Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Find the sum to $n$ terms of the series: $1^2.1+2^2.3+3^2.5+....$ Find the sum to $n$ terms of the series:
$$1^2.1+2^2.3+3^2.5+.....$$
My Attempt:
Here, $n^{th}$ term of $1,2,3,....=n$
$n^{th}$ term of $1^2,2^2,3^2,....=n^2$
Also, $n^{th}$ term of $1,3,5,....=2n-1$
Hence, $n^{th}$ term of the given series is $t_n=n^2(2... | As , $$T_n=n^2(2n-1)$$ Hence , $$S_n=\sum T_n$$ Or , $$S_n=\sum_{k=1}^{n}2k^3-k^2$$ $$S_n=2\sum_{k=1}^{n}k^3-\sum_{k=1}^{n}k^2$$ Hence ,
$$S_n=\frac{(n(n+1))^2}{2}-\frac{n(n+1)(2n+1)}{6}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2680816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Theorem Explanation: Sequences and Convergence I'm currently studying analysis and came across this theorem:
Let $(x_n)$ be a sequence of real numbers and let $x \in {\Bbb R}$. If $(a_n)$ is a
sequence of positive real numbers with $\displaystyle\lim_{n\rightarrow \infty}{a_n} =0$ and if for some constant $C > 0$ and ... | This is a different version of the squeeze theorem. It is basically saying that since we know that $a_n$ gets really small as $n \to \infty$, if we know that there is a constant $C> 0$ such that $|x-x_n| \leq Ca_n$ (the distance between $x_n$ and $x$ is a multiple of how close $a_n$ is to $0$ since $a_n = |a_n - 0|$) t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Rolling dices possibility I must solve given task:
Three dice are rolled. What is possibility the sum of it to be $5$?
I guess I have to use combination formula, but I do not know the basic approach.
Can you please explain. My thoughts:
Sum of $5$ can be only get from numbers: $1$, $2$ and $3$. - $221$ $113$. I guess ... | If the outcome the experience is a triplet$\{d_1,d_2,d_3\}$ with the scores of each of the three dice, you have $6^3$ possible outcomes, all with the same probability.
$3$ of these are a permutation of $1-1-3$ and $3$ are a permutation of $1-2-2$
$6$ favourable outcomes in an universe of $6^3$ possible outcomes gives y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove that $3^{2n-1} + 2^{n+1}$ is always a multiple of $7$. I'm trying to prove the following statement:
$P(n) = 3^{2n-1} + 2^{n+1}$ is always a multiple of $7$ $\forall n\geq1$.
I want to use induction, so the base case is $P(1) = 7$ so that's okay.
Now I need to prove that if $P(n)$ is true then $P(n+1)$ is true.
S... | Alternatively, suppose $3^{2n-1}+2^{n+1}= 7x$.
Then, $x=\frac17\left(3^{2n-1}+2^{n+1}\right)=\frac{1}{21}\left(3^{2n}+6(2^{n})\right)=\frac{1}{21}\left(9^{n}+6(2^{n})\right)\equiv 0 \pmod3$
Now, $9\equiv 2 \pmod7\Rightarrow9^n\equiv 2^n \pmod3$, $6\equiv -1 \pmod7$ and hence, $6\cdot2^n\equiv (-1)\cdot2^n\pmod7$
Theref... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Spectral families of commuting operators Consider two self-adjoint bounded operators $A$ and $B$ on a separable Hilbert space. According to the spectral theorem we can write
$$
A=\int_{-\infty}^{\infty} x d E^{A}_x, \quad B=\int_{-\infty}^{\infty} y d E^{A}_y
$$
where $E^{A}_x$ and $E^{B}_y$ are the spectral families o... | From $AB=BA$, you get $A^nB=BA^n$ for all $n$, and immediately $p(A)B=Bp(A)$ for any polynomial $p$. By Stone-Weierstrass, $f(A)B=Bf(A)$ for any $f\in C(\sigma(A))$.
Now let $$\Sigma=\{\Delta:\ \Delta\ \text{ is Borel and } E^A(\Delta)B=BE^A(\Delta)\}. $$ From the fact that $E^A$ is a spectral measure, it is quickly d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Find a basis for all vectors perpendicular to $x-2y+3z=0$. I am looking at the following question:
Find a basis for all vectors perpendicular to $x-2y+3z=0$.
Clearly $<1,-2,3>$ is a vector that must be in the basis or a scalar multiple of $<1,-2,3>$ must be in the basis. But the solution given says that this vector is ... | The equation $$x-2y+3z=0$$ is the equation of a plain which is a two dimensional subspace in the three dimensional vector space of $\mathbb{R}^3.$
The set of vectors perpendicular to this plain constitute a one dimensional vector space which is just a line, spanned by any non-zero vector on that line.
Since $<1,-2,3>$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
$\sum a_n$ converges conditionally and $\sum b_n$ converges absolutely, then will $\sum a_nb_n$ converge absolutely? Suppose $\sum a_n$ converges conditionally and $\sum b_n$ converges absolutely, then will $\sum a_nb_n$ converge absolutely?
I know that $\sum|a_n|$ does not converge while $\sum a_n$ does
and that $\su... | $|a_{n}|<1$ eventually and so $|a_{n}b_{n}|\leq|b_{n}|$ eventually, but $\displaystyle\sum|b_{n}|<\infty$, then so is $\displaystyle\sum|a_{n}b_{n}|<\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Suppose a large number of N people flip a coin 100 times, how would you find the percentage of them that would get less than 45 heads? Assume that the probability of 100 people is big enough that the trial can be approximated by normal distribution.
In the problem, N is unknown.
I think you would need to use the 68-95... | Let $X$ denote the number of heads we obtain in $100$ flips.
Normal approximation without continuity correction:
$$\mu=np$$
$$\sigma^2=npq$$
where $$\frac{X-\mu}{\sqrt{npq}}\sim N(0,1)$$
Then we have
$$\begin{align*}
P(Z\lt 45)
&=\Phi\left({\frac{45-50}{\sqrt{100\cdot0.5\cdot0.5}}}\right)\\\\
&=\Phi(-1)\\\\
&\approx0.1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681802",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Simplifying $\frac{1}{2\cdot\sqrt{e^x}}\cdot e^x$ We have to simplify:
$$\frac{1}{2\cdot\sqrt{e^x}}\cdot e^x$$
I came to the conclusion that the answer would be:
$$\frac{e^x}{2\cdot\sqrt{e^x}}$$
But I was wrong and it was:
$$\frac{\sqrt{e^x}}{2}$$
Where did I go wrong and why?
Thanks
| $$\frac{1}{2\sqrt{e^x}}\cdot e^x=\frac{1}{2\sqrt{e^x}}\cdot (\sqrt{e^x}\sqrt{e^x})=\frac{\require{cancel}\cancel{\sqrt{e^x}}}{2\cancel{\sqrt{e^x}}}\cdot\sqrt{e^x}=\frac{\sqrt{e^x}}2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2681943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Finding a closed form for $\sum\limits_{\substack{0\le n\le N\\0\le m\le M}}\left|nM-Nm\right|$ I am trying to figure out if there is a closed form for the following sum:
$$
\sum_{\substack{0\le n\le N \\ 0\le m\le M}}\left|(N-n)(M-m)-nm\right|=\sum_{\substack{0\le n\le N \\ 0\le m\le M}}\left|nM-Nm\right|.
$$
Clearly,... | Hint:
The problem is governed by the solutions of $Mn=Nm$, the main diagonal of the rectangle. WLOG $M\ge N$ and let us assume for now that $M,N$ are relative primes, so that equality only occurs at the corners.
By symmetry, we just look below the diagonal and for a given $m$,
$$0\le n\le\left\lfloor{\frac{Nm}M}\right\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
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Why $\frac{1}{z^2+1}$ has simple poles at $\pm i$. $\frac{1}{(z+i)(z-i)}$ has simple poles at $\pm i$. I know that pole is simple if Laurent series has only $a_{-1} \ne 0$. So, i tried to derive the series:
\begin{align*}
\frac{1}{(z+i)(z-i)} &= \frac{1}{2i}\left(\frac{1}{z-i}-\frac{1}{z+i}\right).
\end{align*}
But her... | $\frac{1}{z+i}$ is holomorphic in a neigborhood of $i$, hence
$\frac{1}{z+i}= \sum_{n=0}^{\infty}a_n(z-i)^n$. The Laurent expansion around reads now as follows:
$\frac{1}{z^2+1}= \frac{1}{2i}\left(\frac{1}{z-i}-\frac{1}{z+i}\right)=\frac{1}{2i}\left(\frac{1}{z-i}-\sum_{n=0}^{\infty}a_n(z-i)^n \right)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682156",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Prove that $f$ is uniformly continuous iff there exist sequences $a_n,b_n$ such that if $\lim a_n=\lim b_n \implies \lim f(a_n)=\lim f(b_n)$
Suppose $f$ is a real-valued function, and $a_n,b_n$ are real sequences.
Prove that $f$ is uniformly continuous $\iff (\lim a_n=\lim b_n \implies \lim f(a_n)=\lim f(b_n))$
*... | If you go for contradiction, it would mean whatever $\delta$ you take, we have that $\forall$ $\epsilon$ we have $|x-a_n|< \delta \not \Rightarrow |f(x) - f(a_n)| < \epsilon$. Now take $x=b_n$, and since you can take whatever $\delta$ you want, we have that $\lim a_n = \lim b_n$. Now what does that imply?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682364",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Is $\log|f|$ for a holomorphic function continuous when viewed as a map to the extended real line? Let $f$ be a holomorphic function on $\mathbb{C}$ that is not identically zero, and consider $\log|f|$ and view this as a morphism to $\mathbb{R} \cup \infty \cup -\infty.$ If $f(z)=0$ we of course let $log|f(z)| = -\inft... | $g(z)=\log|f|$ is clearly continuous on $\{z\in\Bbb C:f(z)\ne0\}$. Since $f$ is holomorphic and not identically $0$, its zeroes are isolated. Then, if $f(z_0)=0$, we have $\lim_{z\to z_0}g(z)=-\infty$. Thus, $g\colon\Bbb C\to\Bbb R\cup\{\pm\infty\}$ is continuous. However, $\lim_{z\to\infty}g(z)$ exists if and only if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Examine whether $\sum_{n=1}^\infty{\sin^2\left(\frac{1}{n}\right)}$ converges or not I have the series $\sum_{n=1}^\infty{\sin^2\left(\frac{1}{n}\right)}$ and I'm trying to examine whether it converges or not.
My Attempts:
*
*I first tried finding whether it diverges by checking if $\lim_{n\to\infty}{\sin^2\left(\fr... | Just use the fact that$$(\forall n\in\mathbb{N}):\sin^2\left(\frac1n\right)\leqslant\frac1{n^2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 1
} |
Help me motivate a topic. I'm teaching a topic for the first time, and I'm struggling to motivate it. I usually know where things lead and what they're eventually used for, but in this case I'm a bit stumped. (I suspect Galois Theory?!)
Say you have a cubic equation $ax^3+bx^2+cx+d=0$ with roots $\alpha,\beta$ and $\ga... | As others have noted, a big part of this is the aim to solve polynomial equations (implicitly, usually, by radicals, as opposed to using elliptic functions or modular functions). Also as noted, an interesting tangential point is that symmetric functions in the roots can be expressed in terms of the "standard" ones (and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Sliding mode stability Given the scalar system
$$
\dot{x} = -\text{sgn}(x) \tag{1}
$$
with
$$
\text{sgn}(x) = \begin{cases}
-1 & x < 0 \\
0 & x = 0 \\
1 & x > 0 \,.
\end{cases}
$$
What is an easy method to check for stability? And: Say I use quasi sliding mode with
$$
\dot{x} = -\tanh(a x) \,. \tag{2}
$$
If it is shown... | The answer to the second question is generally NO. Consider a family $f_a(x) = x (x - \frac{1}{a}) (x + \frac{1}{a})$, $a > 0$. One has $\lim\limits_{a \to \infty} f_a(x) = x^3$, uniformly for $x$ in compact subsets of $\mathbb{R}$. The equilibrium $0$ is (even asymptotically) stable for any $\dot{x} = f_a(x)$, wher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682772",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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How to prove an implication within an if and only if Suppose you need to prove that $A\iff (B\implies C)$.
The two ways to prove this are:
(1a): Suppose $A$ and $B$ are true. Prove that $C$ is true.
(1b): Suppose $B$ and $C$ are true. Prove that $A$ is true.
(2a): Suppose $A$ and $B$ are true. Prove that $C$ is true.... | It helps to call $D$ the statement $B\implies C$. One has to prove $A\iff D$. So we need to show that $A$ implies $D$, and $D$ implies $A$. This means again, that, assuming $A$ it must follow $C$ if we assume $B$, and conversely, that whenever $C$ follows from $B$, then $A$ follows. Now check your $4$ statements accord... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Expected number of days until it rains d days in a row when P(rain)=p
What is the expected number of days until it rains $d$ days in a row (including the $d$ days) when the probability it rains on a single day is $p$?
I managed to figure out a solution for the case $d=2, p=1/2$ but don't know how to generalize it to ... | Consider $d=2$ first. Let $M:=E_{days}$ be the requested expected number of days.
Consider recurrent relation for $M$. We are waiting for the first rainy day during the random number of days with geometric distribution and with expected value $\frac1p$.
Next day, we have rainy day with probability $p$ and then the tr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2682996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How do you solve $\dot{X} = UX$? Let $X(t) = \left[\begin{matrix}a(t) & b(t) \\c(t) & d(t)\end{matrix}\right]$ and let $U$ be a nonsingular matrix. How do you solve
$$\frac{d}{dt} X(t)=UX(t)$$
I presume there is some general method to solve these kinds of ODEs but I cannot find anything about it, online.
BTW: --------... | You are correct, it can be solved column-wise, and in analogy to the scalar form of this differential equation.
The solution is
$$X(t)=\exp\left(U t\right) X(0)$$
where $X(0)$ is the initial condition (matrix) and
$$\exp(Ut)=\sum_{n=0}^\infty \frac{(Ut)^n}{n!}$$ is the matrix exponential, which can be computed explic... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Given that $P(x)$ is a polynomial such that $P(x^2+1) = x^4+5x^2+3$, what is $P(x^2-1)$? How would I go about solving this? I can't find a clear relation between $x^2+1$ and $x^4+5x^2+3$ to solve $P(x^2-1)$.
| One more alternative is to start by deducing that $P(x)$ is a quadratic based on the fact that its degree has to be $2$ in order to output a quartic when the argument is a quadratic expression. So let $P(x) = ax^2 + bx + c$.
When $x^2 + 1 = 0, x = \pm i$, so we get From $P(0) = 1 - 5 + 3 = -1$, and also $P(0) = c$ we c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683178",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Breaking up a Stick into different pieces Assume you have a stick of length 1. Choose 5 breaking points randomly along the stick such that the stick is divided into 6 parts. What is the probability that no part is greater than 1/2?
| similar to what @hardmath said
you can break the line into predetermined equal segments for example (10,20,....,40,...100 etc)
Example
lets break the line into 20 segments, the total number of ways 6 line segments can be selected is
total number of positive integers in the equation
$x_1+x_2+x_3+x_4+x_5+x_6 = 20$
= $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Deriving the variance of the Bernoulli distribution For a Bernoulli distribution, $\mu_X = p$. I can easily derive this from the general equation for mean of a discrete random variable:
$$
\mu_X=\sum_{i=1}^kx_iPr(X=x)
$$
$$
\mu_X=1(p)+0(1-p)=p
$$
I know that the variance of the Bernoulli distribution is supposed to be ... | By definition,
$$Var(X)=E(X^2)-E(X)^2%$$
Consider a success being a $1$ and a failure being a $0$.
Then we have
$$\begin{align*}
Var(X)
&=\sum_{x=0}^1(x^2)Pr(X=x)-\left(\sum_{x=0}^1(x)Pr(X=x)\right)^2\\\\
&=\left(1^2\cdot p + 0^2 (1-p)\right)-\left(1 \cdot p + 0 (1-p)\right)^2\\\\
&=p-p^2\\\\
&=p(1-p)
\end{align*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
How can I find the appropiate launching angle given different heights? The problem states the following:
A soccer player $\textrm{20.0 m}$ from the goal stands ready to score.
In the way stands a goalkeeper, $\textrm{1.70 m}$ tall and
$\textrm{5.0 m}$ out from the goal, whose crossbar is at
$\textrm{2.44 m}$ hi... | The initial speed is given as $v_0=18m/s$,
$$y=y_0+(v_0\sin\omega)t-\frac{1}{2}gt^2$$
for $x$,
$$x=x_0+(v_0\cos\omega)t$$
(note both equations contain $v_0$, not $v_{0x}$ and not $v_{0y}$)
Let's put the player at $x=0m$, so $x_0=0m$, the goal at $x=20m$, and the goalkeeper at $x=15m$.
When does the ball pass by the go... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Upper bound on cumulative power of system of limitedly intersecting subsets? We have a set $S$ of power $n$ and $k < n$ subsets $S_1, \ldots, S_k \subseteq S$ such that $|S_i \cap S_j| \le 1$ when $i \ne j$. Is there any nontrivial upper bound on total power of sets $S_1, \ldots, S_k$? In particular, is it true that
$... | The Bonferroni inequalities (basically a truncated version of the inclusion–exclusion principle) can be brought to bear:
$$
\left|\bigcup_{i=1}^kS_i\right| \ge \sum_{i=1}^k |S_i| - \sum_{i<j}\left|S_i\cap S_j\right| \ge \sum_{i=1}^k|S_i| - \frac{k(k-1)}{2},
$$
hence
$$
\sum_{i=1}^k|S_i| \le \left|\bigcup_{i=1}^kS_i\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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How to derive $¬X \lor ¬Y \lor ¬Y$ from $Z ⊃ (¬X \lor ¬Y)$ and $¬Z ⊃ ¬ Y$ I know that one can derive $¬X \lor ¬Y \lor ¬Y$ (which simplifies to $¬X \lor ¬Y$, right?) from $Z ⊃ (¬X \lor ¬Y)$ and $¬Z ⊃ ¬ Y$ but I don't know how you do this.
Maybe one just has to use constructive dilemma?
| If you have Hypothetical Syllogism and Contraposition, you can do it without having to assume $Z \lor \neg Z$:
$1. Z \rightarrow (\neg X \lor \neg Y) \quad Premise$
$2. \neg Z \rightarrow \neg Y \quad Premise$
$3. Y \rightarrow Z \quad Contraposition \ 2$
$4. Y \rightarrow (\neg X \lor \neg Y) \quad Hypothetical \ Syll... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2683827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Probability of symmetric difference How do we prove the following inequality?
$$\mathbb{P}[A\triangle B ]\geq \max\left \{ \mathbb{P}[A-B],\mathbb{P}[B-A] \right \}$$
I already proved that
$$\mathbb{P}[A\triangle B ]=\mathbb{P}[A]+\mathbb{P}[B]-2\mathbb{P}[A\cap B]$$
$$\left | \mathbb{P}[A]-\mathbb{P}[B] \right |\leq \... | We have
$$P(A-B)=P(A)-P(A\cap B)$$
$$P(B-A)=P(B)-P(B\cap A)$$
Using these we can rewrite the first relation as
$$P(A\triangle B)=P(A-B)+P(B-A)$$
Since probabilities are non-negative:
$$P(A\triangle B)\ge\max(P(A-B),P(B-A))$$
| {
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"url": "https://math.stackexchange.com/questions/2683936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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proving that $n^n \le (n!)^2$ I want to prove that $n^n \le (n!)^2$.
Now I tried by induction:
for $n=1$ ,$1=1$ and $P(1)$ is true
I suppose that $P(n)$ is true and I have to demonstrate that $P(n+1)$ is true
$$((n+1)!)^2=(n+1)^2*(n!)^2 \le (n+1)^2*n^n=(n+1)(n+1)*n^n=(n+1)^n$$
But I'm not sure about the last passage
|
Before editing for $n^n \ge (n!)^2$
Note that it is false indeed
$$n^n=\overbrace{n\cdot n \cdot n\cdot...\cdot n} ^{n \, terms} $$
$$(n!)^2=(\overbrace{n\cdot (n-1) \cdot (n-2)\cdot...\cdot 1}^{n \, terms})^2\ge(n\cdot (n-1) \cdot (n-2)\cdot...\cdot 1)\cdot n\ge n^n$$
After editing for $n^n \le (n!)^2$
induction ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that if $f :[0,\infty[ \rightarrow\mathbb{R}$ is continous then $f$ attain a maximum. Show that if $f :[0,\infty[ \rightarrow\mathbb{R}$ is continous with $\lim \limits_{x \to 0} f(x) = 0$ and $\lim\limits_{x \to\infty} f(x) = 0$ then $f$ attain a maximum.
Suppose by contradiction that the real $M=\max_{x\in[0,\in... | HINT.- $f([0,\infty[)$ is connected and $\lim \limits_{x \to 0} f(x) = \lim \limits_{x \to \infty} f(x) = 0$.
| {
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Find a linear transformation such that $S^2 = T$ (general case) Suppose a linear transformation $T:\mathbb{R}^3\rightarrow \mathbb{R}^3$, $T\begin{pmatrix}
x\\
y\\
z
\end{pmatrix} = \begin{pmatrix}
x+y\\
y+z\\
z+x
\end{pmatrix}$ (just an example).
How do i find a linear transformation S such that $S^2=S\circ S=T$ ?... | Here is an explicit counterexample:
Consider the linear transformation $T:\mathbb{R} \to \mathbb{R}$, $x \mapsto -x$. If $S:\mathbb{R} \to \mathbb{R}$, $x \mapsto ax$ (for some $a \in \mathbb{R}$) is a linear transformation satisfying $S \circ S = T$, then $-1 = T(1) = (S \circ S)(1) = a^2 \cdot 1$. But since we are wo... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Complex Analysis Sketch the image under the function $w = e^z$ of each of the following subsets of the z plane: (a) $\{z \in \Bbb C: Re (z) = -2\}$
(b) $\{z \in \Bbb C: Im (z) = 5\pi\}$
(c) $\{z = x + iy : 0 \le x \le 1, 0 \le y \le \pi\}$
(d) $\{z = x + iy : -2 \le x \le -1, -\pi \le y \le 4\pi\}$
(e) $\{z : Im z \ge ... | a)$z = -2 + iy\\
w = e^z = e^{-2} e^{iy} = e^{-2}(\cos y + i\sin y)$
$(\cos y + i\sin y)$ desribes a circle
b) $z = x + 5\pi i\\
w = (e^x)e^{5\pi i} = -e^x$
The image of $w$ is the negative real numbers
c)
$x = e^x(\cos y + i\sin y)$
and I will let you guess what that maps to as the limits of $x,y$ change
| {
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Finding equation and centre of circle through 3 points using matrices Based on answer given here: Get the equation of a circle when given 3 points. We can find equation of circle through points $(1,1), (2,4), (5,3)$ by taking:
$\left|\begin{array}{cccc}
x^2+y^2&x&y&1\\
1^2+1^2&1&1&1\\
2^2+4^2&2&4&1\\... | You’re looking for the coefficients $a$, $b$, $c$ and $d$ of the general equation $$a(x^2+y^2)+bx+cy+d=0 \tag 1$$ of a circle. For each known point on the circle, substituting its coordinates into equation (1) results in a linear equation in these coefficients, so in effect you’re solving the system of linear equations... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A quicker way to do a Lagrange multiplier problem I was working on the problem: minimize $x + 4z$ subject to $x^2 + y^2 + z^2 \le 2 $. I have it solved, I want a faster method for use in standardized exams.
My work:
I tackled this using Lagrange Multipliers, considering the interior by looking for points where all ind... | By Cauchy-Schwarz,
$$|x + 4z| \le \sqrt{1^2 + 4^2} \sqrt{x^2+z^2} \le \sqrt{34}.$$
Then think about when Cauchy-Schwarz attains equality.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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When is it necessary to prove well definedness? Last semester, we tackled group theory. Our professor did not require us to show that a mapping $\phi:G\to G'$ is well-defined before proving that it is a group homomorphism. Now, our new professor requires us to show that $\phi$ is well-defined first before proving that ... | When there's a real chance that the map is not well-defined. For instance, the map $\psi\colon\mathbb{Z}_2\longrightarrow\mathbb Z$ defined by $\psi(\overline x)=x$ is not well-defined, since $\overline 0=\overline 2$ but $\psi(\overline0)\neq\psi(\overline2)$.
| {
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$\exists g\in G: H\cap gPg^{-1}$ is a Sylow $p-$subgroup of $H$ Question:
If $P$ is a Sylow $p-$subgroup of $G$ and $H\leq G$ with $p||H|$ then $\exists g\in G: H\cap gPg^{-1}$ is a Sylow $p-$subgroup of $H$.
Attempt:
We consider the action $H\times G/P\to G/P$ with $(h,xP)\to hxP$ and we know that
$$|G/P|=\sum_{x\in ... | For the "please explain this statement" part: The conjugate $gPg^{-1}$ has the same order with $P$, hence it is a power of $p$. Therefore, $H\cap gPg^{-1}$, being a subgroup of $gPg^{-1}$, has order dividing $|gPg^{-1}|$, therefore, it a power of $p$. Hence, it is a group of order $p^k$ for some $k\neq 0$, hence a $p$-... | {
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Calculete $a$ and $b$ in a limit Calculate $a$ and $b$ in
$$\lim_{x \to 1} \frac{ax^2+(3a+1)x+3}{bx^2+(2-b)x-2} = \frac{3}{2}$$
I tried this
$$\lim_{x \to 1} \frac{(ax+1)(x+3)}{(bx+2)(x-1)} = \frac{3}{2}$$
but I could not see the next step
I tried to look but it did not help. Solve for $a$ and $b$ in a limit
and Find A... | Your first step
$$\lim_{x \to 1} \frac{ax^2+(3a+1)x+3}{bx^2+(2-b)x-2} =\lim_{x \to 1} \frac{(ax+1)(x+3)}{(bx+2)(x-1)}$$
is correct, now observe that you need to cancel out the term $(x-1)$ in order to have a finite limit.
| {
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Understanding $\int_\limits{0}^{1}\frac{1}{|t-s|^{\frac{1}{3}}}ds$ I am trying to understand the following integral $\int_\limits{0}^{1}\frac{1}{|t-s|^{\frac{1}{3}}}ds =\int_\limits{0}^{t}\frac{1}{(t-s)^{\frac{1}{3}}}ds +\int_\limits{t}^{1}\frac{1}{(s-t)^{\frac{1}{3}}}ds=2\int_\limits{0}^{1} \frac{dy}{y^{\frac{1}{3}}}\... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
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What are some good examples of "almost" isomorphic graphs? I'm examining isomorphisms of simple finite undirected graphs.
In order to test whether or not two graphs are isomorphic, there are a lot of "simple" tests one can do, namely, compare the number of vertices, number of edges, degree sequences, and look for edge ... | One class of examples comes from Latin square graphs. If $L$ is an $n\times n$ Latin graphs, take the graph with the $n^2$ triples $(i,j,L_{i,j})$ as vertices, with two triples adjacent if they agree on one of the three coordinates. The graphs from distinct Latin squares are isomorphic if the Latin squares are isotopic... | {
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Evaluating the stochastic integral of Brownian motion without using Ito's formula? I am currently trying to understand some lecture notes (which offer little depth) and the following exercise (without a solution) occurs:
Using the definition
$$
\int_0^t f_s \hspace{1mm} dW_s := \text{l.i.m}_{\lambda \rightarrow 0} \... | The integral value should be:
$$\int_0^tW_s\;dW_s=\frac{W_t^2}{2}-\frac{t}{2}$$
Notice that
$$\sum_iW_{s_i}(W_{s_{i+1}}-W_{s_i})=\frac{1}{2}\sum_i(W_{s_{i+1}}^2-W_{s_i}^2)-\frac{1}{2}\sum_i(W_{s_{i+1}}-W_{s_i})^2$$
The first sum is a telescoping sum that reduces to $\frac{W_t^2}{2}$ (since $W_0 = 0$). The second sum b... | {
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If a Vector Space $V$ has dimension $n$ and $T:V\to V$ is linear, then is the following true? Suppose we have that a basis for $V$ is
$B=\{v_1,v_2,\ldots ,v_n\}$ and $v_1,v_2,\ldots,v_k$ are a basis for $\operatorname{im}(T)$.
Does it follow that $v_{k+1}, v_{k+2},\ldots, v_n$ is a basis for $\ker(T)$? What if $T$ is ... | Suppose $T(v_1) = v_1, \quad T(v_2) = v_2, \quad T(v_3) = v_1+v_2.$ Then $\{v_1,v_2\}$ is a basis of $\operatorname{im}(T)$ but $\{v_3\}$ is not a basis of $\ker(T).$ In this case $\{v_1+v_2-v_3\}$ is a basis of $\ker(T).$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Exact Solution for Simple Sliding Linkage I am working with a linkage that consists of two cylinders with lengths $b$ and $c$ that actuate a platform to height $h$ and angle $\theta$.
The $h$ link is always perpendicular to the bottom two $a$ links which are fixed. The top two $a$ links are collinear. The value of $a$... | I don't know that there is a "pretty" way to solve it symbolically. It is, however, technically possible to separate the variables and derive "exact" equations in either variable alone, which could then be solved numerically to the desired precision. One way to do that:
$$
b^2 - a^2(1-\cos\theta)^2 = {(h+a\sin\theta)}^... | {
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Parity of an odd integer I'm not sure if I'm just tired and I'm missing something obvious, but how come I'm obtaining the following:
$$(2m+1)^n=\sum_{k=0}^n \binom{n}{ k} (2m)^k = 2 \sum_{k=0}^n \binom{n}{ k} 2^{k-1}m^k $$
This seems to imply any power of an odd integer is even, but $3^2=9$ is an obvious counterexampl... | All has been said:
$k=0$ spoils the party:
$\sum_{k=0}^{n} \binom{n}{k}(2m)^k=$
$ \binom{n}{0}(2m)^0 + 2\sum_{k=1}^{n}\binom{n}{k}(2)^{k-1}m^k.$
The first term in the above sum $=1.$
Hence?
| {
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Concerning the invertibility of $T-2I$ and $T-6I$ when $\dim E(8,T) = 4$. Prior to reading the following post please consider the following notation and results.
*
*$\mathbf{F}$ is either $\mathbf{R}$ or $\mathbf{C}$
*$T\in\mathcal{L}(V)$ means the $T$ belongs to the set of all linear transformations over the vecto... | Your proof is correct. However, you made a mistake in the statement of theorem (5.30): it's not “all entries”; it's “all diagonal entries”. Besides, when you applied it, you only mentioned the diagonal entries.
| {
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Tangents of a curve whose trace remains in a half-plane. I need to prove the following result:
If a curve's trace is contained in a half-plane then the tangents at
the intersection of the curve with the line defining the half-plane
$R$ coincide with $R$.
In other words:
If $\alpha:I \to \mathbb{R}^2$ is a regula... | Assume that $R$ does not pass through $0$. For any half-plane that does not contain $0$ there exists a unique vector $x \in \mathbb{R}^2$ such that $$y \in H^+ \Leftrightarrow x.y \geq 1$$ and $$y \in H^- \Leftrightarrow x.y \leq 1.$$ If we assume that $\alpha$ lies in $H^+$ and touches the line $R$ at $t_0$ then $x.\a... | {
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Why: $z^{-1}=\cos (-\phi)+i\sin(-\phi)$ Let's suppose that we have a complex number with
$$r=1 \implies z=\cos\phi+i\sin\phi$$
Then why is $$z^{-1} =\cos (-\phi)+i\sin (-\phi)=\cos\phi-i\sin\phi$$
| As your complex number as $r=1$, you can express it like $z=e^{i\theta}$, where $\theta$ is the argument. Then,
$$z^{-1}=\frac{1}{z}=\frac{1}{e^{i\theta}}=e^{-i\theta}$$
Now, using the trigonometric form of complex numbers,
$$e^{-i\theta}=\cos(-\theta)+i\sin(-\theta)=\cos(\theta)-i\sin(\theta),$$
where we used that $\... | {
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What kind of fraction will have this property? List out i/19:
They form a square, and the sum of each column is the same, 81.
i/7 also have such property:
142857
285714
428571
571428
714285
857142
Any other fractions have such property?
What kind of fractions will have such property?
| You are after numbers that are the decimal period of so-called cyclic numbers.
$7$ and $19$ in this context are called full reptend primes (OEIS A001913):
Primes p such that the decimal expansion of 1/p has period p-1, which
is the greatest period possible for any integer.
The first few are $7, 17, 19, 23, 29, 47,... | {
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Integrating w.r.t. two different variables I can't really follow what's going on here to get from line 2 to line 3:
\begin{align}
& \int^\infty_0 \frac{d^2 p}{(2\pi)^2} \frac{\Lambda^2}{(p^2+\Lambda^2)(p^2+m^2)} \\[10pt]
= {} & \int^1_0 dz \, \frac{d^2 p}{(2\pi)^2}\frac{\Lambda^2}{(p^2+z\Lambda^2+(1-z) m^2)^2} \\[10pt]... | The integral over momentum space is missing in Line 2 of the OP. That is to say that the expression $\displaystyle d^2 p\,\frac{\Lambda^2}{(p^2+z\Lambda^2+(1-z) m^2)^2}$ should read
$$\int_{-\infty}^\infty\int_{-\infty}^\infty \frac{\Lambda^2}{(p_1^2+p_2^2+z\Lambda^2+(1-z) m^2)^2}\,dp_1\,dp_2$$
which after a transform... | {
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Question on evaluating $\int_{C}\frac{e^{iz}}{z(z-\pi)}dz$ without the residue theorem I am trying to figure out how to evaluate the integral $\int_{C}\frac{e^{iz}}{z(z-\pi)}dz$ where $C$ is any circle centered at the origin with radius greater than $\pi$. I can see that $\frac{e^{iz}}{z(z-\pi)}$ is analytic everywher... | By symmetry the residues of $\frac{e^{iz}}{z(z-\pi)}$ at $z=0$ and $z=\pi$ are the same and they equal $-\frac{1}{\pi}$.
It follows that for any $R>\pi$ we have
$$ \oint_{\|z\|=R}\frac{e^{iz}}{z(z-\pi)}\,dz = 2\pi i\cdot\left(-\frac{2}{\pi}\right) = -4i.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2686903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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When an odd polynomial is a one-one map on $\mathbb{R}$ Let $f(x)=c_1x+c_3x^3+c_5x^5+\cdots+c_{2m+1}x^{2m+1}$, $c_1,c_3,c_5,\ldots,c_{2m+1} \in \mathbb{R}$, $m \in \mathbb{N}$,
namely, $f$ is an odd polynomial over $\mathbb{R}$.
When such a polynomial is a one-one map on $\mathbb{R}$?
Is it always one-one?
Example:... | The answer is no, ti is not always one-to-one.
Consider the following odd polynomial
$$P(x)=x(x-5)(x-10)= x^3-15x^2+50x$$
$$ P(0)=P(5)=P(10)=0$$
It is not one-to-one.
However, a function will be one-to-one if it's derivative is always positive or always negative.
| {
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"url": "https://math.stackexchange.com/questions/2687041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to find UMVUE of $\theta^k$ when $x_1, \ldots, x_n$ is a sample from Bernoulli$(\theta)$? Let $x_1, x_2, \ldots, x_n$ be a random sample from the Bernoulli ($\theta$).
The question is to find the UMVUE of $\theta^k$.
I know the $\sum_1^nx_i$ is the complete sufficient statistics for $\theta$.
Is $\left(\frac{\sum_... | Having that
$$\theta^m=P\{ X_1=x_1,X_2=x_2,...,X_m=x_m\}$$
An unbiased estimator for $\theta^m$ is
$$T=
\begin{cases}
1, & if \ \ X_1=X_2= \, ... \,=X_m =1 \\
0, & in \ other \ case
\end{cases}$$
But
$$\begin{align} E[T|S=s] & = P\{X_1=1,X_2=1,...,X_m=1|S=s\}=\frac{P\{X_1=1,X_2=1,...,X_m=1,S=s\}}{P\{S=s\}} = \\\\ & = ... | {
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Summation Manipulation Problem Is the following statement true:
$$\sum^k_{j=0} \left(\sum^j_{i=0}a_i b_{j-i}\right) d_{k-j} = \sum^k_{j=0} \left(\sum^j_{i=0}b_i d_{j-i}\right) a_{k-j}$$
I'm not sure if this is true as I've been unable to prove this directly. However, I think this is true.
If we fix $k$ and then $j$, w... | Both sides equalize: $$\sum_{m+n+o=k}a_mb_nd_o$$ where $m,n,o$ denote nonnegative integers.
The expression: $$\sum_{m+n+o=k}a_mb_nd_o$$ abbreviates: $$\sum_{\langle m,n,o\rangle\in S}a_mb_nd_o$$ where $S=\{\langle m,n,o\rangle\in\mathbb N\mid m+n+o=k\}$ and $\mathbb N=\{0,1,2,\dots\}$.
Further $|S|=\binom{k+2}2$ (which... | {
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Find the number of ways of arranging the letters
Find the number of ways of arranging the letters $\text{AAAAA, BBB, CCC, D, EE & F}$ in a row if no two $\text{C's}$ are together?
My Attempt:
Well, I should I arrive at the answer if I subtract the cases where 3 $\text{C's}$ and 2 $\text{C's}$ appear together from the... | While I prefer the approach explained by Alaleh A and Parcly Taxel, here is how you can solve the problem using the Inclusion-Exclusion Principle.
The number of distinguishable arrangements of $5$ A's, $3$ B's, $3$ C's, $1$ D, $2$ E's, and $1$ F is
$$\frac{15!}{5!3!3!1!2!1!}$$
as you found.
From these, we must subtrac... | {
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"answer_id": 1
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Extension of rational map from normal variety to algebraic group Assume $k$ is an algebraically closed field, $X$ is a normal variety over $k$, $G$ is an algebraic group over $k$. I heard that Weil proved any rational map from $X$ to $G$ which defined over a codimension $>1$ open subset can be extended to the whole $X$... | I'm not too convinced by the statement but, here's something that may be relevant. I'm paraphrasing from page 151-152, Chapter 8 section f of James Milne's ``Algebraic Groups: the theory of group schemes of finite type over a field". The proofs are given in a further reference:
MILNE, J. S. 1986. Abelian varieties, pp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2687792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If matrices $A,B$ similar, find nonsingular $S$ s.t.$B=S^{-1}AS$ Consider the matrices below
$$A=\begin{bmatrix}9&4&5\\-4&0&-3\\-6&-4&-2\end{bmatrix}$$
and
$$B=\begin{bmatrix}2&1&0\\0&2&0\\0&0&3\end{bmatrix}$$
These matrices have the same eigenvalues $\{2,2,-3\}$ and the same Jordan Canonical Form so they are similar. ... | Notice that $B$ is precisely the Jordan form of $A$, so it suffices to find a Jordan basis for $A$.
We have:
$$A - 2I =\begin{bmatrix}7&4&5\\-4&-2&-3\\-6&-4&-4\end{bmatrix},\quad (A - 2I)^2 =\begin{bmatrix}3&0&3\\-2&0&-2\\-2&0&-2\end{bmatrix}$$
so $\ker (A - 2I)^2 = \operatorname{span}\{e_1- e_3, e_2\} = \operatornam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2687921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
How to deal with $(-1)^{k-1}$ It's a problem on mathematical induction.
$$1^2-2^2+3^2-.....+(-1)^{n-1}n^2=(-1)^{n-1}\frac{n.(n+1)}{2}$$
I have proved it for values of $n=1,2$.
Now I assume for $n=k$
$$P(k):1^2-2^2+3^2-.....+(-1)^{k-1}k^2=(-1)^{k-1}\frac{k.(k+1)}{2}$$.
$$P(k+1):1^2-2^2+3^2-.....+(-1)^{k-1}k^... | you have to prove that $$(-1)^k\frac{k(k+1)}{2}+(-1)^k(k+1)^2=(-1)^k\frac{(k+1)(k+2)}{2}$$ or
$$(-1)^{k-1}\frac{k(k+1)}{2}=(-1)^k\frac{(k+1)(k+2)}{2}-(-1)^k(k+1)^2$$
the right-hand side is given by
$$(-1)^k(k+1)\left(\frac{(k+2)}{2}-k-1\right)$$ and this is equal to
$$(-1)^k(k+1)\left(\frac{k+2-2k-2}{2}\right)$$ Can yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
} |
Double Antiderivation problem I have to find $f(x)$ given $f''(x)$ and certain initial conditions.
$$f"(x) = 8x^3 + 5$$ and $f(1) = 0$ and $f'(1) = 8$
$$f'(x) = 8 \cdot \frac{x^4}{4} + 5x + C = 2x^4 + 5x + C$$
Since $f'(1) = 8 \Rightarrow 2 + 5 + C = 8$ so $C = 1$
$$ f'(x) = 2x^4 + 5x + 1$$
$$f(x) = 2 \cdot \frac{x^5}{... | Let check
$$f(x) = \frac{2}{5} \cdot x^5 + \frac{5}{2} \cdot x^2 + x - \frac{39}{10}\implies f'(x)= 2x^4+5x+1\implies f''(x)=8x^3+5$$
$$f(1)=\frac{2}{5}+ \frac{5}{2} + 1 - \frac{39}{10}=\frac{4+25+10-39}{10}=0$$
$$f'(1)=2+5+1=8$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
When point is inside of circle Let we have points $A_1,A_2,A_3,A_4$.In time $t$ point $A_i$ has coordinates $(x_i,y_i) + (v_{xi},v_{yi}) * t$, all parametrs are given. Describe algorithm to find all $t$ when point $A_4$ inside circle circumscribed around $A_1,A_2,A_3$ , or to find first moment when this happens.
| Let $A_i(X_i,Y_i)$ where
$$X_i=x_i+v_{xi}t\quad\text{and}\quad Y_i=y_i+v_{yi}t$$
for $i=1,2,3$ and $4$.
According to MathWorld, the center of the circle passing through three points $(X_1,Y_1),(X_2,Y_2)$ and $(X_3,Y_3)$ is given by
$$\left(-\frac{b}{2a},-\frac{c}{2a}\right)$$
and its radius is given by
$$\sqrt{\frac{b^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Why $A_\varepsilon \subset \bigcup_{\sigma \in \mathfrak{F}} S_\sigma,\;?$ Let $(X,\mu)$ be a measure space and $f_1,\cdots,f_d\in L^\infty(X)$.
Set
$$g:=\displaystyle\sum_{k=1}^d|f_k|^2\;\;\text{and}\;\;c:=\|g\|_\infty.$$
Let $\sigma:=\{a_1,b_1,\cdots,a_d,b_d\}$ be such that $a_i,b_i\in \mathbb{Q}_+$ for all $i$. Set
... | Let $x\in A_\varepsilon$. Choose non negative rational numbers $a_i$ and $b_i$ such that $$ 0\lt \left[\Re(f_i(x))\right]^2- a_i\lt \frac{ \varepsilon}{2d} \mbox{ and } 0\lt \left[\Im(f_i(x))\right]^2 -b_i \lt \frac{ \varepsilon}{2d} .$$
Let $\sigma:=\{a_1,b_1,\cdots,a_d,b_d\}$. Then $\sigma$ belo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Diffeomorphism invariance, Lie derivative There is written in the Hamilton's Ricci flow book about Lie Derivative this:
The Lie derivative, which measures the infinitesimal lack of
diffeomorphism invariance of a tensor with respect to a 1-parameter group of
diffeomorphisms generated by a vector field, has the follo... | To make things simple, let $X,Y$ be vector fields on $\mathbb{R}^n$. Then $\mathcal{L}_XY$ gives the instantaneous rate of change of $Y$ in the direction of the flow $\phi_t$ in which $X$ induces. You can show that,
$$ \mathcal{L}_XY(p) = [X,Y](p) = \lim_{t \to 0} \frac{Y( \phi_t(p)) - Y(p)}{t}$$
Here $\{\phi_t\}$ is t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Homology of the closed topologist sine curve The closed topologist sine curve, $X$, is the subspace of $R^2$ consisting of all the points $(x,\sin(1/x))$ for $x \in (0,1]$, and all points $(0,y)$ for $y \in [-1,1]$ and an arc from $(0,-1)$ to $(1,\sin(1))$ Compute the singular homology of $X$, using a suitable Mayer-Vi... | You do not consider the closed topologist's sine curve, but the Warsaw circle. See To show that Warsaw circle is simply connected.
Let $S = \{(x,\sin\frac{1}{x}):0<x \le 1 \}$, $L = \{0\} \times [-1,1]$ and $T = L \cup S$. This is the closed topologist's sine curve. Your space $X$ is obtained by joining $(0,-1), (1,\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How can i prove this? (infinite summation) I have to prove the following
$$\sum_{k=1}^n \frac{(-1)^{k-1}}{k} {n \choose k} =1+\frac{1}{2}+...+\frac{1}{n}$$
I try to prove it using that $$\sum_{k=0}^n {(-1)^{n-k}} {n \choose k} k^m = \begin{cases}
0, & \text{if $m<n$ } \\
n!, & \text{if $m=n$ }
\end{cases} $$
but my b... | An overkill. By the Melzak's identity with $f\equiv1$ we have $$\sum_{k=1}^{n}\dbinom{n}{k}\frac{\left(-1\right)^{k-1}}{x+k}=\frac{1}{x}-\frac{1}{x\dbinom{x+n}{n}}=\frac{\dbinom{x+n}{n}-1}{x\dbinom{x+n}{n}}$$ then taking $x\rightarrow0$ and recalling that $$\frac{d}{dx}\dbinom{x+n}{n}=\dbinom{x+n}{n}\left(\psi^{\left(0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688718",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Quotient group of $D(2,3,7)$ under the Klein Quartic The Klein Quartic is a quotient space of the hyperbolic plane. Let $k$ be its quotient map. Given two isometries $f,g$ of the hyperbolic plane, we say that $f \cong g$ iff $k \circ f = k \circ g$.
The Von Dyck group $D(2,3,7)$ is a group of isometries of the hyperbol... | The group is isomorphic to $GL(3,2)$. The quotient map $k$ is generated by
$$k(r) = \begin{bmatrix}0&1&1\\0&0&1\\1&0&0\end{bmatrix}$$
$$k(m) = \begin{bmatrix}0&0&1\\0&1&0\\1&0&0\end{bmatrix}$$
These matrices where found by a computer program looking for matrices that satisfied the Von dyck group relations and that gene... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that set $A$ a finite abelian group and $B$ multiplicative group of all homomorphisms from $A$ to units of $C$ are isomorphic
Let $A$ be a finite abelian group and let $B := \{f: A \to \mathbb C^{×} | f$ is a group homomorphism $\}$. It can be easily checked that $B$ is an abelian group via $fg(x) = f(x)g(x)$.
... | The problem with your special case is that we want the multiplicative group of the units of $\mathbb{C}$, so we have that $f(2 \cdot 1) = f(1)^2$, not $f(2 \cdot 1) = 2f(1)$. Since $f$ is a homomorphism, it maps the identity element of $A$ to the identity element of $\mathbb{C}^\times$, and so $f(0) = 1$. Since $f(1)^2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2688992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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$\int_a^\infty f(x) \, dx$ converges $\Rightarrow$ $\lim_{x\to \infty}f(x)=0$ $\int_a^\infty f(x) \, dx$ converges $\Rightarrow$ $\lim_{x\to \infty}f(x)=0$.
Give a proof or counterexample.(Assume $f(x)$ positive and continuous.)
I can show that $\int_R^{R'}f(x)\,dx \approx 0$ for $R,R'$ large. $f(c)(R'-R) \approx 0$ fo... | Construct a piecewise-linear function $f$ such that around every positive integer $n$ the function describes an isosceles triangle with height $1$ and base length of $\frac{1}{2^n}$.
Clearly $\lim_{x\rightarrow \infty} f(x) \nrightarrow 0$, however the integral $\int_0^\infty f(x)dx$ is the sum of the areas of the t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2689297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Derivative of eigenvectors of a symmetric matrix-valued function Given a real symmetric $3\times3$ matrix $\mathsf{A}_{ij}$ and its derivative (w.r.t. some parameter, let's call it time) $\dot{\mathsf{A}}_{ij}$, I want to measure/obtain the rotation (rate and direction) of the eigenvectors (the eigenvectors of a real s... | Suppose that a given (differentiable) matrix-valued function $\mathrm A : \mathbb R \to \mbox{Sym}_n(\mathbb R)$, where $\mbox{Sym}_n(\mathbb R)$ denotes the set of $n \times n$ real symmetric matrices, does have a time-varying spectral decomposition
$$\mathrm A (t) = \mathrm V (t) \, \Lambda (t) \,\mathrm V^\top (t)$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2689374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
} |
Prove that $D_{\mathbf{v}}{f}(\mathbf{x}) = \nabla f(\mathbf{x}) \cdot{} \mathbf{v}$ Given the directional derivative
$$D_{\mathbf{v}}{f}(\mathbf{x}) = \lim_{h \rightarrow 0}{\dfrac{f(\mathbf{x} + h\mathbf{v}) - f(\mathbf{x})}{h}}$$
and that $f$ is differentiable at $\mathbf{x}$, how do I prove that
$$D_{\mathbf{v}}{f}... | Let $g(h):=f(x+hv)$ where $x:=(x_1,...,x_n)$ and $v:=(v_1,...,v_n)$ and $h\in\mathbb{R}$. Then since $f$ is differentiable we have $g(h)$ is differentiable. Therefore
$$g'(h)=\frac{d}{dh}f(x+hv)=\frac{d}{dh}f(x_1+hv_1,...,x_n+hv_n)=\frac{d}{dh}f(y_1,...,y_n)$$
where $y_k:=x_k+hv_k$. Applying the rule of total differen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2689506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Proof verification : $X$ and $Y=g(X)$ are independent random variables $\implies$ $Y$ is degenerate THE PROBLEM :
Source : Alan F. Karr (Probability), p.$96$, problem $3.10.(b)$
MY SOLUTION :
Suppose $B \subset \mathbb{R}$ such that $P\left\{Y \in B\right\}=1$. We want to show that $B$ is singleton. First of all $B \n... | I may be using facts above what you currently know* but here goes:
Y is a distraction. We have that $X$ & $g(X)$ are independent. By definition, $$\sigma(X) \ \text{&} \ \sigma(g(X)) \ \text{are independent.}$$
$$\to \sigma(X) \ \text{&} \ \sigma(X) \ \text{are independent} \tag{Why? Hint: subset}$$
This means that $X$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2689602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Construct example - $A$ and $B$ have strictly positive eigenvalues, but $A+B$ and $AB$ have strictly negative eigenvalues. As a follow-up to this question I asked, I wondered what would happen if I imposed the weaker condition of having positive eigenvalues, rather than being positive definite.
How do I construct an ex... | For an example where $A$ and $B$ have all positive eigenvalues while $AB$ has all negative eigenvalues, consider
$$ A = \pmatrix{10 & 8\cr -9 & -7\cr},\ B = \pmatrix{1 & 0\cr 0 & 2\cr} $$
$A$ and $B$ both have eigenvalues $1$ and $2$, while $AB$ has a double eigenvalue $-2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2689708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
$|\mathbb{Z}|=|\mathbb{N}|$
Hello all, if you could give some critique on my proof I would be grateful.
Show that $|\mathbb{Z}|=|\mathbb{N}|$ for,
$$\displaystyle f(n) = \begin{cases} 2|n|+1, \text{if} & n\leq0 \\ 2n, \text{if} & n>0\end{cases}$$
By the definition of cardinality (def. 2.10b) we note that if a functio... | Using common sense:
*
*send even naturals to positives,
*send odd naturals to negatives.
This is clearly reversible and establishes a bijection. Some care is required in the vicinity of $0$, but if necessary you can adjust by translation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2689829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Can this be done with Stokes' Theorem? (Exercise) Problem description:
Let $\gamma$ denote the curve of intersection of the two surfaces $z=x^2+y^2$ and $z=1+2x$. Calculate the line integral $W=\oint_\gamma \boldsymbol{F}\cdot d\boldsymbol{r}$, where $\boldsymbol{F}=(0,x,-y)$ and $d\boldsymbol{r}=(dx,dy,dz)$.
Now, by e... | If you have an closed path integral, you can use stokes theorem.
$\oint_\gamma F\cdot \ d\gamma = \iint \nabla \times F\cdot dS$
Where $S$ is the elliptical disc on the plane bounded by the paraboloid.
$\nabla \times F = (-1,0,1)$
$dS = (-\frac {\partial z}{\partial x},-\frac {\partial z}{\partial y}, 1)\ dA= (-2,0,1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2689900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Reference request: Hard measure theory / functional analysis problems I want to revise basic measure theory and functional analysis and I was wondering if there's a good source of challenging problems? Ideally I'm hoping these problems will help me go through the material again, as I know that this is the best way to l... | You might try Biler and Witkowski, Problems in Mathematical Analysis
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2690023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Number of divisors of the number $2079000$ which are even and divisible by $15$
Find the number of divisors of $2079000$ which are even and divisible by $15$?
My Attempt: Since they are divisible by $15$ and are even, $2$ and $5$ have to included from the numbers prime factors.
$2079000 = 2^3 \cdot 3^3 \cdot 5^3 \cdo... | Your factorization of $2079000$ is incorrect.
\begin{align*}
20790000 & = 2079 \cdot 1000\\
& = 2079 \cdot 10^3\\
& = 2079 \cdot 2^3 \cdot 5^3\\
& = 3 \cdot 693 \cdot 2^3 \cdot 5^3\\
& = 3 \cdot 3 \cdot 231 \cdot 2^3 \cdot 5^3\\
& = 3 \cdot 3 \cdot 3 \cdot 77 \cdot 2^3 \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2690113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 8,
"answer_id": 1
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Connected Set in $ {R}^{2} $ Is $A=\{(x,y): x^2+y^2=1\}$
is connected in $ℝ^2$?
From its graph, I would conclude that it's not path connected.
| $$
A = \left\{(x,y) : x^2 + y^2 = 1 \right\} = \left\{(x,y) : x = \cos \theta, y = \sin \theta, 0 \leq \theta < 2\pi\right\}
$$
The interval $\Theta = \left[0,2\pi \right)$ is connected, since $A$ is image of continuous map defined on a connected set then it is also connected (as proved here)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Fraction and simplification
solve: $\frac{1}{x(x-1)} + \frac{1}{x} = \frac{1}{x-1}$
What are the possible answers ?
(A) -1 (B) Infinitely Many Solutions (C) No solution (D) 0
The answer from where i've referred this is (B), but when i simplify it I get (D)
My solution:
$$\frac{1}{x(x-1)} + \frac{1}{x} = \frac{1... | You can make it much simpler.
First you have to set the domain of validity: you must have $x\ne 0,1$.
Next, on this domain, remove the denominators multiplying both sides by the l.c.m. of the denominators, and simplify; you get:
$$1+(x-1)=x\iff x=x.$$
Hence any number $x\ne 0,1$ is a solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691116",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Generating multivariate Gaussian samples--Why does it work? I came across the method for generating multivariate normal samples on wikipedia:
https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Drawing_values_from_the_distribution
A widely used method for drawing (sampling) a random vector $X$ from the $N$-... | Simply take the vector you have generated, $\boldsymbol{x}$, and compute its covariance: $$\mathbb E[(\boldsymbol{x}-\boldsymbol{\mu})(\boldsymbol{x}-\boldsymbol{\mu})^T] = \mathbb E[\boldsymbol A\boldsymbol z\boldsymbol z^t\boldsymbol A^t] = \boldsymbol A\mathbb E[\boldsymbol z\boldsymbol z^t]\boldsymbol A^t = \bolds... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Linear combination of non-identically distributed, independent exponential random variables I am working on the following homework assignment:
Under the assumptions of the Normal Simple Linear Regression model, $Y_i|X_i \sim N(\beta_0 + \beta_1 X_i, \sigma^2)$. Consider the model where $Y_i|X_i \sim Exp(\frac{1}{\beta... | Using the moment generating function:
$$M_{\hat{\beta_1}}(t) = E[e^{\hat{\beta_1}t}] = E\left[\prod_{i=1}^ne^{k_i Y_i t}\right] = \prod_{i=1}^n E[e^{k_i Y_i t}] = \prod_{i=1}^n M_{Y_i}(k_i t) = \prod_{i=1}^n (1-k_i \beta_1 x_i t)^{-1}$$
Finally, by substituting $k_i$, I get
$$M_{\hat{\beta_1}}(t) = \left(1 - \dfrac{\be... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to show that $x^2 - 37y^2 =2$ does not have integer solutions We need to prove that $x^2 - 37y^2 =2$ does not have integer solutions.
I have two angles I thought about approaching it from:
*
*Since 37 is prime, I can show that for $x$ not divisible by $37$, we have $x^{36} ≡ 1mod(37)$ but I don't see how that's ... | $x^2 - 37y^2 = 2$ is even. So as $odd \pm even = odd$, $even + even = even$, $odd + odd = even$ we can see that either $x^2$ and $37y^2$ are either both even or both odd and we can pursue that and get a contradiction.
But now would be a nice time to point out that for all integer $m^2 \not \equiv 2 \mod 4$ and $m^2 \n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691514",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Is the function $\sin^{x}(x)$ discontinuous on the entire $\mathbb{R}$? I have come across the function $\sin^{x}(x)$ (or a in a different notation$(\sin(x)^{x})$ while practicing calculating its the first derivative. I then plot it in Desmos since I don't know how to graph this by hands yet. It seems that the graphin... | The canonical way to define a power $a^b$ is to do
$$
a^b:=e^{b\log a}.
$$
This works great, but the problem is that it doesn't make sense when $a<0$. And that's the state of things. There is no natural way to define arbitrary powers of negative numbers (think $(-1)^{1/2}$ for an easy example).
So your function is n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
binomial limit when limit approches to infinity $\displaystyle \lim_{n\rightarrow\infty}\binom{n}{x}\left(\frac{m}{n}\right)^x\left(1-\frac{m}{n}\right)^{n-x}$
solution i try
$\displaystyle \lim_{n\rightarrow\infty}\left(\frac{m}{n}\right)^x\left(1-\frac{m}{n}\right)^{n-x}=\lim_{n\rightarrow\infty}(\frac{m}{n}+1-\frac... | Note that we have
$$\binom{n}{x}\left(\frac{m}{n}\right)^x\left(1-\frac mn\right)^{n-x}=\frac{m^x}{x!}\left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)\cdots\left(1-\frac{x-1}{n}\right)\left(1-\frac mn\right)^{n-x}$$
Therefore, for fixed $x$, we have
$$\lim_{n\to \infty}\binom{n}{x}\left(\frac{m}{n}\right)^x\left(1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
The norm or the singular values of the sum of identity matrix and a rank-$1$ matrix Let $A$ be an $N \times N$ rank-$1$ matrix. I am interested in finding the norm or the maximum singular value of $(A-cI)$ where $I$ is $N \times N$ identity matrix and $c>0$ is a scalar constant.
| If $N=1$ then $\|A-cI\| = |A-c|$.
Assume that $N>1$.
Note that since $\ker A $ is non trivial have $(A-cI)x = -cx$ for some
non zero $x$, and so $\|A-cI\| \ge |c|$.
If $A$ is rank one it can be written as $A=u v^T$ for two vectors $u,v$. Without loss of generality we can take $\|u\| = 1$.
If $u,v$ are colinear, then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How do I find the truth value in this logic problem: $(( p \lor s) \land \neg q) \rightarrow ( r \rightarrow s)$? The problem is as follows:
$$[\left ( p \vee s\right) \wedge \sim q\,] \rightarrow \left ( r \rightarrow s \right )$$
From the preceding statement find the truth value of $\textrm{p, q, r, s}$
The alternati... |
The problem is as follows:
$$[\left ( p \vee s\right) \wedge \sim q\,] \rightarrow \left ( r \rightarrow s \right )$$
From the preceding statement find the truth value of $\textrm{p, q, r, s}$
Your truth table is incorrect. Try this truth table generator: http://web.stanford.edu/class/cs103/tools/truth-table-tool/
Hi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2691983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Dimension of $V(g_1(\mathbf{x})- g_1(\mathbf{y}), ...,g_s(\mathbf{x})- g_s(\mathbf{y}) )$ compared to $V(g_1(\mathbf{x}), ...,g_s(\mathbf{x}) )$? Let $g_1, ..., g_s$ be non-constant homogeneous polynomials in $n$ variables with coefficients in $\mathbb{C}$. Let $\mathbf{x}$ and $\mathbf{y}$ be two sets of $n$ variables... | Here is a heuristic argument for the equality $\dim V=\dim W$ when the $g_i$'s are polynomials with constant term equal to zero (but not necessarily homogeneous).
1) Consider the morphism $g=(g_1,\cdots, g_s):\mathbb A^n\to \mathbb A^s$ and let $$S=V_{\mathbb A^n}(g_1,\cdots, g_s)\subset \mathbb A^n, \;\Sigma=\overli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2692106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is it possible to draw a homomorphism between all groups of the same finite order For example, if I have $2$ groups of order $n$, then could I label their elements $1,2,...,n$ and $1,2,...,m$ and say that $\phi(n)=m$
| Suppose $n$ is 2. We apply your procedure as follows:
*
*For the first group we label the identity as 1 and the other element as 2
*For the second group we label the identity as 2 and the other element as 1
Then $\phi$ is a well-defined function, but it's not a group homomorphism, since it doesn't map the identit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2692276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
What does the notation of “1=/= 0” mean for “Let R be a ring with unity 1=/= 0”? Should I read it literally? Or does it mean that
the multiplicative identity on the left hand side is not equal to the additive identity on the right hand side?
| I think the answers are clear. Here is one more just to be explicit. We are concerned with commutative rings with a multiplicative identity.
Suppose I have a set $S=\{a,b,c\}$ with commutative operations satisfying $a=a+c=b+b=ab$, $b=a+a=b+c=bb$ ,
$c=a+b=c+c=ac=bc=cc.$ Then one way or another you can check the rules f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2692356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Qualitative graph for a function of $x^{3}$. I have this function:
$$f(x)=\frac{x(x^2-3)}{x^3-3x-3}$$
I need to draw its graph. I've tried with a classic study of a function but it cames out a mess. Any idea to simplify the study for draw the graph? Thank you.
| HINT
Note that
$$f(x)=\frac{x(x^2-3)}{x^3-3x-3}=f(x)=\frac{x^3-3x}{x^3-3x-3}=f(x)=1+\frac{3}{x^3-3x-3}$$
For a first sketch
*
*determine domain
*find the value for some "special" and/or "simple" point as x=0,1,etc.
*find the value for which denominator = 0 (and thus vertical asymptothes)
*find limit at $\pm \inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2692471",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Smooth approximation of three-phased linear models I am looking for a smooth (continuous differentiable) approximation of the following two three-phased functions with breakpoints at $B_1$ and $B_2$:$$
y_1(x, B_1, B_2, a, b) = \begin{cases}
a; & x < B_1\\
a + b(x - B_1); & B_1 \leqslant x \leqslant B_2\\
a + b(B_2 - B_... | Ha with the help of my brother found the answer myself in the end.
If we define
\begin{align*}
f(x, b, s) &= \frac{1}{2}\sqrt{\smash[b]{b(4s+bx^2)}}\\
&\mathrel{\phantom{=}}{}
\end{align*}
then my piecewise linear function $y_1(x, a, b, B1, B2, s)$ can be approximated as
\begin{align*}
y_1(x, a,b,B1,B2,s) &= a + f(x-B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2692572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Adjoint of projection onto direct sum of Hilbert spaces Let $K_n$ be Hilbert spaces and define \begin{equation*} K := \bigoplus_{\ell_2} K_n = \left\{ (x_1,x_2,\ldots) \in \bigoplus_{n=1}^\infty K_n : \sum_{n=1}^\infty \|x_n\|^2 < \infty \right\} \end{equation*}
It is easy to see that $K$ is a Hilbert space with the in... | You are right, the adjoint of $\pi_n : K \to K_n$ is the canonical inclusion $\iota_n : K_n \to K$.
Let $(x_1, x_2, \ldots) \in K$ and $y_n \in K_n$. We have:
\begin{align}
\Big\langle \pi_n(x_1, x_2, \ldots), y_n\Big\rangle_K &= \langle x_n, y_n\rangle_{K_n} \\
&= \Big\langle (x_1, x_2, \ldots), (\underbrace{0, \ldots... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2692788",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Isometries of the hyperbolic plane in the Beltrami–Klein disk model I am interested in the isometries of the hyperbolic plane in the Beltrami–Klein disk model. (https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model) The Wikipedia article does not say anything about the structure of the isometries in this model.
Si... | In addition to Lee Mosher's very nice answer, here are some texts that some of my friends have recommended:
*
*John Ratcliffe: Foundations of Hyperbolic Manifolds
*Cannon, Floyd,
Kenyon, Parry: Hyperbolic Geometry
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2692957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How to find the trapping region. Show that the system,
$$x'=-x-y+x(x^2+2y^2)$$
$$y'=x-y+y(x^2+2y^2)$$
has at least one peridic solution.
I know that I need to use the Poincare Bendixon Theorem, but I'm not to sure how to find the trapping region. When my teacher did an example in class he basically did a proof by pic... | Because of the form of the linear part of the vector field it seems advisable to explore the dynamic of the Euclidean radius. For simplicity of computation, use $E=\frac12r^2=\frac12(x^2+y^2)$ to get
$$
\frac{d}{dt}E=x\dot x+y\dot y=-2E+2E(x^2+2y^2)
$$
so that
$$
2E(2E-1)\le\dot E\le 2E(4E-1)
$$
This means that $\dot E... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Distance between a point and a line and between two lines
Let $P = (-5, 3, 4)$, $Q = (-6, 0, 3)$, $R = (-7, 1, 6)$ and $S = (-4, 2, 2)$. Let $A$ be the line passing through $P$ and $Q$, and let $B$ be the line passing through $R$ and $S$.
a) What is the distance between $R$ and $A$?
b) What is the distance between $A$... | Even if you're only in Calc I, you can still do this. Write an equation for the distance from the point to an arbitrary point on the line and then differentiate the equation you come up with with respect to $x$. The value of the derivative will be zero when your equation for the distance from the point to the line is... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Prove that $ \int_{0}^{c} \frac{\sin(\frac{x}{2})x~dx}{\sqrt{\cos(x) - \cos(c)}} = \sqrt{2} \pi \ln(\sec(\frac{c}{2}))$ As the title says, I want to find a way calculate the following integral$ \int_{0}^{c} \frac{\sin(x/2)x~dx}{\sqrt{\cos(x) - \cos(c)}}$, which I know is equal to $\sqrt{2} \pi \ln(\sec(\frac{c}{2}))$.
... | We can transform
\begin{align}
I(c)&=\int_{0}^{c} \frac{\sin(x/2)}{\sqrt{\cos(x) - \cos(c)}}x\,dx\\
&=\frac{1}{\sqrt{2}}\int_{0}^{c} \frac{\sin(x/2)}{\sqrt{\cos^2(x/2) - \cos^2(c/2)}}x\,dx\\
&=2\sqrt{2}\int_0^{c/2}\frac{\sin y}{\sqrt{\cos^2y-\cos^2(c/2)}}y\,dy
\end{align}
Now, denoting $C=\cos (c/2)$ and enforcing the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 1,
"answer_id": 0
} |
Solving the differential equation $2y'\sin x + y\cos x = y^3(x\cos x - \sin x) $ Does anyone know how to solve the following differential equation:
$$2y'\sin x + y\cos x = y^3(x\cos x - \sin x) $$
I tried dividing both sides by sine, then by cosine, which in either case brought me nowhere. I tried isolating $y'$ but ... | Hint
Substitute $u=\frac 1 {y^2}$
Then $$u'=-2 \frac {y'} {y^3}$$
$$2y'\sin x + y\cos x = y^3(x\cos x - \sin x)$$
$$-u' + u\cot (x) = (x\cot(x) - 1)$$
$$ {u'} -u\cot (x) = (1-x\cot(x) )$$
$$.............$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Sum of little o and big o Consider the following expression:
$$o\left(\frac{1}{nh_n^p}\right) + O\left(\frac{1}{n}\right) \ \text{as} \ n \rightarrow \infty$$
where $p$ is a positive integer and $h_n$ is a function of $n$. We assume that $nh_n \rightarrow \infty$ as $n \rightarrow \infty$ and $h_n \rightarrow 0$ as $n ... | Under the assumption that $h_n\to0$ and $p\geq 1$, we have
$$
h_n^p = o(1)
$$
so that $nh_n^p = o(n)$ and therefore $1/nh_n^p = \omega(1/n)$, i.e.,
$$
\frac{1}{n} = o\!\left(\frac{1}{nh_n^p}\right)\,.
$$
It follows that
$$o\!\left(\frac{1}{nh_n^p}\right)+O\!\left(\frac{1}{n}\right) = o\!\left(\frac{1}{nh_n^p}\right)+o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to make a number out of other numbers in mathematica? I’m not entirely sure if this is possible, but I would like to make numbers composed of other numbers. For example, given $a$, $b$, and $c$, I’d like to make a number such that the digits are $a.bc$.
So, if $a = 1$, $b = 2$, and $c = 3$, we would have the numbe... | I think what you want is:
FromDigits[{a, b, c}] // Simplify
100 a + 10 b + c
For your example:
FromDigits[{1, 2, 3}]/100.
1.23
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Quotient space and quotient map Let $X:= [-2,2]$ a topological space with the canonical topology and let $A:=[-1,1]\subset X$ and $f:X \rightarrow X/A$ the quotient map.
How can I find an explicit form of the quotient space $X/A$?
And is $X$ a compact space? (I only know that it is complete, but I don't know if it is ... | The quotient of a compact space is always compact, as $q: X \to X / \sim$ is always continuous (where $q$ is the standard quotient map), and images of compact spaces are compact.
In your case the function $f: X=[-2,2] \to [-1,1]$ where $f(x) = x+1$ for $x\in [-2,-1]$, $f(x) = 0$ for $x \in [-1,1]$ and $f(x) = x-1$ for ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
If $B$ if a finite Boolean algebra, then it contains an atom Let $B=\{a_1,\dots,a_n,0,1\}$ be a finite Boolean algebra. I want to show that there exists an atom $x\in B$. So I want to show that there exists $x\in B$, such that for each $a\in B$ for which $a<x$, we have $a=0$.
I was thinking of maybe looking at $a=\bigw... | Assume that $x\in B$ is not an atom. Then there are $x_a,x_b\in B$ such that $x_a\vee x_b=x$, $x_a,x_b\neq x$, and $x_a,x_b\neq 0$. If $x_a$ is not an atom, then we can find $x_{aa},x_{ab}$ such that $x_{aa}\vee x_{ab}=x_a$, $x_{aa},x_{ab}\neq x_a$, and $x_{aa},x_{ab}\neq 0$. Continuing in this way we find $$x=x_1\vee ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2693996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
the sequence $n!+2,...,n!+n$ is made up of only composite numbers I have found the claim that given that $n\geq2$, we have that the sequence of $n-1$ numbers $n!+2,n!+3,...,n!$ is made up of only composite numbers. Is there a proof of this? I found this pretty fascinating but I am not sure how to go around it. It seems... | $n!$ is a rather special integer. It is composed of 1 factor of each from the list $\{1,2,3, \cdots, n-1, n\}.$
So,
$$n \geq 2 \Rightarrow 2|n! \Rightarrow n!+2 \equiv 0 \mod(2)$$
$$ n \geq 3 \Rightarrow 3|n! \Rightarrow n!+3 \equiv 0 \mod(3)$$
$$\vdots$$
$$ n \geq k \Rightarrow k|n! \Rightarrow n! + k \equiv 0 \mod(k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Is $\lim_{x→a} f(x) $ equivalent to $\lim_{h→0} f(a+h)$? $\lim_{x→a} f(x) = \lim_{h→0} f(a+h)$ does this always hold true? Intuitively these two things seem to describe same concept, and I've seen some algebraic manipulations that implicitly use this identity. Are there any scenarios where this falls apart? Also, if it... | If $\lim_{x\rightarrow a}f(x)=L$, then by $\epsilon$-argument, somehow it is like $0<|x-a|<\delta$, then $|f(x)-L|<\epsilon$, for $0<|h|<\delta$, then $|(a+h)-a|=|h|$, so $0<|(a+h)-a|<\delta$, set $x=a+h$, then $|f(a+h)-L|=|f(x)-L|<\epsilon$, this shows $\lim_{h\rightarrow 0}f(a+h)=L$. Another way is similar.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Approximating the limit of a Cauchy sequence in a Banach space Let $E$ be a Banach space and consider a sequence $(x_n)_n$ in $E$ satisfying the following condition: $$||x_n-x_{n-1}||\leq 3^{-n}\mbox{ for all }n\in\mathbb{N}.$$
Clearly $(x_n)_n$ is a Cauchy sequence and therefore converges to an element $x\in E$.
Ques... | We can prove it using triangle equation:
$$||x-x_n|| \leq \sum_{k=n}^{\infty}||x_{k+1}-x_k|| \leq \sum_{k=n}^{\infty}3^{-k-1}=\frac{3^{-n-1}}{1-\frac{1}{3}} = \frac{3^{-n}}{2}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694316",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Prove that $\lim_{h\to 0} \frac{g(a+h)-2g(a)+g(a-h)}{h^2} = g''(a)$ How do I prove that
$$\displaystyle\lim_{h\to0} \dfrac{g(a+h)-2g(a)+g(a-h)}{h^2} = g''(a),$$
where $g$ is of class $C^2$?
My attempt:
By Mean Value Theorem, there are $c_0\in (a-h, a)$ and $c_1\in(a, a+h)$ such that
$$g(a)-g(a-h) = g'(c_0)\cdot h$$
$... | \begin{align*}
g(a+h)&=g(a)+g'(a)h+\dfrac{1}{2}g''(\xi_{h})h^{2}\\
g(a-h)&=g(a)-g'(a)h+\dfrac{1}{2}g''(\eta_{h})h^{2},
\end{align*}
where $\xi_{h}$ lies in between $a$ and $a+h$, and $\eta_{h}$ lies in between $a$ and $a-h$, so
\begin{align*}
\dfrac{1}{h^{2}}[g(a+h)+g(x-h)-2g(a)]=\dfrac{1}{2}(g''(\xi_{h})+g''(\eta_{h}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Find a matrix A such that $\operatorname{rank}{A} = \operatorname{rank}{A^2} \neq \operatorname{rank}{A^3}$ Let $A$ be a complex square matrix of order 2 ($A \in M_{2,2}$).
Then, does there exist $A$ such that $\operatorname{rank}{A} = \operatorname{rank}{A^2} \neq \operatorname{rank}{A^3}$?
If that doesn't exist, how ... | $\DeclareMathOperator{\rank}{rank}$
In general, for a linear transformation $T: V \to V$, one has $T^{k+1} V \subseteq T^k V$ for all $k \ge 0$. If one ever has equality for a particular $k$, i.e. $T^{k+1} V = T^k V$, then $T^{k + l} V = T^k V$ for all $l \ge 1$. In fact, this holds for $l =1$ by assumption, and if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2694566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
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