Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solve the following quadratic inequalities by graphing the corresponding function Looking at these questions and I am not confident in my abilities to solve them.
Solve the following quadratic inequalities by graphing the corresponding function. Note: a) and b) are separate questions.
$$ a) y \le -2x^2+16x-24\\
b) y... | Step one:
Learn to draw graphs of the equalities.
Suppose we are given
$$
y \le -2x^2+16x-24.
$$
The matching equality is
$$
y = -2x^2+16x-24.
$$
We can factorise this, then graph it.
$$
\begin{align}
y &= -2(x^2-8x+12)\\
&= -2(x-6)(x-2)
\end{align}
$$
This means that the roots of the polynomial are at $x=6$ and $x=2$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/423164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
prime notation clarification When I first learned calculus, I was taught that $'$ for derivatives was only a valid notation when used with function notation: $f'(x)$ or $g'(x)$, or when used with the coordinate variable $y$, as in $y'$.
But I have seen on a number of occasions, both here and in the classroom, where it ... | What you're seeing is a "shorthand" an instructor or such may use in the process of computing the derivative of a function with respect to $x$. Usually when you seem something like $(ax + bx^2)'$, it's assumed from the context that we are taking the derivative of the expression, with respect to $x$. That is, "$(ax + b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/423214",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is the cone locally compact Let $X$ denote the cone on the real line $\mathbb{R}$. Decide whether $X$ is locally
compact. [The cone on a space $Y$ is the quotient of $Y \times I$ obtained by
identifying $Y \times \{0\}$ to a point.]
I am having a hard time showing that there exists a locally compact neighborhood around... | Here is a way of showing that no neighborhood of $r=\Bbb R\times\{0\}\in X$ is compact. The idea is to find in any neighborhood $V$ of $r$ a closed subspace homeomorphic to $\Bbb R$. Since the subspace is not compact, $V$ cannot be compact.
So let $V$ be a neighborhood of $r$ in $X$. Then $V$ contains the image of an o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/423276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$ ?
This means, $gHg^{-1}$ is a proper subgroup of $H$. We know that $H \cong gHg^{-1}$, so if $H$ is finite then we hav... | Let $\mathbb{F}_2 = \langle a,b \mid \ \rangle$ be the free group of rank two. It is known that the subgroup $F_{\infty}$ generated by $S= \{b^nab^{-n} \mid n \geq 0 \}$ is free over $S$. Then $bF_{\infty}b^{-1}$ is freely generated by $bSb^{-1}= \{b^n a b^{-n} \mid n \geq 1\}$, hence $bF_{\infty}b^{-1} \subsetneq F_{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/423328",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
a codeword over $\operatorname{GF}(4)$ -> two codewords over $\operatorname{GF}(2)$ using MAGMA A codeword $X$ over $\operatorname{GF}(4)$ is given. How can I write it as $X= A+wB$ using MAGMA? where $A$ and $B$ are over $\operatorname{GF}(2)$ and $w^2 + w =1$.
Is there an easy way, or do I have to write some for loops... | Probably there is an easier way, but the following function should do the job:
function f4tof2(c)
n := NumberOfColumns(c);
V := VectorSpace(GF(2),n);
ets := [ElementToSequence(c[i]) : i in [1..n]];
return [V![ets[i][1] : i in [1..n]],V![ets[i][2] : i in [1..n]]];
end function;
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/423418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Evaluating $\int_0^\infty \frac{e^{-kx}\sin x}x\,\mathrm dx$ How to evaluate the following integral?
$$\int_0^\infty \frac{e^{-kx}\sin x}x\,\mathrm dx$$
| One more option:$$\begin{align}\int_0^\infty\int_0^\infty e^{-(k+y)x}\sin x\mathrm{d}x\mathrm{d}y&=\Im\int_0^\infty\int_0^\infty e^{-(k+y-i)x}\mathrm{d}x\mathrm{d}y\\&=\int_0^\infty\tfrac{1}{(k+y)^2+1}\mathrm{d}y\\&=[\arctan(k+y)]_0^\infty\\&=\tfrac{\pi}{2}-\arctan k.\end{align}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/423489",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
Let $f \colon \Bbb C \to \Bbb C$ be a complex valued function given by $f(z)=u(x,y)+iv(x,y).$ I am stuck on the following question :
MY ATTEMPT:
By Cauchy Riemann equation ,we have $u_x=v_y,u_y=-v_x.$ Now $v(x,y)=3xy^2 \implies v_x=3y^2 \implies -u_y=3y^2 \implies u=-y^3+ \phi(x) $. Now,I am not sure which way to g... | Hint: you used the second C.R. equation arriving at $u(x,y)=-y^3+\phi(x)$. What does it happen if you apply the other C.R. equation, i.e. $u_x=v_y$, to your result?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/423669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Are there two $\pi$s? The mathematical constant $\pi$ occurs in the formula for the area of a circle, $A=\pi r^2$,
and in the formula for the circumference of a circle, $C= 2\pi r$. How does one prove that these constants are the same?
| One way to see it is if you consider a circle with radius $r$ and another circle with radius $r+\Delta r$ (where $\Delta r\ll r$) around the same point, and consider the area between the two circles.
As with any shape, the area is proportional to the square of a typical length; the radius is such a typical length. That... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/423836",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 2
} |
Nitpicky Sylow Subgroup Question Would we call the trivial subgroup of a finite group $G$ a Sylow-$p$ subgroup if $p \nmid |G|$? Or do we just only look at Sylow-$p$ subgroups as being at least the size $p$ (knowing that a Sylow-$p$ subgroup is a subgroup of $G$ with order $p^k$ where $k$ is the largest power of $p$ th... | For what it is worth, I consider all primes $p$, not just those that divide the group order.
This makes many statements smoother. For instance, the defect group of the principal block is the Sylow $p$-subgroup, and a block is semisimple if and only if the defect group is trivial. Thus the principal block is semisimple ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Maximum cycle in a graph with a path of length $k$ I don't understand why this stands:
Let $G$ be a graph containing a cycle $C$, and assume that $G$ contains a path of length at least $k$ between two vertices of $C$.
Then $G$ contains a cycle of length at least $\sqrt{k}$.
Since we can extend the cycle $C$ with t... | Here is my solution. Let $s$ and $t$ two vertices of $C$ such that
there is a $st$-path $P$ of lenght $k$. If $|V(P) \cap V(C)|\geq \sqrt{k}$ then the proof follows, because the cycle we want is $C$. Otherwise, consider that
$|V(P) \cap V(C)| < \sqrt{k}$. Then, as $|V(P)| \geq k$, by pigeon principle, there is a subpat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Proving a lemma - show the span of a union of subsets is still in the span This is part of proving a larger theorem but I suspect my prof has a typo in here (I emailed him about it to be sure)
The lemma is written as follows:
Let $V$ be a vector space. Let {$z, x_1, x_2, ..., x_n$} be a subset of $V$. Show that if $z ... | We want to show $\operatorname{span}\{z, x_1, \dots, x_n\} = \operatorname{span}\{x_1, \dots, x_n\}$. In general, to show $X = Y$ where $X, Y$ are sets, we want to show that $X \subseteq Y$ and $Y \subseteq X$.
So suppose $v \in \operatorname{span}\{x_1, \dots, x_n\}$. Then, we can find scalars $c_1, \dots, c_n$ suc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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To what extent the statement "Data is normally distributed when mode, mean and median scores are all equal" is correct? I read that normally distributed data have equal mode, mean and median. However in the following data set, Median and Mean are equal but there is no Mode and the data is "Normally Distributed":
$ 1, 2... | It is not correct at all. Any unimodal probability distribution symmetric about the mode (for which the mean exists) will have mode, mean and median all equal.
For the definition of normal distribution, see e.g. Wikipedia.
Strictly speaking, data can't be normally distributed, but it can be a
sample from a normal dist... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Looking for reference to a couple of proofs regarding the Stereographic Projection. I'm looking for a reference to rigorous proofs of the following two claims (if someone is willing to write down a proof that would also be excellent):
*
*The Stereographic Projection is a Homeomorphism between $S^{n}\backslash\left\{... | For the first request just try to write down explicitly the function that defines such a projection, by considering an hyperplane which cuts the sphere along the equator.
Consider $S^n$ in $R^{n+1}$, with $R^n$ as the subset with $x_{n+1}=0$. The North pole is $(0,0,..,0,1)$ and the image of each point is the intersect... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Show that $\frac{x}{1-x}\cdot\frac{y}{1-y}\cdot\frac{z}{1-z} \ge 8$.
If $x,y,z$ are positive proper fractions satisfying $x+y+z=2$, prove that $$\dfrac{x}{1-x}\cdot\dfrac{y}{1-y}\cdot\dfrac{z}{1-z}\ge 8$$
Applying $GM \ge HM$, I get $$\left[\dfrac{x}{1-x}\cdot\dfrac{y}{1-y}\cdot\dfrac{z}{1-z}\right]^{1/3}\ge \dfrac{... | Write $(1-x)=a, (1-y)=b \text { and} (1-z)=c$
$x=2-(y+z)=b+c$
$y=2-(z+x)=a+c$
$z=2-(x+y)=a+b$
Thus we have the same expression in simpler form:
$\dfrac{b+c}{a} \cdot \dfrac{a+c}{b} \cdot \dfrac{a+b}{c}$
Now we have AM-GM:
$b+c \ge 2 \sqrt{bc}$
$a+c \ge 2 \sqrt{ac}$
$b+a \ge 2 \sqrt{ba}$
$\dfrac{b+c}{a} \cdot \dfrac{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 0
} |
Number of distinct points in $A$ is uncountable How can one show:
Let $X$ be a metric space and $A$ is subset of $X$ be a connected set with
at least two distinct points then the number of distinct points in $A$
is uncountable.
| We will show that if $A$ is countable then $A$ is not connected.
Let $a,b$ be two distinct points in $A$ and let $d$ be the metic on $X$. Then, since $d$ is real valued, there are uncountably many $r\in \mathbb R$ such that $0<r<d(a,b)$. Let $r_0$ be such that $\forall x\in A$, $d(a,x)\ne r_0$ and $0<r_0<d(a,b)$. This... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Contour integration with branch cut This is an exercise in a course on complex analysis I am taking:
Determine the function $f$ using complex contour integration:
$$\lim_{R\to\infty}\frac{1}{2\pi i}\int_{c-iR}^{c+iR}\frac{\exp(tz)}{(z-i)^{\frac{1}{2}}(z+i)^{\frac{1}{2}}} dz$$
Where $c>0$ and the branch cut for $z^\frac... | For $t\equiv-\tau<0$, consider a half-circle of radius $M$ centred at $c$ and lying on the right of its diameter that goes from $c-iM$ to $c+iM$. By Cauchy's theorem
$$
\int_{c-iM}^{x+iM}\frac{e^{tz}}{\sqrt{1+z^2}}dz
=\int_{-\pi/2}^{+i\pi/2}\frac{e^{-\tau(c+Me^{i\varphi})}}{\sqrt{1+(c+Me^{i\varphi})^2}}iMe^{i\varphi}d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Integral of polylogarithms and logs in closed form: $\int_0^1 \frac{du}{u}\text{Li}_2(u)^2(\log u)^2$ Is it possible to evaluate this integral in closed form?
$$ \int_0^1 \frac{du}{u}\text{Li}_2(u)^2\log u \stackrel{?}{=} -\frac{\zeta(6)}{3}.$$
I found the possible closed form using an integer relation algorithm.
I fou... | I've decided to publish my work so far - I do not promise a solution, but I've made some progress that others may find interesting and/or helpful.
$$\text{Let } I_{n,k}=\int_{0}^{1}\frac{\text{Li}_{k}(u)}{u}\log(u)^{n}du$$
Integrating by parts gives $$I_{n,k}=\left[\text{Li}_{k+1}(u)\log(u)^{n}\right]_{u=0}^{u=1}-\int_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 4,
"answer_id": 1
} |
Looking for a good counterargument against vector space decomposition. How do I see that I cannot write $\mathbb{R}^n = \bigcup_{\text{all possible }M} \operatorname{span}(M)$, where $M$ runs over the subsets with $n-1$ elements in it of the set of vectors $N=\{a_1,\ldots,a_n,\ldots,a_m\} \in \mathbb{R}^n$, where the t... | Let $V$ be a vector space over an infinite field $F$, and $V_1, \ldots, V_n$ proper subspaces. Then I claim $\bigcup _j V_j$ is not a vector space.
For each $k$ let $u_k$ be a vector not in $V_k$. We then inductively find
vectors $w_k$ not in $\bigcup_{j \le k} V_j$. Namely, if $w_k \notin \bigcup_{j \le k} V_j$, and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424878",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Continuity and Metric Spaces How do I show that the function $f:X \to \mathbb R$ given by $$f(x)=\frac{d(a,x)}{d(a,b)}$$ is continuous.
Given that $(X,d)$ is a metric space, and $a,b$ are distinct points in $X$.
| If $d(x,y)<d(a,b)\cdot\varepsilon$ then
$$
|f(x)-f(y)| = \left|\frac{d(a,x)}{d(a,b)} - \frac{d(a,y)}{d(a,b)}\right| \le \frac{d(x,y)}{d(a,b)}<\varepsilon.
$$
The first inequality follows from two instances of the triangle inequality: $d(a,x)+d(x,y)\ge d(a,y)$ and $d(a,y)+d(y,x)\ge d(a,x)$.
So given $\varepsilon>0$, let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the average weight of a minimal spanning tree of $n$ randomly selected points in the unit cube? Suppose we pick $n$ random points in the unit cube in $\mathbb{R}_3$, $p_1=\left(x_1,y_1,z_1\right),$ $p_2=\left(x_2,y_2,z_2\right),$ etc. (So, $x_i,y_i,z_i$ are $3n$ uniformly distributed random variables between $0... | If $n = 0$ or $n = 1$ the answer obviously is 0. If $n = 2$ we have
$$E\left((x_1 - x_2)^2\right) = E(x_1^2 - 2x_1x_2 + x_2^2) = E(x_1^2) - 2E(x_1)\cdot E(x_2) + E(x_2^2) \\= \frac13 - 2\frac12\cdot\frac12 + \frac13 = \frac16.$$
The same for $y$- and $z$-coordinates. So $E(w_{12}) = \sqrt{\frac16 + \frac 16 + \frac16}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/424995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
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Characterizing continuous exponential functions for a topological field Given a topological field $K$ that admits a non-trivial continuous exponential function $E$, must every non-trivial continuous exponential function $E'$ on $K$ be of the form $E'(x)=E(r\sigma (x))$ for some $r \in K$* and $\sigma \in Aut(K/\mathbb{... | It seems that as-stated, the answer is false. I'm not satisfied with the following counterexample, however, and I'll explain afterwards.
Take $K = \mathbb{C}$ and let $E(z) = e^z$ be the standard complex exponential. Take $E'(z) = \overline{e^z} = e^{\overline{z}}$, where $\overline{z}$ is the complex conjugate of $z$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
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Show that $(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z)$.
If $x>0,y>0,z>0$ and $x+y+z=1$, prove that $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z).$$
Trial: Here $$(1+x)(1+y)(1+z)\ge 8(1-x)(1-y)(1-z) \\ \implies (1+x)(1+y)(1+z)\ge 8(y+z)(x+z)(x+y)$$ I am unable to solve the problem. Please help.
| $$(1+x)(1+y)(1+z) \ge 8(1-x)(1-y)(1-z) \Leftrightarrow $$
$$(2x+y+z)(x+2y+z)(x+y+2z) \ge 8(y+z)(x+z)(x+y)$$
Let $a=x+y, b=x+z, c=y+z$. Then the inequality to prove is
$$(a+b)(a+c)(b+c) \ge 8abc \,,$$
Which follows immediately from AM-GM:
$$a+b \ge 2 \sqrt{ab}$$
$$a+c \ge 2 \sqrt{ac}$$
$$b+c \ge 2 \sqrt{bc}$$
Simplifica... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Calculate the probability of two teams have been drawns If we know that team A had a $39\%$ chance of winning and team B $43\%$ chance of winning, how we can calculate the probability of the teams drawn?
My textbook mention the answer but I cannot understand the logic behind it. The answer is $18\%$. As working is not ... | The sum of all events' probabilities is equal to 1. In this case, there are three disjoint events: team A winning, team B winning or a draw. Since we know the sum of these probabilities is 1, we can get the probability of a draw as follows:
$$
Pr(\text{Draw})=1-Pr(\text{Team A wins})-Pr(\text{Team B wins})=1-0.39-0.43=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425188",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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Where to start Machine Learning? I've recently stumbled upon machine learning, generating user recommendations based on user data, generating text teaser based on an article. Although there are tons of framework that does this(Apache Mahout, Duine framework and more) I wanted to know the core principles, core algorithm... | I would also recommend the course Learning from data by Yaser Abu-Mostafa from Caltech. An excellent course!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/425230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 8,
"answer_id": 6
} |
Question about linear systems of equations Let $X=\{x_1,\cdots,x_n\}$ be a set of variables in $\mathbb{R}$.
Let $S_1$ be a set of linear equations of the form $a_1 x_1+\cdots+a_n x_n=b$ that are independent.
Let $k_1=|S_1|<n$ where $|S_1|$ denotes the rank of $S_1$ (i.e., the number of independent equations).
That is... | Here is a reasonably easy algorithm if you already know some basic stuff about matrices:
Look at the coefficients matrix's rows as vectors in $\;\Bbb R^n\;$ :
$$A:=\{ v_1=(a_{11},\ldots,a_{1n})\;,\;v_2=(a_{21},\ldots,a_{2n})\,\ldots,v_k=(a_{k1},\ldots,a_{kn})\}\;\;(\text{where $\,k=k_1\;$ for simplicity of notation)}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Does $\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2+2}$ converge? I'm trying to find out whether
$$\sum_{n=1}^{\infty} \frac{\ln(n)}{n^2+2}$$
is convergent or divergent?
| Clearly
$$
\frac{\ln(n)}{n^2+2}\leq \frac{\ln(n)}{n^2}
$$
You can apply the integral test to show that $\sum\frac{\ln(n)}{n^2}$ converges. You only need to check that $\frac{\ln(n)}{n^2}$ is decreasing. But, the derivative is clearly negative for $n>e$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/425368",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 1
} |
What is the range of values of the random variable $Y = \frac{X - \min(X)}{\max(X)-\min(X)}$? Suppose $X$ is an arbitrary numeric random variable. Define the variable $Y$ as
$$Y=\frac{X-\min(X)}{\max(X)-\min(X)}.$$
Then what is the range of values of $Y$?
| If $X$ takes values over any finite (closed) interval, then the range of $Y$ is $[0,1]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/425414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Vorticity equation in index notation (curl of Navier-Stokes equation) I am trying to derive the vorticity equation and I got stuck when trying to prove the following relation using index notation:
$$
{\rm curl}((\textbf{u}\cdot\nabla)\mathbf{u}) = (\mathbf{u}\cdot\nabla)\pmb\omega - ( \pmb\omega \cdot\nabla)\mathbf{u}
... | The trick is the following:
$$ \epsilon_{ijk} \frac{\partial u_m}{\partial x_j} \frac{\partial u_m}{\partial x_k} = 0 $$
by antisymmetry.
So you can rewrite
$$ \epsilon_{ijk} \frac{\partial u_m}{\partial x_j} \frac{\partial u_k}{\partial x_m} = \epsilon_{ijk} \frac{\partial u_m}{\partial x_j}\left( \frac{\partial u_k}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 2,
"answer_id": 0
} |
Function problem Show that function $f(x) =\frac{x^2+2x+c}{x^2+4x+3c}$ attains any real value if $0 < c \leq 1$ Problem :
Show that function $f(x)=\dfrac{x^2+2x+c}{x^2+4x+3c}$ attains any real value if $0 < c \leq 1$
My approach :
Let the given function $f(x) =\dfrac{x^2+2x+c}{x^2+4x+3c} = t $ where $t$ is any arbi... | $(2t-1)^2-(t-1)(3tc-c) \geq 0\implies 4t^2+1-4t-(3t^2c-4tc+c)\geq 0\implies t^2(4-3c)+4(c-1)t+(1-c)\geq 0$
Now a quadratic polynomial $\geq 0$ $\forall t\in \Bbb R$ iff coefficient of second power of variable is positive and Discriminant $\leq 0$
which gives $4-3c>0\implies c<\frac{4}{3}$ and $D=16(c-1)^2+4(4-3c)(c-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Quadratic residues, mod 5, non-residues mod p 1) If $p\equiv 1\pmod 5$, how can I prove/show that 5 is a quadratic residue mod p?
2) If $p\equiv 2\pmod 5$, how can is prove/show that 5 is a nonresidue(quadratic) mod p?
| 1) $(5/p) = (p/5)$ since $p$ is $1(mod 5)$ then $(p/5) = (1/5) = 1$. So 5 is a quadratic residue mod p.
2) again $(5/p) = (p/5)$ since $p$ is $2(mod5)$ then $(p/5) = (2/5) = -1$ since 5 is 5(mod8). So 5 is not a quadratic residue
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/425704",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Why does Monte-Carlo integration work better than naive numerical integration in high dimensions? Can anyone explain simply why Monte-Carlo works better than naive Riemann integration in high dimensions? I do not understand how chosing randomly the points on which you evaluate the function can yield a more precise resu... | I think it is not the case that random points perform better than selecting the points manually as done in the Quasi-Monte Carlo methods and the sparse grid method:
http://www.mathematik.hu-berlin.de/~romisch/papers/Rutg13.pdf
Also in Monte Carlo methods one usually uses random numbers to generate an adaptive integrati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425782",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 4,
"answer_id": 3
} |
Help with a conditional probability problem There are 6 balls in a bag and they are numbered 1 to 6.
We draw two balls without replacement.
Is the probability of drawing a "6" followed by drawing an "even" ball the same as the probability of drawing an "even" ball followed by drawing a "6".
According to Bayes Theorem t... | There is no need to compute anything. All orders of drawing the balls are equally likely.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/425869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Sum of greatest common divisors As usually, let $\gcd(a,b)$ be the greatest common divisor of integer numbers $a$ and $b$.
What is the asymptotics of
$$\frac{1}{n^2} \sum_{i=1}^{i=n} \sum_{j=1}^{j=n} \gcd(i,j)$$
as $n \to \infty?$
| Of the lattice points $[1,n] \times [1,n], 1-\frac 1{p^2}$ have no factor $p$ in the $\gcd, \frac 1{p^2}-\frac 1{p^4}$ have a factor $p$ in the $\gcd\frac 1{p^4}-\frac 1{p^6}$, have a factor $p^2$ in the $\gcd, \frac 1{p^6}-\frac 1{p^8}$ have a factor $p^3$ in the $\gcd$ and so on. That means that a prime $p$ contribu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/425954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
$u,v$ are harmonic conjugate with each other in some domain $u,v$ are harmonic conjugate with each other in some domain , then we need to show
$u,v$ must be constant.
as $v$ is harmonic conjugate of $u$ so $f=u+iv$ is analytic.
as $u$ is harmonic conjugate of $v$ so $g=v+iu$ is analytic.
$f-ig=2u$ and $f+ig=2iv$ are a... | Your proof is correct. I add some remarks:
*
*$v$ is a conjugate of $u$ if and only if $-u$ is a conjugate of $v$ (since $u+iv$ and $v-iu$ are constant multiples of each other)
*Since the harmonic conjugate is unique up to additive constant, the assumption that $u$ is a conjugate of $v$ implies (because of 1) tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Cantor's Diagonal Argument Why Cantor's Diagonal Argument to prove that real number set is not countable, cannot be applied to natural numbers? Indeed, if we cancel the "0." in the proof, the list contains all natural numbers, and the argument can be applied to this set.
| How about this slightly different (but equivalent) form of the proof? I assume that you already agree that the natural numbers $\mathbb{N}$ are countable, and your question is with the real numbers $\mathbb{R}$.
Theorem: Let $S$ be any countable set of real numbers. Then there exists a real number $x$ that is not in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426084",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
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Evaluating $\int_0^\infty \frac{dx}{1+x^4}$. Can anyone give me a hint to evaluate this integral?
$$\int_0^\infty \frac{dx}{1+x^4}$$
I know it will involve the gamma function, but how?
| Following is a computation that uses Gamma function:
For any real number $k > 1$, let $I_k$ be the integral:
$$I_k = \int_0^\infty \frac{dx}{1+x^k}$$
Consider two steps in changing the variable. First by $y = x^k$ and then by $z = \frac{y}{1+y}$. Notice:
$$\frac{1}{1+y} = 1 - z,\quad y = \frac{z}{1-z}\quad\text{ and }\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 5
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My text says$ \left\{\begin{pmatrix}a&a\\a&a\end{pmatrix}:a\ne0,a\in\mathbb R\right\}$ forms a group under matrix multiplication. My text says$$\left\{\begin{pmatrix}a&a\\a&a\end{pmatrix}:a\ne0,a\in\mathbb R\right\}$$ forms a group under matrix multiplication.
But I can see $I\notin$ the set and so not a group.
Am I ri... | It's important to note that this set of matrices forms a group but it does NOT form a subgroup of the matrix group $GL_2(\mathbb{R})$ (the group we are most familiar with as being a matrix group - the group of invertible $2\times 2$ matrices) as no elements in this set have non-zero determinant. In particular, we are l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 4,
"answer_id": 0
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Domain and range of a multiple non-connected lines from a function? How do you find the domain and range of a function that has multiple non-connected lines?
Such as, $ f(x)=\sqrt{x^2-1}$. Its graph looks like this:
I'm wanting how you would write this with a set eg: $(-\infty, \infty)$.
P.S. help me out with the ti... | You can find domain of the function by simply analyzing the behavior of the function. For
$$
f(x) = \sqrt{x^2-1}
$$
you can conclude that the expression under the square root must be non-negative. So
$$
x^2-1 \ge 0 \\
(x-1)(x+1) \ge 0 \\
x \in (-\infty, -1] \cup [1, +\infty)
$$
Latter is your domain. $D[f] = (-\infty,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Yet another $\sum = \pi$. Need to prove. How could one prove that
$$\sum_{k=0}^\infty \frac{2^{1-k} (3-25 k)(2 k)!\,k!}{(3 k)!} = -\pi$$
I've seen similar series, but none like this one...
It seems irreducible in current form, and I have no idea as to what kind of transformation might aid in finding proof of this.
| Use Beta function, I guess... for $k \ge 1$,
$$
\int_0^1 t^{2k}(1-t)^{k-1}dt = B(k,2k+1) = \frac{(k-1)!(2k)!}{(3k)!}
$$
So write
$$
f(x) = \sum_{k=0}^\infty \frac{2(3-25k)k!(2k)!}{(3k)!}x^k
$$
and compute $f(1/2)$ like this:
$$\begin{align}
f(x) &= 6+\sum_{k=1}^\infty \frac{(6-50k)k(k-1)!(2k)!}{(3k)!} x^k
\\ &= 6+\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 1,
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Why $x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$?
Why $f(x) = x^2 + 7$ is the minimal polynomial for $1 + 2(\zeta + \zeta^2 + \zeta^4)$ (where $\zeta = \zeta_7$ is a primitive root of the unit) over $\mathbb{Q}$?
Of course it's irreducible by the Eisenstein criterion, however it apparent... | If you don't already know the primitive polynomial, you can find it with Galois theory. The element given is an element of the cyclotomic field, and so it's conjugates are all the roots of the primitive polynomial. In fact, there is only one different conjugate, obtained for example by cubing each primitive root in the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Making exponent of $a^x$ object of the function Is it possible to make a variable the subject of a formula when it is an exponent in the equation? For example:
$$y=a^x\quad a\;\text{is constant}$$
For example, let the constant $a = 5.$
$$
\begin{array}{c|l}
\text{x} & \text{y} \\
\hline
1 & 5 \\
2 & 25 \\
3 & 125 \\
4 ... | Try taking the natural log "ln" of each side of your equation:
$$y = a^x \implies \ln y = \ln\left(a^x\right) = x \ln a \iff x = \dfrac{\ln y}{\ln a}$$
If $a = 5$, then we have $$x = \dfrac{\ln y}{\ln 5}$$
This gives us an equation with $x$ expressed in terms of $y$. $\;\ln a = \ln 5$ is simply a constant so $\dfrac 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Maximal ideals in rings of polynomials Let $k$ be a field and $D = k[X_1, . . . , X_n]$ the polynomial ring in $n$ variables over $k$.
Show that:
a) Every maximal ideal of $D$ is generated by $n$ elements.
b) If $R$ is ring and $\mathfrak m\subset D=R[X_1,\dots,X_n]$ is maximal ideal such that $\mathfrak m \cap R$ is m... | The answer to question a) can be found as Corollary 12.17 in these (Commutative Algebra) notes. The proof is left as an exercise, but the proof of it is just collecting together the previous results in the section.
(As Patrick DaSilva has mentioned, as written your question b) follows trivially from part a). I'm gues... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Substituting an equation into itself, why such erratic behavior? Until now, I thought that substituting an equation into itself would $always$ yield $0=0$. What I mean by this is for example if I have $3x+4y=5$, If I substitute $y=\dfrac {5-3x}{4}$, I will eventually end up with $0=0$. However, consider the equation $\... | You didn't actually substitute anything (namely a solution for $x$) into the original equation; if you would do that the $x$ would disappear. Instead you combined the equation with a modified form of itself to obtain a new equation that is implied by the original one; the new equation may or may not have retained all i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
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how to prove a parametric relation to be a function For example lets suppose that I have given the functions $f:\mathbb{R}\longrightarrow \mathbb{R}$ and $g:\mathbb{R}\longrightarrow \mathbb{R}$. If my relation is $R=\{(x,(y,z))\in \mathbb{R}\times \mathbb{R}^{2}: y=f(x) \wedge z=g(x)\}$ How to prove formally (from a s... | Let's prove a vastly more general statement.
Let $I$ be an index set, and for every $i\in I$ let $f_i$ be a function. Then the relation $F$ defined on $X=\bigcap_{i\in I}\operatorname{dom}(f_i)$ by $F=\{\langle x,\langle f_i(x)\mid i\in I\rangle\rangle\mid x\in X\}$ is a function.
Proof. Let $x\in X$, and suppose tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
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Which of the following groups is not cyclic?
Which of the following groups is not cyclic?
(a) $G_1 = \{2, 4,6,8 \}$ w.r.t. $\odot$
(b) $G_2 = \{0,1, 2,3 \}$ w.r.t. $\oplus$ (binary XOR)
(c) $G_3 =$ Group of symmetries of a rectangle w.r.t. $\circ$ (composition)
(d) $G_4 =$ $4$th roots of unity w.r.t. $\cdot$ (multipli... | Hint: For a group to be cyclic, there must be an element $a$ so that all the elements can be expressed as $a^n$, each for a different $n$. The terminology comes because this is the structure of $\Bbb {Z/Z_n}$, where $a=1$ works (and often others). I can't see what the operator is in your first example-it is some sor... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Need help in proving that $\frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$ We need to prove that $$\dfrac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1} = \frac 1{\sec\theta - \tan\theta}$$
I have tried and it gets confusing.
| $$\frac{\sin\theta-\cos\theta+1}{\sin\theta+\cos\theta-1}$$
$$=\frac{\tan\theta-1+\sec\theta}{\tan\theta+1-\sec\theta}(\text{ dividing the numerator & the denominator by} \cos\theta )$$
$$=\frac{\tan\theta-1+\sec\theta}{\tan\theta-\sec\theta+(\sec^2\theta-\tan^2\theta)} (\text{ putting } 1=\sec^2\theta-\tan^2\theta) $$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/426981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 3
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Drawing points on Argand plane
The points $5 + 5i$, $1− 3i$, $− 4 + 2i$ and $−2 + 6i$ in the
Argand plane are:
(a) Collinear
(b) Concyclic
(c) The vertices of a parallelogram
(d) The vertices of a square
So when I drew the diagram, I got an rectangle in the 1st and 2nd quadrant. So, are they vertices of parallelogr... | Its not collinear, nor a square, nor a parallelogram. Therefore, it must be Concyclic
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/427050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Question on Wolstenholme's theorem In one of T. Apostol's student textbooks on analytic number theory (i.e., Introduction to Analytic Number Theory, T. Apostol, Springer, 1976) Wolstenholme's theorem is stated (Apostol, Chapt. 5, page 116) as follows (more or less):
For any prime ($p \geq 5$),
\begin{equation}
((p - 1)... | In modular arithmetic, you should interpret Egyptian fractions (of the form $\frac 1a$) as the modular inverse of $a \bmod p^2$, in which case this makes perfect sense.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/427123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
prime factors of numbers formed by primorials Let $p,q$ be primes with $p \leq q$. The product $2\cdot3\cdot\dots\cdot p$ is denoted with $p\#$, the
product $2\cdot3\cdot\dots\cdot q$ is denoted with $q\#$ (primorials).
Now $z(p,q)$ is defined by $z(p,q) = p\#+q\#/p\#$
For example $z(11,17) = 2\cdot3\cdot5\cdot7\cdot... | The number $z(p,q)$ is coprime to any of these primes.
It is more likely to be prime, especially for small values, but not necessarily so. For example, $z(7,11) = 13*17$ is the smallest composite example, but one fairly easily finds composites (like $z(11,13)$, $z(5,19)$, $z(3,11)$, $z(13,19)$, $(13,23)$, and $z(13,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does $\sum_{n=1}^{\infty} \frac{\sin(n)}{n}$ converge conditionally? I think that the series $$\sum_{n=1}^{\infty} \dfrac{\sin(n)}{n}$$ converges conditionally, but I´m not able to prove it. Any suggestions ?
| Using Fourier series calculations it follows
$$
\sum_{n=1}^{\infty}\frac{\sin(n x)}{n}=\frac{\pi-x}{2}
$$
for every $x\in(-\pi,\pi)$. Your sum is $\frac{\pi-1}{2}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/427255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Looking for an easy lightning introduction to Hilbert spaces and Banach spaces I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to participate, and who have never taken a course in fu... | I don't know how useful this will be, but I have some lecture notes that motivate the last three things on your list by first reinterpreting the finite dimensional spectral theorem in terms of the functional calculus. (There is also a section on the spectral theorem for compact operators, but this is just pulled from Z... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
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Why is $ \lim_{n \to \infty} \left(\frac{n-1}{n+1} \right)^{2n+4} = \frac{1}{e^4}$ According to WolframAlpha, the limit of
$$ \lim_{n \to \infty} \left(\frac{n-1}{n+1} \right)^{2n+4} = \frac{1}{e^4}$$
and I wonder how this result is obtained.
My approach would be to divide both nominator and denominator by $n$, yieldin... | $$ \lim_{n \to \infty} \left(\frac{n-1}{n+1} \right)^{2n+4} $$
$$ =\lim_{n \to \infty} \left(\left(1+\frac{(-2)}{n+1} \right)^{\frac{n+1}{-2}}\right)^{\frac{-2(2n+1)}{n+1}}$$
$$ = \left(\lim_{n \to \infty}\left(1+\frac{(-2)}{n+1} \right)^{\frac{n+1}{-2}}\right)^{\lim_{n \to \infty}\left(\frac{-4-\frac2n}{1+\frac1n}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
Approximating hypergeometric distribution with poisson I'm am currently trying to show that the hypergeometric distribution converges to the Poisson distribution.
$$
\lim_{n,r,s \to \infty, \frac{n \cdot r}{r+s} \to \lambda} \frac{\binom{r}{k} \binom{s}{n-k}}{\binom{r+s}{n}} = \frac{\lambda^k}{k!}e^{-\lambda}
$$
I k... | This is the simplest proof I've been able to find.
Just by rearranging factorials, we can rewrite the hypergeometric probability function as
$$ \mathrm{Prob}(X=x) = \frac{\binom{M}{x}\binom{N-M}{K-x}}{\binom{N}{K}} = \frac{1}{x!} \cdot \dfrac{M^{(x)} \, K^{(x)}}{N^{(x)}} \cdot \dfrac{(N-K)^{(M-x)}}{(N-x)^{(M-x)}}, $$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Cheating in multiple choice tests. What is the probability that in a multiple choice test exam session, where $k$ people took the test (that contains $n$ questions with 2 possible answers each and where exactly one answer to each question is the correct one) cheating has occurred, i.e. there exists at least two tests t... | It is not reasonable to consider two identical tests as evidence of cheating. And people do not choose answers at random. So let us reword the problem as follows.
We have $k$ people who each toss a fair coin $n$ times. What is the probability that at least two of them will get identical sequences of heads and tails?
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that the interior of a convex set is convex Question: Let $P\subseteq \mathbb{R}^n$ be a convex set. Show that $\text{int}(P)$ is a convex set.
I know that a point $x$ is said to be an interior point of the set $P$ if there is an open ball centered at $x$ that is contained entirely in $P$. The set of all interior ... | I'll give a proof based on the following picture:
Suppose that $x$ and $y$ are interior points of a convex set $\Omega \subset \mathbb R^n$. Let $0 < \theta < 1$. We wish to show that the point $z = \theta x + (1 - \theta) y$ is in the interior of $\Omega$.
There exists an open ball $A$ centered at $x$ such that $A \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
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Clues for $\lim_{x\to\infty}\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2^k-1)x^k}{k k!}$ Some clues for this questions?
$$\lim_{x\to\infty}\sum_{k=1}^{\infty} \frac{(-1)^{k+1} (2^k-1)x^k}{k k!}$$
| Take the derivative and use the exponential series. Thus if the sum is $f(x)$, then
$$x f'(x) = \sum_{n=1}^{\infty} (-1)^{n+1} \frac{2^n-1}{n!} x^n = e^{-x}-e^{-2 x}$$
Then
$$f(x) = \int_0^x dt \frac{e^{-t}-e^{-2 t}}{t}$$
(because you know $f(0)=0$). Thus, using Fubini's theorem, one can show that
$$\lim_{x \to\infty}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427810",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Finite union of compact sets is compact Let $(X,d)$ be a metric space and $Y_1,\ldots,Y_n \subseteq X$ compact subsets. Then I want to show that $Y:=\bigcup_i Y_i$ is compact only using the definition of a compact set.
My attempt: Let $(y_n)$ be a sequence in $Y$. If $\exists 1 \leq i \leq n\; \exists N \in \mathbb N \... | Let $\mathcal{O}$ be an open cover of $Y$. Since $\mathcal{O}$ is an open cover of each $Y_i$, there exists a finite subcover $\mathcal{O}_i \subset \mathcal{O}$ that covers each $Y_i$. Then $\bigcup_{i=1}^n \mathcal{O}_i \subset \mathcal{O}$ is a finite subcover. That's it; no need to deal with sequences.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/427886",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Determine whether $F(x)= 5x+10$ is $O(x^2)$ Please, can someone here help me to understand the Big-O notation in discrete mathematics?
Determine whether $F(x)= 5x+10$ is $O(x^2)$
| It is as $x \to \infty$.
Actually, $5x+10 = o(x^2)$ as $x \to \infty$
(little-oh)
since $\lim_{x \to \infty} \frac{5x+10}{x^2} = 0$.
However,
$5x+10 \ne O(x^2)$ as $x \to 0$,
and $5x \ne O(x^2)$ as $x \to 0$,
because
there is no real $c$ such that
$5x < c x^2$ as $x \to 0$.
Since $x \to 0$ and $x \to \infty$ are the t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/427971",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$ I numerically solved the transcendental equation
$$\ln x-\sqrt{x-1}+1=0$$
and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535...$$
I wonder if it is possible to express the exact ... | Yes, it is possible to express this root in terms of special functions implemented in Mathematica.
Start with your equation
$$\ln x-\sqrt{x-1}+1=0,\tag1$$
then take exponents of both sides
$$x\ e^{1-\sqrt{x-1}}=1.\tag2$$
Change the variable
$$z=\sqrt{x-1}-1,\tag3$$
then plug this into $(2)$ and divide both sides by $2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/428057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
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Boston Celtics VS. LA Lakers- Expectation of series of games?
Boston celtics & LA Lakers play a series of games. the first team who win 4 games, win the whole series.
*
*The probability of win or lose game is equal (1/2)
a. what is the expectation of the number of games in the series?
So i defined an indic... | If we have the family of all length-7 sequences composed of W and L, we see that each of these sequences represent one of $2^7$ outcomes to our task at hand with equal probability. Then, we see that the number of games played is pre-decided for each such given sequence (e.x.: WWWLWLL and WWWLWWW both result in five gam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/428104",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The notations change as we grow up In school life we were taught that $<$ and $>$ are strict inequalities while $\ge$ and $\le$ aren't. We were also taught that $\subset$ was strict containment but. $\subseteq$ wasn't.
My question: Later on, (from my M.Sc. onwards) I noticed that $\subset$ is used for general containme... | This is very field dependent (and probably depends on the university as well). In my M.Sc. thesis, and in fact anything I write today as a Ph.D. student, I still use $\subseteq$ for inclusion and $\subsetneq$ for proper inclusion. If anything, when teaching freshman intro courses I'll opt for $\subsetneqq$ when talking... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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A vector field is a section of $T\mathcal{M}$. By definition, a vector field is a section of $T\mathcal{M}$. I am familiar with the concept of vector field, as well as tangent plane of a manifold.
But such definition is not intuitive to me at all. Could some one give me some intuition? Thank you very much!
| Remember that a point of the tangent bundle consists of pair $(p,v)$, where $p \in M$ and $v \in T_pM$. We have the projection map $\pi: TM \to M$ which acts by $(p,v) \to p$. A section of $\pi$ is a map $f$ so that $\pi \circ f$ is the identity. So for each $p \in M$, we have to choose an element of $TM$ that projects... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/428349",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Hölder- continuous function $f:I \rightarrow \mathbb R$ is said to be Hölder continuous if $\exists \alpha>0$ such that $|f(x)-f(y)| \leq M|x-y|^\alpha$, $ \forall x,y \in I$, $0<\alpha\leq1$. Prove that $f$ Hölder continuous $\Rightarrow$ $f$ uniformly continuous and if $\alpha>1$, then f is constant.
In order to prov... | Hint:
For some $\epsilon>0$ and all $x\ne y$, you have $\Bigl|{f(x)-f(y)\over x-y}\Bigr|\le M|x-y|^\epsilon$ for some $\epsilon>0$.
Why must $f'(x)$ exist? What is the value of $f'(x)$?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How can I measure the distance between two cities in the map? Well i know that the distance between Moscow and London by km it's about 2,519 km and the distance between Moscow and London in my map by cm it's about 30.81 cm and the Scale for my map is 1 cm = 81.865 km but when i tried to measure the distance between oth... | The calculation is somewhat complex. A simplification is to assume that the Earth is a sphere and finding the great-circle distance. A more complex calculation instead uses an oblate spheroid as a closer approximation.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Convergence in $L_1$ and Convergence of the Integrals Am I right with the following argument? (I am a bit confused by all those types of convergence.)
Let $f, f_n \in L_1(a,b)$ with $f_n$ converging to $f$ in $L_1$, meaning
$$\lVert f_n-f \rVert_1 = \int_a^b |f_n(x)-f(x)|dx \rightarrow 0 \ , $$
Then the integral $\int... | Let $f_n \to f$ in $L^p(\Omega)$. Then, we also have that $\|f_n\|_p \to \|f\|_p$. So "something similar" holds.
As for the convergence of $\int f_n$ to $\int f$, this is generally not guaranteed by $L^p$ convergence, unless the measure of the underlying space is finite (like it is in your example).
In that case we hav... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Rees algebra of a monomial ideal User fbakhshi deleted the following question:
Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I=(f_1,\ldots,f_q)$ a monomial ideal of $R$. If $f_i$ is homogeneous of degree $d\geq 1$ for all $i$, then prove that
$$
R[It]/\mathfrak m R[It]\simeq K[f_1t,\ldots, f_q... | $R[It]/\mathfrak mR[It]$ is $\oplus_{n\geq 0}{I^n/\mathfrak mI^n}.$ Let $\phi$ be any homogeneous polynomial of degree $l$. Consider $I_l$ to be the $k$-vector space generated by all $\phi(f_1,\ldots,f_q).$ Then $k[f_1,\ldots,f_q]=\oplus_{l\geq 0}{I_l}.$ Now $\dim_{k}{I_l}=\dim_{k}{I^l/\mathfrak mI^l}.$ Hence $k[f_1,\l... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving the monotone decreasing and find the limit ?? Let $a,b$ be positive real number. Set $x_0 =a$ and $x_{n+1}= \frac{1}{x_n^{-1}+b}$ for $n≥0$
(a) Prove that $x_n$ is monotone decreasing.
(b) Prove that the limit exists and find it.
Any help? I don't know where to start.
| To prove the limit exists use the fact every decreasing bounded below sequence is convergent. To find the limit just assume $ \lim_{n\to \infty} x_n = x = \lim_{n\to \infty} x_{n+1} $ and solve the equation for $x$
$$ x=\frac{1}{1/x+b} .$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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If there is a continuous function between linear continua, then this function has a fixed point? Let $f:X\to X$ be a continuous map and $X$ be a linear continuum. Is it true that $f$ has a fixed point?
I think the answer is "yes" and here is my proof:
Assume to the contrary that for any $x\in X$, either $f(x)<x$ or $f(... | The function $f(x)=x+1$ is a counterexample. Here both sets $A$ and $B$ are open, but one of them is empty :-)
Brouwer fixed point theorem asserts that the closed ball has the property you are looking for: every continuous self-map will have a fixed point. But the proof requires tools well beyond the general topologi... | {
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Prove that $d^n(x^n)/dx^n = n!$ by induction I need to prove that $d^n(x^n)/dx^n = n!$ by induction.
Any help?
| Hint: Are you familiar with proofs by induction? Well, the induction step could be written as $$d^{n+1}(x^{n+1}) / dx^{n+1} = d^n \left(\frac{d(x^{n+1})} {dx}\right) /dx^n $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/429001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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How To Calculate a Weighted Payout Table I am looking to see if there is some sort of formula I can use to calculate weighted payout tables. I am looking for something similar to the PGA payout distribution, but the problem is I want my payout table to be flexible to accommodate a variable or known number of participan... | There are lots of them. You haven't given enough information to select just one. A simple one would be to pick $n$ as the number of players that will be paid and $p$ the fraction that the prize will reduce from one to the next. The winner gets $1$ (times the top prize), second place gets $p$, third $p^2$ and so on. ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $x\not\leq y$, then is $x>y$, or $x\geq y$? I'm currently reading about surreal numbers from here.
At multiple points in this paper, the author has stated that if $x\not\leq y$, then $x\geq y$.
Shoudn't the relation be "if $x\not\leq y$, then $x>y$"?
Hasn't the possibility of $x=y$ already been negated when we sai... | You are correct, if we were speaking of $\leq/\geq$ relations we know and love, as standard ordering relations on the reals: The negation of $x \leq y$ is exactly $x > y$, and that would be the correct assertion if we were talking about a "trichotomous" ordering, where we take that for any two real numbers, one and onl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Hartshorne III 9.3 why do we need irreducibility and equidimensionality? We are trying to prove:
Corollary 9.6: Let $f\colon X \to Y$ be a flat morphism of schemes of finite type over a field $k$, and assume that $Y$ is irreducible. Then the following are equivalent:
(i) every irreducible component of $X$ has dimension... | I was confused by this as well. The desired equalities follow from the general statement
Let $X$ be a scheme of finite type over a field $k$ and let $x\in X$. Then $$\dim \mathcal{O}_x+\dim \{x\}^-=\sup_{x\in V \text{ irreducible component}} \dim V$$
where the sup on the right is taken over all irreducible component... | {
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"timestamp": "2023-03-29T00:00:00",
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Polynomials Question: Proving $a=b=c$. Question:
Let $P_1(x)=ax^2-bx-c, P_2(x)=bx^2-cx-a \text{ and } P_3=cx^2-ax-b$
, where $a,b,c$ are non zero reals. There exists a real $\alpha$ such
that $P_1(\alpha)=P_2(\alpha)=P_3(\alpha)$. Prove that $a=b=c$.
The questions seems pretty easy for people who know some kind... | Hint: if $a=b=c$ then all three polynomials are equal. A useful trick to show that polynomials are equal is the following: if a polynomial $Q$ of degree $n$ (like $P_1-P_2$) has $n+1$ distinct roots (points $\beta$ such that $Q(\beta)=0$) then $Q$ is the zero polynomial. In particular, if a quadratic has three zeroes... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 1
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A basic question on Type and Cotype theory I'm studying basic theory of type and cotype of banach spaces, and I have a simple question. I'm using the definition using averages. All Banach spaces have type 1, that was easy to prove, using the triangle inequality. But I'm having a hard time trying to show that all Banach... | The argument is by induction:
It is trivial for $n=1$. For the case $n=2$ note that we have, by the triangle inequality and the fact that $\|z\|=\|-z\|$,
$$
\| x-y \| + \|x+y\| \geq 2\max\{ \| x\|, \| y\|\},
$$
so that the inequality in this case follows with $C=1$. For the general case consider a vector $\bar{\varepsi... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that the $\max{ \{ x,y \} }= \frac{x+y+|x-y|}{2}$. Show that the $\max{ \{ x,y \} }= \dfrac{x+y+|x-y|}{2}$.
I do not understand how to go about completing this problem or even where to start.
| Without loss of generality, let $y=x+k$ for some nonnegative number $k$. Then,
$$
\frac{x+(x+k)+|x-(x+k)|}{2} = \frac{2x+2k}{2} = x+k = y
$$
which is equal to $\max(x,y)$ by the assumption.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
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"answer_id": 5
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Probability of getting 'k' heads with 'n' coins This is an interview question.( http://www.geeksforgeeks.org/directi-interview-set-1/)
Given $n$ biased coins, with each coin giving heads with probability $P_i$, find the probability that on tossing the $n$ coins you will obtain exactly $k$ heads. You have to write the f... | Consider the function
$[ (1-P_1) + P_1x] \times [(1-P_2) + P_2 x ] \ldots [(1-P_n) + P_n x ]$
Then, the coefficient of $x^k$ corresponds to the probability that there are exactly $k$ heads.
The coefficient of $x^k$ in this polynomial is $\sum_{k-\mbox{subset} S} [\prod_{i\in{S}} \frac{1-p_i}{p_i} \prod_{j \not \in S} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/429698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help to compute the following coefficient in Fourier series $\int_{(2n-1)\pi}^{(2n+1)\pi}\left|x-2n\pi\right|\cos(k x)\mathrm dx$
$$\int_{(2n-1)\pi}^{(2n+1)\pi}\left|x-2n\pi\right|\cos(k x)\mathrm dx$$
where $k\geq 0$, $k\in\mathbb{N} $ and $n\in\mathbb{R} $.
it is a $a_k$ coefficient in a Fourier series.
| Here is the final answer by maple
$$ 2\,{\frac {2\, \left( -1 \right) ^{k} \left( \cos \left( \pi \,kn
\right) \right) ^{2}-2\, \left( \cos \left( \pi \,kn \right)
\right) ^{2}+ \left( -1 \right) ^{k+1}+1}{{k}^{2}}}
. $$
Added: More simplification leads to the more compact form
$$ 2\,{\frac {\cos \left( 2\,\pi \... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A proposed proof by induction of $1+2+\ldots+n=\frac{n(n+1)}{2}$
Prove: $\displaystyle 1+2+\ldots+n=\frac{n(n+1)}{2}$.
Proof
When $n=1,1=\displaystyle \frac{1(1+1)}{2}$,equality holds.
Suppose when $n=k$, we have $1+2+\dots+k=\frac{k(k+1)}{2}$
When $n = k + 1$:
\begin{align}
1+2+\ldots+k+(k+1) &=\frac{k(k+1)}{2}+k+1 ... | Q1: No problems, that's the way induction works.
Q2: go back one step:
$$k(k+1)+2k+2=k(k+1)+2(k+1)=(k+1)(k+2)$$
| {
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"answer_id": 2
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Why does «Massey cube» of an odd element lie in 3-torsion? The cup product is supercommutative, i.e the supercommutator $[-,-]$ is trivial at the cohomology level — but not at the cochain level, which allows one to produce various cohomology operations.
The simplest (in some sense) of such (integral) operations is the ... | Recall that $d(x\cup_1y)=[x,y]\pm dx\cup_1 y\pm x\cup_1dy$.
In particular, in the definition from the question one can take $b=a\cup_1a$. So $\langle a\rangle^3=[a,a\cup_1a]$.
Now $d((a\cup_1a)\cup_1a)=[a,a\cup_1a]+(d(a\cup_1 a))a=\langle a\rangle^3+[a,a]\cup_1 a$. Now by Hirsch formula $a^2\cup_1a=a(a\cup_1a)+(a\cup_1... | {
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Why does the inverse of the Hilbert matrix have integer entries?
Let $A$ be the $n\times n$ matrix given by
$$A_{ij}=\frac{1}{i + j - 1}$$
Show that $A$ is invertible and that the inverse has integer entries.
I was able to show that $A$ is invertible. How do I show that $A^{-1}$ has integer entries?
This matrix is ca... | Be wise, generalize (c)
I think the nicest way to answer this question is the direct computation of the inverse - however, for a more general matrix including the Hilbert matrix as a special case. The corresponding formulas have very transparent structure and nontrivial further generalizations.
The matrix $A$ is a pa... | {
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Is there an inverse to Stirling's approximation? The factorial function cannot have an inverse, $0!$ and $1!$ having the same value. However, Stirling's approximation of the factorial $x! \sim x^xe^{-x}\sqrt{2\pi x}$ does not have this problem, and could provide a ballpark inverse to the factorial function. But can thi... | As $n$ increases to infinity we want to know roughly the size of the $x$ that satisfies the equation $x! = n$. By Stirling
$$
x^x e^{-x} \sqrt{2\pi x} \sim n
$$
Just focusing on $x^x$ a first approximation is $\log n / \log\log n$. Now writing $x = \log n / \log\log n + x_1$ and solving approximately for $x_1$, this ti... | {
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"source": "stackexchange",
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Action of the state I have the following question:
let $A$ be a C*-algebra and let $a$ be a self adjoint element of $A$. Is it true that
for any state $f$ acting on $A$ $$f(a) \in \mathbb{R}.$$
Let me remind that a state is a positive linear functional of norm $1$.
I think it is due to the fact that every state has to ... | Suppose that $a$ is a self-adjoint element in the C$^*$-algebra $A$. Then, by applying the continuous functional calculus, we can write $a$ as the difference of two positive elements $a=a_+ - a_-$ such that $a_+a_-=a_-a_+=0$. See, for example, Proposition VIII.3.4 in Conway's A Course in Functional Analysis, or (*) bel... | {
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How to prove these integral inequalities? a) $f(x)>0$ and $f(x)\in C[a,b]$
Prove $$\left(\int_a^bf(x)\sin x\,dx\right)^2 +\left(\int_a^bf(x)\cos x\,dx\right)^2 \le \left(\int_a^bf(x)\,dx\right)^2$$
I have tried Cauchy-Schwarz inequality but failed to prove.
b) $f(x)$ is differentiable in $[0,1]$
Prove
$$|f(0)|\le \int... | Hint: For part a), use Jensen's inequality with weighted measure $f(x)\,\mathrm{d}x$. Since $f(x)>0$, Jensen says that for a convex function $\phi$
$$
\phi\left(\frac1{\int_Xf(x)\mathrm{d}x}\int_Xg(x)\,f(x)\mathrm{d}x\right)
\le\frac1{\int_Xf(x)\mathrm{d}x}\int_X\phi(g(x))\,f(x)\mathrm{d}x
$$
Hint: For part b), note th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
The continuity of measure Let $m$ be the Lebesgue Measure. If $\{A_k\}_{k=1}^{\infty}$ is an ascending collection of measurable sets, then
$$m\left(\cup_{k=1}^\infty A_k\right)=\lim_{k\to\infty}m(A_k).$$
Can someone share a story as to why this is called one of the "continuity" properties of measure?
| Since $\{A_k\}_{k=1}^\infty$ is an ascending family of sets we can vaguely write that
$$
\lim\limits_{k\to\infty} A_k=\bigcup\limits_{k=1}^\infty A_k \qquad(\color{red}{\text{note: this is not rigor!}})
$$
then this property can be written as
$$
m\left(\lim\limits_{k\to\infty} A_k\right)=\lim\limits_{k\to\infty}m(A_k)
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Check solutions of vector Differential Equations I have solved the vector ODE: $x\prime = \begin{pmatrix}1& 1 \\ -1 &1 \end{pmatrix}x$
I found an eigenvalue $\lambda=1+i$ and deduced the corresponding eigenvector:
\begin{align}
(A-\lambda I)x =& 0 \\
\begin{pmatrix}1-1-i & 1 \\-1& 1-1-i \end{pmatrix}x =& 0 \\
\begin{pm... | Let me work through the other eigenvalue, and see if you can follow the approach.
For $\lambda_2 = 1-i$, we have:
$[A - \lambda_2 I]v_2 = \begin{bmatrix}1 -(1-i) & 1\\-1 & 1-(1-i)\end{bmatrix}v_2 = 0$
The RREF of this is:
$\begin{bmatrix}1 & -i\\0 &0\end{bmatrix}v_2 = 0 \rightarrow v_2 = (i, 1)$
To write the solution, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that $7 \mid( 1^{47} +2^{47}+3^{47}+4^{47}+5^{47}+6^{47})$ I am solving this one using the fermat's little theorem but I got stuck up with some manipulations and there is no way I could tell that the residue of the sum of each term is still divisible by $7$. what could be a better approach or am I on the right tra... | $6^{47} \equiv (-1)^{47} = -1^{47}\mod 7$
$5^{47} \equiv (-2)^{47} = -2^{47}\mod 7$
$4^{47} \equiv (-3)^{47} = -3^{47}\mod 7$
Hence $ 1^{47} +2^{47}+3^{47}+4^{47}+5^{47}+6^{47} \equiv 0 \mod 7$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/430506",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
} |
Linear algebra - Coordinate Systems I'm preparing for an upcoming Linear Algebra exam, and I have come across a question that goes as follows: Let U = {(s, s − t, 2s + 3t)}, where s and t are any real numbers. Find the coordinates of x = (3, 4, 3) relative to the basis B if x is in U . Sketch the set U in the xyz-coord... | This might be an answer, depending on how one interprets the phrase "basis $B$", which is undefined in the question as stated:
Note that $(s, s - t, 2s + 3t) = s(1, 1, 2) + t(0, -1, 3)$. Taking $s = 1$, $t = 0$ shows that $(1, 1, 2) \in U$. Likewise, taking $s = 0$, $t = 1$ shows $(0, -1, 3) \in U$ as well. Incident... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Calculating 7^7^7^7^7^7^7 mod 100 What is
$$\large 7^{7^{7^{7^{7^{7^7}}}}} \pmod{100}$$
I'm not much of a number theorist and I saw this mentioned on the internet somewhere. Should be doable by hand.
| Reading the other answers, I realize this is a longer way than necessary, but it gives a more general approach for when things are not as convenient as $7^4\equiv 1\bmod 100$.
Note that, for any integer $a$ that is relatively prime to $100$, we have
$$a^{40}\equiv 1\bmod 100$$
because $\varphi(100)=40$, and consequent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
Size of new box rotated and the rescaled I have a box of height h and width w. I rotate it to r degrees. Now I resize it so that it can original box in it. What will be the size of newly box.
Original Box:
Box after rotating some degrees.
New box after rescaling.
So my question is what should be the formula to calcu... | Assuming the old rectangle inscribed in the new one, we have the following picture:
Let $\theta$ ($0 \leq \theta \leq \frac{\pi}{2}$) the rotation angle, $w'$ the new width and $h'$ the new height, then we have the following equations:
$$w' = w \cos \theta + h \sin \theta$$
$$h' = w \sin \theta + h \cos \theta$$
The n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430763",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Notation for intervals I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed intervals. I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall someone calling the notation with $[a,b]$ and $]a,b[$ as French notation.
*
*What are the origins ... | As a French student, all my math teachers (as well as the physics/biology/etc. ones) always used the $[a,b]$ and $]a,b[$ (and the "hybrid" $[a,b[$ and $]a,b]$) notations.
We also, for integer intervals $\{a,a+1,...,b\}$, use the \llbracket\rrbracket notation (in LateX, package {stmaryrd}): $[[ a,b ]]$.
I have never see... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430851",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 1
} |
antiderivative of $\sum _{ n=0 }^{ \infty }{ (n+1){ x }^{ 2n+2 } } $ I've proven that the radius of convergence of $\sum _{ n=0 }^{ \infty }{ (n+1){ x }^{ 2n+2 } } $ is $R=1$, and that it doesn't converge at the edges.
Now, I was told that this is the derivative of a function $f(x)$, which holds $f(0)=0$.
My next st... | First, consider
$$
g(w)=\sum_{n=0}^{\infty}(n+1)w^n.
$$
Integrating term-by-term, we find that the antiderivative $G(w)$ for $g(w)$ is
$$
G(w):=\int g(w)\,dw=C+\sum_{n=0}^{\infty}w^{n+1}
$$
where $C$ is an arbitrary constant. To make $g(0)=0$, we take $C=0$; then
$$
G(w)=\sum_{n=0}^{\infty}w^{n+1}=\sum_{n=1}^{\infty}w^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What is the value of the series $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n^2}$? I am given the following series:
$$\sum_{n=1}^\infty \frac{(-1)^n}{n^2}$$
I have used the alternating series test to show that the series converges.
However, how do I go about showing what it converges to?
| Consider the Fourier series of $g(x)=x^2$ for $-\pi<x\le\pi$:
$$g(x)=\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos(nx)+b_n\sin(nx)$$
note $b_n=0$ for an even function $g(t)=g(-t)$ and that:
$$a_n=\frac {1}{\pi} \int _{-\pi }^{\pi }\!{x}^{2}\cos \left( nx \right) {dx}
=4\,{\frac { \left( -1 \right) ^{n}}{{n}^{2}}},$$
$$\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/430973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Vector-by-Vector derivative Could someone please help me out with this derivative?
$$
\frac{d}{dx}(xx^T)
$$
with both $x$ being vector.
Thanks
EDIT:
I should clarify that the actual state I am taking the derivative is
$$
\frac{d}{dx}(xx^TPb)
$$
where $Pb$ has the dimention of $x$ but is independent of $x$. So the whole... | You can always go back to the basics. Let $v$ be any vector and $h$ a real number. Substitute $x \leftarrow x + h v$ to get
$$
(x + hv)(x+hv)^t = (x+hv)(x^t+hv^t) = x x^t + h(xv^t + vx^t) + h^2 vv^t.
$$
The linear term in $h$ is your derivative at $x$ in the direction of $v$, so $xv^t + vx^t$ (which is linear in $v$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
A Covering Map $\mathbb{R}P^2\longrightarrow X$ is a homeomorphism I came across the following problem: Any covering map $\mathbb{R}P^2\longrightarrow X$ is a homeomorphism. To solve the problem you can look at the composition of covering maps
$$
S^2\longrightarrow \mathbb{R}P^2\longrightarrow X
$$
and examine the d... | 1-Prove that $X$ has to be a compact topological surface;
2-Prove that such a covering has to be finite-sheeted;
3-Deduce from 2 and from $\pi_1(\mathbb{R}P^2)$ that $\pi_1(X)$ is finite;
4- Since the map induced by the covering projection on $\pi_1$ is injective you get $\mathbb{Z}/2\mathbb{Z}< \pi_1(X)$;
5-Conclude u... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$ Problem statement:
Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty $.
, $n\in \mathbb{N}$
My progress
LHS is e... | There's a direct proof to the inequality of $\frac{1}{\sqrt{2n+1}}$, though vadim has improved on the bound.
Consider $A = \frac{1}{2} \times \frac{3}{4} \times \ldots \times \frac{2n-1} {2n}$
and $B = \frac{2}{3} \times \frac{4}{5} \times \ldots \times \frac{2n}{2n+1}$.
Then $AB = \frac{1}{2n+1}$. Since each term of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 0
} |
find the power representation of $x^2 \arctan(x^3)$ Wondering what im doing wrong in this problem im ask to find the power series representation of
$x^2 \arctan(x^3)$
now i know that arctan's power series representation is this
$$\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{2n+1} $$
i could have sworn that for solving fo... | The book's answer would be right if it said
$$
\sum_{\text{odd }n\ge 0} (-1)^{(n-1)/2)} \frac{x^{3n+2}}{n}.
$$
That would be the same as your answer, i.e.
$$
\sum_{\text{odd }n\ge 0} (-1)^{(n-1)/2)} \frac{x^{3n+2}}{n} = \sum_{n\ge0} (-1)^n\frac{x^{6n+5}}{2n+1}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/431298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
compute an integral with residue I have to find the value of
$$\int_{-\infty}^{\infty}e^{-x^2}\cos({\lambda x})\,dx$$
using residue theorem. What is a suitable contour? Any help would be appreciate! Thanks...
| Hint: By symmetry, we can let $\gamma$ be the path running along the real axis and get that our integral is just $$\int_\gamma e^{-z^2}e^{i\lambda z} dz.$$ Now what happens when you combine these terms and complete the square? Your answer should become a significantly simpler problem.
But be careful with the resulting ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Number of Spanning Trees in a certain graph Let $k,n\in\mathbb N$ and define the simple graph $G_{k,n}=([n],E)$, where $ij\in E\Leftrightarrow 0 <|i-j|\leq k$ for $i\neq j\in [n]$.
I need to calculate the number of different spanning trees.
I am applying Kirchoff's Matrix Tree theorem to solve this but i am not getting... | The answer seems correct. You can check with a different method in this case, because the graph you are considering is the complete graph minus one specific edge E.
By Cayley's formula, there are $5^3=125$ spanning trees of the complete graph on 5 vertices. Each such tree has four edges, and there are 10 possible edges... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
convergence of a series $a_1 + a_1 a_2 + a_1 a_2 a_3 +\cdots$ Suppose all $a_n$ are real numbers and $\lim_{n\to\infty} a_n$ exists.
What is the condition for the convergence( or divergence ) of the series
$$ a_1 + a_1 a_2 + a_1 a_2 a_3 +\cdots $$
I can prove that $ \lim_{n\to\infty} |a_n| < 1 $ ( or > 1 ) guarantees
a... |
What if $\lim_{n\to\infty}a_n=1$ and $a_n<1$ for all $n$?
Then the series may or may not converge. A necessary criterion for the convergence of the series is that the sequence of products
$$p_n = \prod_{k = 1}^n a_k$$
converges to $0$.
If the $a_n$ converge to $1$ fast enough, say $a_n = 1 - \frac{1}{2^n}$ ($\sum \lv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Show that $(x_n)$ converge to $l$.
Let $(x_n)$ be a sequence of reals. Show that if every subsequence $(x_{n_k})$ of $(x_n)$ has a further subsequence $(x_{n_{k_r}})$ that converge to $l$, then $(x_n)$ converge to $l$.
I know the fact that subsequence of $(x_n)$ converge to the limit same as $(x_n)$ does, but I'm not... | You are correct with your doubts as that argument applies only if you know that the sequence converges in the first place.
Now for a proof, assume the contrary, that is: there exists $\epsilon>0$ such that for all $N\in\mathbb N$ there exists $n>N$ with $|x_n-l|\ge\epsilon$.
For $N\in\mathbb N$ let $f(N)$ denote one su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/431681",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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