Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to do this Math induction problem? Show that:
$$\frac n3 + \frac n9 + \frac {n}{27} + \cdots = \frac n2.$$
When I start with $\frac 13 + \frac 19 + \frac {1}{27}$ it leads to a number close to $.5$ but it's not exactly $.5$.
| You have an infinite geometric series $\frac n3 + \frac n9 + \frac n{27}+\dots=n\sum_{i=1}^\infty\frac 1{3^n}$. Using the terminology in the linked article, we have $a=\frac 13$ (the first term) and $r=\frac 13$ (the ratio between successive terms. As long as $|r| \lt 1$ this infinite sum will converge to $\frac a{1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/504922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What is $\int\log(\sin x)~dx$? I know that the value of the integral of $\cot(x)$ is $\log|\sin x|+C$ .
But what about:
$$\int\log(\sin x)~dx$$
Is there any easy way to find an antiderivative for this? Thanks.
| using $$\ln \sin x =-\ln 2-\sum_{n=1}^{\infty}\frac{\cos(2nx)}{n}, \ x \in [0,\pi].$$
$$\int \ln(\sin x)dx=-\ln(2)\int dx-\sum^{\infty}_{n=1}\frac{1}{n}\int (\cos 2nx)dx$$
$$=-\ln(2)\cdot x-\sum^{\infty}_{n=1}\frac{\sin(2nx)}{2n^2}+\mathcal{C}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/504983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Commutator of $x$ and $p^2$ I have a question:
If I have to find the commutator $[x, p^2]$ (with $p= {h\over i}{d \over dx} $) the right answer is:
$[x,p^2]=x p^2 - p^2x = x p^2 -pxp + pxp - p^2x = [x,p]p + p[x,p] = 2hip$
But why can't I say:
$[x,p^2]=x p^2 - p^2x = - x h^2{d^2 \over dx^2} + h^2 {d^2 \over dx^2}x = 0$ ... | What you describe is a quite common situation which pops up when dealing with commutators of operators. On an appropriate space of functions $\mathcal D$ (like an $L^2$-space or the Schwartz space etc...), the operators $x$ and $p$ are given by
$$x(f)(x):=xf(x), $$
$$p(f)(x):=\frac{h}{i}\frac{df}{dx}, $$
for all $f\in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
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Show that $\gcd(a + b, a^2 + b^2) = 1$ or $2$ if $\gcd(a, b)=1$ Show that $\gcd(a + b, a^2 + b^2) = 1$ or $2$ if $\gcd(a, b)=1$.
I have absolutely no clue where to start and what to do, please provide complete proof and answer.
| We first show that there is no odd prime $p$ that divides both $a+b$ and $a^2+b^2$.
For if $p$ divides both, then $p$ divides $(a+b)^2-(a^2+b^2)$, so $p$ divides $2ab$. Since $p$ is odd, it divides one of $a$ or $b$, say $a$. But then since $p$ divides $a+b$, it must divide $b$. This contradicts the fact that $a$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Use $\epsilon-\delta$ definition of limit to prove that $\displaystyle \lim_{x \to 0} x \lfloor \frac{1}{x} \rfloor = 1$. I was trying to write some nice problems for applying $\epsilon-\delta$ definition to give it to my friend but then I realized that I couldn't solve some of them either. This is one of them:
Use $\... | Sorry, to long for a comment, will delete it in the future. Here is exactly what I mean by my comment.
Assume that on some interval $I=(a-b,a+b)$ around $a$ we have.
$$f(x) \leq g(x) \leq h(x) \, \forall x\in I \backslash \{ a \}$$
and the outside limits are easy, meaning that you can prove with $\epsilon-\delta$ tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Certainty that one has found all of the socks in a pile Suppose that I have a pile of $n$ socks, and, of these, $2k$ are "mine." Each of the socks that is mine has a mate (so that there are $k$ pairs of my socks) I know $n$, but not $k$. Assume that all possible values of $k$ are equally likely.
Now, I draw socks from ... | Suppose after $m$ draws, you might have exactly $j$ pairs of your socks and no singletons. Then, given $K=k,m,n$ and provided that $j \le k$, $2j \le m$, and $2k \le n$, the probability of this event is
$$\Pr(J=j \text{ pairs and no singletons}| K=k,m,n) = {k \choose j}{n-2k \choose m-2j}\big/ {n \choose m}$$
and so ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505264",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why are strict inequalities stronger than non-strict inequalities? I'm working with induction proofs involving inequalities and I am encountering example proofs that wish to show things of the sort, $n!\le\ n^n$ for every positive integer. The proof given in the inductive step is, $(n+1)!$ $=$ $(n+1)\dot\ n!$ $\le$ $(n... | You can change $<$ by $\leq$ but not $\leq$ by $<$.
For example, we have $1 < 2$, so $1 \leq 2$ is also true.
On the other hand, we have $1 \leq 1$, but it's not true that $1 < 1$.
Also, you proved $n! < n^n$ for $n > 1$, but this is not true for every positive integer, since for $n=1$ we get the equality. But it's tru... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Surprising identities / equations What are some surprising equations/identities that you have seen, which you would not have expected?
This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc.
I'd request to avoid 'standard' / well-known results like $ e^{i \pi} + 1 = 0$.
P... | This one really surprised me:
$$\int_0^{\pi/2}\frac{dx}{1+\tan^n(x)}=\frac{\pi}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/505367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "350",
"answer_count": 108,
"answer_id": 24
} |
Surprising identities / equations What are some surprising equations/identities that you have seen, which you would not have expected?
This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc.
I'd request to avoid 'standard' / well-known results like $ e^{i \pi} + 1 = 0$.
P... | Some zeta-identies have been much surprising to me.
Let's denote the value $\zeta(s)-1$ as $\zeta_1(s)$ then
$$ \small \begin{array} {}
1 \zeta_1(2) &+&1 \zeta_1(3)&+&1 \zeta_1(4)&+&1 \zeta_1(5)&+& ... &=&1\\
1 \zeta_1(2) &+&2 \zeta_1(3)&+&3 \zeta_1(4)&+&4 \zeta_1(5)&+& ... &=&\zeta(2)\\
& &1 \zet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "350",
"answer_count": 108,
"answer_id": 56
} |
Surprising identities / equations What are some surprising equations/identities that you have seen, which you would not have expected?
This could be complex numbers, trigonometric identities, combinatorial results, algebraic results, etc.
I'd request to avoid 'standard' / well-known results like $ e^{i \pi} + 1 = 0$.
P... | This is slightly contrived, but consider a situation where you have two balls, of mass $M$ and $m$, where $M=16\times100^N\times m$ for some integer $N$. The balls are placed against a wall as shown:
We push the heavy ball towards the lighter one and the wall. The balls are assumed to collide elastically with the w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "350",
"answer_count": 108,
"answer_id": 88
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Using characteristic function to deduce convergence of Bernoulli random variables Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n =
\sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in distribution and determine the limiting dist... | Bernoulli RVs are bounded, so you have that the convergence even occurs pointwise, hence a.s. pointwise, hence in probability, hence in distribution. Judging by the way the question was asked, I think your instructor wants a very specific answer. The limiting RV in any of the senses above is uniformly distributed on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Last two digits of $2^{11212}(2^{11213}-1)$
What are the last two digits of the perfect numbers $2^{11212}(2^{11213}-1)$?
I know that if $2^n-1$ is a prime, then $2^{n-1}(2^n-1)$ is a perfect number and that every even perfect number can be written in the form $2^n(2^n-1)$ where $2^n-1$ is prime. I'm not sure how to ... | We want the remainder when the product is divided by $100$. Th remainder on division by $4$ is $0$, so all we need is the remainder on division by $25$.
Note that $\varphi(25)=20$. So by Euler's Theorem, $2^{20}\equiv 1\pmod{25}$. It is easier to note that $2^{10}\equiv -1\pmod{25}$.
It follows that $2^{11212}\equiv ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Calculus in ordered fields Is there any ordered field smaller that the set of real numbers in which we can do calculus, also with many restrictions ?
If not why ?
| I remember reading in Körner's book, "A Companion to Analysis", he discusses problems with calculus over $\mathbb{Q}$, and ordered fields it seems. I think there was a problem with continuity over $\mathbb{Q}$ and of course the fact it's not complete. It might be worth a look if you've access to it.
Section 1.3
and Ch... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
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Nilpotent matrix in $3$ dimensional vector space This is a part of a long problem and I'm stuck in two questions of it.
Let $E$ a $3$ dimensional $\mathbb R$- vector space and $g\in\mathcal{L}(E)$ such that $g^3=0$. So the first question is to prove that
$$\dim\left(\ker g\cap \mathrm{Im}\ g\right)\leq 1$$
and the seco... | For the first part dim(im$\,g$)$ + $ dim $(\ker\,g) = 3$.
The possible combinations are $0 + 3$, $1 + 2$, $2 + 1$, $3 + 0$ and you can see that any intersection cannot have dimension greater than 1.
For the second part:
\begin{align*}
g^3(x) &= 0,\, \forall x \in E \\
g^2(g(x)) &= 0 ,\, \forall x \in E \\
&\Rightarrow ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Prove that a function does not have a limit as $x\rightarrow \infty $ Problem statement:
Prove that the function $f(x)=\sin x$ does not have a limit as $x\rightarrow \infty $.
Progress:
I want to construct a $\varepsilon -\delta $-proof of this so first begin by stating that the limit actually exists:
$\lim_{x\rightar... | In terms of writing the logical negation, it is best to use a universal quantifier before the $x$:
$$
\forall \varepsilon >0 \exists K>0: \forall x>K,\ \left | f(x)-l \right |<\varepsilon.
$$
The logical negation of this statement is
$$
\exists \varepsilon >0 \forall K>0: \exists x>K,\ \left | f(x)-l \right |>\varepsil... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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a game of coloring edges of graph I have a clique of size 5 which is partially colored(black or white). I have to color remaining edges so that each of the triangle has either 1 or 3 black edges. How should I go about coloring the graph or how can I tell this is not possible.
| Hint: Given a (complete) coloring of the clique, each vertex of the form $v_{i} v_{i+2}$ is uniquely determined.
It then remains to check that triangles of the form $v_i , v_{i+2}, v_{i+4} $ satisfy the conditions.
Hint: Assume that such a coloring is possible. Label the black edges $-1$ and the white edges $1$. Then,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/505927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that : $\tan 40 + \tan 60 + \tan 80 = \tan 40 \cdot \tan 60 \cdot \tan 80$ I started from Left hand side as 3^1/2 + tan 2(20) +tan 4(20). But that brought me a lot of terms to solve which ends (9 tan 20 - 48 tan^3 20 -50 tan^5 20 - 16 tan^7 20 + tan^9 20)/(1- 7 tan^2 20 + 7 tan^4 20 - tan^6 20), which is very hug... | Generally, if $\alpha+\beta+\gamma=180^\circ$ then $\tan\alpha+\tan\beta+\tan\gamma=\tan\alpha\tan\beta\tan\gamma$.
That is because
$$
0 = \tan180^\circ = \tan(\alpha+\beta+\gamma) = \text{a certain fraction}.
$$
The numerator in the fraction is $\tan\alpha+\tan\beta+\tan\gamma-\tan\alpha\tan\beta\tan\gamma$.
Just appl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/506002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Stuck on a particular 2nd order non-homogeneous ODE ($y'' + 4y' + 5y = t^2$)
$y'' + 4y' + 5y = t^2$
So I solve for $r^2 + 4r + 5 = 0$ returning $r = 2 \pm 2i$. So $y_c = e^{-2t}(C_1\cos(2t) + C_2\sin(2t)$. For $y_p(t)$ I pick $At^2$. So $2A + 8At + 5At^2 = t^2$. I have been stuck on finding a particular solution for ... | You should use method of undetermined coefficients.
Take
$$y_{p}(t)=at^2+bt+c$$
then
$$
y'_{p}(t)=2at+b
$$
$$
y''_{p}(t)=2a
$$
and substitue it in differential equation
$$2a+8at+4b+5at^2+5bt+5c=t^2$$
from equality of polynomials we have
$5a=1,8a+5b=0,2a+4b+5c=0$
from here we can find $a=\frac{1}{5},b=\frac{-8}{25},c=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/506077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Divisibility involving root of unity Let $p$ be a prime number and $\omega$ be a $p$-th root of unity. Suppose $a_0,a_1, \dots, a_{p-1}, b_0, b_1, \dots, b_{p-1}$ be integers such that $a_0 \omega^0+a_1 \omega^1+ \dots a_{p-1} \omega^{p-1}$ and $b_0 \omega^0 + b_1 \omega^1 + \dots b_{p-1} \omega^{p-1}$ are also integer... | Irreducible polynomial over $\mathbb Z[X]$
Consider the factorization $X^p-1=(X-1)\Phi_p$ where $\Phi_p$ is the cyclotomic polynomial defined to contain all primitive $p$'th roots of unity over $\mathbb C$. Dividing both sides by $(X-1)$ we get
$$
\Phi_p=X^{p-1}+X^{p-2}+...+1
$$
Since all cyclotomic polynomials are irr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/506186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is there a finite number of ways to change the operation in a ring and retain a ring? Consider, for instance, the ring $(\mathbb{Z}, +, \cdot)$. We can create a new ring $(\mathbb{Z}, \oplus, \odot)$ by defining
$$ a \oplus b = a + b - 1 \quad \text{and} \quad a \odot b = a + b - ab. $$
There are also other operations... | On $\Bbb {Z/nZ}$ there must be finitely many as there are only finitely many operations on a finite set.
Your examples are derived by considering replacing each element $x$ with $x-1$ and seeing what happens to the operations. The constant is arbitrary, so you have an infinite number of choices on $\Bbb {Z, R}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Differentiability of $\sin ^2(x+y)+i\cos ^2(x+y)$ I want to find the set of all points $z\in\mathbb{C}$ such that $f:\mathbb{C}\to\mathbb{C}$ given by $f(x+iy)=\sin ^2(x+y)+i\cos ^2(x+y)$ is differentiable at $z$.
Isn't it true that $f$ is differentiable at all points of $\mathbb{C}$? Because $\sin$ and $\cos$ are dif... | The total differential is straightforward:
$$ df(z) = 2 \sin(x+y) \cos(x+y) (dx + dy) - 2 i \cos(x+y) \sin(x+y) (dx + dy) $$
Also,
$$ dx = \frac{dz + d\bar{z}}{2} \qquad \qquad dy = \frac{dz - d\bar{z}}{2i} $$
Checking differentiability is the same thing as checking that the coefficient on $d\bar{z}$ is zero, if you wr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/506323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Odds of choosing n members of a x sized subset of a y sized set. Lets say I have a bag of 8 rocks. 3 rocks are red, the rest are black. I choose, randomly, with replacement, 3 rocks. What are the odds that at least one that I choose is red?
It seems in first pass that the odds if I draw just one time, is 3/8. If I draw... | Hint: Consider the complement. What is the probability that all rocks are black?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/506422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Alternative Creative Proofs that $A_4$ has no subgroups of order 6 Since I've been so immersed in group theory this semester, I have decided to focus on a certain curious fact: $A_4$ has no subgroups of order $6$.
While I know how to prove this statement, I am interested in seeing what you guys can offer in terms of un... | So we have a group $G$ of order $12$ with a normal subgroup $H$ of order $4$ and a normal subgroup $K$ of order $6$. Then $H\cap K$ is a normal subgroup. It has order $2$ (since if $H\cap K=1$, then $|G|=24$). Thus, $|G/(H\cap K)|=6$, and it has a unique Sylow 3-subgroup, so it is not $S_3$. The only alternative is tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/506478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Examples of uncountable subgroups, but with countably many cosets. My question is pretty much as stated in the title: what examples are there (or does there exist) uncountable subgroups of a group but which have countably-infinite many cosets. I can only think of examples in the other direction (countable subgroups gi... | The uncountable subgroup of $\Bbb Z^{[0,1]}$ of maps $[0,1]\to\Bbb Z$ vanishing at $0$ has index $|\Bbb Z|=\aleph_0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/506561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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How to find the range of $\sqrt {x^2-5x+4}$ where x is real. How to find the range of $$\sqrt {x^2-5x+4}$$ where $x$ is real.
What I've tried:
Let $\sqrt {x^2-5x+4}=y$, solving for real $x$, as $x$ is real discriminant must be $D\geq0$. Solving I get $y^2\geq\frac{-9}{4}$. Which I suppose implies all real . But on wolf... | Note that the domain of the function is $(-\infty,1]\cup [4, \infty)$, since
$$ y=\sqrt{(x-1)(x-4)}, $$
which tells you that $y\geq 0$
Note: Compare with the function
$$y=\sqrt{x}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/506707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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How is it possible that $\infty!=\sqrt{2\pi}$? I read from here that:
$$\infty!=\sqrt{2\pi}$$
How is this possible ?
$$\infty!=1\times2\times3\times4\times5\times\ldots$$
But
\begin{align}
1&=1\\
1\times2&=2\\
1\times2\times3&=6\\
&~\vdots\\
1\times2\times3\times\ldots\times50&=3.0414093201713376\times10^{64}
\end{alig... | It is taken from
$$ 1\cdot2\cdot3\cdot \ldots \cdot n= n!$$
This is the exponential of
$$ \ln(1)+\ln(2)+\ln(3)+ \ldots + \ln(n) = \ln(n!) $$
Now if you write formally the derivative of the Dirichlet-series for zeta then you have
$$ \zeta'(s) = {\ln(1) \over 1^s}+{\ln(1/2) \over 2^s} +{\ln(1/3) \over 3^s} + \ldots $$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/506781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Support of Convolution and Smoothing
I just want to know how it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in C^{\infty}(V)$ by using the translations, but I'm having difficulty seeing how it extends to $\bar{V}$, since it says that $u_{\epsilon}(x) := u(x^{\epsilon}) \text{ f... | $\DeclareMathOperator \supp {supp}$For functions on $\mathbb R^n$ we have in general $\supp (f*g) \subset \supp f+\supp g$, where $A+B = \{a + b \mid a \in A, b \in B\}$. To see this note that $x$ must be in this set for $f(x-y)g(y)$ to be non-zero.
Thus we have $\supp f^\epsilon \subset \supp f + B(0,\epsilon) = \{x \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expressing the trace of $A^2$ by trace of $A$ Let $A$ be a a square matrix. Is it possible
to express $\operatorname{trace}(A^2)$ by means of $\operatorname{trace}(A)$ ? or at least
something close?
| It is not. Consider two $2\times 2$ diagonal matrices, one with diagonal $\{1,-1\}$ and one with diagonal $\{0,0\}$. They have the same trace, but their squares have different traces.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/506962",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
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How to integrate $\int \frac{1}{\sin^4(x)\cos^4(x)}\,\mathrm dx$? How can I integrate $$\int \frac{1}{\sin^4(x)\cos^4(x)}\,\mathrm dx.$$
So I know that for this one we have to use a trigonometric identity or a substitution. Integration by parts is probably not going to help. Can someone please point out what should I d... | Hint: Substitute $t=\tan x$.
Then $\frac{dx}{\sin^4 x \cos^4 x} = \frac{(1+t^2)^3}{t^4}dt$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/507091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A wrong proof that the kernel and image are always complementary Let $E$ be a vector space, $f\colon E\rightarrow E$ an endomorphism. Let $A=\ker(f)\oplus \operatorname{im}(f)$; that is $$A=\{x\in E\;|\; \text{there exists a unique}\; (a,b)\in \ker(f)\times\operatorname{im}(f), x=a+b\}.$$ I have a proof that $A=E$ whic... | The direct sum of two subspaces of $E$ is an abstract vector space that has a canonical map to $E$, but that map can fail to be injective, and it will precisely when the two subspaces have non-zero intersection. So your mistake is in assuming that the direct sum is a subspace of $E$.
More precisely, if $W_1,W_2$ are su... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Proving convergence Using the definition Use the definition of convergence to prove if $x_n$ converges to $5$, then
$\frac{x_n+1}{\sqrt{x_n-1}}$ converges to $3$.
| There is an almost automatic but not entirely pleasant procedure. Let $y_n=\frac{x_n+1}{(x_n-1)^{1/2}}$. We will need to examine $|y_n-3|$.
Note that
$$y_n-3=\frac{x_n+1}{(x_n-1)^{1/2}}-3=\frac{(x_n+1)-3(x_n-1)^{1/2}}{(x_n-1)^{1/2}}.$$
Multiply top and bottom by $(x_n+1)+3(x_n-1)^{1/2}$. This is the usual "rationalizi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/507230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expected shortest path in random graph Consider all connected graphs with $n$ verticies where each vertex connects to $k$ other verticies. We choose such a graph at random. What is the expected value of the shortest path between two random points? What is the expected value of the maximal shortest path?
If an exact sol... | This seems like a hard problem. For $k=2$ you can do exact calculations because the graph is a single cycle. For a random $k$-connected graph where $ k << n$ and $k > 2$, you can argue that for a given chosen starting vertex $v$, there will be expected $\Theta(k(k-1)^{m-1})$ vertices $w$ that will have a shortest path ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Question about Fermat's Theorem I'm trying to find $2^{25} \mod 21 $. By Fermat's theorem, $2^{20} \cong_{21} 1 $. Therefore, $2^{25} = 2^{20}2^{5} \cong_{21} 2^5 = 32 \cong_{21} 11 $. However, the answer in my book is $2$! What am I doing wrong?
Also, I would like to ask what are the last two digits of $1 + 7^{162} + ... | By Euler's Theorem, we have $2^{12}\equiv 1\pmod{21}$. That is because $\varphi(21)=(2)(6)=12$. Thus $2^{25}=2^{12\cdot 2}\cdot 2^1\equiv 2\pmod{21}$.
If we want to use Fermat's Theorem, we work separately modulo $3$ and modulo $7$.
We have $2^2\equiv 1\pmod{3}$, and therefore $2^{25}=2^{2\cdot 12}\cdot 2^1\equiv 2\pmo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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For any finite group, there is a homomorphism whose image is simple This is for homework. The question asks
"Show that, for any finite group $G$, there is a homomorphism $f$ such that $f(G)$ is simple."
My thought was this. Since $G$ is finite, there are only a finite number of normal subgroups of $G$, call them $N_1... | If $G$ is already simple then any isomorphism will do, so assume $G$ is not simple. Because $G$ is finite, there must exist a maximal nontrivial proper normal subgroup $N$, meaning $N \neq 1$ and whenever
$N\leq H \leq G$ with $H$ normal in $G$ then $H=G$ or $H=N$. Now show that $G/N$ must be simple.
The problem with... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to prove $(1+x)^n\geq 1+nx+\frac{n(n-1)}{2}x^2$ for all $x\geq 0$ and $n\geq 1$? I've got most of the inductive work done but I'm stuck near the very end. I'm not so great with using induction when inequalities are involved, so I have no idea how to get what I need...
\begin{align}
(1+x)^{k+1}&\geq (1+x)\left[1+kx+... | If $n=1$, it is trivial. Suppose it is true for $n$. We will show that this formula is true for $n+1$.
$$
\begin{aligned}
(1+x)^{n+1}&=(1+x)^n(1+x)\\
&\ge \left( 1+nx+\frac{n(n-1)}{2} x^2 \right)(1+x)\\
&= 1+nx+\frac{n(n-1)}{2}x^2 +x+nx^2 +\frac{n(n-1)}{2} x^3\\
&= 1+(n+1)x+\frac{n^2-n+2n}{2}x^2+\frac{n(n-1)}{2} x^3\\
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A probability problem on limit theorem (!) Each of the 300 workers of a factory takes his lunch in one of the three competing restaurants (equally likely, so with probability $1/3$). How many seats should each restaurants have so that, on average, at most one in 20 customers will remain unseated?
How can I approach thi... | That this is an exercise on the limit theorems chapter is actually a clue. Let $n=300$ denote the number of workers, $s$ the number of seats in each restaurant and $X$ the number of workers trying to get a seat in restaurant 1. Then $X-s$ customers remain unseated at restaurant 1 if $X\gt s$, and none if $X\leqslant s$... | {
"language": "en",
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How to calculate a Bell number (Bell[n] mod 95041567) quickly enough? How to calculate a Bell number (Bell[n] mod 95041567) quickly enough?
Here n maybe very big as 2^31.
Bell number is http://oeis.org/A000110
| Wikipedia (https://en.wikipedia.org/wiki/Bell_number) states Touchard's congruence for a prime power $p^m$ as
$$B_{p^m + n} = m B_n + B_{n+1} \pmod{p}$$
Using this and the Chinese remainder theorem you should be able to calculate the numbers pretty quickly.
More details: One straightforward way is to take $m=1$ in the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Showing an integral is unbounded Let $$F(x)=x^2\sin\left(\frac{1}{x^2}\right)\quad\forall x\in(0,1]$$ and $$G(x)=F'(x)=2x\sin\left(\frac{1}{x^2}\right)-2\frac{1}{x}\cos\left(\frac{1}{x^2}\right)\quad\forall (0,1].$$ I want to show that $$\lim_{t\searrow0}\int_t^1\left|G(x)\right|\,\mathrm{d}x=+\infty.$$
I've had a hard... | The $2x\sin\left(\dfrac{1}{x^2}\right)$ part is bounded, and can therefore be ignored when investigating the convergence or divergence of the integral.
So let's look at the integral
$$\int\limits_{a_{k+1}}^{a_k} \left\lvert \frac{2}{x}\cos \left(\frac{1}{x^2}\right)\right\rvert\, dx$$
where the $a_k$ are the successive... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Using the definition of the derivative to prove a constant function I am presented with the following task:
"Let $f$ be a function defined on the interval $I$. All we know about $f$ is that there is a constant $K$ such that
$$|f(a) - f(b)| \leq K|a-b|^2$$
for all $a, b \in I$. Show that $f$ is constant on $I.$ (Hint: ... | Suppose for example that $0\in I$ and $1\in I$.
Then
$$
|f(1)-f(0)|=\bigg|\sum_{k=0}^{n-1} f(\frac{k+1}{n})-f(\frac{k}{n})\bigg|
\leq \sum_{k=0}^{n-1} \big|f(\frac{k+1}{n})-f(\frac{k}{n})\big|
\leq \sum_{k=0}^{n-1} K\frac{1}{n^2}=\frac{K}{n}
$$
Since this holds for any $n$, we deduce $f(1)=f(0)$.
You can do this for an... | {
"language": "en",
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How to prove the limit of a sequence using "$\epsilon-N$" I think I have a proper understanding of the general procedure, but I'm having difficulty manipulating my inequality so that I can isolate $n$ by itself. Sadly I wasn't given many examples to model my answer on.
Prove that $\displaystyle\lim_{n\to\infty}\frac{n... | Since the given sequence is positive for all $n\geq 1$, we can drop the absolute value signs.
Consider the inequality:
\begin{equation*}
\frac{n+1}{n^2+1}<\epsilon
\end{equation*}
This is a messy inequality, as solving for $n$ would be rather difficult. Instead we shall find an upper bound for the numerator $(n+1)$ an... | {
"language": "en",
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If $ a \mid bc $ then $\frac{a}{\gcd(a,b)} \mid c$? Prove or reject this statement:
If $ a \mid bc $ then $\displaystyle \frac{a}{\gcd(a,b)} \mid c$
| Hint: If $a\mid bc$ and $(a,b)=1$ then $a\mid c$. Use this to show:
$$
\displaystyle a \mid bc \implies \frac{a}{\gcd(a,b)} \mid \frac{b}{\gcd(a,b)}\times c \implies \frac{a}{\gcd(a,b)}\mid c
$$
Because $\gcd(\frac{a}{\gcd(a,b)},\frac{b}{\gcd(a,b)})=1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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A round table probability question Hi guys I am writing my P exam for the second time and I remembered two question that confused me when writing the exam. I asked my prof. but it confused him as well. For simplicity I will ask one question here and post the other one after.
So if you could please help me I would reall... | The way you are asking the question it seems to me that whether it is social workers or clients has no relevance. So basically we have 10 people seated around a round table and 3 specific people must not sit next to each other.
So label the seats 1 through 10 as you suggested: You can choose 3 out of 10 in $\binom{10}{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Showing that rationals have Lebesgue measure zero. I have been looking at examples showing that the set of all rationals have Lebesgue measure zero. In examples, they always cover the rationals using an infinite number of open intervals, then compute the infinite sum of all their lengths as a sum of a geometric series.... | Define an interval $(q_n−\frac{ϵ}{2^n},q_n+\frac{ϵ}{2^n})$
around each rational number $q$
For $\epsilon>0$, $\mu((q_n−\frac{ϵ}{2^n},q_n+\frac{ϵ}{2^n}))=2\left(\frac{\epsilon}{2^n}\right)$
and $\sum^{\infty} _{n=1} 2\frac{ϵ}{2^n}=2ϵ$
Since $ϵ$ is arbitrary, so $μ(Q) = 0$.
I think this simple trick could work
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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About a proof that functions differing from a measurable function on a null set are measurable Consider the following problem:
Say $f: M \to \mathbb{R} $ is measurable function, $M$ a measurable set and $g : M \to \mathbb{R}$ such that $ Y = \{ x : f(x) \neq g(x) \} $ is a null set. We want to show $g$ is measurable... | The expression that troubles you:
$$\{x: d(x) > a\} = \begin{cases}N&: a \ge 0\\N^c&: a < 0 \end{cases}$$
is really an abuse of notation. What is meant is something like:
$$\{x: d(x) > a\} = \begin{cases}N_a&: a \ge 0\\N_a^c&: a < 0 \end{cases}$$
because the null set $N_a$ varies with $a$, in the following way:
$$N_a =... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to show that the given equation has at least one complex root ,a s.t |a|>12 How do I show that the equation $x^3+10x^2-100x+1729$$=0$ has at least one complex root $a$ such that $|a|$$>$$12$.
| The function necessarily has three complex roots, which we'll call $\alpha$, $\beta$ and $\gamma$. Hence your polynomial can be factored as
$$(x - \alpha) (x - \beta) (x - \gamma) = x^3 + 10x^2 - 100x + 1729$$
Expanding the left hand side, we find that
$$-\alpha \beta \gamma = 1729$$
If it were true that all three root... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/508416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Range of $S = \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+..............+\frac{1}{\sqrt{n}}$ If $\displaystyle S = \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+..............+\frac{1}{\sqrt{n}}$ and $n\in \mathbb{N}$. Then Range of $S$ is
$\underline{\bf{My \;\; Try}}$:: For Lower Bond:: $\sqrt... | When we say a series is bounded, we mean it is bounded by a CONSTANT. As Gerry wrote above, lower bound is $1$ and there is no upper bound.
If you are familiar with the Harmonic series, you can see each term in $S$ greater equal than each term of the Harmonic series. We all know the Harmonic series diverges and hence ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to prove that $a
Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that
$$a<S_n-[S_n]<b$$
where $[x]$ represents the largest integer not exceeding $x$.
This problem is f... | Let $i\geqslant1/(b-a)$ and $k=\lceil S_i\rceil$. Since the sequence $(S_j)_{j\geqslant1}$ is unbounded, some values of this sequence are greater than $k+a$, hence $n=\min\{j\mid S_j\gt k+a\}$ is well defined and finite. Then $S_{n-1}\leqslant k+a\lt S_{n}$ and $n\gt i$ since $S_i\lt k+a$ hence
$$
S_{n}=S_{n-1}+1/n\lt... | {
"language": "en",
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Proof without using Extreme value theorem? $$\text{Let } f:[a,b]\to \mathbb{R} \text{ be continuous and strictly positive. } $$
$$ \text{prove } c= \inf f(\,(a,b)\,)\neq 0$$
Is there a way other than using the extreme value theorem?
One way might be to show $c= \min f( [a,b] )$, and therefore positive, but I'm not sur... | Suppose that $\inf\{f(x):x\in(a,b)\}=0$. Then, for each positive integer $n$, there is some $x_n\in(a,b)$ such that $$(*)\quad 0<f(x_n)<\frac{1}{n}.$$ Since $[a,b]$ is compact and is the closure of $(a,b)$, the sequence $(x_n)$ has a subsequence $(x_{n_k})$ that converges to some $x^*\in[a,b]$. Since $f$ is continuous,... | {
"language": "en",
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Matrix Algebra Question (Linear Algebra) Find all values of $a$ such that $A^3 = 2A$, where
$$A = \begin{bmatrix} -2 & 2 \\ -1 & a \end{bmatrix}.$$
The matrix I got for $A^3$ at the end didn't match up, but I probably made a multiplication mistake somewhere.
| The matrix $A$ is clearly never a multiple of the identity, so its minimal polynomial is of degree$~2$, and equal to its characteristic polynomial, which is $P=X^2-(a-2)X+2-2a$. Now the polynomial $X^3-2X$ annihilates $A$ if and only if it is divisible by the minimal polynomial $P$. Euclidean division of $X^3-2X$ by $P... | {
"language": "en",
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Connected, but it is not continuous at some point(s) of I I have a mathematics problem, but I have no idea. please help me...
The problem is "Give an example of a function $f(x)$ defined on an interval I whose graph is connected, but is is not continuous at some point(s) of $I$"
In my idea, a solution is topologist's s... | This is a good example. The graph of the topologist's sine curve, which includes the point $(0,0)$ as you have indicated, is indeed connected. However, it is not continuous. To see this, try and produce a sequence of points $x_n$ converging to $0$ for which $\sin(1/x_n) = 1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A problem about dual basis I've got a problem:
Let $V$ be the vector space of all functions from a set $S$ to a field $F$:
$(f+g)(x) = f(x) + g(x)\\
(\lambda f)(x) = \lambda f(x)$
Let $W$ be any $n$-dimensional subspace of $V$. Show that there exist points $x_1, x_2, \dots, x_n$ in $S$ and funtions $f_1, f_2, \dots, f... | This is not a dual basis question. The dual of a vector space $V$ over field $k$ is the set of all linear functions from $V$ into $k$.
In your case the functions do not have to be linear, or even continuous. An example of what you have is $\mathbb R^n$, which can be thought of as the set of all functions from the set $... | {
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If a and b are group elements and ab ≠ ba, prove that aba ≠ identity. Q: If a and b are group elements and ab ≠ ba, prove that aba ≠ identity.
I began by stating my theorem, then assumed ab ≠ ba. Then I tried a few inverse law manipulations, which worked in a sense, however they brought me nowhere, as I couldn't concl... | Try proving the contrapositive; that is, if $aba=e$ then $ab=ba$. First of all, multiply both sides of $aba=e$ on the left by $a^{-1}$ to get $ba=a^{-1}$; now multiply both sides of this on the right by $a^{-1}$ to get $b=a^{-2}$. Now, can you see why $b$ commutes with $a$?
| {
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"timestamp": "2023-03-29T00:00:00",
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Does $\int_a^b f'(\gamma(t))\gamma'(t)\,dt$ depend on the path for meromorphic functions? Given a meromorphic function $f:\mathbb C\to\mathbb C$ and a smooth curve $\gamma:[a,b]\to\Gamma\subset\mathbb C$ with $\gamma(a)\neq\gamma(b)$, I am tempted to think the fundamental theorem of calculus yields
$$\int_a^b\underbrac... | The correction term can be given in terms of the winding numbers $n(k,\gamma)$ of the directed curve $\gamma$ about the poles $k$ of $f$. We have
$$\int_a^b f'(\gamma(t))\gamma'(t) dt = f(\gamma(t)) \bigg\vert_{a}^b +2\pi i \sum_k n(k,\gamma)\mathrm{Res}_k(f').$$
Intuitively, the winding number counts how many times a... | {
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"timestamp": "2023-03-29T00:00:00",
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How to solve a recursive equation I have been given a task to solve the following recursive equation
\begin{align*}
a_1&=-2\\
a_2&= 12\\
a_n&= -4a_n{}_-{}_1-4a_n{}_-{}_2, \quad n \geq 3.
\end{align*}
Should I start by rewriting $a_n$ or is there some kind of approach to solve these?
I tried rewriting it to a Quadratic ... | $\displaystyle{\large a_{1} = -2\,\quad a_{2} = 12,\quad
a_{n} = -4a_{n - 1} - 4a_{n - 2}\,,\quad n \geq 3}$.
$$
\sum_{n = 3}^{\infty}a_{n}z^{n}
=
-4\sum_{n = 3}^{\infty}a_{n - 1}z^{n}
-
4\sum_{n = 3}^{\infty}a_{n - 2}z^{n}
=
-4\sum_{n = 2}^{\infty}a_{n}z^{n + 1}
-
4\sum_{n = 1}^{\infty}a_{n}z^{n + 2}
$$
$$
\Psi\left(z... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that when $BA = I$, the solution of $Ax=b$ is unique I'm just getting back into having to do linear algebra and I am having some trouble with some elementary questions, any help is much appreciated.
Suppose that $A = [a_{ij}]$ is an $m\times n$ matrix and $B = [b_{ij}]$ and is an $n\times m$ matrix and $BA = I$ th... | The usual method for solving uniqueness problems is generally this: assume you have two solutions, say $x$ and $y$. Then do manipulation, use theorems, whatever, and somehow show that $x=y$.
In your case, if $Ax=b$ and $Ay=b$, then multiply both sides on the left by $B$. Then $B(Ax)=Bb$ and $B(Ay)=Bb$. Can you take it ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Newton's Method - Slow Convergence I'm using Newton's method to find the root of the equation $\frac{1}{2}x^2+x+1-e^x=0$ with $x_0=1$. Clearly the root is $x=0$, but it takes many iterations to reach this root. What is the reason for the slow convergence? Thanks for any help :)
| Newton's method has quadratic convergence near simple zeros, but the derivative of $\frac{1}{2}x^2+x+1-e^x$ at $x=0$ is zero and so $x=0$ is not a simple zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/509450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Convergence of partial sums and their inverses
If a sequence $s_{k}$ of partial sums converges to a nonzero limit, and we assume that $s_{k} \neq 0$ for all $k$ $\epsilon$ $\mathbb{N}$, then also the sequence $\left \{ \frac{1}{s_{k}} \right \}$ converges.
In my book, $s_{k}$ is defined as $\sum_{j = 1}^{k}\frac{... | In general, if $\{a_n\}$ is any convergent sequence with a limit $a\neq 0$, then $\dfrac{1}{a_n}$ converges to $\dfrac{1}{a}$.
Proof. Let $\epsilon>0$. Then
$$
\left|\frac{1}{a_n}-\frac{1}{a}\right|=\left|\frac{a-a_{n}}{aa_n}\right|
$$
Since $a_n\to a$ as $n\to\infty$, we can choose a positive integer $N_1$ such that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/509503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Means and Variances For a laboratory assignment, if the equipment is
working, the density function of the observed outcome
$X$ is
$$
f(x) = \begin{cases} 2(1-x), & 0 <x<1, \\
0, & \text{otherwise.} \end{cases}
$$
Find the variance and standard deviation of $X$.
We know that the variance is related to the mean and the s... | The integrals are
$$
\text{mean} = \mu = \int_0^1 x f(x) \, dx
$$
and
$$
\text{variance} = \sigma^2 = \int_0^1 (x-\mu)^2 f(x)\,dx.
$$
In order to evaluate the second integral, one must find $\mu$ by evaluating the first integral.
The second integral is the variance $\sigma^2$; the standard deviation is the square root ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/509574",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Is a chain complete lattice complete? If every chain in a lattice is complete (we take the empty set to be a chain), does that mean that the lattice is complete? If yes, why?
My intuition says yes, and the reasoning is that we should somehow be able to define a supremum of any subset of the lattice to be the same as t... | If $L$ is not complete, it has a subset with no join; among such subsets let $A$ be one of minimal cardinality, say $A=\{a_\xi:\xi<\kappa\}$ for some cardinal $\kappa$. For each $\eta<\kappa$ let $A_\eta=\{a_\xi:\xi\le\eta\}$; $|A_\eta|<\kappa$, so $A_\eta$ has a join $b_\eta$. Clearly $b_\xi\le b_\eta$ whenever $\xi\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof of equivalence of algebraic and geometric dot product? Geometrically the dot product of two vectors gives the angle between them (or the cosine of the angle to be precise). Algebraically, the dot product is a sum of products of the vector components between the two vectors.
However, both formulae look quite diffe... | TL;DR: So, in reality, while they do look different they actually do the exact same thing, thus outputting the same result. It's just that one is in regular coordinates and the other is in polar coordinates, that's all.
Let us consider the 2D case for simplicity.
Imagine we have a vector $\mathbf{r} = (x,y)$. We can r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/509719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 2,
"answer_id": 0
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A continuous function that when iterated, becomes eventually constant Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function, and let $c$ be a number. Suppose that for all $x \in \mathbb{R}$, there exists $N_x > 0$ such that $f^n(x) = c$ for all $n \geq N_x$. Is it possible that $f^n$ (this is $f$ composed... | Take for example
$$f_k(x) = \begin{cases}0 &\text{ for } x \geq -k \\ x+k & \text{ otherwise}\end{cases}$$
which are all continuous and we have $f_1^n = f_n$ which is not constant.
I hope this helps $\ddot\smile$
| {
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"source": "stackexchange",
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If the index $n$ of a normal subgroup $K$ is finite, then $g^n\in K$ for each $g$ in the group. Let $K \unlhd G$ be a normal subgroup of some group $G$ and let $|G/K|=n<\infty$. I want to show that $g^n\in K$ for all $g\in G$.
Let $g\in G$, if $g\in K$, then $g^n\in K$ and we are done. If $g^n\notin K$ then consider th... | You dont need to go that far. Since you know that $|G/K|=n<\infty$ and
$G/K =\{K, gK,g^2K,...,g^{n-1}K\}$. Then for any $gK$ in $G/K$, $(gK)^n= g^nK=K $ which answers your question
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 2
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If $\frac{\cos x}{\cos y}=\frac{a}{b}$ then $a\tan x +b\tan y$ equals If $$\frac{\cos x}{\cos y}=\frac{a}{b}$$ Then $$a \cdot\tan x +b \cdot\tan y$$ Equals to (options below):
(a) $(a+b) \cot\frac{x+y}{2}$
(b) $(a+b)\tan\frac{x+y}{2}$
(c) $(a+b)(\tan\frac{x}{2} +\tan\frac{y}{2})$
(d) $(a+b)(\cot\frac{x}{2}+\co... | So, we have $$\frac a{\cos x}=\frac b{\cos y}=\frac{a+b}{\cos x+\cos y}$$
$$\implies a\tan x+b\tan y=\frac a{\cos x}\cdot\sin x+\frac b{\cos y}\cdot\sin y$$
Putting the values of $\displaystyle\frac a{\cos x},\frac b{\cos y}$
$$a\tan x+b\tan y=(\sin x+\sin y)\frac{(a+b)}{\cos x+\cos y}$$
Now, $\displaystyle \cos x+\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/510108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How many ways to go from $(0,0)$ to $(20,10)$ if precisely $2$ right moves need to be made in a row? Just to clarify: This is the number of ways to go from point $(0,0)$ to point $(20,10)$ if the only directions allowed are right and up. The catch: Each of the ways must include precisely (only) $1$ instance of a "doub... | Imagine that your 'long right' was instead a single rightward step. Then, rather than going from $(0,0)$ to $(20,10)$ instead you would move from $(0,0)$ to $(20-k+1,10)$, where $k$ is the length of the 'long right' — but you would do it without any two consecutive rightward moves. What's more, since there's only one... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Infinite Geometric Series I'm currently stuck on this question:
What is the value of c if $\sum_{n=1}^\infty (1 + c)^{-n}$ = 4 and c > 0?
This appears to be an infinite geometric series with a = 1 and r = $(1 + c)^{-1}$, so if I plug this all into the sum of infinite geometric series formula $S = \frac{a}{1 - r}$, then... | HINT:
As $n$ starts with $1$ not $0$
$\displaystyle a=(1+c)^{-1}=\frac1{1+c},$ not $(1+c)^0=1$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/510311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Low Level Well-Ordering Principle Proof Suppose that I have k envelopes, numbered $0,1,...,k−1$, such that envelope i contains $2^i$ dollars. Using the well-ordering principle, prove the following claim.
Claim: for any integer $0 ≤ n < 2^k$, there is a set of envelopes that contain exactly n dollars between them.
How w... | Let S denote the values in $\{0,1,2,…,2k−1\}$ that cannot be expressed with the envelopes. $0$ is not in $S$, because "no envelopes" has zero dollars. $S$ is well-ordered because it is a subset of the naturals; if it is nonempty it has a smallest element $s$. Hence $0,1,2,…,s−1$ can all be expressed with combinations o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $x^2 + xy + y^2 \ge 0$ by contradiction Using the fact that $x^2 + 2xy + y^2 = (x + y)^2 \ge 0$, show that the assumption $x^2 + xy + y^2 < 0$ leads to a contradiction...
So do I start off with...
"Assume that $x^2 + xy + y^2 <0$, then blah blah blah"?
It seems true...because then I go $(x^2 + 2xy + y^2) - (... | By completing square,
$$x^2+xy+y^2 = x^2+2x\frac{y}{2}+\frac{y^2}{4} + \frac{3y^2}{4} = \left(x+\frac{y}{2}\right)^2+\frac{3y^2}{4}\ge 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/510488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 4
} |
Is there a continuous bijection between an interval $[0,1]$ and a square: $[0,1] \times [0,1]$? Is there a continuous bijection from $[0,1]$ onto $[0,1] \times [0,1]$?
That is with $I=[0,1]$ and $S=[0,1] \times [0,1]$, is there a continuous bijection
$$
f: I \to S?
$$
I know there is a continuous bijection $g:C \to I$ ... | No, such a bijection from the unit interval $I$ to the unit square $S$ cannot exist. Since $I$ is compact and $S$ is Hausdorff, a continuous bijection would be a homeomorphism (see here). But in $I$ there are only two non-cut-points, whereas in $S$ each point is a non-cut-point.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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Why isn't removing zero rows an elementary operation? My prof taught us that during Gaussian Elimination, we can perform three elementary operations to transform the matrix:
1) Multiple both sides of a row by a non-zero constant
2) Add or subtract rows
3) Interchanging rows
In addition to those, why isn't removing zero... | Here is a slightly more long-range answer: a matrix corresponds to a linear operator $T:V\to W$ where $V$ and $W$ are vector spaces with some chosen bases. The elementary row operations (or elementary column operations) then correspond to changing the basis of $W$ or of $V$ to give an equivalent matrix: one which repre... | {
"language": "en",
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"source": "stackexchange",
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Cool mathematics I can show to calculus students. I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen.
Every discussion section I am trying to show my students (give them as a series of exercises that we do on the blackboard) some serious math that ... | I myself did not study graph theory yet but I do know that if you consider the adjacency matrix of a graph, then there are interesting things with its eigenvalues and eigenvectors.
You can also show how to solve a system of ordinary differential equations of the form
$$
{x_1}'(t) = a_{11}x_1(t) + a_{12}x_2(t) + a_{13}x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/510749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Error Term for Fourier Series? Suppose I have a piecewise smooth $2 \pi$-periodic function $f$ on $\mathbf{R}$ with a Fourier series $\sum_{n \in \mathbf{Z}}a_n e^{inx}$, a number $x_0 \in \mathbf R$, and $N>0$. I would like an upper bound for
$|f(x_0)-\sum_{n=-N}^N a_n e^{inx_0}|$. For example, if $f$ is a periodic ... | Fourier series have a spirit quite different from Taylor series, as they are nonlocal. They behavior is affected by everything that goes on in the domain of definition. To get an explicit estimate, you can take a proof of pointwise convergence $s_N(f;x)\to f(x)$ and try to make it quantitative. For example, take Th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is "observation"? Often in mathematical writing one encounters texts like ''..we observe that this-and-that..''. Also one may find a review report basically saying ''..the paper is just a chain of observations...''.
What is an ''observation'' and how it differ from ''true results''? Is it possible to turn a nice t... | Typically, one says that something is observed as a synonym for it being obvious or at least not something that they intend to prove. An observation isn't a result per se, but an implied result often left up to the reader.
Papers can contain a lot of observations, as long as the reviewers don't feel that this is being ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Heat Equation identity with dirichlet boundary condition Show an energy identity for the heat equation with convection and Dirichlet boundary condition.
$$u_t -ku_{xx}+Vu_x=0 \qquad 0<x<1, t>0$$
$$u(0,t) = u(1,t)=0 \qquad t>0$$
$$u(x,0) = \phi(x) \qquad 0<x<1$$
Attempt: I think I can apply maximum principle, but dont k... | You can use the separation of variables techniques, but you should pay more attention about choosing the suitable separation parameter.
Let $u(x,t)=X(x)T(t)$ ,
Then $X(x)T'(t)-kX''(x)T(t)+VX'(x)T(t)=0$
$X(x)T'(t)=kX''(x)T(t)-VX'(x)T(t)$
$X(x)T'(t)=(kX''(x)-VX'(x))T(t)$
$\dfrac{T'(t)}{T(t)}=\dfrac{kX''(x)-VX'(x)}{X(x)}=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/511143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How do I prove this transcendental equation has a solution? I am trying to prove that for the following equation, there is a B that solves it (c is a constant):
$1-B = e^{-cB}$
I understand this is a transcendental equation, but how do I prove there is a B that solves it?
I need a non-zero solution. c > 1
| Note that for $c \gt 1$, for $B$ slightly greater than zero $1-B \gt e^{-cB}$ (you can use the derivatives to show this). At $B=1$ we have $1-B \lt e^{-cB}$ and there must be a point where they cross.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/511222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Number theory problem in induction Without using the fundamental theorem of algebra (i.e. the prime factorization
theorem), show directly that every positive integer is uniquely representable as the product
of a non-negative power of two (possibly $2^0=1$) and an odd integer.
| Existence: Every time we take a positive integer and divide it by $2$, it gets smaller. So by the Well-Ordering Principle (equivalently, by the nonexistence of positive integers $n_1 > n_2 > ... n_k > ...$), eventually this process has to stop: thus we've written $x = 2^a y$ with $y$ not divisible by $2$ (which I assu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/511315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Under what conditions is the identity |a-c| = |a-b| + |b-c| true? As the title suggests, I need to find out under what conditions the identity |a-c| = |a-b| + |b-c| is true.
I really have no clue as to where to start it. I know that I must know under what conditions the two sides of a triangle are equal to the remainin... | if $a<c \implies a<b<c$ and if $c<a \implies c<b<a$ with those condition that identify is always true. So for any b in the interval $(a,c)$ or $(c,a)$ the identity $|a-c| = |a-b| + |b-c|$ is true."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/511485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determining if a sequence of functions is a Cauchy sequence? Show that the space $C([a,b])$ equipped with the $L^1$-norm $||\cdot||_1$ defined by $$ ||f||_1 = \int_a^b|f(x)|dx ,$$
is incomplete.
I was given a counter example to disprove the statement:
Let $f_n$ be the sequence of functions:
$$f_n(x) = \begin{cases} 0 ... | No, that is not correct. You need to be able to make $\lVert f_n - f_m \rVert$ arbitrarily small for sufficiently large $n,m$. $(b-a)/2$ is a fixed number. However, you do have the right idea: try find a bound for $\lVert f_n - f_m \rVert$ for $m \geq n$ by bounding the measure of set on which the difference is nonzero... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving $f(C) \setminus f(D) \subseteq f(C \setminus D)$ and disproving equality Let $f: A\longrightarrow B$ be a function.
1)Prove that for any two sets, $C,D\subseteq A$ , we have $f(C) \setminus f(D)\subseteq f(C\setminus D)$.
2)Give an example of a function $f$, and sets $C$,$D$, for which $f(C) \setminus f(D) \ne... |
Here, $f: A \rightarrow B$ is in green and $\{S_i\} = S_i$ for $S = A, B$ and $i = 1, 2.$
Ignore $\{b_2\} = B_2$ in this picture. This picture proves that
$ f(A_1) - f(A_2) \neq f(A_1 - A_2)$. Incidentally, the same picture works for Is this proof correct for : Does $F(A)\cap F(B)\subseteq F(A\cap B) $ for all functio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/511662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
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How to express a vector as a linear combination of others? I have 3 vectors, $(0,3,1,-1), (6,0,5,1), (4,-7,1,3)$, and using Gaussian elimination I found that they are linearly dependent. The next question is to express each vector as a linear combination of the other two. Different resources say just to use Gaussian el... | Let's look at Gaussian elimination:
\begin{align}
\begin{bmatrix}
0 & 6 & 4 \\
3 & 0 & -7 \\
1 & 5 & 1 \\
-1 & 1 & 3
\end{bmatrix}
\xrightarrow{\text{swap row 1 and 3}}{}&
\begin{bmatrix}
1 & 5 & 1 \\
3 & 0 & -7 \\
0 & 6 & 4 \\
-1 & 1 & 3
\end{bmatrix}\\
\xrightarrow{R_2-3R_1}{}&
\begin{bmatrix}
1 & 5 & 1 \\
0 & -15 & ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/511841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that there exist infinitely many primes of the form $6k-1$ This is a question on the text book that i have no way to deal with. Can anyone help me?
Show that there exist infinitely many primes of the form $6k-1$
| Hint: Suppose $p_1, \dots, p_n$ be all the primes of the form $6k-1$, then $N = 6p_1\dots p_n-1$ is also of the form $6k-1$. If $N$ is divisible by a prime, it must be $3$ or of the form $6k+1$ (why?). Show that these can't actually be factors, so $N$ is prime.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/511955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Creating an alternating sequence of positive and negative numbers
TL; DR -> How does one create a series where at an arbitrary $nth$ term, the number will become negative.
I'm learning a lot of mathematics again, primarily because there are such wonderful resources available on the internet to learn. On this journey,... | $a_n=\frac13-\frac23\cos\frac{2n\pi}3-\frac23\cos\frac{4n\pi}3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/512063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $M\oplus M$ is free, is $M$ free? If $M$ is a module over a commutative ring $R$ with $1$, does $M\oplus M$ free, imply $M$ is free? I thought this should be true but I can't remember why, and I haven't managed to come up with a counterexample.
I apologize if this has already been answered elsewhere.
| This would mean that there is no element of order 2 in a $K$-group, which is clearly not correct. To find an example, you can try to find a ring $R$ such that $K_0(R)$ has an element of order 2. As an example related to my research interest, I can say $K_0(C(\mathbb{RP}^2))= \mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}$, se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/512163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
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Calculating a SQRT digit-by-digit? I need to calculate the SQRT of $x$ to $y$ decimal places. I'm dealing with $128$-bit precision, giving a limit of $28$ decimal places. Obviously, if $\,y > 28$, the Babylonian method, which I'm currently using, becomes futile, as it simply doesn't offer $y$ decimal places.
My questi... | This method is perhaps not very practical, and I don't know how to properly typeset (or even explain) this method, but I will show in an example how to compute $\sqrt{130}$ digit by digit. It is very similar to ordinary long division. (If someone has ideas on how to improve the typesetting, please show me!)
Step 1:
In... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 1
} |
How can I find the formula used to produce this number? In a game, each character has different attributes with values to them. The attributes are things like Strength and Speed and are graded on a scale of 1-100.
The game uses a formula to produce an overall number. I want to make a program that can use the formula to... | That particular function could have been discovered by linear regression, although you might have needed a few data points.
In general, reverse engineering a formula from data is a very hard and involved process, requiring lots of experimentation and some amount of experience in interpreting the results. Although somet... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/512420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Finding the value of $ \sum_{n=5}^{204} (n - 2) $ Is there a generalized formula for finding a sum such as this one? I'm going over an old quiz for a programming class but I'm not able to solve it:
$$ \sum_{n=5}^{204} (n - 2) $$
I know this is probably dead simple, but I'm seriously lacking on the math side of computer... | the only thing that you have to kcon is that :
$$
\sum_{k=0}^{n} k = \frac{n(n+1)}{2}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/512482",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Prove if $a\mid b$ and $b\mid a$, then $|a|=|b|$ , $a, b$ are integers. Form the assumption, we can say $b=ak$ ,$k$ integer, $a=bm$, $m$ integer.
Intuitively, this conjecture makes sense. But I can't make further step.
| From what you wrote, $a=akm$, so $a(1-km)=0$. Also $b=bmk$, so $b(1-km)=0$. Thus either $a=b=0$ (and hence $|a|=|b|$), or $mk=1$. The units in $\mathbb Z$ are of course only $+1$ and $-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/512560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Assignment: Find $a$ and $b$ such that a piecewise function is continuous I'm having trouble solving a problem given in an assignment:
If the following function $f(x)$ is continuous for all real numbers $x$, determine the values of $a$ and $b$.
$$
f(x)=\begin{cases}
a\sin(x)+b~~~~~x\le 0\\
x^2+a~~~~~~~~~~0<x\le 1\\
b\... | Notice that regardless of which values we give to the constants $a$ and $b$, the three functions $f_1(x) = a\sin x + b$, $f_2(x) = x^2+a$, and $f_3(x) = b\cos(2\pi x) + a$ are all continuous, and so the only points at which $f(x)$ can be discontinuous are the points $x = 0$ and $x = 1$ (where $f$ changes from being equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/512653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Independence of a random variable $X$ from itself In our lecture on probability, my professor made the comment that "a random variable X is not independent from itself." (Here he was specifically talking about discrete random variables.) I asked him why that was true. (My intuition for two counterexamples are $X \equiv... | The only events that are independent of themselves are those with probability either $0$ or $1$. That follows from the fact that a number is its own square if and only if it's either $0$ or $1$. The only way a random variable $X$ can be independent of itself is if for every measurable set $A$, either $\Pr(X\in A)=1$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/512755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
"answer_count": 2,
"answer_id": 0
} |
Generating Pythagorean Triples S.T. $b = a+1$ I am looking for a method to generate Pythagorean Triples $(a,b,c)$. There are many methods listed on Wikipedia but I have a unique constraint that I can't seem to integrate into any of the listed methods.
I need to generate Pythagorean Triples $(a,b,c)$ such that:
$$a^2 + ... | We give a way to obtain all solutions. It is not closely connected to the listed methods. However, the recurrence we give at the end can be expressed in matrix form, so has a structural connection with some methods in your linked list.
We want $2a^2+2a+1$ to be a perfect square $z^2$. Equivalently, we want $4a^2+4a+2=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/512807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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} |
What is the purpose of implication in this scenario? Consider the case:
Let:
*
*S(x) = “x is a student”
*F(x) = “x is a faculty member”
*A(x, y) = “x has asked y a question”
*Dx and Dy = Consists of all people associated with your school.
Use quantifiers to express this statement:
Some student has never be... | Your attempted solution is wrong on one point, and equivalent on the other.
The placement of the "All y" part is arbitrary between the two options - as $S(x)$ does not depend on $y$, it can be placed on either side of it (for lack of a better explanation).
For the implication, your implication is inaccurate - it implie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/512861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Literary statements that are false as mathematics I recently wanted to use
the title of the famous short story
"Everything that Rises must Converge"
in a poem of mine.
However, the mathematician in me
insisted on changing it to
"Everything that Rises, if the rise is bounded, must Converge".
Are there other literary quo... | The test instructions, “Draw a circle around the correct answer.” really means, “Draw a Jordan curve around the correct answer.” - or, even more accurately, “Draw an approximation of a Jordan curve around the correct answer.”
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/512915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 16,
"answer_id": 8
} |
Need to prove that $(S,\cdot)$ defined by the binary operation $a\cdot b = a+b+ab$ is an abelian group on $S = \Bbb R \setminus \{-1\}$. So basically this proof centers around proving that (S,*) is a group, as it's quite easy to see that it's abelian as both addition and multiplication are commutative. My issue is find... | I believe you meant to write $S=\mathbb{R}\backslash\{-1\}$
$0$ is indeed the identity element since for any $a\in S$, $a * 0=a+0+a.0=a$
For $b$ to be the inverse of $a$, we require $a * b=0$.
Hence $a+b+a.b=0$
$b+a.b=-a$
$b(1+a)=-a$
$b=\frac{-a}{1+a}$
which is fine, since $a$ can't be $-1$ (since it's not an element o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/513045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
How many 4 vertices connective graphs not including a triangle? How many 4 vertices connective graphs not including a triangle?
I am thinking the answer might be 2, for one is a square, another is straight. But I am not sure.
| I assume that you mean simple graphs, i.e., graphs with no loops or multiple edges. Every tree on $4$ vertices satisfies your condition, because a tree has no cycles at all, and you’ve missed the tree with a vertex of degree $3$. If the graph is not a tree, it must contain a cycle, that cycle must be a $4$-cycle, and i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/513133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
One question to know if the number is 1, 2 or 3 I've recently heard a riddle, which looks quite simple, but I can't solve it.
A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "Yes", "No", or "I don't know," and after the girl answers... | I am thinking of a positive integer. Is your number, raised to my number and then increased in $1$, a prime number?
$$1^n+1=2\rightarrow \text{Yes}$$
$$2^n+1=\text{possible fermat number}\rightarrow \text{I don't know}$$
$$3^n+1=2k\rightarrow \text{No}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/513239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "298",
"answer_count": 39,
"answer_id": 1
} |
One question to know if the number is 1, 2 or 3 I've recently heard a riddle, which looks quite simple, but I can't solve it.
A girl thinks of a number which is 1, 2, or 3, and a boy then gets to ask just one question about the number. The girl can only answer "Yes", "No", or "I don't know," and after the girl answers... | Along the lines of "the open problem" method -
Define $$ f(n) = \pi^{n-1}\mathrm{e}^{\pi(n-1)}.$$
Where $n$ is the number chosen by the girl. The question is, is $f(n)$ irrational?
If $n=1$, $f(1) = 1$, so the answer is "No".
If $n=2$, $f(2) = \pi\mathrm{e}^{\pi}$, the answer is "Yes".
If $n=3$, $f(3) = \pi^2\mathrm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/513239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "298",
"answer_count": 39,
"answer_id": 33
} |
Whether polynomials $(t-1)(t-2),(t-2)(t-3),(t-3)(t-4),(t-4)(t-6)$ are linearly independent. Question is to check if :
$(t-1)(t-2),(t-2)(t-3),(t-3)(t-4),(t-4)(t-6)\in \mathbb{R}[t]$ are linearly independent.
Instead of writing linear combination and considering coefficient equations, I would like to say in the followin... | If you assume that the last one is a linear combination of the first three, that is, that we can write
$$
(t-4)(t-6) = a(t-1)(t-2) + b(t-2)(t-3) + c(t-3)(t-4)
$$
and expand the four polynomials, then you get the three equations
$$
\begin{cases}a + b + c = 0 & \text {from } t^2 \\-3a -5b-7c = -10 & \text{from } t \\ 2a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/513300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
differentiation, critical number and graph sketching Consider the graph of the function $$f(x) = x^2-x-12$$
(a) Find the equation of the secant line joining the points $(-2, -6)$ and $(4, 0)$.
(b) Use the Mean Value Theorem to determine a point c in the interval $(-2, 4)$ such
that the tangent line at c is parallel to... | a) Say the point $(x,y)$ is on the secant line between points $(-2,6)$, $(4,0)$. So the equation of the secant line will be
$$\frac{y-(-6)}{x-(-2)}=\frac{0-(-6)}{4-(-2)}\Rightarrow\frac{y+6}{x+2}=1\Rightarrow y=x-4$$
As you see the slope of the secant line is $1$.
b) If the tangent line is parallel to secant line then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/513359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove any function for $L^\infty$ norm can be approximated Prove that any function, continuous on an interval of $\mathbb R$, can be approximated by polynomials, arbitrarily close for the $L^{\infty}$ norm (this is the Bernstein-Weierstrass theorem). Let $f$ be a continuous function on $[0,1]$. The $n$-th Bernstein pol... | Here we start off with a very simple LOTUS (Law of the unconcious statistician) problem. So first we start off with:
$$\displaystyle S_n(x)=\frac{B^{(n,x)}}{n}$$
Here $\displaystyle B^{(n,x)}$ is a binomial random variable with parameters $n$ and $x$. We must prove that $\displaystyle B_n(x)=\mathbb E(f(S_n(x)))$, or:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/513558",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Expectation values of male-male, male-female, female-female pairs N people are seated at random to form a circle, among whom N1 are male. What are the expected numbers of male-male, male-female, and female-female nearest-neighbor pairs?
| Label the chairs $1$ to $N$, counterclockwise, and assume $N\gt 1$. Let $X_i=1$ if the person in Chair $i$ is male and his neighbour in the counterclockwise direction is male. Let $X_i=0$ otherwise. Then the number $X$ of male-male nearest neighbour (unordered) pairs is given by
$$X=X_1+X_2+\cdots+X_N.$$
The $E(X_i)$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/513720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can the limit of a product exist if neither of its factors exist? Show an example where neither $\lim\limits_{x\to c} f(x)$ or $\lim\limits_{x\to c} g(x)$ exists but $\lim\limits_{x\to c} f(x)g(x)$ exists.
Sorry if this seems elementary, I have just started my degree...
Thanks in advance.
| If you accept divergence as well, $f(x)=g(x)=1/x$ has no limit at $0$, but $1/x^2$ diverges to infinity as $x$ goes to $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/513822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 6,
"answer_id": 4
} |
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