Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Using elementary row operations to solve intersection of two planes The question I am struggling with is the following:
Solve the following using elementary row operations, and interpret each system of equations geometrically:
\begin{align*}
x - 3y + 2z &= 8\\
3x - 9y + 2z &= 4
\end{align*}
The answer given in the book... | $2z = 10$, so $z = 5$.
$2x - 6y = -4 \Rightarrow2x = -4 + 6y \Rightarrow x = -2 + 3y$. Let $y = t$, we get $x = -2 + 3t$, $y = t$, $z = 5$, as desired.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2615659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Confusing description of torsion of a curve? In my textbook (by do Carmo) and both in wikipedia.
There are descriptions of what a torsion is, and both of them says it is a measure of "how fast a curve twists out of the plane of curvature"
I am aware of the definition of torsion which is the magnitude of the derivative ... | When a curve is planar, all osculating planes are equal.
When it is non planar, i.e. has some torsion, the osculating planes stop staying parallel when you move along the curve, and this change of direction is reflected by the binormal.
"Infinitesimal" insight:
Imagine the curve discretized with a fixed step.
Two succ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2615811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
f is a continuous function from (X,$\tau$) to {0,1} with discrete topology, if f non constant then (X,$\tau$) disconnected Let $f$ be a continuous function such that $f : (X,\tau) \rightarrow (\{0,1\},\tau_1\}$. Where $(X,\tau)$ is a generic topological space and $\tau_1$ is the discrete topology. I want to prove that ... | If $X$ is connected and $f:X\to Y$ is continuous then $f(X)$ is connected.
So if moreover $f$ is surjective then $Y=f(X)$ is connected.
In your case non-constant comes to the same as surjective.
Draw conclusions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2615893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Proving that a limit does not exist with absolute values I'm to prove that the following limit does not exist
$\lim_{x\to -2} \frac{\vert 2x +4\vert -\vert x^3 +8 \vert}{x+2}$
From here, I have taken the method of finding $\lim_{x\to -2^+}$ and $\lim_{x\to -2^-}$ to show that they're not equal
However my problem is tha... | Because for $x\rightarrow-2+$ we have
$$\frac{|2x+4|-|x^3+8|}{x+2}\rightarrow2-4-4-4=-10$$ and
for $x\rightarrow-2-$ we have
$$\frac{|2x+4|-|x^3+8|}{x+2}\rightarrow-2+4+4+4=10.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2615944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
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Uniformly minimum variance unbiased estimator of theta. Let $X_1,X_2,....X_n$ be a random sample from a population with probability density function
$$f(x|\theta)=\dfrac{\theta}{2}e^{-\theta|x|} ,-\infty < x < \infty,\theta>0$$
Then a UMVE of $\theta$ is ?
Can someone tell me if i did everything right or not in the f... | Write
$$f(x)=\dfrac{\theta}{2}e^{-\theta|x|}=h(x)g(\theta)\exp\left(\eta(\theta)\cdot T(x)\right)$$
with $h(x) = 1$, $g(\theta) = \theta/2$, $\eta(\theta)=-\theta$, and $T(x) = |x|$.
It follows that $f$ is of the exponential family.
Furthermore, note that the parameter space
$$\Theta=(0, \infty)\supset(0, 1)$$
so $\T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2616101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why this integral diverge? For $D=[0,+\infty)×[0,+\infty)$ and $f(x,y)=(1+x+y)^{-1}$, why does the integral over $D$ of $f(x,y)$ not converge?
It's not like $\rho/(1+\rho|\sin\theta+\cos\theta|)$ which to infinite is like $1/(|\sin\theta+\cos\theta|)$ and, close to $0$ is like $0$?
| Intuitively, the area of the domain grows like $xy$, while the integrand decreases like $\dfrac1{1+x+y}$, which is insufficient to compensate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2616239",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
The energy method for $u_{tt}-du_t-u_{xx}=0, (0,1)\times(0,T) $ It is a problem in Evan's PDE. I want to prove the smooth solution of the following PDE is zero:
$$u_{tt}-du_t-u_{xx}=0, (0,1)\times(0,T) $$$u|_{x=0}=u|_{x=1}=u|_{t=0}=u_t|_{t=0}=0$.
The hint is to use the energy $\frac{1}{2}(||\partial_tu||_{L^2[0,1]}+||... | Let us assume that the solution is sufficiently smooth and $d<0$. The time-derivative of the energy
$E(t) = \frac{1}{2}\left( \|u_t\|_{L^2[0,1]}^2 + \|u_x\|_{L^2[0,1]}^2\right)$ writes
$$
\begin{aligned}
\frac{\text{d}}{\text{d}t}E(t) &= \int_0^1 \left( u_{tt}\, u_{t} + u_{xt}\, u_{x} \right) \text{d}x \\
&= \int_0^1 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2616317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to solve the functional equation $f(x + f(x +y ) ) = f(2x) + y$? Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following equation:
$$ f(x + f(x +y ) ) = f(2x) + y,\quad \forall x,y\in\mathbb{R}$$
The only function I have found is $f(x) = x$, but I think there are more.
| Given $z$, let $x=f(z)$. and $y=z-x.$ Then you get:
$$f(x+f(x+y))=f(2f(z))$$ and $$f(2x)+y=f(2f(z))+z-f(z)$$
From this you get $z=f(z).$
A cute variation of Christian's very nice answer:
$$\begin{align}
2z+f(0)&=f(f(2z))&[x=0,y=2z]\\
&=f(f(z+f(z)))&[x=z,y=0]\\
&=z+f(z)+f(0)&[x=0,y=z+f(z)]
\end{align}$$
So $f(z)=z.$
Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2616572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Examples for $\max\{f(x)+g(x)\}\leq \max{f(x)}+ \max{g(x)}$ and $\min\{f(x)+g(x)\}\geq\min{f(x)}$+min{g(x)}. Can someone give me an example with concrete functions for the following relations $\max\{f(x)+g(x)\}\leq \max{f(x)}+ \max{g(x)}$ and $\min\{f(x)+g(x)\}\geq \min{f(x)}+\min{g(x)}$? I suppose one example would be... | Another solution: $f(x)=g(x)=0$ (or equal to any other constants)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2616736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Parametrization of an arbitrary conic/ellipse I have the coefficients for a conic (I actually know that it is an ellipse) in the form $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F =0$$
Is there an efficient algorithm which returns the parametrization of the eclipse, i.e., $\langle x(t), y(t) \rangle?$
| Based on the comments to your question, it looks like the underlying problem that you’re trying to solve is to determine efficiently whether or not a point lies within the ellipse. A straightforward way to do this is to plug the coordinates of the point into the left-hand side of the general equation. The sign of the r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2616866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Monotone convergence theorem for series (basic proof) My question is how to prove monotone convergence theorem for infinite series without more advanced technique like counting measure. I see this used a lot. But looking through books like Rudin, the theorem for series or elementary proof is not to be found.
The theore... | Here's a constructive $\epsilon$ proof.
Let $\epsilon > 0$ be arbitrary. Consider the the smallest value of $N$ such that the partial sum $$\sum_{n=1}^N
\lim_{m \to \infty} x _{mn}$$ is within $\epsilon/2$ of the actual sum.
Denote $\lim_{m \to \infty} x _{mn}$ by $x _{\infty,n}$ from now on.
Now let $M$ be such that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2616956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Group action on vector space Let $G$ be a group acting on (complex) vector spaces $V, W$ and let $G$ act trivially on the vector space $U$.
Let $Hom^G(V, W)$ denote the linear transformations $T:V \to W$ that respect the group action, that is, $g \cdot T(v) = T(g \cdot v)$ for all $v \in V$.
$W \otimes U$ has a natural... | I don't believe this is true in full generality; one needs $U$ to be finite-dimensional.
In the case where $G$ is the trivial group, this amounts to
$$\text{Hom}(V,W\otimes U)\cong\text{Hom}(V,W)\otimes U.$$
Specialising further to $W=\Bbb C$ (one-dimensional) we get
$$\text{Hom}(V,U)\cong\text{Hom}(V,\Bbb C)\otimes U.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Maximum Area of inscribed rectangle how can we compute the maximum area of a rectangle which can be inscribed in a triangle of area 'M'
I have taken a special case here in the image file to calculate the inscribed rectangle area but how can we calculate it for general case??
| Let $EF=x$.
Thus, since $h_a=\frac{2M}{a}$ and $\Delta ABC\sim \Delta AEF$, we obtain:
$$\frac{x}{a}=\frac{\frac{2M}{a}-EG}{\frac{2M}{a}},$$
which gives $$EG=\frac{2M}{a}\left(1-\frac{x}{a}\right).$$
Id est, by AM-GM $$S_{EFHG}=\frac{2M}{a}\left(1-\frac{x}{a}\right)x=2M\left(1-\frac{x}{a}\right)\frac{x}{a}\leq2M\left(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number theory: find $a, b$ such that $\frac{a}{b} = b.a$ in a general base $\mathcal{B}\neq 10$ I was playing with numbers and thinking about this "coincidence"
$$\frac{5}{2} = 2.5$$
That is, for positive $a$ and $b$ we have
$$\frac{a}{b} = b.a$$
And those questions came into my mind:
1. Could we find all such integers... | My idea would be to do $\frac{a}{b}=b+\frac{a}{10}$ and this leads to $a=\frac{10b^2}{(10-b)}$. I think this may be a possible solution for case 1. Actually I do not have an idea for $\mathcal{B}\neq 10$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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$A+A\subseteq A\times A$ Define $A+A=\{a+b\colon a,b\in A\}$,$A\times A =\{ab\colon a,b\in A\}$.
Does there exist a finite integer set $A\subseteq \mathbb{Z}^+$, such that $|A|>1$ and $A+A\subseteq A\times A$ ?
| We assume such a set exists and derive a contradiction. First, if $\{1,2\} \subset A$ then $3 \in A$ as $1+2=3$ so we must have $1 \cdot 3 \in A \times A$. Also, since $|A| > 1$ then if $1$ or $2$ is not in $A$ then there exist an element of $A$ greater than $2$, so in either case such an element exists.
Edit: the rest... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Embeddings between Hölder spaces $ C^{0,\beta} \hookrightarrow C^{0, \alpha} .$ Let $ \Omega \subset \mathbb R^n $ be an open subset and let $ 0 < \alpha < \beta \leq 1.$ We consider the space of Hölder continuous functions $C^{0, \alpha}$ which is a Banach space endowed with the norm
$$ \| f\|_{C^{0, \alpha}} := \| f ... |
Note that Embeddings between Holder space do not care about the boundedness of the domain
Patently we have $$\sup_{ x,y \in \Omega \\ x \neq y} \frac{ |f(x) -f(y)|}{|x-y|^\alpha}\le \sup_{ x,y \in \Omega \\ |x -y|\le1} \frac{ |f(x) -f(y)|}{|x-y|^\alpha}+\sup_{ x,y \in \Omega \\ |x -y|\ge1} \frac{ |f(x) -f(y)|}{|x-y|^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Evaluate $\lim \limits_{n \to \infty} n \int_{-1}^{0}(x+e^x)^n dx$. Evaluate $\lim \limits_{n \to \infty} n $$\int_{-1}^{0}(x+e^x)^n dx$. The answer should be $\frac{1}{2}$. I tried the substitution $x+e^x=u$ and then using the property that $\lim_{n \to \infty } n \int_{-a}^{1} x^n f(x)dx=f(1)$ but I don't know what ... | Your property says that the limit is
$$
\frac{1}{1+e^{X(1)}},
$$
where $X(u)$ is the solution of $X(u)+e^{X(u)}=u$. Observe that $e^{X(1)}=1-X(1)$, so your limit is
$$
\frac{1}{2-X(1)}.
$$
It suffices to show that $X(1)=0$. But this is clear, since $1=X(1)+e^{X(1)}\ge e^{X(1)}\ge 1$, with equality if and only if $X(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Probability problem involving geometric distribution and expected value We're given the following problem:
"An experiment is conducted until it results in success: the first step has probability $\frac{1}{2}$ to be successful, the second step (only conducted if the first step had no success) has probability $\frac{1}{... | Let $\mu_{0}$ denote the expectation of the cost of the steps yet
to be done if no steps have been made.
Let $\mu_{1}$ denote the expectation of the cost of the steps yet
to be done if step 1 has been made without success.
Let $\mu_{2}$ denote the expectation of the cost of the steps yet
to be done if step 1 and step 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Invertible Linear Map Suppose $V$ is finite-dimensional and $S,T\in \mathcal{L}(V)$. Prove that if $ST$ is invertible, then both $S$ and $T$ are invertible.
This is the partial question retrieved from Linear Algebra Done Right, and I have come out with different solution from the solution guide.
My solution:
Since $... | As José Carlos Santos stated, your proof is not correct. You have to show two things: injectivity and surjectivity of both $S$ and $T$. If you want to use zero element in your proof then injectivity can be shown by proving the following implication $$ f(u) = 0 \Rightarrow u = 0 \ ,$$
that is, you do not assume in the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Why is the inverse of the derivative of f not the actual derivative of the inverse of f? So, I've explored this a little, but it is still confusing. When you calculate the inverse of a function, f, that is one-to-one, the points switch: a point (2,8) on f would be (8,2) on the inverse. So, one would assume that the der... | Intuitive thoughts to reflect on: draw the graph of $f$ and mark a point on it (say $(a,f(a))$). Draw the tangent at that point. It will have slope $f'(a)$.
Now flip the entire plane around the line $y=x$. The graph of $f$ has now become the graph of $f^{-1}$, the marked point has become $$(f(a),a)= (f(a),f^{-1}(f(a)))... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2617954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Showing that $g(x) = \begin{cases} f(x), & \ x \ne 0 \\ c, & \ x = 0 \end{cases}$ is Riemann integrable where $f$ is Riemann integrable Suppose that $f: [-1,1] \to R$ Riemann integrable and let $g:[-1,1]\to R$ be defined
by $g(x) = \begin{cases} f(x), & \ x \ne 0 \\ c, & \ x = 0 \end{cases}$
Show that $g$ is integrabl... | hint
You just need to prove that $f-g $ is integrable.
put $d=|f (0)-c|$,
For a given $\epsilon>0,$ take the partition $$p=\{-1,-\frac {\epsilon}{4d},\frac {\epsilon}{4d},1\} $$
then
$$U (f-g,p)-L (f-g,p)=d\frac {\epsilon}{2d}<\epsilon.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Calculate segment distance to cover before turning via a pivot I'm building a visualisation where I have a body that is moving along-a-path, which is comprised of multiple segments, each with an arbitrary angle.
The body is moving along the path and:
*
*When the body's centre reaches the end of each segment it stop... | You should turn when pivot point $N$ reaches the angle bisector of $\angle ABC$, with a rotation of $2\angle BNO$. But the rotated body is turned by $180°-2\angle BON$ with respect to the direction of the path.
To make this work, then, you must choose your pivot so that $\angle BON=90°$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Easy way to prove $|\text{curl}\ \mathbf n|^2=(\mathbf n\cdot \text{curl}\ \mathbf n)^2+|\mathbf n \wedge\text{curl}\ \mathbf n|^2$? Let $\mathbf n$ be a unit vector field. I would like to show that
$$|\text{curl}\ \mathbf n|^2=(\mathbf n\cdot \text{curl}\ \mathbf n)^2+|\mathbf n \wedge\text{curl}\ \mathbf n|^2$$
holds... | You don't need any properties of curl - for any vectors $u,v$ we have $|u|^2|v|^2 = (u \cdot v)^2 + |u \wedge v|^2,$ so your formula follows from the fact that $|n|=1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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2 different results using associative property in boolean expression I have the following expression that i'm trying to simplify: $Q \vee P \vee Q \vee T$
I simplified this like so: $Q \wedge P \vee Q \vee T \equiv Q \wedge P \vee (Q\vee T) \equiv Q\wedge P \vee T \equiv Q \wedge (P\vee T) \equiv Q\wedge T \equi... | Order of operations matters. $Q\wedge(P\vee Q\vee T)$ and $(Q\wedge P)\vee Q\vee T$ are quite different expressions.
I simplified this like so: $Q \wedge P \vee Q \vee T \equiv Q \wedge P \vee (Q\vee T) \equiv Q\wedge P \vee T \equiv Q \wedge (P\vee T) \equiv Q\wedge T \equiv Q. $
You are treating that as ${Q\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Determine the critical points and identify them as asymptotically stable or unstable? Drawing phase lines? Here's the question:
Determine the critical (equilibrium) points, and classify each one as
asymptotically stable or unstable. Draw the phase line, and sketch
several graphs of solutions in the $ty$-plane
$$dy/dt ... | The only equilibrium point of $$dy/dt = 1 − e^y,\; −∞ < y_0 < ∞.$$ is at $dy/dt =0.$
Thus equilibrium happens at $$ e^y =1 $$ that is $y=0$
This equilibrium is asymptotically stable because $y>0 \implies dy/dx <0$ and $ y<0 \implies dy/dx>0.$
Therefore the solutions approach the equilibrium solution $y=0$ as... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Compare $2^{2016}$ and $10^{605}$ without a calculator So, I am supposed to compare $2^{2016}$ and $10^{605}$ without using a calculator, I have tried division by $2$ on both sides and then comparing $2^{1411}$ and $5^{605}$, and then substituting with $8,16,10$ and then raising to powers and trying to prove that but... | $\log_{10}2^{2016}=2016\log_{10}2\approx 606.88>605$
If calculators are not allowed, we have
\begin{align*}
2^{2016}&=64(1000+24)^{201}\\
&>64\left[1000^{201}+\binom{201}{1}1000^{200}(24)\right]\\
&=64(10^{603})(5.824)\\
&>100(10^{603})\\
&=10^{605}
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618786",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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How to Fourier Transform $\frac{\sin(x)^2}{x^2}$ without Contour Integration. In our lecture we need to Fourier transform $\frac{\sin(x)^2}{x^2}$, i.e. compute the integral:
$$\int_{-\infty}^{\infty} \mathrm e^{-iy x}\frac{\sin(x)^2}{x^2} \mathrm dx$$
Since it's a lecture on partial differential equations and not comp... | Due to parity it is enough to compute
$$\int_{-\infty}^{+\infty}\frac{\sin^2(x)\cos(xs)}{x^2}\,dx \stackrel{\text{def}}{=}\lim_{M\to +\infty}\int_{-M}^{M}\frac{\sin^2(x)\cos(xs)}{x^2}\,dx $$
and by integration by parts the RHS equals
$$ \lim_{M\to +\infty}\int_{-M}^{M}\frac{\frac{1}{2}\cos(xs)-\frac{1}{4}\cos(x(s+2))-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2618903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Meaning of notation $f(x)$ in set theory In my book, a function $f$ is defined as a binary relation such that if $(x,y),(x,z)\in f$ then $y=z$. Moreover, as it is usual, author denotes $(x,y)\in f$ by
$$
y=f(x) .
\tag{1}
$$
So, from this notation, I understand $f(x)$ as the second component of the ordered pair $(x,y)\... | If $A$ is a singleton, $\{a\}$ then $\bigcap A=\bigcup A=a$. Since $f$ is a function, the set $\{y : (x,y)\in f\}$ is a singleton, for a fixed $x$, so the result follows.
The key point to remember is that everything is a set, including $x$ and $y$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Variance of sum of $10$ random variables. Find the variance of sum of $10$ random variables if each has variance $5$ and if each pair has a correlation coefficient $.5$
Let $Y=X_1+X_2+X_3+\ldots+X_{10}$
I tried this problem by calculating variance of first $10$ random variables.
$V(Y)=50$. Then there will be $45$ pair... | $$Y=\sum_{n=1}^{10}X_n\to var(Y)=E(Y^2)-E^2(Y)\\=E(\sum_{n=1}^{10}X^2_n+\sum_{m,n=1\\m\ne n}^{10}X_nX_m)-(\sum_{n=1}^{10}E(X_n))^2\\=50+90\times 2.5-(\sum_{n=1}^{10}E(X_n))^2$$if we take $E(X_n)=0$ we will have:$$var(Y)=275$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Complete graphs in the plane with colored edges where an edge don't cross edges with same color The maximal number of nodes in a complete planar graph is $4$.
Suppose that the edges of the graph can be chosen with $m$ different colors and that edges with different colors are allowed to cross each other. What would be ... | This isn't a complete answer, but the quantity you're asking for is closely related to the thickness of a a graph, which is the minimum number of planar subgraphs which jointly cover the edges of the graph. It is known that the thickness of the complete graph $K_n$ is $\lfloor (n+7)/6 \rfloor$ except at $K_9$ and $K_{1... | {
"language": "en",
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Squared Summation(Intermediate Step) I am studying Economics and are trying to get a firm grasp of summation rules and applications. Looking into the following relation,
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}$
The following "trick" is given below, to u... | I think this may be a question about how telescoping sums work. If so, then
$$\sum_{k=1}^n\left(f(k+1)-f(k)\right)=\sum_{k=1}^nf(k+1)-\sum_{k=1}^nf(k)=\sum_{j=2}^{n+1}f(j)-\sum_{j=1}^nf(j)$$
Where e have made the substitution $k=j-1$ in the first sum and $j=k$ in the second. When $j-1=k=1$, $j=2$ and when $j-1=k=n$, $j... | {
"language": "en",
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Is fraction has the same meaning with rational in number theory? I'm unable to get the difference between fraction and rational, we say $\frac{a}{b}$ is rational number if a and b are two integer with $b\neq 0$, and we can say also $\frac{a}{b}$ is a fraction but i don't know any reason for that, my question here is :I... | Any number written in the form $\frac{a}{b}$ is a fraction provided $b \neq 0$, because division by zero is not defined.
Thus $\frac{2}{3}$, $\frac{221}{5}$ and $\frac{\pi}{2}$ are all fractions but only $\frac{2}{3}$ are $\frac{221}{5}$ rational since the decimal representation either repeats $\frac{2}{3} \approx 0.66... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many years the world's oil reserves will be enough? World oil reserves are estimated at around 240 billion tonnes. Its world production is 4.36 billion tons annually. Calculate how many years the world's oil reserves will be enough: a) if the current level of its production is maintained; b) taking into account the... | I used Excel to calculate this, and found that after 37 years, 235 B will be extracted, so it appears your calculation is correct. And doing a rough estimate, we can use an approximation 2%*26*e = 141%, so in 26 years, production should be about 41% higher than today. But 26 is about half of 55, which would mean that p... | {
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False Dudeney : triangle to quiet a square. ABC is an equilateral triangle.
The same color polygons are isometric.
I can prove that MPQR is a rectangle, but not that it's not a square.
MI = IP and RF' = F'Q = FN.
As MPQR is a rectangle MI = IP = RF' = F'Q = FN.
This equalities are independent of the choice of E and F... | Here is a solution to this problem.
| {
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Maximum possible reputation? (NOT a meta question) I had the following idea about the reputation system of MSE, that led to a math question:
Suppose a certain user on MSE gains an average of $+200$ reputation per day - the daily maximum. Suppose that this reputation comes from one vote from each of $20$ randomly selec... | I believe that your reputation will tend to infinity.
By the way, 200 is not the maximum reputation gain in a day; points from acceptances and bounties don't count toward the cap, and are unlimited. But that doesn't matter because I'm not assuming that you're maxing out your daily reputation gains. I only assume that y... | {
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Finding a representation of $D_n$ on $\mathbb{R}^2$? I am trying to write down a representation of $D_n = \langle \sigma, \tau \mid \sigma^n = \tau^2 =e, \tau \sigma = \sigma^{-1} \tau \rangle$ over $\mathbb{R}^2 \cong \mathbb{C}$ (as an $\mathbb{R}$-vector space).
What I Know:
My representation has to be a group homo... | All you have to do is set
$$\rho(\sigma)=\pmatrix{\cos (2\pi/n)&-\sin(2\pi/n)\\\sin(2\pi/n)&\cos(2\pi/n)}$$
and
$$\rho(\tau)=\pmatrix{1&0\\0&-1}$$
and then verify these preserve the relations, that is $\rho(\sigma)^n=I$
(standard property of rotation matrices), $\rho(\tau)^2=I$ (obvious)
and $\rho(\tau)\rho(\sigma)=\rh... | {
"language": "en",
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"source": "stackexchange",
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Smallest prime $p$ which every integer $< n$ is a primitive root $\mod p$ Another interesting question related to primitive roots is what is the smallest prime $p$ for which the primes less than $n$ are primitive roots $\mod p$. The sequence of primes would be $[p, p_2, p_3,...]$ for every additional prime $k > n$. For... | The following PARI/GP-program does the job :
? k=3;x=primes(k);p=prime(k);gef=0;while(gef==0,p=nextprime(p+1);s=select(n->zno
rder(Mod(n,p))==p-1,x);if(s==x,gef=1));print(p)
53
?
Just change $k$ to get the smallest prime for another $k$.
The smallest primes for $k\le 18$ are :
? for(k=1,20,x=primes(k);p=prime(k);gef=0... | {
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Question about proof of Stein's Lemma by Casella and Berger I am looking at the following proof from Casella and Berger's Statistical Inference. However, I don't understand the final statement. The only condition on $g'$ here is that $E|g'(X)|<\infty$. But how does this ensure that $g(x)e^{-(x-\theta)^2/(2\sigma^2)} \t... | The hypotheses are that, for some nonnegative functions $f$ and $h$, the function $fh$ is integrable on $(0,+\infty)$, that the function $h$ is decreasing on some $(\theta,+\infty)$ and that $h(x)\to0$ when $x\to+\infty$, and the question is whether all this implies that $g(x)h(x)\to0$ when $x\to+\infty$, where $$g(x)=... | {
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Proving that the sequence $\{\frac{3n+5}{2n+6}\}$ is Cauchy. I'm not quite sure how to tackle these kinds of questions in general, but I tried something that I thought could be right. Hoping to be steered in the right direction here!
Let $\{\frac{3n+5}{2n+6}\}$ be a sequence of real numbers. Prove that this sequence i... | As you were told in the comments, you must take some $\varepsilon>0$ and then prove that $\left|\frac{3n+5}{2m+6}-\frac{3n+5}{2n+6}\right|<\varepsilon$ if $m$ and $n$ are large enough.
Note that$$(\forall n\in\mathbb{N}):\frac{3n+5}{2n+6}=\frac{3n+9}{2n+6}-\frac4{2n+6}=\frac32-\frac2{n+3}.$$So, take $p\in\mathbb N$ suc... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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What is the difference between $x$ and $\{x\}$ when $x$ itself is a set? I've already asked this a part of another question, but thought it'd be easier to elaborate a bit more on my concern.
Let $x$ be a set. What is the difference between $x$ and $\{x\}$? I get that the latter is a set consisting of a single element -... | $$\{1\} $$ is a set whose the unique element is the integer $1$
$$\{\{1\}\} $$ is a set whose the unique element is the set $\{1\} $.
| {
"language": "en",
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"source": "stackexchange",
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Each vertex of the square has a value which is randomly chosen from a set. I don't need so much help computing this question (yet). However, I am still at the point of understanding what the question is really asking. If anyone understands it, and can explain it, that would be greatly appreciated!
I understand what pa... |
So say edge 1 is connected to a vertex that randomly selected 2, and another vertex that randomly selected 3 from the set, then the pdf would be $X_1 = 0$ (because they are not the same value).
Instead of fixing values for the vertices, look at the probability that $X_i=1$; let's call this $P(X_i)$. What is $P(X_1)$?... | {
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Induction on summation inequality stuck on Induction step working on a fairly simple induction problem, but stuck when decomposing the m+1^st summation and bringing in the I.H to show it holds.
I am bad with formatting so bare with me.
$$\sum_{k=1}^n \frac1{k^2} \leq 2 - \frac1n$$
Base Case:
Set $n = 1$ and show that t... | i think it must be
$$\sum_{k=1}^m\frac{1}{k^2}+\frac{1}{(m+1)^2}\le 2-\frac{1}{m+1}$$ and it must be
$$\le 2-\frac{1}{m}+ \frac{1}{(m+1)^2}\le 2-\frac{1}{m+1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2620879",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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what does mean a zero eigenvalue in a PDE? I understand that in a PDE, the eigenvalues are some kind of speed of information propagation. For a hyperbolic system of PDE's, with three eigenvalues and one of them being zero, what does this mean? Is it the same than having a system of two equations?
This is the system of ... | If the eigenvalues are the wave propagation speeds, then a zero eigenvalue just says that one wave is stationary
| {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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French playing cards, probability to have a four (of a kind) We have a "French playing cards" composed of $52$ cards. Each kind of cards (Ace, King, Queen, ...) is composed of $4$ cards : There are $4$ Ace, $4$ Jack, etc. We pull $5$ cards among the $52$ cards.
We are a four (of a kind) if among the $5$ cards pulled, ... | Here is a hint:
Given that you have drawn one ace and one king, you will need either to draw the remaining three aces, or draw the remaining three kings, from the remaining $50$ cards.
Can you continue from this point?
(Spoiler answer)
There are two winning ways to draw the remaining three cards, and ${50 \choose 3}$... | {
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Ramification in cyclotomic field I was trying to prove that:
$p$ is a odd prime and $\zeta$ is a primitive $p$-th root of unity and let $p^{*} = (-1)^{\frac{p-1}{2}}p$. Show that $\mathbb{Q}(\sqrt{p^{*}})$ is a quadratic subfield of $\mathbb{Q}(\zeta)$.
This is what I got so far:
The only prime ramified in $\mathbb{Q}(... | Let $p$ be an odd prime, and consider the cyclotomic field $K=\mathbf Q(\zeta_p)$, which is a cyclic extension of degree $(p-1)$. Since $p$ is odd, by the Galois correspondence, $K$ contains a unique quadratic field $k=\mathbf Q (\sqrt d)$, and we want to show that actually $k=\mathbf Q (\sqrt {p^*})$ (in your notatio... | {
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Evaluating the liminf and limsup So I am a little bit stuck on finding the liminf and limsup of the following interval:
$$[(-1/n)^{n} , 2]$$
I know that
$$\liminf_{n\to\infty} A_n = \cup^{\infty}_{n=1}(\cap^{\infty}_{j=n}A_{j})$$
$$\limsup_{n\to\infty} A_n = \cap^{\infty}_{n=1}(\cup^{\infty}_{j=n}A_{j})$$
but I do not... | HINT
Note that $ |a_n|= |(-1/n)^n|\to 0$ is strictly decreasing thus
$$-1\le a_n\le \frac14$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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how to geometrically explain why pell numbers close to sqrt 2
How to graphically explain that the limit of yn/xn = $\sqrt 2$, as n approaches to infinity?
Like, I know how to prove it algebraically.
| Perhaps not quite what you want, but an explanation how the heron-method works.
We want to determine the length of a square with area $2$. We start with the rectangular with sides $1$ and $2$. To get closer to a square, we take the arithmetic mean of the sides (which is $\frac{3}{2}$ and determine the corresponding si... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $\overline S$ is not path connected, where $S=\{x\times \sin(\frac1x):x\in(0,1]\}$
The goal is to show that $\overline S$ is not path connected, where $S=\{x\times \sin(\frac{1}{x}):x\in(0,1]\}$.
Proof.
Suppose there is path $f:[a,c]\to\overline S$ beginning at the origin and ending at a point of $S$.
The... | Let $H=\{(x, sin(1/x)) \;:\; x\in(0,1)\}$ and $T = \{0\} \times [-1, 1]$. Let
$\tag 1 S = H \cup T$
It is easy to show that $S$ is path-connected if and only if $S \cup \{\left(1,sin(1)\right)\}$ is path-connected. This is only mentioned because we modified (ever so slightly) the OP's question to fit the machinery foun... | {
"language": "en",
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"source": "stackexchange",
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$E = \mathbb{R}\times e$ is a n-dimensional measure zero if $e \subset \mathbb{R}^{n-1}$ of (n-1) dimensional measure zero In the book of Mathematical Analysis II by Zurich, at page 116 it is asked that
Show that if a set $E \subset \mathbb{R}^n$ is the direct product of
$\mathbb{R}\times e$ of the line $\mathbb{R... | From the construction of the product measure $\lambda_n$ (Lebesgue measure of $\mathbb{R}^n$),
if $A\in \mathcal{B}(\mathbb{R}^k)$ and $B\in \mathcal{B}(\mathbb{R}^{n-k})$, ($0<k<n$), then
$$ \lambda_n(A\times B) =\lambda_k(A)\lambda_{n-k}(B)$$
So here (using also that if $E_m \uparrow E$, $\lim_{m\rightarrow +\in... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proof by contrapositive: if $a$ and $b$ are consecutive integers then the sum $a + b$ is odd if $a$ and $b$ are consecutive integers then the sum $a + b$ is odd
Proof by contrapositive
Contrapositive form:
if the sum of $a$ and $b$ is not odd then $a$ and $b$ are not consecutive integers
I am struck here, so if $a + b$... | A direct proof is so much clearer:
$b=a\pm1$ implies $a+b= 2a\pm1$, which is odd.
But if you must use contrapositive:
Let $b=a+d$. Then $a+b=2a+d$ is even iff $d$ is even. Therefore, $|a-b|=|d|$ is even and so is never $1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 3
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Number of edges in the simple graph with the given conditions Let $G$ be a simple graph on $n$ vertices $v_1,v_2...v_n$. Let $G-v_i$ is having $m_i$ edges for $1\leq i \leq n$. Then prove that $m=\frac{1}{n-2}\sum_{i=1}^{n}m_i$.
| By handshake lemma we have $$\sum _{i=1}^n d_i = 2m$$
For each $i$ we have $$m_i = m-d_i$$
where $d_i$ is degree of $i$-th node. Now if we sum this over all $i$ we get
$$ \sum _{i=1}^n m_i = n\cdot m -\sum _{i=1}^n d_i = nm -2m = m(n-2)$$
and we are done.
| {
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Determine all ring homomorphisms of $\varphi:\mathbb{Z}_{10} \to \mathbb{Z}_{10}$ This is what I came up with as a solution and I wanted some suggestions on how to proceed with the last part or perhaps a new direction of thinking.
Given that $m=n=10$ we must have $0=m\cdot a=10a$ which is in $\mathbb{Z}_{10}$ $\Longlef... | I was incorrect in assuming there were only two ring homomorphisms. When I calculated $2^{\omega(n)-\omega(n/(m,n))}$ I neglected to realize that it yields 4 instead of 2. The four ring homomorphisms for $\varphi : \mathbb{Z}_{10}\to\mathbb{Z}_{10}$ are in fact:
$$\varphi : \mathbb{Z}_{10}\to\mathbb{Z}_{10};x\mapsto0$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Median estimated from grouped data with a single class Given the formula for grouped median:
$Median = L_m + \left [ \frac { \frac{n}{2} - F_{m-1} }{f_m} \right ] \times c$
Where:
*
*$L_m$: lower boundary of median class
*$c$ : size of the median class
*$F_{m-1}$ : cumulative frequency of the class before median... | When you group data into intervals, information is lost.
So assumptions are made in order to make reasonable estimates of
the sample mean, median, etc.
The assumption of this formula for estimating the median from grouped data is that the data are spread roughly uniformly throughout the interval. Clearly, this assump... | {
"language": "en",
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A set of integers Assume that there is a set of ordered integers initially containing number $1$ to a given $n$. At each step, the lowest number in the set is removed, if the number was odd, then we go to the next step and if it was even, half of that number is inserted into the set and the cycle repeats until the set ... | As in this algorithm, the actual order of the set isn't important, we can assign every $n\in\mathbb{N}$ the number of steps it takes till it's sorted out, which is namely how many times you'll have to divide by two till it's odd.
If we use prime factorization, that means if $z$ takes k steps till it's sorted out, it's ... | {
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Show that $2n\choose n$ is divisible by $2n-1$ I have found many questions asking to prove $2n\choose n$ is divisible $2$, but I also observed by trying the first few "$n$" that $2n\choose n$ is divisible by $2n-1$.
It sure seems true when $2n-1$ is prime, but is it true in general for all $n$?
| Note that $\dbinom{2n}{n}$ is an integer. Note also that $1$ divides $\binom 21 $ and assume $n>1$ hereafter.
$\begin{align}
\dbinom{2n}{n} &= \dfrac{2n!}{n!\cdot n!} \\ &= \dfrac{2n\cdot (2n-1)\cdots (n+1)}{n\cdot (n-1)\cdots 1}\\
&= \dfrac{(2n)(2n-1)}{n\cdot n}\dbinom{2(n-1)}{n-1}\\
&= 2\dfrac{2n-1}{n}\dbinom{2(n-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2622719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding parametrized solutions to a trigonometric equation I want to find what pairs $(x,y)$ satisfy $\cos(x-y) = \cos(x) - \cos(y)$. To start, I defined a function $f(x,y) = \cos(x-y) - \cos(x) + \cos(y)$. I want to find functions $x(t), y(t)$ s.t. $f(x(t), y(t)) = 0$ I didn't really know where to go, so I tried putti... | I'm not sure if your method with exponentials allows you to conclude, but here's an overview of how I'd do it.
Although this isn't linear algebra, you can still get an idea of how to proceed by looking at your number of equations and unknowns. You have two unknowns for one equation, so if you add in one more unknown (p... | {
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Prove that there are no integers $x, y, z$ such that $x+y+z=0$ and $1/x+1/y+1/z=0$
Prove that there are no integers $x, y, z$ such that $x+y+z=0$ and $\frac1x+\frac1y+\frac1z=0$
My thinking was that since the numbers are integers, then there can't be $2$ negative values that cancel out the positive or $2$ positive nu... | Hint: if $x+y+z=0$, then
$$
\begin{align}
\frac1x+\frac1y+\frac1z
&=\frac1x+\frac1y-\frac1{x+y}\\
&=\frac{(x+y)^2+x^2+y^2}{2xy(x+y)}\\
&=-\frac{x^2+y^2+z^2}{2xyz}
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2622967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 9,
"answer_id": 6
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Compute the determinant. If $a$, $b$, and $c$ are roots of unity. Then what is the value of
\begin{vmatrix}
e^a & e^{2a} & (e^{3a}-1) \\
e^b & e^{2b} & (e^{3b}-1) \\
e^c & e^{2c} & (e^{3c}-1)
\end{vmatrix}
I tried expanding it but the expression becomes unmanageable, is there some kind of simplification I can do?
| This is
$$\det\pmatrix{x&x^2&x^3-1\\y&y^2&y^3-1\\z&z^2&z^3-1}
=\det\pmatrix{x&x^2&x^3\\y&y^2&y^3\\z&z^2&z^3}
-\det\pmatrix{x&x^2&1\\y&y^2&1\\z&z^2&1}
$$
for $x=e^a$ etc. Both of these are essentially Vandermonde determinants.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2623058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Assumption of proof with contrapositive I want to prove by proof with contrapositive that
$\left \| u(0) \right \|=0 $ then $\left \| u(t) \right \|=0$ for all t belongs to $\left [ 0,T \right ]$.
Then I write
$\left \| u(t) \right \|\neq 0$ then $\left \| u(0) \right \|\neq 0$ for all t belongs to $\left [ 0,T \right ... | The contrapositive of
If $∥u(0))∥=0$, then $‖u(t))‖=0$ for all $t$ belonging to $[0,T]$.
is the following:
If $‖u(t))‖\ne 0$ for some $t$ belonging to $[0,T]$, then $∥u(0))∥\ne 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2623234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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How to prove that $87!<16! \left(52^{71}\right)$ Prove that
$$87!<16! \left(52^{71}\right)$$
I do not how can i compare between the factorials or what the procedure to solve such questions?
| AM-GM gives:
$$\frac1{71}\sum_{i=1}^{71}(i+16)>\sqrt[71]{\prod_{i=1}^{71}(i+16)}$$
So:
$$\frac1{71}\left(\frac{71(71+1)}{2}+71\cdot 16\right)>\sqrt[71]{\frac{87!}{16!}}$$
or:
$$52^{71}>\frac{87!}{16!}\implies 16!(52^{71})>87!$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2623366",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How do you derive an equation if you know the asymptotes? I am trying to curve fit and let's say you have a data set in which you know the asymptotes are:
$y = 0$
$y = -x + 683$
The data always has a positive $x$ value, and the data converges to the line $y = -x + 683$ and has a parabola looking shape. The data is on ... | A simple function that shoul make the work is the following
$$f(x)=-x+683+\frac{k}{x^a}$$
with $k,a>0$ parameters to fit the data.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2623494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Image of unit ball under twice differentiable function Let $f: \mathbb{R^n} \to \mathbb{R^n}$ be a twice differentiable function such that $f(x) = 0$ outside of the unit ball $B \in \mathbb{R^n}$. I have to show that (the measure is Lebesgue): $$\int_B \det Df = 0$$
I tried to prove that $|f(B)| = 0$ (which should be e... | Let $f=(f_1,\ldots,f_n)$. Consider the form
$$
w=f_1df_2\wedge\ldots \wedge df_n-f_2df_1\wedge df_3\ldots \wedge df_n+\ldots+(-1)^nf_ndf_1\wedge \ldots \wedge df_{n-1}
$$
and apply the Stokes theorem:
$$
\int_{\partial B}w=\int_{B}dw.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2623582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $A \smallsetminus (A \smallsetminus B) = A \cap B$ $A$ and $B$ are any sets, prove that $A \smallsetminus (A \smallsetminus B) = A \cap B.$ This formula makes sense when represented on a Venn diagram, but I am having trouble with proving it mathematically.
I have tried letting $x$ be an element of $A$ and co... | Let $x \in A \setminus (A \setminus B).$
Then $x$ is an element such that $x \in A$ and $x \notin A \setminus B$. But if $x \notin A\setminus B$, with some additional work, we realize this implies that $x \in A$ and $ x \in B$. So $x \in A \cap B$.
Vice-versa: let $x \in A \cap B$ so $x \in A$ and $x \in B$. This imp... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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How to prove that $|\ln(2+\sin(x)) - \ln(2+\sin(y))| <= |x-y| \space \forall \space x,y \in \mathbb R$ Question states
Prove that for all $x$ and $y$ $\in R$, the following inequality is true:
$\lvert \ln(2+\sin(x)) - \ln(2+\sin(y))\rvert \le \lvert x-y\rvert$
i've gotten to the point that
$\frac{y-x}{2+\sin(c)} = \ln... |
I thought it might be instructive to present a way forward that relies on only elementary, pre-calculus tools. To that end we proceed.
First, note that $\log(2+\sin(x))-\log(2+\sin(y))=\log\left(\frac{2+\sin(x)}{2+\sin(y)}\right)$.
Now, in THIS ANSWER, I showed using only the limit definition of the exponential f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2623936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
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"Continuity" of volume function on hyperbolic tetrahedra
Consider a sequence $T_i$ of tetrahedra in $\mathbb H^3$ whose
vertices tend to the vertices of a regular ideal tetrahedron $T$ in
$\partial \mathbb H^3$. Then $$Vol(T_i)\to v_3.$$
This should follow from Lebesgue dominated convergence if $T_i\subseteq T_{i... | There are some explicit formulae for volumes of hyperbolic tetrahedra in terms of dihedral angles which are not just continuous but real-analytic functions, say, one by Ushijima (Theorem 1.1):
A volume formula for generalized hyperbolic tetrahedra.
(See also references to earlier works that he gives in the paper.) Dih... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2624025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Finding integer solutions to $3^y = x^2 + 56$ The original question is asking for the following:
Find all ordered pairs of positive integers
$(x, y)$ for which $3^y = x^2 + 56$.
I've found a solution I believe to be the only solution, but I'm struggling to prove it's the only solution. $(5,4)$
When I see a problem like... | Since $y>0$, $x$ must be odd. Also, as you said, since $3^y\equiv_4(-1)^y\equiv_4 x^2+56\equiv_4 0$, $y$ must be even. Notice that $$3^{y}-x^2=(3^{\frac{y}{2}}-x)(3^{\frac{y}{2}}+x)=56=2^3\cdot7$$
So there are only finitely many solutions, since $3^{\frac{y}{2}}+x>0$ divides $56$ (and take up atmost 8 values, since 56 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2624172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Joining the centers of the edges of Platonic solids We know that taking the centers of the faces of any 3d polyhedron (say, the Platonic solids) produces the dual solid. And repeating this operation gives us back the original solid. Another possible thing we can do is take the centers of the edges. This will produce ot... | The general setup here is dealing with regular polytopes. Then you are asking about taking the centers of k-faces (k-dimensional sub-elements). This process is usually called the k-rectification, cf. https://en.wikipedia.org/wiki/Rectification_(geometry).
When describing regular polytopes by Dynkin diagrams, those are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2624282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\arg(\frac{z_1}{z_2})$ of complex equation If $z_1,z_2$ are the roots of the equation $az^2 + bz + c = 0$, with $a, b, c > 0$; $2b^2 > 4ac > b^2$; $z_1\in$ third quadrant; $z_2 \in$ second quadrant in the argand's plane then, show that $$\arg\left(\frac{z_1}{z_2}\right) = 2\cos^{-1}\left(\frac{b^2}{4ac}\right)^{1/2}$$... | $z_2 = \overline{z_1}\,$ since the quadratic has real coefficients, so $\,\arg\left(\dfrac{z_1}{z_2}\right)=\arg(z_1)-\arg(z_2)=2 \arg(z_1)\,$.
Since $\,a,b,c \gt 0\,$ the roots are in the left half-plane $\,\operatorname{Re}(z_1)=\operatorname{Re}(z_2) = -\,\dfrac{b}{2a} \lt 0\,$, and given the condition that "$z_1\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2624491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For independent, symmetric random variables $(\xi_n)$, $E\left(\left(\sum\limits_n\xi_n\right)^2\land1\right)\le\sum\limits_nE(\xi_n^2\land1)$ The following problem appears as an exercise in the Russian version of Probability, Shiryaev, 2003 edition(it seems that no English version containing this problem is available ... | What is wrong with the following much simpler argument: $$E\left\{\left(\sum {\xi_i}\right)^{2} \wedge 1 \right\} \leq \left\{E\left(\sum \xi_i\right)^{2}\right\}\wedge 1\ = \left\{\sum E(\xi_i)^{2}\right\} \wedge1 \leq \sum \left\{E(\xi_i)^{2} \wedge 1\right\}$$ where we have used the fact that $E\xi_i \xi_j =0$ for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2624583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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There is a natrual connection on the tangent bundle? I come up with a (maybe stupid) question: let $M$ be a smooth manifold, then the exterior differential $d$ is a natural connection on $\Omega^k(M)$, hence by dualizing we get a natural connection on $TM$? How can it be true (without a metric)?
| Since you say "dualizing", I assume you're talking about the case $k=1$. If we view differential forms as alternating tensors, the exterior derivative $\Omega^1(M) \to \Omega^2(M)$ can be viewed as a map $\Gamma(T^*M) \to \Gamma(T^*M \otimes T^*M),$ which has the right "type signature" to be a connection; but it does n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2624748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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What does Morley rank of a quotient group mean? $ \DeclareMathOperator{\RM}{RM} $
I have a problem with understanding part b) of Exercise 6.6.23 in David Marker's "Model Theory: An Introduction": ($ \mathbb M $ is the monster model.)
a) Suppose that $ \mathbb M $ is $ \omega $-stable, $ A , B \subseteq \mathbb M ^ n $... | The algebra question: When $H$ is an arbitrary subgroup of $G$, $G/H$ often denotes the set of left cosets of $H$, $\{gH\mid g\in G\}$. When $H$ fails to be normal, we can't equip $G/H$ with a natural group structure, but we can equip it with the structure of a $G$-set: the action of $G$ on $G/H$ is the obvious one $g'... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2624863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Confusing permutation? Or combination? Is this problem a permutation or combination problem? My idea is that this is permutation but Im not that sure... any idea to solve this problem? I tried putting the first $3$ company in $3^3$ possible ways that is $27$, but i got boggled on the remaining $2$ organizations because... | Since the students are distinguishable, it is a permutaiton problem.
It is just $\frac{5!}{(5-3)!}$.
You can also think of it as first out of the $5$ companies, choose $3$ companies and then rearrange the students.
$$\binom{5}{3} \times 3!.$$
Remark: $3^3$ doesn't seem to avoid the constraint that no $2$ students can ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2624972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving a limit exists using the definition of a limit $ \lim_{x\to\infty} {\sqrt{f(x)+1}}) = {\sqrt{L+1}} $ Let $f(x)$ be a function such that $f(x)\geq -1$ for every $x$. Suppose, $\lim_{x\to\infty} f(x) = L$ and that $L\leq-1$. Prove by using the definition of a limit that:
$ \lim_{x\to\infty} {\sqrt{f(x)+1}}) = {\... | Supposing $L$ is finite and $L\geq -1$: $\displaystyle\lim_{x\to\infty}f(x)=L$ means that (by definition) for every given $\epsilon>0$, there exists a real number $N_1$ (depending on $\epsilon)$ such that the inequality $|f(x)-L|<\epsilon$ is satisfied for all $x\geq N_1$. Also for the same epsilon there is $N_2$ such... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2625128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that if $A,B$ are $2\times2$ matrices, then $(AB-BA)^2$ commutes with all $2\times2$ matrices. I tried to write it all out, but it becomes really messy... Is there a more elegant way to do it?
Note that I don't know about dimensions, vector spaces & bases yet
| $AB-BA$ has trace zero, so has the form $\pmatrix{u&v\\w&-u}$.
What happens when you square such a matrix?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2625217",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Difference between minus one and plus one induction? I recently started a Combinatorics class, in which my teacher (grad student) has instructed us to Prove by induction that $$1^2+2^2+\ldots+n^2 = \frac{n(n+1)(2n+1)}{6} = \frac{2n^3+3n^2+n}{6}$$ this is trivial in the fact that it has been solved many times before, ho... | I would argue that, if anything, there are reasons to prefer $P(n) \Rightarrow P(n+1)$.
Natural numbers can be defined in many ways, but the usual inductive definition is the following:
*
*$0$ is a natural number;
*If $n$ is a natural number, then $s(n)$ is a natural number.
Here $s(n)$ denotes the successor of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2625347",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Where is the error in this proof for showing that $f:\mathbb{R}\rightarrow\mathbb{R},x\mapsto\sin{(2\pi x)}$ is constant? Show that $f:\mathbb{R}\rightarrow\mathbb{R},x\mapsto\sin{(2\pi x)}$ is constant.
$$1=1^x=(e^{2\pi i})^x=e^{2\pi ix}=\cos{(2\pi x)}+i\sin{(2\pi x)}$$
Where is the error in this "proof"? Does it not... | Notice that $(-2)^2 = 4$. So $[(-2)^2]^{\frac 12} = 4^{\frac 12} = 2$.
But $[(-2)^2]^{\frac 12} = (-2)^{2\frac 12} = (-2)^1 = -2 \ne 2$.
What went wrong?
The assumption that $(a^m)^n = a^{mn}$ only holds if either $m, n \in \mathbb Z$ or if $a > 0; a \in \mathbb R; m,n \in \mathbb R$.
If we define $a^m = a*.... *a; ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Modular arithmetic and some applications Show that if $p> 2$ is a prime number, then $p$ divides $(p-2)! - 1$. I have tried using Fermat's Theorem, but I could not solve it.
| This follows from Wilson's theorem, which states that
$$(p-1)!\equiv_p -1$$ if and only if $p$ is a prime
If you accept this, then the rest follows easily since $(p-1)!\equiv_p (p-2)!(-1)\equiv_p-1$ implies: $$(p-2)!\equiv_p 1$$
This is the same as saying that $p$ divides $(p-2)!-1$.
Now for the proof of this theorem... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove the equation $\ln(x) = \frac1 {x-1}$ has exactly 2 real solutions.
Prove the equation $\ln(x) = \frac1 {x-1}$ has exactly 2 real solutions.
Hello all. I thought of defining the function $f(x)=x-e^{\frac 1 {x-1}}$ and showing it has only 2 single roots, though I am not sure on how to show it and I understand it ... | $\ln x$ does not exist for $x \le 0$ and $\frac 1{x-1}$ does not exist for $x = 1$.
So if there are solution they will exist on the intervals $(0,1)$ and $(1, \infty)$.
On these intervals the function $f(x) = \ln x - \frac 1{x-1}$ is continuous.
$f'(x) = \frac 1x - (-1)\frac 1{(x-1)^2} = \frac 1x + \frac 1{(x-1)^2} > 0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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An entire function whose integral is bounded is identically zero Suppose $f$ has a power series at $0$ that converges in all of $\mathbb{C}$ and $$\int_{\mathbb{C}} |f(x+iy)|dxdy$$
Converges. Prove $f$ is identically zero. I don’t know Liouville’s theorem or any integral formulas yet, so I’m a bit stuck on this one.
A... | Let $f(z)=\sum_0 ^{\infty} a_n z^{n}$ be the power series expansion. Write $z=re^{i\theta}$ and integrate with respect to $\theta$ from 0 to $2\pi$. Integrating term by term is permitted because of uniform convergence. You get $2\pi a_0= \int_0 ^{2\pi} f(re^{i\theta}) d\theta$. Note that $a_0 =f(0)$. Multiply both side... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$\lim\limits_{x\to\infty}\frac{\sqrt{4x^2+x^4}+3x^2}{x^2-5x}$ Can anyone help me solve this?
I know the answer is 4, but I don't really know how do I find the biggest power of $x$ when there's a square root.
$$\lim_{x\to\infty}\frac{\sqrt{4x^2+x^4}+3x^2}{x^2-5x}$$
| HINT
Take $x^2$ as a common factor and simplify the quotient.
The limit will be $4$
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Finding the 10th root of a matrix I want to find a $2 \times 2$ matrix, named $A$ in this situation, such that:
$$A^{10}=\begin {bmatrix} 1 & 1 \\ 0 & 1 \end {bmatrix} $$
How can I get started? I was thinking about filling $A$ with arbitrary values $a, b, c, d$ and then multiplying it by itself ten times, then setting ... | Hint: Another approach is to note that
$$
\exp\left(\begin{bmatrix}0&x\\0&0\end{bmatrix}\right)=\begin{bmatrix}1&x\\0&1\end{bmatrix}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2626206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 6,
"answer_id": 5
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Meijer G-function Can you please help me devise a series for the Meijer's G-function (i) with inceces m=3, n=0, p=1 and q=3, for a general real variable?
The first difficulty that I am facing is the proper choice of an integration path, to use the residues' theorem. It seems to me that I may choose between two possibil... | We have
$$G_{1,3}^{3,0}\left( x \middle| {0 \atop z, 0, 0} \right) =
\frac 1 {2 \pi i} \int_{\mathcal L}
\frac {\Gamma(z-y) \Gamma(-y)^2} {\Gamma(-y)} x^y dy.$$
Notice that a pair of the gamma functions cancels out.
To determine the integration contour, you need to analyze the asymptotic behavior of the gamma function... | {
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Random point inside an equilateral triangle Take any equilateral triangle and pick a random point inside the triangle.
Draw from each vertex a line to the random point. Two of the three angles at the point are known let's say $x$,$y$.
If the three line segments from each vertex to the random point were removed out of t... |
As in the attached diagram, let $ABC$ be the original equilateral triangle and let $D$ be a point in $\triangle ABC$.
We let point $E$ be on the opposite side of $BC$ as $D$ such that $\triangle BDE$ is equilateral. Then $BD=BE$, $BA=BC$ and $\angle DBA=\angle EBC=60^{\circ}-\angle DBC$. And therefore $\triangle DBA$... | {
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If $a,b,c$ be in Arithmetic Progression, If $a,b,c$ be in Arithmetic Progression, $b,c,a$ in Harmonic Progression, prove that $c,a,b$ are in Geometric Progression.
My Attempt:
$a,b,c$ are in AP so
$$b=\dfrac {a+c}{2}$$
$b,c,a$ are in HP so
$$c=\dfrac {2ab}{a+b}$$
Multiplying these relations:
$$bc=\dfrac {a+c}{2} \dfrac... | Hint:
Eliminate $c$
$$2ab=(a+b)c=(a+b)(2b-a)$$
Simplify to find $$0=a^2+ab-2b^2=(a+2b)(a-b)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2626517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Proposed definition of a countable set Consider the following proposed definition:
A set $X$ is countable iff $X=\emptyset$ or there exists $x_0\in X$ and $f:X\to X$ such that:
$\forall P\subset X: [x_0\in P \land \forall x\in P: [f(x) \in P] \implies P=X]$
In other words, a set $X$ is countable iff $X$ is empty or ... | We can show that if $X$ satisfies this property, then there is a surjection $g:\mathbb N\rightarrow X$. Thus $X$ is countable in the usual sense. Of course the converse holds, as you already know.
Define $g$ as $g(0)=x_0$ and $g(n+1)=f(g(n)),\forall n\in\mathbb N$. Then by the property in question, the image of $g$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2626767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Number of answers : $f(x)=f^{-1}(x)$
let $f(x)= 1+\sqrt{x+k+1}-\sqrt{x+k} \ \ k \in \mathbb{R}$
Number of answers :
$$f(x)=f^{-1}(x) \ \ \ \ :f^{-1}(f(x))=x$$
MY Try :
$$y=1+\sqrt{x+k+1}-\sqrt{x+k} \\( y-1)^2=x+k+1-x-k-2\sqrt{(x+k+1)(x+k)}\\(y-1)^2+k-1=-2\sqrt{(x+k+1)(x+k)}\\ ((y-1)^2+k-1)^2=4(x^2+x(2k+1)+k^2+k)$$
... | Hint:
Point of intersection of $f(x)$ and $f^{-1}(x)$ while same as that of $f(x)$ and the line $y=x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Does this entangled PDE capture the derivative? In a game-theory textbook, I encountered the following. Suppose we have player $1$ and and $2$ optimizing by playing strategies $x_1$ and $x_2$ $\in \mathbb{R}$. The first order conditions for player $1$ and $2$ are given respectively by:
$$f(x_1,x_2,a,b)=1$$
$$g(x_1,x_2,... | If you differentiate both equations in $a$, you get
$$
f_a+x^1_af_1+x^2_af_2=0,\\
g_a+x^1_ag_1+x^2_ag_2=0,
$$
where I'm denoting $x^j_a=\partial x^j/\partial a$ and $f_j=\partial f/\partial x^j$. If we solve this system for $x^1_a$, we get
$$
x^1_a=\frac{f_2g_a-g_2f_a}{g_2f_1-g_1f_2}.
$$
Using Jacobians, you can expre... | {
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"url": "https://math.stackexchange.com/questions/2627214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to solve $a$ in $\int_a^xf\left(t\right)dt=x^2-2x-3$ How do you solve for $a$ in $\int_a^xf\left(t\right)dt=x^2-2x-3$?
I have differentiated both sides with respect to $x$, but do not know how to continue after this.
| Put $F(x) = \displaystyle \int_{a}^x f(t)dt=x^2-2x-3$. Since $F(a) = 0$, $a^2-2a-3 = 0 \implies (a-1)^2 = 4\implies a - 1 = \pm 2 \implies a = -1, 3$. Thus there are $2$ answers: $a = -1,3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 3
} |
Why does x raised to an odd power result in a unique solution? I don't think I'm phrasing this correctly but anyway, say we are looking for solutions to x for $x^9=1/2$,
why is there only one unique solution as given by $x=(1/2)^{1/9}$
when by raising x to an even value gives two solutions as $\pm$ solution
I think I'm... | The simple answer is there is not only one solution, but only one real solution. For example, the equation $x^3=1$ has the solutions $x=1,e^{\frac{2i\pi}{3}},$ and $e^{\frac{4i\pi}{3}}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Given sequence of nested intervals $I_n[a_n,b_n]$. Show that if $y=\inf\{b_{1},b_{2},b_{3},\ldots\}$ then $y\in[a_{n},b_{n}]$ $\forall n$ Let $I_{1}=[a_{1},b_{1}]$, $I_{2}=[a_{2},b_{2}]$, $I_{3}=[a_{3},b_{3}]$,
$\ldots$ be a sequence of closed bounded nested nonempty intervals
$I_{1}\supseteq I_{2}\supseteq I_{3}\supse... | You have made slight errors in the inequalities in both the cases.
If $k\geq n$, then the nested property gives $[a_n, b_n]\supseteq [a_k, b_k]$ so $a_n\leq a_k\leq b_k$ and therefore $a_n\leq b_k$.
If $k<n$, then $[a_k, b_k]\supseteq [a_n, b_n]$ so $a_n\leq b_n\leq b_k$ and so $a_n\leq b_k$ again.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Value of $\sec^2 a+2\sec^2 b$
If $a,b$ are $2$ real number such that $2\sin a \sin b +3\cos b+6\cos a\sin b=7,$ Then $\sec^2 a+2\sec^2 b$ is
Try: i am trying to sve it using cauchy schwarz inequality
$$\bigg[(\sin b)(2\sin a+6\cos a)+(\cos b)(3)\bigg]^2\leq (\sin^2b+\cos^2b)\bigg((2\sin a+6\cos a)^2+3^2\bigg)$$
Could... | Hint:
$2\sin a+6\cos a=\sqrt{40}\sin (a+x)$ where $x=\sin^{-1}\frac{6}{\sqrt{40}}$.
\begin{align*}
2\sin a \sin b +3\cos b+6\cos a\sin b&=\sqrt{40}\sin (a+x)\sin b+3\cos b\\
&=\sqrt{40\sin^2(a+x)+9}\cdot \sin(b+y)
\end{align*}
where $\sqrt{40}\sin(a+x)=\sqrt{40\sin^2(a+x)+9}\cdot\cos y$ and $3=\sqrt{40\sin^2(a+x)+9}\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Compute the period of this function $f(x)=7+3\cos{(\pi x)}-8\sin{(\pi x)}+4\cos{(2\pi x)}-6\sin{(2\pi x)}$ I'm starting my journey into Fourier Series. I am given this function: $$f(x)=7+3\cos{(\pi x)}-8\sin{(\pi x)}+4\cos{(2\pi x)}-6\sin{(2\pi x)}$$
Following my book, this function has a period of $T=2$ (this is the b... | You need $$7+3\cos{(\pi x)}-8\sin{(\pi x)}+4\cos{(2\pi x)}-6\sin{(2\pi x)}=$$
$$=7+3\cos{(\pi (x+T))}-8\sin{(\pi (x+T))}+4\cos{(2\pi( x+T))}-6\sin{(2\pi (x+T))}$$ for all real $x$, which indeed gives $T=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
$f(x) = x^6, g(x) = x^{10}$ endomorphisms $\implies G$ is abelian
Let $(G, \cdot)$ be a group in which the functions $f: G \to G, f(x) = x^6$ and $g : G \to G, g(x) = x^{10}$ are endomorphisms and $f$ is injective. Prove that $G$ is an abelian group.
We need to prove that $f(xy) = f(yx), \forall x,y \in G$.
Because $... | This is an extended comment rather than an answer.
The assumption that $f$ is injective is necessary.
Indeed, the article Abelian Forcing Sets by Joseph A. Gallian and Michael Reid contains the following result:
Definition: a set of integers $T$ is called abelian-forcing if for
any group $G$, if the map $x\mapsto x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2627964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
Inverse Transform Method for a pdf Given a pdf:
$$f(x)=\tau x \exp\left(\frac{-\tau x^2}{2}\right); \quad x, \tau > 0$$
So I found the corresponsing cdf:
$$F(x)=1 - \exp\left(\frac{-\tau x^2}{2}\right)$$
Then I got given a value for tau: $\tau=0.2$ and I derived the inverse function $F^{-1}_X(u).$:
$$F^{-1}_X(u)=\sqrt{... | The statement is
If $u \sim U[0,1]$ then $F_X^{-1}(u)\sim f$
To show you this I sample a uniform distribution and calculate
$$
x = F^{-1}_X(u) = \sqrt{-\frac{2}{\tau}\ln(1 - u)}
$$
Then I make a histogram and overplot the distribution
$$
f_X(x) = \tau x e^{-\tau x^2/2}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2628070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Linear homogenous second order ODE without constant coefficients I am having trouble finding the general solution of the following second order ODE for $y = y(x)$ without constant coefficients:
$3x^2y'' = 6y$
$x>0$
I realise that it may be possible to simply guess the form of the solution and substitute it back into t... | Hint
Simple way
$$y''-2\dfrac {y}{x^2} = 0$$
$$x^2y''+2xy'-2xy'-2y= 0$$
$$(x^2y')'-2(xy)' = 0$$
Integrate
$$(x^2y')-2(xy) = K_1$$
Divide by $x^4$
$$\dfrac {x^2y'-2xy}{x^4} = \dfrac {K_1}{x^4}$$
$$(\dfrac {y}{x^2})' = \dfrac {K_1}{x^4}$$
integrate again
$$\dfrac {y}{x^2} = \int \dfrac {K_1}{x^4}dx +K_2$$
$$\boxed{y(x)=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2628149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
What is the largest eigenvalue of the following matrix?
Find the largest eigenvalue of the following matrix
$$\begin{bmatrix}
1 & 4 & 16\\
4 & 16 & 1\\
16 & 1 & 4
\end{bmatrix}$$
This matrix is symmetric and, thus, the eigenvalues are real. I solved for the possible eigenvalues and, fortunately, I found that ... | The trick is that $\frac1{21}$ of your matrix is a doubly stochastic matrix with positive entries, hence the bound of 21 for the largest eigenvalue is a straightforward consequence of the Perron-Frobenius theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2628359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "69",
"answer_count": 11,
"answer_id": 3
} |
Basic algebra exercise I'm stuck with this problem. I think that my difficulties are more with dealing with complex numbers then with groups, but still.
Could you please help me?
Let $\mathbb{C}^{*}$ be $\mathbb{C} \setminus \{0\}$, the multiplicative group of the complex numbers without zero.
Let $\rho$ be the equiva... | If we write $a=re^{ix}$ and $b=se^{iy}$, then $a \sim b$ if and only if $e^{2i(x-y)} \in \mathbb R$ For what "angles" does this occur?
Step 1: if $a \sim b$ is it also true that $ac \sim bc$ for any $c \in \mathbb R$?
Step 2: note that $1 \in H$ for any subgroup of $\mathbb C$. We need at least all of the $z \sim 1$. W... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2628447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Strategy for finding integral roots for polynomials with large coefficients I'm trying to find the integer roots for $f(x) = x^5 + 47x^4 + 423x^3 + 140x^2 + 1213x - 420 = 0$. The techniques I'm expected to have at my disposal are:
*
*For a polynomial with integer coefficients $p(x) = a_nx^n + a_{n-1}x^{n-1}+ \dots +... | Try
\begin{align}
x^5 + 47x^4 + 423x^3 + 140x^2 + 1213x - 420
& \equiv (x^2+ax+b)(x^3+cx^2+dx+e) \\
& \equiv x^5+(a+c)x^4+(ac+b+d)x^3 \\
& \quad +(ad+bc+e)x^2+(ae+bd)x+be
\end{align}
with different combinations of $b$ and $e$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2628532",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Absorbing Markov Chain Probabilities Can someone explain to me why the answer to the following question is not 1/2? (The given answer is 1/4, but, after trying various methods, including both argument by symmetry (each absorbing state must get 1/2, as the probability of being absorbed is 1, and the process is symmetric... | I, too, read the question as “What is the probability of reaching state $(\text{AA},\text{AA})$ given that the process starts in state $(\text{Aa},\text{Aa})$?” Your symmetry argument works and is borne out by an explicit calculation. There are some ways to get a probability of $\frac14$, by requiring that the process ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2628627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Measurable subset of Vitaly set has measure zero. Proof. $E_x = \{y \in [0,1]: x-y \in \Bbb{Q}\}$,
$ \varepsilon=\{A \subset [0,1]: \exists x \quad A=E_x\} $ .We chose one element from each set of family $\varepsilon$. This is a Vitaly set $V$.
Prove that if $E$ is measurable and $E \subset V$ then $E$ has measure $0... | Consider
$$E_{\mathbb Q} = \bigcup_{\substack{r \in \mathbb Q \\ -1 \le r \le 1}} (E+r) \subseteq [-1,2]$$
This is a countable infinite union of disjoints subsets, each of those having the measure of $E$. If the measure of $E$ would be strictly positive, $E_{\mathbb Q}$ would have an infinite measure, in contradiction ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2628721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
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