Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
A solution to a Diophantine equation with three unknowns (similar to the Fermat Theorem) I need to either solve the following Diophantine equation with three unknowns:
$n_{1} (n_{1} +1) \pm n_{2} (n_{2} +1) \pm n_{3} (n_{3} +1) = 0$,
where $n_{1,2,3}$ can be positive or negative -- or perhaps prove that this equation d... |
Here is an answer to the question as originally posted. Later on, the OP has somewhat displaced the goalposts.
If $n$ is an integer, no matter what its sign is, then $n(n-1)$ will be either zero or positive, with zero exactly if $n=0$ or $n=1$.
Since neither of the terms in your equation can be negative, the only way... | {
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"timestamp": "2023-03-29T00:00:00",
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Ahlfors' Complex Analysis proof doubt On the third edition of Ahlfors' Complex Analysis, page 121 Lemma 3 it states:
Suppose $\phi(\zeta)$ is continuous on the arc $\gamma$. Then the function
\begin{equation*}
F_n(z)= \int_{\gamma} \frac{\phi(\zeta)}{(\zeta-z)^n} d\zeta
\end{equation*}
is analytic in each of the regio... | Don't let $\delta$ get smaller! Leave $\delta$ fixed. Say $c=\frac1{\delta^2}\int_\gamma|\phi|\,|d\zeta|.$ You have $$|F_1(z)-F_1(z_0)|<c|z-z_0|.$$ Hence $F_1(z)\to F_1(z_0)$ as $z\to z_0$. (For instance, $|z-z_0|<\epsilon/c$ implies $|F_1(z)-F_1(z_0)|<\epsilon$.)
| {
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"question_score": "1",
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"answer_id": 0
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The vector $\mathbf{x}$ is the derivative of $\mathbf{Ax}$ with respect to what? Consider the linear equation
$$\mathbf{A}\mathbf{x}=\mathbf{b},$$
where
*
*$\mathbf{A}=\begin{bmatrix}\mathbf{a}_1 \\ \vdots \\ \mathbf{a}_n\end{bmatrix}$ is an $N \times K$ random matrix
*$\mathbf{x}=\begin{bmatrix}x_1 \\ \vdots \\ x_... | With the usual notations, derivating with respect to a vector increases the dimension by 1, and derivating wrt to a matrix increases the dimension by 2. For example, if $s$ is a scalar, $v$ a vector and $A$ a matrix, $\partial_A s$ is a matrix, $\partial_v s$ is a vector and $\partial_v v$ is a matrix.
In your case, yo... | {
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 1
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To prove following is a metric space I have $X= \mathbb{R^2}$ $d(x,y)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$.
I am having trouble on how to show triangle inequality property while other two are quite trivial. What is method of doing this?
Thanks
| In $\mathbb R^n$, you can use the dot product as a tool to construct this type of proof. The triangle inequality is in fact a direct consequence of the Cauchy-Schwarz inequality $(\mathbf a\cdot\mathbf b)(\mathbf a\cdot\mathbf b)\le(\mathbf a\cdot\mathbf a)(\mathbf b\cdot\mathbf b)$. There are many proofs, but the simp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1892620",
"timestamp": "2023-03-29T00:00:00",
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Stuck in integration: $\int {\frac{dx}{( 1+\sqrt {x})\sqrt{(x-{x}^2)}}}$
$\displaystyle\int {\frac{dx}{( 1+\sqrt {x})\sqrt{(x-{x}^2)}}}$
$=\displaystyle\int\frac{(1-\sqrt x)}{(1+x)\sqrt{x-x^2}}\,dx$
$=\displaystyle\int\frac{(1-\sqrt x+x-x)}{(1+x)\sqrt{x-x^2}}\,dx$
$=\displaystyle\int\frac{\,dx}{\sqrt{x-x^2}}-\disp... | By subtsitution twice we get $t=\sqrt { x } \Rightarrow dt=\frac { dx }{ 2\sqrt { x } } $ $$\int { \frac { dx }{ \left( 1+\sqrt { x } \right) \sqrt { x-{ x }^{ 2 } } } } =2\int { \frac { d\sqrt { x } }{ \left( 1+\sqrt { x } \right) \sqrt { 1-x } } = } 2\int { \frac { dt }{ \left( 1+t \right) \sqrt { 1-{ t }^{ 2 ... | {
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Help solving $1 < \frac{x + 3}{x - 2} < 2$ I worked a lot of inequalities here in MSE and that greatly helped me.
I've seen a similar inequality [here] [1], but the one I have today is significantly different, in that I'll end up with a division by 0, which is not possible.
[1] [Simple inequality
$$1 < \frac{x + 3}{x -... | Break it into two separate inequalities when you get to here:
$$ 0 < \frac{5}{x-2} < 1$$
First let's consider $0 < \dfrac{5}{x-2}$. Since $5 > 0$, this inequality is satisfied when $x-2 > 0$, i.e., when $x > 2$.
Now let's consider $\dfrac{5}{x-2} < 1$. Subtract $1$ from both sides and get a common denominator to get ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the fourier series representation of a function
Consider the function
$f(x) =
\begin{cases}
\frac{\pi}{2}+x & & x \in (-\pi, 0] \\
\frac{\pi}{2}-x & & x \in (0, \pi]\\
\end{cases}$
extended 2$\pi$ periodically to $\mathbb{R}$. Calculate $a_0, a_n, b_n$
I understand how to work out a fourier series but I am ... | Divide it two parts and calculate $$a_{ 0 }=\frac { 1 }{ \pi } \int _{ -\pi }^{ \pi } f\left( x \right) dx=\frac { 1 }{ \pi } \int _{ -\pi }^{ 0 }{ \left( \frac { \pi }{ 2 } +x \right) dx } +\frac { 1 }{ \pi } \int _{ 0 }^{ \pi }{ \left( \frac { \pi }{ 2 } -x \right) dx } =\\ ={ \left( \frac { \pi }{ 2 } x+\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1893008",
"timestamp": "2023-03-29T00:00:00",
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Are representations of finite groups unitary? I'm reading through a proof where $\Psi:G\rightarrow GL(V)$ and $\Phi:G\rightarrow GL(U)$ are representations of a finite group $G$, $a\in U$, $b\in V$, and $R:U\rightarrow V$ is a linear function. Then there is a line in the proof that goes:
$$\sum_{g\in G}\langle b,\Psi(g... | No, it does not. For example, the representation of the $2$-element group $\{e,a\}$ by
$$ 1 \to \pmatrix{1 & 0\cr 0 & 1\cr},\ a \to \pmatrix{1 & 1\cr 0 & -1\cr} $$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Inequality on non-negative reals For non-negative $x,y,z$ satisfy $\frac{1}{2x^2+1}+\frac{1}{2y^2+1}+\frac{1}{2z^2+1}=1$ then show that $x^2+y^2+z^2+6\geq 3(x+y+z)$
Idea how to handle the constraint? I'm unaware .
| Yes, it's true for all reals.
$\sum\limits_{cyc}(x^2-3x+2)=\sum\limits_{cyc}\left(x^2-3x+2-\frac{9}{4}\left(\frac{1}{2x^2+1}-\frac{1}{3}\right)\right)=\sum\limits_{cyc}\frac{(x-1)^2(2x-1)^2}{2(2x^2+1)}\geq0$.
Done!
| {
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Is the absolute value function a metric? I was trying to find out more information about absolute value, and I came upon the fact that AV satisfies a whole set of properties that usually defines a distance function or metric. But in the Wikipedia article on metrics, there's no mention of the AV function, so I'm a bit c... | The absolute value $x\mapsto |x|$ is not a metric but a norm on $\mathbb R$ (or $\mathbb C$), viewed as a one-dimensional vector space. However, from any norm you can derive a metric in a standard way.
In the case of the absolute value, this gives the well-known metric $d(x,y)=|x-y|$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Can someone explain this part of this property of invertible matrices proof? The property states, "A square matrix A is invertible iff it can be written as the product of elementary matrices"
I'm confused on the part of the theorem where they're trying to show that if A is invertible, then it can be written as the prod... | The point is that each step in the process of Gauss-Jordan elimination corresponds to multiplying your matrix on the left by an elementary matrix.
If you start with $[A\mid b]$ (where $A$ is your matrix and $b$ the augmented column), you get $[E_1 A \mid E_1 b$ in the first step, for some elementary matrix $E_1$, then ... | {
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If $x - ε ≤ y$ for all $ε>0$ then $x ≤ y$ I've been asked to prove the following,
if $x - ε ≤ y$ for all $ε>0$ then $x ≤ y$.
I tried proof by contrapositive, but I keep having trouble choosing the right $ε$. Can you guys help me out?
| Suppose $x > y \implies \epsilon = x-y > 0 \implies x = y + \epsilon > y + \dfrac{\epsilon}{2}$, contradiction.
| {
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A convergence problem in real non-negative sequence, $\sum_{n=1}^\infty a_n$. We now have $a_n\geq 0$, $\forall n=1,2,...,$ and $\sum_{n=1}^\infty a_n <\infty$. Then I guess that $\lim_{n\to\infty} a_n \cdot n = 0$. But I realized that it is wrong. Since if we let $a_n = 1/n $ if $n = 2^i$ for some $i=1,2,...$ and $a_... | It is true that $a_n \cdot n \rightarrow 0$ under this condition.
To prove this, observe that if the series is convergent, the sequence of partial sums is a Cauchy sequence. In particular, for every $\varepsilon > 0$, there is some $N_0 = N_0(\varepsilon)$ such that whenever $n > m > N_0$, $\left| \sum_{i=1}^n a_i - \... | {
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Sum/Intersection of Invariant Subspaces that have Invariant Complements Let $V$ be a finite-dimensional vector space, and let $M: V \rightarrow V$ be a linear transformation. Suppose subspaces $S_1,S_2 \subset V$ are both $M$-invariant and have $M$-invariant complements (say $W_1$ and $W_2$, so $V = S_i \oplus W_i$).
Q... | False! Counterexample: $V = \mathbb{R}^4$ and
$$
M = \begin{bmatrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{bmatrix} \,.
$$
Take the two $M$-invariant subspaces $S_i$ (and $M$-invariant complements $W_i$)
$$
S_1 = \mathrm{span} \left\{ \begin{bmatrix} 1\\0\\0\\0 \end{bmatrix}, \begin{bmat... | {
"language": "en",
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Equivalence $\displaystyle((\lnot p \lor q) \land (q \lor r)) \land (p \land \lnot q) ≡ c$ I am trying to prove this. Using only laws. I am getting nowhere. The 3 variables on the left hand side keep tripping me up. I've tried DeMorgans, Distributive, Identity and Negation as a starting point but hit dead ends.
| Distributing,
$$((\lnot p \lor q)\land (q\lor r)) \iff (((\lnot p \lor q)\land q ) \lor ((\lnot p \lor q)\land r)) \iff q\,\lor ((\lnot p \lor q)\land r)$$
$$\iff (\lnot p \lor q)\land (q\lor r) \iff \lnot(p \land \lnot q)\land (q\lor r)$$
Then the original expression is
$$\lnot(p \land \lnot q)\land (q\lor r) \land (... | {
"language": "en",
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How should we calculate difference between two numbers? if we are told to find the difference between 3 and 5,
then we usually subtract 3 from 5 ,5-3=2
and thus, we say that the difference is 2.
but why can't we subtract 5 from 3 ,3-5= -2 to get the
difference -2??
which result is right?
is the d... | Traditionally, the “difference" between two numbers refers to the distance on a number line between the points corresponding to each of the two numbers, a.k.a. the absolute value. Analogously, if you asked “What is the distance from Toronto to Vancouver?” or "What is the distance from Vancouver to Toronto?", you would ... | {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "8",
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Involute of a circle - what is the separation distance? It seems like a simple enough question. For the involute of a circle, what is the separation distance between successive turns?
Is this derivation correct?
Parametric formula for the y-coordinate:
$ y = r(Sin(\theta) - \theta Cos(\theta)) $
Differentiating:
$ \fra... | We can represent the parametric equations of the circle involute, by factoring out the
radius $r$ of the generating circle, which is just a scale factor.
$$
\left\{ \begin{gathered}
x = X/r = \cos \theta + \theta \sin \theta \hfill \\
y = Y/r = \sin \theta - \theta \cos \theta \hfill \\
\end{gathered} \right... | {
"language": "en",
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$f''(x)\ge 0$, $f'(0)=1$ and $f(x)\le 100$. Does such a function exist? $f''(x)\ge 0$, $f'(0)=1$ and $f(x)\le 100$. Does such a function exist?
I can show that $100-e^{-x}$ does not satisfy the given condition. But I have to show that no function can satisfy the initial condition.
I can also say that since $f'$ is inc... | Suppose $f(x) \leq 100$ for all $x$. Now $f'(0) = 1$ and $f''(x) \geq 0$ for all $x$ hence $f'$ is increasing and therefore $f'(x) \geq f'(0) = 1$ for all $x \geq 0$. Hence $f$ is strictly increasing on $[0, \infty)$ and since $f(x) \leq 100$ it follows that $\lim_{x \to \infty}f(x) = L$ exists.
By Mean Value Theorem w... | {
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Pairwise sum and divide I came across a programming task.
Given an array of integer numbers $A\mid A_i\in\mathbb{N}$, one needs to calculate the sum:
$$S=\sum_{i=1}^{N}\sum_{j=i+1}^{N}\left\lfloor\frac{A_i+A_j}{A_i\cdot A_j}\right\rfloor$$
It is the summation of the floor integer values of the fraction $\frac{A_i+A_j}{... | Note that
$$
S = \sum_{i=1}^N\sum_{j=i+1}^N \left\lfloor \frac {1}{A_i} + \frac{1}{A_j} \right \rfloor = \sum_{1 \leq i < j \leq n} \left\lfloor \frac {1}{A_i} + \frac{1}{A_j} \right\rfloor
$$
That is, for every pair of $A_i$, we calculate $f(i,j) = \left\lfloor\frac {1}{A_i} + \frac{1}{A_j} \right\rfloor$, and we add ... | {
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A net $\varphi : [0, \omega_1) \to M$ on a metric space $M$ converges $\iff \varphi$ is eventually constant I want to prove that
If $M$ is a metric space, then a net $\varphi : [0, \omega_1) \to M$ converges if and only if $\varphi$ is eventually constant. ($[0, \omega_1)$ is the set of ordinals less than $\omega_1$,... | For each $n$, the set $\left\{\alpha\in[0,\omega_1):d(\varphi(\alpha),x)>1/n\right\}$ is countable, so $\left\{\alpha\in[0,\omega_1):\varphi(\alpha)\neq x)\right\}$ is also countable, so it is not cofinal in $\omega_1$.
| {
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Example that $R/I$ is not field where $R$ is a commutative ring and $I$ is maximal ideal.
Theorem.
Let $I$ be an ideal in a commutative ring $R$ with identity. Then $I$ is a maximal ideal if and only if the quotient ring $R/I$ is a field.
Above Theorem is very famous theorem. But The Theorem is hold under condition... | $2\mathbb{Z}/4\mathbb{Z}$ is not a field because it has zero multiplication.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1894506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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The large-$N$ limit of eigenvalues of matrices with non-diagonal elements scaling as $1/N$ Define a series of matrices$$H_N=
\begin{bmatrix}
1&1/N&1/N&\cdots&1/N\\
1/N&2&1/N&\cdots&1/N\\
1/N&1/N&3&\cdots&1/N\\
\vdots&\vdots&\vdots&&\vdots\\
1/N&1/N&1/N&\cdots&N
\end{bmatrix}$$
My question is, when $N\to+\infty$, would ... | Nearly a year passed and now I finally got the answer for my problem: In the limit $N\to\infty$, the eigenvalues of $H_N$ won't be different from $\{1,2,\ldots,N\}$.
Proof: The matrix element of $H_N$ is given by $H_{ij}=j\delta_{ij}+1/N$, where we have added an unimportant $1/N$ for the diagonal elements (and in the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is it possible to find a set of 5 integers, such that the 10 sums each have a different last digit? Full question:
If we are given 5 integers, there are $\binom{5}{2} = 10$ different ways to find the sum of 2 of the integers.
Is it possible to find a set of 5 integers, such that the 10 sums each have a different la... | Let $o$ be the number of odd integers in the set. Then you can form $o(5-o)\ne5$ odd sums. Therefore a set of the desired kind does not exist.
| {
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ordinary differential equation of third order using substitution How one can solve ODE in the following form?
$$y''' y +(y'')^{2} =0$$
It looks like some kind of substitution. I did try some substitution but they were useless.
thanks in advance.
| With reference to http://eqworld.ipmnet.ru/en/solutions/ode/ode0503.pdf,
Let $u=\left(\dfrac{dy}{dx}\right)^2$ ,
Then $\dfrac{du}{dx}=2\dfrac{dy}{dx}\dfrac{d^2y}{dx^2}$
$\dfrac{du}{dy}\dfrac{dy}{dx}=2\dfrac{dy}{dx}\dfrac{d^2y}{dx^2}$
$2\dfrac{d^2y}{dx^2}=\dfrac{du}{dy}$
$2\dfrac{d^3y}{dx^3}=\dfrac{d}{dx}\left(\dfrac{du... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1894996",
"timestamp": "2023-03-29T00:00:00",
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Explicit formula for crossed recursion When answering this linked question, I ended up with a formula where two sequences are mutually recurrent :
$$
\begin{cases}
u_0=0, ~~v_0=1\\
\forall n \in \mathbb{N}^*, ~~u_{n+1} = \frac{11}{12}u_n+\frac1{12}v_n\\
\forall n \in \mathbb{N}^*, ~~v_{n+1} = \frac56v_n+\frac16u_n\\
\e... | One approach is to rewrite your recurrences in matrix form as
$$\begin{bmatrix}u_{n+1}\\v_{n+1}\end{bmatrix}=\begin{bmatrix}11/12&1/12\\1/6&5/6\end{bmatrix}\begin{bmatrix}u_n\\v_n\end{bmatrix}\;,$$
so that
$$\begin{bmatrix}u_n\\v_n\end{bmatrix}=\begin{bmatrix}11/12&1/12\\1/6&5/6\end{bmatrix}^n\begin{bmatrix}u_0\\v_0\en... | {
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$x \in (a-r,a+r)$ versus $|x-a| < |r-a|$ From the Princeton book for the GRE Subject Test in Maths:
That part seems to suggest that
if $x \in (a-r,a+r)$ then $f(x) < \frac1q < \varepsilon$ and hence $f$ is continuous at the irrationals in $(0,1)$.
I was thinking to establish continuity we needed to show that
if $|x... | It's a typo. It should have been:
Then, within the open interval $(a - \color{red}{\delta}, a + \color{red}{\delta})$, the value of $f(x)$ will be less than $\color{red}{\text{or equal to}}$ $\frac{1}{q}$
To see why this choice of $\delta$ works, we argue by contradiction.
Suppose instead that there is some $r^* \in ... | {
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Troubles understanding solution to $\cos(\arcsin(\frac{24}{25}))$ I am having troubles understanding how the answer key to my Pre-Calculus and Trigonometry document got to the answer it did from this task/question:
Find the exact value of the expression $$\cos(\arcsin(\frac{24}{25}))$$
At first, I tried to find the val... | The basic definition of "sine" is "opposite side divided by hypotenuse" so they have drawn a right triangle with angle (call it $\theta$) on the left, hypotenuse of length 25 and "opposite side" (the vertical line on the right) of length 24. So $\sin(\theta)= \frac{24}{25}$ and $\theta= \arcsin\left(\frac{24}{25}\righ... | {
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"answer_id": 3
} |
Show if $f,g:S^n \to S^n$ and $|\text{deg}(f)| \neq |\text{deg}(g)|$ there is some $x$ with $f(x), g(x)$ orthogonal. More specifically, I want to show if f and g are maps from $S^{n} \to S^{n}$ with $|\text{deg}(f)| \neq |\text{deg}(g)|$ show that there is some $x \in S^{n}$ with $f(x), g(x)$ orthogonal.
One way I thi... | Suppose for every $x, \langle f(x), g(x)\rangle\neq 0$. Consider $h:S^n\rightarrow R$ defined by $h(x)=\langle f(x), g(x)\rangle$. Since $S^n$ is connected, we have:
*
*For every $x\in S^n$, $h(x)>0$
*For every $x\in S^n, h(x)<0$.
Suppose 1.
Define $H(t,x)={{tf(x)+(1-t)g(x)}\over{\|t(x)+(1-t)g(x)\|}}$.
$H$ is well... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1895364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Two lines through a point, tangent to a curve We are looking for two lines through $(2,8)$ tangent to $y=x^3$. Let's denote the intersection point as $(a, a^3)$ and use the slope equation together with the derivative to get $\frac{a^3-8}{a-2}=3a^2$. This yields a cubic equation. Of course, one of the lines is tangent t... | The tangents must be of the form
$$y-8=m(x-2),$$ and they intersect the cubic $\color{blue}{y=x^3}$ when
$$x^3-8=m(x-2).$$
This equation must have a double root, so that differentiating on $x$, we also have
$$3x^2=m.$$
With the obvious solution $x=2$, we deduce $m=12$ and
$$\color{green}{y-8=12(x-2)}.$$
Otherwise, we m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1895466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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How would I write a function for the following pattern? \begin{align}
Y(0) ={}& 1\\
Y(1) ={}& 2.5\\
Y(2) ={}& 2.5\cdot2.3\\
Y(3) ={}& 2.5\cdot 2.3\cdot 2.1\\
Y(4) ={}& 2.5\cdot 2.3\cdot 2.1\cdot 1.9\\
\vdots\,\,\,
\end{align}
How would I solve for something like $Y(1.3)$ or $Y(2.7)$? How would a function for $Y(x)$ be ... | Notice that
\begin{align*}
Y(x) &= \frac{25}{10} \cdot \frac{23}{10} \cdot \frac{21}{10} \cdots \frac{27 - 2x}{10} \\[5pt]
&= \frac{1}{10^x} \cdot \frac{(26)(25)(24)(23)(22)(21) \cdots (28-2x)(27-2x)}{(26)(24)(22)\cdots (28-2x)} \\
&= \frac{1}{10^x} \cdot
\frac{(26)(25)(24)(23)(22)(21) \cdots (2)(1)}{(26)(24)(22)\cdots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1895551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Unpacking notation with $\inf$, $\sup$ as part of proof of open set $\mathbb{R}$ can be written as countable union of disjoint open intervals Here is a Proposition from my real analysis book.
Proposition. Suppose $G \subset \mathbb{R}$ is open. Then $G$ can be written as the countable union of disjoint open intervals.... | I think it might be easier to understand if it were written as
$$A_x = \inf\{ a : (a,x) \subset G\}$$
$$B_x = \sup\{ d : (x,d) \subset G\}$$
Basically, $A_x$ is the left endpoint of the largest open interval containing $x$ and $B_x$ is the corresponding right endpoint.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1895834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Volume of ellipsoid outside sphere I have the ellipsoid $\frac{x^2}{49} + y^2 + z^2 = 1$ and I want to calculate the sum of the volume of the parts of my ellipsoid that is outside of the sphere $x^2+y^2+z^2=1$
How to do this? I know the volume of my sphere, $\frac{4\pi}{3}$, and that I probably should set up some doubl... | If you do not have a good perception of objects in 3D and want a pure analytical solution:
Let $A(z)$ be the area of a slice of the ellipsoid outside the sphere, at height $z$. At height $z$, the ellipsoid is the ellipse
$$
\frac{x^2}{49(1-z^2)}+\frac{y^2}{1-z^2}=1,
$$
which has area $7(1-z^2)\pi$, and the sphere is t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1895948",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Finding a point on the line perpendicular to a line from another point I'm sorry if this is kind of basic, it's been a while since I took geometry. I did find this answer, but it requires 4 points and I only have 3.
I have three points $A$, $B$ and $C$ that form a non-right triangle. I know the Cartesian coordinates of... | You can solve this fairly easily with a few vector operations.
Finding point $D$ comes down to finding the perpendicular projection of the vector $\vec{AB}$ onto $\vec{AC}$. That’s given by $$\vec{AB}_\parallel={\vec{AB}\cdot\vec{AC}\over\|\vec{AC}\|^2}\vec{AC}$$ and so $D = A+\vec{AB}_\parallel$.
Now, recall that a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Get the number of compositions with restrictions The number of compositions of $n$ is $2^{n-1}$.
For $n=5$, the number of compositions is $2^{5-1}=16$
*
*5
*4 + 1
*3 + 2
*3 + 1 + 1
*2 + 3
*2 + 2 + 1
*2 + 1 + 2
*2 + 1 + 1 + 1
*1 + 4
*1 + 3 + 1
*1 + 2 + 2
*1 + 2 + 1 + 1
*1 + 1 + 3
*1 + 1 + 2 + 1
*1 + 1 +... | The recursion for the unrestricted compositions is:
$F_0=1$ and $F_n=\sum\limits_{j=0}^{n-1} F_j$, it is easy to show with induction that this is equal to $2^{n-1}$.
The recursion for the "restricted" compositions in which every block is in $[a,b]$ is $F_0=1$ and $F_n=\sum\limits_{j=a}^{\min(n,b)}F_{n-j}$. This allows ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
derivative $\frac{\ln{x}}{e^x}$ Im asked to solve find the derivative of: $$ \frac{\ln x}{e^x}$$
my attempt
$$D\frac{\ln x}{e^x} = \frac{\frac{1}{x}e^x + \ln (x) e^x}{e^x} = e^x \frac{\frac{1}{x}+\ln x}{e^{2x}} = \frac{\frac{1}{x}+\ln x}{e^x}$$
But this is apparently wrong and the correct answer is:
$$\frac{\frac{1}{x... | Note that:
$$ \frac{\ln x}{e^x}=e^{-x}\ln {x}$$
And
$$(uv)'=u'v+uv'$$
Thus
$$ (\frac{\ln x}{e^x})'=(e^{-x}\ln {x})'=(-e^{-x}\ln {x})+(\frac{e^{-x}}{x})=\frac{\frac{1}{x} - \ln x}{e^x}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Does Lebesgue integral satisfy Riemann integral properties? From what I did study, I know that Lebesgue integral is more general than Riemann integral.
Then, does Lebesgue integral satisfy all of the Riemann integral properties?
In particular, is the following true?
For some given set X,
$\int_{A}f d\mu + \int_{B}f d\m... | As was stated in the comments, if the domain is compact then the Riemann and Lebesgue integral agree. But one thing Riemann has over Lebesgue is that it allows improper integrals. This question from 2013 gives an example:
Riemann-integrable (improperly) but not Lebesgue-integrable
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve for an integral equation from already having the value of the integral? So I know that $\int_{0}^{m} (mx-x^2)dx$ must equal 8. I also know that $m$ is a positive integer. How do I solve for this without having to use a calculator?
| $$8=\int_0^m (mx-x^2) \mathop{dx} = \left[\frac{1}{2}mx^2 - \frac{1}{3}x^3\right]_{x=0}^m = \frac{1}{6} m^3$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896575",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Gear Ratios in Number Theory? A machine shop is manufacturing a pair of gears that need to be in a ratio as close to
$1.1839323$ as possible, but they can’t make gears with more than $50$ teeth on them. How
many teeth should be on each gear to best approximate this ratio?
I can't figure out a number-theoretic approach ... | Express the value as a continued fraction and then simplify it. The continued fraction for 1.1839323 is:
1; 5, 2, 3, 2, 5, 95, 2, 11, 1, 3, 2
45/38 = [1;5,2,3] = 1.1842105263157894 error +0.0002782263157894427 (0.02350%)
The best ratio with values under 50 is 45 to 38.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The family of non-decreasing functions from the segment info itself endowed with the topology of pointwise convergence is first countable Using the fact that the only type of discontinuity compatible with monotonic function is the jump discontinuity, I showed that $A:=\{a\in[0,1]:f\textrm{ is discontinuous at }a\}$ is ... | Let $f\in X$ be an arbitrary function, $S=\{a_n\}$ be an enumerated countable set containing both the set $A$ of the discontinuity points of the function $f$ and a dense set $D\supset\{0,1\}$ of the segment $[0,1]$ (for instance, its rational points). We claim that the family $$U_n=\{g:\in X: |g(a_i)-f(a_i)|<1/n\mbox{ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If $(a + b\sqrt c)^n = d + e\sqrt c$, then $(a - b\sqrt c)^n = d - e\sqrt c$ I think that:
If $a,b,(c\ge0$ not a prefect square$),d,e,f\in\mathbb Z$ such that for some $n \ge 1$, $(a + b\sqrt c)^n = d + e\sqrt c$, then $(a - b\sqrt c)^n = d - e\sqrt c$
Is this true? Can someone provide a proof or give a hint for how ... | Consider the set
\begin{align*}
A = \{ x + y\sqrt{c}: x, y \in \mathbb{Z} \}
\end{align*}
Consider the map $f: A \rightarrow A$ defined by $f(x+y\sqrt{c}) = x - y \sqrt{c}$. It is easy to see that $f$ is well defined (since $c$ is not a perfect square) and for any two $a_1, a_2 \in A$,
\begin{align*}
f(a_1+a_2) &= f(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896883",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Proof that $\| \varphi \| = \varphi(1)$ for positive linear functionals on operator systems Let $M \subset \mathcal{B}(H)$ be an operator system, i.e., $M$ is a self-adjoint unital subspace, and let $\varphi: M \rightarrow \mathbb{C}$ be a positive linear functional. How does one prove that $\varphi$ is bounded with $\... | Here is an argument. All the time, $\varphi:M\to\mathbb C$ is linear and positive, and $M\subset B(H)$ is an operator system.
Lemma. If $a\in M$ is selfadjoint, then $\varphi(a)\in\mathbb R$. Also, $\varphi(x^*)=\overline{\varphi(x)}$, for all $x\in M$.
Proof. Let $a\in M$ selfadjoint; then, as $1\geq0$ and $a+\|a\|\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1896983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How do I solve a problem consisting of independent events? Question:
A leather bag contains 4 black beads, 3 red beads and three white beads. Inside a plastic bag are 5 black beads, 2 red beads and 3 white beads. Another nylon bag contains 6 black beads, 1 red bead and 3 white beads. One bead is randomly withdrawn from... | We are drawing one bead each from the three bags, and we are drawing them independently. There is a $\frac3{10}$ chance of drawing a white bead from each bag. If we get at least two white beads, we could have got them from
*
*the nylon and plastic bags
*the leather and nylon bags
*the leather and plastic bags
*al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897082",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Multiplication of Rational Matrices Let $\mathbf A(x)$ and $\mathbf B(x)$ be $n \times n$ rational matrices, whose elements are rational functions in the scalar $x \in \mathbb R$. Suppose that $\mathbf A(x) \mathbf B(x)$ is a polynomial matrix in $x$, meaning that the denominators in the elements of $\mathbf A(x)$ and ... | Here is my attempt at a counter example.
$$\left[\begin{array}{cc}
1&1/p(x)\\
0&1
\end{array}\right]\left[\begin{array}{cc}
1&1\\
p(x)&0
\end{array}\right] = \left[\begin{array}{cc}
2&1\\
p(x)&0
\end{array}\right]
$$
$$\left[\begin{array}{cc}
1&1\\
p(x)&0
\end{array}\right]\left[\begin{array}{cc}
1&1/p(x)\\
0&1
\end{ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Calculate the value of the sum $1+3+5+\cdots+(2n+1)$ I have been thinking about this for a long time, may I know which step of my thinking is wrong as I do not seems to get the correct answer. If I am not going towards the right direction, may I get some help thanks!
My attempt:
Let $S = 1+3+5+\dotsb+(2n+1)\label{a}\ta... | Only have $n-1$ numbers of (2n+2)
therefore:
2S = (n-1)(2n+2)
S = (n-1)(n+1)
Reason:
when $n=1$, then first term is 3 rather than 1.
Therefore, you only have $n-1$ numbers of (2n+2).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 3
} |
Please help me understand the definition of straight line given by Euclid. "A straight-line is (any) one which lies evenly with
points on itself."
This is how Euclid defines a straight line but I don't know what it really means.
Is this saying that any point picked on the straight line will have equal distance from eac... | Within the Euclidean system of geometry that is articulated in Book 1 of the Elements, the three fundamental symmetries for objects in two dimensions are reflection, translation and rotation. The idea that a straight line lies evenly between two points can be understood in terms of reflection. If a line A were to be tr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Other formulation of the inverse Galois problem Is it right to say that the inverse Galois problem is equivalent to the following statement:
Does every finite group $G$ occurs as a quotient of $\text{Gal}(\bar{\mathbb Q}/\mathbb Q)$?
I'm not sure if this is "quotient" or "subgroup" that I should write. Thank you for... | You should write quotient: if $K$ is a finite Galois extension of $\mathbb{Q}$, then $\mathrm{Gal}(K/\mathbb{Q})$ is a quotient of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ by the Fundamental Theorem of Galois theory.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is minimum distance between points on concentric ellipses constant? Given an ellipse, $E_1$ with radii $r_x, r_y$ I would like to know whether the minimum distance between any selected point, $P_a$, and $E_1$ is less than, say $D$.
I have seen the related question of finding the distance between a point and an ellipse.... | Pictured below are
*
*[Red] The (degenerate) ellipse $E_1$ with axes $1$ and $0$.
*[Green] The set of points exactly one unit from $E_1$.
*[Blue] The ellipse $E_{1+1}$ with axes $1+1$ and $0+1$.
All points within $E_{1+1}$ are within one unit of $E_1$ , but the converse is false. There are points within one unit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897587",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Minimum and Maximum value of |z| This is a question that I came across today:
If $|z-(2/z)|=1$...(1) find the maximum and minimum value of |z|, where z represents a complex number.
This is my attempt at a solution:
Using the triangle inequality, we can write:
$||z|-|2/z||≤|z+2/z|≤|z|+|2/z|$
Let $|z|=r$ which implies... | Using Complex Inequalities, $$||z|-|w||\le|z+w|\le|z|+|w|$$
$w=-\dfrac2z$ and writing $|z|=r$
$$\left|r-\dfrac2r\right|\le\left|z-\dfrac2z\right|\le r+\dfrac2r$$
$$\implies\left|r-\dfrac2r\right|\le1\le r+\dfrac2r$$
Now as $r>0,$ $$\dfrac{r+\dfrac2r}2\ge\sqrt{r\cdot\dfrac2r}=\sqrt2\iff r+\dfrac2r\ge2\sqrt2>1$$
So, we... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Prove that $n^n>1\times3\times5\times7\times \dots\times(2n-1)$ The main question is :
Prove that $n^n>1\times3\times5\times7\times\dots\times(2n-1)$
My approach :
We can write the R.H.S as,
$$\frac{(2n-1)!}{2\times4\times6\times\dots\times2(n-1)}$$
We can write $(2n-1)!$ as $(2n-3)!(2n-1)2(n-1)$
Thus, in this manner ... | HINT:
Using AM, GM inequality for $r,2n-r>0$
$$\dfrac{r+(2n-r)}2\ge\sqrt{r(2n-r)}$$
Set $r=1,3,5\cdots, 2n-3,2n-1$ and multiply.
Observe that the equality can not be held as $r=2n-r$ will not occur unless $r=n$(which is again possible if $n$ is odd)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Question about Column Space Matrix multiplication properties If say two square matrices A,B have the same column space, will it also hold that some multiplication of these matrices with another matrix will have the same column space?
i.e if Col(A) = Col(B)
does Col(AC) = Col(BC) for some other matrix C of the same ... | If the two matrices have the same column space, it means that the corresponding linear maps have the same image. However they may not necessarily have the same kernel. So a counter example to the statement would be if $C$ maps all the vector to the kernel of $A$ but not for $B$.
E.g. take $$A=\begin{pmatrix} 0 & 1 \\ 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The conjugate of a sylow $p$-subgroup is a sylow $p$-subgroup Second Sylow theorem states that all Sylow $p$-subgroups are conjugate. But reviewing my proof it seems to me that we also prove that all the conjugates of a Sylow $p$-subgroup are Sylow $p$-subgroups.
I can include the proof if needed but can anybody confir... | In general if $H$ is a subgroup of $G$ and $g\in G$, then we have $|gHg^{-1}|=|H|$, so it follows that the conjugates of a Sylow subgroup are also Sylow subgroups.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1897971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Suppose we have two dice one fair and one tha brings $6$ with quintuple probability.Find the probability to throw randomly one die and show $6$ Suppose we have two dice one fair and one that brings $6$ with quintuple probability than the other numbers.We get a die randomly and we throw it.What is the probability to hav... | The probability of rolling 6 on the fair die is obviously $\frac{1}{6}$.
Let $x$ denote the probability of rolling 6 on the non-fair die:
*
*Then $\frac{1}{5}x$ is the probability of rolling each one of the other $5$ values
*Therefore $x+5\cdot\frac{1}{5}x=1$, therefore $2x=1$, therefore $x=\frac{1}{2}$
So the pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A geometry problem based on circles. Question:- Consider three equal circles $S_1$, $S_2$, and $S_3$ each of which passes through a given point $H$. Other than that, $S_1$ and $S_2$ intersect at $A$, $S_2$ and $S_3$ intersect at $B$, $S_3$ and $S_1$ intersect at $C$. Show that $H$ is the orthocenter of triangle $ABC$.... | We have that $H$ is the radical centre of the circles $S_1, S_2, S_3$, so $AH\perp O_2 O_3$ and the midpoint of $AH$ is also the midpoint $M_A$ of $O_2 O_3$. In particular, $H$ is the circumcenter of $O_1 O_2 O_3$, hence it is the orthocenter of its medial triangle $M_A M_B M_C$. Since $ABC$ and $M_A M_B M_C$ are homot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Squaring Infinite Series Expansion Of e^x $Fact$:$$\lim\limits_{n \to \infty}\frac{x^0}{0!}+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^n}{n!}=e^x$$
so
$$\lim\limits_{n \to \infty}\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\dots+\frac{1}{n!}=e$$
also
$$\lim\limits_{n \to \infty}e^2=\frac{2^0}{0!}+... | Since $e^ne^m = e^{n+m}$ you can simply let $x = 2t$, giving you
$$(e^{t})^2 = e^{2t} = 1 + 2t + \frac{4t^2}{2!}+\frac{8t^3}{3!} + \cdots$$
Now, let $t = 1$ and you have
$$e^2 = 1 + 2 + \frac{4}{2!} + \frac{8}{3!} + \cdots = (1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \cdots )^2 = (e^1)^2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898270",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Divergence-free vector field on a 2-sphere. What is the solution of the differential equation
$$\text{div}X=0$$
on the 2-sphere?
| $\def\div{\operatorname{div}}$Let $\nu$ be the volume form on the sphere. The map $X\mapsto\nu(X,\mathord-)$ gives a bijection from vector fields to $1$-forms and, in particular, if $h:S^2\to\mathbb R$ is a smooth function, there is a unique vector field $X_h$ such that $\nu(X_h,Y)=dh(Y)$ for all vector fields $Y$. Usi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898371",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Find the second smallest integer such that its square's last two digits are $ 44 $ Given that the last two digits of $ 12^2 = 144 $ are $ 44, $ find the next integer that have this property.
My approach is two solve the equation $ n^2 \equiv 44 \pmod{100}, $ but I do not know how to proceed to solve that equation.
I t... | If $x^2$ ends with $44$ then $x$ is even. Let $y=2x$. We are trying to solve
$$(2y)^2\equiv 44\pmod{100}$$
and this equation is equivalent to
$$y^2\equiv 11\pmod{25}$$
Since $6^2\equiv 11\pmod{25}$ this equation can be written as
$$(y-6)(y+6)\equiv 0\pmod{25}$$
It is not possible that both $y-6$ and $y+6$ are multiples... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Can one infer independence by simple reasoning/intuition? From my recent experience in probability, it feels as though independence is something we "discover" from the system via the equation:
$$P(A)*P(B)=P(A\cap B)$$
Could one ever conclude independence from the "system" by intuition? Is it wise to conclude independe... | The OP asks for an intuitive understanding of $\mathbb{P}(A)\mathbb{P}(B)=\mathbb{P}(A\cap B)$ for independent events $A$ and $B$. Here it is:
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Assume $\mathbb{P}(B) > 0$. Since $\mathbb{P}(\Omega)=1$, we can write the above equation as
$\displaystyle\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898596",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Find large power of a non-diagonalisable matrix
If $A = \begin{bmatrix}1 & 0 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$, then find $A^{30}$.
The problem here is that it has only two eigenvectors, $\begin{bmatrix}0\\1\\1\end{bmatrix}$ corresponding to eigenvalue $1$ and $\begin{bmatrix}0\\1\\-1\end{bmatrix}$ corresp... | Notice the characteristic polynomial of $A$ is
$$\chi_A(\lambda) \stackrel{def}{=}\det(\lambda I_3 - A) = \lambda^3-\lambda^2-\lambda+1
= (\lambda^2-1)(\lambda-1)$$
By Cayley-Hamilton theorem, we have
$$\chi_A(A) = (A^2 - I)(A-I) = 0 \quad\implies (A^2-I)^2 = (A^2-I)(A-I)(A+I) = 0$$
This means $A^2-I$ is nilpotent. In... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 5,
"answer_id": 1
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Finding $a$ in quadratic equation $2x^2 - (a+1)x + (a-1)=0$ so that difference of two roots is equal to its product Given equation:
$$2x^2 - (a+1)x + (a-1)=0$$
I have to find when the difference of two roots is equal to its product, i.e.:
$$x_1x_2 = x_1 - x_2.$$
From Vieta's formulas we know that:
$$x_1 + x_2 = \frac{a... | Using by the formula $${ \left( { x }_{ 1 }-{ x }_{ 2 } \right) }^{ 2 }+4{ x }_{ 1 }{ x }_{ 2 }={ \left( { x }_{ 1 }+{ x }_{ 2 } \right) }^{ 2 }$$ make be easy
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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When is it true that $a^{n}How can I find the smaller $n\in\mathbb{N}$, which makes the equation true: $$a^{n}<n!$$
For example:
If $a=2$ then $\longrightarrow 2^{n}<n!$ when $n\geq 4$
If $a=3$ then $\longrightarrow 3^{n}<n!$ when $n\geq 7$
If $a=4$ then $\longrightarrow 4^{n}<n!$ when $n\geq 9$
If $a=5$ then $\longri... | I find a equation:
$$ a^{n}<n! \Longleftrightarrow n \geq (2a)+\lfloor\frac{(a-1)}{2}\rfloor$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1898974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Historically key works on structural reliability theory? I want to understand better the structural reliability theory. Is it related to the reliability of structures? Where does it originate and what are the most important work for it?
| this reference may be is useful
Borgonovo E , Iooss B . Moment-independent and reliability-based importance mea- sures. In: Ghanem R, Higdon D, Owhadi H, editors. Springer handbook on uncer- tainty quantification. Springer; 2017. p. 1265–87 .
the book is
Handbook of Uncertainty Quantification by
Roger Ghanem Davi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Is the difference of two irrationals which are each contained under a single square root irrational? Is $ x^\frac{1}{3} - y^\frac{1}{3}$ irrational, given that both $x$ and $y$ are not perfect cubes, are distinct and are integers (i.e. the two cube roots are yield irrational answers)?
I understand that the sum/differen... | Denote the cube roots by $X,Y$, so that $X^3=x$ and $Y^3=y$ with $x,y\in \mathbb Z$.
Suppose, with slightly greater generality, that we have $X-Y-R=0$ where $X^3,Y^3,R\in \mathbb Q$.
We first want to argue that $XY\in \mathbb Q$. To do so, observe the identity:
$$X^3-Y^3-R^3-3XYR=(X-Y-R)(X^2+Y^2+R^2+XY+XR-YR)$$
This i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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Number of arrangements of red, blue, and green balls in which a maximum of three balls of the same color are placed in the five slots
From the picture above I have five slots and a bag of balls (three) of color -red, blue and green...
Question
Now whenever I choose a ball, I note the colour and replace it in the bag, ... | We will solve the problem: "how many words of length $5$ on $\{R,B,G\}$ are there in which no letter appears more than $3$ times".
Easier to work backwards. Without the cap rule there are $3^5$ possible words.
How many of these have exactly $4\;R's$? Well there are $5$ place to put the non-$R$, and $2$ options for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899238",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Probability of combinations that share at least 3 items I have a question about probabilities that I can't get my head around. Suppose I would like to generate a combination of 7 elements, where each element is randomly drawn from a set of n alternatives. For example, the first element has possible alternatives A1, A2,... | This is binomial. The probability the $i^{th}$ character matches is $p=\dfrac{1}{n}$. Thus the probability of exactly $x$ matches in a character length of $k$ is
$$P(X=x) = {k\choose x}\left(\dfrac{1}{n}\right)^x\left(1-\dfrac{1}{n}\right)^{k-x}$$
At least three matches would be
$$P(X\ge 3) = P(X=3)+\cdots+P(X=k) = \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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An accountancy of the natural numbers I guess this formula is known since I believe it's true
$$\sum_{k=2}^\infty\frac{(-1)^{1+\Omega(k)}}{k}=1$$
where $\Omega(k)$ is the number of prime factors of $k$ (not necessary different primes). I can't find it and want a formal proof or a reference.
My intuition about this sum... | We have
$$\sum_{k=2}^{\infty} \frac{(-1)^{1 + \Omega(k)}}{k} = 1 -\prod_p \left(1 - \frac{1}{p} + \frac{1}{p^2} \mp\right) = 1 -\prod_{p} \frac{1}{1 + 1/p}$$
The RHS goes to $1$ since the product goes to zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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PSD rank 1 matrix decomposition into product of two vectors Assume that I'm given with matrix $\textbf{Q} = \textbf{q}\textbf{q}^H$. Assume I only have $\textbf{Q}$, is there any way for me to find vector $\textbf{q}$?
Thanks!
| Let $x$ be such that $x^H Q x > 0$. Then $\dfrac{Qx}{\sqrt{x^H Q x}} = \dfrac{q^H x}{\|q^Hx\|} q = cq$ where $\|c\| = 1.$
Next note if $q \neq 0$ then for some $i$ we must have $ e_i^H Q e_i = || e^H q||^2 = |q_i|^2 > 0$, where $e_i$ is the $i^{\text{th}}$ canonical basis vector and $q=(q_1,\dots,q_n)^T$.
Next note t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
ternary analogues of the Pell equations We know well about the Pell equations: $x^2 -ny^2=1$ and some variants of them. Criterions about the existence of nontrivial solutions of homogeneous equations $ax^2+by^2+cz^2=0$ are also well-known.
Then, how about the 3-v analogues of the Pell equations? I mean, the diophantine... | It turns out that Cassels does this material, pages 301-309. There is quite a big difference based on whether $x^2 - A y^2 - B z^2$ is isotropic or not, meaning there is an integer solution to $x^2 - A y^2 - B z^2=0$ with $x,y,z$ not all equal to zero. When the form is isotropic, pages 301-303, especially the proof of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899643",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Commuting matrix exponentials: necessary condition. In related questions (here, here and here), it was shown that if $A$ and $B$ commute, then $e^A$ and $e^B$ also commute (and incidentally $e^A e^B = e^{(A+B)}$). Here, the commutative property of A and B is a sufficient condition.
Is it also a necessary condition? If ... | It is not a necessary condition. In particular, we can take
$$
A = \pmatrix{2 \pi i & 0\\0&0}, \quad B = \pmatrix{0&1\\0&0}
$$
It's clear that $e^A = I$ commutes with $e^B$, but $A$ does not commute with $B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Understanding of a question The cost of producing a math assessment book is made up of three main components , overhead , type-setting and printing .
In $2009$, the overhead cost for an assessment book is $\$1200$ , the cost of type setting a page is $\$18$ and the cost of printing a book with $120$ pages is $\$1.45$.
... | In the past the term meant, literally, "setting type". That is, in printing newspapers (say) physical letters were positioned as desired in a block. Ink was then applied to the block and if paper was pressed on it, the desired image would pass to the paper. The expression "mind your $p's$ and $q's$" arises from this ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1899945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show $\lim\left ( 1+ \frac{1}{n} \right )^n = e$ if $e$ is defined by $\int_1^e \frac{1}{x} dx = 1$ I have managed to construct the following bound for $e$, which is defined as the unique positive number such that $\int_1^e \frac{dx}x = 1$.
$$\left ( 1+\frac{1}{n} \right )^n \leq e \leq \left (\frac{n}{n-1} \right )^n$... | Let $a_n=\left(1+\frac1n\right)^n$. We need to show first that $\lim_{n\to \infty}a_n$ actually exists.
From the OP, we see that $a_n$ is bounded above by the number $e$, which is defined as $1=\int_1^e \frac1t\,dt$. And in THIS ANSWER, I showed using Bernoulli's Inequality that $a_n$ is monotonically increasing. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 2
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Evaluate the integral $\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$ This integral looks a lot like an elliptic integral, but with cubes instead of squares:
$$I(a,b)=\int_0^\infty \frac{dx}{\sqrt{(x^3+a^3)(x^3+b^3)}}$$
Let's consider $a,b>0$ for now.
$$I(a,a)=\int_0^\infty \frac{dx}{x^3+a^3}=\frac{2 \pi}{3 \sqrt{... | More generally, with $|p-1|<1$, some experimentation shows that,
$$\int_0^\infty \frac{dt}{\sqrt{(t^m+1)(t^m+p)}} = \pi\,\frac{\,_2F_1\big(\tfrac12,\tfrac{m-1}{m};1;1-p\big)}{m\sin\big(\tfrac{\pi}{m}\big)}$$
where the question was just the case $m=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 1
} |
Weak Law of Large Numbers -Special Case For independent and identically distributed random variables $X_1, X_2, \cdots$ with finite mean and variance, by the weak Law of Large Numbers, $\frac{1}{N}\sum_{i=1}^{N} X_i$ converges in probability to $\mathbb{E}[X_i]$:
$$\frac{1}{N}\sum_{i=1}^{N} X_i \xrightarrow{p}\mathbb{E... | We can assume that $|f(N)|\le M$, for all $N$ and certain $M>0$. Moreover, there exist $N_0$ such that, for $N>N_0$, $|f(N)-c|< \varepsilon/(2 E(X_1))$. Since
\begin{align*}
\Big|\frac{f(N)}{N}\sum_{i=1}^N X_i - cE(X_1) \Big| \le \Big|\frac{f(N)}{N}\sum_{i=1}^N X_i - f(N)E(X_1) \Big| + \big|f(N)-c\big|\, E(X_1),
\end{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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pseudo-identities which are not exact but the error is very small i would like to know more example of pseudo identities.. things that there are not equal but the error is about $ 0.01 $
for example $$ \pi ^{4} +\pi ^{5} =e^{6} $$
the error term is about $ 10^{-5} $
where can i see more of this amazing pseudo identitie... | There are many examples given here at MSE. One of my favourites is that
$$
e^{\pi \sqrt{163}}=262 537 412 640 768 743.99999999999925
$$
is very close to an integer, see here. This has some serious number theoretical background, as is explained, and generalised, here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Multi-index sum property Exercise 1.2.3.29 in Donald Knuth's The Art of Computer Programming (3e) states the following property of a multi-indexed sum:
$$
\sum_{i=0}^n \sum_{j=0}^i \sum_{k=0}^j a_ia_ja_k = \frac{1}{3}S_3 + \frac{1}{2}S_1S_2 + \frac{1}{6}S_1^3,
$$
where $S_r = \sum_{i=0}^n a_i^r$.
I tried to prove it an... | For future reference this is the sum over all multisets of size three
chosen from the variables $A_0$ to $A_n$ and evaluated at $a_q.$
Therefore by the Polya Enumeration Theorem it is given by
$$\left.Z(S_3)\left(\sum_{q=0}^n A_q\right)\right|_{A_q=a_q}$$
where $Z(S_3)$ is the cycle index of the symmetric group ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Vector Addition in Euclidean Space I am reading Rudin's Principles of Mathematical Analysis, and I came across something in the proof that I don't quite understand.
Let $x$ and $z$ be vectors in $\mathbb R^k$ for some $k \geq 3$ and $r >0 $ is a real number. Suppose $| z - x | = r$. Then this means that $z = x + ru$ f... | Hint:
$$
z = x + (z-x) .
$$
Then normalize $z-x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Find two constant$c_1, c_2$ to make the inequality hold true for all N Show that the following inequality holds for all integers $N\geq 1$
$\left|\sum_{n=1}^N\frac{1}{\sqrt{n}}-2\sqrt{N}-c_1\right|\leq\frac{c_2}{\sqrt{N}}$
where $c_1,c_2$ are some constants.
It's obvious that by dividing $\sqrt{N}$, we can get $\left|\... | As Hans Engler commented, the problem is not entirely elementary.
Let us check the pieces of the expression $$\frac 1 N \sum_{n=1}^n \frac 1 {\sqrt {\frac n N}}=\frac 1 {\sqrt { N}}\sum_{n=1}^n \frac 1 {\sqrt { n}}=\frac{1}{\sqrt{N}}H_N^{\left(\frac{1}{2}\right)}$$ where appear the generalized harmonic numbers. So, the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Marginal density function based on conditional density example could you please help me to solve this training example for my exam? I need to find marginal density $f_X (x)$ based on knowing conditional density $f_{X|Y}$ and marginal density $f_{Y}$.
$$
f_{X\mid Y} (x\mid y) = \frac{2x}{y^2-1}~\mathbf 1_{1 \le x \le y ... | You were doing well, except that the upper limit $\pi/2$ comes from nowhere. It should be
$$
f_X(x) = \int_x^{2} f_{X,Y}(x,y)~dy = \int_x^{2} \frac{8}{9}xy~dy = \frac{16 x}{9}-\frac{4 x^3}{9}\ ,
$$
for $1\leq x\leq 2$. As a check, you observe that the marginal is correctly normalized:
$$
\int_1^2 dx\ f_X(x)=1\ .
$$
To... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1900951",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Show that the solution for $y'=(2-y)(x^2+2y)$ when $y(0)=3$ has the property $y(x) \in (2,3]$
*
*Without solving the equation, show that the solution for $y'=(2-y)(x^2+2y), y(0)=3$, there exists no $x\in \mathbb{R}$ such that $y(x)=2$.
*Without solving the equation, show that the solution for this problem $y$ has th... | Firs of all, we will show that if $y$ is a solution with $y(0)=3$ then there is no $c\in \mathbb{R}$ such that $y(c)=2.$ Assume that there exists. Then the Cauchy problem $y'=(2-y)(x^2+2y), y(c)=2,$ has two different solutions: the solution with $y(0)=3$ and $y(x)=2.$ This contradicts the unicity of solution.
Now, sinc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Find the value of $\tan A + \tan B$, given values of $\frac{\sin (A)}{\sin (B)}$ and $\frac{\cos (A)}{\cos (B)}$ Given
$$\frac{\sin (A)}{\sin (B)} = \frac{\sqrt{3}}{2}$$
$$\frac{\cos (A)}{\cos (B)} = \frac{\sqrt{5}}{3}$$
Find $\tan A + \tan B$.
Approach
Dividing the equations, we get the relation between $\tan A$ and $... | Although there is something wrong with this question,there is a way which I think maybe a little bit easier to solve this kind of problem.
$$\tan A + \tan B = \frac{\sin(A)\cdot\cos(B) + \sin(B)\cdot\cos(A)}{\cos(A)\cdot\cos(B)} = \frac{{\sin A \over \sin B}+ {\cos A \over \cos B}}{\cos A \over \sin B} $$
then to ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Orthosymplectic Lie Superalgebra I am trying to work out a presentation for the orthosymplectic Lie superalgebra $\mathfrak{osp}(m,2n)$. I am following Musson's book "Lie Superalgebras and Enveloping Algebras".
From what I understand, we can take as the underlying $\mathbb{Z}_2$-graded vector space $k^m\oplus k^{2n}$,... | I think you might have some misunderstanding relating to supertransposition. Here's the definition of supertransposition on wiki:
https://en.wikipedia.org/wiki/Supermatrix#Supertranspose.
As you see, the supertransposition is related to the parity of the supermatrix itself, so this question might not be that easy now. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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how to integrate the beta like $\int_{0}^{1} \frac{y^{n-1}(1-y)^{m-n-1}}{1-xy}dy $ How should I find a closed form of $\displaystyle \int_{0}^{1} \frac{y^{n-1}(1-y)^{m-n-1}}{1-xy}dy $ , Any simple methods ?
| Assuming there are no convergence issues due to the values of $x$, $m$ and $n$,
$$\begin{eqnarray*} \int_{0}^{1}\frac{y^{n-1}(1-y)^{m-n-1}}{1-xy}\,dy &=& \sum_{k\geq 0}x^k \int_{0}^{1}y^{n+k-1}(1-y)^{m-n-1}\,dy\\&=&\sum_{k\geq 0} x^k \frac{\Gamma(n+k)\Gamma(m-n)}{\Gamma(m+k)}\\&=&B(m-n,n)\cdot\phantom{}_2 F_1(1,n;m;x).... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Exclusive Or -Logic Find a formula using only the connectives: conjunction, disjunction, and negation that is equivalent to, P exclusive or Q. I cannot figure out a way to come up with a logical equivalent statement without randomly guessing.
| You want $p$ to be true or $q$ to be true, but not both at the same time, thus all of the below formula would give you the correct result:
*
*"$p$ is true or $q$ is true, and it's not the case that both $p$ and $q$ are true"
$(p \lor q) \land \neg (p \land q)\\$
*"$p$ is true and $q$ is false, or $p$ is false and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 0
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Example of non-isomorphic, Morita-equivalent semisimple Hopf-algebras In the paper http://arxiv.org/abs/1509.01548, section 1.3, I found the following definitions:
Two fusion categories $\mathcal{C}$ and $\mathcal{D}$ are Morita-equivalent if there exists an indecomposable $\mathcal{C}$-module category $\mathcal{M}$ su... | A few weeks ago, a user posted a good hint for the solution, but deleted his answer after only one day (I couldn't even award him with that 50 bounty he deserved). I did the calculations and the following is a good example for my question:
Let $K$ be a field of characteristic zero and $G$ be any finite non-abelian gro... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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How do calculate the derivative of $\text{det } $? Define $\text{det}:M(n)\rightarrow \mathbb{R}$ as the map that sends $A$ to its determinant. This map is clearly smooth and I want to calculate its differential at $I$ (the identity matrix).
I did this for a very special case. Suppose that every eigenvalue of $A$ is r... | The determinant of an $n$-by-$n$ matrix $A=(a_{i,j})$ is
$$\sum_{\sigma\in S_n}\text{sgn}(\sigma)\prod_{i=1}^{n} a_{i,\sigma(i)},$$
where $S_n$ is the set of permutations of $\{1,\ldots, n\}$, and $\text{sgn}$ is the signature of a permutation.
For fixed $A = (a_{i,j})$, $\det(\text{id}+tA)$ is a polynomial in $t$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proof that Rel is a Category The Awodey book about Category Theory gives this definition for Rel:
The objects of $\text{Rel}$ are sets, and an arrow $A → B$ is a
relation from $A$ to $B$, that is, a subset $R ⊆ A×B$.
The equality relation $ \{ \langle a, a\rangle ∈ A×A\;|\; a ∈ A\}$ is the identity
arrow on a set... | What you've quoted is a definition of all the data needed to have a category: objects, morphisms, identity morphisms, and composition operation. To verify you have a category you then just have to check that this data satisfies the axioms for a category: that the identity morphisms are actually identities for the comp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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For $R=I\oplus J$, prove that $I= Re$ and $J= Rf$ where $e$ and $f$ are two idempotent elements Given that $R$ is a commutative unitary ring and
$I$ and $J$ are two ideals of $R$ such that $R=I\oplus J$ (as an internal direct sum), how do you show that there exists two idempotent elements $e$ and $f$ in $R$ such that $... | Yeah, I'll type out all the details. Since $R$ is the internal direct sum of $I$ and $J$, this means that
$$
R = I + J = \{i+j \mid i\in I,\; j\in J\}
\quad\text{and}\quad
IJ = I \cap J =\{0\}
\,.$$
Since $R = I + J$, there must be some aptly named $e \in I$ and $f \in J$ such that $e+f=1$. Note that $f = (1-e)$, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
A limit of a recursion involving the prime number function Suppose $A_1=1$, $B_1=2$ and
$\left\{
\begin{array}{l}
A_{k+1}=A_k\cdot (p_{k+1}-1)+B_k\\
B_{k+1}=B_k\cdot p_{k+1}
\end{array}
\right.
$
where $p_k$ is the $k$-th prime number. I would like a formal proof of that
$\displaystyle\lim_{n\to\infty}\frac{A_n}{B_n}... | Let $r_k=\frac{A_k}{B_k}$. We have $B_k=p_1\cdots p_k$ so
$$r_{k+1}-1=(r_k-1)\cdot\left(1-\frac1{p_{k+1}}\right).$$
It follows that $$r_n=1+\left(1-\frac1{p_1}\right)\cdots\left(1-\frac1{p_n}\right)$$
and it suffices to note that $\prod_p\left(1-\frac1p\right)=0$.
In general, when dealing with a linear recursion (with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Is there a mistake in wikipedia article on interior? In this wikipedia article about the interior, in the section on the interior operator, it is written that $S^{\circ}=X\setminus(X\setminus\bar{S})$, which can't be true since $X\setminus(X\setminus\bar{S})=\bar{S}$.
I think a correct definition would be $S^{\circ}=X\... | Yes, you are right and Wikipedia is wrong.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1901956",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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How do I factorize a polynomial $ax^2 + bx + c$ where $a \neq 1$ How do I factorize a polynomial $ax^2 + bx + c$ where $a \neq 1$?
E.g. I know that $6x^2 + 5x + 1$ will factor to $(3x + 1)(2x + 1)$, but is there a recipe or an algorithm that will allow me to factorize this?
| If you know how to factor polynomials for $a=1$, then simply writing
$$ax^2+bx+c=a(x^2+\frac bax+\frac ca )$$
makes the task immediate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1902077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Inverse of the sum $\sum\limits_{j=1}^k (-1)^{k-j}\binom{k}{j} j^{\,k} a_j$ $k\in\mathbb{N}$
The inverse of the sum $$b_k:=\sum\limits_{j=1}^k (-1)^{k-j}\binom{k}{j} j^{\,k} a_j$$ is obviously
$$a_k=\sum\limits_{j=1}^k \binom{k-1}{j-1}\frac{b_j}{k^j}$$ .
How can one proof it (in a clear manner)?
Thanks in advance.
... | In this proof, the binomial identity
$$\binom{m}{n}\,\binom{n}{s}=\binom{m}{s}\,\binom{m-s}{n-s}$$
for all integers $m,n,s$ with $0\leq s\leq n\leq m$ is used frequently, without being specifically mentioned. A particular case of importance is when $s=1$, where it is given by
$$n\,\binom{m}{n}=m\,\binom{m-1}{n-1}\,.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1902188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
On calculations from $\zeta(3)=\frac{2}{\pi}\sum_{n=1}^\infty\int_0^\infty \frac{\sin ((n+1)x)\sin (nx)}{(xn^2)^2}dx$ I was inspired in a formula that I've found in Internet, page 5 of Jameson's notes about Frullani integrals, to ask to Wolfram Alpha this integral $$\int_0^\infty\frac{\sin ax\sin bx}{x^2}dx=\frac{\pi}{... | $\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1902376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
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If $G$ is compact and $K \subset G$ is closed in $G$, must $G/K$ also be compact? If $G$ is compact and $K \subset G$ is closed in $G$, must $G/K$ also be compact?
The sets $G$ is a topological group for example.
| Yes it is compact if it is endowed with the quotient topology since it is the image of the compact $G$ by the quotient map $G\rightarrow G/K$ and the image of a compact by a continue map is compact.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1902570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If eigenvectors of a matrix are orthogonal, does that imply anything about the matrix (normal, hermitian, etc)? Normally, content related to my question proves the converse. For example, if a matrix is hermitian, then its eigenvectors corresponding to different eigenvalues are orthogonal.
If the eigenvectors of a squar... | Complex case: If $M[e_1 ... e_n] = [e_1 ... e_n] \mbox{ diag } (\lambda_1 ... \lambda_n)$, or $MP = P\Lambda$ then $M=P\Lambda P^{-1} = P\Lambda P^*$. This is Hermetian iff $\Lambda$ is real (i.e. only real eigenvalues). In the real case if you assume that there are $n$ real eigenvectors then implicitly you have that a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1902692",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A set with measure $0$ has a translate containing no rational number.
Suppose $E$ is a set with measure $0$. Show there exists $t\in \mathbb{R}$ such that $E+t$ contains no rational number.
My idea is to find an interval in $E$, then we can get a contradiction. I try to begin with a point in $E$ and then consider if ... | You can also show that $Z:=\bigcup\limits_{q\in\mathbb{Q}}\,(q-E)$ has measure $0$. For a real number $t\notin Z$, can $t+E$ intersect the rationals?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1902794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 0
} |
Birthday Problem: Why isn't the probability 253/365 Consider a set of $23$ unrelated people. Because each 23 pair of people shares the same birthday with probability $1/365$, and there are $\binom{23}2 = 253$ pairs, why isn’t the probability that at least two people have the same birthday equal to $253/365$?
| Let $A$ be the event that some two people have the same birthday. For $i < j$, let $A_{i,j}$ be the event that persons $i$ and $j$ have the same birthday. Then, $\text{Pr}(A_{i,j}) = \frac{1}{365}$, and your calculation is essentially that
$$
\sum_{1 \le i <j \le 23} \text{Pr}(A_{i,j}) = \sum_{i,j} \frac{1}{365} = \fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1902897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Product of roots of $ax^2 + (a+3)x + a-3 = 0$ when these are positive integers There is only one real value of $'a'$ for which the quadratic equation $$ax^2 + (a+3)x + a-3 = 0$$ has two positive integral solutions.The product of these two solutions is :
Since the solutions are positive, therefore the product of roots ... | We need $$\frac{a+3}{a}\in\Bbb{Z}\ , \frac{a-3}{a}\in\Bbb{Z}$$ or $$1+\frac{3}{a}\in\Bbb{Z}\ , \ 1-\frac{3}{a}\in\Bbb{Z}$$ thus $\displaystyle \frac{3}{a}\in\Bbb{Z}$, means that $\displaystyle a=\frac{3}{m}$ where $m\in\Bbb{Z}$.
Now we can write the equation as $$\frac{3}{m}x^2+\left(\frac{3}{m}+3\right)x+\frac{3}{m}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Why is $ (2+\sqrt{3})^n+(2-\sqrt{3})^n$ an integer? Answers to limit $\lim_{n\to ∞}\sin(\pi(2+\sqrt3)^n)$ start by saying that $ (2+\sqrt{3})^n+(2-\sqrt{3})^n $ is an integer, but how can one see that is true?
Update: I was hoping there is something more than binomial formula for cases like $ (a+\sqrt[m]{b})^n+(a-\sqr... | Consider the matrix
$$ A = \begin{bmatrix}
2 & 3\\
1 & 2
\end{bmatrix} $$
Clearly $ A^n $ is a matrix with integer entries, therefore the trace $ \operatorname{tr} A^n $ is an integer. On the other hand, $ A $ is a diagonalizable matrix whose eigenvalues are $ \lambda_1 = 2 + \sqrt{3} $, $ \lambda_2 = 2 - \sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 10,
"answer_id": 7
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How to find the linear transformation associated with a given matrix? Good day, I have a little doubt: It is well known that given two bases (or even one if we consider the canonical basis) of a vector space, every linear transformation $T:V \rightarrow W$ can be represented as a matrix, but since this is an isomorphis... | Suppose that you have an $m,n$ matrix $A$. Choose a basis $B$ of $V$ and another one $B'$ of $W$. The linear transformation associated with $A$ relative to the bases $B$ and $B'$ is $T(v) = Av$, where $v$ is to be written as a column whose entries are the coefficients of $v$ in the basis $B$ and the resulting column $T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
Let $p$ be a prime such that $a^p-1 \equiv 0 \pmod{p}$. Prove that $\gcd(a,p) = 1$.
We know from Fermat's Little Theorem that $a^{p-1}-1 \equiv 0 \pmod{p}$ if and only if $\gcd(a,p) = 1$, but how do we use this to solve the question?
| It is a special case of the general fact that $\,a\,$ invertible mod $\,n\,$ implies $\,\gcd(a,n)= 1.\,$ Indeed, $\,aj\equiv 1\pmod{n}\,\Rightarrow\, aj+kn = 1\ $ so $\ d\mid a,n\Rightarrow\, d\mid 1,\,$ therefore $\,\gcd(a,n) = 1$.
OP is a special case since $\,{\rm mod}\ p\!:\ a^p-1\equiv 0\, \Rightarrow a(a^{p-1})\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 5
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Prove that in set theory $A-B = A - (A \cap B)$ Prove that in set theory $A-B = A - (A \cap B)$
Please give me a hint.
Let $x\in A-B \implies x\in A~ and ~x \notin B\implies x\in A ~and (x\in A ~and~ x \notin B)\implies x\in A - (A \cap B)$
| Let $x \in A - B$. Then, $x$ is in $A$, but $x$ is not in $B$. It follows that $x$ is not in both $A$ and $B$, otherwise $x$ would be in $B$. Hence, $x \in A - (A \cap B)$, so $A-B \subset A - (A\cap B)$. Can you do the other direction using this as a model?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1903386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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