Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Projection Matrix and Orthogonal Complements We have a projection matrix, $P = A(A^T A)^{-1} A$. The columns of $A$, we're given, form a basis for some subspace $W$. I need to prove that for any vector in the orthogonal complement of $W$, if we act on it with this projection matrix, we get that vector.
I'm unsure even... | Note that since $P$ is the projection matrix onto W, for any vector in the orthogonal complement of $W$, if we act on it with this projection matrix, we get zero vector.
What you are claiming is true for the matrix $(I-P)$ that is the projection matrix for the orthogonal complement of $W$.
Refer to How to prove the com... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2735910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How solve the following PDE? I want to solve the following PDE:$$
\begin{cases}
u_{tt}-c^2u_{xx}=0, \quad x\in\mathbb{R},\ t\geq x\\
u(x,x)=φ(x), \quad x\in\mathbb{R}\\
u_t(x,x)=0, \quad x\in\mathbb{R}
\end{cases}
$$
where $φ:\mathbb{R} \to \mathbb{R}$, $φ\in C^1(\mathbb{R})$.
Thanks for your help.
| $\def\d{\mathrm{d}}$Case 1: $c^2 \neq 1$. Make substitution $(y, s) = (x - c^2 t, t - x)$, then$$
u_t = u_s - c^2 u_y, \quad u_{tt} = u_{ss} - 2c^2 u_{ys} + c^4 u_{yy}, \quad u_{xx} = u_{yy} - 2u_{ys} + u_{ss},
$$
and the equations become$$
\begin{cases}
(1 - c^2) u_{ss} - (c^2 - c^4) u_{yy} = 0, \quad y \in \mathbb{R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2736043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Polar Coordinate function of a Straight Line I was having some problem when trying to come out a polar coordinate function with straight line equation.
I know it is not good to post images here, but please bear with me as the question requires us to solve the equation from the straight line in the image.
What I have d... | If $\alpha $ is CCW angle made to the normal then in Cartesian form ( not exactly as shown in your question, it should be normal to the line L.
$$ x \cos\alpha + y \sin \alpha = p $$
and in polar coordinates after transformation
$$ r = p \sec \left( \theta - \tan^{-1} \frac{a}{b} \right ) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2736228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 3
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Find the locus of points $P$ such that the distances from $P$ to the sides of a given triangle can themselves form a triangle.
Find the locus of points $P$ within a given $\triangle ABC$ and such that the distances from $P$ to the sides of the given triangle can themselves be the sides of a certain triangle.
Join $PA... | If the triangle is equilateral, then the sum of the distances from an arbitrary point inside the triangle to the three sides is equal to the height of the triangle, which is a constant. So the three perpendiculars are the lengths of a triangle when all of them are not longer than half the height of the given triangle. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2736311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
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Double integral with strange Change of Variables I am trying to compute the following integral,
$$\iint_{\mathbb{R}^2} \left(\frac{1-e^{-xy}}{xy}\right)^2 e^{-x^2-y^2}dxdy$$
First I tried substituting $x=r\cos{\theta}, y=r\sin{\theta}$ but it didn't really give me anything.
For the second try, I tried $u=x^2+y^2, v=xy$... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2736383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Number of ways to partition a set of $n$ indistinguishable items into $r$ groups I came across the problem where I have to count the number of ways a set of $n$ indistinguishable items can be partitioned into $r$ groups.
For example,
*
*if $n=2$ and $r=2$, I can do partitioning as follows:
$*|*$
*if $n=3$ and... | The number you are looking for is the partition of a number into $k$ positive parts.
There is no closed formula for this, but one can use the recurrence relation:
$$
p_k(n) = p_k(n − k) + p_{k−1}(n − 1)
$$
with
$$
p_0(0) = 1 \text{ and } p_k(n) = 0 \text{ if } n ≤ 0 \text{ or } k ≤ 0.
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof for for convergence of $\int_{-∞}^0\frac{\sqrt{\left|x\right|}}{x^2+x+1}dx$ I want to show that the following integral converges. However I am troubled finding an estimate for this term which is sufficient for the proof. Could I use L'Hospital on this one?
$$\int_{-∞}^0\frac{\sqrt{\left|x\right|}}{x^2+x+1}dx$$
| $$\int_{-\infty}^{0}\frac{\sqrt{|x|}\,dx}{x^2+x+1} = \int_{0}^{+\infty}\frac{\sqrt{x}\,dx}{x^2-x+1} \stackrel{x\mapsto z^2}{=} \int_{0}^{+\infty}\frac{2z^2}{z^4-z^2+1}\,dz$$
equals (by parity)
$$\int_{-\infty}^{+\infty}\frac{dz}{z^2+\frac{1}{z^2}-1}=\int_{\mathbb{R}}\frac{dz}{\left(z-\frac{1}{z}\right)^2+1}\stackrel{\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2736894",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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number of solutions to the equation $x_1+x_2+x_3=2018$ with odd\even condition on $x$. I have to find the number of solutions to $x_1+x_2+x_3=2018$ ,
with the following conditions :
*
*$x_1,x_2,x_3$ are even numbers.
*$x_1$ is even while $x_2$ and $x_3$ are odd.
I guess I have to use stars and bars in order to ... | How many solutions are there for $x_2+x_3=2n$ with both $x_2$ and $x_3$ odd? Well, $x_2$ can range from $1$ to $2n-1$, so there are $n$ solutions. This means that, for a given $x_1=2k$, your equation $x_1+x_2+x_3=2018$ has $\frac{2018-x_1}{2}=1009-k$ solutions. Since $k$ may take values from $0$ to $1009$, the total nu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2736963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 4
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Finding What Percentage a Plane is Covered By Pennies Touching Tangentially The question is the following:
Imagine covering an unlimited plane surface with a single layer of pennies, arranged so that each penny touches six others tangentially. What percentage of the plane is covered?
I can't seem to visually understa... | Consider this image:
The outer six pennies are touching the center penny tangentially. If you filled a plane with pennies like this, the shaded rectangular area would repeat.
If you calculate the area of the shaded rectangle, $A_{shaded}$, and the area covered by the pennies within the shaded rectangle, $A_{covered}$,... | {
"language": "en",
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Prove if $x\gt3$ then $1\ge\frac{3}{x(x-2)}$. I tried to prove it by contradiction.
Suppose it is not true that $1\ge\frac{3}{x(x-2)}$, so $1\lt\frac{3}{x(x-2)}$. Then $\frac{3}{x(x-2)}-1\gt0$. Multiply both sides of $\frac{3}{x(x-2)}-1\gt0$ by ${x(x-2)}$.
$(\frac{3}{x(x-2)}-1\gt0)({x(x-2)}\gt0(x(x-2)$
${3-(x(x-... | $x > 3
\implies x-2 > 1
\implies x(x-2) > 3
\implies 1 > \dfrac{3}{x(x-2)}
$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2737144",
"timestamp": "2023-03-29T00:00:00",
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Finding points on a line that are closest
Find the points that give the shortest distance between the lines$$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}2-t\\-1+2t\\-1+t\end{pmatrix}\\\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}5+3s\\0\\2-s\end{pmatrix}$$
So I subtracted the second line from the first t... | Another approach is to use calculus and minimize the distance between arbitrary points on the lines, which is a function of $s$ and $t$. Distance is non-negative, so it suffices to minimize the squared distance, which is easier to differentiate.
The squared distance $dsq(s,t)$ between two arbitrary points, one on each ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show a normal family $\{f_n\} $ converges uniformly on compacts Let $\{f_n\}$ be a sequence of holomorphic functions on a domain $\Omega\subset \mathbb C$ which is bounded uniformly on compact subsets of $\Omega$. Let $\{z_k\}$ be a sequence of distinct points in $\Omega$.
with lim$_{k\to \infty} z_k = z_0 \in \Omega$.... | It suffices to show that there exists a holomorphic $f$ with the property that every subsequence of $\{f_n\}$ has a further subsequence which converges to $f$: if this were to hold but $f_n\not\to f$, then by Montel’s theorem we can pass to a subsequence which converges to something that is not $f$, however this is imp... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\mathbb{C}^2\setminus \{(z,z)\ :\ z\in \mathbb{C}\}$ is path connected. I am really surprised that why $$ \mathbb{C}^2\setminus \{(z,z)\ :\ z\in \mathbb{C}\} $$ is path connected?
I was thinking that it is same as $\mathbb{R}^4\setminus\mathbb{R}^2.$ But still I am unable to prove that the above space is path connecte... | Like how $\Bbb R^3$ removing a line (an affine subspace of two lower dimension) is path-connected, $\Bbb R^4$ removing a plane (also an affine subspace of two lower dimension) should, by guessing, also be path-connected.
First have a look how $\Bbb R^3$ removing a line is path-connected. Consider $\Bbb R^3\setminus L$,... | {
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Supremum of a sequence - same as supremum of set? This is more of a notational/terminology question than anything. I was refreshing my knowledge on the lemmas in the monotone convergence theorem, and saw the claim "If a sequence of real numbers is increasing and bounded above, then its supremum is the limit." I haven't... | The supremum is the supremum of the range of the sequence $(a_{n})$, that is, $\sup\{a_{n}: n=1,2,...\}$.
So several sequences may have the same supremum, especially when they have the same range:
$(x_{n})=(1,0,1,0,1,0,...)$ and $(y_{n})=(0,1,0,1,0,1,...)$ are different sequences, but they have the same supremum $1$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to predict the maxima and minima of a tidal wave? Assume that a planet of mass $m$ and radius $r$ is orbiting a massive star of mass $M$ at a orbital distance $R$. Assume that the planet is covered in a ideal, non viscous ocean on a friction less surface. Hence the ocean will align itself along the equipotential su... | Equation 20 in that article seems to provide what you are looking for. It describes the displacement (from spherical) of the equipotential at the surface of the planet (where $\phi$ is the polar coordinate with 0 pointing towards the other mass) .
Using the terms you have defined the equation is
$$ dr=\frac{m} {M} \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2738004",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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The "assumption" in proof by induction The second step in proof by induction is to:
Prove that if the statement is true for some integer $n=k$, where
$k\ge n_0$ then it is also true for the next larger integer, $n=k+1$
My question is about the "if"-statement. Can we just assume that indeed the statement is true? If... | The proof by induction is based on the following statement
\begin{aligned}
& \left [\mathcal {P}(0) \land \left( \mathcal {P}(n) \implies \mathcal {P}(n+1)\; \forall n\geq 0\right)\right] \\
& \implies \mathcal {P}(n)\; \forall n\geq 0 \, ,
\end{aligned}
where $\mathcal {P}$ is a predicate over the natural integers $\B... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
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Prove $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},A)\cong A[n]$ by using left-exactness of Hom? Let $A$ be an abelian group, then $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},A)\cong A[n]$, where $A[n]=\{a\in A:na=0\}$. How do I prove this by using left-exactness of Hom? (This is Exercise 2.7 fro... | When I do that I get
$$0\to\text{Hom}(\Bbb Z/n\Bbb Z,A)\to A\to A$$
is exact where the $A\to A$ map is multiplication by $n$. That gives
me $\text{Hom}(\Bbb Z/n\Bbb Z,A)\cong A[n]$ straight away.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to denote an arbitrary expression involving some number of dummy variables? Let me give you an example.
$(n)_{n=0}^{\infty}$ is a reference to a particular Sequence object.
$$ 0, 1, 2, \ldots $$
Is another way to reference the same sequence object.
$a_n = n\forall n\in \mathbb{N}$, $(a_n)_{n=0}^{\infty}$ is yet an... | Hint: Note that sequences $(a_n)_{n=0}^\infty$ are just functions with domain $\mathbb{N}$. So, whatever can be said about functions with domain $\mathbb{N}$ can also be said about sequences.
When considering for instance a real-valued sequence $(a_n)_{n=0}^\infty$ we can equivalently consider a function
\begin{alig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2738426",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Intersection between two lines 3D I'm writing the program for university project on c#. I have to find intersection between two lines in 3d space. Lines are specified by the point lying on the line and by its direction vector. I'm bad in math therefore I don't know in what direction I have to move for solving this task... | Since in the comment I mentioned about something numerical, I would like to provide a method to compute the minimal distance between two lines in 3D. Then, if the distance is zero (or sufficiently small), you could deduce that the two lines intersects.
Let the points $a=(a_x, a_y, a_z)$, $b=(b_x, b_y, b_z)$ on the firs... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does this condition imply that $f$ is locally Lipschitz or Lipschitz? Suppose there are constants $\delta > 0$ and $M < \infty$ such that for all $x \in \mathbb{R}$, $|f(x+t)−f(x)| \leq M|t|$ for all $t \in (−\delta,\delta)$. Then is $f$ locally Lipschitz or Lipschitz on $\mathbb{R}$? And if so, why?
Thank you.
| $f$ is Lipschitz.
To prove this, consider arbitrary $x,y \in \Bbb R$ with $x<y$. Divide the interval $[x,y]$ into $N$ small subintervals $[x_i,x_{i+1}]$ (with $x_{i+1}-x_i < \delta$, $x_0=x$ and $x_N=y$).
Then
$$|f(y)-f(x)| = |f(x_N)-f(x_0)| \le \sum_{i=0}^{N-1} |f(x_{i+1})-f(x_i)| \le M \sum_{i=0}^{N-1} |x_{i+1}-x_i| ... | {
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If M is an oriented topological n-Manifold, is M - {x} oriented? Does removing a point from an oriented topological manifold result in a non-oriented manifold?
I know that if M - {x} is oriented, then M is oriented because we can use the two fold cover given in 3.3 of Hatcher. However, I am not able to see converse is ... | No: you have this the wrong way round: removing a point from an orientable manifold can't make it unorientable (essentially because $\Bbb{R}^n$ and $\Bbb{R}^n \setminus \{0\}$ are both orientable allowing you to convert an oriented atlas for $M$ into an oriented atlas for $M\setminus \{x\}$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2738830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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cancelling out of $\frac{n!}{2n!}$? Perhaps a rather strange question. I am doing an exercise and the last part ends:
$I = \frac{n!}{(2n)!}$
Now I know that:
$$I = \frac{n(n-1)(n-2)(n-3)\cdots}{2n(2n-1)(2n-2)(2n-3)\cdots }$$
Is it possible to cancel out some terms? Or to make it a little bit more 'friendly'?
| Note that $$I = \frac{n(n-1)(n-2)(n-3) \cdots 2 \cdot 1}{2n(2n-1)(2n-2)(2n-3) \cdots 2 \cdot 1} $$$$= \frac{n(n-1)(n-2)(n-3) \cdots 2 \cdot 1}{2^n \times n(2n-1)(n-1)(2n-3)(n-2) \cdots 1 \cdot 1}$$ and after cancelling out we get that$$ I = \frac{1}{2^n (2n-1) (2n-3) \cdots 3 \cdot 1} = \frac{1}{2^n} \prod_{i=1}^n \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2738966",
"timestamp": "2023-03-29T00:00:00",
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Linear Algebra- Subspace proof involving operator Let $\mathbb{R}^\mathbb{R}$ be the real vector space of all functions $f:\mathbb{R}→\mathbb{R}$ and let $∆:\mathbb{R}^\mathbb{R} →\mathbb{R}^\mathbb{R}$ be the linear operator defined by $$∆[f](x) := f(x + 1) − f(x).$$
(a) As usual, $∆^2 := ∆ ◦ ∆$ denotes the compositi... | I'll leave (b) for now, and instead clarify the meaning of $\Delta^2$.
Specifically, let's consider a simple example, $f(x) = x^2$. Then say $g = \Delta f$, so $g(x) = f(x+1) - f(x) = 2x+1$. But what about $\Delta^2$? Well, $$\Delta^2 f= (\Delta \circ \Delta) f = \Delta (\Delta f) = \Delta g,$$
and $\Delta g = g(x+1) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2739108",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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The number $555,555$ can decompose, as the product of two factors of three digits, in how many ways?
The number $555,555$ can decompose, as the product of two factors of three digits, in how many ways?
I've seen the answer to the question, and there is only one way:
Since $555, 555 = 3 \cdot 5 \cdot 7 \cdot 11 \cdo... | We can write $$555,555=5\times111,111$$ and notice that $111=37\times3$, so we have $$555,555=5\times37\times3003$$ and since $1001=7\times11\times13$, the prime factorization is $$555,555=3\times5\times7\times11\times13\times37.$$
If we multiply each of the three combinations $11\times13$, $11\times37$ and $13\times3... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2739192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 4
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Which of the following is/are closed sets Which of the following is/are closed sets
*
*$S=\{(x,x\sin\frac{1}{x}) \mid 0\le x \le1\}\cup\{(0,0)\}$
*$T=\{(x,x^2\sin\frac{1}{x}) \mid 0\le x \le1\}\cup\{(0,0)\}$
My idea
for (1).
$f(x)=x\sin \frac{1}{x}$
then $|f(x)|=|x\sin \frac{1}{x}|\le |x|\cdot|\sin \frac{1}{... | MINOR ISSUE : In the definitions of $S$ and $T$ given in the question, one should ensure $x \neq 0$ in the first component of both set definitions.
I get what you are trying to do, but you have not written things clearly.
The idea in both, is that one should use the limit point definition. You should start with a conv... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If A is similar to B then $A^2$ is similar to $B^2$ im not sure how to begin to prove this, all i know is that for two matrices to be similar, the following equation must be true
$A = PBP^{-1}$
any help will be appreciated
| The definition of $A \sim B$ is that there exists an invertible matrix $P$ of the same dimensions as $A$ and $B$ that satisfies $A = PBP^{-1}$.
You just need to show that:
$$A^{2} = AA = (PBP^{-1})(PBP^{-1}) = PB(P^{-1}P)BP^{-1} = PBIBP^{-1} = PBBP^{-1} = PB^{2}P^{-1}$$
And then conclude that, in fact, there exists an ... | {
"language": "en",
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Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers? I self-realized an interesting property today that all numbers $(a,b)$ belonging to the infinite set $$\{(a,b): a=(2l+1)^2, b=(2k+1)^2;\ l,k \in N;\ l,k\geq1\}$$ have their AM and GM both integers.
Now I wonde... | Expanding on Christian Blatter's answer.
There are a few key points.
*
*The arithmetic mean of two rational numbers is always rational.
*The harmonic mean of two non-zero rational numbers is always rational.
*The geometric mean of two squared positive integers is always an integer.
*For all three types of mean if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2739592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "70",
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Let $f\colon X \rightarrow Y$ be a continuous and surjective function. Show that if $X$ is compact, then so is $Y$. Here is my attempt at answering the above question.
(I feel that there are gaps in my knowledge in this topic and don't have a sound understanding of what a covering actually is but here goes!)
As f is ... | Correct me if wrong:
Hopefully adding a bit of detail to José Carlos' solution .
1) Let $O_i$, $i \in I$, be an open cover of $Y$,
i.e. $(\cup O_i)_{i \in I} =Y.$
$X \subset f^{-1}Y$ , or $X \subset f^{-1}(\cup O_i)_{i \in I}$.
2)$X \subset f^{-1}(\cup O_i)_{ i \in I}=$
$(\cup f^{-1}(O_i))_{i \in I}$.
3) Since $f$ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2739680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Limit of $\lim _{ n->\infty }{ ({ z+{ z }^{ -1 }) }^{ n } } $ I'd like to compute the limit of $\lim _{ n->\infty }{ ({ z+{ z }^{ -1 }) }^{ n } } $ for $z\neq 0$
My attempt was to use the Polar-Cor. representation for complexe numbers.So
${ (z+{ z }^{ -1 })^{n} }$=${ { (r }_{ 1 }(\cos { \theta } +i\sin { \theta )+\fr... | Let
$$z+\frac1z=w.$$
Obviously, $w^n$ converges to zero if $|w|<1$ or to one if $w=1$.
By solving the above identity for $z$,
$$z=\frac{w\pm\sqrt{w^2-4}}2=\frac{re^{i\theta}\pm\sqrt{r^2e^{i2\theta}-4}}2$$ with $r<1$, or
$$z=\frac{1\pm i\sqrt3}2.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2739792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Help Looking for a Function with Particular Properties I'm sorry to bother everyone, but I've been searching for functions that satisfy several properties, and so far, I've yet to be able to think of any! Specifically, the properties needed are:
$f(0)=0$ and $f(1)=1$
And on the interval $[0,1]$:
*
*$f(x)$ is differe... | Consider the polynomial (in particular, infinitely differentiable): $$f(x)=-2.05026 x^4 + 3.74339 x^3 - 2.11177 x^2 + 1.418651 x$$
This satisfies $1\ge f(x)\ge x$ (on $[0,1]$) and has $f(x)-x$ maximized at
$x\approx 0.81$.
$f(x)-x$ was found using alpha by interpolating the five points $$(0,0),(1,0),(0.9,0.05),(0.3,0.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2740058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Homomorphism from $S_n$ to an abelian group Any homomorphism from $S_n$ to an abelian group $G$ is given by $\;f(\sigma) = e$, if $\sigma$ is an even permutation, and $f(\sigma)= a$, where order of $a =2$, if $\sigma$ is an odd permutation.
I can prove that this is a homomorphism, but what guarantees that there is no... | Hint If $f : S_n \to G$ is a group homomorphism and $f((i,j))=a$ than $a^2=e$ in $G$.
Therefore, $f$ takes each transposition in some element of order 2.
Next, if $G$ is abelian, use the fact that
$$(i,j)(1,i)(i,j)=(1,j)$$
And
$$(1,j)(1,i)(1,j)=(i,j)$$
to deduce that
$$f((1,i))=f((i,j))=f((i,j)) \forall (i,j)$$
Final... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2740191",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Finding an invertible matrix P and some matrix C Find an invertible matrix $P$ and a matrix $C$ of the form
$C=\begin{pmatrix}a & -b\\b & a\end{pmatrix}$
such that the given matrix $A$ has the form $A = PCP^{-1}$
$A=\begin{pmatrix}5 & -2\\1 & 3\end{pmatrix}$
The first thing i tried to do was to find the eigenvectors o... | Per this question, $A$ has eigenvalues $4\pm i$, so it is similar to a matrix of the form $$C=\begin{bmatrix}4&-1\\1&4\end{bmatrix}.$$ This answer shows how to construct an appropriate basis without computing any eigenvalues explicitly. Note that the resulting matrix in the question has the opposite signs from what we ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2740287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Finding area of a triangle using equation of a circle **Ignore notes I made they are stupid
Without a calculator
Question reads:
The diagram shows a sketch of the circle with equation $x^2 + y^2 = 5$.
The $y$-coordinate of point $A$ is $-1$.
The tangent to the circle at $A$ crosses the axes at $B$ and $C$ as shown... | Some hints:
*
*Using the $y$ coordinate of $A$, find the $x$ coordinate of $A$.
*Find the slope of line $OA$.
*From that, calculate the slope of the tangent line $BC$.
*Using point-slope form, calculate the intercepts.
*Profit!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2740373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Triangle inequality raised to fractional powers Does the inequality $|x + y|^\alpha \leq |x|^\alpha + |y|^\alpha$ where $\alpha \in [0, 1)$, hold for all $x,y$ ??
| For the $x\ge 0,y\ge 0$. $f : t \mapsto (t + x)^{\alpha} - t^{\alpha}$ is decreasing function on $[0,+\infty)$ (take the derivative $f'(t) = \alpha ((t+x)^{\alpha -1} - t^{\alpha - 1}) < 0$ since $\alpha < 1$) so $$|x|^{\alpha} = f(0) \ge f(y) = |x+y|^\alpha - y^\alpha$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2740533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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For any $n\times n$ matrix $A$, there corresponds a vector $x\neq 0$ such that $\|Ax\|=\|A\|\|x\|$ Prove that for any vector norm and its subordinate matrix norm, and for any $n\times n$ matrix $A$, there corresponds a vector $x\neq 0$ such that $\|Ax\|=\|A\|\|x\|$
I know that $\|Ax\|\leq \|A\|\|x\|$ for all $x\in \mat... | Hint: the map $f(x)=\|A(x)\|$ defined on $S_n=\{x:\|x\|= 1\}$ is continuous, since $S_n$ is compact, there exists $x\in B_n$ such that $f(x)=sup_{y\in S_n}f(y)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2740607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Integral test for series $\sum_{n = 18}^{\infty} \frac{n^2}{(n^3 + 3)^{7/2}}$ I am stuck on how to more so algebraically to solve this problem. I understand that you would rewrite the series as a function of x, and then evaluate the integral from 18 to infinity - but that's all I got. Any pointers? Thank you in advance... | Let $$u=x^3+3$$ to evaluate the $$\int_{18}^{\infty} \frac{x^2}{(x^3 + 3)^{7/2}} dx$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2740707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
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Need hint to solve the following integral We are fixed and need some hint to solve the problem
$$\int\frac{(ax+b)^m}{(cx+d)^n}dx$$ where $m,n\in \mathbb{N}$.
| Let $u=cx+d$.
$$\int\frac{(ax+b)^m}{(cx+d)^n}dx=\frac{1}{c}\int\frac{\left(\frac{au}{c}+\frac{d(c-a)}{c}\right)^m}{u^n}du=\frac{1}{c^{m+1}}\sum_{k=0}^m\binom{m}{k}a^{k}d^{m-k}(c-a)^{m-k}\int u^{k-n}du$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2740967",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Inconsistency for solving $x' = x^{1/2}$ The proposed system $x' = x^{1/2}$ can be solved easily to obtain $x(t) = \frac{1}{4} (t^2 + t c + c^2)$, where $c$ is the integration constant.
However, differentiate the newly-found $x(t)$, one gets: $x(t)' = \frac{1}{2}t+\frac{1}{4}c$. This implies that $x^{1/2} = \frac{1}{2}... | By separation of variables:
$$
\frac{dx}{\sqrt{x}}=dt
$$
so
$$
\sqrt{x}=\frac{1}{2}(t+c)
$$
and finally
$$
x=\frac{1}{4}(t+c)^2=\frac{1}{4}(t^2+2ct+c^2)
$$
Can you spot your error?
$\displaystyle x=\frac{1}{4}(t^2+{\Large\color{red}{2}}ct+c^2)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2741103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Completing a specific matrix to a unitary one Let $n,m,p,k_1,k_2$ be natural numbers.
Given two unitary matrices $U\in\mathbb{U}(n),V\in\mathbb{U}(m)$ and a decomposition of these as follows $$U=\bigg(\begin{matrix}A & C\\B& D\end{matrix}\bigg)~ ~ ~ ~ ~ ~V=\bigg(\begin{matrix}A' & C'\\B'& D'\end{matrix}\bigg)$$ where $... | $A$ and $A'$ each have spectral norm at most $1,$ so $AA'$ also has spectral norm at most $1.$ This means the matrix $I_p-(AA')^*(AA')$ is positive semidefinite, so has a $p\times p$ self-adjoint square root which we can take to be $F.$ This choice ensures $\begin{pmatrix}AA'\\ F\end{pmatrix}^*\begin{pmatrix}AA'\\ F\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2741223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Question on the proof of $e^x>1+x$ for $x>0$
Show that $e^x>1+x$ for $x>0$
Proof: Set $f(x)=e^x-(1+x)$. Show that $f(x)$ is always positive. We know that $f(0)=e^0-(1+0)=0$ and $f'(x)=e^x-1$. When $x$ is positive, $f'(x)$ is positive because $e^x>1$
We know that if $f'(x)>0$ on an interval, then $f(x)$ is increasing o... | For $x>0$, we know $e^x > 1$, so $f'(x) > 0$. So, $f(x)$ is increasing, in particular $f(x) > f(0) = 0$, so $f(x)$ is positive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2741354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 3
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Exists a continuous function $f: \mathbb R \to \mathbb R$ with $f(x_n) = y_n, \, \forall n$ and $f(x) = y$ Let $x_n \to x$ and $y_n \to y$ in $\mathbb R$ such that $x_n \neq x_m, \, \forall n \neq m$.
How can I show the existance of a continuous function $f: \mathbb R \to \mathbb R$ with $f(x_n) = y_n, \, \forall n$ an... | Here's a generalization: Let $X$ be a metric space, $x\in X,$ and $x_n$ is a sequence of distinct points in $X\setminus \{x\}$ converging to $x.$ Let $y,y_n$ be as before. Then there exists a continuous $f:X\to \mathbb R$ such that $f(x_n)=y_n, n=1,2,\dots$ and $f(x)=y.$
Proof (sketch): Let $E=\{x_n\}\cup \{x\}.$ Then... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2741466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Should I apply boundary conditions in the general solution before finding the particular solution? Given a function:
$$ y'' - y' = x$$ I want to find the solution where $x = 1$, $y = 1$, $dy/dx = 2$.
I have managed to find the full form of the equation by first finding the complementary function solution and then the p... | You should apply the boundary conditions to the general solution that you have already determined:
$$y(x)=\underbrace{C_1+C_2e^x}_{y_o}+\underbrace{-\frac{x^2}{2}-x}_{y_p}.$$
So it remains to find $C_1$ and $C_2$ such that $y(1)=1$ and $y'(1)=2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2741610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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conflicting answer to indefinite integral In modelling a mixing problem of a single tank using first order ODE , i ended up with
$$\frac{dy}{dt}=3-0.03y$$
Case 1 : if I do $\frac{dy}{dt}=-0.03(y-100)$ which leads to $\ln|y-100|=-0.03t+c$ and ultimately
$$y=100+Ce^{(-0.03t)}$$
Case 2 : if I do $\frac{dy}{dt... | this is not correct
Case 2 : if I do $\frac{dy}{dt}=0.03(100-y)$ which leads to $\ln|100-y|=0.03t+c$ and ultimately
Substitute $z=100-y$ before to integrate
Second case
$$\frac{dy}{dt}=0.03(100-y)$$
Substitute $z=100-y \implies dz=-dy$
$$-\int \frac{dz}{z}=0.03t+K$$
$$\ln|z|=-0.03t+K$$
$$100-y=e^{-0.03t}K$$
$$y=100-Ke^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2741740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Product of sums which equal to sum of product We can be sure that
$$\left(\sum\limits_{k=0}^{n}\frac{1}{k+1}\right)\left(\sum\limits_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{k+1}\right)= \sum\limits_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{(k+1)^2}$$
Is there any similar identities or some types of generalization to find them?
| Let $e_k=1$. You want sequences $a_k,\,b_k$ with $\sum_k a_k \sum_k b_k =\sum_k a_k b_k$, or in terms of inner products $a\cdot e \, e\cdot b = a\cdot b$. This equation would be easy to satisfy for $3$-dimensional vectors: choose your favourite vector $c$ and take $a=c\times e,\, b=a\times c$ so $a\cdot e =a\cdot b =0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2741897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
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If $x \in \operatorname{cl}(A)$, where $A$ is a connected subspace of a topological space $X$, then $A \cup \{x\}$ is connected.
Prove that if $x \in \operatorname{cl}_X(A)$, where $A$ is a connected
subspace of a topological space $X$, then $A \cup \{x\}$ is connected.
My attempt:
Suppose, in order to find a contr... |
The OP's proof is correct, but by changing the last 2 paragraphs it
can be nailed down in another way.
Suppose, in order to find a contradiction, that $B \cup C = A \cup \{x\}$ where $B,C$ are open in $A \cup \{x\}$ non empty and disjoint.
Then, we have either $A \subseteq B$ or $A \subseteq C$. Indeed, if this w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2742007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In classical geometry why is a line considered to be parallel to itself? A definition in classical geometry (for example, Birkhoff's formulation, but I suppose it could be all of them) is that a line is always considered to be parallel to itself. I understand this is probably for convenience, but in my mind since two d... | The idea is that you want "parallel" to define equivalence classes (called "pencils", cf. Coxeter, Projective Geometry, and Artin, Geometric Algebra), which require the defining relationship to be an equivalence relationship: reflexive, symmetric, and transitive. Those classes then have some nifty uses, like defining ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2742126",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
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The random variable U is uniformly distributed over the interval [3,6]. Find the following probabilities (a) $P[\frac4 5 ≤ U ≤ 4]$ = $$P[\frac4 5 ≤ U ≤ 3]+P[3≤ U ≤ 4]=0+P[U ≤ 4]=\frac{(4-3)}{(6-3)}=\frac1 3$$ Is this even right? I'm just not so confident.
(b) $P[U > 5]$ = $$1-P[U ≤5]=1-\frac{8-3}{6-3}=1-\frac5 3$$
The ... | $(a)$ is correct.
$(b)$ $$1-P[U ≤5]=1-\frac{\textbf{5}-3}{6-3}=\frac1 3$$
$(c)$ $$P[16 ≤ U^2 ≤ 36]= P[4≤ U ≤ 6]+P[-6 ≤ U ≤ -4] = P[4≤ U ≤ 6]+0 = 1-P[U ≤4]=1 - \frac{4-3}{6-3} = \frac2 3$$
$(d)$ $$P[4−2|U|≥−8] = P[-2|U|≥-12]=P[-|U|≥-6] =P[|U|≤6]=P[U≤6]= 1 $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2742266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Fermat generalization, not sure if it's true or how to prove it I have this rule in my notebook, but I don't remember when I took it:
$$a^{p^n}$$ is congruent with a(p) (a modulo p), where p is a prime number, a is an integer and n is a natural number.
Or, in other words, that:
$$p | a^{p^n} - a$$
I can't find this r... | Fermat's (little) theorem is: If $p$ doesn't divide $a$ then $a^{p-1} \equiv 1$ mod $p.$ When both sides multiplied by $a$ it becomes $a^p \equiv a$ mod $p$ and now one can allow $a$ divisible by $p.$ At this point apply an inductive argument, next case being $(a^p)^p \equiv a^p,$ then use inductive hypothesis to fini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2742393",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Probability in a Urn Modell If we have a Urn, with 2 black balls,3 red ones and 4 yellow ones.
I'd like to determine the probability of drawing a black ball before drawing a red ball.
So the probabilies to consider are the 3-tupels
(black,red,random),(random,black,red) and (black,yellow,red)
I tried to split the Set of... | So in our draw of $3$ balls, there's definitely a black ball and a red ball. So the third ball can be a yellow or black or red ball. If it's a yellow ball, the favourable cases are just the number of permutations where black comes before red. This is $2$ (YBR,BRY), so the probability is just $1/12$. If it's a black bal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2742708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Hints on how to solve $(x+y^2)dy = ydx$? I'm looking for hints on how to solve the differential equation: $(x+y^2)dy = ydx$ .
I tried finding an integrating factor and dividing both sides by $y$ but that didn't work.
| In $(x+y^2)dy=ydx$ you can indeed divide by $y^2$ to get
$$
d\left(\frac{x}{y}\right)=\frac{y\,dx-x\,dy}{y^2}=dy
$$
which is directly integrable.
For the first version of the question, $(x^2+y^2)dy=ydx$, observe that
$$
\frac{dx}{dy}=y+\frac{x^2}{y}
$$
looks like a Riccati equation. Set $x(y)=-\dfrac{yu'(y)}{u(y)}$ to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2742859",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
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How can I find all solutions using an iteration formula? Take the equation $x^2+7x+10=0$, which has roots $-2$ and $-5$.
When I use the iteration formula $x_{n+1}=\dfrac{-10}{x_n+7}$, I always converge on $-2$ but not on $-5$.
I have tried starting values of $-6, -4, -1, 1, 2$ and many others, but I only converge on ... | There is nothing wrong, when considering sequences defined by $x_{n+1}=f(x_n)$ to study if $l$ such that $f(l)=l$ is attractive or repulsive you have to consider $|f'(l)|$:
*
*If $|f'(l)|<1$ this is an attractive point.
*If $|f'(l)|>1$ this is a repulsive point (and there is no hope to converge to $l$ if $x_0 \neq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2742975",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Why are these $\sum \cos$ and $\csc$ equivalent? Mathematica 'simplifies' this formula
$$\sum_{k=1}^R \cos \frac{2k \pi x}{R}$$
to this
$$\frac{1}{2} \biggl(\csc \frac{\pi x}{R} \sin \frac{(2R+1) \pi x}{R}-1\biggr)$$
A graphical plot of the two formulae generates two identical continuous curves - but why?
Surely $\csc ... | Assuming you're a bit familiar with complex numbers:
$$e^{ix}=\cos(x)+i\sin(x)\tag{1}$$ Let $a=\frac{2\pi}{R}$, then you want to find the sum $$\sum_{k=1}^{R}\cos(akx)\tag{2}$$
Notice that it is way easier to first compute (which we later can relate to $\cos(akx)$) $$\sum_{k=1}^{R}e^{akix}=\sum_{k=1}^{R}\left (e^{aix}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2743100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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A non-zero ring $R$ is a field if and only if for any non-zero ring $S$, any ring homomorphism from $R$ to $S$ is injective. Show that a non-zero ring $R$ is a field if and only if for any non-zero ring $S$, any unital ring homomorphism from $R$ to $S$ is injective.
I would like to verify my proof, especially the rever... | Yes, this proof looks good to me!
| {
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"url": "https://math.stackexchange.com/questions/2743275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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$f(x+1/2)+f(x-1/2)= f(x)$ Then the period of $f(x)$ is?
$f(x+1/2)+f(x-1/2)= f(x)$. Then the period of $f(x)$ is:
a)$1$
b)$2$
c)$3$
d)$4$?
Attempt:
I substituted $x= x \pm1/2$ but the equations I got didn't help at all.
How do I go about solving such a question? I am just looking for a hint and not the entire soluti... | We may conclude that $f(x)=f(x+3)$ holds for all $x$.
Following @Mike Earnest's suggestion,
\begin{align}
f(x)&=f(x+1/2)+f(x-1/2),\\
f(x-1/2)&=f(x)+f(x-1).
\end{align}
Add up these two equations, and you will have
$$
f(x+1/2)+f(x-1)=0,
$$
or, due to the arbitrariness of $x$,
$$
f(x+3/2)+f(x)=0.
$$
Now let $x\to x+3/2$,... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 0
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If $f$ is differentiable and its derivative is uniformly continuous, then $n \cdot (f(x+\frac{1}{n})-f(x)) \to^{u} f'(x)$ Let $f$ be a differentiable function in $\mathbb{R}$ such that its derivative $f'$ is uniformly continuous in $\mathbb{R}$.
Prove: $$n \cdot (f(x+\frac{1}{n})-f(x)) \to^{u} f'(x)$$ (uniform converg... | Instead of using integration, you can use the mean value theorem as suggested in the comments.
Let $\epsilon > 0$, so we have $\delta > 0$ such that $|x-y|< \delta$, $|f'(x) - f'(y)| < \epsilon$. Now let $n > \frac{1}\delta$ and $x \in \mathbb R$. By the mean value theorem, $\exists \xi \in (x, x+\frac{1}{n})$ such th... | {
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"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Substantial / Total / Material Derivative Contradiction We want to find the material derivative in terms of the spatial representation (classic case).
$\phi$ : Temperature, entropy etc,
$x_{0}$ : Reference coordinate,
$x$ : Spatial / eulerian coordinate
Let $\phi=xt$ and $x= x_{0}(1+t)$ where $x_{0} = c $
$\dfrac{d\phi... | The material derivative is the temporal rate of change of the scalar $\phi$ following the path of a fluid particle. In a Lagrangian framework, the location of a fluid particle at time $t$ initially with coordinate $x_0$ is specified by a smooth function $\xi: \mathbb{R}^2 \to \mathbb{R}$ where
$$x = \xi(x_0,t) = x_0(... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability of a coin stack being greater than a value? What's wrong with my reasoning? Basic probability question.
Consider a pile of 9 coins where each could either be 1 cent or 10
cents and the distribution of the coin combinations is uniform.
Knowing that the upper 4 coins are all 10 cents, what is the
prob... | Well the question does seem to be rather poorly worded. In probability it is important to be precise on what is being conditioned on and the wording of the question was not. I took it to mean that
*
*Each of the 9 coins was pulled out of a large vat w an equal number of pennies and dimes, so that with each pull, th... | {
"language": "en",
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"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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How to find the equation of the tangent to the parabola $y = x^2$ at the point (-2, 4)? This question is from George Simmons' Calc with Analytic Geometry. This is how I solved it, but I can't find the two points that satisfy this equation:
$$
\begin{align}
\text{At Point P(-2,4):} \hspace{30pt} y &= x^2 \\
\frac{dy}{d... | The others did point out your error, so I will just add the way I'd do it:
The tangent line we are looking for is in the form of $$g(x)=ax+b$$ for the function $$f(x)=x^2$$ at $x=-2$. We know that their derivate and their value most be equal at the given point, so we have that $$a=2*(-2)=-4$$ and $$(-2)^2=-4(-2)+b$$ $$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2744119",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Help to solve a system equation: $x-y+xy=-4$; $xy(x-y)=-21$. I need to solve a system equation. Here's how it looks:
$x-y+xy=-4$
$xy(x-y)=-21$
I tried to substitute $x-y$ with $w$ and $xy$ with $t$ to simplify everything. After that I got this system equation:
$w+t=-4$
$tw=-21$
I solved this new system equation and got... | HINTS:
We have the equations $$x-y+xy=-4\implies x-y=-4-xy\tag{1}$$ and $$xy(x-y)=-21\tag{2}$$ so substituting $(1)$ into $(2)$, we get $$xy(-4-xy)=-21\implies(xy)^2+4xy-21=0$$ Let $u=xy$. Then $$u^2+4u-21=0$$ and so...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2744257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Curious integral, $8\pi\int_{0}^{\pi/2}\cos^2x{\ln^2(\tan^2 x)\over [\pi^2+\ln^2(\tan^2 x)]^2}dx=\ln 2-{1\over 4}\zeta(2)$ How to show that,
$$8\pi\int_{0}^{\pi/2}\cos^2x\cdot{\ln^2(\tan^2 x)\over [\pi^2+\ln^2(\tan^2 x)]^2}\mathrm dx=\ln 2-{1\over 4}\zeta(2)$$
This integral is an extract from this paper of Olivier Oloa... | By letting $x=\arctan u$ the original integral is converted into
$$ 32\pi \int_{0}^{+\infty}\frac{\log^2(u)}{(1+u^2)^2\left(\pi^2+\log^2 u\right)^2}\,du\stackrel{u\mapsto e^{\pi x}}{=}32\int_{\mathbb{R}}\frac{e^{\pi x}x^2}{(1+e^{2\pi x})^2(1+x^2)^2}\,dx $$
and by symmetry the RHS collapses into
$$ 16 \int_{0}^{+\infty}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Matrix with nonnegative symmetric part and semisimplicty of the eigenvalue 0 Let B be a real square matrix with non-negative symmetric part, i.e. for all vectors $X$, $X^\top B X\geq 0$. We also assume that $B$ is singular. I am wondering if the eigenvalue $0$ of $B$ is necessary semi-simple, i.e. is the dimension of t... | I came back to this and found the answer: yes, if $B$ is singular and has nonnegative symmetric part, then the eigenvalue $0$ of $B$ is semisimple. This relies on the following observations:
Claim 1: If $0$ is not a semisimple eigenvalue of $B$, then the resolvent $R(z) = (B-z)^{-1}$ has a pole of order at least two at... | {
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"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to see the symmetry in this trigonometric equation Consider the equation $2\cos^2x-\cos x-1=0$. We can factor the LHS to obtain: $$(2\cos x + 1)(\cos x-1)=0,$$ leading to three solutions in the interval $[0,2\pi)$, namely $x=0, \frac{2\pi}{3}, \frac{4\pi}{3}$. If we want all solutions over $\Bbb R$, then we can add... | $2\cos^2x-\cos x-1=(\cos{2x}+1)-\cos x-1=\cos 2x-\cos x$
$\cos 2x -\cos x=-2\sin{\frac{3x}{2}}\sin{\frac{x}{2}}$
Is it OK at this point?
You can play more with the trigonometric identities to see sometimes if it does turn out well.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Using a Taylor polynomial of degree 2, find an approximation for $\sqrt[3]{e}$ I do not understand how to find an approximation using a Taylor polynomial.
Also, I need to find an upper limit to the remainder $|R_2(x)|$.
Excuse me if I have any obvious mistakes, this is my first time solving something like this.
$$R_2(\... | Use the Taylor series for $e^x$ evaluated at $x=\frac{1}{3}$:
$$e^{1/3}=\sum_{n=0}^\infty\frac{1}{3^nn!}=1+\frac{1}{6}+\frac{1}{54}+\cdots$$
which is from the general form
$$e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$$
| {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
A claim from "Example of a linear functional, but not a distribution" I'm studying linear functionals that are not distributions, and I came across this post: link.
In one of the comments it is claimed that the functional $u:C_c^\infty\to\mathbb{C}$ given by
$$ u= \sum_{n\geq 0} \frac{R^n}{n!}\delta^{(n)} $$
is not ev... | Strictly speaking this statement is not true. Fix a smooth function $\phi$ with compact support equal to $1$ on a neighborhood of zero and define
$$f(x)=\begin{cases} \phi(x) e^{-1/x^2} & x \neq 0 \\
0 & x=0 \end{cases}.$$
Then $u(f;R)=0$ for all $R$, but $f$ is not analytic on any neighborhood of zero.
This statement ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If my random variables $X_1,...,X_n$ are i.i.d. $N(\mu,\sigma^2)$, why isn't $\bar{X}\sim N(\mu,0)$? If my random variables $X_1,...,X_n$ are i.i.d. $N(\mu,\sigma^2)$, why isn't $\bar{X}\sim N(\mu,0)$?
In other words, if, as I understand it, $X_1,...,X_n$ all have the same mean, $\mu$, how can there be any variance at ... | You draw a sample consisting of $n$ observations, you can compute a sample mean.
You draw another sample consisting of another $n$ observations, you can compute another sample mean.
We do not expect the first sample mean to be equal to the second sample mean, in fact, it is unlikely that either of them would be equal... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2744905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Abstract index notation vs Ricci Calculus I have come accross some comparison between the abstract index notation and Ricci calculus as it pertains to contraction and what I find is:
The former (abstract notation) indicates that a basis-independent trace operation being applied, which reduces to the aforementioned summ... | A simple example to illustrate the point is an expression of the form $t^a_{\;a}$ where $t^a_{\;b}$ is a tensor (or tensor field) of type (1,1), vulgarly known as an endomorphism. The expression $t^a_{\;a}$ is interpreted in the usual index notation as the sum of the "diagonal" elements and taken literally is dependen... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Chebychev's Inequality Question Not sure if I'm understanding the question wrong, but the prof's notes gave a different answer.
The number of equipment breakdowns in a manufacturing plant averages 5 per week with a standard deviation of 0.8 per week.
Find an interval that includes at least 90% of the weekly figures fo... | The problem is when you write that $$P(|X-E(X)|\geq kSD(X))\leq \frac{1}{k^2}$$
implies $$P(|X-E(X)|< kSD(X))> \frac{1}{k^2}$$
It would rather be $$P(|X-E(X)|< kSD(X))\geq 1- \frac{1}{k^2}$$
Edit: how to get from $1$ to $3$. Call $A$ the event $\{|X-E(X)|\geq kSD(X)\}$.
Then $P(A)+P(A^c)=1$ implies that if $P(A)\leq \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2745222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Identify some Coxeter group As we all know, the weyl group of lie algebra of $B_{2}$ type is $\left\{s_{1},s_{2}|s_{1}^{2}=1, s_{2}^{2}=1, (s_{1}s_{2})^{4}=1\right\}$. How can we identify this with $Z^{2}_{2}\rtimes S_{2}$?
If I choose fundamental roots of $B_{2}$ as basis, then I can identify $s_{1}$ with matrix $\beg... | I will describe this for type $B_n$ in general, since that actually makes it a bit more clear what is going on.
$$W(B_n) = \langle t_0, s_1,\dots, s_{n-1}\mid t_0^2, s_i^2, (t_0s_1)^4, (t_0s_i)^2\mbox{ for }i\geq 2, (s_is_{i+1})^3, (s_is_j)^2\mbox{ for }|i-j|>1\rangle$$
So it has generators $s_i$ for $i\in \{1,\dots, n... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2745515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How to prove $\lim_{x \to \infty} x^{(x+1)}-(x+1)^x = \infty$ I try to prove this by using L'Hospital Rule but it doesn't work.
I know it is infinity from wolframalpha but I don't know how to prove it.
| Use the estimate:
$$n^{n+1}-(n+1)^n>n^n, n>3 \iff n^n(n-1)>(n+1)^n,n>3 \iff \\
n-1>\left(1+\frac 1n\right)^n, n>3.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2745644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Eigenvalues of a block circulant matrix How does one calculate the eigenvalues of a unitary 3 by 3 block circulant matrix where
$$U = -\frac{i}{3} \begin{pmatrix}
\Lambda_{1} & \Lambda_{2} & \Lambda_{3} \\
\Lambda_{3} & \Lambda_{1} & \Lambda_{2} \\
\Lambda_{2} & \Lambda_{3} & \Lambda_{1} \end{pmatrix}$$
whe... | This won't be a full answer, but here's some thoughts that (I think) should move you in the right direction.
Let $P$ denote the matrix
$$
P = \pmatrix{0&1&0\\0&0&1\\1&0&0}
$$
Then, in terms of Kronecker products, we can write your matrix as
$$
3iU = I \otimes \Lambda_1 + P \otimes \Lambda_2 + P^2 \otimes \Lambda_3
$$
I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2745824",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
what is "Minimal Uncountable well-ordered set"? Can anyone make me understand what is "Minimal Uncountable well-ordered set" (Munkres, Topology, Example 2 of the limit point compactness section)?
I know what is Uncountablity and Well ordered set.
Thank You in Advance.
| @cmi ℝ in the usual ordering is not well ordered. And if equipped with proper ordering, ℝ can be isomorphic to the "Minimal Uncountable well-ordered set" (assuming the continuum hypothesis), because under the continuum hypothesis, ℝ has the same cardinality as the "Minimal Uncountable well-ordered set", and therefore a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Prove that $6$ is a divisor of $n^3 - n$ for all natural numbers. How would you approach such a problem? Induction perhaps? I have been studying proof by induction, but so far I have only solved problems of this nature:
$$1 + 4 + 7 +\dots+ (3n-2) = \frac{n(3n-1)}{2}.$$
| To prove by induction take the base case $n=0$ or $n=1$.
Then for the inductive step $$(n+1)^3-(n+1)=n^3+3n^2+3n+1-n-1=(n^3-n)+3(n^2+n)$$and the statement will be true if you can show that $n^2+n=n(n+1)$ is even. You can do that various ways, including a similar induction
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2745984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
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Closed form of $\sum_{k=0}^n\binom{2k}{k}(-1/4)^k$ Loosely related to my last question, I was trying to find a closed form of the finite sum
$$a_n:=\sum_{k=0}^n\binom{2k}{k}\left(-\frac{1}4\right)^k$$
This is not too different from the well-known expression
$$\sum_{k=0}^n\binom{2k}{k}\left(\frac{1}4\right)^k=\binom{n+\... | I think that
\begin{align}
a_n&=\sum_{k=0}^n\binom{2k}{k}\biggl(-\frac14\biggr)^k\\
&=\frac2\pi\int_0^{\pi/2}\frac{(-1)^n \sin ^{2 n+2}x+1}{\sin ^2x+1}\textrm{d}x\\
&=\frac2\pi\int_0^{\infty}\frac{1}{2+x^2}\biggl[1+\frac{(-1)^n}{(1+x^2)^{n+1}}\biggr]\textrm{d}x.
\end{align}
These integral representations imply that
*
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2746097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Paradox vs Tautology. The expression(~p or p )is a Tautology.
Consider this statement(p): This statement is false.
Now here, Statement p is paradoxical.
My question is :- Can we define paradoxes like this as statements which prove Tautologies wrong?
| A statement can be provable or not provable, and it can be sound (something we expect to be true, like 1+1=2) or unsound (we expect it to be false, like 1+1=3).
A paradox arises when a statement is unsound but provable. It indicates an error in the logic.
The claim of the speaker in the liar's paradox is clearly uns... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2746221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
If two random variables have CDFs that have the same value for all x, can we assume the random variables are equal? My text has the following theorem:
Let $X$ have a CDF $F$ and let $Y$ have CDF $G$. If $F(x) = G(x)$ for all $x$, then $\mathbb{P}(X \in A) = \mathbb{P}(Y \in A)$ for all $A$.
I don't see a way that X and... | Elementary probability:
They don't teach this is in elementary probability, but random variables have an explicit representation known as the Skorokhod representation.
Basically, we never really know the formulas for a lot of the $X$'s. We know the $X$'s mainly from the $F_X(x)$'s. It's kinda like talking about $f(x)=x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2746337",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Why worry about commutativity but not associativity in The Fundamental Theorem of Arithmetic? A common statement of The Fundamental Theorem of Arithmetic goes:
Every integer greater than $1$ can be expressed as a product of powers of distinct prime numbers uniquely up to a reordering of the factors.
Now the statement... | While being sloppy, mathematicians create an intuitive and simple language that allows deeper and deeper investigations in an extraordinary creative activity. Keeping all parenthesis and avoid all informal constructions would make mathematics more static.
Of course one can define a normal form which cooperates with the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2746430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 10,
"answer_id": 9
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In which order can you execute the given rotation and the projection successively?
Given is the rotation
$$d:\mathbb{R}^2 \rightarrow \mathbb{R}^2 \mbox{ with }
d:\begin{pmatrix}
x\\
y
\end{pmatrix} \mapsto
\begin{pmatrix}
x \cos \alpha - y \sin \alpha\\
x \sin \alpha + y \cos \alpha
\end{pmatrix}$$
and the proj... | Since
*
*$d:\mathbb{R}^2 \rightarrow \mathbb{R}^2$
and
*
*$p: \mathbb{R}^3 \rightarrow \mathbb{R}^2$
the composition is possible only for projection first and then rotation, that is
*
*$d\circ p: \mathbb{R}^3 \rightarrow \mathbb{R}^2\quad \begin{pmatrix}
x\\
y\\
z
\end{pmatrix} \mapsto \begin{pmatrix}
x ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2746542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If $M=f(N)\oplus K$, then there is a left inverse to $f$ Suppose $f:N\to M$ is an injective $R$-module homomorphism and $f(N)$ is a direct summand of $M: M=f(N)\oplus K$ for a submodule $K\subset M$. I'm trying to show that there is a homomorphism $f':M\to N$ such that $f'(f(n))=n$ for $n\in N$.
It would be natural to ... | We don't need to know what $f$ is, just that it's isomorphic onto its image. We could write the proof as follows:
Theorem: Suppose $f : N \rightarrow M$ is an injective $R$-module homomorphism and $f(N)$ is a direct summand of $M$. Then $f$ has a left inverse.
Proof: Take a submodule $K$ such that $M = f(N) \oplus K$ (... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2746811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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In how many ways can 20 identical chocolates be distributed among 8 students Provided each student gets atleast 1 chocolate and exactly two students get atleast two chocolates each.
We know that if k indistinguishable objects are to be placed in n bins such that each bin contains atleast 1 object- the # of ways we can ... | All $8$ students shall obtain $\geq1$ pieces, and exactly $2$ of them shall obtain $\geq2$ pieces.
Give each student $1$ piece, select the $2$ special students in ${8\choose2}=28$ ways, give the senior of these $s\in[11]$ additional pieces, and the junior the remaining $12-s>0$ pieces. It follows that there are $28\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2746957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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$\tilde{P}$ is a refinement of $P$. $m_j\le \tilde{m_p}$? $\tilde{P}$ is a refinement of $P$. $P=\{x_0,....,x_n\}$ and $\tilde{P}=\{x_{k_0},...,x_{k_n}\}$, and $x_{k_j}=x_j.$ $m_j = \inf\{f(x):x_{j-1}\le x \le x_j\}$, and $\tilde{m_p}=\inf\{f(x_p):x_{k_{j-1}}\le x_p \le x_{k_j}\}$. My textbook says $m_j\le \tilde{m_p... | Let's start with the observation that if $I$ is an interval, and $I' \subset I$ is a subinterval, then $\inf \{f(x) : x \in I\} \leq \inf \{f(x) : x \in I'\}$, since the $\inf$ on the whole interval is clearly smaller than the $\inf$ on the contained interval.
Now let
$$ m_j = \inf\{f(x):x_{j-1}\le x \le x_j\}$$.
If w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2747115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Inner product's property in a Hilbert space on $\mathbb{C}$ Recently, I have just learnt about the concept of a Hilbert space.
As far as I can understand, a Hilbert space is a generalized Euclidean space.
When talking about an Euclidean space $E$, indeed there must be a mapping from $E \times E$ to the scalar field, ca... | It's not a stupid question at all!
One reason is that using conjugate symmetry instead of symmetry allows for a norm to be defined on the space, as with real inner product spaces. We define
$$\|x\| = \sqrt{\langle x, x \rangle},$$
and expect such a thing to be well-defined, real, and non-negative so as to measure dista... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2747274",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find explicitly the positive solutions of $2^x=x^2$
Find explicitly the positive solutions of the equation $2^x=x^2$
I noticed that $x=2$ and $x=4$ are roots of the equation.
How can I prove that they are the only positive ones? Thanks in advance
| They're not the only solutions, if we also allow negative nummbers:. Since $2^0>0^2$ and $2^{-1}<(-1)^2$ there exists a solution $x$ with $-1<x<0$.
Those three are the only real solutions. Let $f(x)=2^x-x^2$. Calculate the third derivative and show that $f'''(x)>0$ for every $x$. If $f$ had four zeroes the Mean Value T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2747413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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if $d\mid n$ then $x^d-1\mid x^n-1$ proof How would you show that if $d\mid n$ then $x^d-1\mid x^n-1$ ?
My attempt :
$dq=n$ for some $q$. $$ 1+x+\cdots+x^{d-1}\mid 1+x+\cdots+x^{n-1} \tag 1$$ in fact, $$(1+x^d+x^{2d}+\cdots+x^{(q-1)d=n-d})\cdot(1+x+\cdots+x^{d-1}) = 1+x+x^2 + \cdots + x^{n-1}$$
By multiplying both sid... | You can always do:
$f(y)=1+y+ \dots +y^{r-1}$ so that $yf(y)=y+y^2+\dots +y^r$ and $yf(y)-f(y)=(y-1)f(y)=y^r-1$
Then put $y=x^d$ with $dr=n$ and obtain $(x^d-1)f(x^d)=x^n-1$ and by construction $f(x^d)$ is a polynomial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2747509",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 2
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Suppose $ x+y+z=0 $. Show that $ \frac{x^5+y^5+z^5}{5}=\frac{x^2+y^2+z^2}{2}\times\frac{x^3+y^3+z^3}{3} $. How to show that they are equal? All I can come up with is using symmetric polynomials to express them, or using some substitution to simplify this identity since it is symmetric and homogeneous but they are still... | Write $S_n=x^n+y^n+z^n$. Then $S_0=3$, you are given $S_1=0$ and are charged to prove that $S_5=(5/6)S_2S_3$. Define
$$F(t)=\sum_{n=0}^\infty S_nt^n.$$
Then
$$ F(t)=\frac1{1-xt} + \frac1{1-yt} + \frac1{1-zt} = \frac{3+2e_1t+e_2t^2}
{1-e_1t+e_2t^2-e_3t^3} $$
where $e_1=x+y+z$, $e_2=xy+xz+yz$ and $e_3=xyz$. Then $e_1=0$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2747626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Finite abelian group generated by two elements Let $G$ be a finite abelian group generated by two elements $a$ and $b$. I am trying to prove that $G$ is isomorphic to the direct product of two cyclic group $C_r$ and $C_s$, where the value of $r$ and $s$ depend on $|a|$, $|b|, d = |\langle a \rangle \cap \langle b \rang... | Here's a different approach:
Theorem: Let $A$ be an abelian group and suppose $A$ is generated by $a, b$, where $a$ and $b$ have finite order. Then $A$ is isomorphic to the product of two cyclic groups.
Proof: Take generators $a$ and $b$ of $A$, and write $C = \langle a \rangle \subset A$. $A /C$ is cyclic with generat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2747715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Out of $8$ points, $4$ points on one branch of a hyperbola and $4$ on the other but no $5$ ever in convex position,is it true? Are any $8$ points, $4$ points on one branch of a hyperbola, $4$ points on the other branch of the same hyperbola always such that no $5$ points are in convex position (form a convex shape)
| More is true: If we have four points on a hyperbola, three of them on the same branch, and one on the other branch, then these four points do not form a convex quadrangle.
Proof. Consider the hyperbola $xy=1$, and assume $P_i=(x_i,y_i)$ $\>(1\leq i\leq3)$ in the first quadrant, with $x_1<x_2<x_3$. The lines $g_{12}=P_1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that if $A$ is diagonally dominant and if $Q$ is chosen as in the Jacobi method, then $\rho(I-Q^{-1}A)<1$ Prove that if $A$ is diagonally dominant and if $Q$ is chosen as in the Jacobi method, then $\rho(I-Q^{-1}A)<1$
I know that $\rho(A)=\inf_{\|.\|}\|A\|$ and \begin{equation} ... | This property can be easily shown with the help of this property of the spectral radius: For the spectral radius the following holds for any matrix $A \in \mathbb R^{n \times n} $ and any (matrix) norm $\Vert \cdot \Vert$:
$$\rho(A) \leq \Vert A \Vert. $$
Thus, for your chosen norm and a strictly diagonally dominant ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2748028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are the rings $\mathbb{Q}[x]$ and $\mathbb{Q}[x,y]$ principal ideal domains? Are the rings $\mathbb{Q}[x]$ and $\mathbb{Q}[x,y]$ principal ideal domains?
I understand what an integral domain is. I know the definitions of ideal and principal but have not ever dealt with principal ideal domains. I know that from the disc... | If $k$ is a field, then $k[x]$ is a PID, for essentially the same reason that $\mathbb{Z}$ is -- in both, we have Euclidean algorithm for division.
On the other hand, $k[x, y]$ is not a PID: consider ideal $I = (x, y)$. Supposed that $I$ is principal, $I = (f)$. Then $x = fg$, $y = fh$ for some $g, h \in k[x, y]$. Sinc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2748211",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Largest integer $n$ such that $3^n$ divides every $abc$ with $a$, $b$, $c$ positive integers, $a^2+b^2=c^2$, and $3|c$
Let $P$ denote the set { $abc$ : $a$, $b$, $c$ positive integers, $a^2+b^2=c^2$, and $3|c$}. What is the largest integer $n$ such that $3^n$ divides every element of $P$?
I first saw that as $3|c\im... | First, convince yourself that every solution of $a^2+b^2=c^2$ has $abc$ a multiple of 3.
Then convince yourself that every solution with $c$ a multiple of 3 must be such that $(a/3)^2+(b/3)^2=(c/3)^2$ with $a,b,c$ all integers (I think you've already done this, so the first sentence above is all you really need).
| {
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Minimum length of the hypotenuse
A point on the hypotenuse of a triangle is at distance a and b from the sides of the triangle. Prove that the minimum length of the hypotenuse is $(a^{2/3}+b^{2/3})^{3/2}$.
My Attempt
$\frac{x}{y}=\frac{a}{CM}=\frac{AN}{b}$
$$
\frac{x}{y}=\frac{AN}{b}\implies y=\frac{xb}{\sqrt{x^2-a^... | we have $$\cos(\alpha)=\frac{a}{x}$$ and $$\sin(\alpha)=\frac{b}{y}$$ so we get
$$1=\frac{a^2}{x^2}+\frac{b^2}{y^2}$$ with this equation you ca eliminate $x$ or $y$ in $$c=x+y$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2748447",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Basis of of subspaces of $R^3$ I'm having a little trouble with this question:
How do I find the basis of the set of vectors lying in the plane 2x − y − z = 0?
I'm stuck on how to start on this question
I tried to start by setting y= 2x-z
I'm not sure where to go from here
| Your start $y= 2x-z$ is good. It means that every vector
$$\begin{pmatrix}x\\2x-z\\z\end{pmatrix}, \quad\text{$x,y$ being real numbers,}$$
belongs to the plane. Hence every point in that plane may be expressed as
$$\begin{pmatrix}x\\2x-z\\z\end{pmatrix}=x\begin{pmatrix}1\\2\\0\end{pmatrix}+
z\begin{pmatrix}0\\-2\\1\e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2748568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Probability in uniform I need help to solve the following problems. Thank you in advance.
Problem 1:
A random variable $X$ is uniform $[0, 1]$. Find the probability that X's 2nd digit is $3$.
As far as I understand it is continuous uniform distribution. Each digit has $1$ chance in $10$ of being a $3$. Does it mean th... | For problem 2, find the intervals of success and divide by the total interval.
The intervals of success (i.e. the first or the second digit is $2$) are:
$$[0.2,0.3); [1.2,1.3); [2,3) \Rightarrow I_S=0.1+0.1+1=1.2$$
The total interval is:
$$[0,3] \Rightarrow I_T=3.$$
Hence:
$$P(D_1=2\cup D_2=2)=\frac{I_S}{I_T}=\frac{1.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2748700",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Compute $\lim_{n\to\infty}\int_\limits{0}^{\infty}\frac{dx}{x^{n}+1}$
Compute the following limit: $$\lim_{n\to\infty}\int_\limits{0}^{\infty}\frac{dx}{x^{n}+1}$$.
So I thought to use the result:
a) If $\{F_n\}$ converges uniformly on $S=[a,b]$ to $F$ and $F_n$ is integrable $\forall n$. Then $\int_\limits{a}^{b}F(... | Instead of working with the limit in the middle integral, treat it like $n$ is large, then approximate the integral (bound the integrand from above by $1$). You should end up with a direct $\delta$ dependence. If you let $\delta $ be arbitrary, what does that tell you the middle integral must be?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2748808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
How to show that $a_n=1+1/\sqrt{2}+\cdots+(1/\sqrt{n-1})-2\sqrt{n}$ has an upper bound. Let $a_n=1+1/\sqrt{2}+\cdots+(1/\sqrt{n-1})-2\sqrt{n}$ while $a_1=-2,n\ge2 $ ,
I need to prove that $a_n$ converges.
I proved that it is monotonically increasing and tried to prove that it is upper-bounded by induction but failed ... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\new... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"answer_id": 3
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Find out how many people from glasses clinking There's a party and there are people clinking glasses. We hear 28 clinks (one person clinks with exactly one person). I have to find out how many people there are at the party.
So I put in this equation:
$\binom{x}{2} = 28$, but I don't know how to find $x$.
| You can solve for $x$ in the polynomial $$28=\binom{x}{2}=\frac{1}{2}x(x-1)$$ as others mention.
However, 28 is small, and $x$ is limited to a positive integer. I'd use trial and error.
When $x=4$ we have $\binom{x}{2}=6$, which is too small.
When $x=5$ we have $\binom{x}{2}=10$, which is too small.
When $x=6$ we have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2749185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Why is no covering space of $\mathbb{R}P^2 \vee S^1$ homeomorphic to an orientable surface? I know that $\pi_1(\Sigma_g) = \langle a_1,b_1,\cdots,a_g,b_g | [a_1,b_1]\cdots[a_g,b_g]\rangle$, where $\Sigma_g$ is the orientable surface of genus $g$, and $[a_i,b_i]$ is their commutator. The idea of my argument for why no s... | Hint: a covering projection is a local homeomorphism. Every point of a surface (oriented or not) has a neighbourhood homeomorphic to $\Bbb{R}^2$. No point of the copy of $S^1$ in $\Bbb{R}P^2 \vee S^1$ has a neighbourhood homeomorphic to $\Bbb{R}^2$. (Since the connected component of a neighbourhood of any such point $x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2749282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Changing index sum of series Let $$f(x)=\sum_{n=1}^{\infty}\frac{\sin(nx)}{n^3}$$.
Now let $f(x)$ be uniform convergence on $\mathbb{R}$
Show that $$\int_0^\pi f(x) \,dx=\sum_{k=0}^{\infty}\frac{2}{(2k+1)^4}$$
I'm not sure how to get past from this point. I was trying to move the index from n=0 but I can't seem to fig... | \begin{align*}
\dfrac{1}{n^{3}}\int_{0}^{\pi}\sin(nx)dx=\dfrac{1}{n^{3}}\cdot\dfrac{-1}{n}\cos(nx)\bigg|_{x=0}^{x=\pi}=\dfrac{-1}{n^{4}}\left((-1)^{n}-1\right),
\end{align*}
now consider even and odd $n$ separately.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2749396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $f:\mathbb{R}^2\to\mathbb{R}^1$ is of class $C^1$, show that $f$ is not one-to-one. If $f:\mathbb{R}^2\to\mathbb{R}^1$ is of class $C^1$, show that $f$ is not one-to-one. [Hint: If $Df(x) = 0$ for all $x$, then $f$ is constant. If $Df(x_0)\neq0$, apply the implicit function theorem.]
Clearly there are two cases, if... | Suppose that $Df(x_0,y_0)\neq 0$. You can suppose without restricting the generality that ${{\partial f}\over{\partial y}}\neq 0$, let $h(x,y)=f(x,y)-f(x_0,y_0), {{\partial h}\over{\partial y}}={{\partial f}\over{\partial y}}\neq 0$, the implicit function theorem implies that there exists a neighborhood $I$ of $y_0$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2749521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
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Find the length and width of rectangle when you are given the area The area of a rectangle is $x^2 + 4x - 12$. What is the length and width of the rectangle?
The solution says the main idea is to factor $x^2 + 4x -12$.
So, since $-12 = -2 \times 6$ and $-2 + 6 = 4$, it can be written as $x^2 + 4x - 12 = (x - 2)(x + 6)... | That is incorrect. Many rectangles, with different lengths and widths, can have the same area. Example: $2\times3 = 1\times6 = \pi\times\frac6\pi$
Also, the area is a function of $x$; if $x$ is not given, then the area is not given!
And width can't be negative.
Where did you find this "solution"? It's very wrong.
EDIT... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2749623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Finding probability that a car experiences a failure
A car is new at the beginning of a calendar year. The time, in years,
before the car experiences its first failure is exponentially
distributed with mean 2. Calculate the probability that the car
experiences its first failure in the last quarter of some calendar
yea... | You need to find this:$$ \sum_{k=1}^\infty e^{-{1\over2}(k-0.25)} -e^{-{1\over2}k} $$
because you're looking for the probability that the car experiences a failure in any year, not just the first year. The summation above will give you the answer in your textbook.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2749746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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