Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How do I prove that $3<\pi<4$? Let's not invoke the polynomial expansion of $\arctan$ function.
I remember I saw somewhere here a very simple proof showing that $3<\pi<4$ but I don't remember where I saw it.. (I remember that this proof is also in Wikipedia)
How do I prove this inequality?
My definition for $\pi$ is th... | I assume that $x\to e^x$ is the unique differentiable function that is its own derivative and maps $0\mapsto 1$.
From this (especially using uniqueness) one quickly establishes $e^{x+y}=e^xe^y$, $e^{\bar z}=\overline{e^z}$ etc.
With the definitions $\cos x=\frac{e^{ix}+e^{-ix}}{2}$, $\sin x=\frac{e^{ix}-e^{-ix}}{2i}$ w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 6,
"answer_id": 1
} |
A proportionality puzzle: If half of $5$ is $3$, then what's one-third of $10$? My professor gave us this problem.
In a foreign country, half of 5 is 3. Based on that same proportion, what's one-third of 10?
I removed my try because it's wrong.
| Given that
$$
\begin{equation}
\tag{1}
\text{half of }5 = 3
\end{equation}
$$
this implies that
$$
\begin{equation}
\tag{2}
\text{half of }10 = 6
\end{equation}
$$
$(2)$ then says that half of $1 = 0.6$ and therefore
$$
\begin{equation}
\tag{3}
\frac{1}{3}\text{ of }1 = \frac{0.33 \times 0.6}{0.5} = 0.396
\end{equati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 14,
"answer_id": 8
} |
Quartets and parity There are 229 girls and 271 boys at a school. They are divided into 10 groups of 50 students each, with numbering 1 to 50 in each group. A quartet consists of4 students from 2 different groups so that there are two pairs of students having identical numbers. Show that the number of quartets with an ... | Let the groups be $G_1,G_2,\ldots,G_{10}$, with respectively $b_1,b_2,\ldots,b_{10}$ boys and $g_1,g_2,\ldots,g_{10}$ girls.
A quartet with an odd number of girls must have either (a) a pair of boys and a boy/girl pair, or (b) a pair of girls and a boy/girl pair. So, from two groups $G_i$ and $G_j$, the number of quar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Rotation of velocity vectors in Cartesian Coordinates I want to rotate a $(X,Y,Z)$ coordinate-system around it $Z$-axis. For the coordinates this can be done with the rotation matrix:
$$
R_Z(\theta)= \begin{pmatrix}
cos \theta & -\sin(\theta) & 0\\
\sin(\theta) & \cos(\theta) & 0 \\
0 & 0 & 1
\end{pmatrix}
$$
($R_Z(\... | Yes, but only because the rotation is about the origin.
The velocity is the time derivative of the position, so if the position at time $t$ is $\vec u(t)$, then the velocity is
$$\vec v(t) = \frac{d}{dt} \vec u(t) = \lim_{h\to 0} \frac1h ( \vec u(t+h) - \vec u(t) ) $$
If we compute this in the rotated coordinate system... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How many $2\times2$ positive integer matrices are there with a constant trace and positive determinant? The trace of a $2\times2$ positive integer matrix is a given constant positive value. How many possible choices are there such that the determinant is greater than 0? Each element of matrix is positive.
| Suppose we're looking for a positive integer matrix with trace $M$. The matrix must have the form
$$
\pmatrix{
M - n & a\\
b & n
}
$$
Where $a,b \geq 1$ and $1 \leq n \leq M-1$. For the determinant to be positive, we must have
$$
(M-n)n \geq ab
$$
So, we may compute the number of matrices with trace $M$ and a positiv... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Paths on a Rubik's cube
An ant is initially positioned at one corner of the Rubik's cube and wishes to go to the farthest corner of the block from its initial position. Assuming the ant stays on the gridlines, how many different paths are possible for it to get to the far corner. Assume that there is no backtracking (... | I think your idea is basically right. Remember, however, that there are two types of paths that have been double-counted: paths that pass through a corner adjacent to the initial corner, and paths that pass through a corner adjacent to the final corner. Therefore you need a factor of $6$ rather than $3$ in your secon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/740960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Hyperbolic Functions (derivative of $\tanh x$) $$\sinh(x) = \frac{1}{2(e^x - e^{-x})}$$
$$\cosh(x) = \frac{1}{2(e^x + e^{-x}}$$
$$\tanh(x) = \frac{\sinh (x)}{\cosh (x)}$$
Prove:
$$\frac{d(\tanh(x))}{dx} = \frac{1}{(\cosh x)^2}$$
I got the derivative for $\tanh(x)$ as:
$$\left[ \frac{1}{2(e^x + e^{-x})}\right]^2 - \fra... | $$\dfrac{d(f/g)}{dx}=\dfrac{gf^\prime-fg^\prime}{g^2}$$
Set $f=\sinh,g=\cosh$ to get
$$\dfrac{d\tanh}{dx}=\dfrac{\cosh\cdot\sinh^\prime-\sinh\cdot\cosh^\prime}{\cosh^2}$$
Now,
$$\sinh^\prime=\dfrac{1}{2}(e^x+e^{-x})=\cosh\\
\cosh^\prime=\dfrac{1}{2}(e^x-e^{-x})=\sinh$$
Thus,
$$\dfrac{d\tanh}{dx}=\dfrac{\cosh^2-\sinh^2}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/741050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
How many times to roll a die before getting $n$ consecutive sixes given $m$ occurences? Given a unbiased dice how to find the average number of rolls required to get $n$ consecutive sixes given already $m$ number of sixes occurred where $m\leq n$. I know how to solve using n consecutive sixes with out any occurrences f... | For $m\in\left\{ 0,\cdots,n\right\} $ let $\mu_{m}$ denote the expectation
of number of rolls required to get $n$ consecutive sixes on the moment
that the throwing of exactly $m$ consecutive sixes has just occurred. Then $\mu_{n}=0$
and for $m<n$ we find the recursion formula:
$$\mu_{m}=1+\frac{1}{6}\mu_{m+1}+\frac{5}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/741149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Limit and integral properties of a continuous function Let $f$ be a continuous function on $[0,\infty)$ such that $\displaystyle\lim_{x \to \infty}f(x)= c$.
Show that $\displaystyle\lim_{x \to \infty} \frac{1}{x}\int_0^x f(s)\;ds = c$.
I've tried splitting the integral into $\int_0^M+ \int_M^x$ but I don't really know ... | If $c \neq 0$, then notice that $\int_0^x f(s)\;ds$ approaches $\pm\infty$ as $x \to \infty$. So your limit is an indeterminate form of type $\frac{\infty}{\infty}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/741312",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Solving a certain differential equation when assuming a surface of revolution is minimal The problem is the following:
Consider the surface of revolution
$$
\textbf{q} (t, \mu) = (r(t)\cos(\mu),r(t)\sin(\mu),t)
$$
If $\textbf{q}$ is minimal, then $r(t) = a\cosh(t)+b\sinh(t)$ for $a,b$ constants.
I'll skip the calcul... | This ODE can be solved by separation of variables, To check that
$$r(t):=a\cosh(\dfrac{t-t_o}{a})$$ for $t_0\in\mathbb{R}$, $a>0$,
solve your ODE is easy.
Indeed
$$a\cosh()\frac{\cosh()}{a}=1+\sinh^2()$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/741402",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Cool property of the number $24$ Recently I've had my 24th birthday, and a friend commented that it was a very boring number, going from 23 which is prime, 25 which is the first number that can be written as the sum of 2 different pairs of squared integers $3^2+4^2 =0^2+5^2 =25$, 24 seems like a very boring number
howe... | 24 is a very special integer number in many regards. John Baez has a nice pdf file about it: [ http://math.ucr.edu/home/baez/numbers/24.pdf ], and there is a very good Youtube video you can watch: https://www.youtube.com/watch?v=vzjbRhYjELo
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/741466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
$g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$ If $g\in G$ is an element of maximal order in a finite abelian group $G$ then exists $H\leq G$ such that $G=\left<g\right>\oplus H$
Attempt: Using fundamental theorem I know that $G=C_{n_1}\times\cdots\times C_{n_k}$ where $n_i|n_{i+1}$. With some work I prov... | Here is a proof using the structure theorem. By taking a primary decomposition of $G$ we can assume that there is a prime $p$ and exponents $a_i \in \mathbb{N}$ such that $n_i = p^{a_i}$ for each $i$. Let
$$ G = \langle g_1 \rangle \oplus \ldots \oplus \langle g_k \rangle $$
where $g_i$ has order $p^{a_i}$ for each $i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/741577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Infinitely many proofs? While compiling a list of my favorite proofs of the infinitude of primes, the following came to mind;
Proposition: There are infinitely many non-isomorphic proofs of the infinitude of primes.
I'm not sure if this is true. Is it? How could one prove (or disprove) this?
I'm worried that because "n... | I think a sensible first step would be to agree on a formal notion of proofs, i.e. on some formally defined system like Natural Deduction. Then you could try to start the proof that there are infinitely many proofs of your proposition on this basis.
If you want to include some notion of non-trivial equivalence of proof... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/741683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Motivation behind steps in proof of Hoeffding Inequality The lemma that is proved for proving Hoeffding's inequality is:
If $a\leq X\leq b$ and $E[X]=0$, $E[e^{tX}] \leq e^{\frac{t^2(b-a)^2}{8}}$
Here's a link to the proof: http://www.stat.cmu.edu/~larry/=stat705/Lecture2.pdf
There's a particular step in the proof the ... | The definition $u:=t(b-a)$ naturally arises from the fact that $a\le X\le b$. $t$ is a reduced variable, independent of the interval length.
The unexpected form of $g(u)$ is a rewrite of $\log(\frac{-a}{b-a}e^{tb}+\frac{b}{b-a}e^{ta})=\log(e^{ta}(\frac{-a}{b-a}e^{t(b-a)}+\frac{b}{b-a}))$, with the intent to bound it fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/741768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Integrating partial fractions I have
$\int{\frac{2x+1}{x^2+4x+4}}dx$
Factorising the denominator I have
$\int{\frac{2x+1}{(x+2)(x+2)}}dx$
From there I split the top term into two parts to make it easier to integrate
$\int{\frac{2x+1}{(x+2)(x+2)}}dx$ = $\int{\frac{A}{(x+2)}+\frac{B}{(x+2)}}dx$
=$\int{\frac{A(x+2)}{(x+2)... | All you need to do is to solve this with respect to polynomials:
$2x+1=Ax+2A+Bx+2B$
$2x+1=x(A+B)+(2A+2B)$
$A+B=2 \rightarrow B=2-A$
$2A+2B=1$
$2A+4-2A=1\rightarrow 4=1$
This is contradiction! You have made an mistake in step where you split the term into two fractions, you should have done it like this:
$\frac{2x+1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/741861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Equation involving cosine I don't really know how to crack this one. Any help appreciated. $$\cos\left(d\sqrt{4-d^2}\right)=-\frac{d}2$$ Why is MSE forcing me to write more? Jeez.
| As Sabyasachi wrote, the domain is $d\in [-2,2]$. Obviously $d=-2$ is a solution but there is another one between $-2$ and $0$ which cannot be expressed analytically (I suppose). So the solution must be found using numerical methods such as Newton, provided a reasonable starting point. By inspection, the value of $$f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/741964",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Optimize function on $x^2 + y^2 + z^2 \leq 1$ Optimize $f(x,y,z) = xyz + xy$ on $\mathbb{D} = \{ (x,y,z) \in \mathbb{R^3} : x,y,z \geq 0 \wedge x^2 + y^2 + z^2 \leq 1 \}$. The equation $\nabla f(x,y,z) = (0,0,0)$ yields $x = 0, y = 0, z \geq 0 $ and we can evaluate $f(0,0,z) = 0$.
Now studying the function on the boun... | Another way:
$f$ is convex in each variable $x, y, z$ so the maximum can be obtained only on the boundary. This means $x^2+y^2+z^2=1$. Further,
$$4 \,f = 4xy(1+z) \leqslant 2(x^2+y^2)(1+z) = 2(1-z^2)(1+z) = (1+z)(2-2z)(1+z)$$
The RHS now is a product of three positive terms with a constant sum, so it gets maximised w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/742031",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Reason what 2^k in a sigma will do I am trying to solve the following calculation without a calculator:
$$\sum_{k=0}^82^k{8\choose k}$$
The first part:
$$\sum_{k=0}^8{8\choose k}$$
is equal to $2^8$. I already know that the answer will be $3^8$. How did the $2^k$ transform the answer from $2^8$ to $3^8$?
| Use that $$
(a+b)^n = \sum_{k=0}^n a^k b^{n-k} \binom{n}{k} \text{.}
$$
For $a=2$, $b=1$ you get the sum you want to compute, and the result is therefore $(2+1)^8$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/742149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Convergence of $\sum_{k=1}^n(1-k/n)a_k$ Assume that the series $\displaystyle \sum_{n=1}^\infty a_n$ converges to a finite number, say $S$. Now let's consider a sequence of modified partial sums $\displaystyle S_n=\sum_{k=1}^n(1-\frac{k}{n})a_k$.
It is easy to see that, if $\displaystyle \sum_{n=1}^\infty |a_n|<\infty... | No additional conditions other than the convergence of the original series are needed.
Since the sum of $a_k$ converges, for any $\epsilon\gt0$, there is an $N$ so that if $m,n\ge N$
$$
\left|\,\sum_{k=m}^na_k\,\right|\le\epsilon/2
$$
Therefore, for $m,n\ge N$,
$$
\begin{align}
\left|\,\frac1n\sum_{k=m}^nka_k\,\right|
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/742251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of automorphisms I'm having difficulties with understanding what automorphisms of field extensions are. I have the splitting field $L=\mathbb{Q}(\sqrt[4]3,i)$ of $X^4-3$ over the rationals.
Now I have to find $\#\mathrm{Aut}(L)$. Is this different from $\#\mathrm{Aut}_\mathbb{Q}(L)$? Also, how do I find the val... | An automorphism is determined by the permutation it induces on the sets $\{\sqrt[4]{3},i\sqrt[4]{3},-\sqrt[4]{3},-i\sqrt[4]{3}\}$ and $\{i,-i\}$. The permutation's restriction to the first set is determined by where it sends $\sqrt[4]{3}$ and how it acts on $i$ in the second set. This yields a list of eight possible ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/742337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How to represent this sequence mathematically? I need to represent the sequence of pairs $$(N,0), (N-1,1), (N-2,2), \ldots , \left( \frac{N}{2}, \frac{N}{2}\right) $$
in a way I can use in a formula. Is there any way to do this? Thanks!
| What about
$$(N-i,i),\quad i\in\{0,1,\ldots N/2\}$$
(assuming $N$ is even)?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/742448",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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ramification index divides $q-1$ (cyclotomic fields) Let $K$ be an abelian extension of $\mathbb{Q}$ with $[K:\mathbb{Q}] = p^m$. Suppose $q$ is a prime $\neq p$ which is ramified in $K$. Let $Q$ be a prime of $K$ lying over $q$.
Prove that $e(Q|q)$ divides $q-1$ and that the $q$th cyclotomic field has a unique subfie... | Since $K/\mathbb{Q}$ is galois, then $e=e(Q/q)$ must divide $[K:\mathbb{Q}](=efg).$ In the other hand by Kronecker-Weber theorem ther is some integre $m$ such that $K\subset\mathbb{Q}(\zeta_m)$ if $q^{\alpha}|| m$ we have $$q\mathcal{O}_K=q\mathbb{Z}[\zeta_m]=(1-\zeta_m)^{\varphi(q^{\alpha})}\;\;\text{and so}\;\;e(\mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/742624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Local max/min points, partial derivatives I'm having a lot of problems with figuring out how to properly do max/min with partial derivatives.
To my knowledge, we have:
$$D(x, y) = f_{xx}(x, y)f_{yy}(x, y) - (f_{xy}(x, y))^{2}$$
With the following conditions:
-If $D > 0$ and $f_{xx} < 0$, the critical point is a local m... | If $f(x,y)=2xye^{-x^2-y^2}$, then $$\frac{\partial f}{\partial x}=2ye^{-x^2-y^2}-4x^2ye^{-x^2-y^2}$$ Setting this equal to 0 gives $$y(1-2x^2)=0$$ and setting $f_y=0$ gives $$x(1-2y^2)=0$$ The possible points are $(0,0),\hspace{2mm}\left(\pm{1\over\sqrt{2}},\pm{1\over\sqrt{2}}\right)$ (the second point is really four d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/742818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Area bounded by two circles $x^2 + y^2 = 1, x^2 + (y-1)^2 = 1$ Consider the area enclosed by two circles: $x^2 + y^2 = 1, x^2 + (y-1)^2 = 1$
Calculate this area using double integrals:
I think I have determined the region to be $D = \{(x,y)| 0 \leq y \leq 1, \sqrt{1-y^2} \leq x \leq \sqrt{1 - (1-y)^2}\}$
Now I can't se... | (I recognize you asked for a method using double integrals; I'm leaving this here as "extra")
Using geometry, the area we want is the area of four one-sixth sectors of a circle with $r = 1$ subtracted by the area of two equilateral triangles of side length $s = 1$. This would be
$$ \frac{2}{3} \pi (1)^2 - 2 \frac{(1)^2... | {
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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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"Rigorous" definition of Cartesian coordinates I, like most, first learned about Cartesian coordinates very early on in my educational career, and so the most instructional way to think about them was that you place down some perpendicular lines and measure the perpendicular distance from each line to get your coordina... | I'm also surprised not to find any definition of Cartesian coordinate, so let's suggest this (personal and most simple) definition:
A cartesian coordinate on a Euclidean space E (finite dimensional $\mathbb{R}$ vector space, with a scalar product) is a "global chart" = Euclidean space isomorphism = isometry from $E$ t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/743087",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 1
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A basic problem on bounded variation If $a > 0$ let
$$f(x) =\left\{\begin{array}{ll}
x^{a} \sin (x^{-a})&\text{if } 0 < x \leq 1\\
0&\text {if }x=0
\end{array}\right.$$
Is it true that for each $0 < \alpha < 1$ the above function satisfies the Lipschitz condition of exponent $\alpha$
$$|f(x) - f(y)| \leq A|x-y|^{\al... | It is just an hint, not a complete proof. I hope it's enough for you. Consider the case $a=1$, the other are very similar. You can consider the following partition of $[0,1]$.
$$[0,1]= \left [\frac{1}{\pi},1 \right ]\cup\ \bigcup_{k=1}^n \left [\frac{1}{(k+1)\pi},\frac{1}{k\pi} \right ]\cup \left [0,\frac{1}{n\pi} \ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/743180",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
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Tell me problems that can trick you I am looking for problems that can easily lead the solver down the wrong path.
For example take a circle and pick $N$, where $N>1$, points along its circumference and draw all the straight lines between them. No $3$ lines intersect at the same point inside the circle. The question is... | Question 1)
You are building a straight fence 100 feet long. There is a fencepost every 10 feet. Fence panels are 20 feet long.
How many fence panels do you need?
How many fenceposts do you need?
Question 2)
You are fencing a rectangular area 100ftx100 ft. There is a fencepost every 10 feet. Fence panels are 20 feet lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/743272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "28",
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How to choose the starting point in Newton's method?
How to choose the starting point in Newton's method ?
If $p(x)=x^3-11x^2+32x-22$
We only learnt that the algorithm $x_{n+1}:=x_n-\frac{f(x_n)}{f'(x_n)}$ converges only in some $\epsilon$-neighbourhood of a root and that if $z$ is a root then $|z|\le 1+\max\limits_{... | The general case is very complicated. See for instance:
*
*Newton fractal
*How to find all roots of complex polynomials by Newton's method by Hubbard et al.
Invent. Math. 146 (2001), no. 1, 1–33. pdf
| {
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Simple inequalities Suppose $l,t\in[0,1]$ and $l+t\leq1$ I want to prove $1+l+t>6lt$. When $t=0$ or $l=0$, it is trivial, so I started with $l,t\neq0$ but I couldn't reach anywhere. I don't have time to write in detail what I have already tried, but I tried to manipulate $(l-t)^2$ mostly. Anyway, if anyone help me with... | If $l+t=a$ then $lt\le\frac{1}{4}a^2$ and we have
$$6lt-(l+t)\le{\textstyle\frac{3}{2}}a^2-a={\textstyle\frac{3}{2}}a(a-{\textstyle\frac{2}{3}})\ .$$
By sketching a graph it is easy to see that for $0\le a\le1$ the right hand side is at most $\frac{1}{2}$.
| {
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What is $\cos(k \pi)$? I want to ask question for which I have been finding answer for.
Please could anyone explain me why $\cos(k \pi) = (-1)^k$ and also explain me same for $\sin(k \pi)$?
| Let $k\in\mathbb Z$. Then
$$\cos(0)=1,~~\text{for}~~k=0,$$
$$\cos(\pi)=\cos(-\pi)=-1,~~\text{for}~~k=\pm 1,$$
$$\cos(2\pi)=\cos(-2\pi)=1,~~\text{for}~~k=\pm 2,$$
and so on, where the first equalities hold as $\cos(\cdot)$ is an even function.
Every time $k$ is even then we get $\cos(k\pi)=1$. When $k$ is odd, then $\co... | {
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Counting divisibility from 1 to 1000 Of the integers $1, 2, 3, ..., 1000$, how many are not divisible by $3$, $5$, or $7$?
The way I went about this was
$$\text{floor}(1000/3) + \text{floor}(1000/5) + \text{floor}(1000/7)-\text{floor}(1000/(3\cdot5)) - \text{floor}(1000/(3\cdot7))-\text{floor}(1000/(5\cdot7))+\text{flo... | The idea is to use the Inclusion/Exclusion principle. Let us first count how many numbers are divisible by $3$, $5$, or $7$. Let set $X$ be the set of all such numbers.
Let $A$ = {Numbers divisible by $3$}
Let $B$ = {Numbers divisible by $5$}
Let $C$ = {Numbers divisible by $7$}
By the inclusion/exclusion principle:
... | {
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Number of primes in $[30! + 2, 30! + 30]$ How to find number of primes numbers $\pi(x)$ in $[30! + 2$ , $30! + 30]$,
where $n!$ is defined as:
$$n!= n(n-1)(n-2)\cdots3\times2\times1$$
Using Fermat's Theorem:
$130=1\mod31$,
(since $31 \in \mathbb{P}$). This implies the above is congruent to $17\mod31$.
This is correct,... | Observe that
$$n!+m$$ is divisible by $m$ for $2\le m\le n$ and integer $n\ge2$
So, we can have an arbitrarily large sequence of composite numbers for an arbitrary large value of integer $n$
| {
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Can this be written in standard "vector calculus notation"? A formula for the gradient of the magnitude of a vector field $\mathbf{f}(x, y, z)$ is:
$$\nabla \|\mathbf{f}\| = \left(\frac{\mathbf{f}}{\|\mathbf{f}\|} \cdot \frac{\partial \mathbf{f}}{\partial x}, \frac{\mathbf{f}}{\|\mathbf{f}\|} \cdot \frac{\partial \math... | $$ \mathrm{d} \|f\|^2 = 2 \|f\| \, \mathrm{d} \|f\|$$
$$ \mathrm{d} \|f\|^2 = \mathrm{d}(f \cdot f) = f \cdot \mathrm{d}f + \mathrm{d}f \cdot f$$
The meaning of most objects involved is clear; e.g.
*
*$f$ is a vector field (i.e. it acts like a column vector-valued function)
*$\|f\|$ is a scalar field
*$d\|f\|$ is ... | {
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showing $\inf \sigma (T) \leq \mu \leq \sup \sigma (T)$, where $\mu \in V(T)$ I am trying to prove the following:
Let $H$ be a Hilbert space, and $T\in B(H)$ be a self-adjoint operator. Then for all $\mu \in V(T)$,
$\inf \left\{\lambda: \lambda \in \sigma (T) \right\}\leq \mu \leq \sup \left\{\lambda : \lambda \in \si... | For selfadjoint operators we know that
$$
\inf\sigma(T)=\inf\{\langle Tx,x\rangle:x\in S_H\}\\
\sup\sigma(T)=\sup\{\langle Tx,x\rangle:x\in S_H\}
$$
It is remains to apply result of this answer.
| {
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What is the easiest way to solve this integral with u-subsitition or what are other methods should be used? How would you calculate this integral?
| Gigantic hint: $$\int_{0}^{\pi/2}\frac{\sin^{45}(x)}{\sin^{45}(x)+\cos^{45}(x)}+\int_{0}^{\pi/2}\frac{\cos^{45}(x)}{\sin^{45}(x)+\cos^{45}(x)}=\frac{\pi}{2}$$ How are those two integrals related? (think u-substitution)
| {
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Counting Shaded Squares In a $4 \times 4$ square, how many different patterns can be made by shading exactly two of the sixteen squares? Patterns that can be matched by flips and/or turns are not considered different.
How many different patterns can be made for a $5\times5$ square?
Would the answers be $15$ and $36$ ... | The analysis is different depending on whether $n$ is even or odd. I will do the case of $n$ even, and leave odd $n$ to you.
To use the Cauchy–Frobenius–Burnside–Redfield–Pólya lemma, we first divide the 8 symmetries of the square into five conjugacy classes, and count the number of colorings that are left fixed by eac... | {
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Is it possible for a set of non spanning vectors to be independent? I was reading about linear spans on Wikipedia and they gave examples of spanning sets of vectors that were both independent and dependent. They also gave examples of non spanning sets of vectors that are dependent. My question is whether it is possible... | You are very close to being right. It is correct that if you have three vectors in $\mathbb R^3$ which do not span $\mathbb R^3$, then they are necessarily dependent. This follows from the dimension theorem, since if they were independent, they would be a proper subset of a basis, which would have more elements than th... | {
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Making $-{{\pi i}\over n} e^{\alpha i}({{1 - e^{2 n \alpha i}\over{1-e^{2 \alpha i}}}})={\pi \over {n sin(\alpha)}}$; $\alpha={{2m+1}\over{2n}} \pi$ As part of a (much) longer problem in complex analysis, I need to show that the equality mentioned in the title makes sense, but I can't seem to find the right algebra tri... | Just note that
$$
\frac{ie^{\alpha i}}{1-e^{2\alpha i}}=\frac{i}{e^{-\alpha i}-e^{\alpha i}}=\frac{-i/2i}{(e^{\alpha i}-e^{-\alpha i})/2i}=\frac{-\frac12}{\sin\alpha}
$$
and that
$$e^{2n\alpha}-1 =e^{(2m+1)\pi}-1=-1-1 =-2$$
and the rest follows immediately.
| {
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Is this set of functions countable? I want to know if the set of functions $F=\{f:\mathbb{Z}\to\mathbb{Z}:f(n)\neq 0 \text{ for finitely many n}\}$.
I haven't done a lot of progress really, but considering define the sets $A_n=\{f: \mathbb{Z}\to \mathbb{Z}\ : f(n)=0\}$. If I could prove that each $A_n$ is countable the... | Hint: First see how many functions do you have satisfying:
$f:\mathbb{Z} \rightarrow \mathbb{Z}$ such that $f(n) \neq 0$ for exactly one $n$. This can easily be seen to be countable. Then $f(n) \neq 0$ for finitely many $n$ is just $f(n) \neq 0$ for exactly 1 $n$ union $f(n) \neq 0$ for exactly 2 $n$ , union.....
Alte... | {
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In $1 < k < n-10^6$, what is $k$? (details in question) This is a homework question of mine, I am not searching for the solution but rather what it means. It seems pretty straight forward but I am a little confused as to what the $k$ in $1 < k < n-10^6$ is supposed to be.
Here is the question:
Consider the number $n... | The statement $1 \lt k \lt n-10^6$ shows the range of $k$. It can range from $2$ to a million and one below $n$. You are given that no $k$ in this range divides $n$.
| {
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What is the formal definition of a variable? What is a variable?
I know that a ($n$-ary) connective can be thought of as a function from $\{ 0,1 \}^n$ to $\{ 0,1 \}$.
And a quantifier over $M$ can be thought of as a set of subsets of $M$.
What is the corresponding way to think of a variable ranging over $M$ ?
| You can see Categorial grammar and, in detail : Sara Negri & Jan von Plato, Structural Proof Theory (2001), Appendix A.2 : CATEGORIAL GRAMMAR FOR LOGICAL LANGUAGES [page 221].
In propositional logic, no structure at all is given to atomic propositions, but these are introduced just as pure parameters $P, Q, R$,..., wi... | {
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Is the fundamental group of a compact manifold finitely presented? Let $X$ be a connected compact smooth manifold. If $X$ is boundaryless, we can choose a Riemannian metric for $X$ so that $\pi_1(X)$ acts geometrically (ie. properly, cocompactly, isometries) on the universal cover $\tilde{X}$. Because it is know that a... | Differentiable manifolds can always be given the structure of PL manifolds, which can be triangulated into simplicial complexes. By shrinking a spanning tree of the 1-skeleton of this simplicial complex, we can obtain a CW complex $X$ with a single $0$-cell. This complex is no longer a manifold, but has the same funda... | {
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Proof in Graph Theory I have a $2D$ undirected graph of size $n \times n$ in which each node is connected to its four neighbours (left,right,top,bottom).
If any general property is true for any nxn graph, what will be the mathematical proof for the property to hold for $(n+1) \times (n+1)$ undirected graph? All nodes ... | What you have defined is a grid graph. You may find the properties of a grid graph in following links:
http://mathworld.wolfram.com/GridGraph.html
http://en.wikipedia.org/wiki/Lattice_graph
| {
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I have a conjecture on local max/min , can any of you propose a contradiction? If $f$ is a non-piecewise function defined continuous on an interval $I$, and within that interval $I$, there exists a value $x$, such that $f`(x)$ (derivative of $f$) does not exist , then at that value $x$, is a local $\max/\min$ value.
| False. Consider, for example,
$$f(x) = \begin{cases} x^3 & \text{ if } x<0 \\ x & \text{ if } x \ge 0\end{cases}.$$
Then $f$ is not differentiable at $0$, but $f$ is strictly increasing around $0$.
| {
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$(a_n)_{n=1}^\infty$ is a convergent sequence and $a_n \in [0,1]$ for all $n$. Proof of limit $(a_n)_{n=1}^\infty$ lies in [0,1]. Textbook question:
$(a_n)_{n=1}^\infty$ is a convergent sequence and $a_n \in [0,1]$ for all $n$. Proof of limit $(a_n)_{n=1}^\infty$ lies in [0,1].
I don't understand the question I suppo... | Assume that the limit of $a_n$ don't live in $[0,1]$. Without loss of generality assume that limit $L$ is $L<0$ (i.e. is at the left of the interval). Convergence says $\forall \epsilon>0 \exists N, \forall n>N |a_n-L|<\epsilon $. Take $\epsilon=|\frac{L}{2}|=-\frac{L}{2}$ then in the interval $\left(\frac{3L}{2},\frac... | {
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Hessian equals zero. I'm currently just working through some maxima/minima problems, but came across one that was a bit different from the 'standard' ones.
So they used the usual procedures and ended up finding that the Hessian is zero at the critical point (0,0).
They set $x=y$, which resulted in $f(x,x)=-x^3$, which... | set x=y means evaluating the function on the line x=y. put it another way, evaluating the function along the direction $\mathbf{v}=[1,1]^T$. Since an extremum on the whole must be an extermum along any direction,. If we can find a direction along which this critical point is not an extremum, then we can assert that thi... | {
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What is the Taylor series of $\frac{1}{\sin(z)}$ about $z_0 = 1$? This was a exam question so I know it cannot take too long to write out the proof. Only I cannot see an answer.
I would imagine you write $\sin(z) = \sin(1+(z-1)) = \sin(1)\cos(z-1) + \sin(z-1)\cos(1)$ and then use the everywhere-defined Taylor series fo... | The function $z\mapsto 1/\sin(z)$ is meromorphic and has simple poles at points of $\pi\Bbb{Z}$. Thus it has a power series expansion $\sum_{n=0}^\infty a_n(z-1)^n$
around $z_0=1$, with radius of convergence $R=d(1,\pi\Bbb{Z})=1$.
Now to determine the coefficients we may can use the identity
$1=\sin(z)\left(\sum_{n=0}... | {
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Arithmetic progression of primes question Is it known whether for all positive integers $k$ there is an integer $a$ such that $a+30n$ is a prime number for all $n\in \{1,\ldots,k\}$?
| $k$ cannot be larger than $6$, since among any seven numbers of the form $b, b + 30, b + 60, \ldots, b + 180$ at least one of them is divisible by $7$, and hence only a prime if it is seven. But if $b = 7$, then $187 = 11\cdot 17$ is not a prime.
That being said, $b = 7, k = 6$ is one maximal example (w.r.t. $k$)as
$$
... | {
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Convert from base $10$ to base $5$ I am having a problem converting $727$(base $10$) to base $5$. What is the algorithm to do it?
I am getting the same number when doing so: $7\times 10^2 + 2\times10^1+7\times10^0 = 727$, nothing changes.
Help me figure it out!
| The trick is to realize that
\begin{align*}
727 &= 625 + 0*125 + 4*25 + 0*5 + 2
\\ &= 5^4 + 0 *5^3 + 4*5^2 + 0*5^1 + 2*5^0.
\end{align*}
So the answer is $10402$.
| {
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null space of an n-by-m matrix I have an $n$-by-$m$ ($n>m$) matrix named $J$. I wanted to find its null space so as I used matrix $M$ defined bellow:
$$JM=0\text{, when } M=I-J^\dagger J$$
$J^\dagger$ is the pseudo inverse of $J$.
Now my question is that am I allowed to choose any ($r=m-n$) columns of $M$ as null space... | Since $M$ is a square matrix, it has linearly independent columns if and only if it is invertible, which happens only if $J$ is the zero matrix.
Indeed, the product $J^\dagger J$ is the orthogonal projection onto $(\ker J)^\perp$, see here. Hence, $M$ is the orthogonal projection onto $\ker J$. So, the columns of $M$ ... | {
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Simplifying a ratio of powers This might sound like a stupid question but when it comes to simplifying when using the ratio test I get confused. Can someone please explain why $$\frac{2^{n+1}}{2^n}=\frac{2}{1}?$$ I think I might be thinking too hard because this confuses me.
| Here, we have
\begin{align}
\frac{2^{n+1}}{2^n}&=2^{(n+1)-n} & \text{using exponent law $x^{a-b}=\frac{x^a}{x^b}$}\\
&=2^1 & \\
&=\frac{2}{1} & \text{because } 2^1=2=\frac{2}{1}
\end{align}
| {
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Can someone explain this? $\sec(x/2) = \cos(x/2)$
I worked on this and got here...
(Let (x/2) = u)
$\cos u - \sec u = 0$
$\cos u(1 - \sec^2u) = 0$
$\cos u[ -1(-1 + \sec^2u)] = 0$
$\cos u(-\tan^2u) = 0$
So, the solutions would be:
$x = pi + 4\pi k, 3\pi + 4\pi k, 0 + 2\pi k$ but the problem is that the first two $(\pi +... | You solved the problem correctly in your original post.
You found all possible potential solutions, and you recognized that some of them are not really solutions. The true solutions are the values $$2\pi k$$ for integral $k$. The false solutions were introduced by setting $\cos u=0$, but that is really implicitly forbi... | {
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Find the limit of the following series As n approaches infinity, find the limit of (n/2n+1).
I know if there was a number in place of infinity, I would plug that in for the "n". But what do I do for the infinity sign?
| Notice $$ \frac{n}{2n +1} = \frac{1}{2 + \frac{1}{n}} \to \frac{1}{2}$$
Since $\frac{1}{n} \to 0 $
Added: if we have
$$ \frac{n}{(2n)^3 + 1} = \frac{ \frac{1}{n^3} }{2^3 + \frac{1}{n^3}} \to 0$$
since $\frac{1}{n^3} \to 0 $
| {
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How much information is in the question "How much information is in this question?"? I'm actually not sure where to pose this question, but we do have an Information Theory tag so this must be the place. The "simple" question is in the title: how do I know how many bits of information is in the question: "How much info... | There is in fact an existing mathematical definition of this exact concept called Kolmogorov complexity, but you will surely be disappointed because there is a serious catch: it is only defined up to a constant additive factor which depends on your model of computation. But aside from that factor, it is an invariant m... | {
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Transition Matrix eigenvalues constraints I have a Transition Matrix, i.e. a matrix whose items are bounded between 0 and 1 and either rows or columns sum to one. I would like to know if it is possible that in any such matrices the eigenvalues or eigenvectors could contain an imaginary part. Thanks in advance.
| I think it is worth pointing out that, having daw's fine upon which to build, we can easily construct a family of irreducible transition matrices which also have a pair of complex conjugate eigenvalues, where by "irreducible" I mean there is a non-zero probability of any state transiting to any other state in one step,... | {
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Integral of 1/(8+2x^2) I have been following a rule saying that
$$\int{\frac{1}{a^2+u^2}}dx = \frac{1}{a}\tan^{-1}(\frac{u}{a})+c$$
The question is asking for the interval of
$$\frac{1}{8+2x^2}$$
Following that rule
$$a=\sqrt8$$
$$u=2x$$
So
$$\int{\frac{1}{8+2x^2}}dx = \frac{1}{\sqrt8}\tan^{-1}(\frac{2x}{\sqrt{8}})+ ... | You can do the change of variable, but use $u=\sqrt{2}x$. And remember that $du=\sqrt{2}dx$.
I think the best way is to factorize out a 2 to get
$$
\frac{1}{2}\int\frac{1}{4+x^2} \, dx
$$
and then use the rule you stated.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
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How do I graduate a cylinder glass in milliliters? Today I decided to cook something, but then I realized there is a critical item missing − a measuring glass. Being a programmer and all, I decided this wouldn't be much of a problem, as I could probably graduate it knowing its radius and height.
Height is 10cm and radi... | Yours calculations are correct, given the measures you've provided us. Maybe 6cm is the diameter of the container, so its radius is 3. That would make the volume you calculated go down by a factor of 4, making it 282.7 ml. Given your constraint that it is definitely below 0.5 liters, that seems correct.
About the incr... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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distance between irreducible elements in a number ring Consider the number ring $\mathbb{Z}[\phi]$ where $\phi$ is the positive root of $X^2-X-1$.
Any of its elements can be written as $a+b\phi$ with $a$ and $b$ integers. There is a norm $N$ such that $N(a+b\phi)=|a^2+ab-b^2|$. The norm is multiplicative and satisfies ... | Call $g(x)=\log_\phi x$ and $f(x)=g(x)\bmod1$ and let $S=\{n\in\Bbb P:(\frac n5)=1\}$ be the set whose image we want to prove dense. It follows from the density of $f(S)$ on $[0,1]$ that $d(f(S),f(S))$ is dense in $[0,\frac12]$ (just pick one element to be fixed and consider its distance to every other element). The de... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Lagrange interpolation, syntax help I am told, the basic interpolation problem can be formulated as:
Given a set of nodes, $ \{x_i, i=0, ..., n\} $ and corresponding data values$\{y_i, i=0, ..., n\}$, find the polynomial $p(x)$ of degree less or equal to $n$ such that $p(x_i)=y_i$.
Which makes sense to me. However, th... | Using the matrix notation for interpolation
$$\mathbf{y}=V\mathbf{a}$$
where $y$ output data vector and $a$ is the coefficients of the polynomial to be resolved. $V$ is a (n+1) by (n+1) Vandermonde matrix whose each row is of the form:
[1, x_i, x_i^2, ...x_i^n];
If you know how to solve the determinant of a Vandermonde... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is $\lim_{n\to\infty} \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$ What is $$\lim_{n\to\infty} \displaystyle \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k}\left(4^{-k}\binom{2k}{k}\right)^{\frac{2n}{\log_2{n}}}\,?$$
| $$
\sum_{k=0}^{\lfloor n/2\rfloor}2^{-2nk}\binom{n}{2k}\binom{2k}{k}^{\large\frac{2n}{\log_2(n)}}
=\sum_{k=0}^{\lfloor n/2\rfloor}\binom{n}{2k}\left[4^{-k}\binom{2k}{k}^{\large\frac2{\log_2(n)}}\right]^{\large n}
$$
and when $n\gt4$, $\frac2{\log_2(n)}\lt1$ and the term in the brackets decays exponentially since
$$
\bi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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$K$ events that are $(K-1)$-wise Independent but not Mutually/Fully Independent I had the following question:
Construct a probability space $(\Omega,P)$ and $k$ events, each with probability $\frac12$, that are $(k-1)$-wise, but not fully independent. Make the sample space as small as possible.
I tried to answer it, ... | Given any $k\ge2$, for $k$ possible events, assume all combinations of an even number of events are equally likely, while an odd number of events has probability 0. One way to construct it, if $X_1,\ldots,X_k\in\{0,1\}$, draw $X_1,\ldots,X_{k-1}$ independently with probability $1/2$, and pick $X_k$ such that $\sum_{i=1... | {
"language": "en",
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"source": "stackexchange",
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How to find the coordinates of intersection points between a plane and the coordinate axes? Can you please explain what I am supposed to do and why that is true?
The equation of the plane is $4x - 3y = 12$.
Is the $z$ coordinate always zero in this plane or not? I mean, it is, if its the
XY-plane, but this doesn't see... | Basically $z$, in the equation $4x-3y=12$ has the potential to be any point. Since we are given an equation with $x$ and $y$ only, and we are graphing this in $3$d space it is going to extend out to an infinite amount of $z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/747101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convex implies not subadditive A function $f: [a,b]\to \mathbb R$ is called convex if for all $x,y \in [a,b], t \in [0,1]$:
$$ f(tx + (1-t)y) \le tf(x) + (1-t)f(y)$$
A function is called subadditive if $f(x+y) \le f(x) + f(y)$.
Is it true that if $f$ is convex then $f$ is not subadditive?
Context: I thought of this qu... | $e^{-x}$ is both convex and subadditive on $[0,\infty)$:
$$
e^{-x-y} = e^{-x}e^{-y} \leq e^{-x}+e^{-x}e^{-y} \leq e^{-x}+ e^{-y}.
$$
EDIT: in fact, any linear function $f:\mathbb{R}\to\mathbb{R}$ is subadditive, superadditive, convex, and concave:
$$\begin{align*}
f(tx + (1-t)y) &= tf(x)+(1-t)f(y)
\\f(x + y) &= f(x)+f(... | {
"language": "en",
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Proving that a function is bijective I have trouble figuring out this problem:
Prove that the function $f: [0,\infty)\rightarrow[0,\infty)$ defined by $f(x)=\frac{x^2}{2x+1}$ is a bijection.
Work: First, I tried to show that $f$ is injective. $\frac{a^2}{2a+1}=\frac{b^2}{2b+1}$
I got $a^2(2b+1)=b^2(2a+1)$. However, I ... | One way to prove injectiveness in this case, is to prove that the function is strictly increasing. This can be done via derivative.
| {
"language": "en",
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Logarithmic Equations and solving for the variable The equation is $\ln{x}+\ln{(x-1)}=\ln{2}$ . I have worked it all the way through, and after factoring the $x^2-1x-2$ I got $x=2$, $x=-1$, but my question is: Can we have both solutions or couldn't we have the negatives?
| The easiest way is indeed to combine them:
$$
\ln \left[x(x-1)/2\right] = 0
$$
implying that $x(x-1) = 2$ or $x^2 - x - 2 = 0$, which indeed has exactly two solutions at $x = 2$ and $x= -1$.
But $x=-1$ cannot be a solution, since the original equation is only defined for $x \ge 1$, so the only solution is $x=2$.
| {
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"source": "stackexchange",
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Proving functions are injective and surjective I am having trouble with the following problem:
For nonempty sets $A$ and $B$ and functions $f:A \rightarrow B$ and $g:B \rightarrow A$ suppose that $g\circ f=i_A$, the identity function of $A$. Prove that $f$ is injective and $g$ is surjective.
Work: Since $g\circ f=i_A$,... | If $f$ weren't injective then $f(x_1)= f(x_2)$ for some $x_1\neq x_2$ in $A$.
So that $x_1=g\circ f\;(x_1)=g\circ f\;(x_2)=x_2$, since $g\circ f$ is the identity function on $A$.
Similarly, if $g$ weren't surjective then for some $a\in A$ there is no $b\in B$ such that $g(b)=a$. But $g\circ f\;(a)=a$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find the general solution of the equation of motion. How do we find the general solution of:
$mu$''+$ku$=0
This is the equation of motion with a damping coefficient of 0.
The characteristic equation is $m$r$^2$+$k$=0.
From here, how do we find the complex roots and get it to look like the following:
$u(t)$=$A$$cos$${w_... | This is the Harmonic Oscillator equation, possibly one of the most common in theoretical physics.
So lets put your equation in the form
$$ u'' + \frac{k}{m} u = 0 $$
and let
$$ \frac{k}{m} = \omega^2 $$
for ease.
As you assumedly did to get that correct characteristic equation, we take a general solution
$$ u(t) = A\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/747594",
"timestamp": "2023-03-29T00:00:00",
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Best strategy for rolling $20$-sided and $10$-sided dice There is a $20$-sided (face value of $1$-$20$) die and a $10$-sided (face value of $1$-$10$) dice. $A$ and $B$ roll the $20$ and $10$-sided dice, respectively. Both of them can roll their dice twice. They may choose to stop after the first roll or may continue to... | This is not really an answer but a long comment. I did not have chance to read the article you posted so not sure what is the indifference method you refer. Perhaps is the same method I have in mind: Assume player 1 rolls for the second time if and only if his first number was below $x$. Given this strategy compute the... | {
"language": "en",
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How to find integers $x,y$ such that $1+5^x=2\cdot 3^y$ Find this equation integer solution
$$1+5^x=2\cdot 3^y$$
I know $$x=1,y=1$$ is such it and $$x=0,y=0$$
I think this equation have no other solution. But I can't prove it.
This problem is from Shanghai mathematics olympiad question in 2014.
| If $x=0$, then $y=0$. If $y=0$, then $x=0$.
Let $x,y\ge 1$. Then $1+(-1)^x\equiv 0\pmod{3}$, so $x$ is odd, so $5^x\equiv 5\pmod{8}$, so $3^y\equiv 3\pmod{8}$, so $y$ is odd. Three cases:
*
*$y=3m$. Then $y\equiv 3\pmod{6}$, so $3^y\equiv -1\pmod{7}$, so $5^x\equiv 4\pmod{7}$, so $x=6t+2$, contradiction (because $x$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Shoelace formula does not work for a given quadrilateral coordinates Given 4 points:
(x0, y0) = (0.34,3.79)
(x1, y1) = (1.09,3.69)
(x2, y2) = (0.44,3.79)
(x3, y3) = (1.19,3.69)
According to formula:
a = x0*y1 + x1*y2 + x2*y3 + x3*y0 - (y0*x1 + y1*x2 + y2*x3 + y3*x0)
area = 0.5 * |a|
However, a = 0. Where do I make mi... | The points need to be ordered either clockwise or anticlockwise. If you plot out the points in the order you've specified them, you'll see you've started with a diagonal.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is $\mathbb{Q}^2$ connected?
Is $(\mathbb Q \times \mathbb Q)$ connected?
I am assuming it isn't because $\mathbb Q$ is disconnected. There is no interval that doesn't contain infinitely many rationals and irrationals.
But how do I show $\mathbb Q^2$ isn't connected? Is there a simple counterexample I can use to show... | Let $X$ be a disconnected topological space. Then $X = A \cup B$, where $A$ and $B$ are open subsets of $X$ and $A \cap B = \varnothing$. But then $X \times X = (X \times A) \cup (X \times B)$. Because $A \cap B = \varnothing$, $(X \times A) \cap (X \times B) = \varnothing$, and these sets are open by definition of the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Difference between tuple and row matrix In Munkres Analysis on Manifolds the author uses the word ''tuple spaces'' to refer to a special sort of vector space. Further down on the same page (page 6) he discusses the linear isomorphism that maps a tuple to a row matrix.
I am greatly confused by this as I do not understa... | I think the point of the author is simply saying that you can understand $\mathbb R^n$ as a set of $n$-tuples, but also as a set of $n$-rows or $n$-columns. It doesn't matter how you look at it, all three views are isomorphic.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/748010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Differences Exponential and Ordinary Generating Functions I am trying to understand conceptually the differences between ordinary generating functions (OGF$=1+x+x^2+\ldots$ ) and exponential generating functions (EGF$=1+x+\frac{x^2}{2!}+\ldots$ ) when it comes to counting objects (e.g. how labeling and ordering come in... | The best explanation of the differences I've seen are in the discussion of the symbolic method, as given by Flajolet and Segdewick in Analytic Combinatorics, and more accessibly by Sedgewick and Flajolet in "Introduction to the Analysis of Algorithms" (Addison Wesley, 2nd edition 2013). This Wikipedia article should be... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $N_n$ be the number of throws before all $n$ dice have shown $6$. Set $m_n := E[N_n]$. Write a recursive formula for $m_n$. Suppose we throw $n$ independent dices. After each throw we put aside the dices
showing $6$ and then perform a new throw with the dices not showing $6$, repeating the process until all dices h... | Two closed formulas, not based on the recursion you suggest:
$$
E(N_n)=\sum_{k\geqslant0}\left(1-\left(1-\left(\frac56\right)^k\right)^n\right)
$$
$$
E(N_n)=\sum_{i=1}^n{n\choose i}\frac{(-1)^{i+1}}{1-\left(\frac56\right)^i}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/748235",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that $G$ is a group if the cancellation law holds when identity element is not sure to be in $G$ Having already read through show-that-g-is-a-group-if-g-is-finite-the-operation-is-associative-and-cancel, however in Herstein's Abstract Algebra, I was required to prove it when we're not sure if identity is in the se... | For each $a\in G$, the map $l_a\colon G\to G, x\mapsto ax$ is injective beacuse $$l_a(x)=l_a(y)\implies ax=ay\implies x=y.$$
As $G$ is finite, $l_a$ is a bijection. Pick $a\in G$ and let $e=l_a^{-1}(a)$.
Then $ae=l_a(e)=a$ and hence for all $x\in G$ we have $ex=x$ because $aex=ax$. The element $e$ is left neutral.
Doi... | {
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"source": "stackexchange",
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Does the improper integral exist? I need to find a continuous and bounded function $\mathrm{f}(x)$ such that the limit
$$ \lim_{T\to\infty} \frac{1}{T}\, \int_0^T \mathrm{f}(x)~\mathrm{d}x$$
doesn't exist.
I thought about $\mathrm{f}(x) = \sin x$ but I am not sure if the fact that we divide by $T$ may some how make ... | The integral has to diverge to infinity if this limit is to not exist, (of course any other limit will give zero overall). Given the type $``\dfrac{\infty}{\infty}"$ limit, if we apply L'Hopitals; $$\displaystyle\lim_{T\to \infty}\dfrac{1}{T}\displaystyle\int_{0}^{T}f(x)\ dx = \lim_{T\to \infty} f(T)$$
This makes it cl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Number theory with positive integer $n$ question If $n$ is a positive integer, what is the smallest value of $n$ such that $$(n+20)+(n+21)+(n+22)+ ... + (n+100)$$ is a perfect square?
I don't even now how to start answering this question.
| Hint: Let $$f(n) = (n+20)+(n+21)+(n+22)+ ... + (n+100)=81n+\frac{100\cdot101}{2}-\frac{19\cdot20}{2}=81n+4860=81(n+60)$$
Now since $81$ is already a perfect square you have to find the smallest $n$ for which $n+60$ is a perfect square too...$\Rightarrow n=4$ and $f(n) = 72^2$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/748485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root
Let $p$ be a prime with $p \equiv 1 \mod 4$, and $r$ be a primitive root of $p$. Prove that $-r$ is also a primitive root of $p$.
I have shown that $-r^{\phi(p)} \equiv 1 \mod p$. What I am having trouble showing, howev... | Result: Let $r$ be a primitive root $\pmod{p}$. Then the order of $r^k \pmod{p}$ is $\frac{p-1}{\gcd(k, p-1)}$.
Proof: Let $m$ be the order of $r^k \pmod{p}$. Then $1 \equiv (r^k)^m \equiv r^{km} \pmod{p}$, so $p-1 \mid km$ as $r$ is a primitive root. Thus $\frac{p-1}{\gcd(k, p-1)} \mid \frac{k}{\gcd(k, p-1)}m$ and $\g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Local Isometry of Sphere How does one show that there exists no neighborhood of a point on a sphere that may be isometrically mapped into a plane? I understand that I can find the first fundamental form of the sphere $(u, v, \sqrt{r^2 - u^2 - v^2})$, for fixed $a>0$, which is given by: $E = 1 + \frac{u^{2}}{r^2 - u^2 ... | If you mean the great circle distances on the sphere, it's fairly simple to brute force it.
Consider a small region on the sphere bounded by a circle. If this were mapped isometrically to the plane, since the center of that spherical region is equidistant from the bounding circle, the same would be true in the plane: ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "3",
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Gaussian curvature $K$ of of orthogonal parametrization $X$ Let $X$ be an orthogonal parametrization of some surface $S$. Prove that the Gaussian curvature $K = - \frac{1}{2 \sqrt{E G}} ((\frac{E_{v}}{\sqrt{E G}})_{v} + (\frac{G_{u}}{\sqrt{E G}})_{u})$, where subscripts denote partial differentiation of the quantity w... | Orthogonal parematrization means that the first fundamental form has $F=0$. We assume sufficient niceness of the surface $S$ (so that we never divide by $0$ and all functions are infinitely differentiable in all arguments, etc.).
We first derive two related results, where $\Gamma_{i,j}^{k}$ denotes Christoffel symbols... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What are elements of a field called From Linear Algebra by Serge Lang we have
"Let K be a field. Elements of K will also be called
numbers (without specification) if the reference to K is made clear by the context, or they will be called scalars."
What is the meaning of "if the reference to K is made clear by the c... | In Linear Algebra, we can sometimes use vector spaces whose elements "look" just like scalars. For example:
Define the vector space $V \subseteq \mathbb{R}^+$ with:
*
*Addition: $x+y = xy$
*Scalar multiplication: $rx = x^r$
Clearly, we could call $x$, $y$, and $r$ "numbers." Yet, in this context, it is unclear w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/749073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that for $0The distribution function for a power family distribution is given by $$F(y)=\begin{cases} 0, & y<0\\ \left(\frac{y}{\theta}\right)^\alpha, &0\le y \le \theta \\ 1, ,&y>\theta,\end{cases}$$ where $\alpha$, $\theta$ > 0. Assume that a sample of size $n$ is taken from a population with a power family dist... | You are asked $\Pr\left[k<\dfrac{Y_{(n)}}{\theta}\le 1\right]$. Now, take a look the part: $k<\dfrac{Y_{(n)}}{\theta}\le 1$. Multiply each side by $\theta$, you will obtain: $k\theta<Y_{(n)}\le\theta$.
Let $Y_1,\cdots, Y_n$ be a random variable from a given power family distribution. Here, $Y_{(n)}$ is $n$-th order st... | {
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"timestamp": "2023-03-29T00:00:00",
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Numerical solution of differential equation Show that the nonlinear oscillator $y" + f(y) =0$ is equivalent to the system
$y'= -z $,
$z'= f(y)$
and that the solutions of the system lie on the family of curves
$2F(y)+ z^2 = constant $
where $F_y= f(y)$. verify that if $f(y)=y$ the curves are circle.
=>
nonlinear oscill... | For the first part, you made a mistake.
You calculated $$\frac{dy}{dt} (F(y)^2 + z^2) = y\frac{dy}{dt} + z\frac{dz}{dt}$$
which is completely untrue.
Instead, try to calculate
$$\frac{d}{dt}\left[2F(y(t)) + z^2(t) \right].$$
If $f(y)=y$, then the differential equation is $y'' + y = 0$, meaning that $y=A\cos x + B\sin x... | {
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"answer_id": 0
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How can you solve this convolution? Let $c$ be a positive constant and let $f(t)=\delta(t-c)$. Compute $f*g$. So setting up the integral, I get $$ (f*g) = \int_0^t \delta (t-\tau-c) g(\tau) d\tau$$ I am unsure of how to take the integral of the delta function. Any help would be greatly appreciated.
| The delta "function" is strictly speaking not a function, but a distribution, that is a continuous linear functional on a space of test functions. It sends a test function $g$ onto its value at $0$:
$$\delta(g)=g(0)$$
You may think of $\delta(g)$ as "$\int \delta(\tau)g(\tau)d\tau$". Be however aware that there does no... | {
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"timestamp": "2023-03-29T00:00:00",
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How show $\mathbb N \cong \mathbb Q$ using Cantor pairing? According to this:
http://en.wikipedia.org/wiki/Cantor_pairing_function#Cantor_pairing_function,
we can show that $\mathbb N\times\mathbb N\cong\mathbb N$. But as for $\mathbb Q$, this is not the exact same case since not all numerators and denominators are cop... | If you don't care about a constructive bijection:
There is a clear injection from $\mathbb{N} \to \mathbb{Q}_+$. If we can show that there is a surjection $\mathbb{N} \to \mathbb{Q}_+$, then this shows (Cantor-Bernstein) that they have the same cardinality.
To construct our surjection, we note that there is a surjectio... | {
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"timestamp": "2023-03-29T00:00:00",
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Finding correlation coefficient Two dice are thrown. $X$ denotes number on first die and $Y$ denotes maximum of the numbers on the two dice. Compute the correlation coefficient.
| If $X \sim DiscreteUniform(1,6)$ and $Z \sim DiscreteUniform(1,6)$ are independent random variables, then their joint pmf, say $f(x,z)$ is:
f = (1/6)*(1/6); domain[f] = {{x, 1, 6}, {z, 1, 6}} && {Discrete};
The desired correlation is then easy to compute using automated tools:
Corr[{x, Max[x, z]}, f]
returns: $3... | {
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Prove that $(2m+1)^2 - 4(2n+1)$ can never be a perfect square where m, n are integers I could prove it hit and trial method. But I was thinking to come up with a general and a more 'mathematically' correct method, but I did not reach anywhere. Thanks a lot for any help.
| If $(2m+1)^2-4(2n+1)$ is a square, then it has the form $q^2$ where $q$ is an odd number (since $(2m+1)^2-4(2n+1)$ is odd). In this case, the quadratic equation
$$x^2-(2m+1)x+(2n+1)=0$$
has a pair of solutions which are integers (since $q$ is odd).
But both the sum $(2m+1)$ and the product $(2n+1)$ of such solutions a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/749756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Real Life Rounding Phenomena When Solving for Variables I have a question that I've been thinking a long time about without being able to come up with an answer and would appreciate some help:
I am attempting to subtract two distinct fees from a total transaction, depending on transaction price.
Fee #1 = 2.9% of transa... | In most generality: Trial and error. Most notably, since $0.129<1$, there are many (about eight) $T$ leading to the same rounded $F$. If you are given $T+F$ instead, i.e. a factor that should equal $1.129>1$, you can determine $T$ uniquely (and there are some values of $T+F$ that cannot legally be obtained): Compute $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/749828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why is the sum over all positive integers equal to -1/12? Recently, sources for mathematical infotainment, for example numberphile, have given some information on the interpretation of divergent series as real numbers, for example
$\sum_{i=0}^\infty i = -{1 \over 12}$
This equation in particular is said to have some im... | Basically, the video is very disingenuous as they never define what they mean by "=."
This series does not converge to -1/12, period. Now, the result does have meaning, but it is not literally that the sum of all naturals is -1/12. The methods they use to show the equality are invalid under the normal meanings of serie... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/749921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Why do some series converge and others diverge? Why do some series converge and others diverge; what is the intuition behind this? For example, why does the harmonic series diverge, but the series concerning the Basel Problem converges?
To elaborate, it seems that if you add an infinite number of terms together, the su... | Here's an intuitive answer
When a series converge it's because that the series goes towards a target, its limit. Likewise a diverging series has no target, it either jumps around in circles or goes to an infinite value. The harmonic series diverges because, even though it increases by smaller and smaller amounts, it wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/749981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Density of cylindrical random variables in classical Wiener space I'm currently working on Malliavin calculus, and a theorem in my class notes is bothering me :
Denote W the Wiener space of continuous functions from $[0,1]$ to $\mathbb{R}$, and $\mu$ the associated Wiener measure. Let also the coordinate random variabl... | I don't know how to use the hint. But you could use the fact that the underlying sigma-algebra is generated by sets of the form $\{w : w_t \in [a,b]\}$ for all $t\in[0,1]$ and $a,b \in \mathbb R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/750049",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Does generalization of axioms apply also to theorems? In Enderton's book "A Mathematical Introduction to Logic" (second edition), he includes six axiom groups, and allows also for a generalization of those axioms such that if $\Psi$ is an axiom then $\forall x \Psi$ is also an axiom.
Is this rule also intended to apply... | I have the first edition of Enderton's book, but I conjecture that what I'm about to write is also true for the second edition. The axioms are defined to be formulas of certain particular forms along with anything obtainable as generalizations of them (i.e., attaching universal quantifiers that govern the whole formul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/750130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Formalism in integration Let's say we have some $y(t)$. The derivative of $y$ along time axis will be $y'(t)=\frac{dy(t)}{dt}=\frac{dy}{dt}$. So I will integrate like this over time:
$\require{cancel}$
$\int_{t=0}^{+\infty}\frac{dy}{\cancel{d\tau}}\cancel{d\tau} = \int_{t=0}^{+\infty}dy=y|_{t=0}^{+\infty}=y(\infty)-y(0... | We chose the fraction-looking notation for derivatives because many "apparent laws", like this cancellation, really do hold. But the notation alone is not a proof of this.
This question is really a question about what many call "u-substition" in disguise. u-substitution is an integral statement of the chain rule for di... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/750243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Systems of Linear Differential Equations - population models I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem:
(Note: $'$ denotes prime)
\begin{align}
x' = -5x - 20y & \text{(Equation 1)} \\
y' = 5x +... | I agree with your auxiliary result.
You could have also written the system as a matrix and used eigenvalues and eigenvectors to solve it.
Using the system matrix approach, we end up finding the solutions (these do not match yours):
$$x(t)=\frac{1}{4} c_1 e^t (4 \cos (8 t)-3 \sin (8 t))-\frac{5}{2} c_2 e^t \sin (8 t) \\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Number of subsets of a nonempty finite set with a given property. Let $S$ be a set with $|S|=n$, where $n$ is a positive integer. How many subsets $B$ of $S\times S$ are there with the property that $(a,a) \in B$ for all $a \in S$ and $(a,b) \in B \implies (b,a) \in B$ for all $a,b \in S$.
So, this is the question I am... | There are $\binom{n}{2}$ ways to choose $2$ (distinct) numbers $a$ and $b$ in $S$. (Note that this is the number of "hands" of $2$, and not a set of ordered pairs.)
For any such distinct numbers $a$ and $b$ we can say Yes if $(a,b)$ and therefore $(b,a)$ will be in our set, and No if $(a,b)$, and therefore $(b,a)$, wi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/750369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maximize the inradius given the base and the area of the triangle BdMO 2013 Secondary:
A triangle has base of length 8 and area 12. What is the radius of the largest circle
that can be inscribed in this triangle?
Let $A,r,s$ denote the area,inradius and semiperimeter respectively.Then we have that
$A=rs$
$\implies... | For b+c to be minimum b=c should hold. So b=c=5.(by simple Pythagoras). So r=1.34.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/750569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Inequality with moments $m(f^3) \le m(f^2) m(f)$ Let $m$ a probability measure, $f$ a positive measurable function (one can assume it is bounded, the existence of the moments is not a problem here).
Is $m(f^3) \le m(f^2) m(f)$?
| No. Actually, for every probability measure $m$ and nonnegative function $f$, $$m(f^3)\geqslant m(f^2)\cdot m(f),$$ with equality if and only if $f$ is ($m$-almost surely) constant.
Hence, checking any example would have shown that the conjecture is wrong.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/750669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Orthocentre of a triangle defined by three lines Problem: If the orthocentre of the triangle formed by the lines $2x+3y-1=0$,$x+2y-1=0$,$ax+by-1=0$ is at the origin, then $(a,b)$ is given by?
I would solve this by finding poins of intersection and the standard text-book methods. But, I was discouraged by the algebraic ... | Since the three lines intersect precisely at the orthocenter = the origin, we have that:
$$\text{intersection of lines I, III}\;:\;\begin{cases}2x+3y=1\\
ax+by=1\end{cases}\implies \begin{cases}\;2ax+3ay=a\\\!\!-2ax-2by=-2\end{cases}\implies$$
$$(3a-2b)y=a-2\implies y=\frac{a-2}{3a-2b}$$
And since we're given $\;y=0\im... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/750741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Unitarily equivalent $C^*$-algebra representations the situation i want to talk about is the following:
$(H_1,\varphi_1),(H_2,\varphi_2)$ irreducible representation of a $C^*$-algebra $A$. A bounded operator $T:H_1\rightarrow H_2$ such that $T\varphi_1(a)=\varphi_2(a)T$ for all $a\in A$
I am asked to prove the followin... | You have
$$
T^*T\varphi_1(a)=T^*\varphi_2(A)T=\varphi_1(a)T^*T.
$$
(for the second equality, note that $\varphi_1(a)T^*=T^*\varphi_2(a)$ by taking adjoints on your original equality).
So $T^*T$ commutes with $\varphi_1(a)$ for all $a\in A$. As $\varphi_1$ is irreducible, $T^*T$ commutes with every operator in $B(H_1)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/750830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Is (the proof of) Fermat's last theorem completely, utterly, totally accepted like $3+4=7$? If a mathematician would/does make use of Fermat's last theorem in a proof in a publication, would s/he still make use of some kind of caveat, like: "assuming Fermat's last theorem is true" or "assuming the proof is correct", or... | Yes, the proof is accepted, and it would be bizarre to write "assuming Fermat's Last Theorem is true" in a paper.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/750932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
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