Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Verify the triple angle formula Verify the triple angle formula
$$\tan(3x) = \frac{3 \tan(x) − \tan^3(x)}{1 − 3 \tan^2(x)}$$
I have tried simplifying the right side by the following
$$\tan(3x) = \frac{\tan(x)(3 − \tan^2(x))}{1 − 3 \tan^2(x)}$$
but then I am getting stuck trying to verify the equation
| $$\tan 3x =\tan(x+2x)= \frac{\tan x + \tan 2x}{1 − \tan x \tan 2x}=$$
$$=\frac{\tan x + \frac{2\tan x}{1-\tan^2x}}{1 − \tan x \frac{2\tan x}{1-\tan^2x}}=\frac{3\tan x- \tan ^3x}{1-3\tan^2x}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Limits at infinity by rationalizing I am trying to evaluate this limit for an assignment.
$$\lim_{x \to \infty} \sqrt{x^2-6x +1}-x$$
I have tried to rationalize the function:
$$=\lim_{x \to \infty} \frac{(\sqrt{x^2-6x +1}-x)(\sqrt{x^2-6x +1}+x)}{\sqrt{x^2-6x +1}+x}$$
$$=\lim_{x \to \infty} \frac{-6x+1}{\sqrt{x^2-6x +1... | It leads to
$$=\lim_{x \to \infty} \frac{-6+(\frac{1}{x})}{\sqrt{1-(\frac{6}{x})+(\frac{1}{x^2})}+1}$$
And so the limit is $-3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1960911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Prove that $max_{J\subset \{ 1,2...n \} } \left| \sum_{j \in J} z_j\right| = max_{t\in[0,2\pi]} \sum_{j=1}^{n} Re^{+}(e^{it}z_j).$ Consider a finite subset of $\mathbb{C}$ $A=\{z_1,z_2,...,z_n\}$ and the function $Re^{+}(z) = Re (z) $ if $ Re(z)>0$ and $0$ if $Re(z)\leq 0$.
I have to prove that:
$max_{J\subset \{ 1,2.... | One direction: $\max_{J\subset\{1,\dots,n\}} |\sum_{j\in J} z_j|\le\max_{t\in[0,2\pi]} \sum_{j=1}^n \Re^+(e^{it}z_j)$:
Choose $J$ so that the max on the LHS is attained. Let $t=-\arg\sum_{j\in J} z_j$.
Then $|\sum_{j\in J} z_j|=\sum_{j\in J} \Re(e^{it}z_j)$. I claim that for each $k\in J$, $\Re(e^{it}z_k)\ge0$ and for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1961159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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10 dice are rolled. What is the probability of getting 6 dice that are even numbers and the other four dice are 3's? Ten dice are rolled. What is the probability of getting six dice that are even numbers and the other four dice are $3$'s?
My approach is that you have $6^{10}$ number of total outcomes, and
*
*$\Pr(\... | Thank you Arthur for correcting me, I got a bit carried away. It's night-time here where I'm at, after all.
Out of the 10 dice you roll, 4 need to be 3's (the other 6 will be even and uniquely determined, then), but there are many ways in which this condition can be satisfied. For example, you can roll $(3, 3, 3, 3, 2,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1961279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Induction Proof: $(1+x)^n = 1+x^n$ for even $n$ in $\mathbb{F}_2[x]$ I'm trying to work on a proof by induction. The statement is:
Let n be even. Then, $(x+1)^n = x^n + 1$ for $n\in\mathbb{N}\cup{0}$ and $(x^n+1) \in \mathbb{F}_2[x]$
Base case: $n=0$ and $n=2$ both satisfy the condition, fairly trivially. I include t... | $(1+x)^6=1+6x+15x^2+20x^3+15x^4+6x^5+x^6$ in $\mathbb{Z}[x]$, hence
$$ (1+x)^6=1+x^2+x^4+x^6 $$
in $\mathbb{F}_2[x]$. So the claim isn't true. It is however the case that $(1+x)^n=1+x^n$ in $\mathbb{F}_2[x]$ if $n$ is a power of $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1961436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Proving $ \overline{E \cup F} = \overline {E} \cup \overline{F}$ and $ \overline{E \cap F} \subset \overline {E} \cap \overline{F}$ Let $E, F \subset X$, prove that $ \overline{E \cup F} = \overline {E} \cup \overline{F}$.
For further clarification: I'm referring to $\overline{E}$ as E closure. E' would be the limit p... | To prove that $\overline{E \cap F} \subset \overline{E} \cap \overline{F}$:
Let $x \in \overline{E \cap F}$. There exists $(x_n)_n \subset E \cap F$ such that $x_n \underset{n \rightarrow \infty}{\rightarrow} x$.
$(x_n)_n \subset E \cap F$ means that $(x_n)_n \subset E$ and also that $(x_n)_n \subset F$. Can you finish... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1961543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Find the limit of $\lim_{n\to\infty}{a^n/n!}$ $$\lim_{n\to\infty}\frac{a^n}{n!}$$
$a>0, n\in N$
If possible, a solution through the squeeze theorem. Not sure how to solve it.
| We know $$\mathrm{e}^x = \sum_{n=0}^{\infty} \frac{ x^n }{n!} $$
Has radius of convergence $R=\infty$. Thus, it better converge for $x=a$. It follows then that
$$ \lim_{n \to \infty} \frac{ a^n }{n!} = 0 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1961648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The need for the Gram–Schmidt process As far as I understood Gram–Schmidt orthogonalization starts with a set of linearly independent vectors and produces a set of mutually orthonormal vectors that spans the same space that starting vectors did.
I have no problem understanding the algorithm, but here is a thing I fail ... | You can also get into function spaces where it's not clear what the basis you can just grab from is. The Legendre polynomials can be constructed by starting with the functions $1$ and $x$ on the interval $x \in [-1,1]$, and using Gram-Schmidt orthogonalization to construct the higher order ones. The second order polyno... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1961727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 6,
"answer_id": 0
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Does the two-element set have a categorical description in the category of (finite) sets? So more or less what I ask in the title: is it possible to identify (uniquely up to bijection) the two-element set in the category of sets as an object that has a particular (categorical) property?
EDIT: Following Hanno's answer, ... | The two-element set, call it $\Omega$, represents the subset-functor: For any set $X$, there is a natural bijection $$(\ddagger):\qquad\text{Hom}_{\textsf{Set}}(X,\Omega)\ \ \cong\ \ \text{Subset}(X).$$
Because of this, it is called a Subobject Classifier. By the Yoneda-Lemma, an object is determined up to unique isomo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1961836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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How do I find the sum of a sequence whose common difference is in Arithmetic Progression? How do I find the sum of a sequence whose common difference is in Arithmetic Progression ?
Like in the following series :-
$1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91$
And also how to find it's $n^{th}$ term ??
| I enjoyed several of the solutions here. However the idea of substituting values and solving for a, b, c seems really long and inefficient.
Below I have an efficient method to do this.
As codetalker pointed out:
a(n+1)-a(n)= some linear term=1+n(for example)
So now what you do is make a telescopic series:
a(2)-a(1)=1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1961952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 3
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The tea bag problem: probability of extracting a single bag of tea Suppose you have a bunch of tea bags in a box, initially in pairs, like these:
Let us suppose the box initially contains only joined pairs of tea bags, say $N_0$ of them (thus making for a total of $2N_0$ tea bags).
Every time you want to make yourself... | (This is a condensed version of @lulu 's answer.)
If in drawing${}_{k+1}\>$ I pick a certain bag $b$ then any other bag, in particular the partner of $b$, is among the $k$ previously drawn bags with probability
$${k\over 2N-1}\ .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1962069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
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If $p(x)$ is a cubical polynomial with $p(1)=3,p(0)=2,p(-1)=4$, then what is $\int_{-1}^{1} p(x)dx$? Q.If $p(x)$ is a cubical polynomial with $p(1)=3,p(0)=2,p(-1)=4$,Then $\int_{-1}^{1} p(x)dx$=__?
My attempt:
Let $p(x)$ be $ax^3+bx^2+cx+d$
$p(0)=d=2$
$p(1)=a+b+c+d=3$
$p(-1)=-a+b-c+d=4$
From them,we get $b=3.5$,$d=2$... | This method will work.
However easier is to apply Simpson's rule for approximating an integral (https://en.wikipedia.org/wiki/Simpson%27s_rule). You have been given all the required information from the polynomial and fortunately Simpson's rule is exact for polynomials up to degree 3!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1962163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Rigidity of surface groups I am interested in the following result:
Theorem: A torsion-free group which contains the fundamental group of a closed surface as a finite index must be the fundamental group of a closed surface.
This is a consequence of a harder theorem stating that PD²-groups are surface groups; see Eck... | You can also use Tukia's theorem, from the 1980's, which shows that a uniform convergence subgroup of the homeomorphism group of the circle is either conjugate to a Fuchsian group or is one of an explicitly described class of groups which are not torsion free. Your group does indeed have a uniform convergence action on... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1962340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Explain solution of this system of non-linear congruence equations So I have a system of non-linear congruence equations (i may be wrong in terms):
\begin{cases} x^3 \equiv 21\ (\textrm{mod}\ 23) \\ x^5 \equiv 17\ (\textrm{mod}\ 23) \end{cases}
Somewhere I've read that to solve this system one should:
*
*Find the so... | Since neither $3$ or $5$ are divisors of $\varphi(23)=22$, both the maps
$$ f:x\mapsto x^3,\qquad g:x\mapsto x^5 $$
are bijective on $\mathbb{F}_{23}$. In particular, since $3^{-1}\equiv 15\pmod{22}$ and $5^{-1}\equiv 9\pmod{22}$,
$$ f^{-1}:x\mapsto x^{15},\qquad g^{-1}:x\mapsto x^9 $$
and $x^3\equiv 21\pmod{23}$ is eq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1962444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Verifying compatibility of symplectic and metric structure of $\mathbb{R}^{2n}$ I was reading about on wikipedia under the Hermitian Manifold page that for a almost-complex structure on a manifold $M$ that we have the following:
$$\omega(\cdot,\cdot) = g(J\cdot,\cdot).$$
I am having trouble with the explicit calculatio... | Using $z_j = x_j + \sqrt{-1} y_j$, the standard symplectic structure on $\mathbb C^n$ is
$$ \omega = \sum_{i=1}^n dx_i \wedge dy_i.$$
Writing $(x, y) = \sum_{i=1}^nx^i e_i + y^i f_i$, $J$ is given by
$$ J e_i = f_i,\ \ \ Jf _j = - e_j.$$
Then by checking directly,
$$\begin{cases} \omega(e_i, e_j) = g(Je_i, e_j) = 0,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1962551",
"timestamp": "2023-03-29T00:00:00",
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Locally compact metric space I'm trying to prove that a metric space is locally compact iff every closed ball is compact, using the more general definition that applies to Hausdorff spaces, that every point has a compact neighbourhood.
So call $X$ my space. The only non trivial thing to prove is that every closed ball ... | If $(X,d)$ is a metric space, then $d'(x,y) = d(x,y)/(1+d(x,y))$ defines a new metric having the same (open or closed) balls as $(X,d)$. But then $X$ is the ball of radius $1$ centered anywhere. Hence if, in addition, $X$ is locally compact but not compact, the metric space $(X,d')$ admits a closed non compact ball.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1962640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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The probability of probability I am trapped by a logic premise. We know coins have no memory and each throw is a independent event. Thus, after 20 faces, it should not really matter what side you bet.
But there is also a normal distribution, and the probability of getting far to the mean. Isn´t there some sort of stati... | Dice and coins indeed have no memory; in fact, if I show you a coin, you even don't know how many tosses before there was a "face". Is the coin smarter than you?
Note that if you throw a die 5 times, then the outcome $(4,2,5,1,6)$ has exactly the same probability as $(6,6,6,6,6)$. The normal distribution can be derived... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1962753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Compute the gradient of mean square error Let $Y = \begin{pmatrix} y_1 \\ \cdots \\ y_N\end{pmatrix}$ and $X = \begin{pmatrix} x_{11} & \cdots & x_{1D} \\ \cdots & \cdots & \cdots \\ x_{N1} & \cdots &x_{ND}\end{pmatrix}$. Let also $e = y - Xw$ and let's write the mean square error as $L(w) = \frac{1}{2N} \sum_{i=1}^{... | Since
$$
L(w) = \frac{1}{2N}\sum_{n=1}^N(y_n - (Xw)_n)^2
$$
it follows that
$$
\frac{\partial L}{\partial w_j} = -\frac{1}{N}\sum_{n=1}^N x_{nj}(y_n - (Xw)_n) = -\frac{1}{N}x_j^Te,
$$
where $x_j$ is the $j$th column of $X$. Therefore,
$$
\nabla L(w) = -\frac{1}{N}X^Te
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1962877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Existence of a local transverse embedded submanifold for a flow Let $M$ be a smooth manifold and $\Phi$ the flow of a non-vanishing vector field. There always exists around every point $x$ at least locally an embedded submanifold $S_x$ that is transversal to $\Phi$ at $x$, right? How can one show this? I thought that s... | You can suppose that $x$ is in the domain of a chart $U$ that you identify with an open subset of $R^n$ and $x=0$. Suppose that $X$ is the vector field which does not vanish, $X(x)$ is a vector of $R^n$, consider an hyperplane (which contains the origin) $H$ defined by $\alpha(u)=0$ which does not contain $X(x)$. The f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1963029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of the two variable function $f(x,y).$ How to show that $$\lim_{(x,y)\to(0,0)}\frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}(x^{4}+y^{2})}=0$$ I tried with different paths as $x=0,y=0, y=x$ its comes to zero but i have no general idea. Please help. Thanks to lot.
| You have
\begin{align}
\left| \frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}(x^{4}+y^{2})}\right|
&=\frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}(x^{4}+y^{2})}
\leq
\frac{x^{2}y^{2}}{\sqrt{x^{2}+y^{2}}\,(y^{2})}\\ \ \\
&=\frac{x^2}{\sqrt{x^2+y^2}}\leq\frac{x^2}{\sqrt{x^2}}=|x|\\ \ \\
&\leq\sqrt{x^2+y^2}
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1963119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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whether this set is closed subset of a Hilbert space Whether the set of sequences $(x_n \in l_2: \sum_{n=1}^{\infty}\frac{x_n}{n}=1 )$ is closed ? How do you find the limit point of a set of sequences? Moreover what is the complement of this set?, and whether that is open? (IIT-GATE 2015)
| We have $f=(1,{1 \over 2},..., {1 \over n},...) \in l_2$, hence
$f^*(x)= \langle f,x\rangle$ is a continuous linear functional, hence the
inverse image of a closed set is closed.
The complement is just the set $\{x | f^*(x) \neq 1 \}$ which is open
because its complement is closed (or, indeed, because $\mathbb{R} \setm... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1963254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Hints on the right-hand side of a combinatorial proof question $\sum_{k=0}^{n} {n \choose k} (n-k)^{n+1}(-1)^{k} = \frac {n(n+1)!} {2}$
So the left-hand side looks so much like inclusion-exclusion principle. The sign changes between - and + depending on whether k is even or odd (due to $(-1)^{k}$) It's like we are sub... | Consider sequences of length $n+1$ using letters from an alphabet of size $n$.
Both sides count the number of such sequences in which one letter appears twice, and all other letters appear exactly once.
Right-hand side:
There are $n$ ways to choose the letter that appears twice. There are $(n+1)!$ ways to order the $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1963324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solutions of $\sin2x-\sin x>0$ with $x\in[0,2\pi]$ What are the solutions of this equation with $x\in[0,2\pi]$?
$$\sin2x-\sin x>0$$
I took this to
$$(\sin x)(2\cos x-1)>0$$
Now I need both terms to be the same sign. Can you please help me solve this?
| When $\sin x>0$, i.e. $x\in(0,\pi)$, the inequation reduces to $\cos x>1/2$, i.e. $x\in(0,\pi/3)$.
When $\sin x<0$, i.e. $x\in(\pi,2\pi)$, the inequation reduces to $\cos x<1/2$, i.e. $x\in(\pi,2\pi-\pi/3)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1963495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Ways of arranging of different nationality persons at a round table
$2$ American, $2$ British, $2$ Chinese, $1$ Dutch, $1$ Egyptian, $1$ French and $1$ German people are to be seated for a round table conference, Then
$(a)$ Then number of ways in which no two persons of same nationality are seated together
$(b)$ The n... | The first problem is an inclusion-exclusion problem. Your $9!-2\cdot8!=357,120$ is the number of arrangements that have at least one of the pairs separated, not the number that have all three pairs separated.
The number of ways in which the two Americans sit together is $2\cdot 8!$: we treat them as a single individual... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1963554",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the goal of harmonic analysis? I am taking a basic course in harmonic analysis right now. Going in, I thought it was about generalizing Fourier transform / series: finding an alternative representation of some function where something works out nicer than it did before.
Now, having taken the first few weeks of ... | Ultimately it helps one to prove theorems (like existence and uniqueness) of partial differential equations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1963688",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 4,
"answer_id": 2
} |
Proof for Riemann's isolated singularity. Let $f$ be complex function has isolated singularity at $z_0$.
Suppose $f$ is bounded on some deleted neighborhood of $z_0.$
Then $f$ is holomorphic and bounded on some deleted neighborhood of $z_0.$
Let $h(z)= \begin{cases} (z-z_0)^2f(z)\mbox{, if z is not z_0 }\\
0 \mbox{, i... | Your argument is correct, but your presentation of it may be considered incomplete. That depends on what you can use without mentioning it.
After you've shown that $h$ is complex differentiable at $z_0$ with $h(z_0) = h'(z_0) = 0$, you assert that $h$ is analytic at $z_0$. That's true, but it doesn't follow from the di... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving $x+x^3=5$ without using the cubic equation. In lessons, I get quite bored and recently throughout these lessons I have been trying to solve for x in:
$$x+x^3=5$$
I've figured out how to do it for squares using the quadratic equation, but the cubic equation looks so dauntingly massive it actually makes my bladde... | Sometimes it is difficult to find roots in closed form, so this answer is in the spirit of numerical values for the roots:
$$x^3+x=5$$
$$x^2+1=\frac5x$$
$$x^2=\frac5x-1=\frac{5-x}x$$
$$x=\sqrt{\frac{5-x}x}$$
Once you solve for $x$, you can employ fixed-point iteration. First, see the root is near $x=1.5$
$$x_0=1.5$$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1964176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
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The Nature of Differentials and Infinitesimals I have been wondering for some time what the limits of Leibniz notation is, and what exactly its meaning is. I learned limits and later learned (to some extent) infinitesimals, but there are some oddities which have me befuzzled. The one person I know who could answer th... | The algebraic definition of the second derivative of $y=f(x)$ is
$$
f''(x)=\frac{d\left[\frac{dy}{dx}\right]}{dx}
$$
This can be expanded using the quotient rule
$$
f''(x)=\frac{d^2y}{dx^2}-\frac{dy\ d^2x}{dx^3}
$$
Furthermore, if you desire to evaluate $d^2y/d^2x$ this can be found by taking algebraic differentials
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1964301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
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Riemann (darboux?) integrating $f: [2,3] \to \mathbb{R} \quad f(x)=\frac{1}{x^2}$? I have function $$ f(x) = \frac{1}{x^2} $$
I want to riemann (I don't know whether what I mean is actually Riemann or Darboux integration) integrate it on the interval $$ x \in \left[2,3\right] $$
What I could do is partition the interva... | Let $I \subset\mathbb{R}$ be a closed interval and $f:I\to\mathbb{R}$ be a bounded function. Let
\begin{eqnarray}
\mathrm{L}f := \sup_{P \text{ is a partition of }I}L_{f,P}\\
\mathrm{U}f := \inf_{P \text{ is a partition of }I}U_{f,P}.
\end{eqnarray}
and $|P| > 0$ be the maximum length among the subinterevals in $P$, wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1964422",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that $u, v, w$ are in the span of $\{u+v, 2u+3v, 4v+6w\}?$ I know this has to do with linear combinations, namely that you would set out to solve the following set of equations to show that $c_{1}, c_{2}$, and $c_{3}$ exist and are not all 0, but I'm unclear as to how I actually solve for those in this case.
Tha... | HINT: What is the rank of the following matrix?
$$
\begin{pmatrix}
1 & 2 & 0 \\
1 & 3 & 4 \\
0 & 0 & 6
\end{pmatrix}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1964527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Range perpendicular to Nullspace I'm stuck in this Linear Algebra problem:
Let $A\in M_n(\mathbb{C})$ with $\mathrm{rank}(A)=k$. Prove that the following are equivalent:
a) $R(A) \bot N(A)$
b) $N(A)=N(A^*)$
c) $R(A)=R(A^*)$
for a) implies b) I should prove double contention, i.e. $N(A)\subset N(A^*)$ and $N(A^*)\subs... | If $A^*y = u \neq 0$ then $(y,Au)=(A^*y,u)>0$, where $(.,.)$ is the inner product. However $Au \in R(A)$ and $y \in N(A)$ by assumption so it must be the case that $(y,Au)=0$, a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1964635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How do I find the length between two circles that have the same tangent line?
P and N are the center of the two circles with radii 50 units and 5 units respectively. TS is the common tangent to the circles at point Z and R. TNP is a straight line and the distance between P and N is 170 units.
Find the length of ZR.
I'... | $$\Delta TNR\sim\Delta TPZ$$
Hence,
$$\frac{NT}{NR}=\frac{NT+PN}{PZ}$$
$$\frac{NT}{5}=\frac{NT+170}{50}$$
$$50NT=5NT+850$$
$$45NT=850$$
$$NT=\frac{850}{45}=\frac{170}{9}$$
$$PT=PN+NT=170+\frac{170}{9}=\frac{1700}{9}$$
Now, use the Pythagoras theorem on $\Delta TPZ$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1964930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Does the sum $\sum_{n \geq 1} \frac{2^n\operatorname{mod} n}{n^2}$ converge? I am somewhat a noob, and I don't recall my math preparation from college. I know that the sum $\displaystyle \sum_{n\geq 1}\frac{1}{n}$ is divergent and my question is if the sum$$\sum \limits _{n\geq 1}\frac{2^n\mod n}{n^2}$$converges. I thi... | In this answer, we prove that
$$ \sum_{n=1}^{\infty} \frac{2^n \text{ mod } n}{n^2} = \infty. \tag{*} $$
Idea. The intuition on $\text{(*)}$ comes from the belief that the sequence $(2^n \text{ mod } n)/n$ is equidistributed on $[0, 1]$, which is quite well supported by numerical computation.
Proving this seems quite ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
"answer_count": 3,
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$\displaystyle \log_a(3)=q$ & $\displaystyle \log_a(2)=p$ Express $log_a 72$ in terms of p & q $\displaystyle \log_a(3)=q$ & $\displaystyle \log_a(2)=p$ Express $log_a 72$ in terms of p & q
Currently I have tried nothing as I cannot even figure out where to begin a demonstration kindly will help much
Many thanks :)
| $ \log_a72 = \log_a(2^3 \cdot 3^2)\\
=\log_a (2^3) + \log_a (3^2)\\
= 3 \log_a2 + 2\log_a3\\
= 3p + 2q$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$A$, $B$ and $C$ can do a ... $A$, $B$ and $C$ can do a piece of work in $30$, $40$ and $50$ days respectively. If $A$ and $B$ work in alternate days started by $A$ and they get the assistance of $C$ all the days, find in how many days the whole work will be finished?
My Attempt:
In $30$ days, $A$ does $1$ work.
In $1$... | A takes-> 30 days, B takes-> 40 days, C takes-> 50 days
Total unit of work they have to finish is(LCM of 30,40,50) i.e 600 units
It means
A does -> 20 unit of work in 1 day, B does -> 15 unit in 1 day and C does 12 units in 1 day.
On 1st day A+C does 32 units out of 600, On 2nd day B+C does next 27 units of remaining ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Show that $2^{n} \geq (n +2)^{2}$ for all $n \geq 6$ Edit: If it is hard to read what I have written the essence of my question is: How come that $2 \times 2^{k} - (k+3)^{2} \geq 2^{k}$ from the assumption that $2^{k} \geq (k+2)^{2}$?
Show that $2^{n} \geq (n +2)^{2}$ for all $n \geq 6$
I have excluded steps:
Assumpt... | Base Case:
$$2^{6} =64\geq (6 +2)^{2}=64$$
Inductive Step:
Assume true for some $k\geq 6$
$$2^{k} \geq (k +2)^{2}$$
Now show true for $n=k+1$ from assumed truth of $n=k$ case.
$$2^{k+1} \geq ((k+1) +2)^{2}=k^2+6k+9$$
so
\begin{align*}
2^{k+1}&=2^k\cdot 2 \geq 2(k+2)^{2}\\
&=2k^2+8k+8\geq ((k+1) +2)^{2}
\end{align*}
for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965308",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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$4x^2y′′-8x^2y′+(4x^2+1)y=0$ solve by Frobenius Method. I would like to ask if someone can explain to me how can we solve the following DE using this method.
$4x^2y′′-8x^2y′+(4x^2+1)y=0$
| Given differential equation $4x^2y^{\prime\prime}-8x^2y^{\prime}+(4x^2+1)y=0$. $P(x)=-2$
and $Q(x)=\frac{1+4x^2}{4x^2}$, since $Q(x)$ is not analytic at $x=0$, we say $x=0$ is a
singular point of this differential equation . Singular points are futher classified into regular singular
and irregular singular points. S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Haven't learned calculus yet but I need this for a proof for Fibonacci numbers and its limit In an infinite series of Fibonacci numbers, is this always true $$\frac{F_{n}}{F_{n+1}}>\frac{F_{n-1}}{F_n}$$? Can you make an argument that in an infinite convergent series, eventually that will be false?
| hint :We know **$\color{red} {F_{n+1}=F_n+F_{n-1}}\\ \to \color{red} {F_{n}=F_{n+1}-F_{n-1}}$
$$\frac{F_{n+1}-F_{n-1}}{F_{n+1}}>\frac{F_{n-1}}{F_{n+1}-F_{n-1}} $$
or see this $$\begin{bmatrix}f_{n+1} & f_n \\f_n & f_{n-1} \end{bmatrix}=\begin{bmatrix}1 & 1 \\1 & 0 \end{bmatrix}^n \to \det(\begin{bmatrix}f_{n+1} & f_n \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Does $\sum_{k=0}^\infty e^{-\sqrt{k}} (-1)^k$ converge faster than $e^{-\sqrt{k}}$? Does $\sum_{k=0}^\infty e^{-\sqrt{k}} (-1)^k$ converge faster than $e^{-\sqrt{k}}$? In particular, is
$$
\lim_{N\to\infty} e^{\sqrt{N}} \sum_{k=N}^\infty e^{-\sqrt{k}} (-1)^k=0?
$$
I know that when the $\sqrt{k}$ and $\sqrt{N}$ are repl... | $$e^{\sqrt{N}}\sum_{k\geq N}e^{-\sqrt{k}}(-1)^k = (-1)^N \sum_{h\geq 0}(-1)^h \exp\left(-\frac{h}{\sqrt{N+h}+\sqrt{N}}\right) $$
is well-approximated by
$$ (-1)^N\sum_{h\geq 0}(-1)^h \exp\left(-\frac{h}{2\sqrt{N}}\right) = \frac{(-1)^N}{1+\exp\left(-\frac{1}{2\sqrt{N}}\right)}$$
hence the given limit does not exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof of Uniqueness of Two Lines I am trying to show a proof of the uniqueness of two equations. If $a,b,c,d\in\mathbb R$ and $ad-bc\neq 0$, then or any $\alpha,\beta \in \mathbb R$ the pair of equations:
$$ax + by = \alpha\\cx + dy = \beta$$
have a unique solution where $x = x_0$ and $y=y_0$ that depends on $a,b,c,d,... | It is more simple and more correct to prove the unicity as a consequence of the unicity of the solution of the equations:
$$(cb-ad)y_0=(c\alpha-a\beta)$$
and
$$\left(da-bc\right)x_0=\left(d\alpha-b\beta\right)$$
Solving this equation you cannot say something as
''Isolate $y_0$ by divining by the coefficients:''
But ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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When should I search for the covariance matrix instead of the variance? Suppose I have a random variable $X$ and $n$ realizations of this variable: $x_1, ..., x_n$. It seems clear to me in that case that if I am interested in knowing the variability I have in my data (realizations) then I should calculate the variance ... | Let $X$ be univariate random variable, say height of an individual. Then
$x_{1},x_{2}\cdots,x_{n}$ be a realization on the variable height, and variance is what we compute.
In multivariate analysis, the components of the random vector $X$ are different variables. In this case we will e computing to know how the diffe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1965871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Dimension of Kernel of Composition of Linear Transformations Take two linear transformations $T \colon U \to V$ and $S \colon V \to W$ where $U$, $V$ and $W$ are finite.
I want to show that
$$
\dim \ker (S \circ T)
\leq \dim \ker S + \dim \ker T.
$$
Attempt: I've been using the dimensional theorem because of ... | We do it like this:
Lemma $1$: $T[\ker (S \circ T)] = \ker S \cap {\rm Im}\,T$.
Proof: you do it, okay?
Lemma $2$: If $F\colon V_1 \to V_2$ is linear and $Z \subseteq V_2$ is a subspace, then $F^{-1}[Z]$ is a subspace of $V_1$ and $\dim F^{-1}[Z] \leq \dim \ker F + \dim Z$.
Proof: I'll check only the formula. Applying... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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What is $\lim\limits_{n\to\infty}\frac {1}{n^2}\sum\limits_{k=0}^{n}\ln\binom{n}{k} $? It was originally asked on another website but nobody has been able to prove the numerical result. The attempts usually go by Stirling's approximation or try to use the Silverman-Toeplitz theorem.
| By Stolz Cezaro
$$\lim\limits_{n\to\infty}\frac {1}{n^2}\sum\limits_{k=0}^{n}\ln\binom{n}{k}=\lim\limits_{n\to\infty}\frac {1}{2n+1} \left(\sum\limits_{k=0}^{n+1}\ln\binom{n+1}{k}- \sum\limits_{k=0}^{n}\ln\binom{n}{k} \right)\\
=\lim\limits_{n\to\infty}\frac {1}{2n+1} \sum\limits_{k=0}^{n}\left(\ln\binom{n+1}{k}- \ln\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1966057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
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Seeking an intuitive explanation of the Mapping Class Group For a surface $S$ the mapping class group $MCG(S)$ of $S$ is defined as the group of isotopy classes of orientation preserving diffeomorphisms of $S$:
$$MCG(S)=Diff^+(S)/Diff_0(S).$$
I understand this definition as well as all of its component pieces. What I d... | If you view an isotopy as a path in the space of diffeomorphisms, each element of the mapping class group corresponds to a path component of the orientation-preserving diffeomorphism group.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1966182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
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Prove $\operatorname{GL}_n(\Bbb{R})/\operatorname{SL}_n(\Bbb{R}) \cong \Bbb{R}^\times$ $$\operatorname{GL}_n(\Bbb{R})/\operatorname{SL}_n(\Bbb{R}) \cong \Bbb{R}^\times$$
This is trivial to prove with the first isomorphism theorem - by using the determinant as a endomorphism, then as $\operatorname{SL}_n(\Bbb{R})$ is $1... | Hint: consider the homomorphism
$$\mathbf{R}^\times \ni \lambda \mapsto (\lambda I_n) SL_n \in GL_n/SL_n$$
where $I_n$ is the $n\times n$ identity matrix and $gSL_n$ is the coset of $g$ in the quotient group.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1966320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Multiplying two logarithms I've searched for some answer already, but couldn't find any solution to this problem. Apparently, there's no rule for the product of two logarithms. How would I then find the exact solution of this problem?
$$
\log(x) = \log(100x) \, \log(2)
$$
| The question does not specify the base $B$ of the logarithm, but it will affect the solution, so we make it explicit:
\begin{align}
\log_B(x)
&= \log_B(100\, x) \, \log_B(2) \\
&= (\log_B(100) + \log_B(x)) \, \log_B(2) \iff \\
(1 - \log_B(2)) \log_B(x)
&= \log_B(2) \log_B(100) \\
\end{align}
For $B = 2$ the LHS vani... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1966921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Diophantine equation $\frac{a+b}{c}+\frac{b+c}{a}+\frac{a+c}{b} = n$ Let $a,b,c$ and $n$ be natural numbers and $\gcd(a,b,c)=\gcd(\gcd(a,b),c)=1$.
Does it possible to find all tuples $(a,b,c,n)$ such that:
$$\frac{a+b}{c}+\frac{b+c}{a}+\frac{a+c}{b} = n?$$
| First note that we can assume $a,b,c \in \mathbb{Q}$, without loss of generality. A rational solution can then be scaled to integers.
Combining into a single fraction gives the quadratic in $a$
\begin{equation*}
a^2+\frac{b^2-nbc+c^2}{b+c}a+bc=0
\end{equation*}
and, for rational solutions, the discriminant must be a ra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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poker probability a pack of poker contains 52 cards and we are going to flip through one by one so the probability of the following events are
*
*a king right after the first ace ?
*an ace right after the first ace ?
*the first ace is the 10-th card?
*the probability the next card is the ace of spades, if the fi... | 1) King follows First Ace.
Let's not worry about suits or values of the other cards. Take a pack of four king of hearts, four ace of spades, and forty four jokers.
There are $52!/(4!4!44!)$ equally probable ways to arrange this deck.
Set aside one king, shuffle the remaining cards, stick that king after the first ace.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Probability that a number is divisible by 11 The digits $1, 2, \cdots, 9$ are written in random order to form a nine digit number. Then, the probability that the number is divisible by $11$ is $\ldots$
I know the condition for divisibility by $11$ but I couldn't guess how to apply it here.
Please help me in this regard... | The rule of divisibility by $11$ is as follows:
The difference between the sum of digits at odd places and the sum of the digits at even places should be $0$ or a multiple of $11$.
We also know that the sum of all the digits will be $45$ as $1 + 2 + ... + 9 = 45$.
Let $x$ denote sum of digits at even position s and $y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 5,
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$\tau = \left(\sum_{n = 1}^\infty f_n\right) d\nu + \sum_{n = 1}^\infty \mu_n$ the Lebesgue decomposition of $\tau$? Assume $\tau_n$ is a sequence of positive measures on a measurable space $(X, \mathcal{F})$ with $\sup_n \tau_n(X) < \infty$ and $\nu$ is another finite positive measure on $(X, \mathcal{F})$. Suppose $\... | Yes.
*
*The composition.
$$\tau (A) = \sum \tau_n (A) = \sum (\int_A f _n d \nu + \mu_n(A)).$$
By monotone convergence $\sum \int_A f_n d \nu = \int_A \sum f_n d\nu$. Let $f=\sum f_n$. Then $f$ is measurable and nonnegative. Also since $\tau$ is finite, $f\in L^1(\nu)$. Define the set function $\mu= \sum \mu_n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967455",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Three vertices of a parallelogram have coordinates (-2,2),(1,6) and (4,3). Find all possible coordinates of the fourth vertex. Do I use the distance formula for the two points and use them to add to each other to get two parallel sides?
| If three points are $P,Q,R$ then $R+(P-Q)$ gives a fourth vertex for a parallelogram. So pick the ordered pair $(P,Q)$ in all six ways, and that gives three parallelograms.
There are actually only three such paralellograms, some using my description being repeats (same vertices). So if $P,Q,R$ are the three given point... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Necessary and Sufficient Conditions for the pair of integers $\{m,n\}$ to generate $\mathbb{Z}$ Let $m,n \in \mathbb{Z}$ be two non zero numbers. I need to find necessary and sufficient conditions on $m$ and $n$ for which the pair $\{m,n\}$ generates the additive group $\mathbb{Z}$.
I made an attempt at a proof, and wo... | Your proof of the first direction looks perfectly fine. Your proof of the second part is definitely on the right track, and as it stands it's not wrong, but it's a little unclear. Somewhat better is to note that, since $\gcd(m,n)$ divides $k$ for every integer $k$, it divides $1$ in particular. Knowing that $\gcd(m,n)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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The space $L^{\infty -}$ and showing it is an algebra. Let
$$\mathcal{A}=L^{\infty -}(\Omega,P):=\bigcap_{1\le p<\infty}L^p(\Omega,P)$$
I've already showed it is a vector space over $\mathbb{C}$. So, with the usual multiplication of (complex-valued) functions operation I want to prove that
$fg\in\mathcal{A}\quad\fora... | Hint: $$|fg|^p = |f|^p |g|^p\le \frac{|f|^{2p} + |g|^{2p}}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The difference between $\mathbb{Z}\times\mathbb{Z}$ and $\mathbb{Z}*\mathbb{Z}$? I have some questions regarding some notations being used here.
I am still relatively new to algebraic topology, so I am a bit confused.
I saw that $\pi_1(S^1\times S^1)\simeq\mathbb{Z}\times\mathbb{Z}$ and $\pi_1(S^1\vee S^1)\simeq\mathbb... | The notation $*$ denotes that you are considering the free product of the two groups. The free product of two groups is basically words in an alphabet given by them. However, we also require that each word is fully reduced (i.e $aa^{-1} = e, aa = a^2$) in the dictionary of words that can be described by the alphabet.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Sequence Question If $(x_n)$ is a sequence of positive values and $\lim_{n\to\infty} n x_n $ exists, prove that $(x_n) \rightarrow 0$.
Since $\lim_{n\to\infty} n x_n $ exists, we know $(nx_n)$ converges to some positive number; call it $x$. Let $\varepsilon > 0$. Then we can find an $N \in \mathbb{N}$ such that $|nx_n ... | Hint. We have that $\frac{x - \varepsilon}{n}\to 0$ and $\frac{x + \varepsilon}{n}\to 0$ as $n$ goes to infinity.
Notice that the limit $x$ is greater or equal to zero (it is not necessarily positive).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1967991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Ways of forming palindromic strings The following problem was given to us in the recruitment test of InMobi.
Given a list of strings $\{a_1, a_2,..,a_n\}$, I want to count the number of ways of forming PALINDROMIC string $S=s_1+s_2+..+s_n$, where $s_i$ represents a non-empty sub-sequence of string $a_i$.
As answer can... | Concatenate all strings and find the number of palindromic subsequences. Now you have to remove those ways in which theres atleast one string that did not participate in palindrome formation. So make all $2^n$ combinations of strings that didn't participate and calculate the number of palindromic subsequences formed by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1968083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How is the Axiom of Choice equivalent to the Banach-Tarski paradox? I've seen many explanations that just state they are equivalent straight away however I don't understand why they're equivalent. As far as I understand, the axiom of choice states that for any indexed family of non empty sets, it is possible to choose ... | Banach Tarski is prooved by picking a representative from an infinite set (orbit equivalence classes). This requires the axiom of choice. However banach tarski does not imply the axiom of choice, they are not equivalent.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1968169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Can two integer polynomials touch in an irrational point? We define an integer polynomial as polynomial that has only integer coefficients. Here I am only interested in polynomials in two variables.
Example:
*
*$P = 5x^4 + 7 x^3y^4 + 4y$
Note that each polynomial P defines a curve by considering the set of points ... | Here's a general way to find such examples where both curves are of the form $y=f(x)$. Notice that $y=f(x)$ and $y=g(x)$ meet at a given value of $x$ iff that value of $x$ is a root of the polynomial $h(x)=f(x)-g(x)$, and they have the same tangent line iff that value is a root of $h(x)$ of multiplicity greater than $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1968254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "35",
"answer_count": 3,
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Increasing $g$ where $g' = 0$ a.e. but $g$ not constant on any open interval? As the question title suggests, does there exist an increasing function $g$ such that $g' = 0$ almost everywhere but $g$ isn't constant on any open interval?
| Yes, Let $\phi(x)$ be Cantor-Lebesgue function for $[0,1]$ and continue it to a function on $\mathbb{R}$ by fixing it $1$ for $x>1$ and $0$ for $x<0$. Let $O_n = (a_n,b_n)$ be an enumeration of all open intervals in $\mathbb{R}$ such that the end-points are of rational value.
Define $ \phi_n(x) = \phi(\frac{x-a_n}{b_n-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1968333",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Matrix calculus in multiple linear regression OLS estimate derivation The steps of the following derivation are from here
Starting from $y= Xb +\epsilon $, which really is just the same as
$\begin{bmatrix}
y_{1} \\
y_{2} \\
\vdots \\
y_{N}
\end{bmatrix}
=
\begin{bmatrix}
1 & x_{21} & \cdots & x_{K1} \\
1 & x_{22} & \cd... | This is not exaclty a proof but rather a way to think about it.
You are trying to minimize a scalar function $F(b)$. Now use the implicit derivative:
$$dF=d(y'y)-2d(b'X'y)+d(b'X'Xb)=-2db'X'y+db'X'Xb+b'X'Xdb.$$
Now transpose the last expression (which is a scalar) and factor $db'$.
$$dF=2db'(-X'y+X'Xb)$$
So the gradient... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1968478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Defining a norm in the quotient space $E/M$ Let $(E,\|\cdot\|)$ a normed space and consider $M \subseteq E$ a closed vectorial subspace. Consider in $E$ the equivalence relation $x \equiv y \iff x-y \in M$, and let $E/M$ the quotient set. The equivalence class of $x$ is the set $x+M = \{ x+m | m \in M \}$. Show that $E... | For positive definiteness you should use that $M$ is assumed closed. So if $\|x+m_k\|\rightarrow 0$, $m_k\in M$ then show (1) that the sequence $(m_k)_k$ is Cauchy, (2) that it therefore converges to some $m\in M$ and (3) finally $x=-m\in M$.
In homogeneity you should distinguish the case $\lambda=0$ and $\lambda\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1968606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Understanding a proof that $2x + 3y$ is divisible by $17$ iff $9x + 5y$ is divisible by $17$ I'm having some trouble understanding a proof on Naoki Sato's notes on Number Theory and I was wondering if you guys could give me some help. The problem is that I don't understand the last implication on the proof for example ... | If $17\mid (26x+39y)$, and $17\mid (-17x-34y)$, then we may add to get $17\mid 9x+5y$. In general the rule is, if $p\mid a$ and $p\mid b$, then $p\mid (a+b)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1968750",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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This question concerns functions $f:\{A,B,C,D,E\}\rightarrow\{1,2,3,4,5,6,7\}$ (counting) Can someone guide me towards a way to count surjective functions of the below question?
This question concerns functions $f:\{A,B,C,D,E\}\rightarrow\{1,2,3,4,5,6,7\}$. How many such functions are there? How many are injective? Su... | There are no surjective functions from a finite set to a bigger finite one.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1968851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What's the difference between continuous and piecewise continuous functions? A continuous function is a function where the limit exists everywhere, and the function at those points is defined to be the same as the limit.
I was looking at the image of a piecewise continuous function on the following page: http://tutori... | A piecewise continuous function doesn't have to be continuous at finitely many points in a finite interval, so long as you can split the function into subintervals such that each interval is continuous.
A nice piecewise continuous function is the floor function:
The function itself is not continuous, but each little s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1968943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Ideals in $S^{-1}A$. I am studying localization of rings and got stuck at a problem. It states that if $S$ is a multiplicatively closed subset of a ring $A$ then fractional ideals of $\ S^{-1}A $ are in bijective correspondence with those of $A$ which do not meet $S$. However, prime ideals of $ S^{-1}A $ are in bijec... | The correspondence $I\mapsto S^{-1}I$ need not be injective. For instance, let $A=k[x,y]$ and $S=\{y^n:y\in\mathbb{N}\}$. Then if $I=(x)$ and $J=(xy)$, $S^{-1}I= S^{-1}J$ even though $I\neq J$ and neither intersects $S$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1969190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A better way to evaluate a certain determinant
Question Statement:-
Evaluate the determinant:
$$\begin{vmatrix}
1^2 & 2^2 & 3^2 \\
2^2 & 3^2 & 4^2 \\
3^2 & 4^2 & 5^2 \\
\end{vmatrix}$$
My Solution:-
$$
\begin{align}
\begin{vmatrix}
1^2 & 2^2 & 3^2 \\
2^2 & 3^2 & 4^2 \\
3^2 & 4^2 & 5^2 \\
\end{vmatrix} &=
(1^2\t... | Using the rule of Sarrus, the computation is really not too long, and we get in general for all $n\ge 1$,
$$
\det \begin{pmatrix} n^2 & (n+1)^2 & (n+2)^2\cr (n+1)^2& (n+2)^2 & (n+3)^2\cr
(n+2)^2& (n+3)^2 & (n+4)^2\end{pmatrix}=-8.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1969290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 6,
"answer_id": 4
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Does $\mathbb{C}\setminus\{0\}$ with those operations constitute a vector space over $\mathbb{C}$? If given set $V = \mathbb{C}\setminus\{0\}$ and field $\mathcal{F} = \mathbb{C}$ with operations defined for all $\vec{x}, \vec{y} \in V, \vec{x} = x \in \mathbb{C}\setminus\{0\}, \vec{y} = y \in \mathbb{C}\setminus\{0\}$... | The answer is yes. The set $\mathbb{C}\backslash \{0\}$ with usual multiplication is an abelian group, so $V$ with $\oplus$ is an abelian group. The problems arise with "scalar multiplication", and although one example is enough, it's also true that distributive laws fail for (almost) any $\alpha,x,y$
$$\alpha\odot (x\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1969392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Equation of tangent to a circle
Find an equation of the tangent to the circle with equation $x^2+y^2-10x+4y+4=0$ at the point $(2,2)$
I have solved up to $4y - 8 = 3x - 6$, but I am not sure whether the final answer should be $3x-4y+2=0$ OR whether it should be $y=\frac{3}{4}x+\frac{1}{2}$.
The solutions say $3x-4y+2... | Actually I'd use neither!
The important parts here are that it passes through a particular point and goes in a particular direction, which means that point-slope form is best:
$$y-2 = \frac{3}{4}(x-2)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1969513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Natural number divisible by $42$? There is a natural number divisible by $42$. The sum of digits which do not take part in the written number is $25$. Prove that there are two identical numerals in the natural number.
| Hint: The sum of the digits of the number has to be a multiple of $3$, because it is divisible by $3$. What is the sum of all the digits there are? If every digit in the number is used only once, what is the sum of its digits?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1969603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is it possible to cover a $8 \times8$ board with $2 \times 1$ pieces?
We have a $8\times 8$ board, colored with two colors like a typical
chessboard. Now, we remove two squares of different colour. Is it
possible to cover the new board with two-color pieces (i.e. domino
pieces)?
I think we can, as after the rem... | Hint: A promising strategy is to prove that the claim
If we remove two opposite-colored squares from a $2m\times 2m$ chessboard, we may tile the remaining part with $2\times 1$ dominoes.
by induction on $m$. The case $m=1$ is trivial. Assume that the claim holds for some $m\geq 1$ and consider a $(2m+2)\times (2m+2)$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "8",
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Integrals of a function which has finitely many discontinuities are not differentiable at the discontinuities I've tried solving an exercise stated below.
$$\text{Suppose that $f\in\mathcal R$ on $[a,b]$ and define $F(x)=\int_{a}^{x}{f(t)dt}$.}\\\text{If $x$ is a point at which $f$ is not continuous, is it still possib... | Hint: Consider the function $f(x) = \sin(1/x), x\ne 0, f(0)=0.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1969840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that every graph with diameter d and girth 2d+1 is regular How to prove that every graph with diameter $d$ and girth $2d+1$ is regular.
I know just relation between diameter and girth. which is given by below formula
$girth(G)\leq 2diam(G)+1$
| The proof of this fact that I know goes as follows. Suppose we have the following claim
Claim 1. If $G$ is a graph of diameter $d$ and girth $2d+1$ then any two vertices $u,v$ at distance $d$ have the same degree.
Once you establish Claim 1 your claim follows easily. If $C$ is a $2d+1$ cycle in $G$ then by Claim 1 a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1969932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Entire function with image contained in "slit plane" is constant Let S be the "slit plane" $S = \mathbb{C} - \{t \in \mathbb{R} : t \leq 0\}$ and deduce that if $f$ is an entire function whose image is contained in $S$, then $f$ is constant.
Clearly $S$ is simply-connected and does not contain zero, so I attempted to ... | Let $g(z) = {1 \over \log f(z) + i (\pi +1) }$, then
$|g(z)| \le 1$ and $g$ is entire, hence constant.
Since $\log f(z) = {1 \over g(0)} -i(\pi+1)$, we see that $f$ is constant too.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Derivatives and Differentiation rules I am currently encountering a math problem that I can't seem to solve on my own and I think it is because I missed the last math lecture.
Usually I am pretty good when it comes to derivatives but this one seems to be my nemesis.
Can somebody maybe help me out?
Thank you guys!
PS Im... | From the chain rule $$F'(a) = \left.(f(x^3))'\right|_a = f'(a^3)\cdot \left.(x^3)'\right|_a = f'(a^3)3a^2 = (4)(3)(4)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1970184",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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George's imagined numbers George pictured 4 natural numbers. He multiplied each of those numbers by three and wrote all four results on a blackboard. He also calculated all possible products of the pairs of the written numbers, he then wrote all 6 products on the blackboard. Prove that (of the ten numbers written on th... | Just to be ornery and different:
Each original number "influences" four of the ten results; once when multiplied by 3, and once when multiplied by each of the three other original numbers.
Given two original numbers each will have four results, but one of the results will be the common result of multiplying those two n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1970320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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Showing $\mathrm{Var}(\min(X,y))$ is increasing in $y$ where $X$ is a random variable For nonnegative random variables, expectation is defined to be the supremum of all expectations of simple random variables $A$ that satisfy $A\leq X$. (For simple random variables, $E(A)=\sum_jc_jP(C_j)$ where $c_j$'s are nonnegative... | Following the hint from @Michael:
$Var(min(X,y))$
= $Var(Z)$
= $\int_0^{\infty} P[Z^2 > t] dt - (\int_0^{\infty} P[Z > t] dt)^2$
= $\int_0^{\infty} P[min(X,y)^2 > t] dt - (\int_0^{\infty} P[min(X,y) > t] dt)^2$
= $\int_0^{y^2} P[X^2 > t] dt - (\int_0^{y} P[X > t] dt)^2$
Then,
$\frac{d}{dy}[Var(min(X,y))]$
= $ 2y P[X^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1970424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Limit points and the trivial topology: A textbook error? I'm reading George L. Cain's Introduction to General Topology, and am confused by the following example. We have:
Pg 32: Definition: Let $(X,T)$ be a topological space. If $S$ is a subset of $X$, a point $p$ in $X$ is a T-limit point of $S$ if every element of $T... | You're both right- in your example, $1$ is indeed a limit point of $S$. But the text doesn't claim that those are the only $T$ limit points of $S$. It just doesn't want to go to deep into the detail of this examples just to say that if $S$ is a singleton, then all points of $X-S$ are $T$ limit points of $S$, and otherw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1970520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$ x\in \left[0,{1\over n-1} \right] \to 1+nx \le (1+x)^n \le {1+x\over 1-(n-1)x}$ (Homework assignement) This is about a homework I have to do. I don't want the straight answer, just the hint that may help me start on this. To give you context, we're now studying integrals.
Now here is the question :
Prove : $ x\in \l... | For the inequality $1+nx\leq (1+x)^n $, just expand using the binomial theorem and notice that all terms are positive.
The other inequality, after some manipulations (note that all terms are positive) looks like $$1-(n-1)x\leq (1+x)^{-(n-1)}. $$
Consider the function $$f (x)=(1+x)^{-(n-1)}+(n-1)x-1.$$ We have $f(0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1970695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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To find $p$ such that max/min of $(\sin p+\cos p)^{10}$ occurs To find max/min of $(\sin p+\cos p)^{10}$. I have to find value of $p$ such that the expression is max/min. I tried to manipulate expression so as to get rid of at least $\sin$ or $\cos$. Then I can put what is left over equals to $1$ to get the maximum. Bu... | $(\sin p + \cos p)^{10} = (\sin^2 p + 2\sin p\cos p + \cos^2 p)^5 = (1+\sin 2p)^5$
Function $x\mapsto (1+x)^5$ is monotone increasing, thus, extremes of $(1+\sin 2p)^5$ are the same as extremes of $\sin 2p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1970786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Tiling a cylindrical piece of paper Imagine a piece of paper. It has a square grid of 1x1 on it, so that every square has an area of 1 cm(squared). That piece of paper was folded into the shape of an (empty, hollow) cylinder whose length is 50 cm and whose base circumference is also 50 cm (look at the picture below). C... | It is impossible :
The number of squares in cylinder is $50^2$
And we color black or white in them, like chess board
Hence at block in b) we have two coloring ways : 3 black and 1 white, 1 black and 3 white If the number of blocks of first type is $x$ and the number of second types is $y$, then $$ 3x+y=50^2/2 $$
$$ x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1970912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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The level curves and the Jacobian How do I have to approach this problem? Intuitionally I can imagine the situation, but I have no idea how to prove this.
Problem : Let $f=(f_1, f_2)$ be a continuously differentiable function defined on an open set $U$ in $R^2$ such that $\nabla f_1$ and $\nabla f_2$ do not vanish at a... | We are given two $C^1$-functions $f_1$, $f_2$ with a common domain $U\subset{\mathbb R}^2$, whereby both $\nabla f_1$ and $\nabla f_2$ do not vanish in $U$.
(a) The condition $$J_f({\bf x})=\nabla f_1({\bf x})\wedge\nabla f_2({\bf x})=0\qquad({\bf x}\in U)$$
means that $\nabla f_1({\bf x})$ and $\nabla f_2({\bf x})$ ar... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to evaluate $\lim\limits_{x\to 0} \frac{\arctan x - \arcsin x}{\tan x - \sin x}$ I have a stuck on the problem of L'Hospital's Rule,
$\lim\limits_{x\to 0} \frac{\arctan x - \arcsin x}{\tan x - \sin x}$ which is in I.F. $\frac{0}{0}$
If we use the rule, we will have
$\lim\limits_{x\to 0} \frac{\frac{1}{1+x^2}-\fra... | Alternatively, one may use standard Taylor expansions, as $x \to 0$,
$$
\begin{align}
\sin x&=x-\frac{x^3}{6}+o(x^4)
\\\tan x&=x+\frac{x^3}{3}+o(x^4)
\\\arctan x&=x-\frac{x^3}{3}+o(x^4)
\\\arcsin x&=x+\frac{x^3}{6}+o(x^4)
\end{align}
$$ giving, as $x \to 0$,
$$ \frac{\arctan x - \arcsin x}{\tan x - \sin x}= \frac{-\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Index of p-subgroup divisible by p implies normalizer index divisible by p
Show that if $H$ is a $p$-subgroup of $G$ with $p$ dividing $[G:H]$, then $[N_G(H):H]$ is divisible by $p$.
By considering the action of $H$ on $G/H$. I've considered the $\varphi:H\to S_{G/H}$ and tried a lot of things, I just fail to see how... | $\newcommand{\Size}[1]{\left\lvert #1 \right\rvert}$$\newcommand{\Set}[1]{\left\{ #1 \right\}}$Consider the action of $H$ by multiplication on the right on the set of cosets
$$
G/H = \Set{ H x : x \in G}.
$$
So $h \in H$ sends the coset $H x$ to the coset $H x h$.
Since $H$ is a (finite) $p$-group, of order $p^{n}$, sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
System of quadratic equations with parameter I'd appreciate your help with this problem:
$p - a^2 = b$
$p - b^2 = c$
$p - c^2 = d$
$p - d^2 = a$
Where $a$, $b$, $c$, $d$ are real numbers, $p$ is a real parameter lower or equal to 1 and greater or equal to 0.
Thank you a lot. I am capable of solving the problem for pos... | *
*Subtracting two adjacent equations,
$$
\left \{
\begin{array}{ccc}
(a-b)(a+b) &=& c-b \\
(b-c)(b+c) &=& d-c \\
(c-d)(c+d) &=& a-d \\
(d-a)(d+a) &=& b-a
\end{array}
\right.$$
Note that
$$a=b \iff b=c \iff c=d \iff a=d$$
Now
$$a^2+a-p=0$$
$$a=b=c=d=\frac{-1\pm \sqrt{1+4p}}{2}$$
*
*Subtrac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
associated homogeneous linear differential equations Can someone please explain how associated homogeneous linear differential equations work with an example?
| Lets say you have the linear differential equation
$y'' + y =3x$;
The associated homogeneous equation is
$y'' + y = 0$
The set of the solutions to the homogeneous equation is {$\alpha \cos +\beta \sin ; \alpha, \beta \in \Bbb R$}.
One particular solution to the innitial equation is 3x.
Thus theset of solutions to the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971539",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to use the binomial theorem to calculate binomials with a negative exponent I'm having some trouble with expanding a binomial when it is a fraction. eg $(a+b)^{-n}$ where $n$ is a positive integer and $a$ and $b$ are real numbers.
I've looked at several other answers on this site and around the rest of the web, but... | Note that $(a+b)^{-n} = \frac{1}{(a+b)^n}$.
Now, apply $(a+b)^n = \sum_{i=0}^n \binom{n}{i} a^i b^{n-i}$ to calculate the denominator.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
Prove function is < 0
For all $x$ if $x^6 + 3x^4 - 3x < 0$ then $0 < x < 1$. Prove this.
(1) Find the negation
(2) Prove
(1) The negation is simply, $\exists x$, $x^6 + 3x^4 - 3x < 0 \wedge (x \le 0 \vee x \ge 1)$
(2) The proof is the difficult part here.
We prove the contrapositive. It is easy to prove it for the co... | If $x <-1$, then $f (x)>f (-x)$ so it will suffice to check the cases for $x>1$. Notice that $x^6>x,$ since $(x\cdot x\dots x>x\cdot 1\dots 1=x $. Can you finish with the same argument for $3x^4$ and $3x $? What can you conclude for values of $f(x) $ outside of the unit interval?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Rational function field of product affine varieties Let $X, Y$ be affine varieties, we know that the coordinate ring of the product variety $X\times Y$ satisfies $k[X\times Y]\cong k[X]\otimes_k k[Y]$.
My question is is it true that for rational function field, we also have $k(X\times Y)\cong k(X)\otimes_k k(Y)$? If n... | It is not true in general that $k(X\times Y)\cong k(X)\otimes_k k(Y)$.
For example, let $k=\mathbb{Q}$, $k[X]=k[Y]=\mathbb{Q}(i)$ then $k(X)=k(Y) = \mathbb{Q}(i)$, but $\mathbb{Q}(i) \otimes_{\mathbb{Q}} \mathbb{Q}(i)$ is not even a field.
It is also not true for product of two affine lines: $k(x) \otimes_k k(y) \subs... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Find the equations of the common tangents to the parabola $y^2=15x$ and the circle $x^2+y^2=16$. The text says:
Find the equations of the common tangents to the parabola $y^2=15x$ and the circle $x^2+y^2=16$.
I tried the approach of the discriminant and also one using the distance from a line but both didn't work for... | Let a common tangent touch the circle at $\displaystyle (a,b)$ and the parabola at $\displaystyle (\alpha, \beta)$, and let it have the equation $\displaystyle y = mx + c$.
Now proceed systematically and list what you know, writing equations along the way.
1) $\displaystyle (a,b)$ satisfies the equation of the tangent.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1971986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Equation in Complex plane I know that $\cos(\theta)=\frac{\exp(i\theta)+\exp(-i\theta)}{2}$ in which $\theta=arg(z)$ for some complex number $z$. Can I assume $\theta$ in above folmula as a comlex number? I mean, $$\cos(z)=\frac{\exp(iz)+\exp(-iz)}{2}$$ You know, I want to do $\cos(z)=2i$. Someone told me that I can, b... | Yes, the formula
$$
\cos(z)=\frac{\exp(iz)+\exp(-iz)}{2}
$$
holds for all complex numbers $z$, including the real values $z = \theta$ that you first saw. This follows from a uniqueness theorem for analytic continuations. The so-called identity theorem states that if two holomorphic functions $f$ and $g$ agree on a se... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1972206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is $(\mathbb{R},*)$, where $a*b=ab+a+b$, a group? I have a problem with this question
is $(\mathbb{R},*)$, when $a*b=ab+a+b$ a group? If not, can you skip any element $a \in\mathbb{R} $ in that way, that $(\mathbb{R}$\{a}$,*)$ is a group?
I can prove, that $(\mathbb{R},*)$ is a binary operation, associative and it's ne... | you can start by looking at which elements lack an invert. Thus, we compute $a^{-1}$:
$aa^{-1}+a+a^{-1}=0\iff a^{-1}=\frac{-a}{a+1}$
So you should delete $-1$ out of $\mathbb{R}$.
However, you have to prove ($\mathbb{R}$\{-1},*) does make a group.
It is closed under multiplication (verify if $a,b\neq-1$ then ($a*b$)$\n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1972316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How do I can evaluate this integral: $\int_{1}^{\infty}\frac{y\cosh(yx)}{\sinh(y\pi)}dy$? I'm interested to know how do I evaluate this integral:
$$\int_{1}^{\infty}\frac{y\cosh(yx)}{\sinh(y\pi)}dy.$$
Wolfram alpha gives the output
$$\frac1{(\pi-x)^2}+\text{Li}(-2e^{-2\pi})-\frac{\log(1-e^{-2\pi})}{\pi}-\frac12+O(\pi-x... | I presume you are actually interested in this integral for some application, and then you might want a convenient expression --- Mathematica does evaluate it in closed form [*], as indicated in the comments, but this involves special functions and might not be of much practical use. Here is what I propose to do. Consid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1972405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Calculate limit involving exponents Calculate:
$$\lim_{x \rightarrow 0} \frac{(1+2x)^{\frac{1}{x}} - (1+x)^{\frac{2}{x}}}{x}$$
I've tried to calculate the limit of each term of the subtraction:
$$\lim_{x \rightarrow 0} \frac{(1+2x)^{\frac{1}{x}}}{x}$$
$$\lim_{x \rightarrow 0} \frac{(1+x)^{\frac{2}{x}}}{x} $$
Each of th... | Using Taylor expansion:
$$(1+2x)^{1/x}\approx e^2-2e^2x+o(x^2)$$
$$(1+x)^{2/x}\approx e^2-e^2x+o(x^2)$$
$$\lim_{x \rightarrow 0} \frac{(1+2x)^{\frac{1}{x}} - (1+x)^{\frac{2}{x}}}{x} \approx \lim _{x\to 0}\left(\frac{e^2-2e^2x+o\left(x^2\right)-e^2+e^2x+o\left(x^2\right)}{x}\right)$$
$$= \lim _{x\to 0}\left(\frac{-e^2x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1972517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Definition of a shadow in space, and how to derive a shadow for a given shape $\newcommand{\Reals}{\mathbf{R}}$I am struggling with the concept of a shadow in $\Reals^3$. My professor provided the class with the following definition:
Given $S \subset \Reals^3$, the Shadow of $S$ in the $XY$ plane is equal to $$\{(x,y,... | $\newcommand{\Reals}{\mathbf{R}}$Imagine a light "at infinity" on the $z$-axis: Its rays travel along lines parallel to the $z$-axis. If $S \subset \Reals^{3}$, and if $(x_0, y_0)$ is a point of $\Reals^{2}$, then the ray of light $\{(x_0, y_0, t): t > 0\}$ touches $S$ if and only if there exists a $z > 0$ such that $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1972777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How does $6^{\frac{5}{3}}$ simplify to $6\sqrt[3]{36}$? I was recently given a problem along the lines of the below:
Simplify $6^{\frac{5}{3}}$ to an expression in the format $a\sqrt[b]{c}$.
The answer, $6\sqrt[3]{36}$, was then given to me before I could figure out the problem myself. I'm wondering what the steps to... | $$6^{5/3} = 6^{1 + 2/3} = 6 \cdot 6^{2/3} = 6 \sqrt[3]{6^2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1972850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to solve this tetration equation $\;^n 2 = \;^2 n $? How would one find all real solutions to the following equation:
$\qquad$ $n^n = 2^{2^{2^{2^{\dots^2}}}} $(where the number of $2$s is equal to $n$)
generalizing to $n$ being a real value. In tetration-notation this is
$\qquad $ find a solution to $\displays... | I wanted to add some graphs to Gottfried's solution. First definitions; $\text{sexp}_2(z)= \;^z 2$ which is extended to the complex plane by Kneser's solution. I wrote a program to calculate the slog; which is the inverse of sexp and has some nice uniqueness properties. The fatou.gp program works for a wide range of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1972958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Tangents to a parabola that go through the same point The question is: The two lines tangent to f (x) = $x^2$ + 4x + 2 through the point (2, -12)have equations y = ax + b and y = cx + d, respectively. What is the value of a + b + c + d?
What I did to solve it:
f '(x) = 2x + 4. The point is (2 -12) so I plugged in two t... | Look at the figure:
The point $A=(2,-12)$ is not a point of the parabola, so the slopes of the tangents to the parabola from this point cannot be be simply derived starting from the derivative of the parabola at $x=2$.
If $P=(X,X^2+4X+2)$ is a point of tangency (the points $C$ and $D$ in the figure), than the slope $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1973094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does the floor of a number preserve order? For example, say you got x < y for some x and y. Then
$\lfloor x \rfloor \geq \lfloor y \rfloor $ ? Is it always the case?
The reason why I am confused on this point is I was reading a solution posted on chegg which doesn't seem convincing.
The question states to prove $\l... | If $x<y$, then $\lfloor x\rfloor\leq\lfloor y\rfloor$. (Your inequality is flipped.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1973244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to solve this linear hyperbolic PDE analytically? Is it possible to solve this equation analytically?
$$
u_t = k u_{xx} + \frac{k}{c} u_{xt}
$$
I attempted to solve it for a finite domain and homogeneous B.C with separation of variables but it got very ugly, with complex eigenvalues. I'm wondering if the equation c... | I realized that the coordinate transformation $x=x_*$ and $t=t_*-\frac{1}{2c} x_*$ transforms this equation into the damped wave equation, which has well-known solutions which can be obtained through separation of variables.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1973318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that if $a \ge c$ for all $c < b$, then $a \geq b$
Let $a$ and $b$ be elements in an ordered field, prove that if $a \ge c$ for every $c$ such that $c \lt b$, then $a\ge b$.
My proof idea below:
Let $S = \{x | x<b\}$. Then $a$ is an upper bound for $S$. If I can show that $b$ is the least upper bound for $S$, t... | Let $S=\{c:c<b\} .$ By hypothesis, $\forall c\in S\;(c<a).$ So if $a\in S$ then $a<a,$ which cannot be, because "$<$" is irreflexive. The whole field is equal to $S\cup \{b\} \cup \{d:d>b\} $ because "$<$" satisfies trichotomy. Since $a\not \in S $ we have $a\in \{b\}\cup \{d:d>b\}.$ QED.
Note that this applies to a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1973432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 3
} |
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