Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Name for probability density function supported on a closed interval and increasing I am looking for names/examples/references for probability density functions which are supported on a closed interval, say $[0,1]$, and increasing there.
If $f(x)$ is positive and increasing on $[a,b]$ and $I=\int_a^b f(x)dx$ then $g=f/... | Fun question. Here are a few distributions that come to mind ... This is illustrative, but should be relatively straightforward to derive first derivatives etc, if needed.
*
*$\text{Beta}(a,b)$ distribution, with parameter $a > 1$ and $b\leq1$
In the above plot, $b=0.97$. The $b = 1$ case is plotted below separ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1571047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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unique fixed point problem Let $f: \mathbb{R}_{\ge0} \to \mathbb{R} $ where $f$ is continuous and derivable in $\mathbb{R}_{\ge0}$ such that $f(0)=1$ and $|f'(x)| \le \frac{1}{2}$.
Prove that there exist only one $ x_{0}$ such that $f(x_0)=x_0$.
| If $f(x)=x$ and $f(y)=y$ for $y>x$, then $$y-x=|f(y)-f(x)|=|\int_{x}^y f'(t)\mathrm{d}t|\leq \int_x^y |f'(t)|\mathrm{d}t<\frac{1}{2}(y-x),$$
a contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1571145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Find limit for $n*C^n$ when $|C|<1$ Find $\lim_{n\to \infty}{n{C^n}}$ when: $|C|<1$
I want to use the squeeze theorem so I bounded it from below with: $C^n\to 0$
But I can't find the upper bound.
| I know thats an old post but here is another solution
Bernoulli inequality states that:
$$\boxed{(1+\alpha)^n \geq n(1+n\alpha) \quad \forall r>-1, \text{ } n\in \mathbb{N^*}}$$
Given than $C<1$:
Let: $C = 1+\alpha \iff \alpha > -1$.
Also $n \in \mathbb{N} $
Therefore the contraints of Bernoulli inequality are satisf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1571231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Computing $(1+\cos \alpha +i\sin \alpha )^{100}$ How to prove that
$$ (1+\cos \alpha +i\sin \alpha )^{100} = 2^{100}\left( \cos \left(\frac{\alpha}{2}\right)\right) ^{100} \left( \cos \left(\frac{100\alpha}{2}\right)+i\sin \left(\frac{100\alpha}{2}\right)\right)$$
I just nee... | $$1+\cos\alpha+i\sin \alpha=1+e^{i\alpha}=e^{\frac{i\alpha}{2}}(e^{-\frac{i\alpha}{2}}+e^{\frac{i\alpha}{2}})=2\cos\left(\frac{\alpha}{2}\right)e^{\frac{i\alpha}{2}}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1571324",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Is it possible that $\frac{|A \cap B|}{|A \cup B|} > \frac 12$ and $\frac{|A \cap C|}{|A \cup C|} > \frac 12$ given that $|B \cap C| = 0$? Given three sets, $A$, $B$, $C$, I have that $B$ and $C$ are disjoint, and $|\cdot|$ represents the number of elements in the set:
$$|B \cap C| = 0$$
Is it possible that both:
$$\... | Disclaimer: I assume $A,B,C$ are finite, otherwise the division makes little sense.
No, this is not possible.
First of all, you can see that if $B$ and $C$ are not subsets of $A$, you can repace them with $B'=B\cap A$ and $C'=C\cap A$, and the quantities you want to be bigger than $\frac12$ will merely increase. In oth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1571522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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How to solve a problem with a variable in both the base and exponent on opposite sides of an equation I am working on systems of equations in Pre-Calculus, and I presented the teacher a question that I had been wondering for a while.
$x^2 = 2^x$
The teacher couldn't figure it out after playing with it for quite a while... | Love the curiosity!
To solve this requires more than regular algebra. It requires use of the Lambert W function.$$f(x)=xe^x$$$$W(x)=f^{-1}(x)$$The solution, is, of course, not solvable. But it does allow us to do some amazing things.
First, let's attempt to solve for $W(x)$ to find its identities.$$x=ye^y$$$$y=W(x)$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1571635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
$\sum a_n $ absolutely convergent and $\sum b_n $ convergent $\implies \sum a_nb_n$ absolutely convergent. Is this true?
$\sum a_n $ absolutely convergent and $\sum b_n $ convergent $\implies \sum a_nb_n$ absolutely convergent.
I don't know how to proceed .Please help.
| Other way:
If $\sum a_n$ and $\sum b_n$ absolutely converge, then $\sum\sqrt{|a_nb_n|}$ converges, as
$$0\le2\sqrt{|a_nb_n|}\le |a_n|+|b_n|$$
And $|a_nb_n|<1$ for $n$ large and so
$$|a_nb_n|<\sqrt{|a_nb_n|}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1571708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Need help proving any subgroup and quotient of a nilpotent group is nilpotent? $G$ is the direct product of its Sylow subgroups $P_i$. Then if $H \le G$, $P_i \cap H \le H$. I'm stuck now on how to proceede to find a terminating central series for $H$.
Also, I know $P_i H / H \le G/H$ so I'm assuming proving the above ... | I posted a proof that a quotient of a nilpotent group is nilpotent recently.
Here is a proof that a subgroup of a nilpotent group is nilpotent.
I will work with the definition that a group $G$ is nilpotent if it has a central series
$$1 = N_0 \leq N_1 \leq \cdots \leq N_r = G$$
where each $N_i \lhd G$ and $N_i / N_{i-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1571815",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Cardinality of the smallest subgroup containing two distinct subgroups of order 2 $G$ is a finite group and $H_1$,$H_2$ are two disjoint subgroups of order $2$. $H$ is the subgroup of smallest order that contains both $H_1$ and $H_2.$ What is the cardinality of $H$ $?$
$A.$ always $2$.
$B.$ a... | Hint: If $G$ is a dihedral group (symmetry group of a regular polygon), then it is generated by two (suitable) reflections.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1571955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Is this (tricky) natural deduction with De Morgan's laws correct? Just a practice question, however just wondering if this natural deduction proof is correct?
I have put brackets in 2.2 and not in 2.3 however this shouldn't make a difference?
| Your proof is wrong because the application of the rule $\lnot_E$ at step 2.4 is not correct.
I assume that the schema of the rule $\lnot_E$ is the following:
\begin{equation}
\frac{\lnot A \to B \quad \lnot A \to \lnot B}{A}
\end{equation}
(this is the way you have applied it at step 4). This schema is one of the pos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Estimate of n factorial: $n^{\frac{n}{2}} \le n! \le \left(\frac{n+1}{2}\right)^{n}$ on our lesson at our university, our professsor told that factorial has these estimates
$n^{\frac{n}{2}} \le n! \le \left(\dfrac{n+1}{2}\right)^{n}$
and during proof he did this
$(n!)^{2}=\underbrace{n\cdot(n-1)\dotsm 2\cdot 1}_{n!} \c... | $$n^{n/2}\le n!\le \left(\frac{n+1}{2}\right)^{n}$$
$$\iff n^n\le (n!)^2\le \left(\frac{(n+1)^2}{4}\right)^n$$
Now, $n\le i((n+1)-i)$ for all $i\in\{1,2,\ldots,n\}$, because this is equivalent to $n(1-i)\le i(1-i)$. If $i=1$, then it's true. If $i\neq 1$, then this is equivalent to $n\ge i$, which is true.
Also $ab\le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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To prove that a vector $x(t)$ lies in a plane. Prove that vector $x(t)=t\,\hat{i}+\left(\dfrac{1+t}{t}\right)\hat{ j}+\left(\dfrac{1-t^2}{t})\right)\hat{k}$ lies in a curve.
I am puzzled. Don't know how to approach it.
| Hint:
Rewrite
$$ x(t) = t \hat{i} + \big{(}\frac{1+t}{t}\big{)}\hat{j} + \big{(}\frac{1-t^2}{t}\big{)}\hat{k} $$
as
$$ x(t) = \hat{j} + t(\hat{i} - \hat{k}) + \frac{1}{t}(\hat{j} + \hat{k}). $$
Now when you revise the definition of plane and study the rewritten expression, you are able to conclude that $x(t)$ li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Is the property "being a derivative" preserved under multiplication and composition? Since differentiation is linear, we therefore have that if $f, g: I\to \mathbb{R}$ is a derivative (where $I\subset \mathbb{R}$ is an interval), then so does their linear combination. What if we consider their multiplication and compos... | Let me address just one of your problems.
Problem. Suppose that $f$ and $g$ are both derivatives. Under
what conditions can we assert that the product $fg$ is
also a derivative?
The short answer is that this is not true in general. In fact even if we assume that $f$ is continuous and $g$ is a derivative th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 3,
"answer_id": 1
} |
Find limit of $f_n(x)= (\cos(x))(\sin(x))^n \sqrt{n+1}$ $f_n(x)$ on $ \Bbb{R}$ defined by
$$f_n(x)= (\cos(x))(\sin(x))^n \sqrt{n+1}$$ Then
Is It converges uniformly ?
I think first we must find limit of $f_n$ , I find limit for 0 and $\frac{\pi}{2}$ ,but I can't find for every point.
| A... If $\sin x=0$ or $\cos x=0$ then $f_n(x)=0$ for every $n.$ If $0<|\sin x|<1$ let $|\sin x|=1/(1+y) .$ Since $y>0$ we have $(1+y)^n\geq 1+n y, $ so $0<|f_n(x)|<|\sin x|^n\sqrt {1+n}<(1+n y)^{-1}\sqrt {1+n}.$ So $f_n(x)\to 0$ as $n\to \infty.$
B... Let $g_n(x)=\cos x \sin^n x.$ For any $x$ there exists $x'\in [0,\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Minimization of integrals in real analysis For $\:a,\:b,\:c\in\mathbb{R}\:$ minimize the following integral:
$$\\\\\int_{-\pi}^{\:\pi}(\gamma-a-b\:\cos\:\gamma\:-c\:\sin\:\gamma)^2\:d\gamma$$
How do we solve this? No idea
| This is not an answer but it is too long for a comment.
Dr. MV gave you the rigorous answer.
If you think about the problem, what it means is that, based on an infinite number of data points, you want to approximate, in the least square sense, $\gamma$ by $a+b\cos(\gamma)+c\sin(\gamma)$ over the range $[-\pi,\pi]$.
For... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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After 6n roll of dice, what is the probability each face was rolled exactly n times? This is closely related to the question "If you toss an even number of coins, what is the probability of 50% head and 50% tail?", but for dice with 6 possible results instead of coins (with 2 possible results). Actually I would like a ... | Think of creating a string of length $6n$, where each letter is one of the numbers on the face of the die. For instance, for $n=1$, we could have the string $351264$. There are $6^{6n}$ such strings, as can be seen by elementary counting methods. The strings you are interested in are ones in which each letter appears $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Consider the following vectors in R3 Consider the following vectors in $\Bbb{R}^3$:
${\bf v}_1= \frac{1}{\sqrt{3}}(1,1,-1)$, ${\bf v}_2 = \frac{1}{\sqrt{2}}(1,-1,0),$ and ${\bf v}_3 = \frac{1}{\sqrt{6}}(1,1,2)$
a) Show that $\{{\bf v}_1, {\bf v}_2, {\bf v}_3\}$ form a basis of $\Bbb{R}^3$. (hint: compute their inner pr... | A set of any 3 linearly independent vectors in $R^3$ is a basis for $R^3.$ If $x_1,x_2,x_3$ are 3 non-zero pairwise-orthogonal vectors then they are linearly independent. Because if $0=a_1x_+a_2x_2+a_3x_3 $, then for $j\in \{1,2,3\}$ we have $0=0\cdot x_j=(a_1x_1+a_2x_2+a_3x_3)\cdot x_j=a_j(x_j\cdot x_j)\implies a_j=0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Ternary strings (combinatorics, recurrence) The questions is: for $A_n$ find all ternary strings of length $n ≥ 0$ that don't include substring $”11”$. Provide answer in form of:
a) recurrence relation
b) combinatorial expression
After that, for $B_n$ take $A_n$ and exclude strings that also have substring $”12”$ and ... | I happened to chance on this, so here is the combinatorial approach that you wanted, which is incidentally much simpler !
Hope you like it !
Combinatorial approach
Any string of $i$ non-$1's$ create $(i+1)$ gaps (including ends) where non-adjacent $1's$ can be put,
thus if string length is $n$, the minimum number of no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are the irrationals as a subspace in the real line and in the plane a connected space? By irrationals, $\color{blue}{\mathbb{I}}$, I mean the set $\color{blue}{\mathbb{R}\setminus \mathbb{Q}}$ and the set $\color{blue}{\mathbb{R^2}\setminus\mathbb{Q^2}}$.
My thought is no in both cases.
For the set $\color{blue}{\math... | I think the confusion comes from that $\mathbb R^2 \setminus \mathbb Q^2$ is not the same thing than $(\mathbb R \setminus \mathbb Q)^2$.
In the first definition these are points of the plane which do not have both coordinates rationnal, thus there can be mixed coordinates, while in the second definition both coordinat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1572927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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What is $1^\omega$? In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as:
If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, $\alpha^\beta$ is the least ordinal greater than any ordinal in the set $\{\alpha^\gamma:\gamma<\beta\}.... | Your reasoning is correct, but Mathworld's definition is wrong: it should specify the least ordinal greater than or equal to all the ordinals in the set $\{ \alpha^\gamma : \gamma < \beta \}$, with the result that $1^\omega = 1$. More generally, $1^\alpha = 1$ for any ordinal $\alpha$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1573099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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How do you prove $\lim_{n\to\infty} 1/n^{1/n}$ using only basic limit theorems? How do you prove $\lim_{n\to\infty} \frac {1}{n^{1/n}}$ using only basic limit theorems? I thought it was $0$, but my book lists the solution as $1$. How come?
| Hint: Use $ \lim_{n\to \infty} (a_n)^{1/n}=l $ if $\lim_{n\to \infty}\frac{a_{n+1}}{a_n}=l$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1573234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
Proving Something about Orthogonal Vectors If $\vec x ,\vec y \in \mathbb R^3$ are orthogonal and $x = \|\vec x\|$ then prove that
$$
\vec x \times \bigl(
\vec x \times \bigl(
\vec x \times (
\vec x \times \vec y
)
\bigr)
\bigr)
=
x^4\vec y
$$
I have no idea how to start this, any tips? (Sorry about... | Here is a step by step approach
$1.$ You can use the famous indentity
$$\vec{x} \times ( \vec{y} \times \vec{z})=(\vec{x} \cdot \vec{z})\vec{y}-(\vec{x} \cdot \vec{y})\vec{z}$$
$2.$ Next consider that
$$\begin{align}
& \quad \,\,\,\vec{x} \times (\vec{x} \times \vec{y}) \\
&=(\vec{x} \cdot \vec{y})x-(\vec{x} \cdot \vec... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1573338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Sum of Sequence Let $c_{n} = \frac{1+(-1)^{n}}{2}$
$S_{n} = c_{1} + c_{2} + c_{3} + ... + c_{n}$
Prove that $\lim \frac{S_{n}}{n} = \frac{1}{2}$
These are my steps
$\rightarrow \frac{S_{n}}{n} = \frac{n}{2n}(\frac{1+(-1)^{n}}{2}) = \frac{1+(-1)^{n}}{4}$
$1+(-1)^{n}$ is $2$ or $0$, so the lim of the sum is $\frac{2}{4... | To summarize the comments:
We are asked about the sequence $$c_n=\{0,1,0,1,\dots\}$$
It is easy to see that the partial sums satisfy $$S_{2n}=n=S_{2n+1}$$
To compute the limit (as $n\to \infty$) of $\frac {S_n}{n}$ it is convenient to distinguish the even indices from the odd.
If $n=2k$ is even we have $S_n=k$ from w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1573420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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ELI5: What are pointwise and uniform convergence and what is the difference? I have been fiddling around with some series of functions and analyzing whether they converge pointwise or uniformly. Furthermore I know that continuity and convergence of integrals does not always follow from pointwise but for uniform converg... | We can start from the definitions. For a sequence $f_n(x)$ with $f_n: S\to \mathbb{R}$, $n\in \mathbb{N}$ we have:
the sequence converge pointwise if :
$$
\left(\forall \epsilon >0 \land \forall x\in S\right) \quad \exists N \in \mathbb{N} \quad such \;that \quad \left(|f_n(x)-f(x)|<\epsilon \;,\;\forall n>N \right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1573510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Dealing with "finding what some vector in the codomain is the image of"?? So I'm having trouble finding a pattern when dealing with these types of questions; I need to find a better way to solve them:
Here's the one i'm currently dealing with:
Find the range space and rank of the map:
$a) f: \mathbb R^2 → P^3$ given by... | Since your map is linear, you can write it in matrix notation as follows:
$$
f(x, y) = \begin{pmatrix}0 & 0\\ 1 & -1 \\ 0 & 3\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix}
$$
The rank of this matrix is 2. You can also notice that it maps your 2D vector onto a plane in $\mathbb{R}^3$. Moreover, from the definition of $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1573592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Divergence of a positive series over a uncountable set.
Let $\Lambda$ be a uncountable set and let $\{a_{\alpha}\}_{\alpha\in\Lambda}$ be such that $a_{\alpha}>0$ for all $\alpha \in\Lambda$ proof that,
$$\sum_{\alpha\in\Lambda}a_{\alpha}$$
diverges.
| One should also define $\sum_{\alpha \in \Lambda}a_{\alpha}$, since this is not a standard thing, perhaps as $\sup \sum_{\alpha \in \Lambda_0}a_{\alpha}$ over all finite $\Lambda_0 \subset \Lambda$. Assume now that $\Lambda' \colon =\{ \lambda\ | \ a_{\lambda}> 0\}$ is uncountable. Now, every positive number is $\ge \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1573710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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find the equations of the tangents at the points P and Q The curve ${y = (x -1)(x^2 + 7)}$ meets the x-axis at P and the y-axis at Q.
Find the equations of the tangents P and Q.
I am not looking for the answer but just some hints to get me started.
I am not sure how to being.
| Hints:
*
*$P$ is where $y=0$, so either $x-1=0$ or $x^2+7=0$.
*Similarly, $Q$ is where $x=0$, in which case $y=(0-1)(0+7) = -7$.
*The tangents will be of the form $y=mx+b$. Since the line is tangent to the curve, you must evaluate the derivative at $P$ and $Q$. Whatever you get for the derivative will be the slope... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1573819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The relationship between sample variance and proportion variance? I'm trying to see the relationship between the sample variance equation
$\sum(X_i- \bar X)^2/(n-1)$ and the variance estimate, $\bar X(1-\bar X),$ in case of binary samples.
I wonder if the outputs are the same, or if not, what is the relationship betw... | \begin{align}
& \sum_{i=1}^n (x_i - \bar x)^2 \\[10pt]
= {} & \sum_{i=1}^n (x_i - 2\bar x x_i + \bar x^2) \\[10pt]
= {} & \left( \sum_{i=1}^n x_i^2 \right) - 2\bar x \left( \sum_{i=1}^n x_i \right) + \left( \sum_{i=1}^n \bar x^2 \right) \\[10pt]
= {} & \left( \sum_{i=1}^n x_i^2 \right) - 2\bar x\Big( n\bar x\Big) + n \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1573947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
difference of characteristic function for measure and random variable Suppose random variable $X$ follow a certain (known) distribution. And I denote the probability measure $\mu$ as the distribution (pushforward measure) of $X$. Is there any difference between $\hat{\mu}(t)=\int_{\mathbb{R}}{e^{itx}\mu(dx)}$ (Fourier ... | The characteristic function of a real random variable $X$ is the characteristic function of its probability distribution, and it is defined as
$$
t\mapsto \operatorname{E}(e^{itX})
$$
where $i=\sqrt{-1}$ is the imaginary unit. The characteristic function is equal to
$$
t\mapsto \int_{\mathbb R} e^{itx}\,d\mu(x)
$$
or i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Limits and rate of change I'm a freshman taking calculus 1 currently studying for finals.
I am reviewing stuff from the beginning of the semester,and I don't remember the proper way to deal with limits like this one.
A ball dropped from a state of rest at time t=0 travels a distance $$s(t)=4.9t^2$$ in 't' seconds. I a... | At $t=2$ the ball is at position $s(t=2)$. At $t=2.5$, the ball has reached $s(2.5)$. Hence the distance traveled is $s(2.5)-s(2)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Induction Proof with Fibonacci How do I prove this?
For the Fibonacci numbers defined by $f_1=1$, $f_2=1$, and $f_n = f_{n-1} + f_{n-2}$ for $n ≥ 3$, prove that $f^2_{n+1} - f_{n+1}f_n - f^2_n = (-1)^n$ for all $n≥ 1$.
| *
*Base Case:
For $n = 1$:
$$f_2^2 - f_2f_1 - f_1^2 = (-1)^1$$
$$(1)^2 - (1)(1) - (1)^2 = -1$$
$$-1 = -1$$
*
*Induction Step:
Assume that the given statement is true. We now try to prove that it holds true for $n+1$:
$$f_{n+2}^2 - f_{n+2}f_{n+1} - f_{n+1}^2 = (-1)^{n+1}$$
Typically you choose one side and try ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574290",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Related Rates using circles can someone please help? I'm taking Calculus, but I'm really having trouble understanding the concept of related rates.
A jogger runs around a circular track of radius 55 ft. Let (x,y) be her coordinates, where the origin is the center of the track. When the jogger's coordinates are (33, 44)... | You did the right thing. It differentiates to:
$$2x\frac{dx}{dt} + 2y\frac{dy}{dt} =0$$
You know x, y, and $\frac{dx}{dt}$. Simply solve for $\frac{dy}{dt}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Prove that $5n^2 - 3$ even $\implies n$ odd I tried to prove this by contradiction. I used contradiction to show that if $n$ is odd then $5n^2 - 3$ is even; but my Professor said this is not a correct answer to the question: you need to prove that if $5n^2 - 3$ is even then $n$ is odd. Why is what I said wrong, and how... | You were asked to prove that if $5n^2-3$ is even, then n is odd. But what you proved was that if n is odd, then $5n^2-3$ is even.
$A \implies B$ is equivalent to $\neg B \implies \neg A $. So if you prove $\neg B \implies \neg A $, then you've proved $A \implies B$. This is an example of proof by contradiction. Assume... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 8,
"answer_id": 6
} |
Prove that for all positive $x,y,z$, $\small\left(2e^x+\dfrac{2}{e^x}\right)\left(2e^y+\dfrac{2}{e^y}\right)\left(2e^z+\dfrac{2}{e^z}\right) \geq 64$ Prove that for all positive $x,y,z$, $\left(2e^x+\dfrac{2}{e^x}\right)\left(2e^y+\dfrac{2}{e^y}\right)\left(2e^z+\dfrac{2}{e^z}\right) \geq 64$
I dont have that much expe... | Let $u = e^x$. Then if $x > 0$, then $u > e^0 = 1$, and consider $$u + u^{-1} - 2 = u - 2u^{-1/2}u^{1/2} + u^{-1} = (u^{1/2} - u^{-1/2})^2 \ge 0.$$ The first equality is because $u^{-1/2}u^{1/2} = 1$, and the inequality is due to the fact that the square of a real number is never negative. Consequently, $$u+u^{-1} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574528",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to use piecewise quadratic interpolation? I'm attempting to get the hang of quadratic interpolation, in MatLab specifically, and I'm having trouble approaching the process of actually creating the spline equations.
For example, I have 9 points that need to be interpolated, so I'll need 8 equations for the whole cur... | Usually the equations look like this (for example with the following set of points):
Points:
$P_0(-1.5|-1.2); P_1(-0.2|0); P_2(1|0.5); P_3(5|1); P_4(10|1.2)$
Equation:
$$f(x) = \begin{cases}-7.4882 \cdot 10^{-2}\cdot x^3 + -3.3697 \cdot 10^{-1}\cdot x^2 + 5.4417 \cdot 10^{-1}\cdot x + 1.2171 \cdot 10^{-1}, & \text{if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What to do when l'Hopital's doesn't work I'm a first time Calc I student with a professor who loves using $e^x$ and logarithims in questions. So, loosely I know L'Hopital's rule states that when you have a limit that is indeterminate, you can differentiate the function to then solve the problem. But what do you do when... | HINT:
As for your problem, divide both numerator and denominator by $e^x$. You'll get your limit as $1$.
In mathematics, logic, representation and arrangement play an extremely vital role. So always check that you have arranged your expression properly. Else repeated applications of several powerful and helpful theore... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Limit exists theoritically,but graphically there is a vertical asymptote there.Why is it so. In finding the limit of $\lim_{x\to 0}\frac{2^x-1-x}{x^2}$
I used the substitution $x=2t$
$\lim_{x\to 0}\frac{2^x-1-x}{x^2}=\lim_{t\to 0}\frac{2^{2t}-1-2t}{4t^2}=\frac{1}{4}\lim_{t\to 0}\frac{2^{2t}-2\times2^t+2\times2^t+1-2-2t... | Your mistake is that you used the limit laws wrongly. The limit of a sum is the sum of the limits if the two separate limits exist. Since you haven't proven that in your usage, and indeed it doesn't, the reasoning is flawed and hence does not contradict the actual fact that the limit does not exist.
Note also that you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
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Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$
Let $m$ be a positive integer and $n$ a nonnegative integer. Prove that
$$(n!)\cdot(m!)^n|(mn)!$$
I can prove it using Legendre's Formula, but I have to use the lemma that
$$ \dfrac{\displaystyle\left(\sum_{i=1}^na_i\right)!}{\displaystyle\prod_{i=1}^na_i!} \in \ma... |
We organise the $m\cdot n$ factors of $(mn)!$ into $n$ blocks of size $m$
\begin{align*}
((j-1)&m+1)((j-1)m+2)\cdots((j-1)m+m)\tag{1}\\
&=((j-1)m+1)((j-1)m+2)\cdots(jm-1)(jm)\qquad 1\leq j \leq n \\
\end{align*}
Since for $0\leq m \leq k$
\begin{align*}
\binom{k}{m}&=\frac{k!}{m!(k-m)!}\\
&=\frac{(k-m+1)\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
"answer_count": 9,
"answer_id": 2
} |
Prove any orthogonal 2-by-2 can be written in one of these forms.... I'd like to prove that any orthogonal $2 \times 2$ matrix can be written
$$\bigg(\begin{matrix}
\cos x & -\sin x \\
\sin x & \cos x
\end{matrix}\bigg) \hspace{1.0em} \text{or} \hspace{1.0em}
\bigg(\begin{matrix}
\cos x & \sin x \\
\si... | Let $M=\begin{pmatrix}a & b \\ c & d \end{pmatrix}$ be orthogonal, then we have $MM^T=I_2$ and thus
$$\begin{cases}a^2+b^2=1 \\ c^2+d^2 = 1 \\ac+bd=0\end{cases}$$
By the $2$ first equations, we know that there exists $\theta,\phi\in [0,2\pi)$ such that
$$ a = \cos(\theta), \quad b = \sin(\theta),\quad c = \sin(\phi), ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1574918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Linear transformation to bilinear form Let there be $f:V\times V \rightarrow F$ over a finite vector space, which for a certain base $(w) = \{ w_1, w_2, ... , w_n \}$ has a nonsingular representing matrix (e.g. $[f]_w$ is nonsingular).
How can I show that for every linear transformation $l:V\rightarrow F$ there exists... | Consider the map $V \to \text{Hom}_F(V,F), v \mapsto (u \mapsto f(u,v))$ and show that your assumption on $f$ implies that this map is an isomorphism. In fact, it suffices to show that this map is injective, as $V$ is assumed to be finite dimensional (and $\text{Hom}_F(V,F)$ has the same dimension as $V$).
Indeed, if $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1575004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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$G$ is a finite abelian group. For every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$. $G$ is a finite abelian group. Assume that for every prime $p$ that divides $|G|$, there is a unique subgroup of order $p$. I'd like to prove that $G$ is cyclic. I'm thinking about the approach of induction b... | Set $G = \{g_{1}, g_{2}, ..., g_{n}\}$, with $|g_{i}| =d_{i}$.
Consider the group $H = \prod_{i=1}^{n}\mathbb{Z}\backslash d_{i}\mathbb{Z}$.
Since $G$ is abelian, the map $f: H \rightarrow G: f(g_{1}, g_{2}, ..., g_{n}) = \prod_{i=1}^{n}g_{i}^{k_{i}}$ is a well-defined surjective group homomorphism so that by one of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1575276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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A name for this property? Let $*$ be an operation such that $(xy)^* = y^*x^*$, e.g. if $x,y$ are $2\times2$ matrices and $*$ is "take the inverse" or if $x,y$ are operators and if $*$ is the adjoint.
Is there a name for such a property ?
| I don't believe this is standard, but my abstract algebra textbook (Contemporary Abstract Algebra by Gallian) refers to the fact that $(ab)^{-1} = b^{-1}a^{-1}$ as the "Socks-Shoes Property" (you put your socks on before your shoes, but if you want to undo that operation, you need to take your shoes off before taking o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1575352",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 5,
"answer_id": 2
} |
Is $(-2)^{\sqrt{2}}$ a real number?
Is $(-2)^{\sqrt{2}}$ a real number?
Clarification: Is there some reason why $(-2)^{\sqrt{2}}$ is not a real number because it doesn't make sense why it shouldn't be a real number.
Mathematically we can define the value of $\sqrt{2}$ in terms of a limit of rationals. But the problem... | No. There are a countably infinite number of possible values of this expression. None of them are real. They are
$$2^{\sqrt{2}}(\cos (2k+1)\pi\sqrt{2} + i\sin(2k+1)\pi\sqrt{2})
$$
The principal value (taking $k=0$) is $2^{\sqrt{2}}(\cos\pi\sqrt{2} + i\sin\pi\sqrt{2})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1575443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 1
} |
Solve $x^2\equiv -3\pmod {\!91}$ by CRT lifting roots $\!\bmod 13\ \&\ 7$ Question 1) Solve $$x^2\equiv -3\pmod {13}$$
I see that $x^2+3=13n$. I don't really know what to do? Any hints?
The solution should be $$x\equiv \pm 6 \pmod {13}$$
Question 2) $\ $ [note $\bmod 7\!:\ x^2\equiv -3\equiv 4\iff x\equiv \pm 2.\,$ Her... | *
*Just try all candidates $0,1,2,3,4,5,6$. You can stop at $6$ because $(13-x)^2\equiv x^2$. This will also give you a second solution if you find one.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1575576",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
What does $\frac{d^6y}{ dx^6}$ mean? The whole question is that
If $f(x) = -2cos^2x$, then what is $d^6y \over dx^6$ for x = $\pi/4$?
The key here is what does $d^6y \over dx^6$ mean?
I know that $d^6y \over d^6x$ means 6th derivative of y with respect to x, but I've never seen it before.
| The symbol "$\frac d{dx}$" is used to indicate a single derivative (with respect to $x$).
We treat repeated application of this operator symbolically as "powers" of the operator (as if it were ordinary multiplication by an ordinary fraction), writing "$\frac{d^n}{dx^n}$" to indicate $n$ successive applications of "$\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1575671",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Divisor of rational function I am finding the divisor of $f = (x_1/x_0) − 1 $ on $C$, where $C = V ( x_1^2 + x_2 ^2 − x_0^ 2 ) ⊂ \mathbb P^ 2 $. Characteristic is not 2.
I am totally new to divisors. So the plan in my mind is first find an open subset $U$ of $C$. In this case maybe it should be the complement of $X={x_... | Here’s how I would do it, but my method and understanding are irremediably old-fashioned, to the extent that they may be of limited assistance to you.
First, I would take the open set where $x_0\ne0$, and dehomogenize by setting $x_0=1$, to get $x_1^2+x_2^2=1$, the unit circle! The function $f$ is now $x-1$, and this c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1575760",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Linearizing a function involving an integral about a point
Find the linearization of $$g(x)= \int_0^{\cot(x)} \frac{dt}{t^2 + 1}$$ at $x=\frac{\pi}{2}$.
I know to find linearization I first plugin the $x$ values into my function $g(x)$: $g(\pi/2)$.
Then as I understand it, I make the necessary conversions to form the... | $$
y = \int_0^u \frac{dt}{t^2 + 1} \quad \text{and} \quad u = \cot x.
$$
$$
\frac {dy}{du} = \frac 1 {u^2+1} \quad\text{and}\quad \frac{du}{dx} = -\csc^2 x.
$$
When $x=\pi/2$ then $-\csc^2 x = -1$ and $\cot x = 0$, so $\dfrac 1 {u^2+1} = \dfrac 1 {0^2+1} = 1$.
Bottom line:
$$
\left. \frac{dy}{dx} \right|_{x=\pi/2} = -1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1575841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluate definite integral using the definition: $\int_{-3}^{1}(x^2-2x+4) dx$? Can someone walk me through finding the value of a definite integral by using the definition itself? In this case:
`The definition of the definite integral: let $f$ be a function that is defined on the closed interval $[a,b]$. The definite i... | Consider $\int_{-3}^{1}x dx$. Divide $[-3, 1]$ in a partition $x_0= -3, x_1=-3 + 4/n, x_2=-3+2(4/n)$ ... $x_n=-3+n(4/n)=1$. The lower sum L and the upper sum S are given by $$L_n=\Sigma_{t=1}^{n}\Big(-3+ (t-1)\frac{4}{n}\Big)\frac{4}{n}$$ $$U_n=\Sigma_{t=1}^{n}\Big(-3+ t\frac{4}{n}\Big)\frac{4}{n}$$ In above the quant... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1575947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do you show $\text{col}A= \text{span}(c)$ and $\text{row }A= \text{span}(r)$ based on the following condition. Let $A=cR$ where $c \ne 0$ is a column in $ℝ^m$ and $R \ne 0$ is a row in $ℝ^n$. Prove $\text{col}A= \text{span}(c)$ and $\text{row }A= \text{span}(R)$.
Could you give me an approach?
| Hint: Write $c = (c_1, \ldots, c_m)^T$ and $R = (r_1, \ldots, r_n)$. Then
$$ cR = \begin{pmatrix} c_1r_1 & \ldots & c_1 r_n \\ c_2r_1 & \ldots & c_2 r_n \\ \vdots & \ldots & \vdots \\ c_mr_1 & \ldots & c_mr_n \end{pmatrix}. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1576029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Arc length of the squircle The squircle is given by the equation $x^4+y^4=r^4$. Apparently, its circumference or arc length $c$ is given by
$$c=-\frac{\sqrt[4]{3} r G_{5,5}^{5,5}\left(1\left|
\begin{array}{c}
\frac{1}{3},\frac{2}{3},\frac{5}{6},1,\frac{4}{3} \\
\frac{1}{12},\frac{5}{12},\frac{7}{12},\frac{3}{4},\fra... | By your definition, $\mathcal{C} = \{(x,y) \in \mathbb{R}^{2}: x^4 + y^4 = r^4\}$. Which can be parametrized as
\begin{align}
\mathcal{C} =
\begin{cases}
\left(+\sqrt{\cos (\theta )},+\sqrt{\sin (\theta )} \right)r\\
\left(+\sqrt{\cos (\theta )},-\sqrt{\sin (\theta )} \right)r\\
\left(-\sqrt{\cos (\theta )},+\sqrt{\sin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1576115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
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Plugging number back into recurrence relation I have this problem that I already solved the recurrence for:
$$T_{n} = T_{n-1} + 3, T_{0} = 1$$
I worked it out to $T_{n-4} + 4[(n-3)+(n-2)+(n-1)+n]$ (where I stopped because I saw the pattern), but I can't figure out how to actually plug in the 1 from the $T_{0} = 1$.
If... | Notice, $$T_n=T_{n-1}+3$$
setting $n=1$, $$T_1=T_0+3=1+3=4$$
$n=2$, $$T_2=T_1+3=4+3=7$$
$n=3$, $$T_3=T_2+3=7+3=10$$
$n=4$, $$T_4=T_3+3=10+3=13$$
$$........................$$
$$........................$$
$$T_n=T_{n-1}+3$$
thus, one should observe that an A.P. is obtained with common difference $d=3$ & the first term $a=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1576271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Determine if the following expressions result in a scalar or vector field. If undefined, please explain why. $F(x,y,z)$ is a vector field in space and $f(x,y,z)$ is a scalar field in space.
*
*$\nabla \times (\nabla(\nabla \cdot F))$
*$\nabla \times (\nabla \cdot (\nabla f))$
*$ \nabla (\nabla \cdot (\nabla \times... | For example, in expression 2:
*
*$f$ is a scalar field
*$\nabla f$ is its gradient: it is a vector field
*$\nabla \cdot \nabla f$ is the divergence of this vector field, it is a scalar field
*$\nabla \times (\nabla \cdot \nabla f)$ is the curl of this scalar field: this is undefined as the curl operator takes a v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1576320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Parabola conic section Two tangents to the parabola $y^2= 8x$ meet the tangent at its vertex in the points $P$ and $Q$. If $|PQ| = 4$, prove that the locus of the point of the intersection of the two tangents is $y^2 = 8 (x + 2)$.
I take the point of intersection $(x,y)$ . But after that how can I write equation of th... | Use the fact that perpendicular from the focus on any tangent to a parabola meets the tangent at the vertex. So let$ty=x+at^2$ be the equation of tangent. Equation of line perpendicular from focus is $y=-tx+ta$ Solve it with the equation of tangent. Get the points. Use the fact that the distance is given and the point ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1576656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Prove that at least one $\ell_i$ is constant. Let $f(x)\in \mathbb{Z}[x]$ be a polynomial of the form
$$f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$$
and let $p$ a prime number such that $p|a_{i}$ for $i=0,1,2,\ldots,n-1$ but $p^3\nmid a_0$. Suppose that $f(x)=\ell_1(x)\ell_2(x)\ell_3(x)$ with $\ell_{i}\in \mathbb{Z}[x]$... | I have one solution, it's similiar to the prove of Eisentein's criterion:
Suppose that $f(x)=\ell_1(x)\ell_2(x)\ell_3(x)$ where $\ell_i(x)\in \mathbb{Z}[x]$ are noconstant polynomials. Then reducing module $p$, I have
$$x^n=\overline{\ell_1(x)}.\overline{\ell_2(x)}.\overline{\ell_3(x)} \;\mbox{ in }\;\mathbb{Z}/(p)[x]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1576730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Triangle inequality for infinite number of terms We can prove that for any $n\in \mathbb{N}$ we have triangle inequality: $$|x_1+x_2+\cdots+x_n|\leqslant |x_1|+|x_2|+\cdots+|x_n|.$$
How to prove it for series i.e. $$\left|\sum \limits_{n=1}^{\infty}a_n\right|\leqslant \sum \limits_{n=1}^{\infty}|a_n|.$$
Can anyone help... | A slightly different proof for the case $\sum_{n=1}^\infty |a_n|< \infty $ (i.e., $\sum_{n=1}^\infty a_n$ is absolutely convergent):
First, we show that $\lim_{N \to \infty} |\sum_{n=1}^N a_n| = | \sum_{n=1}^\infty a_n|$.
Denote the $N$th partial sum by $s_N := \sum_{n=1}^N a_n$.
The absolute convergence of $\sum_{n=1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1576816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 1
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$f:\mathbb R \to \mathbb R$ be differentiable such that $f(0)=0$ and $f'(x)>f(x),\forall x \in \mathbb R$ ; then is $f(x)>0,\forall x>0$? Let $f:\mathbb R \to \mathbb R$ be a differentiable function such that $f(0)=0$ and $f'(x)>f(x),\forall x \in \mathbb R$ ; then is it true that $f(x)>0,\forall x>0$ ?
| Let $y(x)=e^{-x}f(x)$. Then $ f$ (strictly) positive $ \iff y$ (striclty) positive.
$\forall x $, $ y'(x)=e^{-x}(f'(x)-f(x)) \ge 0$ and if $x> 0, y'(x)>0$.
Therefore $y$ is positive for $x \ge 0$.
Now lets suppose it exists $t>0$ such that $y(t)=0$. Then for $\epsilon >0 $ small enough, because $y'(t) >0$, for $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1576874",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
} |
Is there a special value for $\frac{\zeta'(2)}{\zeta(2)} $? The answer to an integral involved $\frac{\zeta'(2)}{\zeta(2)}$, but I am stuck trying to find this number - either to a couple decimal places or exact value.
In general the logarithmic deriative of the zeta function is the dirichlet series of the van Mangolt ... | By differentiating both sides of the functional equation$$ \zeta(s) = \frac{1}{\pi}(2 \pi)^{s} \sin \left( \frac{\pi s}{s} \right) \Gamma(1-s) \zeta(1-s),$$ we can evaluate $\zeta'(2)$ in terms of $\zeta'(-1)$ and then use the fact that a common way to define the Glaisher-Kinkelin constant is $\log A = \frac{1}{12} - \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1576985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
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Can't get to solve this word problem Price of lemon juice bottle is $4$ , price of orange juice bottle is $6$.
A buyer bought $20$ bottles and the total cost is $96$.
How many lemon bottles and orange bottles did the buyer get?
I know the answer but I don't know the steps to get to it.
| The way of getting this system is explained by others, I gonna help with solving that system!
$$
\begin{cases}
4x+6y=96 \\
x+y=20
\end{cases}\Longleftrightarrow
$$
$$
\begin{cases}
4x+6y=96 \\
y=20-x
\end{cases}\Longleftrightarrow
$$
$$
\begin{cases}
4x+6(20-x)=96 \\
y=20-x
\end{cases}\Longleftrightarrow
$$
$$
\begin{c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1577092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Diagonalizable matrices of finite odd order are the identity I want to prove that if $A^n = I$ for some odd $n \geq 1$, and $A$ is diagonalizable, then $A=I$. So if $A$ is diagonalizable, there exists $PAP^{-1}=D$ and also $PA^mP^{-1}=I$. To prove $A=I$, we need $D=I$ but how does it work?
| As JMoravitz noticed, it is false over the complex numbers. Let's assume then that we're talking about reals.
The polynomial $P(X) = X^n - 1$ annihilates $A$, so the eigenvalues of $A$ are roots of $P(X)$, ie are 1 (because $n$ is odd). As $A$ is diagonalizable, $D = 1$, and $A = 1$ follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1577183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Formal proof of Lyapunov stability I was trying to solve the question of AeT. on the (local) Lyapunov stability of the origin (non-hyperbolic equilibrium) for the dynamical system
$$\dot{x}=-4y+x^2,\\\dot{y}=4x+y^2.\tag{1}$$
The streamplot below indicates that this actually is true.
Performing the change of variables ... | *
*OP's streamplot suggests that the line $y=x-4$ is a flow trajectory. If we insert the line $y=x-4$ in OP's eq. (1) we easily confirm that this is indeed the case.
*From now on we will assume that $y\neq x-4$. It is straightforward to check that the function
$$H(x,y)~:=~\frac{xy+16}{x-y-4}-4 \ln |x-y-4| $$
is a fi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1577274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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The infinite sum of integral of positive function is bounded so function tends to 0
Let $f_n(x)$ be positive measurable functions such that
$$\sum_{n=1}^\infty \int f_n \lt \infty.$$
Show that $f_n \to 0$ almost everywhere.
Attempt:
Let $\displaystyle K = \sum_{n=1}^\infty\int f_n$ and $\displaystyle S_m = \sum_... | Define
$$
E_k=\left\{x:\limsup_{n\to\infty}f_n(x)\ge\frac1k\right\}
$$
For each $x\in E_k$, $f_n(x)\ge\frac1{2k}$ infinitely often. Therefore, for each each $x\in E_k$
$$
\sum_{n=1}^\infty f_n(x)=\infty
$$
Thus, if the measure of $E_k$ is positive,
$$
\int_{E_k}\sum_{n=1}^\infty f_n(x)\,\mathrm{d}x=\infty
$$
Therefore,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1577357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
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What's the general mathematical method to go about solving a substitution cipher? Here's a question from a professor's page:
Decipher the following simple-substitution cryptogram, in which every
letter of the message is represented by a number.
8 23 9 26 4 16 24 26 25 8 18 22 24 13 22 7 13 22 8 8 18 19 7
... | Judging by the hint, you need to compute the frequencies of various number-codes and try using the frequent-list-letters instead of them, checking yourself on small words to make sure they are ok... No formal way of doing it, I think
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Analytic functions on a disk attaining the maximum of absolute value at $0$. Find all functions $f$ which are analytic in the region $|z|\le 1$ and are such that $f(0)=3$ and $|f(z)|\le 3$ for all z such that $|z|<1$.
How to do this? I know the maximum principle, which says that the maximum value of $|f(z)|$ is attaine... | I believe due to the maximum modulus Theorem the answer is just f(z)=3 ....since the maximum is not attained on the boundary it must be a constant........
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1577531",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Liouville's Theorem Derivative solving? I believe this is a Liouville's Theorem problem, but I am unsure as to how to solve it.
Assume that $|f(z)|< |z^2 + 3z +1|$ for all $z$, and that $f(1) = 2$. Evaluate $f '(2)$, and explain your answer.
| You are quite correct that this is a Liouville's theorem problem. The function
$$g(z) := \frac{f(z)}{z^2 + 3z + 1}$$
(where $z^2 + 3z + 1 \ne 0$) is bounded and has an analytic extension to all of $\mathbb{C}$; hence, it is constant. This implies that $f(z)$ is a scalar multiple of $z^2 + 3z + 1$, and the fact that $f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1577662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Quartiles when all numbers are same I understand quartiles are to be used in large data sets. But for pedagogical purposes,
What would be the Quartile1,Median and Quartile3 for a set consisting of same numbers?
What would be the quartiles of 7 times 3?
| For corner cases like this, you need to consult your definition. All the definitions out there agree on large continuous data sets, but they differ in detail when individual observations matter. Wikipedia gives three methods for computing the quartiles. If your set is seven samples, each with a value 3, the only thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1577748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Calculation of double integral. Calculate the double integral:
$$\int_{0}^{\frac{\pi^3}{8}}\; dx \int _{\sqrt[3] x}^{\frac{\pi}{2}}\; \cos\frac{2x}{\pi y}\;dy\;.$$
Can someone hint how to approach this as we have to integrate with respect to y but y is in denominator. I think the right approach might be changing it int... | Change the order of integral. We have
$$\int_0^{\pi^3/8} \int_{\sqrt[3]{x}}^{\pi/2} f(x,y)dydx = \int_0^{\pi/2} \int_0^{y^3} f(x,y) dx dy$$
Hope you can finish it from here.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$F$ with Gateaux-derivative $A$, then $pF(u)=A(u)(u)$ Let $F:X\to \mathbb{R}$ Gateaux-differentiable with Gateaux-derivative $A:X\to X^*,$ ($X^*$) is the dual space pf $X$. Let $p\in\mathbb{R}$ such that $$F(\lambda u)=\lambda^pF(u)$$for all $u\in X$ and $\lambda >0$. I already proved the equality $A(\lambda u)=\lambda... | Fix $u \in X$. Define $g \colon \mathbf R \to \mathbf R$ by $g(\lambda) = F(\lambda u)$. Then - as $F$ is Gateaux differentiable - by the chain rule
$$ g'(\lambda) = A(\lambda u)(u) $$
On the other hand $g(\lambda) = \lambda^p F(u)$, hence
$$ g'(\lambda) = p\lambda^{p-1} F(u) $$
Now let $\lambda = 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1578031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is my proof of $C_G(H) \le N_G(H)$ correct? Let $x\in C_G(H)$. This means $xh = hx$ for all $h \in H$. Then $xH = Hx$ (This is that part I'm not so sure about). Hence, $x \in N_G(H)$, so that we have $C_G(H) \le N_G(H)$.
| Yes that's completely correct. To explicitly see the part you're unsure about, since $$xH := \{xh:h\in H\}$$
and $x\in C_G(H)$ so that $xh = hx$ for all $h\in H$, it is indeed the case that
$$xH = \{xh:h\in H\}=\{hx:h\in H\} =Hx.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1578125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Placing a small circle within a large one, trying to maximize the circumference converage Assume you have a circle of radius $1$.
We want to place a smaller circle of radius $r<1$ inside, such that as much of the outer circle's circumference is contained inside the smaller one's area.
How should we place it (how far f... | Let $\theta$ be the angle subtended by the circumference of the big circle (radius $=1$) intercepted by the smaller circle with radius $r(<1)$ then the length of intercepted circumference $$=\text{(radius)} \times \text{angle of aperture }=1\times \theta=\theta$$
Now, the length of common chord of small & big circles ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How is X/Y distributed when X and Y are uniformly distributed on [0,1]? Let $X$,$Y$ be uniformly distributed continuous random variables on [0,1]. How is the random variable $X/Y$ distributed?
| Step 1. Let $A_k$ be the area of the portion of the square $[0,1]\times [0,1]$ for which $y\leq k x$.
If $k\in[0,1]$, $A_k$ is the area of a right triangle with its perpendicular sides having lengths $1$ and $k$. If $k\geq 1$, $A_k$ is the area of the square minus the area of a right triangle with its perpendicular si... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $q, r \in \mathbb{R}, x \in \mathbb{R}^+$ then $(x^q)^r=x^{qr}$ I'm stuck on this exercise from Tao's Analysis 1 textbook:
show that if $q, r \in \mathbb{R}, x \in \mathbb{R}^+$ then $(x^q)^r=x^{qr}$.
DEF. (Exponentiation to a real exponent):
Let $x>0$ be real, and let $\alpha$ be a real number.
We define the quanti... | $ \newcommand{\seqlim}[1]{\lim\limits_{#1\to\infty}} $
I find a simple answer for this old question.
Show that $ (x^s)^r = x^{sr} $ for $ s\in\mathbb{Q},r\in\mathbb{R} $
$$(x^s)^r=\seqlim{n}{(x^s)^{r_n}}=\seqlim{n}{x^{sr_n}}=a^{sr},\quad [\seqlim{n}sr_n=sr] $$
Then we have
$$ (x^q)^r=\seqlim{n}(x^q)^{r_n}=\seqlim{n}x^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1578486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Compute $\sum_{k=-1}^{n=24}C(25,k+2)k2^k$
Compute $\sum_{k=-1}^{n=24}C(25,k+2)k2^k$
Well, I've found a solution for it, but I don't understand the line in the orange rectangle, can anyone exlain it please?
| $$\begin{align*}
\sum_{t=1}^{26}\binom{25}t(t-2)2^{t-2}&=\sum_{t=1}^{26}\left(\binom{25}tt2^{t-2}-\binom{25}t2\cdot2^{t-2}\right)\\
&=\sum_{t=1}^{26}\binom{25}tt2^{t-2}-\sum_{t=1}^{26}\binom{25}t2^{t-1}\\
&=\sum_{t=1}^{26}\left(\binom{25}tt2^{t-2}\cdot 2\cdot\frac12\right)-\sum_{t=1}^{26}\left(\binom{25}t2^{t-1}\cdot 2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $|f'(x)|\leq\sqrt{2MM''}$ Let $f:\mathbb{R}\to\mathbb{R}$ be twice differentiable with
$$|f(x)|\leq M, |f''(x)|\leq M'',\forall x\in\mathbb{R}$$
Prove that $|f'(x)|\leq\sqrt{2MM''},\forall x\in\mathbb{R}$
I am thinking about using Taylor's theorem:
For any $x\in\mathbb{R}$ and $a>0$, by Taylor's theorem $\ex... | A variant which is basically equivalent but with a slightly more geometric touch calculates a minimum bound for M in terms of $f'(x)$.
Assume without loss of generality that $f(a)\ge 0$ and $f'(a) > 0$.
For $x > a$ we find from integrating the lower bound of the second derivative that $$f'(x) \geq f'(a) - M''(x-a)$$
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sequence approaching infimum Given the fact that for a non-empty subset $K ⊂ V$ and $J : V → \mathbb R$ the number $\inf_{v \in K} J(v)$ exists, why does a minimizing sequence always exist? So I read somewhere this: A minimizing sequence of a function $J$ on the set $K$ is a sequence $(u_n)_{n∈ \mathbb N}$ such that $... | Hint:
Gap Lemma: Let $\varnothing \neq A \subseteq \mathbb R$ and $x = \inf A$. Given any $\varepsilon > 0$, there is an $a \in A$ such that $a - x < \varepsilon$.
Then, we can use the gap lemma with $\varepsilon_n = \frac{1}{n}$ to get a sequence $(a_n)_{n \in \mathbb N}$ such that $\lim_{n\to\infty} a_n = x$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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For any events A,B,C is the following true? Is the following statement true? How? I'm having trouble seeing whether not it is true or false.
$$P(A\mid B) = P(A\mid B \cap C)P(C\mid B) + P(A\mid B \cap C')P(C'\mid B)$$
| It is true, generally speaking:
\begin{align}
P(A \mid B) & = \frac{P(A, B)}{P(B)} \\
& = \frac{P(A, B, C)+P(A, B, C')}{P(B)} \\
& = \frac{P(A, B, C)}{P(B)}+\frac{P(A, B, C')}{P(B)} \\
& = \frac{P(A, B, C)}{P(B, C)} \cdot \frac{P(B, C)}{P(B)}
+ \frac{P(A, B, C')}{P(B, C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1578891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Simplifying Dervatives of Hyperbolic functions Last minute Calc I reviews have me stumbling on this question
$$D_x\left[\frac {\sinh x}{\cosh x-\sinh x}\right] $$
I've solved the derivative as
$$ y' = \frac{\cosh x}{\cosh x-\sinh x} -\frac{\sinh x(\sinh x-\cosh x)}{(\cosh x-\sinh x)^2} $$
which is consistent with an o... | Hint
Bringing back to exponential functions simplifies it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Convergence of Definite Improper Integrals of the Form $1/x$ Given a simple integral of the form: $$ \int ^1 _{-1} \frac{1}{x} \, dx =\lim _{a\rightarrow 0} \int ^1 _a \frac{1}{x} \, dx + \int ^a _{-1} \frac{1}{x} \, dx$$
Is it possible to say that this integral converges? I was told explicitly by my professor that thi... | $$
\int_{-1}^{-a} \frac{dx} x + \int_a^1 \frac{dx}x = 0 \to 0.
$$
However:
\begin{align}
& \int_{-1}^{-2a} \frac{dx} x + \int_a^1 \frac{dx}x = \Big(\log|{-2a}| - \log |{-1}|\Big) + \Big( \log 1 - \log a \Big) \\[10pt]
= {} & \log(2a) - \log a = \log \frac{2a} a = \log 2 \approx 0.693\ldots
\end{align}
As $a\downarrow ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579176",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Differentiating $x^2=\frac{x+y}{x-y}$ Differentiate:
$$x^2=\frac{x+y}{x-y}$$
Preferring to avoid the quotient rule, I take away the fraction:
$$x^2=(x+y)(x-y)^{-1}$$
Then:
$$2x=(1+y')(x-y)^{-1}-(1-y')(x+y)(x-y)^{-2}$$
If I were to multiply the entire equation by $(x-y)^2$ then continue, I get the solution. However, if ... | It's worth noting that you haven't actually avoided the quotient rule, at all. Rather, you've simply written out the quotient rule result in a different form. However, we can avoid the quotient rule as follows.
First, we clear the denominator to give us $$x^2(x-y)=x+y,$$ or equivalently, $$x^3-x^2y=x+y.$$ Gathering the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Random Walk on a Cube A particle performs a randowm walk on the vertices of a cube. At each step it remains where it is with probability 1/4, or moves to one of its neighbouring vertices each having probability 1/4.
Let A and D be two diametrically opposite vertices. If the walk starts at A, find:
a. The mean number ... | The critical thing to figure is the probability $p$ it gets to D before it returns to A. Then you have a Markov chain with states $A,D$ and probabilities $p$ for $A \to D$ and $D \to A$ and $1-p$ for $D \to D$ and $A \to A$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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convert differential equation to Integral equation $$ y''(x) + y(x) = x$$
with b.v conditions $$ y(0) = 1, y'(1) = 0 $$
Integrating $$ y'(x) - y'(0) + \int \limits _0 ^x y(x)dx = \frac {x^2} 2$$
$ let y'(0) = c_1 $
$$ y'(x) - c_1 + \int \limits _0 ^x y(x)dx = \frac {x^2} 2$$
$$ y'(x) = c_1 - \int \limits _0 ^x y(x)dx +... | First integral equation must be
$$
y'(x)-y'(0)+\int_{0}^{x}y(s)ds=\frac{x^2}{2}.
$$
Finally you will arrive at
$$
y(x)=y(0)-\frac{1}{2}x+Ax-\int_{0}^{x}\int_{0}^{s}y(t)dtds+\frac{x^{3}}{6},\quad A=\int_{0}^{1}y(s)ds
$$
which satisfies your boundary values.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Coin flipping combined with exponential distribution Let $Z\sim \exp(1)$. Let $X$ be a new random variable (rv) defined as follows: We flip a coin. If we get head, than $X=Z$, and if we get tail than $X=-Z$.
I'm trying to figure whether $X$ is a discrete, continuous or mixed type rv, and to calculate its CDF and PDF (i... | This is a mixture distribution $$F_X(x)=\frac12F_Z(x)+\frac12F_{-Z}(x)$$ for $x\in \mathbb R$, where the weights $1/2$ correspond to the result of the coin flip (assuming a fair coin, $1/2$ head and $1/2$ tail). Now $$F_{-Z}(x)=P(-Z\le x)=P(Z\ge -x)=1-F_Z(-x)$$ so that you can write $F_X(x)$ for $x\in \mathbb R$ as $$F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Show by induction that $(n^2) + 1 < 2^n$ for intergers $n > 4$ So I know it's true for $n = 5$ and assumed true for some $n = k$ where $k$ is an interger greater than or equal to $5$.
for $n = k + 1$ I get into a bit of a kerfuffle.
I get down to $(k+1)^2 + 1 < 2^k + 2^k$ or equivalently:
$(k + 1)^2 + 1 < 2^k * 2$.
A b... | $\displaystyle
2^n = (1+1)^n > \binom{n}{0}+\binom{n}{1}+\binom{n}{2}+\binom{n}{3}>n^2+1
$
iff $n>5$. (*)
The case $n=5$ is proved by inspection.
(*) This seems to be a cubic inequality but it reduces to an easy quadratic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579616",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
The value of double integral $\int _0^1\int _0^{\frac{1}{x}}\frac{x}{1+y^2}\:dx\,dy$? Given double integral is :
$$\int _0^1\int _0^{\frac{1}{x}}\frac{x}{1+y^2}\:dx\,dy$$
My attempt :
We can't solve since variable $x$ can't remove by limits, but if we change order of integration, then
$$\int _0^1\int _0^{\frac{1}{x}... | $$\begin{align}\int_{0}^{1}\int_{0}^{\frac{1}{x}}\frac{x}{1+y^2}\space\text{d}x\text{d}y&=
\int_{0}^{1}\left(\int_{0}^{\frac{1}{x}}\frac{x}{1+y^2}\space\text{d}x\right)\text{d}y\\&=\int_{0}^{1}\left(\frac{1}{1+y^2}\int_{0}^{\frac{1}{x}}x\space\text{d}x\right)\text{d}y\\&=\int_{0}^{1}\left(\frac{\left[x^2\right]_{0}^{\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
If $x+y+z=6$ and $xyz=2$, then find $\cfrac{1}{xy} +\cfrac{1}{yz}+\cfrac{1}{zx}$
If $x+y+z=6$ and $xyz=2$, then find the value of $$\cfrac{1}{xy}
+\cfrac{1}{yz}+\cfrac{1}{zx}$$
I've started by simply looking for a form which involves the given known quantities ,so:
$$\cfrac{1}{xy} +\cfrac{1}{yz} +\cfrac{1}{zx}=\cfra... | $$\cfrac{1}{xy} +\cfrac{1}{yz} +\cfrac{1}{zx}=\cfrac{yz\cdot zx +xy \cdot zx +xy \cdot yz}{(xyz)^2}=\frac{x+y+z}{xyz}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579781",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Second solution of ODE $xy''+y'-y=0$? Suppose we have the following equation $$xy''+y'-y=0$$ where it has a regular singular point at $x=0$ and we want to derive the series solution near $x=0$. We write the ODE in canonical form: $y''+\frac{1}{x}y'-\frac{1}{x}y=0$ and then we set $$y=\sum_{n=0}^{\infty}a_nx^{n+r}.$$ Th... | $xy''+y'-y=0$ is an ODE of the Bessel kind. Some transformations should be necessary to bring it to the standard form. But it isn't what is asked for.
The first solution found by johnny09 is in fact a first Bessel function :
$$y_1=\sum_{n=0}^{\infty}\frac{x^n}{(n!)^2}=J_0 (2\sqrt{x})$$
$J_0(X)$ is the modified Bessel ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579870",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove inverse function theorem (1 dimensional case)
Let $I$ be an open interval in $\mathbb{R}$ that contains point $a$
and let $f:I\rightarrow \mathbb{R}$ be a continuously differentiable
function such that $f'(a)\ne 0$. Then there exist open intervals
$U\subset I$ and $V$ in $\mathbb{R}$ such that restricti... | Let $x$ and $x+h$ belong to the domain of $f$ where inverse exists and $f(x)=y$ and $f(x+h)=y+k$
Also $f^{-1}(y+k)=x+h$ and $f^{-1}(y)=x$
Limit as $h$ tends to zero $\frac {f^{-1}(y+k)-f^{-1}(y)}{k}$
Now $f^{-1}(x+h)$ is $y+k$ and $f^{-1}(x)$ is $y$
so the limit becomes
$= \frac {h}{f(x+h)-f(x)}$ and now take h to the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1579972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why is $-145 \mod 63 = 44$? When I enter $-145 \mod 63$ into google and some other calculators, I get $44$. But when I try to calculate it by hand I get that $-145/63$ is $-2$ with a remainder of $-19$. This makes sense to me, because $63\cdot (-2) = -126$, and $-126 - 19 = -145$.
So why do the calculators give that ... | We say $a \equiv b$ (mod n) if $a-b$ is a multiple of $n$. So notice that:
$$-145-44 = -189 = -3(63)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 1
} |
Is this a proper alternative way for math model for TSP(Travelling Salesman Problem)? I have never seen a model that uses indexing in any article.So I have decided to publish it to be sure. I think indexing model is more suitable for generalizing the model than the subtour elimination method.I have tested the model usi... | A bunch of formulations is described in
A.J. Orman, H.P. Williams, A Survey of Different Integer Programming
Formulations of the Travelling Salesman Problem, Link
I did not check if your version is in there.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $A \subset \mathbb{R}$, $x_0 \in \mathbb{R}$ and $h$ is an affine map, what does $h^{-1}(x_0 - A)$ mean? I am reading a book, and it suddenly says:
A distribution $\nu$ is symmetric around a point $x_0 \in \mathbb{R}$ if $h(\nu) = \nu$ where $h$ is the affine map given by $h(x) = 2x_0 - x$. As $h^{-1}(x_0 - A) = x_... | The affine map $h$ is invertible, with inverse $h^{-1}(x) = h(x)$. So $$h^{-1}(x_0 - A) = h(x_0 - A) = 2x_0 - (x_0 - A) = x_0 + A.$$
Added Later: The question has been edited so that $A \subset \mathbb{R}$ rather than $A \in \mathbb{R}$.
Now $x_0 - A$ denotes the set $\{x_0 - a \mid a \in A\}$ and $x_0 + A$ denotes t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Eigenvectors of special matrix with characteristic polynomial I have a given monic polynomial $P(s)=\sum\limits_{k=0}^N a_ks^k $ of degree $N$, and I construct this matrix which has $P(s)$ for a characteristic polynomial:
$$ M = \begin{bmatrix}
-a_{N-1} & -a_{N-2} & -a_{N-3} & \cdots & -a_2 & -a_1 & -a_0 \\
1... | This matrix is very similar to the Comnpanion matrix, except you have the matrix transposed and flipped upside-down.
The eigenvectors of this matrix should be in the form $\begin{bmatrix}\lambda^{n-1}\\\lambda^{n-2}\\ \vdots \\ \lambda \\ 1\end{bmatrix}$ where $\lambda$ is a root of the characteristic polynomial.
EDIT... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $r\in\mathbb{Q}[\sqrt2]$. Show $\phi(r) = r$ if and only if $r\in\mathbb{Q}$
Let $\mathbb{Q}[\sqrt2]=\{a+b\sqrt2 \mid a,b\in\mathbb{Q}\}$ and define $\phi:R \rightarrow R$ by $\phi(a+b\sqrt2)=a-b\sqrt2$. Show $\phi(r) = r$ if and only if $r\in\mathbb{Q}$.
My approach is that if $r\in\mathbb{Q}$, then the $b\sqrt2... | One direction looks fine, but needs a bit more justification. (You may find a contrapositive approach easier.) For the other, set $$a-b\sqrt2=:\phi(a+b\sqrt2)=a+b\sqrt2$$ and conclude from there.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
$(2n)$-regular graphs don't have bridges Let $G$ be a $k$-regular graph where $k$ is even. I want to prove $G$ doesn't have a bridge. I was thinking I could prove by contradiction and assume that G has a bridge. But from there I have no idea where to go.
| A $2k$-regular graph is Eulerian so removing an edge results in a graph with an Eulerian walk. You could also use the 2-factor theorem to show that the endpoints of any edge a lie on a cycle and therefore there exist at least two paths between them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
average euclidean distance between vector of coin flips I have a biased coin with the probability of flipping heads as $p$. I have a room of $n$ people and they each have this same coin. I ask everybody to flip their coins, and record their results as an $n$-dimensional vector of ones (heads) and zeros (tails) like [1 ... | Let $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ be i.i.d random variables with Bernoulli distribution with parameter $p$. Then
\begin{align*}
\mathbb{E} \left[\sum_{i=1}^n (X_i - Y_i)^2\right]
&= \sum_{i=1}^n \mathbb{E} \left[ (X_i - Y_i)^2 \right] \\
&= \sum_{i=1}^n \mathbb{E} \left[ X_i^2 - 2X_iY_i + Y_i^2\right] \\
&= \sum_{i=1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show $a^p \equiv b^p \mod p^2$ I am looking for a hint on this problem:
Suppose $a,b\in\mathbb{N}$ such that $\gcd\{ab,p\}=1$ for a prime $p$. Show that if $a^p\equiv b^p \pmod p$, then we have: $$a^p \equiv b^p \pmod {p^2}.$$
I have noted that $a,b$ are necessarily coprime to $p$ already, and Fermat's little theorem... | You could generalize this further. Here is one of the Lifting the Exponent Lemmas (LTE):
Define $\upsilon_p(a)$ to be the exponent of the largest prime power of $p$ that divides $a$.
If $a,b\in\mathbb Z$, $n\in\mathbb Z^+$, $a\equiv b\not\equiv 0\pmod{p}$, then $$\upsilon_p\left(a^n-b^n\right)=\upsilon_p(a-b)+\upsilon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Which one of the given sequences of functions is uniformly convergent $?$ Which one of the given sequences of functions is uniformly convergent $?$
$$A.\ \ f_n(x)=x^n;x\in[0,1].$$
$$B.\ \ f_n(x)=1-x^n;x\in\left[{1\over2},1\right].$$
$$C.\ \ f_n(x)={{1}\over{1+nx^2}};x\in\left[0,{1\over 2}\right].$$
$$D.\ \ f_n... | Noting that the uniform limit of continuous functions must be continuous too, only $D$ is possible (as your (correct) calculations have shown).
For the reason why D is continuous on the give interval, you can use Dini's theorem as is indicated in the previous answer. Or more directly, note that
$$|f_n(x)-f_m(x)|=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Counting to 21 game - strategy? In a game players take turns saying up to 3 numbers (starting at 1 and working their way up) and who every says 21 is eliminated. So we may have a situation like the following for 3 players:
Player 1: 1,2
Player 2: 3,4,5
Player 3: 6
Player 1: 7,8,9
Player 2:10,11
Player 3:13,14,15
Playe... | You can use a strategy but only by the means on where you stand in the order of people, there is complex maths involved in probabilities but if numbers chosen ($1$-$3$) by each player are randomised then certain positions have a much lower chance of being forced into saying “$20+1$”
This was figured out using a simulat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Density of a set in $\mathbb R$ Is the set $$K=\{\sqrt{m}-\sqrt{n} : m,n \in \mathbb{N}\}$$ dense in $\mathbb{R}$? It would be appreciated if someone can help me.
| Take $u,v\in \mathbb{R}$ such that $u<v$. Since $\sqrt{m}-\sqrt{m-1}\to 0$ as $m\to +\infty$, there exists $m_0$ such that $0<\sqrt{m}-\sqrt{m-1} <v-u$ for $m\geq m_0$. Now take an $n_1$ such that $u+\sqrt{n_1}>\sqrt{m_0-1}$, and let $m_1$ be such that $\sqrt{m_1-1}\leq u+\sqrt{n_1}<\sqrt{m_1}$. We have $u<\sqrt{m_1}-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why do we take the closure of the support? In topology and analysis we define the support of a continuous real function $f:X\rightarrow \mathbb R$ to be $ \left\{ x\in X:f(x)\neq 0\right\}$. This is the complement of the fiber $f^{-1} \left\{0 \right\}$. So it looks like the support is always an open set. Why then do w... | Given a scheme $X$ there are two notions of support $f\in \mathcal O(X)$:
1) The first definition is the set of of points $$\operatorname {supp }(f)= \left\{ x\in X:f_x\neq 0_x\in \mathcal O_{X,x}\right\}$$ where the germ of $f$ at $x$ is not zero.
This support is automatically closed: no need to take a closure.
2) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1580939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Find the maximum of the $S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}|$ Let $a_1,a_2,\cdots, a_{31} ;b_1,b_2, \cdots, b_{31}$ be positive integers such that
$a_1< a_2<\cdots< a_{31}\leq2015$ , $ b_1< b_2<\cdots<b_{31}\leq2015$ and $a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.$
Find the maximum value of $S=|a_1-b_1|+|a... | This is an outline of my solution.
Suppose $a_1<a_2<a_3<\dots<a_{31},b_1<b_2<b_3<\dots<b_{31}$ satisfies all conditions and maximises the expression.
We sort the ordered pairs $(a_i,b_i)$ in non-decreasing $a_i-b_i$. Let the sorted sequence be relabelled $(a_{\sigma(i)},b_{\sigma(i)})$
Then we generate another sequence... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1581025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 1,
"answer_id": 0
} |
Number of unique faces throwing a dice We select with replacement $N$ times a random number with $M$ options where $M\gg N$. How do I calculate the probability of having $n$ unique numbers $n\ge N$ per selection?
For instance, throwing three times a six-faced dice means $N=3$ and $M=6$. What is the probability of havin... | This is related to Stirling numbers of the second kind and the probability you are looking for is $$\dfrac{\frac{M!}{(M-n)!}S_2(N,n)}{M^N }$$
So in your example where $M=6$ and $N=3$ and $S_2(3,1)=1,S_2(3,2)=3,S_2(3,3)=1$, you would have
*
*$\Pr(n=1)=\dfrac{\frac{6!}{5!}\times 1}{6^3} = \dfrac{1}{36}$
*$\Pr(n=2)=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1581120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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