Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Compute $f'(0)$ and check if $f'$ is continuous or not
Given the function
$$f(x)=\begin{cases}x^{4/3}\sin(1/x)&\text{if $x\neq 0$}\\0&\text{if $x=0$}\end{cases}.$$
*
*Compute $f'(0)$;
*Is $f'$ continuous on $\mathbb{R}$.
I am unsure of how to solve this but will post what I have.
Computing the part 1... | $$f(x)=\begin{cases}x^{4/3}\sin(1/x)&\text{if $x\neq 0$}\\0&\text{if $x=0$}\end{cases}.$$
We have,
$$f'(0)=\lim_{h\to 0}\frac{f(h)-f(0)}{h}=\lim_{x\to 0}h^{1/3}\sin\left(\frac{1}{h}\right) = 0$$
Since $|\sin\left(\frac{1}{h}\right)|\le 1$. Then
$$f'(x)=\begin{cases}\frac43x^{1/3}\sin(1/x) - x^{-2/3}\cos(1/x)&\text{if... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2526370",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Prove that set $\{(x,y,z)\in\mathbb{R}^3: x^2+y^2 \leq z, x+y+z=1\}$ is compact Prove that set $S=\{(x,y,z)\in\mathbb{R}^3: x^2+y^2 \leq z, x+y+z=1\}$ is compact!
I'm not really sure how to go about this. I can prove the set is closed since it's an intersection of preimages (of closed sets) of continuous functions.
I s... | Note that $$
(1-z)^2=(x+y)^2\leqslant 2(x^2+y^2)\leqslant 2z,$$we know $z$ is bounded. Therefore, it follows from the condition $x^2+y^2\leqslant z$ that $x^2+y^2+z^2$ is also bounded.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2526478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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Does $\frac{\langle a_k,b_k\rangle}{\|a_k\|}$ converge, if $(a_k)_k$ and $(b_k)_k$ tend to $0$ and $b\neq 0$ respectively? If $(a_k)_k$ and $(b_k)_k$ both convergent sequences in $\mathbb{R}^2$ such that their limits are $0$ and $b\neq0$ respectively. Does the sequence.
$$\frac{\langle a_k,b_k \rangle}{\|a_k\|}$$ conve... | The answer is no.
Let $$a_n = \left((-1)^n\cdot\frac1n, \frac1n\right), \text{ for } n\in \mathbb{N}$$
and $b_n = (1, 1)$ for $n \in \mathbb{N}$, the constant sequence $(1,1)$.
We have:
$$\frac{\langle a_n, b_n\rangle}{\|a_n\|} = \frac{(-1)^n\cdot\frac1n + \frac1n}{\frac{\sqrt{2}}{n}} = \frac{(-1)^n + 1}{\sqrt{2}}$$
Th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Show recursion in closed form I've got following sequence formula:
$ a_{n}=2a_{n-1}-a_{n-2}+2^{n}+4$
where $ a_{0}=a_{1}=0$
I know what to do when I deal with sequence in form like this:
$ a_{n}=2a_{n-1}-a_{n-2}$
- when there's no other terms but previous terms of the sequence.
Can You tell me how to deal with this ... | $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
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Intuition behind $E(XY) = E(X) E(Y) $ for independent random variables $X,Y$ I have been wondering what's the intuition behind a well known result: $E(XY) = E(X) E(Y) $ for independent random variables $X,Y$
I found this post: here which kinda solves the problem.
But, the explanation given there seems to be not clear ... | It is hard to give precise answer since you are asking for intuition.
Suppose for a certain number b you will compute bX. What’s the expected value of this computation? Well, if the realization of the variable X was done independently of the choice of the number b, then your computation will produce on average b.EX. No... | {
"language": "en",
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Is my professor's explanation of direct sum accurate? This is coming from a graduate-level abstract algebra class, for reference. My professor says that given two groups $G,H$ we say $D$ is the direct sum of $G$ and $H$ and write $C = G \oplus H$ if $G$ and $H$ are disjoint except for zero and $C = G+H = \{g+h | g \in ... | For arbitrary modules $M, N$ over a ring $A$, there's a external direct sum module $M\boxplus N = \{(m, n):\, m\in M, n\in N\}$ with operation $(m, n) + (m', n') = (m + m, n + n')$ and $A$-action $a(n, m) = (an, am)$. The resulting module has $M\boxplus N = M\oplus N$ in your notation, embedding $M$ and $N$ in $M\boxpl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2526923",
"timestamp": "2023-03-29T00:00:00",
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How to figure out if there is an actual horizontal tangent without a graph There is this practice problem that asks to determine the points at which the graph of $y^4=y^2-x^2$ has a horizontal tangent.
So I did implicit differentiation to find that
$$\displaystyle\frac{dy}{dx} = \frac{-x}{2y^3-y}$$
To find the horizon... | If
$y^4=y^2-x^2
$,
then
diffing implicitly,
$4y^3y'
=2yy'-2x
$
or
$2y^3y'
=yy'-x
$.
If $y' = 0$,
then
$x = 0$.
Putting this in,
$y^4 = y^2$
so the possible values are
$y = 0, \pm 1$.
At $x=y=0$,
suppose $y' = c$.
For small $x$ and $y$,
$y \approx cx$
so
$c^4x^4 \approx c^2x^2-x^2$
or,
dividing by $x^2$,
$c^4 x^2 \appro... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2527021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $K = \mathbb{Q}( \sqrt{2}, i)$ be the field generated over $\mathbb{Q}$ by $\sqrt 2$ and $i$ [Delhi-University PhD Screening test]
Let $K = \mathbb{Q}( \sqrt{2}, i)$ be the field generated over $\mathbb{Q}$ by $\sqrt 2$ and $i$. Then the dimension of $\mathbb{Q}( \sqrt2, i)$, as a $\mathbb{Q}$-vector space is equa... | For a given Extension of fields $L/K$ i simply write $[L:K]$ for the degree of $L$ over $K$ and $[a:K]$ for the degree of $a$ over $K$, that is defined as the degree of the minimum polynomial of $a$ over $K$. As we know it holds $[K(a):K] = [a:K]$ for algebraic(over $K$) $a$. Furthermore we need $K(a,b) = K(a)(b) = K(b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2527106",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Direct sum of non-orientable bundles is orientable? Let $M$ be a smooth manifold, and let $E_1,E_2$ be two non-orientable vector bundles over $M$.
Is $E_1 \oplus E_2$ orientable?
I am sure there is an easy answer, but somehow my search didn't result with anything useful.
| As Qiaochu says, the answer is no, in general. However, there is more to be said here.
An orientation on a vector bundle is a global section of the associated orientation bundle. So, a vector bundle is orientable if and only if its associated orientation bundle (which is a double cover) is trivial.
Now, an observation:... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Evaluate $\sum^{n}_{k=1}\sum^{k}_{r=0}r\binom{n}{r}$
Evaluate the summation $$\sum^{n}_{k=1}\sum^{k}_{r=0}r\binom{n}{r}$$
$\bf{Attempt:}$ From $$\sum^{n}_{k=1}\sum^{k}_{r=0}r\binom{n}{r} = \sum^{n}_{k=1}\sum^{k}_{r=0}\left[r\cdot \frac{n}{r}\binom{n-1}{r-1}\right] = n\sum^{n}_{k=1}\sum^{k}_{r=0}\binom{n-1}{r-1}$$
So ... | We will leave out the $r=0$ term since it is $0$.
$$
\begin{align}
\sum_{k=1}^n\sum_{r=1}^kr\binom{n}{r}
&=\sum_{r=1}^n\sum_{k=r}^nr\binom{n}{r}\\
&=\sum_{r=1}^n(n-r+1)r\binom{n}{r}\\
&=\sum_{r=1}^n(n-r)r\binom{n}{r}+\sum_{r=1}^nr\binom{n}{r}\\
&=\sum_{r=1}^nn(n-1)\binom{n-2}{r-1}+\sum_{r=1}^nn\binom{n-1}{r-1}\\[6pt]
&... | {
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"question_score": "2",
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"answer_id": 3
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Show that $\sigma(\mathcal{F})$ coincides with the countable-cocountable $\sigma$-algebra. Let $S$ be a set and let $\mathcal{F} = \{\{s\}:s\in S\}$ be the collection consisting of all sets which contain one element of $S$.
Let $\mathcal{A} = \{A\subseteq S:A \text{ is countable or $A^c$ is countable}\}$ be the countab... | Suppose $\mathcal{S}$ is a $\sigma$-algebra and $\mathcal{F} \subseteq \mathcal{S}$.
We need to see that $\mathcal{A} \subseteq \mathcal{S}$, and we have minimality and so $\sigma(\mathcal{F}) = \mathcal{A}$.
So let $A$ be countable, then enumerate $A$ as $A = \{a_n: n \in \mathbb{N}\}$
Then for all $n$, $\{a_n\} \in ... | {
"language": "en",
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Proving a local minimum is a global minimum. Let $f(x,y)=xy+ \frac{50}{x}+\frac{20}{y}$, Find the global minimum / maximum of the function for $x>0,y>0$
Clearly the function has no global maximum since $f$ is not bounded.
I have found that the point $(5,2)$ is a local minimum of $f$. It seems pretty obvious that this p... | By AM-GM $$f(x,y)\geq3\sqrt[3]{xy\cdot\frac{50}{x}\cdot\frac{20}{y}}=30.$$
The equality occurs for $$xy=\frac{50}{x}=\frac{20}{y}=10,$$
id est, for $(x,y)=(5,2)$, which says that $30$ is a minimal value.
The maximum does not exist. Try $x\rightarrow0^+$.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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Let $K \subset F$ fields. Proof that $F^n$ and $F \,\otimes K^n$ are isomorphic as $F-$vector spaces Let $F$ and $K$ be fields such that $K \subset F$. We can consider the tensor product $F\, \otimes \, K^n$ as $F-$vector space with the operation:
$$ \lambda (a \otimes v) = (\alpha a \otimes v), \, \forall a \in F, \,... | Let $x_i$ be a bsais of $V=K^n$. Then the elements of $F\otimes V$ are of the form $$\sum \lambda_i\otimes x_i.$$ Mapping this to $(\lambda_1,\cdots ,\lambda_n)$ is an isomorphism.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2527797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Evaluate $ \sum_{m > n > 0} \frac{m^2 - n^2}{(m^2 + n^2)^2} $ I have been trying around different ranges of summations:
$$ \sum_{(m,n) \in \mathbb{Z}^2} \frac{m^2 - n^2}{(m^2 + n^2)^2} = 0$$
That's not any good. What about if we restrict to $m, n \in \mathbb{Z}$ as positive integers.
$$ \sum_{m > 0, n > 0} \frac{m^2... | Fix $m>0$ for a moment and consider the sum
$$
H(m):=\sum_{n=0}^{\lfloor m/2\rfloor}\frac{m^2-n^2}{(m^2+n^2)^2}.
$$
Here the numerator of the term, call it $x(n,m)$, is at least $3m^2/4$, and the denomimator is at most $4m^4$. Therefore
$$
x(n,m)\ge\frac{3m^2/4}{4m^4}=\frac{3}{16m^2}
$$
in this range. There are at leas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2527911",
"timestamp": "2023-03-29T00:00:00",
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Find $\lim\limits_{x\to0}\frac 1x(x^{-\sin x}-(\sin x)^{-x})$ The question is to evaluate this limit:$$\lim_{x\to0}\frac{\big(\frac{1}{x}\big)^{\sin x}-\big(\frac{1}{\sin x}\big)^x}{x}$$
I tried using l'Hospital's rule, taking the logarithm, doing some manipulations using known limits, but without success.
| I deleted my previous answer as there were (stupid) mistakes in it, but as it turns out l'Hopital's rule is not helpful, and actually there is a much easier answer. Write the expression as
$$\frac{e^{-(\ln x)(\sin x)}-e^{-x\ln \sin x}}{x}$$
$$=(\sin x)^{-x}\frac{e^{-(\ln x)(\sin x)+x\ln \sin x}-1}{x}$$
Now $(\sin x)^{x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2528080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 1
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Show that the Moore plane is not normal
Definition: A Hausdorff space is normal (or: $T_4$) if each pair of disjoint closed sets have disjoint neighborhoods.
Then, we have
Exercise 5, pg. 158, Dugundji's Topology: Let $X$ be the upper of the Euclidean plane $E^2$, bounded by the $x$-axis. Use the Euclidean topology ... | My preferred way (though the statement that these two sets are closed disjoint sets that cannot be separated is true) is to use a well-known cardinal number fact in normal spaces often called Jones' lemma:
Let $X$ be $T_4$. If $D$ is a dense subset of $X$ and $C$ is a closed and discrete (as a subspace) subset of $X$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2528435",
"timestamp": "2023-03-29T00:00:00",
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Prove $\det(1+tA)=1+t\cdot tr(A)+O(t^2)$ I need help proving
$\det(1+tA)=1+t\cdot \operatorname{tr}(A)+O(t^2)$
I'm not really sure where to start due to the $(1+tA)$, the $1$ is throwing me off.
| Assume $A$ is $M_{n\times n}(\mathbb{C}).$ Suppose that $A$ is diagonalizable, with eigenvalues $\lambda_1,\dotsc,\lambda_n.$ Then $1+tA$ is also diagonalizable, with eigenvalues $1+t\lambda_i.$ The determinant of a diagonalizable matrix is the product of its eigenvalues, so we have
$$
\det(1+tA)=(1+t\lambda_1)\cdot\do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2528609",
"timestamp": "2023-03-29T00:00:00",
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Solution to the heat equation using the method of separation method Solution to the heat equation using the method of separation method
$$u_t=u_{xx},0\le x\le 2\pi, t>0\\\\u(0,t)=0\\u_x(2 \pi, t)=0\\u(x,0)=x=f(x)$$
my attempt:
let us take $u(x,t)=X(x)T(t)$
then i got $X''-kX=0, T'-T=0$
now for $k=0 $ and $k=\lambda^2$... | For first order derivatives, in this case for $t$, the solution is of the form $e^{- \mu t}$ and can be used in the following. For the equation
$$u_{t} = u_{xx}$$ then let $u(x,t) = e^{- \mu t} \, F(x)$ to obtain $F'' + \mu F = 0$. This yields the form $$F(x) = A \, \cos(\mu x) + B \, \sin(\mu x).$$ By using the condi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2528746",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Find all values of : $(2)^{i+1}=?$
Find all values of :
$$(2)^{i+1}=?$$
My Try :
$$i+1=\large\sqrt2e^{\frac{i\pi}4}$$
$$(2)^{\large\sqrt2e^{\frac{i\pi}4}}=?$$
now what ?
| $$ 2^{i+1} = e^{ (i + 1) \ln 2} = e^{\ln 2} e^{i\ln 2} = 2(\cos (\ln 2) + i\sin (\ln 2)) $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2528850",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Is there a formula for coefficients of a given vector written in terms of the basis? Given that $\vec{u}_1,.., \vec{u}_N$ are a basis of $\mathbb{R}^N$, such that we can write any vector $\vec{v}\in\mathbb{R}^n$ as $$\vec{v} = \sum_{n=1}^Nc_n\vec{u}_n$$
Is there a formula to find the coefficients? In an old pdf I found... | That formula holds only if the $u_n$'s are orthogonal. In that case, the coefficients of $v$ are given by the scalar product of $v$ with the normalized basis. This is a basic theorem of Linear algebra. In the general case the formula is not true.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2529007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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General rule for limit of $a_n^{b_n}$? Let $a_n$ and $b_n$ denote two series with well defined limits $a, b \in \mathbb{R}$ for $n\longrightarrow \infty$.
Is it possible to say following?: $$\lim (a_n^{b_n}) = (\lim a_n)^{\lim b_n} = a^b$$
If not, can you give a counterexample?
Edit: Assume that $a$ positive.
| That in general does not provide the result.
Assume that $a = -2$ and $b=1/2$ then
$$\lim (a_n^{b_n}) = (\lim a_n)^{\lim b_n} = a^b =(-2)^{1/2} =\sqrt{-2}$$
Does not make any sense in $\Bbb R.$
But If $a>0$ then , there a certain rang from which $a_n >0$ and you can therefore write
$$\lim (a_n^{b_n}) = \lim \ex... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that there is no positive real number which is less than every positive real number. I think what would happen if there is less than every positive real numbers
How can I prove that?
| Let $r$ be a positive real. Consider the set $S=\{x\in\mathbb{R}~:~x<r\}$. Now this set $S$ is non-empty, as $0\in S$. So it has a lowest upper bound, say $r^*$. $r^*\leq r$ as $r\not<r$. If $r^*<r$, then $r^*<\frac{r+r^*}{2}<r$. Hence $\frac{r+r^*}{2}\in S$, which contradicts our assumption that $r^*$ is the lowest up... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Function is bounded from below if sum of partial derivatives is positive Let $f:\mathbf{R}^n \to \mathbf{R}$ be differentiable, $\sum_{i=1}^n y_i \frac{\partial f}{\partial x_i}(y)\geq 0$ for all $y=(y_1,...,y_n)\in \mathbf{R}^n$. How do I show that $f$ is bounded from below by $f(0)$?
| Hint:
The expression from your question is the derivative of f with respect to the radius squared. So if that is positive, then...
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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How to show the action of $SL_2(\mathbb R)$ on the complex upper half plane is transitive? I've seen other answers here and here but I still am not understanding how to show that this action is transitive. I want to show that for all $z \in \mathbb H$ there exists a matrix $g \in SL_2(\mathbb R)$ such that $gi=z$.
If I... | Here's a hint. You can write
$$ z= a+bi = \frac{b\cdot i + a}{0\cdot i +1} ,$$
which is the transformation corresponding to the matrix
$$\begin{bmatrix} b & a \\ 0 & 1 \end{bmatrix} $$
evaluated on the point $i$ in $\mathbb{H}$. This matrix may not be in $SL_2 \mathbb{R} $, but you can easily fix that.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Find all $C^{1}$ functions $f: (0,+\infty) \to (0, +\infty)$ such that $f(x)^{f'(x)}=x$, $f(1)=1$. As the question title says, I'm trying to find all $C^1$ functions $f:(0, +\infty) \to (0, +\infty)$ which satisfy $f(x)^{f'(x)} = x$, and $f(1)=1$.
I know that $f(x)=x$ is one solution. When I put everything into the exp... | It seems that there is another solution to the equation, but for $0<x<e$, its value decreases from $e$ to $0$, and for $x>e$ the value becomes complex and is no longer real. This means that the derivative does not exist at that point. Other than that, it is a valid solution. Notice that, for $0<x<1$ the $W_0(\_)$ branc... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Usefulness of extended domain of derivative function? Say I have $f(x)=\ln(x)$. We know its domain is $(0,\infty)$, but the domain of $f’(x)$ is $(-\infty,0) \cup (0,\infty)$. Though this is a relatively simple example, is there any application of the derivative extended beyond its function’s real domain? For example, ... | Alex is right that the derivative has the same domain as the function, but what you have discovered is that the derivative may be easier to extend analytically than the original function. In the example of $\log$, one can extend the derivative $1/x$ to all complex numbers, which gives one a hint as to the correct gener... | {
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"source": "stackexchange",
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"answer_id": 1
} |
when can we interchange integration and differentiation Let $f$ be a Riemann Integrable function over $\mathbb{R}^2$. When can we do this?
$$\frac{\partial}{\partial\theta}\int_{a}^{b}f(x,\theta)dx=\int_{a}^{b}\frac{\partial}{\partial\theta}f(x,\theta)dx$$
(Here, $a$ and $b$ are not a function of $\theta$.)
In the prob... | You may interchange integration and differentiation precisely when Leibniz says you may. In your notation, for Riemann integrals: when $f$ and $\frac{\partial f(x,t)}{\partial x}$ are continuous in $x$ and $t$ (both) in an open neighborhood of $\{x\} \times [a,b]$.
There is a similar statement for Lebesgue integrals.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2530213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
"answer_count": 1,
"answer_id": 0
} |
Show that the torque for x = b is $M_y -b\rho S$
The area R on the xy-plane corresponds to a thin metal plate with the
area S and a constant density $\rho$. $M_y$ is the plate's moment
corresponding to the y-axis.
a) Show that the moment corresponding to $x = b$ is $M_y - b\rho S$,
if the plate is right from $x ... | Saying that $M_y$ is the moment corresponding to the $y$-axis (i.e., the line $x = 0$ is saying (according to the definition linked in Sou's comment) that
$$
M_y = \iint_R x^2 \rho~dy~dx
$$
where here $\rho$ is the density, not the "distance to the axis" as it is in the linked wikipedia page.
The thing you're suppose... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2530335",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Find the set of values for $c\in \mathbb R$ that allows real solution for $\sqrt{x}=\sqrt{\sqrt{x}+c}$
Given $\{x,c\}\subset \mathbb R$, $\sqrt{x}=\sqrt{\sqrt{x}+c}~~~~ (1)$.
Find: set of values for $c$ such that $(1)$ has solution in $\mathbb R$.
Question from the Brazilian Math Olympiad 2004. No solution provided.... | Just a slightly different point of view, which may be useful in a different situation. Since it is easier to express $c$ in terms of $x$ rather than the other way around, you might make this argument:
Let $f:\mathbb{R}_{\geq0} \to \mathbb{R}$ be given by $f(x) = x - \sqrt{x}$. Then
$$\exists x\in \mathbb{R} (x\geq 0 \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2530475",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Numerically computing $(X^tX)^{-1}X^t y$ and $(X^tX)^{-1}z$ In some algorithm I need to compute $a=(X^tX)^{-1}X^ty$ and $b=(X^tX)^{-1} z$ in each step, where $X$ is a $n \times p$ non-square matrix ($n \geq p$, $p$ is increased in each step) and $y,z$ are some appropriate vectors, but I'd like to do so in an efficient ... | With $X=QR$ in the small variant where $R$ is square, you get $X^tX=R^tR$ so that $a=R^{-1}Q^ty$ and $b=R^{-1}(R^t)^{-1}z$ where you have only triangular systems to solve.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2530582",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Reference request for Minimal Surfaces. I need books or articles based on minimal surfaces. By minimal surface, I mean a surface with 0 mean curvature.
More specifically, I wish to explore the Plateau's Problem: There exists a minimal surface with a given boundary.
I would also like to see a proof of the fact that a s... | some nice texts are the following
*
*Geometric Measure Theory and Minimal Surfaces - Bombieri, E.
*Lectures on Geometric Measure Theory - Simon, L.
*Minimal Surfaces and Functions of Bounded Variation - Giusti, E.
*Sets of Finite Perimeter and Geometric Variational Problems - Maggi, F.
*Calculus of Variations a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2530707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Are all quadratics factorable into a product of two binomials? I'm learning algebra in school, and my teacher said that all quadratics are factorable into a product of two binomials. I then realized however that some quadratics would have imaginary roots, and therefore wouldn't be able to be put into factored form. Who... | It really depends on whether you want to have complex factors or not. If you can have complex factors, every expression can be. If not, then only if $b^2\ge4ac$ would they be factorable.
Take $x^2+1$, it can be factored into $(x-i)(x+i)$ but none of the factors are real.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2530935",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 4,
"answer_id": 1
} |
Change of variables and circular solutions Consider the equations:
$ \frac{dx}{dt} = y$ and $ \frac{dy}{dt} = -x $
By transforming variables, obtain $ \frac{dr^2}{dt}=0 $ and $ \frac{d \theta}{dt} = -1$
I know that if I say let $r^2 =x^2 +y^2 \cdot \frac{dr^2}{dt} = tx \frac{2x}{dt} + 2y \frac{dy}{dt} =2xy =2xy-2xy =... | Assuming you're using polar coordinates, a change of variables to the polar coordinates system corresponds to $x=r\cosθ,y=r\sin θ,r=x^2+y^2$ :
It is :
$$r^2 = x^2 + y^2 \Rightarrow 2rr' = 2xx' + 2yy' \Rightarrow rr' = xx' + yy' $$
Substituting $x',y'$ from your given system :
$$rr'= xy - yx = 0$$
To find the angle $θ$,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Is this function is one of the inner product in $\mathbb{R}^3$ If $\vec u = (u_1, u_2, u_3)$ and $\vec v = (v_1, v_2, v_3)$ is a vector of $\mathbb{R}^3$, then $f(u, v) =$
$2u_1v_1 + 3u_2v_2 – 2u_2v_2$ is one of the inner product in $\mathbb{R}^3$.
My answer is False, because if we simplify the $f(\vec u,\vec v)$, we c... | To be more clear, note that $$f((0,0,1),(0,0,1))=0$$
but $(0,0,1) \neq (0,0,0)$ violating the positive definiteness.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Confusion about differential of multivariable functions. I have been given a function $F: \mathbb{R²} \to \mathbb{R³}$
and another function $\beta: J \to \mathbb{R³}: t \mapsto F(a + tx)$
where $x = v_1(1,0) + v_2(0,1) = (v_1,v_2)$ and $a$ is fixed. we can assume that all given functions are differentiable on their dom... | $R(t)=(R_{1}(t),R_{2}(t)):=(a_{1}+tx_{1},a_{2}+tx_{2})$, and $\beta=F\circ R$, so $\beta'(0)=\dfrac{\partial F}{\partial x}(R(0))R_{1}'(0)+\dfrac{\partial F}{\partial y}(R(0))R_{2}'(0)$. Note that $R_{1}'(0)=x_{1}$ and $R_{2}'(0)=x_{2}$.
Here $\dfrac{\partial F}{\partial x}=D_{1}F$ and similar to $D_{2}F$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Limit of a trig function. (Without using L'Hopital) I'm having trouble figuring out what to do here, I'm supposed to find this limit:
$$\lim_{x\rightarrow0} \frac{x\cos(x)-\sin(x)}{x^3}$$
But I don't know where to start, any hint would be appreciated, thanks!
| Since$$x\cos(x)-\sin(x)=x\left(1-\frac{x^2}2+\cdots\right)-\left(x-\frac{x^3}6+\cdots\right)=-\frac{x^3}3+\cdots,$$your limit is equal to $-\dfrac13$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Group homomorphisms $S_3\to\mathbb{Z}/2\mathbb{Z}$ Knowing that $S_3=\{\text{id},\sigma,\sigma^2,\tau,\tau\circ\sigma,\tau\circ\sigma^2\}$ ($\tau=(1 2), \sigma=(123)$), why does a group homomorphism $f:S_3\to\mathbb{Z}/2\mathbb{Z}$ satisfy $f(a)=0$ for all $a\in S_3$ such that $a^2\neq e$?
| In $S_3$, for every element $a$ with $a^2\ne e$, we have $a^3=e$. If we map $a$ to $1$ instead of $0$, this creates a problem, because $1+1+1\ne 0$ in $\Bbb Z / 2\Bbb Z$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531523",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Gambler's ruin model In the gambler's ruin model, $X_n$ is a gambling player's fortune after the $n^{th}$ game, when making 1 dollar bets at each game.
Also, for fixed $0<p<1$, we can find random variables $\{Z_i\}$ which are i.i.d. with $P(Z_i=1)=p$ and $P(Z_i=-1)=1-p$. So, we can set $$X_n=a+Z_1+Z_2+...Z_n$$ with $X_... | To get an upper bound, note that if the gambler hasn't hit the boundary by time $n$, then he hasn't had a win or loss streak of length $c$. The latter event is contained in the event that none of the sequences $(Z_1, \ldots, Z_c), (Z_{c+1}, \ldots, Z_{2c}), \ldots, (Z_{(m-1)c+1}, \ldots, Z_{mc})$ are the identically $+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531623",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Does $\bigcup_{n=1}^\infty \left(-\infty, 1-\frac{1}{n}\right) = (-\infty, 1)$? I'm currently trying to think of an example of a proper subset of $\mathbb{R}$ that is not compact in the topological space $(\mathbb{R}, \tau)$, where $\tau = \{(-\infty, a): a\in \mathbb{R}\}\cup\{\emptyset, \mathbb{R}\}$.
I suspect that ... | We wish to show that for any $x \in (-\infty, 1)$, there exists a positive integer $n$ such that $x < 1 - 1/n$, or equivalently, $$n > 1/(1-x).$$ Recall the Archimedean property of the reals, which states that for any reals $a, b > 0$, there exists a positive integer $n$ such that $na > b$. So if we choose $$a = 1, \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Linear Transformation of Dependent set. Let $V$ and $W$ be vector spaces and let $T: V\to W$ be a linear transformation. Let $\{v_1, v_2,\ldots, v_p\}$ be a linearly dependent set of vectors in $V$. Show that $\{Tv_1, Tv_2,\ldots, Tv_p\}$ is also linearly dependent.
| If $\{v_1, v_2,\ldots, v_p\}$ is linearly dependent, there exists $a_1, \cdots, a_p \in \mathbb{R}$, such that
$$a_1v_1+\cdots+a_pv_p =0 $$
where $a_i \neq 0$, for at least one $i \in {1,\cdots,p}$. Now, if we apply $T$, follows that:
$$a_1v_1+\cdots+a_pv_p =0 \Rightarrow T(a_1v_1+\cdots+a_pv_p)=T(0)$$
But, $T$ is lin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
Epsilon-delta derivative proof of $x^n$ I'm currently trying to prove the power rule using the epsilon-delta definition of a derivative. I've already done it for the basic limit definition, but I thought it might be a helpful exercise to test my understanding by doing it this way. However, I'm struggling and would real... | Using
$$(x^n - x_0^n) = (x - x_0)\sum_{k=0}^{n-1} x^k x_0^{n-1-k}$$
One can write
$$
\frac{x^n - x_0^n}{x - x_0} - n x_0^{n-1} = \sum_{k=0}^{n-1} (x^k x_0^{n-1-k} - x_0^{n-1})
$$
Define $M = |x_0| + 1$ and suppose that $|x-x_0|\le 1$, then $|x|\le M$
and $|x_0|\le M$ and
$$
|x^k-x_0^k| \le |x - x_0|\sum_{p=0}^{k-1} |x|... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2531961",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Geometric/arithmetic sequences: $u_{n+1} = \frac12 u_{n} + 3$ I'm having trouble with writing this sequence as a function of $n$ because it's neither geometric nor arithmetic.
$$\begin{cases}
u_{n+1} = \frac12 u_{n} + 3\qquad \forall n \in \mathbb N\\
u_{0} = \frac13
\end{cases}$$
| Let $u_m=v_m+a_0+a_1m+\cdots$
$$6=2u_{n+1}-u_n=2v_{m+1}-v_m+a_0(2-1)+a_1(2(m+1)-m)+\cdots$$
Set $a_0=6,a_r=0\forall r>0$
to find $$v_{n+1}=\dfrac{v_n}2=\cdots=\dfrac{v_{n-p}}{2^{p+1}}$$
Now $v_0+6=u_0\iff v_0=?$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Proving $\sum\limits_{r=1}^n \cot \frac{r\pi}{n+1}=0$ using complex numbers Let $x_1,x_2,...,x_n$ be the roots of the equation $x^n+x^{n-1}+...+x+1=0$.
The question is to compute the expression $$\frac{1}{x_1-1} + \frac{1}{x_2-1}+...+\frac{1}{x_n-1}$$
Hence to prove that $$\sum_{r=1}^n \cot \frac{r\pi}{n+1}=0$$
I ... | Clearly, $x_k\ne1,1\le k\le n$
Let $y_k=\dfrac1{x_k-1}\ne0,1\le k\le n$
$\implies x_k=\dfrac{1+y_k}{y_k}$
Now $\displaystyle0=\sum_{r=0}^nx_k^r=\dfrac{x^{n+1}_k-1}{x_k-1}$
As $x_k-1$ is non-zero finite, $$x^{n+1}_k-1=0$$
$$\implies\left(\dfrac{1+y_k}{y_k}\right)^{n+1}=1$$
$$\binom{n+1}1y_k^n+\binom{n+1}2y_k^{n-1}+\cdot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532206",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Expand function in Legendre polynomials on the interval [-1,1] Expand the following function in Legendre polynomials on the interval [-1,1] :
$$f(x) = |x|$$
The Legendre polynomials $p_n (x)$ are defined by the formula :
$$p_n (x) = \frac {1}{2^n n!} \frac{d^n}{dx^n}(x^2-1)^2$$
for $n=0,1,2,3,...$
My attempt :
we have ... | We know that the Fourier-Legendre series is like
$$
f(x)=\sum_{n=0}^\infty C_n P_n(x)
$$
where
$$
C_n=\frac{2n+1}{2} \int_{-1}^{1}f(x)P_n(x)\,dx
$$
So now we are going to calculate the result of
$$
\frac{2n+1}{2} \int_{-1}^{1}|x|P_n(x)\,dx
$$
As $|x|$ is an even function, and the parity of $P_n(x)$ depends on the pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Is this map continuous? Let $(E,\langle\cdot\;,\;\cdot\rangle)$ be a complex Hilbert space. For $M\in\mathcal{L}(E)^+$ (i.e. $M^*=M$ and $\langle M x\; |\;x\rangle\geq0$, for all $x\in E$). We define
the semi-inner product $\langle x\;,\;y\rangle_M:=\langle Mx\;,\;y\rangle,\; \forall x,y\in E$.
Assume that $\forall ... | Let's write $\|x\|_M=\langle Mx,x\rangle^{1/2}$ for $x\in E$. Then for $x,y,x_0,y_0\in E$ we have
$$\|x\cdot y-x_0\cdot y_0\|_M\leq\|x\|_M\|y-y_0\|_M+\|y_0\|_M\|x-x_0\|_M.$$
Now if $0<\varepsilon<1$, $\|x-x_0\|_M<\varepsilon$ and $\|y-y_0\|_M<\varepsilon$, then $\|x\|_M<\|x_0\|_M+1$. Then we have
$$\|x\cdot y-x_0\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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I wante to find some $v>0$ verifying this property Let us consider a function $f: (0,+∞)→(0,+∞)$. Assuming that $f$ is continuous differentiable and strictly increasing for all $t>0$.
I want to find some $v>0$ verifying this property:
$$f(a)<f(v)≤c$$
where $a>0$ and $c>0$ are given real numbrers. I have no idea how t... | Let $a, c > 0$ such that $f(a) < c$ (otherwise it obviously doesn't hold). Then, because $f$ is continuous at $a$, there exists $\delta > 0$ such that $|v - a| < \delta \implies |f(v) - f(a)| \le c - f(a)$.
For any $v \in \langle a, a + \delta\rangle$ we have $f(a) < f(v)$ so:
$$0 < f(v) - f(a) = |f(v) - f(a)| \le c - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532555",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How many uncountable subsets of power set of integers are there? The question is to determine how many uncountable subsets of ${P(\mathbb Z)}$ are there.
I think that the answer is $2^c$.
Let $A=\{B\in P(P(\mathbb Z)):B \text{ is uncountable}\}$
$P(P(\mathbb Z))$ has $2^c$ elements, so cardinality of $A$ is at most $2... | Start with your definition (after renaming the bound variable) and add a second definition:
$$
A=\{X\in P(P(\mathbb Z)):X \text{ is uncountable}\} \\
B=\{X\in P(P(\mathbb Z)):X \text{ is countable}\}
$$
Consider $f(X) = \overline{X}$ (the complement of $X$ relative to $P(\mathbb Z)$) as a function $f: B \rightarrow A$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 6,
"answer_id": 1
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Let $T:V\rightarrow W$ a linear transformation, $i_w:W\rightarrow W^{**}$ and $i_v:V\rightarrow V^{**}$ canonical morphism of biduality. Let $T:V\rightarrow W$ a linear transformation, $i_w:W\rightarrow W^{**}$ and $i_v:V\rightarrow V^{**}$ canonical morphism of biduality.
Prove $i_w\circ T=T^{**}\circ i_v$
I' m very v... | Let's recall the definitions: if $T : V \to W$, then $T^{*} : W^* \to V^*$ is a linear map defined as $T^{*}(f) = f \circ T$ for all $f \in W^*$.
Then, $T^{**} : V^{**} \to W^{**}$ is defined as $T^{**}(g) = g\circ T$ for all $g \in V^{**}$.
Now, to prove $i_w\circ T=T^{**}\circ i_v$ let's first establish the domain an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532787",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Finding simpler formula I have to find simpler formula to this one:
$$\lnot(p \land q) \lor (\lnot p \land q)$$
I started by using De Morgan's law to get: $$(\lnot p \lor \lnot \lnot q) \lor (\lnot p \land q)$$ then used the Double negation law to get: $$(\lnot p \lor q) \lor (\lnot p \land q)$$
Probably I am missing s... | You have four different cases: $p$ true and $q$ true, $p$ true and $q$ false, $p$ false and $q$ true, $p$ and $q$ both false. Then
\begin{align}
& p \mbox{ true and }q \mbox{ true implies } \lnot p \mbox{ or } \lnot q \mbox{ false, } \lnot p \mbox{ and } q \mbox{ false } \\
& p \mbox{ true and }q \mbox{ false implies }... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Best way to compute the rank of $A$ Let
$$A=\begin{bmatrix}
1&2&3&4&5\\6&7&8&9&10\\11&12&13&14&15\\16&17&18&19&20\\21&22&23&24&25\end{bmatrix}.$$
Which would be the best way to compute its rank?
I first thought about computing the determinant but then it seemed better to find its echelon form which would give me the ... | You can notice that if $a=[1\ 2\ 3\ 4 \ 5]$ and $u=[1\ 1\ 1\ 1\ 1]$, then the matrix is
$$
\begin{bmatrix}
a \\
a + 5u \\
a + 10u \\
a + 15u \\
a + 20u
\end{bmatrix}
$$
and it is clear that the row space is generated by $a$ and $u$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2532993",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
For what values of x does the series $\sum_{n=1}^{\infty} \frac{n^x}{x^n} $ converge?
For what values of x does the series $\sum_{n=1}^{\infty}\frac{n^x}{x^n} $ converge?
I've attempted to solve this problem but I can't finish up my reasoning - I don't know how to "check" the remaining numbers. Namely:
(1) I showed ... | For $x \in (-1, 1), x\neq 0$ we have $x^n \overset{n\to\infty}{\longrightarrow} 0$. So for $x>0$ , we know $\frac{n^x}{x^n}$ does not converge to zero.
For $x<0$ we can write $x=-\frac{1}{y}$ with $y>1$. Then the absolute value of our sequence is
$$\lvert\frac{n^x}{x^n}\rvert = \frac{y^n}{n^\frac1y} \ge \frac{y^n}{n}.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2533090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
What exactly is the $H^{-1/2}$ space? What exactly is the $H^{-1/2}$ space?
Definition for $H^{1/2}$ given e.g. here.
| $Y = \widehat{H^{1/2}(\mathbb{R})}$ is the Hilbert space of functions $f \in L^2(\mathbb{R})$ with the norm $$\|f\|_Y^2 = \int_{-\infty}^\infty (1+|x|) |f(x)|^2dx$$ ie. the inner product $$\langle f, g \rangle_Y = \int_{-\infty}^\infty (1+|x|) f(x)\overline{g(x)}dx$$
Its strong dual is $Y^*=\widehat{H^{1/2}(\mathbb{R}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2533315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Divergence, curl, and gradient of a complex function From an answer here I got
Green's theorem for functions in the complex plane
$$
\oint f(z) \, dz =
i \iint \left( \nabla f \right) \, dx \, dy =
i \iint \left( 1 {\partial f \over \partial x} +
i {\partial f \over \partial y} \right) \, dx \, dy
$$
Which leads t... | I'm not a physicist, but I think that gradient, curl, and divergence are strictly for a real $d$-dimensional environment, in particular for $d=2$ and $d=3$. I have never met your strange complex definition of $\nabla$.
On the other hand it is of course possible to prove the Cauchy integral formula using Green's theorem... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Definite integral of the cube root of $x \ln (x)$ I was trying to solve the 2016 CSE question paper for Math Optional
$$I = \int_{0}^1 \left (x\log \left (\frac{1}{x}\right)\right)^{\frac{1}{3}} dx$$
In my attempt to find $I$, I tried to substitute
$$t = \log \left (\frac{1}{x}\right) \\\ dt = \frac{x}{-x^2}$$
Even tr... | Take $x = e^{-3t/4}$
Then $\log\frac{1}{x} = 3t/4$
And $t$ varies from $0$ to $\infty$
Substitute these values in question
You'll get the answer in gamma function
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do I Factor Odd and even Degree polynomial as a product of irreducible polynomial? I want to factor $x^n -1$ Into a product of irreducible polynomials over The Reals, when n is Odd and when n is even.
I know that The only irreducible polynomials over The Real are first Degree and second Degree polynomials.
But im... | The linear factors are easy because the real roots of $x^n -1$ can only be $\pm 1$.
Irreducible quadratic factors come from complex roots.
The complex roots of $x^n -1$ are the $n$-th roots of unit: $\omega^k$, where $\omega=\exp(\frac{2\pi}{n} i)$.
These roots come in conjugate pairs: $\omega^k, \bar\omega = \omega^{n... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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binormal vectors and generalized helix There's a problem I'm stucked on it. I wonder if anybody is able to help me because I tried almost every idea that I could thougth of. Here it is:
If $ u $ is a fixed direction and for any point $ s $ of a space curve
$ < B(s) , u > = constant $
holds then prove that there's a con... | If $\langle B(s), u\rangle = c$ then $\langle B'(s), u\rangle + \langle B(s), u'\rangle = 0$. But $u$ is constant, so $\langle B(s), u'\rangle = 0$ and therefore $\langle B(s), u\rangle ' = \langle B'(s), u\rangle = 0$.
Using the Frenet Serret formulas, $\langle B'(s), u\rangle = \tau(s) \langle N(s), u\rangle = 0$ , s... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Verifying convergence in probability and distribution. Suppose random variables $\{X_n\}$ are defined on $([0,1],\mathcal{B}([0,1]),\lambda)$ (where $\lambda$ is Lebesgue measure) as follows:
$X_1 = \mathbb{1}_{[0,1]}, X_2 = \mathbb{1}_{[0,1/2]}, X_3 = \mathbb{1}_{[1/2,1]}, X_4 = \mathbb{1}_{[0,1/3]}, X_5 = \mathbb{1}_... | a) Note that
$P(x_n=1)=\frac{1}{k}$ for all $\frac{(k-1)k}{2}<n\leq \frac{k(k+1)}{2}$.
Hence as $n\to\infty$ (because $k\to\infty$)
$P(|X_n|>\epsilon)\to 0$.
b) Here note that
for any $w$,
$X_n(w)=1$ for infinitely many $n\in\mathbb N$.
Hence $X_n(w)\not\to0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2533996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\operatorname{trace}(T^*T) = \|Tu_1\|^2+\cdots+ \|Tu_n\|^2$ Let V be an inner product space with orthonormal basis $(u_1,...,u_n)$,
Show that
$$\operatorname{trace}(T^*T) = \|Tu_1\|^2+\cdots+ \|Tu_n\|^2$$
I was able to solve this in The case $u_i$'s are eigenvectors of the operator $T$. How can do ... | In the $u_i$ basis, we have the matrix of $T^\ast T$:
$[(T^\ast T)_{ij}] = \langle u_i, T^\ast T u_j \rangle = \langle Tu_i, Tu_j \rangle; \tag 1$
thus
$(T^\ast T)_{ii} = \langle Tu_i, Tu_i \rangle = \Vert Tu_i \Vert^2. \tag 2$
In any basis, the trace is the sum of the diagonal entries of the matrix of $T^\ast T$ in t... | {
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Is a quotient of topological vector space a topological vector space I should prove or disprove that quotient of topological vector space is a topological vector space.
I think that it is. May I reduce my prove to proof of following statement?
$X$ - topological vector space, $M \subset X$ - linear subspace of $X$. Let ... | A neighborhood of $0$ in $X/M$ is any set $W$ containing $0+M$ such that $\pi^{-1}(W)$ is a neighborhood of $0$ in $X$, where $\pi\colon X\to X/M$ is the canonical projection.
Let $W$ be a neighborhood of $0$ in $X/M$. Then there exists a neighborhood $V$ of $0$ in $X$ such that $V+V\subseteq \pi^{-1}(W)$.
This implies... | {
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update of centroids in $k$ means clustering I have been reading the lecture notes of Andrew Ng about Clustering techniques available in:
http://cs229.stanford.edu/notes/cs229-notes7a.pdf
but I have a problem in the second for of step $2$ which is the following:
For what I know, the previous step or for each section wh... | The formula means if the $i$-th data point is considered to be group $j$, we should consider that data point in updating our mean.
For example, if we think $x_1, x_2, x_3$ belongs to cluster $1$.
Then $\mu_1 = \frac{x_1+x_2+x_3}{3}.$
The notation $1\{c^{(i)}=j\}$ is an indicator function.
$\sum_{i=1}^m1\{c^{(i)}=j\}$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to solve the equation $(f \circ f)(x)=x$? Solve the equation $(f \circ f)(x)=x$, if $f(x) = \frac {2x+1}{x+2}$ and $x \in \mathbb R$ \ $\{-2\}$.
How would I solve this equation and what does it even mean to be solved in this context?
| How to begin:
$$f\bigl(f(x)\bigr)=\frac{2f(x)+1}{f(x)+2}=\frac{2\cfrac{2x+1}{x+2}+1}{\cfrac{2x+1}{x+2}+2}=\cdots$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the vector notation for $ \epsilon^{ijk} a_i b_j c_k $? If we use Einstein summation notation the upper indices and the lower indices match - then we do summation:
$$ a \cdot b = \langle a | b \rangle = a^i b_i := \sum a^i b_i $$
In the example I'm looking at there is a symbol called $\epsilon^{ijk}$ which is ... | Check: http://internal.physics.uwa.edu.au/~styler/teaching/CM/index.pdf
(2.21)
It is:
$$a\cdot(b\times c)\left(=\det(a,b,c)\right)$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Proof Verification for Connectedness of a Set in a Metric Space While I was reading metric spaces, I came across a theorem statement which had no proof in the book I was reading. Therefore, I tried to construct the proof myself. Given below is the theorem statement as well as the proof that I tried to construct.
Theore... | With the standard definition of connectedness the theorem hardly requires a proof. Here is the proof which uses your definition of connectedness: I start with $X=A \cup B$ with A and B separated. The fact that A and B are separated implies they are disjoint. Since their union is X it follows they are complements of ea... | {
"language": "en",
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Does SO(3) preserve the cross product? Let $g\in SO(3)$ and $p,q\in\mathbb{R}^3$. I wondered whether it is true that $$g(p\times q)=gp\times gq$$
I am not sure how to prove this. I guess I will use at some point that the last row $g_3$ of $g$ can be obtained by $g_3=g_1\times g_2$.
But I assume there is an easier proof... | You may use the scalar triple product formula $r \cdot (p\times q)=\det(r,p,q)$ to prove that
$$
gr \cdot (gp\times gq)=gr \cdot g(p\times q)\tag{1}
$$
($=\det(r,p,q)$) for any vector $r$. Since $g$ is invertible, if $(1)$ holds for every vector $r$, we must have $gp\times gq=g(p\times q)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2534923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 5,
"answer_id": 2
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Expectation of the ratio of dependent random variables where the expectation of the numerator is known to be zero Let $\mathbf{x} = \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix}$ be a complex normal random vector with $\mathbf{x} \sim \mathcal{CN}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\boldsymbol{\mu} \neq \ma... | Based on a classic counter-example for another question:
Let $X$ have a real standard normal distribution $N(0,1)$, and let $Y=X$ when $|X| \lt k$ and $Y=-X$ when $|X| \ge k$, where $k$ is square root of the median of a $\chi^2$ random variable with $3$ degrees of freedom, about $1.538172$. Clearly $Y$ also has a real ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that $d$ defines a metric $d: \mathbb{R}^2 \times \mathbb{R}^2 \rightarrow [0,\infty)$ I have to show that $d((x_1,y_1),(x_2,y_2)) = max\{|3x_1 + y_1 − 3x_2 − y_2|, |y_1 − y_2|\}$ defines a metric.
My approach:
Let $x=(x_1,y_1)$ and $y = (x_2,y_2)$
$M_1: d(x,y) \geq 0$ is clearly satisfied since $d$ chooses the... | To show $d((x_1,y_1),(x_3,y_3)) \leq d((x_1,y_1),(x_2,y_2)) + d((x_2,y_2),(x_3,y_3))$
Note that since the metric is defined as being the max, it is greater than or equal to both options. That is:
$$
\begin{align}
|y_1-y_2|, |3x_1+y_1-3x_2-y_2| \,\leq& \ d((x_1,y_1),(x_2,y_2)) \\
|y_2-y_3|, |3x_2+y_2-3x_3-y_3| \,\leq& \... | {
"language": "en",
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Find an Isometry $ \ S \ $ such that $ \ T=S \sqrt{T^* T} \ $ Let $ \ T : \ \mathbb{C}^2 \to \mathbb{C}^2 \ $ be given by $ \ T(z_1,z_2)=\begin{bmatrix} 2 & -1 \\ 4 & -2 \end{bmatrix} \begin{bmatrix} z_1 \\ z_2\end{bmatrix} $
Find an Isometry $ \ S \ $ such that $ \ T=S \sqrt{T^* T} \ $ , where $ \ T^* \ $ is the con... | You want $T\vec{a_k}=S\sqrt{T^*T}\,\vec{a_k}$ for $k=1,2$. This does not uniquely determine $S$ because $\sqrt{T^*T}\,\vec{a_1}=0$ and $T\vec{a_1}=0$. The condition $S\sqrt{T^*T}\,\vec{a_2}=T\vec{a_2}$ determines $S$ on $\vec{a_2}$. Setting $S\,\vec{a_1}=e^{i\theta}\vec{b_1}$ defines an isometry $S$ if $\vec{b_1}$ is o... | {
"language": "en",
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2,3,5,6,7,10,11 Counting with Restrictions The sequence 2, 3, 5, 6, 7, 10, 11, $\ldots$ contains all the positive integers from least to greatest that are neither squares nor cubes nor perfect fifth powers (in the form of $x^{5}$, where $x$ is an integer). What is the $1000^{\mathrm{th}}$ term of the sequence?
I've see... | Let $B(n)$ be the number of integers in range $\{1,2,\dots, n\}$ that are not squares cubes or fifth powers.
We want to find the first number such that $B(n)=1000$.
A formula for $B(n)$ can be obtained with inclusion exclusion.
We get $B(n)=n- \lfloor \sqrt{n} \rfloor- \lfloor\sqrt[3]{n} \rfloor- \lfloor\sqrt[5]{n} \rf... | {
"language": "en",
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What am I doing wrong with this partial sum formula? I have this series:$$\sum_{n=1}^\infty (1-\frac{7}{4^n})$$
So my strategy is to divide it into two formulas to try and get a geometric formula:$$\sum_{n=1}^\infty 1 - \sum_{n=1}^\infty \frac{7}{4^n}$$
The first sum equals 1 and I simplify the second sum to a geometri... | $$\sum_{n=1}^\infty 1 = \infty$$
not one.
You can also observe that
$$\lim_n(1-\frac{7}{4^n})=1 \neq 0$$
therefore, the series is divergent.
| {
"language": "en",
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Reference Triangle Issue - Trig Evaluate $\cos(2\cot^{-1}(32/49))$
What I did for this problem was:
*
*Turn $\cot$ into $\tan(49/32)$
*And I am not sure what I do with the $2*$
I tried multiplying $49$ by $2$ and plugging the values into a reference triangle.
After I had all three sides I just found the $\cos$ o... | By a well-known formula you have
$$\sin\left(\cot^{-1}x\right)=\frac1{\sqrt{1+x^2}}$$
And, of course, the double angle formula
$$\cos2x=1-2\sin^2x$$
Now your solution is straightforward
$$\begin{align}\cos\left(\cot^{-1}\frac{32}{49}\right)&=1-2\sin^2\left(\cot^{-1}\frac{32}{49}\right)\\[10pt] &=1-2\left(\frac1{\sqrt{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2535902",
"timestamp": "2023-03-29T00:00:00",
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Showing convergence and integrability from $\sum_{n=0}^\infty \int_X |f_n| \, d \mu \lt \infty$ I have seen this question here but I didnt find a comprehensible answer so ill try my luck.
Let $(f_n)$ be a sequence of almost everywhere integrable functions defined on X with $$\sum_{n=0}^\infty \int_X |f_n| \, d \mu \lt ... | Let
$$
F_n= \sum_{k=0}^{n}f_k(x); \quad F(x)= \sum_{k=0}^{\infty}f_k(x)
$$
Clearly, $F_n\in L^1$. Further
$$
|F_n|\leq \sum_{k=0}^{n} |f_k(x)|\leq \sum_{k=0}^{\infty} |f_k(x)|\in L^1\tag{1}
$$
since
$$
\int_X \sum_{k=0}^{\infty} |f_k(x)|=\sum_{k=0}^{\infty}\int_X |f_k| \,d \mu <\infty
$$
by the Monotone convergence th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simplifying an equation (mod, floor) In what ways can I simplify the equation $$y=(1-\lfloor \bmod(x,3)\rfloor)(\bmod(x,1))+\frac{\lfloor \bmod(x,3)\rfloor-\frac{1}{2}}{2|\lfloor \bmod(x,3)\rfloor-\frac{1}{2}|}+\frac{1}{2}$$ or at least make it look nicer?
| Notice that
*
*$\text{mod}(x+3,3)=\text{mod}(x,3)$
*$\text{mod}(x+3,1)=\text{mod}(x,1)$
So the function is periodic with period $p=3$.
So you can describe the function on the interval $[0,3)$ in a piecewise fashion, then use that to obtain a general piecewise formula.
The graph looks like this:
On the interval $... | {
"language": "en",
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Show a non-constant, continuous function $f:\bar{D}\rightarrow \bar{D}$ is such that $f(\partial D)=\partial D$ Suppose $\bar{D}= \{z:|z|\leq 1\}$ and assume we have a non-constant, continuous function $f:\bar{D}\rightarrow \bar{D}$, that is holomorphic on the interior of $\bar{D}$ and such that $f(\partial D)\subset \... | Hint. Let $\mathcal{B}$ be the set of non-constant, continuous functions $f:\bar{D}\rightarrow \bar{D}$, holomorphic in $D$ and such that $f(\partial D)\subset \partial D$ where $D=\{z: |z|<1\}$.
1) If $f\in \mathcal{B}$ then $0\in f(D)$.
If not $1/f\in \mathcal{B}$ and $1/|f|$ has a maximum in $\partial D$, which ... | {
"language": "en",
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"source": "stackexchange",
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show that $\int_{(0,1)} f dx = \infty$ Anyone knows how to show that $\int_{(0,1)} f dx = \infty$
where
$$
f(x)=
\begin{cases}
0 & \text{if $x \in \mathbb{Q}$}\\
[\frac{1}{x}]^{-1} & \text{if $x \notin \mathbb{Q}$}
\end{cases}
$$
| Edit: There were too many mistakes in the original answer. Thanks to Peter Melech for pointing them out.
The integral is actually $\frac{\pi^2}{6} - 1$, and not $+ \infty$.
Denote by $\mu$ the Lebesgue measure on $\mathbb{R}$.
Consider the simple functions
$$f_n = \sum\limits_{k = 1}^{n} \frac {1}{k} \cdot \chi_{\left(... | {
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Second Cohomology of the modular group What is known about $H^2(SL(2,\mathbb{Z}),\mathbb{Z})$?
Anyone knows some references about that?
| Yes, this is known, see the first lines here: "Our goal is to compute its cohomology groups with trivial coefficients, i.e. $H^q (SL_N(\mathbb{Z}),\mathbb{Z})$. The case $N = 2$ is well-known and follows from the fact that $SL_2(\mathbb{Z})$ is the amalgamated product of two finite cyclic groups ([21], [6], II.7, Ex.3,... | {
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Indefinite Integral: $\int{x\cos{(x+1)}}dx$ I am trying to solve the following indefinite integral:
$$\int{x\cos{(x+1)}}dx$$
But I keep running into problems as I am thinking of solving it using parts in which I did the following:
$$
u = x+1 \\
du = 1\ dx \\
\int{x\cos{(x+1)}}dx = \int{x\cos{u}}du
$$
Then from $u = x+1... | Your substitution is correct: the easiest way of proceeding with your method is to notice that $\int{(u-1)\cos{u}}\;du$ can be divided into two integrals
$$\int{(u-1)\cos{u}}\;du=\int u\cos{u}\;du-\int{\cos{u}}\;du$$
The first integral of the two can be found by integrating by parts
$$\int u\cos{u}\;du=u\sin u-\int \s... | {
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Prove or disprove: If $P = x...y$ is a path in a $2$-connected graph $G$ then there is another $xy$-path $P'$, which is internally disjoint from $P$. Prove or disprove: If $P = x...y$ is a path in a $2$-connected graph $G$ then $G$ contains another $xy$-path $P'$, which is internally disjoint from $P$.
I'm not sure whe... | You can prove this by induction on the distance of $x$ and $y$.
Base case: If $dist(x,y)=1$, then $\{x,y\}$ is an edge. Let $z$ be a vertex different from $x$ and $y$. Since the graph is 2-connected, the removal of $x$ (respectively $y$) does not disconnect the graph, therefore there is a $(x,z)-$path $P_1$ which does ... | {
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Graph Coloring Clique bound: Find a graph G such that ω(G) < χ(G) = 4 For the life of me I can't seem to think of anything that works. I tried randomly combining triangle graphs and taking edges away from $K_4$, but every time I think I found a way to force the chromatic number to be 4, I find that there's actually a s... | Take a pentagon $C_5$. You need $3$ colors already. Add a vertex, and link it to all vertices of the pentagon, so that it cannot take any of these $3$ colors. You end up with $\chi=4$, and $\omega=3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2536929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Platonic solids calculating the relations between the diameter of the sphere and the sides lengths Let D be a diameter of a sphere and let s be the side of an inscribed regular polyhedron. Show that the d and s are related as follows. This is a book called Geometry by Hartshorne exercise 44.8
I) For an inscribed Tertra... | A general formula for $\frac{d^2}{s^2}$ for a Platonic solid with $f$ faces and $e$ edges per face is:-
$$\frac{\sec^2(\pi/e)}{\tan^2(\pi/e)-\tan^2(\pi(f-2)/ef)}$$
This was obtained using solid angles as follows.
Let $O$ be the centre of a face. The circumradius of the face, $r$ is given by$$s=2r\sin(\pi/e)$$ The dist... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is there any way to solve this without using a graphing calculator? Solve for $x$:
$$x=2^{8-2x}$$
Whenever I've seen solutions to this question, they have always been through plotting the two graphs and finding their intersection point. But, is there any other way to solve this (perhaps in a more algebraic way)?
| Hint
When you have linear and exponential function in the same equation, then use Lambert W function defined as
$$e^{W(x)}W(x)=x$$
Your equation becomes
$$4^xx=256$$
which is equivalent to
$$e^{x\ln4}x\ln4=256\ln4$$
Can you continue from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2537583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Find the value of $\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots + \frac{1}{1+2+3 +\ldots+2015}$ The question:
Find the value of $$\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+\ldots + \frac{1}{1+2+3 +\ldots +2015}$$
If this is a duplicate, then sorry - but I haven't been able to find this question yet. To start, I n... |
And I'm not sure if this is right. How does one check whether their summation is correct?
Replace "2015" with "10", do it by hand, and see if your results match with your general formula.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2537684",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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The meaning of denotation E[X|Y]? Assume we have 2-dim data $(x,y): {(a_1,b_1),...,(a_n,b_n)}$; where $X, Y$ are random variables.
The conditional expectation is $E(X|Y=b_j)=\sum_{i}a_iP(X=a_i|Y=b_j)$
There is a theorem: $E(E(X|Y))=E(X)$, but what does $E(X|Y)$ exactly mean? In the former fomula, $Y=b_j$ is a condition... | $\mathbb E\left(X\mid Y\right)$ is by definition a random variable satisfying:
*
*It is measurable with respect to the $\sigma$-algebra generated by rv $Y$.
*$\int_{A}X\left(\omega\right)\rm P\left(d\omega\right)=\int_{A}\mathbb E\left(X\mid Y\right)\left(\omega\right)\rm P\left(d\omega\right)$
for each set $A$ in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2537929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Knight on chess board aperiodic? So suppose I have a knight on the corner of a chess board. I have to work out the average return time and also the limit of the probability that the knight is back in the corner after n steps as n tends to infinity. I have done this using a model of vertex and edges and using the statio... | It's true that the knight's walk is periodic; specifically, it has period 2. As such, any statement you can make about its limiting distribution will have to be one about the walk when sampled after an even number of steps; that is, you can address the distribution of $Y_n := X_{2n}$, where $X_n$ is the position of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2538030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Direct sum and Hom Let $R$ be a commutative ring with unity and let $M$ be a finitely generated noetherian R-module.
Can someone tell me how the isomorphism
$$
Hom_R(R\oplus R, M)\simeq Hom_R(R,M)\oplus Hom_R(R,M)\simeq M\oplus M
$$
is given? I know how the second isomorphism is given but I don't know about the first... | The universal property of the biproduct is precisely the existence of the first isomorphism (assuming it is natural in $M$).
More generally, the coproduct of $X \amalg Y$ of two objects in any category is the object that satisfies an isomorphism natural in $Z$:
$$ \hom(X \amalg Y, Z) \cong \hom(X, Z) \times \hom(Y, Z)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Tangent plane of the surface: $z-g(x,y)=0$ in $(x_0, y_0, z_0)$ How can I determine the equation of tangent plane of the surface:
$$z-g(x,y)=0$$
in the point:
$$(x_0, y_0, z_0)$$
in terms of implicit derivatives?
| The gradient of a function is normal to its level curves. Proof:
Suppose $\vec r(t)=\langle x(t),y(t),z(t) \rangle$ parametrizes the curve $f(x,y,z)=c$, where $c \in \mathbb{R}$ is some constant. Then,
$$f(x(t),y(t),z(t))=c$$
Since $\vec r(t)$ is on the curve. We also have,
$$\frac{d}{dt}f(x(t),y(t),z(t))=0$$
By the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2538298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Prove that $\gcd(a,b) = \gcd (a+b, \gcd(a,b))$ I started by saying that $\gcd(a,b) = d_1$ and $\gcd(a+b,\gcd(a,b)) = d_2$
Then I tried to show that $\ d_1 \ge d_2, d_1 \le d_2$.
I know that $\ d_2 | \gcd(a+b, d_1)$ hence $\ d_2 \le d_1 $.
How do I prove that $\ d_2 \ge d_1$ ?
| Don't worry about size so much as what they divide.
1) $\gcd(a,b)=d_1$ by definition divides $a$ and $b$ and thus $a+b$. And everything divides itself. So $\gcd(a,b)$ is a common divisor of $a+b$ and $\gcd(a,b)$ (and thus by definition is less or equal to the greatest common divisor of $a+b$ and $\gcd(a,b)$. $d_1 \l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2538383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 3
} |
Since the limit of the sequence $b^n$ where $0The question is in the title. For a fixed number $b \in \mathbb{R}$ where $0<b<1$, I can prove that the sequence $b^n$ converges to zero.
Proof
Choose $K(\epsilon) = \lfloor \frac{\ln \epsilon}{\ln b} \rfloor + 1$. Then for all $n \geq K$ we have $$ \ln b^n < \ln \epsilo... | Since $0\lt\left(\frac12\right)^{1/n}\lt1$ the logic in your argument could be used to show that
$$
\lim_{n\to\infty}\left(\left(\frac12\right)^{1/n}\right)^n=0
$$
However, for each $n$, $\left(\left(\frac12\right)^{1/n}\right)^n=\frac12$. The theorem you state assumes a fixed $b$; it does not necessarily hold for a va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2538534",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Permutations of 9 total letters, using at least 7 letters. Original Question:
How many different strings can be made from the letters in "EVERGREEN"
using at least 7 of it's letters.
Note that the 4 "E"s and 2 "R"s are indistinguishable.
I understand how to use permutations with repetition to determine the number... | using 9 letters the number of different words is $\binom{9!}{4!2!}$
to find how many words you can form with 8 letters you have to perform the same calculation on each of the following set of letters
EEEVRRGN, EEEERRGN, EEEEVRGN, EEEEVRRN, EEEEVRRG
e.g. for the first one the number of words is $\binom{8!}{3!2!}$, and s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2538636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Inverse of matrix with nonnegative entries I am interested in matrices with the property that both $A$ and $A^{-1}$ have nonnegative entries. The only such matrices I could construct were diagonal matrices, and my question is whether these are the only such examples.
What I can say about such matrices is that they mus... | Of course, il maestro @Qiaochu Yuan is right; and, of course, he knows that there exists an elementary proof !
Let $A=[a_{p,q}],A^{-1}=[b_{p,q}]$. We consider the $i^{th}$ row of the matrix $A$ and we assume that there are $j\not= k$ s.t. $a_{i,j},a_{i,k}\not= 0$. Then $AA^{-1}=I$ implies that, for every $p\not= i$, $b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2538937",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Need help in understanding the proof for Theorem 11.17 in Baby Rudin
I don't understand why the first two equality signs hold in the proof.
| $g(x) > a \iff \sup_n f_n(x) > a$, i.e. there is some $n$ for which $f_n(x) > a$. So, $$\{x \mid g_n(x) > a\} = \bigcup_{n=1}^\infty \{x \mid f_n(x) > a\}.$$
The second equality sign follows by definition of the $\limsup$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Best approximation to $t^2$ in first-dgree polynmial in $L^1[0,1]$ Let $u(t)=t^2$. Find the best approximation $v(t)$ in the form of $v(t)= ct+d $ (with $c,d\in\mathbb{R}$) to $u(t)$ in $L^1[0,1]$.
So we need to find
$$\inf\limits_{c,d\in\mathbb{R}} \int_0^1 \left|t^2-ct-d\right|dt$$
I've tried to approach this problem... | As user7530 commented, consider the roots
$$t^2-ct-d=0 \implies t_{1,2}=\frac{1}{2} \left(c\pm\sqrt{c^2+4 d}\right)$$ So
$$I=\int_0^1 \left|t^2-ct-d\right|\,dt=\int_0^{t_1}(t^2-ct-d)\,dt-\int_{t_1}^{t_2}(t^2-ct-d)\,dt+\int_{t_2}^{1}(t^2-ct-d)\,dt$$
Compute each of these three integrals; for sure, the formulae are not ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539208",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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The closure of a subspace
Let $X$ be a topological space. If $A$ is a subspace of $X$ we denote its closure by $\overline A$. For each point $x \in X$ the family $N_x$ of neighbourhoods of $x$ is a filter on $X$, the $\textit{neighbourhood filter}$ of $x$.
This is from Bell and Slomson (1969) Models and Ultraproducts... | When you write $\overline{A}$ in your last sentence, do you mean the closure in the subspace topology? This would be A, as you said and your argumentation is correct. Or the closure in the topology of X? In this case: $\overline{A} = \{a,b,c,e\}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539380",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Prob. 4, Chap. 7, in Baby Rudin: For what values of $x < 0$ does this series converge (absolutely)? Here is part of Prob. 4, Chap. 7, in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition:
Consider
$$ f(x) = \sum_{n=1}^\infty \frac{1}{1+n^2 x }. $$
For what values of $x$ does the series c... | For any $x\neq 0$ we have $|a_n|=\left|\dfrac 1{1+xn^2}\right|\sim \dfrac 1{|x|n^2}$ which is a term of a convergent series, so the initial series is absolutely convergent.
Of course this criteria operates for values $n\gg1$, but as kolobokish noticed, there is an issue when $x\in A=\{-\frac{1}{k^2}\mid k\in\mathbb N^*... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Normal operator close in norm to projection, what about the spectrum? Let $T$ be a normal operator on a Hilbert space and suppose $T$ is close in norm to a projection $P$. Can I say that the spectrum of $T$ is contained in small balls around $0$ and $1$?
| First of all $\rho(T)=||T||$ because $T$ is normal ($\rho$ is the spectral radius). So by the triangle inequality $\rho(T)\leq1+c$, given $||T-P||\leq c$.
Now the formula in the comment
$$(T-\lambda I)^{-1}=((T-P)(P-\lambda I)^{-1}+I)^{-1}(P-\lambda I)^{-1}$$
makes sense, using the Von Neumann series, if $(T-P)(P-\lam... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Are all connected and locally integral affine schemes globally integral? In these notes, on p. 2 Section 4, Kedlaya claims an affine scheme is integral if and only if it is connected and every local ring is an integral domain. But elsewhere I have seen that this requires a Noetherian condition on the affine scheme, e.g... | The error is in the exercise referred to in the first paragraph: it is not necessarily true that the set of $x$ such that $f$ is nonzero in $O_{X,x}$ is open. In fact, this can fail even if $A$ is Noetherian. For instance, let $k$ be a field and take $A=k[x,y]/(xy)$, so $X$ is the union of two lines intersecting at a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
$ \lim_{n \to \infty} \sqrt[n]{a^n+1}$ with $a \ge 0$ I'm a bit rusty with limits
$$ \lim_{n \to \infty} \sqrt[n]{a^n+1}$$ with $a \ge 0$.
The solution in my book is $max \left \{ 0,1 \right \}$ but
my final results are:
1) $+\infty$ if $0<a<1$
2) $1$ if $a>1$
3) $+\infty$ if $a=1$
| The correct result is $\max\{a, 1\}$, so I suppose it's a typo or a miscopied expression. To see this, note that
*
*If $a \le 1$, then $1 \le a^n + 1 \le 2$ for all $n$. Taking $n$-th roots and a limit gives limit $1$.
*If $a > 1$, then
$$\left(a^n + 1\right)^{1/n} = a \left(1 + \frac{1}{a^n}\right)^{1/n}$$
Now app... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
How to visualise the curve $y=\ln x$ rotating around its $y$-axis
Let the area limited by the curve $y=\ln x$, the line $x=e$ and the $x$-axis rotate around the $y$-axis. Decide the volume of the resulting rotational body.
First thing, I drew the graph:
But then I got stuck on how to imagine/visualise the curve rota... | take a look here with wolfy for $$f=log(\sqrt{x^2+y^2}$$
http://m.wolframalpha.com/input/?i=plot+log%28%28x%5E2%2By%5E2%29%5E.5%29
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2539911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Without using a calculator and logarithm, which of $100^{101} , 101^{100}$ is greater? Which of the following numbers is greater? Without using a calculator and logarithm.
$$100^{101} , 101^{100}$$
My try : $$100=10^2\\101=(100+1)=(10^2+1)$$
So :
$$100^{101}=10^{2(101)}\\101^{100}=(10^2+1)^{100}=10^{2(100)}+N$$
Now w... | You want to determine if $\left(\frac{101}{100}\right)^{100}\geq 100$. But we know that $ \left(1+\frac{1}{n}\right)^n$ is always less than $e$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2540063",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 5,
"answer_id": 1
} |
Parametric Equation for A Wiggly Tube I need to form a shape where the side view in the $xz$-plane is parallel inverse sines, and the surface is a pipe with circular cross-sections. Is there a name for this shape?
I tried messing around with ParametricPlot3D in Mathematica, but couldn't figure it out. I tried messing a... | Since the cross section is a circle and the cross section from the side are parallel arc sine functions, we can form the shape as a continuous sequence of circles whose height changes as we go along the x-axis. With circles lying in the xz-plane and the arcsines in flat in the xz-plane we get this parametric function:
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2540190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Understanding Legendre-fenchel Transform, looking for an easy example and intuition Looking for help in understanding this transform. I have no background in real analysis but need this stuff for my research.
I hope someone can give me some light on the intuition behind this transform and better if you can provide some... | Will answer my own question since I found a very good reading material which helped me to fully understand the topic.
If interested, take a look here.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2540332",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
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