Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
One lily pad, doubling in size every day, covers a pond in 30 days. How long would it take eight lily pads to cover the pond?
A lily pad sits on a pond. It doubles in size every day. It takes 30
days for it to cover the pond. If you start with 8 lily pads instead,
how many days does it take to cover the pond?
I t... | Hint $\#1$:
At the end of the $30$ days with one lilypad, the doubling means that the lilypad now encompasses the area of $2^{30}$ of the original lilypads.
In that light, starting with $2^3 = 8$ lilypads and each one doubling in size per day, how many doublings will it take for you to get to $2^{30}$?
(I know you've a... | {
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"url": "https://math.stackexchange.com/questions/3067148",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Proof of Bertrand's postulate The following proof is from the 19th page of
Everest, Graham; Ward, Thomas, An introduction to number theory, Graduate Texts in Mathematics 232. London: Springer (ISBN 1-85233-917-9/hbk). x, 294 p. (2005). ZBL1089.11001.
In fact, I think this proof is not finished. For the red line, only $... | The idea is to notice that
$$\log(N) \leq \sum_{p|N}{\log(p)} + \sum_{k(p) \geq 2}{k(p)\log(p)}.$$
The first term is dealt with by $(1.16)$, the second one by the last estimate of the image before last.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Nature of the series $\sum 1+(-1)^{n+1} (2n+1)$ . The series $\sum 1+(-1)^{n+1} (2n+1)$ .is
1. Convergent
2. Oscillates finitely
3. Divergent
4. Oscillates infinitely
I found first few terms of this series, which are 4-4+8-8+...
So it seems like I will get such pairs if I expand the series more. But what can we concl... | Suppose for contradiction that the series $\sum 1+(-1)^{n+1} (2n+1)$ converges, then the sequence $$1+(-1)^{n+1} (2n+1)\to 0,\;\;\text{ as }n\to\infty.$$
However, \begin{align} c_n:= 1+(-1)^{n+1} (2n+1)=\begin{cases}2n+2,&\text{if}\;n\;\text{is even,}\\-2n,&\text{if}\;n\;\text{is odd.}\end{cases} \end{align}
\begin{a... | {
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Order of integration in triple integral Is there any hard and fast rule for what order you integrate for triple integrals. I know of Fubini's theorem but surely this doesn't cover all cases of triple integrals.
Say for example I have,
$$\int_{0}^{1} \int_{0}^{1-r^{2}} \int_{0}^{2 \pi} r^{3} d\theta dz dr $$
Why is it t... | For the integral $\int_0^{2\pi}d\theta$, it is completely independent, as you said, from the other variables, so you can evaluate it at any time and multiply the resulting double integral by its results.
For the integral $\int_0^{1-r^2}dz$, although the integrand is just $1$, the limits on the integral depend on the ot... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Get the work required to lift a chain A 3-m chain with linear mass density p(x)=?kg/m lies on the ground. Calculate the work required to lift the chain until it's fully extended.
My question is that, is the work that lift the chain from bottom equal to the work that lift the chain from top?
My understanding is that if ... | One way to calculate the work is to look at all the small bits of chain $dm$. Each bit is raised to a certain height $h$, so the bit of work is $gh\ dm$. Now integrate along the chain using the known density per unit length, so $dm=\rho(x)dx$ and you get the total work to be $\int_0^3\rho(x)gx\ dx$. This applies whe... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that the expected total present value of the bonds > purchased by time $t$ is $1000\lambda(1-e^{-rt})/r.$
Investors purchase $1000$ dollar bonds at the random times of a
Poisson process with parameter $\lambda$. If the interest rate is
$r$, then the present value of an investment purchased at time $t$ is
$1... | Let $T_i$ be the purchasing time of the $i$-th bond with value $P$. The total present value of the bond purchased up to time $t$ is
$$ V_t = \sum_{i=1}^{I_t} Pe^{-rT_i}$$
with the convention that $V_t = 0$ when $I_t = 0$. Then the expected value is
$$\begin{align}
E[V_t]
&= PE\left[\sum_{i=1}^{I_t} e^{-rT_i}\right] \\... | {
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"url": "https://math.stackexchange.com/questions/3068077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many sequence of length twelve are there consisting of eight ones and four zeros, such that there are no two consecutive zeros. I'm working through this problem and I haven't been able to make any progress. The textbook provides the answer of $ {9 \choose 4}$ but I'm not sure as to how they got this result.
| Here is a more sophisticated way to solve this, using DFAs. We can construct a state machine accepting the language of all strings without two consecutive zeroes as follows:
*
*There are two states, $q_0$ and $q_1$.
*The initial state is $q_1$.
*At state $q_0$, there is a $1$-transition leading to $q_1$.
*At stat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Operator norm of $T:l^{2}\rightarrow l^{1}$ where $Tx=(x_{1},x_{2}/2,x_{3}/3,x_{4}/4,...)$ As the title states, I need to compute the operator norm of a linear operator
$T:l^{2}\rightarrow l^{1}$, where
$$Tx=\left(x_{1},\frac{x_{2}}{2},\frac{x_{3}}{3},\frac{x_{4}}{4},... \right)$$
Using Holder's inequality for any se... | Let $x_i=\frac c i, i=1,2\cdots$ where $c$ is such that $c\sum\limits_{i=1}^{\infty} x_i^{2}=1$. In other words, $c=\frac {\sqrt 6} {\pi}$. Then $\|T(x_i)\|=\sum\limits_{i=1}^{\infty} \frac c {i^{2}}$ which is exactly $\frac {\pi} {\sqrt 6}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3068318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to rewrite matrix formula for Diagonalizable matrix $A=PDP^{-1}$ I am working on an old exam containing a question about Diagonalizable matrix, I am quite confident about the subject overall but there is one simple thing that bothers me, a lot!
We are given the formula $A=PDP^{-1}$ I know from my memory that this c... | You just have to take into account that matrix multiplication is not commutative.
So from $A=PDP^{-1}$, just multiply with $P^{-1}$ on the left, and with $P$ on the right. As matrix multiplication is associative, you obtain
$$P^{-1}AP=P^{-1}(PDP^{-1})P=(P^{-1}P)D(P^{-1}P)=D.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3068451",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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A question on binomial theorem If $C_0$, $C_1$, $C_2$,...$C_n$ are the coefficients in the expansion of $(1+x)^n$, where $n$ is a positive integer, show that $$C_1- {C_2\over 2}
+{C_3\over 3}-...+{(-1)^{n-1} C_n\over n}=1+ {1\over 2}+ {1\over 3}+...+{1\over n}$$
| In other words, you are saying that
\begin{equation}
\sum_{k=1}^n \dfrac{\left(-1\right)^{k-1}}{k} \dbinom{n}{k} = \dfrac{1}{1} + \dfrac{1}{2} + \cdots + \dfrac{1}{n}
\end{equation}
(because your $C_k$ are precisely the binomial coefficients $\dbinom{n}{k}$).
This is a fairly known identity. The one place I remember se... | {
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"url": "https://math.stackexchange.com/questions/3068565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A simple integral with one question
Question is: For $x$ equals $4$ and $9$, why is $t$ not $\pm2$ and $\pm3$ but just $2$ and $3$ ?
| By convention, $\sqrt{a}$ represents the non-negative square root, or the principal square root, of $x$. Hence, the only case in which there are two opposite solutions is when you have $\pm\sqrt{a}$. (Note the extra $\pm$ sign.) Hence is important to note that $x = \sqrt{a}$ (one non-negative solution). should not be c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are some applications of mathematics whose objectives are not computations? In mathematics education, sometimes a teacher may stress that mathematics is not all about computations (and this is probably the main reason why so many people think that plane geometry shall not be removed from high school syllabus), but... | Would you count these sculptures by Bathsheba Grossman as non-computational? Maths for the sake of beauty. (Also they include a Klein Bottle Opener!)
A nice but technical example I remember from electronics electronics at university was the proof that a filter which perfectly blocks a particular frequency range but let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3068748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Comparison of integrals by algebraic means $$ \begin{align}A&:=\int_0^1\frac1{\sqrt{x(1-x)}}\ \mathrm dx \\ B&:=\int_0^1\sqrt{x(1-x)}\ \mathrm dx \end{align} $$
My CAS tells me that $A = \pi$ and $B = \frac18\pi$.
How can one prove that $A=8B$ using just basic rules of integration such as the chain rule?
Trigonometric ... |
If integration by parts is an acceptable approach, then we can proceed as follows.
First, let $B$ be the integral defined as
$$B=\int_0^1 \sqrt{x(1-x)}\,dx\tag1$$
Integrating by parts with $u=\sqrt{x(1-x)}$ and $v=x$ in $(1)$, we obtain
$$B=\frac12 \int_0^1 x\left(\frac{\sqrt x}{\sqrt{1-x}}-\frac{\sqrt{1-x}}{\sqrt x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3068890",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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Positive Definite Matrices and eigenvalues Problem:
Let $A$ ∈ ${C}^{n×n}$ be, such that for every x ∈ $C^n$ <$A$x$, x$> ≥ 0
Show that all eigenvalues of $A$ are positive or zero
I suppose that from the standart inner product in the problem we can say that $A$ is a positive definite matrix and therefore follows that t... | It's a straightforward computation. If $\lambda$ is an eigenvalue of $A$, choose an eigenvector $v$ with $\langle v,v\rangle=1$. Then
$$
\lambda=\langle \lambda v,v\rangle=\langle Av,v\rangle\geq0.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3069034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Show that $MNPQ$ is a square Let $ ABCD $ a quadrilateral s.t. $AC=BD $ and $m (\angle AOD)=30°$ where $O=AC\cap BD $.
Let $\triangle ABM, \triangle DCN, \triangle ADN, \triangle CBQ $ equilateral triangles with $Int (\triangle ABM)\cap Int (ABCD)=\emptyset$, $Int (\triangle DCP)\cap Int (ABCD)=\emptyset$, $Int (\tri... | Let $R^{\alpha}_O$ be a rotation in the plain by an angle $\alpha$ around a point $O$.
Easy to see that to rotate a vector by an angle $\alpha$ it's the same to rotate this vector around his tail.
Now, by using the beautiful Daniel Mathias's picture we obtain:
$$R^{90^{\circ}}\left(\vec{NM}\right)=R^{30^{\circ}}\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3069153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Homeomorphism between region above parabola and $ \mathbb R ^2$
Define the set $ X = \{ (x,y) \in \mathbb R ^2 : x^2 < y \}. $
Construct a homeomorphism between $ X $ and $ \mathbb R ^2 $.
Graphically, $ X $ is the region above the parabola $y = x^2$, not including the boundary lines. Since it's contained entirely... | Hint: For the strategy $(x,y)\mapsto(x,\star)$ to work, you'll want a function of both $x$ and $y$ in the $\star$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3069277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to add vector with itself tranposed? So I'm solving basic linear algebra questions as part of review.
$$v=\begin{bmatrix} 1 & 2 & 3 \\ \end{bmatrix}$$
When I do the operation $v+v^T$ according to matlab, numpy and wolfram alpha
spits out
$$v+v^T=\begin{bmatrix} 2 & 3 & 4\\ 3 & 4 & 5\\ 4 & 5 & 6\\ \end{bmatrix}$$
O... |
Can someone explain to be how this makes any sense?
It doesn't make any sense.
If you're talking about tansposition, then you're implicitly viewing your vectors as matrices with one of the dimensions being $1$.
Then your $v$ is a $1\times 3$ matrix and $v^T$ is a $3\times 1$ matrix.
Addition of matrices that don't ha... | {
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Proof that $K_{3,3}$ is non planar using Euler's formula. I'm struggling to understand the proof that $K_{3,3}$ is nonplanar. Using Euler's formula we know that $3f \leq 2e$. The proof goes like this:
If we had drawn the graph in the plane, there would be no triangles: this is because in any triangle either two wells ... | As $K_{3,3}$ is a bipartite graph, each face is bounded by an even number of edges, so at least four. If there are $f$ faces, then the total number of edges
in their boundaries is $\ge 4f$, but that total number is $2e$ as each edge
is in two faces, so $2e\ge 4f$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3069511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $P[x]+\langle x\rangle $ is a prime ideal of $R[x]$.
Show that if $P$ is a prime ideal of a commutative ring $R$ with unity then $P[x]+\langle x\rangle $ is a prime ideal of $R[x]$.
Here $P[x]$ consists of all polynomials whose coefficients are in $P$.
I tried to show it using the fact that if $f(x)g(x)\in... | Consider the map
$$
\varphi\colon R[x]\to R/P, \qquad \varphi(f)=f(0)+P
$$
and prove it is a surjective ring homomorphism.
What's its kernel?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is the Lie bracket always invariant under coordinate transformations? This is my first question on StackExchange. I think it's probably quite easy, but it's been baffling me for a while.
I'm doing computations to determine invariant properties of the quantity $X\circ Y= \nabla_Y X$ where $X$ and $Y$ are vector fields,... | The mistake is in the change of coordinates. If $x=\phi(y)$, then
$$dx = d\phi = \frac{d\phi}{dy}(y)\,dy$$
is the change of coordinates for one-forms, not vectors. For vectors you have the dual formula
$$\frac{\partial}{\partial_x} = \left(\frac{d\phi}{dy}(y)\right)^{-1}\frac{\partial}{\partial_y}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3069963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Is $\{0,1\}$ a subgroup of $\mathbb{R}$ under multiplication? I am playing around with subgroups and I was wondering if the group $\{0,1\}$ is a subgroup of $\mathbb{R}$ with respect to multiplication. It seems to fit the criteria but the inverse property is making me worry because $0$ is its own inverse. I vaguely re... | Hint: The Cayley table of $\{0, 1\}$ under multiplication is
$$\begin{array}{c| c c}
\times & 0 & 1\\
\hline
0 & 0 & 0 \\
1 & 0 & 1.
\end{array}$$
What do you know about Latin squares?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $X : \Omega \to \Bbb R$ is a random variable if $X$ is constant. I have to prove this:
Let $(\Omega, \mathcal F)$ be a measurable space, $\mathcal F=\{\emptyset, \Omega\}$ prove that $X : \Omega \to \Bbb R$ is a random variable if and only if $X$ is constant.
I've tried using that $X$ is a random varia... | Use the second definition. Let $c=\sup \{x:\{X\leq x\} \neq \Omega\}$. Verify that $-\infty<c<\infty$. Note that $\{X\leq x\}=\Omega$ for $x>c$ and conclude that $X\leq c$. Next,note that $\{X\leq x\}=\emptyset$ for all $x<c$. Conclude that $X=c$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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let $f(x)$ be rational non constant polynomial and $f\circ f(x)=3f(x)^4-1$ then find the $f(x)$ let $f(x)$ be rational non constant polynomial and $f\circ f(x)=3f(x)^4-1$
then find the $f(x)$ .
My Try :
$$f(f(x))=3f(x)^4-1$$
Let $f(x)=ax^n+g(x)$ so :
$$a(ax^n+g(x))^n+g(ax^n+g(x))=3(ax^n+g(x))^4-1$$
$$a^2x^{n^2}+h(x)+k(... | Note that, because I think that you supposed deg $g(x)\leq n-1$, from
$$
a_n(a_n x^n+g(x))^n+g(a_n x^n+g(x))=
3(a_n x^n+g(x))^4-1$$
the degree of the LHS is $n^2$, while on the other hand the degree of the RHS is $4n$. This is an equality between two polynomials, hence their degree must be the same. Then $n=4$. Moreove... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Partial derivative of coordinates with respect to function Let $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$. Then
$$\frac{\partial f^i}{\partial x^j} = (\nabla f)^i_j$$
where $\nabla f$ is the Jacobian matrix of $f$. When reading this paper I came across the expression
$$\frac{\partial x^i}{\partial f^j}$$
Should I inte... | Imagine you can invert the problem $x^i = x^i(f)$. Clearly
$$
x^i = x^i(f^1(x),\cdots,f^n(x))
$$
Now apply the chain rule
$$
\frac{\partial x^i}{\partial x^j} = \frac{\partial x^i}{\partial f^k} \frac{\partial f^k}{\partial x^j} = (\nabla_f x)^{i}_{\;k}(\nabla_x f)^{k}_{\;j} = \delta^i_j
$$
That means that
$$
\mathbb{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3070450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Understanding kinematics formula in two dimensions Determine the angle of projection of a projectile if its speed at maximum height is $\sqrt{\frac{2}{5}}$ of its speed at half the maximum height.
My solution:
$$H_{max}=\frac{{v_0}^2\sin^2(\theta)}{2g}\implies \frac{1}{2}H_{max}=\frac{{v_0}^2\sin^2(\theta)}{4g}\\v_{x}=... | You know that
$$v_y^2 = v_{0y}^2 + 2a_y\Delta y$$ and that
$$v_x^2 = v_{0x}^2 + 2a_x\Delta x.$$
First,
$$v^2 = \textbf{v}\cdot\textbf{v} = \left(v_x \hat{\textbf{x}} + v_y \hat{\textbf{y}}\right)\cdot\left(v_x \hat{\textbf{x}} + v_y \hat{\textbf{y}}\right) = v_x^2 + v_y^2.$$
Second,
$$v_0^2 = \textbf{v}_0\cdot\textbf{v... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does the real projective plane / Boy surface look like this?
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901
My question is, you can see that the Boy surface is made up of three identical parts. But how does the number $3$ come up? I canno... | 3 occurs in the usual definition of $RP^2$ as the set of lines in $R^3$. That is, the quotient space of $R^3-0$ that identifies $x\sim cx$ for all nonzero $x\in R^3$ and nonzero real $c$. The homeomorphism $(x_1,x_2,x_3)\to(x_2,x_3,x_1)$
for example induces a threefold symmetry of $RP^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3070745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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algebraic topology - hatcher 3.3 exercise 17 The following is a question from Hatcher's "Algebraic Topology"
“show that homology commutes with direct limits. “
I have tried to solve this problem but I can’t .
| Here is another easy counterexample: take $S^1$ and consider $\{S_i\subset S^1:S_i\ \text {is countable}\}.$ Then, $S_i$ is totally disconnected, so $H_1(S_i)=0$. And $\varinjlim S_i=S$ but $H_1(S)=\mathbb Z$.
| {
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Showing that $Y\cong W$ but $X/Y\not \cong X/W$. I was trying to solve the following question:
Let $X=\mathbb{Z}_4\times\mathbb{Z}_2$, $Y=\{0,2\}\times\{0\}$ and $W=\{0\}\times \mathbb{Z}_2$. Show that $Y\cong W$ but $X/Y\not \cong X/W$.
I guess I must verify manually the Isomorphism theorems (link), but how should I... | Hint: Show that $Y\cong W\cong\Bbb Z_2$.
Secondly, $X/Y\cong V_4=\Bbb Z_2×\Bbb Z_2$, but $X/W\cong\Bbb Z_4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3070973",
"timestamp": "2023-03-29T00:00:00",
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How do I evaluate $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} e^{-(3x^2+2 \sqrt 2 xy+3y^2)} \mathrm dx\,\mathrm dy$?
Evaluate $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \exp\left(-3x^2-2 \sqrt 2 xy - 3y^2\right) \, \mathrm dx\,\mathrm dy$$
I first evaluate
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} ... | $$3x^2+2\sqrt{2} xy + 3y^2
=\begin{bmatrix}x & y \end{bmatrix}
\begin{bmatrix} 3 & \sqrt{2} \\ \sqrt{2} & 3 \end{bmatrix}
\begin{bmatrix}x \\ y \end{bmatrix}$$
so the integrand is
$$\exp(- v^\top \Omega v/2)$$
where $v = \begin{bmatrix}x \\ y \end{bmatrix}$ and $\Omega = 2\begin{bmatrix} 3 & \sqrt{2} \\ \sqrt{2} & 3 \e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3071214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Show that $(Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt$ is Compact linear operator on $C([0,1])$ Show that
\begin{equation}
(Tu)(x)=\int_{\alpha(x)}^{\beta(x)} u(t)dt
\end{equation}
is Compact linear operator on $C([0,1],R)$ where $\alpha, \beta:[0,1]\rightarrow [0,1]$ are continuous.
My Attempt
$T$ is obviously linear... | The integral from $a$ to $b$ can be viewed as $\int_a^bf=\int_{0}^1\chi_{[a,b]}f$. Recall the relation
$$
|\chi_A-\chi_B| = \chi_{A \Delta B}
$$
where $A\Delta B$ is the symmetric difference of $A$ and $B$. It is also easy to verify that
$$
[a,b] \Delta [c,d] \subset [a,c] \cup[c,a]\cup[b,d]\cup[d,b]
$$
where $[x,y]=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3071336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Showing existence of irreducible polynomial of degree 3 in $\mathbb{F}_p$ I'am trying to show that for every p$ \in \mathbb{N}$ where p is prime,
there is an irreducible polynomial of degree 3 in $\mathbb{F}_p$.
I've found too general answers for that question, but I want to show it in the most simple way.
I know to do... | If a degree $3$ polynomial is reducible over a field, then it has a root. So you (just) need a degree $3$ polynomial without a root.
There are $\frac{p^3-p}3$ monic irreducible polynomials of degree $3$, according to this argument.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3071406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Seeking an efficient way to calculate $\sum_{k=2}^n\frac{1}{a_k}$ in a computer program, where the $a_k$ are integers stored in a vector I need an efficient way to compute sums of reciprocal of numbers using a computer program. Currently, I have a set of integers $\{a_{0}, a_{1}, \ldots a_{n}\}$, and I want to compute ... | To repeat myself, from another site, on a different problem:
The way I'd do it? Build a class representing fractions. Obviously, this class would have two integer fields for the numerator and denominator.
Methods to implement:
*
*Reduction to lowest terms. Find the gcd of the numerator and denominator, and divide bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3071570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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How do you simplify $3^{\frac{(-1)^n + 1}{2}}$ I am solving non linear recursive relations and I stumbled upon this: $$x(n)*x(n+1)=3,\ x(0)=3$$
If you start calculating the values after $0$ you will notice that a pattern emerges: $$x(0) = 3$$ $$x(1) = 1$$ $$x(2) = 3$$ $$x(3) = 1$$ $$...$$
From this I concluded that $x(... | This is of the same type of situation as saying that $2$ and $1 + 1$ are the same. They are different expressions for the same value.
In your case, let $f\left(n\right) = \left(-1\right)^n + 2$, for $n \ge 0 \text{ and } n \in N$. For even values of $n$, the value is $1 + 2 = 3$ and for odd values of $n$, the value is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3071679",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Can we simplify this logarithm? if so, please provides some tips ${|x|^{11/10}} \log_{|x|^{{1/10}}}|x|$.
I only know doing the first step, not sure if it is correct
$\log_{|x|^{{1/10}}}(|x|^{|x|^{11/10}})$
as got stuck following this proof. Please help understand how we can get step two from step one.
| Consider
$$y=x^a\log_{x^b} (x)$$ Using the laws of logarithms
$$y=\frac{x^a \log (x)}{\log \left(x^b\right)}=\frac{x^a \log (x)}{b\log \left(x\right)}=\frac 1b x^a$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3071779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Prove that a function is smooth if it is smooth in almost all directions Question
So suppose we have a function $f:\mathbb R^2\to \mathbb R$ for which it is given that $x\mapsto f(x,g(x))$ is smooth (i.e., $C^\infty$) for all smooth functions $g:\mathbb R\to\mathbb R$. Can we prove that $f$ is smooth as well?
I don't ... | It need not even be continuous. Let $f(x,y) = \frac{xy^2}{x^2+y^4}$ for $(x,y)\neq (0,0)$ and $f(0,0) = 0$. This is discontinuous, since $\lim_{t\rightarrow 0} f(t^2,t) = \frac{t^4}{t^4+t^4} = \frac{1}{2} \neq 0$.
Now, $f(x,g(x))$ is clearly smooth whenever $g(0) \neq 0$, so it remains to check the case $g(0)=0$. Then,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3071867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Convergence/Divergence of $\int_0^{+\infty} {\frac {\sin x}{x\ln^2 (x^2+2)}} \mathrm{d}x$ Let's call $f(x)= \frac {\sin x}{x\ln^2(x^2+2)}$ and split our integral: $$\int_0^{+\infty} {\frac {\sin x}{x\ln^2 (x^2+2)}} \mathrm{d}x={\int_0^2 {\frac {\sin x}{x\ln^2 (x^2+2)}} \mathrm{d}x}+\int_2^{+\infty} {\frac {\sin x}{x\ln... | Hint. Note that $f(x)=\frac{\sin(x)}{x\ln^2(x^2+2)}$ is continuous in $(0,2]$ and its limit at $0^+$ is $1/\ln^2(2)$. So it can be extended to a continuous function in $[0,2]$. Moreover, as you already remarked, for $x\in [2,+\infty)$,
$$|f(x)|=\frac{|\sin(x)|}{x\ln^2(x^2+2)}\leq \frac{1}{4x\ln^2(x)}$$
and the integra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3072046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to properly represent a matrix function. Given the function $f_{h}(x,y,z)=(x-z,y+hz,x+y+3z)$, what is the correct way to represent the matrix function in respect to the standard basis?
With the representation theorem, I would write the matrix in columns as:
$$F_{h|S_3}=(f_h(e_1)|{S_3} \quad f_h(e_2)|{S_3} \quad f_... | The correct way to represent the function $f_h$ in matrix form depends on the convention that you want to use to represent it. Let $(x,y,z) \in \mathbb{R}^3$. The matrix which you computed is useful for expressing $f_h$ as $$f_h(x,y,z) = \begin{bmatrix} x& y & z \end{bmatrix} \begin{bmatrix}
1& 0 & 1 \\
0 & 1& 1\\
-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3072171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Minimum variance unbiased estimator of exponential distribution The given model is $\text{Exp}(\mu,\sigma),\;\mu\in\Bbb{R},\sigma\gt0$ whose pdf is
$f(x\text{;}\theta)={1\over \sigma}e^{-{{(x-\mu)}\over \sigma}}I_{(\mu,\infty)}(x)$
I easily found $(X_{(1)},\bar{X}-X_{(1)})'$ is CSS for $\theta=(\mu,\sigma)'$ with the s... | I will use the more common notations, i.e., $1/\sigma = \lambda$ and $\mu = \gamma$, hence
$$
\mathbb{P}(X>a)= \exp\{-\lambda(a-\gamma)\},
$$
hence the MLE is
$$
\hat{P}=\exp\{-\frac{1}{\bar{X}_n}(a-X_{(1)})\}.
$$
This is a biased estimator, so you can find its expectation using the joint probability function of $\bar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3072303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Dick throws a die once. If the upper face shows $j$, he then throws it a further $j−1$ times and adds all $j$ scores shown. If this sum is $3$ . . . Dick throws a die once. If the upper face shows $j$, he then throws it a further $j − 1$ times and adds all $j$ scores shown. If this sum is $3$, what is the probability t... | In the first case, it's not possible as for the die to be only thrown altogether once, $j$ would necessarily have to be $1$. So the sum $3$ with one throw is not possible.
In the second case, we start with $j=2$. This means that we throw the dice once more $(j-1=2-1=1)$. Now for the sum to be $3$, this number has to be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3072457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How can I calculate $\int\frac{x-2}{-x^2+2x-5}dx$? I'm completely stuck on solving this indefinite integral:
$$\int\frac{x-2}{-x^2+2x-5}dx$$
By completing the square in the denominator and separating the original into two integrals, I get:
$$-\int\frac{x}{x^2-2x+5}dx -\int\frac{2}{(x-1)^2 + 4}dx$$
The second one is tri... | Another plan that may be useful: once we see that form with the completed square, we make a simple substitution - not the whole thing, but just the part inside the square.
\begin{align*}I &= -\int \frac{x-2}{(x-1)^2+4}\,dx\\
&\phantom{|}^{u=x-1}_{du=dx}\\
&= \int -\frac{u-1}{u^2+4}\,du = \int\frac{-u}{u^2+4}\,du+\int\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3072589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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What is the distribution of $X|W=w$? Let $X$ and $Y$ be independent random variables with uniform distribution between $0$ and $1$, that is, have joint density $f_{xy}(x, y) = 1$, if x $\in$ $[0,1]$ and y $\in [0,1]$ and $f_{xy} (x, y) = 0$, cc. Let $W = (X + Y) / 2$:
What is the distribution of $X|W=w$?
I started cons... | The conditional pdf is given by
$$f_{X | W = w}(x) =
\frac {f_{X, W}(x, w)} {f_W(w)}.$$
$f_W$ is the pdf of a sum of two independent uniformly distributed r.v.:
$$f_W(w) = 4 w \left[0 < w \leq \frac 1 2 \right] +
4 (1 - w) \left[\frac 1 2 < w < 1 \right].$$
The transformation $(x,w) = (x, (x + y)/2)$ maps the square $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3072720",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Solving $8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$
Solve the equation
$$8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$$
I have this idea: set $$\sqrt{x+2}=a , x+2=a^2 , \sqrt{x-1}=b.$$
So $$x-1=b^2 , 2a^2+6b^2 =8b-4$$ and $$x^2+x-2 =a^2b^2$$ and then I'd simplify, but it's still very hard to solve.
Any hint is a... | The domain gives $x\geq1$ and we need to solve that
$$\sqrt{x+2}-\sqrt{x-1}=7\sqrt{x^2+x-2}-8x+3.$$
Now, since $\sqrt{x+2}-\sqrt{x-1}\geq0,$ we obtain
$$7\sqrt{x^2+x-2}-8x+3\geq0$$ or
$$\frac{97-\sqrt{2989}}{30}\leq x\leq\frac{97+\sqrt{2989}}{30}.$$
Thus, we need to solve
$$\left(\sqrt{x+2}-\sqrt{x-1}\right)^2=\left(7\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3072817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Count conditional probability of winning a game
In a certain game of tennis, Alice has a 60% probability to win any
given point against Bob. The player who gets to 4 points first wins
the game, and points cannot end in a tie. What is Alice's probability
to win the game?
When solving in terms of a random walk it... | Just as a generalization for the answer above, formula for getting N points in a game first
$\sum_{i=0}^{N-1}{N-1+i\choose i}\cdot p^N\cdot (1-p)^{i}$
So in our case we have N=4, p=0.6, q=1-p=0.4
$\sum_{i=0}^{3}{2+i\choose i}\cdot 0.6^4\cdot 0.4^{i}$
which gives us 0.71 as a result.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What's the difference between "relation", "mapping", and "function"? I think that a mapping and function are the same; there's only a difference between a mapping and relation. But I'm confused. What's the difference between a relation and a mapping and a function?
| Mathematically speaking, a mapping and a function are the same. We called the relation
$$
f=\{(x,y)\in X\times Y : \text{For all $x$ there exists a unique $y$ such that $(x,y)\in f$} \}
$$
a function from $X$ to $Y$, denoted by $f:X\to Y$. A mapping is just another word for a function, i.e. a relation that pairs exact... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3073075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 4,
"answer_id": 2
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Proof of first Fundamental theorem of calculus Can you please, check if my proof is correct?
Suppose that $f:[a,b]\to \Bbb{R}$ is continuous and $F(x)=\int^{x}_{a}f(t)dt$, then $F\in C^{1}[a,b]$ and
$$\dfrac{d}{dx}\int^{x}_{a}f(t)dt:=F'(x)=f(x)$$
MY PROOF: Credits to Aweygan for the correction
Let $x_0\in[a,b]$ a... | It's essentially correct, but you should either split up the last part of the proof into the cases where $x<x_0$ and $x_0<x$, or write $x_1=\min\{x,x_0\}$, $x_2=\max\{x,x_0\}$ and do the following:
\begin{align}
\left| \dfrac{F(x)-F(x_0)}{x-x_0}-f(x_0) \right|&= \left| \dfrac{1}{x-x_0}\int^{x}_{x_0}(f(t)-f(x_0))dt \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3073225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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$f$ differentiable $5$ times around $x=a,\ f'(a)=f''(a)=f'''(a)=0,\ f^{(4)}(x) <0 \Rightarrow x=a$ is either a local minimum or a local maximum point So I've been trying to prove the following statement:
Let $f$ be a function such that it is differentiable $5$ times around $x=a$. Prove or disprove, that if $f'(a)=f''(... | The following theorem was proved by Colin Maclaurin in 1742.
Let $f$ be a real-valued function defined on an open interval $J$ which is $(n-1)$-times continuously differentiable in a neighborhood of a point $a \in J $ and for which moreover $f^{(n)}(a)$ exists.
Assume $f'(a) = f''(a) = \dots f^{(n-1)}(a) = 0$ and $f^{(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3073390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are there any obvious reasons $g_2^3-27g_3^2$ a cusp form? Recall that $G_k(\tau)$ are Eisenstein series$(k\geq 2)$ defined over upper half plane $\mathcal{H}$. Now define $g_2=60G_4,g_3=140G_6$ and $\Delta=(g_2)^3-27(g_3)^2$. Note that $\Delta$ corresponds to elliptic curve's discriminant defined by Wierstrass form $\... | There is a complex torus/elliptic curve reason.
*
*$z \mapsto (\wp_\tau(z),\wp_\tau'(z))$ is an isomorphism $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z}) \to E_\tau/\mathbb{C} : y^2 = 4x^3-g_2(\tau) x-g_3(\tau)$ (with $20 g_2(\tau),28 g_3(\tau)$ the coefficients of $z^2, z^4$ in the Laurent expansion of $\wp_\tau(z)$ at ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3073576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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About pseudo-differential operators Let $\Omega$ be an open and connect subset of $\mathbb{R}^2$,we denote by $\partial \Omega$ its boundary the latter is supposed to be smooth ($\mathcal{C}^\infty)$, its outword normal vector is denoted by $n$. Let $f: \Omega \mapsto \mathbb{R}$ such that $f(x)\geq \alpha > 0$. Now, ... | Finally, The answer is yes. In deed the result remains true if we remplace $div(f \nabla .)$by any other second order elliptic opertor. Morover, $A$ is of order one. The proof can be found here : https://arxiv.org/pdf/1212.6785.pdf
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Support of Variance of Random IID Sample (Bounded)
Let $X_i$, $i\in\{1, 2, ..., n\}$ be independent and identically distributed random variables with bounded support $[\alpha, \beta]$, with $\alpha,\beta \in \mathbb{R}$.
What is the support of the random variable that corresponds to the variance of this sample? That ... | Largest value is obtained when all $X_i$ have the distribution $P(X_i=\alpha)=P(X_i=\beta)=\frac{1}{2}$. The mean for one varialbe is $\frac{\alpha+\beta}{2}$. The variance for one variable is $(\frac{\beta-\alpha}{2})^2$ so the variance for $n$ independent variables is $\frac{(\beta-\alpha)^2}{4n}$. The proof may ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3073814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Do I have something wrong when solving $y'+2y=6$? Solve $$y'+2y=6.$$
When I do $$y'=2(3-y)\implies\int\frac{\mathrm dy}{3-y}=2\int\mathrm dx\implies-\ln{|3-y|}=2x+c\implies3-y=ke^{-2x}\therefore y=\boxed{3-ke^{-2x}},\quad c,k\in\Bbb R.$$ It satisfies the ODE because $$2ke^{-2x}+6-2ke^{-2x}=6=6.$$ However, when I try a... | Both solutions are the same, $k$ is a real number, you can write $3+(-k)e^{-2x}$.
| {
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"url": "https://math.stackexchange.com/questions/3073929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Envelope Question: Five letters addressed to individuals 1-5 are randomly placed in five addressed envelopes, one letter in each envelope. I'm trying to find the probability of:
*
*Exactly three letters go in the correct envelopes.
*Exactly two letters go in the correct envelopes
*No letters go in the correct enve... | There are not $2!$ ways to organize the lat two letters. There is only $1$ way. Because the second way of organizing them would be to put them in their correct envelopes, which wouldn't match up with the constraint of having exactly $3$ letters getting sent correctly.
A similar mistake was made in the second problem.
T... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there a right triangle with angles $A$, $B$, $C$ such that $A^2+B^2=C^2$? A right angle triangle with vertices $A,B,C$ ($C$ is the right angle), and the sides opposite to the vertices are $a,b,c$, respectively.
We know that this triangle (and any right angle triangle) has the following properties:
*
*$a^2+b^2=c^2... | We have $C=\pi/2,B=\pi/2-A$, so we need to solve the quadratic equation$$A^2+(\pi/2-A)^2=\pi^2/4\\\implies2A^2=\pi A$$giving $A=0,\pi/2$ which is not possible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3074189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Can we simplify $ A^{-1}Bx = x$ where $A$ is a block matrix with each block being diagonal and half the blocks of $B$ are zero? I have the following eigenvalue problem involving block matrices $A$ and $B$:
$$
A^{-1}Bx = x. \quad \quad \quad \quad (*)
$$
$A$ and $B$ have special structures. I would like to reduce/simpli... | $\bigg(I - A(\omega)^{-1}B(\omega)\bigg)x = 0 \implies (A(\omega)-B(\omega) )x=0$
so $A(\omega)-B(\omega)$ has 0 as an eigenvalue; so find an $\omega$ such that $det(A(\omega)-B(\omega))=0$
Now you can use $det \begin{pmatrix}
A & B \\
C & D
\end{pmatrix} = det\left ( A -BD^{-1}C \right )det(D)$
| {
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What function $f(n)$ is defined by $f(1)=2$ and $f(n+1)=2f(n)$ for $n\geq 1$ I have to 2 qusetions in a mathematical induction homework:
1-What function $f(n)$ is defined by $f(1)=2$ and $f(n+1)=2f(n)$ for $n\geq 1$
My attempt:
$f(1)=2$
$f(2)=2f(1)=2(2)=2^2$
$f(3)=2f(2)=2(2^2)=2^3$
$f(4)=2f(3)=2(2^3)=2^4$
.
.
.
Thus,... | For the first one, you are correct. $f(n)=2^n$. You still have to prove that $f(n)=2^n$, but you are on the right track.
For the second one, if you only need to calculate $g(4)$, you are done. If you need a more general expression of $g(n)$ think about it this way:
$$g(4)=2^{g(3)} = 2^{2^{g(2)}} = 2^{2^{2^{2}}}$$ so $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3074413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I use the sum of squares formula to create blocks of a certain dimension?
The previous expression was just the sum of consecutive squares so $1^2+2^2+...+n^2 = \frac{n(n+1)(2n+1)}{6}$
I know how to derive this formula but can someone please explain the claim that
"This expression says that a box with dimensio... | I can make 6 pyramid like structures starting with a base of 4 x 4, then on top 3 x 3, then 2 x 2 then 1 x 1. then I can combine them all. I found this from
https://ckrao.wordpress.com/2012/03/14/the-sum-of-consecutive-squares-formula/
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3074562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Characterizing points by their distance to the unit ball Let $x,y\in\mathbb{R}^n$. Assume that, for all $z$ in the unit ball, $|x-z|=|y-z|=d_z$. From this we can deduce that $|x| = |y|$ since $0$ is in the unit ball. How can we show that $x=y$? I think it must be true but I cannot show it easily.
| You can use the fact that $\bar x=x/|x|$ is the unique best approximation of $x\in \mathbb{R}^n\setminus B$ in $B$.
Since
$$
|y-\bar x|=|x-\bar x|<|x-z|=|y-z|
$$
for $z\in B\setminus\{\bar x\}$ by assumption, $\bar x$ is also a best approximation of $y$ in $B$. By uniqueness it follows that $\bar x=\bar y$. As you alre... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3074680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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quadratic programming /symmetric matrix I have a quadratic program with $ F: \mathbb{R^n} \rightarrow \mathbb{R}, F(x)=x^TQx$ I want to find a symmetric matrix M for Q, such that $F(x)=x^TMx$ holds for all x.
I can write Q as sum of symmetric matrices and antisymmetric matrices:
$ Q = \frac{1}{2}(Q + Q^T) +\frac{1}{2} ... | Yes.
Moreover, the antisymmetric matrix does not contribute. After all, $x^T Q x$ is a scalar and therefore $x^T Q x = (x^T Q x)^T = x^T Q^T x$ for any $x$. Consequently $x^T(Q-Q^T)x=0$.
So we can write:
$$M=\frac 12(Q+Q^T)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the Laurent Series around a given point While studying I got this exercise:
Find the Laurent Series expansion valid for $0 < |z - i| < \sqrt2$ for the following function:$$f(z) = \frac{1}{(z-i)^8(z+1)}$$
So I have to get a series expansion around the point $i$.
I tried using $w = z - i$ to see if I could get so... | As I've updated the main post I just want to explain my confusion, which came from mostly seeing problems where I had fractions that looked like $$\frac{1}{a - r}$$ where a was simply a real number.
The solution came from, as AndreasBlass pointed, using a complex number instead of the real number, which putting it in e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Irrationality of $(a_1+\sqrt{b_1})(a_2+\sqrt{b_2})$ Sorry, for a rather silly question.
Suppose $a_1$, $b_1$, $a_2$, $b_2$ are integers, all different from zero, while $b_1$ and $b_2$ are co-prime positive integers, neither being a complete square.
Is there an elementary proof that $(a_1+\sqrt{b_1})(a_2+\sqrt{b_2})$ is... | $(a + \sqrt b)(c + \sqrt d) = k \in \mathbb Q$ would mean
$\sqrt{b} = \frac k{c+\sqrt d} - a$
$b = ( \frac k{c+\sqrt d} - a)^2 \in \mathbb Q$ which can probably be proven false.
Indeed $( \frac k{c+\sqrt d} - a)^2 = $
$\frac {k(c - \sqrt d)}{c^2 - d} -a)^2 =$
$(m\sqrt d - n)^2$ where $m = \frac k{c^2-d}\in \mathbb Q$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is it possible for the sum of two independent random variables (where at least one of them is not normal) to sum to a normal random variable? Let $X$ be a normal random variable. Suppose we have the following decomposition:
$$
X = Y + Z
$$
where $Y$ ad $Z$ are independent. Is it possible for either $Y$ or $Z$ to be not... | No, it is not possible.
It is a famous result of Cramér that if the sum of two independent random variables $X + Y$ is a normal random variable, then $X$ and $Y$ are normally distributed as well.
This is a difficult result whose proof uses the machinery of complex analysis. The original paper can be found here and the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can every monoid be turned into a ring? It’s well-known that $\mathbb{Q} / \mathbb{Z}$ is an example of an abelian group which is not isomorphic to the additive group of any ring. But my question is, does there exist a monoid which is not isomorphic to the multiplicative monoid of any ring?
Or can you always find a bi... | Well, an obvious necessary condition for a monoid $M$ to admit a ring structure is the existence of an element $0\in M$ such that $0\cdot x=x\cdot 0=0$ for all $x\in M$ (an absorbing element). This is not true of most monoids (for instance, it is not true of any nontrivial group).
Even this condition is not sufficient... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How would you calculate this limit? $\lim\limits_{n \rightarrow\infty}\frac{\pi}{2n}\sum\limits_{k=1}^{n}\cos\left(\frac{\pi}{2n}k\right)$ I decided to calculate $\int_{0}^{\pi/2}cos(x)dx$ using the sum definition of the integral. Obviously the answer is $1$ . I managed to calculate the resulting limit using the geomet... | HINTS:
(1)
\begin{equation}
\cos\left(\frac{\pi}{2n}\cdot k\right) = \Re\left[\exp\left(\frac{\pi}{2n}\cdot k i \right) \right]
\end{equation}
(2)
\begin{equation}
\exp\left(\frac{\pi}{2n}\cdot k i \right) = a^k, \quad a = \exp\left(\frac{\pi}{2n}i \right)
\end{equation}
(3)
\begin{equation}
\sum_{k = 1}^{n} a^k = \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Solving an ODE by deriving it to get a nice form I came across this ODE:
$$2xy'(x)-y(x)=\log(x),\quad x>0 \tag{1}$$
Now one sees that the inhomogeneous part is an Euler-Differential-Equation. In fact, the nice way to solve this would probably be to use the substitution $x=e^t$ and $h(t)=y(e^t)$ since the problem here s... | After substitution $x=e^t,\;t=\log{x}$ we have ODE with constant coefficients:
$$2y'-y=t$$
Solution is
$$y=Ce^{t/2}-t-2=C\sqrt{x}-\log{x}-2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3075576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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differential forms- $\omega $ closed but not exact let be
$$ \omega= |x|^{-3} \left(x_1 dx_2 \wedge dx_3+x_2dx_3 \wedge dx_1 + x_3dx_1 \wedge dx_2\right) $$
and $G:= \mathbb{R}^3 \backslash \{ 0 \} $
I want to prove, that $ \omega$ is closed, but not exact
That $ \omega $ is closed, I can prove it by looking if $ d\o... | Switch to spherical coordinates and integrate $\omega$ on the unit sphere $r = 1$. Let $\theta,\phi$ denote the azimuth and polar angles, respectively; then $$dx = \cos \theta \sin \phi \, dr - \sin \theta \sin \phi \, d\theta + \cos\theta \cos \phi \, d\phi$$
$$ dy = \sin \theta \sin \phi \, dr + \cos \theta \sin\phi ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $\varphi \in E'$ and if $E$ is a Banach space then $\varphi \in E$
Problem: Let $E$ be a normed space over field $\mathbb{C}$. Fix a continuous function $f: \left[ a,b \right] \rightarrow E$ with $\left[ a,b \right] \subset \mathbb{R}$. Consider $\varphi: E' \rightarrow \mathbb{C}$ given by $\varphi(y) := \d... | $y\mapsto \varphi(y)=\int_a^b(y\circ f)(t)dt$ is linear by the linearity of the integral. It is continuous because $y_n\to y$ in $E'$ means uniform convergence on the unit ball of $E$ and hence on every multiple on the unit ball, and continuity of $f:[a,b]\to E$ implies that the range is compact and hence bounded in $E... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$n\in \mathbf{N}$ such that a solution of $X^4+nX^2 +1$ is a root of unit Consider $f_n(X)=X^4+nX^2 +1$ in $\mathbf{Q}[X]$. I found that for all natural $n$ such that $n\neq 2-m^2$ for a natural $m$, $f_n(X)$ is irreducible in $\mathbf{Q}$.
Consider $K_n=\mathbf{Q}(x)= \mathbf{Q}[X]/(f_n(X))$.
Using Dirichlet Unit t... | If some root $r$ of $X^4+nX^2+1$ is a root of unity, then $r^2$ is a root of unity and vanishes $X^2+nX+1$. Thus $-n$ is double the real part of $r^2$ (because $r^2$ and its complex conjugate are the roots of $X^2+nX+1$, since the product of roots is $1$), thus $|n| \leq 2$.
If $|n|=2$, your polynomial is $(X^2 \pm 1)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the sufficient statistic for a beta distribution? Let {$X_1,\ldots,X_n$} be a random sample from the $beta(\alpha,\beta)$ distribution.
Below is the beta distribution with the parameters referred to:
$$f_X(x;\alpha,\beta)=\frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)(\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}$$
H... | When $\beta$ is known, you can take $T(x) = \prod _i x_i$ and $g_\alpha(T(x)) = \frac{1}{B^n(\alpha,\beta)} \left(\prod_i x_i \right)^{\alpha - 1}$ in the factorization theorem for $\alpha$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Primitive of a function with $\sin \frac{1}{x}$ I have the next integral:
$$\int\biggl({\frac{\sin \frac{1}{x}}{x^2\sqrt[]{(4+3 \sin\frac{2}{x})}}}\biggr)\,dx ,\;x\in \Bigl(0,\infty\Bigr)$$
I used the substitution $u=\frac{1}{x}$ and I got $$-\int\biggl({\frac{\sin u}{\sqrt[]{(4+3 \sin2u)}}}\biggr)\,du$$
Can somebody ... | As $(\sin v\pm\cos v)^2=1\pm\sin2v$
$$\int\dfrac{2\sin v\ dv}{f(\sin2v)}=\int\dfrac{(\sin v-\cos v)\ dv}{f(\sin2v)}+\int\dfrac{(\sin v+\cos v)\ dv}{f(\sin2v)}=I+J$$ where $f(\sin2v)$ is a function of $\sin2v$
As $\displaystyle\int(\sin v-\cos v)=-(\sin v+\cos v)+C,$ set $\sin v+\cos v=y$ for
$$I=\int\dfrac{(\sin v-\cos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3076240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"Standard reference" for $C_c^\infty(\mathbb R)$ is dense in $C_c(\mathbb R)$ $C_c^\infty(\mathbb R)$ is dense in $C_c(\mathbb R)$. This can be shown by mollification. This is a well-known, widely used fact. However, I wasn't able to find any book which I could point in a reference to. Is there any kind of "standard re... | I don't have a reference at hand. But one can prove this without too much trouble using the Weierstrass approximation theorem.
Suppose $f\in C_c$ with support contained in $[a,b].$ Let $\epsilon>0.$ Choose $a'<a$ and $b'>b.$ Then there exists a polynomial $p$ such that $|p-f|<\epsilon$ on $[a',b'].$
Now there exists ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $A^2 + B^2 = O_2$ given conditions Show that, given $A,B$ second order matrices with real entries, such that $AB = BA$, $\det{(A + iB)} = 0$ and $4 \det{A} >( \text{tr}{A} )^2$, then $A^2 + B^2 = O_2$.
My progress:
Considering the polynomial $\det(A + xB)$, since $i$ is a root, $-i$ is also a root, and thus ... | The condition $4\det A>(\operatorname{tr} A)^2$ implies that $x^2-(\operatorname{tr} A)x+\det A$, the characteristic polynomial of $A$, has not any real root. Therefore $A$ is non-singular. Let $X=A^{-1}B$. Since $AB=BA$, the statements $A^2+B^2=0$ and $I+X^2=0$ are equivalent. Also, $\det(A+iB)=0$ implies that $\det(I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3076499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Which point of the graph of $y=\sqrt{x}$ is closest to the point $(1,0)$? This problem was assigned for an AP Calculus AB class and was not allowed a calculator:
Which point of the graph of $y=\sqrt{x}$ is closest to the point $(1,0)$?
We are not given answers and the teacher will be absent for $2$ weeks. I need to che... | An algebra-free approach:
In order to draw the tangent from a point $P$ on a parabola, it is sufficient to project $P$ on the axis, reflect this point with respect to the vertex and join the new point with $P$. Since the tangent drawn from $P=\left(\frac{1}{2},\frac{1}{\sqrt{2}}\right)$ is orthogonal to the line joini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3076633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
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Coordinate independence of connections So I am trying to prove the following:
Let $V \rightarrow M$ be a vector bundle $\nabla$ a connection on $V$. Then there is a unique sequence of linear maps
$$ \Omega^0(M;V) \xrightarrow{\nabla} \Omega^1(M;V) \xrightarrow{\nabla} \cdots $$
such that $\nabla$ coincdies... | Since $ A_{ij} dw_i + w_i dA_{ij} = d(w_i A_{ij})$ for all $i$ and $j$, the expression on the right-hand side reduces to
$$ \sum_{i}\sum_j \left( d(w_i A_{ij}) \otimes f_j + (w_i A_{ij}) \nabla f_j \right),$$
which is precisely the expression you want.
Edit: If the $w_i$'s are $k$-forms (rather than just smooth functi... | {
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Find any local max or min of $x^2+y^2+z^2$ s.t $x+y+z=1$ and $3x+y+z=5$
Find any local max or min of
\begin{align}
f(x,y,z)=x^2+y^2+z^2 && (1)
\end{align}
such that
\begin{align}
x+y+z=1 && (2)\\
3x+y+z=5 && (3)
\end{align}
My attempt. Let
$L(x,y,z,\lambda_1, \lambda_2)= f(x,y,z)+\lambda_2 (x+y+z-1) + \lambda_... | An option:
1) $x+y+z=1$; and
2) $3x+y+z=5$;
$2$ planes , their intersection is a straight line.
Subtract: 2)-1):
$2x=4$; $x=2$ ;and
$y+z=-1$;
$d^2=x^2+y^2+z^2$ .
Minimal distance of line from origin:
$d^2= 4 +y^2+z^2.$
2D problem:
Minimize $y^2+z^2$ with constraint $y+z=-1$.
$d_2^2= $
$[-(1+z)]^2+z^2=2z^2+2z+1=$
$2(z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3076938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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minimum value of $(8a^2+b^2+c^2)\cdot (a^{-1}+b^{-1}+c^{-1})^2$
If $a,b,c>0.$ Then minimum value of
$(8a^2+b^2+c^2)\cdot (a^{-1}+b^{-1}+c^{-1})^2$
Try: Arithmetic geometric inequality
$8a^2+b^2+c^2\geq 3\cdot 2\sqrt{2}(abc)^{1/3}$
and $(a^{-1}+b^{-1}+c^{-1})\geq 3(abc)^{-1/3}$
so $(8a^2+b^2+c^2)\cdot (a^{-1}+b^{-1}+c... | Hint: Apply $AM \ge GM$ not to $8a^2 + b^2 + c^2$, but to
$$
(2a)^2 + (2a)^2 + b^2 + c^2
$$
and $HM \le GM$ not to $a^{-1}+b^{-1}+c^{-1}$, but to
$$
\frac{1}{2a} + \frac{1}{2a} + \frac{1}{b} + \frac{1}{c}
$$
The “partitions” are chosen in such a way that equality can hold simultaneously in both estimates, in this case... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Invertible elements of $\mathbb{Z}_3[x] / (x^4+x^3-1)^3$ Let $F=\mathbb{Z}/3\mathbb{Z}$, $h(x)=x^4+x^3-1$, $R = F[x]/(h(x)^3)$.
I know $R$ has $4$ ideals and $1$ maximal ideal. Let $M$ be the maximal ideal $(h(x))/(h(x)^3)$
I need to find the number of invertible elements of $R$ and in order to do so I need the number ... | Your procedure is entirely valid; you can verify that $|R/M|=3^4$ by simply noting that $R/M$ is a vector space over $F$ with basis $\{1,x,x^2,x^3\}$. In fact, whatever argument you use to prove that $|R|=3^{12}$ undoubtedly also proves that $|R/M|=3^4$. Indeed it follows that $|M|=3^8$, and the solution given to you i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Sum of geometric series where $x=\sin^{-1}{\frac{7}{8}}$ This problem is from a local contest.
The series $\sin{x}$, $\sin{2x}$, $4\sin{x}-4\sin^3{x}$, $...$ is a geometric series if $x=\sin^{-1}{\frac{7}{8}}$. Compute the sum of the geometric series.
I did not compute the solution in time but this was my reasoning.
... | A sequence $(a_n)$ is a geometric series if $a_{n+1}/a_n=c$, for every $n$. Thus the first two terms suffice to determine it:
$$
a_0=\sin x,\qquad a_1=\sin2x=2\sin x\cos x
$$
Then
$$
c=\frac{a_1}{a_0}=2\cos x
$$
and indeed
$$
a_2=4\sin x-4\sin^3x=4\sin x\cos^2x=a_1\cdot 2\cos x
$$
Thus you have
$$
a_n=2^n\cos^nx\sin x
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3077517",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding the tenth derivative of $f(x) = e^x\sin x$ at $x=0$ I came across this Question where I have to find $$f^{(10)}$$ for the following function at $x = 0$
$$f(x) = e^x\sin x$$
I tried differentiating a few times to get a pattern but didn’t get one, can someone provide the solution.
| Hint:
$$f(x)=e^x\sin x$$
$$f'(x)=e^x(\sin x +\cos x)$$
$$f''(x)=e^x(\sin x+\cos x)+e^x(\cos x -\sin x)=2e^x(\cos x)$$
$$f'''(x)=e^x(2\cos x)-e^x(2\sin x)=2e^x(\cos x-\sin x)$$
$$f^{IV}(x)=2e^x(\cos x-\sin x)-2e^x(\cos x+\sin x)=-4e^x(\sin x)=-4f(x)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3077641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 10,
"answer_id": 1
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Simple series question. $\sum_{n=0}^\infty \frac{1}{((n^5)+1)^\frac{1}{3}}$ I think this converges due to direct comparison with $\frac{1}{n^{5/3}}$ but I can't double check this anywhere.
| Right.
$(n^5+1)^{1/3}
\gt (n^5)^{1/3}
=n^{5/3}
$
so the sum converges by the $p$-test:
$\sum \dfrac1{n^p}$
converges for
$p > 1$
(easily proved by the
integral test)
and diverges for
$p \le 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3077730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Counterexample for Converse of Surjective Homomorphisms In universal algebra, I am trying to find a counterexample using groups for the converse of the following:
If $\mathcal{A},\mathcal{B}$ are algebras, and $\phi:\mathcal{A}\to \mathcal{B}$ is a surjective homomorphism and the identity $\mathbf{s}=\mathbf{t}$ holds... | For any kind of algebra (regardless of the operations it has) the converse of that result is indeed false.
Just consider that if $\mathbf A$ is an algebra, then $\theta = A^2$ is a congruence on $\mathbf A$, and the quotient $\mathbf A/\theta$ is a one-element algebra.
Now, one-element algebras satisfy the equation $x=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3077816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Markov matrices with no ones on off-diagonals don't have -1 as an eigen value Every time I saw a Markov matrix with an Eigen value of -1, it had some 1's on its off-diagonals. The most obvious example is a simple permutation matrix:
$$M = \left(\begin{array}{ccc}0&1\\1&0\end{array}\right)$$
With eigen values 1 and -1. ... | Too long for a comment.
The only thing that can derail my argument in the blog is if the matrices have an eigen value, -1
Then eigenvalue $−1$ of the matrix $M$ does not cause big problems for the investigation, because to it correspond an eigenvalue $1$ of the matrix $M^2$, and we can consider the convergence of odd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3077955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show $S^2$ with 2 cells attached is equivalent to a wedge of spheres Show that a space obtained from $S^2$ by attaching n 2-cells along any collection of n circles in $S^2$ is homotopy equivalent to the wedge of n+1 spheres.
I'm a little confused here. I'm imagining a sphere, and putting 2 dimensional discs inside of i... | If you form $Z= S^2 \cup_f e^2$ then the attaching map- $f: S^1 \to S^2$ is null homotopic, as $S^2$ is simply connected. So $Z \simeq S^2 \vee S^2 $. The general result you need is that if $Z= B \cup_f X$ where $f: A \to B$, and the inclusion $i : A \to X$ is a closed cofibration, then the homotopy type of $Z$ depend... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3078377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integrate $\int\frac{\cos^2(x)-x^2\sin(x)}{(x+\cos(x))^2}dx$ I had to integrate the following integral:
\begin{equation}
\int\frac{\cos^2(x)-x^2\sin(x)}{(x+\cos(x))^2}dx
\end{equation}
but I can't find a suitable substitution to find a solution. Nothing I try works out and only seems to make it more complicated. Does a... | Note that the derivative of $x + \cos(x)$ in the denominator is $1-\sin(x)$ . We can try to make this term appear in the numerator and then integrate by parts. We have
$$ \frac{\cos^2(x)-x^2\sin(x)}{(x+\cos(x))^2} = \frac{\cos^2(x) - x^2 + x^2(1-\sin(x))}{(x+\cos(x))^2} = \frac{\cos(x) - x}{x+\cos(x)} + x^2 \frac{1-\si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3078486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Solving $(\ln(x)-1)y'' - \frac{1}{x}y' + \frac{1}{x^2}y = \frac{(\ln(x) - 1)^2}{x}$ On my exam I had to solve the following differential equation.
\begin{equation}
(\ln(x)-1)y'' - \frac{1}{x}y' + \frac{1}{x^2}y = \frac{(\ln(x) - 1)^2}{x^2}
\end{equation}
Which is a differential equation of the form:
\begin{equation}
y'... | $$(\ln(x)-1)\frac{d^2y}{dx^2} - \frac{1}{x}\frac{dy}{dx} + \frac{1}{x^2}y = \frac{(\ln(x) - 1)^2}{x^2}$$
Change of variable : $t=\ln(x)-1\quad;\quad x=e^{t+1}\quad;\quad dx=x\:dt$
$\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=\frac{1}{x}\frac{dy}{dt}$
$\frac{d^2y}{dx^2}=-\frac{1}{x^2}\frac{dy}{dt}+\frac{1}{x}\frac{d^2y}{dt... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3078569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How do complex number exponents actually work? I know Euler's formula and how to take complex exponents, but in it it's $e$ to an imaginary angle, not a number, it seems. From my understanding pi itself is not an angle, but $\pi$ radians is. And since cosine can only take in an angle, or at least a representation of on... | Just a little note that I hope can be of some use. In order to avoid confusion when dealing with angles and unit of measures of them, one can directly define the trigonometric functions starting with the arc lenght measurement.
Consider the semicircle of equation
$$f(x) = \sqrt{1-x^2}, \ \ x\in [-1,1].$$
It is relativ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3078651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
} |
Determine where peak or valley of polynomial graph without calculus I notice that if I graph $y=(x+2)^2(x-4)^2$ that the midpoint of the roots occurs at $x=1$. I note that this is also where the local maximum occurs.
If I graph $y=(x-5)^2(x^2)$ the midpoint of the roots is 2.5 where the local maximum occurs.
I also kn... | There is no precise relation because you can keep the extrema fixed while the roots are moving. Consider the cubic
$$x^3-3x+c,$$ that always has a maximum at $x=-1$ and a minimum at $x=1$.
We can place a root wherever we want, say at $x_0$, just by setting
$$c=-x_0^3+3x_0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3078771",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find projection-valued measure associated with parity operator Let's define parity operator as follows: $$\pi:L^2(\mathbb{R})\to L^2(\mathbb{R})$$ $$\psi(x)\mapsto \psi(-x)$$
It's easy to show that $\pi$ is a self-adjoint operator and its spectrum is just $\sigma(\pi)=\{-1,+1\}$. According to spectral theorem there is ... | Observe for any $\psi\in L^2(\mathbb{R})$ we see that
\begin{align}
\psi(x) = \frac{\psi(x)+\psi(-x)}{2}+ \frac{\psi(x)-\psi(-x)}{2}=: \psi_\text{even}(x)+\psi_\text{odd}(x)
\end{align}
then
\begin{align}
\pi(\psi)(x) = \psi_\text{even}(-x)+\psi_\text{odd}(-x)=\psi_\text{even}(x)-\psi_\text{odd}(x).
\end{align}
In sho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3078915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Norm in a Vector Space A vector space with norm $\parallel\cdot\parallel$ Satisfy for two vectors the following
$\parallel x+y\parallel=\parallel x\parallel +\parallel y\parallel$
i need to proof the fallowing statement
$\parallel \alpha x+\beta y\parallel=\alpha\parallel x\parallel +\beta\parallel y\parallel$
for all ... | I have changed $\alpha, \beta$ to $a,b$. Assume without loss of generality that $b\leq a$. Then $\left\Vert ax+by\right\Vert =\left\Vert
a(x+y)+(b-a)y\right\Vert \geq a\left\Vert x+y\right\Vert -(a-b)\left\Vert
y\right\Vert =a(\left\Vert x\right\Vert +\left\Vert y\right\Vert
)-(a-b)\left\Vert y\right\Vert $
$=a\left\Ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3079023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Walter Rudin "Principles of Mathematical Analysis" Definition 3.16, Theorem 3.17. I cannot understand. I am reading Walter Rudin's "Principles of Mathematical Analysis".
There are the following definition and theorem and its proof in this book.
Rudin didn't prove that $E \neq \emptyset$.
Why?
Rudin wrote "If $s^*... | Every sequence in $\overline{\mathbb{R}}$ has a convergent subsequence.
If the sequence is bounded, this is trivial by Bolzano's theorem.
Otherwise, the sequence is unbounded. If it is unbounded above, you can find a subsequence that converges to $+ \infty$. If it is unbounded below, you can find a subsequence that con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3079188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Homeomorphism between the unit disc and the unit square I know that the function that describes the homeomorphism is:
$f(x,y) =
\begin{cases}
\frac{x^{2}+y^{2}}{\max(|x|,|y|)}(x,y) &\quad\text{if } (x,y)\ne (0,0) \\
\text{} (0,0) &\quad\text{if } (x,y) = (0,0) \\
\end{cases}$
But it's impossible... | You do not mention whether you consider the closed or open unit disk resp. unit square. Here, let us consider the closed unit disk $D$ and the closed unit square $Q$.
Both receive their topologies as a subspaces of $\mathbb{R}^2$ which carries its standard topology. There are various equivalent descriptions of this top... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3079355",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $U=Y - E[Y|X]$ and $X$ are uncorrelated
Let $U = Y - E[Y|X]$. How can I prove that $U$ and $X$ are not correlated?
I've been doing a lot of things but when I calculate $\text{cov}(U,X)$ I finish with $EXY - EXEY$ and not $0$ which would be the result.
Any help, guys?
Thanks
| $$E[XU] = E[X(Y - E[Y|X])] = E[XY - XE[Y|X]] = E[XY] - E[XE[Y|X]]$$
Because $X$ is a function of $X$, we can pull out $X$: $XE[Y|X] = E[XY|X]$. Then $E[XE[Y|X]] = E[E[XY|X]]$, so
$$E[XU] = E[XY] - E[E[XY|X]]$$
Then $E[E[XY|X]] = E[XY]$. You can say this follows from tower property, but you can just say this is total ex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3079460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Determine $P(X > n +k\mid X > n)$
Bob is at the shooting range.
With probability $\frac{1}{3}$ Bob hits the target.
Every shot is independent of the previous ones.
Bob starts and keeps shooting until he hits the target.
Let the random variable $X$ be the number of the shot that first hits the target.
Fo... | The way I understand this is:
What's the probability that it will take at least n+k shots to git the target given that it will take at least n shots?
So I imagine this scenario:
Bob already took n shots (and missed), what's the probability that he'll have to take at least k more shots before he hits the target?
Note t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3079575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
True or False: Entries on the main diagonal of matrix A Q. If $A = [a_{ij}]$ is an $m \times n$ matrix which satisfies $A^T = -A$, then the entries on the main diagonal of $A$ are all equal to $0$.
I don't see how $A^T = -A$ can be true for a $m \times n$ matrix. Also, two matrices are only equal if they have the same ... | Firstly, for this question to make any sense (that is, that the statement $A^T=-A$ is meaningful) we must have $m=n$. Now let's examine the diagonal entries, $a_{ii}$. The $i, j$-th element of $A^t$ is the $j, i$-th element of $A$. In particular (for $i=j$) the diagonal entries of $A^t$ are the same as the diagonal ent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3079635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Error in calculation of intersection of three cylinders Set $A:=\{(x,y,z)\in \mathbb R^{3}: x^2+y^2\leq 1, x^2+z^2\leq 1, y^2+z^2 \leq 1\}$
I want to find the volume. I have to use symmetry and without polar coordinates.
My idea: Using symmetry we can look at the first octant and can restrict $x,y,z \geq 0$
$\lambda^{d... | $4\pi$ is quite clearly too much: we should expect a volume close to (and a bit larger than) the volume of a unit sphere, i.e. $\frac{4}{3}\pi$. Assume that the value of $z\in[-1,1]$ has been fixed. The $z$-section of our body is shaped as
$$ \left\{\begin{array}{rcl}x^2+y^2&\leq& 1\\x^2,y^2&\leq &1-z^2\end{array}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3079769",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Bounding the determinant of a matrix with bounded coefficients Suppose that $A$ is a real matrix of dimension $n \times n$ and that its coefficients are bounded by $c\ge0$ ($\vert a_{ij} \vert \le c$ for all $1\le i,j \le n$).
How to prove that
$$\vert \det A \vert \le c^n n^{n/2}$$
| Since
is ture for every element in the matrix, let us consider a matrix where all of the coefficients are the value of c.
Now factor out the c throughout the entire matrix.
Using a formula we obtain
Which reduces to
Using another formula
Reduces to
Since all elements in the matrix where of value c then all of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3079857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to define a function with summation? Suppose we have two sets: $A=\{a_1,a_2,a_3\}$ and $B=\{b_1,b_2,b_3\}$.
Is there a way to define a function that simply adds/subtracts the elements of these two sets? For example,
$$\mu(\cdot)=\sum_{i\in\mathbb{N}:\;a_i\in A}a_i+\sum_{j\in\mathbb{N}:\;b_j\in B}b_j.$$
I am not sur... | Hint: The cartesian product
\begin{align*}
A\times B&=\{(a_1,b_1),(a_1,b_2),(a_1,b_3),\\
&\qquad(a_2,b_1),(a_2,b_2),(a_2,b_3),\\
&\qquad(a_3,b_1),(a_3,b_2),(a_3,b_3)\}
\end{align*}
is not appropriate as domain for $\mu$, since a function $f:A\times B\to \mathbb{R}$ can only map elements of $A\times B$ i.e. pairs $(a_j,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3079995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
$ f_i=\sum a_{ij}e_j $ form a base for a free submodule $ K $ of $ R^{(n)} $ if and only if $ \det A $ is not a zero-divisor.
Let $ R $ be commutative and let $ (e_1, ..., e_n) $ be a base for $ R^{(n)} $. Put $ f_i=\sum a_{ij}e_j $ where $ A=(a_{ij})\in M_n(R) $. Show that the $ f_i $ form a base for a free submodule... | Your argument that if $A$ is injective then $\det A$ is a nonzero divisor is incorrect. The problem is precisely what you point out later in the paragraph. Namely that we don't know that there exists $B$ with $AB=1$, since that asserts that $A$ is surjective as well, which is certainly not always the case. Consider for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3080098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Radon Nykodym derivative process Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and a random variable $Z$ satisfying $\mathbb{E}Z=1$. Define the R-N random process $$Z(t)=\mathbb{E}[Z|\mathcal{F(t)}]$$For $0\leq s\leq t \leq T$, $$\mathbb{E}[Z(t)|\mathcal{F(s)}]=\mathbb{E}[\mathbb{E}[Z|\mathcal{F(t)}]|\... | This is a basic property of conditional expectations: if $\mathcal G_1 \subset\mathcal G_2$ then $E(Z|\mathcal G_1)=E(E(Z|\mathcal G_2)|\mathcal G_1)$. You can prove it easily using definition of conditional expectation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3080202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solving ordinary differential equations of order 2 I'm struggling solving this linear ODE:
$$y''(x) + 2y'(x) = -4$$
I solved the homogeneous solution for this equation: $c_1 + c_2 e^{-2x}$ for some constants $c_1$, $c_2$;
But I'm struggling with the non-homogeneous part:
If I choose $y_p(x) = C$ for some constant $C$... | Hint. Since $0$ is a solution of multiplicity $1$ of the characteristic equation $z^2+2z=0$ and $-4$ is a polynomial of zero degree then, by the Method of undetermined coefficients, you should try as a particular solution the following form
$$y_p(x)=x\cdot C$$
where $C$ is a constant to be determined.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3080341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove the following logarithm inequality.
If $x, y \in (0, 1)$ and $x+y=1$, prove that $x\log(x)+y\log(y) \geq \frac {\log(x)+\log(y)} {2}$.
I transformed the LHS to $\log(x^xy^y)$ and the RHS to $\log(\sqrt{xy})$, from where we get that $x^xy^y \ge \sqrt{xy}$ beacuse the logarithm is a monotonically increasing fun... | Since $x-y$ and $\log x-\log y$ have the same sign, we have
$$
(x-y)(\log x-\log y)\ge 0
$$ or equivalently
$$
x\log x+y\log y\ge y\log x+x\log y.
$$ Hence it holds that
$$
2x\log x+2y\log y\ge (x+y)\log x+(x+y)\log y=\log x+\log y.
$$ This proves
$$
y\log y+x\log x\ge \frac{\log x+\log y}{2}.
$$
Note: As @Martin R p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3080415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Composition of discontinuous functions Let $f(x) = [x]$ and $$ g(x)=\begin{cases} 0&\text{if}\;x \in \Bbb Z\\x^2&\text{otherwise}\end{cases}$$
Is $g\circ f$ continuous?
I know conditions of continuity but in case of composition of discontinuous functions shouldn't the composition be always discontinuous since the do... | A simple example of the composition of two discontinuous functions giving a continuous one.
$$f(x)=\{x\}$$
and
$$g(x)=x(1-x)+\lfloor x\rfloor.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3080559",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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