Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Number of possible combinations of the Enigma machine plugboard This is a question about basic combinatorics.
I recently watched again a youtube video about the Enigma cipher machine (in the Numberphile channel, https://www.youtube.com/watch?v=G2_Q9FoD-oQ), where the Enigma machine is briefly analyzed. In particular, t... | Here is one way to do it: arrange the alphabet in such a way that the pairs come first and the unpaired letters last. For example
$$\hbox{BGSI . . . UAWEXPTL}$$
means that B and G are paired, S and I are paired, . . . U and A are paired, while W, E, X, P, T and L are unpaired. There are $26!$ ways to do this. Howeve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1095042",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Text for study of subgroup lattices of finite abelian groups. I want to study the subgroup lattice of a finite abelian group. I have found a text on the subject: Subgroup Lattices of Groups by Roland Schmidt, de Gruyter 1994. This book is about subgroups of any group, not just finite abelian groups.
Is this text a good... | Subgroup lattices of finite abelian groups have been studied from various points of view, see the following reference:
Vogt, Frank : Subgroup lattices of finite Abelian groups: Structure and cardinality.
In: Lattice theory and its applications. Hrsg.: K.A. Baker, R. Wille. S. 241-259. Heldermann , Berlin .
[Buchkapitel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1095151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Simple solution to system of three equations I've been given the question;
$$xy = \frac19$$
$$x(y+1) = \frac79$$
$$y(x+1) = \frac5{18}$$
What is the value of $(x+1)(y+1)$?
Of course, you could solve for $x$ and $y$, then substitute in the values. However, my teacher says there is a quick solution that only requ... | Multiply the second by third then substitute the first.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1095273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Showing that a function is in $L^1$ I need to prove the following statement or find a counter-example:
Let $u\in L^1\cap C^2$ with $u''\in L^1$. Then $u'\in L^1$.
Unfortunately, I have no idea how to prove or disprove it, since the $|\bullet|$ in the definition of $L^1$ is giving me huge problems. I found counter-examp... | It is useful to recall the following lemma:
If $f$ is a twice differentiable function over $I=[a,b]$ and
$$M_0=\sup_{x\in I}|f(x)|,\quad M_1=\sup_{x\in I}|f'(x)|,\quad M_2=\sup_{x\in I} |f''(x)|,$$ then $M_1^2\leq 4M_0 M_2$.
It is a well-known exercise from baby Rudin's: you can find a proof of it here.
By the Cau... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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If $AB = 0$, prove that the columns of matrix $B$ are vectors in the kernel of $A$
Let $A,B$ be $n\times n$ matrices.
If $AB=0$, prove that the columns of matrix $B$ are vectors in the kernel of $Ax=0$.
I'm not sure how to approach this. I know that if $B = 0$ and $A$ isn't, then $Ax=0$ is when $x=0=B$. But what if $... | It results from the very definition of matrix multiplication: to obtain column $j$ in the product $AB$, you multiply each line of $A$ with column $j$ of $B$, i.e. you multiply $A$ by column $j$ of $B$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding a differentiable function with a particular property If $f:\mathbb{R}^2 \to \mathbb{R}$ is differentiable and satisfies $$2 \frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=0,$$ show that there exists a differentiable function $\widetilde{f}:\mathbb{R} \to \mathbb{R}$ such that $f(x,y)=\widetilde{f}(... | Hint: As you observed, we may write $f(2u + v, u + v) = g(v)$ for some differentiable function $g$ (because it is independent of $u$). Now if you find $u,v$ such that $2u + v = x$ and $u + v = y$, then you can write $f(x,y) = g(v)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1095553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to find a transcendental number where no two adjacent decimal digits are equal? By using WolframAlpha, I couldn't find any transcendental number without equal adjacent digits among the numbes $\tan(n)$, $\sin(n)$, $\cos(n)$, $\sec(n)$, $\cot(n)$, $\csc(n)$, $e^n$, and $ \log(n)$, where $n$ is an integer number.
H... | Start with Liouville's constant $0.11000100000000000000000100 \dots$ which is known to be transcendental and add $\frac{2}{99} = 0.02020202 \dots$. The resulting number is transcendental (because it differs from Liouville's constant by a rational) and has no identical adjacent digits.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1095645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Innocent looking open problems in real analysis Are there any apparently easy problems or conjectures in basic real analysis (that is, calculus) that are still open? By apparently easy, I mean: so much so, that, if it was for the statement alone, they could be part of a calculus book for undergraduates?
| Look for questions marked as open-problem or open-problem-list on mathoverflow. I guess you will find some open problems there.
Here a list of some open questions i have found there:
*
*Convergence of a series and this question
*Gourevitch's conjecture
*Cover of the unit square by rectangles
*...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1095743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Confusion with Algebra (Substitution) I apologize for the title, I'm not sure what category this would fall under.
The advertisement read Buy $3$ tires at the regular price and get a fourth tire for only $3$ dollars. Carol paid $\$240$ for a set of $4$ tires. What is the regular price of a tire?
So I came up with $3x +... | Your mistake is that dividing by three means you divide the entire left side, so you'd end up with: $$3x+3=240,$$$$\frac{3x+3}{3}=\frac{240}{3},$$$$x+1=80,$$$$x=\$79.$$ You omitted the $3x$ term in your calculations.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1095984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the coefficient of $x^{18}$ in the expansion of $(x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6})^{4}$? How to approach this type of question in general?
*
*How to use binomial theorem?
*How to use multinomial theorem?
*Are there any other combinatorial arguments available to solve this type of question?
| We really seek the coefficient of $x^{14}$, factoring out an $x$ from each term in the generating function. Then observe that:
$(1 + x + x^{2} + x^{3} + x^{4} + x^{5}) = \frac{1-x^{6}}{1-x}$
Now raise this to the fourth to get: $f(x) = \left(\frac{1-x^{6}}{1-x}\right)^{4}$.
We have the identities:
$$(1-x^{m})^{n} = \su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Sign of a Permutation using Polynomials One way to define sign of permutation is to consider the polynomial
$P(x_1,x_2,\cdots, x_n)=\prod_{1\leq i<j\leq n} (x_i-x_j)$ and define
$$Sign(\sigma)= \frac{P(x_{\sigma(1)},x_{\sigma(2)},\cdots, x_{\sigma(n)})}{P(x_1,x_2,\cdots, x_n)}.$$
I want to see the proof of $Sign(\sig... | Hint:
The definition
$$
Sign(\sigma){}={}\dfrac{P(x_{\sigma(1)},x_{\sigma(2)},\cdots, x_{\sigma(n)})}{P(x_1,x_2,\cdots, x_n)}
$$
is valid for any initial order of $x_1,\ldots,x_n\,$.
For example,
$$
\dfrac{(x_1-x_2)}{(x_2-x_1)}{}={}\dfrac{(x_2-x_1)}{(x_1-x_2)}{}={}-1\,.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Squeeze Theorem Question My question is from this Video
In the last example
He says that
$$\lim_{x \to 0} x^2 \cos(\frac{1}{x^2}) = 0$$
Squeeze Theorem:
$$g(x) \leq f(x) \leq h(x)$$
Given:
$$-1 \leq \cos(x) \leq 1$$
he confusing gets
$$-x^2 \leq x^2\cos(\frac{1}{x^2})\leq x^2$$
and finds the limits with that.
How doe... | I don't know whether there is any special way to find $g(x)$ and $h(x).$ It's mostly intuition and using some known inequality. Here are some examples:
(1). $\lim_{x \to 0} \sin x = 0.$ In this case we use the fact that $|\sin x | \leq 1.$ So we get $-x \leq \sin x \leq x.$
(2). $\lim_{x \to 0} \cos x = 1.$ In this cas... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Power set of $\{\{\varnothing\}\}$ $$\mathcal{P}(x)=\{y\mid y\subseteq x\}$$
$$\mathcal{P}(\varnothing)=\{\varnothing\}$$
$$\mathcal{P}(\{\varnothing\})=\{\varnothing,\{\varnothing\}\}$$
$$\mathcal{P}(\{a,b\})=\{\varnothing,\{a\},\{b\},\{a,b\}\}$$
For $\mathcal{P}(\{\{\varnothing\}\})$ we have: $$\varnothing\subseteq\{... | This one $$\{\varnothing\}\subseteq\{\{\varnothing\}\}$$ is incorrect.
$A\subseteq B$ means that for every $x\in A$ it is true that $x\in B$.
$\{\varnothing\}$ contains one element: $\varnothing$, but $\{\{\varnothing\}\}$ doen not contain $\varnothing$ so $$\{\varnothing\}\not\subseteq\{\{\varnothing\}\}$$
The other t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why do you reject negative base solution for Logs? $log_x64=2$ translates to $x^2=64$
This solves to $x=\pm8$
Why do you reject the solution of $x=-8$ ?
Doesn't it successfully check?
$log_{-8}64=2$ means "The exponent for -8 to get 64 is 2"
which is a true statement, no ?
| $\log_{-8}x$ would be an inverse function of $(-8)^x$ but this function does not behave well at all. What would be $(-8)^π$ for example?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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If $f \in L^1 \cap L^2$ is $L^2$-differentiable, then $Df \in L^1 \cap L^2$ Working with the definition that $f \in L^2(\mathbb{R})$ is $L^2$-differentiable with $L^2$-derivative $Df$ if
$$ \frac{\|\tau_hf-f-hDf\|_2}{h} \to 0 \text{ as } h \to 0 $$
(where $\tau_h(x) = f(x+h)$), I want to try and show that
If $f \in L^... | I don't think that what you are trying to prove is true.
Consider e.g.
$$
f(x) = \frac{1}{x^2} \cdot \sin(x^2) \text{ for large } x,
$$
i.e. truncate $f$ somehow near the origin.
We then have $f \in L^1 \cap L^2$ with
$$
f'(x) = -\frac{1}{x^3} \cdot \sin(x^2) + \frac{2}{x} \cdot \cos(x^2).
$$
We have $f' \in L^2$, but ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096544",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Relation between Frobenius norm and eigenvalues I'm considering a stochastic multivariate process, the stability of which implies that
*
*all eigenvalues $\lambda_i$, $i = \overline{1,n}$ of a certain square real-valued matrix $A$ lie within the unit circle.
Besides that we know nothing about $A$. But I also nee... | Let $\lambda$ be an eigenvalue of $A$ and $v$ an associated eigenvector. We may suppose without loss of generalities that $\|v\|_2 =1$. Thus we have $|v_i|\leq 1$ for every $i$ and there exists $j \in \{1,\ldots,n\}$ such that $|v_j|\geq n^{-1/2}$ since
$$ 1 = \left(\sum_{k=1}^n |v_k|^2\right)^{1/2} \leq \left(n\max_{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Probability of Sets I need some help on this one:
We have sets $X$ and $Y$ chosen independently and uniformly at random from among all subsets of $\{1,2,\ldots,100\}$. Determine the probability that $X$ is a subset of $Y$.
| how many pairs $(X,Y)$ satisfy $X\subseteq Y$? classify according to the size of $Y$.
So we get $\sum\limits_{k=0}^n\binom{n}{k}2^k$.
Since there are $2^n\times 2^n$ possible pairs $(X,Y)$ you want $$\frac{\sum\limits_{k=0}^n\binom{n}{k}2^k}{2^{2n}}$$
Now notice that $\sum\limits_{k=0}^n\binom{n}{k}2^k$ is equal to $3^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096727",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Example of a unbounded projection Let $H$ be a Hilbert space over $\mathbb{K}$.
Let $T:H\rightarrow H$ be a linear transformation such that $T^2=T$.
What is an example of $T$ such that $T$ is unbounded?
| For example: let $H = \ell^2$. Define the transformation
$$
(x_1,x_2,\dots) \mapsto
\left(\sum_{k=1}^\infty kx_k, 0,0,\dots \right)
$$
Note, however, that this operator is not defined over all of $\ell^2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Brownian Motion and Continuity Consider a Brownian Motion $(B_{t})_{t\geq0}$. In my lecure notes it says, without proof, that $\mathbb{P}\left(\sup_{t,s\leq N}\left\{ \left|B_{t}-B_{s}\right|:\left|t-s\right|<\delta\right\} <\varepsilon\right)$ converges to $0$ for $\delta$ tending to $0$.
I think it is a consequence o... | First there's a typo: you should have $>\varepsilon$ or should say that the probability converges to to $1$, not $0$.
Anyway, here's my proof. It's informal and skips over details; you should fill these in.
Fix $t$ and $s$ for a moment. Then your probability is the probability that a normal variable centered at zero wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1096969",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Describe explicitly all inner products on $\mathbb{R}$ and $\mathbb{C}$ I know this is an elementary question, however I am really lost as to where to start.
Since both $\mathbb{R}$ and $\mathbb{C}$ are finite-dimensional I think the inner product will be completely determined by the basis $\{1\}$.
I am not sure where... | It seems the following.
We shall consider these fields as vector spaces over itself.
Let $x,y\in\Bbb R$. Then $(x,y)=xy(1,1)$, so an inner product on $\Bbb R$ is completely determined by the value $(1,1)$. Conversely, it is easy to check that
for each $c>0$ the function $f_c:\Bbb R\times \Bbb R$, $f(x,y)=xyc$ is an... | {
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"timestamp": "2023-03-29T00:00:00",
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why is $\sqrt{-1} = i$ and not $\pm i$? this is something that came up when working with one of my students today and it has been bothering me since. It is more of a maths question than a pedagogical question so i figured i would ask here instead of MESE.
Why is $\sqrt{-1} = i$ and not $\sqrt{-1}=\pm i$?
With positive... | Well, if you are considering that $y=\sqrt{x}$ is the relation $y^2=x$, then, yes, $\pm i$ are both solutions to $\sqrt{-1}$. However, this is not usually how square roots are defined. Typically we say:
$$\sqrt{1}=1$$
Not plus or minus $1$ - just $1$. This means that $\sqrt{x}$ is a "right inverse" of $x^2$ - that is, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1097134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 7,
"answer_id": 5
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Math, how do we know if a substitution is true? For instance, in calculus we often do u-substitutions. Quite often, we do trignometric substitutions to solve integrals. For instance, if we have the following relation
$y=\sqrt{1-x^2}$
And we substitute $x = \sin u$, for $x \in [-1, 1]$; how do we know that our substit... | In general, you're allowed to do substitutions whenever you're able to undo them. Precisely speaking, the substitution $$y = g(x)$$
has to be such that $g$ is a bijective function. What this means is that:
*
*Two values $x_1\neq x_2$ cannot be mapped to the same value, i.e. $f(x_1) \neq f(x_2)$.
*Every $y$ has to b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1097237",
"timestamp": "2023-03-29T00:00:00",
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If $x,y,z \geq 1/2, xyz=1$, showing that $2(1/x+1/y+1/z) \geq 3+x+y+z$ If $x,y,z \geq 1/2, xyz=1$, showing that $2(1/x+1/y+1/z) \geq 3+x+y+z$
I tried Schturm's method for quite some time, and Cauchy Schwarz for numerators because of the given product condition.
| Let $x+y+z=3u$, $xy+xz+yz=3v^2$ and $xyz=w^3$.
Hence, our inequality is equivalent to $f(v^2)\geq0$, where $f$ is a linear increasing function.
Hence, $f$ gets a minimal value, when $v^2$ gets a minimal value, which happens for equality of two variables or maybe one of them equal to $\frac{1}{2}$.
*
*$y=x$, $z=\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1097317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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A conical tent is $8$ $m$ high and the radius of its base is $6$ $m$. A conical tent is $8$ $m$ high and the radius of its base is $6$ m. Find
(i) Slant height of the tent
(ii) Cost of the canvas required to make the tent, if the cost of $1$ $m^2$ canvas is $\$70$.
What I've tried so far,
Height=$8$ $m$
Radius=$6$ $m$
... | Lets start by deriving the Surface area of a cone ignoring the base
Let the height be $h$
We can see by Pythagoras that the slant height $s = \sqrt{h^2+r^2}$
The shape of the material would, when flattened out look like this
Which we can see could be cut from a circle of radius $s$
We know the formula for the area o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1097429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to say if Eigenvectors of A are orthogonal or not? without computing eigenvectors I am give matrix :
$$A=\begin{bmatrix} 0&-1 & 2 \\ -1 & -1 & 1 \\ 2 & 1 &0
\end{bmatrix}
$$
*1. Without finding the eigenvalues and eigenvectors, determine whether the eigenvectors are orthogonal or not. Justify your answer
*2. Ex... | Let $v$ be an eigenvector correspond to $\lambda$ and let $w$ be an eigenvector correspond to $\delta$. Then
$$\lambda \langle v,w \rangle= \langle \lambda v,w \rangle=\langle Av,w \rangle=\langle v,A^tw \rangle=\langle v,\delta w \rangle=
\delta \langle v,w \rangle\Rightarrow (\lambda-\delta)\langle v,w \rangle
\Right... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1097520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do you find this limit $\lim_{n\to +\infty} \int _{\frac{1}{n}}^{n} \frac{|\sin x|^n}{x^{\alpha}}\,dx $ I don't know how to solve the limit
$$\lim_{n\to +\infty} \int _{\frac{1}{n}}^{n} \frac{|\sin x|^n}{x^{\alpha}}\,dx $$
for each $\alpha>1$.
My attempt:
$\displaystyle f_n(x)=\frac{\chi_{[\frac{1}{n},n]}(x) |\sin ... | Let be: $$f_n(x)=\chi_{[\frac{1}{n},1]}\frac{|sin(x)|^n}{x^\alpha}+\chi_{[1,n]}\frac{|sin(x)|^n}{x^\alpha}$$ and let be$$\chi_{[\frac{1}{n},1]}\frac{|sin(x)|^n}{x^\alpha}=g_n(x)$$ $$\chi_{[1,n]}\frac{|sin(x)|^n}{x^\alpha}=h_n(x)$$
so we have:
$$g_n(x)<\chi_{[0,1]}sin(1)^nn^\alpha\to0$$$$\int{\chi_{[0,1]}sin(1)^nn^\alph... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1097685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Calculate Angle between Two Intersecting Line Segments Need some help/direction, haven't had trig in several decades.
On a 2 dimensional grid, I have two line segments.
The first line Segment always starts at the origin $(0,0)$, and extends to $(1,0)$ along the $X$-axis.
The second line Segment intersects the first at ... | Not sure EXACTLY what you are asking, but I will answer this to the best of my ability. If you could include a visual that would greatly help me.
When we have two intersecting line segments like this
finding any single value (a, c, b, d) will reveal all other values. For example, if we have the value of a, then c = a,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1097779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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} |
How does one expression factor into the other? How does $$(k+1)(k^2+2k)(3k+5)$$ factor into $$(k)(k^2-1)(3k+2) + 12k(k+1)^2$$
| Well, $RHS=k(k^2-1)(3k+2) + 12k(k+1)^2 = k(k+1)((k-1)(3k+2)+12(k+1))=k(k+1)(3k^2+2k-3k-2+12k+12)=k(k+1)(3k^2+11k+10)=k(k+1)(k+2)(3k+5)=LHS$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Arrange soccer fixtures with correct home - away alternation for each team I am trying to do as the title says. I have 10 teams in the same group. Every team must play the rest once each but each of them will always alternate home and away. This means that if they play at home their first game, the second MUST be away,... | Yes, it is possible.
By way of illustration... here's how my solution starts
$$\begin{array} \\
1 & BvA & DvC & FvE & HvG & JvI \\
2 & AvD & CvF & EvH & GvJ & \\
3 & DvB & FvA & HvC & JvE & IvG \\
4 & BvF & AvH & CvJ & EvI & \\
5 & FvD & HvB & JvA & IvC & GvE \\
\end{array}$$
(it might be that my interest in bell-rin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1097923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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How can I show the two limits How can I show the two limits
$$
\displaylines{
\mathop {\lim }\limits_{x \to + \infty } \frac{{x^2 e^{x + \frac{1}{x}} }}{{e^{ - x} \left( {\ln x} \right)^2 \sqrt x }} = \mathop {\lim }\limits_{x \to + \infty } \left( {\frac{{x^{\frac{3}{4}} e^{\left( {x + \frac{1}{{2x}}} \right)} }}{... | $$\frac{x^2e^x}{e^{-x} (\ln x)^2\sqrt x} < \frac{x^2e^{x + \frac1x}}{e^{-x} (\ln x)^2\sqrt x} \text{ for large $x$ since } e^{\frac1x} > 1$$
$$$$
$$\lim_{x \to \infty} \frac{x^2e^x}{e^{-x} (\ln x)^2\sqrt x} = e^{2x}\frac{x^{1.5}}{(\ln x)^2} = +\infty$$
$$$$
$$\text{Thus, the right hand side at the top should diverge to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Showing $\left\{ \frac{1}{n^n}\sum_{k=1}^{n}{k^k} \right\}_{n \in \mathbb{N}}\rightarrow 1$ I would like to prove:
$$\left\{ \frac{1}{n^n}\sum_{k=1}^{n}{k^k} \right\}_{n \in \mathbb{N}}\rightarrow 1$$
I found a proof applying Stolz criterion but I need to use the fact that:
$$\left\{\left(\frac{n}{n+1}\right)^n\right\}... | Obviously
$$\frac{1}{n^n}\sum_{k=1}^{n}k^k \geq 1,$$
while:
$$\frac{1}{n^n}\sum_{k=1}^{n}k^k\leq \frac{1}{n^n}\sum_{k=1}^{n}n^k\leq\frac{1}{n^n}\cdot\frac{n^n}{1-\frac{1}{n}}=\frac{n}{n-1}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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how can I prove that $\frac{\arctan x}{x }< 1$? I have got some trouble with proving that for $x\neq 0$:
$$
\frac{\arctan x}{x }< 1
$$
I tried doing something like $x = \tan t$ and playing with this with no success.
| We want to show that $\arctan(x) \leq x$ for all positive x (or vice-versa for negative x). Notice that at $x=0$, we can evaluate $\arctan(x) = 0$, so the functions are equal. Now, the derivative of $\arctan$ is $1/(1+x^2) < x' = 1$, and paired with our former observation, by a well-known theorem from calculus, this me... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to find the solution for $n=2$? Let $f: \mathbb R^n \setminus \{0\} \to \mathbb R^n \setminus \{0\}$ be a function only depending on the distance from the origin, $f = f(r)$, where $r = \sqrt{\sum_{i=1}^n x_i^2}$.
I calculated
$$ \Delta f = {n-1\over r}f_r + f_{rr}$$
and I am trying to determine which $f$ satisfy... | You're not showing your detailed reasoning, but I imagine the penultimate step must have been
$$ f'(r) = K_0 r^{1-n} $$
from which you get by indefinite integration
$$ f(r) = \frac{K_0}{2-n} r^{2-n} + K_1 $$
When $n\ne 2$, the division by $2-n$ can be absorbed into the arbitrary constant, but for $n=2$ you end up divid... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to explain this formally? This is an exercise of my homework:
Let $K\subset A\subset \mathbb{R}^N$ such that $N\in\mathbb{N}^*$, $K$ is compact and $A$ is open. Show that there is an $K_1$ compact such that $K\subset {K_1}^o \subset K_1 \subset A$
(Where ${K_1}^o$ is the set of interior points of $K_1$)
My str... | HINT: For each $x\in K$ there is an $\epsilon_x>0$ such that $B(x,\epsilon_x)\subseteq\operatorname{cl}B(x,\epsilon_x)\subseteq A$. Each of the sets $\operatorname{cl}B(x,\epsilon_x)$ is closed and bounded in $\Bbb R^N$, so it’s compact. Let $\mathscr{U}=\{B(x,\epsilon_x):x\in K\}$; $\mathscr{U}$ is an open cover of $K... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098427",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Find all optimal solutions by Simplex Let "stable operation" be an operation on a simplex tableau such that the entering variable has a reduced cost of 0. Recall that a pivoting operation will not change the objective value if either the reduced cost (i.e. in the $\bar c$ row shown below) is 0, or if the leaving variab... | Note that the "unstable operation" above won't give you a different solution. You've just replaced the basic variable $x_1 = 0$ by a nonbasic variable $x_4 = 0$, but the solution shown in the above optimal simplex tableau is still $(x_1,x_2,x_3,x_4,x_5,x_6)^T = (0,2,3,0,0,0)^T$.
Therefore, only "stable operations" wit... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Examples of bounded continuous functions which are not differentiable Most often examples given for bounded continuous functions which are not differentiable anywhere are fractals.If we include probabilistic fractals exact self-similarity is not required. Are their examples of functions which are bounded ,continuous, n... | It might be worth pointing out that the typical function, in the sense of Baire category, has this property. More specifically, Let $X=C([0,1])$ denote the set of all continuous, real valued functions defined on the unit interval endowed with the sup norm. Let $S\subset X$ denote the set of all functions that nowhere... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Mistake in my proof: what is the normalisation factor of the surface integral of a sphere? I was trying to prove
$$ {1\over \varepsilon} \int_{\partial B(a,\varepsilon)} f dS = {1\over r} \int_{\partial B(a,r)} f dS$$
where $0<\varepsilon < r$ and $f$ is harmonic on $\mathbb R^2$ and $n$ is the normal vector to the sp... |
(4) I put (1)-(3) together so that
$$
0 = \int_{B(a,r) \setminus B(a,\varepsilon)} (f \Delta G - G \Delta f)\, dV
= \int_{\partial B(a,r) \sqcup \partial B(a,\varepsilon)} f\, dS
$$
It appears you've lost the normal derivative of $g$. That is, the right-hand integral should be
$$
0 = \int_{B(a,r) \setminus B(a,\v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Number of handshakers in a conference I was given the following problem.
In a conference where n representatives attend, if 1 of any 4 of the attendants shake hands with the other
3, prove that 1 of any 4 of the attendants shake hand with the rest of the n − 1 attendants.
I'm familiar with the handshaking theorem and m... | Suppose this is not true. Pick a person $v$. He doesn't know someone (call him $w$). If $w$ does not know anyone we are done since if we pick $w$ and any other three persons no one will know $w$. Now pick a friend $x$ of $w$ such that $x$ knows $v$. (If no person exists who knows $w$ and $v$ we are also done since if w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098785",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The roots of a certain recursively-defined family of polynomials are all real Let $P_0=1 \,\text{and}\,P_1=x+1$ and we have $$P_{n+2}=P_{n+1}+xP_n\,\,n=0,1,2,...$$
Show that for all $n\in \mathbb{N}$, $P_n(x)$ has no complex root?
| Interlacing is a good hint, but let we show a brute-force solution. By setting:
$$ f(t) = \sum_{n\geq 0}P_n(x)\frac{t^n}{n!} \tag{1}$$
we have that the recursion translates into the ODE:
$$ f''(t) = f'(t) + x\, f(t) \tag{2}$$
whose solutions are given by:
$$ f(t) = A \exp\left(t\frac{1+\sqrt{1+4x}}{2}\right) + B\exp\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1098889",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z$ Give a direct proof of the fact that $a^2-5a+6$ is even for any $a \in \mathbb Z.$
I know this is true because any even number that is squared will be even, is it also true than any even number multiplied by 5 will be even?? is this direc... | Here is a 'logical' way to prove this, without using modular arithmetic, and without case distinctions.
$
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\calcop}[2]{\\ #1 \quad & \quad \unicode{x201c}\text{#2}\unicode{x201d} \\ \quad & }
\newcommand{\endcalc}{\end{align}}
\newcommand{\ref}[1]{\text{(#1)}}
\newco... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 7,
"answer_id": 6
} |
How find a closed form for the numbers which are relatively prime to $10$, Interesting Question
Let $a_n$ be the positive integers (in order) which are relatively prime to $10$.
Find a closed form for $a_n$.
I know $$a_{1}=1,a_{2}=3,a_{3}=7,a_{4}=9,a_{5}=11,a_{6}=13,a_{7}=17,a_{8}=19,a_{9}=21,\cdots$$
It is said th... | That solution to question 1 looks quite complex. One could also just say
$$ a_n = 2n + 2\left\lfloor\frac{n+1}4\right\rfloor - 1 $$
which uses that the first differences 2,4,2,2,2,4,2,2,2,4,2,2,2,4,... have a particularly simple structure in this case.
As a more immediately generalizable solution one could say
$$ a_n =... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Finding the formula of sum $(k+1)^2+(k+2)^2+...(k+(n-1))^2$ I know the sum of square of numbers which stars from $1$
but I don't know what the formula becomes when the first term is not $1$ as follow
$$(k+1)^2+(k+2)^2+...(k+(n-1))^2$$
| Hint
$$\sum_{i = k}^n i^2 = \sum_{i=1}^n i^2 - \sum_{i=1}^{k-1}i^2$$
$\text{Now substitute using the fact that }$
$$\sum_{i=1}^n i^2 = \frac{n(n + 1)(2n + 1)}{6}$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Application of the mean value theorem to find $\lim_{n\to\infty} n(1 - \cos(1/n))$ While reading Heuser (2009) "Lehrbuch der Analysis Teil I" on page 286, I got this question:
Find $$\lim\limits_{n \rightarrow \infty} n\Big(1 - \cos\Big(\frac{1}{n}\Big)\Big)$$ with the help of the Mean Value Theorem.
How do you apply t... | Obviously, $$\left|1-\cos \left(\frac{1}{n} \right) \right| = \left|\cos(0)-\cos \left(\frac{1}{n} \right) \right|.$$ By the mean value theorem there exist $a_n \in (0,1/n)$ such that
$$\left|1-\cos \left(\frac{1}{n} \right) \right| = |\cos'(a_n)| \left( \frac{1}{n} - 0 \right) = \sin(a_n) \frac{1}{n}.$$
Since $\sin(0)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1099353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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"answer_id": 1
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Mean value theorem with trigonometric functions Let $f(x) = 2\arctan(x) + \arcsin\left(\frac{2x}{1+x^2}\right)$
*
*Show that $f(x)$ is defined for every $ x\ge 1$
*Calculate $f'(x)$ within this range
*Conclude that $f(x) = \pi$ for every $ x\ge 1$
Can I get some hints how to start? I don't know how to start provin... | Hint
Recall that the $\arctan$ function is defined on $\Bbb R$ while the $\arcsin$ function is defined on $[-1,1]$. Compute the derivative $f'(x)$ and prove that it's equal to $0$. Conclude that $f$ is a constant which we can determinate by taking the limit of $f$ at $+\infty$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Mathematic correct plotting rational function What is range of $$-\frac{1-4\epsilon+3\epsilon^2}{1-\epsilon^2}$$
assuming $\epsilon\in(0,1)$?
It looks like $$\lim_{\epsilon\rightarrow1}-\frac{1-4\epsilon+3\epsilon^2}{1-\epsilon^2}=\lim_{\epsilon\rightarrow1}-\frac{-4+6\epsilon}{-2\epsilon}=1$$
However plotting on mathe... | There is no problem in Mathematica!
If you define
$$T=1-\frac{1-4\epsilon+3\epsilon^2}{1-\epsilon^2},$$
And use the following command:
T//Simplify,
the output generated is
$$\frac{4\epsilon}{1+\epsilon}$$
Mathematica will, in general, not automatically simplify your expression.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1099513",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Simplifying $\int_0^t\frac{1}{1+Au^b}du$ I'm trying to simplify
$$
\int_0^t\frac{1}{1+Au^b}du,\quad A>0,b>0,t\in[0,1].
$$
It looked simple at first but after trying a bit, I actually don't know how to tackle this. I entered the integral into Mathematica but it gave me back the same expression without any further simp... | A simple closed form in terms of elementary functions probably does not exist, but the integral can be expressed in terms of the Incomplete Beta Function, $\mathbf{B}\left(x;\alpha,\beta\right)$.
$$\int_{0}^{t}\frac{1}{1+Au^b}\operatorname{d}u = \frac{1}{bA^{\frac{1}{b}}}\mathbf{B}\left(\frac{At^b}{1+At^b};\frac{1}{b},... | {
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"timestamp": "2023-03-29T00:00:00",
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Prove $1! + 2! + 3! + \ldots + n! =y^3$ has only one solution in the set of natural numbers? I actually know that the above equation is true for $n=1$ and $y=1$ but am unable to prove it for the entire set of natural numbers. Can anyone please help me solve this in a simple way?
| Hint. Let
$$a_n = \sum_{k=1}^{n} k!.$$
Then $a_n$ is divisible by 3 for all $n \geq 2$, and we have $a_n \equiv 0 \pmod{27}$ only when $n=7$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1099744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Need help in metric spaces proving this statement! Please if someone could help me prove this rather annoying statement.
Let $C(0,1)$ be the set of continuous functions on the open interval $(0,1) \subset \mathbb R$. Fro any two functions $x(t), y(t) \in C(0,1)$ define the set $E(x,y)=\{t \in (0,1) | x(t) \neq y(t)\}$.... | Addition to the answer by Henno Brandsma: Take the function $h:(0,1)\rightarrow \mathbb R$ with $h(t) = x(t)-y(t)$, which is continuously (because $x$ and $y$ are continously). You have $$E(x,y) = h^{-1}(\mathbb R\setminus\{0\})$$ and thus $E(x,y)$ is open ($E(x,y)$ is the inverse image of the open set $\mathbb R \setm... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are birational morphisms stable under base change via a dominant morphism Let $f: X \to Y$ be a birational morphism of integral schemes and $g: Z \to Y$ a morphism of integral schemes which maps the generic point of $Z$ to the generic point of $Y$, i.e., the morphism $g$ is dominant.
Is then $X \times_Y Z \to Z$ birati... | Since $f: X \to Y$ is birational, we can find some open subsets $U \subset Y$ and $V \subset X$ so that $f$ restricts to an isomorphism $f: V \to U$. Then $g: W = g^{-1}(U) \to U$ is still dominant (really dominance of $g$ here just guarantees that $W$ is nonempty for any open $U$ we may need to restrict to).
Then the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Why is unitary diagonalization works? I've been told that if you take an Hermitian matrix, find it's eigenvectors, normalize them and write them as columns of a matrix, $P$ then:
$$P^{-1}AP = D$$
Where (Magically) $D = \text{Diag}(\lambda_1,\ldots,\lambda_n)$ ($\lambda_i$ is an eigenvalue of $A$).
So I really want to u... | By Schur decomposition, every square matrix $A$ is unitarily similar to an upper triangular matrix $U$. In particular if $A$ is Hermitian, then
$$U^* = (Q^* A Q)^* = Q^* A^* Q = Q^* A Q = U$$
which shows that $U$ is a diagonal matrix.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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mean value theorem question I was trying to solve the following:
Given: $0 < a < b$ and $n>1$,
prove:
$$na^{n-1}(b-a) < b^n-a^n < nb^{n-1}(b-a)$$
I managed to get this far using the mean value theorem:
$$a^n(b-a)<b^n -a^n<b^n (b-a)$$
Any idea how to continue?
| Let $f(x)= x^n-a^n$ using the Mean Value Theorem
$$f'(c) = \frac{f(b)-f(a)}{b-a} = \frac{b^n - a^n}{b-a}$$ for some $c \in (a,b)$.
But $f'(c)= n \cdot c^{n-1}$ so
$$\frac{b^n - a^n}{b-a} = n \cdot c^{n-1}$$
Now note that $n \cdot a^{n-1} < n \cdot c^{n-1} < n \cdot b^{n-1}$ so
$$ n \cdot a^{n-1} < \frac{b^n - a^n}{b-a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that there is a symmetric matrix B, such that BX=Y Let $X,Y$ be two vectors in ${\mathbb C}^n$ and assume that $X≠0$. Prove that there is a symmetric matrix $B$ such that $BX=Y$.
| Ok, the last answer was not detailed enough, so here is another approach, which is constructive.
Pick some orthogonal matrix $Q$, hence $QQ^T = I$ and build $B$ as $B = Q\Lambda Q^T$, where $\Lambda$ is diagonal with it's entries $\Lambda_{jj}$ still left to determine. Then,
$BX =Y \Leftrightarrow Q\Lambda Q^TX =Y \Le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Sum: $\sum_{n=1}^\infty\prod_{k=1}^n\frac{k}{k+a}=\frac{1}{a-1}$ For the past week, I've been mulling over this Math.SE question. The question was just to prove convergence of
$$\sum\limits_{n=1}^\infty\frac{n!}{\left(1+\sqrt{2}\right)\left(2+\sqrt{2}\right)\cdots\left(n+\sqrt{2}\right)}$$
but amazingly Mathematica to... | The idea of this solution is to appeal to the Beta function and then to exchange the order of integration and summation (made possible by Fubini's theorem).
$$\begin{align}
\sum\limits_{n=1}^\infty\prod\limits_{k=1}^n\frac{k}{k+a}&=\sum\limits_{n=1}^\infty\frac{n!}{(1+a)(2+a)\cdots(n+a)}
\\&=\sum\limits_{n=1}^\infty\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Is there a law that you can add or multiply to both sides of an equation? It seems that given a statement $a = b$, that $a + c = b + c$ is assumed also to be true.
Why isn't this an axiom of arithmetic, like the commutative law or associative law?
Or is it a consequence of some other axiom of arithmetic?
Thanks!
Edit: ... | This is an axiom of predicate logic. For example, check out this list of the axioms in predicate calulus, intended to be an ambient logic for ZFC set theory. Note axioms 13 and 14:
$$\vdash x=y\to (x\in z\to y\in z)$$
$$\vdash x=y\to (z\in x\to z\in y)$$
In set theory, the only basic atomic formulas are of the form $x=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "30",
"answer_count": 9,
"answer_id": 6
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Unitary matrix matrix problem I need to proof this question Show that for any
Recall that the unitary group $U(n)$ consists of all $A \in M_n(C)$ with $A^*A = I$. Show that a matrix $A \in M_n(C)$ is in $U(n)$ if and only if $\langle Ax,Ayi\rangle = \langle x,y\rangle$ for all $x, y \in C$.
So I just need the $\Leftarr... | Hints:
1.) For $T,S\in M_n(\Bbb C)$ we have $\ T=S\ \iff\ \forall x,y:\langle Tx,y\rangle=\langle Sx,y\rangle$.
2.) Use the adjointness property: $\langle z,Ay\rangle=\langle A^*z,y\rangle$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100651",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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'Distributive' property for a function mod m What properties must some function $f(n)$ have for it to be the case that: $f(n) = (n + 3) \mod m = (n \mod m) + (3 \mod m)$?
Similarly, what if $f(n) = (n + 3) \mod m = (n \mod m + 3)?$
Is this something that is well studied? Where might I go to find more information?
Supp... | Let's assume that $m$ is a positive integer and look at your first equation.
Your equation $(n+3)\mod m=(n\mod m)+(3\mod m)$ is true for all $n$ if $m$ divides $3$: i.e. $m$ is $1$ or $3$. In those cases, both sides of the equation are the same as $n\mod m$: adding the $3$ does nothing.
In all other cases, your equatio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all invariant subspace of $A$ Consider a matrix mapping $A: V \to V$ for a vertor space V. Matrix $A$ has 3 eigenvalues are distinct : $\lambda_1,\lambda_2,\lambda_3$ and $v_1,v_2,v_3$ are vectors-corresponding .Find all invariant subspace of $A$
| Hint: note that each $\text{span}(v_i)$ is a invariant vector space.
Also note that $\{0\}$ and $V$ are invariant subspaces. Furthermore note that if $A$ is an invariant subspace and $B$ is an invariant subspace then so is $A + B$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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(Dis)prove that $\sup(A \cap B) = \min\{\sup A, \sup B\}$ Just beginning real analysis so I'm having some trouble with disproving this statement:
$$\sup(A \cap B) = \min\{\sup A, \sup B\}$$
Initially it asks whether it's true or false and to provide a counterexample if false, which by basic intuition to me it is. How... | Let $A = \{1\}$, $B = \{2\}$. Note that $\sup A \cap B = -\infty$.
If you want to avoid infinities, try $A = \{1, 2\}$, $B = \{1, 3\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100899",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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When are graphs deceiving? What are some examples of functions or quantities relating to functions (e.g., limits) $f:A \to B$ where $A$, $B \subseteq \mathbb{R}$ that require by-hand, "analytical" methods for analysis which are seemingly contradicted by a graph generated by software?
For instance, I recall that a Pre-c... | Almost any example of catastrophic cancellation plus enough zoom will do the trick. Two cases:
A simplification of your example (simpler function, bigger zoom):
$$\frac{(1-\cos(x^2))}{x^4},\qquad x\in[-0.001,0.001]$$
A rational function with a removable discontinuity:
$$\frac{x^{50}-1}{x-1},\qquad x\in[0.999999999,1.0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1100960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
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Guidelines for choosing integrand to get a sum.
The idea was to find:
$$\sum_{n=1}^{\infty} \frac{\coth(n\pi)}{n^3}$$
As you see in the solution, they conveniently choose a $f(z)$ they chose:
$$f(z) = \frac{\pi \cot(\pi z)\coth(\pi z)}{z^3}$$
That eventually led to their goal.
What are the guidelines for choosing such... | First of all, i'm not an expert in this kind of problems, but i think you should have always three things in mind:
1.) Some of the poles of your generating function should render the sum that you are looking for
2.) The residues of all the other poles should be:
2a) As Easily obtainable as possible and
2b) be of fini... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1101062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to derive the equation of a parabola from the directrix and focus Could someone please offer me proof and explanation of the following? - I am just having trouble with finding the '$a$' part of the equation.
"The leading coefficient '$a$' in the equation
$$y−y_1 = a(x−x_1)^{2}$$
indicates how "wide" and in what ... | The parabola is defined by the locus of the points that are at equal distance from the focus and a line, called the directrice.
Let's call $d$ the distance between the focus $F$ and the directrix $D$.
Then $a=\dfrac{1}{2d}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1101195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Prove that $[ab]=[[a][b]]$
Let $a,b,N\in\mathbb{Z}$, where $N>0$. Prove that $[ab]=[[a][b]]$, where $[x]$ denotes the remainder of $x$ after division by $N$.
Here's my attempt:
Proof. Since $x\equiv{[x]}_c\pmod c$, we know $a\equiv{[a]} \pmod{N}$ and $b\equiv{[b]} \pmod{N}$. By Proposition 1.3.4 in the book,
$$ab\equ... | Hint: Since the remainder of something upon division by $N$ is an element of $\{0,1,2,\ldots,N-1\}$, it is sufficient to show that $[ab] \equiv [[a][b]] \mod N$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1101299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Finding $ \int \sin^2(2x)/[1+\cos(2x)]dx$. I am surprisingly having a bit of difficulty with an indefinite integral which is interesting since the integral I solved before is
$$ \int \frac{1+\cos2x}{\sin^2(2x)} dx$$
The integral I am currently working on is
$$ \int \frac{\sin^2(2x)}{1+\cos2x} dx$$
I first divided out ... | The function $\sin^2(2x)/(1+\cos(2x))$ can be simplified to $1-\cos 2x$:
$$
\frac{\sin^2(2x)}{1+\cos(2x)}=\frac{1-\cos^2(2x)}{1+\cos(2x)}=\frac{(1-\cos(2x))(1+\cos(2x))}{1+\cos(2x)}=1-\cos(2x).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1101382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Prove that for an arbitrary (possibly infinite) language, that for a finite L-structure $M$, if $M \equiv N$ then $ M \cong N$
Prove that for an arbitrary (possibly infinite) language, that for a finite L-structure $M$, if $M \equiv N$ then $ M \cong N$
I'm struggling to think of what to do, I presume the best thing ... | Let $a=a_1,\dots,a_k$ be an enumeration of $M$ and let $p(x)={\rm tp}_M(a)$. By construction $p(x)$ is consistent in $M$, then $p(x)$ is finitely consistent in any $N\equiv M$. As $N$ is also finite, $p(x)$ is realized in $N$, say by the tuple $b$. Then $a\mapsto b$ is the required isomorphism.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1101460",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate $\iint\limits_D {\sqrt {{x^2} - {y^2}} }\,dA$ ...
Calculate $$\iint\limits_D {\sqrt {{x^2} - {y^2}} }\,dA$$ where $D$ is
the triangle with vertices $(0,0), (1,1)$ and $(1,-1)$.
I get the following integral $$I = 2\int\limits_0^1 {\int\limits_0^x {\sqrt {{x^2} - {y^2}} } \,dydx} $$ I would appreciate some ... | Write $D=\{(x,y)\in \mathbb R^2\colon 0\leq x\leq 1\land -x\leq y\leq x\}$.
The integral then comes equal to $\displaystyle \int \limits_0^1\int \limits _{-x}^x\sqrt{x^2-y^2}\mathrm dy\mathrm dx$.
You can get away with the one dimensional substitution $y=x\sin(\theta)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1101572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Inequality $\arctan x ≥ x-x^3/3$ Can you help me prove $\arctan x ≥ x-x^3/3$? I have thought of taylor but I have not come up with a solution.
| Have you tried derivatives?
$$(\arctan x-x+x^3/3)'=\frac 1{1+x^2}-1+x^2=\frac{x^4}{1+x^2}\ge0$$
So the difference is an increasing function. This fact, together the equality when $x=0$ means that
$$\arctan x\ge x-x^3/3\text{ when }x\ge 0\\\arctan x\le x-x^3/3\text{ when }x\le 0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1101658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Prove $\mathbb{P}( k < l/2 ) \geq \frac{l}{2} \times \mathbb{P}( k = l/4 ) $ for binomial variable $k$ Suppose we have a Binomial variable:
$$
k \sim Bin(l,\alpha)
$$
Is it possible to prove/disprove that:
$$
\mathbb{P}( k < l/2 ) \geq \frac{l}{2} \times \mathbb{P}( k = l/4 )
$$
EDIT: it's been used in 2nd line of ... | As stated in the comments, the inequality holds vacuously if $l$ is not a multiple of four.
Assuming $l=4n$, if we prove that the pdf of $\operatorname{Bin}(\alpha,4n)$ is convex on the interval $[0,2n]$, then the inequality follows from Jensen's inequality. Notice that the inflection points of the pdf of the standard ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1101748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that two different representations to the base $k$ represent two different integers I would like to show:
Given two distinct, positive, integer representations in base $k$, say $\sum_{i=0}^na_ik^i$ and $\sum_{i=0}^mb_ik^i$ where $a_n \neq 0 \neq b_m$ and $a_i,b_i \in \{0,1,\ldots , k-1 \}$, prove that $$\sum_{i=0... | This is a comment,
not an answer,
but the comment space is too small
to hold this.
I proved this result on
representation in general bases
over 40 years ago:
Let $\mathbb{B} =(B_j)_{j=0}^{\infty}$
be an increasing series
of positive integers
with $B_0 = 1$.
A positive integer $n$
is represented in
$\mathbb{B}$ if
$n$ c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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Find the point on the curve farthest from the line $x-y=0$. the curve $x^3-y^3=1$ is asymptote to the line $x-y=0$. Find the point on the curve farthest from the line $x-y=0$.can someone please explain it to me what the question is demanding?
I cant think it geometrically as I am not able to plot it
Also is there any s... | Rotate by $-\pi/4$. This corresponds to $x\mapsto x-y$ and $y\mapsto x+y$ (up to a scalar factor). Then your line becomes $x-y-(x+y)=0\iff y=0$ and the equation of the curve becomes
$$(x-y)^3-(x+y)^3=1\iff -2y^3-6yx^2=1\iff x^2=-\frac{1+2y^3}{6y}$$
as $y$ can't be $0$. You want to find the farthest point from the line ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
$\lim_{n \rightarrow \infty} \frac{1-(1-1/n)^4}{1-(1-1/n)^3}$ Find
$$\lim_{n \rightarrow \infty} \dfrac{1-\left(1-\dfrac{1}{n}\right)^4}{1-\left(1-\dfrac{1}{n}\right)^3}$$
I can't figure out why the limit is equal to $\dfrac{4}{3}$ because I take the limit of a quotient to be the quotient of their limits.
I'm taking th... | $$\lim_{n \rightarrow \infty} \dfrac{1-\left(1-\dfrac{1}{n}\right)^4}{1-\left(1-\dfrac{1}{n}\right)^3}
\stackrel{\mathscr{L}}{=}\lim_{n \rightarrow \infty} \dfrac{4\left(1-\dfrac{1}{n}\right)^3 \dfrac{1}{n^2}}{3\left(1-\dfrac{1}{n}\right)^2\dfrac{1}{n^2}}
=\lim_{n \rightarrow \infty} \dfrac{4\left(1-\dfrac{1}{n}\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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About the rank of a small square matrix An interesting question which hit me just now: Suppose we have a square matrix, for instance, a 3 by 3 one. Each of its entry is a randomly assigned integer from 0 to 9, then whats' the probability that it becomes a singular matrix? Or, in general cases, what if the matrix is n b... | Are you familiar with the theorem that states: In the set of all nxn (in this case n=3) matrices, the set of all singular matrices has Lebesgue measure zero?
If you restrict the space to only entries from 0 to 9, the space itself has Lebesgue measure zero, and thus, if you were to equip this space with probability meas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102248",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What does a prime (apostrophe) mean before a predicate? I found this statement in a paper by John McCarthy:
$$
\forall x.ostrich\ x \supset species\ x ={}^\prime{}ostrich
$$
I can't figure out what the prime indicates.
| It seems to me that it is used as an "operator" for nominalization, in place of the standard $\lambda$ operator.
See :
*
*Nino Cocchiarella, Conceptual Realism and the Nexus of Predication : Lecture Five (Rome, 2004), page 14 :
Consider, for example, the predicate phrase "is famous", which can be symbolized as a $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102392",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Quick way to express percent from negative interval I have some data which looks like this:
I get a number in the -6/+4 range and I need to express it with percent.
If I get 4 of course the result is 100%, If I get -6 is 0%, etc.
Is there a quick formula I can use to obtain the percent value of, say, -3.25?
| For $x \in [-6, 4]$ the function
$$p(x) = \frac{x+6}{10}$$
is what you are looking for. You should note however that this gives numbers between $0$ and $1$. If you want the actual percentages, just use the function
$$q(x) = p(x) * 100 = 10 (x + 6)$$
instead.
In general for an interval $[a, b]$ you can use the formulas
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How likely is it to guess three numbers? In the Irish lottery if you guess three numbers correctly you win 576x your original stake and there are 12 draws a week.
My questions is: How likely is it, over the course of two years (104 weeks or 1248 draws) that I will guess the numbers correctly?
The rules are as follows:
... | The chance of getting the first number right is $\frac6{49}$, the second one is $\frac5{49}$ and third one is $\frac4{49}$. the product of these 3 fractions gives you the probability of winning a draw on a single attempt. If you have $1248$ draws, multiply this fraction by the same to get the probability of winning $1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Find an explicit atlas for this submanifold of $\mathbb{R}^4$ I'm having a hard time coming up with atlases for manifolds. I can prove using the implicit function theorem that
$M = \{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4:x_1^2+x_2^2=x_3^2+x_4^2=1\}$
is a $2$-dimensional manifold. I would like to find an explicit atlas for t... | OK, so based on the comments, I think this should be the answer:
Let
$U_1=\{(x_1,x_2,x_3,x_4)\in M: x_1>0\}, \phi_1:U_1\to\mathbb{R}$ defined as $\phi(x_1,x_2,x_3,x_4)=x_2$,
$U_2=\{(x_1,x_2,x_3,x_4)\in M: x_2>0\}, \phi_2:U_2\to\mathbb{R}$ defined as $\phi(x_1,x_2,x_3,x_4)=x_1$,
$U_3=\{(x_1,x_2,x_3,x_4)\in M: x_1<0\}, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102768",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Suppose $\int_{[a,b]}f=0,\text{ then }f(x)=0 \forall x\in[a,b]$
Let $a<b$ be real numbers. Let $f:[a,b]\to\mathbb{R}$ be a continuous non-negative function. Suppose $\int_{[a,b]}f=0,\text{ then }f(x)=0 \forall x\in[a,b]$
Proof:
Suppose for the sake of contradiction $\exists x_0\in[a,b] \text{ such that } f(x_0)= D> ... | This isn't exactly what you're asking for, but this is how I would approach this problem:
If $f$ is Riemann integrable, then it is Lebesgue integrable and the two integrals are equal. So if $\int_{[a,b]}f\mathrm dm=0$ where $m$ is Lebesgue measure, then $f=0$ almost everywhere (since $f$ is nonnegative). Since $f$ is c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Unusual mathematical terms From time to time, I come across some unusual mathematical terms. I know something about strange attractors. I also know what Witch of Agnesi is. However, what prompted me to write this question is that I was really perplexed when I read the other day about monstrous moonshine, and this is so... | I always wanted to get a room at the Hilbert Hotel.
I also love working with annihilators....
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102872",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "97",
"answer_count": 39,
"answer_id": 22
} |
Is it an axiom that the inequalities are translation-invariant or can we prove it? I was thinking about the inequalities on the set of real numbers. To me and everyone else, it's been taught that an inequality is translation-invariant, i.e.:
$x < y \implies x + c < y + c \quad \forall c \in \mathbb{R}$
But I've been tr... | The reals can either be defined axiomatically or constructed from (usually) a model of the rationals, in various ways. Axiomatically, the reals are an ordered field, which means that translation invariance property you refer to is an axiom. So, if your approach to the reals is that they are simply a model (one of many,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1102902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How do I solve this, first I have to factor $ 2x\over x-1$ + $ 3x +1\over x-1$ - $ 1 + 9x + 2x^2\over x^2-1$? I am doing calculus exercises but I'm in trouble with this
$$\frac{ 2x}{x-1} + \frac{3x +1}{ x-1} - \frac{1 + 9x + 2x^2}{x^2-1}$$
the solution is
$$ 3x\over x+1$$
The only advance that I have done is fact... | Multiply the first two terms by $x+1$ in nominator and denominator. Then add all three terms and you obtain an expression
$$\frac{f(x)}{(x-1)(x+1)}. $$
Now see how to factor $f(x)=3x^2-3x$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1103001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
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Find the value of $\lim_{x \to +\infty} x \lfloor \frac{1}{x} \rfloor$
Determine if the following limits exist
$$\lim_{x \to +\infty} x \lfloor \frac{1}{x} \rfloor$$
note that $$\frac{1}{x}-1 <\lfloor \frac{1}{x}\rfloor \leq \frac{1}{x}$$
$$1-x <x\lfloor \frac{1}{x}\rfloor \leq 1$$
i'm stuck here
| Observe that $\lfloor\frac1x\rfloor=0$ for $x>1$, hence $x\lfloor\frac1x\rfloor$ is identically zero on $]1,+\infty[$. Hence the limit is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1103101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove that $e^x = 8x^3$ has only one solution in $[0,1]$? Prove that $e^x = 8x^3$ has only one solution in $[0,1]$.
If we define $f(x) = e^x - 8x^3$ then by mean value theorem there exists at least one solution. But $f$ is not strictly decreasing/increasing. How do I continue?
| $f$ is concave on $[0.1,1]$, so $f(0.1)>0$ and $f(1)<0$ imply that the root from $(0.1,1)$ is unique. To check that there is no root on $(0,0.1)$, note that $f'(x)$ is positive here and $f(0)>0$, $f(0.1)>0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1103182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Finding parameters for a quotient of a polynomial ring
Let $a,b \in \mathbb{R}, T_{a,b} := \mathbb{R}[x] \ /\langle x^2+ax+b\rangle$, where $\langle x^2+ax+b\rangle$ is the ideal generated by $x^2+ax+b$.
1) for which $a,b \in \mathbb{R}$ is $T_{a,b}$ a field?
2) for which $a,b \in \mathbb{R}$ is $x \in T_{a,b}$ invert... | 2) We have
$$
x(x+a)=x^2+ax=-(ax+b)+ax=-b.
$$
Thus
$$
\frac{1}{x}=\frac{x+a}{x(x+a)}=\frac{x+a}{-b}
$$
| {
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"url": "https://math.stackexchange.com/questions/1103255",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Compute the derivative $ \frac{d}{dR}\iiint_{\{(x,y,z)\in\textbf{R}^3: \sqrt{x^2+y^2+z^2} \leq R\}}f(x,y,z)\,dx\,dy\,dz. $ Let the function f and its first-order partial derivatives be continuous in $\textbf{R}^3$. Suppose that
$$
\iiint_{\textbf{R}^3}|f(x,y,z)|\,dx\,dy\,dz < \infty.
$$
Compute the derivative
$$
\f... | In general moving the derivative in or outside an integral is not mathematically valid, but in your case you can. It makes no difference since the integrals with respect to $\phi$ and $\theta$ are not dependant on $R$, thus differentiating the results of these integrals with respect to $R$ will yield $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1103356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Preimages of intersections/unions Let $f(x) = x^2$ and suppose that $A$ is the closed interval $[0, 4]$ and $B$ is the closed interval $[−1, 1]$. In this case find $f^{−1}(A)$ and $f^{−1}(B)$.
Does $f^{−1}(A\cap B) = f^{−1}(A) \cap f^{−1}(B)$ in this case?
Does $f^{−1}(A \cup B) = f^{−1}(A) \cup f^{−1}(B)$?
| I will assume that $f: \mathbb{R} \rightarrow \mathbb{R}$.
If $F: U \rightarrow V$ and $Y \subset U$ then $f^{-1}(Y)= \{x \in U : F(x) \in Y \}$
So $f^{-1}(A)=f^{-1}([0,4])=\{x \in \mathbb{R} : x^2 \in [0,4] \}= [-2,2]$
and $f^{-1}(B)=f^{-1}([-1,1])=\{x \in \mathbb{R} : x^2 \in [-1,1] \}= [0,1]$ as complex solutions ar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1103439",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What significant differences are there between a Riemannian manifold and a pseudo-Riemannian manifold? I am reading John Lee's book Riemannian Manifolds. On page 91, he begins a chapter called "Geodesics and Distance," which is I think the first chapter that seriously addresses geodesics.
I was very surprised when I c... | For one thing, a Lorentz-signature metric on a compact manifold can fail to be geodesically complete. If memory serves, Chapter 3 of Einstein Manifolds by Besse contains an example of a metric on a torus where a finite-length geodesic "winds" infinitely many times.
Generally, the "unit sphere" in a tangent space is non... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1103533",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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"answer_id": 0
} |
How to iterate through all the possibilities in with this quantifier? This is a problem from Discrete Mathematics and its Applications
My question is on 9g.
Here is my work so far
I am struggling with the exactly one person part. The one person whom everybody loves is pretty straight forward ( ∃ x∀y(L(y,x)). I am try... | Following the approach in the link, you would write $\forall y L(y,x) \wedge \forall z L(z,w) \implies w=x$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1103597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to visualize the gradient as a one-form? I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I still visualize gradients as vector fields instead of the level sets associated with dual vectors. How to visualize the gradient as a... | To clear up some confusion in the comments: when Carroll refers to the gradient of a function $f$ as a $1$-form he probably intends to refer to the exterior derivative $df$ of $f$. This is a $1$-form containing all of the information which is contained in the gradient, but which can be defined in the absence of a (pseu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1103667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Finding a particular solution to a differential equation what is the particular solution for the following differential equation?
$$D^3 (D^2+D+1)(D^2+1)(D^2-3D+2)y=x^3+\cos\left(\frac{\sqrt{3}}2x \right)+xe^{2x}+\cos(x)$$
I tried Undetermined Coefficients and it took so long to solve it,not to mention it was on an exam... | For the term $x^3$, use the indeterminate coefficients method ($6^{th}$ degree polynomial).
For the terms $\cos\left(\frac{\sqrt{3}}2x \right)$ and $\cos(x)$ use the complex exponential form and keep the real part of $e^{i\lambda x}/Y(i\lambda)$.
For the term $xe^{2x}$, also use indeterminate coefficients. Rewriting th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1103754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Can I use the index of a series for help with divergence? I was studying this series:
$$\sum_{n=2}^{\infty}\dfrac{5}{7n+28}$$
I know that it's an increasing, monotone sequence.
Also, I know I can rewrite as:
$$\sum_{n=2}^{\infty}\dfrac{5}{7(n+4)} = \dfrac{5}{7} \cdot \sum_{n=2}^{\infty}\dfrac{1}{n+4}$$
Also, I know tha... | Notice that by changing the index we have
$$\sum_{n=2}^\infty\frac1{n+4}=\sum_{n=6}^\infty\frac1n$$
so the series is divergent. Notice also that the nature of a series doesn't depend on the first few terms which means that the two series $\sum\limits_{n\ge1}u_n$ and $\sum\limits_{n\ge n_0}u_n$ (for any $n_0$) have the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104112",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
} |
Is there a solution for y for ${{dy}\over dx} = axe^{by}$ I have come up with the equation in the form $${{dy}\over dx} = axe^{by}$$, where a and b are arbitrary real numbers, for a project I am working on. I want to be able to find its integral and differentiation if possible. Does anyone know of a possible solution f... | I think there are some scenarios to consider if $a$ and or $b$ is equal to zero.
Case 1: $a=0$
Then $$\frac{dy}{dx} = 0 \implies y = C$$
Case 2: $a \neq 0, b=0$.
$$\frac{dy}{dx} = ax \implies y = \frac{ax^2}{2}+C$$
Case 3: $a \neq 0 \neq b$.
Then $$\frac{dy}{dx} = axe^{by} \implies e^{-by}dy = axdx \\ \implies \int e^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Why is Hawaiian earring not semilocally simply connected? Let $H$ denote the Hawaiian earring:
We defined a space $X$ to be semilocally simply connected if every point in $X$ has a nbhd. $U$ for which the homomorphism from the fundamental group of $U$ to the fundamental group of $X$, induced by the inclusion map, is t... | Consider any neighborhood of the point where the circles accumulate. At least one of the circles is completely contained in that neighborhood. A loop going around that circle is a non-trivial representative of the fundamental group of $U$ and it gets mapped by the inclusion to a non-trivial loop in $X$.
To show that th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104275",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Integer solutions of the following question: How many integer solutions are there in this equation:
$$x_1 + x_2 + x_3 + x_4 + x_5 = 63, \quad x_i \ge 0, \quad x_2 ≥10$$
I got $C(56,3)$.
Is that correct?
| Substitute $y_1 = x_1 + 1, y_2 = x_2 - 9, y_3 = x_3 + 1, y_4 = x_4 + 1, y_5 = x_5 + 1$. This yields the equivalent problem
$$
(y_1 - 1) + (y_2 + 9) + (y_3 - 1) + (y_4 - 1) + (y_5 - 1) = 63; y_i \ge 1
$$
i.e.
$$
y_1 + y_2 + y_3 + \cdots + y_5 = 58
$$
By a classic stars-and-bars argument (writing $58$ in unary) the answ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Separable diff. eqn: $(1+x^2)y' = x^2y^2, x > 0$ I have been given a step-by-step answer which I just cannot understand or follow.
$\begin{eqnarray}
&(1+x^2)y' &= x^2y^2 + y\cdot1 \\
\iff& \frac{1}{y^2} &= \frac{x^2}{1+x^2}
\end{eqnarray}$
From there on it's a matter of integrating and using a intitial value I was give... | First, if you differentiate $G(y(x))$ with respect to $x$ you get $\cfrac {dy}{dx} \cdot \cfrac {dG}{dy}$
So if you have an equation $G(y)=G(y(x))=F(x)$ and differentiate it, you get$$y'\frac {dG}{dy}=\frac {dF}{dx}$$
Going into reverse, if you have $y'g(y)=f(x)$ then you can put $G(y)=F(x)+c$ where $G(y)=\int g(y)dy$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Do 'symmetric integers' have some other name? $-1 \cdot -1 = +1$, but there seems to me to be no reason we couldn't define a number system where negative number's and positive numbers were completely symmetric.
Where:
$$1 \cdot 1 = 1$$
$$-1 \cdot -1 = -1$$
I understand that in order to do this, multiplication could no ... | Note that elements of a ring which satisfy $x^2=x$ are called idempotents, and these become important in some contexts - for example a square matrix with $1$ as the top left entry and zero everywhere else is a non-trivial idempotent. Such things become significant, for example, in representation theory.
[note $0,1$ are... | {
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"url": "https://math.stackexchange.com/questions/1104501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 5,
"answer_id": 1
} |
Finding determinant of $n \times n$ matrix I need to find a determinant of the matrix:
$$
A = \begin{pmatrix}
1 & 2 & 3 & \cdot & \cdot & \cdot & n \\
x & 1 & 2 & 3 & \cdot & \cdot & n-1 \\
x & x & 1 & 2 & 3 & \cdot & n-2 \\
\cdot & \cdot & \cdot & \cdot & \cdot & \cdot & \cdot \\
\cdot & \cdot & \cdot & \cdot & \cdot... | Multiply the last row by $\frac{1-x}{x}$; this means that the determinant you want will be the determinant of the changed matrix times $-\frac{x}{x-1}$. Now subtract $r_1$ from $r_n$
leaving
$$r_n = (0, -x, -x, -x, \cdots, -x, \frac{(x-1)^2 - x^2}{x})
$$
where I have intentionally written
$$
\frac{1-x}{x} -1 = \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
If M is a manifold of dimension $ n \neq0$ then M has no isolated points. I am in doubt whether the following statement is true or false:
"If M is a manifold of dimension $ n \neq0$ then M has no isolated points."
The idea that made me find the true statement was as follows:
If $ p \in M $ is an isolated point, conside... | This was given as a comment, but the question needs closure, so I will answer it.
Yes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
Finding total after percentage has been used? Tried my best with the title. Ok, earlier while I was on break from work (I have low level math, and want to be more fond of mathematics)
I work retail, and 20% of taxes are taken out, and I am wanting to find out how much I made before 20% is taken out.
So I did some scrib... | $80\% = 80/100$. Since we've multiplied this into the gross pay to get the net pay, we have to do the opposite, which is to say divide by this, to go from net pay to gross pay. Dividing by $a/b$ is the same as multiplying by $b/a$, so you have to multiply your net pay by $100/80=5/4=125/100$ to get your gross pay bac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 7,
"answer_id": 3
} |
2nd order differential equation with missing y' I have the following 2nd order differential equation:
$$y'' + p(x) y =0, \tag{1}$$
where $p(x)$ involves only first order of $x$, for example, $p(x)=ax+b$. Any suggestion how to obtain or guess a solution for (1)? Thanks.
| This does not seem (for me) to be so easy to solve in a nice way. I suppose there might be a substitution, but I can't think of any. This, however, can most certainly be solved by assuming that $y(x)$ is analytic on reasonable domain, and so with that assumption we can write out $y(x)$ as a power series and $y'(x)$ as ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1104944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to transform $ \int_a^b f(x^2+y(x)^2)\sqrt{1+y'(x)^2}\;dx$ into polar coordinates I have the following homework problem (from Calculus of Variations course) :
Show that if in
$$ \min \int_a^b f(x^2+y(x)^2)\sqrt{1+y'(x)^2}\;dx$$
polar coordinates are used, then the problem will be converted into
one that contai... | Suppose the graph of the given curve $x \mapsto y(x)$, $a \leq x \leq b$, can be written as a polar curve, say, as $\theta \mapsto r(\theta)$, $\alpha \leq \theta \leq \beta$.
Now, note that $$\sqrt{1 + y'(x)^2} dx$$ is just the arc length element (mnemonically, this is $\sqrt{\left[1 + \left(\frac{dy}{dx}\right)^2\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Specific treatment for the first and last element of sequence in a function? Let $A = \langle a_1,\dots,a_n \rangle$ be a sequence. I have a function that given any element $a_k$ it will return the values of $a_{k-1}+a_k+a_{k+1}$ with the exception of the first and last element that is going to return $a_k + a_{k+1}$ f... | I'd write this as
For $1\le k\le n$ let $f(k)=a_{k-1}+a_k+a_{k+1}$, with the convention that $a_0=a_{n+1}=0$.
Note that this formulation even covers the case $n=1$ correctly (which the cases-statement does not).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1105202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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