Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Given $\frac {a\cdot y}{b\cdot x} = \frac CD$, find $y$. That's a pretty easy one... I have the following equality : $\dfrac {a\cdot y}{b\cdot x} = \dfrac CD$ and I want to leave $y$ alone so I move "$b\cdot x$" to the other side
$$a\cdot y= \dfrac {(C\cdot b\cdot x)}{D}$$
and then "$a$"
$$y=\dfrac {\dfrac{(C\cdot b... | No mistake was made. Observe that:
$$
y=\dfrac{\left(\dfrac{Cbx}{D}\right)}{a}=\dfrac{Cbx}{D} \div a = \dfrac{Cbx}{D} \times \dfrac{1}{a}=\dfrac{Cbx}{Da}=\dfrac{bCx}{aD}
$$
as desired.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/415232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Representation problem: I don't understand the setting of the question! (From Serre's book) Ex 2.8 of Serre's book "Linear Representations of Finite Groups" says: Let $\rho:G\to V$ be a representation ($G$ finite and $V$ is complex, finite dimensional) and $V=W_1\oplus W_1 \oplus \dotsb \oplus W_2 \oplus \dotsb W_2\opl... | I'm not sure I understand what is worrying you, but each $W_{i}$ is a $G$-submodule, for any $w \in W_{i}, w\rho_{s} \in W_{i}.$ Then we can apply the map $h,$ s $(w\rho_{s})h$ is an element of $V_{i}.$ On the other hand, $wh \in V_{i}.$ We know that $V_{i}$ is also a $G$-submodule, so $(wh)\rho_{s}$ is an element of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Projections Open but not closed I often read that projections are Open but generally not closed. Unfortunately I do not have a counterexample for not closed available. Does anybody of you guys have?
| The example which is mentioned by David Mitra shows that the projections are not closed:
The projection $p: \mathbb R^2 \rightarrow \mathbb R$ of the plane $ \mathbb R^2 $ onto the $x$-axis is not closed. Indeed, the set $\color{red}{F}=\{(x,y)\in \mathbb R^2 : xy=1\}$ is closed in $\mathbb R^2 $ and yet its image $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Logic Circuits And Equations Issue - Multiply Binary Number By 3 I am trying to build a logic circuit that multiplies any 4 digit binary number by 3.
I know that if I multiply/divide number by 2 it moves left/right the digits, but what I'm
doing with multiply by 3?
*
*How to extract the equations that multiply the... | The easiest way I see is to note that $3n=2n+n$, so copy $n$, shift it one to the left, and add back to $n$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/415440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Recursive Sequence Tree Problem (Original Research in the Field of Comp. Sci) This question appears also in https://cstheory.stackexchange.com/questions/17953/recursive-sequence-tree-problem-original-research-in-the-field-of-comp-sci. I was told that cross-posting in this particular situation could be approved, since t... | This is more of a comment, but it's too big for the comment block. An interesting note on Kaya's matrix $\mathbf{M}$: I believe that it can be defined recursively for any value of $p$. (I should note here that this is my belief. I have yet to prove it...)
That is, let $\mathbf{M}_p$ be the matrix for the value of $p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Is the set $\{(x, y) : 3x^2 − 2y^ 2 + 3y = 1\}$ connected? Is the set $\{(x, y)\in\mathbb{R}^2 : 3x^2 − 2y^ 2 + 3y = 1\}$ connected?
I have checked that it is an hyperbola, hence disconnected am i right?
| Your set $S$ contains the points $(0,{1\over2})$ and $(0,1)$, but does not contain any points on the line $y={3\over4}$. Therefore $$S\cap\{(x,y)|y<{3\over4}\},\quad S\cap\{(x,y)|y>{3\over4}\}$$
is a partition of $S$ into disjoint (relatively) open subsets of ${\mathbb R}^2$. This shows that $S$ is not connected.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finding a counterexample; quotient maps and subspaces Let $X$ and $Y$ be two topological spaces and $p: X\to Y$ be a quotient map.
If $A$ is a subspace of $X$, then the map $q:A\to p(A)$ obtained by restricting $p$ need not be a quotient map. Could you give me an example when $q$ is not a quotient map?
| M.B. has given an example that shows that the restriction of a quotient map to an open subspace need not be a quotient map. Here is an example showing that the restriction to a saturated set also need not. Restrictions to an open saturated set however always works.
Let $X=[0,2],\ \ B=(0,1],\ \ A=\{0\}\cup(1,2]$ with th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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$\vec{u}+\vec{v}-\vec{w},\;\vec{u}-\vec{v}+\vec{w},\;-\vec{u}+\vec{v}+\vec{w} $ are linearly independent if and only if $\vec{u},\vec{v},\vec{w}$ are I'm consufed: how can I prove that $$\vec{u} + \vec{v} - \vec{w} , \qquad \vec{u} - \vec{v} + \vec{w},\qquad - \vec{u} + \vec{v} + \vec{w} $$ are linearly independent ve... | Call the three vectors $a, b, c$ then you find $2u=a+b, 2v=a+c, 2w=b+c$
If there is a linear dependence in $a, b, c$ the equations translate into a (nontrivial) dependence in $u, v, w$ and vice versa - the two sets of vectors are related by an invertible linear transformation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/415729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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describe the domain of a function $f(x, y)$ Describe the domain of the function $f(x,y) = \ln(7 - x - y)$. I have the answer narrowed down but I am not sure if it would be $\{(x,y) \mid y ≤ -x + 7\}$ or $\{(x,y) \mid y < -x + 7\}$ please help me.
| Nice job: on the inequality part of things; the only confusion you seem to have is with respect to whether to include the case $y = x-7$. But note that $$f(x, y) = f(x, x - 7) = \ln(7 - x - y) = \ln \left[7 - x -(7 - x)\right] = \ln 0$$ but $\;\ln (0)\;$ is not defined, hence $y = x - 7$ cannot be included domain!
So ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/415800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove: $D_{8n} \not\cong D_{4n} \times Z_2$.
Prove $D_{8n} \not\cong D_{4n} \times Z_2$.
My trial:
I tried to show that $D_{16}$ is not isomorphic to $D_8 \times Z_2$ by making a contradiction as follows:
Suppose $D_{4n}$ is isomorphic to $D_{2n} \times Z_2$, so $D_{8}$ is isomorphic to $D_{4} \times Z_2$. If $D_{1... | $D_{8n}$ has an element of order $4n$, but the maximal order of an element in $D_{4n} \times \mathbb{Z}_2$ is $2n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/415879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Conditional Expected Value and distribution question
The distribution of loss due to fire damage to a warehouse is:
$$
\begin{array}{r|l}
\text{Amount of Loss (X)} & \text{Probability}\\
\hline
0 & 0.900 \\
500 & 0.060 \\
1,000 & 0.030\\
10,000 & 0.008 \\
50,000 & 0.001\\
100,000 & 0.001 \\
\end{array}
$$
Given that ... | You completely missed the word "given". That means you want a conditional probability given the event cited. In other words, your second option is right.
For example the conditional probability that $X=500$ given that $X>0$ is
$$
\Pr(X=500\mid X>0) = \frac{\Pr(X=500\ \&\ X>0)}{\Pr(X>0)} = \frac{\Pr(X=500}{\Pr(X>0)} =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Help with a trig-substitution integral I'm in the chapter of trigonometric substitution for integrating different functions. I'm having a bit of trouble even starting this homework question:
$$\int \frac{(x^2+3x+4)\,dx} {\sqrt{x^2-4x}}$$
| In order to make a proper substitution in integral calculus, the function that you are substituting must have a unique inverse function. However, there is such a case where the the derivative is present and you can make what I refer to as a "virtual substitution". This is not exactly the case here, we have to do other ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A recursive formula for $a_n$ = $\int_0^{\pi/2} \sin^{2n}(x)dx$, namely $a_n = \frac{2n-1}{2n} a_{n-1}$ Where does the $\frac{2n-1}{2n}$ come from?
I've tried using integration by parts and got $\int \sin^{2n}(x)dx = \frac {\cos^3 x}{3} +\cos x +C$, which doesn't have any connection with $\frac{2n-1}{2n}$.
Here's my d... | Given the identity $$\int \sin^n x dx = - \frac{\sin^{n-1} x \cos x}{n}+\frac{n-1}{n}\int \sin^{n-2} xdx$$ plugging in $2n$ yields $$\int \sin^{2n} x dx = - \frac{\sin^{2n-1} x \cos x}{2n}+\frac{2n-1}{2n}\int \sin^{2n-2} xdx$$
Since
$$\int_0^{\pi/2} \sin^{2n} x dx = - \frac{\sin^{2n-1} x \cos x}{2n}|_0^{\pi/2}+\frac{2n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Has $X\times X$ the following property? Let $X$ be a topological space that satisfies the following condition at each point $x$:
For every open set $U$ containing $x$, there exists an open set $V$ with compact boundary such that $x\in V\subseteq U$.
Does $X\times X$ also have that property?
| Try to Proof: For every open set $W$ of $X \times X$, there exists $U_1$ and $U_2$ of $X$ such that $U_1 \times U_2 \subset W.$ Then there exist $V_1$ and $V_2$ such that there compact boundary are contained in $U_1$ and $U_2$ respectively.
Next we prove that the compact boundary of $V_1 \times V_2$ are contained in $U... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416270",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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If $f^2$ is differentiable, how pathological can $f$ be? Apologies for what's probably a dumb question from the perspective of someone who paid better attention in real analysis class.
Let $I \subseteq \mathbb{R}$ be an interval and $f : I \to \mathbb{R}$ be a continuous function such that $f^2$ is differentiable. It ... | To expand on TZakrevskiy's answer, we can use one of the intermediate lemmas from the proof of Whitney extension theorem.
Theorem (Existence of regularised distance) Let $E$ be an arbitrary closed set in $\mathbb{R}^d$. There exists a function $f$, continuous on $\mathbb{R}^d$, and smooth on $\mathbb{R}^d\setminus E$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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How to show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$? I really think I have no talents in topology. This is a part of a problem from Topology by Munkres:
Show that if $A$ is compact, $d(x,A)= d(x,a)$ for some $a \in A$.
I always have the feeling that it is easy to understand the problem emotionall... | Let $f$ : A $\longrightarrow$ $\mathbb{R}$ such that a $\mapsto$ d(x, a), where $\mathbb{R}$ is the topological space induced by the $<$ relation, the order topology.
For all open intervals (b, c) in $\mathbb{R}$, $f^{-1}((b, c))$ = {a $\in$ A $\vert$ d(x, a) $>$ b} $\cap$ {a $\in$ A $\vert$ d(x, a) $<$ c}, an open set... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$ How
$\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$
i have tried to do that by the following procedure:
$\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2$
=$\frac{1}{n}(\sum_{i=1}^n X_i^2 - n\bar X^2)... | I think it is cleaner to expand the right-hand side. We have
$$(X_i-\bar{X})^2=X_i^2-2X_i\bar{X}+(\bar{X})^2.$$
Sum over all $i$, noting that $2\sum X_i\bar{X}=2n\bar{X}\bar{X}=2n(\bar{X})^2$ and $\sum (\bar{X})^2=n(\bar{X})^2$.
There is some cancellation. Now divide by $n$ and we get the left-hand side.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/416581",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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lim sup and sup inequality Is it true that for a sequence of functions $f_n$
$$\limsup_{n \rightarrow \infty }f_n \leq \sup_{n} f_n$$
I tried to search for this result, but I couldn't find, so maybe my understanding is wrong and this does not hold.
| The inequality
$$\limsup_{n\to\infty}a_n\leq\sup_{n\in\mathbb{N}}a_n$$
holds for any real numbers $a_n$, because the definition of $\limsup$ is
$$\limsup_{n\to\infty}a_n:=\lim_{m\to\infty}\left(\sup_{n\geq m}a_n\right)$$
and for any $n\in\mathbb{N}$, we have
$$\left(\sup_{n\geq m}a_n\right)\leq\sup_{m\in\mathbb{N}}a_n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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recurrence relations for proportional division I am looking for a solution to the following recurrence relation:
$$ D(x,1) = x $$
$$ D(x,n) = \min_{k=1,\ldots,n-1}{D\left({{x(k-a)} \over n},k\right)} \ \ \ \ [n>1] $$
Where $a$ is a constant, $0 \leq a \leq 1$. Also, assume $x \geq 0$.
This formula can be interpreted a... | For $x>0$ the answer is $$D(x,n)=\frac{x(1-a)(2-a)\ldots (n-1-a)}{n!}$$
Proof: induction on $n$. Indeed,
$$D(x,n)=\min_{k=1,\ldots,n-1}\frac{x(k-a)(1-a)\ldots (k-1-a)}{k!n}$$
$$=\frac{x}{n}\min_{k=1,\ldots,n-1}\frac{(1-a)\ldots (k-a)}{k!}$$
andf it remains to prove
$$\frac{(1-a)\ldots (k-a)}{k!}>\frac{(1-a)\ldots (k+1-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How to find out X in a trinomial How can I find out what X equals in this?
$$x^2 - 2x - 3 = 117$$
How would I get started? I'm truly stuck.
| Hint:1.$$ax^2 +bx +c=0 \to D=b^2-4ac\ge0 $$$$\ \to\begin{cases}
\color{green}{x_1=\frac{-b+\sqrt{D}}{2a}} \\
\color{red}{x_2=\frac{-b-\sqrt{D}}{2a}} \\
\end{cases} $$$$$$ 2. $$x^2 +bx +c=(x+\frac{b}{2})^2=\frac{b^2}{4}-c\ge0\quad \text{then} x=\pm\sqrt{\frac{b^2}{4}-c}-\frac{b}{2}$$
3. find $x_1$and $x_2 $by solving ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Limit of two variables, proposed solution check: $\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt{x^2+y^2}}$ Does this solution make sense,
The limit in question:
$$
\lim_{(x,y)\to(0,0)}\frac{xy}{\sqrt{x^2+y^2}}
$$
My solution is this:
Suppose, $$ \sqrt{x^2+y^2} < \delta $$ therefore $$xy<\delta^2$$ So by the Squeeze Theorem the l... | Here's a more direct solution.
We know $|x|,|y|\le\sqrt{x^2+y^2}$, so if $\sqrt{x^2+y^2}<\delta$, then
$$\left|\frac{xy}{\sqrt{x^2+y^2}}\right|\le\frac{\big(\sqrt{x^2+y^2}\big)^2}{\sqrt{x^2+y^2}}=\sqrt{x^2+y^2}<\delta.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/416925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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Telephone Number Checksum Problem I am having difficulty solving this problem. Could someone please help me? Thanks
"The telephone numbers in town run from 00000 to 99999; a common error in dialling on a
standard keypad is to punch in a digit horizontally adjacent to the intended one. So on a
standard dialling keypad... | Code 1 can detect horizontal error. Let $c$ be the correct misdialed digit (order doesn't matter because they're added) and let $c'$ be the mistaken misdialed digit. For any two horizontally adjacent digits consist of an odd and even number. That is, $c-c'\equiv 1 \pmod 2$. It follows that the sum of the new digits... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/416974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Probability to open a door using all the keys I have 1 door and 10 keys. What is the probablity to open the door trying all the keys? I will discard every single key time to time.
| Solution 1:
Hint: How many permutations are there for the order of the 10 keys?
Hint: How many permutations are there, where the last key is the correct key?
Solution 2:
Hint: The probability that the last key is the correct key, is the same as the probability that the nth key is the correct key. Hence ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/417055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why is $\log z$ continuous for $x\leq 0$ rather than $x\geq 0$? Explain why Log (principal branch) is only continuous on $$\mathbb{C} \setminus\{x + 0i: x\leq0\}$$ is the question.
However, I can't see why this is. Shouldn't it be $x \geq 0$ instead?
Thanks.
| The important thing here is that the exponential function is periodic with period $2\pi i$, meaning that $e^{z+2\pi i}=e^z$ for all $z\in\Bbb C$.
If you imagine any stripe $S_a:=\{x+yi\mid y\in [a,a+2\pi)\}$ for any $a\in\Bbb R$, then we get that $z\mapsto e^z$ is actually one-to-one, restricted to $S_a$, (and maps ont... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Where's the error in this $2=1$ fake proof? I'm reading Spivak's Calculus:
2 What's wrong with the following "proof"? Let $x=y$. Then
$$x^2=xy\tag{1}$$
$$x^2-y^2=xy-y^2\tag{2}$$
$$(x+y)(x-y)=y(x-y)\tag{3}$$
$$x+y=y\tag{4}$$
$$2y=y\tag{5}$$
$$2=1\tag{6}$$
I guess the problem is in $(3)$, it seems he tried to divide b... | We have $x = y$, so $x - y = 0$.
EDIT: I think I should say more. I'll go through each step:
$x = y \tag{0}$
This is our premise that $x$ and $y$ are equal.
$$x^2=xy\tag{1}$$
Note that $x^2 = xx = xy$ by $(0)$. So completely valid.
$$x^2-y^2=xy-y^2\tag{2}$$
Now we're adding $-y^2$ to both sides of $(1$) so comple... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
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Is this Expectation finite? How do I prove that
$\int_{0}^{+\infty}\text{exp}(-x)\cdot\text{log}(1+\frac{1}{x})dx$
is finite? (if it is)
I tried through simulation and it seems finite for large intervals. But I don't know how to prove it analytically because I don't know the closed form integral of this product.
I am ... | If we write $$\int_{0}^{+\infty}=\int_0^1+\int_1^{\infty}$$ then we see that $$\lim_{x\to 0^+} x^{1/2}f(x)=0<\infty\Longrightarrow\int_0^1f(x)dx~~\text{is convergent}$$ and $$\lim_{x\to +\infty} x^{2}f(x)=0<\infty\Longrightarrow\int_1^{\infty}f(x)dx~~\text{is convergent}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/417410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
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If $\gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c\ $ [Euclid's Lemma] Well I thought this is obvious. since $\gcd(a,b)=1$, then we have that $a\not\mid b$ AND $a\mid bc$. This implies that $a$ divides $c$. But apparently this is wrong. Help explain why this way is wrong please.
The question tells you give me... | It seems you have in mind a proof that uses prime factorizations, i.e. the prime factors of $\,a\,$ cannot occur in $\,b\:$ so they must occur in $\,c.\,$ You should write out this argument very carefully, so that it is clear how it depends crucially on the existence and uniqueness of prime factorizations, i.e. FTA = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 8,
"answer_id": 0
} |
how to determine the following set is countable or not? How to determine whether or not these two sets are countable?
The set A of all functions $f$ from $\mathbb{Z}_{+}$ to $\mathbb{Z}_{+}$.
The set B of all functions $f$ from $\mathbb{Z}_{+}$ to $\mathbb{Z}_{+}$ that are eventually 1.
First one is easier to determi... | Let $B_n$ be the set of functoins $f\colon \mathbb Z_+\to\mathbb Z_+$ with $f(x)\le n$ for all $x$ and $f(x)=1$ for $x>n$.
Then $$B=\bigcup_{n\in\mathbb N}B_n$$
and $|B_n|=n^n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/417587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The derivative of a linear transformation Suppose $m > 1$. Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ be a smooth map. Consider $f + Ax$ for $A \in \mathrm{Mat}_{m\times n}, x \in \mathbb{R}^n$. Define $F: \mathbb{R}^n \times \mathrm{Mat}_{m\times n} \rightarrow \mathrm{Mat}_{m\times n}$ by $F(x,A) = df_x + A$.
So ... | Since, by your definition, $F$ is a matrix-valued function, $DF$ would be a rank-3 tensor with elements
$$
(DF)_{i,j,k} = \frac{\partial^2 f_i(x)}{\partial x_j \partial x_k}
$$
Some authors also define matrix-by-vector and matrix-by-matrix derivatives differently be considering $m \times n$ matricies as vectors in $\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does $((x-1)! \bmod x) - (x-1) \equiv 0\implies \text{isPrime}(x)$ Does $$((x-1)! \bmod x) - (x-1) = 0$$
imply that $x$ is prime?
| I want to add that this is the easy direction of Wilson's theorem. I have often seen Wilson's Theorem stated without the "if and only if" because this second direction is so easy to prove:
Proof: If $x > 1, x \ne 4$ were not prime, then the product $(x-1)!$ would contain two factors multiplying to get $x$, and thus we... | {
"language": "en",
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"source": "stackexchange",
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Linear equation System What are the solutions of the following system:
$ 14x_1 + 35x_2 - 7x_3 - 63x_4 = 0 $
$ -10x_1 - 25x_2 + 5x_3 + 45x_4 = 0 $
$ 26x_1 + 65x_2 - 13x_3 - 117x_4 = 0 $
*
*4 unknowns (n), 3 equations
$ Ax=0: $
$ \begin{pmatrix} 14 & 35 & -7 & -63 & 0 \\ -10 & -25 & 5 & 45 & 0 \\ 26 & 65 & -13 ... | Yes, there are infinitely many real solutions. Since there are more unknowns than equations, this system is called underdetermined. Underdetermined systems can have either no solutions or infinitely many solutions. Trivially the zero vector solves the equation:$$Ax=0$$
This is sufficient to give that the system must ha... | {
"language": "en",
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"source": "stackexchange",
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Showing that $\{x\in\mathbb R^n: \|x\|=\pi\}\cup\{0\}$ is not connected I do have problems with connected sets so I got the following exercise:
$X:=\{x\in\mathbb{R}^n: \|x\|=\pi\}\cup\{0\}\subset\mathbb{R}^n$. Why is $X$ not connected?
My attempt: I have to find disjoint open sets $U,V\ne\emptyset$ such that $U\cup V... | A more fancy approach: it will suffice to say that the singleton $\{0\}$ is a clopen (=closed and open) set. It is closed, because one point sets are always closed. It is open, because it is $X \cap \{ x \in \mathbb{R}^n \ : \ || x || < 1 \}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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find $\frac{ax+b}{x+c}$ in partial fractions $$y=\frac{ax+b}{x+c}$$ find a,b and c given that there are asymptotes at $x=-1$ and $y=-2$ and the curve passes through (3,0)
I know that c=1 but I dont know how to find a and b?
I thought you expressed y in partial fraction so that you end up with something like $y=Ax+B+\fr... | The line $x =-1$ is assymptote, in that case we must have:
$$\lim_{x\to -1}\frac{ax+b}{x+c}=\pm\infty$$
I've written $\pm$ because any of them is valid. This is just abuse of notation to make this a little faster. For this to happen we need a zero in the denominator at $x=-1$ since this is just a rational function. In ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In linear logic sequent calculus, can $\Gamma \vdash \Delta$ and $\Sigma \vdash \Pi$ be combined to get $\Gamma, \Sigma \vdash \Delta, \Pi$? Linear logic is a certain variant of sequent calculus that does not generally allow contraction and weakening. Sequent calculus does admit the cut rule: given contexts $\Gamma$, $... | The rule you suggest is called the "mix" rule , and it is not derivable from the standard rules of linear logic. Actually, what I know is that it's not derivable in multiplicative linear logic; I can't imagine that additive or exponential rules would matter, but I don't actually know that they don't. Hyland and Ong c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/417991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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What are the main relationships between exclusive OR / logical biconditional? Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain.
Its well known that both exclusive OR and logical biconditional make $\mathbb{B}$ into an Abelian group (in the former case the identity is $0$, in the latter the identity is $1$).
Further... | You probably already know this, but the immediate connection between them is $(x\oplus y) \leftrightarrow \neg(x \leftrightarrow y)$. Then the exclusive OR reduces trivially to the biconditional, and vice versa.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/418064",
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"source": "stackexchange",
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Calculating $\int_{\pi/2}^{\pi}\frac{x\sin{x}}{5-4\cos{x}}\,\mathrm dx$
Calculate the following integral:$$\int_{\pi/2}^{\pi}\frac{x\sin{x}}{5-4\cos{x}}\,\mathrm dx$$
I can calculate the integral on $[0,\pi]$,but I want to know how to do it on $[\frac{\pi}{2},\pi]$.
| $$\begin{align}\int_{\pi/2}^\pi\frac{x\sin x}{5-4\cos x}dx&=\pi\left(\frac{\ln3}2-\frac{\ln2}4-\frac{\ln5}8\right)-\frac12\operatorname{Ti}_2\left(\frac12\right)\\&=\pi\left(\frac{\ln3}2-\frac{\ln2}4-\frac{\ln5}8\right)-\frac12\Im\,\chi_2\left(\frac{\sqrt{-1}}2\right),\end{align}$$
where $\operatorname{Ti}_2(z)$ is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/418134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why differentiating a function is similar to differentiating its log function? $f(6;p)=\binom{25}{6}p^6(1-p)^6,\quad 0\le p\le1$
I have three questions:
$(1)$To find the relative maxima of p, the process is to take the derivative of the function with respect to $p$ equal to $0$ and solving the resulting equation for $... | There is a typo in the post, presumably you mean $p^6(1-p)^{19}$.
Since $\log$ is an increasing function, finding the maximum of $p^6(1-p)^{19}$ and finding the maximum of its logarithm are equivalent problems. There is a bit of a question mark about the endpoints $p=0$ and $p=1$, but the maximum is clearly not there, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Basis for $L^2(0,T;H)$ Given a basis $b_i$ for the separable Hilbert space $H$, what is the basis for $L^2(0,T;H)$? Could it be $\{a_jb_i : j, i \in \mathbb{N}\}$ where $a_j$ is the basis for $L^2(0,T)$?
| You are not far from correct result. The desired basis is a family of fnctions
$\{f_{i,j}:i,j\in\mathbb{N}\}$ defined as
$$
f_{i,j}(t)=a_j(t)b_j
$$
The deep reason for this is the following. Since we have an identification.
$$
L_2((0,T), H)\cong L_2(0,T)\otimes_2 H
$$
it is enough to study to study bases of Hilbert t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that a function f is continuous (1) $f:\mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(x) =
\begin{cases}
x \sin(\ln(|x|))& \text{if $x\neq0$} \\
0 & \text{if $x=0$} \\
\end{cases}$$
Is $f$ continuous on $\mathbb{R}$?
I want to use the fact that 2 continuous functions:
$$f:I \rightarrow J ( \subset \mathbb{... | For all $x\in (-\infty;0)\cup(0;+\infty)$ is function continuous since it is composition of continuous functions (I think it is necessary to show it in this task, as mentioned in comments, the real problem is $x=0$).
By definition, function is continuous at $x_0$, if $$\lim_{x\rightarrow x_0}f(x)=f(x_0)$$
In this case:... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Counter example of upper semicontinuity of fiber dimension in classical algebraic geometry We know that if $f : X\to Y$ is a morphism between two irreducible affine varieties over an algebraically closed field $k$, then the function that assigns to each point of $X$ the dimension of the fiber it belongs to is upper sem... | I got my answer on MO, here : mathoverflow.net/questions/133567/…
| {
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"url": "https://math.stackexchange.com/questions/418469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Distinguishable telephone poles being painted Each of n (distinguishable) telephone poles is painted red, white, blue or yellow. An odd number are painted blue and an even number yellow. In how many ways can this be done?
Can some give me a hint how to approach this problem?
| Consider the generating function given by
$( R + W + B + Y )^n$
Without restriction, the sum of all coefficients would give the number of ways to paint the distinguishable posts in any of the 4 colors. We substitute $R=W=B=Y = 1$ to find this sum, and it is (unsurprisingly) $4^n$.
Since there are no restrictions on $R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/418520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are there any integer solutions to $\gcd(\sigma(n), \sigma(n^2)) = 1$ other than for prime $n$? A good day to everyone!
Are there any integer solutions to $\gcd(\sigma(n), \sigma(n^2)) = 1$ other than for prime $n$ (where $\sigma = \sigma_1$ is the sum-of-divisors function)?
Note that, if $n = p$ for prime $p$ then
$$\... | Let $n=pq$ where $p,q$ are two distinct primes.
Then
$$\sigma(n)=(p+1)(q+1)$$
$$\sigma(n^2)=(1+p+p^2)(1+q+q^2)$$
$$\sigma(6)=12$$
$$\sigma(6^2)=7 \cdot 13 \,.$$
Note that as long as $1+p+p^2$ and $1+q+q^2$ are primes, then for $n=pq$ we have $gcd(\sigma(n), \sigma(n^2))=1$.
Actually you only need
$$gcd(p+1,q^2+q+1)=\g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Smooth Pac-Man Curve? Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ which Wolfram Alpha shows to look like this:
The curve is not defined at $\theta=0$, but we can aug... | Not a very good one: $r(\theta) = e^{-\dfrac{1}{20 \theta^2}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/418641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Need explanation of passage about Lebesgue/Bochner space From a book:
Let $V$ be Banach and $g \in L^2(0,T;V')$. For every $v \in V$, it holds that
$$\langle g(t), v \rangle_{V',V} = 0\tag{1}$$
for almost every $t \in [0,T]$.
What I don't understand is the following:
This is equivalent to $$\langle g(t), v(t) \r... | For $(2)$ implies $(1)$, consider the function $v\in L^2(0,T;V)$ defined by $$v(t)=w, \forall\ t\in [0,T]$$
where $w\in V$ is fixed. Hence, you have by $(2)$ that $$\langle g(t),v(t)\rangle=\langle g(t),w\rangle=0$$
for almost $t$. By varying $w$, you can conclude.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/418693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In diagonalization, can the eigenvector matrix be any scalar multiple? One can decompose a diagonalizable matrix (A) into A = C D C^−1, where C contains the eigenvectors and D is a diagonal matrix with the eigenvalues in the diagonal positions. So here's where I get confused. If I start with a random eigenvector matrix... | Well:
$$(xA)^{-1}=\frac{1}{x}A^{-1}\qquad x\neq 0$$ so the result is actually the same. Eigenvectors are vectors of an eigenspace, and therefore, if a vector is an eigenvector, then any multiple of it is also an eigenvector. When you build a matrix of eigenvectors, you have infinite of them to choose from, that program... | {
"language": "en",
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minimum value of a trigonometric equation is given. the problem is when the minimum value attains Suppose the minimum value of $\cos^{2}(\theta_{1}-\theta_{2})+\cos^{2}(\theta_{2}-\theta_{3})+\cos^{2}(\theta_{3}-\theta_{1})$ is $\frac{3}{4}$.
Also the following equations are given
$$\cos^{2}(\theta_{1})+\cos^{2}(\theta... | As $\sin2x=2\sin x\cos x,$
$$\cos\theta_1\sin\theta_1+\cos\theta_2\sin\theta_2+\cos\theta_3\sin\theta_3=0$$
$$\implies \sin2\theta_1+\sin2\theta_2+\sin2\theta_3=0$$
$$\implies \sin2\theta_1+\sin2\theta_2=-\sin2\theta_3\ \ \ \ (1)$$
As $\cos2x=\cos^2x-\sin^2x,$
$$\cos^2\theta_1+\cos^2\theta_2+\cos^2\theta_3=\frac32=\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/418812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Two propositions about weak* convergence and (weak) convergence Let $E$ be a normed space. We have the usual definitions:
1) $f, f_n \in E^*$, $n \in \mathbb{N}$, then $$f_n \xrightarrow{w^*} f :<=> \forall x \in E: f_n(x) \rightarrow f(x)$$ and in this case we say that $(f_n)$ is $weak^*$-$convergent$ to $f$.
2)$x, x_... | a)
$$
\|f_n(x_n) - f(x) \| \leq \|f_n(x_n) - f_n(x)\| + \| f_n(x) - f(x) \| \leq \|f_n\| \|x_n - x\| + \| f_n(x) - f(x) \|.
$$
By the principle of uniform boundedness, $\sup_n \|f_n\|$ is finite, so you're done.
b) Not true in general. Let $\{x_n\}$ be an orthonormal basis in some Hilbert space. Then $x_n \rightarro... | {
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solvable subalgebra I want to show that a set $B\subset L$ is a maximal solvable subalgebra.
With $L = \mathscr{o}(8,F)$, $F$ and algebraically closed field, and $\operatorname{char}(F)=0$ and
$$B= \left\{\begin{pmatrix}p&q\\0&s \end{pmatrix}\mid p \textrm{ upper triangular, }q,s\in\mathscr{gl}(4,F)\textrm{ and }p^t=-... | As far as I can think, use the Thm c at page 84, however i have no idea of how to compute this $B(triangular)=H+\cup L_{\alpha}$ which are the positive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/418954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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single valued function in complex plane Let
$$f(z)=\int_{1}^{z} \left(\frac{1}{w} + \frac{a}{w^3}\right)\cos(w)\,\mathrm{d}w$$
Find $a$ such that $f$ is a single valued function in the complex plane.
| $$
\begin{align}
\left(\frac1w+\frac{a}{w^3}\right)\cos(w)
&=\left(\frac1w+\frac{a}{w^3}\right)\left(1-\frac12w^2+O\left(w^4\right)\right)\\
&=\frac{a}{w^3}+\color{#C00000}{\left(1-\frac12a\right)}\frac1w+O(w)
\end{align}
$$
The residue at $0$ is $1-\frac12a$, so setting $a=2$ gives a residue of $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/419006",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Does every infinite group have a maximal subgroup?
$G$ is an infinite group.
*
*Is it necessary true that there exists a subgroup $H$ of $G$ and $H$ is maximal ?
*Is it possible that there exists such series $H_1 < H_2 < H_3 <\cdots <G $ with the property that for every $H_i$ there exists $H_{i+1}$ such that $H_i ... | Rotman p. 324 problem 10.25:
The following conditions on an abelian group are equivalent:
*
*$G$ is divisible.
*Every nonzero quotient of $G$ is infinite; and
*$G$ has no maximal subgroups.
It is easy to see above points are equivalent. If you need the details, I can add them here.
| {
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Finding $a$ s.t the cone $\sqrt{x^{2}+y^{2}}=za$ divides the upper half of the unit ball into two parts with the same volume My friend gave me the following question:
For which value of the parameter $a$ does the cone
$\sqrt{x^{2}+y^{2}}=za$ divides $$\{(x,y,z):\,
x^{2}+y^{2}+z^{2}\leq1,z\geq0\}$$ into two parts wi... | Check your bounds again. I believe they should be
$$\begin{align}0<&z<\frac{1}{\sqrt{a^2+1}}\\az<&r<\sqrt{1-z^2}\end{align}$$
Finishing the integral with these bounds should yield $a=\sqrt3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/419163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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truncate, ceiling, floor, and...? Truncation rounds negative numbers upwards, and positive numbers downwards. Floor rounds all numbers downwards, and ceiling rounds all numbers upwards. Is there a term/notation/whatever for the fourth operation, which rounds negative numbers downwards, and positive numbers upwards? Tha... | The fourth operation is called "round towards infinity" or "round away from zero". It can be implemented by
$$y=\text{sign} (x)\text{ceil}(|x|)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/419250",
"timestamp": "2023-03-29T00:00:00",
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Usefulness of the concept of equivalent representations Definition: Let $G$ be a group, $\rho : G\rightarrow GL(V)$ and $\rho' : G\rightarrow GL(V')$ be two representations of G. We say that $\rho$ and $\rho'$ are $equivalent$ (or isomorphic) if $\exists \space T:V\rightarrow V'$ linear isomorphism such that $T{\rho_g}... | This is just the concept of isomorphism applied to representations, i.e. $T$ is providing an isomorphism between $V$ and $V'$ which interchanges the two group
representations. So all your intuitions for the role of isomorphisms in group theory should carry over.
Why are you having trouble seeing that properties of $\... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof for Sum of Sigma Function How to prove:
$$\sum_{k=1}^n\sigma(k) = n^2 - \sum_{k=1}^nn\mod k$$
where $\sigma(k)$ is sum of divisors of k.
| $$\sum_{k=1}^n \sigma(k) = \sum_{k=1}^n\sum_{d|k} d = \sum_{d=1}^n\sum_{k=1,d|k}^{n}d = \sum_{d=1}^n d\left\lfloor \frac {n} {d}\right\rfloor$$
Now just prove that $$d\left\lfloor \frac n d\right\rfloor = n-(n\mod d)$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Why does the result of the Lagrangian depend on the formulation of the constraint? Consider the following maximization problem:
$$ \max f(x) = 3 x^3 - 3 x^2, s.t. g(x) = (3-x)^3 \ge 0 $$
Now it's obvious that the maximum is obtained at $ x =3 $. In this point, however, the constraint qualification
$$ Dg(x) = -3 (3-x)^2... | Yes, reformulating the constraint might change the validity of the CQ. This is quite natural, since the Lagrangian expresses optimality via derivatives of the constraints (and the objective). Changing the constraint may render its derivative useless ($0$ in your case).
| {
"language": "en",
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Endomorphisms of a semisimple module Is there an easy way to see the following:
Given a $k$-algebra $A$, with $k$ a field, and a finite dimensional semisimple $A$-module $M$. Look at the natural map $A \to \mathrm{End}_k(M)$ that sends an $a \in A$ to
$$
M \to M: m \mapsto a \cdot m.
$$
Then the image of $A$ is a fin... | Here's one way to look at it: Notice that the kernel of the map is exactly $ann(M)$, which necessarily contains the Jacobson radical $J(A)$ of $A$. Since $A/J(A)$ and all of its quotients are semiprimitive, it follows that $A/ann(M)$ is semiprimitive.
Now $M$ as a faithful $A/ann(M)$ module. Since the simple submodule... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Question about order of elements of a subgroup Given a subgroup $H \subset \mathbb{Z}^4$, defined as the 4-tuples $(a,b,c,d)$ that satisfy $$ 8| (a-c); a+2b+3c+4d=0$$
The question is: give all orders of the elements of $\mathbb{Z}^4 /H$.
I don't have any idea how to start with this problem. Can anybody give some hints,... | As Amir has said above, consider the homomorphism $\phi:\oplus_{i=1}^{4}\mathbb{Z}\rightarrow \mathbb{Z}_{8}\oplus\mathbb{Z}$. You can check that as Amir pointed out $\operatorname{im}(\phi)\cong (\oplus_{i=1}^{4}\mathbb{Z})/H$, so what is $\operatorname{im}(\phi)$? If you wanted, you could work this out, but you are o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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number of errors in a binary code To transmit the eight possible stages of three binary symbols, a code appends three further symbols: if the message is ABC, the code transmits ABC(A+B)(A+C)(B+C). How many errors can it detect and correct?
My first step should be to find the minimum distance, but is there a systematic ... | The code is linear and there are seven non-zero words, so the brute force method of listing all of them is quite efficient in this case. You can further reduce the workload by observiing that the roles of $A,B,C$ are totally symmetric. What this means is that you only need to check the weights of the words where you as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/419914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can $\lim_{n\to \infty} (3^n + 4^n)^{1/n} = 4$? If $\lim_{n\to \infty} (3^n + 4^n)^{1/n} = 4$, then $\lim_{n\to \infty} 3^n + 4^n=\lim_{n\to \infty}4^n$ which implies that $\lim_{n\to \infty} 3^n=0$ which is clearly not correct. I tried to do the limit myself, but I got $3$. The way I did is that at the step $\lim_... | $\infty-\infty$ is not well-defined.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/419999",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that: $\sup_{z \in \overline{D}} |f(z)|=\sup_{z \in \Gamma} |f(z)|$ Suppose $D=\Delta^n(a,r)=\Delta(a_1,r_1)\times \ldots \times \Delta(a_n,r_n) \subset \mathbb{C}^n$
and
$\Gamma =\partial \circ \Delta^n(a,r)=\left \{ z=(z_1, \ldots , z_n)\in \mathbb{C}^n:|z_j-a_j|=r_j,~ j=\overline{1,n} \right \}$.
Let $f \in... | Since it's homework, a few hints:
First assume that $f$ is holomorphic on a neighbourhood of $\bar D$. Then use the maximum principle in each variable separately to conclude that it's true in this case. Finally, approximate your $f$ by functions that are holomorphic on a neihbourhood of $\bar D$.
More details I'll do ... | {
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"source": "stackexchange",
"question_score": "1",
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What is the difference between isomorphism and homeomorphism? I have some questions understanding isomorphism. Wikipedia said that
isomorphism is bijective homeomorphism
I kown that $F$ is a homeomorphism if $F$ and $F^{-1}$ are continuous. So my question is: If $F$ and its inverse are continuous, can it not be bije... | Isomorphism and homeomorphism appear both in topology and abstract algebra. Here, I think you mean isomorphism and homeomorphism in topology. In one word, they are the same in topology.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/420110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Is there a terminological difference between "sequence" and "complex" in homology theory Suppose you are given something like this:
$\dots \longrightarrow A^n \longrightarrow A^{n+1} \longrightarrow \dots$
People tend to talk about "chain complexes" but about "short exact sequences". Is there any terminological differe... | To say that the sequence is a chain complex is a less imposing condition: it simply says that if you compose any two of the maps in the sequence, you get $0$. But to say the sequence is exact says more: it says this is precisely (or, if you rather, exactly) the only way you get something mapping to zero.
The first sta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Need help solving the ODF $ f''(r) + \frac{1}{r}f'(r) = 0 $ I am currently taking complex analysis, and this homework question has a part that requires the solution to a differential equation. I took ODE over 4 years ago, so my skills are very rusty. The equation I derived is this:
$$ f''(r) + \frac{1}{r}f'(r) = 0 $$
I... | You now have:
1/v dv = -1/r dr
We can integrate this:
ln(v) = -ln(r) + C1
v = e^(-ln(r) + c1)
v = 1/r + c1
Thus we have:
df/dr = 1/r + c1 (since v = f'(r) = df/dr)
df = (1/r + c1) dr
f = ln(r) + c1(r) + c2
And thats our answer!
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is the outer boundary of a connected compact subset of $\mathbb{R}^2$ an image of $S^{1}$? A connected compact subset $C$ of $\mathbb{R}^2$ is contained in some closed ball $B$. Denote by $E$ the unique connected component of $\mathbb{R}^2-C$ which contains $\mathbb{R}^2-B$. The outer boundary $\partial C$ of $C$ is de... | No; the outer boundary of any handle body is a $n$-holed torus for some $n$. But all sorts of other things can happen; if $C$ is the cantor set, then the boundary is the Cantor set. So it can be all sorts of things.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/420347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
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Sets question, without Zorn's lemma Is there any proof to $|P(A)|=|P(B)| \Longrightarrow |A|=|B|$ that doesn't rely on Zorn's lemma (which means, without using the fact that $|A|\neq|B| \Longrightarrow |A|<|B|$ or $|A|>|B|$ ) ?
Thank you!
| Even with Zorn's Lemma, one cannot (under the usual assumption that ZF is consistent) prove that if two power sets have the same cardinality, then the sets have the same cardinality.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $?
If $G$ is a finite group, $H$ is a subgroup of $G$, and $H\cong Z(G)$, can we conclude that $H=Z(G) $?
I learnt that if two subgroups are isomorphic then it's not true that they act in the same way when this action ... | No. Let me explain why. You should think of $H\cong K$ as a statement about the internal structure of the subgroups $H$ and $K$. The isomorphism shows only that elements of $H$ interact with each other in the same way that elements of $K$ interact with each other. It doesn't say how any of these elements behave with... | {
"language": "en",
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"source": "stackexchange",
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What's the difference between Complex infinity and undefined? Can somebody please expand upon the specific meaning of these two similar mathematical ideas and provide usage examples of each one? Thank you!
| I don't think they are similar.
"Undefined" is something that one predicates of expressions. It means they don't refer to any mathematical object.
"Complex infinity", on the other hand, is itself a mathematical object. It's a point in the space $\mathbb C\cup\{\infty\}$, and there is such a thing as an open neighborh... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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kaleidoscopic effect on a triangle Let $\triangle ABC$ and straightlines $r$, $s$, and $t$. Considering the set of all mirror images of that triangle across $r$, $s$, and $t$ and its successive images of images across the same straightlines, how can we check whether $\triangle DEF$ is an element of that set?
Given:
*... | Create an image of your coordinate system, your three lines of reflections, and your original triangle. You can draw in mirror triangles pretty easily, and with a few of these, you will probably find a pattern. (I created my illustration using Cinderella, which comes with a tool to define transformation groups.)
As yo... | {
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"source": "stackexchange",
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Evaluating $\lim\limits_{x\to0}\frac{1-\cos(x)}{x}$ $$\lim_{x\to0}\frac{1-\cos(x)}{x}$$
Could someone help me with this trigonometric limit? I am trying to evaluate it without l'Hôpital's rule and derivation.
| There is also a fairly direct method based on trig identities and the limit $ \ \lim_{\theta \rightarrow 0} \ \tan \frac{\theta}{2} \ = \ 0 , $ which I discuss in the first half of my post here.
[In brief,
$$ \lim_{\theta \rightarrow 0} \ \tan \frac{\theta}{2} \ = \ 0 \ = \ \lim_{\theta \rightarrow 0} \ \frac{1 - \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/420698",
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Convergence of increasing measurable functions in measure? Let ${f_{n}}$ a increasing sequence of measurable functions such that $f_{n} \rightarrow f$ in measure.
Show that $f_{n}\uparrow f$ almost everywhere
My attempt
The sequence ${f_{n}}$ converges to f in measure if for any $\epsilon >0$ there exists $N\in \mathb... | If $f_n$ converges to $f$ in measure, we have a subsequence $f_{n_k}$ converging to $f$ almost everywhere. As $f_n$ is increasing, we know that $f_n$ converges at every point or every subsequence increases to $\infty$. But as the subsequence $f_{n_k}$ converges to $f$ at almost every point, we have that $f_n$ converges... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A directional derivative of $f(x,y)=x^2-3y^3$ at the point $(2,1)$ in some direction might be: A directional derivative of
$$
f(x,y)=x^2-3y^3
$$
at the point $P(2,1)$ in some direction might be:
a) $-9$
b) $-10$
c) $6$
d) $11$
e) $0$
I'd say it's $-9$ for sure, but what about $0$ (the direction would be $<0,0>$)?
Are... | $$D_{\vec u}f(\vec x)=\nabla f_{(2,1)}\frac{\vec u}{||\vec u||}\cdot=4u_1-9u_2\;\;\wedge\;\;u_1^2+u_2^2=1$$
so you get a non-linear system of equations
$$\begin{align*}\text{I}&\;\;4u_1-9u_2=t\\\text{II}&\;\;\;\;u_1^2+\;u_2^2=\,1\end{align*}$$
and from here we get
$$u_1^2+\left(\frac{4u_1-t}{9}\right)^2=1\implies 97u_1... | {
"language": "en",
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Whether $L=\{(a^m,a^n)\}^*$ is regular or not? I am condidering the automatic structure for Baumslag-Solitar semigroups. And I have a question. For any $m,n \in Z$, whether the set $L=\{(a^m,a^n)\}^*$ is regular or not. Here a set is regular means it can be recognized by a finite automaton.
Since the operations:union, ... | As Boris has pointed out in the comments, so far your question doesn't make complete sense, because you haven't specified a finite alphabet $\Sigma$ such that $L\subseteq \Sigma^*$.
What I think you probably want is to consider $L$ as a language over $\Sigma = (A\cap\{\epsilon\})\times(A\cap\{\epsilon\})$, where we are... | {
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Symmetric positive definite with respect to an inner product Let $A$ be a SPD(symmetric positive-definite) real $n\times n$ matrix. let $B=LL^T$ be also SPD. Let $(,)_B$ be an inner product given by $(x,y)_B=x \cdot By=y^T Bx$. Then $(B^{-1}Ax,y)_B=(x,B^{-1}Ay)_B$ for all $x,y$. Show that $B^{-1}A$ is SPD with respect ... | It means that in the definition of positive definite matrix you replace a standard euclidean scalar product by an inner product generated by another matrix.
A matrix $C$ is positive definite with respect to inner product $(,)_B$ defined by a positive definite matrix $B$ iff
$$\forall v\ne 0\, (Cv,v)_B = (BCv,v)>0$$
or,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove $n\mid \phi(2^n-1)$ If $2^p-1$ is a prime, (thus $p$ is a prime, too) then $p\mid 2^p-2=\phi(2^p-1).$
But I find $n\mid \phi(2^n-1)$ is always hold, no matter what $n$ is. Such as $4\mid \phi(2^4-1)=8.$
If we denote $a_n=\dfrac{\phi(2^n-1)}{n}$, then $a_n$ is A011260, but how to prove it is always integer?
Thanks... | I will use Lifting the Exponent Lemma(LTE).
Let $v_p(n)$ denote the highest exponent of $p$ in $n$.
Take some odd prime divisor of $n$, and call it $p$.
Let $j$ be the order of $2$ modulo $p$.
So, $v_p(2^n-1)=v_p(2^j-1)+v_p(n/j)>v_p(n)$ as $j\le p-1$.
All the rest is easy. Indeed, let's pose $n=2^jm$ where $m$ is odd.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/421151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "44",
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Closed subset of $\;C([0,1])$
$$\text{The set}\; A=\left\{x: \forall t\in[0,1] |x(t)|\leq \frac{t^2}{2}+1\right\}\;\;\text{is closed in}\;\, C\left([0,1]\right).$$
My proof:
Let $\epsilon >0$ and let $(x)_n\subset A$, $x_n\rightarrow x_0$. Then for $n\geq N$
$\sup_{[0,1]}|x_n(t)-x_0(t)|\leq \epsilon$ for some $N$ ... | Yes, it is correct. An "alternative" proof would consist in writing $A$ as the intersection of the closed sets $F_t:=\{x,|x(t)|\leqslant \frac{t^2}2+1\}$. It also work if we replace $\frac{t^2}2+1$ by any continuous function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/421216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Efficient way to compute $\sum_{i=1}^n \varphi(i) $ Given some upper bound $n$ is there an efficient way to calculate the following:
$$\sum_{i=1}^n \varphi(i) $$
I am aware that:
$$\sum_{i=1}^n \varphi(i) = \frac 12 \left( 1+\sum_{i=1}^n \mu(i) \left \lfloor \frac ni \right\rfloor ^2 \right) $$
Where:
$\varphi(x) $ is ... | If you can efficiently compute $\sum_{i=1}^n \varphi(i)$, then you can efficiently compute two consecutive values, so you can efficiently compute $\varphi(n)$. Since you already know that $\varphi(n)$ isn't efficiently computable, it follows that $\sum_{i=1}^n \varphi(i)$ isn't either.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/421274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
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How many of these four digit numbers are odd/even? For the following question:
How many four-digit numbers can you form with the digits $1,2,3,4,5,6$ and $7$ if no digit is repeated?
So, I did $P(7,4) = 840$ which is correct but then the question asks, how many of those numbers are odd and how many of them are even.... | We first count the number of ways to produce an even number. The last digit can be any of $2$, $4$, or $6$. So the last digit can be chosen in $3$ ways.
For each such choice, the first digit can be chosen in $6$ ways. So there are $(3)(6)$ ways to choose the last digit, and then the first.
For each of these $(3)(6)$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Change of variables in $k$-algebras Suppose $k$ is an algebraically closed field, and let $I$ be a proper ideal of $k[x_1, \dots, x_n]$. Does there exist an ideal $J \subseteq (x_1, \dots, x_n)$ such that $k[x_1, \dots, x_n]/I \cong k[x_1, \dots, x_n]/J$ as $k$-algebras?
| This is very clear from the corresponding geometric statement: Every non-empty algebraic set $\subseteq \mathbb{A}^n$ can be moved to some that contains the zero. Actually a translation from some point to the zero suffices.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/421476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating $\lim_{x\to0}\frac{x+\sin(x)}{x^2-\sin(x)}$ I did the math, and my calculations were:
$$\lim_{x\to0}\frac{x+\sin(x)}{x^2-\sin(x)}= \lim_{x\to0}\frac{x}{x^2-\sin(x)}+\frac{\sin(x)}{x^2-\sin(x)}$$ But I can not get out of it. I would like do it without using derivation or L'Hôpital's rule .
| $$\lim_{x\to0}\;\frac{\left(x+\sin(x)\right)}{\left(x^2-\sin(x)\right)}\cdot \frac{1/x}{1/x} \quad =\quad \lim_{x\to 0}\;\frac{1+\frac{\sin(x)}x}{x-\frac{\sin(x)}x}$$
Now we evaluate, using the fact that
$$\lim_{x \to 0} \frac {\sin(x)}x = 1$$
we can see that:
$$\lim_{x\to 0}\;\frac{1+\frac{\sin(x)}x}{x-\frac{\sin(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/421553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is $\frac{1}{\frac{1}{X}}=X$? Can someone help me understand in basic terms why $$\frac{1}{\frac{1}{X}} = X$$
And my book says that "to simplify the reciprocal of a fraction, invert the fraction"...I don't get this because isn't reciprocal by definition the invert of the fraction?
| Well, I think this is a matter of what is multiplication and what is division.
First, we denote that
$$\frac{1}{x}=y$$
which means
$$xy=1\qquad(\mbox{assuming $x\ne0$ in fundamental mathematics where there isn't Infinity($\infty$)})$$
Now,
$$\frac{1}{\frac{1}{x}}=\frac{1}{y}$$ by using the first equation. Here,
by che... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/421620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 12,
"answer_id": 2
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Neighborhoods in the product topology. In the last two lines thread, it is said that $N \times M \subseteq A\times B$ is a neighborhood of $0$ if and only if $N, M$ are neighborhoods of $0$. Here $A, B$ are topological abelian groups. How to prove this result? I searched on the Internet, but was not able to find a proo... | If $A,B$ are topological spaces,then we define the product topology by means of the topologies of $A$ and $B$ in such a way that $U\times V$ is open whenever $U\subseteq A$ and $V\subseteq B$ are open. More proecisely, we take the smallest topology on $A\times B$ such that these $U\times V$ are all open. This is precis... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to convert a permutation group into linear transformation matrix? is there any example about apply isomorphism to permutation group
and how to convert linear transformation matrix to permutation group and convert back to linear transformation matrix
| It seems that for each natural number $n$ there is an isomorphic embedding $i$ of the permutation group $S_n$ into the group of all non-degerated matrices of order $n$ over $\mathbb R$, defined as $i(\sigma)=A_\sigma=\|a_{ij}\|$ for each $\sigma\in S_n$, where $a_{ij}=1$ provided $\sigma(i)=j$, and $a_{ij}=0$ in the op... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/421846",
"timestamp": "2023-03-29T00:00:00",
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$\mathbb Q$-basis of $\mathbb Q(\sqrt[3] 7, \sqrt[5] 3)$. Can someone explain how I can find such a basis ? I computed that the degree of $[\mathbb Q(\sqrt[3] 7, \sqrt[5] 3):\mathbb Q] = 15$. Does this help ?
| Try first to find the degree of the extension over $\mathbb Q$. You know that $\mathbb Q(\sqrt[3]{7})$ and $\mathbb Q(\sqrt[5]{3})$ are subfields with minimal polynomials $x^3 - 7$ and $x^5-3$ which are both Eisenstein.
Therefore those subfields have degree $3$ and $5$ respectively and thus $3$ and $5$ divide $[\mathb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/421941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $x+{1\over x} = r $ then what is $x^3+{1\over x^3}$? If $$x+{1\over x} = r $$ then what is $$x^3+{1\over x^3}$$
Options:
$(a) 3,$
$(b) 3r,$
$(c)r,$
$(d) 0$
| $\displaystyle r^3=\left(x+\frac{1}{x}\right)^3=x^3+\frac{1}{x^3}+3(x)\frac{1}{x}\left(x+\frac{1}{x}\right)=x^3+\frac{1}{x^3}+3r$
$\displaystyle \Rightarrow r^3-3r=x^3+\frac{1}{x^3}$
Your options are incorrect.For a quick counter eg. you can take $x=1/2$ to get $r=\frac{5}{2}$ and $x^3+\frac{1}{x^3}=\frac{65}{8}$ but n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/421995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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eigenvalue and independence Let $B$ be a $5\times 5$ real matrix and assume:
*
*$B$ has eigenvalues 2 and 3 with corresponding eigenvectors $p_1$ and $p_3$, respectively.
*$B$ has generalized eigenvectors $p_2,p_4$ and $p_5$ satisfying
$Bp_2=p_1+2p_2,Bp_4=p_3+3p_4,Bp_5=p_4+3p_5$.
Prove that $\{p_1,p_2,p_3,p_4,p_5... | One can do a direct proof. Here is the beginning:
Suppose $0=\lambda_1p_1+\dots+\lambda_5p_5$. Then
$$\begin{align*}
0&=(B-3I)^3(\lambda_1p_1+\dots+\lambda_5p_5)\\
&=(B-3I)^3(\lambda_1p_1+\lambda_2p_2)\\
&=(B-2I-I)^3(\lambda_1p_1+\lambda_2p_2)\text{ now one can apply binomial theorem as $B-2I$ and $-I$ commute}\\
&= (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Fixed points for $k_{t+1}=\sqrt{k_t}-\frac{k_t}{2}$ For the difference equation
$k_{t+1}=\sqrt{k_t}-\frac{k_t}{2}$
one has to find all "fixed points" and determine whether they are locally or globally asymptotically stable.
Now I'm not quite sure what "fixed point" means in this context. Is it the same as "equilibrium... | (I deleted my first answer after rereading your question; I thought perhaps I gave more info than you wanted.)
Yes, you can read that as "equilibrium points". To find them, just let $k_t = \sqrt{k_t} - \frac{1}{2}k_t$. Solving for $k_t$ will give you a seed $k_0$ such that $k_0 = k_1$. As you wrote, letting $k_0 = 0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422140",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Testing convergence of $\sum_{n=0}^{\infty }(-1)^n\ \frac{4^{n}(n!)^{2}}{(2n)!}$ Can anyone help me to prove whether this series is convergent or divergent:
$$\sum_{n=0}^{\infty }(-1)^n\ \frac{4^{n}(n!)^{2}}{(2n)!}$$
I tried using the ratio test, but the limit of the ratio in this case is equal to 1 which is inconclusi... | By Stirling's approximation $n!\sim\sqrt{2\pi n}(n/e)^n$, so
$$\frac{4^{n}(n!)^{2}}{(2n)!}\sim \frac{2\pi n 4^{n} (n/e)^{2n}}{\sqrt{4\pi n}(2n/e)^{2n}} =\sqrt{\pi n}.$$ Thus, the series diverges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/422208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
A line through the centroid G of $\triangle ABC$ intersects the sides at points X, Y, Z. I am looking at the following problem from the book Geometry Revisited, by Coxeter and Greitzer. Chapter 2, Section 1, problem 8: A line through the centroid G of $\triangle ABC$ intersects the sides of the triangle at points $X, Y... |
a single line through the centroid can only intersect all three sides of a triangle at $X,Y,Z$ if two of these points are coincident at a vertex of the triangle, and the third is the midpoint of the opposite side
The sides of the triangle are lines of infinite length in this context, not just the line segments termin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to prove that the set $\{\sin(x),\sin(2x),...,\sin(mx)\}$ is linearly independent? Could you help me to show that the functions $\sin(x),\sin(2x),...,\sin(mx)\in V$ are linearly independent, where $V$ is the space of real functions?
Thanks.
| If $\{ \sin x, \sin 2x, \ldots, \sin mx\}$ is linear dependent, then for some $a_1,\ldots,a_m \in \mathbb{R}$, not all zero, we have:
$$\sum_{k=1}^m a_k \sin kx = 0, \text{ for all } x \in \mathbb{R}$$
This in turn implies for every $z \in S^1 = \{ \omega \in \mathbb{C} : |\omega| = 1\}$,
if we write $z$ as $e^{ix}$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 2
} |
Why does the Tower of Hanoi problem take $2^n - 1$ transfers to solve? According to http://en.wikipedia.org/wiki/Tower_of_Hanoi, the Tower of Hanoi requires $2^n-1$ transfers, where $n$ is the number of disks in the original tower, to solve based on recurrence relations.
Why is that? Intuitively, I almost feel that it ... | Your intuition is right. All but the bottom disk must be moved TWICE, so you should expect (one more than) twice the number of transfers for one fewer disk. We have $$2(2^n-1)+1=2^{n+1}-1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/422409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Does $\sum_{k=1 }^ n (x-1+k)^n=(x+n)^n$ have integer solution when $n\ge 4$? from a post in SE, one says $3^2+4^2=5^2,3^3+4^3+5^3=6^3$,that is interesting for me. so I begin to explore further, the general equation is
$\sum_{k=1}^n (x-1+k)^n=(x+n)^n$,
from $n \ge 4$ to $n=41$, there is no integer solution for $x$.
... | Your problem has been studied, and it is conjectured that only $3,4,5$ for squares and $3,4,5,6$ for cubes are such that all the numbers are consecutive, and the $k$th power of the last is the sum of the $k$th powers of the others ($k>1$).
I've spent a lot of time on this question, and no wonder did not get anything fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Quadratic form $\mathbb{R}^n$ homogeneous polynomial degree $2$ Could you help me with the following problem?
My definition of a quadratic form is: it is a mapping $h: \ V \rightarrow \mathbb{R}$ such that there exists a bilinear form $\varphi: \ V \times V \rightarrow \mathbb{R}$ such that $h(v)=\varphi(v,v)$.
Could y... | Choose a basis for $V$, call it $\{v_1,\dots,v_n\}$. Then leting $v = \sum_{i=1}^n a_i v_i$
$$
h(v) = \varphi(v,v) = \sum_{i=1}^n \sum_{j=1}^n a_i a_j \varphi(v_i,v_j).
$$
Therefore $h(v)$ is a polynomial whose terms are $a_i a_j$ (i.e. degree $2$ in the coeficients of $v$) and the coefficient in front of $a_i a_j$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Convert two points to line eq (Ax + By +C = 0) Say one has two points in the x,y plane. How would one convert those two points to a line? Of course I know you could use the slope-point formula & derive the line as following:
$$y - y_0 = \frac{y_1-y_0}{x_1-x_0}(x-x_0)$$
However this manner obviously doesn't hold when $x... | Let $P_1:(x_1,y_1)$ and $P_2:(x_2,y_2)$. Then a point $P:(x,y)$ lies on the line connecting $P_1$ and $P_2$ if and only if the area of the parallellogram with sides $P_1P_2$ and $P_1P$ is zero. This can be expressed using the determinant as
$$
\begin{vmatrix}
x_2-x_1 & x-x_1 \\
y_2-y_1 & y-y_1
\end{vmatrix} = 0 \Longle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Question in do Carmo's book Riemannian geometry section 7 I have a question. Please help me.
Assume that $M$ is complete and noncompact, and let $p$ belong to $M$.
Show that $M$ contains a ray starting from $p$.
$M$ is a riemannian manifold. It is geodesically and Cauchy sequences complete too. A ray is a geodesic... | Otherwise suppose every geodesic emitting from p will fail to be a segment after some distance s. Since the unit sphere in the tangent plane that parameterizing these geodesics is compact, s has a maximum $s_{max}$. This means that the farthest distance from p is $s_{max}$, among all points of the manifold. So the diam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
How to prove that the following function has exactly two zeroes in a particular domain? I am practicing exam questions for my own exam in complex analysis. This was one I couldn't figure out.
Let $U = \mathbb{C} \setminus \{ x \in \mathbb{R} : x \leq 0 \} $ en let $\log : U \to \mathbb{C} $ be the usual holomorphic bra... | Your estimate $|h(z)|\le 0$ (when $|z-2|=1$) cannot possibly be true: a nonconstant holomorphic function cannot be equal to zero on a circle. I marked the incorrect steps in red:
$$|\log(z)| \color{red}{\leq} \log|z| = \log|2+e^{it}| \leq \log(|2| + |e^{it}|) \color{red}{=} \log|2|\cdot \log|e^{it}| = \log(2) \cdot \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Generating all coprime pairs within limits Say I want to generate all coprime pairs ($a,b$) where no $a$ exceeds $A$ and no $b$ exceeds $B$.
Is there an efficient way to do this?
| If $A$ and $B$ are comparable in value, the algorithm for generating Farey sequence might suit you well; it generates all pairs of coprime integers $(a,b)$ with $1\leq a<b\leq N$ with constant memory requirements and $O(1)$ operations per output pair. Running it with $N=\max(A,B)$ and filtering out pairs whose other co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 2
} |
$l^2$ is not compact
Prove that the space $l^2$ (of real series $a_n$ such that $\sum_{i=1}^{\infty}a_i^2$ converges) is not compact.
I want to use the open cover $\{\sum_{i=1}^{\infty}a_i^2<n\mid n\in\mathbb{Z}^+\}$ and show that it has no finite subcover. To do that, I must prove that for any $n$, the set $\{\sum_{... | There're two easy ways to prove that $\ell_2$ is not compact. First, is to say that its dimension is infinite, hence closed unitary ball is not compact, hence the space itself is not compact.
Another way to see this is to find a bounded sequence which doesn't have a convergent subsequence; as a matter of fact, a sequen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Finding the definite integral $\int_0^1 \log x\,\mathrm dx$ $$\int_{0}^1 \log x \,\mathrm dx$$
How to solve this? I am having problems with the limit $0$ to $1$. Because $\log 0$ is undefined.
| $\int_0^1 \log x dx=\lim_{a\to 0^+}\int_a^1\log x dx=\lim_{a\to 0^+}(x\log x-x|_a^1)=\lim_{a\to 0^+}(a-1-a\log a)=\lim_{a\to 0^+}(a-1) -\lim_{a\to 0^+}a\log a=-1-\lim_{a\to 0^+}a\log a$
Now $$\lim_{a\to 0^+}a\log a=\lim_{a\to 0^+}\frac{\log a}{1/a}$$
Using L' hopital's rule( which is applicable here), $$\lim_{a\to 0^+}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/422970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 9,
"answer_id": 4
} |
Two students clean 5 rooms in 4 hours. How long do 40 students need for 40 rooms? A class decides to do a community involvement project by cleaning classrooms in a school. If 2 students can clean 5 classrooms in 4 hours, how long would it take for 40 students to clean 40 classrooms?
| A student-hour is a unit of work. It represents 1 student working for an hour, or 60 students working for one minute, or 3600 students working for 1 second, or ...
You're told that cleaning 5 classrooms takes 2 students 4 hours, or $8$ student-hours. So one classroom takes $\frac{8}{5}$ or $1.6$ student-hours.
So the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/423044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Archimedean Proof? I've been struggling with a concept concerning the Archimedean property proof. That is showing my contradiction that For all $x$ in the reals, there exists $n$ in the naturals such that $n>x$.
Okay so we assume that the naturals is bounded above and show a contradiction.
If the naturals is bound... | I dont think you need the fact that $u\in N$(the fact is even not true) .And for the second difficulty the fact follows from the supremum property.As $u-1$ is not an upper bound so there exists a natural number greater than it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/423107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
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