Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Counting labelled graphs with k edges and n vertices Is there a way to count labelled graphs (simple graphs - without loops and without multiple edges) with k edges and n using combinatorics methods without having to draw them?
For example - How many labelled graphs are there with 3 edges over the vertices {a, b, c, d,... | HINT: An edge is between two vertices. And assuming the graph is simple, we cannot choose the same vertex pair twice. Then first of all, how many unordered vertex pairs are there when we have $6$ vertices? (Unordered means $\{a,b\} = \{b,a\}$ for all $a,b$) Secondly, how many ways are there of choosing three distinct v... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2628819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving Probability Statement via Axioms of Probability and Set Identities I am given the following problem and I am trying to figure out the last step.
Using the axioms of probability and set identities, prove that
if $(B \cap C) \subset A$, then $P(A) \geq P(B) + P(C) - 1$
The axioms:
*
*$P(A) \geq 0 \text{ for ... | Hint: You don't need $\mathsf P(B\cup C)=1$, just: $$-\mathsf P(B\cup C)\geq -1$$
So $\mathsf P(A)~{\geq \mathsf P(B)+\mathsf P(C)-\mathsf P(B\cup C)\\\geq \mathsf P(B)+\mathsf P(C)-1}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629077",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Why are $-1$ and $1$ generators for the Set of integers under addition? I'm reading my textbook and I'm confused why $-1$ and $1$ are generators for the group of integers under addition.
For example for 1:
we have $1, 1+1=2, 1+1+1=3, 1+1+1+1=4,$ etc.
So shouldn't $1$ be a generator for only the group of positive intege... | In group theory, the word "generates" has the following meaning: a set $S$ generates a group $G$ if every element of $G$ can be written as a string over $S\cup S^{-1}$ (here, if $S=\{a, b, \ldots\}$ then $S^{-1}=\{a^{-1}, b^{-1}, \ldots\}$). Equivalently, $G$ is the minimal subgroup of $G$ containing all the elements o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 2
} |
Three circles have the same radical axis?
Given three circles $\bigcirc O_1$, $\bigcirc O_2$, $\bigcirc O_3$, let $A$, $B$, $C$ be three points on $\bigcirc O_3$. If we have
$$
\frac{\operatorname{power}(A, \bigcirc O_1)}{\operatorname{power}(A, \bigcirc O_2)}=
\frac{\operatorname{power}(B, \bigcirc O_1)}{\operator... | I'll change your notations to make it more comfortable for me.
Let $\mathscr C_i$ be the circle centered at $O_i$ with radius $r_i$, for $1\le i\le 3$.
Denote by $P(X,\mathscr C)$ the power of point $X$ to circle $\mathscr C$.
Let $\alpha$ such that $\alpha=\frac{P(X,\mathscr C_1)}{P(X,\mathscr C_2)}$ for $X\in\{A,B,C\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Convergence of fixed-point in a gauss-seidel style I had a problem in the form
$$ \left(
\begin{array}
xx_1 \\
x_2 \\
... \\
x_n
\end{array}
\right)
=
\left(
\begin{array}
FF_1(x_2,...,x_n) \\
F_2(x_1,x_3...,x_n) \\
... \\
F_n(x_2,...,x_{n-1})
\end{array}
\right)$$
and I tried to solve this problem using fixed-point ... | The procedure described in this question is correct, the reason for such non convergent behavior was that the fixed point was far from the position at which the norm does not provide convergence. Indeed, even if chosen as starting point, the method may initially diverge. This cause the method to go far from the initial... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629458",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find all functions $ f:\mathbb{R} \mapsto \mathbb{R} $ such that $f[x^2+f(y)] = y +[f(x)]^2$ I read this question in the book 'Problem Solving Strategies ' by Arthur Engel. This question was asked in IMO of 1992. Here's how it goes
Find all the functions $ f:\mathbb{R} \mapsto \mathbb{R} $ that satisfy
$f[x^2+f(y)] = ... | This partially answers the amended question with a different requirement: $$f(x^2+f(y)) = y+f(x)^2$$
Let $x=0$ to obtain $f(f(y)) = y + f(0)^2$. Let $y=0$ to obtain $f(x^2+f(0)) = f(x)^2$.
Suppose $f$ has a fixed point: $y=f(y)$. Then $f(x^2+y) = y+f(x)^2$, so $f(y) = y+f(0)^2$ by letting $x=0$, and hence $f(0) = 0$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 0
} |
Model given for a formula The formula in question
$$ \forall x \exists y [\ \ P(x,y) \rightarrow \exists z \forall u (\lnot P(u,z))\ \ ] $$
according to my text-book, has the following model
$ \text{Domain} = \mathbb{N} $
$ \lvert P\rvert = \{ (n,m): n,m \ \in \mathbb{N},\ \ n =2m \} $
According to its description, I'... | Every $z$ belongs to a pair $(2z,z)$. Thus (as you say) there is no $z$ such that $\lnot P(u,z)$ does hold for every $u$.
Thus, up to now: $∃z∀u(¬P(u,z))$ is false, and this is independent of $x$ and $y$.
Consider now $P(x,y)$; obviously, the pair $(x,x+1) \notin |P|$ and thus, we have that for every $x$ there is an $y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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General solution to simultaneous equations - Resistors in parallel I'm designing a system that uses a grid of resistive sensors, and I'm having trouble figuring out the solution to a set of equations that is the output of this system. I haven't done serious maths for a long time, so go easy on me!
The equations are:
$... | Let $\;\dfrac{1}{R} = \dfrac{1}{R_1}+\dfrac{1}{R_2}+\dfrac{1}{R_3}+\ldots+ \dfrac{1}{R_x}+\dfrac{1}{A}\;$ then the equations can be written as:
$$\require{cancel}
\dfrac{\;\;\dfrac{1}{\dfrac{1}{R}-\dfrac{1}{R_k}}\;\;}{\;\;R_k+\dfrac{1}{\dfrac{1}{R}-\dfrac{1}{R_k}}\;\;} = Y_k \;\;\iff\;\; R_k\left(\dfrac{1}{R}-\cancel{\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Existence of partial derivatives & Cauchy-Riemann does not imply differentiability example I learned about the Cauchy-Riemann equations today, and my instructor used the following example to show that differentiability is not guaranteed if the partial derivatives are not continuous.
Let
$$
f(z)=f(x+iy)=
\begin{cases}
\... | You're right -- it does have to do with the definition of $f$ at $(0,0)$.
By the definition of partial derivative,
$$ \begin{align} u_x(0,0) &= \lim_{h\to0}\dfrac{u(h,0)-u(0,0)}h \\ &= \lim_{h\to0} \dfrac{\frac{h^20}{h^2+0^2}-0}h \\ &= \lim_{h\to0}\dfrac0h \\ &= 0
\end{align} $$
Notice how we had to use the definition ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Then $E(X_1+X_2+X_3+X_4)^4$ equals? Let $X_1, X_2, X_3, X_4$ are i.i.d random variable taking values $1$ and $-1$ with probability $1/2$ each. Then $E(X_1+X_2+X_3+X_4)^4$ equals?
I see that each $X_i$ is standard normal and so $X_1+X_2+X_3+X_4$ is a normal variable with mean $0$ and variance $4.$ I find the answer $... | This seems one of those problems where just enumerating the possibilities is quick (even if dirty):
X1 X2 X3 X4 Sum Sum^4
-1 -1 -1 -1 -4 256
-1 -1 -1 1 -2 16
-1 -1 1 -1 -2 16
-1 -1 1 1 0 0
-1 1 -1 -1 -2 16
-1 1 -1 1 0 0
-1 1 1 -1 0 0
-1 1 1 1 2 16
1 -1 -1 -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2629949",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If an operator preserves divisibility, does that imply that it preserves multiplicability? Specifically, if it were true that
$$\int_a^b \frac{f(x)}{g(x)} \, dx = \frac{\int_a^b f(x) \, dx}{\int_a^b g(x) \, dx}\,,$$ then would that imply
$$ \int_a^b f(x) \cdot g(x) \, dx = \int_a^b f(x) \, dx \cdot \int_a^b g(x) \, dx\... | Are you sure about integration? It looks to me like you are assuming $\int{1/f}$ = $1/\int{f}$, which I don't think it true (for integrals).
As for your more general question, "preserves division" and "preserves inverses" would imply "preserves multiplication", which you should be able to prove easily, but "definite in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Existence of a normalized vector, far from proper closed subspace. Say we have a normed vector space $(E, \|\cdot\|)$, and a proper closed subspace $F\subsetneq E$, I want to prove the following:
$$(\forall \varepsilon > 0), (\exists e \in E) : \|e\|=1 \text{ and } d(e,F) \geq 1 - \varepsilon $$
Where $d(e,F)= \inf... | I managed to follow trough with David's help.
So we choose $y=\frac{\bar{x}-f}{\|\bar{x}-f\|}$, where f is given as in the property stated in the question body, for $\alpha =\frac{1}{1-\varepsilon}$.
Clearly $\|y\|=1$, so we compute it's distance to $F$.
$$d(y,F) = \inf\limits_{g \in F} \|y-g\| = \inf\limits_{g \in F} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630169",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Problem about continuous functions and the intermediate value theorem
Let $S^{1} := \lbrace(\cos\alpha, \sin\alpha) \subset \mathbb{R}^{2} | \alpha \in \mathbb{R}\rbrace$ be the circumference of radius $1$ and $f: S^{1} \to \mathbb{R}$ a continuous function. Prove that there exist two points
diametrically opposed at w... | The idea is good, but the
intermediate value theorem
applies to functions mapping a closed interval $I \subset \Bbb R$ to $\Bbb R$.
It would be possible to formulate a similar statement for functions $\phi: S^1 \to \Bbb R$, but it is simpler to consider
$$
\phi: [0, \pi] \to \Bbb R, \quad
\phi(\alpha) = f(\cos\alpha, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Equation of Normal I'm struggling to get the same answer as the book on this one. I wonder if someone could please steer me in the right direction?
Q. Find the equation of the normal to $y=x^2 + c$ at the point where $x=\sqrt{c}$
At $x = \sqrt{c}$ then $y = 2c$, so the point given is $(\sqrt{c},2c)$
The gradient of the... | Your statement about the gradient of the normal should have been as follows:
The gradient of the tangent is $2x$ so the gradient of the normal is $ \frac {−1}{2x}$ which will be $ \frac {−1}{2\sqrt c}$
You will get the book's answer with this new slope.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 2
} |
Prove $\sum_{n=1}^\infty \text{Ci}(\pi n)=\frac{\ln(2)-\gamma}{2}$ I'm trying to prove that
$$\sum_{n=1}^\infty \text{Ci}(\pi n)=\frac{\ln(2)-\gamma}{2}$$
I've tried parametrizing the sum by replacing $\pi$ with $x$ and differentiating, but this creates to a divergent series (whose partial sum is too messy to integrate... | An overkill. Let $\mathfrak{M}\left(*,s\right) $ the Mellin transform. Using the identity $$\mathfrak{M}\left(\underset{k\geq1}{\sum}\lambda_{k}g\left(\mu_{k}x\right),\, s\right)=\underset{k\geq1}{\sum}\frac{\lambda_{k}}{\mu_{k}^{s}}\mathfrak{M}\left(g\left(x\right),s\right) $$ we have $$\mathfrak{M}\left(\sum_{n\geq1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Probability that $a^2+b^2+c^2$ divisible by $7$
Three numbers $a,b,c\in\mathbb{N}$ are choosen randomly from the set of natural numbers. The probability that $a^2+b^2+c^2$ is divisible by $7$ is
Try:any natural number when divided by $7$ gives femainder $0,1,2,3,4,5,6$
So it is in the form of $7k,7k+1,7k+2\cdots ,7k... | Hint
Any perfect square $x$ Is
$x\equiv 0\pmod 7$
Or
$x\equiv 1\pmod 7$
Or
$x\equiv 2\pmod 7$
Or
$x\equiv 4\pmod 7$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630660",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Commutivity of Endomorphism with Jordan Decomposition I asked this question earlier, but am still a bit stuck on the following question, and would appreciate any help.
Let $\mathbb{F}$ be algebraically closed. If $T=D+N$ is a Jordan decomposition of a linear operator, ($D$ is diagonal and $N$ is nilpotent), prove for $... | Hint In fact, use - as you say - that $D$ and $N$ are polynomials in $T$. There is a lemma :
Lemma : The commutant of $S$,
$$
C(S):=\{S\mid ST=TS\}
$$
is an algebra (i.e. closed by products and linear combinations).
Hence, if $S$ commutes with $T$, it commutes with all powers of $T$ and polynomials $P(T)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Normal Coordinates on a Riemannian Manifold Consider Riemann Normal coordinates on a manifold. Consider a point other than the origin. Given that the metric has vanishing derivatives at this point, is it correct to deduce that the metric is Euclidean at this point? If the deduction is correct, how to prove/ argue this?... | Clearly: $x_k=x^ig_{ik}$. Consider differentiating on both sides: $\partial_j x_k=(\partial_j x^i)g_{ik}+x^i\partial_j g_{ik}$. Now if the metric derivatives vanish then the last term is zero and we have: $\partial_j x_k=(\partial_j x^i)g_{ik}\Rightarrow\delta_{jk}=\delta_j^ig_{ik}\Rightarrow\delta_{jk}=g_{jk}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Finding the minimum value of a complex number If $z$ is a complex number satisfying $|z^2+1| = 4|z|$ . Then prove that the minimum value of $|z|$ is $4$
This is how I attempted the problem ,
$\frac{|z^2+1|}{|z|} = 4$
Therefore ,
$|z + \frac{1}{z}| = 4$
How do I proceed from here ?
According to the solution of the abo... | This follows from a version of the reverse triangle inequality, in your case $$\lvert\lvert z\rvert-\lvert\frac{1}{z}\rvert\rvert\leq\lvert z+\frac{1}{z}\rvert.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2630990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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What substitution would I make to integrate this? Problem:
The integral is gonna be:
$$\int^{1}_{0}\int^{1}_{0}4xy\sqrt{x^2+y^2} dy \, dx$$
But I'm quite rusty with my calculus, and this is mainly for a statistics course. I know that $x$ and $y$ are bounded within a unit square region in the $xy$-plane, and I was cons... | An alternative approach:
$$ \begin{eqnarray*}\iint_{(0,1)^2}4xy\sqrt{x^2+y^2}\,dx\,dy&=&\iint_{(0,1)^2}\sqrt{X+Y}\,dX\,dY\\&=&2\iint_{0\leq Y\leq X\leq 1}\sqrt{X+Y}\,dX\,dY\\&=&2\iint_{(0,1)^2}X\sqrt{X}\sqrt{K+1}\,dX\,dK\\&=&2\int_{0}^{1}X\sqrt{X}\,dX\int_{0}^{1}\sqrt{K+1}\,dK\end{eqnarray*}$$
by exploting the substitu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2631229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the limit of $\lim\limits_{x\to0^+}\frac{1}{e} \frac{e- e^ \frac{\ln (1+x)}{x}}{x}$ as $x$ Find the limit of $\frac{1}{e} \frac{e- e^ \frac{\ln (1+x)}{x}}{x}$ as $x$ approaches right of zero.
The answer is $\frac{1}{2}$ but I keep getting 1. Here's what I have:
Since $\lim_{x\to0^+} \frac{\ln (1+x)}{x}$ is of inde... | $$\lim_{x\rightarrow0}\frac{e- e^ \frac{\ln (1+x)}{x}}{ex}=-\lim_{x\rightarrow0}\frac{e^ \frac{\ln (1+x)}{x}\left(\frac{x}{1+x}-\ln(1+x)\right)}{ex^2}=-\lim_{x\rightarrow0}\frac{x-(1+x)\ln(1+x)}{x^2}=$$
$$=-\lim_{x\rightarrow0}\frac{1-\ln(1+x)-1}{2x}=\frac{1}{2}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2631343",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
$\pi (x) = -1 + \pi(\sqrt x) + \sum \mu (d)\lfloor \frac {x}{d}\rfloor$, Sorry about the title i didn't know what to call this other than "analytic number theory"
I am asked to show that $\pi (x) = -1 + \pi(\sqrt x) + \sum \mu (d)\lfloor \frac {x}{d}\rfloor$, where the sum is over all d for which ever prime factor is l... | This should be Legendre’s formula for computing $\pi(x)$:
$$
\pi(x)=-1+\pi(\sqrt{x})+\lfloor x \rfloor-\sum_{p_i\le a}\left\lfloor \dfrac{ x }{(p_i)}\right\rfloor+\sum_{p_i<p_j\le a}\left\lfloor\dfrac{ x}{(p_ip_j)}\right\rfloor-\sum_{p_i<p_j<p_k\le a}\left\lfloor \dfrac{x}{(p_ip_jp_k)}\right\rfloor+\dots
$$
This is pro... | {
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"url": "https://math.stackexchange.com/questions/2631463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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6 tests in one month, each must be separated from other tests by 2 free days in between A university is determining the dates for tests in January. There can be a test on every day in January(all $31 $of them), but each two tests have to have at least $2$ free days in between them. (so if there was a test on Monday, th... | Misunderstood Question
Before, I noticed that the title specified $6$ test, I computed all the possible test days.
Use atoms of $\left(x+x^3\right)$ representing a free day, $x$, or a test and two free days, $x^3$.
We also need to note that we can end in two or more free days, $1$, or one free day, $x^2$, or no free da... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2631622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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If some complex numbers lie on the one side of a line which goes through 0, then the sum of them are not 0 Let me describe the problem more specifically.
Suppose that $z_{1},z_{2},\cdots,z_{n}$ are all complex number and they all lie on one side of a straight line passing through $0$. Then $z_{1}+z_{2}+\cdots+z_{n}\neq... | Well, you have to give a characterization of those numbers. Since the lines passes through $0$, let's suppose it forms with the positive $x$ semiaxis and angle $\alpha$, what can you say of the $\arg(z_i)$ for all the points on one side and what for those on the other side?
| {
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"url": "https://math.stackexchange.com/questions/2631752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Plotting $\{z\in \mathbb{C}\mid |z| > \Re(z)-2\}$ How to plot the set of complex numbers
$$\{z\in \mathbb{C}\mid |z| > \Re(z)-2\}$$
I know that $ |z|$ should be a circle centred at $(0,0)$, but I don't know what would be its radius.
| Your condition is, with $\;z=x+iy\;,\;\;x,y\in\Bbb R\;$ :
$$\sqrt{x^2+y^2}>x-2\implies x^2+y^2>x^2-4x+4\implies y^2>-4(x-1)$$
You have a parabola there...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2631862",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Main theorem of Dedekind
Main theorem of Dedekind: It claims, that for every cut $A|A'$ in the set of real numbers there exists a number $b$ that forms the cut. This number $b$ will be $1)$ the largest in the lower class or $2)$ the smallest in the higher class.
I can't get the proof. The proof in the book was given ... | This is the original proof by Dedekind (also repeated by Hardy in his A Course of Pure Mathematics) and to understand it properly you need to be very clear of the fact there are are two types of Dedekind cuts being used here. One is the cut $(A, A') $ which involves partitioning the set of reals into two sets $A, A'$. ... | {
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"url": "https://math.stackexchange.com/questions/2632021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Assumption made in Proof of System of ODEs with Repeated Roots In a system of ODEs with a repeated root, what brings one to assume that the form of the equation is
\begin{align}
y(t)=v(t)e^{rt}
\end{align}
where $r$ is the repeated root. I get that multiplying $ce^{rt}$ by $t$ yields a second, linearly independent solu... | In ODE and PDE research, sometimes a "guess and check" method is employed. Multiplying by $ce^{rt}$ works nicely.
Consider the ODE $y'=y$ with initial condition $y(0)=k$. Without knowing the solution is the exponential, we want to find a function (more generally, a family of functions) that is equal to its own derivat... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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limit without expansion I was solving this limit instead of using sum of expansion i did this and got zero but answer is 1/5 by expansion why this is wrong
$$\lim_{n\rightarrow \infty } \frac{1^4 + 2^4 + \ldots + n^4}{n^5}$$
$$\lim_{n\rightarrow \infty } \frac{n^4( \frac{1^4}{n^4} + \frac{2^4}{n^4} +\ldots + 1 )}{n... | The problem is that while each term tends to zero, the number of terms tends to infinity at the same time.
The rule “the limit of a sum is the sum of the limits” only applies if you have a finite sum with a fixed number of terms (each of which has a finite limit).
(Also, you can't really write
$$\lim_{n \to \infty} (\d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2632317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
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Find at least one solution or prove that it does not exist Please help me to find solution or prove that it does not exist.
$$
C^{2^k}_{2^n} < 2^{2^k (n - k)}, 1<k<n, k \in \mathbb{N}, n \in \mathbb{N}
$$
I tried to find solution numerically, but when $n$ > 20, the numbers grows very fast, so it seems like it has not s... | I've found an answer.
Firstly we can introduce $N = 2^n$, $K = 2^k$. So the inequality will be rewritten as
$C_N^K < 2^{K(\log_2(N)-\log_2(K))}$
or
$\log_2(C_N^K) < K \log_2(N/K)$.
Secondly, according to Best upper and lower bound for a binomial coeficient we can use inequality
$C_N^K \ge (N / K)^K$.
It shows that the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2632414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Which statement is false ?(Linear algebra problem) Let $P=\dfrac{xx^{T}}{x^{T}x}$ be an a square matrix of order n where $x$ is a non zero column vector. Then which one of the following statement is False.
$(A)$ P is idempotent
$(B)$ P is orthogonal
$(C)$ P is symmetric
$(D)$ Rank of P is one
In this question i o... | A matrix of rank $1 $ cannot be orthogonal. Orthogonal matrices have maximal rank.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2632516",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A sufficient condition for $x^p-x-a$ to be primitive Let $p$ ba a prime number and $F_p$ be the finite field with $p$ elements. Characterize the set of $a\in F_p$ such that $f=x^p-x-a$ is a primitive polynomial i.e. $x$ generates the multiplicative group of $F_p[x]/(f)$.
| Further to my hint above, that
The polynomial $x^p − x − a \in \mathbb{F}_p[x]$ is primitive if and only if $a$ is primitive in $\mathbb{F}_p$.
I found the result of Cao to be of use, I think, from $2010$
On the Order of the Polynomial $x^p − x − a$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2632612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Is this $\binom{n}{p}$ for $p>n$ make a sense in mathematics or it is $0$ by convention? It is well known that gamma function is not defined at negative integers , but my question is to know how i take the value of $\binom{n}{p}$ for $p>n$ then is this make a sense or it is $0$ by convention ?
|
A common definition of the binomial coefficient with $\alpha\in\mathbb{C}$ and integer values $p$ is
\begin{align*}
\binom{\alpha}{p}=
\begin{cases}
\frac{\alpha(\alpha-1)\cdots(\alpha-p+1)}{p!}&p\geq 0\\
0&p<0
\end{cases}
\end{align*}
From this we conclude $\binom{n}{p}=0$ if $p>n \ \ (n,p\in\mathbb{N})$.
Hint... | {
"language": "en",
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Calculate domain $f(x)=x^{\frac{x+1}{x+2}}$ I have the following function:
$$f(x)=x^{\frac{x+1}{x+2}}$$
I tried to calculate the domain, which seems easy, and my result is: $D(f)=(0,\infty)$.
When I tried to calculate it, by using Wolfram-Alpha, I obtain: $D(f)=[0,\infty)$.
Can someone explain me the reason, or if it i... | I think Wolfram is wrong and you are right.
If we write $(f(x))^{g(x)}$ then the domain it's
$$D(g)\cap\{x|f(x)>0\},$$
where $d(g)$ it's the domain of $g$.
I think it's better to define such that even $0^{\frac{1}{2}}$ does not exist, but $\sqrt0=0.$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How many ways are there to have a collection of $10$ fruits +I am just getting started in my combinatorics class and I came across the following problem in my textbook that I am looking for some help with, thanks!
How many ways are there to have a collection of $10$ fruits from a large pile of identical oranges, apples... | Since your collection needs to include exactly $2$ kinds of fruits, this means that you have $C(5,2)$ possible choice for the fruits that will be part of your collection. Recall that $C(5,2)$ means that you pick two kind of fruits out of $5$ possible choices. But this doesn't take into account all possible collections ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the domain of $f(x)=x^{2/4}$ What is the domain of $f(x)=x^{2/4}$
Is $f(x)= (\sqrt[4]x)^2 $ with $\operatorname{dom} (f) = [0,\infty) $
or $\sqrt[4]{x^2}$ with $\operatorname{dom} (f) = (-\infty,\infty)$
or is $f(x)=\sqrt x$ with $\operatorname{dom} (f)=[0,\infty)$
I have tried searching the internet but cou... | since $$x^{2/4}=x^{1/2}$$ we have $$x\geq 0$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How many ways can $4$ couples be seated around a table if ...? I'd like to know if my solution is plausible.
4 couples are to be seated around a circular table. How many ways can they be seated if each person is sitting directly across from their spouse (i.e. there are three people between them and their spouse on eith... | Here's a different approach. There are four surnames A B C and D. If you draw a dividing line across the middle of the table, then on one side you need to arrange the letters A B C D which can be done in $4!$ ways.
For each of these surnames, there is a husband or a wife who occupies this position, with their spouse au... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is $\cfrac 1n$ in this expression?
I bought a T-shirt at Kenendy Space Center a while back that had this expression on it:
$$
B > \frac1n\sum_{i=1}^n x_i
$$
And below it is captioned "Be greater than average."
I can see the "Be greater than" and the "average" in the expression, but I do not understand what the $... | The term $\dfrac 1n$ is part of the average. If we want to find the average of two numbers $a$ and $b$, the average would be $\dfrac{a + b}{2} = \dfrac 12(a + b)$. The denominator is equal to the amount of variables we have. In this case, we have two variables $a$ and $b$, so we add them together and divide it by two. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2633520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $X$ and $Y$ be i.i.d. Expo($\lambda$). Find the conditional distribution of $X$ given $X < Y$. (a) by using calculus to find the conditional PDF.
My Solution:
$$P(X \leq t \mid X < Y) = \dfrac{P(X \leq t, X < Y)}{P(X < Y)}\\
= \frac{P(X \leq t, X < Y \mid Y \leq t)P(Y \leq t) + P(X\leq t, X < Y\mid Y > t)P(Y>t)}{... | Rather, try this:
$$\begin{align}\mathsf P(X\leqslant t\mid X\lt Y) &= \dfrac{\mathsf P(X\lt Y, X\leqslant t)}{\mathsf P(X\lt Y)}\\[1ex] &=\dfrac{\mathsf P(X\lt Y, Y\leqslant t, X\leqslant t)+\mathsf P(X\lt Y, X\leqslant t, Y>t)}{\mathsf P(X\lt Y)}\\[1ex] &=\dfrac{\mathsf P(X\lt Y\mid X\leqslant t, Y\leqslant t)\,\math... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove by induction that $(k + 2)^{k + 1} \leq (k+1)^{k +2}$
Prove by induction that $$ (k + 2)^{k + 1} \leq (k+1)^{k +2}$$ for $ k > 3 .$
I have been trying to solve this, but I am not getting the sufficient insight.
For example, $(k + 2)^{k + 1} = (k +2)^k (k +2) , (k+1)^{k +2}= (k+1)^k(k +1)^2.$
$(k +2) < (k +1)^... | Try taking log of both sides and prove $\frac{\log x}x$ is decreasing.
Or by induction try to show $(\frac{k+1}k)^k\leq k$:
$$(1+1/k)^k\leq \sum_{i=0}^k \binom ki k^{-i}<\sum_{i=0}^k 1=k+1$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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How to prove sin(nx) has no pointwise convergent subsequence without prior knowledge of Lebesgue's Theory? In Baby Rudin 7.20 example, the author mentions to prove that the function sequence
$$f_n(x):=\sin(nx) \qquad0(\leq x\leq 2\pi)$$has no pointwise convergent subsequence would be troublesome without Lebesgue's Theo... | Here's an argument my officemate and I came up with that should work (while avoiding Lebesgue theory). Given a subsequence $\sin(n_kx)$, it constructs a sub-subsequence $\sin(n_{k_m}x)$ and a point $y$ such that $\sin(n_{k_m}y)$ fails to converge.
Suppose a subsequence $\sin(n_kx)$ is given. Then, we can find some c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2633927",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Does there exist an integrable function f such that norm of f*g is equals that of g for all g? Does there exist $f \in L^{1} (\mathbb R)$ such that $||f*g||_1 =||g||_1$ for all $g \in L^{1} (\mathbb R)$? I read somewhere (long ago) that no such function exists. It is easy to see that $L^{1} (\mathbb R)$ has no unit un... | No. If $\tau_x f(t)=f(t-x)$ we know that $$\lim_{x\to0}||f-\tau_xf||_1=0.$$It follows easily that if $$g_n=n(\chi_{(0,1/n)}-\chi_{(1/n.2/n)})$$then $$\lim_{n\to\infty}||f*g_n||_1=0.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2634026",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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To test whether $\sum_{n=1}^\infty\frac{n+2}{2^n+3}\sin\left[(n+\frac12)\pi\right]$ converges To determine whether the following sequence converges or divergence
$$\sum_{n=1}^\infty\frac{n+2}{2^n+3}\sin\left[(n+\frac12)\pi\right]$$
I don't know which test to use here, but my guess is it may be a comparison test but how... | Hint: I assume you mean $$\sin\left[(n+1/2)\pi\right]=(-1)^{n}$$
By using this you can see that this is an alternating series. Use the Leibniz criterion to rule out convergence.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to prove this definite integral does not depend on the parameter? I am working on some development formulas for surfaces and as a byproduct of abstract theory i get that:
$$\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\frac{1+\sin^2\theta}{(\cos^4\theta+(\gamma\cos^2\theta-\sin\theta)^2)^\frac{3}{4}}d\theta$$
is independent ... | Put
\begin{equation*}
I = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{1+\sin^2\theta}{(\cos^4\theta +(\gamma\cos^2\theta-\sin \theta)^2)^{\frac{3}{4}}}\, d\theta
\end{equation*}
If $x = \dfrac{\sin\theta}{\cos^2\theta}$, $\, y = \gamma-x$ and $y = \sqrt{z}$ then
\begin{equation*}
dx = \dfrac{\cos^2\theta+2\sin^2\theta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2634277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Chain rule problem: given $f(x)=\sqrt{4x+7}$ and $g(x)=e^{x+4}$, compute $f(g(x))'$. Question:
Given the functions $f(x)=\sqrt{4x+7}$ and $g(x)=e^{x+4}$, compute $f(g(x))'$.
My Approach:
I have found that found that $f(g(x))=\sqrt{4e^{x+4}+7}$. Should I now just differentiate it to get my answer or is there any simpler... | $f(g(x))'=f'(g(x))\cdot g'(x)=\frac{4}{2\sqrt{4g(x)+7}}\cdot e^{x+4}=\frac{2e^{x+4}}{\sqrt{4e^{x+4}+7}}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2634379",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Let $f: [-1, 1] \longrightarrow [-1, 1]$ such that $f\in C^{1}$. Prove that there's exist $x_{0} \in [-1, 1]$ such that $|f'(x_{0})| \leq 1$
Let $f: [-1, 1] \longrightarrow [-1, 1]$ such that $f$ is a class $C^{1}$ function. Prove that there's exist $x_{0} \in [-1, 1]$ such that $|f'(x_{0})| \leq 1$.
I know that $f'(... | Assume your claim is false. Your claim combined with the continuity of $f'$ implies that either $f'(x)>1$ is always true or that $f'(x)<-1$ is always true. Also, since $f'$ is continuous, we can take its integral: $$f(1)-f(-1) = \int_{-1}^1 f'(x)\,dx$$
If $f'(x)>1$ then $f(1)-f(-1)>2$. Similarly if $f'(x)<1$ then $f(1)... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Elementary question on pointwise convergence and norm continuity
Let $\;a_n \in \mathbb R\;$ be a bounded sequence, then by
Bolzano-Weierstrass Theorem it follows there exists a subsequence
$\;a_{n_k}\subset a_n\;$ such that $\;a_{n_k} \to a \in \mathbb R\;$.
If $\;f:\mathbb R \to \mathbb R^m\;$ a continuous ... | Consider $y_{n_k}=x-a_{n_k}$ and $y =x-a $ for any fixed $x\in \mathbb{R}$, then $y_{n_k} \rightarrow y $ in $\mathbb{R}$. Now as both $f$ and $g$ is continuous in $\mathbb{R},$ you can have both the convergence. Also you have to use the fact $||$ is continuous and composition of two continuous maps is continuous to g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Why does a function $f(x)$ have to be bijective in order to have a $f^{-1}(x)$ (in Euclidean plane)? Yes, the title explains itself. I have no take on this one, or insights for that matter. For example, why isn't it enough for a certain function to be injective to have its $f^{-1}(x)$?
| It is the matter of the nuances of definitions.
If a function from a domain $A$ to a codomain $B$ is merely a subset of $A\times B$, as in the Wikipedia definition, then the function does not "encode" $B$. You end up with the same set of pairs regardless of whether you view $f$ as a function of $A$ to $B$, or of $A$ to... | {
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Proving this limit as $x \rightarrow ∞$? I'm having trouble proving this using epsilon-delta:
$$\lim_{x\to\infty} \left|\frac{x}{x+1}\right|=1$$
I translated this into:
$$\forall \epsilon>0,∃\delta\in\mathbb R,x>\delta\implies \left|\frac{x}{x+1}-1\right|<\epsilon$$
I don't really know where to go from here. Any help w... | Try observing that
$$
\left|\frac{x}{x+1}-1\right|=\left|\frac{x-x-1}{x+1}\right|
=\left|\frac{1}{x+1}\right|
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2634759",
"timestamp": "2023-03-29T00:00:00",
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Limit of $[\ln (3+x^2) - \ln (2+x)]$ as $x$ approaches $\infty$ I got an answer as $\infty$ but I need more clear explanation.
Please help me!
| Hint:
You can use equivalents, and the properties of log:
$$\ln(3+x^2)-\ln(1+x)=\ln \frac{x^2+3}{x+1}.$$
On the other hand,
$$\frac{x^2+3}{x+1}\sim_\infty \frac{x^2}x=\ln x\enspace\text{so}\quad\ln(3+x^2)-\ln(1+x)\sim_\infty \ln x.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2634850",
"timestamp": "2023-03-29T00:00:00",
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When does the cyclist get overtaken This question is not limited to any particular method; all methods accepted. This is a homework question, stated exactly as:
A cyclist starts off in a bike at an average speed of 16 km/h. 15 minutes later, a motorcyclist sets off on the same trail with an average speed of 48 km/h. H... | The speed of the cyclist is $3$ times slower than the speed of the motorcyclist. The motorcyclist, however goes after $15$ minutes from the same point. You can use number sense to figure this.
Or, when $x=$ amount in hours,
$16x=48(x-0.25)$, since the cyclist sets in $15$ minutes before the motorcyclist.
$0.25$ hours =... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2634974",
"timestamp": "2023-03-29T00:00:00",
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Could someone check if I did this correctly? Homogeneous Differential Equation
$$y' = \frac{-y-x}{x}, y' = \frac{-y}{x} -1$$
$$F(v) = -v -1$$ Since $$y = xv$$ then $$y' = v + xv'$$
Therefore:
$$v+xv' = -v-1$$
$$xv' = -2v-1$$
$$\frac{-dv}{2v+1} = \frac{dx}{x}$$
$$-\frac{1}{2}ln\bigg |1+2v\bigg|=ln|x| + C$$
$$ln|1+2v| =... | $$y'x+y+x=0$$
$$(xy)'+x=0$$
Simply integrate
$$xy =-\int xdx=-\frac {x^2} 2 +K$$
$$y =-\frac {x} 2 +\frac Kx$$
then since $y(1)=1 \to K= \frac 32$
$$y =-\frac {x} 2 +\frac {3}{2x}$$
So your answer is correct..
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Iterative sequence converging Consider the sequence
$$1^4,2^4,3^4,\ldots,k^4,\ldots$$
Form a new sequence, whose terms consist of the difference of the above sequence.
$$2^4-1^4,3^4-2^4,4^4-3^4,\ldots$$
Repeat the process with the terms of this new sequence. When this is done sufficiently many times, you will even... | This is the calculus of finite differences. You are starting
with a function $f(x)$ (here $f(x)=x^4$) and defining a new one by $g(x)=f(x+1)-f(x)$, then iterating the procedure.
In general if $f(x)=a_nx^n+\cdots+a_0$ is a polynomial of degree $n$,
then $g(x)=na_nx^{n-1}+\cdots$ is also a polynomial. Iterating then give... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does absolute consistency imply consistency? As I understand a set of sentences $\Phi$ in first-order logic is consistent iff for all $\psi$ either $\Phi \vdash \psi$ or $\Phi \vdash \neg \psi$ is false.
On the other hand $\Phi$ is absolute consistent iff there exists a sentence $\psi$ such that $\Phi \vdash \psi$ is f... | Yes, in first order logic these concepts are equivalent. I wull use $T$ to stand for a set of sentences, rather than $\phi$, which should stand for a single sentence.
If $T$ proves both $\psi$ and $\lnot \psi$ for some $\psi$, then $T$ proves $\rho$ for all $\rho$. So if $T$ is not consistent it is not absolutely co... | {
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Proving sequence is Cauchy sequence I need to examine if this sequence is Cauchy sequence using definition.
$$a_n= 1/3 + 2^2/3^2+...+n^2/3^n$$
I start with $m,n \in N, n>m$
$$ \vert(a_n - a_m)\vert = \vert(m+1)^2/3^{(m+1)}+...+n^2/3^n\vert$$
And I don't know what is bigger than this expression and what to compare it t... | An idea to make things simpler: take $\;n,\,\,m=n+p\;,\;\;p\in\Bbb N\;$ , so
$$\left|a_{n+p}-a_n\right|=\frac{(n+1)^2}{3^{n+1}}+\ldots+\frac{(n+p)^2}{3^{n+p}}\le p\frac{n^2}{3^n}$$
Since for any $\;p\in\Bbb N\;$ we have that $\;\lim\limits_{n\to\infty}\;p\,\cfrac {n^2}{3^n}=0\;$ , we get that for any $\;\epsilon>0\;$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2635434",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is probability that a line joining two randomly selected coordinates from set form an angle $45°$ with one of axes? What is probability that a line joining two randomly selected coordinates from a set form an angle $45°$ with one of the axes?
For example, if we have set of coordinates $$\{(1, 2), (1, 3), (3, 3),... | For any two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ to form a $45^\circ$ angle with two axes, the equivalent condition is$$
x_1 + y_1 = x_2 + y_2 \ \text{or}\ x_1 - y_1 = x_2 - y_2. \tag{1}
$$
Note that to compute the probability, it suffices to count the number of pairs satisfying (1) and then divide it by $\bin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2635595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does $M:=\{x \in \mathbb{Q}: x^2<7\}$ $M\subseteq\mathbb{R}$ have an infimum, supremum, min, max? $M\subseteq\mathbb{R}$
$M:=\{x \in \mathbb{Q}: x^2<7\}$
Does $M$ have an infimum, supremum, min, max?
My answer would be that it doesn't because $\sqrt{7}$ and $-\sqrt{7}$ are $\not\in \mathbb{Q}$
Is that correct?
| Definition: An upper bound $u$ of $M \subset \mathbb{R}$ is an element $u \in \mathbb{R}$ such that $u \geq m$ for all $m \in M$.
Definition: A supremum of $M \subset \mathbb{R}$ is an element $x \in \mathbb{R}$ such that
*
*$x$ is an upper bound of $M$
*for each upper bound $u$ of $M$, $x \leq u$
According to th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2635730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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$a_n $ is a positive integer for any $n\in \mathbb {N} $. Let $(a_n)_{n\geq 1}$ be a sequence defined by $a_{n+1}=(2n^2+2n+1)a_n-(n^4+1 )a_{n-1} $.
$a_1=1$, $a_2=3$.
I have to show that $a_n $ is a positive integer for any $n\in \mathbb {N}, n\geq 1$.
I tried to prove it by induction but it doesn't work.
| This answer provides partial results.
For each $n$ put $b_n=\frac {a_{n+1}}{a_n}$. Then $b_1=3$ and
$$b_n=2n^2+2n+1-\frac{n^4+1}{b_{n-1}}.$$
It suffices to prove that $b_n>0$. Computer evaluation suggests that a sequence $\{c_n=b_n-n^2-2n\}$ decreases and converges to about $-5.78734$. We have $c_1=0$ and
$$c_n=n^2+1... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Finding Parametric Equations for the right side of Hyperbola I have this equation $x^2-y^2=1$.
I can understand that it is a Hyperbola. But my question is, how can I create/find a parametric equation for the right side of this hyperbola? It seems it doesn't work, using $\sin$ or $\cos$.
Thanks
| Let's find functions $x(t), y(t)$ that satisfy $x^2 - y^2 = 1$. Perhaps, if we rewrite as $x^2 = 1 + y^2$, that'll help.
This reminds me of Pythagorean identities. In particular, it reminds me of $1 + \tan^2(t) = \sec^2(t)$.
Trying $(x(t), y(t)) = (\sec(t), \tan(t))$ has the unfortunate side effect that we may get nega... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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x odd then exists y such that $x^2 = 8y+1$ How can I algebraically show that if $x$ is odd, then $x^2 = 8y + 1$?
Making sure it is a true statement, I tabulated some values of $x,y$ pairs
$(1,0)$, $(3,1)$, $(5,3)$, $(7,6)$, $(9,10)$
I let $x = 2w + 1$ (since $x$ is odd)
Then $(2w + 1)^2 = 4w^2 + 4w + 1$
I don't think I... | If $x$ is odd, it is congruent to $\pm 1$ or $\pm 3$ modulo 8. In both cases, its square is congruent to $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2636159",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $\lim_{x\to+\infty}f'(x)>0$, does that mean that $\lim_{x\to+\infty}f(x)=+\infty$ Let's assume some function $f: \mathbb{R}\to\mathbb{R}$, which is differentiable in some region $(a, +\infty)$, where $a\in(0,+\infty)$. It makes sense (to me) that the following holds true:
$$\lim_{x\to+\infty}f'(x)\in(0, +\infty) \Ri... | Sketch:
Assuming that $\lim_{x\rightarrow\infty}f'(x)$ exists and equals $L>0$, then there is some $M$ such that for all $x>M$, $|f'(x)-L|>\frac{L}{2}$. Therefore, $f'(x)>\frac{L}{2}$ for $x>M$.
Observe that
$$
f(x)=f(M)+\int_M^xf'(t)dt.
$$
For $x>M$, it follows that
$$
f(x)=f(M)+\int_M^xf'(t)dt\geq f(M)+\int_M^x\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2636251",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Is it possible to prove associativity from $a(bc)=a(cb)$? With the given information, I'm trying to prove that an operation is both associative and commutative.
Let $(A,*)$ be a set and a binary operation such that for all $a,b,c \in A$
$$a*(b*c) = a*(c*b),$$
and
for some $e \in A$ and for any $a \in A$ $$e*a = a*e = ... | It is not possible. Consider the operation of combining unordered binary trees, where a * b is the binary tree whose root's unordered pair of children are the roots of a and b; toss in by fiat also an identity element for this operation. This operation is clearly commutative and thus satisfies your properties, but does... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Polynomial such that $f''(x) \rightarrow2$ as $x\rightarrow\infty$ given some values what is $f(1)$? Let $f$ be a polynomial such that $f''(x) \rightarrow2$ as $x\rightarrow\infty$, the minimum of f is attained at $3$, and $f(0)=3$, Then $f(1)$ equals.
$(A) \ 1$
$(B) \ 2$
$(C) -1$
$(D) -2$
I am not sure how to deal wi... | Suppose $f$ has degree $n$, then we can write $$f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0$$
We are given that $f''(x)\to 2$ as $x\to\infty$, so what could $n$ possibly be?
Suppose $n>2$, then $\deg(f''(x))=n-2>0$, so $f''(x)\to\pm\infty$ as $x\to\infty$.
Suppose $n<2$, then $f''(x)\equiv 0$ , so clearly $f''(x)\to 0\neq 2... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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if $g$ be a continuous function not differentiable at $0$ Let $g$ be a continuous function not differentiable at $0$ with $g(0)=8$. Let $f(x)=x\,g(x)$ .Find $f'(0)$
a)$0$
b)$4$
c)$2$
d)$8$
I am getting that $f'(x)=g(x)+x\,g'(x)$. But since $g'(x)$ doesn't exist for $x=0$, hence $f'(x)=8$. Please help whether it is... | It's wrong. You should try from the definition of differentiability.
Hint: $f^{'}(0)=\lim_{h \to 0} \frac{f(0+h)-f(0)}{h}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let $A,B$ be some sets such that $|A|=a, |B|=b$. Prove $\binom{a}{b}=|P_b(A)|$ is a well defined expression.
Let $A,B$ be some sets such that $|A|=a, |B|=b$. Prove $\binom{a}{b}=|P_b(A)|$(=the set of all subsets of A of cardinality $b$) is a well defined expression.
Hello all. In this question I need to show that th... | Using binomial theorem could work. You could also consider the "definition" (or interpretation) of the number $\binom{a}{b}$:
$\binom{a}{b}$ is the number of ways to choose $b$ elements from a set with $a$ elements.
This is, in a very literal sense, the number of subsets of size $b$ from a set of size $a$.
This is c... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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If $a,b,c$ are the roots of $x^3-px+q=0$, then what is the determinant of the given matrix in $a, b, c$?
If $a,b,c$ are the roots of $x^3-px+q=0$, then what is the determinant of
$$
\begin{pmatrix}
a & b & c \\
b & c & a \\
c & a & b \\
\end{pmatrix} \,\,?
$$
(A) $p^2+6q \quad$
(B) $1 \quad$... |
In this equation given we have product of eigenvalues given as $−q$ and we know product of eigenvalues is determinant then why isn't the determinant is $−q$?
The statement you're thinking of is that the product of eigenvalues of a matrix $A$ is equal (up to sign, depending on parity) to the constant term of its chara... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2636987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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Lebesgue integrable implies Riemann integrable? I'm self studying measure theory by Bartle's book and there he defined integrability for non-negative functions as follow
Definition: Let $f$ a non-negative measure function, then the integral of $f$ is
$$\int f = \sup \left\{ \int \phi : \ \phi \leq f \right\},$$
where... |
It's clear that this definition is the same of lower integral for Riemann integral (...)
That is false. As a counter-example, the function $\mathbf{1}_{\mathbb{[0,1] \backslash Q}}: [0,1] \to \mathbb{R}$ has lower integral for Riemann integral equal to $0$, and "lower" integral according to the Lebesgue definition eq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Differential equations theorems(Pure mathematics) I'm currently doing some graduate work and came upon some problems. The content of the course is of a pure form with topics such as
*
*Existence and Uniqueness of solutions
*linear system of 1st order ODE
*asympotitic behaviour of soltuions and stability analysis
... | This book by Walter covers the theoretical aspects of ordinary differential equations quite well and was used for my graduate course.
Also the book by Coddington is often suggested as a good source for rigorous existence and uniqueness theorems.
I personally find the book by Hale to be rigorous yet not too hard to re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2637204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Determining Spot Rates A three-year, 4%, par-value bond with annual coupons sells for $990$, a two-year, $1000$, 3% bond with annual coupons sells for $988$, and a one-year, zero-coupon, $1000$ bond sells for $974$. Determine the spot rates $r_1$, $r_2$ and $r_3$.
This comes from Mathematical Interest Theory textbook s... | For the one year bond we have:
$$
974=\frac{1000}{1+r_1}\qquad\Longrightarrow\qquad r_1=\frac{1000}{974}-1\approx 2.66940\%
$$
For the two years bond we have the coupon $3\%\times 1000=30$ and
$$
988=\frac{30}{1+r_1}+\frac{1030}{(1+r_2)^2}
$$
Observing that $\frac{1}{1+r_1}=\frac{974}{1000}=0.974$ we have
$$
988=\unde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2637391",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Show that the collection is a basis for a topology on $\mathbb{R}$ We are given the collection, $$\mathcal{B}=\{[a,b]\mid a,b\in\mathbb{R}\}.$$ We want to show that this collection is a basis for a topology on $\mathbb{R}$. To show this we show the following:
First, for all $x\in\mathbb{R}$ there exists an element in $... |
Is it okay for me to treat the element $[a,a]=\{a\}$ as an element of $\mathcal{B}$ in this proof?
Yes, that's totally fine. If you want to avoid it, you could also write
$$
\{a \} = [a-1,a] \cap [a, a+1].
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2637487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Is there a formula to calculate the area of a trapezoid knowing the length of all its sides? If all sides: $a, b, c, d$ are known, is there a formula that can calculate the area of a trapezoid?
I know this formula for calculating the area of a trapezoid from its two bases and its height:
$$S=\frac {a+b}{2}×h$$
And I ... | This problem is more subtle than some of the other answers here let on. A great deal hinges on whether "trapezoid" is defined inclusively (i.e. as a quadrilateral with at least one pair of parallel sides) or exclusively (i.e. as a quadrilateral with exactly one pair of parallel sides). The former definition is widely... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2637690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 11,
"answer_id": 4
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Set-theoretic equality of splitting fields within a fixed algebraic closure Let $F$ be a field and let $f(x)\in F[x]$ be a polynomial. Recall the following two facts:
(1) algebraic closures are unique up to isomorphism
(2) splitting fields are unique up to isomorphism
Fix an algebraic closure $\overline F$ of $F$. Is i... | I don't think they are equal as sets. Remember that in our way to get splitting field for $\;f(x)\in F[x]\;$ , we first get (assuming $\;f\;$ is irreducible, otherwise we take one of its irreducible factors) the quotient ring (field) $\;K:=F[x]/\langle f(x)\rangle\;$ .
Here, we already have no more $\;F\;$ but an iso... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove inequality using the Mean Value Theorem I'm trying to hone my problem-solving skills using the Mean Value Theorem and in one exercise, where $x \in (0, +\infty)$, I have to prove that:
*
*$(1+x)^a>ax+1$, if $a > 1$.
*$(1+x)^a<ax+1$, if $a \in (0, 1)$.
What I've tried:
I've tried to solve this problem us... | Your “instinct” is correct, and it requires only small additions to
make it a full proof.
The mean value theorem implies that for $x > 0$
$$
(1+x)^\alpha = 1 + \alpha x (1+k)^{\alpha-1}
$$
for some $k \in (0, x)$. It is relevant that $k$ is strictly positive,
so that one can continue to argue
$$
\alpha > 1 \Longright... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Interpretation of Sampling Distribution and Relationship to Test Statistics Explain what a sampling distribution is and why it is important to understand the sampling distribution of a test statistic.
This is the way that I understand it:
A sampling distribution is the probability distribution of a statistic that comes... | It is often impossible to test every member of a population, but if it were, one can know for certain the characteristics of the population. Since it is not practical to test the entire population it is often the case that a randomly chosen sample, or subset, of the population is analyzed. The sample is ran through tes... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find limit of $\lim_{x\to 0}\frac{\sin^{200}(x)}{x^{199}\sin(4x)}$, if it exists I'm practising solving limits and the one I'm currently struggling with is the following: $$\ell =\lim_{x\to 0}\frac{\sin^{200}(x)}{x^{199}\sin(4x)}$$
What I've done:
*
*Since this is an obvious $0/0$ , I tried using de L'Hospital's Rul... | Since $\sin x= x+o(x)$ we have, $$\frac{\sin^{200}x}{x^{199}\sin(4x)}= \frac{x^{200}+o(x^{200})}{x^{199}(4x+o(x))}=\frac{x^{200}+o(x^{200})}{4x^{200}+o(x^{200})}=\frac{1+o(1)}{4+o(1)} =\to\frac{1}{4}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2638190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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What is the value of $f(100)$?
We have $f:\Bbb R\to \Bbb R^*$, a function that admits primitives and admits the relations $$\cos \left(f(x)\right)=1,\ ∀x\in \Bbb R, \quad\text{and}\quad|f(\pi )−\pi |≤\pi .$$
What is the value of $f(100)$?
My thought. We obviously have
$$\cos (f(100)) =1\overset{?}{\implies} f(100) =... | Clearly $f(x)=2\pi k_x$ where $k_x\in\mathbb Z$ Besides
$$|2\pi k_x-\pi|\le\pi\iff-\pi\le2\pi k_x-\pi\le\pi\Rightarrow 0\le k_x\le1$$
Since $k_x$ is an integer the problem gives two solutions $f(x)=0$ and $f(x)=2\pi$ but because of the function is from $\Bbb R$ to $\Bbb R^*$ the only solution is
$$f(x)=2\pi$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2638315",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 1
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The sequence satisfying $a_2^{a_3^{\dots^{a_n}}} = n $ $a_2 = 2$
$a_2^{a_3 } = 3$
So $a_3= \ln(3)/\ln(2)$.
I wonder about all solutions $a_n$ such that
$a_2^{a_3^{\dots^{a_n}}} = n$
For all $n$.
How does $a_n$ behave? What are the best asymptotics?
Of course $a_n$ goes quickly towards values between $\exp(1/e)$ and $1... | We can't have $a_n\to a<\exp(1/e)$. If we did, then there must exist some $N$ such that $a_n<\exp(1/e)$ for all $n>N$, and then we'd have
$$a_2^{a_3^{\dots^{a_n^{a_{n+1}^{\dots}}}}}<a_2^{a_3^{\dots^{a_n^{\exp(1/e)^{\dots}}}}}=a_2^{a_3^{\dots^{a_n^e}}}$$
which is bounded, while $n$ is unbounded.
We also can't have $a_n\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2638828",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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On tensor product $U\otimes V$, $0\otimes w = 0$ for any $w\in V$ I've tried using the definition of tensor product: $U\otimes V = \mathcal{F}_{U\times V}/\langle S\rangle$, where
$$S = \{(\lambda_1u_1+\lambda_2u_2,v)-\lambda_1(u_1,v)-\lambda_2(u_2,v); (u,\lambda_1v_1+\lambda_2v_2)-\lambda_1(u,v_1)-\lambda_2(u,v_2)\}$... | Note that $0\otimes w = (0+0)\otimes w = 0\otimes w + 0 \otimes w$ by bilinearity,
so you get that $0\otimes w$ is the zero element of $V\otimes W$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2638955",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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The system of exponential equations How do you solve:
$$\begin{cases}
x\cdot2^{x-y}+3y\cdot2^{2x+y-1}=1 \hspace{0.1cm},\\ x\cdot2^{2x+y+1}+3y\cdot8^{x+y}=1\\
\end{cases}$$
I subtracted the equations, factorized by grouping, and got two terms equal zero. When I tried to use one of the terms, by expressing one variable t... | We obtain $$x(2^{x-y}-2^{2x+y+1})+3y(2^{2x+y-1}-2^{3x+3y})=0$$ or
$$x2^{x-y}(1-2^{x+2y+1})+3y(1-2^{x+2y+1})=0,$$ which gives
$$x2^{x-y}+3y=0$$ or
$$x+2y+1=0.$$
Can you end it now?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2639071",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to take the conjugate of a number with more than 2 square roots I was doing some abstract algebra and I came across the problem of figuring out if $\mathbb{Q}(\sqrt{2},\sqrt{3})=\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6} : a,b,c,d \in \mathbb{Q}\}$ is a ring and further if it is a field. Part of this is proving that every e... | The background story here is that your field is a splitting field of the polynomial $(x^2-2)(x^2-3)$, and as such the Galois group $\operatorname{Gal}(\mathbb Q(\sqrt 2, \sqrt 3):\mathbb Q)$ acts on it. It turns out that this group has four elements:
*
*$I$: Identity,
*$\Phi_2$: Automorphism that maps $\sqrt 2$ to ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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One-to-One function? In text it states that a one to one function is “A function f is one to one if for any two range values f(u) and f(v), f(u)=f(v) implies that u=v. What exactly does this mean? I thought if there are two equal y values it is NOT a one to one function?
| One to one means: For every output there is exactly one input.
Or in other words, If two times you got the same out put, then you must have had the same input.
Or in other words. If you got $f(u) = f(v)$ that means $u = v$.
=====
Or... one to one means. If $u \ne v$ then $f(u) \ne f(v)$.... so if you ever DO find y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that completely regular is a topological property Show that completely regular is a topological property.
Let $ X $ be a completely regular space and let $ h:X \rightarrow Y $ a homeomorphism. We will prove that $ Y $ is completely regular.
My Attempt
Suppose that $ C $ is a closed subset of $ Y $ and that $ y \... | To prove $Y$ is completely regular, you need to prove there exists a continuous function $g:Y\to [0,1]$ such such that $g(y)=0$ and $g(c)=1$ for all $c\in C$. You haven't actually exhibited such a function, so you haven't proved that $Y$ is completely regular. However, you've done most of the work needed to do so. B... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2639627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What is the derivative of $(2x)^{4x}$? This is quite simple. I know. I am having a problem when comparing my answer to online calculators like Symbolab and such.
$$\begin{array}{rll} y &= **(2x)^{4x}** & \text{equation}\\
\ln(y) &= \ln((2x)^{4x}) &\text{ take ln of both sides to bring 4x out front}\\
\ln(y) &= 4x \ln... | Given,
$$ f(x) = y = (2x)^{4x}$$
Taking natural log on both sides,
$$ \ln y = 4x \ln 2x$$
Now differentiating w.r.t. x,
$$ \frac{1}{y}\frac{dy}{dx} = 4x.\frac{1}{2x}.2 + ln 2x . 4$$
$$ \frac{dy}{dx} = y \,(4+4\ln 2x)$$
$$ = 4(2x)^{4x}(1+\ln 2x)$$
$$ = 2^{(4x+2)} . x^{4x} (1+ \ln 2x) $$
That's even the answer given i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2639778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
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How to study the strong and weak convergence of this sequence of functions? Discuss the strong and weak convergence of the sequence
$$u_n(x)=\sin(x)+\frac{1}{n}\sin^2(nx)$$
in the Sobolev space $W^{1,2}(0,1)$.
I know that a function $u(x)$ belong to $W^{1,p}(a,b)$ iff the $L^p$ norms of $u(x)$ and $u'(x)$ are finite. I... | The $L^2$ norm of $u_n(x)$ is not so hard to deal with. Remember, we just have to show that it is finite:
\begin{align}
\int_0 ^1\left|\sin(x)+\frac{1}{n}\sin^2(nx)\right|^2\;dx &\leq \int_0^1\left(|\sin(x)| + \frac{1}{n}\sin^2(nx)\right)^2\;dx\\
&=\int_0^1\sin^2(x)+\frac{2}{n}|\sin^3(x)|+\frac{1}{n^2}\sin^4(x)\;dx \\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2639921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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How many numbers from $1$ to $99999$ have the sum of the digits $= 15$? The problem:
How many numbers from $1$ to $99999$ have the sum of the digits $= 15$?
I thought of using the bitstring method and $x_1 + x_2 + x_3 + x_4 + x_5$ will be the boxes therefor we have $4$ zeros and $15$ balls. I'd say the answer would... | Your answer is incorrect since you have not considered the restriction that a digit in the decimal system cannot exceed $9$.
We want to find the number of positive integers between $1$ and $99~999$ inclusive that have digit sum $15$. Since $0$ does not have digit sum $15$, we get the same answer by considering nonnega... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640046",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Polynomials: gaining irreducibility by adding a constant EDITED:
Let $f\in\mathbb{Q}[X]$. I'm interested in the factoring properties of the family $F:=\{f+c\}_{c\in\mathbb{Q}}$ of rational polynomials that differ from $f$ only by a constant. Specifically:
1) I think there must be at least one irreducible polynomial in ... | The case of degree $2$ polynomials in $\mathbb{R}$ or $\mathbb{Q}$ can be handled. Let $f(x) = ax^2 + bx$. By the quadratic formula, if $b^2 - 4ac < 0$, then we have $f(x) + c$ irreducible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
How to Find the number of tangents to the curve y=$f\left(x\right)$ parallel to line $x+y=0$
Question For $x$$>0,$ let
$f\left(x\right)=\int_{1}^{x}\left(\sqrt{\log t}-\frac{1}{2}\log\sqrt{t}\right)dt$
The number of tangents to the curve y=$f\left(x\right)$parallel to
line x+y=0 is _________________
MY approach $x... | Hint: Finding the number of tangents of $f(x)$ parallel to line $y=-x$ is equivalent to finding the number of roots of the equation $f'(x)=-1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Is the set of terms of a sequence countable? According to Rosen, an infinite set A is countable if $|A|= |\mathbb{Z}^+|$ which in turn can be established by finding a bijection from A to $\mathbb{Z}^+$.
Also, a sequence is defined as a function from $\mathbb{Z}^+$ (or $\{0\} \cup \mathbb{Z}^+$) to some set.
With the ab... | Every sequence has a countable or a finite set of values.
Besides, you are mixing two ideas : a sequence $(u_n)_n$ is a function $n\mapsto u_n\in F$ ($F$ being any possible set) and almost never a bijection, but the set of all its values are finite or countable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640477",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is the automorphism group $Aut_{\mathbb{Q}} \mathbb{Q}(\sqrt{3}+\sqrt{2})$? I am trying to understand the automorphism group $Aut_{\mathbb{Q}} \mathbb{Q}(\sqrt{3}+\sqrt{2})$, aka the the automorphism group of $\mathbb{Q}(\sqrt{3}+\sqrt{2})$ over the rationals $\mathbb{Q}$. I know the following:
*
*$\mathbb{Q}(\... | What you are missing is the fact that a field automorphism for an algebraic extension always permutes the zeros of the minimal polynomial - but the permutations allowed are not arbitrary, they are those which always yield consistent results when applied to ANY sum or product of field elements. For example, in $\mathbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Determine all pairs of positive integers $(m,n)$ that satisfies $m!+n!=m^n$ Determine all pairs of positive integers $(m,n)$ that satisfies $m!+n!=m^n$
I found easily the pairs $(2,2)$ and $(2,3)$ but i can't prove that these are the only pairs possibles.
Any hints?
| This is not quite a full answer but it has the main ideas you'll need.
First, let us get the ability to assert a lower bound on $m$ after checking only finitely many pairs. This can be done by noting that $m^n>n!$, so by Stirling's approximation, $n<em$. Thus we can divide the problem between $m<m^*$ and $m \geq m^*$ b... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Are imaginary numbers really incomparable? If we really don't know which is bigger if $ i $ is greater or $ 2i $ or so on then why do we plot $ i $ first then $ 2i $ and so on, on the imaginary axis of the Argand plane? My teacher said that imaginary numbers are just points and all are dimensionless so they are incompa... | I've come to think of i as a rotation operator, so ordering may not make any sense. But these are the thoughts of an old man, and I'm not sure there is any basis to this? This probably ought to be a comment, but you can't comment on a question until you have a reputation level of 50+.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640860",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "47",
"answer_count": 9,
"answer_id": 8
} |
All prime ideals of $\mathbf{R}[x,y]$ and $\mathbf{C}[x, y]$. What are the all the prime ideals of $\mathbf{R}[x,y]$ and $\mathbf{C}[x, y]$? (Also, how do you prove that you've found all of them?) I'm trying to understand what the $\mathbf{R}$-algebraic vs $\mathbf{C}$-algebraic subsets of $\mathbf{C}^2$ are, defined a... | Here's an answer for $\mathbf C[x,y]$: Hilbert's Nullstellensatz, asserts the maximal ideals have the form $(x-\alpha,y-\beta)$ for some $\;\alpha, \beta\in \mathbf C$.
On the other hand, $\mathbf C[x,y]$ is a U.F.D. of (Krull) dimension $2$. So the prime ideals of height $1$ are principal, generated by irreducible p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2640985",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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$s + \frac{1}{n}$ is an upper bound for $A$ and $s - \frac{1}{n}$ is not an upper bound for $A$. Show $s = \sup A.$
Let $A \subset \mathbb R$ be non empty and bounded above, and let $s \in \mathbb R$ have the property that for all $n \in \mathbb N$, $s + \frac{1}{n}$ is an upper bound for $A$ and $s - \frac{1}{n}$ is ... | Pure definitions.
The sup must exist as the reals have the least upper bound property.
$s - \frac 1n$ is not an upper bound for all $n$ so $\sup A > s + \frac 1n$ or all natural $n$.
All $s + \frac 1n$ is an upper bound so $\sup A \le s+ \frac 1n$ for all natural $n$.
So $s-\frac 1n < \sup A \le s+ \frac 1n$ for all na... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2641128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
What does it mean for a function to be continuous on its domain? I never understood the phrase "continuous on its domain."
Isn't everything continuous on its own domain, since the domain are all the $x$ values that we can plug into $f(x)$ and get a defined $y$ value back? i.e. doesn't the domain by definition tell you ... | A function is said to be continuous if it continues at each point. This means that over the domain. Functions that are not continuous do not exist for every x value over the domain. For example if a function is defined near an open interval (the circle that is not shaded on a graph) then the function is discontinuous. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2641243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
The number of continuous functions $f:[0,1]\to\mathbb R$ that satisfy $\int_0^1xf(x)\,dx=\frac13+\frac14\int_0^1(f(x))^2\,dx$
90) The number of continuous functions $f:[0,1]\to\mathbb R$ that satisfy
$$\int_0^1xf(x)\,dx=\frac13+\frac14\int_0^1(f(x))^2\,dx$$
is
A) 0
B) 1
C) 2
D) $\infty$
How to approach th... | Another way:
$$\int^{1}_{0}4xf(x)dx-\int^{1}_{0}(f(x))^2dx=\frac{4}{3}$$
$$\int^{1}_{0}f(x)\bigg(4x-f(x)\bigg)dx\leq \frac{1}{4}\int^{1}_{0}\bigg[f(x)+4x-f(x)\bigg]^2dx=\frac{4}{3}.$$
Equality hold when $f(x)=2x$
In $2$ line earlilier i have used the inequality $$ab\leq \frac{(a+b)^2}{4}$$ equality hold when $a=b.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2641474",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
permutation of 5 digit numbers divisible by 3 "The total number of possible combination of 5 digits numbers formed from the digits(0,1,2,3,4,5,6,7,8,9) which are divisible by 3?"
This was the question given to me by my mathematics teacher during out permutation and combination lessons;I was able to solve this with ease... | Here is a correct solution along the lines of Manthanein's answer:
There are $9\cdot8\cdot7\cdot 6 \cdot5=15\,120$ strings of length $5$ not containing a repeat or zero. Adding $1$ mod $9$ to each digit in such a string changes its sum by $2$ mod $3$. From this we can conclude that exactly one third of these string... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2641627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Kids in wagons probability $11$ kids get on a train with $3$ wagons. What's the probability that in the $1$st wagon there are exactly $3$ kids? Isn't this the same with saying $x_1+x_2+x_3=11$ and $x_1=3$? If yes, how could one solve this?
| Well, yes, it's the same, but neither question has enough information to answer it. You don't give the underlying probability distribution, so we can't evaluate the probability.
The "natural" choice of underlying distribution might be different depending on your two presentations of the question. In the $x_1 + x_2 + x_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2641744",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Is it possible to "re-normalize" a Dirac delta function? The delta function in spherical coordinates is given by:
$$\delta(\vec{r}_0-\vec{r})=\frac{1}{r^2}\delta(r_0-r)\delta(\cos\theta_0-\cos\theta)\delta(\phi_0-\phi),$$
(The ordering of the coordinates inside the $\delta$'s isn't important). If I have a particular lo... | It seems OP wants to avoid having poles inside & outside the Dirac delta distribution. Using a hopefully obvious notation
$$s~\equiv~ \sin\theta,\qquad c~\equiv~ \cos\theta,$$
one possibility is to replace
$$ \frac{c}{s} \delta(c-c_0)~=~ \frac{sc}{|f^{\prime}(c)|} \delta(c-c_0)~=~sc~\delta(f(c)-f(c_0)), $$
where $$f(c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2641826",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Why the hexadecimal numbers can be converted directly into binary numbers so cleanly? Suppose we have F9 hex. If we want to convert it into binary, we just replace the hexadecimal numbers with their corresponding binary numbers. Like 9 has1001 in binary and F has 1111 in binary. By combining it becomes 1111 1001. But w... | Suppose we have a number $X$ which is written as $X_2X_1X_0$ in hexidecimal. Then we have $X = 16^0X_0 + 16^1X_1 + 16^2X_2$.
$16 = 2^4$ so we rewrite that as $X = 2^0X_0 + 2^4X_1 + 2^8X_2$. Now let's suppose $X_2$ can be written $x_3x_2x_1x_0$ in binary, then $X_2 = 2^3x_3 + 2^2x_2 +2^1x_1 +2^0x_0$. Substituting we ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2642058",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Basic counting trouble Suppose you have $8$ people. How many possible ways can you seat these people if
a) two persons, A and B, must sit together?
b) $4$ men and $4$ women, but no $2$ men and no $2$ women can sit together
c) there are $5$ men who must all sit together
d) there are $4$ married couples that must sit... | Are these circular tables or tables with a distinct orientation.
Based on the way you answered the first question, I am going to guess that these tables have an orientation.
a) 1 group of 2 and 6 groups of 1 is 7 groups total.
$2\cdot 7!$
b) You can but a man at the head or you can put a woman a the head. Once you hav... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2642146",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Piecewise functions in MATLAB. Help! I am trying to plot a pretty repetitive function in MATLAB. The values of $f$ are the same each $y$ for a designated season in the year. We have a function $f(t)$ and:
$$ f(t) = \begin{cases}
500 & \text{ if } 0 \le x < 91 \\
1500 & \text{ if } 91 \le x < 182 \\
500 & \text{ if } ... | It looks like you have 12 segments, so I'd make an input num_segments=12; and an input t_length=92; as finally an array y_result=[500,1500,500,0]; (and so on). Then, in your function, you have, say, a for loop, with i=0:num_segments-1 and then on each iteration, programmatically define t_lengthi<=t(i+1) and extract y_r... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2642225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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