Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. so I'm supposed to let $A$ be a square real matrix with the property that every nonzero vector in $\mathbb{R}^n$ is an eigenvector of $A$. And I'm supposed to show that $A=\lambda I$ for a constant $\lambda$ in $\mathbb{... | If $n=1$, there's nothing to prove, as all matrices are scalar multiples of the identity.
If $n\ge 2$, take two linearly independent vectors $v_1,v_2$ with corresponding eigenvalues $\lambda_1$, $\lambda_2$. Since every vector is an eigenvector, so is $v_1-v_2$ with corresponding eigenvalue $\lambda_3$. So then $A(v_1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1126028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Limit to zero of the $p$-norm I have the $p$-norm defined as $$\|x\|_p=\left(\sum_{k=1}^n|x_k|^p\right)^\frac{1}{p}.$$ I am trying to find the limit as $p\to0^+$ of $\|x\|_p$. I've seen it defined as $\{x_k:x_k\neq0\}$. Why is this true? Is there a way to prove this?
| Let $m=\min\{|x_i|:\ x_i\neq0\}$, and $k$ the number of non-zero $x_i$. Then
$$||x||_p=m\cdot\left(\sum_{i=1}^{n}\left|\frac{x_i}{m}\right|^{p}\right)^{1/p}\to\begin{cases}0&\text{ if }k=0\\m&\text{ if }k=1\\\infty&\text{ if }k>1\end{cases}$$
The reason for the value when $k>1$ is that the $\sum$ above has $k$ terms $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1126145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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multivariable limit problem I have a confusion regarding this problem.
Problem: $\displaystyle f(x,y)=\frac{\sin^2|x+2y|}{x^2+y^2}$ is continuous for all $(x,y)\neq (0,0)$. True or false?
I think that the limit does not exist so the function is not continuous.
How to prove that limit does not exist?
Any help will be a... | Try approaching the limit by the line $y=k\cdot x$.
Applying the standard limit $$\lim_{t\rightarrow 0} \frac{\sin(t)}{t}$$
You can prove that the limit depends on the constant K, hence does not exist
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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A digraph is a graph where every edge is directed. How many digraphs on $n$ vertices are there? So far I have that between any two vertices (say $j$ and $k$) there are 3 options.
*
*there is no edge between $j$ and $k$
*there is an edge directed from $j$ to $k$
*there is an edge directed from $k$ to $j$
And so f... | In a directed graph, there are $n(n-1)$ distinct edges, each of these can be present or not. Thus the number of digraphs with $n$ vertices should be
$$2^{n(n-1)}$$
Note that this allows two vertices to be connected by directed edges in both ways. This coincides with the definition on wikipedia (see the example image).
... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Chinese Remainder Theorem for non prime-numbers. Let's say I want to find x such that x leaves remainder 2 when divided by 3 and x leaves remainder 3 when divided by 5.
x % 3 = 2
x % 5 = 3
We break down the problem to:
x % 3 = 1
x % 5 = 0
Therefore,
5k % 3 = 1
2k % 3 = 1
k = 2
10, when remainder = 1
20, when remainder ... | Exactly the same way. The equation $x\equiv 3 \mod 7$ tells you that $x=3+7y$ .
Plugging this into the second equation gives you $3+7y\equiv 2 \mod 4$, that is $-y\equiv -1 \mod 4$, so $y=1+4z$, and $x=10+28z$, i.e. $x\equiv 10 \mod 28$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1126463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Pattern Recognition - How to solve this problem? A general term of the sequence
$$
-8,-7,-10,-1,-28,53,\ldots
$$
can be expressed as $\dfrac{a-b^{n-1}}{4}$, where $a$ and $b$ are integers.
What is the value of $ab$?
Teach me how to solve this problem.
| If we call this sequence $x_n$, then $a-4x_n$ is the geometric sequence $b^{n-1}$. So the first element should be $1$, meaning $a-4(-8)=1$, and the second element should be $b$, so $b=a-4(-7)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1126682",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Show that $\mathrm{gcd}(x+4,x-4)$ divides $8$ for all integers $x$. I want to prove that $\mathrm{gcd}(x-4,x+4)$ divides $8$ for all $x\in \mathbb{Z}$
Since they are both polynomials of degree $1$, it suggests that the $\mathrm{gcd}$ is a constant.
Using Euclidean Algorithm, I get: $(x+4) = 1(x-4) + 8$, so $\mathrm{gcd... | Your approach is correct. Another way of proving it is the following:
$$
d\mid x-4\text{ and }d\mid x+4\implies d\mid(x+4)-(x-4)=8.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1126779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Adjoint Transformations and Self-Adjoint Operators I don't quite understand the whole adjoint and self adjoint thing. I know their definitions:
Given a linear transformation $A:\mathbb{R}^d \to \mathbb{R}^m$, its adjoint >transformation, $A^*:\mathbb{R}^m \to \mathbb{R}^d$ is defined by $$(Ax,y)=(x,A^*y).$$
A linear o... | We can't say that any $B$ is defined by $(Bx,y) = (x,By)$. However, if $B$ is self-adjoint, then $B$ has the property that $(Bx,y) = (x,By)$ for every $x,y \in \Bbb R^k$.
Here are some examples with matrices: recall that the usual inner product for $x,y \in \Bbb R^n$ is given by
$$
(x,y) = x^Ty
$$
Define the transfor... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges. Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges.
I tried few things but it wouldn't work out. I would appreciate your help.
| $\require{cancel}$
Using the limit comparison test with the harmonic series we see
$$\lim_{n\to\infty} {\cancel{(1/n)}\cdot 1\over \cancel{(1/n)}\sqrt[n]{n}}=1$$
hence the series diverges since the harmonic series does.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1126941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is this theorem equivalent to "existential instantiation" rule? In Enderton's, There is a theorem called "existential instantiation", it says:
Assume that the constant symbol $c$ does not occur in $\alpha ,\beta , \Gamma$ and that:
$$ \Gamma\cup\{\alpha^x_c\}\vdash\beta$$
Then
$$\Gamma\cup\{\exists\alpha\}\vdas... | If you can infer $A$ from $B$, and you have $\Gamma \cup \{A\} \vdash \beta$, then you can conclude that $\Gamma \cup \{B\} \vdash \beta$. (The left-side of a turnstile is like the left-hand side of an implication: strengthening an assumption preserves validity.) Hence there is no conflict between Enderton's rule that ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Can you recommend a book with techniques for solving hard algebra/arithmetic problems? I'm a university student who never really studied maths in high school (I did the basic courses, but because I'm dyslexic I was to embarrassed to try the harder courses) now I'm getting back into it, and, while I'm perfectly fine wit... | Have a look at 'How to solve it - George Polya'. It's more general but quite helpful.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Area enclosed by cardioid using Green's theorem Let $$\gamma(t) = \begin{pmatrix} (1+\cos t)\cos t \\ (1+ \cos t) \sin t \end{pmatrix}, \qquad t \in [0,2\pi].$$ Find the area enclosed by $\gamma$ using Green's theorem.
So the area enclosed by $\gamma$ is a cardioid, let's denote it as $B$. By Green's theorem we have fo... | As a rule of thumb, it's better to use $$\frac12\int_{\partial B}(-y\,dx+x\,dy)$$ whenever trig functions are involved. That will give a simpler expression once you use standard trig identities.
Comment: It's more straight forward in this particular case to do the area integral directly in polar coordinates: $$\int_0^{... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove $f(x) = \sum_{n=1}^\infty \frac{\sin nx + \cos nx}{n^3}$ is well-defined and $C^1$.
Prove $f(x) = \sum_{n=1}^\infty \frac{\sin nx + \cos nx}{n^3}$ is well-defined and $C^1$.
First of all I need to prove that $f(x)$ is well-defined. I'm not so sure what does it mean.
Basically I can claim that:
$$ f(x) = \sum_{n... | Here's some remarks:
*
*You said that "$f(x)$ converges uniformly to some continuous $f(x)$" which is misspoke and we should write "the series converges uniformly..."
*You said "... Hence, we may take the derivative of the term...": I don't see the implication: how if $f$ is continuous then you may take the derivat... | {
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Proof that these two definitions are equivalent
Where can I find a proof of or how can I prove that these two definitions are equivalent?
Definition 1: The Lie algebra of a Lie group $G \subset GL_n$ is the tangent space at $I$.
Definition 2: The Lie algebra of a Lie group $G \subset GL_n$ is all elements $g\in G$ wi... | Definition 2 is incorrect: In general, the Lie algebra of $G\subset GL_n$ is not a subset of $G$ (or even of $GL_n$). For example, the zero matrix is an element of the Lie algebra, but it's not an element of $GL_n$.
In place of Definition 2, you could write
Definition 2$'$: The Lie algebra of a Lie group $G \subset GL... | {
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"timestamp": "2023-03-29T00:00:00",
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How can I prove every Riemann sum of x^2 of [a,b] is the integral? I find that to prove $\int_a^b x\,dx = (b^2-a^2)/2$ (using the $\epsilon - \delta$ definition of Riemann integrable) is pretty straightforward, and after manipulating some sums, I end up with an expression like
$\left| \sigma - \frac{b^2 - a^2}{2} \rig... | Let $\{I_i\}_{i=1}^n$ be a partition of $[a,b]$ with $n$ sub intervals. Let us choose the tag of the interval $I_i=[x_{i-1},x_i]$ to be the mid point $q_i=\dfrac{1}{2}(x_{i-1}+x_i)$.
Then the contribution of this term to the Riemann sum corresponding to the tagged partition
$Q=\{(I_i,q_i)\}_{i=1}^n$ is
$h(q_i)(x_i-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1127756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Euler's formula, is this true? *I've changed this question as below.
Let me have a function such as $ f(k) = \exp(j 2 \pi k ) $, where $k$ is real value.
Using Euler's formula, we can write $f(k)$ as below,
$$ f(k) = \exp(j 2 \pi k ) = \cos(2\pi k)+j \sin(2\pi k).$$
If $k$ is integer, this always goes to 1.
Up to he... | Following on from Yves Daoust:
$$\left(z^x\right)^y$$
$$=\left(e^{\log \left(z^x\right)}\right)^y=\left(e^{x\log \left(z\right)}\right)^y$$
$$=e^{\log \left(\left(e^{x\log \left(z\right)}\right)^y\right)}=e^{y\log \left(e^{x\log \left(z\right)}\right)}$$
$$= e^{y\left(x\log \left(z\right)+i\left(2nπ\right)\right)}$$
$$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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A property of the fourier series Show that any periodic function $f(x)$ with period $2\pi$ which is both odd and satisfies $f(\pi-x)=f(x)$ has $b_{n}=0$ for $n$ even and so has a fourier series of the form $$f(x) = \sum^{\infty}_{m=0} c_{m}\sin{(2m+1)x}.$$
So we know $f$ is odd and periodic with period $2\pi$ thus it ... | $$f(\pi-x) = \sum_{n=1}^{\infty} b_n \sin{n(\pi-x)} = \sum_{n=1}^{\infty} (-1)^{n+1} b_n \sin{n x}$$
The only way $f(\pi-x)=f(x)$ is when $b_n=0$ when $n$ is even.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Understanding graphically the convergence of alternating harmonic and divergence of harmonic I understand the rules of convergence of a series so that I know that $\sum \frac 1 n$ (the harmonic series) diverges and $\sum \frac 1 {n^2}$ squared converges. It doesn't make sense graphically to me. One just gets to $0$ a l... | The Cauchy condensation test may help here. For a non-increasing sequence $(f(n))_{n\in\mathbb{N}}$ of non-negative real numbers we have (from Oresme's proof of the divergence of the harmonic series) :
\begin{align}
\tag{1}\sum_{k=1}^\infty f(k)&\le \underbrace{f(1)}_{f(1)}+\underbrace{(f(2)+f(2))}_{2\times f(2)}+\unde... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Can the null space be empty? I was reading a proof of the theorem that the range of a linear map $T$ is always a subspace of the target space, and when the author was showing that the $0$ vector was included in the range, he made an appeal to a previous theorem which says that the null space of $T$ is always a subspace... | Let $T:V\to W$ be a linear map. Then
$$
T(\mathbf{0})=T(\mathbf{0}-\mathbf{0})=T(\mathbf{0})-T(\mathbf{0})=\mathbf{0}
$$
This proves that $\mathbf 0$ is always in the nullspace of $T$. Hence the nullspace of $T$ cannot be empty.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1128244",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Are $\emptyset$ and $X$ closed, open or clopen? It is indeed a very basic question but I am confused:
(1) In an 2013 MSE posting under general topology here, I was told that $\emptyset$ is an open set and therefore I assume $X$ must be open too.
(2) But in Wikipedia page on clopen set here, it says "In any topological ... | By the first axiom in the definition of a topology, $X$ and $\emptyset$ are open. However, closed sets are precisely those whose complements are open, by definition. Hence the empty set and $X$, being each others complements, are also closed. So, they are clopen.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1128323",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Integration inequality question help: Sketch the curve y=1/u for u > 0... Sketch the curve $y=\frac{1}{u}$ for $u > 0$. From the diagram, show that $\int\limits_1^{\sqrt{x}}\frac{du}{u}< \sqrt x-1$, for x > 1. Use this result to show that $0 < \ln(x) < 2(\sqrt{x}-1)$ and hence that $\frac{\ln x}{x}\to0$ as $x\to\infty$... | Big Hint:
We know by the fundamental theorem of calculus that
$$\int_{1}^{\sqrt{x}} \frac{du}{u} = \frac{ln(x)}{2}.$$
However,
$$\int_{1}^{\sqrt{x}} \frac{du}{u} < \sqrt{x} -1.$$
Thus, we can say $$0<\frac{ln(x)}{2} < \sqrt{x} -1 \text{ or } 0<lnx<2(\sqrt{x} -1).$$
Now, if we divide by $x$ and apply Squeeze Theorem... | {
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"timestamp": "2023-03-29T00:00:00",
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Does $A\setminus B = A\setminus C$ imply $B=C$? Let $A, B, C$ be sets with $B \subset C$ and $C \subset A$. Does $A\setminus B = A\setminus C$ imply $B=C$?
I am not sure what the "\" means, so I don't know how to solve this.
| Yes it does, since, if $B\subset A$, then $A\setminus (A\setminus B)=B$. So $A\setminus B = A\setminus C$ implies $A\setminus (A\setminus B) = A\setminus (A\setminus C)$, i.e. $\,\,B=C$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1128474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Proving that $\sin^7\theta + \cos^7\theta <1$ using basic trigonometry and identities How do I prove $\sin^7\theta + \cos^7\theta < 1$ for an angle between $(0,\pi/2)$?
| Hint: If $\theta \in (0, \pi/2)$, we have $$0 < \sin \theta < 1 \text{ and }0 < \cos \theta < 1,$$ which gives us that $\sin^7 \theta < \sin^2\theta$ and $\cos^7\theta < \cos^2\theta$. So...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1128570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to connect a line between 4 randomly placed points on a plane such that the line does not cross itself You get 4 coordinates of points on a plain. You need to connect them all with a line. The line must not cross itself.
What's your strategy?
| Let $${\bf z}_k=(x_k,y_k)\qquad(1\leq k\leq 4)$$
be the four points, arranged such that
$$x_1\leq x_2\leq x_3\leq x_4\ .$$
This means that the leftmost point gets number $1$, the point with next-larger $x$-coordinate gets number $2$, and so on. The rightmost point gets number $4$.
It is assumed that no three of the fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1128668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How many 0's are in the end of this expansion? How many $0's$ are in the end of:
$$1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4.... 99^{99}$$
The answer is supposed to be $1100$ but I have absolutely NO clue how to get there.
Any advice?
| There as many zeros as the number of the factors of 5 in the product:
So we need to check the subproduct
$$
5^5\cdot 10^{10}\cdots 95^{95}
$$
We have $5+10+15+\cdot+95=5(1+2+\cdots+19)=950$. And we have missed the case where we have two factors: $25,50,75$. So we have another $25+50+75=150$. Altogether $950+150=1100$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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How can I see that the space curve given by $\gamma(s) = (\frac 4 5 \cos s, 1 - \sin s, - \frac 3 5 \cos s)$ forms a circle in space ? Let $\gamma: \mathbb R \rightarrow \mathbb R^3$ be a space curve given by $\gamma(s) = (\frac 4 5 \cos s, 1 - \sin s, - \frac 3 5 \cos s)$.
How do I see that $\gamma$ has image in $\mat... | Since you seem to see what values the coordinates "go between", then you can already make an educated guess: what about the circle with center $\;(0,1,0)\;$?
$$x^2+(y-1)^2+z^2=\frac{16}{25}\cos^2x+\sin^2s+\frac9{25}\cos^2s=1$$
and there you go.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1128869",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to prove this: $a^4+b^4+2 \ge 4ab$? How to prove this: $a^4+b^4+2 \ge 4ab$?
$a$ and $b$ are reals.
| Another possible method is generation of a function and calculating its minimum:
$$f(a,b)=a^4+b^4+2-4 a b$$
$$D_a f(a,b)=4 a^3-4 b = 0$$
$$ a^3=b$$
$$ f(a)=a^4+a^{12}+2-4a^4=a^{12}+2-3a^4$$
$$D_a f(a)=12a^{11}-12a^3=0$$
$$a^3 (a^8-1)=0$$
Minimum must be at one of:
$$f(0,0)=2$$
$$f(1,1)=0$$
$$f(-1,-1)=0$$
so
$$f(a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1128929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number theory: show that ${ 1^2, 2^2, 3^2,... , m^2}$ cannot be a complete residue system Is this an acceptable answer?
Question: show that ${ 1^2, 2^2, 3^2,... , m^2}$ cannot be a complete residue system.
Since the above has $m$ elements, one must show it cannot be a complete residue system
modulo $m$. Consider the... | Let's start with an example. Below is a congruence table modulo &11&.
\begin{array}{rrr}
k & k^2 & k^2 \pmod{11} \\
\hline
0 & 0 & 0\\
1 & 1 & 1\\
2 & 4 & 4\\
3 & 9 & 9\\
4 & 16 & 5\\
5 & 25 & 3\\
6 & 36 & 3\\
7 & 49 & 5\\
8 & 64 & 9\\
9 & 81 & 4\\
10 & 100 & 1\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1128989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
} |
How to solve this ordinary differential equation? I am just trying to find general solution
$$\frac{dy}{dx} = 1 + \sqrt{1 - xy}$$
| Let $u=-\sqrt{1-xy}$ ,
Then $y=\dfrac{1-u^2}{x}$
$\dfrac{dy}{dx}=\dfrac{u^2-1}{x^2}-\dfrac{2u}{x}\dfrac{du}{dx}$
$\therefore\dfrac{u^2-1}{x^2}-\dfrac{2u}{x}\dfrac{du}{dx}=1-u$
$\dfrac{2u}{x}\dfrac{du}{dx}=u-1+\dfrac{u^2-1}{x^2}$
Approach $1$:
$\dfrac{2u}{x}\dfrac{du}{dx}=\dfrac{(u-1)x^2+(u+1)(u-1)}{x^2}$
$(x^2+u+1)\dfr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 1
} |
Closure, interior and boundary of $(0, 1)$ with Zariski topology If we consider $\mathbb{R}$ together with the Zariski topology, what is the closure, interior and boundary of $(0,1)$?
A set is closed iff it is either finite or $\mathbb{R}$ under this topology, so since the closure of $(0,1)$ must be closed am I right i... | The closure is the smallest closed superset, so in the case is $\Bbb R$. The interior is the greatest oper subset, so in this case is... And the border is...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129130",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
$d(n)$ is odd if and only if $n = k^2$ Can someone help me prove that $d(n)$ is odd if and only if $n = k^2$ for some integer $k$?
For reference:
$d(n)$ gives the number of positive divisors of $n$, including $n$ itself.
| Hint : A number with the prime factorization $p_1^{a_1}\times ...\times p_n^{a_n}$ has
$(a_1+1)\times ...\times (a_n+1)$ divisors. When is the latter product odd ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Show that the arithmetic mean is less or equal than the quadratic mean I tried to solve this for hours but no success.
Prove that the arithmetic mean is less or equal than the quadratic mean.
I am in front of this form:
$$
\left(\frac{a_1 + ... + a_n} { n}\right)^2 \le \frac{a_1^2 + ... + a_n^2}{n}
$$
With rewriting ... | This is kind of a simplistic and brute-force approach:
$$\bigg(\frac{\sum_{i = 1}^n a_i}{n}\bigg)^2 \leq \frac{\sum_{i=1}^n a_i^2}{n} \iff %
\bigg(\sum_{i = 1}^n a_i\bigg)^2 \leq n \sum_{i=1}^n a_i^2$$
Now open the bracket on the left of the inequality and simplify and note that the result is equivalent to:
$$\sum_{1 \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129300",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 5
} |
Find $\int_{-1}^3xf(x)\,dx$ where $f(x)=\min(1,x^2)$ Find:
$$\int_{-1}^3xf(x)\,dx,$$
where $f(x)=\min(1,x^2)$.
I thought about solving it like this:
$$\int_{-1}^1 x^3\,dx + \int_{1}^3x\,dx = \cdots = 4.$$
But the solution is $\frac{26}{3}$
and I don't understand how they got it.
| My attempt::
Assuming $f:\Bbb R \to \Bbb R$,
$$f(x) = \min\Big(1\ ,\ x^2\Big) =
\begin{cases}
x^2\quad,\ x \in [-1,1] \\
1 \quad,\ x \in (-\infty,-1)\cup(1, \infty)
\end{cases}$$
The graphs of $ y = \min(1, x^2)$ and $y = x\min(1,x^2)$ are respectively as follows:
Now,
$$\int x\cdot f(x)\ \mathrm dx =
\begin{cases}
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "16",
"answer_count": 3,
"answer_id": 2
} |
How many $6$ digit numbers there are, where first and last digits and also consecutive digits are not equal? We have a 6 digit number
$a_1a_2a_3a_4a_5a_6$ and $a_1 \neq 0$
also $a_1 \neq a_2, a_2 \neq a_3, a_3 \neq a_4, a_4 \neq a_5, a_5 \neq a_6, a_6 \neq a_1$
All of numbers where $a_1 \neq a_2, a_2 \neq a_3, a_3 \neq... | Here's another approach. We will do cases based on the digits in positions $1$, $3$, and $5$.
Case 1:
$a$, ___ , $a$, ____, $a$, ____.
There are $9$ ways to select the digit $a$, and $9$ ways to fill each blank. So $9^4$ numbers of interest in this case.
Case 2.1, 2.2, 2.3:
$a$, ___ , $a$, ____, $b$, ____
and $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Show that if the sum of an diverges, no discrete probability space can contain independent events Suppose that $0\leq p_n\leq 1$, and put $a_n= \min \{p_n, 1-p_n\}$. Show that if
$\sum a_n$ diverges, then no discrete probability space can contain independent events $A_1, A_2, \ldots$ such that $A_n$ has probability $p... | Let's do not consider the $P_n=0$ cases, because if the empty set is independent with any sets.
So , consider the discrete probability space, contains at most countable many points with each point assigned with a positive probability.
All the probabilities sum up to 1.
Now $\sum a_n$ diverge,
suppose $A_n$ are the even... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to prove the cofactor of a matrix is related to it's minor? How do I prove $A_{ij}$, the cofactor of $a_{ij}$ of a matrix $D$, has the property $A_{ij}=(-1)^{i+j}M_{ij}$, where $M_{ij}$ is the minor obtained by deleting the $i$th row and $j$th column?
| For the case of $i=j=1$, $A_11$ will be the sum of the terms $(1)^{N(\alpha_1=1,\alpha_2,\dots,\alpha_n)} a_{\alpha_2 2} \cdots a_{\alpha_n n}$. Observe that there is no transposition (or inversion) that contains $a_{11}$. So $N(\alpha_1=1,\alpha_2,\dots,\alpha_n)=N(\alpha_2,\dots,\alpha_n)$. Then we have the sum of te... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can this be solved for $f(n)$? While working upon a partial sum formula for the harmonics, I came across a necessity for the function defined below
$f\left(\frac{n(n-1)}{2} +1\right) = n!$
Can it be solved for $f(n)$?
| Let your formula define $f(t)$ for all integer $t$ on the form $t=\frac{n(n-1)}{2}+1$ with $n$ integer. For other $t$ define $f$ to be whatever you want. The resulting function will satisfy your equation. This means that there is no unique function satisfying your equation.
However, there is a 'natural' function. Solvi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A nonmetrizable image of a metrizable space It is well known that a hausdorff continuous image of a compact metric space is metrizable. What is a counterexample for noncompact case?
| Let $X$ be an infinite set, let $\mathcal{T}_\text{d}$ be the discrete topology on $X$, and $\mathcal{T}$ any non-metrizable Hausdorff topology on $X$. Then the identity mapping $\operatorname{id}_X : X \to X$ is a continuous function from $( X , \mathcal{T}_{\text{d}} )$ onto $( X , \mathcal{T} )$, and the discrete to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1129897",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$ A_n $ =$[\frac{n}{n+1},\frac{n+1}{n+2}] $ be closed subsets find $\bigcup_{n=1}^\infty A_n $ Let $ A_n =\frac{n}{n+1},\frac{n+1}{n+2}] $, $n=1,2,3...$ be closed subsets of real line R. Then $\bigcup_{n=1}^\infty A_n $ is
*
*(1/2,1)
*[1/2,1)
*(1/2,1]
*[1/2,1]
My attempt : think it could be [1/2,1) since $\lim ... | Hint: Since $f(x)=\frac{x}{x+1}$ is monotonically increasing, we have that for any $0\le x\lt1$,
$$
n=\left\lfloor\frac{x}{1-x}\right\rfloor\iff n\le\frac{x}{1-x}\lt n+1\iff\frac{n}{n+1}\le x\lt\frac{n+1}{n+2}
$$
Thus, for any $\frac12\le x\lt1$, if $n=\left\lfloor\frac{x}{1-x}\right\rfloor$, then $n\ge1$ and $x\in\lef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130000",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Does $\sum_{n\ge0} \cos (\pi \sqrt{n^2+n+1}) $ converge/diverge? How would you prove convergence/divergence of the following series?
$$\sum_{n\ge0} \cos (\pi \sqrt{n^2+n+1}) $$
I'm interested in more ways of proving convergence/divergence for this series.
My thoughts
Let
$$u_{n}= \cos (\pi \sqrt{n^2+n+1})$$
Let's firs... | A hint:
$$\sqrt{n^2+n+1}-\left(n+{\textstyle{1\over2}}\right)={3/4 \over
\sqrt{n^2+n+1}+\left(n+{1\over2}\right)}=:\ \alpha_n\searrow\ 0\qquad(n\to\infty)\ .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130091",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Why do we pretend + and x not to be addition and multiplication while eventually they are considered so while explaining groups, rings and fields? Aren't commutative, associative properties result of addition and multiplication? Then why most of the definitions of Groups and Fields contain + and x but don't accept that... | As an exemple of how '' addition'' and ''multiplication'' can be different from the usual operations in $\mathbb{R}$, take the ring $(M_2(\mathbb{R},\oplus, \odot)$ of $2\times 2$ matrices with real entries.
Here the $\oplus$ is defined as
$$
A\oplus B=
\left(
\begin{array}{ccccc}
a_1&a_2 \\
a_3&a_4
\end{array}
\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do i find P(2)? Let $P(x)$ be the polynomial of least degree with real coefficients such that $P(i)=P(1+i)=0$ and $P(1)=4$. Find $P(2)$.
Here i is the complex number. I have done such questions with real numbers but I don't know why I am not getting it.
| Hint: $P$ has real coefficients. So, if $P(i) = 0$, then $P(-i) = 0$. If $P(1+i) = 0$, then $P(1 - i) = 0$.
Can you construct the smallest complex polynomial that has these zeros?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Weird integration issue: $\ln(x+1)=\ln(2x+2)$ ?! Weird integration issue: Using $(\ln[f(x)])'=\frac {f'(x)}{f(x)}$ we get that $\int \frac{2\,dx}{2x+2}=\ln(2x+2)$. Yet, $\int \frac{2\,dx}{2x+2}= \int\frac{dx}{x+1}=\ln(x+1)$ using the same rule as earlier.
What is wrong here?
| Just add a constant term and also notice that $\ln2$ is also a constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
dimension of inverse image Let $f : V \to W$ be a linear transformation between two vector spaces. Let $E$ be a subspace of $V$.
Assume $V$ to be finite dimensional. Is there a way to obtain the dimension of $f^{-1} (f(E))$ in terms of the dimension of $E$, the dimension of $ \mathrm{im}f$ and the dimension of $\mathrm... | One has $f^{-1}(f(E))=E+\ker(f)$, which sum need not be direct, so the general formula applies
$$
\dim(f^{-1}(f(E)))=\dim(E+\ker(f))=\dim(E)+\dim(\ker f)-\dim(E \cap\ker(f)).
$$
So apart from the dimension of $E$ and of the image of $f$ (from which you can deduce that $\ker f$), you need to know hoe $E$ intersects $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why is a circle 1-dimensional? In the textbook I am reading, it says a dimension is the number of independent parameters needed to specify a point. In order to make a circle, you need two points to specify the $x$ and $y$ position of a circle, but apparently a circle can be described with only the $x$-coordinate? How i... | That is a bad definition of dimension. By this definition, the (filled-in) unit square in the real coordinate plane is one-dimensional, since you only need one number to describe a point $(x,y)$. An easy way to do this is to write out the decimal expansions of $x$ and $y$:
$$x=0.a_1a_2a_3a_4a_5\dots$$
$$y=0.b_1b_2b_3b_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "71",
"answer_count": 17,
"answer_id": 1
} |
Can the cohomology ring of the two-fold torus be calculated abstractly? In our lectures, we are given an unusual definition of cohomology and cup products which makes explicit calculations a bit tedious (that is, even more tedious than usual).
For the $n$- and $m$-spheres $S^n$ and $S^m$, I know that there’s an isomorp... | I think Qiaochu Yuan's post (https://qchu.wordpress.com/2013/10/12/the-cohomology-of-the-n-torus/) is quite relevant. In particular the first part gives a derivation of the cohomology (group) of the $n$-torus using Mayer–Vietoris . Of course you need more tools to compute the ring structure, but I think Qiaochu did a q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Do you know any almost identities? Recently, I've read an article about almost identities and was fascinated. Especially astonishing to me were for example $\frac{5\varphi e}{7\pi}=1.0000097$ and $$\ln(2)\sum_{k=-\infty}^{\infty}\frac{1}{\left(\sqrt{2}+\frac{1}{\sqrt{2}}\right)^k}=\pi+5.3\cdot10^{-12}$$ So I thought it... | There is a substantial list on Wikipedia:
http://en.wikipedia.org/wiki/Mathematical_coincidence
Some of the more interesting (imo) examples are:
\begin{equation}
\pi\approx\frac{4}{\sqrt{\varphi}}\\
\pi^4+\pi^5\approx e^6\\
\frac{\pi^{(3^2)}}{e^{(2^3)}}\approx10\\
e^{\pi}-\pi\approx20
\end{equation}
There's also the cl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130823",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
$X+Y$ is closed $\Leftrightarrow$ $\|x\|\leq c\|x+y\|$ for all $x\in X$ and all $y \in Y$. The problem says
Let $(Z,\|\cdot\|)$ be a Banach space. Let $X$ and $Y$ be two closed subspaces of $Z$ such that $X\cap Y=\{0\}$. Prove that $X+Y$ is closed if, and only if, there exists $c\geq 0$ such that
$$\|x\|\leq c\|x+y\... | For the inverse,
consider
$X+Y$ is closed, but (1) is not sure.
This means
$\forall c\geqslant 0$ we can find a $x\in X$ and $y\in Y$ such that,
$\Vert x\Vert>c\Vert x+y\Vert $
divided by $\Vert x\Vert$ we can assume that $x$ has norm 1.
then for any $\frac{1}{c}>0$ we can find
$\Vert x+y\Vert<\frac{1}{c}$,
now
$\vert... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Find a simple expression for $\sum_{k=0}^{n} k(k-1)\binom{n}{k}$ for $n\geq0$ Find a simple expression for $\sum_{k=0}^{n} k(k-1)\binom{n}{k}$ for $n\geq0$.
My intuition is to somehow make use of the fact that $\binom{n}{k} = \frac{n}{k}\binom{n - 1}{k - 1}$. Can I simply replace the $\binom{n}{k}$ with this identity a... | Let $X \sim \operatorname{Bin}(n,\tfrac{1}{2})$ be a binomial random variable. You are after $$2^n\mathbb{E}[X(X-1)] = 2^n(\mathbb{E}[X^2]-\mathbb{E}[X]) = 2^n(\mathbb{V}[X] + \mathbb{E}[X]^2 - \mathbb{E}[X]).$$ Since expectation and variance are additive, it is easy to calculate $\mathbb{E}[X] = \tfrac{n}{2}$ and $\ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1130937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Differential calculus: integrate $\frac{1}{x \log^3 (x)}$ I would like a step by step description of how to integrate $$\frac{1}{x \log^3 (x)}$$
*
*I know that the answer is - $\frac{1}{2\log^2(x)}$
*and that the integral of $\frac{1}{\log^2(x)}$ is $\frac{1}{\log(x)}$
*and that the integral of $\frac{1}{\log(x)}... | For this problem:$$\int\frac{1}{x\log^3(x)}dx\tag{1}$$we can use the substitution:$$u=\log^3(x)\tag{2}$$$$\therefore \log(x)=u^{\frac{1}{3}}\tag{3}$$which leads to:$$du=3\log^2(x)\times\frac{1}{x}dx$$$$=3(u^{\frac{1}{3}})^2\times\frac{1}{x}dx=\frac{3u^{\frac{2}{3}}}{x}dx$$$$\therefore \frac{dx}{x}=\frac{du}{3u^{\frac{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1131048",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
proof $z \mid b$ and $w\mid b$ Question I'm working on: Let $a,b$ be integers with $b$ not equal to $0$. suppose $x^2+ax+b=0$ and $x=z,w$. If $z,w$ are integers, show that $z\mid b$ and $w\mid b$.
Is it sufficient for me to show that $x^2+ax+b=(x-z)(x-w)$ so $b=zw$?
| Here is a better approach:
We need to show $z|b$ and $w|b$. What does it mean for $z$ to "divide" $b$? It means there is some integer $k$ such that $z \cdot k = b$. Well, what are we given? We are given that if $x = z$ or $x = w$, then $x$ satisfies $x^{2} + ax + b = 0$.
But that means $z^{2} + az + b = 0$, or $z^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1131120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Partial Derivatives and the Fundamental Theorem of Calculus I am being asked to evaluate the 1st-order partial derivatives $-$ $f_{x}$(x,y) and $f_{y}(x,y)$ $-$ of the following multi-variable function: $f(x,y) = \int_{y}^{x} cos(-1t^2 + 3t -1) dt$. Any help is appreciated.
| $$
f(x,y)=\int_y^x g(t) dt=G(t)|_y^x=G(x)-G(y) \text{ where } G'=g
$$
so differentiating w.r.t $x$ on both sides of $f(x,y)=G(x)-G(y)$ gives
$$
f_x(x,y)=(G(x)-G(y))_x$$
and by difference rule that is
$$
f_x=(G(x))_x = g(x).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1131297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Using two coins to select a person fairly. Good evening, I would like to know if the solution to this problem, I know it can be solved because it is from a Hungarian Olympiad. The problem is as follows: You need to fairly select a person between $n$ persons using two unfair coins for which you get to decide the odds.
... | Here's a quite straightforward method with two coins, bounded number of throws, and both coins having rational values. First coin is unbiased; second coin has the following probability for a Head:
$$p = \frac{2^m}{n!}; m = \lfloor{\log_2(n!)}\rfloor$$
With the unbiased coin you can emulate a biased coin which has $\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1131386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 0
} |
Doubts on solving an initial value problem The problem is : $y(\ln y)'= \frac{2t}{y}$ with $y(0)=0$. How to obtain $y'$ ?
| $y(\ln y)'=y(\dfrac{1}{y}\dfrac{dy}{dt})=\dfrac{dy}{dt}.$
So the problem is $\dfrac{dy}{dt}=\dfrac{2t}{y}\implies ydy=2t dt$
So, $\dfrac{y^2}{2}=t^2+C,$ thus with the initial condition $y(0)=0$, we get $C=0$
ie, $y^2=2t^2\implies y=\pm \sqrt2t \square$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1131472",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Binomial expansion (sort of ) rearrangement Let
$$
x_n=\sum_{i=1}^n\binom{n}{i}y_iz_{n-i} \qquad n=1,\ldots,k
$$
For general $k$, can you find an explicit expression for $y_k$ only in terms of $x_1,\ldots,x_k$ and $z_1,\ldots,z_k$?
For example, if $k=3$, then
\begin{align*}
x_3 &=y_3+3y_2z_1+3y_1z_2+z_3 \\
x_2 &=y_2+2... | Given a sequence $s=(s_1,s_2,\ldots)$, it is useful to consider the exponential generating function
$$
f_s(T):=1+\sum_{n\geq 1}s_n\frac{T^n}{n!}.
$$
The relationship between the $x=(x_n)$, $y=(y_n)$, and $z=(z_n)$ can be expressed as
$$
f_x(T) =f_y(T)\cdot f_z(T).
$$
We want to solve for $y$ in terms of $x$ and $z$, so... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1131610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determining whether equation $\ln x=\frac{1}{3}x$ has at least 3 solutions I'm trying to say whether
$$\ln x=\frac{1}{3}x$$
Has at least 2 solutions, and after that the problem asks us to tell if the third solution exists...
So I managed to find 1 solution: $f(x)=\ln x - \frac{1}{3} x$, $f(e)>0, f(1)=-\frac{1}{3} < 0$,... | If you look at the graph of the given equation, the plot of ln(x) bends towards X-axis after the second point of intersection while the plot of $\frac13x$ continues straight above. So, there won't be a third solution of the equation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1131693",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Bisector and symmetric point I can't solve the following problem and need help.
Given an acute-angled triangle ABC.CL is the bisector of $\angle C$ where $L \in AB$. CL instersects the circumcircle of the triangle at point D. O is the center of the circumcircle and I is the incenter of the triangle. If $P$ is the symme... | Since $\widehat{ACD}=\widehat{DCB}$, $D$ is the midpoint of the arc $AB$ in the circumcircle $\Gamma$ of $ABC$, hence $D,P,O$ are collinear and, obviously, $OC=OD$.
Moreover $\widehat{DOC}=\widehat{BOC}+\widehat{BOD}=2\widehat{A}+\widehat{C}$, so $CD = 2R\sin(\widehat{A}+\widehat{C}/2) $.
Now it is easy to check tha... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Number of digits $d$ in $d^k$ The title says it all really. For example, how many occurences of $6$ are there in $6^k$? It starts $6, 36, 216, \dots$ so $1, 1, 1,\dots$ The question can now be generalized into any digit or group of digits.
| Here is a general formula for base $10$ and $d\in[1,10-1]$:
$$\sum\limits_{n=1}^{k}1-\left\lceil\left(\frac{\left\lfloor\frac{d^k}{10^{n-1}}\right\rfloor-10\left\lfloor\frac{d^k}{10^n}\right\rfloor-d}{\left\lfloor\frac{d^k}{10^{n-1}}\right\rfloor-10\left\lfloor\frac{d^k}{10^n}\right\rfloor+d}\right)^2\right\rceil$$
He... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1131854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Minimum value of trigonometry function How can I get the minimum value of function
$$f(x) = (2 + \sin x)(5 - \sin x)$$
I have used the differential ways but the answer was not match with the key answer.
By the way the key answer is $6$.
| You can use the change of variable $t=\sin x$ and remember that $-1\le t\le 1$.
Then
$$(2+t)(5-t)=10+3t-t^2$$
has the derivative $3-2t$, which cancels at $t=\frac32$, out of the domain of $t$.
The minimum can only arise at one of the ends of that domain,
$$f(-1)=\color{green}6$$or $$f(1)=12.$$
No trigonometry !
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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connected and compact subset of $\mathbb{C}$ is either a singleton or has uncountable boundary Let's say that $A\subseteq\mathbb{C}$ is connected and compact but not a singleton. I want to conclude that $\partial A$ is uncountable. Can anyone help me find a reference to cite?
I can see how to prove this in a routine ... | The boundary of such a set is a perfect set (closed with no isolated points). Such set in $\Bbb R^k$ is uncountable. See Rudin's Principles of Mathematical Analysis Theorem 2.43.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Modularity and prime number sequence I tried to solve this modular equation involving first $n$ prime numbers. And it is:
$$2^{3+5+7+11+13+.....+p_{n-3}+p_{n-2}}\equiv p_{n-1}\ \left(\text{mod }p_{n}\right),$$
where $p_{n}$ is the $n$-th prime number.
I couldn't find any solution for this equation until first $300$ pr... | For what is worth, next solutions $\{n,p_n\}$ are:
$$
\{306\,311,4\,353\,467\}\\
\{859\,825,13\,174\,621\}\\
\{1\,700\,098,27\,291\,793\}\\
$$
There are no more with $n\le10^7$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1132216",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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matrix multiplication by column vectors If $A$ is an $n \times n$ matrix, and $x$ and $y$ are column vectors in $R^n,$ show that $(A x)\cdot y=x \cdot (A^t y),$ and then that $(A x) \cdot (A y)= x(A^t A y).$ I tried to take $2 \times 2$ or $3 \times 3$ matrices and it worked but i did not know how to generalize it. tha... | hint. here are things you need to know:
(a) $x \cdot y = x^Ty = y^Tx,$
(b) $(Ax)^T = x^TA^T$
(c) $A(BC) = (AB)C$
i will do the first one. $Ax \cdot y = (Ax)^Ty = x^TA^Ty = x.A^Ty$
can you see how to do the second one?
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that the series $\sum_{k=0} ^\infty (-1)^k \frac{x^{2k+1}}{2k+1}$ converges for $|x|<1$ and that it converges to $\arctan x$ Show that the series $\sum_{k=0} ^\infty (-1)^k \dfrac{x^{2k+1}}{2k+1}$ converges for $|x|<1$ and that it converges to $\arctan x$
I tried using the ratio test but I got that it equals 1, so... | Modifying Jack D'Aurizio's answer,
since
$\sum_{k=0}^n x^k = \dfrac{1-x^{n+1}}{1-x}$,
putting $-x^2$ for $x$ we get
$\sum_{k=0}^n (-1)^kx^{2k}
= \dfrac{1-(-1)^{n+1}x^{2n+2}}{1+x^2}
= \dfrac1{1+x^2}+\dfrac{(-1)^{n}x^{2n+2}}{1+x^2}
$
or
$\dfrac1{1+x^2}
=\sum_{k=0}^n (-1)^kx^{2k} -\dfrac{(-1)^{n}x^{2n+2}}{1+x^2}
$.
Integ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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if Y1 and Y2 are independent, does it follow that U and X are independent, where U = f1(Y1, Y2) and X= f2(Y1, Y2) If X, Y are iid rvs, and U and Z are r.v.s that can each be written in terms of X and Y, does that mean that U and Z are independent?
| No. Consider i.i.d. standard normal random variables, $\,X,Y\sim\mathcal{N}\left(0,1\right)\,.$ Let $\,\,U{}={}a_1X+a_2Y\,\,$ and $\,\,Z{}={}b_1X+b_2Y$ for non-zero real numbers $a_1, a_2, b_1$ and $b_2$.
Then, see that
$$
\mathbb{C}ov\left(U,Z\right){}={}a_1b_1+a_2b_2\ne0\,,
$$
therefore $U$ and $Z$ can't be indepe... | {
"language": "en",
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Why must a locally compact second countable Hausdorff space be second countable to imply paracompactness? The textbook version of the result I've seen states: A locally compact second countable Hausdorff space is paracompact. Is the property of being second countable needed, or have I missed something?
My thinking:
... | Without the second countable condition the result is not true. The long line is locally compact Hausdorff and is not paracompact because it is locally metrizable but not metrizable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1132608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Intersection of context-free language and its reversal I know that intersection of two context-free languages is not always context-free and the following problem:
Given two context-free languages A and B, is $A \bigcap B \neq \emptyset$ ?
is undecidable. But is that true in particular case when we know that $B = \{ ... | If $A$ contains a palindromic string $w$ then $w^R=w$. So in this case, $A \cap B \neq \varnothing$. But if $A$ doesn't contain any palindromic string, we cannot find a common element of $A$ and $B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1132725",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that Möbius transformations that preserve the unit disk are of the matrix form $\tiny \begin{bmatrix}a & b \\ \bar{b} & \bar{a} \end{bmatrix}$ Show that Möbius transformations that preserve the unit disk are of the matrix form $$\begin{bmatrix}a & b \\ \bar{b} & \bar{a} \end{bmatrix},$$ where $|a|^2 - |b|^2 = 1$ a... | First note that your statement of the result is not quite accurate. For example, the Mobius transformation
$$f(z)=\frac{az+b}{cz+d}=\frac{iz+0}{0z+i}=z$$
clearly preserves the unit disc even though the corresponding matrix
$$\pmatrix{i&0\cr0&i\cr}$$
does not have the form you state. However, multiplying $a,b,c,d$ by ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Interpreting a group homomorphism $f: \mathbb{Z}_{12} \to \mathbb{Z}_{3}$ visually I am having a hard time studying and I am a visual learner. How could I visually imagine a (group) homomorphism
$$\mathbb{Z}_{12} \to \mathbb{Z}_3?$$
Also, if the question states that the map $f$ is a group homomorphism such that $f(1)=2... | I find it useful to imagine the group $\mathbb{Z}_n$ a group of rotations (of, e.g., the plane) by multiples of $\frac{1}{n}$ revolution, that is, by multiples of $\frac{2\pi}{n}$ radians, so that $[k] \in \mathbb{Z}_n$ corresponds to a (say, anticlockwise) rotation by $\frac{2 \pi k}{n}$ radians.
Since $\mathbb{Z}_n$ ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Equation of normal vector pointing away from ellipse Assuming that I have an ellipse, centered at $(h,k)$ of type:
$$\left(\frac {x-h}{a}\right)^2 + \left(\frac {y-k}{b}\right)^2 = 1$$
The gradient of the normal is:
$$\frac{a^2(y-k)}{b^2(x-h)}$$
From this I can get a vector representation of the normal, but my question... | I think the simplest way is look at the coordinate $(x,y)$ on the ellipse.
For example, if $x-k > 0$, then choose your $x$-component to be positive. Same for $y$-component
explicitly, $(\frac{x-h}{a^2},{\frac{y-k}{b^2}})$
For gradient, do you mean
$$\nabla\left(\left(\frac{x-h}{a}\right)^2+\left(\frac{y-k}{b}\right)^2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove or disprove absolute convergence Prove or Disprove:
if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty} max\{a_n,\frac{1}{n^2}\}$ is also absolutely convergent.
| ok, I think I just missed something pretty simple:
The sum of two convergent series is also convergent.
$\sum_{n=1}^{\infty} max\{a_n,\frac{1}{n^2}\} < \sum_{n=1}^{\infty} |a_n| + \sum_{n=1}^{\infty} |\frac{1}{n^2}| $
therefore by the first comparison test, the original series is absolutely convergent.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove $|f|$ is constant implies $f$ is constant Let $f$ be an entire function (differentiable everywhere over $\mathbb{C})$. Suppose that $|f|$ is constant. Prove that $f$ is constant.
Hint: $|f|\equiv c$ implies that $u^2+v^2\equiv c^2$. Take partial derivatives and apply the Cauchy-Riemann equations.
Proof: Suppose ... | A short, but different proof can be given by using the fact that a non-constant holomorphic function is an open map: By assumption $f(\mathbb C)$ is contained in a circle in $\mathbb C$ with radius $c$. Any non-empty subset of such circle is not open in $\mathbb C$, hence $f$ is constant, q.e.d.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1133234",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Is it possible for a relation to be transitive and symmetric but not reflexive with only one element? E.g. On the set $A = \{1,2,3,4,5,6\}$, is the relation set $R = \{(1,1)\}$ a transitive and symmetric relation but not reflexive?
| Yes. The relation $R$ that you have given is transitive and symmetric, but not reflexive. To see this, note that $(2,2)\not\in R$, but by definition, $R$ is reflexive if $(a,a)\in R$ for all $a\in A$. It's easy to check that $R$ is symmetric and transitive. For example, note that for all $(a,b)\in R$, we also have $(b,... | {
"language": "en",
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Using first order sentences, axiomize the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes. Problem
Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes.
Current Solution... | You have axiomatized the theory asserting that $E$ is an equivalence relation with infinitely many classes, ALL of which are infinite. Your statement $\psi_n$ says that every class has at least $n$ elements.
But the theory mentioned in the title problem was just that $E$ should have infinitely many infinite classes (a... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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Moving from Finitely Additive to Countably Additive I have a question on basic measure theory here:
Let $(X, \mathscr A)$ be measurable space and $\mu$ is a non-negative set function that is finitely additive and such that $\mu (\emptyset) = 0$. Suppose that whenever $A_i$ is an increasing sequence of sets in $\mathsc... | The idea is to turn a countable disjoint union into an increasing union. To do that, you'll need to use different sets. So suppose $\{ A_n \}_{n=1}^\infty$ is a disjoint collection of measurable sets. Consider $\{ B_m \}_{m=1}^\infty$ defined by $B_m = \bigcup_{n=1}^m A_n$. This is an increasing union, so it has the li... | {
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"timestamp": "2023-03-29T00:00:00",
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An example of an infinite open cover of the interval (0,1) that has no finite subcover I've been having a hard time solving this problem that I was given in class. The problem states
" Give an example of an infinite open cover of the interval (0,1) that has no finite subcover."
I know that the set has to be compact an... | You can take the open cover $\{(0+\frac{1}{n}, 1-\frac{1}{n})\mid n>2\} $. It has no finite subcover because if it does simply take the largest $n$ and $\frac{1}{2n}$ is not included.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 2
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Prove or disprove : if $x|y^2 $then $x|y$ How can I prove such statement?
I think that if $x|y^2$ then $x|(y*y)$ so $x|y$ or $x|y$ which means that in any case $x|y$. Am I correct?
I ask this question as such template because I think that it doesn't matter which $x$ or $y$ represents.
However, the original question is:... | The claim $x \mid y^2 \implies x \mid y$ is clearly false. Take $x = y^2 > 1$ a perfect square and see for yourself. What is true is: $$\begin{cases} x \mid yz \\ \gcd(x,y)=1\end{cases} \implies x \mid z.$$
Having this in mind, if $\gcd(x,y)=1$, and taking $y = z$...
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve $y''+Ay'+By=a+bt+ct^{2}$? I know that the solution of this second-order non-homogeneous will be of the form $y = y_{p} + y_{c}$.
First I find the solution to the homogeneous DE: $y''+Ay'+By=0$, with the characteristic equation $r^{2}+Ar+B=0$. However, it's hard to actually get the roots since I don't know ... | to find the particular soultion,we assume
$$y_p=A_1t^2+B_1t+C_1$$
$$y'_p=2A_1t+B_1$$
$$y''_p=2A_1$$
substitute in D.E to get
$$2A_1+A(2A_1t+B_1)+B(A_1t^2+B_1t+C_1)=a+bt+ct^2$$
$$BA_1=c$$
$$A_1=c/B$$
so the $c$,$B$ are known
$$2AA_1+BB_1=b$$
$$B_1=(b-2A(c/B))/B$$
$$2A_1+AB_1+BC_1=a$$
$$C_1=(a-2A_1-AB_1)/B$$
If you want... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do I prove, using the definition, that the nth root is a continuous function? Stuff I've tried:
\begin{align}
|x^{\frac 1 n}-a^{\frac 1 n}| &< \epsilon
\\
|(x^{\frac 1 n}-a^{\frac 1 n})^n| &< \epsilon ^n
\\
\left|\sum_{i=0}^{n}C(n,i)x^{\frac in}(-a)^{1-\frac i n}\right| &< \epsilon ^n
\end{align}
(it looks like a o... | With some accessible work, you can also show the inequality
$$
\left|x^{1/n}-y^{1/n}\right|\le|x-y|^{1/n}
$$
showing Hölder continuity and uniform continuity on $[0,\infty)$.
The proof starts with considering $0\le y\le x$, so that $x^{1/n}=y^{1/n}+h$, $h\ge0$ and uses only the binomial theorem,
$$
x-y=(y^{1/n}+h)^n-y... | {
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"timestamp": "2023-03-29T00:00:00",
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Solving natural logarithms with absolute value Question from my text: $e^{4x-2014} - 7 = |-3|$. I've never seen this before and my text is useless! Thank you!
| Hint: First of all, what is $|-3|$? Second, sum $7$ on both sides of the equation. Third, apply $\ln$ in both sides of the equation. Fourth: sum $2014$ on both sides of the equation. And finally divide both sides of the equation by $4$. Carry the steps carefully and you'll see what's happening.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Finding a paper/reference where a solution also exists I have already calculated the following sum:
$$\sum_{k=0}^\infty{B_2(k)}=\frac{\pi^2}{3}-2\approx1.2898681336964$$
where $B_2(k)$ is the 2nd hypergeometric Bernoulli number of order $k$. These numbers are defined with the following generating function:
$$\frac{x^N... | Observe that we are trying to compute the sum
$$2\sum_{k=0}^\infty
[x^2] \frac{x^k/k!}{e^x-1-x-\cdots-x^{k-1}/(k-1)!}$$
which is
$$2\sum_{k=0}^\infty
[x^2] \frac{x^k/k!}{\sum_{q=k} \frac{x^q}{q!}}.$$
Use the Cauchy Product to extract the coefficient:
$$\frac{x^k}{k!}
= \left(\sum_{q\ge k} \frac{x^q}{q!}\right)
\times... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1133997",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Find a solution: $3(x^2+y^2+z^2)=10(xy+yz+zx)$ I'm something like 90% sure that this diophantine equation has nontrivial solutions:
$3(x^2+y^2+z^2)=10(xy+yz+zx)$
However, I have not been able to find a solution using my calculator. I would greatly appreciate if someone could try to find one using a program. Or maybe yo... | As far as I understand - this is the site for solving the problem. Programming and calculation using the computer is not mathematics. If you want to calculate - there is a special section. https://mathematica.stackexchange.com/questions
Here it is necessary to solve the equations.
For the equation:
$$3(x^2+y^2+z^2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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First examples for topology of non-Hausdorff spaces I have absolutely no intuition about non-Hausdorff spaces. I would like to understand the topology of non-Hausdorff spaces (in particular spaces obtained by "bad" group actions).
As a first example, I would like to calculate the fundamental group of $\mathbb R^2 / \ma... | Yes, it is trivial.
To get some intuition for non-Hausdorff spaces, consider this fact: if you take a simplicial complex and quotient every open simplex to a point, you get a space that has the same homotopy groups as the original space. For instance, you can take the boundary of a triangle (which is homeomorphic to a... | {
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} |
Integral Power Rule Step-by-step Real quick, two things: I'm sorry if my notation or terminology is incorrect, and I know what I'm asking isn't strictly necessary for my studies, but writing something out step by step helps me to understand math intuitively.
I have spent a while thinking about this and cannot figure ou... | Here is what the problem boils down to: You need to have a closed formula for summing $$\sum_{i=1}^ni^k$$ I know of such a formula for $k=1,2,3$ and a buddy of mind derived one for $k=4$, but beyond that I don't know that there is a general formula for any $k$. For $k=1$ we know $$\sum_{i=1}^ni^1 = \frac{n(n+1)}{2}$$ S... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134280",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Using Euler's formula for cos/sin 12 degrees and cos/sin 48 degrees $(\cos12^\circ+i\sin12^\circ+\cos48^\circ+i\sin48^\circ)$. Using Euler's Formula, turn this into exponential form (i.e. something like $e^{i\frac{5\pi}{12}}$).
Would I need to use the $\cos$ and $\sin$ sum and difference formulas? I tried doing that a... | Observe
\begin{align*}
\cos 12^{\circ}=\cos(30^{\circ}-18^{\circ})&=\cos30^{\circ}\cos18^{\circ}+\sin30^{\circ}\sin18^{\circ} \\
\sin 12^{\circ}=\sin(30^{\circ}-18^{\circ})&=\sin30^{\circ}\cos18^{\circ}-\cos30^{\circ}\sin18^{\circ}\\
\cos 48^{\circ}=\cos(30^{\circ}+18^{\circ})&=\cos30^{\circ}\cos18^{\circ}-\sin30^{\cir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Showing some Function $fg$ is Integrable when $g(x)\le e^{-\vert x\vert }$ and given some Conditions on $f$. Suppose $\int_a^b \vert f \vert <\infty$ for all real $a,b$ and that $$\int_{-r}^r \vert f \vert \le (r+1)^a$$ for all real $r$ some real $a$, and that $$g(x)\le e^{-\vert x \vert}$$ I want to show $$f(x)g(tx)\i... | Motivation for this solution: we're given information about the integral of $f$, so we want to bring that function into the bounds somehow. That suggests integration by parts.
Let's just work on $[0,\infty)$ (the other half is similar) and assume $t>0$. Define $F(x) = \int_0^x |f(y)|\,dy$. Then
$$
\int_0^r |f(x)g(tx)|\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134464",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let $x,y,z>0,xyz=1$. Prove that $\frac{x^3}{(1+y)(1+z)}+\frac{y^3}{(1+x)(1+z)}+\frac{z^3}{(1+x)(1+y)}\ge \frac34$ Let $x,y,z>0$ and $xyz=1$. Prove that $\dfrac{x^3}{(1+y)(1+z)}+\dfrac{y^3}{(1+x)(1+z)}+\dfrac{z^3}{(1+x)(1+y)}\ge \dfrac34$
My attempt:
Since it is given that $xyz=1$, I tried substituting $x=\dfrac{a}{... | As another approach, you could prove it as follows.
Due to Hölder's inequality, we have:
$$
\left(\sum_{cyc} \frac{x^3}{(1+y)(1+z)}\right)\cdot\left(\sum_{cyc} (1+y)\right)\cdot\left(\sum_{cyc} (1+z)\right)\ge(x+y+z)^3\iff \sum_{cyc} \frac{x^3}{(1+y)(1+z)}≥\frac{(x+y+z)^3}{\left(\sum_{cyc} (1+y)\right)\cdot\left(\sum_{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Requirement for Hall-$\pi$-subgroups to be conjugate (Wielandt) I am reading through Isaacs Theory of finite groups and towards the end of chapter 3 on "split extensions" there is mention of a theorem of Wielandt which states that if a finite group $G$ contains a nilpotent Hall-$\pi$-subgroup then all the Hall-$\pi$-su... | You can look at the Theorem and proof in Derek J.S. Robinson's "A course in the Theory of Groups" Chapter 9 (p. 258-259). It says: Let the finite group $G$ possess a nilpotent Hall $\pi$-subgroup $H$. Then every $\pi$-subgroup of $G$ is contained in a conjugate of $H$. In particular all Hall $\pi$-subgroups of $G$ are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Algebraic characterization of commutative rings with Krull dimension=1,2, or 3 A commutative ring $R$ (with $1$) is $0$-dimensional if and only if $R/\sqrt 0$ is von Neumann regular. Besides this result, there is a wealth of information about zero-dimensional rings.
I could not find any information about rings of highe... | What about the following result due to Coquand and Lombardi
(A short proof for the Krull dimension of a polynomial ring.
Amer. Math. Monthly 112 (2005), no. 9, 826–829.)
Let $R$ be a commutative unital ring. For $N\in \mathbb N$,
the following assertions are equivalent:
(1)The Krull dimension of $R$ is at most $N$.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding circle of a sphere through two points We have two points $P_1, P_2$ on a sphere $S$ of radius $R$. Suppose for $r \ll R$, the distance between $P_1$ and $P_2$ is less than $2r$.
Then, $P_1$ and $P_2$ both lie on exactly two radius-$r$ circles of the sphere. This is clear: of the set of planes that contain $P_... | Here is a partial answer, if you don't mind a bit of linear algebra.
Let $u_1$ and $u_2$ be the vectors from the center of the sphere to $P_1$ and $P_2$ respectively. These vectors have length $R$. The points you are looking for will each correspond to vectors of length $R$ as well; let us call either of them $x$.
So y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Non-linear ODE $1+y'^2=yy'^2$, $y(0)=b,y(a)=c$ How to solve
$$1+y'^2=yy'^2?$$
The initial condition is $y(0)=b,y(a)=c$.
It is a non-linear ODE.
Some hint is also helpful! Thanks for your help!
| We have $$1+(y')^2=y(y')^2$$
Divide by $(y')^2$ to get $$\frac{1}{(y')^2}=y-1$$
Let us solve for $x=x(y)$ instead of for $y=y(x)$. Then we get the equation:
$$x'=\sqrt{y-1}\ \ \ \ \ \ \text{We need to do the other sign of the square root too.}$$
Separate variables now: $$\begin{align}dx&=\sqrt{y-1}dy\\x+C&=\frac{2}{3}(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1134980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $4^n > n^4$by induction for $n\ge 5$ That is a simple question and I can't start a simple desenveloment.
Just $k=5$ we have $4^5 = 1024 > 5^4 = 625$
for $k+1$: $4^{k+1} > (k+1)^4\Rightarrow 4^k > (k+1)^4/4$
And How can i proceed after that?
Best,
| Hint:
For $k \geq 5$ the following is true:
$$k\sqrt2 > k+1$$
As:
$$5(\sqrt2 -1)>5\times \frac25 =2> 1$$
We can show that $\sqrt2 -1 >\frac25$ as such:
$$\sqrt2>\frac75$$
Now as both $\sqrt2$ and $\frac75$ are bigger than zero, the inequality is persevered by squaring.
$$2=\frac{50}{25}>\frac{49}{25}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
How to prove there always exists a two variables or one variable equation corresponding to a curve drawn on a 2D plane? How to prove there always exists a two variables or one variable equation corresponding to a curve drawn on a 2D plane?
For example, a circle 's equation is $x^2+y^2=r^2$.
| Since Victor didn't specify what he means by "a curve drawn ...", I'll assume that
this is a parametric curve $p = \gamma(t)$, $0 \le t \le 1$, with $\gamma: [0,1] \to \mathbb R^2$ continuous. The image $C = \gamma([0,1])$ is then a compact set in the plane, and $F(p) = \text{dist}(p, C)$ (i.e. the distance from $p$ t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135185",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Closed and Bounded but not compact Let $(C[0,1];d_{\infty})$ the metric space with $C[0,1]$ the continuous functions on $[0,1]$ and $d_{\infty}(f,g)=\max_{x \in[0,1]}|f(x)-g(x)|$
Prove that $$S=\{f:|f(x)| \leq 1\}$$ is closed and bounded but no compact.
My attempt: I've already proved that is bounded using a ball with ... | It is closed because the uniform limit of continuous functions is continuous, and the limit function will belong to S, because it is also the pointwise limit.
For non-compactness, since we are in a metric space, it is enough to find a sequence in S which (you can show) has no convergent subsequence. $\{x^n\}_{n\geq 1}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 0
} |
Sum of entries made one Assume that I have performed some loop and summed values of $x_i$ and the sum is as follows $$S= \sum_{i=1}^n (x_i *0.0001) = 0.9999$$
However I would like to have the sum to be equal to 1 i.e,
$$\sum_{i} (x_i *0.0001)= 1$$
How much should I add to each of the entires $x_i$ so that I end with ... | $$I_1=\sum _{i=1}^n{x_i}*u=X_1$$
$$I_2=\sum _{i=1}^n{(x_i+ \delta)}*u=X_2$$
$$I_2-I_1=\sum _{i=1}^n{(\delta)}*u=X_2-X_1=\Delta$$
$$ n*\delta={\Delta \over u}$$
$$\delta = { \Delta \over (nu)}$$
For $\Delta=0.0001$ and $u=0.0001$
$$\delta = { 1 \over n}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135420",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Does the convergence of $\sum_{n = 0}^{\infty} |a_{n}| \Rightarrow |a_{n}| \rightarrow 0$ Consider the power series $\sum_{n = 0}^{\infty} a_{n} x^{n}$.
If $\sum_{n = 0}^{\infty} |a_{n}|, n \in \mathbb{Z}$ is absolutely convergent, does that necessarily imply $|a_{n}| \rightarrow 0$?
When I think about it, it seems tr... | Suppose $|a_n|$ converged to some nonzero number $L>0$. That implies that for any $\epsilon>0$, there is a natural number $N$ such that $L-\epsilon<|a_n|$ for all $n\ge N$. But then we would have
$$\sum_{n=0}^{\infty}|a_n|\ge \sum_{n=N}^{\infty}|a_n|>\sum_{n=N}^{\infty}(L-\epsilon)=\infty.$$
Contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Solving for one of variables locally when the Implicit Function Theorem does not apply I'm having trouble deciding whether certain functions can be locally solved. I have some examples:
*
*Can $xye^{xz} - z\log y =0$ be locally solved in $(0,1,0)$ for x? y? z?
In this case, I used Implicit Function Theorem to answ... | When implicit function theorem does not apply, the approach is usually ad hoc, based on the details of the function. Two things to try:
*
*Change the variables. Some substitutions, like $u=x^3$, can reduce the function to a form in which the implicit function theorem applies.
*Restrict $f$ to some line parallel to t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135656",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove $\int_t^{\infty} e^{-x^2/2}\,dx > e^{-t^2/2}\left(\frac{1}{t} - \frac{1}{t^3}\right)$ How to formally prove the following inequality -
$$\int_t^{\infty} e^{-x^2/2}\,dx > e^{-t^2/2}\left(\frac{1}{t} - \frac{1}{t^3}\right)$$
| For a better lower-bound you may use the following proof by @robjohn:
$$x\int_x^\infty e^{-t^2/2}\,\mathrm{d}t \le \int_x^\infty e^{-t^2/2}\,t\,\mathrm{d}t =e^{-x^2/2}$$
Integrate both sides of the preceding:
$$
\begin{align}
\int_s^\infty e^{-x^2/2}\,\mathrm{d}x
&\ge\int_s^\infty x\int_x^\infty e^{-t^2/2}\,\mathr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Confusion on how to calculate mean value I've done this type of thing in multiple classes over multiple years since high school, and still, when it's presented to me, I fumble around like a dope.
Consider a gas of $N_0$ non-interacting molecules enclosed in
a container of volume $V_0$. Focus attention on any sub-vo... | The probability of ONLY two molecules being in the volume is
$p^2(1-p)^{N-2}$ times the combinatorial (n, 2). Where $p$ is V/V0.
Etc Etc
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135821",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How can I solve a difference of inverse sines? I'm trying to solve the equation
$$\arcsin(\lambda_1 / d) - \arcsin(\lambda_2 / d) = \pi / 6$$
for $d$, where $\lambda_1$ and $\lambda_2$ are known.
I don't really know where to start here.
I don't know any identities for inverse trigonometric functions akin to the ones li... | Note that if $\sin x=y$, then since
$$\sin^2 x+\cos^2 x=1$$
we have that $\cos x=\pm \sqrt{1-y^2}$. Use this together with the sum formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1135990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Asymptotic Notation Analysis Problem I'm new here. I have some question on asymptotic analysis
I am trying to calculate the Big-O of these five functions and rank them up:
a: $$2^{\log(n)}$$
b: $$2^{2\log(n)}$$
c: $$n^{5\over2}$$
d: $$2^{n^2}$$
e: $$n^2\log(n)$$
These are my approaches:
a: $$O(n)$$ because I log the bo... | I don't see anything wrong if it is $\log_2$. In general $2^{\log_b(n)} = n^{\log_b(2)} = n^{\log_2(b)^{-1}}$ and so the asymptotic behaviour would depend on $b$.
Wait your reasoning for (d) is wrong. $2^{n^2} = 2^{(n^2)} \ne (2^n)^2$... Though it really is the biggest, much bigger than what you thought.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1136083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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