Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Probability of a probability to happen I'm trying to solve this problem but I'm struggling to.
A university has 300 students enrolled. You sample 30 students and see that 20 are boys and 10 are girls. If the university had 150 male students and 150 females, what would be the probability of your observation? More genera... | To solve this problem you can use the Hypergeometric distribution. With the variables
*
*$N$ the population size ($300$)
*$K$ number of success states in the population ($150$)
*$n$ the number of draws ($30$)
*$k$ the number of observed successes (boys, $20$)
Then if $X$ is the number of boys drawn in the sampl... | {
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Product of units modulo m Help!!! I can find units in a given $Z_m$ manually by testing out every single value which is relative to m,,, but in general I do not know a way how to find such units... hence I cant answer this question: Prove that the product of units mod m is congruent to $\pm 1$ mod m.
Any idea?
| The product of all units mod $m$ can be written as $P=\prod_{x^2\ne1} x \prod_{y^2=1} y$.
The first product is $1$ because each $x$ is paired with $x^{-1}$ and they are different.
Therefore, $P=y_1 \cdots y_n$ with $y_i^2=1$.
If $n=1$, then $P=1$ because $y=1$ is certainly there.
If $n=2$, then $P=-1$ because $y=-1$ is... | {
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Show that $\exists m \in(k,\ell):f''(m)+f(m)=0$ Let $f$ be twice differentiable on $\Bbb R$ and $f'(x)\not=0$, $f'(k)=f(\ell)$ and $f'(\ell)=f(k)$. Show that $\exists m \in(k,\ell):f''(m)+f(m)=0$.
The only thing that I was able to do is to show that $\exists n \in (k,\ell):f'(n)-f(n)=0$. I'd like a hint.
| Hint: consider
$$
h(x)=(f(x))^2+(f'(x))^2
$$
and see what the mean value theorem implies.
| {
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Prove that every terms of a sequence defined by a recurrence relation is a perfect square.
I would be happy if you let me know how to tackle this problem.
Thanks.
| The idea is to look for a linear recurrence relation among the square roots. Say we have a relation $b_n=c_1b_{n-1}+c_2b_{n-2}$. Assuming the roots of the characteristic polynomial are distinct, the general formula should be $b_n=d_1\lambda_1^n+d_2\lambda_2^n$ where $d_1,d_2$ are constants and $\lambda_1,\lambda_2$ a... | {
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Can we say Dirac delta function is zero almost surely? It is known that $\delta(x) = \infty $ if $x = 0$ and $=0$ if $x\ne 0 $ and we also know that $\int_{-\infty}^{\infty}\delta(x)dx=1$. However, if we consider a Lebesgue integration, $\delta(x)$ is zero almost surely so that we can get $\int_{-\infty}^{\infty}\delt... | Because that is a bad "definition" of the "Dirac function"! The "Dirac function" is not a function at all, it is a "distribution" or "generalized function", a functional that assigns a number to every function. Specifically, the "Dirac function" assigns the number f(0) to every function, f.
| {
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Find all integers $x$ such that $\frac{2x^2-3}{3x-1}$ is an integer. Find all integers $x$ such that $\frac{2x^2-3}{3x-1}$ is an integer.
Well if this is an integer then $3x-1 \mid 2x^2-3$ so $2x^2-3=3x-1(k)$ such that $k\in \mathbb{Z}$ from here not sure where to go I know that it has no solutions I just can't see th... | Let $k=3x-1$ then $$3x\equiv_k 1$$ and $$2x^2-3 \equiv _k0$$ Multiplying last equation with 9 we get $$0\equiv _k2(3x)^2-27\equiv _k -25$$ So $$3x-1\mid 25 \implies 3x-1\in\{\pm 1,\pm5,\pm 25\}$$ and thus $$x\in\{0,2,-8\}$$
| {
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Solving $n\times n$ determinant using triangular shape I have just started learning to solve nth order determinants by getting it into the triangular shape ( in this way the determinant is equal to the multiple of main or additional diagonal + the determination of the sign ). I have solved a couple of easy ones, but go... | First, you can subtract the last line from the others which gives you :$$\begin{vmatrix}
2 & 0 &\ldots&0&-3 \\
0 & 3 &\ddots&\vdots& \vdots \\
\vdots &\Large{0}&\ddots&0& \vdots \\
0&\ldots&0&3&-3\\
3&\ldots&\ldots&3&6
\end{vmatrix}$$
Then subtract $\frac{3}{2} L_{1}$ from $L_{n}$:
$$\begin{vmatrix}
2... | {
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What is $\int \frac{x}{x^5-7} dx$? I have tried out many trigonometric substitution like $x=\sin^{\frac{2}{5}}z$. But it did not work.
| To make the problem more general, consider
$$I_n=\int\frac x {x^n-A}\,dx=A^{\frac 2n-1}\int\frac t {t^n-1}\,dt=A^{\frac 2n-1}\int\frac t {\prod_{i=1}^n(t-r_i)}\,dt$$ where $r_i$ are the roots of unity. Using partial fraction decomposition, you will end with
$$I_n=A^{\frac 2n-1}\sum _{i=1}^n\int \frac {\alpha_i}{t-r_i}\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Verifying a long polynomial equation in (the reciprocal of) the Golden Ratio I'm trying to show that the following equation holds true:
$$4\sigma^{12}+11\sigma^{11}+11\sigma^{10}+9\sigma^9+7\sigma^8+5\sigma^7+3\sigma^6+\sigma^5+\sigma^4+\sigma^3+\sigma^2+\sigma = 1 + 2\sigma$$
where $\sigma$ is the reciprocal of the g... | Work with $x$ instead of $\sigma$. You have $$x^2=-x+1$$ so $$x^3=x-x^2=+2x-1$$ $$x^4=x^2-x^3=-3x+2$$ and you will find (without surprise) that the Fibonacci numbers appear, so you can write down $$x^5=+5x-3, x^6=-8x+5 \dots$$ and this may be an efficient way of reducing to a linear expression.
| {
"language": "en",
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A Square Inside A Triangle (but with a twist) I have a right $\triangle ABC$ and I want to find the side length of one of the legs, $x$, when the square is at its max area, as shown in
I am given that $AB + BC$ is $10$.
Here's what I tried so far:
*
*I found ratios between the sides using similarity, but I wasn't a... | The area of the square would be maximal, when $l$ would be maximal.
Let $AB=a$ and $AC=b$.
Thus, $$\frac{a-l}{a}=\frac{l}{b},$$
which by AM-GM gives:
$$l=\frac{ab}{a+b}\leq\frac{\left(\frac{a+b}{2}\right)^2}{a+b}=\frac{5}{2}.$$
The equality occurs for $a=b=5$.
Thus, the area of the square gets a maximal value for $x=5$... | {
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Dual codes over finite commutative rings Let $R$ be a finite commutative ring and $C$ be a linear code of length $n$ over $R$, i.e., an $R$-submodule of $R^n$. We define $C^{\perp}$ by
$C^\perp=\{v\in R^n~|~[v,w]=0~~\text{for all}~w\in C\}$
where $[v,w]=\sum_{i=1}^nv_iw_i\in R$ is the Euclidean inner product. I read in... | You're right. It's not true, for example, using the ring $R=F_2[x,y]/(x^2,xy,y^2)$, which is not quasi-Frobenius.
If you take $C=(x)\subseteq R^1$, then $C^\perp=(x,y)$ and $C^{\perp\perp}=(x,y)$. A ring which is quasi-Frobenius cannot suffer this problem.
You could make an example for any length $n$ over $R$ by using ... | {
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intuition behind discrepancy in expected number of coin tosses Yesterday evening I read a very interesting article on prime numbers which contained the following paragraph on coin tosses:
If Alice tosses a coin until she sees a head followed by a tail, and
Bob tosses a coin until he sees two heads in a row, then on ... | Alice and Bob toss their coins until they see their first head. The expected tosses until this happens is the same for each. Then things are different.
Alice just needs to continue until she finally sees a tail.
However, if Bob doesn't immediately see a head on the next throw, he needs to start over from the beginn... | {
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finding remainder by using fermat's little theorem or euler's totient I am trying to carry on for the following equation but I stuck with a big number that still needs to be smaller to calculate without calculator.
$\ 331^{51}\mod 49$
Since the $\phi(49)=42$, I carried on the problem as follows:
$=\ 331^{42}.331^{9}\mo... | Hint:
Using binomial theorem,
$$(-12)^7=(2-14)^7\equiv2^7\pmod{49}$$
and $$12^2\equiv-3\pmod{49}$$
| {
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How to prove $(a_1+a_2+\dots a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\right)\ge n^2$? let $a_1,a_2,\dots ,a_n$ be a positive real numbers . prove that
$$(a_1+a_2+\dots a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\right)\ge n^2$$
how to prove this inequality ..is this result prove by Induc... | Another proof: See Art of Problem Solving about Chebyshev's inequality.
There are two different versions of it. Use the second version.
If $a_1\ge a_2\ge \ldots \ge a_n$ and $b_n\ge b_{n-1}\ge \ldots \ge b_1$, then $$n\left(\sum_{i=1}^{n} a_ib_i\right)\le \left(\sum_{i=1}^{n}a_i\right)\left(\sum_{i=1}^{n}b_i\right)$$
I... | {
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24 different complex numbers
There are $24$ different complex numbers $z$ such that $z^{24}=1$. For how many of these is $z^6$ a real number?
This is one of the AMC problems from this year. I've been trying to solve it, but I couldn't and a peek at the answers (not recommended, I know) talked about Euler's theorem et... | Let's say $w=z^6$. We know that $w^4=1$, so $w=\pm 1,\pm i$. Each of these four numbers has $6$ distinct sixth roots.
Does that help?
| {
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Is this proof style legitimate? Normally for direct proof of equality we have the form:
Prove
$$a = b$$
Proof (Style):
We start with $a$ (or $b$) and show through a sequence of logically connected steps that $a$ is $b$ (or the other way around). $_{_\square}$
But, since I'm not great with proofs I just wanted to... | Instead of starting with $x=x$, start with $a=a$ or $b=b$
But yes, sometimes that's exactly what you need to do. For example, suppose I have that $b=a$ and I want to show that $a=b$. Then I can do:
*
*$b=a \quad $ Premise
*$b=b \quad $ = Intro
*$a=b \quad$ = Elim 1,2
| {
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How to show that $\sum_{n=1}^{\infty}{2^n\over (2^n-1)(2^{n+1}-1)(2^{n+2}-1)}={1\over 9}?$ How to show that
$${2\over (2-1)(2^2-1)(2^3-1)}+{2^2\over (2^2-1)(2^3-1)(2^4-1)}+{2^3\over (2^3-1)(2^4-1)(2^5-1)}+\cdots={1\over 9}?\tag1$$
We may rewrite $(1)$ as
$$\sum_{n=1}^{\infty}{2^n\over (2^n-1)(2^{n+1}-1)(2^{n+2}-1)}={1\... | Hint: We can express
$$\frac{1}{2^n-1}=\sum_{k=1}^{\infty} \frac{1}{2^{kn}}.$$
| {
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Why is the degree of a rational map of projective curves equal to the degree of the homogeneous polynomials? Let $C_1 \subseteq \mathbb{P}^m$ and $C_2 \subseteq \mathbb{P}^n$ be projective curves, and let $\phi : C_1 \rightarrow C_2$ be a nonconstant rational map given by $\phi = \left[ f_1, \ldots, f_n \right]$ for ... | The fact is that deg$(\phi)$ can be anything between $1$ and $d^n$. $1$ is the birational case, and $d^n$ if the sequence $f_0,\cdots,f_n$ is a regular sequence.
For example degree of $(s^5:s^3t^2:s^2t^3:t^5):\mathbb{P}^1\to \mathbb{P}^3$ is $1$ and degree of $(s^6:s^4t^2:s^2t^4:t^6):\mathbb{P}^1\to \mathbb{P}^3$ is $... | {
"language": "en",
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A question on generic point and induced map between spectrum We let $\varphi:A \rightarrow A'$ be a ring homomorphism and $\varphi^*:Spec\,A' \rightarrow Spec\,A$ the corresponding induced map between spectrum. Now we suppose that $Spec\,A'$ is irreducible and let $x$ be its (unique) generic point (i.e. $\overline{\{x\... | Your first claim about $\overline{\phi^{\ast}(\textrm{Spec}(A'))}$ being irreducible is a general fact from topology: a continuous image of an irreducible space is irreducible, and the closure of irreducible subset remains irreducible.
Your next claim is false without some further assumptions.
| {
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Prove that: $\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\ge\frac{2}{3}$ I've got three inequalities:
$\forall n\in\mathbb N:$
$$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n} \ge\frac{1}{2}$$
$$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n} \ge\frac{7}{12}$$
$$\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2n}\ge\frac{2}{3}... | Hint: $$\frac{1}{n}\ge\ln(n+1)-\ln(n)$$
| {
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“D-module” or “$D$-module”? Disclaimer: This question is not transcendental at all, so go easy on me.
Starting with a (for simplicity) commutative unital ring $R$, we define a $R$-module. Obviously, since the name of the ring was italicized, the same must be done each time we write “$R$-module”, but if we are talking a... | I am working with $D$-modules and I as you can see I am using the italic version. That is because it feels more natural to use the italic version as for other rings $R$ and $R$-modules. I am only working with one specific ring of differential operators though, which is the Weyl-Algebra $D = D_n$. So for me a $D$-module... | {
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Notion of uniquely transportable Categories I do not fully understand the notion of uniquely transportable categories. In the “Abstract and Concrete Categories” book one says that, for example, concrete category $(\mathbf G\mathbf r\mathbf p,U)$ over $\mathbf X$ with forgetful functor is uniquely transportable.
But tak... | Given any group $D$ and a set $E$ with a bijection $f:D\to E$, make $E$ a group using the operation $*$ defined by $x*y=f(f^{-1}(x)\cdot f^{-1}(y))$, where $\cdot$ is the multiplication of $D$. The fact that $(E,*)$ is a group follows easily from the fact that $(D,\cdot)$ is a group and $f$ is a bijection. And essent... | {
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The difference between ∈ and ⊂ I had a task where I had to figure if the argument was true or not.
$A=\{ n ∈ ℤ \mid n^2 < 5 \}, \quad B=\{ 7, 8, \{2\}, \{2, 7, 8\}, \{\{7\}\} \}$
The first one was $\{-1, 2\} ∈ A$ and the answer to this was not true
since the set is not an integer.
The second one was $\{-1, 2\} ⊂ A$ a... | $\in$ stands for "belongs to".For eg. an element may belong to a set.
$\subset$ is the symbol for subset .For eg. one set can be a subset of another set if all elements of that set is included in the later set.
You must be careful on what you are applying $\in$ or $\subset$.
| {
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Let $p$ be a prime that decomposes completely in $O_K$. Show that $\#\text{Hom}(O_K, \mathbb{F}_p)=n$ Let $K$ be a number field of degree $n$. We say a prime $p$ decomposes completely in $O_K$ if $pO_K = \mathfrak{p}_1...\mathfrak{p}_n$ for some prime ideals $\mathfrak{p}_i$. I want to show that if $p$ is a prime that ... | Here are a few off the cuff comments that may help you on your way. First under any map $\mathcal{O}_K \rightarrow \mathbb{F}_p$, the set $p\mathcal{O}_K$ is sent to zero. So look at $\mathcal{O}_K/p\mathcal{O}_K$. This is isomorphic to $$\prod \mathcal{O}_K/\mathfrak{p}_i.$$ Maybe this is a good point for me to stop.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Density of a subset of $\mathbb{R}$ Let $\alpha$ be an irrational number.
Is it true that the set $\{\frac{m}{\alpha} + n | m,n \in \mathbb{Z} \}$ dense in $\mathbb{R}$?
If it is, how do we prove it?
| Yes it is
Lemma: Let $H$ be a subgroup of $\mathbb{R}$, then $H$ is dense if and only if there exists a convergence sequence $h_n\rightarrow \xi$ for some $\xi\in \mathbb{R}$ and $h_n\in H$ with the property that $h_n$ is not eventually constant.
Proof: The first direction is obvious, For the other direction, let $h_n\... | {
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Though $0^0$ is an indeterminate form of limits, is it undefined? I’ve done a my research, though I have not been able to find an adequate explanation as to whether or not $$0^0$$ exists as a real number, and why or why not? I must credit this question to “Question on the controversial ‘undefined’ $0^0$.”
This Wikipedi... | Zero to zeroth power is often said to be "an indeterminate form", because it could have several different values.
Since $x^0$ is $1$ for all numbers $x$ other than $0$, it would be logical to define that $0^0 = 1$.
But we could also think of $0^0$ having the value $0$, because zero to any power (other than the zero p... | {
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binary relation that is both symmetric and irreflexive I was asked to find a binary relation on $A$ that is symmetrical and irreflexive, and which is also a function from $A$ to $A$ where $A=\{1,2,3,4\}$
so i don't know if its correct but i came up with these $\{<1,2>, <1,3>, <1,4>, <2,1>, <2,3>, <2,4>, <3,1>, <3,2>, <... | hint:
rephrase as find a function which has no fixed points and whose square is the identity.
Your example is not a function since it relates the same element to multiple other elements.
| {
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Linear feedback shift register sequences cipher breaking The premise here is that somehow it is known that a cipher is linear feedback shift register sequence as well as a bit of plaintext and ciphertext, such that it can be found that the first several bits of key are 1 0 1 0 0 1 1 1 0 1 0 0 1 1 1. So then it's possi... | 101001110100111 is really 1010011 being repeated, so then m=7.
$x_{1+7} = c_{0}x_{1+0}+c_{1}x_{1+1}+c_{2}x_{1+2}+c_{3}x_{1+3}+c_{4}x_{1+4}+c_{5}x_{1+5}+c_{6}x_{1+6}$
$x_{8}=c_{0}x_{1}+c_{1}x_{2}+c_{2}x_{3}+c_{3}x_{4}+c_{4}x_{5}+c_{5}x_{6}+c_{6}x_{1}$
$=c_{0}*1+c_{1}*0+c_{2}*1+c_{3}*0+c_{4}*0+c_{5}*1+c_{6}*1=c_{0}+c_{2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2462491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Normed linear space and non-zero linear bounded functional
Let $V$ be a normed linear space and $l$ be a non-zero linear bounded functional on $V$.
If $d=\inf\{\|v\|:l(v)=1\}$, show that $\|l\|=\frac{1}{d}$.
| You must have $\forall v \in V,|l(v)| \leq ||l||.||v||$. Without loss of generality you can assume that $l(v)=1$ (using the linearity of $l$ on the LHS and the absolute homogeneity of the norm on the RHS; of course in this case $||v|| \neq 0$).
Your equation becomes $\forall v \in V, \frac{1}{||v||} \leq ||l||$ meanin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2462595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can someone tell me what should i do next? Should I use the inequality between Arithmetic and Geometric mean? Problem 1: Let $a,b,c> 0$
$ab+3bc+2ca\leqslant18$
Prove that:
$\frac{3}{a} + \frac{2}{b}+ \frac{1}{c}\geqslant 3$.
I started on this way:
$\frac{3}{a} + \frac{2}{b}+ \frac{1}{c}\geqslant 3$
$\frac{3bc+2ac+ab}{... | By Holder:
$$18\left(\frac{3}{a}+\frac{2}{b}+\frac{1}{c}\right)^2\geq(ab+3bc+2ac)\left(\frac{3}{a}+\frac{2}{b}+\frac{1}{c}\right)\left(\frac{2}{b}+\frac{1}{c}+\frac{3}{a}\right)\geq$$
$$\geq\left(\sqrt[3]{ab\cdot\frac{3}{a}\cdot\frac{2}{b}}+\sqrt[3]{3bc\cdot\frac{2}{b}\cdot\frac{1}{c}}+\sqrt[3]{2ac\cdot\frac{1}{c}\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2462742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving that $f(x)=x^3+x+2$ is bijective without calculus I want to prove that $f(x)=x^3+x+2$, $f: \mathbb R \rightarrow \mathbb R$ is bijective without calculus. My attempts at showing to prove that it' injective and surjective are written below:
$1)$ Injectivity:
I want to show that $\forall a,b \in \mathbb R$ $f(a)... | As for the injectivity, if $a>b$, then $a^3>b^3$, so $f(a)=a^3+a+2>b^3+b+2=f(b)$.
As for the surjectivity, there is a formula to find a real solution of an equation of degree 3; it is not very nice, but you can use it. You can find it here
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2462852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Prove that if and only if $(A,B)$ is controllable, then $(A-BK,B)$ is also controllable Proof that the rank of $(A,B)$ is the same as $(A-BK,B)$
The image is taken from "Optimal Control Methods for Linear Discrete-Time Economic Systems" by Y. Murata
Could someone please explain why the sum of columns of $B, AB$ and so ... | The proof is based on the following fact (which is easy to show): If $w_1,\ldots,w_m\in\operatorname{span}\{v_1,\ldots,v_n\}$ then
$$
\operatorname{span}\{v_1,\ldots,v_n,(u_1+w_1),\ldots,(u_m+w_m)\} = \operatorname{span}\{v_1,\ldots,v_n,u_1,\ldots,u_m\}.
$$
In particular,
$$
\dim\operatorname{span}\{v_1,\ldots,v_n,u_1+... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Logic of the implication in $ε$-$δ$ proofs Im confused by why epsilon delta proofs logically work.
An example is
Proof:
Given $ε>0$, choose $δ = {ε\over3}$. For all $x$, if $0<|x−2|<δ$ then $|(3x−1)−5| < ε$.
That last part if $0<|x−2|<δ$ then $|(3x−1)−5| < ε$ LOOKS A LOT LIKE $P\to Q$ because of the "if then" but ye... | Let's just run through the proof.
We want to prove that, for any $\epsilon > 0$, there exists $\delta > 0$ such that for all $x$, $0 < |x-2| < \delta \implies |(3x-1)-5|<\epsilon$.
Hence, we take an arbitrary $\epsilon > 0$. We now want to prove the existence of a $\delta > 0$ such that the statement "for all $x$ (...)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 6,
"answer_id": 2
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Sum the series $(1^2+1)1! + (2^2+1)2! + (3^2+1)3! \cdots + (n^2+1)n!$ Problem:
Sum the series: $$(1^2+1)1! + (2^2+1)2! + (3^2+1)3! \cdots + (n^2+1)n!$$
Source: A book on algebra.I came across this interesting looking series and decided to tackle it.
My try :
All I have tried is taking the $r^{th}$ term and summing it,... | Hint. Note that
$$(n^2+1)n!=n(n+1)!-(n-1)(n)!$$
Therefore the sum can be written as
$$(1(2)!-(0)(1)!)+(2(3)!-(1)(2)!)+(3(4)!-(2)(3)!)+\dots +(n(n+1)!-(n-1)(n)!).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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A quotient of infinite direct product of field Let $R$ be a ring of countable infinite direct product of a field $F$ and $I$ be the countable infinite direct sum of $F$,clearly $I$ is an essential ideal of $R$.
Is $R$/$I$ an indecomposable $R$-module?
Is $R$/$I$ a cyclic $R$-module?
Is $I$ a maximal ideal in $R$?
I wil... | As it is a countable product we will asume that it is indexed in the natural.
1) We will put $e\in R$ such that $e_i=1_F$ if $i=2k$ for some $k\in\mathbb{N}$ and $0_F$ otherwise, and $f\in R$ such that $f_i=1_F$ if $i=2k+1$ for some $k\in\mathbb{N}$ and $0_F$ otherwise. It is clear that $1_R=e+f$, $e$ and $f$ are cent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Solve a nonlinear system of equations in 3 variables I need to solve this system of equations
$$\frac 1x+\frac{1}{y+z}=-\frac2{15}$$
$$\frac 1y+\frac{1}{x+z}=-\frac2{3}$$
$$\frac 1z+\frac{1}{x+y}=-\frac1{4}$$
I've tried to express $x$ in terms of $y$, then $y$ in terms of $z$. But this leads to nothing good. I think I ... | The solution is given by
$$
(x,y,z)=(5,-1,-2)
$$
This follows by multiplying with the common denominator, which gives three polynomial equations in $x,y,z$, which can be easily solved using resultants. The first polynomial equation, for example, is $x(2y + 2z + 15) + 15(y + z)=0$. One of the resultant equations is, for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463492",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the volume between $z=4-x^2-y^2$ and $z=4-2x$ as a triple integral So the volume of $z=4-x^2-y^2$ and $z=4-2x$ as a triple integral shall look similar to $$\int^2_0\int^{y=?}_{y=?}\int^{4-x^2-y^2}_{4-2x} dz dy dx$$ but how do I find the limits on $y$?
| The point is that you are integrating over a domain in which $4-2x<4-x^2-y^2$. In that domain,
$$
-2x < -x^2-y^2\\
2x>x^2+y^2\\
y^2 < 2x -x^2 \\
-\sqrt{2x-x^2} < y < +\sqrt{2x-x^2}
$$
So the integral is
$$
\int_{x=0}^2 \int_{y=-\sqrt{2x-x^2}}^{-\sqrt{2x-x^2}} (4- x^2 -y^2 -(4-2x) )dy\,dx = \int_{x=0}^2 \int_{y=-\sqrt{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Consider a box containing four balls: one red, one green, one blue, and one tricolor (=red,green and blue) Consider a box containing four balls: one red, one green, one blue, and one tricolor (=red,green and blue).
You draw one ball from the box.
Consider the three events:
R = {the drawn ball contains red}
G = {the ... | Using conditional probability formulas:
$$P(R|G)=\frac{P(R\cap G)}{P(G)}=\frac{1/4}{2/4}=\frac12=P(R) \Rightarrow Independent;$$
$$P(R|Y)=\frac{P(R\cap Y)}{P(Y)}=\frac{1/4}{1/4}=1\ne \frac12=P(R) \Rightarrow Dependent.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Redundant finite character family definition Definition: Let $F$ be a family of sets. $F$ is called of finite character if for each set $A$, we have that:
$A\in F\iff$ Each finite subset of $A$, is also in $F$.
I can't see the point of this definition, I find it redundant.
It seems trivial, if $A$ is in $F$ then every ... | As Somos pointed out in the comments, this is not "redundant". Let's consider the vector space $\mathbb{R}^{\alpha}$ for some dimension (possible infinite) $\alpha \ge 2$ and let $F_1$ be the collection of all linearly independent subsets $A \subseteq \mathbb R^{\alpha}$. $F_1$ is of finite character since a subset $A ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2463804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Necessity of absolute value in The Fubini–Tonelli theorem? In The Fubini–Tonelli theorem(reference in wiki):
What's the point of taking absolute value in $\int_{X\times Y}|f(x,y)|\,d(x,y)$? Isn't $f$ integrible in $X\times Y$ automatically implies $|f|$ integrable in $X\times Y$? So we only need $f$ integrable to apply... | No .The theorem holds for all non-negative measurable functions; you compute the repeated iterated integral of the absolute value of of f and if it comes out finite you CONCLUDE the $f$ is integrable so you may then apply Fubini to $f$ to calculate the integral of $f$ by iterated integrals -very useful result
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Summing up $3+5+9+17+...$
Find the sum of sum of $3
+5+9+17+...$ till $n$ terms.
Using Method of differences, the sum of the series is
$$\sum\limits_{j=1}^n 2^{j-1}+n$$
I am facing difficulty in evaluating $$\sum\limits_{j=1}^n 2^{j-1}$$. How do I do that?
Now I have $2^0 + 2^1 + 2^2 ... 2^{n-1}$
The sum of this ser... | Without words:
$$n+2^{n+1}-2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2464118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Product of polynomes in extensions Sorry if this is an elementary question. Let $K$ be a field and $L$ an extension of $K$. Let $P,Q\in{K[X]}$, so that $Q$ divides $P$ in $L[X]$, i.e. so that there exists a polynome $R\in{L[X]}$ so that $P=QR$. Then $R\in{K[X]}$.
I think this is a consequence of the unique factorisatio... | The relation $P=QR$ gives a linear system for the coefficients of $R$.
This system has entries in $K$ because $P,Q \in K[x]$.
If this system has a solution in $L$, then this solution can be found using Gaussian elimination. Therefore, the solution is in $K$ because Gaussian elimination only uses rational operations on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2464228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How can I find the equation of a circle given two points and a tangent line through one of the points? I was wondering whether it was possible to find the equation of a circle given two points and the equation of a tangent line through one of the points so I produced the following problem:
Find the equation of the circ... | Hint. The center of such circle is on the line which is orthogonal to tangent line and passes through the point of tangency.
Therefore, in your case, the coordinate of the center is $C=(1+2t,7-9t)$ for some $t\in \mathbb{R}$. In order to find $t$, impose that $C$ has the same distance from the given points $P=(1,7)$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2464364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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The sum of the real roots of $x^4+x^3+x^2+x-1$ Consider a polynomial $f(x)=x^4+x^3+x^2+x-1$. The sum of the real roots of $f(x)$ lies in the interval ...
*
*$(0,1)$
*$(-1,0)$
*$(-2,-1)$
*$(1,2)$
Using Intermediate Value property, I know that one root exists between $0$ and $1$, but I am stuck here and can't do ... | Let $$f(x)=x^4+x^3+x^2+x-1.$$
Thus, $$f(0.5)f(0.6)<0$$ and $$f(-1.3)f(-1.2)<0,$$
which says that there are two roots:
$$x_1\in(0.5,0.6)$$ and
$$x_2\in(-1.3,-1,2)$$ and
$$x_1+x_2\in(-0.8,-0.6).$$
But $$f''(x)=12x^2+6x+2>0,$$
which says that $f$ is a convex function.
Thus, our equation has at most two roots and the answe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2464457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The number of terms in the Multinomial Expansion $(x+\frac{1}{x}+x^2+\frac{1}{x^2})^n$ I am aware that there is a formula to calculate the number of terms in a multinomial expression $(x_1+x_2+x_3+...x_r)^n$, i.e. $^{n+r-1}C_{r-1}$. However, this is in the case when the terms $x_1, x_2, x_3 ... x_r$ are different varia... |
We obtain
\begin{align*}
\color{blue}{\left(x+\frac{1}{x}+x^2+\frac{1}{x^2}\right)}&=\frac{1}{x^{2n}}(1+x+x^2+x^3)^n\\
&=\frac{1}{x^{2n}}\left(1+x+x^3\left(1+x\right)\right)^n\\
&=\frac{1}{x^{2n}}(1+x)^n(1+x^3)^n\\
&\color{blue}{=\frac{1}{x^{2n}}\sum_{j=0}^n\binom{n}{j}x^j\underbrace{\sum_{k=0}^n\binom{n}{k}x^{3k}}_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find all primes $p$ such that $p+1$ is a perfect square. Find all primes $p$ such that $p+1$ is a perfect square.
All primes except for 2 (3 is not a perfect square, so we can exclude that case) are odd, so we can express them as $2n+1$ for some $n\in\mathbb{Z}_{+}$. Let's express the perfect square as $a^2$, where $a\... | If a prime $p$ is of the form of $n^2-1$, then
$$p=(n+1)(n-1)$$
and so $n-1$ must be 1.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2464649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Combinatorial proof that the Stirling numbers of second kind satisfy $S(m+n+1,m) = \sum_{k=0}^{m} kS(n+k,k)$ Give a combinatoric proof for the identity
$S(m+n+1,m) = \sum_{k=0}^{m} kS(n+k,k)$
LHS gives us a way to partition $m+n+1$ elements into $m$ blocks. How will we interpret $k$?
| Note that we can rewrite this as
$$
S(n+m+1,m) = \sum_{k=1}^{m}k \, S(n+k,k) = \sum_{s=n+2}^{n+m+1}(s-n-1)\, S(s-1,s-n-1) = \sum_{s=n+2}^{n+m+1} (m - (n+m+1-s)) \, S(s-1,m-(n+m+1-s)),
$$
by substituting $s = n+k+1.$
Let $P$ be a partition of the set $\{1, \dots, n+m+1\}$ into $m$ parts. Let $s$ be the largest element o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2464779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Solve integral $\int\frac{dx}{\sin x+ \cos x+\tan x +\cot x}$ I need to find:
$$\int\frac{1}{\sin x+ \cos x+\tan x +\cot x}\ dx$$
My attempts:
I have tried the conventional substitutions. I have tried the $\tan(x/2)$ substitutions, tried to solve it by quadratic but nothing has worked so far.
| Partial Solution
$$\begin{aligned}
&\int\frac{\mathrm{d}x}{\sin(x)+\cos(x)+\tan(x)+\cot(x)}
\\&=\int\frac{sc}{s^2c+c^2s+1}\,\mathrm{d}x\quad *
\\&=\int sc\sum_{n\geq0}(-1)^n(s^2c+c^2s)^n\,\mathrm{d}x\quad\text{(Binomial series)}**
\\&=\int sc\sum_{n\geq0}(-1)^n\left(\sum_{0\leq k\leq n}\binom{n}{k}(s^2c)^{k}(c^2s)^{n-k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2464865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Segment trisection without compass I'm trying to figure out how to trisect a segment using only pen and ruler.
There is a parallel line provided.
No measurement is allowed.
| lets mark ends of the segment as A and B.
*
*mark points C and D on the parallel line.
*draw lines BD and AC, mark the cross point as E
*draw lines BC and AD, mark the cross point as F
*draw line EF, mark its cross point with line CD as G
*draw line BG, mark its cross point with line AD as H
The line EH cuts s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2464986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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For what pair of integers $(a,b)$ is $3^a + 7^b$ a perfect square. Problem: For what pair of positive integers $(a,b)$ is $3^a + 7^b$ a perfect square.
First obviously $(1,0)$ works since $4$ is a perfect square, $(0,0)$ does not work, and $(0,1)$ does not work, so we can exclude cases where $a$ or $b$ are zero for the... | Just to register, from the comment by Daniel, there are just two possibilities, if the result is $x^2,$ either
$$ 1 + 2 \cdot 3^c = 7^d, $$ or
$$ 1 + 2 \cdot 7^e = 3^f. $$
I would guess that an elementary method shown in an answer by
Gyumin Roh to http://math.stackexchange.com/questions/1551324/exponential-diophanti... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Alternative & Null Hypothesis formulation Situation: A t-test is being used, population mean is known as well as sample mean, sample standard deviation and n = 31
.
The statement is "Is the true mean goals per game µ for soccer players in the 2014 - 2015 season still 12.5"
Is the null hypothesis, HO, µ = 12.5
How do I ... | Yes. $H_0: \mu = 12.5$ against $H_a: \mu \ne 12.5.$
You would use the test statistic $T = \frac{\bar X = \mu_0}{s/\sqrt{n}}.$
And you would reject $H_0$ at level $\alpha = 5\%$if $|T| > q^*,$ where
$q^*$ cuts probability $0.025$ from the upper tail of Student's t distribution
with $n-1$ degrees of freedom. With $n = 3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove existence and uniqueness of Convex Hull containing compact set I want to prove the existence and uniqueness of the convex set described below, which is the convex hull. My thinking is that I'm to generate a set containing all the convex sets containing $A$ and take their intersection. Then pointing out that the... | To formalize: let $\mathscr C$ be the collection of sets (so it is a subset in the power set of $\mathbb R^n$):
$$\mathscr C:= \{ B\subset \mathbb R^n : B \text{ is convex and } A\subset B\}.$$
Then the set you want is
$$ A^h := \bigcap _{B\in \mathscr C} B.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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Removing points from a triangular array without losing information I'm trying to find insights about the following puzzle, to see if I can find it on the OEIS (and add it if it's not already there):
Suppose I give you a triangular array of light bulbs with side length $n$:
o
o o
o o o
o o o o
o o o o o
o... | A stronger upper bound on $a(n)$ comes from the following observation: if we choose any two missing lights from the same row, then they are part of a triangle with at most one light still working. By pigeonhole, there can be at most $\binom{n+1}{2} + 1 - a(n) \le \binom{n+1}{2}$ such triangles, otherwise two of them wo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465571",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
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Calculus of variation: Lagrange's equation A particle of unit mass moves in the direction of $x$-axis such that it has the Lagrangian $L= \frac{1}{12}\dot x^4 + \frac{1}{2}x \dot x^2-x^2.$ Let $Q=\dot x^2 \ddot x$ represent a force (not arising from a potential) acting on the particle in the $x$-direction. If $x(0)=1$ ... | well, continue
\begin{align}
\frac{d}{dt}(x \dot x ) &=\frac{1}{2} \dot x^2-2x \\
\implies \dot{x}^2+x \ddot{x} &= \frac{1}{2}\dot{x}^2-2x \\
\implies 0 &= x \ddot{x}+\frac{1}{2}\dot{x}^2+2x \\
&\vdots
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Does the existence of the Levi-Civita connection depend on whether or not we define a metric on our smooth manifold? The Christoffel symbols of the Levi-Civita connection are calculated through the metric, but does that necessarily mean that its existence depends on whether or not we have a metric?
Specifically, th... | I think maybe you're misinterpreting the phrase "the Levi-Civita connection." Despite the definite article, there isn't just one such connection -- every Riemannian metric has its own Levi-Civita connection, uniquely determined by the metric. So the question in your title doesn't make sense -- without a specific choice... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2465947",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
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Probability of three dice in n rolls Consider the bet that all three dice will turn up sixes at least once in n rolls of three dice. Calculate $f(n)$, the probability of at least one triple-six when three dice are rolled n times. Determine the smallest value of n necessary for a favorable bet that a triple-six will occ... | The probability $\frac{5^3}{6^3}$ is the probability of the event "you get no sixes at all (in a particular throw)".
The probability for not getting "a triple six" (in one throw) is $1-\frac{1}{6^3}$. So
$$f(n) = 1 - (1-\frac{1}{6^3})^n$$
The first value of $n$ for which this is larger than $0.5$ is $n=150$, so I would... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Spectral Theorem - $AB = BA \implies B\Phi(f) = \Phi(f)B$ Consider the construction of the Borel Functional Calculus for a self adjoint operator $A$ as descibred here:
Continuity of the functional calculus form of the Spectral Theorem
or better yet, here:
http://www.math.mcgill.ca/jakobson/courses/ma667/mendelsontombe... | I think my last answer has a problem in it (see comments), I'll leave it there in case someone comes upon this. Here's my new suggested solution:
Step 1 is to prove $\forall x \in H (x, B\Phi(f) x) = (x, \Phi(f) B x)$.
Step 2 is to notice that the above implies $\forall x,y \in H$ $(x, B\Phi(f) y) = (x, \Phi(f) B y)$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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How to show that $- \log_b x = \log_{\frac{1}{b}} x$ I saw the following log rule and have been struggling to show it's true, using the change of base rule. Any hints for proving it would be much appreciated.
$- \log_b x = \log_{\frac{1}{b}} x$
I get as far as showing that $- \log_b x = \log_b \frac{1}{x}$ and think t... | $$\log_{1/b} x=\frac{\log_b x}{\log_b \frac{1}{b}}=\frac{\log_b x}{-1}=-\log_b x$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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What is the identity arrow in this category? This is taken from An Introduction to Category Theory by Harold Simmons (Example 1.3.1) (slightly changed for formatting reasons).
The objects are the finite sets. An arrow $f$ with source $A$ and target $B$ is a function $$f : A\times B \to \mathbb{R}$$ with no imposed co... | Writing down the relation that the identity $I_A$ would have to satisfy on one side : $$f(a,a') = (f \circ I_A)(a,a') = \sum_{a'' \in A} f(a,a'')I_A(a'',a'),$$
so a natural candidate would be $I_A(a'',a') = 1$ if $a'' = a'$ and $0$ otherwise. (This is known as the Kronecker delta function.)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
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How do I find the mean and variance of $Y_{(1)}$ in a random sample? I have that $Y_1, \dots, Y_n$ is a random sample from a uniform population on $(\theta, \theta+1)$. I need to show that $Y_{(1)}-\frac{1}{n+1}$ is an unbiased estimator of $\theta$. I know that would mean that its mean is should equal $\theta$ but I'm... | \begin{align}
F_{Y_{(1)}} (x)=\Pr(Y_{(1)} \le x) = 1-\Pr(Y_{(1)} > x) & = 1 - \Pr(Y_1>x\ \&\ \cdots\ \&\ Y_n>x) \\[10pt]
& = 1-\Pr(Y_1>x)\cdots\Pr(Y_n>x) \\[10pt]
& = 1-\Big(\Pr(Y_1>x)\Big)^n.
\end{align}
And then $f_{Y_{(1)}}(x) = \dfrac d {dx} F_{Y_{(1)}}(x).$ Once you have the density, you can evaluate an integral f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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show that $\left(\sum_{i=0}^\infty\frac{a^i}{i!}\right)\left(\sum_{j=0}^\infty\frac{b^j}{j!}\right) = \sum_{k=0}^\infty\frac{(a+b)^k}{k!}$ I need to show that if $\sum_{i=0}^\infty \frac{a^i}{i!}$ is absolutely convergent for all $a\in\mathbb{R}$, then
$$\left(\sum_{i=0}^\infty\frac{a^i}{i!}\right)\left(\sum_{j=0}^\in... | We have that
\begin{align*}
\left(\sum_{i=0}^\infty\frac{a^i}{i!}\right)\left(\sum_{j=0}^\infty\frac{b^j}{j!}\right)&=\sum_{k=0}^{\infty}\sum_{(i,j)\in S_k}\frac{a^i}{i!}\cdot \frac{b^j}{j!}\\
&=\sum_{k=0}^{\infty}\sum_{(i,j)\in S_k}\frac{1}{(i+j)!} \binom{i+j}{i}a^ib^j\\
&=\sum_{k=0}^{\infty}\frac{1}{k!} \sum_{i=0}^k\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
$\limsup$ and $\liminf$ in Extended Reals Is it true that any sequence $X_n$ converges to $x$ in the extended real number system if and only if $\limsup X_n \le \liminf X_n$?
| Strictly speaking, what you wrote is not quite true. Indeed, consider the constant sequence $x_n=0$ and set $x:=1$. Then $\liminf x_n=0=\limsup x_n$, but $x_n\not\to x$. However, it's clear that you meant to ask:
If $(x_n)$ is a sequence of real numbers, then $(x_n)$ converges to an extended real number $x$ iff $\lims... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Embedding of central simple algebras A paper I am reading has the following statement (without proof):
If $B$ and $A$ are central simple algebras with the same center $K$
where $K$ is a local or a global field, then $B$ can be embedded into $A$
if and only if [B : K] is relatively prime with $[A : K]/[B : K]$.
The nece... | This sounds very unlikely to me. What if $A$ is a division algebra
and $B$ is a matrix algebra over $K$? (When $K=\Bbb Q_p$, theses exist with any given square degree over $K$.)
The other way round sounds fishy too. Over $K=\Bbb Q_p$ there are
division algebras $A$ and $B$ with $|A:K|=16$ and $|B:K|=4$
and with $B$ a s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How do I solve this kind of differential equation? $ $ $\frac{dy}{dx} + ay^2+b = 0$ How do I solve this kind of differential equation?
$$\frac{dy}{dx} + ay^2+b = 0$$
I'm not seeing how to deal with the $y^2$ part. $ $ I suppose there's a simple technique.
| $$\frac{dy}{dx}= -(ay^2+b)$$
$$\frac{y'}{(ay^2+b)}= -1$$
This is now a standard integral that can be solved using substitution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2466996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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To prove given function is constant function Suppose $f,g,h$ are functions from the set of positive real numbers into itself satisfying
$$f(x)g(y)=h\left(\sqrt {x^2+y^2} \right)\ \ \forall \ x,y\in (0,\infty )$$
Show that the three functions $\frac {f(x)}{g(x)},\frac{g(x)}{h(x)},\frac{h(x)}{f(x)}$ are all constants.
I ... | So, by dividing by $f(y)g(x)$, you get $f(x)/g(x)=f(y)/g(y)$ for all positive $x,y$. So then obviously $f/g$ is constant.
Ok, here is my try: $f(x)g(1)=h(\sqrt{x^2+1})$ and $f(x)g(\sqrt{2})=h(\sqrt{x^2+2})=f(\sqrt{x^2+1})g(1)$.
So,$f(x)g(\sqrt{2})=f(\sqrt{x^2+1})g(1)$. i.e. $f(x)=f(\sqrt{x^2+1})g(1)/g(\sqrt{2})$. Henc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2467098",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Product and sum of line bundles on circle Given nontrivial one-dimensional vector bundle $E$ on circle (which is of course Möbius strip). I need to find out if $E \oplus E \oplus ... \oplus E$ (n times) would be trivial or not (my guess is that it never happens) and the same question about $E \otimes E \otimes ... \o... | Real line bundles $L \to B$ are classified by their first Stiefel-Whitney class $w_1(L) \in H^1(L; \mathbb{Z}_2)$, in particular, $w_1(L) = 0$ if and only if $L$ is trivial. Therefore, if $E$ denotes the non-trivial line bundle on $S^1$, we see that $w_1(E)$ is the unique non-zero element of $H^1(S^1; \mathbb{Z}_2) \co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2467238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Assuming that $g(z)=\frac{f(z)}{z-z_0}$ is continuous at $z_0$, prove that $\int_\gamma \frac{f(z)}{z-z_0}dz=0$. Suppose that $f$ is analytic in some domain D containing the unit circle $\gamma$ and that $z_0$ is a point in D not on $\gamma
$. Assuming that $g(z)=\frac{f(z)}{z-z_0}$ is continuous at $z_0$, prove that $... | Since $g(z)=\frac{f(z)}{z-z_0}$ is given to be continuous at $z_0$. Then
\begin{eqnarray*}
g(z_0)&=&\lim_{z \to z_0}\frac{f(z)}{z-z_0}\\
&=&\frac{\lim_{z \to z_0}f(z)}{\lim_{z \to z_0} (z-z_0)} \\
\lim_{z \to z_0}f(z)&=&g(z_0) \lim_{z \to z_0}(z-z_0)\\
f(z_0)&=&0
\end{eqnarray*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2467359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Proof in Fitch and counterexample in Tarski's World - From $A \to B$, infer $A \to (B \land C)$. Good official morning community,
Strengthening the Consequent: From A→B
A→B
, infer $A \to (B \land C)$.
I know that this proof is invalid and I want to make a counter example to prove that. How can I write $A \to (B \lan... | Create $1$ object in Tarski's World, make it a small cube, and label it $a$.
Then $Cube(a) \rightarrow Small(a)$ will evaluate to True, and $Cube(a) \rightarrow (Small(a) \land Tet(a))$ will evaluate to False.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2467515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Existence of integers $a$ and $b$ such that $p = a^2 +ab+b^2$ for $p = 3 $ or $p\equiv 1 \mod 3$ **Eisenstein primes, existence of integers **
I am working in subring $$R = \{a + b\zeta : a,b \in \mathbb{Z}\}$$ of $\mathbb{C}$ where $\zeta = \frac{1 + \sqrt{-3}}{2} \in \mathbb{C}$. I want to show that there exist integ... | Consider $p = a^2 +ab+b^2$.
If $p$ divides $a$, then $p$ divides $b$, and so $p^2$ divides the RHS but not the LHS.
So, $p$ does not divide either $a$ or $b$.
Then, $p$ divides $a^3-b^3=(a-b)(a^2 +ab+b^2)$, that is $a^3 \equiv b^3 \bmod p$.
If $bc \equiv 1 \bmod p$, then $(ac)^3 \equiv 1 \bmod p$.
If $ac \equiv 1 \bmod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2467631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Trick to calculate integrals such as $\int_0^\infty \left(\frac {2x^3}{3K}\right) e^{-x}dx$? Besides integration by parts, is there any trick to calculate integrals such a$$\int_0^\infty \left(\frac {2x^3}{3K}\right) e^{-x}dx=1 $$
I got $K=4$ by using integration by parts multiple times, but I think there should be som... | $$
\int_0^\infty \left(\frac {2x^3}{3K}\right) e^{-x} \, dx = \frac 2 {3K} \int_0^\infty x^3 e^{-x} \, dx.
$$
The integral here is $\displaystyle\int_0^\infty x^3 e^{-x}\, dx;$ the fraction $\dfrac 2 {3K}$ can be attended to after evaluating the integral.
You say you did integration by parts multiple times. Now suppos... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2467696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Epsilon- Delta Proof $$
\lim_{x\to 1} \frac{1}{x^2+2} = \frac{1}{3}
$$
I'm having a problem proving this using $\epsilon-\delta$ proofs. For some reason When I solve for $\epsilon$, I get a negative number. Since this value is supposed to equal $\delta$ and $\delta$ can't be negative I'm not sure how to move forward.
... | When $x, y\in [0, 2]$, one has
$$\left|\frac{1}{x^2+2} - \frac{1}{y^2+2}\right|
= \left|y-x\right| \frac{y+x}{(x^2+2)(y^2+2)}\le |y-x|\frac{4}{4} \le |y-x|
$$
Take $y = 1$ and $\delta = \epsilon$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2467948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
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Proving a loop is non-trivial using van Kampen's theorem So I want to prove the following statement in algebraic topology.
Let $X=A \cup B$ be a topological space with $A$, $B$, and $A \cap B$ being open and path-connected. Suppose $\gamma$ is a loop in $A\cap B$, and $\gamma$ is not homotopic to the trivial loop in bo... | Group-theoretically, consider the amalgamated free product $(\langle a\rangle *\langle b\rangle)/N$ where $N$ is the normal closure of the subgroup $\langle ab,ab^{-1}\rangle$. The product is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ since $a=b$ and $a^2=1$. The word $a^2$ is non-trivial in the "intersection" and on "ei... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Generating a certain number after $k$ operations. Obtain a formula for this number as a function of $k$ There are n cookies on a table. Adam did this series of steps:
In the 1st step he put 1 cookie in the middle of every two neighbouring cookies, in the 2nd step he put 2 cookies in the middle of every two neighbouring... | *
*Instead of considering previous pairs of cookies, ignore the one on the far right, so you start with $n-1$ cookies
*and say that at the $k$th step you put $k$ cookies to the immediate right of each previous cookie, in effect multiplying the number of cookies by $k+1$ on the $k$th step
*so from the start you mult... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can we see $ax^2-bx-\bar b x+c,a>0,c \ge 0 $ attains minimum when $x=\frac{\Re b}{a}$ How can we see $ax^2-bx-\bar b x+c, a>0,c\ge 0$ attains minimum (just minimize it over the set of all possible $x$ s.t. the quadratic function is real) when $x=\frac{\Re b}{a}$ where $\Re b$ is the real part of $b$? I know that $... | You know that $ax^2-bx+c$ attains minimum at $x=\frac{b}{2a}$ (assuming $a,b,c\in\Bbb R$). You can then see that your equation reduces to $ax^2-2\Re bx+c$, which satisfies the assumptions, allowing you to apply the same rule to reach the desired conclusion.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
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Prove a 3 parameter integral identity I have stumbled upon the following identity:
$$\int_0^1 \frac{(1-t)^c}{(1-z t)^b} dt=\int_0^1 \frac{(1+t)^{b-c-2}}{(1+(1-z) t)^b} dt+\int_0^1 \frac{t^c(1+t)^{b-c-2}}{(1-z+t)^b} dt$$
It appears to work for all $b \in \mathbb{R}$, $c \geq 0$ and $|z|<1$.
The identity is related to ... | We want to show
\begin{eqnarray*}
\int_0^1 \frac{(1-t)^c}{(1-z t)^b} dt=\int_0^1 \frac{(1+t)^{b-c-2}}{(1+(1-z) t)^b} dt+\int_0^1 \frac{t^c(1+t)^{b-c-2}}{(1-z+t)^b} dt.
\end{eqnarray*}
Make the substitution $t=\frac{1}{u}$ into the third integral ($dt=-\frac{du}{u^2}$)
\begin{eqnarray*}
\int_0^1 \frac{t^c(1+t)^{b-c-2}}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show $\max_{i\leq n} |X_i|^p/n$ converges to $0$ in probability $X_i$'s are i.i.d random variables and $\mathbb E(|X_i|^p)=k<\infty$ for some $p,k$. I want to show $Z= \max_{i\leq n} |X_i|^p/n$ converges to $0$ in probability.
Here is what I have tried.
$$\mathbb P(Z\geq \epsilon)=1-P(Z< \epsilon)=1-\prod_{i\leq n}\ma... | Let $Y_n=\max_{1\le i\le n}|X_i|^p$. Then $Y_n/n\xrightarrow{p}0$ iff $n\mathsf{P}(|X_1|^p>n\epsilon)\to 0$ as $n\to \infty$ for all $\epsilon>0$. Since $\mathsf{E}|X_1|^p<\infty$,
$$
n\mathsf{P}(|X_1|^p>n\epsilon)\le \epsilon^{-1}\mathsf{E}[|X_1|^p 1\{|X_1|^p>n\epsilon\}]\to 0
$$
as $n\to\infty$ by the dominated conve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468473",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A curve where all tangent lines are concurrent must be straight line I'm trying to solve this question in the classical Do Carmo's differential geometry book (page 23):
*A regular parametrized curve $\alpha$ has the property that all its tangent lines pass through a fixed point. Prove that the trace of $\alpha$ is... | Let's assume that the tangents pass trough the point $c$ in $\mathbb{R}^2$. Then we the vectors $\alpha(s) - c$ and $\alpha'(s)$ are proportional, that is
$(\alpha(s)-c) + \lambda(s) \alpha'(s)=0$
for some scalar function $\lambda(s)$. Replacing $\alpha(s)$ with $\alpha(s) - c$, we may assume that $c=0$, that is
$$\al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
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Induction - why do we allow $k=1$ in the second step? Having proved that the property holds for the first case (most often 1), in the second step we need to assume that it's true for some $k \ge 1$. Why not $k>1$
| It's because if you don't you loose the connection with the first case. The idea is that you should in principle be able to step from the first case to any integer.
If you don't assume it's only valid for some $k>1$ and prove that it's valid for $k+1$ then that step would only be valid if you step from $2$ and higher. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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What is the integral of $e^{\cos x}$ Question:
Find out $\displaystyle{\int e^{\cos x}~dx}$.
My Attempt:
Let $\cos x = y$. Hence $-\sin x\ dx = dy$ or $$dx = \displaystyle{\frac{-dy}{\sin x}=\frac{-dy}{\sqrt{1-\cos^2x}}=\frac{-dy}{\sqrt{1-y^2}}}$$ So
$$\begin{align}\int e^{\cos x}~dx &= \int e^y\left(\frac{-dy}{\sqrt... | $\int e^{\cos x}~dx$
$=\int\sum\limits_{n=0}^\infty\dfrac{\cos^{2n}x}{(2n)!}dx+\int\sum\limits_{n=0}^\infty\dfrac{\cos^{2n+1}x}{(2n+1)!}dx$
$=\int\left(1+\sum\limits_{n=1}^\infty\dfrac{\cos^{2n}x}{(2n)!}\right)dx+\int\sum\limits_{n=0}^\infty\dfrac{\cos^{2n+1}x}{(2n+1)!}dx$
For $n$ is any natural number,
$\int\cos^{2n}x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 5,
"answer_id": 3
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Locus problem need help
My try
Let $(x,y)$ place on the demanded graph
$\sqrt {x^{2}+y^{2}}-1=\sqrt {x^{2}+(y+3)^{2}}$
$-2\sqrt {x^{2}+y^{2}}= 6y+8$
$-\sqrt {x^{2}+y^{2}}=3y+4$
$x^{2}+y^{2}= 16+9y^2+24y$
What should I do now?
| Well, you should start with formulating the "distance to a circle" and "distance to a line", which probably means "distance to the closest point on a ..."
Let's start with the line. The line is horizontal, so only the distance from the y-coordinate to $-3$ matters. This expression becomes the distance between $y$ and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2468960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Why is the lim inf the union of intersections For my statistics class we had elementary set theory.
It was stated that:
$$\inf_{k\geq n } A_k = \bigcap\limits_{k=n}^{\infty} A_k$$
and
$$\sup_{k\geq n } A_k = \bigcup\limits_{k=n}^{\infty} A_k$$
From this was deduced that:
$$\lim\limits_{n\to\infty} \inf A_k = \bigcup\l... | It is clear that for real numbers, e.g. $3$ and $8$, that the minimum is the smaller one, i.e. $3$. There is a way to talk about smaller sets: a set $A$ is "smaller" than a set $B$ if it is completely contained in it, i.e. $A\subseteq B$. This is natural somehow, as e.g. $\{1,2,3\}$ is smaller than $\{1,2,3,4,5,6\}$. I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2469060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Differentiate hadamard product of square matrix $S \odot VV^T \in R^{n \times n} $ over rectangular matrix $V \in R^{n \times r}$ I want to differentiate $f = \log\det(L)$ over $V$ where $L = S \odot VV^T$.
The thing that I know is $df = L^{-T} : dL = (S \odot VV^T) : (dS \odot VV^T+ S \odot (VdV^T+dVV^T ))$ where $:... | The Hadamard and Frobenius products commute with each other, i.e.
$$A\odot B:C = A:B\odot C$$
This property can be used to complete your derivation (assuming $dS$ is zero)
$$\eqalign{
df
&= L^{-T}:dL \cr
&= L^{-T}:S\odot(dV\,V^T+V\,dV^T) \cr
&= S\odot L^{-T}:(dV\,V^T+V\,dV^T) \cr
&= (S\odot L^{-T}+S^T\odot L^{-1}):... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2469224",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a generalization of Brahmagupta–Fibonacci identity for cubic? I was wondering if there was a cubic version of Brahmagupta–Fibonacci identity .
I looked everywhere but I didn't find anything ,except the splendid website of Tito Piezas (see here)
Furthermore I know the Gauss identity wich is :
$$a^3+b^3+c^3-3abc... | There is, provided your are prepared to have 3 letters in the identity. It requires consideration of multiplication in a cubic field. See http://www.mathjournals.org/jrms/2017-032-004/2017-032-004-001.html
Let $$N_j = \left ( \begin {array} {ccc} u_j & -a d y_j & -a d x_j - b d y_j \\ x_j & u_j - b x_j -c y_j & - c x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2469516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
existence of a ball in a measurable set Let $\lambda$ be a Lebesgue-measure on $\mathbb{R}^n$ and let $A\subset \mathbb{R}^n$ be Lebesgue-measurable set with $\lambda(A)>0$. I know that the difference set $A - A$ contains an open ball $B_r(x)$. My question is if the set A also contains an open ball with $B_r(x) \subset... | No, not necessarily. One interesting counterexample is a "fat" Cantor set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2469647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find all integer solutions to $y^3 = x^6 + 6x^3 + 13$. Find all ordered pairs of integers $(x, y)$ that satisfy
$$y^3 = x^6 + 6x^3 + 13.$$
I've found the solutions $(-1, 2)$ and $(2, 5)$. I believe that these are all the integer solutions, but I don't know how to prove it. Could someone please help?
| As Peter said in the comments, this is a special case of the Mordell equation $y^3=z^2+4$ but we can use the fact that $y$ is very nearly $x^2$ to obtain an elementary proof.
$$\frac y{x^2}=\left(1+\frac 6{x^3}+\frac{13}{x^6}\right)^{1/3}$$
$$1-\frac 3{|x|^3}<\frac y{x^2}<1+\frac 3{|x|^3}$$
$$|y-x^2|<\frac 3{|x|}$$
An... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2469764",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$ \sum_{n=1}^{\infty} \frac{ x^{n^2} (1 + x^n) - x^n}{1 -x^n} = 0.$ ?? While studying theta functions I noticed
$$ \sum_{n=1}^{\infty} \frac{ x^{n^2} (1 + x^n) - x^n}{1 -x^n} = 0.$$
Why is that so ??
Is there a similar case with a term $x^{n^3}$ ??
| I am not sure what you mean by "similar case", but the infinite sum is zero because of the equation
$$ \sum_{n=1}^\infty x^n/(1-x^n) =
\sum_{n=1}^\infty x^{n^2}(1+x^n)/(1-x^n) =
\sum_{n=1}^\infty \sigma_0(n)x^n \tag{1}$$
which is the generating function of the number of divisors of $n$. This is OEIS sequence A000005 w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2469886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Pigeonhole principle explanation Let $A$ be any set of $10$ positive integers.
Prove that there must exist at least $11$ subsets of $A$ having their element-sum with the same last $2$ digits.
(Here element-sum means sum of all elements in the set)
Answer: Let $A$ be a set of any $10$ positive integers.
So $|A| = 10$.... | The element sum is used indirectly, in the fact that, we only care for it's last two digits. There are less than one tenth the possible 2 digit endings, as there are subsets. Pigeonhole principle then implies that at least 11 subsets go to one of the subset sum endings.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2469976",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Semifactorial Identity I was wondering if anyone had any insight on how to prove the following identity:
For all $m \in \mathbb{N}$ $$ \frac{1}{2m-1} + \frac{2m-2}{(2m-1)(2m-3)} + \frac{(2m-2)(2m-4)}{(2m-1)(2m-3)(2m-5)} + \cdots + \frac{(2m-2)!!}{(2m-1)!!} = 1$$
I attempted to rewrite and simplify the left hand side ... | $$\scriptsize\begin{align}
&\;\;\;\frac 1{2m-1}+\frac {2m-2}{(2m-1)(2m-3)}+\frac{(2m-2)(2m-4)}{(2m-1)(2m-3)(2m-5)}+\cdots+\frac {(2m-2)!!}{(2m-1)!!}\\\\
&=\frac 1{2m-1}\left(1+\frac {2m-2}{2m-3}\left(1+\frac {2m-4}{2m-5}\left(1+\;\;\cdots\;\;\ \left(1+\frac 65\left(1+\frac 43\left(1+\frac 21\right)\right)\right)\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2470085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Closed form of the elliptic integral $\int_0^{2\pi} \sqrt{1+\cos^2(x)}\,dx $ I want to prove the closed form shown in Wikipedia for the arc length of one period of the sine function.
Source of wikipedia
$$\int_0^{2\pi} \sqrt{1+\cos^2(x)} \ dx= \frac{3\sqrt{2}\,\pi^{\frac32}}{\Gamma\left(\frac{1}{4}\right)^2}+\frac{\Gam... | $$\int_0^{2\pi} \sqrt{1+\cos^ 2 x} dx = 4 \int_0^1 \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} dx = 4\int_0^1 \frac{1+x^2}{\sqrt{1-x^4}} dx $$
Now use the beta function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2470259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How to calculate the grey area of this irregular five-pointed star?
The corners of a five-pointed star lie on the sides of a square ABCD with the side length 1, that two corners of the star coincide with the corner points A and D. Further corner points of the star lie in each case in the interior of the edges AB, BC a... | The idea is to use the several way to compute the grey areas.
I don't want to multiply the notations to name all the lines intersections.
I will call however $A_g$ the sum of all grey areas, $A_{ext}$ the sum of all white exterior areas and $A_m$ the area of the middle pentagon.
If you calculate the area of each of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2470374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solve $x \ddot {x} + {\dot {x}}^2=0$. I solved a physics problem and I got this equation, but I don't know how to proceed. Could you solve for $x (t)$ this equation:
$x \ddot {x} + {\dot {x}}^2=0$
| $(xx')'=xx''+x'^2=0$ and $2xx'=2C$ then $(x^2)'=2C$ so $x^2=2Ct+D$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2470429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Prove that $\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta$
Prove that $$\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}=\sin\theta+i\cos\theta$$
I tried to rationalize the denominator but I always end up with a large fraction that won't cancel out. Is there something I'm m... | We know that, $\sin^2\theta+\cos^2\theta = 1$ and
$a^2-b^2=(a-b)(a+b$
then
\begin{split} \frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta} &= &\frac{\color{red}{\sin^2\theta+\cos^2\theta} +\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta} \qquad\quad\\\\&=& \frac{\color{red}{\sin^2\theta+(-i\cos\theta)(i\cos\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2470541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 8,
"answer_id": 4
} |
$\sum c_nn^{-s_0}$ converges implies $\sum c_nn^{-s}$ cannot have a pole in the line $\Re{s}=\sigma_0$ If the Dirichlet series $\sum c_nn^{-s}$ converges at $s_0=\sigma_0+it_0$, prove that the function defined by the series for $\Re{s}>\sigma_0$ cannot have a pole on the line $\Re{s}=\sigma_0$.
I hope the question mean... | One cannot show that for every $s$ with $\operatorname{Re} s = \sigma_0$ there is a sequence $s_k \to s$ on which the sum function is bounded. For example the series
$$P(s) = \sum_{p \text{ prime}} \frac{1}{p^s}$$
converges for all $s$ with $\operatorname{Re} s \geqslant 1$, except for $s = 1$ (see e.g. here). Since
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2470657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find two arithmetic progressions of three square numbers I want to know if it is possible to find two arithmetic progressions of three square numbers, with the same common difference:
\begin{align}
\ & a^2 +r = b^2 \\
& b^2 +r = c^2 \\
& a^2 +c^2 = 2\,b^2 \\
\end{align}
and
\begin{align}
\ & d^2 +r = e^2 \\
& e... | There are infinitely many solutions to the system,
\begin{align}
\ & a^2 +r_1 = b^2 \\
& b^2 +r_1 = c^2 \\
& a^2 +c^2 = 2b^2 \\
\hline
\ & d^2 +r_2 = e^2 \\
& e^2 +r_2 = f^2 \\
& d^2 +f^2 = 2e^2 \\
\end{align}
with $\color{blue}{r_1=r_2}$. Eliminating $r_1$ between the first two equations (and similarly for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2470772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Defining dihedral groups $\{\sigma \in S_n: $ something $\}$ I am trying to understand hos one can define the dihedral groups $D_n$. I have seen the "definition" that just says this is the group of symmetries of an $n$-polygon. So you have rotations and reflections. But I feel this definition is a bit vague. I asked ar... | Yes. Label the vertices of your regular $n$-gon with $1, 2, 3, \ldots, n$. Keep track of where the vertices go after the rotation or reflection is done. The result is a literal permutation on $\{1, 2, 3, \ldots, n\}$ and the operation is still composition, so $D_n$ is a subgroup of $S_n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2470880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Example of element of double dual that is not an evaluation map It's well known that if $V$ is a vector space over a field $F$, then there is a natural injection from $V$ to the double dual $V^{**}$, which associates to every $v \in V$ the evaluation map $\phi \mapsto \phi(v)$, where $\phi: V \to F$ is an arbitrary fun... | Quick example, using the axiom of choice: take $V$ to be the set of polynomials with real coefficients. Let $S$ denote the subspace of $V^*$ consisting of those functionals such that
$\lim_{n \to \infty} f(x^n)$ exists. With the axiom of choice, there necessarily exists a complementary subspace $S'$ such that $V = S \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2471015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
probability of a simple children’s race game Suppose that a simple children’s race game is played as follows, using an urn containing some blue balls and some red balls, and a “track” having positions numbered consecutively from 0 to 10. A blue “racer” starts at position 0 and moves one space forward (towards position ... | You should note that order is critical here. As such, it may be harder to simplify the calculation (note that yours is slightly lacking, again due to the ordering issue - also, as there are less blue balls than red ones, it should come to attention that the probability of the blue racer winning should be less than 0.5)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2471160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Derivatives of composite functions How would I solve a problem that is asking me to find the derivative of $F$ when $$F(x)=f\left(\frac{x+2}{x+4}\right)$$ and $f$ is differentiable.
Not asking for the answer here obviously, just the steps needed to get off the ground.
| You use the chain rule.
If $F(x) = f(g(x))$
then $F'(x) = g'(x)f'(g(x))$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2471293",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Differential equations for chemical reaction $\mathrm{A + 2B \to 3C}$
In a chemical reaction $\mathrm{A + 2B \to 3C}$, the concentrations $a(t)$, $b(t)$ and $c(t)$ of the three substances A, B and C measure up to the differential equations
$$
\begin{align}
\frac{da}{dt} &= -rab^2\tag{1}\\
\frac{db}{dt} &= -2rab^2\ta... | Notice that, by your equations,
$\dfrac{d(2a - b)}{dt} = 2\dfrac{da}{dt} - \dfrac{db}{dt} = -2rab^2 -(-2rab^2) = 0; \tag 1$
hence $2a(t) - b(t)$ is constant. Now
$2a(0) - b(0) = 2(1) - 2 = 0; \tag 2$
the desired result follows.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2471462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Vertex Cover without integer programming Is there a way to formulate an LP(linear programming) for minimum vertex cover problem without forcing the variables to be integers (no integer programming)? The number of variables are not restricted as long as they are finite. I googled it for so long, but all I could find was... | To add to Mark's answer:
It has been shown here that the factor 2 cannot be improved.
On the other hand, if your graph is bipartite, the relaxed version of the ILP gives integer solutions, since then the constraint matrix is totally unimodular.
So, this version of the problem can be solved efficiently.
See e.g. here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2471543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
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