Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
calculate $\lim\limits_{x \to 1}(1 - x)\tan \frac{\pi x}{2}$ I need to calculate $$\lim_{x \to 1}\left((1 - x)\tan \frac{\pi x}{2}\right)$$.
I used MacLaurin for $\tan$ and got $\frac{\pi x} {2} + o(x)$. Then the full expression comes to $$\lim_{x \to 1}\left(\frac {\pi x} {2} - \frac {\pi x^2} {2} + o(x)\right) = 0$$B... | $L=\lim\limits_{x \to 1}((1 - x)\tan \frac{\pi x}{2})$
$L=\lim\limits_{(x-1) \to 0}((1 - x)\cot(\frac{\pi }{2}- \frac{\pi x}{2}))$
$L=\frac{2 }{\pi}\lim\limits_{\frac{\pi }{2}(x-1) \to 0}(\frac{\pi }{2}(1 - x)(-)\frac{cos\frac{\pi }{2}(x- 1)}{sin\frac{\pi }{2}(x- 1)})$
$L=\frac{2 }{\pi}\lim\limits_{\frac{\pi }{2}(x-1) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1537118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How many complex roots does this polynomial $x^3-x^2-x-\frac{5}{27}=0$ have? The fundamental theorem of algebra states that any polynomial to the $n^{th}$ will have $n$ roots (real and complex). But I know that complex roots only come in pairs because they are conjugates of each other. According to the theorem, $x^3-x^... | Counting repeated roots, it will, of course, be $3$.
To find out how many roots are repeated roots: If $p(x)$ has repeated roots, they are also roots of $p'(x)$ of one less degree. So compute the GCD of $p(x)$ and $p'(x)$ and subtract the degree of the result from the degree of $p(x)$ to get the number of unique roots... | {
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"timestamp": "2023-03-29T00:00:00",
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Is the set of all function with this condition... countable? Let $A=\{f|f:\Bbb{N} \to \{0,1\},lim _ {n\to\infty} f(n) =0\}$, now is A countable ?
I think it is countable because if we let $K_n=\{f|f:\Bbb{N} \to \{0,1\}, for \ every \ n \le m f(m) =0\}$ then every $K_n $ is not empty,countable and $A=\bigcup _{n\in \Bbb... | If $f:\mathbb{N}\to \{0,1\}$ has the property $\lim_{n\to\infty} f(n)=0$, then this means that $f$ is eventually $0$.
More precisely, there exists $N$ such that $f(n)=0$ for all $n\ge N$.
This means that $A$ is countable, because it is in bijection with the set of finite $\{0,1\}$ sequences.
| {
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How do I do this integral? How do I integrate $$\int\limits_{-c}^c \frac{e^{-\gamma\, x^2}}{1-\beta \, x}\, dx $$, under the constraints (if necessary) that $\beta < 1\, /\, c$ and $\beta \ll 1$ ?
| You'll have a bear of a time looking for exact solutions, since the indefinite integral will be in terms of the exponential integral function (Ei(t) = -$\int_{-t}^\infty \frac{e^x}{x} dx$), which is not an elementary function.
You can plug your integral into something like Mathematica to get $$- \frac{e^{ -\frac{\gamm... | {
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Hausdorff Spaces Subsets open? 'Given a Hausdorff space $X$ with finitely many elements, show all subsets of $X$ are open in $X$.'
I let $U$ be an arbitrary subset of $X$
Since every subset of a Hausdorff space is a Hausdorff space, then $U$ is also Hausdorff.
This means that any two points in $U$ $x \ne y$ have disjo... | Let $X=\{x_1,...,x_n\}$. Given $x_i$, for $j\ne i$ let $U_{i,j}$ be an open neighborhood of $x_i$ that doesn't contain $x_j$. Then $U_i = \bigcap_{j\text{: } j\ne i}U_{i,j}$ is a finite intersection open neighborhood of $x_i$ that contains no other $x_j$, so it's open. But $U_i = \{x_i\}$.
So all singletons $\{x_i\}$ ... | {
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"timestamp": "2023-03-29T00:00:00",
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Limit of $\lim\limits_{n\to\infty}\frac{\sum_{m=0}^n (2m+1)^k}{n^{k+1}}$ I wanted to find the limit of: ($k \in N)$
$$\lim_{n \to \infty}{\frac{1^k+3^k+5^k+\cdots+(2n+1)^k}{n^{k+1}}}.$$
Stolz–Cesàro theorem could help but $\frac{a_n-a_{n-1}}{b_n-b_{n-1}}$ makes big mess here:
$$\lim_{n \to \infty}{\frac{-0^k+1^k-2^k+3^... | Using Faulhaber's formula,
$\lim_{n \to \infty}{\frac{1^k+2^k+3^k+\cdots+n^k}{n^{k+1}}} = \frac{1}{k+1}$.
Then,
$\lim_{n \to \infty}{\frac{1^k+3^k+5^k+\cdots+(2n+1)^k}{n^{k+1}}} = \lim_{n \to \infty}{\frac{1^k+2^k+3^k+\cdots+(2n+1)^k}{n^{k+1}}} - 2^k \lim_{n \to \infty}{\frac{1^k+2^k+3^k+\cdots+n^k}{n^{k+1}}}$
$ = \li... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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What probability p corresponds to an expected number of 10 turns The problem below is from a problem set for a Game Theory course. We never really touched on much probability, probability distributions, etc so I was surprised when I saw this question...
"In probability theory, if an event occurs with fixed probability ... |
Okay, so, I have no idea what I am being asked here. All I can think of doing is plugging $v=10$ to get $p=0.9$, but that seems all too easy for a question for a graduate level course.
Yes, that is exactly what is needed. You were being asked: "Given that $v = \frac 1{1-p}$, when $v=10$, what does $p$ equal?" Eve... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Limits with L'Hôpital's rule Find the values of $a$ and $b$ if $$ \lim_{x\to0} \dfrac{x(1+a \cos(x))-b \sin(x)}{x^3} = 1 $$
I think i should use L'Hôpital's rule but it did not work.
| The easiest way (in my opinion) is to plug in the power series expansion of $x(1+a\cos x)-b\sin x$ around zero. Then, the limit becomes
$$\lim_{x\rightarrow 0} \frac{x(a-b+1)+x^3(b-3a)/6+O(x^5)}{x^3}=1.$$
Now you have two equations involving $a$ and $b$ to satisfy (can you figure out what those equations are?)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Taylor series representation problem During an exam recently, I was asked to prove that $-\ln(2) = \sum_{n=1}^\infty \dfrac{(-1)^n}{n\cdot2^n}$. But after fiddling around a lot, I kept reaching an argument that the sum actually equals $-\ln(1.5)$.
Was the exam incorrect?
| Indeed, we have $\ln(2)=\displaystyle\sum_{n=1}^\infty\dfrac1{n\cdot2^n}~,~$ while the expression they gave is $\ln\dfrac23~.$
| {
"language": "en",
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Integrating a function using residues theorem Let $f(z) = \frac{1}{e^z - 1 }$. I want to compute $\int_{\gamma} f dz $ where $\gamma$ is the circle of radius $9$ centered at $0$.
Try:
I know $z = 0$ is a singularity of $f(z)$. Let $f(z) = g(z)/h(z)$ where $g(z) = 1 $ and $h(z) = e^z - 1$. We know $h(0) = 1 \neq 0 $ and... | Yes it is right. To see this explicitly,
$$\frac{1}{e^z - 1} = \frac{1}{z(1 + z/2 +z^2/3! + \dots)} = \frac{1}{z}(a_0 + a_1z + a_2z^2 + \dots)$$
By comparison, $a_0 = 1, a_1 = -1/2, \text{and} \;a_2 = 1/12$. The analytic part of the series will of course go to $0$, leaving you to integrate $$\int_\gamma \frac{1}{z} dz... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1538079",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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determinant constraint on the dimension of SO(n) It is very well known that the dimension of $SO(n)$ is $n(n-1)/2$, which is obtained by the number of independent constraint equations we have from the fact that the matrix is orthogonal.
However, it is a little puzzling to me why the determinant constraint does not affe... | You can vary $n(n-1)/2$ parameters continuously and keep the determinant equal to $1$, or equal to $-1$, but to get from one to the other you have to make a discontinuous jump.
The split of $O(n)$ into $SO(n)$ and its complement is a little bit like how the equation $x^2=1$ in the $(x,y)$-plane defines two lines $x=1$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1538187",
"timestamp": "2023-03-29T00:00:00",
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Surjectivity of a continuous map between $\mathbb{R}^d$s Let $f:\mathbb{R}^{d}\to\mathbb{R}^d$ be a continuous map.
Show that if $\displaystyle\sup_{x\in\mathbb{R}^d}|f(x)-x|<\infty$, then $f$ is surjective.
I encountered this problem more than 3 years ago in my class of exercises in calculus when I was an undergradu... | Let
$$c > \sup_{x\in \mathbb R^d} \|f(x) - x\|$$
be fixed. For each $y\in \mathbb R^d$ (the image), let $D$ be large so that $D \ge 2c, 2\|y\|$. Restrict $f$ to $S_D = \{x\in \mathbb R^d: \|x\| = D\}$. Then if $x\in S_D$,
$$\|f(x)\| = \|f(x) - x +x\| \le c+D \le \frac 32 D.$$
On the other hand,
$$D=\|x\| = \|x - f(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1538328",
"timestamp": "2023-03-29T00:00:00",
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What's the order class of T(n) = n(T(n−1) + n) with T(1)=1? This recurrence basically comes from the typical solution to N-queens problem. Some people say the complexity is O(n) while giving recurrence T(n) = n(T(n−1) + n). But from what I can tell, this concludes O((n+1)!), which is surely different from O(n!) asympto... | Let $U(k)=\frac{T(k)}{k!}$. Then
$$U(n)=U(n-1)+\frac{n}{(n-1)!}=U(n-1)+\frac1{(n-1)!}+\frac1{(n-2)!}\\
<U(1)+2e-1\\
T(n)<2e(n!)$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Basic question: Global sections of structure sheaf is a field
$X$ is projective and reduced over a field $k$ (not necessarily algebraically closed). Why is $H^0(X,\mathcal{O}_X)$ a field?
Are there any good lecture notes on this (valuative criteria, properness, projectiveness, completeness)? I really don't have time ... | This is not true. The scheme $X=\mathrm{Spec}(k\times k)$ is projective and reduced over $k$ but the global sections of the structure sheaf do not form a field. You need to assume $X$ is connected too. In any case, by properness, $H^0(X,\mathscr{O}_X)$ is a finite-dimensional $k$-algebra (this is the hard part), hence ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1538563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Proving two matrices are equal A friend and I are having some trouble with a linear algebra problem:
Let $A$ and $B$ be square matrices with dimensions $n\times n$
Prove or disprove:
If $A^2=B^2$ then $A=B$ or $A=-B$
It seems to be true but the rest of my class insists it's false - I can't find an example where this is... | $\begin{pmatrix} 0&1\\0&0\end{pmatrix}^2=\begin{pmatrix} 0&2\\0&0\end{pmatrix}^2=\begin{pmatrix} 0&0\\0&0\end{pmatrix}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1538641",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Inverse equivalence of categories What is an "inverse equivalence" between two categories and how is it different from the regular equivalence notion?
My book says that two functors are an "inverse equivalence"... I thought this means they are one the inverse of the other, but isn't that a regular equivalence?
Note tha... | "Functors F and G are inverse equivalences" just means that not only do F and G have inverses, they are each other's inverses.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1538713",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why do we take $x=\cos t$, $y=\sin t$ for a parametric circle when we can take the opposite? Both $x = \cos t$, $y = \sin t$ and $x = \sin t$, $y = \cos t$ describe a circle
So why is the first parameterization so commonly used in mathematics, and not the second?
| Because we usually
start the circle
at the right.
There,
the coordinates are
$(1, 0)$,
which is
$(\cos(0), \sin(0))$.
If we started it at the top,
we would probably use
$(\sin(t), \cos(t))$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1538852",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Distinction between "if any" and "if every". Today in my math class I presented a counter example to the theorem:
"if any infinite sequence in X has an adherent point in X, then X is compact."
Let $X=(-1,2)$. Choose $\{X_{n}\}= \frac{1}{n} = \{1,\frac{1}{2}, \frac{1}{3},\cdots \}$.
Then $0$ is an adherent point of $\{ ... | If you are in doubt, sometimes, putting parentheses around logical statements can help make their meaning clearer. In this instance:
if (any infinite sequence in $X$ has an adherent point in $X$) then ($X$ is compact)
What the first bracketed statement essentially means is "choose any infinite sequence in $X$ and it wi... | {
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Show that $f=0$ almost everywhere on E Let $f$ be a non-negative bounded measurable function on a set of finite measure $E$. Assume that $\int f=0$ over $E$. Show that $f=0$ a.e. on E.
I want to prove this without using Chebychev's inequality proof. any idea
by the way i already solve it, it is just that i am curious i... | For any $\epsilon>0$, set $E_\epsilon=\{x\in E \colon f(x)\geq\epsilon\}$. Then $E_\epsilon$ is measurable since $f$ is, and $\mu(E_\epsilon)=0$ for all $\epsilon>0$ since otherwise (if,say, $\mu(E_\epsilon)=\delta>0$) we would have $\int_Efd\mu\geq\int_{E_\epsilon}fd\mu\geq\delta\epsilon$.
| {
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Number of Non-Abelian Groups of order 21 My goal is to count the number of non-abelian groups of order 21, up to isomorphism. I also need to show their presentations. This is a homework assignment, so I would appreciate leads rather than answers as much as possible. However, since we have to do this for all groups up t... | If you replace $b$ with $b^{-1}$ in the first presentation, you get the second presentation, since $a\mapsto a^4$ is the inverse of the automorphism $a\mapsto a^2$ of a cyclic group of order $7$. So they're actually the same, with just a different choice of generators.
| {
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"url": "https://math.stackexchange.com/questions/1539183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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The limit of a sequence of uniform bounded variation functions in $L_1$ is almost sure a bounded variation function Let $\{f_n\} $be a sequence of functions on $[a,b] $ that $\sup V^b_a (f_n) \le C$,
if $f_n \rightarrow f $ in $L_1$ ,Prove that $f $ equals to a bounded variation function almost every where.
I really ... | I wouldn't care much to do this question without relying on Helly's selection theorem but, then, I'm pretty lazy and am quite satisfied with proofs that take a single paragraph.
Theorem (Helly) Let $f_n:[a,b]\to\mathbb{R}$ be a sequence of
functions of bounded variation with
(i) $V(f_n,[a,b])\leq C$ for all $n$ (... | {
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A general approach for this type of questions I've come across multiple questions like these. $\left(\iota=\sqrt{-1}\right)$
If $f\left(x\right)=x^4-4x^3+4x^2+8x+44$, find the value of $f\left(3+2\iota\right)$.
If $f\left(z\right)=z^4+9z^3+35z^2-z+4$, find $f\left(-5+2\sqrt{-4}\right)$.
Find the value of $2x^4+5... | Unless there is some noticeable relation between f and x(for example if $ f'(x)=0$ ) or some simplifying expression for f (for example if $f(x)=1+x+x^2+x^3 +x^5$ and $x\ne 1$), or some special property of x (for example if $x^3=i$), all you can do is compute the complex arithmetic.There is considerable computer-scien... | {
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What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$? What is the value of $\int_{0}^{2\pi}(x-\pi)^2 (\sin x) dx$?
AFAIK : $f(x)$ is odd function $(x-\pi)^2$ should be even because of square, and it's odd because of $(\sin x)$.
Can you explain in formal way please?
| Hint: Take, $y=x-\pi$. So, now the $f(y)=y^2\sin(y+\pi)=-y^2\sin y$ . So, this is your odd function. Current upper and lower limit of your integral will be $\pi$ and $-\pi$ respectively.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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topos defined by sets Let $C=Sets$ be the category/site of sets, equipped with the topology defined by surjective families. Why is the associated topos $T$ equivalent to the punctual topos $Sh(pt)\simeq Sets$? (This is claimed in Luc Illusie's article "What is a topos?").
I reason that sheaves on the site $C$ are all p... | Let $\mathcal{E}$ be a Grothendieck topos. It is a standard fact that the category of sheaves on $\mathcal{E}$ with respect to the canonical topology on $\mathcal{E}$ is equivalent to $\mathcal{E}$ (via the Yoneda embedding): see here. (The canonical topology on $\mathcal{E}$ has as its covering families all jointly ep... | {
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How to manually calculate the sine? I started studying trigonometry and I'm confused.
How can I manually calculate the sine? For example: $\sin(\frac{1}{8}\pi)$?
I was told to start getting the sum of two values which will result the sine's value. For $\sin(\frac{7}{12}\pi)$, it would be $\sin(\frac{1}{4}\pi + \frac{1}... | For $\frac{\pi}{8}$ you need to use double angle formula.
Recall that $\sin[2\theta)=2\sin\theta\cos\theta=2\sin\theta\sqrt{1-\sin^2\theta}$
Then let $\theta=\frac{\pi}{8}$:
$$\sin\left(\frac{\pi}{4}\right)=2\sin\frac{\pi}{8}\sqrt{1-\sin^2\frac{\pi}{8}}$$
$$\frac{1}{2}=4\sin^2\frac{\pi}{8}\left(1-\sin^2\frac{\pi}{8}\ri... | {
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Nilradical equals intersection of all prime ideals Suppose we have a commutative ring R with identity 1, and A is an ideal of R. Also, we have $B=\{{x\in R\mid x^n\in A }$ for some natural number n$\}$.
Show that B is the intersection of all prime ideals that contain A.
I know that we can use Zorn's lemma to prove it, ... | Let $\mathfrak{p}$ be a maximal element (maybe the symbol is a bit presumptuous). You should first tell me why $\mathfrak{p}$ isn't the whole ring. Now take two elements $a,b$ not contained in $\mathfrak{p}$. Then, for instance, the ideal $\mathfrak{p} + Ra$ properly contains $\mathfrak{p}$ and hence can't be one of th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Unclear about matrix calculus in least squares regression The loss function of a Least Squares Regression is defined as (for example, in this question) :
$L(w) = (y - Xw)^T (y - Xw) = (y^T - w^TX^T)(y - Xw)$
Taking the derivatives of the loss w.r.t. the parameter vector $w$:
\begin{align}
\frac{d L(w)}{d w} & = \frac{d... | Let $z=(Xw-y)$, then the loss function can be expressed in terms of the Frobenius norm or better yet, the Frobenius product as
$$L=\|z\|^2_F = z:z$$
The differential of this function is simply
$$\eqalign{
dL &= 2\,z:dz \cr
&= 2\,z:X\,dw \cr
&= 2\,X^Tz:dw \cr
}$$
Since $dL=\frac{\partial L}{\partial w}:dw,\,$ the ... | {
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Showing bilinearity of the tensor product The tensor product for vector spaces $U,V$ is the initial object in the category of bilinear functionals on $U \times V$.
Spelt out, this means there is a vector space $U \otimes V$, and a bilinear fuctional $i: U \times V \rightarrow U \otimes V$
Such that for any bilinear fun... | Take the universal bilinear map $i$ and get your universal $i':U\otimes V\to U\otimes V$. This map is the identity since such a map is unique and the identity satisfies the condition. We have
$$i(au,v)=ai(u,v)=i'(au\otimes v)=ai'(u\otimes v)$$
so
$$au\otimes v=a(u\otimes v)$$
Similarly $u\otimes av=a(u\otimes v)$, s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1540147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Confused regarding the interpretation of A in the least squares formula A^T A = A^T b So I'm was watching gilbert strang's lecture to refresh my memory on least squares, and there's something that's confusing me (timestamp included).
In the 2D case he has $A=[1,1;1,2;1,3]$. In the lecture he talks about how A is a subs... | You have an $n\times 2$ matrix:
$$
A = \begin{bmatrix}
1 & a_1 \\ 1 & a_2 \\ 1 & a_3 \\ \vdots & \vdots \\ 1 & a_n
\end{bmatrix}
$$
The two columns span a $2$-dimensional subspace of an $n$-dimensional space. Your scatterplot is $\{(a_i,b_i) : i = 1,\ldots, n\}$; it is a set of $n$ points in a $2$-dimensional space. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1540251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Entire functions with finite $L^1$ norm must be identically $0$ Another question from Complex Variables: An Introduction by Berenstein and Gay.
Show that an entire function has finite $L^1$ norm on $\mathbb{C}$ iff $f\equiv0$. Does this also hold true for $L^2, L^{\infty}$?
So, the $L^{\infty}$ case I think just foll... | If I remember my complex analysis, the real and imaginary parts of $f(z)$, denoted $u(x,y)$ and $v(x,y)$ are harmonic on $\mathbb{R}^{2}$. If $\|f\|_{L^{1}(\mathbb{R}^{2})}$ is finite, then $\max\{\|u\|_{L^{1}},\|v\|_{L^{1}}\}<\infty$. By the mean value property, we have for any fixed $(x,y)\in\mathbb{R}^{2}$,
$$|u(x,y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1540383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Ring where $s^3-s=0$. Show $6=0$. Let $S$ be a ring in which every element $s$ satisfies $s^3-s=0$.
Show that $6=0$ in $S$.
I'm not sure if I should relate this to a concrete example involving integers, or attempt to prove it completely abstractly. I believe there is a proof by Jacobson that such a ring is commutative.... | Hint:$s^3-s=s(s-1)(s+1)=0, s=2$, $2(2-1)(2+1)=6=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1540498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $T$ injective or $T$ surjective, what is the composition $T^\ast T$? (where $T^\ast$ denotes adjoint of linear map $T$) Let $T:V \to W$ be a linear transformation between inner product spaces. Then $T^\ast: W \to V$ denotes the linear transformation with the property that for every $v \in V$ and $w \in W$, $$\langle... | start with
$$T^*\times T=T\times T^*=\det(T) I_n$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1540600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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Proof: Fibonacci Sequence (2 parts) Part a) Prove or Disprove: There are only finitely many even Fibonacci numbers.
I think I want to disprove this, as I know that every 3rd Fibonacci number is even, and thus there will be infinitely many even Fibonacci numbers. I am having trouble deciding how to technically prove thi... | For the first one, we have $F_0=0$ and $F_1 = 1$ with $F_n=F_{n-1}+F_{n-2}$
Keep in mind that in arithmetic we have that $E+E=E$, $E+O=O$ and $O+O=E$ with $O$ being some odd number and $E$ being an even number. If there are finitely many even numbers that means that at some $M$ we have that for all $n>M$ that $F_n$ is ... | {
"language": "en",
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"source": "stackexchange",
"question_score": "2",
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Proof limit by definition
Prove: $\lim_{x \to 2}x^2=4$
We need to find $\delta>0$ such that for all $\epsilon>0$ such that if $|x-2|<\delta$:
$$|x^2-4|=|(x-2)(x+2)|=|x+2|\cdot|x-2|\implies|x-2|<\frac{\epsilon}{|x+2|}$$
We will take $\delta=\frac{\epsilon}{|x+2|}$
is it valid?
| You can cheat a little bit, as long as it doesn't appear in your final solution: At $x = 2$, the derivative of $x^2$ is $4$. That means that $\delta$ must be, for small $\epsilon$, less than $\frac\epsilon4$.
For small $\delta$, $x$ is close to $2$, which makes $|x+2|$ close to $4$. That means you're on the right track... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1540825",
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Converting a sum of trig functions into a product Given,
$$\cos{\frac{x}{2}} +\sin{(3x)} + \sqrt{3}\left(\sin\frac{x}{2} + \cos{(3x)}\right)$$
How can we write this as a product?
Some things I have tried:
*
*Grouping like arguments with each other. Wolfram Alpha gives $$\cos{\frac{x}{2}} + \sqrt{3}\sin{\frac{x}{2}... | First of all
$$
A\cos\alpha+B\sin\alpha=\sqrt{A^2+B^2}\left(\frac{A}{\sqrt{A^2+B^2}}\cos\alpha+\frac{B}{\sqrt{A^2+B^2}}\sin\alpha\right)\\
=\sqrt{A^2+B^2}(\sin\beta\cos\alpha+\cos\beta\sin\alpha)=\sqrt{A^2+B^2}\sin(\beta+\alpha)
$$
you only have to find $\beta$ such that
$$
\left\{
\begin{align}
\sin\beta&=\frac{A}{\sq... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1540940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why I am getting wrong answer to this definite integral? $$\int_0^{\sqrt3} \sin^{-1}\frac{2x}{1+x^2}dx $$
Obviously the substitution must be $x=tany$
$$2\int_0^{\frac{\pi}{3}}y\sec^2y \ dy $$
Taking $u=y $, $du=dy;dv=sec^2y \ dy, v=\tan y $
$$2\Big(y\tan y+\ln(\cos y)\Big)^{\frac{\pi}{3}}_{0} $$
Hence $$2\frac{\pi}{\s... | $$I=\int_{0}^{\sqrt{3}} \sin^{-1} \frac{2x}{1+x^2} dx$$
Notice that $$\frac{d}{dx} \sin^{1} \frac{2x}{1+x^2}= \frac{2}{1+x^2}~~ \frac{|1-x^2|}{(1-x^2)}= \frac{2}{1+x^2}, ~if~~ x^2<1,~~\frac{-2}{1+x^2}~if~ x^2>1.$$
So let us integrate by parts taking 1 as the second function. Then
$$I=\left ( x \sin^{-1} \frac{2x}{1+x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1541284",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Common area of two squares Two squares are inscribed in a circle of diameter $\sqrt2$ units.prove that common region of squares is at least 2$(\sqrt2-1)$
This question is in my calculus book but seems to be algebraic.but frankly speaking i have no idea how to proceed .
| Here is one way.
I drew two squares inscribed in the circle centered at the origin with radius $\frac{\sqrt 2}2$. The equation of the line containing the upper side of the red square, which is parallel to the axes, is $y=\frac 12$. The blue square is at an angle of $\alpha$ to the red square. The overlap area is eight... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1541369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Elementary combinatorics... non-intuitive nPr/nCr? problem. Of course, when I say non-intuituve, it's completely subjective.
Suppose there are 12 woman and 10 men on the faculty. How many ways are there to pick a committee of 6?
For this portion of the problem, I simply did $\binom{22}{6} = 74613$ different ways to arr... | The easiest way would just be to do complimentary counting and do $\binom{22}{6} - \binom{20}{4}$. The other way would be to do cases.
Take three cases. One where Janet is on and Jane is off and vice versa and the case where none are on.
Case $1$: Janet on and Jane off: There are $20$ people left on the committee and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1541490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Closed form of the integral ${\large\int}_0^\infty e^{-x}\prod_{n=1}^\infty\left(1-e^{-24\!\;n\!\;x}\right)dx$ While doing some numerical experiments, I discovered a curious integral that appears to have a simple closed form:
$${\large\int}_0^\infty e^{-x}\prod_{n=1}^\infty\left(1-e^{-24\!\;n\!\;x}\right)dx\stackrel{\c... | To be somewhat explicit. One may perform the change of variable, $q=e^{-x}$, $dq=-e^{-x}dx$, giving
$$
{\large\int}_0^\infty e^{-x}\prod_{n=1}^\infty\left(1-e^{-24\!\;n\!\;x}\right)dx={\large\int}_0^1 \prod_{n=1}^\infty\left(1-q^{24n}\right)dq\tag1
$$ then use the identity (the Euler pentagonal number theorem)
$$
\prod... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1541601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Outer measure allowing only finite covers Let $\lambda$ be the usual length function of intervals of $\mathbb R$ and set
$\overline \lambda(A) = \inf \{\sum^m_{n=1} \lambda(I_n): A_n \text{ interval and } A \subset \bigcup^m_{n=1} I_n\}$.
Consider the set $A = \mathbb Q \cap [0,1]$. The usual definition of outer measur... | HINT: Use induction on the number of open intervals in your cover . . .
Specifically, prove the following statement by induction on $n$:
If $\mathcal{C}$ is a cover of $\mathbb{Q}\cap (a, b)$ by $n$ intervals, then the sum of the diameters of the elements of $\mathcal{C}$ is $>b-a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1541689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What are your favorite relations between e and pi? This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for mathematics due to some personal stuff in his life. (I will NOT discuss his p... | I honestly just like the fact that $e + \pi$ might be rational. This is the most embarrassing unsolved problem in mathematics in my opinion. It's clearly transcendental and we have no idea how to prove that it's even irrational.
They're so unrelated additively that we can't prove anything about how unrelated they are. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1541939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
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Clarification wanted: Let $T$ be the set of all infinite sequences of $0$'s and $1$'s with finitely many $1$'s. Prove that $T$ is denumerable. I think I'm misunderstanding the following proposition.
Let $T$ be the set of all infinite sequence of $0$'s and $1$'s with only finitely many $1$'s. Prove that $T$ is denumera... | Use the fact that "a countable union of countable sets is countable," several times.
*
*The set of binary strings with finitely many 1s is the union of binary strings with 1 1, 2 1s, 3 1s, 4s, etc.
*The set of binary strings with $n$ 1s is the union of binary strings with $n$ 1s where the greatest 1 is in the $k$ p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1542018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Placing Books on Shelves Permutation Question Question: There are five distinct A, three distinct B
books, and two distinct C books. In how many ways can these
books be arranged on a shelf if one of the B books is on the
left of all A books, and another B book is on
the right of all the A books?
This is what I've done:... | *
*Choose an order for the A books . . . . . $5!$ ways
*Choose the leftmost B book . . . . . $3$ ways
*Choose the rightmost B book . . . . . $2$ ways
*At this stage we have B A A A A A B. Choose one of the spaces to take the last B book . . . . . $6$ ways.
*We have now a row of $8$ books; choose a place for the f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1542163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Continuity in Probability
Could someone give me a simple explanation of property 4?
Context: Currently learning a bit of financial mathematics on my own to see if it is a field I really want to get into. The first 3 properties make it a stochastic process but the 4th condition I am unfamiliar with. The fourth property... | I interpret it the way that Henry does in the comments on Graham Kemp's answer: Heuristically, any change in infinitesimal time is almost surely infinitesimal (i.e., is infinitesimal with probability one).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1542258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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What is $\frac{x^{10} + x^8 + x^2 + 1}{x^{10} + x^6 + x^4 + 1}$ given $x^2 + x - 1 = 0$?
Given that $x^2 + x - 1 = 0$, what is $$V \equiv \frac{x^{10} + x^8 + x^2 + 1}{x^{10} + x^6 + x^4 + 1} = \; ?$$
I have reduced $V$ to $\dfrac{x^8 + 1}{(x^4 + 1) (x^4 - x^2 + 1)}$, if you would like to know.
| HINT: Note that $$\color{Green}{x^2=-x+1}.$$
By multiplying the given expression by $x,$ we can obtained $$\color{Green}{x^3=2x-1}.$$ Again by multiplying $x$ we have $$\color{Green}{x^4=-3x+2}$$ and so on..
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1542378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 3
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Two riders and their distance Two riders have distance between them $118$ km and they are moving towards each other to meet . B starts an hour later by A. A travels $7$km in hour while B travels $16$km every three hours. How many kilometers(km) will A have already traveled, once they meet each other?
Could anyone hel... | Let us assume the time taken by them to meet for A=$x$ and for B=$y$. So now we know speed.time=distance so now we can generate 2 equations $7x+\frac{16y}{3}=118$...(1) and B starts 1 hour late $x-y=1$ ..(2) solving them we get $x=10,y=9$ . Hope its clear.so distance travelled by $A=7.t=7.10=70km$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1542655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to show that $\int_M(\Delta f-|\nabla f|^2)(2\Delta f-|\nabla f|^2)e^{-f} dV=\int_M-\nabla_if\nabla_i(2\Delta f-|\nabla f|^2)e^{-f}dV$ $M$ is a compact Riemannian manifold. $f$ is function on $M$
How to show that $\int_M(\Delta f-|\nabla f|^2)(2\Delta f-|\nabla f|^2)e^{-f} dV=\int_M-\nabla_if\nabla_i(2\Delta f-|\na... | This is integration by part, (or divergence theorem):
Write $H = 2\Delta f-|\nabla f|^2$, then
$$\begin{split}
\int_M \Delta f H e^{-f} dV & = -\int_M \nabla f\cdot \nabla(He^{-f} )dV \\
&= -\int_M \nabla f\cdot (e^{-f}\nabla H+ H \nabla (e^{-f}) )dV \\
&= -\int_M \nabla f \cdot \nabla H e^{-f} dV - \int_M \nabla f \cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1542784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Convergence in $L^p$ spaces Let $f_{n} \subseteq L^{p}(X, \mu)$, $1 <
p < \infty$, which converge almost everywhere to a function $f$ in $L^{p}(X, \mu)$ and suppose that there is a constant $M$ such that $||f_{n}||_{\infty} \leq M$ for all $n$. Then for each function $g \in L^{q}(X, \mu)$ (where $q^{-1} + p^{-1} = 1$)... | If $\mu(X)=\infty$, the result is not true in general. Let $X=[1,\infty)$ with $\mu$ Lebesgue measure and let $f_n$ be the characteristic function of the interval $[n,2\,n]$. Then $f_n\in L^p$, $\|f_n\|_\infty=1$ for all $n$ and $f_n(x)$ converges point wise to $f(x)=0$ for all $x\in X$. Let $g(x)=1/x$. Then $g\in L^q$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to find the unique sums in the values 1,2,4,8,16, 32 I apologize but I'm not sure what you would even call this problem. I have some data that provide a numeric code for race as follows:
hispanic(1) + american_indian_or_alaska_native(2) + asian(4) + black_or_african_american(8) + native_hawaiian_or_other_pacific_i... | Open MS Excel. Type $DEC2BIN(5)$, for example, if $5$ is the number of the person. Each $1$ or $0$ tells you whether the person belongs to the race or not.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1542949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 4
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$I+B$ is invertible if $B^{k} = 0$ If $B$ is nilpotent and $B^{k} = 0$ (and B is square), how should I go around proving that $I + B $ is invertible? I tried searching for a formula - $I = (I + B^{k}) = (I + B)(???)$
But I didn't get anywhere :(
| Essentially if $B$ is idempotent of order $n$ you can find a basis where
$$B=\left(\begin{array}{ccccc}
0 & b_{12} & 0 & \cdots & 0\\
0 & 0 & b_{23} & & 0\\
0 & 0 & 0 & & 0\\
\vdots & & & \ddots & b_{n-1n}\\
0 & 0 & 0 & \cdots & 0
\end{array}\right)$$
In this form you can easly see what's happening when you calcul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1543041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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Limit with trigonometric function
Find $$\lim_{x \to \pi/4}\frac{1-\tan(x)}{\cos(2x)}$$
without L'Hôpital.
$$\lim_{x \to \pi/4}\frac{1-\tan(x)}{\cos(2x)}=\lim_{x \to \pi/4}\frac{\cos^2(x)+\sin^2(x)-\frac{\sin(x)}{\cos(x)}}{\cos^2(x)-\sin^2(x)}$$
How can I continue?
| $\lim\limits_{x \to \frac{\pi}{4}}\frac{(1-tanx)}{cos2x}=
\lim\limits_{x \to \frac{\pi}{4}}\frac{(cosx-sinx)}{cosx(cos^2x-sin^2x)}=
\lim\limits_{x \to \frac{\pi}{4}}\frac{(cosx-sinx)}{cosx(cosx-sinx)(cosx+sinx)}=
\lim\limits_{x \to \frac{\pi}{4}}\frac{1}{cosx(cosx+sinx)}=\frac{1}{\frac{1}{\sqrt{2}}(\frac{1}{\sqrt{2}}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1543143",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Proof that $\sqrt x$ is absolutely continuous.
I want to prove that $f(x)=\sqrt x$ is absolutely continuous.
So I must show that for every $\epsilon>0$,there is a $\delta>0$ that if $\{[a_k,b_k]\}_1^n$ is a disjoint collection of intervals that $\sum_{k=1}^n (b_k-a_k)<\delta$, then $\sum_{k=1}^n \left(\sqrt{b_k}-\sqr... | Hint: You can prove that by the following steps:
*
*$\sqrt{x}$ is continuous on $[0,1].$
*for any $\epsilon \in [0,1]$, $\sqrt{x}$ is ac on $[\epsilon,1].$
*$\sqrt{x}$ is ac on $[0,1]$ provided it is increasing.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1543221",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
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"answer_id": 3
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Maximization problem on an ellipsoid for three variables,
$$\max f(x,y,z)= xyz \\ \text{s.t.} \ \ (\frac{x}{a})^2+(\frac{y}{b})^2+(\frac{z}{c})^2=1$$
where $a,b,c$ are constant
how to solve the maximization optimization problem?
thank you for helpin
| Hint: One way is symmetrization via change of variable $s=\frac{x}{a},t=\frac{y}{b},u=\frac{z}{c}$ and then optimize.
$\max f(s,t,u)=(abc)stu\,\,\text{subject to: } s^2+t^2+u^2=1.$
You can either use calculus or the fact that optimal point is of form $(\lambda,\lambda,\lambda)$. Thus $s=t=u=\frac{1}{\sqrt 3}$ so $(x... | {
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Alternative definition of Gamma function. Show that $ \lim_{n \to \infty} \frac{n! \; n^m}{m \times (m+1) \times \dots \times (m+n)} = (m-1)!$ Alternative definition of Gamma function on Wikipedia has it defined as a limit.
$$ \Gamma(t) = \lim_{n \to \infty} \frac{n! \; n^t}{t \times (t+1) \times \dots \times (t+n)}$$ ... | I just love little challenges...
$\Gamma(t)=\lim \limits_{n \to \infty}\frac{n!n^t}{t(t+1)(t+2)\ldots(t+n)}$
Note the denominator of the fraction is in itself a factorial of sorts.
$\lim \limits_{n \to \infty}\frac{n!n^t}{t(t+1)(t+2)\ldots(t+n)}=\lim \limits_{n \to \infty}\frac{n!n^t}{\frac{(t+n)!}{(t-1)!}}=\lim \limit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1543501",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 2
} |
Wilson's Theorem: (n-1)! is congruent to -1(mod n) implies that n is prime. I have researched Wilson's theorem several times over stack exchange. I would only like to prove one direction. This seems to be a good explanation: Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime
However, on their explanation, the auth... | This isn't true in general. For example, $2 | (3-1)! = 2$ but $2$ is not congruent to $1 \pmod 3$. What the author on the question you referenced said was that from the conditions in Wilson's Theorem, $k | (n-1)!$ and also $k$ is congruent to $1 \pmod n$, not that $k | (n-1)!$ implies $k$ is congruent to $1 \pmod n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1543710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Solving $a_n=3a_{n-1}-2a_{n-2}+3$ for $a_0=a_1=1$ I'm trying to solve the recurrence $a_n=3a_{n-1}-2a_{n-2}+3$ for $a_0=a_1=1$. First I solved for the homogeneous equation $a_n=3a_{n-1}-2a_{n-2}$ and got $\alpha 1^n+\beta 2^n=a_n^h$. Solving this gives $a_n^h =1$. The particular solution, as I understand, will be $a... | Hint: Look for a particular solution of the shape $a_n^\ast=Bn$. You should get $B=-3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1543804",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Roots of quadratic equation by completing the square or other method? I'm trying to find solution(s) to the following equation:
$x^2 - 5x + 3 = 0$
It seems like it can't be factored normally so I tried solving by completing the square:
$x^2-5x=-3$
$x^2-5x+6.25=-0.5$
$(x-2.5)^2 = -0.5$
That's where I get stuck since you... | $(x-2.5)^2+3-2.5^2=0$
$x=(2.5)+(3.25)^{1/2}$
$x=(2.5)-(3.25)^{1/2}$
Also, since you're asking for another method, try the quadratic formula.
$x = \frac{-b\pm \sqrt{b^{2} - 4 ac}}{2a}$
where $a, b, c$ are the coefficients of $ax^2+bx+c=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1543900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Finding a basis for the intersection of two vector subspaces.
Suppose:
$V_1$ is the subspace of $\mathbb{R}^3$ given by $V_1 =
\{(2t-s,t,t+s)|t,s\in\mathbb{R}\}$
and
$V_2$ is the subspace of $\mathbb{R}^3$ given by $V_2 =
\{(s,t,s)|t,s\in\mathbb{R}\}$.
How could a basis and dimension for $V_1 \cap V_2$ be determine... | Here is one way:
The idea is to write $V_k$ as $\ker u_k^T$ for some $u_k$,
then
$v \in V_1 \cap V_2$ iff $v \in \ker \begin{bmatrix} u_1^T \\ u_2^T \end{bmatrix}$. If $u_k$ are linearly independent, then if $u_3\neq 0$ is
orthogonal to $u_1,u_2$, we see that $V_1 \cap V_2 = \operatorname{sp} \{ u_3 \}$.
In the follo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1544001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can i change the order of quantifiers in this case? In this sentence:
"No hero is cowardly and some soldiers are cowards."
Assuming
h(x) = x is a hero
s(x) = x is a soldier
c(x) = x is a coward.
So the sentence is like this i think:
($\forall x\ (h(x) \longrightarrow \neg C(x)) \land (\exists y\ (s(y) \land C(y))$
I... | To clarify the other responses here,
$\forall x\ (h(x) \longrightarrow \neg C(x)) \land (\exists y\ (s(y) \land \neg C(y))$ is equivalent to $\forall x\ \exists y\ (\neg h(x) \lor \neg C(x)) \land ((s(y) \land \neg C(y))$, but not equivalent to $\exists x\ \forall y\ (\neg h(x) \lor \neg C(x)) \land ((s(y) \land \ne... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1544099",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Consecutive rolls of a die whose face count increases by $1$ with each roll. The game starts with a dice with (let's say) $1000$ faces. Every roll, the die magically gets another face. So, after $1$ roll it has $1001$ faces, after another $1002$, and so on. Now the question is:
What is the probability of rolling at le... | Compute the complement of the propability to not get a 1, i.e.
$$p(x)=1-\prod_{k=1}^x(1-\frac{1}{999+k})= \frac{x}{999+x}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1544237",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that for any real number c, it is possible to rearrange the terms of the series $\sum_{n=1}^\infty (-1)^{n+1} (1/n)$ Show that for any real number c, it is possible to rearrange the terms of the series $\sum_{n=1}^\infty (-1)^{n+1} (1/n)$ so that the sum is exactly c.
| Hints:
Suppose $c>0$. Add the positive terms of the series, one by one, i.e. do the addition $1+1/3+1/5+...$ and stop till this rises above $c$.
Then, start adding the negative terms i.e. $-1/2-1/4-1/6-...$ to the above so that the result just drops below $c$.
Repeat this step with the remaining positive and negative t... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that sequence $a_{n+1} = \frac{1}{1+a_n}$ is bounded? Consider sequence;
$$a_1 = 1$$
$$a_{n+1} = \frac{1}{1+a_n}$$
Prove that sequence is bounded.
It is a question from my test. I couldn't figure out how to solve it.
| If $a_n$ is positive then
$$a_{n+1} = \frac{1}{1 + a_n} > 0.$$
Also, if $a_n$ is positive, then
$$1 < 1 + a_n \implies 1 > \frac{1}{1 + a_n} \implies 1 > a_{n+1}.$$
Therefore, if $a_{n} > 0$ then
$$0 < a_{n + 1} < 1.$$
That is, if all the elements of the sequence are positive then the sequence is bounded. Since the fir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1544443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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The sum of the first $n$ squares $(1 + 4 + 9 + \cdots + n^2)$ is $\frac{n(n+1)(2n+1)}{6}$ Prove that the sum of the first $n$ squares $(1 + 4 + 9 + \cdots + n^2)$ is
$\frac{n(n+1)(2n+1)}{6}$.
Can someone pls help and provide a solution for this and if possible explain the question
| By mathematical induction, say $P(n):1+4+9+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$
Now $P(1)$ is true as L.H.S. $= 1$ and R.H.S. $= 1$
Say $P(k)$ is true for some $k \in \mathbb{N}$ and $k>1$.
Therefore $1+4+9+\cdots+k^2=\frac{k(k+1)(2k+1)}{6}$
Now for $P(k+1)$,
\begin{align}
& 1+4+9+\cdots+k^2+(k+1)^2 \\[10pt]
= {} & \fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1544526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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New angle formed after rotating pipe I am having a bit of an issue with a problem (home maintenance) and would need to figure out a new angle formed after rotating a pipe. I will try to be as descriptive as possible:
This diagram shows the top view and the side view, so the pipe is both going down AND to the side, for... | This is not an answer but is a little long for a comment.
I don't think there's quite enough information to answer the question. As I understand it, you've stated the apparent angle that a revolving pipe makes when viewed from one particular direction. You then ask for the apparent angle after rotation about an axis p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1544734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 1
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Distance between points - equation of a line I have worked on this particular example:
The distance between the point $M_1(3,2)$ and $A$ is $2\sqrt5$ and the distance between the point $M_2(-2, 2)$ and $B$ is $\sqrt5$. Come up with a equation for the line going through $A$ and $B$.
I have tried playing around with th... | Hint:
As noted in the other answer your intuition that there are infinitely many solutions is correct.
All these soution can be represented as a family of straight lines dependent on a parameter in this way:
1) Represent the points $P_1$ of the circle of center $O_1=(\alpha_1,\beta_1)$ and radius $r_1$ in parametric ... | {
"language": "en",
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Show a group of order 60 is simple. I am given that $G$ is a group of order $60$, with $20$ elements of order $3$, $24$ elements of order $5$ and $15$ elements of order $2$. I have to show that $G$ is isomorphic to $A_5$.
I think that the best way to go about this is to prove that $G$ is itself simple, rather than list... | (1) There are more than one Sylow-$2$ subgroups (order $4$). [Hint: see no. of elements of order $2$].
(2) Let $H_1,H_2$ be two (abelian) Sylow-$2$ subgroups. Suppose they intersect non-trivially. Then $|H_1\cap H_2|=2$.
*
*(2.1) Let $x\in H_1\cap H_2$ with $x\neq 1$. Show: $|C_G(x)|\geq 8$. [Hint $C_G(x)\supseteq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1544902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proof of $\sum \frac 1 {p_{2n}} = \infty$ I found the following
Theorem
Let $p_n$ denote the $n$-th prime number.
$S_1= \sum_{n \in \Bbb N} \frac 1 {p_{2n}} = \infty$ and $S_2=\sum_{n \in \Bbb N} \frac 1 {p_{2n+1}} =\infty$.
Proof
If one of $S_1,S_2$ converges, then so does the other, but then $S = \sum_{n=1}^\infty... | As $p_{2n+1}>p_{2n}$, we have $S_1\ge S_2$, so if $S_1$ converges then so does $S_2$.
For the other direction, note
$$S_2=\sum_{n=1}^\infty\frac{1}{p_{2n+1}}\ge\sum_{n=1}^\infty\frac{1}{p_{2n+2}}=\sum_{n=1}^\infty \frac{1}{p_{2(n+1)}}=\sum_{n=2}^\infty\frac{1}{p_{2n}}=S_1-\frac{1}{p_1}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1544990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Why isn't $GL_n(\mathbb{R})$ a ring? Definition : A ring R is a set R equipped with two binary operations $+$ and $\cdot$ on R satisfying the following conditions:
(1) R is an abelian group under $+$.
(2) $\cdot$ is associative and has an identity element.
(3a) $\cdot$ is left-distributive over $+$, that is, $a \cdo... | Yes, property (1) fails, because if $+$ denotes the usual addition of matrices, $GL_n(\mathbb{R})$ isn't even closed under $+$. For instance, if $I$ is the identity matrix, then $I\in GL_n(\mathbb{R})$ and $-I\in GL_n(\mathbb{R})$, but $I+(-I)=0\not\in GL_n(\mathbb{R})$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1545107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 6,
"answer_id": 1
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How can I find $\cos(\theta)$ with $\sin(\theta)$?
If $\sin^2x$ + $\sin^22x$ + $\sin^23x$ = 1, what does $\cos^2x$ + $\cos^22x$ + $\cos^23x$ equal?
My attempted (and incorrect) solution:
*
*$\sin^2x$ + $\sin^22x$ + $\sin^23x$ = $\sin^26x$ = 1
*$\sin^2x = 1/6$
*$\sin x = 1/\sqrt{6}$
*$\sin x =$ opposite/hypoten... | $\sin^2x+\sin^22x+\sin^23x=1\implies(1-cos^2x)+ (1-\cos^22x) + (1-cos^23x)=1$
Rearranging terms we get $\cos^2x+\cos^22x+\cos^23x=2$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove diophantine equation $S^2+R^2+(r_1-r_2)^2 = 2R(r_1+r_2)$ has at most one solution Given this diophantine equation:
$$S^2+R^2+(r_1-r_2)^2 = 2R(r_1+r_2)$$
$S,r_1,r_2$ are variables. $R$ is a given constant. all values are positive integers.
How do I prove that there's at most one solution, not counting solutions wh... | For the equation.
$$S^2+R^2+(x-y)^2=2R(x+y)$$
You can set some numbers infinitely different way.
$$R=(a-b)^2+(c-d)^2$$
Then decisions can be recorded.
$$S=2(cb-ad)$$
$$x=b^2+d^2$$
$$y=a^2+c^2$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1545282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Decreasing by $\sqrt{n}$ every time We start with a number $n>1$. Every time, when we have a number $t$ left, we replace it by $t-\sqrt{t}$. How many times (asymptotically, in terms of $n$) do we need to do this until our number gets below $1$?
Clearly we will need more than $\sqrt{n}$ times, because we need exactly $\... | To attack this problem, let me first give a good heuristic by passing the problem to a continuous analog which is easier to solve. We will then modify the heuristic to give formal proof of our asymptotic. Firstly, consider what happens if we consider a sequence $a_n$ defined as
$$a_{n+1}=a_n-\sqrt{a_n}$$
starting with ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the probability that he counts head more that tails when tosses a coin 6 times Suppose kitty tosses a coin 6 times. What is the probability that she counts more "heads" than tails?
This is a question from the test and I got only partial marks so I am wondering what is the solution to this.
What I did was I assu... | Add up the following:
*
*The probability of getting $4$ heads and $2$ tails is $\dfrac{\binom64}{2^6}=\dfrac{15}{64}$
*The probability of getting $5$ heads and $1$ tails is $\dfrac{\binom65}{2^6}=\dfrac{6}{64}$
*The probability of getting $6$ heads and $0$ tails is $\dfrac{\binom66}{2^6}=\dfrac{1}{64}$
Hence the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1545488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Need help with this problem on the applications of the Hahn-Banach Theorem for normed spaces. If $p$ is a defined on a vector space $X$ and satisfies the properties
$p(x+y) \le p(x) + p(y)$ and $p(\alpha x) = |\alpha|p(x)$,
how do I show that for any $x_0 \in X$ there exists some linear functional $\bar f$ such that $\... | Start with the one dimensional subspace $W$ spanned by $x_0$ and define $f:W\to k$ by
$$
f(\alpha x_0) = \alpha p(x_0)
$$
This is a well-defined linear functional that satisfies $|f(x)| \leq p(x)$ for all $x\in W$. Now simply apply Hahn-Banach to get $\overline{f}$ defined on the whole space.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1545629",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find the characteristic polynomial of $(M^{-1})^3$ Given that M is a square matrix with characteristic polynomial
$p_{m}(x) = -x^3 +6x^2+9x-14$
Find the characteristic polynomial of $(M^{-1})^3$
My attempt:
x of $(M^{-1})^3$ is $1^3$, $(-2)^3$ , $7^3$ = $1$ , $-8$ , $343$
$p_{(m^-1)^3} = (x-1)(x+8)(x-343)$
or $-(x-1)(x... | Note that $-x^3+6x^2+9x-14=-(x-1)(x+2)(x-7)$, so we may assume that $M$ is the diagonal matrix with entries $1,-2,7$ on the diagonal. Then it's easy to see that the characteristic polynomial of $M^{-3}$ is given by $(x-1)(x+(1/2)^3)(x-(1/7)^3)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1545732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Value of sine of complex numbers I stumbled upon a problem with evaluating the sine function for complex arguments.
I know that in general I can use
$$
\sin(ix)=\frac{1}{2i}(\exp(-x)-\exp(x))=i\sinh(x).
$$
But I could also write the sine function as the imaginary part of the exponential function as
$$\sin(ix)=\text{I... | First, note that
$$\mbox{for }y\in\mathbb{R}\ \mbox{we have}\ \sin{y}=\mathrm{Im}\{e^{iy}\}.$$
Stating $\mathrm{Im}\left\{\mathrm{exp}(-x)\right\}=0$ means you assume $x\in\mathbb{R}$. However, then $ix$ is complex, such that you cannot apply the above rule.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1545882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Cauchy sequences the space of binary sequences with the metric $\sum 2^{-k}|x_k-y_k| $ Let's consider the following metric space $(X,d)$, where:
$X = \{ \ x = (x^1,x^2,x^3,\ldots,x^k,\ldots)\ \mid \ x^j \in \{0,1\}\ \forall j \geq 1\ \}$
$d(x,y) = \sum\limits_{k=1}^{\infty} \frac{1}{2^{k}} | x^{k} - y^{k} |$
This is th... | KoliG gave an example of a nonconstant Cauchy sequence: let $x_n$ be the sequence with $n$th entry $1$ and other entries $0$. Since $d(x_n,x_m)=2^{-n}+2^{-m}$ for $n\ne m$, this is a Cauchy sequence. It converges to the zero sequence.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1545998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A set contains $0$, $1$, and all averages of any subset of its element, prove it contains all rational number between 0 and 1 Assume we have a set $S$ $\subseteq \mathbb{R}$, we know $0,1 \in S$. Also, $S$ has the property that if $A \subseteq S$ is finite, the average of elements in $A$ is in $S$. We want to prove tha... | Here is a more explicit construction than given in other answers.
As other answers, first notice that it’s quite easy to get all dyadic numbers in $[0,1]$, i.e. numbers of the form $\frac{n}{2^a}$, where $0 \leq n \leq 2^a$.
Now suppose you want to get $\frac{p}{q}$. If you can find $q$ distinct dyadic numbers $x_1$, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1546051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
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"answer_id": 3
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Optimization: Finding Volume of a cubic rectangle.
Question: According to postal regulations, a carton is classified as "oversized" if the sum of its height and girth (the perimeter of its base) exceeds 118 in. Find the dimensions of a carton with square base that is not oversized and has maximum volume.
Attempt: Sin... | The volume of such a container is:
$$V= s^2h$$
Where V is volume, s is one side of the square base, and h is the height. Obviously the maximum volume will use the maximum side lengths, so we have:
$$4s+h = 118$$
$$h = 118-4s$$
We want the volume equation to be in terms of just one variable, so we plug in the new valu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1546106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to eliminate $\theta$? While doing a sum I was stuck in a particular step:
$$r_1= \frac{4a \cos \theta }{\sin^2 \theta}$$ and $$r_2=\frac{4a \sin \theta }{\cos^2 \theta}$$
How to eliminate $\theta$ ?
| $$\begin{cases}
\displaystyle
r_1=\frac{4a\cos\theta}{\sin^2\theta}\\
\displaystyle
r_2=\frac{4a\sin\theta}{\cos^2\theta}\\
\end{cases}$$
Try to find $\sin\theta$ and $\cos\theta$ as following:
$$r_1\sin^2\theta=4a\cos\theta\tag{1}$$
$$r_2\cos^2\theta=4a\sin\theta\tag{2}$$
To equation $(1)$ plug computed $\sin\theta$ f... | {
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"url": "https://math.stackexchange.com/questions/1546217",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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If the columns of an $n \times n$ matrix are linearly independent, then the columns span $\mathbb{R}^{n}$ My textbook says that its true but I can't find a proof of this on the internet.
"If $A$ is an $n \times n$ matrix with linearly independent columns, then the columns of $A$ span $\mathbb{R}^{n}$."
| The dimension of a finite dimensional vector space is the maximal number of linearly independent vectors, and it is also the minimal number of a system of generators. A maximal system of linearly independent vectors is a basis. A minimal system of generators is a basis.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1546317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
How to prove an inequality using the Mean value theorem I've been trying to prove that $\frac{b-a}{1+b}<\ln(\frac{1+b}{1+a})<\frac{b-a}{1+a}$ using the Mean value theorem. What I've tried is setting $f(x)=\ln x$ and using the Mean value theorem on the interval $[1,\frac{1+b}{1+a}]$. I managed to prove that $\ln(\frac{... | You apply the mean value theorem to the function $f : x \mapsto \ln (1+x)$ on the interval $[a,b]$. Note that $f$ fulfills the hypothesis of the theorem : being continuous on $[a,b]$ and differentiable on $]a,b[$.
On this interval the function $f'$ is bounded below by $\frac{1}{1+b}$ and above by $\frac{1}{1+a}$, so t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1546431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Zero-Diagonal Matrix and Positive Definitness? Can a $n \times n$ symmetric matrix $A$ with diagonal entries that are all equal to zero, be positive definite (or negative definite)?
Thanks in advance!
| The answer is negative and actually even more is true: the matrix cannot be positive definite if there is at least one diagonal element which is equal to $0$.
By definition, an $n\times n$ symmetric real matrix $A$ is positive definite if
$$
x^TAx>0
$$
for all non-zero $x\in\mathbb R$.
Suppose that $a_{ii}=0$ for some ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1546515",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding the limit of $\frac {\sin(2x)} {8x}$ I am taking an online course in Calculus from Ohio State and am just being introduced to the concept of limits. One of the exercises given to me is to find the limit of the following
$$\lim_{x \rightarrow 0}\frac {\sin({2x})}{8x} $$
Using what I have so far been taught, I de... | Here's a case where you use L'Hôpital's rule.
Given two functions $f(x)$ and $g(x)$, if
$$\lim_{x\to c}\frac{f(x)}{g(x)}$$
is indeterminate (for all intents and purposes, undefined in this case), then
$$\lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f'(x)}{g'(x)}$$
and the same for higher derivatives.
Therefore, in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1546696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Show that $E[\sum_{i=1}^{N}X_i] =E(N)E(X_1)$
Let $(X_n)$ be a sequence of random variables that are independent and identically distributed, with $EX_n < \infty$ and $EX_n^2 < \infty$. Let N be a random variable with range $R \subseteq \Bbb{N}$, independent of $(X_n)$, with $EN < \infty$ and $EN^2 < \infty$. Show that... | Hint:
$$E \left [ \sum_{i=1}^N X_i \right ] = \sum_{n=1}^\infty E \left [ \left. \sum_{i=1}^N X_i \right | N=n \right ] P(N=n) \\
= \sum_{n=1}^\infty \sum_{i=1}^n E[X_i|N=n] P(N=n).$$
Can you compute the inner expectation?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1546834",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Property of function $\varphi(x)=|x|$ on $\mathbb{R}$
Define $\varphi(x)=|x|$ on $[-1,1]$ and extend the definition of $\varphi(x)$ to all real $x$ by requiring that $\varphi(x+2)=\varphi(x).$ How do you prove that for any $s,t$
$$
|\varphi(s)-\varphi(t)|\leqslant |s-t|?
$$
I was going to do the following: For a... | Note that $\phi(x) = \min_{k \in \mathbb{Z}} |x-2k|$.
We have $|x-2k| \le |y-2k| + |x-y|$. Hence
$\phi(x) \le |y-2k| + |x-y|$, and since this holds for all $k$ we have
$\phi(x) \le \phi(y) + |x-y|$. Repeating this with $x,y$ interchanged
gives the desired result.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1546979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Converting Polar Equation to Cartesian Equation problem So I have
1. $$\frac{r}{3\tan \theta} = \sin \theta$$
2. $$r=3\cos \theta$$
What would be the Cartesian equation???
| First note that
$$\left\{ \matrix{
x = r\cos \theta \hfill \cr
y = r\sin \theta \hfill \cr} \right.$$
the general approach will be to solve for $r$ and $\theta$ and replace in your polar equation. However, in most times there are some shortcuts. See the following for the second one
$$\eqalign{
& r = 3\cos \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1547097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Dominated convergence theorem and uniformly convergence I try to solve the following task:
Let $(\Omega,\mathfrak{A},\mu)$ be a measurable space and $\mu(\Omega)<\infty$. Let $(f_n)_{n\geq1}$ be a sequence of integrable measurable functions $f_n:\Omega \rightarrow [-\infty,\infty]$ converging uniformly on $\Omega$ to a... | Fix an $\varepsilon > 0$, by uniform convergence, we know that there exists $N \in \mathbb{N}$ such that for $n \geq N$,
\begin{equation*}
|f_n| = |f_n - f + f| \leq |f_n - f| + |f| < |f| + \varepsilon
\end{equation*}
Then define the function $g : \Omega \to \mathbb{R}$ by $g(\omega) = |f(\omega)| + \varepsilon$. Then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1547218",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
How can I show that the composition of two coverings is also a covering? I'm trying to prove the following:
Let $\varpi ' : X'' \to X'$ and $\varpi : X' \to X$ be two coverings
and let $X$ be a locally simply connected space. Prove that $\varpi \circ \varpi ' : X'' \to X$ is also a covering.
I am completely stuck a... | Firts assume that $X$ is simply connected. Then $X'= X\times D$, $D$ a space with the discrete topology. Therefore $X"= X\times D'\times D$ for another discrete space $D'$, and the result follows. The general case follows from the hypothesis : let $x\in X$ and $U$ a simply connected neigbourhood of $X$, the previous a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1547297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Choosing numbers at random - expected value calculation From set $\{1,2,\ldots,49 \}$ we choose at random 6 numbers without replacing them. Let X denotes quantity of odd numbers chosen. Find $\mathbb{E}X$, how to find that? I have no idea whatsoever.
EDIT:: still looking for the sufficient explanation.
| Let $X_1, X_2, X_3, X_4, X_5, X_6$ be $0$ or $1$ according to whether the $i$th choice is even or odd. So $X=X_1+\cdots+X_6$. You can now use linearity of expectation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1547386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Fair die: Probability of rolling $2$ before rolling $3$ or $5$ Independent trials consisting of rolling a fair die are performed, what is the probability that $2$ appears before $3$ or $5?$
There are $36$ cases if we take two trials like $11 12 13 14 15 16
..21 22 23 24 25 26..31 32 33 34 35 36$ like this . But two ha... | Hint: The process is a renewal process since it starts over after each roll (or terminates). So, let $X_1$ denote the result of the first draw and $p$ the required probability. Then you have that $$p=P(X_1=2)+P(X_1=1,4 \text{ or }6)p+P(X_1=3 \text{ or }5)\cdot0$$
(can you see why?).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1547480",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Can there be a month with 6 Mondays? (or 6 of any other days?) This year, this month (November) we've 5 Mondays so I get 5 pay checks on 2nd, 9th, 15th, 22nd, 29th (which feels very good - I get paid one extra check :-).
I wonder if it's possible some years to have 6 Mondays? I haven't seen it but I wonder if there wil... | If the 1st is a Monday, being the most hopeful case, then so are the 8th, 15th, 22nd, 29th and 36th days of the month. The 36th is at least the 5th of the next month.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1547537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 4
} |
Bounded function - Proving $f(x)=0$ for all $x$ Let $f$ be a bounded function on $\mathbb R$
such that
$f(x) = \frac{1}{4}(f(\frac{x}{2})+f(\frac{x+1}{2}))$ for all $x$
Prove that $f(x)=0$ for all x
I let $|f(x)|≤M$ where $M$ is fixed then showed $\frac{M}{2^k}$ is a bound and this tends to $0$ as $k$ tends to $\infty$... | Another way:
If $M = \sup \{|f(x)|: x \in \mathbb R\} > 0$, there is some $x$ for which
$|f(x)| > M/2$. But
$$\max\left(\left|f\left(\frac{x}{2}\right)\right|, \left|f\left(\frac{x+1}{2}\right)\right|\right) \ge \frac{1}{2} \left( \left|f\left(\frac{x}{2}\right)\right|+ \left|f\left(\frac{x+1}{2}\right)\right|\right)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1547640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Show that there exists $a$, $b$ in $G$ such that $|a| = p$ and $|b| = q$ Show that there exists $a$, $b$ in $G$ such that $|a| = p$ and $|b| = q$, where $G$ is a non-abelian group with $|G| = pq$ where $2 < p, q$ are distinct primes.
Is there a way to do this without Sylow or Cauchy Theorem? Using the conjugacy class e... | Pick any element $g$ different from 1 and consider the cyclic subgroup $C$ it generates. In view of Lagrange's theorem this has order $p$ or $q,$ so the problem is solved for $a$ or for $b.$ Without loss of generality assume $|C|=p$ so we are done for $a.$
There are $q$ distinct left cosets of $C$ in $G.$
Pick any elem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1547832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Expected Value After n Trials Suppose there is a game where a player can press a button for a chance to win a cash prize of $50,000. This can be done as many times as he/she wishes. The catch is that the chance of winning is 14/15 each time. A loss, which happens 1/15 times at random, will void all winnings and remove ... | Let $x$ be the number of times the player plans to press the button. The probability that the player wins is ${(\tfrac{14}{15})}^x$ and the expected amount won is $x\cdot\$50000\cdot {(\tfrac{14}{15})}^x \color{silver}{+ 0\cdot \big(1-{(\tfrac{14}{15})}^x\big)}$.
You want to maximise $x\cdot {(\tfrac{14}{15})}^x$
$\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1547950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Determine all of the monic irreducible polynomials in $\mathbb Z_3 [x]$ of degree $4.$
Determine all of the monic irreducible polynomials in $\mathbb Z_3 [x]$ of degree $4$. Prove that you have found them all and that the ones you found are irreducible.
I am looking for some sort of way to figure this out without ha... | I don’t see how to do it without any listing at all. I would do it this way, but I’d be using a fairly primitive symbolic-computation package to help me. First, I’d work over $k=\Bbb F_9$, and find an element $z\in S=\Bbb F_{81}\setminus\Bbb F_9$. Then I would do a listing of the elements of $S$, namely all $a+bz$ with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1548113",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How to solve the functional equation $f(x+a)=f(x)+a$ Are there any other solutions of the functional equation $f(x+a)=f(x)+a$ ($a=\mathrm{const}$, $a\in\mathbb{R}\setminus\left\{0\right\}$) apart from $f(x)=x+C$ ($C=\mathrm{const}$)?
Edit: $a$ is a fixed number here.
| There are continuous functions apart from $x+C$.
Given $a\not=0$ let $f(x)=x+\sin(\dfrac{2\pi x}{a})$.
Then $f(x+a)=x+a+\sin(\dfrac{2\pi (x+a)}{a})=x+a+\sin(\dfrac{2\pi x}{a}+2\pi)= x+\sin(\dfrac{2\pi x}{a}) +a=f(x)+a$.
As indicated in another answer you may get many other (discontinuous) functions by partitioning ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1548240",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
A continuous function $f$ such that $\frac{\partial f}{\partial x}$ does not exists but $ \frac{\partial^2 f}{\partial x\partial y}$ exists Does there exist a continuous function $f:\mathbb R^2\to \mathbb R$ such that $\displaystyle \frac{\partial f}{\partial x}$ does not exists but $\displaystyle \frac{\partial^2 f}{\... | The function $f: R^2 \to R$ where $f(x, y) = |x|+xy$ is an example
See that for the following function $$f_{yx} = 1$$ for every $(x, y) \ \epsilon \ R^2$
But $f_x$ doesn't exist at the following set of points $\{ (0, y) \ \epsilon \ R^2 \ | \ y \ \epsilon \ R \} $
Thus for the set $\{ (0, y) \ \epsilon \ R^2 \ | \ y ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1548314",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find change of basis matrix I'm asked to find the change of basis matrix from basis $\underline{e}$ to $\underline{f}$ given the following information:
The coordinate relationship is given by:
$$3y_1 = -x_1 + 4x_2 + x_3$$
$$3y_2 = 2x_1 + x_2 + x_3$$
$$3y_3 = 0 + -3x_2 + 0$$
The coordinates $x_i$ belongs to basis $\unde... | I will explain change of basis with a simpler example, perhaps it is easier to see how this inverting comes into being. This may be a bit easier to follow if you accept the philosophy that vectors exist in the space regardless of any bases or coordinate grids. When we choose a basis in the space, the vectors get an alg... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1548443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Solve exponenital integral equation $$\frac{1}{\sqrt{2\pi}\sigma_1 }\int_x^\infty\exp(-\frac {t_1^2-1}{2\sigma_1^2})dt_1 + \frac{1}{\sqrt{2\pi}\sigma_2 }\int_x^\infty\exp(-\frac {t_2^2-1}{2\sigma_2^2})dt_2 = a $$
$$\sigma_1 , \sigma_2 \gt 0$$
Is there a way to solve for $x$ !?
please help
| Let $X_i\sim\mathcal N(1,\sigma_i^2)$ be independent for $i=1,2$, then the expression above is
$$\mathbb P(X_1>x) + \mathbb P(X_2>x) = a. $$
After standardizing we find that
$$\Phi\left(\frac{x-1}{\sigma_1}\right) + \Phi\left(\frac{x-1}{\sigma_2}\right)=a, $$
where $\Phi$ is the distribution function of the normal dist... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1548565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Laplace $2$-D Heat Conduction Consider the following steady state problem
$$\Delta T = 0,\,\,\,\, (x,y) \in \Omega, \space \space 0 \leq x \leq 4 ,\space \space \space\space 0 \leq y \leq 2 $$
$$ T(0,y) = 300, \space \space T(4,y) = 600$$
$$ \frac{\partial T}{\partial y}(x,0) = 0, \space \space \frac{\partial T}{\parti... | Sort of guide
*
*Transform the equation such that it'll have homogeneous boundary conditions.
I suggest to use $W(x, y) = 100 + 50x$ : this is the simplest function that has $W(0, y) = 100$, $W(2, y) = 200$ and by the way $\frac{\partial W}{\partial y} \equiv 0$. What will happen to solutions of original equation if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1548667",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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