Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Specializations of elementary symmetric polynomials Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$
indeterminates. The $h^{th}$elementary symmetric polynomial is the
sum of all monomials with $h$ factors
\begin{eqnarray*}
e_{h}(\mathcal{S}_{x}) & = & \sum_{1\leqslant i_{1}<i_{2}<\ldots<i_{h}\leqsla... | You have mentioned what are known as the (stable) principal specializations of the ring of symmetric functions $\Lambda$. If you haven't already, you should check out section 7.8 of Stanley's Enumerative Combinatorics Vol. II, where he summarizes the specialization you have have mentioned. In particular, using the spe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801399",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
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How find this sum $\sum_{k=0}^{p}t^k\binom{n}{k}\binom{m}{p-k}$ Find the closed form
$$\sum_{k=0}^{p}t^k\binom{n}{k}\binom{m}{p-k}$$
since
$$\binom{n}{k}\binom{m}{p-k}=\dfrac{n!}{(n-k)!k!}\cdot\dfrac{m!}{(p-k)!(m-p+k)!}$$
then I can't
| Let's find the generating function $F(z):=\sum_p a_pz^p$, where $a_p$ is your sum.
Notice that your sum is a convolution of $t^k\binom{n}{k}$ and $\binom{m}{k}$.
Therefore $$\begin{align}F(z)&=\left(\sum_k t^k\binom{n}{k}z^k\right)\left(\sum_k\binom{m}{k}z^k\right)\\&=(1+tz)^n(1+z)^m\end{align}$$
Notice that for $m=0$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Deterministic Push-Down Automata Does there exist Deterministic Push-Down Automata for the language below.
Any kind of answer will be highly appreciated!
$$L =ba^nb^n U bba^nb^{2n}$$
| Probably $U$ denotes union, and you mean $L = \{ ba^nb^n \mid\ n\ge 0\} \cup \{ bba^nb^{2n} \mid\ n\ge 0\}$.
Yes that can be done by a deterministic PDA. The first two letters of the string decide how to handle the remainder of the string. Pushing the $a$'s and popping the $b$'s in appropriate ratio is a standard task.... | {
"language": "en",
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"question_score": "1",
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Equivalence of different definitions of continuity Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a real function. $f$ is continuous at point $c$ iff
$$(\forall\epsilon>0)(\exists\delta>0)(\forall x)(|x-c|<\delta\Rightarrow|f(x)-f(c)|<\epsilon)$$
Continuity is also defined at a point $c$ if $\lim\limits_{x\rightarrow c}f(x... | Take the first definition as given. Then clearly the second condition follows, since the range of $x$ values being considered is a subset of the range provided by the first definition.
If we take the second definition as given, we have to show that the weaker condition $$0<|x-c|<\delta\implies |f(x)-f(c)|<\epsilon$$ im... | {
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Does this sum converge or diverge? Does the infinite sum $\large{\sum_{n=1}^\infty \frac{1}{n^{x_{\small{n}}}}}$ converge if $x_n$ is a random variable (generated within each term) that takes values between $0$ and $2$ with equal probability converge or diverge? I have a suspicion that it diverges, but I don't know how... | I don't know if I exactly have a proof, but here's a thought. The infinite series of reciprocals of the prime numbers diverges. How likely is it that $ \ x_n \ > \ 1 \ $ "often enough" to produce a series with terms that can bring the series to convergence despite that? That is, can there be a high enough "density"... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/801709",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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How to convert expectation to integration $S: \{1,-1\}^n \rightarrow \{0,1\}$ and $E(S(x))=p$,
Where $E$ denotes the expectation, and is taken over $x$ , where $x$ is uniformly distributed on $\{-1,1\}^n$.
Then how to prove the following,
\begin{equation*}
E_x\Bigg[S(x) \sum_{i=1}^n x_i \Bigg] \leq \int_{0}^{\infty}... | Answer: Using the fact that, for every nonnegative random variable $Y$,
$$
E(Y)=\int_0^\infty P(Y\gt y)\,\mathrm dy=\int_0^\infty P(Y\geqslant y)\,\mathrm dy.
$$
Proof:
$$
Y=\int_0^Y\mathrm dy=\int_0^\infty \mathbf 1_{Y\gt y}\,\mathrm dy=\int_0^\infty \mathbf 1_{Y\geqslant y}\,\mathrm dy.
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve for $x$ in $2\log(x+11)=(\frac{1}{2})^x$ Solve for $x$.
$$2\log(x+11)=(1/2)^x$$
My attempt:
$$\log(x+11)=\dfrac{1}{(2^x)(2)}$$
$$10^{1/(2^x)(2)}= x+11$$
$$x=10^{1/(2^x)(2)}-11$$
I'm not sure what to do next, because i have one $x$ in the exponent while the other on the left side of the equation.
| Hints:
$$2\log(x+11)=\frac14\implies \log(x+11)=\frac18\implies \color{red}{x+11=e^{1/8}}\ldots$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/801885",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Natural Deduction Given the following premises:
P AND Q 1.
P IMPLIES R 2.
Q IMPLIES R 3.
I need to demonstrate that this entails:
Q AND R
The way I tackled the problem was:
Q 4. AND ELIMINATION on Line 1
R 5. IMPLICATION ELIMINATION on Lines 3, 4
Q AND R 6. AND INTRODUCTION on Lines 4, 5
However, the textbook soluti... | Proof :
$$\begin{align}
(1) & P \land Q && [\text{assumed}] \\
(2) & P && [\land \text{-elim(1)}] \\
(3) & Q && [\land \text{-elim(1)}] \\
(4) & P \rightarrow R && [\text{assumed}] \\
(5) & R && [\rightarrow \text{-elim}(2,4)] \\
(6) & Q \land R && [\land\text{-intro}(3,5)] \\
\end{align}$$
Thus we have proved :
... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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cross product in cylindrical coordinates Hi i know this is a really really simple question but it has me confused.
I want to calculate the cross product of two vectors
$$
\vec a \times \vec r.
$$
The vectors are given by
$$
\vec a= a\hat z,\quad \vec r= x\hat x +y\hat y+z\hat z.
$$
The vector $\vec r$ is the radius vec... | The radius vector $\vec{r}$ in cylindrical coordinates is $\vec{r}=\rho\hat{\rho}+z\hat{z}$. Calculating the cross-product is then just a matter of vector algebra:
$$\vec{a}\times\vec{r} = a\hat{z}\times(\rho\hat{\rho}+z\hat{z})\\
=a(\rho(\hat{z}\times\hat{\rho})+z(\hat{z}\times\hat{z}))\\
=a\rho(\hat{z}\times\hat{\rho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/802077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Intricate proof by induction: $2+8+24+64+...+(n)(2^n)=2(1+(n-1)(2^n))$ Help the King out...
$$2+8+24+64+...+(n)(2^n)=2(1+(n-1)(2^n))$$
I am at the step where I am proving $P(k+1)$ to be true:
$$2(1+(k-1)(2^k))+(k+1)((2)^{k+1}))=2(1+((k+1)-1)(2^{k+1}))$$
| See part I of my answer here for the background to the following systematic approach.
We have here $f(k) = k\cdot2^k$ and $g(n) = 2 + (n-1)\cdot 2^{n+1}$
Inductive step:
1: Assume true for $n$, that is $\sum_{k=1}^{n}f(k) = g(n)\tag{1}$
2: Let $m = n + 1$
$\begin{align}f(m) &= m\cdot2^m\\\\ g(m) - g(m-1) &= \left(2 +... | {
"language": "en",
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When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?
When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?
Often times I "feel" as if I can write a proof to an exercise but most... | Use a computer with automated proof checking software, also called a proof assistant or interactive theorem prover. Typically you will need to write your proof in a special, machine readable format (be careful for translation/copy errors), but past that point this field is well studied and computer-based proof checking... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "46",
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If $(B \cap C) \subset A$, then $(C\setminus A) \cap (B\setminus A) = \emptyset$ Question:
Prove/disprove: For all sets $A,B,C$, if $B \cap C \subset A$, then $(C \backslash A) \cap (B \backslash A) = \emptyset$
I'm a bit confused about the question, or where to start. When we learned how to prove these, the examples ... | $
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\calcop}[2]{\notag \\ #1 \quad & \quad \text{"#2"} \notag \\ \quad & }
\newcommand{\endcalc}{\notag \end{align}}
$ (This is not a direct answer, but an alternative approach.)
I would prefer a more 'logical' approach, where you start with the most complex side, $\;... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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"answer_id": 2
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Why is $S_{\ast}\left(X,A\right)$ free? Why is $S_{\ast}\left(X,A\right)$ free? it is the quotient of two free groups $S_{\ast}\left(X\right)$ & $S_{\ast}\left(A\right)$
| The quotient is free because the smaller group is generated by a subset of a basis of the larger one.
Indeed, SX is freely generated by all singular simplices in X, and SA is generated by the set of simplices in X whose image contained in A.
| {
"language": "en",
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Find matrix determinant How do I reduce this matrix to row echelon form and hence find the determinant, or is there a way that I am unaware of that finds the determinant of this matrix without having to reduce it row echelon form given this is all I know and there exists no additional information.
$\left[
\begin{arra... | Assume $f(x)=\Delta$. It's a fourth degree polynomial.
$C_1\to C_1+C_2+C_3+C_4\implies x+10 $ is a factor.
$f(0)=0\implies 0$ is a root.// Repeated Rows
$f'(0)=0+0+0+0\implies 0$ is again repeated.// Repeated Rows
$f''(0)=0\implies 0$ is again repeated.// Repeated Rows for last time.
This all $\implies f(x)=x^3(x+10)$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 2
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Dropping letters in post boxes In how many different ways can 5 letters be dropped in 3 different post boxes if any number of letters can be dropped in all of the post boxes?
| In general number of droppings of $k$ letters in $m$ boxes is
$$\sum_{x_1+x_2+...+x_m=k,0\leq x_i\leq k}1=\binom{m+k-1}{k}$$
in our case $m=3,k=5$
$$\sum_{x_1+x_2+x_3=5,0\leq x_i\leq 5}1=\binom{5+3-1}{5}=21$$
Below is the list of all droppings
$$(5,0,0),(0,5,0),(0,0,5)$$
$$(4,0,1),(4,1,0),(0,1,4),(0,4,1),(1,0,4),(1,4,0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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there is any relation between $\pi$, $\sqrt{2}$ or a generic polygon? I'm a programmer, I'm always looking for new formulas and new way of computing things, to satisfy my curiosity I would like to know if there are any formulas, or I should say equalities, that make use of both $\pi$ and $\sqrt{2}$ .
I would also like ... | Stirling's approximation:
$$
n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/802751",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 9,
"answer_id": 2
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A problem of diagram chasing Consider the following diagram of functions between sets:
I know that the $4$ inner triangles (i.e. $\{X,X',Z\}$,$\{X',Y',Z\}$...) are all commutative diagrams and moreover that $f_1$ and $f_3$ are bijective functions.
Can I conclude that the outer square $\{X,X',Y',Y\}$ is a commutative... | I think that you can't. Take for example $X=X'=Z=\{a\}$, $Y=Y'=\{a,b\}$, $f_1,f_3,a,b$ the identity maps. Then set $f_2\colon a\mapsto a$ and $f_4\colon a\mapsto b$. Then every triangle commute but the big square doesn't.
More generally, you can take $X=X'$, $Y=Y'$, $f_1,f_3$ the identity maps. Then you pick two differ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that convergence of double sequence Suppose $f:X\rightarrow \mathbb R$ has property
$$sup\left \{ {\sum_{a\in F}^{}} \left |f(a) \right | \right \}< \infty$$ :F is finite subset of X.
1.Show that $\left \{ \ a \in X : f(a)\neq 0 \right \}$ is countable set
2.If $a_{kj}\in R$, show that $$\sum_{k=1}^{\infty... | Let $S$ be the supremum of $\sum_{a\in F}|F(a)|$, taken over all finite subsets
$F \subset A$. Then for each $m \in \mathbb N$, the set
$$
A_m = \left \{ \ a \in X : |f(a)| \ge \frac 1m \right \}
$$
has at most $m \cdot S$ elements. Therefore
$\left \{ \ a \in X : f(a)\neq 0 \right \} = \cup A_m$
is countable a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to think when solving $3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$? Solve this differential equation
$$3\frac{\partial f}{\partial x}+5\frac{\partial f}{\partial y}=0$$
Usually, when we get these problems, they tell us what variable change is smart to do and we just chunk through the chain rule... | Basically, when given a change of variables $u=u(x,y),v=v(x,y)$, you will use the Chain rule to transform your equation with unknowns $\dfrac{\partial f}{\partial x}$ and $\dfrac{\partial f}{\partial y}$ (i.e. the gradient $\nabla f(x,y)$) into an equation with unknowns $\dfrac{\partial f}{\partial u}$ and $ \dfrac{\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/802954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
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Möbius transformations lines and circles I am looking for a basic outline of a proof
I know that all MT's are of the form $\frac{ax+b}{cx+d}$
For $c=0$, I know that lines/circles are preserved because translations and dilations do not change a line/circle from being a line/circle
But I am not sure how to prove this for... | To begin with, if $c\neq0,$ then put $f_1(z)=z+d/c,$ $f_2(z)=1/z,$ $f_3(z)=\frac{bc-ad}{c^2}z,$ and $f_4(z)=z+a/c.$ Then $$(f_4\circ f_3\circ f_2\circ f_1)(z)=\frac{az+b}{cz+d}.$$ So, it suffices to show that each of $f_1,f_2,f_3,f_4$ maps generalized circles to generalized circles. The fact that $ad-bc\ne0$ will be es... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof for $\sin(x) > x - \frac{x^3}{3!}$ They are asking me to prove $$\sin(x) > x - \frac{x^3}{3!},\; \text{for} \, x \, \in \, \mathbb{R}_{+}^{*}.$$ I didn't understand how to approach this kind of problem so here is how I tried:
$\sin(x) + x -\frac{x^3}{6} > 0 \\$
then I computed the derivative of that function to d... | You just have to prove your inequality when $x\in(0,\pi)$, since otherwise the RHS is below $-1$. Consider that for any $x\in(0,\pi/2)$,
$$ \sin^2 x < x^2 \tag{1}$$
by the concavity of the sine function. By setting $x=y/2$, $(1)$ gives:
$$ \forall y\in(0,\pi),\qquad \frac{1-\cos y}{2}<\frac{y^2}{4}\tag{2}, $$
so:
$$ \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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"answer_id": 1
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Determine the region bounded by the inequalities
Determine the region bounded by the inequalities:
$$
0 \leq x + y \leq 1 \\
0 \leq x - y \leq x + y
$$
I don't know what to solve for first, so I just added them:
$$
0 \leq x \leq 1 + x + y \\
$$
I guess I can subtract $x$:
$$
-x \leq 0 \leq 1 + y \\
$$
Or:
$$
-y - ... | Note that you can also write your pair of inequalities as a single linear chain of inequalities:
$$0 \leq x-y \leq x+y \leq 1.$$
So all the information you need is contained in the three inequalities of the chain:
$$\begin{cases}0 \leq x-y,\\x-y \leq x+y,\\x+y \leq 1.\end{cases}
\iff \begin{cases}y \leq x,\\0 \leq y,\\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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I am having problems proving that the limit of a certain multivariable function is equal to 0. What I need to prove is the following:
$$\lim_{(x,y)\rightarrow(0,0) }xy^2e^{x^2/y^4}=0$$
for $x,y \in D=\{(x,y):0\leq y \leq 1, 0\leq x\leq y^2\}$.
I tried solving the problem using the 'sandwich'theorem and ended up with t... | This inequality is false:
$$\lim_{(x,y)\rightarrow(0,0)} y^4e^{x^2/y^4}\leq\lim_{(x,y)\rightarrow(0,0) }y^4e^{x^2}.$$
However, you can say that $e^{x^2/y^4}\le e^1$ (by assumption $x/y^2\le 1$) and write
$$\lim_{(x,y)\rightarrow(0,0)} y^4e^{x^2/y^4}\leq e \lim_{(x,y)\rightarrow(0,0) }y^4 =0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/803325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Integral $I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3}$ Hi how can we prove this integral below?
$$
I:=\int_0^1 \frac{\log^2 x}{x^2-x+1}\mathrm dx=\frac{10\pi^3}{81 \sqrt 3}
$$
I tried to use
$$
I=\int_0^1 \frac{\log^2x}{1-x(1-x)}\mathrm dx
$$
and now tried changing variables to $y=x(1-x)$ ... | Hint:
Consider the change of variable $x=1/t$ hence you have
$$2I = \int^\infty_0 \frac{\log^2(t)}{t^2-t+1}\,dt$$
Now integrate the function
$$f(z) =\frac{\log^3(z)}{z^2-z+1}$$
Along a key hole contour indented at 0
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/803389",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "31",
"answer_count": 8,
"answer_id": 2
} |
Optimal Strategy for Rock Paper Scissors with different rewards Imagine Rock Paper Scissors, but where winning with a different hand gives a different reward.
*
*If you win with Rock, you get \$9. Your opponent loses the \$9.
*If you win with Paper, you get \$3. Your opponent loses the \$3.
*If you win with Sciss... | If "optimal" means Nash equilibrium (i.e. a state that is stable wrt. small perturbations of strategies), than it can be computed. If you assume that $x_1$ is the probability of first player to play Rock, $x_2$ his probability to play Scissors and $1-x_1-x_2$ his probability to play Paper, and similarly for $y_i$, then... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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A fair coin is flipped 2k times. What is the probability that it comes up tails more often than it comes up heads? I'm studying for a probability exam and came across this question. I watched the video solution to it but I don't really understand it. I was hoping someone could explain this problem to me. Are there diff... | Hint:
*
*Fair coin $\implies$ Probability of tails occurring more $=$ probability of heads occurring more $= p$, say.
*Probability of exactly equal number of heads and tails $=1-2p$. Can you find this one?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Sentence $\varphi$ of set theory that is satisfied by all well-founded models of ZFC, but which is not a theorem of ZFC. I think I read somewhere the following.
If a first-order sentence $\varphi$ in the language of set theory holds for every well-founded model of ZFC, then nonetheless:
*
*$\varphi$ may fail for a n... | Every statement which is in its essence a true [first-order] number theoretic statement in the universe must be true for every well-founded model. The most striking example for these statements are consistency of various theories.$\DeclareMathOperator{\con}{con}$
For example, if there are well-founded models of $\sf ZF... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why does the imag. part of the graph of $\zeta(n^{ix})$ resemble the tangent function? If you input $\zeta(n^{ix})$ into the Wolfram Alpha search bar, in the plot, you get an infinitely repeating sinusoidal curve, which resembles the real part, and you get an infinitely repeating tangent curve, which resembles the imag... | Two facts explain the qualitive picture from your link. First: $\zeta(z)\;$ has a pole at $z=1= e^{i\cdot 0}\;$ with $\zeta(e^{ix})=-\frac{i}{x} + \dots\;$ for $x\approx 0,\;$ second: $e^{ix}=e^{i(x+2\pi)},\;$ this gives the periodic structure.
Further: Since $\zeta(-1)=-\frac{1}{12}\;$ you have $\Im \zeta(e^{i\pi})=0.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Surely You're Joking, Mr. Feynman! $\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx$
Prove the following
\begin{equation}\int_0^\infty\frac{\sin^2x}{x^2(1+x^2)}\,dx=\frac{\pi}{4}+\frac{\pi}{4e^2}\end{equation}
I would love to see how Mathematics SE users prove the integral preferably with the Feynman way (other methods a... | This integral is readily evaluated using Parseval's theorem for Fourier transforms. (I am certain that Feynman had this theorem in his tool belt.) Recall that, for transform pairs $f(x)$ and $F(k)$, and $g(x)$ and $G(k)$, the theorem states that
$$\int_{-\infty}^{\infty} dx \, f(x) g^*(x) = \frac1{2 \pi} \int_{-\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/803954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 3,
"answer_id": 2
} |
get an element by finitely generated set I want to know the method to get a element in a finitely generated group by its generated set, is there a general way to calculate? For example, $SL(2,\mathbb{Z})=<a,b|a=\begin{pmatrix}0 &1\\-1 &0\end{pmatrix}, b=\begin{pmatrix}1&1\\-1&0\end{pmatrix}>$, how to write $\begin{pmat... | The question in your first paragraph does not quite make sense: how is the element to be given in general, if not by a product of generators? In specific instances that question could make sense, such as the instance of $SL_2(\mathbb{Z})$ where elements are given by matrices.
For the special case of $SL(2,\mathbb{Z})$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that two paths on opposing corners of the unit square must cross. I'm looking for a simple argument to the following:
Given two (continuous) paths on the unit square, one from (0,0) to (1,1) and the other from (1,0) to (0,1), prove that the paths cross at some point $(x_0, y_0)$.
I have constructed a topological ... | The result is Lemma 2 of this paper by Maehara, which uses the Brouwer Fixed Point Theorem. (I'm not sure if this argument qualifies as "simple" since this theorem is usually proved using homology.) See also this MO thread.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/804150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How can we prove that the rank of a matrix is a non-convex function of that matrix? How can we prove that $\operatorname{rank}(\mathbf{X})= 1$ is a non-convex function of $\mathbf{X}$.
| It seems pretty clear that if we take $X = \begin{bmatrix} 1 & 0 \\ 0 & 0\end{bmatrix}$ and $Y = \begin{bmatrix} 0 & 0 \\ 0 & 1\end{bmatrix}$, then $\operatorname{rank}(tX+(1-t)Y) = 2$ for $t\ne 0,1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/804246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Checking if a monic polynomial can be decomposed into linear factors I have questions about how to determine if a polynomial can be decomposed into linear factors. If it is not solvable over radicals by Galois Theory, then I am done. But do I have to resort to Galois Theory?
Let the polynomial be:
$$f(x) = x^5 + a x... | There are general formulas for the solutions of equations such as:
$ax + b = c$
$ax^2 + bx + c = 0 $
$ax^3 + bx^2 + cx + d = 0$
$ax^4 + bx^3 + cx^2 + dx + e = 0$
In other words, general radical solutions exist up to fourth order solutions. Whether you need to use Galois theory actually depends on the polynomial itse... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804401",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Do these integrals have a closed form? $I_1 = \int_{-\infty }^{\infty } \frac{\sin (x)}{x \cosh (x)} \, dx$ The following integrals look like they might have a closed form, but Mathematica could not find one. Can they be calculated, perhaps by differentiating under the integral sign?
$$I_1 = \int_{-\infty }^{\infty } \... | For $I_2$, we can use a well-known result:
$$ \int_{-\infty }^{\infty } \frac{\sinh (ax)}{\sinh(bx)}dx=\frac{\pi}{b}\tan\frac{a\pi}{2b}. $$
Note $\sinh(ix)=\sin(x), \tanh(ix)=\tan(x)$. Thus
$$ \int_{-\infty }^{\infty } \frac{\sin (ax)}{\sinh(bx)}dx=\int_{-\infty }^{\infty } \frac{\sinh (iax)}{\sinh(bx)}dx=\frac{\pi}{b}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 5,
"answer_id": 0
} |
How to prove Ass$(R/Q)=\{P\}$ if and only if $Q$ is $P$-primary when $R$ is Noetherian? Let $R$ be a Noetherian ring, $P$ be a prime ideal, and $Q$ an ideal of $R$. How to prove that
$$
\text{Ass}(R/Q)=\{P\}
$$
if and only if $Q$ is $P$-primary?
Update
In fact, I have proved that if $Q$ is primary, then Ann$(R/Q)$ is ... | Here's a relatively elementary proof, which is (in my opinion) one of many extremely beautiful proofs in the theory of associated primes and primary decomposition:
An ideal $Q$ is primary iff every zerodivisor in $R/Q$ is nilpotent, i.e. the set of zerodivisors is equal to the nilradical. Since zerodivisors are a union... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to find determinant of this matrix? Is there a manual method to find $\det\left(XY^{-1}\right)$ ?
Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\
1 & 3 & 3^2 & \cdots & 3^{2012} \\
1 & 4 & 4^2 & \cdots & 4^{2012} \\
\vdots & \vdots & \vdots & \cdots & \vdots \\
1 & 2014 & 2014^2 & \cdo... | Consider something a bit more general. Let $$X=\left[ {\begin{array}{cc} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\
1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\
1 & x_3 & x_3^2 & \cdots & x_3^{n-1} \\
\vdots & \vdots & \vdots & \cdots & \vdots \\
1 & x_n & x_n^2 & \cdots & x_n^{n-1} \\ \end{array} } \right], $$
and
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/804657",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Very good linear algebra book. I plan to self-study linear algebra this summer. I am sorta already familiar with vectors, vector spaces and subspaces and I am really interested in everything about matrices (diagonalization, ...), linear maps and their matrix representation and eigenvectors and eigenvalues. I am looking... | S. Winitzki, Linear Algebra via Exterior Products (free book, coordinate-free approach)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/804716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "58",
"answer_count": 12,
"answer_id": 9
} |
How to verify whether a solution to an optimization problem is correct. Consider a general optimization problem
min f(x)
subject to g(x) <=0
h(x)=0,
where x denotes a vector and the functions are $R^n$ -> $R^n$.
suppose somebody gave me a solution x*, how can I verify whether this solution is correct?
One strai... | You could try and plot it (using some mathematics software) and see if the solution is actually a minimum within the constrains.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/804801",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Any open interval in R is union of intervals of the form (a,b] As part of a proof that the Borel Set $B\mathbb(R)$ is generated by the collection of subintervals of the reals of the form $(a,b]$, my measure theory textbook (Cohn) asserts that any open interval $(x,y)$ can be written as the union of a sequence of sets o... | Hint: Consider $\bigcup_{n\in\mathbb{N}}{(a,b-\frac{1}{n}]}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/804893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Sage usage to calculate a cardinality I would like to compute the cardinality of an elliptic curve group over the finite field $\mathbb{F}_{991}$. I'm trying to use sage but I still have an error in the syntax (I never used it before and I tryed to adapt a code). Here is what I have:
sage: E = EllipticCurve(GF(991))
sa... | Alternatively, you can use MAGMA online. I usually do it like this:
K:=GF(991);
g:=Generator(K);
E:=EllipticCurve([0,0,0,446*g,471*g]);
#E;
and MAGMA returns 984.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/805075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Domain of the Function Square Root of 12th Degree Polynomial Find the Domain of $$f(x)=\frac{1}{\sqrt{x^{12}-x^9+x^4-x+1}}$$
My Try: The Domain is given by
$$x^{12}-x^9+x^4-x+1 \gt 0$$ $\implies$
$$x(x-1)(x^2+x+1)(x^8+1)+1 \gt 0$$
Please help me how to proceed further..
| If $x$ is out of the interval $[0,1]$ all the facotrs in $$x(x-1)(x^2+x+1)(x^8+1)+1$$
are positive so we have $$x(x-1)(x^2+x+1)(x^8+1)+1\gt 0$$
But for $x\in [0,1],$ $$x(x-1)\geq -\frac{1}{4}$$
So
$$x(x-1)(x^2+x+1)(x^8+1)+1\geq -\frac{1}{4}\cdot (x^2+x+1)(x^8+1)+1\geq -\frac{1}{4}\cdot\frac{3}{4}.1+1=\frac{13}{16}$$
i.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Decimal form of irrational numbers In the decimal form of an irrational number like:
$$\pi=3.141592653589\ldots$$
Do we have all the numbers from $0$ to $9$. I verified $\pi$ and all the numbers are there. Is this true in general for irrational numbers ?
In other words, for an irrational number
$$x=\sum_{n\in \mathbb{Z... | This gives a nice opportunity to use cardinality arguments to show that the answer is negative.
Pick two digits $n,k$ such that $\{n,k\}\neq\{0,9\}$. There is a bijection between the numbers in $[0,1]$ whose decimal form includes only $n$ and $k$, and the set of infinite binary sequences.
Therefore the set $\{x\in[0,1]... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 3
} |
Proving that the harmonic p series converges for p>1 and diverges for p<=1 Can someone please check if I have done this correctly?
The harmonic p-series:
$$ \sum_{n=1}^\infty \frac{1}{n^p}$$
$$ let $$
$$f(n)=\frac{1}{n^p}$$
$$ f(x)=\frac{1}{x^p}$$
Since f(x) is a positive, decreasing, continuous function, applying the ... | You're almost correct but take care with the limits of the integral:
the given series has the same nature (being convergent or divergent) as the integral
$$\int_{\color{red}{\pmb1}}^\infty\frac{dx}{x^p}$$
which's convergent if and only if $p>1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/805477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the inverse of the $\mbox{vec}$ operator? There is a well known vectorization operator $\mbox{vec}$ in matrix analysis.
I've vectorized my matrix equations, did some transformation of vectorized equations and now I want to get back to the matrix form. Is there special operator for it?
| Adding to the excellent answer by Rodrigo de Azevedo, I would like to point out that there is an explicit formula for the inverse $\operatorname{vec}_{m\times n}^{-1}$, given by
$$
\mathbb{R}^{mn}\ni x \mapsto \operatorname{vec}_{m\times n}^{-1}(x) = \big((\operatorname{vec} I_n)^\top \otimes I_m\big)(I_n \otimes x) \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 2,
"answer_id": 0
} |
The ability of a logical statement to represent a two-place truth function. How can i determine which two-place truth functions can be represented using a logical statement built out of a subset of two logical connectors in $ \{\rightarrow, \wedge, \vee ,\equiv \}$ ?
for example $\{\rightarrow, \wedge\}$
| For any two place truth function X, we can write it's truth table as follows:
p q X(p, q)
0 0 ?1
0 1 ?2
1 0 ?3
1 1 ?4
where, of course, ?1, ?2, ?3, and ?4 belong to {0, 1}. Notice that all wffs of propositional logic can get built up from the variables and the connectives. For example (using Poli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805634",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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find all values of k for which A is invertible $\begin{bmatrix}
k &k &0 \\
k^2 &2 &k \\
0& k & k
\end{bmatrix}$
what I did is find the det first:
$$\det= k(2k-k^2)-k(k^3-0)-0(k^3 -0)=2k^2-k^3-k^4$$
when $det = 0$ the matrix isn't invertible
$$2k^2-k^3-k^4=0$$
$$k^2(k^2 +k-2)=0$$
$$k^2+k-2=0$$
$$(k+2)(k-1)=0$$
$k... | Almost all square matrices are invertible. It is very special, i.e. singular, for a square matrix to be non-invertible. As you say, $\det = 2k^2-k^3-k^4$. This factorises to give $k^2(2+k)(1-k)$.
Your matrix is invertible for all values of $k$ except $k=0$, $k=-2$ or $k=1$. The matrix is invertible for all of the other... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Markov property for a Stochastic Process My question: Every Stochastic Process $X(t), t\geq 0$ with space states $\mathcal{S}$ and independent increments has the Markov property, i.e, for each $\in \mathcal{S}$ and $0\leq t_0\leq< t_1<\cdots <t_n<\infty$ we have
$$
P[X(t_n)\leq y|X(t_0),X(t_1), \ldots, X(t_{n-1})] =P[X... | For every nonnegative $s$ and $t$, let $X^t_s=X_{t+s}-X_t$, then the hypothesis is that the processes $X^t=(X^t_s)_{s\geqslant0}$ and ${}^t\!X=(X_s)_{s\leqslant t}$ are independent.
For every $s\geqslant t$, $X_s=X_t+X^t_{s-t}$ hence the process $(X_s)_{s\geqslant t}$ is a deterministic function of the random variable... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
A logarithmic integral $\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx$ How to prove the following
$$\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx=\frac{\pi^2}{2}$$
I thought of separating the two integrals and use the beta or hypergeometric functions but I thought these are ... | After this corps has rised from the dead anyway, let me give an additional solution which is based on complex analysis but avoids the use of branch cuts, so it should be viewed as complementary to @RandomVariables approach.
First, perform an subsitution $x\rightarrow\sin(t)$ which brings our integral into the form
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805893",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 10,
"answer_id": 5
} |
What does the dot product of two vectors represent? I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent.
The product of two numbers, $2$ and $3$, we say that it is $2$ added to itself $3$ times or something like that.
But when it c... | First of all, if we write $\vec{a} = a \vec{u}$ and $\vec{b} = b \vec{v}$,
where $a$ and $b$ are the length of $\vec{a}$ and $\vec{b}$ respectively,
then $$\vec{a} \cdot \vec{b} = (a \vec{u})\cdot (b \vec{v})
= ab \,\, \vec{u} \cdot \vec{v};$$
this is a pretty natural
property for a product to have.
Now as for $\vec{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/805954",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "124",
"answer_count": 12,
"answer_id": 0
} |
Is injective function $f:A \to A$ always surjective? Ok so while browsing a book(namely Herbert Endertons book "Elements of set theory") I have stumbled upon a curiosity which provoked me to try to prove this.Here is how I went about it,but I do not think my solution is correct.
All answers as well as corrections are m... | You statement works for finite $A$. If
$$
f: A\to A
$$
is injective then it is surjective.
This is not true when $A$ is infinite. Consider for example the funcstion $f : \mathbb{Z} \to \mathbb{Z}$ given by $f(x) = 2x$. This function is injective, but not surjective,
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/806016",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Help Understanding Evaluation of Integral Please help me to understand the evaluation of this integral.
$$\int_0^1\int_u^{\mathrm{min(1,u+z)}} 2\;dv\;du$$
I know that the correct answer is
$$
f(z) = \left\{
\begin{array}{lr}
1 & & z \geq 1\\
2z & & z \leq 0\\
-(z-2)z && \mathrm{else}
... | Note that the limits of integration on $u$ are $u \in [0,1]$. So for $u$ in this interval, and for $z \ge 1$, it follows that $u + z \ge 1$, hence $\min\{1, u+z \} = 1$, and the integral becomes $$\int_{u=0}^1 \int_{v = u}^1 2 \, dv \, du = \int_{u=0}^1 2(1-u) \, du = 1.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/806110",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integral $\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a}$ I am trying to prove this interesting integral$$
\mathcal{I}:=\int_0^{\pi/4}\frac{dx}{{\sin 2x}\,(\tan^ax+\cot^ax)}=\frac{\pi}{8a},\qquad \mathcal{Re}(a)\neq 0.
$$
This result is breath taking but I am more stumped than usual. It truly is ... | $\newcommand{\+}{^{\dagger}}
\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
\newcommand{\dd}{{\rm d}}
\newcommand{\down}{\dow... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/806195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 2
} |
Using Integration By Parts results in 0 = 1 I've run into a strange situation while trying to apply Integration By Parts, and I can't seem to come up with an explanation. I start with the following equation:
$$\int \frac{1}{f} \frac{df}{dx} dx$$
I let:
$$u = \frac{1}{f} \text{ and } dv = \frac{df}{dx} dx$$
Then I find... | The problem here is that when you applied the By Parts Formula
$$uv-\int \frac{du}{dx}vdv $$
you took $u=\frac 1f$ and $dv=\frac {df}{dx}dx$ Now when you use by parts formula your first task is to get $v$ and for that you need to do this $\int dv$, right? You have done all righty, but the problem comes when you writ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/806254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 5,
"answer_id": 4
} |
Prove that $A^k = 0 $ iff $A^2 = 0$ Let $A$ be a $ 2 \times 2 $ matrix and a positive integer $k \geq 2$. Prove that $A^k = 0 $ iff $A^2 = 0$.
I can make it to do this exercise if I have $ \det (A^k) = (\det A)^k $. But this question comes before this.
Thank you very much for your help!
| The solution using the minimal polynomial and Cayley-Hamilton is a bit of an over-kill and somewhat of a magic solution. I prefer the following non-magic solution (for the non-trivial implication, and there is no need to assume the matrix is $2\times 2$).
Think of $A$ as a linear operator $A: V \to V$ with $V$ an $n$-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/806394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 8,
"answer_id": 2
} |
Sum of products of binomial coefficient $-1/2 \choose x$ I am having trouble with showing that
$$\sum_{m=0}^n (-1)^n {-1/2 \choose m} {-1/2 \choose n-m}=1$$
I know that this relation can be shown by comparing the coefficients of $x^2$ in the power series for $(1-x)^{-1}$ and $(1+x)^{-1/2} (1+x)^{-1/2}$.
| Consider the sum
\begin{align}
S_{n} = (-1)^{n} \sum_{m=0}^{n} \binom{-1/2}{m} \binom{-1/2}{n-m}.
\end{align}
Using the results:
\begin{align}
\binom{-1/2}{m} &= \frac{(-1)^{m}(1/2)_{m}}{m!} \\
\binom{-1/2}{n-m} &= \frac{(-1)^{n-m} (1/2)_{n-m}}{(n-m)!} = (-1)^{n+m} \frac{(1/2)_{n} (-n)_{m}}{(1)_{n} (1/2-n)_{m}} \\
\end... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/806605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is a proof still valid if only the writer understands it? Say that there is some conjecture that someone has just proved.
Let's assume that this proof is correct--that it is based on deductive reasoning and reaches the desired conclusion.
However, if he/she is the only person (in the world) that understands the proof,... | There only appears to be a problem because we are using the same word for closely-related but distinct concepts (not an uncommon situation in philosophy), namely
*
*"proof" as in formal proof, which Wikipedia defines as
a finite sequence of sentences each of which is an axiom or follows from the preceding sentences... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "52",
"answer_count": 16,
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Sum of these quotient can not be integer Suppose $a$ and $b$ are positive integers such that are relatively prime (i.e., $\gcd(a,b)=1$).
Prove that, for all $n\in \mathbb{N}$, the sum
$$
\frac{1}{a}+\frac{1}{a+b}+\frac{1}{a+2b}+\cdots+\frac{1}{a+nb}
$$
is not an integer.
I think I have tried many w... | A good way to demonstrate it could be to find an integer K such that, multiplying the sum by K, the result is not an integer. Finding this K would clearly imply that the sum is not an integer.
For example, we could choose as K a value obtained starting from the product $a(a+b)(a+2b)(a+3b)...(a+nb)$ and eliminating one ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/806711",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
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Comparing two linear functions Let $X$ be a Banach space and let $h:X\to\Bbb C$ and $f:X\to\Bbb C$ be two bounded linear functions such that if for some $x\in X$ we have $f(x)=0$ then $h(x)=0$. Prove that there exists a $\lambda\in \Bbb{C}$ such that for any $x\in X$ we have $h(x)=\lambda f(x)$.
| If $f=0$ then choose $\lambda=0$. If $f\neq0$ then let $x_0\in X$ such that $f(x_0)\neq0$ or we may suppose that $f(x_0)=1$. For any $x\in X$ we have $f(x-f(x)x_0)=0$ and so $h(x-f(x)x_0)=0$. This means $h(x)=f(x)h(x_0)$. Thus it's enough to set $\lambda=h(x_0)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Propositional logic De-morgans theorem question the theorem states that $(A\wedge B) = \neg (\neg A\vee \neg B)$, where $A$ and $B$ are propositional formulas.
Can't I turn $\neg (\neg A\vee \neg B)$ to $(\neg \neg A\vee \neg \neg B)$ then cancel the double negations so its $(A\vee B)$, because that seems to be allowed... | $$\neg(\neg A \vee \neg B) \iff \neg\neg A \wedge \neg\neg B \iff A \wedge B $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/806897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Derive a transformation matrix that mirrors the image over a line passing through the origin with angle $\phi$ to the $x$-axis. Question: Using homogeneous coordinates, derive a $3$x$3$ transformation matrix $M$ that mirrors an image over a line passing through the origin, with angle $\phi$ to the $x$-axis.
Comment: Th... | If vector $A$ is reflected across vector $B$ to create vector $C$,
*
*The midpoint of $A$ and $C$ is along $B$:
$C + (A - C)/2 \in kB$, so $A + C \in kB$
*Length is preserved: $|A| = |C|$
Your $B$ vector is $\begin{bmatrix} \cos(\phi) \\ \sin(\phi)\end{bmatrix}$
First consider the x-axis unit: $e_1 = \begin{bmat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/807031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Equivalent norm in Sobolev space Let $\rho\in H^{1}(0,\pi)$ be a function, and consider the functional
$$
I(\rho)=\bigg(\int_{0}^{\pi}{\sqrt{\rho^2(t)+\dot\rho^2(t)}\,dt}\bigg)^2.
$$
I'm asking if it is equivalent to the norm
$$
\lVert \rho \rVert_{H^1}=\lVert \rho \rVert_{L^2}+\lVert \dot\rho \rVert_{L^2}
$$
on $H... | To be precise, you are asking if $\sqrt{I(\rho)}$ is equivalent to $\|\rho\|_{H^1}$. As you noted, $\sqrt{I(\rho)}$ is dominated by $\|\rho\|_{H^1}$. However, the converse fails.
Consider $\rho(x)=\sqrt{x+\epsilon}$. Since $\rho'(x) = \dfrac{1}{2\sqrt{x+\epsilon}}$, we have $\|\rho\|_{H^1}\to\infty$ as $\epsilon \to... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculus Proof Unsure Let $$g(x)=x^2\sin\left(\frac{1}{x}\right)+\frac{1}{2}x$$ Show that $g'(0)>0$ but there is no neighborhood of $0$ on which $g$ is increasing. (More precisely, every interval containing $0$ has sub intervals on which g is decreasing).
For the first part, I used the limit definition of the derivati... | This is a very good question. Clearly $$g'(x) = 2x\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right) + \frac{1}{2},\,\,\text{if }x \neq 0$$ and $$g'(0) = \lim_{x \to 0}\frac{g(x) - g(0)}{x} = \lim_{x \to 0}x\sin\left(\frac{1}{x}\right) + \frac{1}{2} = \frac{1}{2}$$ Thus we have $g'(0) > 0$. But if we see the f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/807259",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Do we have such kind of estimates? Let $0<a_0\leq a(x)$ be a smooth function on $\mathbb{T=[0,2\pi]}$, and $a(0)=a(2\pi)$, then whether it holds that
$$
\int_{\mathbb{T}}a(x)|\partial_x\phi|^2 dx\geq \int_{\mathbb{T}}|\partial_xa|^2|\phi|^2 dx
$$
for all $\phi\in H_{per}^1(\mathbb{T})$ ? More precisely,
$$
\phi(0)=\p... | Controlling $|\phi|$ by $|\phi'|$ sounds reasonable. But we can't control $|a'|$ by $|a|$. Example: let $a(x) = 2+\sin nx$, where $n$ is large. As $n\to \infty$,
$$\int_{\mathbb{T}}a(x)|\partial_x\phi|^2 dx$$ stays bounded but $$\int_{\mathbb{T}}|\partial_xa|^2|\phi|^2 dx$$ blows up (unless $\phi\equiv 0$).
You may... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Permutation and Combination Puzzle - Spy Keypad Keypad
1 2 3
4 5 6
7 8 9
J. Bond has to break into the headquarters of an evil organization
and steal important documents. The documents are in a safe that
can only be opened by entering the correct code into the keypad,
which is a 3 × 3 grid as shown on the right.
Bond ... | First of all, note that there are $3$ classes of numbers: corner ($1,3,7,9$), edge ($2,4,6,8$) and centre ($5$). Now clearly a corner number can be followed by any $1$ of $2$ edge numbers. An edge number can be followed by any $1$ of $2$ corner numbers or a centre number. And a centre number can only be followed by $1$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Explaining something to the half I'm a private tutor in my free time, teaching some basic high school mathematics and I've often been asked: ''Why is something to the half equal to the root of that something?''.
And I'm having problems explaining it. I have an idea of why in my head but obviously this idea is not stron... | We have for $x\in\Bbb{R}_{>0}$ the functional equation $x^ax^b =x^{a+b}$, so $x^{\frac{1}{2}}x^{\frac{1}{2}}=x^{\left(\frac{1}{2}+\frac{1}{2}\right)}=x^{1}$. Since finding a square root of $x$ is equivalent to finding an $y\in\Bbb{R}$ with $y\cdot y=x$, we can conclude $\sqrt{x}=x^{\frac{1}{2}}$ (for the standard bran... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/807541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integration $\int_{-1}^1 \sqrt{\frac{r^2-x^2}{1-x^2}}dx$ I am interested in the following integral: (r is a constant)
$$\int_{-1}^1 \sqrt{\frac{r^2-x^2}{1-x^2}}dx$$
Initially I thought of a trigonometric substitution, or a substitution like $z^2=r^2-x^2$, but to no avail. Is it possible to find an analytical solution? ... | The antiderivative involves elliptic integrals (which are not the nicest I know). From there, the integral is given by
$$\int_{-1}^1 \sqrt{\frac{r^2-x^2}{1-x^2}}dx=2 r E\left(\frac{1}{r^2}\right)$$
provided that $\Re(r)\geq 1\lor \Re(r)\leq -1\lor r\notin \mathbb{R}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$ Let $m\in \mathbb{N}$ and $c_1,c_2,\ldots,c_m \in \mathbb{R}_+$. Show that $$\lim_{n\rightarrow \infty} \sqrt[n]{c_1^n+c_2^n+\ldots+c_m^n} = \max\{c_1,c_2,\ldots,c_m\}$$
My attempt: Since $$\lim_{n\rightarrow \infty} ... | You can see there is an error in your approach if you consider a simple example. Let $c_1=2$ and $c_2=\cdots=c_m=0$. Then
$$\lim_{n\to\infty}\sqrt[n]{c_1^n+c_2^n+\cdots+c_m^n}=\lim_{n\to\infty}\sqrt[n]{2^n}=2$$
but
$$\lim_{n\to\infty}\sqrt[n]{\max\{c_1,c_2\ldots,c_m\}}=\lim_{n\to\infty}\sqrt[n]{2}=1$$
so your first ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that the gradient of a unit vector equals 2/magnitude of the vector Let $\vec r=(x,y,z)$
Firstly find $\vec \nabla (\frac 1 r)$ where r is the magnitude of $\vec r$.
I think I've done this correctly to get $-x(x^2+y^2+z^2)^{-\frac32} \hat i-y(x^2+y^2+z^2)^{-\frac32} \hat j-z(x^2+y^2+z^2)^{-\frac32} \hat k$
Second... | Let $\vec r=(x,y,z)=v_x\hat{i}+v_y\hat{j}+v_z\hat{k}$ so that $v_x=x$, $v_y=y$ and $v_z=z$.
Note that the magnitude of the vector $\vec r$ is given by $$r=(v_x^2+v_y^2+v_z^2)^{1/2}=(x^2+y^2+z^2)^{1/2}$$
For the second part, the divergence operator on vector $\vec r$ (denoted by $\nabla\cdot\vec r$) results in a signe... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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in triangle abc the measure of angle b is 90o ac is 50o and bc is 14o which ratio represents the tangent of angle a Need help answering this question. in triangle abc the measure of angle b is 90o ac is 50o and bc is 14o which ratio represents the tangent of angle a
| Since you have a right triangle, you can use the pythagorean theorem to find side ab .
$ab^2+bc^2=ac^2$
$ab^2=500^2-140^2$.
Tangent is opposite over adjacent, so it would be $\frac{bc}{ab}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/807950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Subgroups of GL(2,C) isomorphic to Z Let $\mathbb Z\to \mathrm{GL}_2(\mathbb C)$ be an injective homomorphism. I'm wondering about the possibilities for the image of $\mathbb Z$.
I think the image is always conjugate to a subgroup of matrices of the form $$\left( \begin{array}{cc} \lambda_1 & b \\ 0 & \lambda_2\end{arr... | The Jordan normal form of the image of $1\in\mathbb Z$ is either
$$\begin{pmatrix}\lambda_1&0\\0&\lambda_2\end{pmatrix}, $$
which is the case you handled, or it is
$$\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}. $$
In the second case,
$$\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}^n=\begin{pmatrix}\lambda^n&... | {
"language": "en",
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Examples of quasigroups with no identity elements If you scroll to the bottom of this page, there is a table claiming quasigroups have divisibility but not identity (in general).
What would be some examples of quasigroups without an identity element?
| A finite quasigroup is essentially a Latin square used as a "multiplication" table. Consider for $n \gt 2$ a Latin square, and label the rows (resp. columns) with a permutation of the symbols not appearing in any row (resp. in any column). This determines a quasigroup without identity, if the entries of the Latin squ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why does an $n \times n$ rotation matrix have $\frac{1}{2}n(n-1)$ undetermined parameters? Consider an orthogonal transformation between Cartesian coordinate systems in $n$-dimensional space. The $n \times n$ rotation matrix
$$R = \left(a_{ij}\right)$$
has $n^2$ entries. These are not independent; they are related by... | If you want something a bit more concrete, pick one row at a time. The first row is any norm 1 vector, so that is $n-1$ parameters. The second row is any norm 1 vector perpendicular to the first row, so that is $n-2$ parameters. The $k$ row is any norm 1 vector perpendicular to the first $k-1$ columns, so that is $n... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why are integrals called integrals? What is the historical background for this term?
I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve. It seems to me a strange choice of word.
| "I cannot quite see what is *integral* about an integral"
From your statement above, it appears you are thinking of an alternate meaning of the word "integral." Specifically, A is integral to B if it is a necessary component of B (e.g., "this scene is integral to the plot").
But that is not how it is used in mathemati... | {
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"timestamp": "2023-03-29T00:00:00",
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Is there a standard recurrence relation to solve this? I have infinite supply of $m\times 1$ and $1\times m$ bricks.I have to find number of ways I can arrange these bricks to construct a wall of dimensions $m\times n$.
My problem is how can I approach the question? Is there a recurrence relation to describe the proble... | Hint. A recurrence relation would be a good idea.
*
*Place an $m\times1$ brick along the side of length $m$. What do you now have to do to complete the construction of the wall?
*Are there any other ways that you could have started?
Good luck!
| {
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"timestamp": "2023-03-29T00:00:00",
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Confusion over Matrix rotation I want to make a function in C++ that accepts an angle 'a', and a vector 'v' as arguments and returns a matrix. 'a' should represent the amount that is rotated around vector 'v', an arbitrary axis, and the matrix returned should contain values that will make the rotation around v by cert... | See Quaternion-derived rotation matrix here
$a=(a_x,a_y,a_z)$ is the axis, $c=\cos \theta$, $s=\sin \theta$ where $\theta$ is the angle of rotation.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Degrees of maps in algebraic topology Please can I have some tips on how to construct maps between topological spaces of a given degree? For example, how would you go about building a map of degree $3$ from $\mathbb{CP}^1\times\mathbb{CP}^2 \to \mathbb{CP}^3$? Or a map from $S^2\times S^2 \to \mathbb{CP}^2$ of even deg... | In general it is hard to write down maps of a given degree, or even to determine whether such maps exist.
I don't know if there is a map $\mathbb{CP}^1\times\mathbb{CP}^2 \to \mathbb{CP}^3$ of degree three, but there are maps $S^2\times S^2 \to \mathbb{CP}^2$ of even degree. In fact, every map $S^2\times S^2 \to \mathb... | {
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An equation that generates a beautiful or unique shape for motivating students in mathematics
Could anyone here provide us an equation that generates a beautiful or unique shape when we plot? For example, this is old but gold, I found this equation on internet:
$$
\large\color{blue}{ x^2+\left(\frac{5y}{4}-\sqrt{|x|... | Here is a way to generate bunches of intriguing (most often) periodic curves drawn by adding unit length complex numbers of the form
$$e^{2\pi i m} \ \ \ \text{with} \ \ \ m:=\dfrac{n}{a}+\dfrac{n^2}{b}+\dfrac{n^3}{c}$$
for $0 \le n < abc$, where $a,b,c$ are fixed positive integers.
Here are displayed some of them wit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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$SL(2, \mathbb F_3)$ does not have a subgroup of order $12$ Using the characteristic polynomial I can prove that $SL(2, \mathbb F_3)$ does not has an element of order $12$, but how can I prove that $SL(2, \mathbb F_3)$ does not has a subgroup of order $12$?
| Outline for a proof:
With the characteristic polynomial, you can see that $A^3 = I$ in $G = \operatorname{SL}(2,3)$ if and only if $tr(A) = -1$. Count that there are $8$ elements of order $3$ in $G$.
A subgroup $H$ of order $12$ would have to contain every element of order $3$. Conclude $H \cong A_4$.
On the other hand... | {
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Questions--Heat equation with $x>0,t>0$ I have the following problem:
$$u_t=u_{xx}, x>0, t>0$$
$$u(x=0,t)=0 , t>0$$
$$u(x,t=0)=f(x), x>0$$
The solution of the problem is:
$$u(x,t)=\int_0^{+\infty} a(k) \sin(kx) e^{-k^2t} dk$$
$$u(x,0)=f(x)=\int_0^{+\infty} a(k) \sin(kx) dk$$
$$\sin(k'x) f(x)= \sin(k'x) \int_0^{+\infty... | Of course use separation of variables:
Let $u(x,t)=X(x)T(t)$ ,
Then $X(x)T'(t)=X''(x)T(t)$
$\dfrac{T'(t)}{T(t)}=\dfrac{X''(x)}{X(x)}=-k^2$
$\begin{cases}\dfrac{T'(t)}{T(t)}=-k^2\\X''(x)+k^2X(x)=0\end{cases}$
$\begin{cases}T(t)=c_3(k)e^{-tk^2}\\X(x)=\begin{cases}c_1(k)\sin xk+c_2(k)\cos xk&\text{when}~k\neq0\\c_1x+c_2&\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 2
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Finding pure strategy and pay off matrix in game theory "A two person games begins with the random selection of an integer $x$ from the set {$1,2,3$}, each choice is equally likely. Then the two players, not knowing the value of $x$, simultaneously select integers from {$1,2,3$}. Each players objective is to choose an ... | You already did the hardest part, now write 3 payoff matrices.
The first matrix $M1$ corresponds to the game if Nature chose 1:
$$ \begin{array}{c|c|c|c|} P1\backslash P2\\\hline\\& 1 & 2 &3\\
\hline 1 & 0 & 1 & 1\\
\hline 2 &-1 &0 &1 \\
\hline 3 & -1 &-1 &0 \\ \hline\end{array} \tag{M1}$$
The second matrix $M2$ corre... | {
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$\alpha$ is a plane curve if and only if all its osculator planes intersect at one point Let $\alpha$ be a regular curve. Prove that $\alpha$ is plane if and only if all the osculator planes intersect at one point.
I know that $\alpha$ is plane iff the binormal vector is constant, or iff the osculator plane is the same... | HINT: Say all the osculating planes pass through the origin. This means that $$\alpha(s)=\lambda(s)T(s)+\mu(s)N(s)$$
for some functions $\lambda$, $\mu$. Now differentiate and use Frenet.
| {
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Find a formula for $1 + 3 + 5 + ... +(2n - 1)$, for $n \ge 1$, and prove that your formula is correct. I think the formula is $n^2$.
Define $p(n): 1 + 3 + 5 + \ldots +(2n − 1) = n^2$
Then $p(n + 1): 1 + 3 + 5 + \ldots +(2n − 1) + 2n = (n + 1)^2$
So $p(n + 1): n^2 + 2n = (n + 1)^2$
The equality above is incorrect, so e... | The issue here is that $p(n+1)$ is note the statement
$$
1+3+5+\cdots+(2n-1)+2n=(n+1)^2;
$$
it is the statement
$$
1+3+5+\cdots+(2n-1)+(2n+1)=(n+1)^2.
$$
Why? The left side of your formula is the sum of all odd numbers between $1$ and $2n-1$. So, when you replace $n$ by $n+1$, you get the sum of all odd numbers b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Algebra Manipulation Contest Math Problem The question was as follows:
The equations $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ have two roots in common. Compute the product of these common roots.
Because $x^3+Ax+10=0$ and $x^3+Bx^2+50=0$ it means that $x^3+Ax+10=x^3+Bx^2+50$
Take $x^3+Ax+10=x^3+Bx^2+50$ and remove $x^3$ from ... | Hint: The common roots must be both roots of
$$- (x^3 + Ax +10 ) + (x^3 + Bx^2 + 50) = Bx^2 - Ax + 40 $$
Let this quadratic polynomial be denoted by $f(x)$.
Hint: We have
$$ f(x) ( \frac{1}{B} x + \frac{5}{4} ) = x^3 + Bx^2 + 50. $$
This gives $B^2 = 4A$ and $160=5AB$, so $5B^3 = 640 $. This gives $B = 4 \sqrt[3]{2} $,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809170",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Limit with a big exponentiation tower
Find the value of the following limit:
$$\huge\lim_{x\to\infty}e^{e^{e^{\biggl(x\,+\,e^{-\left(a+x+e^{\Large x}+e^{\Large e^x}\right)}\biggr)}}}-e^{e^{e^{x}}}$$
(original image)
I don't even know how to start with. (this problem was shared in Brilliant.org)
Some of the ideas I ... | Let $ A=\exp\left(x+e^{-\left(a+x+e^x+e^{e^x}\right)}\right) $ and $B=\exp(x) $. Then we can easily conclude that $A/B$ tends to $1$, but a little more analysis allows us to infer that $A-B$ also tends to $0$.
We have $$A-B=B\cdot\frac{\exp(\log A-\log B) - 1}{\log A-\log B} \cdot(\log A-\log B) $$ and the middle fract... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
"answer_count": 4,
"answer_id": 3
} |
On proving events have nonempty intersection if the sum of their complement is smaller than 1 Suppose for Events $A_1, A_2,\ldots,A_n$ we have that:
$$\sum\limits_{i=1}^n {\mathbb P}(A^{c}_i) < 1 $$
Does this imply:
$$\bigcap_{i=1}^{n} A_i \neq \emptyset $$
I think it does, but I couldn't manage to prove it, anybody pl... | Here's a hint:
Start with n = 2. Suppose that
$$\sum\limits_{i=1}^2 {\mathbb P}(A^{c}_i) = 1$$
We also know that
$$P(A_1^c \cup A_2^c)=P(A_1^c)+P(A_2^c)-P(A_1^c \cap A_2^c) $$
Since probability is bounded at 1, it is clear that $P(A_1^c \cap A_2^c)$ must equal 0.
By De Morgan's laws, we know that
$$P(A_1^c \cap A_2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809322",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Computing eigenvalues with characteristic polynomial I have two questions about computing eigenvalues with the characteristic polynomial.
*
*Eigenvalues exist if and only if I can factor the polynomial?? For example, I know i can calculate the roots of $ t^2 - 3t + 3 $ but I would use a quadratic formula for that.... | Eigenvalues may be imaginary, if you calculate the characteristic polynomial and set it equal to zero to find the roots then just as with a quadratic which may have no real solutions the characteristic polynomial may have imaginary roots. Take a rotation matrix as an example Eigenvalues of a rotation
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/809410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Matrix multiplication: $X_{r \times c}$ and $Y_{c \times d}$ Matrix $X$ has $r$ rows and $c$ columns, and matrix $Y$ has $c$ rows and $d$ columns, where $r, c$, and $d$ are different. Which of the following must be false?
*
*The product $YX$ exists
*The product of $XY$ exists and has $r$ rows and $d$ columns
*Th... | It helps to visualize matrix multiplication:
(courtesy of Wikimedia)
The number of columns of the first multiplicand has to match the number of rows of the second multiplicand.
Looking at the three choices:
*
*The product YX exists
This would require that $d$ the number of columns in Y equals the number $r$ of
row... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to solve this quadratic congruence equation? Well, we have :
$$n^2+n+2+5^{4n+1}\equiv0\pmod{13}$$
i'm little bit confused, I think i can solve this using the reminders of $n^2$, $n$ and $5^{4n+1}$ over $13$, by the way I have no idea about the Chinese Reminder Theorem no need to use it. and thanks in advance
edit:
... | It is easy to see that
$$5^2\equiv -1 \pmod{13}$$
So, we have
$$5^4\equiv 1 \pmod{13}$$
Therefore, for any n, we have
$$5^{4n+1}\equiv 5\pmod{13}$$
So, the equation simplifies to
$$n^2+n+7\equiv 0\pmod{13}$$
Considering vieta, we check the factors of 7,20,33,...
Looking at the factors of 20 , we notice that 2 and 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809774",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Why is the Koch curve homeomorphic to $[0,1]$? Henning Makholm has provided a nice proof that the limiting curve is a continuous function from $[0,1]$ to the plane. I was curios if the function is homeomorphism. A quick search gave me many sources mentioning that $[0,1]$ and the Koch curve are indeed homeomorphic (as a... | The map from the interval onto the Koch curve is a continuous bijection from a compact space to a Hausdorff one. So it's closed since closed subsets of compact spaces are compact, images of compact spaces are compact, and compact subspaces of Hausdorff spaces are closed, and thus a homeomorphism.
To check this is indee... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809873",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
can $L^p$ norm convergence and pointwise monotonic imply pointwise convergence? Let $(f_n)_{n=1}^\infty$ be a sequence of measurable function such that $\lim_{n\to\infty}||f_n-f||_p=0$. If for any $x\in \Omega$, $\{f_{n}(x)\}_{n=1}^\infty$ is a monotonic sequence, can we deduce that $f_n\to f$ almost everywhere?
| For any $x\in\Omega$, the monotonic sequence $(f_n(x))$ has a limit, possibly equal to $-\infty$ or $+\infty$; so the sequence $(\vert f_n(x)-f(x)\vert)$ has a limit in $[0,\infty]$. Let $g$ be the measurable function (with values in $[0,\infty]$) defined by $g(x)=\lim_{n\to\infty} \vert f_n(x)-f(x)\vert^p$. By Fatou's... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/809949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Isn't it possible that $D_4$ has also a subgroup with $1$ element?? A consequence of the Lagrange theorem:
Let $G$ a finite group and $H$ a subgroup of G. Then $|H| \mid |G|$.
is that each subgroup $\neq <i_d>$ of $D_4$, which has $8$ elements , has either $2$ or $4$ elements..
But.... $1 \text{ divides also }8$..Is... | $D_4$ has a subgroup with one element as does every group: the trivial group $\langle id\rangle$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/810032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to find the Taylor series of $f(x)=\arctan x$. I want to find the Taylor series of $f(x)=\arctan x,\; x\in[-1,1],\;\xi=0$.
That's what I have tried do far:
$$f'(x)=\frac{1}{1+x^2}=\frac{1}{1-(-x^2)}=\sum_{n=0}^{\infty} (-x^2)^n.$$
How can I continue?
| Thus, we have
$$f(x)=\int f'(x)\,dx=\int\sum_{n\ge 0}(-1)^nx^{2n}\,dx\ =\\
=\ \sum_{n\ge 0}(-1)^n\frac{x^{2n+1}}{2n+1}\ +C$$
Then find $C$ by plugging in $x=0$.
What will you get if you plug in $x=1$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/810121",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How do i prove this?
Bartle - Introduction to Analysis p.275
Define $x_n = \frac{a(a+1)\cdots(a+n)b(b+1)\cdots(b+n)}{n!c(c+1)\cdots(c+n)}$.
Show there $\sum x_n$ converges if $c>a+b$ and diverges if $c<a+b$.
How do i prove this?
I was trying to apply Raabe's Test, so i found that $\lim n(|\frac{x_{n+1}}{x_n}| - 1) = ... | You have
$$\frac{x_n}{x_{n-1}}=\frac{n^2+(a+b)n+ab}{n^2+cn},\tag{1}$$
hence if $a+b\geq c$ you have:
$$ \frac{x_n}{x_{n-1}}\geq 1+\frac{ab}{n^2+cn}, \tag{2}$$
so:
$$ x_n \geq x_0\prod_{k=1}^{n}\left(1+\frac{ab}{n^2+cn}\right),\tag{3}$$
and since the product on the right is convergent $x_n$ cannot be infinitesimal, henc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810196",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Constructing Galois extensions. In Wiki Page I found the following statement.
A result of Emil Artin allows one to construct Galois extensions as follows: If $E$ is a given field, and $G$ is a finite group of automorphisms of $E$ with fixed field $F$, then $E$ over $F$ is a Galois extension.
Somebody please explain m... | First we show that $E/F$ is finite, of degree less than or equal to $m = |G|$.
Let $\alpha \in E$. Let $\alpha=\alpha_1,\dots\alpha_r$ be the orbit of $\alpha$ under $G$, $r \leq m$. Then define $f(X) = \Pi(X-\alpha_i)$. $f$ is separable by definition and is fixed by every element of $G$, since any element of $G$ simpl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How come the function and the inverse of the function are the same? What is the inverse of the function:
$$f(x)=\frac{x+2}{5x-1}$$
?
Answer:
$$f^{-1}(x)=\frac{x+2}{5x-1}$$
Can one of you explain how the inverse is the same exact thing as the original equation?
| The inverse is not in general "the same exact thing as the original equation".
Generally, $f(x)\ne f^{-1}(x)$, but this is not always true. For example, consider the function $f(x) = -x$. This function is just the function that negates its input. Of course, if your negate your input twice, you get the original input. P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 1
} |
optimal way to approximate second derivative Suppose there is a function $f: \mathbb R\to \mathbb R$ and that we only know $f(0),f(h),f'(h),f(2h)$ for some $h>0$. and we can't know the value of $f$ with $100$% accuracy at any other point.
What is the optimal way of approximating $f''(0)$ with the given data?
I'd say th... | Let's say that $$y_k=\{f(0),f(h),f'(h),f(2h)\}$$ are given and
$$x_k=\{f(0),f'(0),f''(0),f'''(0)\}$$ are unknown. Taylor's theorem
gives a way of writing each $y_k$ as a linear combination of $x_j$'s, dropping all the terms starting with $f^{(4)}(0)$. For example:
$$f(2h)=f(0)+f'(0)(2h)+\frac12f''(0)(2h)^2+\frac16f'''(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810454",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
translate sentences into logic If I would like to translate the English sentence below into a predicate logic formula
"The parents of a green dragon are green"
Using predicates dragon, childOf and green, how would I go about this?
I understand that it may help to work the sentence into something that looks like logic,... | Being a parent is a relation. Let $D$ be the predicate of being a dragon, $G$ being the predicate of being green and $P$ being the "is a parent of" relation.
Then we have: $\forall x\forall y (G(x)\wedge D(x)\wedge P(y,x))\implies G(y)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/810541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Partial fractions on $(cx^2+dx+e)^n$ If I have
$$\frac{ax+b}{(cx^2+dx+e)^n}$$
with real coefficients and $(cx^2+dx+e)$ has complex roots, what does
$$\frac{ax+b}{[c(x-\alpha)(x-\alpha^*)]^n}$$
turn into, in terms of partial fractions?
| Based on differentiating this answer w.r.t $x$, you have:
$$ \frac{1}{ \left( x-\mu \right) ^
{1+n} \left( x-\nu \right) ^
{1+n} }=\sum _{m=0}^{n}{2\,n-m\choose n} \left( -\nu+\mu
\right) ^{m-1-2\,n} \left( {\frac { \left( -1 \right) ^{n+m}}{
\left( -x+\nu \right) ^{m+1}}}-{\frac { \left( -1 \right) ^{n}}{
\left( -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Arrangements of children Can somebody please double check my work?
$n$ children must be arranged in a line. $k$ pairs of children want to be next to each other, and each member of the pair will be unhappy if they are not next to each other. The other children don't care where they are (i.e., they will be happy anywhe... | We need to choose where the leftmost of each pair of fussy children will sit. Write down $n-k$ stars, where the other $n-k$ children will sit, like this
$$\ast\quad\ast\quad\ast\quad\ast\quad\ast\quad\ast\quad\ast\quad\ast\quad\ast\quad\ast$$
There are $n-k$ positions where these leftmost children can sit: In one of th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Computing the coordinates of a Bezier Curve I just started messing with Bezier Curves over the past couple days and I'm trying to get some of the basics down. I have this problem.
Consider a quadratic Bezier curve with control points (0, 0), (2, 2), and (4, 0).
What are the coordinates of the curve at t = 0.3?
How... | The curve point $\mathbf{C}(t)$ at parameter value $t$ is given by the standard formula
$$
\mathbf{C}(t) = (1-t)^2\mathbf{P}_0 + 2t(1-t)\mathbf{P}_1 + t^2\mathbf{P}_2
$$
In our case, we have $\mathbf{P}_0 = (0,0)$, $\mathbf{P}_1 = (2,2)$, $\mathbf{P}_2 = (4,0)$, and we're interested in the parameter value $t = 0.3$. Pl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/810926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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