Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Invertability of a linear transformation Given $T : \mathbb{R}^3 \to \mathbb{R}^3$ such that $T(x_1,x_2,x_3) = (3x_1,x_1-x_2,2x_1+x_2+x_3)$ Show that $(T^2-I)(T-3I) = 0$.
Solution 1: I can very easily write down the matrix representing $T$, calculate each of the terms in each set of parenthesis, and multiply the two ma... | Let $A=T^2-I$ and $B=T-3I$. Proving that $A\cdot B = 0$ is the same as showing that $\text{ker}(A\cdot B)=\text{ker}(A)\cup\text{ker}(B)=\Bbb R^3$.
An easy calculation shows that $\text{ker}(B)=\langle (1,-2,3) \rangle$. Now observe that
$$
\{v_1=(1,-2,3),v_2=(0,1,0),v_3=(0,0,1)\}
$$
is a basis for $\Bbb R^3$. Then it ... | {
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"timestamp": "2023-03-29T00:00:00",
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What is $\int\frac{dx}{\sin x}$? I'm looking for the antiderivatives of $1/\sin x$. Is there even a closed form of the antiderivatives? Thanks in advance.
| Hint: Write this as $$\int \frac{\sin (x)}{\sin^2 (x)} dx=\int \frac{\sin (x)}{1-\cos^2(x)} dx.$$ Now let $u=\cos(x)$, and use the fact that $$\frac{1}{1-u^2}=\frac{1}{2(1+u)}+\frac{1}{2(1-u)}.$$
Added: I want to give credit to my friend Fernando who taught me this approach.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/331307",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 2
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Arrange $n$ men and $n$ women in a row, alternating women and men. A group contains n men and n women. How many ways are there to arrange these people in a row if the men and women alternate?
I got as far as:
There are $n$ [MW] blocks. So there are $n!$ ways to arrange these blocks.
There are $n$ [WM] blocks. So there ... | You added where you needed to multiply. You're going to arrange $n$ men AND $n$ women in a row, not $n$ men OR $n$ women, so you've got $n!$ ways to do one task and $n!$ ways to do the other, making $(n!)^2$.
But after that there's this issue: Going from left to right, is the first person a man or a woman? You can do... | {
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"source": "stackexchange",
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The epsilon-delta definition of continuity As we know the epsilon-delta definition of continuity is:
For given $$\varepsilon > 0\ \exists \delta > 0\ \text{s.t. } 0 < |x - x_0| < \delta \implies |f(x) - f(x_0)| < \varepsilon $$
My question: Why wouldn't this work if the implication would be:
For given $$\varepsilon > ... | Consider the implications of using this definition for any constant function (which should all be continuous, if any function is to be continuous).
*
*In particular, for $c \in \mathbb{R}$ consider the constant function $f(x) = c$. Given $x_0 \in \mathbb{R}$, taking $\varepsilon = 1$, note that for any $\delta > 0$ ... | {
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Missing dollar problem This sounds silly but I saw this and I couldn't figure it out so I thought you could help.
The below is what I saw.
You see a top you want to buy for $\$97$, but you don't have any money so you borrow $\$50$ from your mom and $\$50$ from your dad. You buy the top and have $\$3$ change, you give ... | top = 97
from each parent: 48.50
48.50 + 48.50 = 97
remainder from each parent = 1.50 x 2 = 3.00
3.00 - (1.00 to each parent = 2) - (1.00 for yourself = 1) = 0
owed to each parent: 48.50 [97]
giving one dollar to each parent: 49.50 [99]
check your pocket for the remaining dollar. [100]
| {
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"timestamp": "2023-03-29T00:00:00",
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Question 2.1 of Bartle's Elements of Integration The problem 2.1 of Bartle's Elements of Integration says:
Give an exemple of a function $f$ on $X$ to $\mathbb{R}$ which is not
$\boldsymbol{X}$-mensurable, but is such that the function $|f|$ and
$f^2$ are $\boldsymbol{X}$-mensurable.
But, if one define $f^{+}:= \... | Take any non-empty measurable set $U$ in $X$, and any non-empty non-measurable set $V$ in $U$. Let $f$ be the function that sends $U$ to $1$, except for the points in $V$, which are sent to $-1$. In other words, $f(x) = 1_{U\backslash V} - 1_V$.
Now $|f| =f^2 = 1_U$.
I see now that Bunder has written a similar idea (th... | {
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Cantor set on the circle Draw a Cantor set C on the circle and consider the set A of all the chords between points of C. Prove that A is compact.
| $C$ is compact as it's closed and bounded. Then, $A$ is compact as it's the image of the compact set $C\times C\times [0,1]$ under the continuous map $\phi: {\Bbb R}^2\times {\Bbb R}^2\times [0,1]\to {\Bbb R}^2$ given by $\phi(x,y,\lambda)= \lambda x + (1-\lambda )y$.
| {
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"timestamp": "2023-03-29T00:00:00",
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What Kind of Geometric Object Is Represented By An Equation I'm trying to understand the solution to a particular problem but can't seem to figure it out.
What kind of geometric object is represented by the equation:
$$(x_1, x_2, x_3, x_4) = (2,-3,1,4)t_1 + (4,-6,2,8)t_2$$
The answer is: a line in (1 dimensional subspa... | The first one is a line because the vector $(4,-6,2,8)$ is twice the vector $(2,-3,1,4)$. Thus your collection of points is just the collection of all points of the form $(t_1+2t_2)(2,-3,1,4)$. So it is the collection of all points of the form $t(2,-3,1,4)$. The multiples of a non-zero vector are just a line through t... | {
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Limit involving a hypergeometric function I am new to hypergeometric function and am interested in evaluating the following limit:
$$L(m,n,r)=\lim_{x\rightarrow 0^+} x^m\times {}_2F_1\left(-m,-n,-(m+n);1-\frac{r}{x}\right)$$
where $n$ and $m$ are non-negative integers, and $r$ is a positive real constant.
However, I do... | OK, let's start with an integral representation of that hypergeometric:
$$_2F_1\left(-m,-n,-(m+n);1-\frac{r}{z}\right) \\= \frac{1}{B(-n,-m)} \int_0^1 dx \: x^{-(n+1)} (1-x)^{-(m+1)} \left[1-\left(1-\frac{r}{z}\right)x\right]^m$$
where $B$ is the beta function. Please do not concern yourself with poles involved in gam... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove inequality: $28(a^4+b^4+c^4)\ge (a+b+c)^4+(a+b-c)^4+(b+c-a)^4+(a+c-b)^4$ Prove: $28(a^4+b^4+c^4)\ge (a+b+c)^4+(a+b-c)^4+(b+c-a)^4+(a+c-b)^4$ with $a, b, c \ge0$
I can do this by: $EAT^2$ (expand all of the thing)
*
*$(x+y+z)^4={x}^{4}+{y}^{4}+{z}^{4}+4\,{x}^{3}y+4\,{x}^{3}z+6\,{x}^{2}{y}^{2}+6\,{
x}^{2}{z}^{2... | A nice way of tackling the calculations might be as follows:$$~$$
Let $x=b+c-a,y=c+a-b,z=a+b-c.$ Then the original inequality is just equivalent with
$$\frac74\Bigl((x+y)^4+(y+z)^4+(z+x)^4\Bigr)\geq x^4+y^4+z^4+(x+y+z)^4.$$
Now we can use the identity
$$\sum_{cyc}(x+y)^4=x^4+y^4+z^4+(x+y+z)^4-12xyz(x+y+z),$$
So that it... | {
"language": "en",
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Why is the tensor product constructed in this way? I've already asked about the definition of tensor product here and now I understand the steps of the construction. I'm just in doubt about the motivation to construct it in that way. Well, if all that we want is to have tuples of vectors that behave linearly on additio... | When I studied tensor product, I am lucky to find this wonderful article by Tom Coates. Starting with the very trivial functions on the product space, he explains the intuition behind tensor products very clearly.
| {
"language": "en",
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How to find non-cyclic subgroups of a group? I am trying to find all of the subgroups of a given group. To do this, I follow the following steps:
*
*Look at the order of the group. For example, if it is $15$, the subgroups can only be of order $1,3,5,15$.
*Then find the cyclic groups.
*Then find the non cyclic gro... | In the $n=15=3\cdot 5$ case, recall that every group of order $p$ prime is cyclic. This leaves you with the subgroups of order $15$. How many are there?
Of course, this is not as easy in general. For general finite groups, the classification is a piece of work. Finite Abelian groups are easier, as they fall in the clas... | {
"language": "en",
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Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$. Given the function
$$
f(x) = \left\{\begin{array}{cc}
e^{- \frac{1}{x^2}} & x \neq 0
\\
0 & x = 0
\end{array}\right.
$$
show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.
So I have to show that nth derivative is always equal to zero... | What about a direct approach?:
$$f'(0):=\lim_{x\to 0}\frac{e^{-\frac{1}{x^2}}}{x}=\lim_{x\to 0}\frac{\frac{1}{x}}{e^{\frac{1}{x^2}}}\stackrel{\text{l'Hosp.}}=0$$
$$f''(0):=\lim_{x\to 0}\frac{\frac{2}{x^3}e^{-\frac{1}{x^2}}}{x}=\lim_{x\to 0}\frac{\frac{2}{x^4}}{e^\frac{1}{x^2}}\stackrel{\text{l'Hosp.}\times 2}=0$$
........ | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find all the symmetries of the $ℤ\subset ℝ$. Find all the symmetries of the $ℤ\subset ℝ$.
I'm not sure what is meant with this.
My frist thought was that every bijection $ℤ→ℤ$ is a symmetry of $ℤ$.
My second thought was that if I look at $ℤ$ as point on the real line, then many bijections would screw up the distance ... | Both of you thoughts are correct, but in different contexts. If we regard $\mathbb{Z}$ as just a set with no extra structure then the symmetries of that set are just the bijections, (as in this case, we are thinking as $\mathbb{Z}$ as a bag of points). If on the other hand, we add the extra structure of the distances, ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Does π depends on the norm? If we take the definition of π in the form:
π is the ratio of a circle's circumference to its diameter.
There implicitly assumed that the norm is Euclidian:
\begin{equation}
\|\boldsymbol{x}\|_{2} := \sqrt{x_1^2 + \cdots + x_n^2}
\end{equation}
And if we take the Chebyshev norm:
\begin{... | Under Euclidean metric there are number of constants that their values coincide and are collectively denoted by the symbol $\pi$.
How ever some of the coinciding values are independent from the metric and some are coupled with the metric and the geometry under consideration.
In your example, would the calculation of ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is this proof using the pumping lemma correct? I have this proof and it goes like this:
We have a language $L = \{\text{w element of } \{0,1\}^* \mid w = (00)^n1^m \text{ for } n > m \}$.
Then, the following proof is given:
There is a $p$ (pumping length) for $L$. Then we have a word $w = (00)^{p+1}1^p$ and $w$ elemen... | Yes, the proof is bogus, as Brian M. Scott's answer expertly explains.
It is easier to prove that the reverse of your language isn't regular, or (by minor adjusting the proof of the lemma to place the pumped string near the end, not the start) that it isn't regular.
| {
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Is the Picard Group countable? Is the Picard group of a (smooth, projective) variety always countable?
This seems likely but I have no idea if it's true.
If so, is the Picard group necessarily finitely generated?
| No. Even for curves there is an entire variety which parametrizes $Pic^0$(X). It is called the jacobian variety. The jacobian is g dimensional (where g is the genus of the curve), so in particular if g > 0 and you are working over an uncountable field the picard group will be uncountable.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Set Theory and surjective function exercise Let $E$ a set and $f:E\rightarrow P(E)$ any function. If $$A=\{a\in E:a\notin f(a)\}$$ Prove that $A$ has no preimage under $f$.
| Suppose $\exists a\in E$, $f(a)=A$. Then, is $a\in f(a)$?
If $a\in f(a)$, that means $a\notin A$, contradicts to $f\in f(a)=A$.
If $a\notin f(a)$, that means $a\in A$.
Thus, such $a$ does not exists i.e. $f(a)$ has no preimage.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/332453",
"timestamp": "2023-03-29T00:00:00",
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Understanding a Theorem regarding Order of elements in a cyclic group This is part of practice midterm that I have been given (our prof doesn't post any solutions to it) I'd like to know whats right before I write the midterm on Monday this was actually a 4 part question, I'm posting just 1 piece as a question in its o... | For the theorem in your textbook, note that $(a^d)^{m/d}=a^m$ and $(a^m)^{b}=a^d$ where $b$ is the integer that $bm+cn=d$, which you can find using Euclidean Algorithm.
This means they can generate each other. Therefore, the groups they generate are the same.
You can use this to prove the question.
The question says $... | {
"language": "en",
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Can SAT instances be solved using cellular automata? I'm a high school student, and I have to write a 4000-word research paper on mathematics (as part of the IB Diploma Programme). Among my potential topics were cellular automata and the Boolean satisfiability problem, but then I thought that maybe there was a connecti... | If you could find an efficient way to solve SAT, you'd become very rich and famous. That's not likely to happen when you're still in high school. What you might be able to do, though, is get your cellular automaton to go through all possible values of the variables, and check the value of the Boolean expression for e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $S$ be a linear operator on $W=P_3(\mathbb R)$ defined by $S(p(x)) = p(x) - \frac {dp(x)} {dx}$ Let S be a linear operator on W = $P_3(\Bbb R)$ defined by S(p(x)) = $p(x) - $$\frac {dp(x)} {dx}$.
(a) Find nullity (S) and rank (S)
(b) Is S an isophormism? If so, write down a formula for $S^{-1}$.
Please help me corr... | (a) Looks fine, but instead of saying "since there are 4 column in the matrix" (what matrix?), I would say "since $\dim P_3(\mathbb{R})=4$".
(b) You have already found $S^{-1}$, haven't you? According to your calculation, $S^{-1}(h+ex+fx^2+gx^3)=gx^3 + (f+3g)x^2 + (e+2f+6g)x + (h+e+2f+6g)$.
BTW, you can show that $S$ i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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defining equation of general projective line Let $\mathbb{P}$ be a projective $n$-space.
For $p=[a_0,\cdots,a_n], q=[b_0,\cdots,b_n]$ I know that the line pass through $p$ and $q$ is defined by the set $\{ [xa_0+yb_0,\cdots, xa_n+yb_n] | [x,y] \in \mathbb{P}^1\}$
I wonder the defining equation of this projective line. ... | I am not sure if this is what you want. If $n=2$, you can think of the projective points $p,q$ are two vectors in the corresponding vector space. Then the projective line through these two points corresponds to the class of planes parallel to these two vectors. By taking cross product of these two vectors, you then fin... | {
"language": "en",
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If $C\subseteq B \subseteq A$ then $A\backslash C = (A\backslash B)\cup (B\backslash C)$ Is it true that if $C\subseteq B \subseteq A$ then $A\backslash C = (A\backslash B)\cup (B\backslash C)$? By drawing Venn diagrams this clearly seems so, but I can't seem to prove it.
Similarly, is it true that if $C\subseteq A$ an... | Let $x\in A-C$. Then $x\in A$ and $x\not\in C$. If $x\in B$, then $x\in B$ and $x\not\in C$, so $x\in B-C$, while if $x\not\in B$, then $x\in A$ and $x\not\in B$, so $x\in A-B$. This proves that $A-C\subset (A-B)\cup (B-C)$.
Now suppose that $C\subset B\subset A$. Let $y\in (A-B)\cup (B-C)$. If $y\in A-B$, then $y\in A... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find lim sup $x_n$ Let $x_n = n(\sqrt{n^2+1} - n)\sin\dfrac{n\pi}8$ , $n\in\Bbb{N}$
Find $\limsup x_n$.
Hint: lim sup $x_n = \sup C(x_n)$.
How to make it into a fraction to find the cluster point of $x_n$?
| Expand: $n · \left( \sqrt{n²+1} - n \right) = n · \tfrac{1}{\sqrt{n²+1} + n}$ for all $n ∈ ℕ$.
Since $\tfrac{\sqrt{n²+1} + n}{n} = \sqrt{1+\tfrac{1}{n²}} + 1 \overset{n → ∞}{\longrightarrow} 2$, you have $n · \left( \sqrt{n²+1} - n \right) \overset{n → ∞}{\longrightarrow} \tfrac{1}{2}$.
Now $x_n = n · \left( \sqrt{n²+... | {
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An increasing probability density function? Could anyone come up with a probability density function which is:
*
*supported on [1,∞) (or [0,∞))
*increasing
*discrete
| I was having the same question, but in a bounded environment, so the distribution would be increasing with support a and b. It grows like a slow exponential, actually is the the distribution of the following:
Take N repetitions of 4 uniformly generated values between A and B. Make the histogram of the max in each repet... | {
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Calculating determinant of a block diagonal matrix Given an $m \times m$ square matrix $M$:
$$
M = \begin{bmatrix}
A & 0 \\
0 & B
\end{bmatrix}
$$
$A$ is an $a \times a$ and $B$ is a $b \times b$ square matrix; and of course $a+b=m$. All the terms of A and B are known.
Is there a way of calculating determinant of $M$ b... | Hint: It's easy to prove that
$$\det M=\det A\det B$$
| {
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Intersection of irreducible sets in $\mathbb A_{\mathbb C}^3$ is not irreducible I am looking for a counterexample in order to answer to the following:
Is the intersection of two closed irreducible sets in $\mathbb
A_{\mathbb C}^3$ still irreducible?
The topology on $\mathbb A_{\mathbb C}^3$ is clearly the Zariski o... | Choose any two irreducible plane curves, they will intersect in a finite number of points.
| {
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"url": "https://math.stackexchange.com/questions/332973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the general solution of $ y'''- y'' - 9y' +9y = 0 $ Find the general solution of $ y'''- y'' - 9y' +9y = 0 $
The answer is
$y=c_{1}e^{-3x}+c_{2}e^{3x}+c_{3}e^{x}$
how do i approach this problem?
| Try to substitute $e^{\lambda x}$ in the equation and solve the algebraic equation in $\lambda$ as is usually done for second order homogeneous ODEs.
| {
"language": "en",
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convergence and limits How can I rewrite $\frac{1}{1+nx}$ and prove it's absolute convergence as $n \rightarrow \infty$? Given $\epsilon > 0$, should I define $f_n(x)$ and $f(x)$?
Any help is hugely appreciated. Thank you
| It seems you're being given $$f_n(x)=\frac{1}{1+nx}$$ and asked to find its (pointwise) limit as $n \to \infty$. Note that if $x>0$, $\lim f_n(x)=0$. If $x=0$, $f_n(x)=1$ for each $n$, so $\lim f_n(x)=1$. There is a little problem when $x<0$, namely, when $x=-\frac 1 n $, so I guess we just avoid $x<0$. Thus, you can s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Find the angle between the main diagonal of a cube and a skew diagonal of a face of the cube I was told it was $90$ degrees, but then others say it is about $35.26$ degrees. Now I am unsure which one it is.
| If we assume the cube has unit side length and lies in the first octant with faces parallel to the coordinate planes and one vertex at the origin, then the the vector $(1,1,0)$ describes a diagonal of a face, and the vector $(1,1,1)$ describes the skew diagonal.
The angle between two vectors $u$ and $v$ is given by:
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Why does the power rule work? If $$f(x)=x^u$$ then the derivative function will always be $$f'(x)=u*x^{u-1}$$
I've been trying to figure out why that makes sense and I can't quite get there.
I know it can be proven with limits, but I'm looking for something more basic, something I can picture in my head.
The derivativ... | This may be too advanced for you right now,
but knowing about the derivative of the log function
can be very helpful.
The basic idea is that
$(\ln(x))' = 1/x$,
where $\ln$ is the natural log.
Applying the chain rule,
$(\ln(f(x))' = f'(x)/f(x)$.
For this case,
set $f(x) = x^n$.
Then $\ln(f(x)) = \ln(x^n) = n \ln(x)$.
Ta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 1
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Question about integration (related to uniform integrability) Consider a probability space $( \Omega, \Sigma, \mu) $ (we could also consider a general measure space). Suppose $f: \Omega -> \mathbb{R}$ is integrable. Does this mean that
$ \int |f| \chi(|f| >K) d\mu $ converges to 0 as K goes to infinity? N.B. $\chi$ is... | Let $f \in L^1$, then we have $|f| \cdot \chi(|f|>k) \leq |f| \in L^1$ and $$|f| \cdot \chi(|f|>k) \downarrow |f| \cdot \chi(|f|=\infty)$$ (i.e. it's decreasing in $k$) since $\chi(|f|>n) \leq \chi(|f|>m)$ for all $m \leq n$. Thus, we obtain by applying dominated convergence theorem $$\lim_{k \to \infty} \int |f| \cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Formula for Product of Subgroups of $\mathbb Z$, Problem What is the product of $\mathbb{Z}_2$ and $\mathbb{Z}_5$ as subgroups of $\mathbb{Z}_6$?
Since $\mathbb{Z}_n$ is abelian, any subgroup should be normal. From my understanding of the subgroup product, this creates the following set: $\{ [0], [1], [2], [3], [4], [5... | @Ittay Weiss, made you a complete illustration, but for noting a good point about the subgroups of $\mathbb Z$, we memorize:
If $m|n$ then $n\mathbb{Z}\leq m\mathbb{Z}$ (or $n\mathbb{Z}\lhd m\mathbb{Z}$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/333374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Need to prove the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges I need to prove that the sequence $a_n=1+\frac{1}{2^2}+\frac{1}{3^2}+\cdots+\frac{1}{n^2}$ converges. I do not have to find the limit. I have tried to prove it by proving that the sequence is monotone and bounded, but I am hav... | Notice that $ 2k^2 \geq k(k+1) \implies \frac{1}{k^2} \leq \frac{2}{k(k+1)}$.
$$ \sum_{k=1}^{\infty} \frac{2}{k(k+1)} = \frac{2}{1 \times 2} + \frac{2}{2 \times 3} + \frac{2}{3 \times 4} + \ldots $$
$$ \sum_{k=1}^{\infty} \frac{2}{k(k+1)} = 2\Big(\, \Big(1 - \frac{1}{2}\Big) + \Big(\frac{1}{2} - \frac{1}{3} \Big) + \Bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333417",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
"answer_count": 8,
"answer_id": 5
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Finding the Expected Value and Variance of Random Variables This is an introductory math finance course, and for some reason, my prof has decided to ask us this question. We haven't learnt this type of material yet, and our textbook is close to NO help. If anyone has a clue on how to solve this problem, PLEASE help me!... | Well, just to expand on mne__povezlo's answer, I guess a more complete (and useful, in your case) formula for variance would be:
$$\mathrm{Var}\left(\sum_{i=1}^{n}a_{i}X_{i}\right)=\sum^n_{i=1}a_{i}^2\mathrm{Var}X_{i}+2\underset{1\le{i}<j\le{n}}{\sum\sum}a_{i}a_{j}\mathrm{Cov}\left(X_i,X_{j}\right)$$
Now what's left is... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Rings and unity The set $R = {([0]; [2]; [4]; [6]; [8])}$ is a subring of $Z_{10}$. (You do not need to
prove this.) Prove that it has a unity and explain why this is surprising.
Also, prove that it is a field and explain why that is also surprising.
This sis a HW Question.
The unity is not [0] is it ?? Could I get a h... | Notice that $[6]\times[a]=[a]$ for $a=0,2,4,6,$ and $8$. See if you can do the rest!
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Evaluating $\int \frac{1}{{x^4+1}} dx$ I am trying to evaluate the integral
$$\int \frac{1}{1+x^4} \mathrm dx.$$
The integrand $\frac{1}{1+x^4}$ is a rational function (quotient of two polynomials), so I could solve the integral if I can find the partial fraction of $\frac{1}{1+x^4}$. But I failed to factorize $1+x^4$.... | Without using fractional decomposition:
$$\begin{align}\int\dfrac{1}{x^4+1}~dx&=\dfrac{1}{2}\int\dfrac{2}{x^4+1}~dx
\\&=\dfrac{1}{2}\int\dfrac{(x^2+1)-(x^2-1)}{x^4+1}~dx
\\&=\dfrac{1}{2}\int\dfrac{x^2+1}{x^4+1}~dx-\dfrac{1}{2}\int\dfrac{x^2-1}{x^4+1}~dx
\\&=\dfrac{1}{2}\int\dfrac{1+\dfrac{1}{x^2}}{x^2+\dfrac{1}{x^2}}~d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333611",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Find $\int_0^\infty \frac{\ln ^2z} {1+z^2}{d}z$ How to find the value of the integral $$\int_{0}^{\infty} \frac{\ln^2z}{1+z^2}{d}z$$ without using contour integration - using usual special functions, e.g. zeta/gamma/beta/etc.
Thank you.
| Here's another way to go:
$$\begin{eqnarray*}
\int_0^\infty dz\, \dfrac{\ln ^2z} {1+z^2}
&=& \frac{d^2}{ds^2} \left. \int_0^\infty dz\, \dfrac{z^s} {1+z^2} \right|_{s=0} \\
&=& \frac{d^2}{ds^2} \left. \frac{\pi}{2} \sec\frac{\pi s}{2} \right|_{s=0} \\
&=& \frac{\pi^3}{8}.
\end{eqnarray*}$$
The integral $\int_0^\infty... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 0
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Modular homework problem Show that:
$$[6]_{21}X=[15]_{21}$$
I'm stuck on this problem and I have no clue how to solve it at all.
| Well, we know that $\gcd\ (6,21)=3$ which divides $15$. So there will be solutions:
$$
\begin{align}
6x &\equiv 15 \pmod {21} \\
2x &\equiv 5 \pmod 7
\end{align}
$$
because that $2\times 4\equiv 1 \pmod 7$, thus:
$$
\begin{align}
x &\equiv 4\times 5 \pmod 7\\
&\equiv 6 \pmod 7
\end{align}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Every invertible matrix can be written as the exponential of one other matrix I'm looking for a proof of this claim: "every invertible matrix can be written as the exponential of another matrix".
I'm not familiar yet with logarithms of matrices, so I wonder if a proof exists, without them. I'll be happy with any proof... | I assume you are talking about complex $n\times n$ matrices. This is not true in general within real square matrices.
A simple proof goes by functional calculus. If $A$ is invertible, you can find a determination of the complex logarithm on some $\mathbb{C}\setminus e^{i\theta_0}[0,+\infty)$ which contains the spectrum... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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How to combine Bézier curves to a surface? My aim is to smooth the terrain in a video game. Therefore I contrived an algorithm that makes use of Bézier curves of different orders. But this algorithm is defined in a two dimensional space for now.
To shift it into the third dimension I need to somehow combine the Bézier ... | I would not use Bezier curves for this. Too much work to find the end-points and you end up with a big clumsy polynomial.
I would build a linear least squares problem minimizing the gradient (smoothing the slopes of hills).
First let's split each pixel into $6\times 6$ which will give the new smoothed resolution (just ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/333991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why this function is not uniformly continuous? Let a function $f: \mathbb R \rightarrow \mathbb R$ be such that $f(n)=n^2$ for $n \in \mathbb N$.
Why $f$ is not uniformly continuous?
Thanks
| Suppose it is. i.e. for every $\varepsilon>0$ there is some $\delta>0$ such that if $|x-y| \leq \delta$ then $|f(x)-f(y)| \leq \varepsilon$. Any interval $[n,n+1]$ can be broken into at most $\lceil \frac{1}{\delta} \rceil$ intervals with length$<\delta$ using some partition $x_0=n<x_1<\ldots<n+1=x_k$. Using the triang... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to evaluate $\int^\infty_{-\infty} \frac{dx}{4x^2+4x+5}$? I need help in my calculus homework guys, I can't find a way to integrate this, I tried use partial fractions or u-substitutions but it didn't work.
$$\int^\infty_{-\infty} \frac{dx}{4x^2+4x+5}$$
Thanks much for the help!
| *
*Manipulate the denominator to get $(2x+1)^2 + 4 = (2x+1)^2 + 2^2$.
*Let $u = 2x+1 \implies du = 2 dx \implies dx = \frac 12 du$,
*$\displaystyle \frac 12 \int_{-\infty}^\infty \dfrac{du}{u^2 + (2)^2} $
*use an appropriate trig substitution which you should recognize:
$$ \int\frac{du}{{u^2 + a^2}} = \frac{1}{a}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Bernstein Polynomials and Expected Value The first equation in this paper
http://www.emis.de/journals/BAMV/conten/vol10/jopalagzyl.pdf
is:
$$\displaystyle B_nf(x)=\sum_{i=0}^{n}\binom{n}{i}x^i(1-x)^{n-i}f\left(\frac{i}{n}\right)=\mathbb E f\left(\frac{S_{n,x}}{n}\right)$$
where $f$ is a Lipschitz continuous real functi... | $$\mathbb E(g)=\sum_{i}\mathbb P\left(S_{n,k}\right)g(i)$$
where:
$$g(S_{n,x})=f\left(\frac{S_{n,x}}{n}\right)$$
Therefore we have:
$$\mathbb E\left(f\left(\frac{S_{n,x}}{n}\right)\right)=\sum_{i=0}^{n}\mathbb P(S_{n,x}=i)f\left(\frac{i}{n}\right)$$
But $$\mathbb P(S_{n,x}=i)=\binom{n}{i}x^i(1-x)^{n-i}$$
Therefore
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Need to prove the sequence $a_n=(1+\frac{1}{n})^n$ converges by proving it is a Cauchy sequence
I am trying to prove that the sequence $a_n=(1+\frac{1}{n})^n$ converges by proving that it is a Cauchy sequence.
I don't get very far, see: for $\epsilon>0$ there must exist $N$ such that $|a_m-a_n|<\epsilon$, for $ m,n>... | We have the following inequalities:
$\left(1+\dfrac{1}{n}\right)^n = 1 + 1 + \dfrac{1}{2!}\left(1-\dfrac{1}{n}\right)+\dfrac{1}{3!}\left(1-\dfrac{1}{n}\right)\left(1-\dfrac{2}{n}\right) + \ldots \leq 2 + \dfrac{1}{2} + \dfrac{1}{2^2} + \dots =3$
Similarly,
$
\begin{align*}
\left(1-\dfrac{1}{n^2}\right)^n &= 1 - {n \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Does there exist a bijective mapping $f: \Delta\to\mathbb{C} $ Does there exist a bijective mapping $f: \Delta\to\mathbb{C} $?
Yet I have not found such example. Is it false (why?),please help me.
$ \Delta $ is the unit disk in $\mathbb {C}$.
| I would do this radius by radius. So in case we were talking about the open disk, all we’d need to do is get a nice map from $[0,1\rangle$ onto $[0,\infty\rangle$, like $x/(1-x)$. In case we were talking about the closed disk, you need a (discontinuous) map from $[0,1]$ onto $[0,1\rangle$, and follow it by the map you ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Expected value of a minimum
Under a group insurance policy, an insurer agrees to pay 100% of the medical bills incurred during the year by employees of a company, up to a max of $1$ million dollars. The total bills incurred, $X$, has pdf
$$
f_X(x) =
\begin{cases}
\frac{x(4-x)}{9}, & \text{for } 0 < x < 3\\
0& \text{... | For amounts greater than one million, they still pay out a million.
Let $x$ be the total bills in millions. For $x \in [0,1]$ the payout is $x$, for $x \in [1,\infty)$, the payout is $1$. Hence the payout as a function of $x$ is $p(x) = \min(x,1)$, and you wish to compute $Ep$.
\begin{eqnarray}
Ep &=& \int_0^\infty p(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334494",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Evaluate the Integral using Contour Integration (Theorem of Residues) $$
J(a,b)=\int_{0}^{\infty }\frac{\sin(b x)}{\sinh(a x)} dx
$$
This integral is difficult because contour integrals normally cannot be solved with a sin(x) term in the numerator because of singularity issues between the 1st quadrant to the 2nd quadra... | You can change the integration $\int_{0}^{\infty}$ to $\frac{1}{2}\int_{-\infty}^{\infty}$ . Since $\cos$ part will disappear,
Rewrite the integrand as
$$
\frac{e^{ibx}}{e^{ax}-e^{-ax}}=\frac{\exp\{(a+ib)x\}}{\exp(2ax)-1}
$$
Take the contour $[-R,R]$, $[R,R+\frac{i\pi}{a}]$, $[R+\frac{i\pi}{a},-R+\frac{i\pi}{a}]$, $[-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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irreducibility of polynomials with integer coefficients Consider the polynomial
$$p(x)=x^9+18x^8+132x^7+501x^6+1011x^5+933x^4+269x^3+906x^2+2529x+1733$$
Is there a way to prove irreducubility of $p(x)$ in $\mathbb{Q}[x]$ different from asking to PARI/GP?
| This polynomial has the element $\alpha^2+\beta$ described in this question as a root. My answer to that question implies among other things that the minimal polynomial of that element is of degree 9, so this polynomial has to be irreducible.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Solvability of $S_3\times S_3$ I know that the direct product of two solvable groups are solvable. The group $S_3$ is solvable, so $S_3\times S_3$ is solvable. But how am I going to establish the subnormal series of $S_3\times S_3$?or is there any simpler way to show its solvability?
Thanks.
| Perhaps this is one of those cases where you understand things better by looking at a more general setting.
Let $G, H$ be soluble groups, and let $G.H$ be any extension of $G$ by $H$. Then $G.H$ is soluble.
Start with a subnormal series with abelian factors that goes from $\{1\}$ to $G$. Then continue with a subnormal ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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Value of $\sum\limits^{\infty}_{n=1}\frac{\ln n}{n^{1/2}\cdot 2^n}$ Here is a series:
$$\displaystyle \sum^{\infty}_{n=1}\dfrac{\ln n}{n^{\frac12}\cdot 2^n}$$
It is convergent by d'Alembert's law. Can we find the sum of this series ?
| Consider
$$f(s):=\sum_{n=1}^\infty \frac {\left(\frac 12\right)^n}{n^s}=\operatorname{Li}_s\left(\frac 12\right)$$
with $\operatorname{Li}$ the polylogarithm then (since $\,n^{-s}=e^{-s\ln(n)}$) :
$$f'(s)=\frac d{ds}\operatorname{Li}_s\left(\frac 12\right)=-\sum_{n=1}^\infty \frac {\ln(n)}{n^s}\left(\frac 12\right)^n$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334742",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Are all $n\times n$ invertible matrices change-of-coordinates matrices in $\mathbb{R}^n$? More precisely, I'm trying to prove the following problem:
Assume that $\text{Span}\{\vec{v}_{1},\dots,\vec{v}_{k}\}=\mathbb{R}^{n}$
and that A
is an invertible matrix. Prove that $\text{Span}\{A\vec{v}_{1},\dots,A\vec{v}_{k}\... | Actually, the opposite is true: the problem in your question is half of the proof of the proposition in the title, namely, that $\{Av_1,\dotsc,Av_k\}$ is a spanning set whenever $\{v_1,\dotsc,v_k\}$ is. So, let $x \in \mathbb{R}^n$. Then, of course, $A^{-1}x \in \mathbb{R}^n$, so $A^{-1}x = \sum_{i=1}^k \alpha_i v_i$ f... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Variation of Pythagorean triplets: $x^2+y^2 = z^3$ I need to prove that the equation $x^2 + y^2 = z^3$ has infinitely many solutions for positive $x, y$ and $z$.
I got to as far as $4^3 = 8^2$ but that seems to be of no help.
Can some one help me with it?
| the equation:
$X^2+Y^2=Z^3$
Has the solutions:
$X=2k^6+8tk^5+2(7t^2+8qt-9q^2)k^4+16(t^3+2qt^2-tq^2-2q^3)k^3+$
$+2(7t^4+12qt^3+6q^2t^2-28tq^3-9q^4)k^2+8(t^5+2qt^4-2q^3t^2-5tq^4)k+$
$+2(q^6-4tq^5-5q^4t^2-5q^2t^4+4qt^5+t^6)$
..................................................................................................... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/334839",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 8,
"answer_id": 5
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How do I show $\sin^2(x+y)−\sin^2(x−y)≡\sin(2x)\sin(2y)$? I really don't know where I'm going wrong, I use the sum to product formula but always end up far from $\sin(2x)\sin(2y)$. Any help is appreciated, thanks.
| Just expand, note that $(a+b)^2-(a-b)^2 = 4 ab$.
Expand $\sin (x+y), \sin (x-y)$ in the usual way. Let $a = \sin x \cos y, b= \cos x \sin y$.
Then $\sin^2(x+y)−\sin^2(x−y)= 4 \sin x \cos y \cos x \sin y$.
Then note that $\sin 2 t = 2 \sin t \cos t$ to finish.
Note that the only trigonometric identity used here is $\sin... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Functions question help me? a) We have the function $f(x,y)=x-y+1.$ Find the values of $f(x,y)$ in the points of the parabola $y=x^2$ and build the graph $F(x)=f(x,x^2)$ .
So, some points of the parabola are $(0;0), (1;1), (2;4)$. I replace these in $f(x,y)$ and I have $f(x,y)=1,1,-1\dots$. The graph $f(x,x^2)$ must ha... | (a)
*
*To find points on $f(x, y)$ that are also on the parabola $y = f(x) = x^2$: Solve for where $f(x, y) = x - y + 1$ and $y = f(x) = x^2$ intersect by putting $f(x, y) = f(x)$:
$$ x^2 = x - y + 1$$ and and express as a value of $y$:
$$y = 1 + x - x^2\;\;\text{ and note}\;\; y = F(x) = f(x, x^2)\tag{1}$$
*
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $B$ is a basis of the space $V$. Let $V,W$ be nonzero spaces over a field $F$ and suppose that a set $B =\lbrace v_1, . . . , v_n \rbrace \subset V$ has the following property:
For any vectors $w_1, . . . ,w_n \in W$, there exists a unique linear transformation $T : V \rightarrow W$ such that $T(v_i) = w_i$ ... | Choose $w \in W$, $w \neq 0$. Define $T_k$ by $T_k(v_i) = \delta_{ik} w$. By assumption $T_k$ exists.
We want to show that $v_1,...,v_n$ are linearly independent. So suppose $\sum_i \alpha_i v_i = 0$. Then $T_k(\sum_i \alpha_i v_i) = \alpha_k w = 0$. Hence $\alpha_k = 0$.
Now we need to show that $v_1,...,v_n$ spans $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the complete solution to the simultaneous congruence. I'm having trouble understanding the steps involved to do this question so any step by step reasoning in solving the solution would help me study for my exam.
Thanks so much!
$$x\equiv 6 \pmod{14}$$
$$x\equiv 24 \pmod{29}$$
| Applying Easy CRT (below), noting that $\rm\displaystyle\, \frac{-18}{29}\equiv \frac{-4}{1}\ \ (mod\ 14),\ $ quickly yields
$$\begin{array}{ll}\rm x\equiv \ \ 6\ \ (mod\ 14)\\ \rm x\equiv 24\ \ (mod\ 29)\end{array}\rm \!\iff\! x\equiv 24\! +\! 29 \left[\frac{-18}{29}\, mod\ 14\right]\!\equiv 24\!+\!29[\!-4]\equiv -92... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof involving modulus and CRT Let m,n be natural numbers where gcd(m,n) = 1. Suppose x is an integer which satisfies
x ≡ m (mod n)
x ≡ n (mod m)
Prove that x ≡ m+n (mod mn).
I know that since gcd(m,n)=1 means they are relatively prime so then given x, gcd(x,n)=m and gcd(x,m)=n. I have trouble getting to the next step... | From the Chinese Remainder Theorem (uniqueness part) you know that, since $m$ and $n$ are relatively prime, the system of congruences has a unique solution modulo $mn$.
So we only need to check that $m+n$ works. For that, we only need to verify that $m+n\equiv n\pmod{m}$, and that $m+n\equiv m\pmod{n}$. That is very ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Convert segment of parabola to quadratic bezier curve How do I convert a segment of parabola to a cubic Bezier curve?
The parabola segment is given as a polynomial with two x values for the edges.
My target is to convert a quadratic piecewise polynomial to a Bezier path (a set of concatenated Bezier curves).
| You can do this in two steps, first convert the parabola segment to a quadratic Bezier curve (with a single control point), then convert it to a cubic Bezier curve (with two control points).
Let $f(x)=Ax^2+Bx+C$ be the parabola and let $x_1$ and $x_2$ be the edges of the segment on which the parabola is defined.
Then $... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding the derivatives of sin(x) and cos(x) We all know that the following (hopefully):
$$\sin(x)=\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n+1}}{(2n+1)!}\ , \ x\in \mathbb{R}$$
$$\cos(x)=\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n}}{(2n)!}\ , \ x\in \mathbb{R}$$
But how do we find the derivates of $\sin(x)$ and $\cos(x)$ by usin... | The radius of convergence of the power series is infinity, so you can interchange summation and differentiation.
$\frac{d}{dx}\sin x = \frac{d}{dx} \sum^{\infty}_{n=0}(-1)^n \frac{x^{2n+1}}{(2n+1)!} = \sum^{\infty}_{n=0}(-1)^n \frac{d}{dx} \frac{x^{2n+1}}{(2n+1)!} = \sum^{\infty}_{n=0}(-1)^n \frac{x^{2n}}{(2n)!} = \co... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/335284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 3
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Determinant of the transpose of elementary matrices Is there a 'nice' proof to show that $\det(E^T) = \det(E)$ where $E$ is an elementary matrix?
Clearly it's true for the elementary matrix representing a row being multiplied by a constant, because then $E^T = E$ as it is diagonal.
I was thinking for the "row-additio... | You can use the fact that switching two rows or columns of a matrix changes the sign of the determinant. Switching two rows of $E$ makes it diagonal, then switch the corresponding columns and you have $E^T$
| {
"language": "en",
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degree of the extension of the normal closure So I am trying to do the following problem: if $K/F$ is an extension of degree $p$ (a prime), and if $L/F$ is a normal closure of $K/F$, then $[L:K]$ is not divisible by $p$.
This is what I tried. If $F$ is separable, then primitive element theorem holds and we get $K=F(\a... | If $[K:F]=P$, then for any $\alpha\in K\setminus F$, we get $K=F(\alpha)$.
Let $f(x)$ be the minimal polynomial of $\alpha$ over $F$, then the normal closure $L/F$ of $K/F$ is the splitting field of $f(x)$ over $K$. If we let $f(x)=g(x)(x-\alpha)$, then $L$ is the splitting field of $g(x)$ over $K$, but $\deg(g)=p-1... | {
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halfspaces question How do I find the supporting halfspace inequality to epigraph of
$$f(x) = \frac{x^2}{|x|+1}$$
at point $(1,0.5)$
| For $x>0$, we have
$$
f(x)=\frac{x^2}{x+1}\quad\Rightarrow\quad f'(x)=\frac{x^2+2x}{(x+1)^2}.
$$
Hence $f'(1)=\frac{3}{4}$ and an equation of the tangent to the graph of $f$ at $(1,f(1))$ is
$$y=f'(1)(x-1)+f(1)=\frac{3}{4}(x-1)+\frac{1}{2}=\frac{3}{4}x-\frac{1}{4}.$$
An inequality defining the halfspace above this ta... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is 1 divided by 3 equal to 0.333...? I have been taught that $\frac{1}{3}$ is 0.333.... However, I believe that this is not true, as 1/3 cannot actually be represented in base ten; even if you had infinite threes, as 0.333... is supposed to represent, it would not be exactly equal to 1/3, as 10 is not divisible by 3.
0... | The problematic part of the question is "no matter how many ones you add, 0.111... will never equal precisely 1/9."
In this (imprecise) context $0.111\ldots$ is an infinite sequence of ones; the sequence of ones does not terminate, so there is no place at which to add another one; each one is already followed by anothe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove this function is quasi-convex/concave? this is the function:
$$\displaystyle f(a,b) = \frac{b^2}{4(1+a)}$$
| For quasi convexity you have to consider for $\alpha\in R$ the set
$$\{(a,b)\in R^{2}: f(a,b)\leq \alpha\}
$$
If this set is convex for every $\alpha \in R$ you have quasi convexity.
So we obtain the equality
$$4(1+a)\leq \alpha b^{2}.$$
If you draw this set as set in $R^{2}$ for fixed $\alpha\in R$, this should give y... | {
"language": "en",
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$\int \frac{dx}{x\log(x)}$ I think I'm having a bad day. I was just trying to use integration by parts on the following integral:
$$\int \frac{dx}{x\log(x)}$$
Which yields
$$\int \frac{dx}{x\log(x)} = 1 + \int \frac{dx}{x\log(x)}$$
Now, if I were to subtract
$$\int \frac{dx}{x\log(x)}$$
from both sides, it would seem... | In a general case we have
$$\int\frac{f'(x)}{f(x)}dx=\log|f(x)|+C,$$
and in our case, take $f(x)=\log(x)$ so $f'(x)=\frac{1}{x}$ to find the desired result.
| {
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2D Partial Integration I have a (probably very simple problem): I try to find the variational form of a PDE, at one time we have to partially integrate:
$\int_{\Omega_j} v \frac{\partial}{\partial x}E d(x,y)$ where v is our testfunction and E ist the function we try to find. We have $v, E: \mathbb{R}^2\rightarrow \mat... | For the following, I refer to Holland, Applied Analysis by the Hilbert Space Method, Sec. 7.2. Let $D$ be a simply connected region in the plane, and $f$, $P$, and $Q$ be functions that map $D \rightarrow \mathbb{R}$. Then
$$\iint_D f \left ( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right ) \,dx\... | {
"language": "en",
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Intersecting set systems and Erdos-Ko-Rado Theorem Suppose you have an $n$-element set, where $n$ is finite, and you want to make an intersecting family of $r$-subsets of this set. Each subset has to intersecting each other subset.
We may assume $r$ is not larger than $n/2$, because that would make the problem trivial... | If I understand your question correctly, you can't do better. Indeed from the proof of Erdos-Ko-Rado you can deduce that only the stars have size equal to $\binom{n-1}{r-1}$, when $2r<n$. In the exceptional case $r=n/2$ you can take one of every pair of complementary sets $A,A^c$.
EDIT: The intended question seems to b... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help me to prove this integration Where the method used should be using complex analysis.
$$\int_{c}\frac{d\theta}{(p+\cos\theta)^2}=\frac{2\pi p}{(p^2-1)\sqrt{p^2-1}};c:\left|z\right|=1$$
thanks in advance
| Hint:
$$\cos \theta = \frac {z + z^{-1}} 2$$
Plug this in and refer to classic tools in complex analysis, such as the Cauchy formula or the residue theorem.
| {
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$e^{\large\frac{-t}{5}}(1-\frac{t}{5})=0$ solve for t I'm given the following equation an i have to solve for $"t"$ This is actually the derivative of a function, set equal to zero:
$$f'(t) = e^\frac{-t}{5}(1-\frac{t}{5})=0$$
I will admit im just stuck and im looking for how to solve this efficiently.
steps $1,\;2$ - r... | $e^{\dfrac{-t}{5}}\neq0$. (Cause: $\frac{1}{e^\frac{t}{5}}$ is a real number)
Therefore, the other term has to be zero.
Now you can solve it !
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $|Im(z)|\le |\cos (z)|$ for $z\in \mathbb{C}$. Let $z\in \mathbb{C}$ i.e. $z=x+iy$. Show that $|Im(z)|\le |\cos (z)|$.
My hand wavy hint was to consider $\cos (z)=\cos (x+iy)=\cos (x)\cosh (y)+i\sin (x)\sinh(y)$ then do "stuff".
Then I have $|\cos (z)|=|Re(z)+iIm(z)|$ and the result will be obvious.
Thanks in adv... | $$\cos(z) = \cos(x) \cosh(y) + i \sin(x) \sinh(y) $$
Taking the norm squared:
$$ \cos^2(x) \cosh^2(y) + \sin^2(x) \sinh^2(y)$$
We are left with:
$$ \cos^2(x) \left(\frac{1}{2}\cosh(2y) + \frac{1}{2}\right) + \sin^2(x) \left(\frac{1}{2}\cosh(2y) - \frac{1}{2}\right)$$
Simplifying, we get:
$$ \frac{1}{2} \left(\cosh(2y) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/336077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the property that allows $5^{2x+1} = 5^2$ to become $2x+1 = 2$? What is the property that allows $5^{2x+1} = 5^2$ to become $2x+1 = 2$? We just covered this in class, but the teacher didn't explain why we're allowed to do it.
| $5^{(2x+1)} = 5^2$
Multiplying by $1/5^2$ om both sides we get,
$\frac{5^{(2x+1)}}{5^2} = 1$
$\Rightarrow 5^{(2x+1)-2} = 1$
Taking log to the base 5 on both sides we get $2x+1-2=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/336133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$ Let $a,b,c$ are non-negative numbers, such that $a+b+c = 3$.
Prove that $\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
Here's my idea:
$\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab + bc + ca$
$2(\sqrt{a} + \sqrt{b} + \sqrt{c}) \ge 2(ab + bc + ca)$
$2(\sqrt{a} + \sqrt... | I will use the following lemma (the proof below):
$$2x \geq x^2(3-x^2)\ \ \ \ \text{ for any }\ x \geq 0. \tag{$\clubsuit$}$$
Start by multiplying our inequality by two
$$2\sqrt{a} +2\sqrt{b} + 2\sqrt{c} \geq 2ab +2bc +2ca, \tag{$\spadesuit$}$$
and observe that
$$2ab + 2bc + 2ca = a(b+c) + b(c+a) + c(b+c) = a(3-a) + b(... | {
"language": "en",
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Nth number of continued fraction Given a real number $r$ and a non-negative integer $n$, is there a way to accurately find the $n^{th}$ (with the integer being the $0^{th}$ number in the continued fraction. If this can not be done for all $r$ what are some specific ones, like $\pi$ or $e$. I already now how to do this ... | You can do it recursively: $$\eqalign{f_0(r) &= \lfloor r \rfloor\cr
f_{n+1}(r) &= f_n\left( \frac{1}{r - \lfloor r \rfloor}\right)\cr}$$
Of course this may require numerical calculations with very high precision.
Actually, if $r$ is a rational number but you don't know it, no finite precision
numerical calculation wil... | {
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After $t$ hours, the hour hand was where the minute hand had been, and vice versa
On Saturday, Jimmy started painting his toy helicopter between 9:00 a.m. and 10:00 a.m. When he finished between 10:00 a.m. and 11:00 a.m. on the same morning, the hour hand was exactly where the minute hand had been when he started, an... | I'm not sure what you mean by "complete the above solution". The above attempt is actually pretty far from actually finding out what $m$ and $h$ are. However, you do get $t$ as a function of $m$ and $h$, which will be needed to compute the time elapsed.
To actually solve this problem, you have to make use of the fact... | {
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Help solving $\sqrt[3]{x^3+3x^2-6x-8}>x+1$ I'm working through a problem set of inequalities where we've been moving all terms to one side and factoring, so you end up with a product of factors compared to zero. Then by creating a sign diagram we've determined which intervals satisfy the inequality.
This one, however, ... | Hint: you may want to remove the root first, by noticing that $f(y)=y^3$ is a monotonic increasing function.
| {
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Trignometry - Cosine Formulae c = 21
b = 19
B= $65^o$
solve A with cosine formulae
$a^2+21^2-19^2=2a(21)cos65^o$
yield an simple quadratic equation in variable a
but, $\Delta=(-2(21)cos65^o)^2-4(21^2-19^2) < 0$ implies the triangle as no solution?
How to make sense of that? Why does this happen and in what situation? P... | Here is an investigation without directly using sine law:
From A draw a perpendicular line to BC. Let the intersection be H
BH is AB*cos(65) which is nearly 8.87
From Pythagoras law, AH is nearly 19.032
AC is 19 But we arrived at a contradiction since hypotenuse is less than another side which then immediately implies ... | {
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If $E$ is generated by $E_1$ and $E_2$, then $[E:F]\leq [E_1:F][E_2:F]$? Suppose $K/F$ is an extension field, and $E_1$ and $E_2$ are two intermediate subfields of finite degree. Let $E$ be the subfields of $K$ generated by $E_1$ and $E_2$. I'm trying to prove that
$$ [E:F]\leq[E_1:F][E_2:F].$$
Since $E_1$ and $E_2$ ar... | The sum of the degrees in general is not going to be an upper bound. Consider $K = \Bbb{Q}(\sqrt[3]{2},e^{2\pi i/3})$. This is a degree $6$ extension of $\Bbb{Q}$. Take $E_1 = \Bbb{Q}(\sqrt[3]{2})$ and $E_2 = \Bbb{Q}(e^{2 \pi i/3})$. Then
$$[E_1 : \Bbb{Q}] + [E_2:\Bbb{Q} ] = 3 + 2 = 5$$
but $E = K$ that has degree $6$ ... | {
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Eigenvalues of $A$ Let $A$ be a $3*3$ matrix with real entries. If $A$ commutes with all $3*3$ matrices with real entries, then how many number of distinct real eigenvalues exist for $A$?
please give some hint.
thank you for your time.
| If $A$ commute with $B$ then $$[A,B] = AB - BA = 0$$
$$AB=BA$$
$$ABB^{-1}=BAB^{-1}$$
$$A=BAB^{-1}$$
If the $B$ has inverse matrix $B^{-1}$, then sets of eigenvalues not change.
Matrix $3\times{3}$ has 3 eigenvalues.
| {
"language": "en",
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Balls, Bags, Partitions, and Permutations We have $n$ distinct colored balls and $m$ similar bags( with the condition $n \geq m$ ). In how many ways can we place these $n$ balls into given $m$ bags?
My Attempt: For the moment, if we assume all the balls are of same color - we are counting partitions of $n$ into at most... | This is counting the maps from an $n$-set (the balls) to an $m$-set (the bags), up to permutations of the $m$-set (as the bags are similar). This is just one of the problems of the twelvefold way, and looking at the appropriate section you find that the result can only be expressed as a sum of Stirling numbers of the s... | {
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Easy volumes question Note: I am trying to prepare, i already see the answer to this question but i dont understand it.
It says volume of earth used for making the embarkment = $\pi (R^2 - r^2)h$
But i dont understand, what R? What r?The whole part solves out to 28$\pi$ but i dont understand where the two Rr come from ... | The total volume of earth taken out is $r^2\pi\cdot h=3^2\pi\cdot14$. The volume of the new pile is $(R^2-r^2)\pi\cdot H=(4^2-3^2)\pi H$, where you need to find $H$. Compare the two, and express $H$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Counting Couples A group of m men and w women randomly sit in a single row at a theater. If a man and woman are seated next to each other, they form a "couple." "Couples" can overlap, which means one person can be a member of two "couples."
Question: What is the expected number of couples?
Comment:
I have a hard time ... | Continuing the computation we can calculate $E[C(C-1)]$.
We have $$\left.\left(\left(\frac{d}{dz}\right)^2 (P+Q)\right)\right|_{z=1} =
2\,{\frac {uv \left( -2\,uv-u+{u}^{2}+{v}^{2}-v \right) }{ \left( v-1+u \right) ^{3}}}.$$
After a straightforward calculation this transforms into
$$ E[C(C-1)] =
{\frac {2\,mw \,\left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/336896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
How can I prove that $\|Ah\| \le \|A\| \|h\|$ for a linear operator $A$? On http://www.proofwiki.org/wiki/Definition:Norm_(Linear_Transformation) , it is stated that $||Ah|| \leq ||A||||h||$ where $A$ is an operator.
Is this a theorem of some sort? If so, how can it be proved? I've been trying to gather more informatio... | By definition:
$$
\|A\|=\sup_{\|x\|\leq 1}\|Ax||=\sup_{\|x\|= 1}\|Ax||=\sup_{x\neq 0}\frac{\|Ax\|}{\|x\|}.$$
In particular,
$$
\frac{\|Ax\|}{\|x\|}\leq \|A\|\qquad\forall x\neq 0\quad\Rightarrow\quad\|Ax\|\leq \|A\|\|x\|\qquad \forall x.
$$
Note that $\|A\|$ can alternatively be defined as the least $M\geq 0$ such tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337040",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
How to prove inverse of Matrix If you have an invertible upper triangular matrix $M$, how can you prove that $M^{-1}$ is also an upper triangular matrix? I already tried many things but don't know how to prove this. Please help!
| The co-factor of any element above the diagonal is zero.Resaon :The matrix whose determinant is the co-factor will always be an upper triangular matrix with the determinant zero because either its last row is zero or its first column is zero or one of its diagonal element is zero.This can easily be verified. This will ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337161",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 2
} |
How can I calculate or think about the large number 32768^1049088? I decided to ask myself how many different images my laptop's screen could display. I came up with (number of colors)^(number of pixels) so assuming 32768 colors I'm trying to get my head around the number, but I have a feeling it's too big to actually... | Your original number is
$2^{15*2^{20}}
<2^{2^{24}}
< 10^{2^{21}}< 10^{3*10^6}
$
which is certainly computable
since it has less than
3,000,000 digits.
The new, larger number is
$2^{24*2^{20}}
<2^{2^{25}}
< 10^{2^{22}}< 10^{6*10^6}
$
which is still computable
since it has less than
6,000,000 digits.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/337227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Analytic Geometry question (high school level) I was asked to find the focus and diretrix of the given equation: $y=x^2 -4$. This is what I have so far:
Let $F = (0, -\frac{p}{2})$ be the focus, $D = (x, \frac{p}{2})$ and $P = (x,y)$ which reduces to $x^2 = 2py$ for $p>0$. Now I have $x^2 = 2p(x^2 - 4)$ resulting in $ ... | I seem to remember the focal distance $p$ satisfies $4ap=1$ where the equation for the parabola is $y = ax^2 + bx + c$. Your focus will be $1/4$ above your vertex, and the directrix will be a horizontal line $1/4$ below your vertex.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/337338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Generating function with quadratic coefficients. $h_k=2k^2+2k+1$. I need the generating function $$G(x)=h_0+h_1x+\dots+h_kx^k+\dots$$ I do not have to simplify this, yet I'd really like to know how Wolfram computed this sum as $$\frac{x(2x^2-2x+5}{(1-x)^3}$$ when $|x|<1$. Rewrite Wolfram's answer as $$x(2x^2-2x+5)(1+x+... | $$\sum_{k=0}^{\infty} x^k = \frac{1}{1-x}$$
$$\sum_{k=0}^{\infty}k x^k = \frac{x}{(1-x)^2}$$
$$\sum_{k=0}^{\infty}k^2 x^k = \frac{x(1+x)}{(1-x)^3}$$
$$G(x) = \sum_{k=0}^{\infty} (2 k^2+2 k+1) x^k$$
Combine the above expressions as defined by $G(x)$ and you should reproduce Wolfram.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/337400",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Is the sum or product of idempotent matrices idempotent? If you have two idempotent matrices $A$ and $B$, is $A+B$ an idempotent matrix?
Also, is $AB$ an idempotent Matrix?
If both are true, Can I see the proof? I am completley lost in how to prove both cases.
Thanks!
| Let $e_1$, $e_2 \in \mathbb{R}^n$ be linearly independent unit vectors with $c := \left\langle e_1,e_2\right\rangle \neq 0$, viewed as column vectors. For $i=1$, $2$, let $P_i := e_i e_i^T \in M_n(\mathbb{R})$ be the orthogonal projection onto $\mathbb{R}e_i$. Thus, $P_1$ and $P_2$ are idempotents with
$$
P_1 e_1 = e_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
A question about topology regarding its conditions A question from a rookie.
As we know, $(X, T)$ is a topological space, on the following conditions,
*
*The union of a family of $T$-sets, belongs to $T$;
*The intersection of a FINITE family of $T$-sets, belongs to $T$;
*The empty set and the whole $X$ belongs to ... | The intersection of intervals $(-1/n,\ 1/n)$ for $n\in\mathbb N$ is only $\{0\}$ which is not open in $\mathbb R$ with the euclidean topology.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/337540",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Calculate:$\lim_{x \rightarrow (-1)^{+}}\left(\frac{\sqrt{\pi}-\sqrt{\cos^{-1}x}}{\sqrt{x+1}} \right)$ How to calculate following with out using L'Hospital rule
$$\lim_{x \rightarrow (-1)^{+}}\left(\frac{\sqrt{\pi}-\sqrt{\cos^{-1}x}}{\sqrt{x+1}} \right)$$
| Let $\sqrt{\arccos(x)} = t$. We then have $x = \cos(t^2)$. Since $x \to (-1)^+$, we have $t^2 \to \pi^-$. Hence, we have
$$\lim_{x \to (-1)^+} \dfrac{\sqrt{\pi} - \sqrt{\arccos(x)}}{\sqrt{1+x}} = \overbrace{\lim_{t \to \sqrt{\pi}^-} \dfrac{\sqrt{\pi} - t}{\sqrt{1+\cos(t^2)}}}^{t = \sqrt{\arccos(x)}} = \underbrace{\lim_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337603",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Calculating line integral I'm working on this problem:
Calculate integral
\begin{equation}
\int_C\frac{z\arctan(z)}{\sqrt{1+z^2}}\,dz + (y-z^3)\,dx - (2x+z^3)\,dy,
\end{equation}
where the contour $C$ is defined by equations
$$
\sqrt{1-x^2-y^2}=z, \quad 4x^2+9y^2 = 1.
$$
Seems to me that I know the solution, b... | It's absolutely fine to exploit the symmetries in the given problem. But we need a clear cut argument. Observing that a "variable goes up and down" doesn't suffice.
Relevant are the following symmetries in the parametrization of $C$:
$$x(\phi+\pi)=-x(\phi),\quad y(\phi+\pi)=-y(\phi),\quad z(\phi)=z(-\phi)=z(\phi+\pi)\ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How fast can you determine if vectors are linearly independent? Let us suppose you have $m$ real-valued vectors of length $n$ where $n \geq m$.
How fast can you determine if they are linearly independent?
In the case where $m = n$ one way to determine independence would be to compute the determinant of the matrix whos... | Please use the following steps
*
*Arrange the vectors in form of a matrix with each vector representing a column of matrix.
*Vectors of a matrix are always Linearly Dependent if number of columns is greater than number of rows (where m > n).
*Vectors of a matrix having number of rows greater than or equal to numb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337739",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 2
} |
How to calculate the number of pieces in the border of a puzzle? Is there any way to calculate how many border-pieces a puzzle has, without knowing its width-height ratio? I guess it's not even possible, but I am trying to be sure about it.
Thanks for your help!
BTW you might want to know that the puzzle has 3000 piece... | Obviously, $w\cdot h=3000$, and there are $2w+h-2+h-2=2w+2h-4$ border pieces. Since $3000=2^3\cdot 3\cdot 5^3$, possibilities are \begin{eqnarray}(w,h)&\in&\{(1,3000),(2,1500),(3,1000),(4,750),(5,600),(6,500),\\&&\hphantom{\{}(8,375),(10,300),(12,250),(15,200),(20,150),(24,125)\\ &&\hphantom{\{}(25,120),(30,100),(40,75... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 0
} |
Find the necessary and sufficient conditions for all $ 41 \mid \underbrace{11\ldots 1}_{n}$, $n\in N$. Find the necessary and sufficient conditions for all $ 41 \mid \underbrace{11\ldots 1}_{n}$, $n\in N$. And, if $\underbrace{11\ldots 1}_{n}\equiv 41\times p$,
then $p$ is a prime number.
Find all of the possible valu... | The first sentence is asking that $41|\frac {10^n-1}9$. This is just the length of the repeat of $\frac 1{41}$. The second statement forces $n$ to be the minimum value. Without it, any multiple of $n$ would work, but $p$ would not be prime.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/337872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
totally ordered group Suppose a no trivial totally ordered group .This group has maximum element?
A totally ordered group is a totally ordered structure (G,∘,≤) such that (G,∘) is a group.I couldnt find a more exact definition
| I assume you want the ordering to be compatible with the group operation, such that if $a \geq b$ and $c\geq d$ then $ac\geq bd$.
In this case, the group cannot have a maximal element, which we can see as follows: Assume $g$ is such a maximal element and let $h\in G$ with $h\geq 1$.
Now we have that $g\geq g$ and $h\ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/337933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
volume of "$n$-hedron" In $\mathbb{R}^n$, why does the "$n$-hedron" $|x_1|+|x_2|+\dots+|x_n| \le 1$ have volume $\cfrac{2^n}{n!}$? I came across this fact in some of Minkowski's proofs in the field of geometry of numbers.
Thank you.
| The $2^n$ comes because you have that many copies of the simplex $0 \le x_i \le 1$
The $n!$ comes from integrating up the volume. The area of the right triangle is $\frac 12$, the volume of the tetrahedron is $\frac 12 \cdot \frac 13$ and so on.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/337988",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Find a polynomial as $2836x^2-12724x+16129$ I found a polynomial function with integer coefficients:$f(x)=2836x^2-12724x+16129$
and $f(0)=127^2,f(1)=79^2,f(2)=45^2,f(3)=59^2,f(4)=103^2,f(5)=153^2.$
My question is:can we find a polynomial function with integer coefficients,called $f(x)$,which has no multiple roots,and $... | One such quadratic
$$p(t)=-4980t^2+32100t+2809$$
$p(0)=53^2,p(1)=173^2,p(2)=217^2,p(3)=233^2,p(4)=227^2,p(5)=197^2,p(6)=127^2$
Source : Polynomials E.J Barbeau
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/338037",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
inscribed angles on circle
That's basically the problem. I keep getting $\theta=90-\phi/2$. But I have a feeling its not right. What I did was draw line segments BD and AC. From there you get four triangles. I labeled the intersection of BD and AC as point P. From exterior angles I got my answer.
| One way would be to let $E$ be the center of the circle. A standard result in geometry tells you that $AEC=2\theta$. And the two sides $AE$ and $CE$ are of equal lengths, and there are right angles at $A$ and $C$, and the sides $AD$ and $CD$ are also of equal lengths. So the triangle $EAD$ is right triangle congruen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Surface integral over ellipsoid I've problem with this surface integral:
$$
\iint\limits_S {\sqrt{ \left(\frac{x^2}{a^4}+\frac{y^2}{b^4}+\frac{z^2}{c^4}\right)}}{dS}
$$, where
$$
S = \{(x,y,z)\in\mathbb{R}^3: \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}= 1\}
$$
| Let the ellipsoid $S$ be given by
$${\bf x}(\theta,\phi)=(a\cos\theta\cos\phi,b\cos\theta\sin\phi,c\sin\theta)\ .$$
Then for all points $(x,y,z)\in S$ one has
$$Q^2:={x^2\over a^4}+{y^2\over b^4}+{z^2\over c^4}={1\over a^2b^2c^2}\left(\cos^2\theta(b^2c^2\cos^2\phi+a^2c^2\sin^2\phi)+a^2b^2\sin^2\theta\right)\ .$$
On the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 3,
"answer_id": 1
} |
Integrating $x/(x-2)$ from $0$ to $5$ How would one go about integrating the following?
$$\int_0^5 \frac{x}{x-2} dx$$
It seems like you need to use long division, split it up into two integrals, and the use limits. I'm not quite sure about the limits part.
| Yes, exactly, you do want to use "long division"...
Note, dividing the numerator by the denominator gives you:
$$\int_0^5 {x\over{x-2}} \mathrm{d}x = \int_0^5 \left(1 + \frac 2{x-2}\right) \mathrm{d}x$$
Now simply split the integral into the sum of two integrals:
$$\int_0^5 \left(1 + \frac 2{x-2}\right) \mathrm{d}x \qu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/338221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 1
} |
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