Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.
Show that in any group of order $23 \cdot 24$, the $23$-Sylow subgroup is normal.
Let $P_k$ denote the $k$-Sylow subgroup and let $n_3$ denote the number of conjugates of $P_k$.
$n_2 \equiv 1 \mod 2$ and $n_2 | 69 \implies n_2= 1, 3, ... | As commented back in the day, the OP's solution is correct. Promoting a modified (IMHO simplified) form of the idea from Thomas Browning's comment to an alternative answer.
The group $G$ contains $23\cdot24-24\cdot22=24$ elements outside the Sylow $23$-subgroups. Call the set of those elements $X$. Clearly $X$ consists... | {
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"url": "https://math.stackexchange.com/questions/463823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $S=\{\frac{p}{2^i}: p\in\Bbb Z, i \in \Bbb N \}$ is dense in $\Bbb R$. Show that $S=\{\frac{p}{2^i}: p\in\Bbb Z, i \in \Bbb N \}$ is dense in $\Bbb R$.
Just found this given as an example of a dense set while reading, and I couldn't convince myself of this claim's truthfulness. It kind of bugs me and I wonde... | I like to think of the answer intuitively. Represent $p$ in binary (base 2). Then $\frac{p}{2^i}$ is simply a number with finitely many binary digits. Conversely, any number whose binary representation has finitely many digits can be written as $\frac{p}{2^i}$.
To show a set is dense, we have to show that given an elem... | {
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"url": "https://math.stackexchange.com/questions/463884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof read from "A problem seminar" May you help me judging the correctness of my proof?:
Show that the if $a$ and $b$ are positive integers, then
$$\left(a+\frac{1}{2}\right)^n+\left(b+\frac{1}{2}\right)^n$$
is integer for only finintely many positive integers $n$
We want $n$ so that
$$\left(a+\frac{1}{2}\right)^n+\le... | Our expression can be written as
$$\frac{(2a+1)^n+(2b+1)^n}{2^n}.$$
If $n$ is even, then $(2a+1)^n$ and $(2b+1)^n$ are both the squares of odd numbers.
Any odd perfect square is congruent to $1$ modulo $8$. So their sum is congruent to $2$ modulo $8$, and therefore cannot be divisible by any $2^n$ with $n\gt 1$.
So w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Quartic Equation having Galois Group as $S_4$ Suppose $f(x)\in \mathbb{Z}[x]$ be an irreducible Quartic polynomial with Galois Group as $S_4$. Let $\theta$ be a root of $f(x)$ and set $K=\mathbb{Q}(\theta)$.Now, the Question is:
Prove that $K$ is an extension of degree $\mathbb{Q}$ of degree 4 which has no proper Subfi... | As has been remarked, the non-existence of intermediate fields is equivalent to $S_{3}$ being a maximal subroup of $S_{4}.$ If not, then there is a subgroup $H$ of $S_{4}$ with $[S_{4}:H] = [H:S_{3}] = 2.$ Now $S_{3} \lhd H$ and $S_{3}$ contains all Sylow $3$-subgroup of $H.$ But $S_{3}$ has a unique Sylow $3$-subgroup... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding an orthogonal basis of the subspace spanned by given vectors
Let W be the subspace spanned by the given vectors. Find a basis for $W^\perp$.
$$v_1=(2,1,-2) ;v_2=(4,0,1)$$
Well I did the following to find the basis.
$$(x,y,z)*(2,1,-2)=0$$ $$(x,y,z)*(4,0,1)=0$$
If you simplify this in to a Linear equation
... | Since you work in $\mathbb R^3$ so simply take $v_3=v_1\wedge v_2=(1,-10,-4)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is there never a proof that extending the reals to the complex numbers will not cause contradictions? The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot give rise to ... | There are several ways to introduce the complex numbers rigorously, but simply postulating the properties of $i$ isn't one of them. (At least not unless accompanied by some general theory of when such postulations are harmless).
The most elementary way to do it is to look at the set $\mathbb R^2$ of pairs of real numbe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464262",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "59",
"answer_count": 8,
"answer_id": 6
} |
Please help on this Probability problem
A bag contains 5 red marbles and 7 green marbles. Two marbles are drawn randomly one at a time, and without replacement. Find the probability of picking a red and a green, without order.
This is how I attempted the question: I first go $P(\text{Red})= 5/12$ and $P(\text{Green})... | Very nice and successful attempt. You recognized that there are two ways once can draw a red and green marble, given two draws: Red then Green, or Green then Red. You took into account that the marbles are not replaced. And your computations are correct: you multiplied when you needed to multiply and added when you nee... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464344",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Mean value of the rotation angle is 126.5° In the paper
"Applications of Quaternions to Computation with Rotations"
by Eugene Salamin, 1979 (click here),
they get 126.5 degrees as the mean value of the rotation angle of a random rotation (by integrating quaternions over the 3-sphere).
How can I make sense of this r... | First, SO(3) of course has its unique invariant probabilistic measure. Hence, “random rotation” is a well-defined SO(3)-valued random variable. Each rotation (an element of SO(3)) has an uniquely defined rotation angle θ, from 0 to 180° (π) because of axis–angle representation. (Note that axis is undefined for θ = 0 an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464419",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to find the degrees between 2 vectors when I have $\arccos$ just in radian mode? I'm trying to write in java a function which finds the angles, in degrees, between 2 vectors, according to the follow equation -
$$\cos{\theta} = \frac{\vec{u} \cdot \vec{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}$$
but in java the Math.acos... | You can compute the angle, in degrees, by computing the angle in radians, and then multiplying by
$\dfrac {360}{2\pi} = \dfrac {180\; \text{degrees}}{\pi\; \text{radians}}$:
$$\theta = \arccos\left(\frac{\vec{u} \cdot \vec{v}}{\|\mathbf{u}\|\|\mathbf{v}\|}\right)\cdot \frac {180}{\pi}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/464557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expected number of people sitting in the right seats. There was a popular interview question from a while back: there are $n$ people getting seated an airplane, and the first person comes in and sits at a random seat. Everyone else who comes in either sits in his seat, or if his seat has been taken, sits in a random un... | I found this question and the answer might be relevant.
Seating of $n$ people with tickets into $n+k$ chairs with 1st person taking a random seat
The answer states that the probability of a person not sitting in his seat is $\frac{1}{k+2}$ where $k$ is the number of seats left after he takes a seat. This makes sense be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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"answer_id": 2
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What does it mean "adjoin A to B"? What does it mean that we can obtain $\mathbb{C}$ from $\mathbb{R}$ by adjoining $i$?
Or that we can also adjoin $\sqrt{2}$ to $\mathbb{Q}$ to get $\mathbb{Q}(\sqrt{2})=\{a+b \sqrt{2}\mid a,b \in \mathbb{Q}\}$?
| It means exactly what you have written there. Let $F$ be a field and $\alpha$ be a root of a polynomial $f(x)$ that is irreducible of degree $d$ over $F[x]$. Then we say we can adjoin $\alpha$ to $F$ by considering all linear combinations of field elements of $F$ with scalar multiples of powers of $\alpha$ up to $d-1$,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Laplace operator's interpretation (Laplacian) What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in the given spot?
I can grasp the meaning of gradient and divergence. But vi... | It's enlightening to note that the adjoint of $\nabla$ is $-\text{div}$, so that $-\text{div} \nabla$ has the familiar pattern $A^T A$, which recurs throughout linear algebra. Hence you would expect (or hope) $-\text{div} \nabla$ to have the properties enjoyed by a symmetric positive definite matrix -- namely, the eig... | {
"language": "en",
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Classification of all semisimple rings of a certain order I'd appreciate it if you tell me where to begin in order to solve this question:
Classify (up to ring isomorphism) all semisimple rings of order 720.
Could the Wedderburn-Artin Structural Theorem be applicable?
| Yes, you should definitely apply Artin-Wedderburn.
The thing you gain from knowing the ring is finite is that the ring will be a product of matrix rings over fields, since finite division rings are fields. Hopefully you know that all finite fields are of prime power order.
Now then, an n by n matrix ring over a field w... | {
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"source": "stackexchange",
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When polynomial is power $P(x)$ ia a polynomial with real coefficients, and $k>1$ is an integer. For any $n\in\Bbb Z$, we have $P(n)=m^k$ for some $m\in\Bbb Z$. Show that there exists a real coefficients polynomial $H(x)$ such that $P(x)=(H(x))^k$, and $\forall n\in\Bbb Z,$ $H(n)$ is an integer.
This is an old quest... | The result is Corollary 3.3 in this paper.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/464902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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$\sum_{k=1}^nH_k = (n+1)H_n-n$. Why? This is motivated by my answer to this question.
The Wikipedia entry on harmonic numbers gives the following identity:
$$
\sum_{k=1}^nH_k=(n+1)H_n-n
$$
Why is this?
Note that I don't just want a proof of this fact (It's very easily done by induction, for example). Instead, I w... | I suck at making pictures, but I try nevertheless. Write $n+1$ rows of the sum $H_n$:
$$\begin{matrix}
1 & \frac12 & \frac13 & \dotsb & \frac1n\\
\overline{1\Big\vert} & \frac12 & \frac13 & \dotsb & \frac1n\\
1 & \overline{\frac12\Big\vert} & \frac13 & \dotsb & \frac1n\\
1 & \frac12 & \overline{\frac13\Big\vert}\\
\vdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/464957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Spectrum and tower decomposition I'm trying to read "Partitions of Lebesgue space in trajectories defined by ergodic automorphisms" by Belinskaya (1968). In the beginning of the proof of theorem 2.7, the author considers an ergodic automorphism $R$ of a Lebesgue space whose spectrum contains all $2^i$-th roots of unity... | Let $R$ be an ergodic automorphism of a Lebesgue space. Let $\omega$ be a root of unity in the spectrum of $R$, and $n$ be the smallest positive integer such that $\omega^n = 1$.
Let $f$ be a non-zero eigenfunction corresponding to the eigenvalue $\omega$. Then:
$f^n \circ R = (f \circ R)^n = (\omega f)^n = f^n,$
and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Mathematical Analysis advice Claim: Let $\delta>0, n\in N. $ Then $\lim_{n\rightarrow\infty} I_{n} $exists, where $ I_{n}=\int_{0}^{\delta} \frac{\sin\ nx}{x} dx $
Proof: $f(x) =\frac{\sin\ nx}{x}$ has a removable discontinuity at $x=0$ and so we let $f(0) =n$
$x = \frac{t}{n}$ is continuous and monotone on $t\in[0,... | I can suggest a alternative path. Prove that $$\lim_{a\to 0^+}\lim_{b \to\infty}\int_a^b\frac{\sin x}xdx$$
exists as follows: integrating by parts
$$\int_a^b \frac{\sin x}xdx=\left.\frac{1-\cos x}x\right|_a^b-\int_a^b\frac{1-\cos x}{x^2}dx$$
Then use $$\frac{1-\cos h}h\stackrel{h\to 0}\to 0$$ $$\frac{1-\cos h}{h^2}\sta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Parametric equations, eliminating the parameter $\,x = t^2 + t,\,$ $y= 2t-1$ $$x = t^2 + t\qquad y= 2t-1$$
So I solve $y$ for $t$
$$t = \frac{1}{2}(y+1)$$
Then I am supposed to plug it into the equation of $x$ which is where I lose track of the logic.
$$x = \left( \frac{1}{2}(y+1)\right)^2 + \frac{1}{2}(y+1) = \frac{1... | Let's assume you are walking on an xy-plane. Your x-position (or east-west position) at a certain time t is given by $x = t^2 + t$, and your y-position is $y = 2t - 1$.
If you want to know what the whole path you remained is, without wanting to know when you stepped on where? Eliminate t:
$$x = \frac{1}{4} y^2 + y + \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Showing probability no husband next to wife converges to $e^{-1}$ Inspired by these questions:
*
*Probability of Couples sitting next to each other (Sitting in a Row)
*Probability question about married couples
*Four married couples, eight seats. Probability that husband sits next to his wife?
*In how many ways c... | I observe that each term with $i$ fixed approaches a nice limit. We have
$$ 2^i \frac{n(n-1)(n-2)\cdots(n-i+1)}{i!} \frac1{(2n-i+1)(2n-i+2)\cdots(2n)} $$
or
$$ \frac1{i!} \frac{2n}{2n} \frac{2(n-1)}{2n-1} \cdots \frac{2(n-i+1)}{(2n-i+1)} \sim \frac 1{i!} $$
This gives you the series, assuming the limits (defining terms... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof that equality on categorical products is componentwise equality I want to proof that in the categorical product as defined here it holds that
for $x,y \in \prod X_i$ then
$$
x = y \textrm{ iff } \forall i \in I : \pi_i(x) = \pi_i(y).
$$
The direction from left to right is trivial, but the other, that iff the com... | In arbitrary categories, there is a notion of "generalized element": A generalized element of an object $A$ is any morphism into $A$ (from any object of the category). A morphism $A\to B$ can be applied to a generalized element $Z\to A$ just by composing them to get a generalized element $Z\to B$. In these terms, the r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465398",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Induced homomorphism between fundamental groups of a retract is surjective I'm trying to understand why the induced map $i_*: \pi_1(A) \rightarrow \pi_1(X)$ is surjective, for $A$ being a retract of $X$ and $i: A \rightarrow X$ being the inclusion map? For homotopy retracts it's obvious, but for retracts it seems I mis... | Any loop in $A$ is also a loop in $X$. What does $f_*$ do to an element of $\pi_1(X)$ that is a loop in $A$?
More categorically, if $i:A\to X$ is the inclusion map (so that $f\circ i=\mathrm{id}_A$), then $f_*\circ i_*=\mathrm{id}_{\pi_1(A)}$ because $\pi_1$ is a functor. Since $\mathrm{id}_{\pi_1(A)}$ is surjective we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $A$ is compact and $B$ is Lindelöf space , will be $A \cup B$ Lindelöf I have 2 different questions:
As we know a space Y is Lindelöf if each open covering contains a countable subcovering.
(1) :If $A$ is compact and $B$ is Lindelöf space , will be $A \cup B$ Lindelöf?
If it is right, how can we prove it?
(2) : A ... | I am facing a notational problem. What is a $KC$ space? Answer of your first question is the following.
Lindelof Space: A space $X$ is said to be Lindelof is every open cover of the space has a countable subcover.
Consider an open cover $P = \{P_{\alpha}: \alpha \in J, P_{\alpha}$ is open in $A \cup B\}$
Now $P$ will g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can a 2D person walking on a Möbius strip prove that it's on a Möbius strip? Or other non-orientable surface, can a 2D walker on a non-orientable surface prove that the surface is non-orientable or does it always take an observer from a next dimension to prove that an entity of a lower dimension is non-orientable? So i... | If he has a friend then they both can paint their right hands blue and left hands red.
His friend stays where he is, he goes once around the strip, now his left hand and right hand are switched when he compares them to his friends hands.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/465594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 4,
"answer_id": 0
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Expected value of game involving 100-sided die The following question is from a Jane Street interview.
You are given a 100-sided die. After you roll once, you can choose to either get paid the dollar amount of that roll OR pay one dollar for one more roll. What is the expected value of the game? (There is no limit on ... | Let $v$ denote the expected value of the game. If you roll some $x\in\{1,\ldots,100\}$, you have two options:
*
*Keep the $x$ dollars.
*Pay the \$$1$ continuation fee and spin the dice once again. The expected value of the next roll is $v$. Thus, the net expected value of this option turns out to be $v-1$ dollars.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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What's 4 times more likely than 80%? There's an 80% probability of a certain outcome, we get some new information that means that outcome is 4 times more likely to occur.
What's the new probability as a percentage and how do you work it out?
As I remember it the question was posed like so:
Suppose there's a student, T... | The only way I see to make sense of this is to divide by $4$ the probability it does not happen. Here we obtain $20/4=5$, so the new probability is $95\%$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/465718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "146",
"answer_count": 6,
"answer_id": 5
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Evaluating $\int_0^\infty \frac{1}{x+1-u}\cdot \frac{\mathrm{d}x}{\log^2 x+\pi^2}$ using real methods. By reading a german wikipedia (see here) about integrals, i stumpled upon this entry
27 1.5
$$ \color{black}{
\int_0^\infty \frac{1}{x+1-u}\cdot \frac{\mathrm{d}x}{\log^2 x+\pi^2} =\frac{1}{u}+\frac{1}{\log(1-u)}\,... | I'm not sure about the full solution, but there is a way to find an interesting functional equation for this integral.
First, let's get rid of the silly restriction on $u$. By numerical evaluation, the integral exists for all $u \in (-\infty,1)$
Now let's introduce the more convenient parameter:
$$v=1-u$$
$$I(v)=\int... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Can this difference operator be factorised? If a difference operator is defined as $$LY_i=\left(-\epsilon\dfrac{D^+ -D^-}{h_1}+aD^-\right)Y_i,\quad 1\leq i\leq N$$ Suppose $Y_N$ and $Y_0$ are given and that the difference operators are defined as follows $D^+V_i=(V_{i+1}-V_i)/h_1$, $D^-V_i=(V_i-V_{i-1})/h_1$. How is it... | $-\frac{\epsilon}{h^2} (Y_{i+1}-2Y_i + Y_{i-1}) + \alpha(Y_i-Y_{i-1}) = 0 \;\;\; (1)$
$-\frac{\epsilon}{h^2} Y_{N+1} + \left ( \frac{2\epsilon}{h^2} + \alpha \right )Y_N - \left ( \frac{\epsilon}{h^2} + \alpha \right ) Y_{N-1} = 0$
$\dots$
$Y_{N} = \frac{\epsilon Y_{N+1}}{2\epsilon + \alpha h^2}+\frac{\epsilon + \alpha... | {
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"timestamp": "2023-03-29T00:00:00",
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Is there a simpler way to express the fraction $\frac{x}{x+y}$? Can I simplify this expression, perhaps into two expressions $\frac{x}{x+y}$ or is that already simplified as much as possible?
| The given expression uses two operations (one division and one addition). If we judge simplicity by the number of operations, only an expression with one operation would be simpler, but the expression equals none of $x+y$, $x-y$, $y-x$, $xy$, $\frac xy$, $\frac yx$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/465932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Evaluating a 2-variable limit Could you help me evaluating this limit?
$$
\lim_{x\to 0}\frac{1}{x}\cdot\left[\arccos\left(\frac{1}{x\sqrt{x^{2}-
2x\cdot \cos(y)+1}}-\frac{1}{x}\right)-y\right]
$$
| Notice: I changed what I think a typo otherwise the limit is undefined.
By the Taylor series we have (and we denote $a=\cos(y)$)
$$\frac{1}{\sqrt{x^{2}-2xa+1}}=1+xa+x^2(\frac{3}{2}a^2-\frac{1}{2})+O(x^3)$$
so
$$\frac{1}{x\sqrt{x^{2}-2xa+1}}-\frac{1}{x}=a+x(\frac{3}{2}a^2-\frac{1}{2})+O(x^2)$$
Now using
$$\arccos(a+\al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/465973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Does $\det(A + B) = \det(A) + \det(B)$ hold? Well considering two $n \times n$ matrices does the following hold true:
$$\det(A+B) = \det(A) + \det(B)$$
Can there be said anything about $\det(A+B)$?
If $A/B$ are symmetric (or maybe even of the form $\lambda I$) - can then things be said?
| Although the determinant function is not linear in general, I have a way to construct matrices $A$ and $B$ such that $\det(A + B) = \det(A) + \det(B)$, where neither $A$ nor $B$ contains a zero entry and all three determinants are nonzero:
Suppose $A = [a_{ij}]$ and $B = [b_{ij}]$ are 2 x 2 real matrices. Then $\det(A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/466043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 5,
"answer_id": 1
} |
Three quotient-ring isomorphism questions I need some help with the following isomorphisms.
Let $R$ be a commutative ring with ideals $I,J$ such that $I \cap J = \{ 0\}$. Then
*
*$I+J \cong I \times J$
*$(I+J)/J \cong I$
*$(R/I)/\bar{J} \cong R/(I+J) \quad \text{where} \quad \bar{J}=\{x+I \in R/I: x \in J \}$
... | The line
$ x \in \ker(\phi) \ \iff \ x+ I \in I+J \iff x+I \in \bar{J} $
is wrong, because $x+I$ could be principally no element of $I+J$ since $I+J$ is an ideal which contains elements of $R$, and $x+I$ is a left coset of an ideal and hence also a set of elements of $R$. You could write instead
$ x + I \in \ker(\ph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/466170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How find this $3\sqrt{x^2+y^2}+5\sqrt{(x-1)^2+(y-1)^2}+\sqrt{5}(\sqrt{(x-1)^2+y^2}+\sqrt{x^2+(y-1)^2})$ find this follow minimum
$$3\sqrt{x^2+y^2}+5\sqrt{(x-1)^2+(y-1)^2}+\sqrt{5}\left(\sqrt{(x-1)^2+y^2}+\sqrt{x^2+(y-1)^2}\right)$$
I guess This minimum is $6\sqrt{2}$
But I can't prove,Thank you
| If $v_1 = (0,0), v_2 = (1,1), v_3 = (0,1)$, and $v_4 = (1,0)$ and $p = (x,y)$, then you are trying to minimize $$3|p - v_1| + 5|p - v_2| + \sqrt{5}|p - v_3| + \sqrt{5}|p - v_4|$$Note that if $p$ is on the line $y = x$, moving it perpendicularly away from the line will only increase $|p - v_1|$ and $|p - v_2|$, and it i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/466244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Partition Topology I am trying to prove the following equivalence:
"Let $X$ be a set and $R$ be a partition of $X$, this is:
i) $(\forall A,B \in R, A \neq B) \colon A \cap B = \emptyset$
ii) $ \bigcup_{A \in R} A = X$
We say that a topology $\tau$ on $X$ comes from is a partition topology iff $\tau = \tau(R)$ for some... | Alternative hint: $R$ consists of the closures of the one-point sets.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/466321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How can some statements be consistent with intuitionistic logic but not classical logic, when intuitionistic logic proves not not LEM? I've heard that some axioms, such as "all functions are continuous" or "all functions are computable", are compatible with intuitionistic type theories but not their classical equivalen... | If $A$ is a sentence (ie has no free variables), then your reasoning is correct and in fact $\neg (A \vee \neg A)$ is not consistent with intuitionistic logic.
However, all the instances of excluded middle that are contradicted by the statements "all functions are continuous" and "all functions are computable" are for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/466501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
A gamma function identity I am given the impression that the following is true (for at least all positive $\lambda$ - may be even true for any complex $\lambda$)
$$ \left\lvert \frac{\Gamma(i\lambda + 1/2)}{\Gamma(i\lambda)} \right\rvert^2 = \lambda \tanh (\pi \lambda) $$
It would be great if someone can help derive t... | Using the Euler's reflection formula
$$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z},$$
we get (for real $\lambda$)
\begin{align}
\left|\frac{\Gamma\left(\frac12+i\lambda\right)}{\Gamma(i\lambda)}\right|^2&=
\frac{\Gamma\left(\frac12+i\lambda\right)\Gamma\left(\frac12-i\lambda\right)}{\Gamma(i\lambda)\Gamma(-i\lambda)}=\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/466614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Showing that the function $f(x,y)=x+y-ye^x$ is non-negative in the region $x+y≤1,x≥0,y≥0$ ok, since it's been so long when I took Calculus, I just wanna make sure I'm not doing anything wrong here.
Given $f:\mathbb{R}^2\rightarrow \mathbb{R}$ defined as $f(x,y)=x+y-ye^x$. I would like to show that the function is nonn... | I'll try using Lagrange multiplier:
The function is:
$$f(x,y) = x + y - ye^x$$
and constraint are:
$$g(x,y) = x+y \leq 1$$
$$h(x) = x \geq 0$$
$$j(y) = y \geq 0$$
So using Lagrange multiplier now we have:
$$F(x,y,\lambda,\lambda_1,\lambda_2) = x + y - ye^x - \lambda(x+y-1) - \lambda_1(x) - \lambda_2(y)$$
Now we take pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/466695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Why is $\pi r^2$ the surface of a circle Why is $\pi r^2$ the surface of a circle?
I have learned this formula ages ago and I'm just using it like most people do, but I don't think I truly understand how circles work until I understand why this formula works.
So I want to understand why it works and not just how.
Ple... | The simplest explanation is that the area of any shape has to be in units of area, that is in units of length squared. In a circle, the only "number" describing it the the radius $r$ (with units of length), so that the area must be proportional to $r^2$. So for some constant $b$,
$$A=b r^2$$
Now, to find the constant $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/466762",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 0
} |
Primes between $n$ and $2n$ I know that there exists a prime between $n$ and $2n$ for all $2\leq n \in \mathbb{N}$ . Which number is the fourth number that has just one prime in its gap? First three numbers are $2$ , $3$ and $5$ . I checked with computer until $15000$ and couldn't find next one. Maybe, you can prove th... | There is no other such $n$.
For instance,
In 1952, Jitsuro Nagura proved that for $n ≥ 25$, there is always a prime between $n$ and $(1 + 1/5)n$.
This immediately means that for $n \ge 25$, we have one prime between $n$ and $\frac{6}{5}n$, and another prime between $\frac{6}{5}n$ and $\frac65\frac65n = \frac{36}{25}n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/466844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 1
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prove $\sum\limits_{n\geq 1} (-1)^{n+1}\frac{H_{\lfloor n/2\rfloor}}{n^3} = \zeta^2(2)/2-\frac{7}{4}\zeta(3)\log(2)$ Prove the following
$$\sum\limits_{n\geq 1}(-1)^{n+1}\frac{H_{\lfloor n/2\rfloor}}{n^3} = \frac{1}{2}\zeta(2)^2-\frac{7}{4}\zeta(3)\log(2)$$
I was able to prove the formula above and interested in what ... | The chalenge is interresting, but easy if we know some classical infinite sums with harmonic numbers : http://mathworld.wolfram.com/HarmonicNumber.html
( typing mistake corrected)
I was sure that the formula for $\sum\frac{H_{k}}{(2k+1)^3}$ was in all the mathematical handbooks among the list of sums of the same kind.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 1,
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$O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb R).$
Let $M(n,\mathbb R)$ be endowed with the norm $(a_{ij})_{n\times n}\mapsto\sqrt{\sum_{i,j}|a_{ij}|^2}.$ Then the set $O(n,\mathbb R)$ of all orthogonal matrices is a closed subset of $M(n,\mathbb R).$
My Attempt: Let $f:M(n,\mathbb R)... | It would be quicker to observe that $f$ is a vector of polynomials in the natural coordinates, and polynomials are continuous, so $f$ is continuous.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/467089",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
The origin of $\pi$ How was $\pi$ originally found?
Was it originally found using the ratio of the circumference to diameter of a circle of was it found using trigonometric functions?
I am trying to find a way to find the area of the circle without using $\pi$ at all but it seems impossible, or is it?
If i integrate th... | to answer at
"Is it at all possible to find the exact area of the circle without using π?"
hello, $A=CR/2$
"How was π originally found?"
maybe Pythagore and euclide with a²+b²=c² found the area of squares.
Then Archimede found $3+10/71<pi<3+1/7$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/467149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Abelian Groups and Number Theory What is the connection between "Finite Abelian Groups" and "Chinese Remainder Theorem"?
(I have not seen the "abstract theory" behind Chinese Remainder Theorem and also its proof. On the other hand, I know abstract group theory and classification of finite abelian groups. Please, give... | Let $m$ and $n$ be coprime, and let $a$ and $b$ be any integers. According to the Chinese remainder theorem, there exists a unique solution modulo $mn$ to the pair of equations
$$x \equiv a \mod{m}$$
$$x \equiv b \mod{n}$$
Now the map $(a,b) \mapsto x$ is an isomorphism of rings from $\mathbb{Z}/m\mathbb{Z} \oplus \mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
singleton null vector set linearly dependent, but other singletons are linearly independent set Why the set $\{\theta_v\}$ where $\theta_v$ is the null vector of a vector space is a dependent set intuitively (what is the source of dependence) and the singleton vector set which are non-null are independent sets ? (btw, ... | The intuition is the following: the null vector only spans a zero-dimensional space, whereas any other vector spans a one-dimensional space. This is captured by the following thought:
A set of vectors $\{ \bar v_1, \bar v_2, ..., \bar v_n\}$ is linearly independent iff $span(\{ \bar v_1, ..., \bar v_n\})$ is not spanne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Integral $ \lim_{n\rightarrow\infty}\sqrt{n}\int\limits_0^1 \frac {f(x)dx}{1 + nx^2} = \frac{\pi}{2}f(0) $
Show that for $ f(x) $ a continuous function on $ [0,1] $ we have
\begin{equation}
\lim_{n\rightarrow\infty}\sqrt{n}\int\limits_0^1 \frac {f(x)dx}{1 + nx^2} = \frac{\pi}{2}f(0)
\end{equation}
It is obvious th... | Hint: Make the change of variables $ y=\sqrt{n}x .$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/467562",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Is a locally compact space a KC-space if and only if it is Hausdorff? A topological space is called a $US$-space provided that
each convergent sequence has a unique limit.
We know that for locally compact spaces, $ T_{2} \equiv KC$.
We have:
$ T_2 \Rightarrow KC \Rightarrow US\Rightarrow T_1 $.... | GEdgar has given one example in the comments. Start with the ordinal space $\omega_1$, and add two points, $p$ and $q$. For each $\alpha<\omega_1$ let $U_\alpha(p)=\{p\}\cup(\alpha,\omega_1)$ and $U_\alpha(q)=\{q\}\cup(\alpha,\omega_1)$, and take $\{U_\alpha(p):\alpha<\omega_1\}$ and $\{U_\alpha(q):\alpha<\omega_1\}$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A sum of zeros of an infinite Hadamard product I was experimenting with pairs of zeros of the following function ($i$ = imaginary unit):
$\displaystyle \xi_{int}(s) := \xi_{int}(0) \prod_{n=1}^\infty \left(1- \frac{s}{+ ni} \right) \left(1- \frac{s}{{- ni}} \right) = \frac{\sinh(\pi s)}{s}$
and plugged these zeros into... | If $s\ne e^{2n\pi},\quad n\in \mathbb{Z}$, you have $$f(s)=i\ln\left(\frac{1-s^i}{1-s^{-i}}\right)=i\ln\left(\frac{1-s^i}{1-s^{-i}}\right)=i\ln(-s^i)\\ =i\ln((se^{(2k+1)\pi})^i)=i\ln\left(\left(r^ie^{(-\theta+i(2k+1)\pi)}\right)\right),\quad (k\in \mathbb{Z})\\= -((2k+1)\pi+\ln r)-i\theta$$where $s=re^{i\theta}$. So, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simple integration (area under the curve) - help I'm currently doing a simple integration question:
Here is my working/solution so far:
I have calculated this several times and only be seem to be getting a negative number as the final result. I know this is wrong as it is an area that needs to be calculated and ther... | It seems your work is in an image which is appearing weirdly on my screen. I'll just outline my work.
$ \displaystyle\int \left( 5 + \dfrac{5}{4\sqrt{x}} - x^4 \right) \, \mathrm{d}x = \displaystyle\int 5 \, \mathrm{d}x + \displaystyle\int 4 \cdot x^{\frac{1}{2}} \, \mathrm{d}x - \displaystyle\int x^4 \, \mathrm{d}x $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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What is the main use of Lie brackets in the Lie algebra of a Lie group? I am beginner in Lie group theory, and I can't find the answer a question I am asking myself : I know that the Lie algebra $\mathfrak g$ of a Lie group $G$ is more or less the tangent vector of $G$ at the identity, so that $\mathfrak g$ have a very... | A good question. There are many aspects of the situation... At least one fundamental structure can be understood in the following way. First, imagining that $t$ is an "infinitesimal", so that $t^3=0$ (not $t^2=0$!) (or equivalent...), and imagining that elements of the Lie group near the identity are $g=1+tx$ and $h=1+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 0
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Function that is identically zero Is it true that:
Any rational function $f$ on $\mathbb{C}^2$ that vanishes on $S=\{(x,y)\in\mathbb{C}^2 : x=ny \text{ for some } n \in \mathbb{Z}\}$ must be identically zero.
I have a theorem that says any rational function that vanishes on an open set in Zariski topology must be iden... | If the rational function $f = \frac{p}{q}$ vanishes on $S$, then at each point of $S$, so does either the polynomial $p$ or the polynomial $q$. Which means that the polynomial $pq$ vanishes on the whole of $S$. However, if this polynomial is non-zero, this means that $(x-ny)$ is a factor of $pq$ for all $n$, and theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/467933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Evaluating an improper integral using complex analysis I am trying to evaluate the improper integral $I:=\int_{-\infty}^\infty f(x)dx$, where
$$
f(z) := \frac{\exp((1+i)z)}{(1+\exp z)^2}.
$$
I tried to do this by using complex integration. Let $L,L^\prime>0$ be real numbers, and $C_1, C_2, C_3, C_4$ be the line segment... | \begin{eqnarray*}
\int_{-\infty}^{\infty}
{{\rm e}^{\left(1\ +\ {\rm i}\right)x} \over \left(1 + {\rm e}^{x}\right)^2}\,{\rm d}x
& = &
\int_{0}^{\infty}\left\lbrack%
{{\rm e}^{\left(-1\ +\ {\rm i}\right)x} \over \left(1 + {\rm e}^{-x}\right)^2}
+
{{\rm e}^{-\left(1\ +\ {\rm i}\right)x} \over \left(1 + {\rm e}^{-x}\righ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468019",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
Plotting large equation in mathematica I have this rather large equation which needs to be solved with respect to I1 so that I can plot it against X:
-1 - (0.742611 I1 (1/(-(-14 + 16/I2)^2 + (1 - 15/I1 + 16/I2)^2) - (
30 (1 - 15/I1 + 16/I2))/(
I1 (-(-14 + 16/I2)^2 + (1 - 15/I1 + 16/I2)^2)^2)) X^1.5)/((1.36-
... | Suppose we define equation as follows:
equation=yourBigEquation;
Now you can solve it numerically using NSolve producing a table of values (I select only $I1\in\mathbb{R}$ here; start from $X=10^{-10}$ because for $X=0$ there're no usable solutions):
sol=I1/.Table[NSolve[equation, I1, Reals], {X, 10^-10, 1, 1/500}];
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468175",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Do there exist some relations between Functional Analysis and Algebraic Topology? As the title: does there exist some relations between Functional Analysis and Algebraic Topology.
As we have known, the tools developed in Algebraic Topology are used to classify spaces, especially the geometrical structures in finite dim... | See Atiyah–Singer index theorem:
http://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/468272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 1
} |
Find the 12th term and the sum of the first 12 terms of a geometric sequence. A geometric series has a first term $\sqrt{2}$ and a second term $\sqrt{6}$ . Find the 12th term and the sum of the first 12 terms.
I can get to the answers as irrational numbers using a calculator but how can I can obtain the two answers in ... | So, the common ratio $=\frac{\sqrt6}{\sqrt2}=\sqrt3$
So, the $n$ th term $=\sqrt2(\sqrt3)^{n-1}\implies 12$th term $=\sqrt2(\sqrt3)^{12-1}=\sqrt2(\sqrt3)^{11}$
Now, $\displaystyle(\sqrt3)^{11}=\sqrt3 \cdot 3^5=243\sqrt3$
The sum of $n$ term is $\displaystyle \sqrt2\cdot\frac{(\sqrt3)^n-1}{\sqrt3-1}$
$\implies 12$th ter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Solve $\int \sqrt{7x + 4}\,dx$ I need to solve the following integral
$$\int \sqrt{7x + 4}\,dx$$
I did the following steps:
\begin{align}
\text{Let} \, u &= 7x+4 \quad \text{Let} \, du = 7 \, dx \\
\int &\sqrt{u} \, du\\
&\frac{2 (7x+4)^{3/2}}{3}
\end{align}
The solution is: $\frac{2 (7x+4)^{3/2}}{21}$. I am having so... | When you made the u-substitution, you took $u=7x+4$ and hence $du=7 dx$. You forgot this factor of 7! In particular, $dx=du/7$.
It helps to write out the $dx$ in the integral:
$$\int \sqrt{7x+4} dx=\int \frac{\sqrt{u}}{7} du.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/468397",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Trigonometric Identities Like $A \sin(x) + B \cos(y) = \cdots$ Are there any identities for trigonometric equations of the form:
$$A\sin(x) + B\sin(y) = \cdots$$
$$A\sin(x) + B\cos(y) = \cdots$$
$$A\cos(x) + B\cos(y) = \cdots$$
I can't find any mention of them anywhere, maybe there is a good reason why there aren't ide... | $A \, \cos(x) + B \, \cos(y)= C \, \cos(z)$, where,
$$ C = \sqrt{(A \, \cos(x) + B \, \cos(y))^2 + (A \, \sin(x) + B \, \sin(y))^2}, $$
and
$$ z = \tan^{-1}\left(\frac{A \, \sin(x) + B \, \sin(y)}{A \, \cos(x) + B \, \cos(y)}\right). $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/468475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Why is the Lebesgue-Stieltjes measure a measure? I'm having difficulty convincing myself the Lebesgue-Stieltjes measure is indeed a measure. The Lebesgue-Stieltjes measure is defined as such:
Given a nondecreasing, right-continuous function $g$, let $\mathcal{H}_1$ denote the algebra of half-open intervals in $\mathbb... | Firstly, note that the measure defined here is a Radon measure (that is $\lambda(B)<\infty$ for any bounded borel set $B$). hence it is also $\sigma$-finite (Because $\mathbb{R}=\bigcup_{n\in\mathbb{Z}}(n,n+1]$).
So if I can only show that the measure $\lambda$ is $\sigma$-additive on the semifield $\{(a,b]:-\infty\leq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468545",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Distance Between Subsets in Connected Spaces Suppose $\langle X, d \rangle$ is a metric space. For any two sets $F,G \subseteq X$, by abuse of notation define $d(F,G) = \inf \{ d(f,g): f \in F, g \in G \}$.
Let $\rho > 0$, $x \in X$, and $E \subseteq X$ be such that the open ball of radius $\rho$ centered at $x$ has ... | Counterexamples for connected spaces have already been given by Daniel Fischer and Stefan H.
It turns out that connectedness is somewhat tangential to the issue. The property you are after is inherited by dense subspaces, so it also applies to $\mathbb{Q}^n$ for example. That means it makes sense to look for condition... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Evaluating $\int_0^{\infty} {y^2 \cos^2(\frac{\pi y}{2}) \over (y^2-1)^2} dy$ I´m having trouble with the following integral
$$
\int_0^{\infty} {y^2 \cos^2(\frac{\pi y}{2}) \over (y^2-1)^2} dy
$$
I have tried lots of approaches and nothing works. Mathematica says it does not converge but that is not true. It appears in... | \begin{align*}
I
&=
{1 \over 2}\int_{-\infty}^{\infty}
{y^2 \cos^{2}\left(\pi y/2\right)
\over
\left(y^{2} - 1\right)^{2}}\,{\rm d}y
=
{1 \over 8}\int_{-\infty}^{\infty}y\cos^{2}\left(\pi y/2\right)\left\lbrack%
{1 \over \left(y - 1\right)^{2}}
-
{1 \over \left(y + 1\right)^{2}}
\right\rbrack
\,{\rm d}y
\\[5mm]&=
{1 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/468664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Calculate $\int \frac{dx}{x\sqrt{x^2-1}}$ I am trying to solve the following integral
$$\int \frac{dx}{x\sqrt{x^2-1}}$$
I did the following steps by letting $u = \sqrt{x^2-1}$ so $\text{d}u = \dfrac{x}{\sqrt{{x}^{2}-1}}$ then
\begin{align}
&\int \frac{\sqrt{x^2-1} \, \text{d}u}{x \sqrt{x^2-1}} \\
&\int \frac{1}{x} \tex... | $$
\begin{aligned}\int \frac{d x}{x \sqrt{x^{2}-1}} =\int \frac{1}{x^{2}} d\left(\sqrt{x^{2}-1}\right) =\int \frac{d\left(\sqrt{x^{2}-1}\right)}{\left(\sqrt{x^{2}-1}\right)^{2}+1} =\tan ^{-1}\left(\sqrt{x^{2}-1}\right)+C
\end{aligned}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/468727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 4
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Are the square roots of all non-perfect squares irrational? I was asked to define a non-perfect square. Now obviously, the first definition that comes to mind is a square that has a root that is not an integer. However, in the examples, 0.25 was considered a perfect square. And the square itself + its root were both no... | In the integers, a perfect square is one that has an integral square root, like $0,1,4,9,16,\dots$ The square root of all other positive integers is irrational. In the rational numbers, a perfect square is one of the form $\frac ab$ in lowest terms where $a$ and $b$ are both perfect squares in the integers. So $0.25... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Check condition normal subgroup in these three examples Is the subgroup H of G is a normal subgroup of G, for:
$$ i)\ G = S_5, \ H = \{id, (1,2)\} $$
$$ ii) \ G = (Sym(\mathbb{N}), \circ), \ H = \{f\in Sym(\mathbb{N}) : f(0) = 0 \}$$
$$ iii) \ G = S_4, \ H = \{id, (1,2,3), (1,3,2) \} $$
I know, that subgroup is normal,... | While the condition for a subgroup being normal is correct, it is often not the most efficient or at least intuitive to check this via using it.
Multiplying by $g^{-1}$ you get equivalently $gHg^{-1} = H$ for all $g \in G$.
So, to get you started you could just calculate $ghg^{-1}$ for a couple of $g\in G$ and $h \in ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why is lasso not strictly convex I know a nonmonotonic convex function which attains its minimum value at a unique point only is strictly convex. I didn't get how lasso is not strictly convex. For eg if I consider two dimensional case.
$||x||_1$ attains its minimum value at (0,0) which is unique
|
I thought that a strictly convex function is a convex function which has a unique minimizer. I am saying this was the wrong definition.
Indeed, this was quite wrong, and the source of confusion here. What is true is that strict convexity is related to uniqueness of minimizer: if a strictly convex function attains it... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Eigenvector/value in linear transformation. I came across this problem
For (a), I wrote $$T(3x,4y) = \lambda (3x, 4y)$$
Since $T$ is a reflection, $\lambda = 1$
That is as far as I got
(b) I simply have no idea. I only know how to find $A$ through brute force.
| Since the point $(4,3)$ is on the line, the reflection will take this point to itself; so this will be an eigenvector corresponding to $\lambda =1$.
Similarly, the point $(3,-4)$ is on the line through the origin perpendicular to the given line; so the reflection will take $(3,-4)$ to its negative $(-3,4)$
and so $(-3,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The unsolved mathematical light beam problem I have the following problem:
Imagine that you have a sphere sitting at the interface of two media(like water and oil). And the position(the heigth) of the interface to the center of the sphere is fixed and known(I drew a picture for two different situations). Now think abou... | Let us assume that the radius of the sphere is $R$, the height of the center above the separation plane is $h$ (if the center is below the plane, $h$ is negative, and the angle the rays make with the vertical line is $\beta\in(0,\pi/2)$ (it is $\beta=\frac \pi 2-\alpha$ on your picture).
Only the case $|h|<R$ is intere... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/469263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Need help creating a Context-Free Grammar I'm trying to generate a CFG for the following $L$, but I'm stuck on how to do this.
$$L = \{0^i1^j2^k\mid i<j+k\}$$
| Since I don't know anything about context-free grammars, I'll feel free to give what might be a full solution to half the problem, or might be completely wrong. Remember that last bit: might very well be completely wrong. The good news is that I would venture to guess that even if it's wrong, it's probably not complete... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/469367",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does the following inequality holds for any weak solution $u\in C^1(B_1)$ of uniformly elliptic equation $D_i(a_{ij}D_ju)=0$? Now I'm studying "Elliptic Partial Differential Equations" by Q.Han and F. Lin. Throughout the section 5 of the chapter 1, $u\in C^1(B_1)$ is a weak solution of
$$D_i(a_{ij}D_j u)=0$$
where... | Let $f(r)=\int_{B_r} u^2$. We can forget the whole PDE thing and just work with this nonnegative increasing function of $r$, which satisfies $$f(r/2)\le \theta f(r),\quad 0\le r\le 1\tag1$$
Given $0<\rho<r\le 1$, let $k$ be the largest integer such that $2^k\rho\le r$. (It's possible that $k=0$.) Applying the inequal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/469443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Are these proofs on partial orders correct? I'm trying to prove whether the following are true or false for partial orders $P_1$ and $P_2$ over the same set $S$.
1) $P_1$ ∪ $P_2$ is reflexive?
True, since $P_1$ and $P_2$ contains all the pairs { ($x,x$) : $x$ in $S$}, we know the union also does. Thus $P_1$ ∪ $P_2$ is ... | 1) Your answer is correct. One can just as easily say something stronger: if $R_1$ is a reflexive relation on a set $X$ and $R_2$ is any relation on $X$ containing $R_1$, then $R_2$ is also reflexive.
2) Your conclusion is correct, but you should nail it down by giving a specific counterexample. Can you find one? ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is the regularization of an otherwise diverging two-sided sum always equal to zero? As a first example, take the divergent series of all powers of two $1+2+4+8+...=\sum\limits_{k=0}^\infty 2^k$ which can be regularized by using the analytical continuation of the geometric series $\sum\limits_{k=0}^\infty q^k = \frac1{1... | It's too much to ask that the regularization of any two-sided divergent series be equal to zero. Clearly there is an extra symmetry in the examples you picked, both sides being given by the same expression. Otherwise, one could define either side separately to be any arbitrary divergent series and get all kinds of answ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Number of solutions of exponential equation Can anyone tell me how to find number of solutions $(x+a)^x=b$? For example $(x+1)^x=-1$ has four complex solutions, $(x+3)^x=10$ has two solutions,one positive one negative, and $(x-4)^x=-10$ hasn't any solutions.
PS.Sorry for my bad English,I hope you understand my question... | How to solve $(x+a)^x=b$?
Well, to begin, as you already gather, we must allow for $x \in \mathbb{C}$. Then the question begs the question how do we define $z^x$ for complex $x$. Use the complex exponential function:
$$ z^x = \exp (x \log (z)) $$
clearly $z=0$ is a problem. Moreover, this is a set of values due to the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/469675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How can I prove that a set of natural numbers always have a minimum? Let's say I have a finite not-empty set named A, which is a set of natural numbers.
How do I prove it has a minimum?
(In Calculus)
| You can't. Unless you add the condition that $A\ne \emptyset$.
Assume $A\subseteq \mathbb N$ has no monimal element.
Then prove by induction that $$\{1,\ldots,n\}\cap A=\emptyset$$ holds for all $n\in \mathbb N$.
The induction stecp $n\to n+1$ goes as follows: $\{1,\ldots,n\}\cap A$. If $n+1\in A$, this would imply tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating $E[(X-E[X])^3]$ by mgf
Calculate by mgf $E[(X-E[X])^3]$ where
a. $X\sim B(n,p)$
b.$X\sim N(\mu,\sigma)$
Before I begin I thought symbolizing $Y=X-E[X]$ and then I'd derivative $M_Y(t)$ three times substitute $t=0$ and solve both questions but I'm not sure about the distribution of Y.
My question is "Does ... | Yes, it changes the distribution. For one thing, the mean changes by that constant. In the binomial case, the distribution is no longer binomial. In the normal case, the new distribution is normal, mean $\mu-c$, variance $\sigma^2$, where $c$ is the constant you subtracted.
We will look at the problem in two ways. Th... | {
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"source": "stackexchange",
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I've solved this problem, but why is this differentiable? Let $\alpha:\mathbb R\rightarrow \mathbb R^3$ be a smooth curve (i.e., $\alpha \in C^\infty(\mathbb R)$). Suppose there exists $X_0$ such that for every normal line to $\alpha$, $X_0$ belongs to it. Show that $\alpha$ is part of a circumference.
Part of solution... | You can write
$$\lambda(s)=\langle X_0-\alpha(s), n(s)\rangle$$
supposing $\|n(s)\|=1$ (otherwise you divide by this norm).
That expression is obtained with sums and products of smooth functions (at most divisions by non-vanishing smooth functions), hence it is smooth.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Fundamental group of a complex algebraic curve residually finite? Is the analytic fundamental group of a smooth complex algebraic curve (considered as a Riemann surface) residually finite?
| Yes. Recall that topologically such a surface is a $g$-holed torus minus $n$ points. Except in the cases $(g, n) = (1, 0), (0, 0), (0, 1), (0, 2)$ such a surface, call it $S$, has negative Euler characteristic, so by the uniformization theorem its universal cover is the upper half plane $\mathbb{H}$. Since the action o... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $X$ be such that $S=e^x$. You are given that $M_X(u)=e^{5u+2u^2}$ Suppose, for the stock market, the price of a certain stock S has density function $f_S(s)=\frac{1}{ts\sqrt{2\pi}} e^{\frac{-1}{2}\left(\frac{\ln(s)-m}{i}\right)^2}$ where $S>0$ and $-\infty<m<\infty $ and $t>0$ are constants.
Let $X$ be such that $... | We are told that $X\gt \ln(50)$, and want to find the probability that $\ln(70)\lt X\lt \ln(90)$.
Let $A$ be the event $\ln(70)\lt X\lt \ln(90)$, and $B$ the event $X\gt \ln(50)$. We want $\Pr(A|B)$.
This is $\dfrac{\Pr(A\cap B)}{\Pr(B)}$.
Note that in our case we have $A\cap B=A$. So we need to find $\Pr(A)$ and $\Pr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof: for a pretty nasty limit Let $$ f(x) = \lim_{n\to\infty} \dfrac {[(1x)^2]+[(2x)^2]+\ldots+[(nx)^2]} {n^3}$$.
Prove that f(x) is continuous function.
Edit: $[.] $ is the greatest integer function.
| In general,
if $[z]$ is the integer part
of $z$,
$D(n)
=\sum_{k=1}^n f(kx)
-\sum_{k=1}^n [f(kx)]
=\sum_{k=1}^n (f(kx)-[f(kx)])
$.
Since $0 \le z-[z] < 1$,
$0 \le D(n) < n$.
For your case,
$f(x) = x^2$,
so
$0 \le \sum_{k=1}^n (kx)^2
-\sum_{k=1}^n [(kx)^2]
< n$
or
$0 \le \dfrac1{n^3}\sum_{k=1}^n (kx)^2
-\dfrac1{n^3}\sum_... | {
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"url": "https://math.stackexchange.com/questions/470247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Continuous bounded functions in $L^1$ Is a continuous function in $L^1$bounded?
I know that continouos functions are always bounded on a compact intervall.
But how do I prove it?
| If we study $L^1(0,\infty)$ or $L^1(\Bbb R)$, then you can have an unbounded continuous integrable function.
We can build it in the following way: let's start with $f= 0$. Then we add positive continuous "bumps" at $n=1,2$,etc with each "bump" being higher, say, of magnitude $n$, but adding only $1/n^2$ to the integra... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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how many elements are there in this field $\mathbb Z_2[x]/\langle x^3+x^2+1\rangle $, I understand it is a field as $\langle x^3+x^2+1\rangle $ ideal is maximal ideal as the polynomial is irreducible over $Z_2$.
but I want to know how many elements are there in this field and how to find out that?
| You have $x^3=x^2+1$ so you can eliminate powers of $x$ greater than $2$ and every element of the field is represented by a polynomial of order less than $3$.
Such polynomials have form $ax^2+bx+c$ and there are two choices in $\mathbb Z_2$ for each of $a,b,c$ so eight candidates. It remains to confirm that the eight e... | {
"language": "en",
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A proof in vectors If it is given that:
$$ \vec{R} + \dfrac{\vec{R}\cdot(\vec{B}\times(\vec{B}\times\vec{A}))} {|\vec{A} \times \vec{B} |^2}\vec{A} + \dfrac{\vec{R}\cdot(\vec{A}\times(\vec{A}\times\vec{B}))} {|\vec{A} \times \vec{B} |^2}\vec{B} = \dfrac{K(\vec{A}\times\vec{B})} {|\vec{A} \times \vec{B} |^2} $$
then pro... | Let $\vec{r} = x \vec{a}+y \vec{b}+ z \left(\vec {a} \times \vec {b}\right)$
If we wish to compute, say $x$, we need to take dot product with a vector that is perpendicular to $\vec{b}$ as well as $\left(\vec {a} \times \vec {b}\right)$ and that vector is $\vec{b} \times \left(\vec {a} \times \vec {b}\right)$
We will n... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Does a graph contain a 3-cycle or a 4-cycle Given a graph $\mathscr G$, that has 100 nodes each with a degree can you show that this graph contains a 3-cycle and/or a 4-cycle?
The graph in question represents 100 people at an event, and they each know $\ge 70$ people. I need to prove that there a 3 people there who kn... | Divide the group in half. The fifty people in one half must each know at least 20 people in their half. The maximal girth 5 cage graph for 50 vertices is the Hoffman-Singleton graph, with degree 7.
If each person knows more than 57 people, a 3 or 4 cycle is also forced.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/470509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Are there any integer solutions to $2^x+1=3^y$ for $y>2$? for what values of $ x $ and $ y $ the equality holds
$2^x+1=3^y$
It is quiet obvious the equality holds for $x=1,y=1$ and $x=3,y=2$.
But further I cannot find why $x$ and $y$ cannot take higher values than this values.
| I am assuming that $x$ and $y$ are supposed to be integers.
The claim follows from the fact that if $x>3$, then the order of the residue class of $3$ in the group $\mathbb{Z}_{2^x}^*$ (sometimes denoted by $U_{2^x}$)
is $2^{x-2}$. In other words: for $3^y-1$ to be divisible by $2^x$ the exponent $y$ has to be a multipl... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Counting votes, as long as one has more votes all the way through.
*
*Two competitors won $n$ votes each.
How many ways are there to count the $2n$ votes, in a way that one competitor is always ahead of the other?
*One competitor won $a$ votes, and the other won $b$ votes. $a>b$.
How many ways are there to count the... | We can use a similar idea to the solution to #1:
The total number of ways the votes can be counted is $a+b \choose a$, so we have to subtract the number of ways competitor B can get ahead of competitor A:
In any count where B gets ahead of A, change all the votes up to and including the first vote where B takes the le... | {
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"timestamp": "2023-03-29T00:00:00",
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Calculate the determinant of a matrix multiplied by itself confirmation If $ \det B = 4$ is then is $ \det(B^{10}) = 4^{10}$?
Does that also mean that $\det(B^{-2}) = \frac{1}{\det(B)^2} $
Or do I have this completely wrong?
| $det(B^2)$ means $ det(B*B)$ means at first, By multiplying two matrices you are getting a matrix and then you are finding determinant of that matrix
Similarly, $det(B^{10})$ means at first you are multiplying 10 matrices , that will give you a matrix and then finding the determinant
But in the case of the negative pow... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Generalised composition factors Let $A$ be a semiprimary ring.
A simple module $L$ is said to be a generalised composition factor of $M$ if there are $M'$ and $M''$, $M'' \subset M'$, submodules of $M$, such that $M'/M'' \cong L$.
Suppose $L$ is a generalised composition factor of $M$. Is it possible to have a submodul... | I feel very foolish for just noticing this now...
The answer to the question is no.
Let $P$ be the projective cover of the simple module $L$ (and let $Q$ be its injective hull). It is easy to see that $L$ is a generalised composition factor of a module $M$ if and only if $\operatorname{Hom}_A(P,M)\neq 0$ (if and only i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/470763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finite set of congruences Is it true that for every $c$ there is a finite set of congruences
$a_i(mod\,\,n_i) , c = n_1<n_2<n_3<...........<n_k \,\,\, (1)\\
$
So that every integer satisfies at least one of the congruence (1)
| You are referring to Covering Systems of congruences. The link, and the name, will let you explore what is a quite large literature. You may also want to look at this survey by Carl Pomerance. If the $n_i$ are strictly increasing, they cannot be chosen arbitrarily.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/470845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that the Euler characteristic of $O[3]$ is zero.
Show that the Euler characteristic of $O[3]$ is zero.
Consider a non zero vector $v$ at the tangent space of identity matrix. Denote the corresponding matrix multiplication by $\phi_A$. Define the vector field $F$ by $F(A)=(\phi_A)_*(v)$. Where $\phi_*$ is the der... | Are you familiar with the theory of Lie Groups? You can just take any non-zero vector at the identity and translate it everywhere, generating a non-vanishing smooth vector field on $O(3)$. From here it's easy with Poincare-Hopf.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/470903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Limit of $x \log x$ as $x$ tends to $0^+$ Why is the limit of $x \log x$ as $x$ tends to $0^+$, $0$?
*
*The limit of $x$ as $x$ tends to $0$ is $0$.
*The limit of $\log x$ as $x$ tends to $0^+$ is $-\infty$.
*The limit of products is the product of each limit, provided each limit exists.
*Therefore, the limit of ... | By $x=e^{-y}$ with $y \to \infty$ we have
$$x \log x =-\frac y {e^y} \to 0$$
which can be easily proved by the definition of $e^y$ or by induction and extended to reals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/470952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "45",
"answer_count": 5,
"answer_id": 4
} |
Solving Riccati-like matrix inequality How can I find P2 in the following inequality?
$$S_{ai} = Q + G P_2 G'- P_2 - \bar{\gamma}^2 G P_2 C' R^{-1} C P_2 G' \prec 0$$
where $R = \beta C P_2 C'+ V$, $\beta=0.9$, $\bar{\gamma} = 0.9$ and
G =
0.5437 0.0768
0.2040 -1.1470
C =
1 0
0 1
Q =
0.1363 0.1527
0.1... | $$
\begin{align}
Q + G P_2 G'- P_2 - \bar{\gamma}^2 G P_2 C' R^{-1} C P_2 G' &\prec 0 \\[5mm]&\Updownarrow \textrm{if }R\prec 0\\[5mm]
\begin{pmatrix}Q + G P_2 G'- P_2 &\bar{\gamma} G P_2 \\ \bar{\gamma}P_2 G'& \beta(C'C)P_2(C'C)\end{pmatrix}&\prec 0
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/471040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can I decompose a compact set in a finite number of convex set? My problem is in a finite-dimensional space. I look at $\mathcal{X}$ the support of a function $f$, that is continuous and has bounded support.
\begin{eqnarray}
\mathcal{X}_o & = & \{x \in \Omega, f(x)>0 \} \\
\mathcal{X} & = & \bar{\mathcal{X}}_o \\
\lVe... | As Daniel Fischer said (in a comment, unfortunately), the answer is negative. One counterexample is a round annulus $1\le |x|\le 2$ in dimension $2$ (or higher).
Any convex subset of that can meet at most one point of the inner bounding circle.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/471204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Defining $W^{k,p}(M)$ for non-integers $k$ and $p$ and manifold $M$ For $k$ and $p$ not necessarily integer, and on a smooth manifold $M$, how to define the Sobolev space $W^{k,p}(M)$? I've only seen definitions for $p=2$.
| For an integer $k$, one defines $W^{k,p}$ to be (roughly speaking) the space of $L^p$ functions whose $k$-th derivative is $L^p$.
More precisely, the Fourier transform takes a degree-$k$ differential operator to a degree-$k$ polynomial, so we can use weak derivatives to reconceptualize $W^{k,p}$ (for integer $k$ still)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
equivalence between problem and his variational formulation let $\Omega$ an open bounded and regular in $\mathbb{R}^n$, and let $\overline{\Omega}_1$ and $\overline{\Omega}_2$ and partition to $\Omega:$ $\Omega = \overline{\Omega_1} \cup \overline{\Omega_2}.$ We put $\Gamma=\partial \Omega_1 \cap \partial \Omega_2$ the... | Once you are sure your weak formulation is right, suppose your weak solution is regular enough that you can integrate by parts, then do it and choose your test functions $v$ in spaces of smooth functions. You'll probably need to do this several times, once for each boundary condition.
For example you may first try with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability defective items Two shipments of parts are received. The first shipment contain 1000 parts with 10% defective and the second contain 2000 parts with 5% defectives. Two parts are selected from a shipment selected at random and they are found to be good. Find the probability that the tested parts were from 1s... | Let $A$ be the event the parts were selected from the first shipment, and let $G$ be the event they are both good.
We want the probability they are from the first shipment, given they are both good. So we want $\Pr(A|G)$.
By the definition of conditional probability, we have
$$\Pr(A|G)=\frac{\Pr(A\cap G)}{\Pr(G)}.\tag... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471403",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Property of cyclic quadriterals proof! http://en.wikipedia.org/wiki/Cyclic_quadrilateral
This article states that: "Another necessary and sufficient condition for a convex quadrilateral ABCD to be cyclic is that an angle between a side and a diagonal is equal to the angle between the opposite side and the other diagona... |
The proof is based on the following theorem:
The angles subtended by a chord at the circumference of a circle are
equal, if the angles are on the same side of the chord
Hence $\angle{ACB} = \angle{ADB}-----------> 1$
similarly $ \angle{CBA} = \angle{CDA} ------------------>2$
Adding 1 and 2 .
$\angle{ACB} + \angle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Prove AB is hermitian if A is hermitian and B is hermitian If $A$ and $B$ are two hermitian transformations, prove that $AB$ is hermitian if $AB = BA$, knowing that a hermitian transformation is one such that $(T(f), g) = (f, T(g))$ and basic axioms for inner products: $(x,y) = (y,x)$, $(x,x) > 0$, $(cx, y) = c(x,y)$, ... | $$\langle ABf, g \rangle = \langle Bf, Ag \rangle = \langle f, BAg \rangle = \langle f, ABg \rangle.$$
Here
*
*I "=" $\longleftarrow$ $A$ is Hermitian,
*II "=" $\longleftarrow$ $B$ is Hermitian,
*III "=" $\longleftarrow$ $AB = BA$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/471546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Archimedean property concept I want to know what the "big deal" about the Archimedean property is. Abbott states it is an important fact about how $\Bbb Q$ fits inside $\Bbb R.$
First, I want to know if the following statements are true:
The Archimedean property states that $\Bbb N$ isn't bounded above--some natural nu... | Your statements, strictly interpreted,
are not true.
You need to change the order of quantifiers.
You say
"The Archimedean property states that $\Bbb{N}$ isn't bounded above--some natural number can be found such that it is greater than any real number."
There is no natural number "such that
it is greater than any real... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 2
} |
Probability of being up in roulette A player bets $\$1$ on a single number in a standard US roulette, that is, 38 possible numbers ($\frac{1}{38}$ chance of a win each game). A win pays 35 times the stake plus the stake returned, otherwise the stake is lost.
So, the expected loss per game is $\left(\frac{1}{38}\right)(... |
This is perhaps surprising as it seems to suggest you can win at
roulette if you play often enough.
The most optimal way of "winning" at roulette is bold play. Bet your entire fortune or the amount desired to "end up," whichever is less.
However, in other sub-fair games when $p > 1/2$ then timid play is optimal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/471659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
When colimit of subobjects is still a subobject? What are the conditions on a category (or on a certain object) that will guarantee that the colimit of a family of subobjects of a given object is a subobject of the same object?
Update: To clarify the question - let $C$ be a category with arbitrary colimits. Consider a ... | I assume that your question is: If $\{X_i \to X\}$ is a diagram of subobjects of $X$, and $\mathrm{colim}_i X_i$ is a colimit in the ambient category, when is the induced morphism $\mathrm{colim}_i X_i \to X$ again a monomorphism and therefore exhibits the colimit as a subobject of $X$?
Well without restrictions, of co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/471809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Prove that the sequence $(3n^2+4)/(2n^2+5) $ converges to $3/2$
Prove directly from the definition that the sequence $\left( \dfrac{3n^2+4}{2n^2+5} \right)$ converges to $\dfrac{3}{2}$.
I know that the definition of a limit of a sequence is $|a_n - L| < \varepsilon$ .
However I do not know how to prove this using thi... | Hints: for an arbitrary $\,\epsilon>0\;$ :
$$\left|\frac{3n^2+4}{2n^2+5}-\frac32\right|=\left|\frac{-7}{2(2n^2+5)}\right|<\epsilon\iff2n^2+5>\frac7{2\epsilon}\iff$$
$$(**)\;\;2n^2>\frac7{2\epsilon}-5\ldots$$
Be sure you can prove you can choose some $\,M\in\Bbb N\;$ s.t. for all $\,n>M\;$ the inequality (**) is true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/471896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 2
} |
Sphere-sphere intersection is not a surface In my topology lecture, my lecturer said that when two spheres intersect each other, the intersecting region is not a surface. Well, my own understanding is that the intersecting region should look like two contact lens combine together,back to back. The definition of a surfa... | In general 2 spheres on $\mathbb{R}^3$ intersect on a circle which is a curve as you can simply imagine. You don't need to see a picture to visualize that I think..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/471970",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How does definition of nowhere dense imply not dense in any subset? In some topological space $X$, a set $N$ is nowhere dense iff $\text{Int}\left(\overline{N}\right)=\emptyset$,
where Int is the interior, and overbar is closure.
How can I show this is equivalent to the statement "any non-empty
open subset of $X$ conta... | Note that $\overline N^c \text{ is dense in }X \Longleftrightarrow \overline N^c\cap U\neq \emptyset \text{ for all }U\neq \emptyset \text{ open in }X$. Note further that $\overline N^c$ is open. Now, let $U$ be a non-empty open subset of $X$, by the above $V=\overline N^c\cap U$ is a non-empty open subset of $U$ conta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/472074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
How to solve this difficult system of equations? $$1+4\lambda x^{3}-4\lambda y = 0$$
$$4\lambda y^{3}-4\lambda x = 0$$
$$x^{4}+y^{4}-4xy = 0$$
I can't deal with it. How to solve this?
| As noted in several comments, the second equation yields $x=y^{3}$. Hence, by the third equation, $$y^{12}=3y^{4} \implies y^{8}=3$$
Assuming $y$ is real, we get $y=\pm 3^{1/8}$ and $x=\pm3^{3/8}$. By the first equation, then, (taking the positive roots): $$1+4\lambda3^{9/8}-4\lambda3^{1/8}=0 \implies 8\lambda3^{1/8}=-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/472125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
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