Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Ways of writing sets My English is not very good so I hope you understand .
I don't know how to say it in English but we studied that there are 2 ways to write a set.
*
*The first way is just to list these {you list the elements here and you separate with commas}
*The second way is to find a special property that ... | The solutions are as follows:
*
*1: $\{ x \in \mathbb{Z}_{\geq 3} | x = 0 \, \mathrm{mod}\, 3 \, \mathrm{or}\, x = 0 \, \mathrm{mod}\, 5, \, \max(x) = 24 \} -$ See this
*3: $\{x(n) \in \mathbb{Z}_{> 0} | 2\leq n \leq 11, x(n) = x(n - 1) + x(n - 2), x(0) = 0, x(1) = 1\} -$ See this
The second series contains prim... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find a quadratic equation with integral coefficients whose roots are $\frac{α}{β}$ and $\frac{β}{α}$ The roots of the equation $2x^2-3x+6=0$ are α and β. Find a quadratic equation with integral coefficients whose roots are $\frac{α}{β}$ and $\frac{β}{α}$.
The answer is $4x^2+5x+4=0$
I don't know how to get to the answe... | As $\alpha+\beta=\dfrac32, \alpha\beta=\dfrac62$
let $y=\dfrac\alpha\beta\iff y+1=\dfrac3{2\beta}\iff\beta=\dfrac3{2(y+1)}$
But as $\beta$ is a root of $$2x^2-3x+6=0$$
$$2\left(\dfrac3{2(y+1)}\right)^2-3\left(\dfrac3{2(y+1)}\right)+6=0$$
As $y+1\ne0,$ multiply both sides by $\dfrac{2(y+1)^2}3$ to find $$0=3-3(y+1)+4(y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2446243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 4
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Solve PDE: $xu_x + yu_y + u_z = u , u(x,y,0)=h(x,y)$ Here is problem: solve PDE, quasilinear, problem
$xu_x + yu_y + u_z = u , u(x,y,0)=h(x,y)$.
Here what I did: Given $\Gamma: <x=s, y=s, z =0, u=h(s)>$
$dx/dt =x$, $dy/dt = y$, $dz/dt=1$ and $du/dt = u$
$x=se^t$, $y=se^t$, $z=t$ , $u=h(s)e^t$. Now I am stuck. I do not ... | The characteristic equations are given by:
$$ dx/x=dy/y=dz/1=du/u.$$
$dx/x=dy/y \implies \ln(x)=\ln(y)+\ln(c_1) \implies x/y = c_1$
$dy/y =dz/1 \implies \ln(y)=z+c_2 \implies c_2 = \ln(y)-z$
$dz/1=du/u \implies z=ln(u)-\ln(c_3) \implies u = c_3e^{z}$
Now use that $c_3=F(c_1,c_2)$, hence:
$u=F(c_1,c_2)e^{z}=F(x/y,\ln(y)... | {
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If α and β are the roots of the equation $3x^2+5x+4=0$, find the value of $α^3+β^3$ If α and β are the roots of the equation $3x^2+5x+4=0$, find the value of $α^3+β^3$
How can I factorize the expression to use the rule of sum and product of roots?
The answer is $\frac{55}{27}$
| Using the general form of quadritic equation, $x^2 -(a+b) x + ab$,
we get the values of $a+b$ and $ab$.
Now, the expression $a^3+b^3$ can be reduced to $(a+b)^3 -3ab(a+b)$.
Substitute the value of $a+b$ and $ab$ in the above equation.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Constructing a divergent series Please would you help me with this question? I've been thinking about it for ages but I've made very little headway, so if possible a hint would be ideal.
Let $\sum_{n=1}^∞{x_n}$ be a divergent series, where $x_n > 0$ for all $n$. Show that there is a divergent series $\sum_{n=1}^∞{y_... | Based on this result:
If the positive series $\sum a_n$ diverges and $s_n=\sum\limits_{k\leqslant n}a_k$ then $\sum \frac{a_n}{s_n}$ diverges as well
Let's take $S_n=\sum\limits_{k=1}^n x_n\to+\infty\quad$ then $\quad\displaystyle y_n=\frac{x_n}{S_n}$ agrees with your requirements.
*
*$x_n>0\implies S_n>0\implies y_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integral Representation of the Dottie Number I noticed that a lot of commonly-used mathematical constants that can't be expressed in closed-form can be expressed by integrals, such as
$$\pi=\int_{-\infty}^\infty \frac{dx}{x^2+1}$$
and
$$\frac{1}{1+\Omega}=\int_{-\infty}^\infty \frac{dx}{(e^x-x)^2+\pi^2}$$
I was wonderi... | You seem to have provided an answer to your own question. Thank you. I will bookmark this post; it may help me in some of my own work. I'd like to add a little more information; it may suggest other ways to tackle your goal, perhaps direct you to a more concise solution.
The Dottie Number (D) also happens to be the sol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2446725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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Certain property of finite field $F_p$ Let $F_p$ be a finite field of order $p$ where $p$ is a prime number. Let $\{ \alpha_1, \ldots, \alpha_{p-1} \}$ be a multi-set with $\alpha_i \in F_p$ and each $\alpha_i$ non-zero.
I want to show that $$\sum_{i\in K} \alpha_i = -1$$ for some subset $K \subseteq \{1,\ldots, p-1\}... | *
*Since any element $\alpha \in \mathbf{F}_p^*$ generates $(\mathbf{F}_p,+)$, the only subset $S \subset \mathbf{F}_p$ such that $S= S \cup (\alpha+S) \bmod p$ is $\mathbf{F}_p$.
*If all the $\alpha_i$ are the same element $\alpha$, take $b \equiv -\alpha^{-1} \bmod p$ so that $\sum_{i=1}^b \alpha_i = -1$.
*Otherw... | {
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"question_score": "2",
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Proving that this series has a finite sum Consider the following series
$$\sum_{n=1}^{\infty}\dfrac{\log n}{n(n-1)}$$
I have tried to use the ratio test, but then I would get
$$\dfrac{(n-1)\log(n+1)}{(n+1)\log n}$$
And taking the limit as $n \to \infty$ would yield 1 so I don't think it would help.
| Note that $\log(n)\le \sqrt{n}$. Hence, we have
$$\left|\frac{\log(n)}{n(n-1)}\right|\le \frac{1}{n^{1/2}(n-1)}$$
Can you finish now?
| {
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First-order sentence involving only a symmetric binary relation with only infinite models It's known that there are sentences of first-order logic which only have infinite models, even if our language consists only of a binary relation $R$. An example of such a sentence is
$$\forall x \exists y Rxy \wedge \forall x \f... | Let us define the formulas:
$$\kappa_3(x)=\exists u\exists v(Rxu\land Rxv\land Ruv)$$
$$\kappa_4(x)=\exists u\exists v\exists w(Rxu\land Rxv\land Rxw\land Ruv\land Ruw\land Rvw)$$
$$\alpha(x)=\neg\kappa_3(x)$$
$$\beta(x)=\kappa_3(x)\land\neg\kappa_4(x)$$
$$\gamma(x)=\kappa_4(x)$$
$$\sigma(x,y)=\alpha(x)\land\alpha(y)\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2447174",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "6",
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If a $10$-element subset whose sum is 155 is removed from $\{1,2,...,30\}$, then it's always possible to split the remaining set into equal parts. In other words, I'd like to prove that, given the set $\{1,2,...,30\}$, if we designate a $10$-element subset whose sum is 155 (for example $\{1,2,3,4,5,26,27,28,29,30\}$), ... | So first pair up the numbers as $r, 31-r$ and note that any such pair can replace any other without changing the sum of a subset.
You are given a set of ten elements adding to $155$. Consider constructing a second set as made up of elements $31-r$ where $r$ is in the first set. The sum of elements in both is then $310... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2447321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Card guessing game There is a pile of $52$ cards with $13$ cards in each suit (diamonds, clubs, hearts, spades). The cards are turned over one at a time. At any time, the player must try to guess its suit before it is revealed. If the player guesses the suit that has the most cards and if there is more than one suit wi... | This is based on 5xum's answer but I think a better explanation...
Initially there are four suits that have equal number of cards. Let us assume we actually are very unlucky and guess incorrectly until there is only one suit left with 13 cards in. We will then keep guessing that suit until the first card in that suit i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2447434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
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Separable algebras over a non-commutative ring
What is the 'correct' definition of a separable algebra over a non-commutative ring? Are there known results about such algebras? Examples?
Recall that one of the equivalent definitions of a separable algebra $A$ over a commutative ring $R$ says that $A$ is a projective... | The standard definition is: an inclusion of algebras $R\subseteq A$ is a separable extension of algebras if the map $\mu:A\otimes_RA\to A$ induced by the multiplication of $A$ is split as a map of $A$-bimodules.
This is used in many places and has many applications. For example, if $G$ is a finite group, $k$ a field of... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove that $f(n) \geq n$ Let $f$ be a function from $\mathbb{N}$ to $ \mathbb{N}$ such that $\forall n \in \mathbb{N}$, $f(f(n)) <f(n+1)$.
Prove that $\forall k \geq n$, $f(k) \geq n$.
I've put much time and effort to solve this but unfortunately couldn't.
I tried to prove a simpler version $\forall n\geq 0$, $f(n) \ge... | I think that the simpler version is actually harder to prove than the original question. We can prove the original statement with induction to $n$:
Base case: For $n = 0$ it clearly holds, because $f(k) \geq 0$ for all $k \in \mathbb{N}$.
Inductive step: Suppose that it holds for all $n = m$ with $m \in \mathbb{N}$. Le... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How many times must I toss a coin in order that the odds are more than 100 to 1 that I get at least one head? How many times must I toss a coin in order that the odds are more than 100 to 1 that I get at least one head?
I believe that it is 10 times as if the coin is flipped ten times there is only ten outcomes that in... | The chance of getting at least one head is $1 - (\frac{1}{2})^n$
This equation has to equal $99 \%$, which gives us the following:
$$1 - \Big(\frac{1}{2}\Big)^n = 0.99$$
$$\Big(\frac{1}{2}\Big)^n = 0.01$$
$$n \log\frac{1}{2} = \log0.01$$
$$n = \frac{\log0.01}{\log\frac{1}{2}}$$
$$n = 6.6438...$$
Obviously we can't have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2447854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Maximal real subfield of a number field Let $L\subset \mathbb{C}$ be a number field such that $L / \mathbb{Q}$ is a Galois extension, then is it true that $[L:L\cap \mathbb{R}]\leq 2$?
Thanks very much!
| Yes. If $c$ is complex conjugation, $c$ acts on $L$, as an automorphism of order $1$ or $2$, so its fixed field, $L\cap\Bbb R$, is a subfield of $L$ of index $1$ or $2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2448011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simple humps of a continuous function Suppose $y=f(x)$ is a continuous function and $f(x)=f(x')$ with $x≠x'$. Can we always find a sub-interval of the interval $[x, x']$ where $f$ is a simple hump or trough? By a simple hump, I mean a curve that rises monotonically from a certain height $y=k$, reaches a maximum, and th... | No, we can't necessarily do that. Take, for instance, the Weierstrass function, whose graph is a fractal, going up and down infinitely many times on any interval.
Note that if your function is indeed a bumb, meaning that it is first increasing then decreasing, then it is necessarily differentiable almost everywhere.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2448122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Image sets in Complex Analysis I was given the problem "what is the image set of the first quadrant in the $z$ plane under the mapping $w=z^4$, but I have no idea how to even think about this.
The most I did was write $w=|z|^4 (\cos(4\theta) +i\sin(4\theta))$
But again, Im not sure if that's helpful or not. How do we p... | HINT
The complex number $(r, \theta)$ is mapped to $\left(r^4, 4 \theta\right)$.
What is the range of $r$ and $\theta$? What is the range of $r^4$ and $4 \theta$? What is the resulting range for $\left(r^4, 4 \theta\right)$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2448249",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is $N$ normal to $HN$ if $H$ subgroup and $N$ normal subgroup of the group $G$ If $H$ is a subgroup of $G$ and $N$ is a normal subgroup of $G$, then whats the relation between $N$ and the subgroup $HN$ in respect to normality, i.e. must $N$ be normal to $HN$ ?
| Given:
*
*$G$ is a group.
*$H\lt G$.
*$N$ normal to $G$.
To Show: $N$ normal to $H\circ N$.
Possible Proof:
Now,
$H\circ N$ = {$h\circ n | h\in H, n\in N$}$ = H\cup N$.
Since, $H\lt G$ and $N$ normal to $G$ $\implies H, N \lt G \implies (H\cup N)\subseteq G\implies H\circ N\subseteq G$.
Now,
$N$ normal to $G$
... | {
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"timestamp": "2023-03-29T00:00:00",
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Discovering Quadratic Reciprocity Is there anything similar to this (page written by Field Medalist Timothy Gowers) for quadratic reciprocity ?
I mean, the link there explains how you can figure out the solution of cubic equation by yourself without having a suddent flash of inspiration/ genius genes. Is there some si... | The first part of this paper http://www.math.ubc.ca/~belked/lecturenotes/620E/Frei%20-%20The%20Reciprocity%20Law%20from%20Euler%20to%20Eisenstein.pdf shows you how the QR law was discovered historically and shows the path all the way back to Diophantus.
It seems that the first question explicitly stated that is equiva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2448519",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is there a $2 \times 2$ real matrix $A$ such that $A^2=-4I$? Does there exist a $2 \times 2$ matrix $A$ with real entries such that $A^2=-4I$ where $I$ is the identity matrix?
Some initial thoughts related to this question:
*
*The problem would be easy for complex matrices, we could simply take identity matrix multi... | $A = \left [\begin{array}{ccc}
0 & 2 \\
-2 & 0 \\
\end{array} \right ]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2448604",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Alternatives to Fano's Axiom in Projective Space In Projective Geometry, Fano's Axiom says:
The three diagonal points of a complete quadrangle are never collinear.
I would like to prove this from more basic Axioms within three-dimensional Projective Space. The theorem of Desargues is non-trivial in plane geometry, bu... | You can't prove Fano's axiom from 3-dimensional geometry because the projective plane over the field $F_2$ with two elements does not satisfy Fano's axioms. Recall that the projective plane can be defined starting with 3-dimensional space over $F_2$ by a suitable equivalence relation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2448707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Closed form solution for $\int_{0}^{2\pi}|\cos^N x||\sin^M x|\cos^n x\sin^m x dx$ I am trying to calculate Fourier series coefficients (by hand) and the integrals I need to solve are of the following type
$$I(N,M,n,m)=\int_{0}^{2\pi}|\cos^N x||\sin^M x|\cos^n x\sin^m x dx,$$
in which $N,M,n,m \in \{0,1,2,3\}$. I tried ... | Hint: An easy way is to use the following identity which is quite easy to prove
which is an easy consequence of Integration, trigonometry, gamma/beta functions
| {
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"timestamp": "2023-03-29T00:00:00",
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Existence of a subsequence such that $\lim_{n\to \infty} \|x_{n_k}-x\|=\ell>0$ in a Banach space? Let $x_n$ be a sequence in a uniformly convex Banach space $E$ such that $x_n \to x$ weakly in $E$ and $\|x_n\|_E \to \|x\|_E$. Then $x_n \to x$ strongly in $E$.
To show this by contradiction a proof I'm reading states tha... | The point at which you seem to have a question is not about Banach spaces; it's about sequences of real numbers. You have a sequence $\{a_n\}_{n=1}^\infty$ of real numbers for which
$$
\limsup_{n\to\infty} a_n >0
$$
and the question is: how does this imply that there is a subsequence $\{a_{n_k}\}_{k=1}^\infty$ for whic... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove or give a counterexample
There exists a non-negative number s such that for all non-negative numbers t, the inequality s $\geq$ t holds
Don't really know if this statement is true or false. I can see it being either, but can't really approach a proof or a counterexample for it.
Thanks.
| The quantifiers are in the order "first $s$, then $t$". Since $t$ may depend on $s$, we should think of $t$ as a function of $s$. Can we write constraints for $t$ in terms of $s$ that make the inequality true?
We find that $0 \leq t$ because $t$ is nonnegative and $s \geq t$ is our inequality, so we must have $t \in ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Definite integral on $[0,\pi]$ How to calculate the following integral if $\varepsilon \in (0,1)$:
$$\int \limits_{0}^{\pi}\frac{d\varphi}{(1+\varepsilon\cos \varphi)^2}$$
| Hint:
Use the substitution
$$
s=\tan{\frac{\varphi}{2}}, \quad \sin{\varphi}=\frac{2s}{s^2+1}, \quad \cos{\varphi} = \frac{1-s^2}{s^2+1}.
$$
(Apparently more details are given in this answer.)
| {
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Converting riemann sum to definite integral Another Riemann sum I'm struggling to convert to a definite integral... $\lim_{n\to \infty}$ $\sum_{i=1}^n$ $\frac{6n}{9n^2+4i^2}$. Any ideas as to what my $x_i$ should be in this case?
| Let $x_i=\dfrac{i}{n}$ then $x_1=\dfrac{1}{n}\to0$, $x_n=\dfrac{n}{n}\to1$and $\Delta x=\dfrac1n$ so
$$\lim_{n\to\infty}\sum_{i=1}^n\dfrac{6}{9+4(\dfrac{i}{n})^2}\frac1n=\int_0^1\dfrac{6}{9+4x^2}dx$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Example of a semigroup S with no identity element and a subgroup G of S I need an example of a semigroup S without an identity element and a subgroup G of S.
I have found it easy to find/make semigroups without identities but then making a subgroup from it has not been fruitful. An example or a hint would be much appre... | Take your favorite group $G$. Let $S$ consist of $G$ together with two additional elements $a$ and $b$, and extend the multiplication in $G$ by defining $as = sa = bs = sb = b$ for all $s \in S$. You can confirm this operation is associative, and clearly it has no identity since any product with $a$ is $b$. But $G$ ... | {
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Finding the polar cone of the given cone Given a closed convex cone $D$ in $\mathbb{R}^{n}$, the cone $K_{2} \in \mathbb{R}^{m}$ is defined by $$ K_{2} = \{ y = (y^{1}, y^{2}, \cdots , y^{m}): y^{i} \in \mathbb{R}^{n},\, i= 1, \cdots , m, \, y^{1} + y^{2} + \cdots + y^{m} \in D \} $$
I need to describe its polar cone $... | Here's an initial observation, but not a full solution. Changing notation slightly,
$$K_2 = \{ Y = \begin{bmatrix} y_1 & \cdots & y_m \end{bmatrix} \in \mathbb R^{n \times m} \mid y_1 + \cdots + y_m \in D \}.$$
(Here $y_i$ is the $i$th column of the matrix $Y$.) A matrix $X \in \mathbb R^{m \times n}$ belongs to $K_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2449707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of functions $f : A \to A$ such that $f(f(x))=f(x)$ If $A=\left\{1,2,3,4,5\right\}$ Then
Find Number of functions $f : A \to A$ such that $f(f(x))=f(x)$
Case $1.$ if $f$ is injective then $f(f(x))=f(x)$ $\implies$ $f(x)=x$, hence there is only one injective function which is an identity function.
Case $2.$ Whe... | There are other possibilities. Here is one:
1 goes to 2
2 goes to 2
3 goes to 5
4 goes to 5
5 goes to 5
Expanding on this answer: The framework for these functions is to have some number (at least one) of fixed points, and have everything else map into the set of fixed points.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2449839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 1
} |
How to find minimum and maximum value of x, if x+y+z=4 and $x^2 + y^2 + z^2 = 6$? I just know that putting y=z, we will get 2 values of x. One will be the minimum and one will be the maximum. What is the logic behind it?
| $z=4-x-y$
$x^2+y^2+(4-x-y)^2-6=0$
Differentiate wrt $x$ and $y$
$2x-2(4-x-y)=0$
$2y-2(4-x-y)=0$
Gives $x=y=4/3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2449942",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $\Delta \le \frac {\sqrt{abc(a+b+c)}}{4}$ If $\Delta$ is the area of a triangle with side lengths a, b, c, then show that: $\Delta \le \frac {\sqrt{abc(a+b+c)}}{4}$. Also show that equality occurs in the above inequality if and only if a = b = c.
I am not able to prove the inequality.
| One has $\Delta = \frac{1}{4}\sqrt{(a+b+c)(b+c-a)(c+a-b)(a+b-c)}$ (See this link).
Moreover, one has $(a+b-c)(b+c-a) \leq (\frac{a+b-c + b+c-a}{2})^2 = b^2$
So $[(a+b+c)(b+c-a)(c+a-b)(a+b-c)]^2 \leq a^2b^2c^2$.
Thus, $\Delta \leq \frac{1}{4}\sqrt{abc(a+b+c)}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2450103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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One dimensional martingale is recurrent? Consider a sequence of independent and identically distributed random variables $X_i$ with $P(X_i >1 )>0$ and $E(X_i) = 0$.
therefore $M_n = \sum_{i=1}^n X_i$ is a martingale.
I would like to prove that $M_n\geq 0$ infinitely often with probability $1$.
I thought about using up... | No, your reasoning does not work. The martingale covergence theorem requires $M_n(\omega) \geq 0$ for all $\omega \in \Omega$ (and not just $M_n(\omega) \geq 0$ for some $\omega \in \Omega$). To fix this gap in your reasoning you have to show that
$$\mathbb{P} \left( \limsup_{n \to \infty} M_n < 0 \right) \in \{0,1\}.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2450225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
for $x^2+y^2=a^2$ show that $y''=-(a^2/y^3)$ For $x^2+y^2=a^2$ show that $y''=-(a^2/y^3)$
I got that
$y^2=a^2-x^2$
$y'=-x/y$
$y''=(-1-y'^2)/y$
But then I get stuck.
| Implicit differentiation gives
$$
2x+2yy'=0 \tag{*}
$$
Differentiate again (after removing the common factor $2$):
$$
1+(y')^2+yy''=0 \tag{**}
$$
Now (*) implies $y'=-x/y$, so you can substitute in (**):
$$
1+\frac{x^2}{y^2}+yy''=0
$$
Isolate $y''$ and go on:
$$y''=-\frac{y^2+x^2}{y^3}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2450343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Does $\limsup_{x\to\infty}\int_x^{x+h}f'(y)dy = 0$, $\forall h>0$, imply $\lim_{x\to\infty}\int_x^{x+h}f'(y)dy = 0$, $\forall h>0$? I have an absolutely continuous function $f:[0,\infty)\to[0,\infty)$ that satisfies $\limsup_{x\to\infty}\int_x^{x+h}f'(y)dy = 0$ for all $h>0$. I need to check if it is true or false that... | Sketch: Choose $N\in \mathbb N$ such that $2^n +n^2+n < 2^{n+1}$ for $n\ge N.$ Define
$$g = \sum_{n=N}^{\infty}\left ( \frac{1}{n}\chi_{(2^n,2^n+n^2)} - \chi_{(2^n+n^2,2^n+n^2+n)}\right ).$$
Now define $f(x) = \int_0^x g.$ Then $f$ is a counterexample. I'll leave it here for now. Ask if you have questions.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finitely many conjugacy classes of finite subgroups of given order Let $G$ be a periodic locally soluble group with finite Sylow p-subgroups for all primes p.
It is know that in these conditions $G$ is residually finite. Moreover it can be proved that $G$ has only finitely many conjugacy classes of finite subgroups of... | You know that there are only finitely many conjugacy classes of subgroups of $HN$ that are isomorphic to $H$.
Supose $g_1,g_2 \in N_G(L)$ and $H^{g_1}$ and $H^{g_2}$ are in the same conjugacy class in $HN$. Since $g_1 \in N_G(L)$, we have $HN=H^{g_1}N$, so there exists $n \in N$ with $H^{g_1n}=H^{g_2}$, so $g_1ng_2^{-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2450617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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On the foliation determined by isometric flow on homogeneous manifold Let $(M,g)$ be a non-compact homogeneous, 1-connected Riemannian manifold with $G := Isom(M,g)$ and let $v \in \Gamma (M,TM)$ be a $G$-invariant, nowhere-vashining vector field, such that $||v|| = 1$ everywhere. Then $v$ determines a global flow $\ph... | Consider the 3-dimensional round sphere and its isometry group $O(4)$. This group contains the subgroup $U(2)$, whose center is isomorphic to $U(1)$ (scalar unitary matrices). Of course, $O(4)\ne U(2)$, but one can modify the constant curvature metric on $S^3$ making it a Berger sphere $B=S^3_{t,\epsilon}$, whose isome... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2450710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Showing that $f(x,y)=\dfrac{x}{y}$ is continuous when $x>0$ and $y>0$. So I am hoping to show that $f(x,y)=\dfrac{x}{y}$ is continuous when $x>0$ and $y>0$.
I am not sure how to approach this problem. My idea was that taking $$\dfrac{\partial f}{\partial x}=\dfrac{1}{y}$$ and $$\dfrac{\partial f}{\partial y}=\dfrac{-x}... | For $f(x)$ and $g(y)$ continuous then in $(x_0,y_0)$ where $g(y_0)\neq0$ we have:
*
*$\forall \varepsilon>0,\exists \delta_1>0\mid |x-x_0|<\delta_1\implies |f(x)-f(x_0)|<\varepsilon$
*$\forall \varepsilon>0,\exists \delta_2>0\mid |y-y_0|<\delta_2\implies |g(y)-g(y_0)|<\varepsilon$
Since $g(y_0)\neq 0$ it is possi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2450838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove square root of 3 is irrational I have read several articles on math.stackexchange.com, and also this article: https://www.grc.nasa.gov/www/k-12/Numbers/Math/Mathematical_Thinking/irrationality_of_3.htm
I still can't quite understand why one of the numbers can't be even.
Especially this part:
"Since any choice of ... | Your citation establishes that $a$ and $b$ must either both be even, or both be odd:
*
*$b$ is either even or odd (2 cases)
*
*If $b$ is odd, then $b^2$ is odd. Hence $3b^2$ is odd, being the product of two odd numbers. But $3b^2 = a^2$, and so $a^2$ is odd. And this means $a$ must be odd, since the square of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2450949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Eigenbundles of an self-adjoint endomorphism of complex vector bundles Let $L\colon E\to E$ be an endomorphism of complex Hermitian vector bundle over a smooth manifold $M$. Suppose $L$ is self-adjoint and its eigenvalues are constant. How do I see that $E$ can be decomposed into a direct sum of eigenbundles.
For each ... | If I understand well, you assume that the eigenvalues $a_1,\dots,a_k$ of $L_x$ are independent of the point $x$. Then you can sort things out by using the the projection onto an eigenspace of a diagonalizable linear operator can be written as a polynomial in the operator. For each $i=1,\dots,k$, define
$$
P_i:=(\prod_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $A$ and $B$ have $n$ and $m$ elements respectively with $A\cap B=\emptyset$. Prove that $A\cup B$ has $m+n$ elements. If $A$ and $B$ have $n$ and $m$ elements respectively with $A \cap B=\emptyset$. Prove that $A \cup B$ has $m+n$ elements.
The solution in the book starts by letting $f$ be a bijection from $\{$1,..m... | Since $A$ has $m$ elements and $B$ has $n$ elements, there are bijections $f:\{1,\ldots,m\}\to A$ and $g:\{1,\ldots,n\}\to B$.
Now define $h:\{1,\ldots,m+n\}\to A\cup B$ by
$$
h(i) = \begin{cases}
f(i) & \text{if}\ i\in\{1,\ldots,m\}, \\
g(m-i) & \text{if}\ i\in\{m+1,\ldots,m+n\}.
\end{cases}
$$
First you should convin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Volume of a bounded solid in R3 What is the volume of the solid in xyz-space bounded by \begin{align}
y = 2 - x^2 \\
y = x^2 \\
z = 0 \\
z = y + 3 ?
\end{align}
I have formatted the problem as follows:
$$\iiint 1 \,dx\,dy\,dx$$
\begin{align}
-1 ≤ x ≤ 1 \\
2 - x^2 ≤ y ≤ x^2 \\
0 ≤ z ≤ y + 3 \\
\end{align}
When I ... | The volume is symmetric with respect to the $yz$-plane, so in the integral:
$$
\int _{-1}^1\int_{x^2}^{2-x^2}\int_0^{y+3} dzdydx
$$
the two parts $\{-1<x<0\}$ and $\{0<x<1\}$ have opposite sign and its sum is null. You have to express the volume as:
$$
V=2\int _{0}^1\int_{x^2}^{2-x^2}\int_0^{y+3} dzdydx
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451286",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Real Analysis: density of rationals and reals My textbook in real analysis proves that the rationals are dense in the reals by using first the archemidean principle and then constructing a fairly long and contrived proof.
What I am wondering is, is that could you not first show that the sum of two rationals divided by... | You can do that if you know about decimal representations (yes, you know, but it looks like you are there building reals from the axioms).
If you were to define rationals as periodic decimals and irrationals as non-peridic, then you would have, for irrational $\alpha=a_0.a_1a_2...a_n...$, the rational $a_0.a_1a_2,...a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $\limsup\limits_{n\to\infty}a_n=+\infty$ and $\limsup\limits_{n\to\infty}b_n\in\mathbb{R}$, then $\limsup\limits_{n\to\infty}(a_n+b_n)=+\infty$? There is the following exercise in a book:
$\limsup \limits_{n \to \infty} (a_n + b_n) \leq \limsup \limits_{n \to \infty} a_n + \limsup \limits_{n \to \infty} b_n $
And ... | The first is easy to prove:
Since,for all $m \in \mathbb{N}$:
$$
\limsup_{n\to\infty}a_n \leq \sup_{k\geq m} a_k
$$
If:
$$
\limsup_{n\to\infty}a_n = +\infty
$$
Then:
$$
\sup_{k\geq m} a_k = +\infty
$$
Therefore if:
$$
\limsup_{n\to\infty}a_n = +\infty\\
\limsup_{n\to\infty}b_n = +\infty
$$
Then, for all $k \in \mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451497",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Endomorphisms: if $Im(g)$ is contained in $Im(f)$ then necessarily $g=f \phi$? Let $f$ and $g$ be endomorphisms of a vector space $V$.
If $Im(g)$ is contained in $Im(f)$ then does there necessarily exist an endomorphism $\phi$ of $V$ such that $g=f \phi$?
Do we also have that $Ker(g)$ contained in $Ker(f)$ implies th... | Hint : The hypothesis already tells you that for all $v\in V$, $g(v)\in Im(f)$, i.e. there exist $w\in V$ such that $f(w)=g(v)$. Use this to construct first a function $\phi$ on a basis of $V$, and then extend that function to a linear application defined on $V$.
For the second part, first take a basis $\mathcal{B}_1$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451590",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Symplectic group $Sp(n)$ acts transitively on the unit Sphere $S^{4n-1}$ I'm trying to prove that the symplectic group $Sp(n)$ acts transitively on the sphere $S^{4n-1}$, and as a consequence $Sp(n)/Sp(n-1)$ is homeomorphic to $S^{4n-1}$. To me $Sp(n)$ is the group of $2n\times 2n$ unitary complex matrices satisfying ... | First, let's figure out a nice description of $Sp(n)$. For $A\in Sp(n)$, write it in the block form $A = \begin{bmatrix} B & C\\ D & E\end{bmatrix}$ where each block is $n\times n$. Then a simple calculation shows that $AJ = J\overline{A}$ iff $A$ has the form $A = \begin{bmatrix} B & -\overline{D}\\ D & \overline{B}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451678",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Convex function and conditional expectation I have the following question. I hope someone has encountered this or can point me in a direction.
Probability space $(\Omega,\mathcal{F}, \mathbb{P})$. Let $X:\Omega\to\mathbb{R}$ be a random variable and $f:\mathbb{R} \times \mathbb{R} \to \mathbb{R}$ be a function. Suppose... | A simple counterexample. Let $X\sim N(0,1)$ and $f(x,y)=(x^2-1)y^2$. It follows that
$$
E\left [f(X,y)\right ]=0,
$$
which is convex in $y$. If $\mathcal{G}=\{\varnothing, \Omega\}$,
$E(X|\mathcal{G})=EX=0$ and so
$$
E\left [f(E(X|\mathcal{G}),y)\right ]=-y^2,
$$
which is not convex. Unfortunately, I have no idea w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451777",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Tossing coins, one fair, one unfair: how to represent the probability tree? Suppose I have a fair coin and an unfair coin. The fair coin has head, tail, the unfair coin has both heads.
You pick one coin at random and toss them two times and observe the outcomes.
Which of the figure below is a better probability tree re... | I think you interpreted it wrong for unfair coin probablity of getting head is greater than or less than half not equal to half also it may not be equal to one so your second tree emphasize on only one condition that coin is unfair with all head with probablity 1 which not seems to look correct.This is my thought maybe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2451867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Approximation for the sum of primes I have attempted to put together an approximation for the sum of primes.
I've used the much simplified $$\operatorname{li}(x)=\frac{x}{\log(x)-1}$$ combined with $$\frac{x}{2}$$ to give:
$$\frac{x^2}{2(\log(x)-1)}$$
The only thing is it is not accurate so:
1) I wonder if I've gone w... | I'm not sure why you're using $x/(\log x - 1)$ instead of $x/ \log x$ (which is what's given in the standard prime number theory), but if you were to use $x/\log x$ you would actually get the correct asymptotic: $x^2 / 2 \log x$. This is a pretty slowly converging asymptotic: the error term is on the order of $O(x^2/\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2452000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why does $arg(z^{2})\neq 2arg(z)$? I was reading a text that i found about the argument of a complex number (http://scipp.ucsc.edu/~haber/ph116A/arg_11.pdf) but I dont truly understand the proof given in that text about why is it that $arg(z^{2})\neq 2arg(z)$, so if it is true, can you give me an idea of why does it ha... | The property $\arg(z_1z_2)=\arg(z_1)+\arg(z_2)$ is not true in general.
Assuming $\arg$ takes its values on $[0,2\pi)$, then what happens when $z_1=z_2=e^{\frac{3}{2}\pi i}=-i$?
If by $\arg$ the multivalued function is meant, then it depends on how you define $2\arg(z)$.
More generally, one can define $A+B$ for two su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2452144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Two $p$-norms are not equivalent for different $p$ on $\ell_1$ Given $1\le p < r < \infty$, prove that $\|\cdot\|_p$ and $\|\cdot\|_r$ are not equivalent.
The approach I was trying is as follows:
-
Want to show that there do not exist $m,M>0$ such that $$m\|x\|_r\le\|x\|_p\le M\|x\|_r$$
for all sequences $(x_k)\in \ell... | Let $x_n\in l_1$ where the first $n$ co-ordinates of $x_n$ are each equal to $1$ and the remaining co-ordinates are all $0.$ Let $1\leq p_1<p_2<\infty.$
For brevity let $q=(1/p_1+1/p_2)/2$ and $r=(1/p_1-1/p_2)/2 .$
Let $y_n=x_n/n^q.$ Then $$\|y_n\|_{p_1}=n^r\geq 1 \; \text { and }\; \|y_n\|_{p_2} =n^{-r}\to 0 \;\tex... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Nonlinear Programming by Wiley 3rd ed, 1.2 a,b (I don't understand obj function) The following is from page 30, chapter 1 of Wiley's Nonlinear Programming: Theory and Algorithms (third edition). This is for a graduate course in Optimization Theory. My problem is that I do not understand how they came up with the object... | I am mostly comfortable with the objective function now, and propose the following solution:
Let $C(Q_j)=T[d_j k_j/Q_j + c_j d_j + Q_j h_j/2]$, under the (possibly erroneous) assumption that only $Q_j$ varies and the rest of the unknowns are constant. Under the simplification that all other unknowns are constant, let $... | {
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"timestamp": "2023-03-29T00:00:00",
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A Hard Inequality Given that $x,y,z$ are positive real numbers such that $2x+4y+7z=2xyz$, find the minimum of $L=x+y+z$.
Does anybody have a solution that is purely algebraic?
I was only able to solve it with Lagrange multipliers.
Also, how would you show that the solution given by Lagrange multipliers is in fact a glo... | For $x=3$, $y=2.5$ and $z=2$ we get the value $7.5$.
We'll prove that it's a minimal value.
Indeed, let $x=3a$, $y=2.5b$ and $z=2c$.
Thus, the condition gives
$$3a+5b+7c=15abc$$ and we need to prove that
$$6a+5b+4c\geq15$$ or
$$(6a+5b+4c)^2(3a+5b+7c)\geq15^3abc,$$ which is true by AM-GM:
$$(6a+5b+4c)^2(3a+5b+7c)\geq\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find all roots of $2x^3+16$ in $\mathbb C$
Find all roots of $p(x) = 2 x^3 + 16$ in $\mathbb C$.
I found my answers to be x = 2, -1+i$\sqrt{3}$, -1+i$\sqrt{3}$.
But when I put the expression into Symbolab, it gives me -2, 1+i$\sqrt{3}$, 1+i$\sqrt{3}$ as roots of p(x) in C.
Can someone explain where I went wrong?
This... | Another way you can calculate this is to use De Moivre's theorem, which states that if $z = r(\cos \theta + i\sin \theta)$, the $n$th roots of $z$ are $$r^{1/n}\left(\cos\frac{\theta+2\pi k}{n} + i \sin \frac{\theta+2\pi k}{n}\right)$$
From $(1)$, which is $x^3=-8$, we find that $z = 8(\cos \pi + i \sin \pi)$, so $r=8$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2452715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Show determinant is non-negative Let $A,B \in M_{2}(\mathbb R)$ . Show that $\det((AB+BA)^4 + (AB-BA)^4)\geq 0$
My attempt: expression becomes $\det(2(M-N)^2+16MN)$ where $M=(AB)^2$ and $N=(BA)^2$.
Not sure how to continue from here.
Any hints appreciated.
| Let $d=\det(AB-BA)$ and $\lambda_1,\lambda_2$ be the two eigenvalues of $AB+BA$. Since $X^2=-\det(X)I_2$ and in turn $X^4=\det(X)^2I_2$ for any traceless $2\times2$ matrix $X$, we get
$$
\det\left[(AB+BA)^4 + (AB-BA)^4\right]
=\det\left[(AB+BA)^4 + d^2I_2\right]
=(\lambda_1^4+d^2)(\lambda_2^4+d^2).
$$
As $AB+BA$ is rea... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2452842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Solving linear differential equation $y'+\frac{1}{3}sec(\frac{t}{3})y=4cos(\frac{t}{3})$ using integrating factor Given \begin{array}{l} y^{\prime} +\dfrac{1}{3}\,\sec\left(\dfrac{t}{3}\right) y=
4\, \cos\left(\dfrac{t}{3}\right) \\ y(0)=3 \end{array} where $ 0<\dfrac{t}{3}<\dfrac{\pi}{2}$, I must find the general so... | compute $$\mu(t)=e^{\int\frac{1}{3}\sec(t/3)dt}=\frac{\sin(t/6)+\cos(t/6)}{\cos(t/6)-\sin(t/6)}$$
and multiply both sides with $$\mu(t)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2452936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I find the limit of the function? How do I find the limit for the function $(1 + h)^{\frac{1}{h}}$ as $h$ goes to $0$? I do not know where to start. We just started using Logs.
| Write $$(1+h)^{\frac{1}{h}}=e^{\frac{1}{h} \ln{(1+h)}}$$
So what is the limit $\frac{1}{h}\ln{(1+h)}$ as $h \to 0$?
Take the function $f(h)=\ln(1+h)$ and we have $f(0)=0$
Then $$\lim_{h \to 0} \frac{\ln(1+h)}{h}=\lim_{h \to 0}\frac{f(h)-f(0)}{h-0}=f'(0)=1$$
Thus the general limit is $e$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A question about analytic geometry (vectors) Let $a , b, c$ be three vectors such that $a+3b+c = o$ where $o$ is $(0,0,0)$ vector. we also know $|a| = 3 , |b| = 4 , |c| =6$. we want to find $a.b + a.c + b.c$
I know we can solve it by saying $a+b+c = -2b$ and then squaring the sides. and at last we get the answer 3/2 . ... | As Lord Shark of the Unknown pointed out in the comments, it's not possible to have three such vectors sum up to zero. But nevertheless the calculation is possible: it is just a series of logical passages which do not yet lead to a contradiction, but surely enough if you continued to use these data in other you might e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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how is the minus sign understood in set theory, is it similar to the complement (\) function. Assume:
$A = \{1,2,3\},$
$B = \{2,3,4\}$
Is $A - B = \{1\}$, or is it $\{1\}$ plus the piece of $\{4\}$ that you 'owe', assuming in Venn Diagram you are subtracting a piece of $B$ from $A$ itself that do not contain the elemen... | $A-B$ is alternative notation for $A \setminus B$. They both mean the elements that are in $A$ but not in $B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Distributing balls in boxes There are five balls of identical sizes but different colors. One of the balls is red, one is blue, one is green and the other two are yellow. Moreover, there are three boxes which are numbered 1, 2 and 3. There are two more boxes but both of them are numbered 4. In how many different ways c... | One way to do it is to "temporarily mark" the identical balls and boxes so that all balls and boxes are distinguishable. That is, we imagine that we put a sticker on one of the yellow balls and on one of the boxes labeled $4.$
With the stickers, we have five distinguishable balls in five boxes
and can apply the known f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The union of two open sets is open (In metric Spaces) Let $X$ a set not empty and $(X,d)$ a metric space. Prove he union of two open sets is open.
My proof:
Let $A_1,A_2$ open sets, we need to prove $A_1\cup A_2$ is open.
As $A_1,A_2$ are open set, then for all $a_1,a_2\in A_1,A_2$ respectively we have $r_1,r_2>0$ ... | Considering $a_1,a_2$ is confusing. In fact, the union of any collection of open sets is open. Let $A=\bigcup_t A_t$, where all $A_t$ are open. Let $x\in A$. So, there is at least one $t$ for which $x\in A_t$. Therefore for some $r>0$ we have $B(x,r)\subset A_t\subset\bigcup_t A_t=A$.
This proof works in any topologica... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Why is $\left\{ n\in \mathbb{N}:n^2 \right\}$ nonsense, $ $ but $\left\{ n^2: n\in \mathbb{N} \right\}$ correct? Why is $\left\{ n\in \mathbb{N}:n^2 \right\}$ nonsense, $ $ but $\left\{ n^2: n\in \mathbb{N} \right\}$ correct?
From my understanding, $\left\{ n\in \mathbb{N}:n^2 \right\}$ should be read as:
"The s... | To confirm what everybody else has said, after the colon, you need a sentence that may or may not be true. Since there’s no verb in what follows the colon, your formulation is bad grammar.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 5
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Given a Markov chain $X \rightarrow Y \rightarrow Z$, under what condition $I(X;Y) = I(X;Z)$ A theorem (The Data Processing Inequality) states that
if $X \rightarrow Y \rightarrow Z$, then $I(X ; Y ) \geq I ( X ; Z )$
Question: I was wondering under what conditions $I(X;Y) = I(X;Z)$?
The proof:
Using chain rule of ... | As stated in Cover and Thomas's Elements of Information theory 2e (in the discussion of Theorem 2.8.1, the data processing inequality), you have equality iff $X \to Z \to Y$ is a Markov chain (think of why this is equivalent to $I(X;Y|Z) =0$ and the joint distribution of $X,Y$ given $Z$ under the markovian assumption I... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing isomorphism between normal subgroups. This a fairly vague and basic example but i am struggling with it.
Suppose G is a group and it is the internal direct product of two subgroups H and K. $H\times K=G$ write down a explicit isomorphism $\phi :K \to G/H$ and prove that K is isomorphic to $G/H$.
Let $ k \in K $... | $a\in G/H$ then $a= xH$, for $x\in G$. Note that $G = H\times K$, then there exists $h\in H$, $k\in K$ such that $x =kh$. So one has $\phi(k) = kh = kH = a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453871",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of ways of arranging ten girls and three boys if the boys separate the girls into groups of sizes $3, 3, 2, 2$ Ten girls are to be divided into groups of sizes $3,3,2,2$. Also, there are $3$ boys. Number of ways of linear sitting arrangement such that between any two groups of girls, there is exactly one boy (no... | For $10$ girls we have $10!$ permutations. We have $3!$ for boys. We just put the boys in the right positions. Thus, the result is $10! \times 3!$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2453992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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What is the area of canvas required to make a conical surface tent with height $35~\text{m}$ and radius of base $84~\text{m}$?
A conical circus tent is to be made of canvas. The height of the tent is $35~\text{m}$ and the radius of the base is $84~\text{m}$. What is the area of canvas required?
I calculated slant hei... | The slant height is $a=\sqrt{h^2+r^2}=91m$
$$S=\pi r a \approx 3.14\cdot 84 \cdot 91\approx 24024m^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2454084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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what is the image of the set $ \ A=\{(x,y)| x^2+y^2 \leq 1 \} \ $ under the linear transformation what is the image of the set $ \ A=\{(x,y)| x^2+y^2 \leq 1 \} \ $ under the linear transformation $ \ T=\begin{pmatrix}1 & -1 \\ 1 & 1 \end{pmatrix} \ $
Answer:
From the given matrix , we can write as
$ T(x,y)=(x-y,x+y) ... | Another way to look at this question is to look at the action of $T$ on the complex plane $\mathbb{C}$. If we consider $\mathbb{C}$ as a vector space over $\mathbb{R}$, it is a $2$-dimensional space with (standard) basis $(1, i)$. Then a complex number $x + iy$ in cartesian form maps to $\begin{pmatrix} x\\ y \end{pmat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2454179",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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What is $\Omega$ in the context of Poisson's equation? I have recently started a new course on PDE's and have already stumbled on an example that I'm struggling to understand:
The main aspect of this example that I am struggling to wrap my head around is not in the proof, it's in the way that $\Omega$ is used in the s... | For PDEs, it is very common to denote the domain of the PDE with $\Omega$.
Common examples are a circle/ball
$$
\Omega = B_1(0) := \{ x\in \mathbb R^n : \|x\| < 1 \}
$$
or a square
$$
\Omega = (0,1)\times (0,1)
$$
In this case $\partial\Omega$ denotes the boundary of the domain, and not partial derivatives!
For exam... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Throw 10 dice, probability of having 6 the $6$th time at the $6$th throw
We throw a die $10$ times. What is the probability of having the number 6 appear for the first time at the $6$th throw?
I have tried $$\frac{5}{6} \cdot \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{5}{6} \cdot \frac{1}{6} \cdot (... | *
*For your first question:
$$\left(\frac56\right)^5\cdot\frac16$$
You want to get anything but six on the first five rolls and six on the sixth roll. After that, you do not care whether you get a six or not, so the probability of succeeding is one after the sixth throw. You don't need to use the factorial at all in t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2454388",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Sequence of Measurable Functions Converging Pointwise
Let $\{f_n\}$ be a sequence of measurable functions on $E$ that converges pointwise almost every on $E$ to the function $f$. Then $f$ is measurable.
Proof:
Let $g : E \to \Bbb{R}$ be defined by $g(x) = \lim_{n \to \infty} f_n(x)$ for every $x \in E$. Then $f = g$... |
Proposition
Let $f_n:E \to \Bbb{R}$ a sequence of measurable functions such that $f_n(x) \to f(x)$ pointwise $\forall x \in E$.Then $f$ is measurable.
Proof
If $f_n$ are measurable ,then $\limsup_n f_n,\liminf_n f_n$ are also measurable.
But $f_n$ converges to $f,\forall x \in E\Rightarrow f(x)=\limsup_nf_n(x)=\liminf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2454639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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ln(1+x) maclaurin series I found the first four derivatives of
$ f(x) = ln(1+x) $
Then for all n > 1, $$ f^n(x) = \frac{(-1^{n+1})(n-1)!}{(1+x)^n } $$
So, $$ f^n(0) = (-1^{n+1})(n-1)! $$
By definition Maclaurin Series are defined as:
$$\sum_{n=0}^{\infty} \frac{f^n(0)}{n!}x^n$$
*Since the $ f^n(0) $ is only true when ... | If $f(x)=\log(1+x)$, then we have
$$\begin{align}
f^{(1)}(x)&=(1+x)^{-1}\\\\
f^{(2)}(x)&=-(1)(1+x)^{-2}\\\\
f^{(3)}(x)&=(-1)(-2)(1+x)^{-3}\\\\
\vdots
f^{(n)}(x)&=(-1)(-2)\cdots (-(n-1))(1+x)^{-n}\\\\
\end{align}$$
Therefore, we can write
$$\begin{align}
f(x)&=\log(1+x)\\\\
&=\sum_{n=0}^\infty f^{(n)}(0)\frac{x^n}{n!}\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2454725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that there are infinitely many positive integers n so that 2^n starts with d Given the studies of rotational transformations and measurable transformations, let d be any positive integer. Show that there are infinitely many positive integers n so that $2^n$ starts with d.
| The claim follows from the following Lemma in dynamics:
Let $\alpha$ be an element in $S^1$ of infinite order (i.e $\alpha^n\not=1$ for every $n$). Then the set $\{\alpha^n : n\in\mathbb{N}\}$ is dense in $S^1$.
How can we use the lemma? the number $2^n$ starts with $d$ if and only if there exists $k$ so that $d\cdot 1... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Union on the empty set and the set containing the empty set I'm trying to get a clearer sense of some of the consequences the axiom of unions has on the empty set. I understand that $\emptyset = \{\} \not= \{\emptyset\}$.
But assuming the following identities are correct, I don't understand why $\bigcup\emptyset = \bi... | $z \in \bigcup A$ iff there exists $y \in A$ for which $z \in y$. No such $z$ exists for $A = \emptyset$ or $A = \{ \emptyset \}$.
Indeed, for the former, we have no $y$; for the latter, there is a $y$, but it's empty, so there's no $z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2454969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solving $\frac{4x}{x+7}I know how to solve the problem. The reason I post the problem here is to see whether there is a quick approach, rather than a traditional method, to solve the problem.
The problem is: find $x$ that satisfy $\frac{4x}{x+7}<x$
I considered two cases: $x+7>0$ and $x+7<0$, and then went through... | Here is another way--whether or not it is simpler depends on the problem and your previous experience.
First solve the equality
$$\frac{4x}{x+7}=x$$
I'm sure you can do that fairly quickly, getting $x=0$ or $x=-3$. Then find the values of $x$ where one or both of the sides of the equation are undefined. In your problem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2455049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Distributing gifts so that everybody gets at least one So I was in class discussing the following problem:
We have $20$ different presents to distribute to $12$ children. It is not required that every child get something; it could even happen that we give all the presents to one child. In how many ways can we distrib... | It is actually quite a bit more simple than the previous posters have mentioned - no need to use the binomial and you will end up with a correct answer if you do
I believe the correct answer is
(k-1)C(n-1) thus 19 choose 11
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2455121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
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Show that for any bounded shape, you can make a straight cut in any direction to create halves of equal area On an infinite frying pan, there is a bounded pancake. Prove that one can make a straight cut in any given direction (that is, parallel to a given line) that splits the pancake in halves of equal area.
I think t... | Here's the outline of a solution. For every nonzero vector $v \in \mathbb{R}^n$ and $\lambda \in \mathbb{R}$ consider the half-space defined as $H_{\lambda} = x \cdot v \geq \lambda$. For any bounded region $B$, let $B_{\lambda} = H_{\lambda} \cap B$. This corresponds to the part of $B$ on side of a cut that runs paral... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2455251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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First-order set theory : What is the class of all sets in ZFC? What do we mean when we let the universal set be the class of all sets? How do I intuitively think about this? Do I just think of it as a collection of all sets? Also, is the Axiom of Regularity a part of ZFC?
I have to prove a statement in the class of al... | The following is meant to be a clarification of Ross Millikan's post:
In most commonly used set theories, we define classes (sometimes known as virtual classes) as a collection of sets satisfying some property. More precisely: Let $\phi(x, y_1, \ldots, y_n)$ be a formula in the language of set theory and let $p_1, \ldo... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why is base-10 decimal? A way to write a number in various bases is: 1012, 3Fa16, 51910.
The thing here, is that we apparently specify the base in decimal by default. This makes sense in everyday life, since we're not really doing base conversion when grocery shopping. But 10 is ambiguous when working with different ba... | Yes, you could indicate the base of the base too, but eventually you need a base-indicator that is specified in some "default" base. We normally use base ten so if nothing else is stated that is what the base that is assumed.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Does their exist a real continuous function other than $f(x)=0$ such that $f(2x) = -2f(x)$? I have a gut feeling it doesn't exist but I'm not sure how to prove/disprove it.
My attempt: Suppose there exists $a \in \mathbb{R}\setminus\left\{0\right\}$ such that $f(a) \neq 0$ . Define $x_n = \frac{a}{2^n}$
$f(x_{n+1}) = ... | (Rewriting achille hui's comment as an answer.)
Yes, there are other functions satisfying that equation. One such function is $f(x) = x \sin(\pi \log_2(x))$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2455708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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3D application of basic trigonometry https://imgur.com/a/D3ANJ (can't be uploaded because of size)
I first thought that ON//BC, because B is due east of O and C is due north of B.
However, that results in me getting an incorrect value of OT(correct value = 39.3)
| Let $OB=a$.
Thus, $$OT=a\tan40^{\circ}$$ and
$$OC\tan25^{\circ}=a\tan40^{\circ},$$
which gives
$$OC=\frac{a\tan40^{\circ}}{\tan25^{\circ}}$$ and by Pythagoras theorem we obtain:
$$70^2+a^2=\left(\frac{a\tan40^{\circ}}{\tan25^{\circ}}\right)^2$$ and from here we can find a value of $a$:
$$a=\frac{70}{\sqrt{\frac{\tan^24... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Conditional probability of multivariate gaussian I'm unsure regarding my (partial) solution/approach to the below problem. Any help/guidance regarding approach would be much appreciated.
Let $\mathbf{X} = (X_1, X_2)' \in N(\mu, \Lambda ) $ , where
$$\begin{align}
\mu &= \begin{pmatrix}
1 \\
1
... | The covariance between $X_1 + \lambda (3 X_1 + X_2)$ and $3 X_1 + X_2$ is $10 + 35 \lambda$, therefore if we take $\lambda = -2/7$, we get
$$\operatorname{P}(X_1 \geq 2 \mid 3 X_1 + X_2 = 3) =
\operatorname{P} \left(
X_1 -\frac 2 7 (3 X_1 + X_2 - 3) \geq 2 \mid 3 X_1 + X_2 = 3 \right) = \\
\operatorname{P} \left( X_1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2455972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Why quotient ring $R/R$ is zero ring $\{ 0\}$? There is already similar question.
Factor rings $R/R$ and $R/0$
Of course, I read it. However I still don't know, how and why the quotient ring $R/R$ is zero ring. So, I ask your help to check my thinking logic.
In my opinion, it seems to be $R/R=R$.
The definition of the... | The elements of the quotient ring $R/P$ are equivalence classes of elements in $R$. That is, two different elements $r_1, r_2 \in R$ produce the same class in $R/P$ if $r_1 - r_2 \in P$. This is equivalent to the definition you wrote above.
Now $R/0 = R$ because $r_1, r_2 \in R$ are in the same class in $R/0$ only if ... | {
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"source": "stackexchange",
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} |
Alternate proof that $1+x+...+x^{p-1}$ is irreducible for prime $p$
For $p$ prime, $P(x)=1+x+...+x^{p-1}$ is irreducible in $\mathbb{Z}[x]$.
This is a classic problem to which there exists a clever solution which applies Eisenstein's criterion to $P(x+1)$.
However I believe I have another solution, but I wish to make ... | This does not quite work; if it did, the same logic should hold for $x^{p^p}-1$; for instance, say $x^{27}-1$. This should imply $$
1+x+\dots+x^{26}
$$
is irreducible, but it is not, as can be checked by wolfram. The problem is as @Wojowu points out in the comments.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Optimal route consisting of rowing then walking Problem
You're in a boat on point A in the water, and you need to get to point B on land. Your rowing speed is 3km/h, and your walking speed 5km/h.
See figure:
Find the route that takes the least amount of time.
My idea
I started by marking an arbitrary route:
From here... | *
*a) the solution
The formula has already been indicated by wgrenard and AdamBL
$$
T = {1 \over 3}\sqrt {36 + \left( {9 - W} \right)^{\,2} } + {1 \over 5}W
$$
differentiating that
$$
{{dT} \over {dW}} = {{5W + 3\sqrt {36 + \left( {9 - W} \right)^{\,2} } - 45} \over {15\,\sqrt {36 + \left( {9 - W} \right)^{\,2} } }}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
pointwise product of two characters of $G$ is a character of $G$ Let $\phi: G \to GL_{n}(\mathbb{C})$ and $\rho: G \to GL_{m}(\mathbb{C})$ be representations. Let $V = M_{mn}(\mathbb{C})$. Define the representations $\tau : G \to GL(V)$ by $\tau_{g}(A) = \rho_{g}A\phi_{g}^{T}$.
I know that $\chi_{\tau}(g) = \chi_{\rho}... | Have you learned about the tensor product of two representations? The character of $\phi\otimes\rho$ is the pointwise product of the characters of $\phi$ and $\rho$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Anyone have some information about this identity/identities for regular polygons? I was looking at some equilateral triangles and started drawing up some equations and I came across the following (it may take some time for my actual question to come):
Suppose that we put a point $D$ anywhere within the triangle. Then ... | Essentially the same proof shows a corresponding theorem for a regular tetrahedron -- the altitude is the sum of the length of the perpendicular dropped to each of the four sides (from any interior point). This is, in fact, one starting point for the notion of "barycentric coordinates", which might be a starting place ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
$\sigma(f)=f(\Omega)$ for $f\in C(\Omega)$, $\Omega$ a Compact, Hausdorff Topological Space Let $\Omega$ be a compact, Hausdorff, topological space; let $A=C(\Omega)$, the unital, Banach algebra of continuous functions from $\Omega$ to $\mathbb C$; let
$$\text{Inv}(A)=\{f\in A:g(\omega)f(\omega)=f(\omega)g(\omega)=1\te... | I don't think there's anything wrong with your argument.
It is just that $C(\Omega)$, for non-compact $\Omega$, is not an interesting algebra. In particular, it is not a Banach algebra, not with the natural norm, because you would need your functions to be bounded.
Also, on "arbitrary" $C(\Omega)$, the spectrum becom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Proving that a cyclic group is generated by a single element. I am currently reading "The Theory of Finite Groups" by Kurzweil and Stellmacher.
I am already stuck on page 4.
On page 3, a cyclic group is defined as:
The group G is cyclic if every element of G is a power of a fixed element g.
Then on page 4 a proof is ... | By definition, if $G$ is cyclic with generator $g$, then every element of $G$ is a power of $g$. What 1.1.2 is saying is that if $G$ has order $n$, then more specifically every element of $G$ is equal to $g^m$ for some $m$ such that $0\leq m<n$. This is stronger than the definition, because of the restriction that $0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2456979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Affine subspaces and parallel linear subspaces Let $\mathcal{X}$ be a real vector space and $C\subset \mathcal{X}$ an affine subspace of $\mathcal{X}$, i.e. $C\neq\emptyset$ and $C=\lambda C + (1-\lambda)C$ for all $\lambda\in\mathbb{R}$. In the text I am reading, they have defined the linear subspace parallel to $C$ t... | I am not sure if you believe me or not! but the only reason that people like to write parallel subspace in the way $V=C-C$ is that it looks NICER.
But what you said is correct both representations are equivalent. it is a simple exercise .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\sqrt[n]{n} > \sqrt[n+1]{n+1}$ without calculus? I'm stuck with this sample RMO question I came across:
Determine the largest number in the infinite sequence $\sqrt[1]{1}$, $\sqrt[2]{2}$, $\sqrt[3]{3}$, ..., $\sqrt[n]{n}$, ...
In the solution to this problem, I found the solver making the assumption,
$\s... | We wish to compare $\sqrt[n]n \lessgtr \sqrt[n+1]{n+1}$. Raise each side to the $n(n+1)$th power to get
$$ n^{n+1} \lessgtr (n+1)^n $$
and use the binomial theorem on the right-hand side:
$$ n\cdot n^n \lessgtr \underbrace{n^n+\binom n1 n^{n-1} + \binom n2 n^{n-2} + \cdots + \binom n{n-1} n^1}_{n\text{ terms}} + 1 $$
B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
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How do we conclude using De Morgan's laws that these two are equal? Question: (p∧q)→(p∨q)≡¬(p∧q)∨(p∨q)
Which steps should I take to derive the equation to the right from the equation to the left? In the book, it just shows this equation but doesn't answer how did they actually get it. Since this example in the book sho... | If both $p$ and $q$ are true, then at least one of either $p$ or $q$ will be true.
Since this example in the book shows up under the De Morgan's laws section, I rightfully considered De Morgan's laws could help us to solve this problem.
Yes, but it is too soon. The step you have applies Material Implication.
*
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Method to solve A = mod( Bn, C) How would I go by solving an equation of the general form.
A = mod ( Bn, C )
Solve for n knowing A, B and C
Where B and C are Natural numbers and A and n are whole numbers. Also the greatest common denominator between B and C is 1.
| If $A \not \in [0, C-1]$ then clearly there is no solution. Otherwise, we have
$$A \equiv Bn \pmod C.$$
Since $(B,C) = 1$, the inverse $B^{-1}$ exists modulo $C$, and we can multiply it on both sides to get:
$$AB^{-1} \equiv n \pmod C.$$
So your problem is simply to find the inverse $B^{-1}$, which can be done using th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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} |
Computing $\lim_{\varepsilon\to 0^{+}}\psi(\varepsilon)/\Gamma(\varepsilon)$ with asymptotic expansions I have the following limit of which I want to compute:
\begin{equation}
\lim_{\varepsilon\to 0^{+}} \frac{\psi(\varepsilon)}{\Gamma(\varepsilon)}.
\end{equation}
For $\varepsilon\approx 0$ and $\varepsilon\neq 0$ I h... | One could take the following method:
\begin{align}
\frac{\psi(x)}{\Gamma(x)} &= \frac{\Gamma(x) \, \psi(x)}{\Gamma^{2}(x)} = \frac{\Gamma'(x)}{\Gamma^{2}(x)} \\
&= - \frac{d}{dx} \left(\frac{1}{\Gamma(x)}\right).
\end{align}
Now make use of the Taylor series expansion of the inverse of the Gamma function, namely,
$$\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What are some examples in which the introduction of nets helped understand a concrete topological space? I've had the chance to learn about nets, though every statement I was exposed to didn't seem to be useful in practice.
For example, the fact $x \in \bar{A}$ iff there exists some net $(x_\alpha)_{\alpha \in J}$ such... | You can prove Tychonoff's theorem using nets (that's how it's done in Folland's Real Analysis, for instance), but that can't be done with sequences, even for concrete spaces.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Solving $\tau(n)+\phi(n)=n$ for $n\in\mathbb{N})_{\ge 1}$.
Let $\tau(n)$ denote the number of divisors of a positive integer $n$,
and let $\phi(n)$ be Euler's totient function, i.e. the number of
positive integers less than and coprime to $n$. I'd like to find all
$n$ such
$$ \tau(n)+\phi(n)=n.$$
*
*Th... | $$n-\tau(n) = \sum_{a=2}^n 1_{a \,\nmid \, n}$$
$\sum_{a=2}^n 1_{a \,\nmid \, n} = \phi(n) = 1+\sum_{a=2}^n 1_{(a,n)=1}$ means there is exactly one $a \in [2,n]$ such that $(a,n)\ne 1$ and $a\nmid n$.
Take a prime $p | (a,n)$. Thus $n = p d$ and for any $b = p r,r \in [2,d], (r,d)=1$ we have $(b,n)\ne 1$, $b\nmid n$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How are $z$ and $z^*$ independent? I have been told that a complex number $z$ and its conjugate $z^*$ are independent. Part of me understands this, since for two independent variables $x$ and $y$ we can always define new independent variables $x' = \alpha x + \beta y$ and $y' = \alpha x - \beta y$.
However, this contra... | It is true that there is a one-to-one map between $z$ and $z^*$, it's just the reflection about the $x$-axis of the complex plane. Therefore, it is certainly not true that $z$ and $z^*$ are independent. However, if we consider $z^*$ as a function of $z$, so that $z^* = f(z)$, then it turns out that $f(z)$ is not a "nic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2457972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 2
} |
Let $f : I \to \mathbb R$ be continuous. For any compact interval $J \subseteq f(I), \exists$ a compact interval $K \subseteq I$ with $f(K)=J.$
Let $f \colon I \to \mathbb R$ be continuous where $I$ is an interval. For any compact interval $J \subseteq f(I)$ there exists a compact interval $K \subseteq I$ such that $f... | WLOG $p\leq q.$ (Otherwise study the function $g(t)=-f(t)$...). We have $$f([r,s])\supset [f(r),f(s)]=[p,q]$$ by the IVP, because $f$ is continuous.
If $t\in (r,s)$ and $f(t)<p\leq q=f(s)$ then by the IVP there exists $t'\in (t,s)$ with $f(t')=p,$ contrary to the def'n of $r.$
If $t\in (r,s)$ and $f(t)>q\geq p=f(r)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2458102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Proving that every reduced residue class contain at least one prime I don't know if I expressed this clearly, but I want to know if the following is true and also some help proving it in case it is.
$\forall a,b \in \mathbb{N} , \gcd{(a,b)} = 1 \Rightarrow \exists p \equiv b \pmod{a}$
Where $p$ is a prime and $b < a$.... | This is a consequence of Dirichlet's theorem. I don't know a way to prove your theorem without it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2458224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
SAT II Geometry Find the missing side length
I'm thinking the answer is choice A but I want someone to back up my reasoning/check. So since DE and DF are perpendicular to sides AB and AC respectively that must make EDFA a rectangle. Therefore AF must be 4.5 and AE must be 7.5. Since they state AB = AC that must mean E... | Here is my thoughts on this one. It is indeed clear that triangles BED and CDF are similar. If $BD = y$ then DC is $24-y$. Using similarity we get the ratio $y/4.5 = (24-y)/7.5$ from which follows $y=9$ and so $CD=15$. With Pythagorean theorem you find 12.99
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2458313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Convexity of logistic loss How to prove that logistic loss is a convex function?
$$f(x) = \log(1 + e^{-x})?$$
I tried to derive it using first order conditions, and also took 2nd order derivative, though I do not see neither $f(y) \geq f(x) + f'(x)(y-x)$, nor positive definiteness (aka always positive second derivative... | You can simplify the given function and then take $2$nd order derivative:
$$y=\ln{(1 + e^{-x})}=\ln{\frac{e^x+1}{e^x}}=\ln{(e^x+1)-x}.$$
$$y'=\frac{e^x}{e^x+1}-1,$$
$$y''=\frac{e^x(e^x+1)-e^{2x}}{(e^x+1)^2}=\frac{e^x}{(e^x+1)^2}>0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2458438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Two points on curve that have common tangent line Find the two points on the curve $y=x^4-2x^2-x$ that have a common tangent line.
My solution: Suppose that these two point are $(p,f(p))$ and $(q,f(q))$ providing that $p \neq q$. Since they have a common tangent line then: $y'(p)=y'(q),$ i.e. $4p^3-4p-1=4q^3-4q-1$ and ... | Complete the square :
your curve is $y = x^4 - 2x^2 - x = (x^2 - 1)^2 + (-x-1)$.
So the curve stays above the line $y = -x-1$, and is tangent to it when it touches it, that is when $x^2-1 = 0$, so that's when $x=1$ and $x=-1$ :
The line $y=-x-1$ is tangent to the curve at those two points $(1,-2)$ and $(-1,0)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2458594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
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