Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
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Convergence of series given a monotonic sequence Let $(x_n)_{n \in \Bbb N}$ be a decreasing sequence such that its series converges, want to show that $\displaystyle \lim_{n \to \infty} n x_n = 0$.
Ok I don't even know where to start.
I need a direction please!
Thankyou!
| Just another approach. Since $\{x_n\}_{n\in\mathbb{N}}$ id decreasing and its associated series converges, we have $x_n\geq 0$ for every $n\in\mathbb{N}$ (otherwise, $\lim_{n\to +\infty}x_n < 0$ and the series cannot converge).
Assume now the existence of a positive real number $\alpha$ such that
$$ n\, x_n \geq \alpha... | {
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"url": "https://math.stackexchange.com/questions/223452",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$a^2-b^2 = x$ where $a,b,x$ are natural numbers Suppose that $a^2-b^2 =x$ where $a,b,x$ are natural numbers.
Suppose $x$ is fixed. If there is one $(a,b)$ found, can there be another $(a,b)$?
Also, would there be a way to know how many such $(a,b)$ exists?
| You want $x = a^2 - b^2 = (a-b)(a+b)$. Let $m = a-b$ and $n = a+b$, then note that $a = (m+n)/2$ and $b = (n-m)/2$. For these to be natural numbers, you want both $m$ and $n$ to be of the same parity (i.e., both odd or both even), and $m \le n$. For any factorization $x = mn$ satisfying these properties, $a = (m+n)/2$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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sines and cosines law Can we apply the sines and cosines law on the external angles of triangle ?
| This answer assumes a triangle with angles $A, B, C$ with sides $a,b,c$.
Law of sines states that $$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$
Knowing the external angle is $\pi - \gamma$ if the angle is $\gamma$, $\sin(\pi-\gamma) = \sin \gamma$ because in the unit circle, you are merely reflecting the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/223583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What type of singularity is this? $z\cdot e^{1/z}\cdot e^{-1/z^2}$ at $z=0$.
My answer is removable singularity.
$$
\lim_{z\to0}\left|z\cdot e^{1/z}\cdot e^{-1/z^2}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{z-1}{z^2}}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{-1}{z^2}}\right|=0.
$$
But someone says it is an essential si... | $$ze^{1/z}e^{-1/z^2}=z\left(1+\frac{1}{z}+\frac{1}{2!z^2}+...\right)\left(1-\frac{1}{z^2}+\frac{1}{2!z^4}-...\right)$$
So this looks like an essential singularity, uh?
I really don't understand how you made the following step:
$$\lim_{z\to 0}\left|z\cdot e^{\frac{z-1}{z^2}}\right|=\lim_{z\to0}\left|z\cdot e^{\frac{-1}... | {
"language": "en",
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Math induction ($n^2 \leq n!$) help please I'm having trouble with a math induction problem. I've been doing other proofs (summations of the integers etc) but I just can't seem to get my head around this.
Q. Prove using induction that $n^2 \leq n!$
So, assume that $P(k)$ is true: $k^2 \leq k!$
Prove that $P(k+1)$ i... | The statement you want to prove is for all $n\in\mathbb{N}$ it holds that $n^2\leq n!$ (you called this $P(n)$. So lets first prove $P(4)$ i.e. $4^2\leq 4!$ but since $16\leq 24$ this is clear. So lets assume $P(n)$ and prove $P(n+1)$.
First note that for $n\geq 2$ it holds that
$$ 0\leq (n-1)^2+(n-2)=n^2-2n+1+n-2=n^2... | {
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"url": "https://math.stackexchange.com/questions/223718",
"timestamp": "2023-03-29T00:00:00",
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Are Trace of product of matrices- distributive/associative? Is $\operatorname{Tr}(X^TAX)-\operatorname{Tr}(X^TBX)$ equal to $\operatorname{Tr}(X^TCX)$, where $C=A-B$ and $A$, $B$, $X$ have real entries and also $A$ and $B$ are p.s.d.
| Yes, as
$$X^t(A-B)X=X^t(AX-BX)=X^tAX-X^tBX,$$
using associativity and distributivity of product with respect to the addition. The fact that the matrices $A$ and $B$ are p.s.d. is not needed here.
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "3",
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Does a tower of Galois extensions in $\mathbb{C}$ give an overall Galois extension? If $L/K$ and $F/L$ are Galois extensions inside $\mathbb{C}$, must $F/K$ be a Galois extension?
| Consider the extension $\mathbb Q\subset\mathbb Q(\sqrt[4]{2})\subset \mathbb Q(\sqrt[4]{2},i) $. You have that $\mathbb Q(\sqrt[4]{2})/\mathbb Q$ is not Galois since it is not normal. Yu have to enlarge $\mathbb Q(\sqrt[4]{2})$ over $\mathbb Q$ in order to get Galois extension.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/223866",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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generalized MRRW bound on the asymptotic rate of q-ary codes Among the many upper bounds for families of codes in $\mathbb F _2 ^n$, the best known bound is the one by McEliece, Rodemich, Rumsey and Welch (MRRW) which states that the rate $R(\delta)$ corresponding to a relative distance of $\delta$ is such that:
\begin... | The source is formula (3) on page 86 in the artice
Aaltonen, Matti J.: Linear programming bounds for tree codes. IEEE Transactions on Information Theory 25.1 (1979), 85–90,
doi: 10.1109/tit.1979.1056004.
According to the article, there is the additional requirement $0 < \delta < 1 - \frac{1}{q}$.
Instead of comparing t... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Interior Set of Rationals. Confused! Can someone explain to me why the interior of rationals is empty? That is $\text{int}(\mathbb{Q}) = \emptyset$?
The definition of an interior point is "A point $q$ is an interior point of $E$ if there exists a ball at $q$ such that the ball is contained in $E$" and the interior set ... | It is easy to show that there are irrational numbers between any two rational numbers. Let $q_1 < q_2$ be rational numbers and choose a positive integer $m$ such that $m(q_2-q_1)>2$. Taking some positive integer $m$ such that $m(b-a)>2$, the irrational number $m q_1+\sqrt{2}$ belongs to the interval $(mq_1, mq_2)$ and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is the difference of the natural logarithms of two integers always irrational or 0? If I have two integers $a,b > 1$. Is
$\ln(a) - \ln(b)$
always either irrational or $0$. I know both $\ln(a)$ and $\ln(b)$ are irrational.
| If $\log(a)-\log(b)$ is rational, then $\log(a)-\log(b)=p/q$ for some integers $p$ and $q$, hence $\mathrm e^p=r$ where $r=(a/b)^q$ is rational. If $p\ne0$, then $\mathrm e=r^{1/p}$ is algebraic since $\mathrm e$ solves $x^p-r=0$. This is absurd hence $p=0$, and $a=b$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/225039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Homology of pair (A,A) Why is the homology of the pair (A,A) zero?
$$H_n(A,A)=0, n\geq0$$
To me it looks like the homology of a point so at least for $n=0$ it should not be zero.
How do we see this?
| Let us consider the identity map $i : A \to A$. This a homeomorphism and so induces an isomorphism on homology. Now consider the long exact sequence of the pair $(A,A)$: We get
$$\ldots \longrightarrow H_n(A)\stackrel{\cong}{\longrightarrow} H_n(A) \stackrel{f}{\longrightarrow} H_n(A,A) \stackrel{g}{\longrightarrow} ... | {
"language": "en",
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How many ways to reach $1$ from $n$ by doing $/13$ or $-7$? How many ways to reach $1$ from $n$ by doing $/13$ or $-7$ ?
(i.e., where $n$ is the starting value (positive integer) and $/13$ means division by $13$ and $-7$ means subtracting 7)?
Let the number of ways be $f(n)$.
Example $n = 20$ , then $f(n) = 1$ since $1... | Some more thoughts to help:
As $13 \equiv -1 \pmod 7$, you can only get to $1$ for numbers that are $\equiv \pm 1 \pmod 7$. You can handle $1$ and $13$, so you can handle all $k \equiv \pm 1 \pmod 7 \in \mathbb N $ except $6$.
Also because $13 \equiv -1 \pmod 7$, for $k \equiv -1 \pmod 7$ you have to divide an odd num... | {
"language": "en",
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Constructing a triangle given three concurrent cevians? Well, I've been taught how to construct triangles given the $3$ sides, the $3$ angles and etc. This question came up and the first thing I wondered was if the three altitudes (medians, concurrent$^\text {any}$ cevians in general) of a triangle are unique for a par... | It is clear that the lengths of concurrent cevians cannot always determine the triangle. Indeed, they probably never can. But if it is clear, we must be able to give an explicit example.
Cevians $1$: Draw an equilateral triangle with height $1$. Pick as your cevians the altitudes.
Cevians $2$: Draw an isosceles tria... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/225255",
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Logic for getting number of pages If a page can have 27 items printed on it and number of items can be any positive number then how can I find number of pages if I have number of items, I tried Modulus and division but didn't helped.
FYI, I am using C# as programming platform.
| If I understand the question correctly, isn't the answer just the number of total items divided by $27$ and then rounded up?
If you had $54$ total items, $54/27=2$ pages, which doesn't need to round.
If you had $100$ total items, $100/27=3.7$ which rounds up to $4$ pages.
If you had 115 total items, $115/27=4.26$ which... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Squeeze Theorem Problem I'm busy studying for my Calculus A exam tomorrow and I've come across quite a tough question. I know I shouldn't post such localized questions, so if you don't want to answer, you can just push me in the right direction.
I had to use the squeeze theorem to determine:
$$\lim_{x\to\infty} \dfrac{... | I assume you meant $$\lim_{x \to \infty} \dfrac{2x^3 + \sin(x^2)}{1+x^3}$$ Note that $-1 \leq \sin(\theta) \leq 1$. Hence, we have that $$\dfrac{2x^3 - 1}{1+x^3} \leq \dfrac{2x^3 + \sin(x^2)}{1+x^3} \leq \dfrac{2x^3 + 1}{1+x^3}$$
Note that
$$\dfrac{2x^3 - 1}{1+x^3} = \dfrac{2x^3 +2 -3}{1+x^3} = 2 - \dfrac3{1+x^3}$$
$$\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/225374",
"timestamp": "2023-03-29T00:00:00",
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Derivative of a split function We have the function:
$$f(x) = \frac{x^2\sqrt[4]{x^3}}{x^3+2}.$$
I rewrote it as
$$f(x) = \frac{x^2{x^{3/4}}}{x^3+2}.$$
After a while of differentiating I get the final answer:
$$f(x)= \frac{- {\sqrt[4]{\left(\frac{1}{4}\right)^{19}} + \sqrt[4]{5.5^7}}}{(x^3+2)^2}$$(The minus isn't behind... | Let $y=\frac{x^2\cdot x^{3/4}}{x^3+2}$ so $y=\frac{x^{11/4}}{x^3+2}$ and therefore $y=x^{11/4}\times(x^3+2)^{-1}$. Now use the product rule of two functions: $$(f\cdot g)'=f'\cdot g+f\cdot g'$$ Here $f(x)=x^{11/4}$ and $g(x)=(x^3+2)^{-1}$. So $f'(x)=\frac{11}{4}x^{7/4}$ and $g'(x)=(-1)(3x^2)(x^3+2)^{-2}$. But thinking... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/225438",
"timestamp": "2023-03-29T00:00:00",
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Normal subgroups of the Special Linear Group What are some normal subgroups of SL$(2, \mathbb{R})$?
I tried to check SO$(2, \mathbb{R})$, UT$(2, \mathbb{R})$, linear algebraic group and some scalar and diagonal matrices, but still couldn't come up with any. So can anyone give me an idea to continue on, please?
| ${\rm{SL}}_2(\mathbb{R})$ is a simple Lie group, so there are no connected normal subgroups.
It's only proper normal subgroup is $\{I,-I\}$
| {
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Almost A Vector Bundle I'm trying to get some intuition for vector bundles. Does anyone have good examples of constructions which are not vector bundles for some nontrivial reason. Ideally I want to test myself by seeing some difficult/pathological spaces where my naive intuition fails me!
Apologies if this isn't a pa... | Fix $ B = (-1,1) $ to be the base space, and to each point $ b $ of $ B $, attach the vector-space fiber $ \mathcal{F}_{b} \stackrel{\text{def}}{=} \{ b \} \times \mathbb{R} $. We thus obtain a trivial $ 1 $-dimensional vector bundle over $ B $, namely $ B \times \mathbb{R} $. Next, define a fiber-preserving vector-bun... | {
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for $\nu$ a probability measure on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ the set ${x\in \mathbb{R} ; \nu(x) > 0}$ is at most countable Given a probability measure $\nu$ on $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$, how do I show that the set (call it $S$) of all $x\in \mathbb{R}$ where $\nu(x)>0$ holds is at most count... | Given $n\in\mathbb N$, consider the set
$$A_n=\{x\in\mathbb R:\nu(\{x\})\geq\tfrac{1}{n}\}$$
It must be finite; otherwise, the probability of $A_n$ would be infinite since $\nu$ is additive. Thus, $A=\cup_{n\in\mathbb N}A_n$ is countable as a countable union of finite sets, but it is clear that
$$A=\{x\in\mathbb R:\nu... | {
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Are these two predicate statements equivalent or not? $\exists x \forall y P(x,y) \equiv \forall y \exists x P(x,y)$
I was told they were not, but I don't see how it can be true.
| Take the more concrete statements: $\forall {n \in \mathbb{N}}\ \exists {m \in \mathbb{N}}: m > n$ versus $\exists {m \in \mathbb{N}}\ \forall {n \in \mathbb{N}}: m > n$. Now the first statement reads: For every natural number there's a bigger one. The second statement reads: There's a biggest natural number.
Quantifie... | {
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Hatcher: Barycentric Subdivision At the bottom of page 122 of Hatcher, he defines a map $S:C_{n}(X)\rightarrow C_{n}(X)$ by $S\sigma=\sigma_{\#}S\Delta^{n}$. What is the $S$ on the right hand side and how does it act on the simplex $\Delta$? I'm having trouble deconstructing the notation here.
| It's the $S$ he defines in (2) of his proof: barycentric subdivision of linear chains.
I think you will benefit from reading the following supplementary notes.
update: here's the idea: first he deals with linear chains so he knows how to define the map $S: LC_n(\Delta^n) \to LC_n(\Delta^n)$, i.e. he knows how to find ... | {
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The dual of a finitely generated module over a noetherian integral domain is reflexive. As posted by user26857 in this question the dual of a finitely generated module over a noetherian integral domain is reflexive. Could you tell me how to prove it?
| I think I have a proof.
I will use this theorem:
Let $A$ be a Noetherian ring and $M$ a finitely generated $A$-module. Then
$M^*$ is reflexive if and only if $M^*_P$ is reflexive for every $P$ such that $\mathrm{depth}\;A_P=0$; in short, a dual is reflexive if and only if it is reflexive in depth $0$.
If the ring ... | {
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"timestamp": "2023-03-29T00:00:00",
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Find the smallest positive integer satisfying 3 simultaneous congruences $x \equiv 2 \pmod 3$
$2x \equiv 4 \pmod 7$
$x \equiv 9 \pmod {11}$
What is troubling me is the $2x$.
I know of an algorithm (not sure what it's called) that would work if all three equations were $x \equiv$ by using euclidean algorithm back subst... | To supplement the other answers, if your equation was
$$ 2x \equiv 4 \pmod 8 $$
then the idea you guessed is actually right: this equation is equivalent to
$$ x \equiv 2 \pmod 4 $$
More generally, the equations
$$ a \equiv b \pmod c $$
and
$$ ad \equiv bd \pmod {cd}$$
are equivalent. Thus, if both sides of the equation... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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An example for a calculation where imaginary numbers are used but don't occur in the question or the solution. In a presentation I will have to give an account of Hilbert's concept of real and ideal mathematics. Hilbert wrote in his treatise "Über das Unendliche" (page 14, second paragraph. Here is an English version -... | The canonical example seems to be Cardano's solution of the cubic equation, which requires non-real numbers in some cases even when all the roots are real. The mathematics is not as hard as you might think; and as an added benefit, there is a juicy tale to go with it – as the solution was really due to Scipione del Fer... | {
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"url": "https://math.stackexchange.com/questions/225922",
"timestamp": "2023-03-29T00:00:00",
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Adding sine waves of different phase, sin(π/2) + sin(3π/2)? Adding sine waves of different phase, what is $\sin(\pi/2) + \sin(3\pi/2)$?
Please could someone explain this.
Thanks.
| Heres the plot for $\sin(L)$ where $L$ goes from $(0, \pi/2)$
Heres the plot for $\sin(L) + \sin(3L)$ where $L$ goes from $(0, \pi/2)$
I hope this distinction is useful to you.
This was done in Mathematica.
| {
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Showing that $1/x$ is NOT Lebesgue Integrable on $(0,1]$ I aim to show that $\int_{(0,1]} 1/x = \infty$. My original idea was to find a sequence of simple functions $\{ \phi_n \}$ s.t $\lim\limits_{n \rightarrow \infty}\int \phi_n = \infty$. Here is a failed attempt at finding such a sequence of $\phi_n$:
(1) Let $A_... | I think this may be the same as what Davide Giraudo wrote, but this way of saying it seems simpler. Let $\lfloor w\rfloor$ be the greatest integer less than or equal to $w$. Then the function
$$x\mapsto \begin{cases} \lfloor 1/x\rfloor & \text{if } \lfloor 1/x\rfloor\le n \\[8pt]
n & \text{otherwise} \end{cases}$$ is... | {
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Using induction to prove $3$ divides $\left \lfloor\left(\frac {7+\sqrt {37}}{2}\right)^n \right\rfloor$ How can I use induction to prove that $$\left \lfloor\left(\cfrac {7+\sqrt {37}}{2}\right)^n \right\rfloor$$ is divisible by $3$ for every natural number $n$?
| Consider the recurrence given by
$$x_{n+1} = 7x_n - 3 x_{n-1}$$ where $x_0 = 2$, $x_1 = 7$.
Note that $$x_{n+1} \equiv (7x_n - 3 x_{n-1}) \pmod{3} \equiv (x_n + 3(2x_n - x_{n-1})) \pmod{3} \equiv x_n \pmod{3}$$
Since $x_1 \equiv 1 \pmod{3}$, we have that $x_n \equiv 1 \pmod{n}$. Ths solution to this recurrence is give... | {
"language": "en",
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Integration with infinity and exponential How is
$$\lim_{T\to\infty}\frac{1}T\int_{-T/2}^{T/2}e^{-2at}dt=\infty\;?$$
however my answer comes zero because putting limit in the expression, we get:
$$\frac1\infty\left(-\frac1{2a}\right) [e^{-\infty} - e^\infty]$$ which results in zero?
I think I am doing wrong. So how can... | $I=\int_{-T/2}^{T/2}e^{-2at}dt=\left.\frac{-1}{2a}e^{-2at}\right|_{-T/2}^{T/2}=-\frac{e^{-aT}+e^{aT}}{2a}$
Despite that $a$ positive or negative, one of the exponents will tend to zero at our limit, so we can rewrite it as :
$\lim_{T\rightarrow\infty}\frac{I}{T}=\lim_{T\rightarrow\infty}\left(-\frac{e^{-aT}+e^{aT}}{2aT... | {
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The positive integer solutions for $2^a+3^b=5^c$ What are the positive integer solutions to the equation
$$2^a + 3^b = 5^c$$
Of course $(1,\space 1, \space 1)$ is a solution.
| There are three solutions which can all be found by elementary means.
If $b$ is odd
$$2^a+3\equiv 1 \bmod 4$$
Therefore $a=1$ and $b$ is odd.
If $b>1$, then $2\equiv 5^c \bmod 9$ and $c\equiv 5 \bmod 6$
Therefore $2+3^b\equiv 5^c\equiv3 \bmod 7$ and $b\equiv 0 \bmod 6$, a contradiction.
The only solution is $(a,b,... | {
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"timestamp": "2023-03-29T00:00:00",
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Why $0!$ is equal to $1$? Many counting formulas involving factorials can make sense for the case $n= 0$ if we define $0!=1 $; e.g., Catalan number and the number of trees with a given number of vetrices. Now here is my question:
If $A$ is an associative and commutative ring, then we can define an
unary operation on... | As pointed out in one of the answers to this math.SX question, you can get the Gamma function as an extension of factorials, and then this falls out from it (though this isn't a very combinatorial answer).
| {
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Signs of rates of change when filling a spherical tank. Water is flowing into a large spherical tank at a constant rate. Let $V\left(t\right)$ be the volume of water in the tank at time $t$, and $h\left(t\right)$ of the water level at time $t$.
Is $\frac{dV}{dt}$ positive, negative, or zero when the tank is one quarte... | Uncookedfalcon is correct. You're adding water, which means that both the volume and water depth are increasing (that is, $\frac{dV}{dt}$ and $\frac{dh}{dt}$ are positive) until it's full. In fact, $\frac{dV}{dt}$ is constant until it's full.
| {
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"timestamp": "2023-03-29T00:00:00",
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is a non-falling rank of smooth maps an open condition? If $f \colon M \to N$ is a smooth map of smooth manifolds, for any point $p \in M$, is there an open neighbourhood $V$ of $p$ such that $\forall q \in V \colon \mathrm{rnk}_q (f) \geq \mathrm{rnk}_p (f)$?
| Yes. Note that if $f$ has rank $k$ at $p$, then $Df(p)$ has rank $k$. Therefore in coordinates, there is a non-vanishing $k \times k$-minor of $Df(p)$. As having a non-vanishing determinant is an open condition, the same minor will not vanish in a whole neighbourhood $V$ of $p$, giving $\operatorname{rank}_q f \ge k$, ... | {
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Describing the intersection of two planes Consider the plane with general equation 3x+y+z=1 and the plane with vector equation (x, y, z)=s(1, -1, -2) + t(1, -2 -1) where s and t are real numbers. Describe the intersection of these two planes.
I started by substituting the parametric equations into the general equation... | As a test to see if the planes are parallel you can calculate the normalvectors for the planes {n1, n2}.
If $abs\left (\frac{(n1\cdot n2)}{\left | n1 \right |*\left | n2 \right |} \right )==1$ the planes are parallel.
| {
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Example where $\int |f(x)| dx$ is infinite and $\int |f(x)|^2 dx$ is finite I read in a book that the condition $\int |f(x)|^2 dx <\infty$ is less restrictive than $\int |f(x)| dx <\infty$. That means whenever $\int |f(x)| dx$ is finite, $\int |f(x)|^2 dx$ is also finite, right?
My understanding is that $|f(x)|$ may h... | The most noticeable one I think is the sinc function $$\mathrm{sinc}(x)=\frac{\sin(\pi x)}{\pi x}$$
| {
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Which letter is not homeomorphic to the letter $C$? Which letter is not homeomorphic to the letter $C$?
I think letter $O$ and $o$ are not homeomorphic to the letter $C$. Is that correct?
Is there any other letter?
| There are many others $E$ or $Q$ for example. The most basic method I know of is by assuming there is one, then it restricts to the subspace if you take out one (or more) points. Then the number of connected components of this subspace is an invariant.
| {
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"question_score": "5",
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How do I prove that $S^n$ is homeomorphic to $S^m \Rightarrow m=n$? This is what I have so far:
Assume $S^n$ is homeomorphic to $S^m$. Also, assume $m≠n$. So, let $m>n$.
From here I am not sure what is implied. Of course in this problem $S^k$ is defined as:
$S^k=\lbrace (x_0,x_1,⋯,x_{k+1}):x_0^2+x_1^2+⋯+x_{k+1}^2=1 \rb... | Hint: look at the topic Invariance of Domain
| {
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Trace of $A$ if $A =A^{-1}$ Suppose $I\neq A\neq -I$, where $I$ is the identity matrix and $A$ is a real $2\times2$ matrix. If $A=A^{-1}$ then what is the trace of $A$? I was thinking of writing $A$ as $\{a,b;c; d\}$ then using $A=A^{-1}$ to equate the positions but the equations I get suggest there is an easier way.
| From $A=A^{-1}$ you will know that all the possible eigenvalues are $\pm 1$, so the trace of $A$ would only be $0$ or $\pm 2$. You may show that all these three cases are realizable.
| {
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"answer_id": 3
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Newton-Raphson method and solvability by radicals If a polynomial is not solvable by radicals, then does the Newton-Raphson method work slower or faster? I don't know how to approach this.
| The speed of Newton-Raphson has [EDIT: almost] nothing to do with solvability by radicals. What it does have to do with is $f''(r)/f'(r)$ where $r$ is the root: i.e. if $r$ is a root of $f$ such that $f'(r) \ne 0$ and
Newton-Raphson starting at $x_0$ converges to $r$, then
$$\lim_{n \to \infty} \dfrac{x_{n+1} - r}{(x... | {
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If $b$ is a root of $x^n -a$, what's the minimal polynomial of $b^m$? Let $x^n -a \in F[x]$ be an irreducible polynomial over $F$, and let $b \in K$ be its root, where $K$ is an extension field of F. If $m$ is a positive integer such that $m|n$, find the degree of the minimal polynomial of $b^m$ over $F$.
My solution:
... | Hint Let $km = n$ then if $b^{mk} - a = 0$ then $(b^m)^k - a = 0$ so maybe $x^k-a$ is the minimal polynomial?
Hint' Show that if $x^k-a$ has a factor then so does $x^{mk} - a$.
Given a field extension $K/L$ then $K$ is a vector space with coefficients in $L$ of dimension $\left[K:L\right]$ which is called the degree o... | {
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Describe a group homomorphism from $U_8$ to $S_4$ Im in an intro course to abstract algebra and we have been focusing completely on rings/the chinese remainder theorem and this question came up in the review and totally stumped me (we only have basic definitions of groups and subgroups and homomorphisms).
I think that ... | A unit mod $8$ is a congruence class mod $8$ which is invertible, i.e., a class $[a]$ such that there exists $[b]$ with $[a][b] = [1]$, or equivalently $ab +8k = 1$ for some integer $k$. Now any number dividing both $a$ and $8$ would also divide $ab+8k=1$, so this implies that $[a]$ being a unit implies $(a,8)=1$ (wher... | {
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Expectations with self-adjoint random matrix So, we have a square matrix $A=(a_{ij})_{1 \leq i,j \leq n}$ where the entries are independent random variables with the same distribution. Suppose $A = A^{*}$, where $A^{*}$ is the classical adjoint. Moreover, suppose that $E(a_{ij}) = 0$, $E(a_{ij}^{2}) < \infty$. How can ... | Clearly
$$
\operatorname{Tr}(A^2) = \sum_{i,j} a_{i,j} a_{j,i} \stackrel{\rm symmetry}{=}\sum_{i,j} a_{i,j}^2
$$
Thus
$$
\mathbb{E}\left(\operatorname{Tr}(A^2) \right) = \sum_{i,j} \mathbb{E}(a_{i,j}^2) = n^2 \mathbb{Var}(a_{1,1})
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/227279",
"timestamp": "2023-03-29T00:00:00",
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Is there a Math symbol that means "associated" I am looking for a Math symbol that means "associated" and I don't mean "associated" as something as complicated as isomorphism or anything super fancy.
I am looking for a symbol that means something like "$\triangle ABC$ [insert symbol] $A_{1}$" (as in triangle ABC "asso... | In general, you can use a little chain link symbol since the meaning behind "associated" is "connection" where you are not specifying the type of connection or how they are connected. That will reduce your horizontal space and make sense to people.... ~ is the NOT symbol in logic so never use that! Don't use the squigg... | {
"language": "en",
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Coercion in MAGMA In MAGMA, if you are dealing with an element $x\in H$ for some group $H$, and you know that $H<G$ for some group $G$, is there an easy way to coerce $x$ into $G$ (e.g. if $H=\text{Alt}(n)$ and $G=\text{Alt}(n+k)$ for some $k\geq 1$)? The natural coercion method $G!x$ does not seem to work.
| G!CycleDecomposition(g);
will work
| {
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Do an axis-aligned rectangle and a circle overlap? Given a circle of radius $r$ located at $(x_c, y_c)$ and a rectangle defined by the points $(x_l, y_l), (x_l+w, y_l+h)$ is there a way to determine whether the the two overlap? The square's edges are parallel to the $x$ and $y$ axes.
I am thinking that overlap will occ... | No. Imagine a square and enlarge its incircle a bit. They will overlap, but wouldn't satisfy neither of your requirement.
Unfortunately, you have to check all points of the circle. Or, rather, solve the arising inequalities (I assume you are talking about filled idoms):
$$\begin{align} (x-x_c)^2+(y-y_c)^2 & \le r \\
x... | {
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How to solve second order PDE with first order terms. I know we can transform a second order PDE into three standard forms. But how to deal with the remaining first order terms?
Particularly, how to solve the following PDE:
$$
u_{xy}+au_x+bu_y+cu+dx+ey+f=0
$$
update:
$a,b,c,d,e,f$ are all constant.
| Case $1$: $a=b=c=0$
Then $u_{xy}+dx+ey+f=0$
$u_{xy}=-dx-ey-f$
$u_x=\int(-dx-ey-f)~dy$
$u_x=C(x)-dxy-\dfrac{ey^2}{2}-fy$
$u=\int\left(C(x)-dxy-\dfrac{ey^2}{2}-fy\right)dx$
$u=C_1(x)+C_2(y)-\dfrac{dx^2y}{2}-\dfrac{exy^2}{2}-fxy$
Case $2$: $a\neq0$ and $b=c=0$
Then $u_{xy}+au_x+dx+ey+f=0$
Let $u_x=v$ ,
Then $u_{xy}=v_y$
$... | {
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How many numbers between $1$ and $6042$ (inclusive) are relatively prime to $3780$? How many numbers between $1$ and $6042$ (inclusive) are relatively prime to $3780$?
Hint: $53$ is a factor.
Here the problem is not the solution of the question, because I would simply remove all the multiples of prime factors of $378... | $3780=2^2\cdot3^3\cdot5\cdot7$
Any number that is not co-prime with $3780$ must be divisible by at lease one of $2,3,5,7$
Let us denote $t(n)=$ number of numbers$\le 6042$ divisible by $n$
$t(2)=\left\lfloor\frac{6042}2\right\rfloor=3021$
$t(3)=\left\lfloor\frac{6042}3\right\rfloor=2014$
$t(5)=\left\lfloor\frac{6042}5\... | {
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$n$ Distinct Eigenvectors for an $ n\times n$ Hermitian matrix? Much like the title says, I wish to know how it is possible that we can know that there are $n$ distinct eigenvectors for an $n\times n$ Hermitian matrix, even though we have multiple eigenvalues. My professor hinted at using the concept of unitary transfo... | You can show that any matrix is unitarily similar to an upper triangular matrix over the complex numbers. This is the Schur decomposition which Ed Gorcenski linked to. Given this transformation, let $A$ be a Hermitian matrix. Then there exists unitary matrix $U$ and upper-triangular matrix $T$ such that
$$A = UTU^{\dag... | {
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Is there a name for this ring-like object? Let $S$ be an abelian group under an operation denoted by $+$. Suppose further that $S$ is closed under a commutative, associative law of multiplication denoted by $\cdot$. Say that $\cdot$ distributes over $+$ in the usual way. Finally, for every $s\in S$, suppose there exist... | This is a pseudo-ring, or rng, or ring-without-unit. The article linked in fact actually mentions the example of functions with compact support. The fact that you have a per-element neutral element is probably not sufficiently useful to give a special name to pseudo-rings with this additional property.
| {
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Simple Characterizations of Mathematical Structures By no means trivial, a simple characterization of a mathematical structure is a simply-stated one-liner in the following sense:
Some general structure is (surprisingly and substantially) more structured if and only if the former satisfies some (surprisingly and superf... | A natural number $p$ is prime if and only if it divides $(p-1)! + 1$ (and is greater than 1).
| {
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Are there values $k$ and $\ell$ such that $n$ = $kd$ + $\ell$? Prove. Suppose that n $\in$ $\mathbb Z$ and d is an odd natural number, where $0 \notin\mathbb N$. Prove that $\exists$ $\mathcal k$ and $\ell$ such that $n =\mathcal kd +\ell$ and $\frac {-d}2 < \ell$ < $\frac d2$.
I know that this is related to Euclidean... | We give a quite formal, and unpleasantly lengthy, argument. Then in a remark we say what's really going on. Let $n$ be an integer. First note that there are integers $x$ such that $n-xd\ge 0$. This is obvious if $n\ge 0$. And if $n \lt 0$, we can for example use $x=-n+1$.
Let $S$ be the set of all non-negative integer... | {
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Minimal surfaces and gaussian and normal curvaturess If $M$ is the surface $$x(u^1,u^2) = (u^2\cos(u^1),u^2\sin(u^1), p\,u^1)$$ then I am trying to show that $M$ is minimal. $M$ is referred to as a helicoid.
Also I am confused on how $p$ affects the problem
| There is a good reason that the value of $p$ does not matter, as long as $p \neq 0.$
If you begin with a sphere of radius $R$ and blow it up to a sphere of radius $SR,$ the result is to multiply the mean curvature by $\frac{1}{S}.$ This is a general phenomenon. A map, which is also linear, given by moving every point ... | {
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Proj construction and fibered products How to show, that
$Proj \, A[x_0,...,x_n] = Proj \, \mathbb{Z}[x_0,...,x_n] \times_\mathbb{Z} Spec \, A$?
It is used in Hartshorne, Algebraic geometry, section 2.7.
| Show that you have an isomorphism on suitable open subsets, and that the isomorphisms glue. The standard ones on $\mathbb{P}^n_a$ should suffice. Use that $$\mathbb{Z}[x_0, \ldots, x_n] \otimes_\mathbb{Z} A \cong A[x_0,..., \ldots, x_n].$$ Maybe you could prove the isomorphism by using the universal property of project... | {
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Show that $\left(\frac{1}{2}\left(x+\frac{2}{x}\right)\right)^2 > 2$ if $x^2 > 2$ Okay, I'm really sick and tired of this problem. Have been at it for an hour now and we all know the drill: if you don't get to the solution of a simple problem, you won't, so ...
I'm working on a proof for the convergence of the Babyloni... | First, swap $x_n^2$ for $2y$, just to make it simpler to write. The hypothesis is then $y > 1$, and what we want to show is
$$
\frac{2}{4}y + \frac{1}{2y} > 1
$$
$$
y + \frac{1}{y} > 2
$$
Multiply by $y$ (since $y$ is positive, no problems arise)
$$
y^2 -2y + 1 > 0
$$
$$
(y-1)^2 > 0
$$
which is obvious, since $y > 1$.... | {
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Convergence of Lebesgue integrable functions in an arbitrary measure. I'm a bit stuck on this problem, and I was hoping someone could point me in the right direction.
Suppose $f, f_1, f_2,\ldots \in L^{1}(\Omega,A,\mu)$ , and further suppose that $\lim_{n \to \infty} \int_{\Omega} |f-f_n| \, d\mu = 0$. Show that $f_n \... | A bit late to answer, but here it is anyways.
We wish to show that for any $\epsilon > 0,$ there is some $N$ such that for all $n \geq N, \mu(\{x : |f_n(x) - f(x)| > \epsilon\}) < \epsilon.$ (This is one of several equivalent formulations of convergence in measure.)
If this were not the case, then there'd be some $\ep... | {
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Is this the category of metric spaces and continuous functions?
Suppose the object of the category are metric spaces and for $\left(A,d_A\right)$ and $\left(B,d_B\right)$ metric spaces over sets A and B, a morphisms of two metric space is given by a function between the underlying sets, such that $f$ presere the metri... | I don't think there's really one the category of metric spaces. The fourth axiom here gives you a category of metric spaces and (uniformly) continuous functions. The other axioms are implied by the assumptions. Allowing $\delta$ to depend on $x$ gives you the category of metric spaces and (all) continuous functions.
On... | {
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Counting permutations of students standing in line Say I have k students, four of them are Jack, Liz, Jenna and Tracy. I want to count the number of permutations in which Liz is standing left to Jack and Jenna is standing right to Tracy. I define $A = ${Liz is left to Jack} so $|A| = \frac{k!}{2}$. The same goes for $B... | The order relationship between Liz and Jack is independent of that between Jenna and Tracy. You already know that there are $k!/2$ permutations in which Liz stands to the left of Jack. In each of those Jenna can be on the right of Tracy or on her left without affecting the order of Liz and Jack, so exactly half of thes... | {
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Does a natural transformation on sites induce a natural transformation on presheaves? Suppose $C$ and $D$ are sites and $F$, $G:C\to D$ two functors connected by a natural transformation $\eta_c:F(c)\to G(c)$.
Suppose further that two functors $\hat F$, $\hat G:\hat C\to\hat D$ on the respective categories of presheave... | Recall: given a functor $F : \mathbb{C} \to \mathbb{D}$ between small categories, there is an induced functor $F^\dagger : [\mathbb{D}^\textrm{op}, \textbf{Set}] \to [\mathbb{C}^\textrm{op}, \textbf{Set}]$, and this functor has both a left adjoint $\textrm{Lan}_F$ and a right adjoint $\textrm{Ran}_F$. Now, given a natu... | {
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How to solve/transform/simplify an equation by a simple algorithm? MathePower provides an form. There you can input a formula (1st input field) and a variable to release (2nd input field) and it will output a simplified version of that formula.
I want to write a script which needs to do something similar.
So my questio... | This is generally known as "computer algebra," and there are entire books and courses on the subject. There's no single magic bullet. Generally it relies on things like specifying canonical forms for certain types of expressions and massaging them. Playing with the form, it seems to know how to simplify a rational e... | {
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"url": "https://math.stackexchange.com/questions/228453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Maximum based recursive function definition Does a function other than 0 that satisfies the following definition exist?
$$
f(x) = \max_{0<\xi<x}\left\{ \xi\;f(x-\xi) \right\}
$$
If so can it be expressed using elementary functions?
| Since we cannot be sure if the $\max$ exists, let us consider $f\colon I\to\mathbb R$ with
$$\tag1f(x)=\sup_{0<\xi<x}\xi f(x-\xi)$$
instead, where $I$ is an interval of the form $I=(0,a)$ or $I=(0,a]$ with $a>0$.
If $x_0>0$ then $f(x)\ge (x-x_0)f(x_0)$ for $x>x_0$ and $f(x)\le\frac{f(x_0)}{x_0-x}$ for $x<x_0$.
We can c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/228608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Alice and Bob Game Alice and Bob just invented a new game.
The rule of the new game is quite simple. At the beginning of the game, they write down N
random positive integers, then they take turns (Alice first) to either:
*
*Decrease a number by one.
*Erase any two numbers and write down their sum.
Wh... | The complete solution to this game is harder than it looks, due to complications when there are several numbers $1$ present; I claim the following is a complete list of the "Bob" games, those that can be won by the second player to move. To justify, I will indicate for each case a strategy for Bob, countering any move ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/228696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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How to undo linear combinations of a vector If $v$ is a row vector and $A$ a matrix, the product $w = v A$ can be seen as a vector containing a number of linear combinations of the columns of vector $v$. For instance, if
$$
v = \begin{bmatrix}1, 2\end{bmatrix}, \quad
A = \begin{bmatrix}0 & 0 & 0 \\ 1 & 1 & 1\end{bmat... | Clearly
$$ \begin{bmatrix}0 & 2\end{bmatrix}
\begin{bmatrix}0 & 0 & 0 \\ 1 & 1 & 1\end{bmatrix}
= \begin{bmatrix}2 & 2 & 2\end{bmatrix}$$
So $v'=[0 \; 2]$ is a solution.
So we can suppose than any other solution can look like
$v'' = v' + [x \; y]$.
\begin{align}
(v' + [x \; y])A &= [2 \; 2 \; 2] \\
v'A + [... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/228809",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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*Recursive* vs. *inductive* definition I once had an argument with a professor of mine, if the following definition was a recursive or inductive definition:
Suppose you have sequence of real numbers. Define $a_0:=2$ and $a_{i+1}:=\frac{a_i a_{i-1}}{5}$. (Of course this is just an example and as such has only illustrati... | Here is my inductive definition
of the cardinality
of a finite set
(since,
in my mind,
finite sets are built
by adding elements
starting with the
empty set):
$|\emptyset|
= 0
$.
$|A \cup {x}|
=
\begin{cases}
x \in A
&\implies |A|\\
x \not\in A
&\implies |A|+1\\
\end{cases}
$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/228863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "34",
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"answer_id": 4
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Prove that $mn|a$ implies $m|a$ and $n|a$ I am trying to prove this statement about divisibility: $mn|a$ implies $m|a$ and $n|a$.
I cannot start the proof. I need to prove either the right or left side. I don't know how to use divisibility theorems here. Generally, I have problems in proving mathematical statements.
Th... | If $mn|a$ then $a=kmn$ for some integer $k$. Then $a=(km)n$ where $km$ is an integer so that $n|a$. Similarly, $a=(kn)m$ where $kn$ is an integer so that $m|a$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/228934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Evaluating a double integral: $\iint \exp(\sqrt{x^2+y^2})\:dx\:dy$? How to evaluate the following integral? $$\iint \exp\left(\sqrt{x^2+y^2} \right)\:dx\:dy$$
I'm trying to integrate this using substitution and integration by parts but I keep getting stuck.
| If you switch to polar coordinates, you end out integrating $re^r \,dr \,d\theta$, which you should be able to integrate over your domain by doing the $r$ integral first (via integration by parts).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/228995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Non-closed compact subspace of a non-hausdorff space I have a topology question which is:
Give an example of a topological (non-Hausdorff) space X and a a non-closed compact subspace.
I've been thinking about it for a while but I'm not really getting anywhere. I've also realised that apart from metric spaces I don't re... | Here are some examples that work nicely.
*
*The indiscrete topology on any set with more than one point: every non-empty, proper subset is compact but not closed. (The indiscrete topology isn’t good for much, but as Qiaochu said in the comments, it’s a nice, simply example when it actually works.)
*In the line with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/229064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Maps of maximal ideals Prove that $\mu:k^n\rightarrow \text{maximal ideal}\in k[x_1,\ldots,x_n]$ by $$(a_1,\ldots,a_n)\rightarrow (x_1-a_1,\ldots,x_n-a_n)$$ is an injection, and given an example of a field $k$ for which $\mu$ is not a surjection.
The first part is clear, but the second part needs a field $k$ such that... | At Julian's request I'm developing my old comment into an answer. Here is the result:
Given any non algebraically field $k$, the canonical map $$k\to \operatorname {Specmax}(k[x]):a\mapsto (x-a)$$ is not surjective.
Indeed, by hypothesis there exists an irreducible polynomial $p(x)\in k[x]$ of degree $\gt 1$.
Thi... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number of squares in a rectangle Given a rectangle of length a and width b (as shown in the figure), how many different squares of edge greater than 1 can be formed using the cells inside.
For example, if a = 2, b = 2, then the number of such squares is just 1.
| In an $n\times p$ rectangle, the number of rectangles that can be formed is $\frac{np}{4(n+1)(p+1)}$ and the number of squares that can be formed is $\sum_{r=1}^n (n+1-r)(p+1-r)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/229182",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
For which values of $\alpha \in \mathbb R$ is the following system of linear equations solvable? The problem I was given:
Calculate the value of the following determinant:
$\left|
\begin{array}{ccc}
\alpha & 1 & \alpha^2 & -\alpha\\
1 & \alpha & 1 & 1\\
1 & \alpha^2 & 2\alpha & 2\alpha\\
1 & 1 & \alpha^2 & -\alpha
\en... | Let me first illustrate an alternate approach. You're looking at $$\left[\begin{array}{ccc}
\alpha & 1 & \alpha^2\\
1 & \alpha & 1\\
1 & \alpha^2 & 2\alpha\\
1 & 1 & \alpha^2
\end{array}\right]\left[\begin{array}{c} x_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{c} -\alpha\\ 1\\ 2\alpha\\ -\alpha\end{array}\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/229254",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Combinatorics: When To Use Different Counting Techniques I am studying combinatorics, and at the moment I am having trouble with the logic behind more complicated counting problems. Given the following list of counting techniques, in which cases should they be used (ideally with a simple, related example):
*
*Repeat... | Let me address some of the more general techniques on your list, since the specific ones just appear to be combinations of the general ones.
Repeated Multiplication: Also called "falling factorial", use this technique when you are choosing items from a list where order matters. For example, if you have ten flowers and ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve $2a + 5b = 20$ Is this equation solvable? It seems like you should be able to get a right number!
If this is solvable can you tell me step by step on how you solved it.
$$\begin{align}
{2a + 5b} & = {20}
\end{align}$$
My thinking process:
$$\begin{align}
{2a + 5b} & = {20} & {2a + 5b} & = {20} \\
{0a + 5b} & =... | Generally one can use the Extended Euclidean algorithm, but that's overkill here. First note that since $\rm\,2a+5b = 20\:$ we see $\rm\,b\,$ is even, say $\rm\:b = 2n,\:$ hence dividing by $\,2\,$ yields $\rm\:a = 10-5n.$
Remark $\ $ The solution $\rm\:(a,b) = (10-5n,2n) = (10,0) + (-5,2)\,n\:$ is the (obvious) part... | {
"language": "en",
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Strict Inequality for Fatou Given $f_n(x)=(n+1)x^n; x\in [0,1]$
I want to show $\int_{[0,1]}f<\liminf\int_{[0,1]}f_n$, where $f_n$ converges pointwise to $f$ almost everywhere on $[0,1]$.
I have found that $\liminf\int f_n = \int f +\liminf\int|f-f_n|$, but I'm not sure how to use this, and I don't even know what $f_n$... | HINT
Consider $a \in [0,1)$. The main crux is to compute $$\lim_{n \to \infty} (n+1)a^n$$
To compute the limit note that $a = \dfrac1{1+b}$ for some $b > 0$.
Hence, $$a^n = \dfrac1{(1+b)^n} < \dfrac1{\dfrac{n(n-1)}2 b^2}\,\,\,\,\,\,\,\,\, \text{(Why? Hint: Binomial theorem)}$$ Can you now compute $\lim_{n \to \infty} (... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$G$ finite group, $H \trianglelefteq G$, $\vert H \vert = p$ prime, show $G = HC_G(a)$ $a \in H$ Let $G$ be a finite group. $H \trianglelefteq G$ with $\vert H \vert = p$ the smallest prime dividing $\vert G \vert$. Show $G = HC_G(a)$ with $e \neq a \in H$. $C_G(a)$ is the Centralizer of $a$ in $G$.
To start it off, I ... | Since $N_G(H)/C_G(H)$ injects in $Aut(H)\cong C_{p-1}$ and $p$ is the smallest prime dividing $|G|$, it follows that $N_G(H)=C_G(H)$. But $H$ is normal, so $G=N_G(H)$ and we conclude that $H \subseteq Z(G)$. In particular $G=C_G(a)$ for every $a \in H$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/229543",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $A=\left(\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right)$ is not invertible $$A=\left(\begin{array}{ccc}1&2&3\\4&5&6\\7&8&9\end{array}\right)$$
I don't know how to start. Will be grateful for a clue.
Edit: Matrix ranks and Det have not yet been presented in the material.
| Note that $L_3-L_2=L_2-L_1$. What does that imply about the rank of $A$?
| {
"language": "en",
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Finding common terms of two arithmetic sequences using Extended Euclidean algorithm I have a problem which could be simplified as: there are two arithmetic sequences, a and b. Those can be written as
a=a1+m*d1
b=b1+n*d2
I need to find the lowest term, appearing in both sequences. It is possible to do by brute force, ... | Your equations:
$$a(m) = a_1 + m d_1$$
$$b(n) = b_1 + n d_2 $$
You want $a(m) = b(n)$ or $a(m)-b(n)=0$, so it may be written as
$$(a_1-b_1) + m(d_1) +n(-d_2) = 0$$ or $$ m(d_1) +n(-d_2) = (b_1-a_1) $$
You want $n$ and $m$ minimal and that solves that. This is of course very similar to the EGCD, but that the value of $b... | {
"language": "en",
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If $d^2|p^{11}$ where $p$ is a prime, explain why $p|\frac{p^{11}}{d^2}$. If $d^2|p^{11}$ where $p$ is a prime, explain why $p|\frac{p^{11}}{d^2}$.
I'm not sure how to prove this by way other than examples. I only tried a few examples, and from what I could tell $d=p^2$. Is that always the case?
Say $p=3$ and $d=9$.... | Hint $\ $ It suffices to show $\rm\:p^{11}\! = c\,d^2\Rightarrow\:p\:|\:c\ (= p^{11}\!/d^2).\:$ We do so by comparing the parity of the exponents of $\rm\:p\:$ on both sides of the first equation. Let $\rm\:\nu(n) = $ the exponent of $\rm\,p\,$ in the unique prime factorization of $\rm\,n.\:$ By uniqueness $\rm\:\colo... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Net Present Worth Calculation (Economic Equivalence) I'm currently doing some work involving net present worth analyses, and I'm really struggling with calculations that involve interest and inflation, such as the question below. I feel that if anyone can set me on the right track, and once I've worked through the ful... | Using the discount rate (interest rate) calculate the present value of each payment. Let A = 100,000, first payment a = 5,000 (annual increase), and r = 2.5%, interest rate to simply the formulae.
$$
PV_1 = A/(1+r) \\
PV_2 = (A+a)/(1+r)^2 \\
... \\
PV_{10} = (A+9a)/(1+r)^{10}
$$
The total present value (PV) is just t... | {
"language": "en",
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Noetherian rings and modules A ring is left-Noetherian if every left ideal of R is finitely generated. Suppose R is a left-Noetherian ring and M is a finitely generated R-module. Show that M is a Noetherian R-module.
I'm thinking we want to proceed by contradiction and try to produce an infinitely generated ideal, but ... | If $\{x_i\mid 1\leq i\leq n\}$ is a set of generators for $M$, then the obvious map $\phi$ from $R^n$ to $M$ is a surjection. Since $R^n$ is a Noetherian left module, so is $R^n/\ker(\phi)\cong M$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does $f(x)$ is continuous and $f = 0$ a.e. imply $f=0$ everywhere? I wanna prove that
"if $f: \mathbb{R}^n \to \mathbb{R}$ is continuous and satisfies $f=0$ almost everywhere (in the sense of Lebesgue measure), then, $f=0$ everywhere."
I am confident that the statement is true, but stuck with the proof. Also, is the s... | A set of measure zero has dense complement. So if a continuous function zero on a set of full measure, it is identically zero.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/231103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Find a basis of $\ker T$ and $\dim (\mathrm{im}(T))$ of a linear map from polynomials to $\mathbb{R}^2$ $T: P_{2} \rightarrow \mathbb{R}^2: T(a + bx + cx^2) = (a-b,b-c)$
Find basis for $\ker T$ and $\dim(\mathrm{im}(T))$.
This is a problem in my textbook, it looks strange with me, because it goes from polynomial to $\... | You can treat any polynomial $P_n$ space as an $R^n$ space.
That is, in your case the polynomial is $P_2$ and it can be converted to $R^3$.
The logic is simple, each coefficient of the term in the polynomial is converted to a number in $R^3$.
In the end of this conversion you'll get an isomorphics spaces/subspaces.
In... | {
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"8 Dice arranged as a Cube" Face-Sum Problem I found this here:
Sum Problem
Given eight dice. Build a $2\times 2\times2$ cube, so that the sum of the points on each side is the same.
$\hskip2.7in$
Here is one of 20 736 solutions with the sum 14.
You find more at the German magazine "Bild der Wissenschaft 3-1980".
... | Regarding your reference request:
The site of the magazine offers many of their articles online starting from 1997, so you cannot obtain the 1980 edition online (although you can likely buy a used print version).
Most good libraries in German-speaking countries do have this magazine, so, depending on your country, you ... | {
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Show $W^{1,q}_0(-1,1)\subset C([-1,1])$ I need show that space $W^{1,q}_0(-1,1)$ is a subset of $C([-1,1])$ space. How I will able to doing this?
| If $q=+\infty$, and $\varphi_n$ are test functions such that $\lVert \varphi_n-u\rVert_{\infty}\to 0$, then we can find a set of measure $0$ such that $\sup_{x\in [-1,1]\setminus N}|\varphi_n(x)-u(x)|\to 0$, so $u$ is almost everywhere equal to a continuous function, and can be represented by it.
We assume $1\leqslan... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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The degree of a polynomial which also has negative exponents. In theory, we define the degree of a polynomial as the highest exponent it holds.
However when there are negative and positive exponents are present in the function, I want to know the basis that we define the degree. Is the order of a polynomial degree ex... | For the sake of completeness, I would like to add that this generalization of polynomials is called a Laurent polynomial. This set is denoted $R[x,x^{-1}]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/231357",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
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Leslie matrix stationary distribution Given a particular normalized Perron vector representing a discrete probability distribution, is it possible to derive some constraints or particular Leslie matrices having the given as their Perron vector?
There is a related question on math overflow.
| I have very little knowledge about demography. Yet, if the Leslie matrices you talk about are the ones described in this Wikipedia page, it seems like that for any given $v=(v_0,v_1,\ldots,v_{\omega - 1})^T$, a corresponding Leslie matrix exists if $v_0>0$ and $v_0\ge v_1\ge\ldots\ge v_{\omega - 1}\ge0$.
For such a v... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Infinitely many primes p that are not congruent to 1 mod 5 Argue that there are infinitely many primes p that are not congruent to 1 modulo 5.
I find this confusing. Is this saying $p_n \not\equiv 1 \pmod{5}$?
To start off I tried some examples.
$3 \not\equiv 1 \pmod{5}$
$5 \not\equiv 1 \pmod{5}$
$7 \not\equiv 1 \pmod... | You can follow the Euclid proof that there are an infinite number of primes. Assume there are a finite number of primes not congruent to $1 \pmod 5$. Multiply them all except $2$ together to get $N \equiv 0 \pmod 5$. Consider the factors of $N+2$, which is odd and $\equiv 2 \pmod 5$. It cannot be divisible by any p... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Showing $f:\mathbb{R^2} \to \mathbb{R}$, $f(x, y) = x$ is continuous Let $(x_n)$ be a sequence in $\mathbb{R^2}$ and $c \in \mathbb{R^2}$.
To show $f$ is continuous we want to show if $(x_n) \to c$, $f(x) \to f(c)$.
As $(x_n) \to c$ we can take $B_\epsilon(c)$, $\epsilon > 0$ such that when $n \geq$ some $N$, $x_n \in ... | There's a bit of repetition when you say $x_n \in B_\epsilon(c) \implies f(x_n) \in f(B_\epsilon(c))$. While this is true as you define it, repeating it doesn't add to the proof. What you need to show is that the image of $B_\epsilon(c)$ is itself an open neighborhood of $f(c)$.
Another look, which uses uniform continu... | {
"language": "en",
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What does $ ( \nabla u) \circ \tau \cdot D \tau $ and $ \nabla u \cdot (D \tau_\gamma)^{-1} $ mean? To understand the question here
$\def\abs#1{\left|#1\right|}$
\begin{align*}
F(u_\gamma) &= F(u \circ \tau_\gamma^{-1})\\
&= \int_\Omega \abs{\nabla(u \circ \tau_\gamma^{-1})}^2\\
&= \int_\Omega \abs{(\... |
$\nabla ( u \circ\tau )= ( \nabla u) \circ \tau \cdot D \tau$. I'd like to understand this equality or this notaition.
Think of the chain rule: derivative of composition is the product of derivatives. On the left, $u\circ \tau $ is composition (not Hadamard product, as suggested in the other answer). On the right, w... | {
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"answer_id": 0
} |
continued fraction expression for $\sqrt{2}$ in $\mathbb{Q_7}$ Hensel's lemma implies that $\sqrt{2}\in\mathbb{Q_7}$. Find a continued
fraction expression for $\sqrt{2}$ in $\mathbb{Q_7}$
| There's a bit of a problem with defining continued fractions in the $p$-adics. The idea for finding continued fractions in $\Bbb Z$ is that we subtract an integer $m$ from $\sqrt{n}$ such that $\left|\sqrt{n} - m\right| < 1$. We can find such an $m$ because $\Bbb R$ has a generalized version of the division algorithm: ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/231754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proof of $\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n \text{Pr}(X>n)$ $X$ is a random variable defined in $\mathbb N$. How can I prove that for all $n\in \mathbb N$?
*
*$ \text E(X) =\sum_{k=0}^n k \text{Pr}(X=k) = \sum^{n-1}_{k=0} \text{Pr}(X>k) -n \text{Pr}(X>n)$
*$\text E(X) =\sum... | For part $a)$, use Thomas' hint. You get
$$
\sum_{i=0}^{n}k(P(X>i-1)-P(X>i)).
$$
This develops as $P(X>0)-P(X>1)+2P(X>1)-2P(X>2)+3P(X>2)-3P(X>3)+\cdots nP(X>n-1)-nP(X>n)$
for part $b)$:
In general, you have
$\mathbb{E}(X)=\sum\limits_{i=1}^\infty P(X\geq i).$
You can show this as follow:
$$
\sum\limits_{i=1}^\infty P(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/231832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Expectation and Distribution Function? Consider X as a random variable with distribution function $F(x)$. Also assume that $|E(x)| < \infty$. the goal is to show that for any constant $c$, we have:
$$\int_{-\infty}^{\infty} x (F(x + c) - F(x)) dx = cE(X) - c^2/2$$
Does anyone have any hint on how to approach this?
Than... | Based on @DilipSarwate suggestion, we can write the integral as a double integral because:
$\int f(y)dy = F(y)$ so, we can write:
$ \int_{-\infty}^{\infty} x (F(x + c) - F(x)) dx = \int_{-\infty}^{\infty} x \{\int_{x}^{x + c} f(y)dy\} dx = \int_{-\infty}^{\infty} \{\int_{x}^{x + c} xf(y)dy\} dx = \int_{-\infty}^{\inft... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/231907",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Proof of sigma-additivity for measures I understand the proof for the subadditivity property of the outer measure (using the epsilon/2^n method), but I am not quite clear on the proof for the sigma-additivity property of measures. Most sources I have read either leave it an exercise or just state it outright.
From what... | As far as I'm aware, that's the standard approach. The method I was taught is here (Theorem A.9), and involves showing countable subadditivity, defining a new sigma algebra $\mathcal{M}_{0}$ on which countable additivity holds when the outer measure is restricted to $\mathcal{M}_{0}$ (by showing superadditivity), and t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/231995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
convergence tests for series $p_n=\frac{1\cdot 3\cdot 5...(2n-1)}{2\cdot 4\cdot 6...(2n)}$ If the sequence:
$p_n=\frac{1\cdot 3\cdot 5...(2n-1)}{2\cdot 4\cdot 6...(2n)}$
Prove that the sequence
$((n+1/2)p_n^2)^{n=\infty}_{1}$ is decreasing.
and that the series $(np_n^2)^{n=\infty}_{1}$ is convergent.
Any hints/ answe... | Hint 1:
Show that (n+1/2)>=(n+1.5)(2n+1/2n+2)^2 for all positive integers n, then use induction to show that the first sequence is decreasing
Hint 2:
show that 1/2n<=p(n), thus 1/2n<= np(n)^2 therefore the second series diverges
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/232092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Prove $\frac{1}{a^3} + \frac{1}{b^3} +\frac{1}{c^3} ≥ 3$ Prove inequality $$\frac{1}{a^3} + \frac{1}{b^3} +\frac{1}{c^3} ≥ 3$$ where $a+b+c=3abc$ and $a,b,c>0$
| If $a, b, c >0$ then $a+b+c=3abc \ \Rightarrow \ \cfrac 1{ab} + \cfrac 1{bc}+ \cfrac 1{ca} = 3$
See that $2\left(\cfrac 1{a^3} +\cfrac 1{b^3}+ \cfrac 1{c^3}\right) +3 =\left(\cfrac 1{a^3} +\cfrac 1{b^3}+ 1\right)+\left(\cfrac 1{b^3} +\cfrac 1{c^3}+ 1\right)+\left(\cfrac 1{c^3} +\cfrac 1{a^3}+ 1\right) $
Use $AM-GM$ in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/232171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show that the unit sphere is strictly convex I can prove with the triangle inequality that the unit sphere in $R^n$ is convex, but how to show that it is strictly convex?
| To show that the closed unit ball $B$ is strictly convex we need to show that for any two points $x$ and $y$ in the boundary of $B$, the chord joining $x$ to $y$ meets the boundary only at the points $x$ and $y$.
Let $x,y \in \partial B$, then $||x|| = ||y|| = 1.$ Now consider the chord joining $x$ to $y$. We can param... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/232276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Does this change in this monotonic function affect ranking? I need to make sure I can take out the one in $(1-e^{-x})e^{-y}$ without affecting a sort order based on this function. I other words, I need to prove the following:
$$
(1-e^{-x})e^{-y} \ >= \ -e^{-x}e^{-y}\quad\forall\ \ x,y> 0
$$
If that is true, then I can ... | Looking at the current version of your post, we have
$$(1-e^{-x})e^{-y}=e^{-y}-e^{-x}e^{-y}>-e^{-x}e^{-y},$$ since $e^t$ is positive for all real $t$. However, we can't take the logarithm of the right-hand side. It's negative.
Update:
The old version was $$(1-e^{-x_1})(1-e^{-x_2})e^{-x_3}=e^{-x_3}-e^{-x_1-x_3}-e^{-x_2-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/232360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
definition of morphism of ringed spaces I've recently started reading about sheafs and ringed spaces (at the moment, primarily on wikipedia). Assuming I'm correctly understanding the definitions of the direct image functor and of morphisms of ringed spaces, a morphism from a ringed space $(X, O_X)$ to a ringed space $(... | Think about what it means to give a morphism from $\mathcal O_Y$ to $f_* \mathcal O_X$: it means that for every open set $V \subset Y$, there is a map
$$\mathcal O_Y(V) \to \mathcal O_X\bigl( f^{-1}(V) \bigr).$$
If you imagine that $\mathcal O_X$ and $\mathcal O_Y$ are supposed to be some sorts of "sheaves of function... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/232431",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 2
} |
Is $G/pG$ is a $p$-group? Jack is trying to prove:
Let $G$ be an abelian group, and $n\in\Bbb Z$. Denote $nG = \{ng \mid g\in G\}$.
(1) Show that $nG$ is a subgroup in $G$.
(2) Show that if $G$ is a finitely generated abelian group, and $p$ is prime,
then $G/pG$ is a $p$-group (a group whose order is a power of $p$... | $G/pG$ is a direct sum of a finite number of cyclic groups by the fundamental theorem of finitely generated abelian groups. Since every non-zero element of $G/pG$ is of order $p$.
It is a direct sum of a finite number of cyclic groups of order $p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/232526",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
} |
Bernoulli Polynomials I am having a problem with this question. Can someone help me please.
We are defining a sequence of polynomials such that:
$P_0(x)=1; P_n'(x)=nP_{n-1}(x) \mbox{ and} \int_{0}^1P_n(x)dx=0$
I need to prove, by induction, that $P_n(x)$ is a polynomial in $x$ of degree $n$, the term of highest degree ... | Recall that $\displaystyle \int x^n dx = \dfrac{x^{n+1}}{n+1}$. Hence, if $P_n(x)$ is a polynomial of degree $n$, then it is of the form $$P_n(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$ Since $P_{n+1}'(x) = (n+1) P_n(x)$, we have that $$P_{n+1}(x) = \int_{0}^x (n+1) P_n(y) dy + c$$
Hence, $$P_{n+1}(x) = \i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/232594",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
f, g continuous for all rationals follow by continuous on all reals?
Possible Duplicate:
Can there be two distinct, continuous functions that are equal at all rationals?
Let $f, g:\Bbb{R}\to\Bbb{R}$ to be continuous functions such that $f(x)=g(x)\text{ for all rational numbers}\,x\in\Bbb{Q}$. Does it follow that $f(... | Hint: prove that if $\,h\,$ is a real continuous function s.t. $\,h(q)=0\,\,,\,\,\forall\,q\in\Bbb Q\,$ , then $\,h(x)=0\,\,,\,\,\forall\,x\in\Bbb R\,$
Further hint: For any $\,x\in\Bbb R\,$ , let $\,\{q_n\}\subset\Bbb Q\,$ be s.t. $\,q_n\xrightarrow [n\to\infty]{} x\,$ . What happens with
$$\lim_{n\to\infty}f(q_n)\,\,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/232735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
67 67 67 : use 3, 67's use any way how to get 11222 I need to get 11222 using three 67 s (Sixty seven)
We can use any operation in any maner
67 67 67
use 3, 67's use any way but to get 11222.
| I'd guess this is a trick question around "using three, sixty-sevens" to get $11222$.
In particular, $67 + 67 = 134$, which is $11222$ in ternary (base $3$).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/232800",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Which player is most likely to win when drawing cards? Two players each draw a single card, in turn, from a standard deck of 52 cards, without returning it to the deck. The winner is the player with the highest value on their card. If the value on both cards is equal then all cards are returned to the deck, the deck is... | If the second player were drawing from a full deck, he would draw each of the $13$ ranks with equal probability. The only change when he draws from the $51$-card deck that remains after the first player’s draw is that the rank of the first player’s card becomes less probable; the other $12$ ranks remain equally likely.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/232963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 1
} |
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