Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Explanation of an integration trick for $\int \frac{a_1 \cos x + b_1 \sin x}{a\cos x + b\sin x}dx$ I'm not sure how to formulate my question correctly. Basically it comes from solving the integral:
$$
\int \frac{a_1 \cos x + b_1 \sin x}{a\cos x + b\sin x}dx\\
a^2 + b^2 \ne 0
$$
I haven't been able to solve the integral... | Any linear combination of sine waves with the same period and different phase shifts can be written as a single sine wave with that same period and a suitable phase shift.
\begin{align}
& A\cos(x+\varphi) + B\cos(x+ \psi) \\[8pt]
= {} & A\big(\cos x\cos\varphi - \sin x \sin\varphi\big) \\
& {} + B\big(\cos x\cos\psi - ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3416775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Question about Answer to limsup of $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ here's the relevant question: If $\sigma_n=\frac{s_1+s_2+\cdots+s_n}{n}$ then $\operatorname{{lim sup}}\sigma_n \leq \operatorname{lim sup} s_n$
In the accepted answer, doesn't the last inequality only work if $\sup_{l\geq k}s_l$ is nonnegative?... | You have that
$$ \tag{*}
\sigma_n\geqslant \frac 1n\sum_{j=1}^ks_j+\frac{n-k}n\inf_{l\geqslant k}s_l
$$
and you are right that this is $\ge \frac 1n\sum_{j=1}^ks_j+\inf_{l\geqslant k}s_l$ only if $\inf_{l\geqslant k}s_l \le 0$.
But that estimate is actually not needed: For fixed $k$ you can take the $\liminf_{n \to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3416895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Complex numbers question - sum of three complex numbers Seems an easy one but i can't figure it out:
$z_1+z_2+z_3=0$
$|z_1|=|z_2|=|z_3|=1$
Need to prove the following:
$z_1^2+z_2^2+z_3^2=0$
Thanks!
| Conjugate $z_1+z_2+z_3=0$ and get $\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}=0$ which simplifies to $z_1z_2+z_1z_3+z_2z_3=0$. Now square the original equation and you are done!
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3417029",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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Integral over Modified Bessel Functions I've stumbled upon this integral
\begin{equation}
\mathscr{I}_\nu^k(a,b)=\int_0^\infty dr\frac{r^k}{r^2+m^2}e^{-ar^2}I_\nu(br)
\end{equation}
(where $I_\nu(x)$ is the modified Bessel function) during some QFT research, but I cannot seem to crack it. I'm specifically interested in... | Let's do the $(1,2)$ case. First let's denote
$$J(a,b) = \int_0^\infty \frac{r^2}{r^2+m^2}e^{-ar^2}I_1(br)dr = \int_0^\infty e^{-ar^2}I_1(br)dr - \int_0^\infty \frac{m^2}{r^2+m^2}e^{-ar^2}I_1(br)dr$$
Which means we have that
$$\partial_a J(a,b) = \int_0^\infty -r^2e^{-ar^2}I_1(br)dr + \int_0^\infty \frac{m^2r^2}{r^2+m^... | {
"language": "en",
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Why does exponentiating group elements by integers always make sense? Let $G$ be a group.
For any $g\in G$, we can define a mapping $\mathbb Z\to G$ via $n\mapsto g^n$.
This map is a homomorphism between the additive group $\mathbb Z$ and $G$.
On the one hand, it's obvious why we can always do this. We just define $g^n... | First, for each $g$ your map is really a homomorphism of the integers $\mathbb{Z}$ to $G$. You map $-1$ to $g^{-1}$. You don't want to restrict the map to the positive integers.
All you are really saying is that any group has lots of cyclic subgroups - every element generates one. Any any cyclic group is a homomorphic ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Do bijections like the ones defined in this question exist?
Do everywhere discontinuous bijections $b: \mathbb R \to \mathbb R$ such that $b(\mathbb Q) \cap \mathbb Q= \emptyset$ exist?
| Yes, example: $f(x)=x+\sqrt{2}$.
Edit: for the function to be discontinuous everywhere:
$$
f(x) =
\begin{cases}
x+\pi & \text{if } x \in \mathbb{Q} \\
x+\pi+1 & \text{if } x \in \mathbb{R} \backslash \mathbb{Q}
\end{cases}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Finding a basis of solutions to a linear homogeneous system with a matrix with non-constant entries. Is there a way to find a basis of solutions to the following linear homogeneous system using the eigenvector method?
\begin{array}{l} \\ {\qquad \boldsymbol{y}^{\prime}(t)=A(t) \boldsymbol{y}(t), \quad A(t)=\left[\begin... | No, it's not a matrix exponential, though it's sometimes called a "time-ordered exponential", and eigenvectors/eigenvalues won't help. In general there's no way to do it in closed form, though as you mentioned, in this case you can solve it by doing $y_1$ first and then $y_2$.
| {
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Prove that $f(x)=e^{-1/x^2}$ for $x\neq 0$ is continuous.
Prove that $f(x)=e^{-1/x^2}$ for $x\neq 0$ is continuous.
So I know I need to use the epsilon-delta limit definition, but when I do I end up needing to show that
$e^{-1/a^2}\left|\dfrac{2}{x^3}e^{-1/x^2+1/a^2}-\dfrac{2}{a^3}\right|<\epsilon\; \forall \epsilon>... | We may evaluate the continuity of $e^{\frac{-1}{x^2}}$ straightforwardly using theorems regarding the continuity of composed and elementary functions.
Let $f(x)=\frac{1}{x},$ $g(x) = e^x$, and $h(x)=\frac{1}{x^2}$.
Then $$e^{\frac{-1}{x^2}} = \frac{1}{e^{\frac{1}{x^2}}} = \frac{1}{(g \circ h)(x)}= (f \circ (g \circ h))... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3417826",
"timestamp": "2023-03-29T00:00:00",
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a question on Bernoulli function in the book of Tenenbaum In section 0.2 of Introduction to Analytic and Probabilistic Number Theory by Gérald Tenenbaum, I read that "One easily verifies that these assumptions imply the identity...". I started from the left hand side of the series taking first finite terms and using i... | Hint: Let $f(x,y)=\sum _{r\geq 0}b_r(x)\frac{y^r}{r!}$ Take $\frac{d}{dx}f(x,y)$ to get
$$\frac{d}{dx}f(x,y)=\frac{d}{dx}(1+\sum _{r>0}b_r(x)\frac{y^r}{r!})=\sum _{r>0}b_{r-1}(x)\frac{y^r}{r!}=yf(x,y),$$ hence $$\frac{\frac{df}{dx}}{f}=y$$ and so
$$f(x,y)=ce^{xy},$$ where $c$ does not depend on $x.$ Now, take
$$\int ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3417984",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How many numbers have the form $10n+d$ where $d$ is a non-zero digit? How many numbers that do not end in a series of zeros are such that if we erase the last digit, the resulting number will divide the original?
I was even thinking of excluding the possibility of being multiples of 5, and by the divisibility criterion... | Let the number be $(a_{1}a_{2}.....a_{n})_{10}$$=k$
so $k=$$(a_{1}a_{2}.....a_{n})_{10}=$$10$$(a_{1}a_{2}.....a_{n-1})_{10}$$+a_{n}$
now $(a_{1}a_{2}.....a_{n-1})_{10}$ divides $(a_{1}a_{2}.....a_{n})_{10}$
which means $(a_{1}a_{2}.....a_{n-1})_{10}$ divides $a_{n}$ which is only possible for 2 digit numbers
so just a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3418080",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 2
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Show that $G = M \circledast N$ has a diagonal subgroup iff $M$ is isomorphic to $N$. A subgroup $D$ of $G = M \circledast N$ is a diagonal subgroup provided:
$$D \cap M = 1 = D \cap N$$
$$DM = G = DN$$
(Where $\circledast$ is denoting the internal direct product of $M$ and $N$.)
$$$$
GOAL: Show that $G$ has a diagonal... | The other direction is easy. When $M \simeq N$, then $G \simeq M \times M$ and $\{(m,m), m \in M\}$ is a diagonal subgroup (you are just writing elements of $M$ twice, so every needed check follows from this). Can you explain this? I don't get it.
Now, suppose that $G$ has a diagonal subgroup $D$. As Arturo suggested i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3418185",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "6",
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Locations of root theorem confusion Theorem: if f is continuous in [a, b] and f(a) < 0, f(b) > 0, then there exists c in [a, b] such that f(c) = 0.
The most popular proof on the website is proof by contradiction. Thus, we have two cases, f(c) < 0 or f(c) > 0.
first, I suppose f(c) > 0. How can I use continuity to prov... | In the next section we a prove a proposition by contradiction that can be used by the OP.
Proposition 1: Let $g:[a,b] \to \Bbb R$ be a function satisfying $g(x) \lt 0$ for $x \lt b$.
If $g$ is continuous at $x = b$ then $g(b) \lt 0$ or $g(b) = 0$.
Proof
To arrive at a contradiction, assume that $g(b) \gt 0$.
Let
$\qua... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Related Rates (point moving along a curve) Consider a point moving along the curve $$f(x) = \sqrt x$$.
a). Find the position of the point on the curve where both coordinates of the point are changing at the same rate.
b). If $\dfrac{dx}{dt}$ is $2 \text{ m/sec}$ at the point $(4,f(4))$, how fast is the point moving awa... | a) No. You're not looking for the point where the x and y have the same value, you're looking for the point where the values are changing at the same rate. Whenever you see "change" in calculus, that is a free clue that you should be thinking about the derivative. So you are looking for the point where $\frac{dy}{dx... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3418724",
"timestamp": "2023-03-29T00:00:00",
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Proving a Supremum of a Set The Question:
Find the supremum of the set $${\{\sqrt[4]{n^4+n^3}-n:n\in \mathbb{N}\}}$$
And then it tells us to plug large values of n to determine a suitable guess, show that is an upper bound and then prove it is the smallest upper bound.
I followed the question, finding a suitable gues... | Let
$$
f(n) = \sqrt[4]{n^4+n^3}-n = n ((1+\frac{1}{n})^{1/4}-1)
$$
Now by Bernoulli's inequality, $(1+\frac{1}{n})^{1/4} \le 1+\frac{1}{4n}$, so
$$
f(n) \le n ((1+\frac{1}{4n})-1) = \frac{1}{4}
$$
Now we need to show that this is the smallest upper bound. To do so,
let $x = \frac{1}{n}$ and re-consider Bernoulli's... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $T$ is continuous. I am having a really hard time trying to solve this problem:
Let $X$ and $Y$ be normed linear spaces and $T : X \to Y$ a linear operator with closed graph and finite-dimensional range $R(T)$. Prove that T is continuous.
Obviously the closed graph theorem cannot be applied since our spaces ... | Assume $T$ is not continuous at zero. Then there exist a sequence $(x_n)\subset X$ with $x_n\to 0$ and $\epsilon>0$ such that $\|Tx_n\|\ge\epsilon$ for $n\in\mathbb N$. Set $u_n := \frac{x_n}{\|Tx_n\|}\in X$. Then $u_n\to 0$ and $\|Tu_n\|=1$ for all $n\in\mathbb N$. Since $R(T)$ is finite-dimensional, there exists a su... | {
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If $ u_t=u_{xx} $ and $u(x,0)=4x(1-x),~x\in [0,1],~~u(0,t)=u(1,t)=0,~t\geqslant 0$, prove $00$
Let $u$ be the solution of the heat equation initial and boundary value problem:
$$\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2},~~x\in (0,1),~t>0,$$
and:
$$u(x,0)=4x(1-x),~x\in [0,1]~~ \textrm{and} ~~... | The strong maximum principle states that if the solution attains its maximum $M$ in some $0 < \bar{x} <1$ and $\bar{t} > 0$, then the solution $u(t, x) \equiv M$ for $0 \le x \le 1$ and $0 \le t \le \bar{t}$. Since the initial condition is not constant, you have a contradiction. (Similarly for the minimum)
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that the map $c:{\mathbb {T}}^{3}=S^{1}\times S^{1}\times S^{1}\setminus\ \Delta\longrightarrow \{\pm 1\}$ is continuous. Prove that the map $c:{\mathbb {T}}^{3}=S^{1}\times S^{1}\times S^{1}\setminus \Delta\longrightarrow \{\pm 1\}$ is continuous where $\Delta$ is the diagonal
$\Delta=\{(g_i,g_j,g_k)\}$ for $i... | You can choose such a $c$ to be continuous. Note that means that the 3-torus without the fat diagonal is at least two connected components. How can we see that?
First, instead of trying to imagine a 3-torus, imagine three, possibly with multiplicity, marked points (which are colored/distinguished from each other) on th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3419267",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Non-zero admissible representation of $sl_\infty$ Does someone has an example of a non-zero admissible representation of $sl_\infty$ ?
| I think I found one.
We consider a vector space $V_m = \mathbb{C}V_0 \oplus \cdot\cdot\cdot \oplus \mathbb{C} V_m$ and the vector space $V^{(\mathbb{Z})}_n$ of $\mathbb{Z}$ indexed sequences of $V_m$ with a finite number of non zero terms.
Then, for $N$ in $\mathbb{N}$ we introduce the basis $(v^i_p)$ where $0 \leq p ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that a function has a derivative at $x=0$ Okay, we have this function:
$f(x)= |x|^\alpha \sin(\frac{1}{x})$, if $x\neq0$
$f(x)= 0$, if $x=0$
The question is, at point x=0:
1) At which value of $\alpha$ does $f(x)$ have a derivative?
2) At which value of $\alpha$ does $f(x)$ have a continuous derivative?
The answe... | For question $1$ we need that the following limit exists
$$\lim_{h\to0} \frac{|h|^\alpha \sin(\frac{1}{h})}{h}$$
and since $\sin(\frac{1}{h})$ oscillates and is bounded we need that
$$\lim_{h\to0} \frac{|h|^\alpha }{h}=0 \implies \alpha>1$$
For question $2$ we need to check that the derivative is continuous and since w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Difference between topologically complete space and complete metric space Definition: Topological space $(X,\tau)$ is called topologically complete if there is metric $d$ on $X$ which induces the topology $\tau$ of $X$ and $(X,d)$ is complete metric space.
Also the following fact is true: If $f:X\to Y$ where $f$ is hom... | A preferred definition of topologically complete is
S is a topologically complete topological space
when S is homeomorphic to A complete metric space.
Clearly, complete metric spaces are topologically complete topological spaces. In particular, R with the usual metric is topologically complete, a topologically compl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$X_n$ is bounded in probability and $Y_n$ converges to 0 in probability then $X_nY_n$ congerges to probablity with 0 I want to show : $X_n$ is bounded in probability and $Y_n \rightarrow 0$ in probability then $X_nY_n \rightarrow 0 $ in probablity.
I know the following definitions that is
Definition 2.17 : We say th... | $P(|X_nY_n| >\epsilon) \leq P(|X_n| \leq M, |X_nY_n| >\epsilon)+P(|X_n| > M, |X_nY_n| >\epsilon)\leq P(|Y_n| >\frac {\epsilon} M)+[1-P(|X_n|<M)]$. For $n >N$ the second term is less than $\epsilon$ and the first term tends to $0$ as $n \to \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3419796",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Artin Schreier equation Let $K$ be the field obtained by adjoining to $\mathbb{Q}_p$ a root $\alpha$ of the polynomial $f(x)=x^p-x-\dfrac{1}{p}$. I should prove that $K \supset \mathbb{Q}_p$ is a Galois extension of degree $p$. I managed to prove that $K$ has degree $p$:taking $\dfrac{1}{\alpha}$ we see that is a root ... | It's just Hensel's lemma.
First note that the $p$-adic valuation of $\alpha$ is $-1/p$: from the identity $\alpha^p-\alpha=1/p$, it follows that $v_p(\alpha) <0$, and then by triangle inequality $v_p(\alpha^p)=v_p(\alpha^p-\alpha)=-1$.
Putting $K = \mathbb{Q}_p(\alpha)$ and $y = x - \alpha$, we want to show that the po... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3419972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Prove that $n+2$ points in $\Bbb R^n$ cannot all be at a unit distance from each other. I saw this on the FB group "Actually good math problems" where one solution was an illegible photo and the other involved intersecting spheres. I have a solution. I'd like to see how many good answers I get before I post it.
If $S$ ... | Let us assume that $\{P_1,\ldots,P_{n+2}\}$ is a set of points in $\mathbb{R}^n$ such that $\|P_i-P_j\|=1$ for any $i\neq j$.
We may assume without loss of generality that $P_{n+2}=O$. From the polarization formula
$$ 2\cos\theta_{ij}=2\langle P_i, P_j \rangle = \|P_i-P_j\|^2 + \|P_i+P_j\|^2 = 1+\|P_i+P_j\|^2\geq 1$$
i... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Determining decreasing sequence For a sequence $x_{n+1}=4x_n-x_{n-1}$, $x_1=4, x_2=15$, show that the sequence $\frac{x_{n+1}}{x_n}$ is decreasing
I know from calculating that $\frac{x_{n+1}}{x_n}-\frac{x_{n}}{x_{n-1}}=\frac{-1}{x_nx_{n-1}}$, but I can't seem to prove that.
Any tips please?
| $$x_{n+1}=4x_n-x_{n-1}$$
divide $x_n$ through out
$\dfrac{x_{n+1}}{x_n}=4-\dfrac{x_{n-1}}{x_n}$
$\implies \dfrac{x_{n+1}}{x_n} - \dfrac{x_{n}}{x_{n-1}} =4-\dfrac{x_{n-1}}{x_n} - \dfrac{x_{n}}{x_{n-1}} = 4-\dfrac{x_n^2+{x_{n-1}}^2}{x_nx_{n-1}} \le 0$
Because $a^2+b^2\ge 4ab$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Matrix with $i,j$ entry $a_i/(a_i + a_j)$ Suppose $a_1 > a_2 > \cdots > a_n > 0$, consider an $n\times n$ matrix $A$ with $i,j$ entry $\frac{a_i}{a_i + a_j}$. I am wondering if the matrix has a name/ any insight about the eigenvalue/eigenvectors? All I can observr is $A - 1/2$ is a skew-symmetric matrix, but nothing el... | $A$ is the product of two positive definite matrices $\operatorname{diag}(\mathbf a)$ and $\left(\frac{1}{a_i+a_j}\right)_{i,j\in\{1,2,\ldots,n\}}=\int_0^\infty e^{-x\mathbf a}e^{-x\mathbf a^\top}dx$, where $\mathbf a=(a_1,\ldots,a_n)^\top$ and $e^{\mathbf v}$ denotes the entrywise exponential of a row/column vector $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3420441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Which of the following relations on $\{1,2,3\}$ is an equivalence relation? $$\begin{array}{l}{R_{1}=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}} \\ {R_{2}=\{(1,1),(2,2)\}} \\ {R_{3}=\{(1,2),(2,3),(3,1)\}}\end{array}$$
$$\begin{array}{l}{R_{4}=\{(1,2),(2,1),(1,3),(3,1),(2,3),(3,2)\}} \\ {R_{5}=\{(1,1),(2,2),(3,3),(1,2),(2,3),(3,... | For transitivity you need for all $a,b,c\in \{1,2,3\}$:
If $a$ is related to $b$ and $b$ is related to $c$, then $a$ is related to $c$. You can indeed check for $R_{1}$ that this is true, since if you pick $a = 3$, then the statement is trivial. Since it only satisfies the "if" condition if you pick $b = 3$, since $3$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3420588",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
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If $x=9$, then can we write $\sqrt{x}=\pm3$ or only $\sqrt{x}=3$ If $x=9$, then can we write $\sqrt{x}=\pm3$ or only $\sqrt{x}=3$.
I am confused as square root always gives positive number.
But the irony is that if we have $a^2=9$, then we write $a=\pm3 \text { where $a=\sqrt{a^2}$ }$
| You are right that often $\sqrt{n}$ is meant as the positive root. As a result, it is very common to see the solutions to $x^2=n$ expressed as $x=\pm\sqrt{n}$ rather than $\pm x=\sqrt{n}$. While they are equivalent mathematically, the latte
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3420992",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
$\text{tr}(X)=\text{tr}(A^{-1}B)$ for $AX+XA=2B$ Suppose X is a solution to following equation over positive definite matrices $A$,$B$
$$XA+AX=2B$$
The following seems to hold (numerically)
$$\text{tr}(X)=\text{tr}(A^{-1} B)$$
Can anyone see the way to prove this?
| Since $A$ is positive definite, it has an inverse. Hence
$$ +=2 \\ \Rightarrow
A^{-1}XA +X=2A^{-1}B.$$
Taking $Tr(\cdot)$ on both sides leads to
$$Tr(A^{-1}XA) + Tr(X) = 2Tr(A^{-1}B)\\
\Rightarrow Tr(AA^{-1}X) + Tr(X) = 2Tr(A^{-1}B)\\
\Rightarrow Tr(X) + Tr(X) = 2Tr(A^{-1}B) \\
\Rightarrow Tr(X) = Tr(A^{-1}B) $$
The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3421167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that $(11 \cdot 31 \cdot 61) | (20^{15} - 1)$ Prove that
$$ \left( 11 \cdot 31 \cdot 61 \right) | \left( 20^{15} - 1 \right) $$
Attempt:
I have to prove that $20^{15}-1$ is a factor of $11$, $31$, and $61$. First, I will prove
$$ 20^{15} \equiv 1 \bmod11 $$
Notice that
$$ 20^{10} \equiv 1 \bmod 11$$
$$ 20^{5} ... | $$20\equiv3^2\pmod{11}$$
$20^{15}\equiv(3^2)^{15}\equiv(3^{10})^3\equiv1^3$ by Fermat's Little Theorem
$$20=2^2\cdot5,\implies20^{15}\equiv2^{30}5^{15}$$
By Fermat's Little Theorem
$$2^{30}\equiv1\pmod{31},5^3\equiv1\pmod{31}\implies5^{15}=(5^3)^5\equiv1^5$$
Again $5^3\equiv3\pmod{61},2^6\equiv3$
$$20^{15}\equiv(2^6)^5... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Problem of Harsthorne page 35 problem 5.1. Q Which is which in Figure?
a)$x^2=x^4+y^4$
b)$xy=x^6+y^6$
c)$x^3=y^2+x^4+y^4$
d)$x^2y+xy^2=x^4+y^4$
a)$x^2=x^4+y^4$
This is invariant under the transformation $x\mapsto -x$ and $y\mapsto -y$. Thus it is Tacnode.
b)$xy=x^6+y^6$
It is invariant under the map $(x,y) \mapsto (... | Hint: The homogeneous term of lowest degree tells you the tangent directions at the origin. For instance, the lowest order term of $x^2 - x^4 - y^4 = 0$ is $x^2 = x \cdot x$, so the line $x=0$ is a double tangent line at the origin. Only one of your graphs has this property--can you see which one?
You can match the oth... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
All subsets of $\{1,\{\}\}$ I'm trying to figure out what all subsets of the set $A \colon= \{1,\{\}\}$ are.
I am not sure if the answer is:
*
*$P(A) = \{ \{ \}, \{1\}, \{1;\{\}\}, \{\{\}\} \}$ or
*$P(A) = \{ \{\}, \{1\}, \{1; \{\}\} \}$
| Recall that the power set is the set of all subsets of A, including the empty set and A itself therefore in this case
$$P(A) = \Big\{ \{\}, \{1\}, \{\{\}\}, \{1,\{\}\} \Big\}$$
which indeed, since $|A|=n=2$, has $2^n=4$ elements.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Prove that the convex hull is the union of all the triangles determined by triples of points from X I'm trying to prove that the convex hull of a set X of three or more points in the plane is the union of all the triangles determined by triples of points from X, however I can't think of the meaningful approach to go wi... | (1) Every triangle must be a subset of the convex hull. (2) Suppose some edge of the convex hull doesn't belong to any triangle -> contradiction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3421726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Confusing proof for $\sqrt{2}$ being irrational $\sqrt{2}$ is irrational using proof by contradiction.
say $\sqrt{2}$ = $\frac{a}{b}$ where $a$ and $b$ are positive integers.
$b\sqrt{2}$ is an integer. ----[Understood]
Let $b$ denote the smallest such positive integer.----[My understanding of this is
that were are go... | Your initial premise is that $b$ is defined to be the smallest positive integer such that $b\sqrt 2$ is a positive integer. This is equivalent to reducing $\frac ab$ to its lowest terms (that is, make $a$ and $b$ coprime).
Obviously $b^\mathrm * = b\sqrt 2 - b = a-b$ will be a positive integer such that $b^*<b$.
And w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\int_{a}^{b}xf(x)dx=a\int_a^cf(x)dx + b\int_c^bf(x)dx$ $f:[a,b]\to\mathbb{R}$ continous. Show that existe c in $[a,b]$ such that $$\int_{a}^{b}xf(x)dx=a\int_a^cf(x)dx + b\int_c^bf(x)dx.$$
I already tried integration by parts, and mean value theorem for integrals... I need a light
| One can proceed as in Prove for continous function $f$, $\int_0^1 xf (x) dx = \int_c^1 f (x) dx $, where the case $[a,b]=[0, 1]$ is handled.
Define $F(x) = \int_a^x f(t) \, dt$ and integrate by parts:
$$
\int_a^b xf(x) \,dx = xF(x) \bigr]_a^b - \int_a^b F(x) \, dx = bF(b) - \int_a^b F(x) \, dx\, .
$$
From the mean val... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3422060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How I can solve this : $-2^x+3^{x/2}+1=0$ without using numerical ways? This equation : $-2^x+3^{x/2}+1=0$ is confusing me , However it has only one solution which it is an integer $ x=2$ , But i can't resolve it using clear way , I have used the varibale change $y= x/2$ in order to transforme it in equation of degree... | Using the equivalent equation as of the hint of zeraoulia rafik, one observes that the equation
$$
-1+\left(\frac{\sqrt3}2\right)^x+\left(\frac12\right)^x=0
$$
is constant or strictly monotonously falling in all terms on the left side, so that also the whole left side as their sum is strictly falling.. Which proves tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3422149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Unifying the product of odd powers of function with the product of even powers of the same function in one product. Assume that
\begin{equation}
\begin{split}
f_n(x)&= \begin{cases} \big(g(x)\big)\cdot \big(g(x)\big)^3\cdot \big(g(x)\big)^5\cdots \big(g(x)\big)^{n-1},& n \ \text{even},\\
\big(g(x)\big)^2\cdot \big(g(x)... | you can do much better actually. Let me start with $n=2N+1$ odd, then
$$
f_n(x)=g(x)^{\sum_{k=0}^{N}(2k)}=g(x)^{2\sum_{k=0}^{N-1}k}=g(x)^{N(N+1)}
$$
and for $n=2N$ even
$$
f_n(x)=g(x)^{\sum_{k=0}^{N-1}(2k+1)}=g(x)^{2\sum_{k=0}^{N-1}k+N}=g(x)^{N(N+2)}
$$
There are many ways to rewrite this in a single line, one of them... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
Proving $\int_{\Bbb R} \frac{\cos x}{(x^2+t^2)^2}dx=\frac{\pi(t+1)}{2t^3e^t}, t>0$
I want to prove that $$\int_{\Bbb R}\frac{\cos x}{(x^2+t^2)^2}dx=\frac{\pi(t+1)}{2t^3e^t},t > 0$$
Integral calculator with steps gives the following answer.
$\displaystyle\int_{\Bbb R}\frac{\cos{(x)}}{(x^2+t^2)^2 }dx=\frac{(t*\cosh{(t)... | Another way to do it is to note that$$\frac{1}{x^2+t^2}=\frac{1}{2it}\left(\frac{1}{x-it}-\frac{1}{x+it}\right)\implies\frac{1}{(x^2+t^2)^2}=\frac{-1}{4t^2}\left(\frac{1}{(x-it)^2}+\frac{1}{(x+it)^2}+\frac{i}{t}\left(\frac{1}{x-it}-\frac{1}{x+it}\right)\right).$$The $e^{ix}$ in $\cos x$ doesn't diverge when $x=i\infty$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3422707",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
An interesting problem of polynomials
In the polynomial
$$
(x-1)(x^2-2)(x^3-3) \ldots (x^{11}-11)
$$
what is the coefficient of $x^{60}$?
I've been trying to solve this question since a long time but I couldn't. I don't know whether opening the brackets would help because that is really a mess. I have run out of... | Hint : $1+2+3 +...+11= \frac {11×12}{2} =66 $ so we must find how we can construct number $6=6+0=5+1=4+2=3+3=1+2+3$ and note that $3+3$ impossible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3422830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 1
} |
$\sum_{n=1}^{\infty} 1/\sqrt[n]{n}$ converge Does the series:
$$
\sum_{n=1}^{\infty} \frac{1}{\sqrt[n]{n}}
$$
converge or diverge?
I'm unsure where to start with this question. I know that $n$th root of $n$ converges to $1$ but not sure about its reciprocal.
| If a series $\sum_{n=0}^\infty a_n$ converges, then $\lim_na_n=0$. Therefore your series diverges.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3423011",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
integration the very concept so i was just taught about integration and one thing i do not understand is that say we integrate x dx from 1 to 2 ... there are an infinite number of numbers between them ... so how does the sum turn out to be finite ?
| Integration has got nothing to do with adding up numbers, let alone all numbers between two bounds which would be clearly impossible. It is rather about computing the area of the region bounded by the curve of the function, the $x$-axis, and the two vertical lines defined by $x=1$ and $x=2$.
This computation can be do... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3423164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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How to check if a subset is open in Zariski I'm having troubles determining if a given subset of $\operatorname{Spec}A$ is open or not. The contest is not trivial.
I have to consider a morphism of finitely generated $k$-algebras $A\rightarrow B$, which are also integral domains. We assume that the map induced on the fr... | Here's a suggestion.
If you can prove that for any $q\in U$, there is some $f\in A\setminus q$ such that $B\otimes_A A_f$ is finite as an $A_f$-module, then you are done. Do you see why this would mean that you are done?
This is proving that every point in $U$ has a standard neighborhood, $q\in D(f)\subseteq U$. To s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3423295",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Basics of manifolds and derivatives
I'm really struggling with the very basics definitions of this course on Riemannian geometry and I was wondering if someone could point me in the right direction in regard to this question (Please don't just tell me the answer)
I think I mainly know how to do part (a) because identi... | Hints:
Step 1. Compute $c'(t)$ for general $t$ as a function of $t$.
Step 2. Compute $h=\varphi\circ \pi$.
Step 3. Compute both $x_1, x_2$ -components of the vector field (along $\varphi\circ \gamma$) $Dh(c'(t))$ as functions of $t$.
Step 4. These will be $\alpha_1(t)$ and $\alpha_2(t)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3423463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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How to show that $I = \langle x^2+2\rangle \subseteq \mathbb{Z}[x]$ is a prime ideal WITHOUT proving that $x^2+2$ is irreducible?
How to show that $I = \langle x^2+2\rangle \subseteq \mathbb{Z}[x]$ is a prime ideal WITHOUT proving that $x^2+2$ is irreducible in $\mathbb{Z}[x]$?
I know how to show it by showing that i... | Consider the ring homomorphism $f:\mathbb Z[x] \to \mathbb C$ induced by $x \mapsto \sqrt{2}\,i$. Then $\mathbb Z[x]/\ker f$ is certainly a domain and so $\ker f$ is a prime ideal. Prove that $\ker f = \langle x^2+2\rangle$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Endomorphism ring of an irreducible module is irreducible Let $M$ be an $A$-module and define $B:=End_A (M)$. Prove that if $M_A$ is irreducible (that is, the only decomposition of $M_A$ in direct sumands is the trivial decomposition) then $B_{B}$ is irreducible.
I have no idea on how to tackle this. I tried this.
Cons... | You’re on the right track.
From where you left off, $\pi(M)\oplus (1-\pi)(M)=M$ is a direct decomposition of $M$. Since one must be zero, you have $\pi=1$ or $\pi=0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3423752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does the tensor product respect semidefinite ordering in this way? I'll use $\succeq$ to denote the positive semidefinite ordering: for square matrices $X,Y$, one has $X \succeq Y$ iff $X - Y$ is positive semidefinite. It's a well known fact that if $X, Y \succeq 0$ then $X \otimes Y \succeq 0$. However, if one has two... | \begin{aligned}
&X\otimes X'-Y\otimes Y'\\
&=\left[(X-Y)+Y\right]\otimes\left[(X'-Y')+Y'\right]-Y\otimes Y'\\
&=\left[(X-Y)\otimes(X'-Y')+Y\otimes(X'-Y')+(X-Y)\otimes Y'+Y\otimes Y'\right]-Y\otimes Y'\\
&=(X-Y)\otimes(X'-Y')+Y\otimes(X'-Y')+(X-Y)\otimes Y'.
\end{aligned}
Now the result follows because $X-Y,\,X'-Y',\,Y$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Is there a difference between "Linear System of Equations" and "System of linear Equations"? We are translating German math videos to English and stumbled over the fact that there are two translations of "Lineare Gleichungssysteme":
*
*Linear System of Equations
*System of linear Equations (e.g. Wikipedia)
From m... | The two terms are interchangeable and mean the same thing. The term "linear system of equations" should be broken up as "(linear) (system of equations)", meaning that the system of equations is linear, i.e. that it involves equations which are linear. This is the same as a "system of linear equations", of course. You c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3424071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integral of $\int_0^1 \frac{\sin(a \cdot \ln(x))\sin (b \cdot \ln(x))}{\ln(x)} dx$ I'm trying to calculate the integral $$\int_0^1 \frac{\sin\Big(a \cdot \ln(x)\Big)\cdot \sin \Big(b \cdot \ln(x)\Big)}{\ln(x)} dx, $$
but am stuck. I tried using Simpsons' rules and got here:
$$\int_0^1 \frac{\cos\Big((a+b) \cdot \ln(x)\... | For $c \in \mathbb{R}$ we have
\begin{align}
\int \limits_0^\infty \frac{1 - \cos(c t)}{t} \, \mathrm{e}^{-t} \, \mathrm{d} t &= \int \limits_0^\infty \int \limits_0^c \sin(u t) \, \mathrm{d} u \, \mathrm{e}^{-t} \, \mathrm{d} t = \int \limits_0^c \int \limits_0^\infty \sin(u t) \mathrm{e}^{-t} \, \mathrm{d} t \, \mat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3424189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Functional notation usage for arbitrary relation? Question about questionable notation used in a homework assignment.
The assignment defines a relation f := { (0,1), (1,3), (2,1) }. It then asks to "show" this is a function. IMO strictly (i.e. Bourbaki) speaking it is not but so be it. The intention is clear.
It furthe... | The inverse of a function f is f$^{-1}$ when the inverse of f exists.
The inverse of a relation R is R$^{-1}$ = { (x,y) : yRx }.
The set extensions of a function f are f[A] = { f(x) : x in A }
and f$^{-1}$[A] = { x : f(x) in A }.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3424311",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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The number of real roots of the equation $5+|2^x-1|=2^x(2^x-2)$ I'm trying to find the number of real roots of the equation $5+|2^x-1|=2^x(2^x-2)$.
Let $2^x=a$
$$|a-1|=a^2-2a-5$$
Then there are two cases
$$a-1=a^2-2a-5$$
And $$a-1=-a^2+2a+5$$
Solving both equations
$$a=1,-4,-2,3$$
Now -4 and -2 can be neglected so th... |
Then there are two cases
$$a-1=a^2-2a-5$$
$$a-1=-a^2+2a+5$$
Solving both equations
$$a=1,-4,-2,3$$
You made a small mistake in solving these.
The first equation is $a^2 - 3a - 4 = 0$, so $(a - 4)(a + 1) = 0$, so $a = 4$ or $a = -1$.
Looks like you solved the second equation correctly. You should have
$$
a = -1,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3424491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
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How these 3 vectors form a right triangle? $ A = 2\hat{i} -2\hat{j} + 3\hat{k}$
$ B = 2\hat{i} -\hat{j} + 3\hat{k}$
$ C = \hat{i} -\hat{j} - \hat{k}$
My textbook demands that these 3 vectors form right triangle. Firstly, I think these 3 vectors don't form a triangle.
Actual Condition for triangle : $ A\pm{B}\pm{C} = 0... | The given vectors are position vectors. To get the vectors representing the sides of the triangle do $A-B, B-C, C-A$. You will find that $\angle B = 90^{\circ}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3424658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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Evaluating $ \lim_{x\to +\infty} x\left(\frac{\pi}{4} - \arctan\left(\frac{x}{x+1}\right)\right) $
$$ \lim_{x\to +\infty} x\left(\frac{\pi}{4} - \arctan\left(\frac{x}{x+1}\right)\right) $$
I tried to do this with some kind of substitution but failed miserably. Any hints or help?
| Substitute $y:=\frac{1}{x+1}$ to rewrite your limit as$$\lim_{y\to0}(1-y)\frac{(\arctan 1-\arctan(1-y))}{y}=\lim_{y\to0}(1-y)\cdot\arctan^\prime1=\frac12.$$Or if we take @DinnoKoluh's approach,$$\arctan1-\arctan\frac{x}{x+1}=\arctan\frac{1}{2x+1}\approx\frac{1}{2x}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3424752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Finding the complete Taylor expansion of $\frac{1}{1+z^2}$ around $z=0$. For an exercise, I need to find the complete Taylor expansion for $(1+z^2)^{-1}$ around $z=0$. I have tried decomposing first $(1+z^2)^{-1}$ into partial fractions. Since $1+z^2=0$ gives $z=\pm i$, the partial fractions are:
$$\frac{1}{1+z^2} = \f... | Your approach is all correct. Note that your sum can be rewritten as $$\frac{1}{2}\sum_{n=0}^\infty i^n ((-1)^n+1)z^n.$$
For an odd index value $n$, the term equals zero, and for even - it equals $2(-1)^kz^{2k}$, $k\in \mathbb{Z}$.
Therefore
$$\frac{1}{2}\sum_{n=0}^\infty i^n ((-1)^n+1)z^n=\frac{1}{2} ( 2-2z^2+2z^4-2z^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3424898",
"timestamp": "2023-03-29T00:00:00",
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Disproving $ 0^0 $ by binomial theorem For fun, is something like this true?
Let
\begin{equation} \nonumber
\begin{split}
k &= 0^0 = (a-a)^{a-a} = \frac{(a-a)^{a}}{(a-a)^{a}} \\
&= \frac{\binom{a}{0}a^a(-a)^0 + \binom{a}{1}a^{a-1}(-a)^1 + ... +
\binom{a}{a-1}a^{1}(-a)^{a-1} + \binom{a}{a}a^{0}(-a)^{a}}{\binom{... | You are not disproving anything. $0^0$ is a mathematical expression with no agreed-upon value. The most common possibilities are $k=1$ or leaving the expression undefined, with justifications existing for each, depending on context.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3425066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$a_{n}$ geometric sequence, prove that : $a_{1}+a_{2}+a_{3}+...+a_{n}\|\ a_{1}^{k}+a_{2}^{k}+a_{2}^{k}+...+a_{n}^{k}$ with $(n,k)=1$ Problem :
Let $a_{n}$ be a geometric sequence of the integer numbers $a_{n}\in \mathbb Z$ for all $n\in \mathbb N$. Prove that:
$$a_{1}+a_{2}+a_{3}+...+a_{n} \mid a_{1}^{k}+a_{2}^{k}+a... | Show that the greatest common divisor of $r^k-1$ and $r^n-1$ is $r-1$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculate $\lim_{x\to\infty}\biggr(x\sqrt{\frac{x}{x-1}}-x\biggr)$ $$\lim_{x\to\infty}\biggr(x\sqrt{\frac{x}{x-1}}-x\biggr)$$
I know this limit must be equal to $\frac{1}{2}$ but I can't figure why. This is just one of the thing I tried to solve this limit:
$$\lim_{x\to\infty}\biggr(x\sqrt{\frac{x}{x-1}}-x\biggr)$$
$$... | By application of L' Hopital's rule:
$$\lim_{x\to +\infty}\frac{\sqrt{\frac{x}{x-1}}-1}{\frac{1}{x}}=\left(\frac{0}{0}\right)=
\lim_{x\to +\infty}\frac{\left(\sqrt{\frac{x}{x-1}}-1\right)'}{\left(\frac{1}{x}\right)'}=
\ldots= \frac{1}{2}\lim_{x\to +\infty}\frac{\frac{x^2}{(x-1)^2}}{\sqrt{\frac{x}{x-1}}}=\frac{1}{2}\cd... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3425638",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 4
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Indexing twice for conditional summation Clearly it is permitted to put restrictions on the indices in a summation, and
One often sees generalizations of this notation in which an arbitrary logical condition is supplied, and the sum is intended to be taken over all values satisfying the condition. Here are some common... | You don't actually need the condition in the summation, nor the verbose use of Iverson brackets:
$$1\over\displaystyle\sum_{i=1}^n\frac1{E_i}[E_i\neq 0]$$
This succinctly expresses the idea of summing the reciprocals of the non-zero elements. Now, you might counter that this involves the undefined $1/0$ when $E_i=0$, b... | {
"language": "en",
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Continuous function and IVT. Let $f:[0,1]->\mathbb R$ be a continuous function which satisfies $f(0)=f(1)$. Prove that there exists a number $a \in [0,1/2]$ such that $f(a)=f(a+1/2)$.
I know that i'll use IVT theorem but i confused a bit.Can you help me ?
| Consider the function $g: [0, \frac{1}{2}] \to \mathbb{R}$ given by
$$
g(x) = f(x) - f\left(x + \frac{1}{2}\right).
$$
Then $g$ is continuous, and $g(0) = -g\left(\frac{1}{2}\right)$. It follows from the intermediate value theorem that $g$ has a zero somewhere.
| {
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Solution of simple algebraic equations
Suppose $a=\alpha+2\beta$ and $b=3\alpha+5\beta$. Then by solving this, we get $\alpha=2b-5a$ and $\beta=3a-b$.
I tried substitution method to solve this, but i couldn't get the above solutions viz., $\alpha$ and $\beta$.
I am afraid if this is too basic and it might not be appr... | Since $a=\alpha+2\beta$ and $b=3\alpha+5\beta$,$$b-3a=(3\alpha+5\beta)-3(\alpha+2\beta)=-\beta.$$So, $\beta=3a-b$ indeed. And now it is easy to get $\alpha$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Irreducible polynomial in integer polynomial ring with complex root is a reciprocal polynomial I am working on this problem:
Let $p(x) \in \mathbb{Z}[x]$ be an irreducible polynomial. Show that $p(x)$ is a reciprocal polynomial (i.e., that its coefficients equidistant from either end are equal) if one of its roots is ... | The following proof is due to a classmate of mine:
Let $K$ be the splitting field of $p(x)$ over $\Bbb Z$ and $\sigma\in Aut(K/\Bbb Z)$ such that for every root $\alpha$ of $p(x), \sigma(\alpha)=\dfrac1{\alpha}$. But $\sigma$ sends any root of $p$ to some root of $p$, which means that the set of all roots of $p$ is the... | {
"language": "en",
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Why is $\lim\limits_{n\to \infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$ not equal to $1$? So I saw the limit $\lim\limits_{n\to \infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}$ here the other day:
Evaluating $\lim\limits_{n\to\infty} e^{-n} \sum\limits_{k=0}^{n} \frac{n^k}{k!}$
and when I saw it, I right away thought the answer... | What you're doing is taking the identity
$$
\lim_{n\to\infty}\sum_{k=0}^n\frac{x^k}{k!}=e^x\tag1
$$
and plugging in $x=n$ to obtain the (false) statement
$$
\lim_{n\to\infty}\sum_{k=0}^n\frac{n^k}{k!}=e^n.\tag2
$$
Why is (2) false? Setting $x=n$ in (1) is illegal because the $n$ in (1) is busy being used as the label f... | {
"language": "en",
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Simplifying ArcSine Function I was wondering if there is a nice formula (or approximation) for $\arcsin(x)$ which is defined $[-1,1]$?
| As noted there is no way to explicitly express $\arcsin(x)$ in terms of e.g. the $\sin,\cos,\tan,\exp,\log$ functions. After all, if there were, why would we need to invent the new notation $\arcsin$? There are however nice approximations to the function. The most obvious one which comes to mind is the Taylor approxima... | {
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Finding the Inverse A function $g$ is defined by $g(t) = 0.3(1 - \cos(2\pi t))$.
For $n \in \mathbb{N}$, $I_n = [\frac{n}{2}, \frac{n+1}{2}]$. I have found that on this $n$-interval, $g$ is one-to-one meaning we can take the inverse of $g$ when restricted to this domain.
For $n \in \mathbb{N}$, $h_n$ is the inverse o... | The problem here is that when you try to invert the function $g(t)$, it will automatically appear to be in the interval $\left [ 0, \frac{1}{2}\right ]$ for even $n$, or $\left [ -\frac{1}{2}, 0\right ]$ for odd $n$. To amend this, simply define a translation function $f_{even}: \left [ \frac{n}{2}, \frac{n+1}{2}\righ... | {
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Minimum value of Complex Trigonometric Expression
Minimum value of of $\displaystyle f(\theta) = \frac{a}{\cos \theta}+\frac{b}{\sin \theta}+\sqrt{\frac{a^2}{\cos^2 \theta}+\frac{b^2}{\sin^2 \theta}}.$
Where $\displaystyle a,b>0, \theta \in \bigg(0,\frac{\pi}{2}\bigg).$
what i try
$$f(\theta)=\frac{2(a\sin \theta +... | \begin{align*}
f'(\theta) &= a \tan \theta \sec \theta + b \tan \theta \sec \theta \hfill \\
&\quad {}+ \frac{2 a^2 \tan \theta \sec^2 \theta + 2 b^2 \tan \theta \sec ^2\theta }{2 \sqrt{a^2 \sec ^2 \theta + b^2 \sec^2 \theta}} \\
&= \tan \theta \sec \theta \left(\cos \theta \sqrt{\left(a^2+b^2\right) \sec^2 \theta }+... | {
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"timestamp": "2023-03-29T00:00:00",
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Show that if $f : [1, 2] →\Bbb R$ is a continuous then there exists $\gamma\in (1, 2)$ with $f(\gamma) = \frac{1}{1 − \gamma}+ \frac{1}{2 − \gamma}.$ Show that if $f : [1, 2] →\Bbb R$ is a continuous function then there exists $\gamma\in (1, 2)$ such that $$f(\gamma) = \frac{1}{1 − \gamma}+ \frac{1}{2 − \gamma}.$$
I ha... | Consider the function $$g:x\mapsto f(x)-\frac{1}{1 − x}- \frac{1}{2 − x},1<x<2.$$ Since $\lim_{x\to 1+}f(x)=f(1)$ and $\lim_{x\to 2-}f(x)=f(2)$ we have, $$\lim_{x\to 1+}g(x)=+\infty,$$$$\lim_{x\to 2-}g(x)=-\infty.$$ So by intermediate value theorem we have $\gamma\in (1,2)$ such that, $g(\gamma)=0$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Automorphism with no square root Related to Every normal operator on a separable Hilbert space has a square root that commutes with it
Does it exist an automorphism $f$ in a separable $\mathbb C$ Hilbert space, such that $f$ has no square root?
If so, a concrete example would be useful.
| Yes, such operators exist. This was proven by Halmos, Lumer, and Schäffer, Proc. AMS, 4, 1 (1953), 142-149.
Concretely, given a domain $D\subset\mathbb C$ define
$$
D^{1/2}=\{\lambda\in\mathbb C:\ \lambda^2\in D\}.
$$
They proved that the multiplication operator $M_z\in B(L^2(D))$ given by $(M_zf)(z)=zf(z)$ has a sq... | {
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Determining Weight function in Sturm Liouville problem By choosing the proper weight function $\sigma (x) $ solve the Sturm-Liouville problem and determine its eigenvalues and eigenfunctions.
$$ \frac{d}{dx}\left[x\frac{dy(x)}{dx}\right] + \frac{2}{x}y(x) +\lambda \sigma (x)y(x)=0,\; y'(1)=y'(2)=0,\; 1 \leq x \leq 2. $... | For generic 2nd order (homogeneous) ODE:
$ a(x)y''(x) + b(x)y'(x) +c(x)y(x) = 0 $
To transform it to Sturm Liouville form you need:
$ a(x)>0$ and $a,b,c$ to be on continuous on the interval of definition.
Now the weight function is defined as follows:
$w(x) = \dfrac{1}{a(x)}e^{\int \frac{b(x)}{a(x)}dx}$
We now set:
$ p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Could the set composed of real quaternion be represented by a Hilbert space? Assume we have "real" quaternions: $Q = a+bi+cj+dk$ where $a,b,c,d$ are real numbers. The dot product between any two real quaternions is an inner product, and we can define the length of a quaternion $Q$ as $|Q| = \sqrt{<Q,Q>}$.
I am still v... | The key challenge is choosing a field for $\Bbb H$ to be a space over. (Bear in mind $\Bbb H$, unlike fields, isn't commutative.) One way to do this is to consider $\Bbb H$ a $2$-dimensional Hilbert space over $\Bbb C$ with basis $1,\,j$, so $a+bi+cj+dk=(a+bi)1+(c+di)j$ is a unique decomposition. Then $$\langle a+bi+cj... | {
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Is it possible to determine the given matrix is positive semidefinite under these conditions? Suppose I have a $2^n$ by $2^n$ symmetric matrix M. I know the following facts are true about $M$.
*
*The diagonal of M is $n+1$, which is strictly larger than any other non-diagonal entry.
*The sum of each row of the matr... | The hypoteses given on $M$ don't allow to conclude that $M$ is positive semidefinite. The following is a counterexample for $n=3$. Define:
$$A:=\begin{bmatrix}
4& -2& -2& -2\\
-2& 4& -2& -2\\
-2& -2& 4& -2\\
-2& -2& -2& 4\\
\end{bmatrix}
\qquad
B:=\begin{bmatrix}
3 & 3 & 2 & 2\\
3 & 3 & 2 & 2\\
2 & 2 & 3 & 3\\
2 & 2 &... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why does the Well Ordering Theorem imply decreasing ordinals go to zero? The Well Ordering Theorem states that any set can be well ordered. But in this PBS Infinite Series video, Kelsey states the theorem as, "Any decreasing sequence of ordinals eventually goes to zero." This isn't an obvious equivalence to me. Does de... | If the sequence is decreasing, it has to be finite, since a well-ordering does not have any infinitely decreasing sequences. But that means that we have to stop somewhere, and if you take "decreasing" to mean "continue to decrease as long as you can" that means that the last element of the sequence must be the minimum,... | {
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Find a restricted domain so that the function is injective The sinus function is not injective when the domain is the whole $\mathbb{R}$.
To find a restricted domain where the function is injective, can we do that only using the graph or is there also an other way?
Such a resticted domain is for example $\left[-\frac... | Let $f: A \to B$ be any function.
The restriction of $f$ to the empty set is injective.
Let $f: A \to B$ be any function defined on a nonempty set $A$.
If $a \in A$ then $f$ restricted to the singleton set $\{a\}$ is injective.
Let $f: A \to B$ be any function defined on a nonempty set $A$. Let the range of $f$ be de... | {
"language": "en",
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How to prove using only set identities? I have given three sets A, B and C and I need to prove that following to statements are equivalent:
S1 = (( − ) − ) ∪ ( − ( − ( ∪ ))) − ( ∩ ( ∩ ))
S2 = ( ∩ ) − ( ∩ ( ∩ )) ∪ ( − )
prove that S1 is equivalent to S2 without using venn diagram.
How can I prove this?
| Using Boolean algebra you consider the Boolean variables
*
*$a,b,c$ with $a=1$ ($1$ stands for $True$) iff $x \in A$. Similarly, with $b$ and $c$.
Now, write for each set the corresponding Boolean expressions
*
*$x \in S_1 = (cb')a' + b(b(a+c)')'(abc)'$
*$x \in S_2 = ab(abc)'+ca'$
Now, the only thing you need ... | {
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Which ring is $R[X,Y,Z,T]/(X-Y^2,T-Y^4,T^3-Z)$ isomorphic to?
Which ring is $R[X,Y,Z,T]/(X-Y^2,T-Y^4,T^3-Z)$ isomorphic to?
I already did substitution for $X$ so we get the ring $R[x,x^{1/2},x^6,x^2]$ but I don't know to which ring this is isomorphic.
| Note that $T^3 - Y^{12}$ divisible by $T- Y^4$, so we have the equality of ideals
$$\langle T- Y^4, T^3 - Z\rangle = \langle T- Y^4, T^3 - Y^{12}, T^3 - Z\rangle =\langle T- Y^4, Z- Y^{12}\rangle$$
and so
$$\langle X-Y^2, T- Y^4, T^3 - Z\rangle = \langle X-Y^2, T- Y^4, Z- Y^{12}\rangle$$
Now the evaluation map
$R[Y,X,... | {
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"timestamp": "2023-03-29T00:00:00",
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Is there any relation between $Rank(A^2)$ and $Rank(A^3)$ if $Rank(A)=Rank(A^2)$? It is a question from my textbook :
$A$ is a square matrix of order $n\times n$. If $Rank(A)=Rank(A^2)$ then verify whether $Rank(A^2)=Rank(A^3)$ or not.
It is definite that $Rank(A^3)\leq Rank(A^2)$ but after that I cannot proceed.
Ple... | In terms of triangularisation of $A$ which does not change the rank you may suppose $A$ upper triangular with the last eigenvalues the zero ones, if algebraic multiplicity and geometric multiplicity of $0$ as eigenvalue are equal, $\textrm{rank}(A^n)$ is constant for all $n$, now if not $A$ as upper triangular has its ... | {
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The Closed-Form of $\displaystyle \int_{0}^{1}\mathrm{li}(x)\ln\Big(\ln\Big(\frac{1}{x}\Big)\Big) \mathrm{d}x$ $$\mathrm{Prove \;that } \int_{0}^{1}\mathrm{li}(x)\ln\Big(\ln\Big(\frac{1}{x}\Big)\Big) \mathrm{d}x\;\;=\;\; \frac{1}{2}\zeta(2)+\frac{1}{2}\ln^2(2)+\gamma \ln(2)$$
I found this problem on a group in facebook... | To solve this problem I will use a very nice result
$$\int_{0}^{1}\mathrm{li}(x)\sin\Big(u\ln\Big(\frac{1}{x}\Big)\Big) \mathrm{d}x \;\;=\;\; \frac{1}{u^2+1}\Bigg(\tan^{-1}\Big(\frac{u}{2}\Big)-\frac{u\ln(u^2+4)}{2}\Bigg)$$
Multiply both sides by $\displaystyle \frac{\ln(u)}{u} $ and integrate both sid... | {
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Why does $\left(x \cdot \tan\left(\frac{1}{x}\right)-1\right)^{-1}$ asymptotically approach $3x^2 - 6/5$? I noticed that $\lim_{x \to \infty}\tan\left(\frac{1}{x}\right)*x = 1$ and I was wondering how fast it approaches $1$. I looked at $\frac{1}{\tan\left(\frac{1}{x}\right)*x-1}$ and found that this grows slower than ... | We have that by Taylor's series
$$\tan\left(\frac{1}{x}\right)=\frac1x+\frac1{3x^3}+o\left(\frac1{x^3}\right)$$
and therefore
$$\left(\tan\left(\frac{1}{x}\right)\cdot x-1\right)\cdot x^2=\left(1+\frac1{3x^2}+o\left(\frac1{x^2}\right)-1\right)\cdot x^2=\frac 13+o(1) \to \frac 13$$
and since
$$\tan\left(\frac{1}{x}\righ... | {
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"timestamp": "2023-03-29T00:00:00",
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Tetration convergence: prove $\lim_{x\rightarrow0} {}^{n}x = \begin{cases} 1, & n \text{ even} \\ 0, & n \text{ odd} \end{cases}$ I'm a computer student, learning math just for fun. Today I was graphing for fun that I found something strange! I noticed that that wired function ${x^{x^{\cdot^{\cdot^{x}}}}}$ in zero, see... | Sorry, I'm leaving an answer so I can show an image. Note the red colored large kidney in the center of the image. It is the location of period one convergence and is referred to as the Shell-Thron Region (STR). Immediately to left of the center of the STR is a small yellow disk of period two convergence. Note that $^... | {
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Can you go from the integral of x squared to the summation of x squared? To be specific, I was wondering if you could go from the indefinite integral of $x^2$, mainly
$\int$ $x^2$$dx = \frac{x^3}{3}$, to the summation of $x^2$, $\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$?
| The difference between the integral and the actual sum is the difference of $n^3/3$ and $\frac {n(n+1)(2n+1)}{6}$ that is $$\frac{(n+1)^2}{6}$$
Which grows very fast with $n$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Any clues on how to do this modular arithmetic proof? Assume:
*
*$2x^3 - 8x^2 + 8y^3 - 12y^2 -10 \equiv 0 \mod 10$.
*$2y^3 - 8y^2 + 8z^3 - 12z^2 -10 \equiv 0 \mod 10$.
WTP:
*
*$2x^3 - 8x^2 + 8z^3 - 12z^2 -10 \equiv 0 \mod 10$.
I'm not sure where to start. How should I go about this?
| Reducing $\pmod {10}$ gives:
$2x^3 +2x^2 - 2y^3 - 2y^2 \equiv 0 \pmod {10}$
So:
$2x^3 +2x^2 \equiv 2y^3 + 2y^2 \pmod {10}$
Similarly from eqn. 2 we get:
$2z^3 +2z^2 \equiv 2y^3 + 2y^2 \pmod {10}$
The equivalence relation is transitive, therefore:
$2z^3 +2z^2 \equiv 2x^3 + 2x^2 \pmod {10}$
and so eqn. 3 is true.
| {
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Finite Complement Topology and Local Path Connectedness I'm having trouble with deciding whether or not a given space is locally path connected.
Let $(R,F)$ denote the finite complement topological space over the real numbers.
a) Determine the connected and path-connected components of (R,F).
b) Is (R,F) locally path-... | In the finite complement topology, and infinite subset of the same size is homeomorphic to the whole space (any bijection between finite complement topologies is a homeomorphism).
And injective mapping from $[0,1]$ (usual topology) into a space with the finite complement topology is continuous.
The second fact implies ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3429481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
Picking a popular vs unpopular friend There are 2 types of people on a social media platform: type A has 80 friends, type B has 20 friends.
We assume that half of the people are Type A, and half of the people are Type B.
The expected number of friends for a person is then $\frac{1}{2} \times 80 + \frac{1}{2} \times 2... | In the overall population if a person is selected at random, they are equally likely to be Type A or Type B. However, your friends circle is not representative of the overall population.
It's like say half of all people in the world are rich and half are poor. A guy walks up to you in the street and gives you one dolla... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3429630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Number in binary as a product I agree that any binary number that consists of $n$ ones (and no zeros) has as its decimal equivalent the number $2^n - 1$. However, the author of the book I'm reading next makes the following claim, which I don't quite see. He says that as a consequence of the fact above, it follows that ... | Since your binary number starts with $n$ ones followed by $n-1$ zeros, the number has $n+(n-1) = 2n-1$ binary digits. Therefore its decimal value is
$$
\begin{align}
& 0 \times 2^{0} + 0 \times 2^{1} + \cdots + 0 \times 2^{n-2} + 1 \times 2^{n-1} + 1 \times 2^{n} + 1 \times 2^{n+1} + \cdots + 1 \times 2^{2n-3} + 1 \t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3429924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
General formula for $e^x+\cos(x)$, $e^x+\sin(x)$, $e^x-\sin(x)$, $e^x-\sin(x)$ I have been able to derive the formal series for these four functions:
$e^x+\sin(x) = 1+2x+\dfrac{x^2}{2!}+\dfrac{x^4}{4!}+\dfrac{2x^5}{5!}+\dfrac{x^6}{6!}+\dfrac{x^8}{8!}+\dfrac{2x^9}{9!}+...$
$e^x+\cos(x) = 2+\dfrac{x^3}{3!}+\dfrac{2x^4}{4... | Formal power series can be added and then their area of convergence is limited by the most restricted one.
So one easy way would be to
*
*find separately expansions for $\{e^{x},\cos(x),\sin(x)\}$
*add (or subtract) them to each other
*put them under the same $\sum$ and then
*try to simplify the expression you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3430117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Generators of $\text{GL}_2(\mathbb{Z})$ and $\text{SL}_2(\mathbb{Z})$. We denote by $(a,b,c)$ the integral binary quadratic form $q(x,y)=ax^2+bxy+cy^2$. Also, we denote by $\sim$ (resp. $\sim_+$) $\text{GL}_2(\mathbb{Z})$ equivalence (resp. $\text{SL}_2(\mathbb{Z})$ equivalence).
I know that for all $a,b,c\in\mathbb{Z}... | It is true that $\mathrm{SL}_2(\mathbb Z)$ is generated as a monoid by
$$S = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}, \qquad T = \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix},\qquad T^{-1} = \begin{bmatrix}1 & -1 \\ 0 & 1\end{bmatrix}$$
and thus $\mathrm{GL}_2(\mathbb Z)$ is generated by those matrices together with ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3430200",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Solve system solutions The number of actual system solutions
$ \begin{cases}
a^2=b+2\\
b^2=c+2 \\
c^2=a+2\\
\end{cases}$
is equal to:
Solution:
$\cos 2\theta=2\cos^2\theta-1\implies 2\cos 2\theta=(2\cos\theta)^2-2$.
Using this results in all $8$ solutions to the system.
$(2,2,2)$, $(-1,-1,-1)$, and cyclic permutations ... | $$a^2=b+2 \tag1$$
$$b^2=c+2 \tag2$$
$$c^2=a+2 \tag3$$
By successive eliminations
$$(1) \implies b=a^2-2$$
$$(2) \implies c=a^4-4a^2-2$$
$$(3) \implies a^8-8 a^6+20 a^4-16 a^2-a+2=0\tag 4$$
$(4)$ can be factorized as
$$(a-2) (a+1) \left(a^3-3 a+1\right) \left(a^3+a^2-2 a-1\right)=0$$ Each cubic equation has three real r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3430353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Computational complexity of calculating the $n$th derivative of $f(x)=\exp\bigg(\frac{1}{\ln(x)}+\frac{1}{\ln(1-x)}\bigg)$? Computational complexity of calculating $$ f^{(n)}(x)? $$ where $f(x)=\exp\bigg(\frac{1}{\ln(x)}+\frac{1}{\ln(1-x)}\bigg)$
I don't know much about computational complexity but I do know that the d... | The answer by Steven Stadnicki isn't quite right. For each term with exponents $(a,b,c,d)$, you get six terms in the next derivative:
*
*$(a-1,b,c,d)$
*$(a-1,b-1,c,d)$
*$(a,b,c-1,d)$
*$(a,b,c-1,d-1)$
*$(a-1,b-2,c,d)$
*$(a,b,c-1,d-2)$
The first four come from differentiating $x$, $\ln x$, $(1-x)$, $\ln (1-x)$ re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3430547",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove that $f$ is an increasing function if $f'(x)$ is more than zero for all real values for $x$ I'm having some difficulty with proving this theory. What I do know is how to prove that it is a constant function when $f'(x) = 0$ by simplying assuming that f is not constant and contradict the supposition. In this case,... | $f'(x) > 0$ everywhere means $f$ is continuous everywhere.
And the mean value theorem says that for any $a, b; a < b$ that there is a $c: a < c < b$ where $f'(c)=\frac {f(b)-f(a)}{b-a}$.
But we know $f'(c) > 0$ so $f(b) > f(a)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3430659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Taking a bizarre limit Consider the set of integers, $\Bbb{Z}$. Now consider the sequence of sets which we get as we divide each of the integers by $2, 3, 4, \ldots$.
Obviously, as we increase the divisor, the elements of the resulting sets will get closer and closer.
Question: In the limit as $\text{divisor}\to\infty$... | The typical way to define limits of sets is via
$$\liminf_{n\to\infty} A_n = \bigcup_{n\geq 1} \bigcap_{k \geq n} A_k \\ \limsup_{n\to\infty} A_n = \bigcap_{n\geq 1} \bigcup_{k\geq n} A_k$$
Using these and $A_n = f_n(\mathbb{Z})$ where $f_n(x) = x/n,$ we have
$$\liminf_{n\to\infty} A_n = \mathbb{Z} \\ \limsup_{n\to\inf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3430812",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 0
} |
Continuity of the stochastic process $X_t=\int_{0}^{t}(a+b\frac{u^n}{t^n} ) dW_{u} $ I am wondering about the continuity of the stochastic process
$$X_t=\int_{0}^{t}(a+b\frac{u^n}{t^n} ) dW_{u} $$
Where n=1,2,..
At t=0, there seems to be a discontinuity except for b=0 . Is there a discontinuity?
| This problem comes down to showing the continuity of
$$\frac{1} {t^n}\int_0^t u^n dW_u $$
From Itô's formula, we have
$$\frac{1} {t^n}\int_0^t u^n dW_u =W_t-\frac{n}{t^n}\int_{0}^{t}W_u u^{n-1}du$$
So essentially we have to evaluate the limit
$$\lim_{t\to 0} \frac{1}{t^n}\int_{0}^{t}W_u u^{n-1}du$$
Where n is any nat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3430941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Span of Density operators (Positive Semi-definite matrices of Trace one) While reading basic-mathematics of quantum mechanics I came across a statement -
"For every complex euclidean space $\cal X$ there exists spanning sets
of the space $L({\cal X})$ consisting of only density operators".
Here,
*
*$\cal X$ is... | Given any $T\in L(\mathbb C^n)$, you can write
$$
T=\frac{T+T^*}2+i\,\frac {T-T^*}{2i}
$$
so $T$ is a linear combination of selfadjoints. And for each selfadjoint you have the Spectral Theorem saying that they are a linear combination of rank-one projections (which are positive semidefinite of trace one).
The result ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3431129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Are these infinite groups decomposable? I have been asked to:
Decide whether the following groups are decomposable:
(a) - $(\mathbb{R^*}, \cdot)$
(b) - $(\mathbb{C}, +)$
(c) - $(\mathbb{Q^*}, \cdot)$
(d) - $(\mathbb{Q}, +)$
I would like a hint for item (a). I believe I was able to do itens (b), (c) and (d).
Regarding... | For item (a), you can decompose where one factor is the sign and the other is the absolute value. Notice that you probably should do (c) the same way. Your decomposition is for the multiplicative group of positive rational numbers.
It turns out $(\mathbb R, +) $ is decomposable as well, but it's not quite as easy to s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3431485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Differentiability of the operator norm My question is simple. Given finite-dimensional real Banach spaces $V, W$, is the operator norm on $\mathcal{L}(V, W) \setminus \{ 0 \}$ differentiable?
I know the standard Euclidean norm would be, but I don’t know what to do with this.
| Consider $V=W=\mathbb R^n$ with $\|\cdot\|$ as the spectral norm on matrices (induced by the 2-norm on vectors). If $\|\cdot\|$ is differentiable, then for any $A,B$, the one-sided directional derivatives
$$\nabla_B\|A\|=\lim_{h\to0^+}\frac{\|A+hB\|-\|A\|}{h}$$
should satisfy $\nabla_B\|A\|=-\nabla_{-B}\|A\|$.
However,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3431615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Can't solve a difficult limit I need to solve this limit
${\lim_ {x\to {+∞}}}{\frac{{x}(\sqrt{x^2 + x} - x) +\cos(x)\ln(x)}{\ln(1+\cosh(x))}}$
I've tried to use Taylor's Theorem with Peano's Form of Remainder, but first time I forgot that ${x\to{+∞}}$, so I made a substitution ${t=\frac{1}{x}}$, then I just didn't get... | Hint:
First the expression in two, and use equivalents:
We have $\cosh x\sim_{+\infty}\frac12\mathrm e^x$, so $1+\cosh x\sim \frac12\mathrm e^x$, and finally
$$\ln(1+\cosh x)\sim_{+\infty}x-\ln 2\sim_{+\infty} x$$
. On the other hand,
$$x(\sqrt{x^2 + x} - x)=\frac{x(\not x^2 + x - \not x^2)}{\sqrt{x^2 + x} + x}\sim_{+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3431716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Is this integral evaluation legitimate? I would like to evaluate the following integral:
$$\int_{-1}^{1}\frac{1}{\sqrt{1-x^{2}}}\cos (2\arccos (x))\cos (3\arccos (x)){\mathrm{d} x}$$
Some experience from taking a Numerical Methods course gives me the idea that the integrand can be thought of as some weight function $w(... | Hint :
Both function $\cos(2\arccos x)$ and $\cos(3\arccos x)$ are Chebyshev polynomials and
$$
\cos(2\arccos x)=2x^2-1 \\
\cos(3\arccos x)=4x^3 -3x
$$
Let $$f(x)=\frac{\cos(2\arccos x)\cos(3\arccos x)}{\sqrt{1-x^2}} \\
=\frac{(2x^2-1)(4x^3 -3x)}{\sqrt{1-x^2}}$$
Then $f\left(\frac{\sqrt{3}}{2}\right) = 0$ and $\forall... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3431860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What's the probability that I draw at least 1 white card when drawing 3 cards from 3 decks of 15 cards, 2 of which are white? I haven't seen quite this scenario on a card drawing problem on here. I'm trying to figure out the probabilities for a card game I'm developing. There are 3 separate decks with 15 cards each. In... | So I was messing around with it more and talking to a friend about it. Can anyone tell me if this is correct...
$P(W) = 1 - P(W')$
$ = 1 - \left(\frac{^{13}C_3}{^{15}C_3}\right)^3 $
Which comes out to $P(W) = 0.752$
It just seems quite high to me.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3431985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
$u(t,B_t)$ is a martingale if $u(t,x)$ is polynomial in each variables and satisfies the heat equation I want to show that for $u(t,x)$ which is a polynomial in $t$ and $x$ such that $$\frac{\partial u}{\partial t} + \frac{1}{2}\frac{\partial^2 u}{\partial x^2}$$
we have $u(t,B_t)$ is a martingale where $B_t$ stands fo... | A mean-constant process with markov property is a martingale.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3432101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Define rank of matrix by reduced row echelon form - well-defined? In order to define the rank of a matrix, I want to use reduced row echelon form (rref). I have an ugly proof that the rref is unique in the following sense: If $A,B$ are in rref and $A=ZB$ with invertible $Z$ then $A=B$.
However, the claim is too strong:... | Assume $q<r$. Let me partition $R=\pmatrix{ R_{11} & R_{12}\\ R_{21} & R_{22}}$ with $R_{11}\in K^{r,q}$. Then
$$
\pmatrix{ I_r& 0 \\ 0&0} R= \pmatrix{ I_r& 0 \\ 0&0}\pmatrix{ R_{11} & R_{12}\\ R_{21} & R_{22}} =\pmatrix{ R_{11} & R_{12}\\ 0&0}.
$$
Due to the assumption, the last $n-q$ columns are zero, so $R_{12}=0$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3432263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The number of real roots of the equation
$$e^{\sin x}-e^{-\sin x}-4=0$$
Let $e^{\sin x}=y$
Then $$y-\frac 1y -4=0$$
$$y^2-4y-1=0$$
$$y=2+\sqrt 5 , 2-\sqrt 5$$
How should I solve further ?
| Your method is fine, now observe that
*
*$e^{\sin x}=2+\sqrt 5 \implies \sin x=\log(2+\sqrt 5)>1$
which is not possible and
*
*$e^{\sin x}=2-\sqrt 5<0 $
which is not possible, therefore there are not real solutions.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3432487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
For rational numbers $a, b$, what is the range of $b$ such that $\lceil a + b \rfloor = \lceil a \rfloor$ holds? For rational numbers $a, b$, what is the range of $b$ such that $\lceil a + b \rfloor = \lceil a \rfloor$ holds?
Clearly, b=0 gives us the result.
What are the lower and upper bounds of $b$?
$\lceil \cdot ... | For positive $a$, $\lceil a \rfloor = \lfloor a + \frac 12 \rfloor$, being the floor function. If $a = a_q+a_r$ where $a_q$ is the whole number part and $a_r$ is the fractional part, $\lceil a \rfloor = a_q + \lfloor a_r + \frac 12 \rfloor$. Similarly, if $b$ (also positive) is $b_q + b_r$ then
\begin{align}
\lceil a +... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3432605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
question about sets and subsets using the definition
Assume $A$ and $B$ are two sets such that: $Card(A)=n$ and $Card(B)=m$, also $B⊆A$,Clearly $n\ge
m$.
define: $$\mathscr{P}(A:B)=\left\{X∈\mathscr{P}(A):B⊆X\right\}$$
find the number of elements of $\mathscr{P}(A:B)$
I just know $Card\left(B\right)\le Card\left... | If we assume $Card(A)=n$ and $Card(B)=m$ then
$$\mathscr{P}(A:B)=\sum_{k=m}^{n}{{n}\choose{k}}{{k}\choose{m}}$$
For example if $A=\left\{1,2,3\right\}$, clearly $Card(A)=3$,and take $Card(B)=2$ then:
$\mathscr{P}(A)=\left\{\left\{\right\},\left\{1\right\},\left\{2\right\},\left\{3\right\},\left\{1,2\right\},\left
\{1,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3432737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How to find the singular solution of $y'=\frac{2x+y}{y-x}$ $y'$ and $y$ occur linearly in the ODE
$$y'=\frac{2x+y}{y-x}.$$
yet it is a first order non-linear ODE. I can find a family solutions of this homogeneous ODE by using $y=vx \implies \frac{dy}{dx}= v+x \frac{dv}{dx}$. I want to know if there is(are) singular so... | $$ y'=\frac{x(2+\frac{y}{x})}{x(\frac{y}{x}-1)} $$
then you can make a replacement $$ t=\frac{y}{x} \Rightarrow \frac{dy}{dx}=\frac{dt}{dx}x+t \Rightarrow \frac{dt}{dx}x+t=\frac{2+t}{t-1} $$
It remains only to divide the variables and take the integral :)
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3432861",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
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