Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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The $1997$ IIT JEE problem
Let $S$ be a square of unit area. Consider any quadrilateral whose $4$ vertices lie on each side square $S$. Let the length of the sides of this quadrilateral be $a,b,c,d$. Then prove that $$2 \leq a^2+b^2+c^2+d^2 \leq 4$$
This problem appeared in IIT JEE $1997$ (re-exam).I really do not ... | Consider this:
You want to find the minimum of:
$$2(a^2+b^2+c^2+d^2-a-b-c-d)+4$$
Or
$$-2.\sum(x(1-x)) +4 $$
Here x, represents a,b,c or d
Notice that it's a quadratic, so maximum of x(1-x) happens at x = 0.5 as it happens between x = 1 and x =0
So max of quadratic is 1/4
Hence,
$$-2.\sum(x(1-x)) +4 = -2.4.1/4 +4 = 2$... | {
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"url": "https://math.stackexchange.com/questions/2996633",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show continuous functions need not be open maps and open maps need not be continuous. A function from one metric space to another is said to be an open map
if it maps open sets to open sets. Similarly one can define a closed
map.
1-Provide a continuous function which does not map an open set to another open set?
2- Pr... | A) To violate openness you can give any map which is not bijective.
B) Open maps which are not continuous
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof in The Integral Test
Theorem: Assume there is an $N\in\mathbb{N}$ so that $f:[N,\infty)\to \mathbb{R}$ is non-negative, continuous and decreasing. Define $a_n=f(n)$ for $n\in \mathbb{N}$ with $n\geq N$. Then, $\sum_{n=N}^{\infty}a_n$ converges if and only if $\int_{N}^{\infty}f(x)\,dx$ converges.
Proof: Since $... | (1) For your second question in the edit: suppose there is a real number $A \geq N$ such that
$$
\int_N^{A+1} f(x)\, dx \geq K.
$$
Since $f$ is non-negative, the integral of $f$ from $A+1$ to $\lceil A+1 \rceil$ is non-negative as well. Therefore
$$
\int_N^{\lceil A+1\rceil} f(x)\, dx = \int_N^{A+1} f(x)\, dx + ... | {
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total of non zero value I want to write a math formula that represents the average of non zero value e.g. if I have 4 numbers, one of them is zero, then the sum of the numbers will be divided by 3. Is it correct to say:
$$
\frac {\sum_{k=1}^{4} x_{k}} {|x_{k} > 0|}
$$
| I think what you want is the following:
$$\frac{\sum_{k=1}^{4} x_{k}}{\sum_{k:x_{k} \neq 0} 1}.$$
You could alternatively define a set $S$ such that
$$S = \{k: x_{k} \neq 0\}$$
and then sum over this set so that you would have
$$\frac{\sum_{k=1}^{4} x_{k}}{\sum_{S} 1}.$$
Note that adding zeros to the sum will not cha... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Approximation of the quadratic formula with straightedge and compass Given a directrix and a focus (blue), we can define a parabola as illustrated below. We suppose the parabola intersecting the $x$-axis in correspondence of the red dots.
We draw the line perpendicular to the $x$-axis and passing through the focus. Th... | Given $y=ax^2+bx+c$.
You know y is a parabola.
It's vertex is $(x_0,y_0)=(\frac{-b}{2a},\frac{-b^2}{4a}+c)$.
From geometry, we know we have no roots if $y_0>0$ and $a>0$, or if $y_0<0$ and $a<0$. In the first case, all the values of y are positive and therefor never zero. In the second case, all the values are negati... | {
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Proof verification: the image of a continuous function on a connected domain is itself connected. Take the definition of a connected topological space to be one that has no clopen sets other than the space itself or the empty set.
I claim to prove that if $f: X \to Y$ is a continuous function between topological spaces... | You're almost good;
You cannot state that $f^{-1}(U)$ is clopen because you don't know that $f(X)$ is open or closed in $Y$!
To overcome this difficulty simply notice that the restriction map $f:X \to f(X)$ is itself continuous, when $f$ is. Then the proof becomes quite trivial. Any separation of $f(X)$ projects back t... | {
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"source": "stackexchange",
"question_score": "2",
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Formula when index of sigma is negative We're currently learning series and sigma notation
We've been given the formulas for $\sum_{k=1}^{n}k$, $\sum_{k=1}^{n}k^2$, and $\sum_{k=1}^{n}k^3$ plus the properties on how to break them apart etc, place the constant c in front and multiply by the resulting sum.
Now I've bee... | I give you below the general formulae
$$\sum_{k=m}^{n}k=\frac{1}{2} (n-m+1) (n+m)$$
$$\sum_{k=m}^{n}k^2=\frac{1}{6} (n-m+1) \left(2 m^2+2 m n-m+2 n^2+n\right)$$
$$\sum_{k=m}^{n}k^3=\frac{1}{4} (n-m+1) (m+n) \left(m^2-m+n^2+n\right)$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Double checking a coefficient for a Laurent series, the series for $f(z) = \frac{z^2 + 1}{(z - i)^2}dz$ So we seek the coefficient $a_{-1}$ of the Laurent series for $f(z) = \frac{z^2 + 1}{(z - i)^2}dz$, where Laurent series is denoted by
$$f(z) = \sum_{n = -\infty}^\infty a_n(z - z_0)^n$$
when the series is given abou... | Observe: $f(z) = \frac{z^2+1}{(z-i)^2} = \frac{(z+i)(z-i)}{(z-i)^2} = \frac{z+i}{z-i} = \frac{(z-i) +i +i}{z-i} = \frac{2i}{z-i} + 1.$ What is the coefficient for $a_{-1}$ given this observation?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Supremum and infimum of a set I wanted to check my reasoning for the following question:
Determine minimum, maximum, supremum and infimum of the set:
$$B=\left\{ -\frac{1}{n} \in \mathbb{Q}: n \in \mathbb{N}_+ \right\}$$
Whenever $n=1$ we have that $-\frac{1}{1}=-1$, this is the minimum element of the set, since ... | You are correct!
Indeed, if you feel uncomforatble to work with negative signs, just compute $\inf A$ and $\sup A$ where $A=\{1/n:n \in \Bbb N\}$ and then use $$\sup(-A)=-\inf A\\\inf(-A)=-\sup A$$
| {
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"question_score": "2",
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Check linear Independence of vectors with a non-square matrix In a ℝ¹⁵ space, I take two vector that I would like to check if they are linear independent. They are:
-0.0049 0.0000
-0.0085 0.0000
0.3555 0.0000
0.4364 0.3921
0.4267 -0.2660
-0.3448 0.1596
-0.3215 -0.3921
-0.3694 0.2660
-0.2737 ... | These two vectors are linearly independent. The first coordinate is $0$ for the right vector but non-zero for the left vector whereas the forth is non-zero for both.
In general, if $v_1$ is the first vector and $v_2$ is the second vector, they are linearly dependent if you can find a real number $\alpha\neq 0$ such th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Taylor series of functions Consider the Taylor series of the function
$$\frac{2e^x}{e^{2x}+1} = \sum_{n=0}^{\infty} \frac{E_n}{n!} x^n$$
Prove that $E_0 = 1, E_{2n-1} = 0$ and, for $n \ge 1$, $$E_{2n} = - \sum_{l=0}^{n-1} C_{2l}^{2n} {E_{2l}}$$
The hint given was to consider $\cosh(x)$.
I understand that the function ... | We use the Taylor series expansion of $\cosh x$ which you can find e.g. here.
\begin{align*}
\cosh x = \frac{e^{2x}+1}{2e^x} = \sum_{l=0}^\infty \frac{x^{2l}}{(2l)!}
\end{align*}
We obtain
\begin{align*}
\color{blue}{1}&=\left(\sum_{k=0}^\infty\frac{E_k}{k!}x^k\right)\frac{e^{2x}+1}{2e^x}\tag{1}\\
&=\left(\sum_{k=0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2998157",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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coloring with a dihedral group $D_n$ with n prime I need to find out how many different colorings you can make with 2 colors in a dihedral group $D_n$ with $n$ prime and $m$ black and $p-m$ white beads. So first I compute the cycle index:
The cycle index of a dihedral group with $n$ prime (odd) is equal to:
$$Z(D_n) = ... | Cycle index.
$$Z(D_p) = \frac{1}{2p}
\left(a_1^{p} + (p-1) a_p + p a_1 a_2^{(p-1)/2}\right)$$
We are interested in
$$[B^m W^{p-m}] Z(D_p; B+W).$$
This has three components.
First component.
$$[B^m W^{p-m}] \frac{1}{2p} (B+W)^p
= \frac{1}{2p} {p\choose m}.$$
Second component.
$$[B^m W^{p-m}] \frac{p-1}{2p} (B^p+W^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2998245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Meaning of a transformation with respect to 1 or 2 bases? So let's say I have:
$$A =\begin{bmatrix}
1 & 2 & 1\\
-1 & 1 & 0
\end{bmatrix}$$
$A$ represents a transformation $L: R^3 \rightarrow R^2$ with respect to bases $S$ and $T$ where:
$$S = \begin{bmatrix} -1\\1\\0\end{bmatrix},
\begin{bmatrix} 0\\1\\1\... | From the description you have provided, the matrix $A$ is intended to represent a linear mapping $L$ from $\mathbb{R}^3$ to $\mathbb{R}^2$ that takes as input a vector of co-ordinates relative to basis $S$ in $\mathbb{R}^3$ and outputs a vector of co-ordinates relative to basis $T$ in $\mathbb{R}^2$. There is nothing a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Geometric Meaning of the Jacobian of a Linear Transformation Consider the multivariable function $f(x, y) = \begin{bmatrix}-y \\ x \end{bmatrix}$, whose geometry is shown here. For any point $(x,\ y)$, this function's Jacobian matrix is always $\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix}$. There is no need to even pl... | Really it is the mapping
$$
x \mapsto f(x_0) + f'(x_0)(x - x_0)
$$
which approximates $f$ well when you zoom in near the point $x_0$. (Here $x_0$ is a point in $\mathbb R^n$, and the $n \times n$ matrix $f'(x_0)$ is the derivative (i.e., Jacobian) of $f$ at $x_0$.) This approximation
$$
f(x) \approx f(x_0) + f'(x_0)(x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2998672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Approximating midpoint of a curve I am wondering whether there is any general method of approximating the midpoint of a given curve, given the coordinates of the endpoints and the equation of the curve. I know the calculus method of finding it exactly, but I am looking for a purely algebraic method that works for all e... | If you are trying to find the midpoint of the arc from the (upper, lower, eastern or western) most point of a circle to any adjacent most point of a circle given the radius and center of a circle, then you just divide the radius of the circle by the square root of 2 to find the distance the x and y travelled along the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2998832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the probability of three consecutive results X and two results Y in an event? I have N number of days where three different events X,Y,Z can occur in each day. A is a set of possible occurrences of length N. I want to calculate the number of ways where:
*
*Y does NOT happen twice or more in these number of d... | a) Y cannot appear two or more times
Then
- if Y does not appear, we are left with a binary (X,Z) string of length $n=N$;
- if Y appears once, by removing it, we are left with two binary (X,Z) string of length $n$ and $N-n-1$, with $0 \le n \le N-1$.
b) The string does not contain one (or more) runs of three (o... | {
"language": "en",
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Is there a way to solve this equation arising from a weighted sum? If I have a weighted finite sum that is equal to a known real value $x$, and I also know the real-valued and non-negative weights $a_i$, and I have unknown but also real and non-negative elements $b_i$ :
$x = \sum a_ib_i$
is it possible to extract the v... | No. Consider $x=3$, $a_1=1$, $a_2=2$.
$$3=1\cdot1+2\cdot1=1\cdot\frac12+2\cdot\frac54$$
$$1+1\neq\frac12+\frac54$$
So $\sum b_i$ is not determined by $\sum a_ib_i$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Reduction formula for $\int\frac{dx}{(ax^2+b)^n}$ I recently stumbled upon the following reduction formula on the internet which I am so far unable to prove.
$$I_n=\int\frac{\mathrm{d}x}{(ax^2+b)^n}\\I_n=\frac{x}{2b(n-1)(ax^2+b)^{n-1}}+\frac{2n-3}{2b(n-1)}I_{n-1}$$
I tried the substitution $x=\sqrt{\frac ba}t$, and it ... | Hint The appearance of the term in $\frac{x}{(a x^2 + b)^{n - 1}}$ suggests applying integration by parts with $dv = dx$ and thus $u = (a x^2 + b)^{-n}$. Renaming $n$ to $m$ we get
$$I_m = u v - \int v \,du = \frac{x}{(a x^2 + b)^m} + 2 m \int \frac{a x^2 \,dx}{(a x^2 + b)^{m + 1}} .$$
Now, the integral on the right ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2999219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What does the word "extend" mean in the context of model theory? Consider the following two problems:
(1) Let $L=\{E\}$ be a language consisting one binary relation symbol. Let $T$ be the $L$-theory saying that $E$ is an equivalence relation with infinitely many classes. Prove that there are infinitely many inequivalen... | "Extend" just means "be a superset of" in this context. So a theory $T'$ extends a theory $T$ if $T\subseteq T'$, and a filter $F'$ extends a filter $F$ if $F\subseteq F'$.
(The phrasing "extends to" in statement (2) is an error and should be just "extends". Indeed, saying "$F$ extends to $F'$" would normally mean th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2999347",
"timestamp": "2023-03-29T00:00:00",
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Limits - Calculating $\lim\limits_{x\to 1} \frac{x^a -1}{x-1}$, where $a \gt 0$, without using L'Hospital's rule
Calculate $\displaystyle\lim\limits_{x\to 1} \frac{x^a -1}{x-1}$, where $a \gt 0$, without using L'Hospital's rule.
I'm messing around with this limit. I've tried using substitution for $x^a -1$, but it d... | For $a$ rational, let $a=\dfrac pq$. We set $x=t^q$, and
$$\frac{x^{p/q}-1}{x-1}=\frac{t^p-1}{t^q-1}=\frac{\dfrac{t^p-1}{t-1}}{\dfrac{t^q-1}{t-1}}$$
which tends to $\dfrac pq$.
For irrational $a$, the result could be extended using continuity, but this is more technical and depends on your definition of the powers.
| {
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How to find $\lim_{n\to \infty} (\frac {n}{\sqrt{n^2+n}+\sqrt{n}})$? I am trying to solve this : $\lim_{n\to \infty} (\frac {n}{\sqrt{n^2+n}+\sqrt{n}})$ but I always end up with $\frac {\infty}{\infty}$ which is undefined
I tried for eg $$\lim_{n\to \infty} (\frac {n}{\sqrt{n^2+n}+\sqrt{n}}) (\frac {\sqrt{n^2+n}-\sqrt{... | You are almost done indeed
$$\lim_{n\to \infty} \frac {n^2(\sqrt{1+\frac1n}-(\frac 1n)^\frac12)}{n^2}= \frac {(\sqrt{1+0}-(0)^\frac12)}{1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2999625",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Proving by Contrapositive (specific question within) I'm having issues coming up with a contrapositive proof for the following question.
As far as I know, a proof by contraposition is based on the following :
$\overline Q \to \overline P \equiv P \to Q$
or that's where I'm mistaken?
The question is:
X,Y,Z are natura... | Following the comments, if we have $X, Y$ and $Z$ none of which are divisible by 3 and such that $X^3 + Y^3 = Z^3$, we may assume that $X,Y \equiv 1$ (mod 3) and $Z \equiv 2$ (mod 3). So, we may assume that there are $a$ $b$ and $c$ such that $X = 3a+1$, $Y = 3b+1$ and $Z = 3c+2$. By applying the binomial expansion to ... | {
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"url": "https://math.stackexchange.com/questions/2999739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does a closed right ideal of a C$^*$-algebra have a C$^*$-algebra? $A$ is an infinite dimensional C$^*$-algebra and $J\subset A$ is a closed right ideal. $A$ and $J$ are infinite dimensional(as a vector space). I want to find an infinite dimensional C$^*$-algebra subset of $J$. How can I find it?
I know an infinite dim... | You can't do that in general. Many C$^*$-algebras are simple. In such a case if $B\subset J$ is a C$^*$-algebra, then
$$
B=B\cap B^*\subset J\cap J^*=\{0\},
$$
since $J\cap J^*$ is a closed ideal.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2999914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can area of rectangle be greater than the square of its diagonal?
Q: A wall, rectangular in shape, has a perimeter of 72 m. If the length of its diagonal is 18 m, what is the area of the wall ?
The answer given to me is area of 486 m2. This is the explanation given to me
Is it possible to have a rectangle of diago... | No. Using Pythagoras and a simple inequality we get
$$d^2=a^2+b^2\geq 2ab\geq ab$$
If $a,b$ are the sides and $d$ the diagonal
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3000024",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "50",
"answer_count": 12,
"answer_id": 5
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Prove that a sequence is divergent (By definition - Epsilon-N Way) First, this is the question:
Prove (using epsilon-N definition) that the sequence $ a_n = \left<\sqrt{n}\right> $ is divergent.
Note: $ \left<x\right> = x- \lfloor x \rfloor$
My question:
I proved it by splitting it into cases: $L=0$ and $ L \neq 0 $
I ... | For each $n \ge 1$ you have
$$\left( n + 1 - \frac 1n \right)^2 < n^2 + 2n < (n+1)^2$$
so that
$$ n + 1 - \frac 1n < \sqrt{n^2 + 2n} < n+1$$
and consequently
$$1 - \frac{1}{n} < \langle \sqrt{n^2 + 2n} \rangle < 1$$
for all $n$. Thus $L = \lim a_n$, if it exists, must equal $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3000222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Help calculating $\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$ I need some help calculating this limit:
$$\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$$
I know it's equal to 1 but I have no idea how to get there. Can anyone give me a tip? I can't use l'Ho... | By Lagrange's theorem, $a>b>0$ ensures $\sqrt{a}-\sqrt{b} = (a-b)\frac{1}{2\sqrt{c}}$ with $c\in(b,a)$.
If we let $a=x+\sqrt{x}$ and $b=x-\sqrt{x}$ we get
$$ \sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}} = \frac{2\sqrt{x}}{2\sqrt{c}},\quad c\in(x-\sqrt{x},x+\sqrt{x})$$
and since $\sqrt{x\pm\sqrt{x}}=\sqrt{x}(1+o(1))$ the outcome... | {
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"timestamp": "2023-03-29T00:00:00",
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} |
Evaluate $\lim \limits_{n \to \infty\ } \Biggl( \frac{2,7}{(1+\frac{1}{n})^n}\Biggr)^n=0?$ $\lim \limits_{n \to \infty\ } \Biggl( \frac{2,7}{(1+\frac{1}{n})^n}\Biggr)^n$
I would like to replace $(1+\frac{1}{n})^n$ by $e$, and then $\frac{2,7}{e}<1$, so $\lim \limits_{n \to \infty\ } \Biggl( \frac{2,7}{(1+\frac{1}{n})^n... | You're right – one cannot replace only a part of an expression with its limit.
The simplest way consists in determining the limit of the log, using Taylor's formula at order $12$:
\begin{align}
n\log(2.7)-n^2\log\Bigl(1+\frac1n\Bigr)&=n\log(2.7)-n^2\biggl(\frac1n-\frac1{2n^2}+o\Bigl(\frac1{n^2}\Bigr)\biggr) \\
&=n\log... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3000438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
Solving $\int_{0}^{\infty} \frac{\sin(x)}{x^3}dx$ In my attempt to solve the this improper integral, I employed a well known improper integral (part of the Borwein family of integrals):
$$ \int_{0}^{\infty} \frac{\sin\left(\frac{x}{1}\right)\sin\left(\frac{x}{3}\right)\sin\left(\frac{x}{5}\right)}{\left(\frac{x}{1}\rig... |
$$-\int_{0}^{\infty} \frac{\sin\left(\frac{7}{15}x\right)}{x^3}\:dx + \int_{0}^{\infty} \frac{\sin\left(\frac{13}{15}x\right)}{x^3}\:dx +
\int_{0}^{\infty} \frac{\sin\left(\frac{17}{15}x\right)}{x^3}\:dx -
\int_{0}^{\infty} \frac{\sin\left(\frac{23}{15}x\right)}{x^3}\:dx
= \frac{2\pi}{15}$$
You cannot expand the inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3000733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Four dice are thrown simultaneously
Four dice are thrown simultaneously. The probability that $4$ and $3$ appear on two of the dice given that $5$ and $6$ appear on the other two dice is:
a) $1/6$
b) $1/36$
c) $12/51$
d) None of these
Since the events are independent, I feel the probability is $1/6 \times 1/6 = 1/36... | The tricky part of the solution is to find the number of outcomes such that 2 dice land $5$ and $6$; it can be done using inclusions/exclusions.
Let $\Omega$ be the set of all outcomes, of size $6^4$.
Let $S_5$ be the set of outcomes that contain no $5$'s, of size $5^4$.
Let $S_6$ be the set of outcomes that contain no... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3000843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that in $\triangle ABC$ ,$QP\parallel BC$
In the triangle $\triangle ABC$ , the point $M$ is between $B$ and $C$. And also the lines $MP$ and $MQ$ are bisectors of $\angle AMC$ and $\angle AMB$. It means that: $$\angle AMP=\angle PMC$$
$$\angle AMQ=\angle QMB$$
and
$$BM=MC$$
So now the puzzle tells us to prove ... | By the Angle Bisector Theorem, $$\frac{AQ}{QB}=\frac{AM}{MB}\text{ and }\frac{AP}{PC}=\frac{AM}{MC}\,.$$
Since $M$ is the midpoint of $BC$, we have $MB=MC$, whence
$$\frac{AQ}{QB}=\frac{AP}{PC}\,.$$
Therefore, $PQ\parallel BC$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3000966",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a better concept than expectation for one time play? Given a simple lottery game like
*
*Guess the right (random generated) number $\in [0,1000]$.
*Stake = 1€
*Win= 2001€
the expected outcome is $\frac{1}{1001}\cdot2001 + \frac{1000}{1001}\cdot(-1) = 1$.
Hence in the limit, you will win 1€ per play. So ... | For your first part:
The mathematical concept you are looking for is "most probable outcome". The most probable outcome is $-1$. Why would we need another term to describe the most probable outcome? The term we have is perfectly fine. It's descriptive, not too long, and very understandable.
For the second part:
The con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3001074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Systems of Linear Equations: Antifreeze Drain and Replace Problem
The radiator in your car contains 4 gallons of antifreeze and water. the mixture is 45% antifreeze. How much of the mixture should be drained and replaced with pure antifreeze in order to have a 60% antifreeze solution? Setup as a system of linear equat... | x = amount drained from radiator.
0.45(4 - x) + x = 0.6×4.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3001250",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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General solution or approximate solution Is there a known general or approximate explicit solution for $\xi$ in
$$(1+\xi)^m (1-\xi)^n = C$$
where $m$ and $n$ positive fractions and $C$ being constant?
| Since $m$ and $n$ are fractions, you can take the LCM of their
denominators and write
$$
\left\{ \matrix{
m = {p \over L}\quad n = {q \over L}\quad \left| {\;p,q,L \in N} \right. \hfill \cr
\left( {1 + x} \right)^{\,p} \left( {1 - x} \right)^{\,q} = C^{\,L} \hfill \cr} \right.
$$
so that we can always reduce th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3001346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 1
} |
Gauge transformation of differential equations I This is a follow-up question to Gauge transformation of differential equations. .
Let $y(x)$ be a solution to the following ODE:
\begin{eqnarray}
y^{''}(x) + a_1(x) y^{'}(x)+a_0(x) y(x)=0
\end{eqnarray}
Now define:
\begin{equation}
g(x):= \frac{y(x)+ r(x) y^{'}(x)}{r(x) ... | You can think further about e.g. the effect the below approach apply on HEUN-type ODEs, or some superposition approaches with e.g. Solutions in terms of the hypergeometric functions etc.
Hopefully someone can get challenge on some quite advanced ODEs like:
$\dfrac{d^2u}{dr^2}+\left(\dfrac{1}{2(r+1)}+\dfrac{1}{2(r-1)}-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3001530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find a formula a_n Find a formula an for the nth term of the arithmetic sequence whose first term is $a1=1$ such that $a_{n+1} - a_n=17$ for $n≥1$.
I am not sure on the process for solving this. Is it simply solving for $a_n$ so it would give me the result $a_n = a_{n-1} + 17$
| Observe that
$$
a_n=a_1+\sum_{k=1}^{n-1} (a_{k+1}-a_k) \quad (n>1)
$$
by telescoping sum whence
$$
a_n=1+17(n-1)\quad (n>1)
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3001814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Determine all homomorphisms from $Q$ to $Q_{>0}^\times$. This question is from a past year paper: Determine all homomorphisms from $Q$ to $Q_{>0}^\times$.
Let $Q$ denote the group of rationals under addition.
Let $Q_{>0}^\times$ denote the group of positive rationals under multiplication.
(a) Determine all homomorphism... | The reasoning for the first question is correct : for $a^{\frac mn}$ to remain rational for all $m,n$, we must have $a = 1$. Hence any such homomorphism is trivial.
For the other direction, any element of $\mathbb Q^+_{>0}$ is of the form $2^{n_1}3^{n_2}5^{n_3}7^{n_4}...$ where $n_i$ is an eventually zero sequence of i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3001923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is polynomial in general the same as polynomial function? The algebra text book says, a polynomial in one variable over $\mathbb{R}$ is given by,
$$f(x)= a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1x + a_0$$
Where $x$ is an unknown quantity which commutes with real numbers, called "indeterminate".
So I have a few question... | No, a polynomial is not a function. However, for each polynomial $p(x)=a_0+a_1x+\cdots+a_nx^n$ you may consider the polynomial function$$\begin{array}{rccc}p\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&a_0+a_1x+\cdots+a_nx^n.\end{array}$$And distinct polynomials will be associated with distinct functions. How... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3002099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
If $z = cis(2k\pi/5)$, $z \neq 1$, then what is $(z+1/z)^2+(z^2 + 1/z^2)^2=$? question 20, part c in the picture:
I substituted the first time as $4 \cos^2(2k \pi/5)$ and the second term as $4 \cos^2(4k \pi/5)$, and then tried writing one term in terms of the other using the identity $\cos 2a = 2 \cos^2 a- 1$. I even t... | The sum is $4+z^2+z^{-2}+z^4+z^{-4}$.
Show that it equals $4+z+z^2+z^3+z^4$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3002199",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Prove $V^+\oplus V^-=V \Longleftrightarrow f^2=1_V$ Let $V$ be a $\mathbb{R}$ vector space, let $f\in \operatorname{End}(V)$. We define subsets of $V$ as follows:
$V^+=\{v\in V:f(v)=v\}$ and $V^-=\{v\in V:f(v)=-v\}$
We know that $V^+$ and $V^-$ are vector subspaces of V, and that their intersection is the zero vector.
... | Note that the polynomial $X^2-1$ kills $f$, and this factors as $(X-1)(X+1)$. Now we can write $\frac 1 2 ((X+1)-(X-1)) = 1$, and we observe that for any $v$, this means we can then write
$v= v^+ + v^-$ where $v^+ = \frac 12 (f(v)+v)$ and $v^- = \frac 12 ((f(v)-v)$. Then check that this gives the desired decomposition.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3002432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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bilinear transformation $\phi U\times V\to W$ such that $Im(\phi)=\{\phi(u,v): u\in U, v\in V\}$ is not a subspace of $W$ Find a bilinear transformation $\phi U\times V\to W$ such that $Im(\phi)=\{\phi(u,v): u\in U, v\in V\}$ is not a subspace of $W$
I truly don't have an idea otherwise to brute force lots of tries and... | As far as I know there is no technique, but you might want to consider the case $U=V=\Bbb{R}^2$ and the map $\phi$ that sends a pair to the four coordinate products. That is to say
$$\phi:\ \Bbb{R}^2\times\Bbb{R}^2\ \longrightarrow\ \Bbb{R}^4: ((x_1,y_1),(x_2,y_2))\ \longmapsto\ (x_1x_2,x_1y_2,y_1x_2,y_1y_2).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3002589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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What is an example of a proof by minimal counterexample? I was reading about proof by infinite descent, and proof by minimal counterexample. My understanding of it is that we assume the existance of some smallest counterexample $A$ that disproves some proposition $P$, then go onto show that there is some smaller counte... | Fundamental examples in number theory arise via descent by (Euclidean) division with remainder (or, equivalently in $\Bbb Z$, by repeated subtraction), as in the following basic result.
Lemma $\ $ Let $\,S\,$ be a nonempty set of positive integers that is closed under subtraction $> 0,\,$ i.e. for all $ \,n,m\in S,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3002706",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
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Pushout of unital non commutative algebras I like to know if there is a pushout in the category of non commutative alegbras with unit and if the answer is "yes", who is it?
| Given unital $R$-algebras $A\leftarrow B\to C$, the pushout $A \star_B C$ is generated as an $R$-algebra by generators of $A$ and of $C$, modulo the union of the relations in $A$ and in $C$, as well as further relations identifying the two resulting images of each element of $B$. This immediately gives the canonical ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3002797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Norm of $f$ in a dual space space $(\ell^{\infty})^{*}$ For $x \in \ell^{\infty}$, let $f(x)= \sum_{n \in \mathbb{N}}x_{n}2^{-n}$ determine the norm of $f$ in $(\ell^{\infty})^{*}$ (the dual space of $(\ell^{\infty})$.
Notes: I think I need to related this back to the fact that $(\ell^{\infty})^{*}$ is isometrically is... | We know that $f$ is the image of $(a_n)$, where $a_n=2^{-n}$, under the isometric embedding $i:\ell^1\to (\ell^\infty)^*$, so we simply need to compute $\|f\|_1$. This is $1$.
Alternatively, we know that $\|f\|\ge |f(1)|=1$, and then you just need an upper bound by $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3002900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show $\lim_{x \to x_0^+} f(x)(x-x_0) =0$ when $f(\mathbb{R}) \subset \mathbb{R}^+$ & monotone increasing.
Show $\lim_{x \to x_0^+} f(x)(x-x_0) =0$ when $f(\mathbb{R}) \subset \mathbb{R}^+$ & monotone increasing.
Try
I need to show,
$$
\forall \epsilon >0, \exists \delta >0 : x \in (x_0, x_0 + \delta) \Rightarrow |f... | Use $M=f(x_0+1)$ and cosider $\delta=\min\{\frac{1}{2},\frac{\epsilon}{2M}\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3003033",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Is there a function $g$ such that $\int_0^1 x^n g(x) \, \mathrm d x$ is $1$ if $n=0$ and $0$ for $n \in \mathbb N_{\ge 1}$? Is there a function $g:[0,1]\to \mathbb R$ such that $$\int_0^1 x^n g(x) \, \mathrm d x$$ is equal to $1$ if $n=0$ and equal to $0$ for $n=1,2,3, \ldots$ ?
If there is, what would be an example of... | Assuming $g\in L^2(0,1)$ we are allowed to write
$$ g(x) \stackrel{L^2}{=} \sum_{n\geq 0} c_n P_n(2x-1),\qquad c_n=(2n+1)\int_{0}^{1}g(x)P_n(2x-1)\,dx.$$
Our constraints give $c_0=1$ and
$$ c_n = (2n+1)\int_{0}^{1}g(x)\left[(-1)^n+x q_n(x)\right]\,dx = (-1)^n (2n+1) $$
so, formally,
$$ g(x) \stackrel{L^2}{=}\sum_{n\geq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3003123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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A question about continuity of a specific function with probability measure Let $X$ be a compact metric space, and $\Theta$ be a finite space, endowed with their own $\sigma$-algebra.
Let $f \colon X \times \Theta \to \mathbb{R}$ be a Caratheodory function such that
(1) for each $x \in X$, the function $f(x, \cdot) \co... | I think your integral is
$$ h(x) = \sum_{\theta\in \Theta} f(x,\theta) \pi(\{\theta\}|x) \quad \forall x \in X $$
if $\pi(\{\theta\}|x) = \pi(\{\theta\})$ for all $x \in X$ then this is a sum of a finite number of functions that are continuous in $x$, and hence is continuous in $x$. More generally, if $\pi(\{\theta\}|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3003313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $\forall t \in \mathbb R$ the set $f^{-1}(\{t\})$ is a hyperplane of $X$ Exercise :
Let $X$ be a vector space and $f:X \to \mathbb R$ be a linear functional. Show that for all $t \in \mathbb R$, the set $f^{-1}(\{t\})$ is a hyperplane of $X$.
Attempt :
I have proved a slightly different example, showing th... | Assume that $f$ is nonzero. Then $f$ must be surjective, so $f^{-1}(t)$ is non empty. Pick any $v\in f^{-1}(t)$. Then we have $f^{-1}(t)=v + \ker f$: Clearly the right hand side is contained in the left hand side. Conversely, for any $w\in f^{-1}(t)$ we have $v-w \in \ker f$ by linearity of $f$, hence the left hand sid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3003440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Find the range of values which has no real solutions I would like to know how to solve the following problem:
Find the range of values of the parameter $m$ for which the equation $2x^2 - mx + m = 0$ has no real solutions.
I know I have to use the quadratic formula and the response is $0 < m < 8$.
But what I don't kno... | Guide:
*
*A quadratic equality has no real solution if and only the discriminant is negative.
*First, find the discriminant, find out when is it negative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3003610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Measure on a sigma-algebra with integral Let $\mu$ be a measure on $(X, \mathcal{A})$ and a measurable function $f:X \to \mathbb{R}, \ f \geq 0$.
Define $\mu_f(E): \mathcal{A} \to \mathbb{R}, \ \mu_f(E):=\int_E f \ d\mu$ for $E \in \mathcal{A}$.
How to prove that $\mu_f$ is a measure on the sigma-algebra $\mathcal{A}$?... | $$\mu_f(\varnothing)=\int_{\varnothing}f\;d\mu=\int\mathbf1_{\varnothing}f\;d\mu=\int0\;d\mu=0$$
Further be aware that we always have $\int\sum_{i=1}^{\infty}g_i\;d\mu=\sum_{i=1}^{\infty}\int g_i\;d\mu$ if the $g_i$ are measurable and nonnegative.
By disjoint and measurable $E_i$ moreover we have $\mathbf1_{\bigcup_{i=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3003719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Difficult inequality with three real variables For any real $e, t, \sigma$ such that
\begin{aligned}
\label{s}
0&<e<1\,,\\
0&<t<\pi\,,\qquad\qquad\qquad(1)\\
-\pi/2&\leqslant\sigma\leqslant\pi/2
\end{aligned}
the inequality
\begin{aligned}
(1-e^2\cos^2\sigma)\eta&+(1+e\cos(t-\sigma))(e\eta\cos\sigma\cos t
... | Continuity of the functions
\begin{align}
f(e, t, \sigma) &\stackrel{\mathrm{def}}{=} \mathrm{LHS}(2) =
\tau\eta+e\gamma\delta_1,\\
g(e, t, \sigma) &\stackrel{\mathrm{def}}{=} \mathrm{RHS}(2) =
\gamma\sqrt{1-e^2\delta_2^2},\\
h(e, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3003881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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The many ways in which to express a plane There are many ways to express a plane of $R^3$. I am focusing on two of them.
The first is the cartesian equation $Ax + By + Cz + D = 0$.
The second is to give two direction vectors $u$ and $v$ and a point $P$ of the plane.
My question is: how can I obtain two ortogonal direct... | From $$Ax+By+Cz+D=0$$
you get first the normal vector to the plane $n=(A,B,C)$.
then you can take
$$u=(0,C,-B)$$
and
$v$ as the vectorial product of $n$ by $u$.
To get the cartesian equation from two vectors $u,v$ and a point $P$,
$$det(PM,u,v)=0$$
with $M=(x,y,z)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3003986",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Generalized Schanuel Lemma This is on page 128, ex 3.15, of Rotman's AIHA,
(Schanuel) Let $B$ be a left $R$-module over some ring $R$ consider two exact sequences,
$$ 0 \rightarrow K \rightarrow P_n \rightarrow \cdots \rightarrow B \rightarrow 0 $$
$$ 0 \rightarrow K' \rightarrow P'_n \rightarrow \cdots \right... | I found the following proof in Lectures on Modules and Rings by T. Y. Lam.
We do an induction on $n$. Assume the claim is true for $n-1$. Write $f$ and $g$ for the arrows $P_{0}\to B$ and $Q_{0}\to B$.
Applying the usual version of Schanuel's lemma to the sequences
\begin{gather*}
0\to\ker f\to P_{0}\to B\to0,\\
0\to\k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3004103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Determine the number of integer solutions to $x_1 + x_2 + x_3 + x_4 = 19$, where $−5 \le x_i \le 10$ for all $1 \le i \le 4$ What I have so far:
Goal: Using the inclusion exclusion I want to find
$|\overline A_{1}\cap \overline A_{2} \cap \overline A_{3} \cap \overline A_{4}| = |U| - S_{1} + S_{2} - S_{3} + S_{4}$
$S_... |
I have incremented the values of i by 5 so that the range can start from zero like this: $
x_1+x_2+x_3+x_4=24$ with
$0≤x_i≤15$
Are you sure this is correct? You might need to check that inequality and the equation preceding it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3004323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
nuclear $C^*$ algebra If $(A_i)$ is a sequence of nuclear $C^*$ algebras,Is $\oplus_{c_0}A_i$ ($c_0$ direct sum)and $\prod A_i$($\ell ^\infty $ direct sum) also nuclear?
| Fix $a\in \bigoplus_nA_n$ and $\varepsilon>0$. For each $n$, there exist ucp maps $\varphi_n:A_n\to M_{k(n)}(\mathbb C)$ and $\psi_\varepsilon:M_{k(n)}(\mathbb C)\to A_n$ such that $\|\psi_n\circ\varphi_n(a_n)-a_n\|<\varepsilon$.
There is also $m$ such that $\|a_n\|<\varepsilon$ for all $n\geq m$. Write $a_0$ for the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3004484",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
What proportion of the quarter circle is shaded?
Interesting yet challenging quiz I found on a website. My answer is a $\frac{1}{ \sqrt{2}}$.
After I assumed the semicircle has radius $r\sin{45}$, where $r$ is the radius of the quarter circular part.
Any objections or comment?
| Let $R$ be the radius of the outer circle and $M=(r,r)$ be the center of the brown semidisc. Then $|OM|=\sqrt{2} r$ and therefore $R^2=3r^2$. The ratio of the areas then comes to ${2\over3}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3004641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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If $\lambda_n = \int_{0}^{1} \frac{dt}{(1+t)^n}$, for $n \in \mathbb{N}$, then $\,\lim_{n \to \infty} (\lambda_{n})^{1/n}=1.$ If $\displaystyle\lambda_n = \int_{0}^{1} \frac{dt}{(1+t)^n}$ for $n \in \mathbb{N}$. Then prove that $\lim_{n \to \infty} (\lambda_{n})^{1/n}=1.$
$$\lambda_n=\int_{0}^{1} \frac{dt}{(1+t)^n}= \f... | Actually,
$$
\int_0^1 \frac{dt}{(1+t)^n}=\left.\frac{1}{1-n}\frac{1}{(1+t)^{n-1}}\,\right|_0^1=\frac{1}{n-1}-\frac{2^{-n+1}}{n-1}
$$
and hence, for all $n>1$
$$
\frac{1}{2(n-1)}<\int_0^1 \frac{dt}{(1+t)^n}<\frac{1}{n-1}.
$$
Next, observe that
$$
\lim_{n\to\infty}\left(\frac{1}{2(n-1)}\right)^{1/n}=\lim_{n\to\infty}\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3004767",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to transform this limit expression as a limit of $e$ I have the following expression:
$\displaystyle\lim _{x\to \infty }\left(\dfrac{x+2}{\:x-6}\right)^\left(\dfrac{x}{\:4}\right)$
I'm studying Calculus I and our lector has shown us ways of transforming such limits to:
$\displaystyle\lim_{x\to \infty }\left(1+\frac... | You may proceed as follows:
*
*Set $y = x-6$
$$\left(\frac{x+2}{x-6} \right)^{\frac{x}{4}} = \left(1 +\frac{8}{y} \right)^{\frac{y+6}{4}} = \left(1 +\frac{2}{\frac{y}{4}} \right)^{\frac{y}{4}}\cdot \left(1 +\frac{8}{y} \right)^{\frac{3}{2}} \stackrel{y \to \infty}{\longrightarrow}e^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3004929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
An element of a set with a finite cover must be an element of at most two open intervals in a subcover? Prove:
If a set $A\subseteq\mathbb{R}$ has a cover consisting of a finite number of open intervals, then A has a subcover such that for each $x\in A$, x is an element of at most two of the open intervals in the subco... | What if you argued by contradiction? This is a super crude discussion on how I'm thinking one could proceed:
Suppose the statement is false. So assume that for every subcover $T'$, there exists an element $x\in A$ such that $x$ is in at least $3$ of the open intervals of the arbitrary subcover $T'$. Without loss of gen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3005012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Prove that in similar triangles ratio of correspondent medians is same as ratio of correspondent sides I had a math exam today about geometry and similar triangles.
One of our math puzzle wanted us to proves something. Now I’ll explain that for you and if you help me I won’t lose 2 points of my midterm exam! So imagine... | I think you have a reasonable idea here, but your proof is incomplete. In order to apply Thales' theorem in this way, you need to know that $A$, $O$, and $H$ are collinear, and you haven't given any reason why they should be.
Notice that you haven't ever used the fact that $O$ and $H$ are midpoints. This is what you wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3005145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $n$ is an integer , find all the possible values for $(8n+6,6n+3)$ I have got 2 questions which I could not solve:
1) if $n$ is an integer , find all the possible values for $(8n+6,6n+3)$
2)if $n$ is an integer, find all possible values of $(2n^2+3n+5,n^2+n+1)$
| Let $d=\gcd(8n+6,6n+3)$, then $$d\mid 8n+6$$
$$d\mid 6n+3$$
so $$d\mid 6(8n+6)-8(6n+3)= 12$$
so $d\in \{1,2,3,4,6,12\}$ Since $6n+3$ is odd $d$ can not be $2,4,6$ or $12$ so $d=1$ or $d=3$ (which is realised at $n=3k$ for some integer $k$)
For second one:
Let $d=\gcd(2n^2+3n+5,n^2 + n+1)$, then $$d\mid 2n^2+3n+5$$
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3005287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Explanation of the metric tensor Before anything, I just want to say that I'm studying mathematics/physics in a different language (Serbian) so some of the English terms that I use might be a bit wonky. Just ask if a term makes no sense.
So, this is the first time we're being introduced to the concept of a metric tenso... | What they mean is that $m_{ij}$ is the value of the scalar product of $v_i$ and $v_j$, so $\left<v_i,v_j\right> = m_{ij}$. They didn't "get" $m_{ij}$ from anywhere...it's just given to you, as the definition of the scalar product. Let's do a concrete example. Let's say for $\Bbb{R}^2$, with basis $e_1,e_2$. For example... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3005435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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I need help finding the general solution to the differential equation $y''(t)+7y'(t)=-14$ What I've tried:
I have the inhomogeneous differential equation:
$$y''(t)+7y'(t)=-14$$
I find the particular solution to be on the form $$kt$$
by inserting the particular solution in the equation
$$(kt)''+7(kt)'=-14$$
and isolatin... | You went wrong when you thought what Maple wrote is different from your solution in any significant way.
Maple has swapped the roles of $c_1$ and $c_2$ compared to you. And Maple's $c_1$ is seven times larger than your $c_2$ and has opposite sign, but since the constants are arbitrary anyways this doesn't matter. So yo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3005544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Exact Sequence of Galois Groups Let $E_1/F$, $E_2/F$ be Galois extensions. Then $E_1E_2/F$ and $E_1\cap E_2/F$ are Galois extensions. Supposedly there is a short exact sequence
$$1\to \mathrm{Gal}(E_1E_2/F) \xrightarrow{\varphi} \mathrm{Gal}(E_1/F)\times \mathrm{Gal}(E_2/F) \to \mathrm{Gal}(E_1\cap E_2/F) \to 1$$
where... | $(\sigma,\tau) \mapsto (\sigma - \tau)|_{E_1\cap E_2}$ works: it's surjective (take any $\sigma$ in the target, extend it to some $\bar{\sigma}$ on $E_1$ any way you like, and then $(\bar{\sigma},0)\mapsto \sigma$), and its kernel is the set of all pairs of maps which agree on $E_1\cap E_2$, which clearly includes the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3005663",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Trisecting $2\pi/5$, is this possible? I guess that the answer is no, even knowing that $cos(2\pi/5)$ is constructible since the $5$th root o unity is construtctible.
But when I use the trick for finding the minimal polynomial of $3\theta=2\pi/5$ I get that $\theta$ is the root of
$p(x)=4x^3 - 3x - cos(2\pi/5)$
and th... | The minimal polynomial for an $n$-th root of unity has degree $\phi(n)$ and the field has an abelian Galois group. The real subfield containing $y=2\cos(2\pi/n)$ has degree $\phi(n)/2$ is also abelian. Hence since $\phi(15)/2=4$ the field generated by $y$ is constructible. (In fact, the minimal polynomial for $y$ is $y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3005820",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
System of equations with three variables Characterize all triples $(a,b,c)$ of positive real numbers such that
$$ a^2-ab+bc = b^2-bc+ca = c^2-ca+ab. $$
This is the equality case of the so-called Vasc inequality. I think the answer is that $a=b=c$ or $a:b:c = \sin^2(4\pi/7) : \sin^2(2\pi/7) : \sin^2(\pi/7)$ and cyclic e... | If $c=0$ then $a^2-ab=b^2=ab$, which gives $a=b=c=0.$
Let $abc\neq0$ and $a=xb$.
Thus, from the first equation we obtain:
$$a^2-ab-b^2=(a-2b)c.$$
If $a=2b$ so $a=b=c=0$, which is impossible here.
Thus, $c=\frac{a^2-ab-b^2}{a-2b}$ and from the second equation we obtain:
$$b^2+\frac{(a-b)(a^2-ab-b^2)}{a-2b}=\frac{(a^2-ab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3005940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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How to find the Newton polygon of the polynomial product $ \ \prod_{i=1}^{p^2} (1-iX)$ How to find the Newton polygon of the polynomial product $ \ \prod_{i=1}^{p^2} (1-iX)$ ?
Answer:
Let $ \ f(X)=\prod_{i=1}^{p^2} (1-iX)=(1-X)(1-2X) \cdots (1-pX) \cdots (1-p^2X).$
If I multiply , then we will get a polynomial of deg... | Partial Answer: regarding the coefficients of the polynomial:
Fix one term in the brackets, say $Y=(1-5X)$. In order for the coefficient $5$ to contribute to $a_j$, we have to multiply $Y$ with $j-1$ other brackets, since this is the only way of getting a power of $j$ for $X$. This corresponds to choosing a subset $S ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3006046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How do we prove or visualize $[x+(x-2)]=[2+(x-2)]$ the same way we prove or visualize $0+5\mathbb Z = 5 + 5\mathbb Z$? Denote $\langle x-2\rangle$ as the principal ideal generated by $x-2$ in the polynomial ring $\mathbb R[x]$.
*
*$[x+\langle x-2\rangle]$ and $[2+\langle x-2\rangle]$ are elements of the quotient rin... | The analogy should be $5+5\mathbb Z=0+5\mathbb Z$ and $[x-2+\langle x-2 \rangle] = [0+\langle x-2 \rangle]$:
$$5+5\mathbb Z = 5+\{...,-5,0,5,...\} = 0+\{...,0,5,10,...\}=0+5\mathbb Z$$
or
$$5+5\mathbb Z = 5+\{5m\} = 0+\{5+5m\}=0+\{5(m+1)\}=0+\{5(n)\}$$
Similarly,
$$x-2+\langle x-2 \rangle = x-2+\{(x-2)(p)\} = 0+\{(x-2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3006193",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Lineintegral $\int_{\gamma}|z|^2dz$ over ellipse Let $a,b\in\mathbb{R}_{>0}$ and $\gamma: [0,2\pi]\rightarrow\mathbb{C},t\mapsto a\cos(t)+ib\sin(t)$
calculate the line integral
$\int_{\gamma}|z|^2dz$
My calculation turns out to be really ugly. Is there maybe a "nice" way to calculate this integral?
| The integrals are not ugly at all. You obtain
$$\int_\gamma|z|^2\>dz=\int_{\omega-\pi}^{\omega+\pi}\bigl(a\cos^2 t+b^2\sin^2 t\bigr)(-a\sin t+ib\cos t)\>dt\ ,$$
whereby $\omega$ can be chosen at will, due to periodicity. Choose $\omega:=0$ for the real part, then $\omega:={\pi\over2}$ for the imaginary part, and note t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3006483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}(2H_{2k}+H_k)=\frac{\pi^3}{32}-2G\ln2$ How to prove
$$\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}(2H_{2k}+H_k)\stackrel ?=\frac{\pi^3}{32}-2G\ln2,$$
where $G$ is the Catalan's constant.
Attempt
For the first sum,
$$\sum_{k=1}^{\infty}\frac{(-1)^k}{(2k+1)^2}H_{2k}=\Re\... | The series involving $H_k$ and $H_{2k}$ can be studied in a similar way: since
$$ \frac{-\log(1-x)}{1-x} = \sum_{n\geq 1} H_n x^{n} $$
we have $ \frac{-\log(1+x^2)}{1+x^2} = \sum_{n\geq 1} H_n(-1)^n x^{2n} $ and
$$ \sum_{k\geq 1}\frac{(-1)^k}{(2k+1)^2}H_k = \int_{0}^{1}\frac{\log(1+x^2)\log(x)}{1+x^2}\,dx$$
boils down ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3006595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
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Find coordinate in first quadrant which tangent line to $x^3-xy+y^3=0$ has slope 0 Find coordinate in first quadrant which tangent line to $x^3-xy+y^3=0$ has slope 0
First, I do implicit differentiation:
$\frac{3x^2-y}{x-3y^2}=y'$
so I look at the numerator and go hmmm if i put in (1,3) that makes the slope 0.
But then... | You solved only half of the problem. You have that the derivative is $0$, but you also need to use the fact that the point is on the graph of your line. You have two equations with two unknowns. Since they are not linear equations, you might have multiple solutions.
Just plug in $y=3x^2$ into your original equation, an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3006952",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Does $\int_{-\infty}^\infty f(x) dx < \infty$ where $f \ge 0$ implies $\sup_{x \in \mathbb R} f(x)Question. Does $\,\int_{-\infty}^\infty\, f(x)\, dx < \infty,\,$ where $\,f \ge 0,\,$ imply that $\,\,\mathrm{ess}\sup_{x \in \mathbb R}\, f(x)<\infty\,? $
My attempt
Since $f \ge 0$ and $\int_{-\infty}^\infty f(x) dx < \i... | Neither if $f$ is continue, let $\rho:\mathbb R\rightarrow [0, +\infty[$ a continue non identically zero function such that $\rho(x)\neq 0$ only inside $[-1, 1]$ and
$$
\int^{+\infty}_{-\infty}\rho(x)dx = 1
$$
Then let
$$
f(x)=\sum^{+\infty}_{i=1}2^i\rho\left(i+4^ix\right)
$$
$f$ is continuous and
$$
\int^{+\infty}_{-\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Why does the "$i$" in $f=u+vi$ not affect the measurability of $f$? I stumbled across the following result in Rudin's Real and Complex Analysis:
He says in part (c) that "the complex case then follows from (a) and (b)." To me, this is saying that because $f$ and $g$ are sums of real-valued functions, i.e. $$f=u_1+iv_... | Per part (b), $u_1, u_2, v_1$, and $v_2$ are real-measurable functions on $X$, and thus $u_1 + u_2$ and $v_1 + v_2$ are real-measurable functions on $X$. Per part (a), this implies that $f+g = (u_1 + u_2) + i(v_1 + v_2)$ is a complex-measurable function on $X$. In particular, the $i$ is baked in to the statement of par... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given that $X \sim \operatorname{Binomial}(n,p)$, Find $\mathbb{E}[X(X-1)(X-2)(X-3)]$
Given that $X \sim \operatorname{Binomial}(n,p)$, Find $\mathbb{E}[X(X-1)(X-2)(X-3)]$.
It is suggested that I can transform it into
\begin{align}
\mathbb{E}[X(X-1)(X-2)(X-3)]
&=\sum_{k=0}^n k(k-1)(k-2)\mathbb{P}\{X=k\}\\
&=\sum_{k... | Start as suggested, and write down what the probability mass function (pmf) of the Binomial actually is:
$$\begin{align*}
\mathbb{E}[X(X-1)(X-2)(X-3)]
&= \sum_{k=0}^n k(k-1)(k-2)(k-3)\mathbb{P}\{X=k\}\\
&= \sum_{k=4}^n k(k-1)(k-2)(k-3)\mathbb{P}\{X=k\}\\
&= \sum_{k=4}^n k(k-1)(k-2)(k-3)\binom{n}{k}p^k(1-p)^{n-k}\\
&= \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Symmetry group of equilateral triangle I have read at some places that the symmetry of equilateral triangle is C3v
as well as some places mention it to be D3.
The group tables for these two groups differ, hence they are not isomorphic.
Yet both these groups define symmetry of same shape.
Please, explain what is going... | The symmetry group of an equilateral triangle is the dihedral group $D_3$ with $6$ elements. It is a non-abelian group and hence isomorphic to $S_3$, since $C_6$ is abelian and there are only two different groups of order $6$. So there is one and only one symmetry group of the regular $3$-gon up to isomorphism. In part... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Term for this concept in category theory? Suppose we have three objects $X,Y,Z$, and a morphism $m:X\to Y$.
Moreover, this morphism has the following property:
For any morphism $f:Z\to Y$, there exists a morphism $f_X:Z\to X$, such that $m\circ f_X=f$.
Intuitively, this seems to me to capture the notion that “any infor... | The natural way to name this property is "the object $Z$ has the left lifting property with respect to the morphism $m$". Indeed, if the category has an initial object, then the property you mentioned is equivalent to the left lifting property between $i_Z$ and $m$, where $i_Z$ is the unique morphism from the initial o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Does this specific SO(4) matrix have to be block-diagonal? So I have a specific real $4\times4$ matrix $\mathbf{P}$ given by
\begin{align}
\mathbf{P}=
\begin{pmatrix}
p_{11} & -p_{21} & p_{13} &-p_{23}\\ p_{21} & p_{11} & p_{23} & p_{13}\\p_{31} & -p_{41}& p_{33} & -p_{43}\\p_{41} & p_{31} & p_{43} & p_{33}.
\end{pmat... | Skip to the end for the big reveal, or read through this for the "how I got there" version.
Let's rewrite that as
\begin{align}
\mathbf{P}=
\begin{pmatrix}
a & -b & p &-q\\
b & a & q & p\\
c & -d& r & -s\\
d & c & s & r.
\end{pmatrix},
\end{align}
Orthogonality of the first and third and first and 4th columns tells ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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What is $\lim_{x \to 3} (3^{x-2}-3)/(x-3)(x+5)$ without l'Hôpital's rule? I'm trying to solve the limit $\lim_{x \to 3} \frac{3^{x-2}-3}{(x-3)(x+5)}$
but I don't know how to proceed: $\lim_{x \to 3} \frac{1}{x+5}$ $\lim_{x \to 3} \frac{3^{x-2}-3}{x-3}$ = $1\over8$ $\lim_{x \to 3} \frac{\frac{1}{9}(3^{x}-27)}{x-3}$
Any... | Perhaps use definition of $3^x$ ... namely $3^x = e^{x\log 3}$. Instaed of $x \to 3$ write $y=x-3$ and do $y \to 0$.
$$
\lim_{x \to 3} \frac{3^{x-2}-3}{(x-3)(x+5)} = \lim_{y \to 0}\frac{3^{y+1}-3}{y(y+8)}
=3 \lim_{y \to 0}\frac{3^{y}-1}{y(y+8)}
\\
3^y = \exp(y\log 3) = 1 + y\log 3 + o(y)
\\
3^y-1 = y\log 3 + o(y)
\\
y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3007918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Stuck with matrix equation I'm trying to solve a matrix equation problem and I can't work out the correct form for the equation for it to be valid.
The matrices given are:
A= $\begin{bmatrix}
1 & -1 & 3\\
4 & 1 & 5\\
0 & 0 & 0\\
\end{bmatrix}$, B= $\begin{bmatrix}
1 & -1\\
3 & 6\\
1 & 0\\
\end{b... | $X=(A+I)^{-1}(C-B )=\begin{bmatrix}\frac14 &\frac18 &\frac{-11}{8}\\\frac{-1}{2} &\frac14 &\frac14\\0 &0 &1\end{bmatrix}\begin{bmatrix}-2 &1\\2 &0\\-1 &1\end{bmatrix}=\begin{bmatrix}\frac98 &\frac{-9}{8}\\\frac54 &\frac{-1}{4}\\-1 &1\end{bmatrix}$
We get two independent solutions for $X$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3008049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let $A$ and $B$ be well-ordered sets, and suppose $f:A\to B$ is an order-reversing function. Prove that the image of $f$ is finite. Let $A$ and $B$ be well-ordered sets, and suppose $f:A\to B$ is an
order-reversing function. Prove that the image of $f$ is finite.
I started by supposing not. Then we must have that the ... | Let $C=f(A)$ the image of $f$ with the order induced by $B$. Every non-empty subset of $C$ has minimum and maximum. This implies that every element in $C$ distinct from the minimum has immediate predecessor and every element distinct from the maximum has immediate successor. Let $c_1$ the minimum of $C$. For every natu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3008162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Given $5$ white balls, $8$ green balls and $7$ red balls. Find the probability of drawing a white ball then a green one. Given $5$ white balls, $8$ green balls and $7$ red balls in an urn. Find out the probability to draw a white ball and then a green one if the drawing is done consecutively and after drawing the ball ... | First you calculate the probability of getting a white ball. $$P(white)=\frac{Number\ of\ white\ balls}{Total\ number\ of\ balls} = \frac{5}{20}$$
Then the probability of getting a green ball is $$P(green)=\frac{Number\ of\ green\ balls}{Total\ number\ of\ balls} = \frac{8}{20}$$
Because you put the ball back in the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3008279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
For $n,k \in {\mathbb{Z}}^{+}$ (excluding $n=1$), does $\frac{(n+k)!}{n!}$ ever equal $n!$ While investigating an integer sequence, I came across the following two OEIS entries:
*
*A094331: Least k such that n! < (n+1)(n+2)(n+3)...(n+k).
*A075357: a(n) = smallest k such that (n+1)(n+2)...(n+k) is just >= n!.
The ge... | With the exception of $N=1$, $N!$ is never a square. This is because, by Bertrand's Postulate, there is always a prime between $N$ and $\lfloor N/2\rfloor$ (to be precise, a prime $p$ satisfying $\lfloor N/2\rfloor\lt p\le N$), and such a prime can only divide $N!$ once. So if we take $N=n+k$ with positive integers $n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3008628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Eigenspaces of an orthogonal projection So for an orthogonal projection $P:V\rightarrow U $, the task is to find the eigenvalues and eigenspaces of P.
I have found that $\lambda = 0,1$
Then $E_0 = ker(P) = U^{\perp}$
but for $E_1$ i'm not sure
$E_1 = ker(P-Id)$
I feel like the solution for this is somewhat trivial and ... | Let $x \in E_1$. Then, $x=P(x) \in P(V)$. Hence, $E_1 \subseteq P(V)$.
Let, $y \in P(V)$. Then, $y=P(X)$ for some $x \in V$. Hence, $P(y)=P^2(x)=P(x)=y$ i.e. $y \in E_1$. $\therefore P(V) \subseteq E_1$.
Combining, we get, $P(V)=E_1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3008736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that $A_{1},B_{1},C_{1}$ are on a straight line which is perpendicular to $OI$ Triangle $ABC$ has a circumcircle $(O)$ and a incircle $(I)$. The external bisectors of $\angle A, \angle B,\angle C$ cut $BC,CA,AB$ at $A_{1},B_{1},C_{1}$. Show that $A_{1},B_{1},C_{1}$ are on a straight line which is perpendicular to ... |
PART ONE: Let us first prove that points $A_1$, $B_1$ and $C_1$ are collinear.
By applying law of sines to triangle $\triangle ACC_1$:
$${AC_1 \over AC}={\sin\angle ACC_1 \over \sin\angle AC_1C}={\sin(90^\circ-\frac\gamma2) \over \sin\angle AC_1C}\tag{1}$$
By applying law of sines to triangle $\triangle BCC_1$:
$${BC... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3008882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding the intersection points of a line with a cube The following is an old high school exercise:
Let $A = (5, 4, 6)$ and $B = (1,0,4)$ be two adjacent vertices of a cube in $\mathbb{R}^3$. The vertex $C$ lies in the $xy$-plane.
a) Compute the coordinates of the other vertices of the cube such that all $x$- and $z$-... | a) finding the edges and vertices
Given
$$
A = \left( {5,4,6} \right)\quad B = \left( {1,0,4} \right)
$$
then
$$
\overline {BA} = 6\quad \mathop {BA}\limits^ \to = \left( {\matrix{ 4 \cr 4 \cr 2 \cr } } \right)\quad
{\bf u} = {{\mathop {BA}\limits^ \to } \over {\overline {BA} }}
={1 \over 3} \left( {\matrix{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3009020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Degree of polynomial interpolating the primes The polynomial $p_3(x)$ passes through the points
$(1,2), (2,3), (3,5)$, where $2,3,5$ are the first three primes:
$$
p_3(x) = \frac{x^2}{2}-\frac{x}{2}+2 \;.
$$
Similarly, one can form an interpolating polynomial $p_n(x)$ that
passes through the first $n$ primes.
For examp... | The degree of $p_n(x)$ is always $n-1$. The proof is by induction.
Note that $p_1(x) = 2$ has degree $0$. Now assume that $p_{n}(x)$ has degree $n-1$. We want to prove that $p_{n+1}(x)$ has degree $n$. Assume otherwise, so $p_{n+1}(x)$ also had degree at most $n-1$. Then since $p_{n+1}(x)$ and $p_n(x)$ agree on the fir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3009163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
} |
Proving such a function is always constant Let $I \subset R$ be an interval. Let $f : I \to R$ be a continuous function.
Assume that $I := [a, b]$. Assume that for all $c, d \in [a, b]$ such that $c < d$, there exists $e \in [c,d]$ such that $f(e) = f(a)$ or $f(e) = f(b)$. Prove that $f$ is a constant.
Consider this st... | The set $D:=f^{-1}(f(a))\cup f^{-1}(f(b))$ is closed by continuity of $f$ and dense in $I$ by the special property. Clearly $D$ does not intersect the open set $I\setminus D$. By defnition of dense, this means that $I\setminus D$ is empty. Hence $I=D$. This makes $I$ the union of the two non-empty closed sets $f^{-1}(f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3009279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Probability of choosing the basket-i Here is the question:
There are N+1 baskets 0,1,2,3,...,N. Where each basket i has i white balls and N-i black balls.
We choose randomly a basket and take out a ball after another with returning.
a)If we know that the first two withdraws we got a black and a white ball (not necessar... | another approach:(using bayes`s equation)
let:
A=choosing basket i
B=two two ball withdrawn are blak and white
P(A|B)=$ \frac{P(B|A)*P(A)}{P(B)} $= $ \frac{2*( \frac{i}{N} * \frac{N-i}{N})*( \frac{1}{N+1}) }{\frac{1}{2}} $ =4 * $\frac{i*(N-i)}{N^3+N^2}$
But the sum of i-es does not equal 1.I think I am wrong somewhere... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3009436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Solve $\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$ I am having great problems in solving this:
$$\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$$
I am trying to solve this for hours, no solution in sight. I tried so many ways on my paper here, which all lead to nonsense or to nowhere. I conclude... | Consider the function $f(x)=x^{1/3}$. By the mean value theorem there's a number $y\in (n, n+\sqrt n)$ such that
$$
f(n+\sqrt n) - f(n) = f'(y)(n+\sqrt n - n)= \frac{y^{-2/3}}{3}\sqrt n<n^{-2/3}\sqrt n=n^{-1/6}\to 0.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3009543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 4
} |
How to prove $ \left\{ t^2,t^3 \right\}$ equals the vanishing set of $y^2-x^3$? Exercise 3.2 in Hartshorne is about proving that morphisms of varieties may be underlain by homeomorphisms without being isomorphisms of varieties. The morphism in consideration is $\varphi:t\mapsto (t^2,t^3)$ where the image is to be thoug... | Suppose $x^3=y^2$. If $x=0$ then $y=0$ so that $(x,y)=(t^2,t^3)$ for $t=0$.
Otherwise $x\ne0$. We can then define $t=y/x$. Then $t^2=y^2/x^2=x^3/x^2=x$
and $t^3=t^2t=x(y/x)=y$. So $(x,y)=(t^2,t^3)$.
Not a square or cube root in sight!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3009721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maclaurin series for $\arctan^{2}(x)$ I have a question here that requires me to find the Maclaurin series expansion of
$\arctan^{2}(x)$. Now I know how to find it for $\arctan(x)$, by taking the derivative, expanding it into a series, and integrating back (given x is in the interval of uniform convergence), But apply... | We can try to obtain the series in the following way:
$$f(x)=\arctan^2 x=x^2 \int_0^1 \int_0^1 \frac{du~dv}{(1+x^2u^2)(1+x^2v^2)}$$
It's easier to consider:
$$g(x)=\int_0^1 \int_0^1 \frac{du~dv}{(1+x^2u^2)(1+x^2v^2)}$$
Let's use partial fractions:
$$\frac{1}{(1+x^2u^2)(1+x^2v^2)}=\frac{u^2}{(u^2-v^2)(1+x^2u^2)}-\frac{v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3009865",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 0
} |
A definite integral inequality Suppose $f(x)$ has continuous derivative on $[-\pi, \pi]$, $\,f(-\pi)=f(\pi)\,$ and $\,\int_{-\pi}^{\pi}\, f(x)\, dx=0$. Then prove that:
$$
\int_{-\pi}^{\pi} [\,f'(x)]^2\, dx \ge \int_{-\pi}^{\pi} f^2(x)\, dx,
$$
with the equal sign holding if and only if $\,f(x)=A\cos x+B\sin x$.
Thanks... | If $f: [-\pi,\pi]\to\mathbb R$ is continuously differentiable, then $f$ and $f'$ are also $L^2$, and hence they are expressed as
$$
f(x)=\sum_{k\in\mathbb Z}\hat f_k\,\mathrm{e}^{ikx} \quad \text{while}\quad
f'(x)=\sum_{k\in\mathbb Z}ik\,\hat f_k\,\mathrm{e}^{ikx},
$$
and we have that
$$
\int_{-\pi}^\pi|\,f(x)|^2\,dx=2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3009987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Existence of a vector $v$ in $V$ such that the $T$-annihilator of $v$ is the minimal polynomial for $T$.
Definition: $T$-annihilator of a vector $\alpha$ (denoted as $p_\alpha$) is the unique monic polynomial which generates the ideal
such that $g(T)\alpha = 0$ for all $g$ in this ideal.
I'm trying to prove the belo... | Let's first show the result when the minimal polynomial has the form $p^n$, with $p$ irreducible.
We know that $p(T)^n=0$ , $p(T)^{n-1}\neq0$ so there exist a vector $\alpha \in V$ such that $p(T)^{n-1}\alpha\neq0$, $p(T)^n\alpha=0$. Thus the T-annihilator $g$ of $\alpha$ divides $p^n$ and since $p(T)^{r}\alpha\neq0$ f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3010121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Find the area of the surface formed by revolving the given curve about $(i)x$-axis and $(i)y$-axis
Q:Find the area of the surface formed by revolving the given curve about $(i)x-axis$ and $(i)y-axis$
$$x=a\cos\theta ,y=b\sin\theta,0\le\theta\le2\pi$$
About $x-$axis is, $S=2\pi\int_0^{2\pi}b\sin\theta \sqrt{a^2(\sin\t... | The limits of integration need some correction. While finding the surface area about the $x$-axis, $x$ ranges from $-a$ to $a\implies\theta$ ranges from $\pi\rightarrow 2\pi$, not $0\rightarrow 2\pi$. For the surface area about the $y$-axis, $\theta$ ranges from $-\pi/2 \rightarrow +\pi/2$, or from $3\pi/2\rightarrow 2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3010259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proof that sum of two subspaces is another subspace $U_1,U_2$$⊂V$ be subspaces of V (a vector space). Define the subspace sum of $U_1,$ and $U_2$ be defined as the set:
$U_1 + U_2$ $=$ {$u_1 + u_2 : u_1 ∈ U_1, u_2 ∈ U_2$}.
Let $A$ denote the set $U_1+ U_2$
A is a subspace if a meets all the criteria of a subspace, that... | Yes your proof is fine as a minor issue I would prefer to present the second and third properties in that way
*
*$u_1 + u_2\in U_1 + U_2 \implies a(u_1 + u_2)=au_1 + au_2$ with $au_1\in U_1 + U_2$ and $au_2\in U_2 $
and
*
*$(u_1 + u_2) + (w_1 + w_2)\in U_1 + U_2 \implies (u_1 + u_2) + (w_1 + w_2)=(u_1+w_1)+(u_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3010404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$\iint e^{x-y}$ over the triangle with vertices at $(0,0),(1,3),(2,2)$ $\displaystyle\iint e^{x-y}$ over the triangle with vertices at $(0,0),(1,3),(2,2)$
I tried the following change of variables. let $u=x-y$, $v=3x-y$. In $(u,v)$, we get the triangle with vertices at $(0,0),(-2,0),(0,4)$.The Jacobian i calculated is ... | First of all, I assume you mean the region $R$ enclosed by the triangle, not the triangle itself. Otherwise the integral is just zero.
Instead of bothering with Jacobians, which I personally find super annoying, it may be best simply to break it up into two pieces. Thus we have
$$\iint\limits_R e^{x-y}\;dA=\int_0^1\i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3010507",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability that the first 2 letters are consonants when the letters of the word 'equilibrium' are rearranged Here's what I tried:
The total number of ways is $\dfrac{11!}{3!\cdot2!}$.
The consonants can be together in $\dfrac{6(5)(9!)}{3!\cdot 2!}$ ways. When I divide, I get $\dfrac 3{11}$ but the answer is $\dfrac2{1... | It's all about trying possible paths/combinations and their probabilities. For the first example, proceed sequentially:
*
*Probability that the first letter that you pick is a consonant: $\frac{5}{11}$ (5 consonants over eleven letters)
*Probability that the second letter that you pick is a consonant: $\frac{4}{10}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3010711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is there an intuitive way to understand $Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$? After a lot of effort I discovered some pattern that the function:
$$Pair(x,y) = \frac{(x+y)(x+y+1)}{2} + x$$
essentially zigzags along the grid with $\mathbf N$ vs $\mathbf N$ (natural numbers). So intuitively, given any P(x,y) we follow... | Zigzagging is kind of obvious if you look at it in the right way. Pair of $x$ and $y$ is equal to the $(x+y)$th triangle number, plus $x$. The $y$ term is guaranteeing that the "plus $x$" term on the end will not cause the traversal to run more than the full length of the hypotenuse. So $y$ is effectively telling you h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3010935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to show that $\langle a,b \mid aba^{-1}ba = bab^{-1}ab\rangle$ is not Abelian? I'd like to show that
$$
G = \langle a,b \mid aba^{-1}ba = bab^{-1}ab\rangle
$$
is non-Abelian.
I have tried finding a surjective homomorphism from $G$ to a non-Abelian group, but I haven't found one. The context is that I would like to ... | Since you tag this with knot-theory and knot-invariants, it looks like you are trying to show the fundamental group of the knot complement $S^3-4_1$ is nonabelian.
One of the "obvious" things to try is Fox $n$-coloring, since they yield a homomorphism (usually surjective) to a dihedral group. There is a $5$-coloring (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
How many selections of three cards can be made from cards bearing the letters of the word EXAMINATION if ...? $11$ cards each bear a letter, and together they can be made to spell the word "EXAMINATION". $3$ Cards are selected from the $11$ cards and the order of selection is not important. Find how many selections can... | The first part of your answer is right. You have 3 ways to choose from $A, I, N$. However, after that you're only considering picking from a group of letters which occur only once, in this case $E, X, M, T, O$. This would lead you to miss a case like $AAN$ where the second letter could also come from one of the repeate... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding a formula for a sequence Let $a_n$ be a sequence such that $a_0=5$ and $a_{n}+a_{n+1}=3$ for all $n$ greater than $0$.
I defined a function $f(x)$ whose coefficients are same of that mentioned sequence.
I was able to get $$f(x)=\displaystyle \sum_{n=0}^\infty [ 5 (-1)^n x^n + 3 (-1)^n x^{n+1} + 3 x^{2n+1} ] $$
... |
Let an be a sequence such that $a_0=5$ and $a_n+a_{n+1}=3$ for all $n$ greater than $0$.
First note: $a_0+a_1=3 \Rightarrow 5+a_1=3 \Rightarrow a_1=-2$.
It looks you are trying to use the generating function $f(x)=\sum_{n=0}^{\infty} a_nx^n$. Here are the steps:
$$\sum_{n=0}^{\infty} a_nx^{n+1}+\sum_{n=0}^{\infty} a_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3011336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
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