a 50/50 chance. What are the odds that I'll be correct 7 times in a row or 20 or simply X times?
... and can the answer be ... | You guys are making it too complicated. It goes like this:
double your last...
1st time: 1 in 2 - (win a single coin toss)
2nd time: 1 in 4 - (win 2 consecutive coin tosses) . . .etc.
3rd time: 1 in 8
4th time: 1 in 16
5th time: 1 in 32
6th time: 1 in 64 . . . etc . . . 1 in 2 to the nth. If you wanna get complic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 3
} |
How to determine the arc length of ellipse? I want to determine the length of an arc from the ellipse in the picture below:
How can I determine the length of $d$?
| Giving a Mathematica calculation. Same result as coffeemath (+1)
In[1]:= ArcTan[3.05*Tan[5Pi/18]/2.23]
Out[1]= 1.02051
In[2]:= x=3.05 Cos[t];
In[3]:= y=2.23 Sin[t];
In[4]:= NIntegrate[Sqrt[D[x,t]^2+D[y,t]^2],{t,0,1.02051}]
Out[4]= 2.53143
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/433094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 4,
"answer_id": 1
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divergence of $\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$ i ran into this question and im sitting on it for a long time.
why does this integral diverge:
$$\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$$
thank you very much in advance.
yaron.
| $$\int_{2}^{\infty}\frac{dx}{x^{2}-x-2} = \int_{2}^{\infty}\frac{dx}{(x - 2)(x+1)}$$
Hence at $x = 2$, the integrand is undefined (the denominator "zeroes out" at $x = 2, x = -1$. So $x = 2$ is not in the domain of the integrand. Although we could find the indefinite integral, e.g., using partial fractions, the indefin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Valid Alternative Proof to an Elementary Number Theory question in congruences? So, I've recently started teaching myself elementary number theory (since it does not require any specific mathematical background and it seems like a good way to keep my brain in shape until my freshman college year) using Elementary Numbe... | I don't think there is anything wrong with your proof, but put it more simply, the congruence
$$
p \equiv 1 \pmod{2}
$$
is equivalent to
$$
\text{either}\quad
p \equiv 1 \pmod{4},
\quad\text{or}\quad
p \equiv 3 \pmod{4}.
$$
This is simply because if $p \equiv 0, 2 \pmod{4}$, then $p \equiv 0 \pmod{2}$.
An then, note ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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An odd integer minus an even integer is odd. Prove or Disprove: An odd integer minus an even integer is odd.
I am assuming you would define an odd integer and an even integer. than you would use quantifiers which shows your solution to be odd or even. I am unsure on how to show this...
| Instead of a pure algebraic argument, which I don't dislike, it's also possible to see visually. Any even number can be represented by an array consisting of 2 x n objects, where n represents some number of objects. An odd number will be represented by a "2 by n" array with an item left over. Odd numbers don't have "ev... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The image of $I$ is an open interval and maps $\mathbb{R}$ diffeomorphically onto $\mathbb{R}$? This is Problem 3 in Guillemin & Pallock's Differential Topology on Page 18. So that means I just started and am struggling with the beginning. So I would be expecting a less involved proof:
Let $f: \mathbb{R} \rightarrow \... | If $f$ is a local differeomorphism then the image must be connected, try to classify the connected subsets of $\mathbb{R}$ into four categories. Since $f$ is an open map, this gives you only one option left. I do not know if this is the proof the author has in mind.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/433357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Probability Question about Tennis Games! $2^{n}$ players enter a single elimination tennis tournament. You can assume that the players are of equal ability.
Find the probability that two particular players meet each other in the tournament.
I could't make a serious attempt on the question, hope you can excuse me this ... | We will make the usual unreasonable assumptions. We also assume (and this is the way tournaments are usually arranged) that the initial locations of the players on the left of your picture are chosen in advance. We also assume (not so reasonable) that the locations are all equally likely.
Work with $n=32$, like in you... | {
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"url": "https://math.stackexchange.com/questions/433430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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Regression towards the mean v/s the Gambler's fallacy Suppose you toss a (fair) coin 9 times, and get heads on all of them. Wouldn't the probability of getting a tails increase from 50/50 due to regression towards the mean?
I know that that shouldn't happen, as the tosses are independent event. However, it seems to go ... | This is interesting because it shows how tricky the mind can be. I arrived at this web site after reading the book by Kahneman, "Thinking, Fast and Slow".
I do not see contradiction between the gambler´s fallacy and regression towards the mean. According to the regression principle, the best prediction of the next mea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
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Is a semigroup $G$ with left identity and right inverses a group? Hungerford's Algebra poses the question: Is it true that a semigroup $G$ that has a left identity element and in which every element has a right inverse is a group?
Now, If both the identity and the inverse are of the same side, this is simple. For, inst... | If $cc=c$ than $c$ is called idempotent element.
Semigroup with left unit and right inverse is called left right system or shortly $(l, r)$ system.
If you take all the idempotent elements of $(l,r)$ system they also form $(l,r)$ system called idempotent $(l,r)$ system. In such system the multiplication of two elements ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
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Solving for X in a simple matrix equation system. I am trying to solve for X in this simple matrix equation system:
$$\begin{bmatrix}7 & 7\\2 & 4\\\end{bmatrix} - X\begin{bmatrix}5 & -1\\6 & -4\\\end{bmatrix} = E $$ where $E$ is the identity matrix.
If I multiply $X$ with $\begin{bmatrix}5 & -1\\6 & -4\\\end{bmatrix}$ ... | Since $\begin{pmatrix}7&7\\2&4\end{pmatrix}-X\begin{pmatrix}5&-1\\6&-4\end{pmatrix}=\begin{pmatrix}1&0\\0&1\end{pmatrix}$, we obtain:
$\begin{pmatrix}6&7\\2&3\end{pmatrix}=\begin{pmatrix}5x_1+6x_2&-x_1-4x_2\\5x_3+6x_4&-x_3-4x_4\end{pmatrix}$, where $X=\begin{pmatrix}x_1&x_2\\x_3&x_4\end{pmatrix}$.
Now you can multiply ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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How does one combine proportionality? this is something that often comes up in both Physics and Mathematics, in my A Levels. Here is the crux of the problem.
So, you have something like this :
$A \propto B$ which means that $A = kB \tag{1}$
Fine, then you get something like :
$A \propto L^2$ which means that $A = k'L^2... | Suppose a variable $A$ depends on two independent factors $B,C$, then
$A\propto B\implies A=kB$, but here $k$ is a constant w.r.t. $B$ not $C$, in fact, $k=f(C)\tag{1}$
Similarly, $A\propto C\implies A=k'C$ but here $k'$ is a constant w.r.t. C not $B$, in fact, $k'=g(B)\tag{2}$
From $(1)$ and $(2)$,
$f(C)B=g(B)C\implie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/433754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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pressure in earth's atmosphere as a function of height above sea level While I was studying the measurements of pressure at earth's atmosphere,I found the barometric formula which is more complex equation ($P'=Pe^{-mgh/kT}$) than what I used so far ($p=h\rho g$).
So I want to know how this complex formula build up? I c... | If
$\frac{dP}{dh} = (-\frac{mgP}{kT})$,
then
$\frac{1}{P} \frac{dP}{dh} = (-\frac{mg}{kT})$,
or
$\frac{d(ln P)}{dh} = (-\frac{mg}{kT})$.
Integrating with respect to $h$ over the interval $[h_0, h]$ yields
$ln(P(h)) - ln(P(h_0)) = (-\frac{mg}{kT})(h - h_0)$,
or
$ln(\frac{P(h)}{P(h_0)}) = (-\frac{mg}{kT})(h - h_0)$,
or
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof of Riemann-Lebesgue lemma: what does "integration by parts in each variable" mean? I was reading the proof of the Riemann-Lebesgue lemma on Wikipedia, and something confused me. It says the following:
What does the author mean by "integration by parts in each variable"? If we integrate by parts with respect to $... | This question was answered by @danielFischer in the comments.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving an operator is self-adjoint I have a linear operator $T:V \to V$ (where $V$ is a finite-dimensional vector space) such that $T^9=T^8$ and $T$ is normal, I need to prove that $T$ is self-adjoint and also that $T^2=T$.
Would appreciate any help given.
Thanks a million!
| Hint. Call the underlying field $\mathbb{F}$. As $T$ is normal and its characteristic polynomial can be split into linear factors $\underline{\text{over }\, \mathbb{F}}$ (why?), $T$ is unitarily diagonalisable over $\mathbb{F}$. Now the rest is obvious.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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domain of square root What is the domain and range of square $\sqrt{3-t} - \sqrt{2+t}$? I consider the domain the two separate domain of each square root.
My domain is $[-2,3]$. Is it right? Are there methods on how to figure out the domain and range in this kind of problem?
| You are right about the domain. As to the range, use the fact that as $t$ travels from $-2$ to $3$, $\sqrt{3-t}$ is decreasing, and $\sqrt{2+t}$ is increasing, so the difference is decreasing.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\sum^{\infty}_{n=1} \frac {x^n}{1+x^{2n}}$ interval of convergence I need to find interval of convergence for this series:
$$\sum^{\infty}_{n=1} \frac {x^n}{1+x^{2n}}$$
I noticed that if $x=0$ then series is convergent. Unfortunately, that’s it.
| If $\left| x\right|<1$ then
$$
\left| \frac{x^n}{1+x^{2n}} \right|\leq \left| x\right|^n
$$
because $1+x^{2n}\geq1$. Now $\sum_{n=1}^{\infty} \left| x\right|^n$ is a finite geometric series.
Similarly, if $\left| x\right|>1$ then
$$
\left| \frac{x^n}{1+x^{2n}} \right|\leq \frac{1}{ \left| x\right|^n}
$$
because $1+x^{2... | {
"language": "en",
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Why does $\oint\mathbf{E}\cdot{d}\boldsymbol\ell=0$ imply $\nabla\times\mathbf{E}=\mathbf{0}$? I am looking at Griffith's Electrodynamics textbook and on page 76 he is discussing the curl of electric field in electrostatics. He claims that since
$$\oint_C\mathbf{E}\cdot{d}\boldsymbol\ell=0$$
then
$$\nabla\times\mathbf{... | The infinitesimal area $d\mathbf{a}$ is arbitrary which means that $\nabla\times\mathbf{E}=0$ must be true everywhere, and not just locally.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/434291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 4
} |
simple limit but I forget how to prove it I have to calculate the following limit
$$\lim_{x\rightarrow -\infty} \sqrt{x^2+2x+2} - x$$
it is in un undeterminated form.
I tried to rewrite it as follows:
$$\lim_{x\rightarrow -\infty} \sqrt{x^2+2x+2} - \sqrt{|x|^2}$$
but seems a dead road.
Can anyone suggest a solution?
... | Clearly
$$\lim_{x\rightarrow -\infty} \sqrt{x^2+2x+2} - x=+\infty+\infty=+\infty$$
But
\begin{gather*}\lim_{x\rightarrow +\infty} \sqrt{x^2+2x+2} - x="\infty-\infty"=\\
=\lim_{x\rightarrow +\infty} \frac{(\sqrt{x^2+2x+2} - x)(\sqrt{x^2+2x+2} + x)}{\sqrt{x^2+2x+2} + x}=\lim_{x\rightarrow +\infty} \frac{2x+2}{\sqrt{x^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$d(f \times g)_{x,m} = df_x \times dg_m$?
(a) $d(f \times g)_{x,m} = df_x \times dg_m$?
Also,
(b) does $d(f \times g)_{x,m}$ carry $\tilde{x} \in T_x X, \tilde{m} \in T_x M$ to the tangent space of $f$ cross the tangent space of $g$?
Furthermore,
(c) does $df_x \times dg_m$ carry $\tilde{x} \in T_x X, \tilde{m} \... | I don't know what formalism you are adopting, but starting from this one it can be done intuitively. I will state the construction without proof of compatibility and so.
An element of $T_x X$ is a differential from locally defined functions $C^{1}_x(X)$ to $\mathbb{R}$ induced by a locally defined curve $\gamma_x(t):(-... | {
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Inequality involving Maclaurin series of $\sin x$ Question:
If $T_n$ is $\sin x$'s $n$th Maclaurin polynomial.
Prove that $\forall 0<x<2, \forall m \in \Bbb N,~ T_{4m+3}(x)<\sin(x)<T_{4m+1}(x)$
Thoughts
I think I managed to prove the sides, proving that $T_3>T_1$ and adding $T_{4m}$ on both sides, but about the middle... | Your thoughts are not quite right. Firstly, $T_{3} < T_{1}$ for $0 < x < 2$, and for that matter showing $T_{3} > T_{1}$ would be antithetical to what you are looking to prove. Secondly, truncated Taylor series are not additive in the way that you are claiming, i.e. it is NOT true that $T_{3} + T_{4m} = T_{4m + 3}$.
H... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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A function/distribution which satisfies an integral equation. (sounds bizzare) I think I need a function, $f(x)$ such that $\large{\int_{x_1}^{x_2}f(x)\,d{x} = \frac{1}{(x_2-x_1)}}$ $\forall x_2>x_1>0$. Wonder such a function been used or studied by someone, or is it just more than insane to ask for such a function? I ... | No such $f(x)$ can exist. Such a function $f(x)$ would have antiderivative $F(x)$ that satisfies $$F(x_2)-F(x_1)=\frac{1}{x_2-x_1}$$ for all $x_2>x_1>0$.
Take the limit as $x_2\rightarrow \infty$ of both sides; the right side is 0 (hence the limit exists), while the left side is $$\left(\lim_{x\rightarrow \infty} F(x... | {
"language": "en",
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Soccer betting combinations for accumulators I would like to bet on soccer games, on every possible combination. For example, I bet on $10$ different games, and each soccer game can go three ways: either a win, draw, or loss.
How many combinations would I have to use in order to get a guaranteed win by betting $10$ ma... | If you bet on every possibility, in proportion to the respective odds, the market is priced so that you can only get back about 70% of your original stake no matter what.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Ultrafilter condition: If A is a subset of X, then either A or X \ A is an element of U My confusion concerns ultrafilters on sets that are themselves power sets.
If $X=\{\emptyset,\ \{1\},\ \{2\},\ \{3\},\ \{4\},\ \{1,2\},\ \{1,3\},\ \{1,4\},\ \{2,3\},\ \{2,4\},\ \{3,4\},\ \{1,2,3\},\ \{1,2,4\},\ \{1,3,4\},\ \{2,3,4\... | It is fine to have filters on a power set of another set. But then the filter is a subset of the power set of the power set of that another set; rather than our original set which is the power set of another set.
But in the example that you gave, note that $W\in U$ if and only if $\{1\}\in W$. Indeed $\{1\}\notin A$, b... | {
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Proving $\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$ Wikipedia informs me that
$$S = \vartheta(0;i)=\sum_{n=-\infty}^\infty e^{-\pi n^2} = \frac{\sqrt[4] \pi}{\Gamma\left(\frac 3 4\right)}$$
I tried considering $f(x,n) = e^{-x n^2}$ so that its Mellin transform becomes $\m... | I am not sure if it will ever help, but the following identity can be proved:
$$ S^2 = 1 + 4 \sum_{n=0}^{\infty} \frac{(-1)^n}{\mathrm{e}^{(2n+1)\pi} - 1}. $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "52",
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"answer_id": 2
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countability of limit of a set sequence Let $S_n$ be the set of all binary strings of length $2n$ with equal number of zeros and ones. Is it correct to say $\lim_{n\to\infty} S_n$ is countable? I wanted to use it to solve this problem. My argument is that each of $S_n$s is countable (in fact finite) thus their union wo... | The collection of all finite strings of $0$'s and $1$'s is countably infinite. The subcollection of all strings that have equal numbers of $0$'s and $1$'s is therefore countably infinite.
I would advise not using the limit notation to denote that collection. The usual notation for this kind of union is $\displaystyle\... | {
"language": "en",
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How to construct a non-piecewise differentiable S curve with constant quick turning flat tails and a linear slope? I need to find an example of a non-piecewise differentiable $f:\mathbb{R}\to\mathbb{R}$ such that
$$
\begin{cases}
f(x)=C_1 &\text{ for } x<X_1,\\
C_1 < f(x) < C_2 &\text{ for } X_1 < x < X_2,\\
... | After a bit of trying to mix various segments I came up with the following for my problem
f(x) = (1/(1+e^(100*(x-1))))(1/(1+e^(-100(x+1))))x-(1/(1+e^(100(x+1))))+(1/(1+e^(-100*(x-1))))
Which atleast seems to be in the right direction to meet all the requirements
*
*Has a slope between -1 to +1.
*Flatens quickly > 1... | {
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Is this true? $f(g(x))=g(f(x))\iff f^{-1}(g^{-1}(x))=g^{-1}(f^{-1}(x))$.
Is this true?
Given $f,g\colon\mathbb R\to\mathbb R$. $f(g(x))=g(f(x))\iff f^{-1}(g^{-1}(x))=g^{-1}(f^{-1}(x))$.
I met this problem when dealing with a coding method, but I'm really not familiar with functions. Please help.
Thank you.
| Of course that is only true if $f^{-1}$ and $g^{-1}$ exist. But then, it's easy to show:
Be $y=f^{-1}(g^{-1}(x))$. Then obviously $g(f(y))=g(f(f^{-1}(g^{-1}(x)))) = g(g^{-1}(x)) = x$.
On the other hand, by assumption $f(g(y))=g(f(y))=x$. Therefore $g^{-1}(f^{-1}(x)) = g^{-1}(f^{-1}(f(g(y)))) = g^{-1}(g(y)) = y = f^{-1}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Numerical solutions to wave equation Does the wave equation always have an analytical solution given well-behaved boundary/initial conditions? If not, under what conditions does the wave equation need to be solved numerically? This figure of a simple 1D-problem seems to have been generated numerically.
Any recommended... | Derivation
Solutions
It has a general solution but numerical solutions can still be an interesting exercise. Numerical solutions are useful when you are solving some variation of the wave equation with an additional term in it which makes it unsolvable analytically.
| {
"language": "en",
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How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent? How to show that this limit $\lim_{n\rightarrow\infty}\sum_{k=1}^n(\frac{1}{k}-\frac{1}{2^k})$ is divergent?
I applied integral test and found the series is divergent. I wonder if there exist easier solutions.
| Each partial sum of your series is the difference between the partial sum of the harmonic series, and the partial sum of the geometric series. The latter are all bounded by 1. Since the harmonic series diverges, your series does also.
| {
"language": "en",
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Definable sets à la Jech Jech in Set Theory, p. 175 defines definable sets over a given model $(M,\in)$ (where $M$ is a set) as those sets (= subsets of $M$) $X$ with a formula $\phi$ in the set of formulas of the language $\lbrace \in \rbrace$ and some $a_1,\dots,a_n \in M$ such that
$$ X = \lbrace x \in M : (M,\in) ... | If $M$ is transitive and $a \in M$ then $a \subseteq M$ and moreover $$a = \{x \in M : (M,{\in}) \vDash x \in a\}.$$ So $a \in \operatorname{def}(M)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$A$ is a subset of $B$ if and only if $P(A) \subset P(B)$ I had to prove the following for a trial calculus exam:
$A\subset B$ if and only if $P(A) \subset P(B)$ where $P(A)$ is the set of all subsets of $A$.
Can someone tell me if my approach is correct and please give the correct proof otherwise?
$PROOF$:
$\Big(\Long... | $(\Rightarrow)$ Given any $x\in P(A)$ then $x\subset A$. So, by hypothesis, $x\subset B$ and so $x\in P(B)$.
The counter part is easier, as cited by @Asaf, below.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
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Algebraic Solutions to Systems of Polynomial Equations Given a system of rational polynomials in some number of variables with at least one real solution, I want to prove that there exists a solution that is a tuple of algebraic numbers. I feel like this should be easy to prove, but I can't determine how to. Could any... | Here's a thought. Let's look at the simplest non-trivial case. Let $P(x,y)$ be a polynomial in two variables with rational (equivalently, for our purposes, integer) coefficients, and a real zero.
If that zero is isolated, then $P$ is never negative (or never positive) in some neighborhood of that zero, so the graph of... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Edge coloring in a graph How do I color edges in a graph?
I actually want to ask you specifically about one method that I've heard about - to find a dual (?) graph and color its vertices. What is the dual graph here? Is it really the dual graph, or maybe something different? If so, what is this?
The graph I'm talking ... | One method of finding an edge colouring of a graph is to find a vertex colouring of it's line graph. The line graph is formed by placing a vertex for every edge in the original graph, and connecting them with edges if the edges of the original graph share a vertex.
By finding a vertex colouring of the line graph we obt... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $(\sup\{y\in\Bbb R^{\ge0}:y^2\le x\})^2=x$ This question is a simple one from foundations of analysis, regarding the definition of the square root function. We begin by defining
$$\sqrt x:=\sup\{y\in\Bbb R^{\ge0}:y^2\le x\}:=\sup S(x)$$
for $x\ge0$, and now we wish to show that it satisfies its defining prop... | Since you have the l.u.b. axiom, you can use that, for any two bounded sets $A,B$ of nonnegative reals, we have
$$(\sup A) \cdot (\sup B)=\sup( \{a\cdot b:a \in A,b \in B\}).$$
Applying this to $A=B=S(x)$ we want to find the sup of the set of products $a\cdot b$ where $a^2\le x$ and $b^2 \le x.$ First note any such pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Comparing Areas under Curves I remembered back in high school AP Calculus class, we're taught that for a series:
$$\int^\infty_1\frac{1}{x^n}dx:n\in\mathbb{R}_{\geq2}\implies\text{The integral converges.}$$
Now, let's compare $$\int^\infty_1\frac{1}{x^2}dx\text{ and }\int^\infty_1\frac{1}{x^3}dx\text{.}$$
Of course, th... | $$\text{The main reason that: }\int^\infty_{1/2}\frac{1}{x^2}dx=\int^\infty_{1/2}\frac{1}{x^3}dx$$
$$\text{is because although that }\int^\infty_1\frac{1}{x^2}dx>\int^\infty_1\frac{1}{x^3}dx$$
$$\text{remember that }\int^1_{1/2}\frac{1}{x^2}dx<\int^1_{1/2}\frac{1}{x^3}dx\text{.}$$
$$\text{So, in this case: }\int^\infty... | {
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Notation of random variables I am really confused about capitalization of variable names in statistics.
When should a random variable be presented by uppercase letter, and when lower case?
For a probability $P(X \leq x)$, what do $x$ and $X$ mean here?
| You need to dissociate $x$ from $X$ in your mind—sometimes it matters that they are "the same letter" but in general this is not the case. They are two different characters and they mean two different things and just because they have the same name when read out loud doesn't mean anything.
By convention, a lot of the t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/435846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Complex integration: $\int _\gamma \frac{1}{z}dz=\log (\gamma (b))-\log(\gamma (a))?$ Let $\gamma$ be a closed path defined on $[a,b]$ with image in the complex plan except the upper imaginary axis, ($0$ isn't in this set).
Then $\frac{1}{z}$ has an antiderivative there and it is $\log z$. Therefore $\int _\gamma \frac... | The denominator in the second expression should be $e^{it}+3$ instead of $e^{it}$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Find maximum and minimum of $f(x,y) = x^2 y^2 - 2x - 2y$ in $0 \leq x \leq y \leq 5$.
Find maximum and minimum of $f(x,y) = x^2 y^2 - 2x - 2y$ in $0 \leq x \leq y \leq 5$.
So first we need to check inside the domain, I got only one point $A(1,1)$ where $f(1,1) = -3$. and after further checking it is a saddle point.
N... | Well, we have $f(x,y)=x^2y^2-2x-2y$ considered on the following green area:
So $f_x=2xy^2-2,~~f_y=2x^2y-2$ and solving $f_x=f_y=0$ gives us the critical point $x=y=1$ which is on the border of the area. Now think about the border:
$$y=5,~~ 0\le x\le 5 \;\;\;\; x=0,~~0\le y\le 5 \;\;\;\; y=x,~~ 0\le x\le 5$$
If $y=5,~... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Only 'atomic' vectors as part of the base of a vector space?
Given a vector subspace $U_1=${$\left(\begin{array}{c}
\lambda+µ \\
\lambda \\ µ
\end{array}\right)\in R^3$: $\lambda,µ \in R$ }
Determine a possible base of this vector subspace.
As far as I know, the base of a vector space is a n... | The first thing you need to know is that a subspace's dimension cannot exceed the containing space's dimension. Since the number of vectors constituting a base is equal to the dimension, your first suggestion is wrong, as it suggests that the subspace is of dimension 4 in $\mathbb{R}^3$, which is only of dimension 3.
T... | {
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Limits and exponents and e exponent form So I know that $\underset{n\rightarrow \infty}{\text{lim}}\left(1+\frac {1}{n}\right)^n=e$
and that we're not allowed to see it as $1^\infty$ because that'd be incorrect.
Why is then that we can do the same thing with (for example): $$\lim_{n\rightarrow \infty} \left(1+\sin\le... | In fact there's no difference between the two examples, indeed if you have a function $h$ such that $h(n)\to\infty$ then
$$\lim_{n\to\infty}\left(1+\frac{1}{h(n)}\right)^{h(n)}=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}=e$$
and ofcourse if we have another function $f$ such that $f(n)\to a\in\mathbb R$ then
$$\lim_... | {
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"url": "https://math.stackexchange.com/questions/436122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
How to show that right triangle is intersection of two rectangles in Cartesian coordinates? I am trying to do the following.
Given the triangle
$$T:=\left\{(x,y)\mid 0\leq x\leq h,0\leq y\leq k,\frac{x}{h}+\frac{y}{k}\leq1\right\}$$
find two rectangles $R$ and $S$ such that $R\cap S=T$, $\partial R\cap T$ is the two le... | Note that $\frac{x}{h}+\frac{y}{k}=1$ is equivalent to $y=-\frac{k}{h}x+k$.
Define
$$H_1=\left\{(x,y)\mid y\leq-\frac{k}{h}x+k\right\}\\
H_2=\left\{(x,y)\mid y\leq\frac{h}{k}x+k\right\}\\
H_3=\left\{(x,y)\mid y\leq\frac{h}{k}x-\frac{h^2}{k}\right\}\\
H_4=\left\{(x,y)\mid y\leq-\frac{k}{h}x\right\}.$$
Show that $H_1\cap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/436202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Testing polynomial equivalence Suppose I have two polynomials, P(x) and Q(x), of the same degree and with the same leading coefficient. How can I test if the two are equivalent in the sense that there exists some $k$ with $P(x+k)=Q(x)$? P and Q are in $\mathbb{Z}[x].$
| The condition of $\mathbb{Z}[x]$ isn't required.
Suppose we have 2 polynomial $P(x)$ and $Q(x)$, whose coefficients of $x^i$ are $P_i$ and $Q_i$ respectively. If they are equivalent in the sense of $P(x+k) = Q(k)$, then
*
*Their degree must be the same, which we denote as $n$.
*Their leading coefficient must be th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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solving an equation of the type: $t \sin (2t)=2$ where $0Need to solve:
How many solutions are there to the equation,
$t\sin (2t)=2$ where $0<t<3 \pi$
I am currently studying calc 3 and came across this and realized i dont have a clue as to how to get started on it.
| As an alternate approach, you could rewrite the equation as $$\frac{1}{2}\sin{2t}=\frac{1}{t}$$ and then observe that since $\frac{1}{t}\le\frac{1}{2}$ for $t\ge2$,
the graph of $y=\frac{1}{t}$ will intersect the graph of $y=\frac{1}{2}\sin{2t}$
twice in each interval $[n\pi,(n+1)\pi]$ for $n\ge1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/436376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
calculate the derivative using fundamental theorem of calculus This is a GRE prep question:
What's the derivative of $f(x)=\int_x^0 \frac{\cos xt}{t}\mathrm{d}t$?
The answer is $\frac{1}{x}[1-2\cos x^2]$. I guess this has something to do with the first fundamental theorem of calculus but I'm not sure how to use that to... | The integral does not exist, consequently it is not differentiable.
The integral does not exist, because for each $x$ ($x>0$, the case of negative
$x$ is dealt with similarly) there is some $\varepsilon > 0$ such that
$\cos(xt)> 1/2$ for all $t$ in the $t$-range $0\le t \le \varepsilon$. If one splits the integral $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/436567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Algebra and Substitution in Quadratic Form―Einstein Summation Notation
Schaum's Outline to Tensor Calculus ― chapter 1, example 1.5 ―――
If $y_i = a_{ij}x_j$, express the quadratic form $Q = g_{ij}y_iy_j$ in terms of the $x$-variables.
Solution: I can't substitute $y_i$ directly because it contains $j$ and there's alre... | In equation (1) $a_{js}$ and $x_r$ commute because these are just regular (reals or complex) numbers using standard multiplication which is commutative.
Equation (2) $h_{rs}$ is defined to save space more than anything, it's the coefficients of the polynomial in $x_i$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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A simple inequality in calculus? I have to solve this inequality:
$$\left(\left[\dfrac{1}{s}\right] + 1 \right) s < 1,$$
where $ 0 < s < 1 $.
I guess that $s$ must be in this range: $\left(0,\dfrac{1}{2}\right]$.But I do not know if my guess is true. If so, how I can prove it?
Thank you.
| try $$\left(\left[\dfrac{1}{s}\right] \right) < \frac{1}{s}-1$$
| {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
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p-adic Eisenstein series I'm trying to understand the basic properties of the p-adic Eisenstein series.
Let $p$ be a prime number.
Define the group $X =
\begin{cases}
\mathbb{Z}_p\times \mathbb{Z}/(p-1)\mathbb{Z} & \mbox{if }p \neq2 \\
\mathbb{Z}_2 & \mbox{if } p=2
\end{cases}$
where $\mathbb{Z}_p$ is the ring of $... | Like your previous question, there's a slight philosophical issue: the question should not be "is $d^{k-1} \in \mathbb{Z}_p$", but "when and how is $d^{k-1}$ defined"? It's far from obvious what the definition should be, but once you know what the conventional definition is, the fact that it gives you something in $\ma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Partial fraction integration $\int \frac{dx}{(x-1)^2 (x-2)^2}$ $$\int \frac{dx}{(x-1)^2 (x-2)^2} = \int \frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x-2}+\frac{D}{(x-2)^2}\,dx$$
I use the cover up method to find that B = 1 and so is C. From here I know that the cover up method won't really work and I have to plug in values... | To keep in line with the processes you are learning, we have:
$$\frac{1}{(x-1)^2 (x-2)^2} = \frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{C}{x-2}+\frac{D}{(x-2)^2}$$
So we want to find $A, B, C, D$ given $$A(x-1)(x-2)^2 + B(x-2)^2 + C(x-1)^2(x-2) + D(x-1)^2 = 1$$
As you found, when $x = 1$, we have $B = 1$, and when $x = 2$, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How can I calculate this determinant? Please can you give me some hints to deal with this :
$\displaystyle \text{Let } a_1, a_2, ..., a_n \in \mathbb{R}$
$\displaystyle \text{ Calculate } \det A \text{ where }$ $\displaystyle A=(a_{ij})_{1\leqslant i,j\leqslant n} \text{ and }$ $\displaystyle \lbrace_{\alpha_{ij}=0,\te... | Hint: The matrix looks like the following (for $n=4$; it gives the idea though):
$$
\begin{bmatrix}
0 & 0 & 0 & a_1\\
0 & 0 & a_2 & 0\\
0 & a_3 & 0 & 0\\
a_4 & 0 & 0 & 0
\end{bmatrix}
$$
What happens if you do a cofactor expansion in the first column? Try using induction.
| {
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Sequence of natural numbers Numbers $1,2,...,n$ are written in sequence. It's allowed to exchange any two elements. Is it possible to return to the starting position after an odd number of movements?
I know that is necessarily an even number of movements but I can't explain that!
| Basically, if you make an odd number of switches, then at least one of the numbers has only been moved once (unless you repeat the same switch over and over which is an easy case to explain). But if you start in some configuration and move a number only once and want to return to the start, you must move again.
Try in... | {
"language": "en",
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"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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Solution to $y'' - 2y = 2\tan^3x$ I'm struggling with this nonhomogeneous second order differential equation
$$y'' - 2y = 2\tan^3x$$
I assumed that the form of the solution would be $A\tan^3x$ where A was some constant, but this results in a mess when solving. The back of the book reports that the solution is simply $y... | Have you learned variation of parameters? This is a method, rather than lucky guessing :)
http://en.wikipedia.org/wiki/Variation_of_parameters
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solving for the integrating factor in a Linear Equation with Variable Coefficients So I am studying Diff Eq and I'm looking through the following example.
Solve the following equation:
$(dy/dt)+2y=3 \rightarrow μ(t)*(dy/dt)+2*μ(t)*y=3*μ(t) \rightarrow (dμ(t)/dt)=2*μ(t) \rightarrow (dμ(t)/dt)/μ(t)=2 \rightarrow
(d/dt)\... | Method 1: Calculus
We have: $y' + 2y = 3$.
Lets use calculus to solve this and see why these statements are okay. We have:
$$\displaystyle \frac{\dfrac{dy}{dt}}{y - \dfrac{3}{2}} = -2$$
Integrating both sides yields:
$$\displaystyle \int \frac{dy}{\left(y - \dfrac{3}{2}\right)} = -2 \int dt$$
We get: $\ln\left|y - \dfr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists? I would like to investigate the convergence of
$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$
Or more precisely, let $$\begin{align}
a_1 & = \sqrt 1\\
a_2 & = \sqrt{1+\sqrt2}\\
a_3 & = \sqrt{1+\sqrt{2+\sqrt 3}}\\
a_4 & = \sqrt{1+\sqrt{2+\sqrt{3+\... | For any $n\ge4$, we have $\sqrt{2n} \le n-1$. Therefore
\begin{align*}
a_n
&\le \sqrt{1+\sqrt{2+\sqrt{\ldots+\sqrt{(n-2)+\sqrt{(n-1) + \sqrt{2n}}}}}}\\
&\le \sqrt{1+\sqrt{2+\sqrt{\ldots+\sqrt{(n-2)+\sqrt{2(n-1)}}}}}\\
&\le\ldots\\
&\le \sqrt{1+\sqrt{2+\sqrt{3+\sqrt{2(4)}}}}.
\end{align*}
Hence $\{a_n\}$ is a monotonic ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "63",
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"answer_id": 3
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How many cones pass through a given conic section? Given a conic section in the $xy$-plane, how many cones (infinite double cone) in the surrounding 3D space intersect the $xy$-plane at that conic? Is the family continuous, with a nice parametization?
At least one must exist, and I expect symmetry in the conic to give ... | To take the simplest case, take the circle to be centred at $(0, 0, 0)$ in the $xy$-plane; and now take any point $(0, 0, z)$. Then plainly there is a double cone of rays which pass through $(0, 0, z)$ and some point on the circle (and this is a right circular cone). So there are continuum-many distinct such cones (i.e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What went wrong? Calculate mass given the density function
Calculate the mass:
$$D = \{1 \leq x^2 + y^2 \leq 4 , y \leq 0\},\quad p(x,y) = y^2.$$
So I said:
$M = \iint_{D} {y^2 dxdy} = [\text{polar coordinates}] = \int_{\pi}^{2\pi}d\theta {\int_{1}^{2} {r^3sin^2\theta dr}}$.
But when I calculated that I got the a... | You have the set-up correct, but you have incorrectly computed the integral
Let's work it out together.
$\int_{\pi}^{2\pi}d\theta {\int_{1}^{2} {r^3\sin^2\theta dr}}$
$\int_{\pi}^{2\pi} {\int_{1}^{2} {r^3\sin^2\theta drd\theta}}$
$\int_{\pi}^{2\pi} \sin^2\theta d\theta {\int_{1}^{2} {r^3dr}}$
$\int_{\pi}^{2\pi} \sin^2\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Lang $SL_2$ two formulas for Harish transform Let $G = SL_2$ and give it the standard Iwasawa decomposition $G = ANK$. Let: $$D(a) = \alpha(a)^{1/2} - \alpha(a)^{-1/2} := \rho(a) - \rho(a)^{-1}.$$ Now, Lang defines ($SL_2$, p.69) the Harish transform of a function $f \in C_c(G,K)$ to be $$Hf(a) := \rho(a)\int_Nf(an)dn ... | This equality is not at all obvious. Just before that section, it was proven that
$$
\int_{A\backslash G} f(x^{-1}ax)\;dx\;=\; {\alpha(a)\over |D(\alpha)}
\int_K\int_N f(kank^{-1})\;dn\;dk
$$
for arbitrary $f\in C_c(G)$. For $f$ left and right $K$-invariant, the outer integral goes away, leaving just the integral over ... | {
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convergence to a generalized Euler constant and relation to Zeta serie Let $0 \leq a \leq 1$ be a real number. I would like to know how to prove that the following sequence converges:
$$u_n(a)=\sum_{k=1}^n k^a- n^a \left(\frac{n}{1+a}+\frac{1}{2}\right)$$
For $a=1$:
$$u_n(1)=\sum\limits_{k=1}^{n} k- n \left(\frac{n}{1+... | From this answer you have an asymptotics
$$
\sum_{k=1}^n k^a = \frac{n^{a+1}}{a+1} + \frac{n^a}{2} + \frac{a n^{a-1}}{12} + O(n^{a-3})
$$
Use it to prove that $u_n(a)$ converges.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Minimum and maximum of $ \sin^2(\sin x) + \cos^2(\cos x) $ I want to find the maximum and minimum value of this expression:
$$ \sin^2(\sin x) + \cos^2(\cos x) $$
| Your expression simplifies to
$$1+\cos(2\cos x)-\cos (2\sin x).$$
We optimize of $1+\cos u-\cos v$ under the constraint $u^2+v^2=4$.
$\cos$ is an even function, so we can say that we optimize $1+\cos 2u-\cos(2\sqrt{1-u^2})$, $u\in [0,1]$, which should be doable.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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I need a better explanation of $(\epsilon,\delta)$-definition of limit I am reading the $\epsilon$-$\delta$ definition of a limit here on Wikipedia.
*
*It says that
$f(x)$ can be made as close as desired to $L$ by making the independent variable $x$ close enough, but not equal, to the value $c$.
So this means tha... | If you are a concrete or geometrical thinker you might find it easier to think in these terms. You are player $X$ and your opponent is player $Y$.
Player $Y$ chooses any horizontal lines they like, symmetric about $L$, but not equal to it.
You have to choose two vertical lines symmetric about $c$ - these create a recta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is Dirichlet function Riemann integrable? "Dirichlet function" is meant to be the characteristic function of rational numbers on $[a,b]\subset\mathbb{R}$.
On one hand, a function on $[a,b]$ is Riemann integrable if and only if it is bounded and continuous almost everywhere, which the Dirichlet function satisfies.
On t... | The Dirichlet function $f : [0, 1] → \mathbb R$ is defined by
$$f(x) = \begin{cases}
1, & x ∈ \mathbb Q \\
0, & x ∈ [0, 1] - \mathbb Q
\end{cases}$$
That is, $f$ is one at every rational number and zero at every irrational number.
This function is not Riemann integrable. If $P = \{I_1, I_2, . . . , I_n\}$ is a parti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "32",
"answer_count": 4,
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Evaluating $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\ $ using polar coordinates Use polar coordinates to evaluate $\int_0^1 \int_0^{\sqrt{1-x^2}}e^{-(x^2+y^2)} \, dy \, dx\
$
I understand that we need to change $x^2+y^2$ to $r^2$ and then we get $\int_0^1 \int_0^{\sqrt{1-x^2}} e^{-(r^2)} \, dy \, dx\
$.... | Hints:
$$\bullet\;\;\;x=r\cos\theta\;,\;\;y=r\sin\theta\;,\;0\le\theta\le \frac\pi2\;\text{(why?). The Jacobian is}\;\;|J|=r$$
So the integral is
$$\int\limits_0^1\int\limits_0^{\pi/2}re^{-r^2}drd\theta$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/437758",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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The preimage of continuous function on a closed set is closed. My proof is very different from my reference, hence I am wondering is I got this right?
Apparently, $F$ is continuous, and the identity matrix is closed. Now we want to show that the preimage of continuous function on closed set is closed.
Let $D$ be a clos... | Yes, it looks right. Alternatively, given a continuous map $f: X \to Y$, if $D \subseteq Y$ is closed, then $X \setminus f^{-1}(D) = f^{-1}(Y \setminus D)$ is open, so $f^{-1}(D)$ is closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/437829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 3,
"answer_id": 0
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The tangent plane of orthogonal group at identity. Why the tangent plane of orthogonal group at identity is the kernel of $dF_I$, the derivative of $F$ at identity, where $F(A) = AA^T$?
Thank you ~
| $\exists$
Proposition Let $Z$ be the preimage of a regular value $y\in Y$ under the smooth map $f: X \to Y$. Then the kernel of the derivative $df_x:T_x(X) \to T_y(Y)$ at any point $x \in Z$ is precisely the tangent space to $Z, T_x(Z).$
Proof:
Since $f$ is constant on $Z$, $df_x$ is zero on $T_x(Z)$. But $df_x: T_x(... | {
"language": "en",
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problem of probability and distribution Suppose there are 1 million parts which have $1\%$ defective parts i.e 1 million parts have $10000$ defective parts. Now suppose we are taking different sample sizes from 1 million like $10\%$, $30\%$, $50\%$, $70\%$, $90\%$ of 1 million parts and we need to calculate the probabi... | There was an earlier problem of which this is a variant. In the solution to that problem I did a great many computations. For this problem, the computations are in the same style, with a different value of $p$, the probability that any one item is defective.
Some of the computations in the earlier answer were done for ... | {
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motivation of additive inverse of a Dedekind cut set My understanding behind motivation of additive inverse of a cut set is as follows :
For example, for the rational number 2 the inverse is -2. Now 2 is represented by the set of rational numbers less than it and -2 is represented by the set of rational numbers less th... | I think the confusion arises when we are trying to identify a rational number say $2$ with the cut $\{ x\mid x \in \mathbb{Q}, x < 2\}$. When using Dedekind cuts as a definition of real numbers it is important to stick to some convention and follow it properly. For example to represent a real number either we choose
1)... | {
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Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$ Is there any field of characteristic two that is not $\mathbb{Z}/2\mathbb{Z}$?
That is, if a field is of characteristic 2, then does this field have to be $\{0,1\}$?
| To a beginner, knowing how one could think of an answer is at least as important as knowing an answer.
For examples in Algebra, one needs (at least) two things: A catalogue of the basic structures that appear commonly in important mathematics, and methods of constructing new structures from old. Your catalogue and cons... | {
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Prove that $\frac{100!}{50!\cdot2^{50}} \in \Bbb{Z}$ I'm trying to prove that :
$$\frac{100!}{50!\cdot2^{50}}$$
is an integer .
For the moment I did the following :
$$\frac{100!}{50!\cdot2^{50}} = \frac{51 \cdot 52 \cdots 99 \cdot 100}{2^{50}}$$
But it still doesn't quite work out .
Hints anyone ?
Thanks
| We have $100$ people at a dance class. How many ways are there to divide them into $50$ dance pairs of $2$ people each? (Of course we will pay no attention to gender.)
Clearly there is an integer number of ways. Let us count the ways.
We solve first a different problem. This is a tango class. How many ways are there to... | {
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"timestamp": "2023-03-29T00:00:00",
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Proving by induction: $2^n > n^3 $ for any natural number $n > 9$ I need to prove that $$ 2^n > n^3\quad \forall n\in \mathbb N, \;n>9.$$
Now that is actually very easy if we prove it for real numbers using calculus. But I need a proof that uses mathematical induction.
I tried the problem for a long time, but got stuc... | For your "subproof":
Try proof by induction (another induction!) for $k \geq 7$
$$k^3 > 3k^2 + 3k + 1$$
And you may find it useful to note that $k\leq k^2, 1\leq k^2$
$$3k^2 + 3k + 1 \leq 3(k^2) + 3(k^2) + 1(k^2) = 7k^2 \leq k^3 \quad\text{when}??$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/438260",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 0
} |
Norm inequality (supper bound) Do you think this inequality is correct? I try to prove it, but I cannot. Please hep me.
Assume that $\|X\| < \|Y\|$, where $\|X\|, \|Y\|\in (0,1)$ and $\|Z\| \gg \|X\|,\|Z\| \gg \|Y||$.
prove that
$$\|X+Z\|-\|Y+Z\| \leq \|X\|-\|Y\|$$
and if $Z$ is increased, the left hand side become sma... | The inequality is false as stated. Let
$$ \begin{align}
X &= (0.5,0)\\
Y &= (-0.7,0)\\
Z &= (z,0), 1 \ll z
\end{align}$$
This satisfies all the conditions given. We have that
$$ \|X + Z\| - \|Y + Z\| = z + 0.5 - (z - 0.7) = 1.2 \not\leq -0.2 = \|X\| - \|Y\| $$
From the Calculus point of view, in $n$ dimensions, we ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438319",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the difference between exponential symbol $a^x$ and $e^x$ in mathematics symbols? I want to know the difference between the exponential symbol $a^x$ and $e^x$ in mathematics symbols and please give me some examples for both of them.
I asked this question because of the derivative rules table below contain both ... | The two are essentially the same formula stated in different ways. They can be derived from each other as follows:
Note that $$\frac{d}{dx}(e^x)=e^x \ln(e) = e^x$$ is a special case of the formula for $a^x$ because $e$ has the special property that $\ln (e) =1$
Also $a^x=e^{\ln(a) x}$, which is another way into the der... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Proving a set of linear functionals is a basis for a dual space I've seen some similar problems on the stackexchange and I want to be sure I am at least approaching this in a way that is sensible.
The problem as stated:
Let $V= \Bbb R^3$ and define $f_1, f_2, f_3 \in V^*$ as follows:
$f_1(x,y,z)= x-2y ,\; f_2(x,y,z)... | What about a direct approach? Suppose $\,a,b,c\in\Bbb R\,$ are such that
$$af_1+bf_2+cf_3=0\in V^*\implies\;\forall\,v:=(x,y,z)\in\Bbb R^3\;,\;\;af_1v+bf_2v+cf_3v=0\iff$$
$$a(x-2y)+b(x+y+z)+c(y-3z)=0\iff$$
$$\iff (a+b)x-(2a-b-c)y+(b-3c)z=0$$
As the above is true for all $\;x,y,z\in\Bbb R\,$ , we must have
$$\begin{alig... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
} |
Can anyone provide me a step-by-step proof for proving a function IS onto/surjective? I've seen the definition, I've seen several examples and anti-examples (e.g. the typical x squared example). I get the idea, but I can't seem to find a proof for proving that a function IS onto, with proper explanation start to finis... | What you need to do to prove that a function is surjective is to take each value $y$ and find - any way you can - a value of $x$ with $f(x)=y$. If you succeed for every possible value of $y$, then you have proved that $f$ is surjective.
So we take $x=-\cfrac 13y+ \cfrac 43$ as you suggest. This is well-defined (no divi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Integral representation of cosh x On Wolfram math world, there's apparently an integral representation of $\cosh x$ that I'm unfamiliar with. I'm trying to prove it, but I can't figure it out. It goes \begin{equation}\cosh x=\frac{\sqrt{\pi}}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty} \frac{ds}{\sqrt{s}}\,e^{s+\frac... | Expand the difficult part of the exponential in power series, the integral equals
$$ I = \sqrt\pi \sum_{k\geq0} \frac{(x^2/4)^k}{k!} \frac{1}{2\pi i}\int_{\Re s=\gamma} s^{-k-1/2}e^{s}\,ds. $$
The integral here is the inverse Laplace transform of the function $s^{-k-1/2}$ evaluated at the point $t=1$, given by
$$ \math... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Trigonometry Equations. Solve for $0 \leq X \leq 360$, giving solutions correct to the nearest minute where necessary,
a) $\cos^2 A -8\sin A \cos A +3=0$
Can someone please explain how to solve this, ive tried myself and no luck. Thanks!
| HINT: $\cos^2 A=\frac{1+\cos 2A}{2},$
$\sin A\cos A=\frac{\sin 2A}{2}$
and $\sin^2 2A+\cos^2 2A=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/438648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
$\int \frac{dz}{z\sqrt{(1-{1}/{z^2})}}$ over $|z|=2$ I need help in calculating the integral of $$\int \frac{dz}{z\sqrt{\left(1-\dfrac{1}{z^2}\right)}}$$ over the circle $|z|=2$. (We're talking about the main branch of the square root).
I'm trying to remember what methods we used to calculate this sort of integral in m... | $$\frac{1}{z \sqrt{1-\frac{1}{z^{2}}}} = \frac{1}{z} \Big( 1 - \frac{1}{2z^{2}} + O(z^{-4}) \Big) \text{for} \ |z| >1 \implies \int_{|z|=2} \frac{1}{z \sqrt{1-\frac{1}{z^{2}}}} \ dz = 2 \pi i (1) = 2 \pi i $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/438714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Congruence in rings Let $R$ be a commutative (and probably unitary, if you like) ring and $p$ a prime number. If $x_1,\ldots,x_n\in R$ are elements of $R$, then we have $(x_1+\cdots+x_n)^p\equiv x_1^p+\cdots+x_n^p$ mod $pR$. Why is this true? I tried to show that in $R/pR$ their congruence classes are equal, but withou... | Just compute ;-) ... we have - as $R$ is commutative - by the multinomial theorem
$$ (x_1 + \cdots + x_n)^p = \sum_{\nu_1 + \cdots + \nu_n = p} \frac{p!}{\nu_1! \cdots \nu_n!} x_1^{\nu_1} \cdots x_n^{\nu_n} $$
If all $\nu_i <p $, the denominator contains no factor $p$ (as $p$ is prime), hence $\frac{p!}{\nu_1! \cdots ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
sum of exterior angles of a closed broken line in space I am looking for a simple proof of the following fact:
The sum of exterior angles of any closed broken line in space is at least $2 \pi$. I believe it equals $2 \pi$ if and only if the closed broken line equals a polygon.
| Quoting Curves of Finite Total Curvature by J. Sullivan:
Lemma 2.1. (See[Mil50, Lemma 1.1] and [Bor47].)
Suppose $P$ is a polygon in $\mathbb E^d$. If $P'$ is obtained from $P$ by deleting one vertex $v_n$ then $\operatorname{TC}(P')\leq\operatorname{TC}(P)$. We have equality here if $v_{n-1}v_nv_{n+1}$ are collinea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438909",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that the matrix $A+E$ is invertible. Let $A$ be an invertible matrix, and let $E$ be an upper triangular matrix with zeros on the diagonal. Assume that $AE=EA$. Show that the matrix $A+E$ is invertible. WLOG, we can assume $E$ is Jordan form. If $A$ is Jordan form, it's trivial. If $A$ is not Jordan form, how to u... | $E^n=0$ and since $A,E$ commute you have
$$A^{2n+1}=A^{2n+1}+E^{2n+1}=(A+E)(A^{2n}-A^{2n-1}E+...+E^{2n})$$
Since $A^{2n+1}$ is invertible, it follows that $A+E$ is invertible.
P.S. I only used in the proof that $E$ is nilpotent and commutes with $A$, so more generally it holds that in (any ring), if $A$ is invertible, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/438976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Solving Bessel integration What would be the solution of the bessels equation,
$$b=k A(t)\int_0^{\infty} J_0 (k \rho) e^ \frac{-\rho^2}{R^2} \rho d \rho$$
Can I sove that by using this formulation?
$$c= \int_0^{\infty}j_0(t) e^{-pt} dt= \frac{1}{\sqrt{1+p^2}}$$
| According to Gradshteyn and Ryzhik, we have:
$$\int_0^{\infty}x^{\mu}\exp(-\alpha x^2)J_{\nu}(\beta x)dx = \frac{\beta^{\nu}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{1}{2}\right)}{2^{\nu+1}\alpha^{\frac{1}{2}(\mu+\nu+1)}\Gamma(\nu+1)}\mbox{}_1 F_1\left(\frac{\nu+\mu+1}{2};\mbox{ }\nu+1;\mbox{ }-\frac{\beta^2}{4\a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/439034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
An odd question about induction. Given $n$ $0$'s and $n$ $1$'s distributed in any manner whatsoever around a circle, show, using induction on $n$, that it is possible to start at some number and proceed clockwise around the circle to the original starting position so that, at any point during the cycle, we have seen at... | Alternatively: Count the total number of ways to arrange $0$s and $1$s around a circle ($2n$ binary digits in total), consider the number of Dyck words of length $2n$, i.e., the $n$th Catalan number, and then use the Pigeonhole Principle. "QED"
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/439105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Uniform grid on a disc Do there exist any known methods of drawing a uniform grid on a disk ? I am looking for a map that converts a grid on a square to a grid on a disk.
| There are many possibilities to map a square on a disk. For example one possibility is:
$$
\phi(x,y) = \frac{(x,y)}{\sqrt{1+\min\{x^2,y^2\}}}
$$
which moves the points along the line trough the origin.
If you also want the map to mantain the infinitesimal area, it's a little bit more complicated. One possibility is to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/439161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove that $\vdash p \lor \lnot p$ is true using natural deduction I'm trying to prove that $p \lor \lnot p$ is true using natural deduction. I want to do this without using any premises.
As it's done in a second using a truth table and because it is so intuitive, I would think that this proof shouldn't be too difficul... | If you have the definition Cpq := ANpq and you have that A-commutes, and Cpp, then you can do this really quickly. Since we have Cpp, by the definition we have ANpp. By A-commutation we then have ApNp.
More formally, I'll first point out that natural deduction allows for definitions as well as uniform substitution on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/439291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
Given that $\cos x =-3/4$ and $90^\circGiven that $\;\cos x =-\frac{3}{4}\,$ and $\,90^\circ<x<180^\circ,\,$ find $\,\tan x\,$ and $\,\csc x.$
This question is quite unusual from the rest of the questions in the chapter, can someone please explain how this question is solved? I tried Pythagorean Theorem, but no luck. I... | As $90^\circ< x<180^\circ,\tan x <0,\sin x>0$
So, $\sin x=+\sqrt{1-\cos^2x}=...$
$\csc x=\frac1{\sin x}= ... $
and $\tan x=\frac{\sin x}{\cos x}=...$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/439369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
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