Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Constant of integration
Consider the integral $$\int x \arctan (x) \;dx$$ Evaluate this integral using integration by parts. Then find a constant of integration that makes the last integration trivial. Compare the answers and explain any differences.
I know how to integrate by parts, I got $$\frac{x^2 \arctan (x) + ... | $$
\begin{aligned}
\int x \arctan (x) \;dx
&=
\int \frac 12(x^2+1)' \arctan (x) \;dx
\\
&=
\frac 12(x^2+1)\arctan (x)
-
\int \frac 12(x^2+1) \arctan' (x) \;dx
\\
&=
\frac 12(x^2+1)\arctan (x)
-
\int \frac 12(x^2+1) \cdot\frac 1{x^2+1} \;dx
\\
&=
\frac 12(x^2+1)\arctan (x)
-
\int \frac 12\;dx
\\
&=
\frac 12(x^2+1)\arcta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2967916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Proof verification of $x_n = \sqrt[3]{n^3 + 1} - \sqrt{n^2 - 1}$ is bounded
Let $n \in \mathbb N$ and:
$$
x_n = \sqrt[^3]{n^3 + 1} - \sqrt{n^2 - 1}
$$
Prove $x_n$ is bounded sequence.
Start with $x_n$:
$$
\begin{align}
x_n &= \sqrt[^3]{n^3 + 1} - \sqrt{n^2 - 1} = \\
&= n \left(\sqrt[^3]{1 + {1\over n^3}} - \sqrt{... | Your prove is fine but a lot more work than necessary.
As $n \ge 1$ we have
$n = \sqrt[3]{n^3} < \sqrt[3]{n^3 + 1} < \sqrt[3]{n^3 + 3n^2 + 3n + 1} = \sqrt[3]{(n+1)^3} = n+1$
and
$n = \sqrt{n^2} > \sqrt{n^2 -1 } = \sqrt{n^2 - 2 + 1} \ge \sqrt{n^2 - 2n + 1} = \sqrt{(n-1)^2} = n-1$.
So $0 = n - n < \sqrt[3]{n^3 + 1} - \s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2968028",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
} |
Combinatorial proof that $\sum_{i=0}^n 2^i\binom{n}{i}i!(2n-i)! = 4^n(n!)^2$ I'm looking for a combinatorial proof that $$\sum_{i=0}^n 2^i\binom{n}{i}i!(2n-i)! = 4^n(n!)^2.$$
My thoughts so far: the RHS counts the number of pairs of permutations on $n$ elements along with an $n$-tuple whose entries come from 4 choices.... | Alternatively, a more combinatorial flavor approach:
Multiply both sides by $\binom{2n}{n}$ so you get
$$\sum_{i=0}^n 2^i\binom{2n}{\color{red}{n}}\binom{\color{red}{n}}{i}i!(2n-i)! = 4^n(n!)^2\frac{(2n)!}{n!^2},$$
then, using that $\binom{a}{b}\binom{b}{c}=\binom{a}{c}\binom{a-c}{b-c},$ we get
$$\sum_{i=0}^n 2^i\binom... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2968147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Finding real part of complex number in exponential form in fraction Given this complex number,
$$e^{9ix/2} \frac{\sin 4x }{ (\sin (x/2) }$$
The real part of this complex number can be worked out easily, by replacing the $e^{9ix/2}$ with $\cos(9x/2)$
However if I'm given the complex number,
$$\frac{3} {3 - e^{ix} }$$
I... | HINT
Use that
$$\frac{3} {3 - e^{ix} }=\frac{3} {3 - e^{ix} }\frac{3 - e^{-ix} } {3 - e^{-ix} }=\frac{9 - 3e^{-ix} } {10 -3 (e^{ix}+e^{-ix}) }$$
then recall that $z+\bar z= 2\Re(z)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2968265",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Using $ \sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}=\frac1e$ evaluate first $3$ decimal digits of $1/e$.
Using the series $\displaystyle \sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}=\frac{1}{e}$, evaluate the first $3$ decimal digits of $1/e$.
Attempt. In alternating series $\displaystyle \sum_{k=0}^{\infty}(-1)^{k+1}\alpha_n$... | Note
$$\bigg|\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}-\frac{1}{e}\bigg|=\bigg|\sum_{k=n+1}^{\infty}\frac{(-1)^{k}}{k!}\bigg|\le\sum_{k=n+1}^\infty\frac{1}{3^{k}}\le\frac{1}{2\cdot3^{n}}.$$
Let $\frac{1}{2\cdot3^{n}}<0.001$ and then $n>\frac{\ln500}{\ln 3}\approx5.65678$. Now set $n=6$ and then
$$ \bigg|\sum_{k=0}^{6}\frac{(-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2968360",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 5,
"answer_id": 4
} |
Global optimum in a convex space/set
Recall that a set $S⊂\mathbb R^n$ is said to be convex if for any $x,y∈S$, and any $λ∈[0, 1]$, we have $λx+(1−λ)y∈S$.
Let $f:\mathbb R^n→\mathbb R$ be a convex function and let $S⊂\mathbb R^n$ be a convex set. Let $x^∗$ be an element of $S$. Suppose that $x^∗$ is a local optimum ... | Suppose $x^*$ is not a global minimizer; there is some $x$ with $f(x) < f(x^*)$. Then every point $y$ on the line segment from $x^*$ to $x$ has $f(y) < f(x^*)$, too. Points on the line segment get arbitrarily close to $x^*$, so $x^*$ is not a local minimizer, either.
To prove the inequality $f(y) < f(x^*)$, apply conve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2968529",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Finding the Probability from the sum of 3 random variables Let $X_1, X_2$ and $X_3$ be three independent normal random variables having mean $\mu= 0$ and variance $\sigma^2=16.$
Compute $P(X_1^2+X_2^2+X_3^2>8).$
Hint: First transform the random variables to standard normal.
I transformed the random variables to $Z$ st... | Suggested outline.
(1) Use MGFs (or a transformation method) to show that each of the three $Z_i,$ for $i=1,2,3,$ has a chi-squared distribution with $1$ degree of freedom.
(2) Use MGFs to show that $Q = Z_1^2 + Z_2^2 + Z_3^2$
has a chi-squared distribution with $3$ degrees of freedom.
(3) Use software or printed tabl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2968630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Name of the following structure? I need the name of the algebraic structure that is like a vector space, but the vectors form a monoid, not a group; the field and the scalar multiplication stays the same.
| I think the closest thing to what you are talking about is a semimodule over a semifield which is a commutative monoid acted upon by a semifield. For both the semimodule and the semiring, we've dropped the condition that what used to be an abelian group now does not necessarily have inverses for its elements.
The situa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2968902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Can i determine if a function will increase in the future? Is it possible to determine whether a function will have increased in the future relative to starting points, given a sample of the first $m$ points?
For example, given the 4 first values of a function $(f(1), f(2), f(3), f(4) )= (2, 1, 4, 5 )$ can I with some ... | If there is a causal relation between the successive function values, you can hope for reliable extrapolation. For instance by linear prediction.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
$G$ a group of order 36, $P$ is the only Sylow 3-subgroup, which is normal, prove the existence of a homomorphism $\phi:G\rightarrow S_4$ Specifically the question I was asked was to prove that there was either a normal subgroup of $G$ of index 4 or that there was a non-trivial homomorphism $\phi:G\rightarrow S_4$, i.e... | If $P$ is the unique Sylow subgroup it must be normal (a conjugate of $P$ would be a different Sylow subgroup).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969147",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Annihilators and semisimple rings Let $R$ be a semisimple ring and let $I$ be a left ideal of $R$. Denote $\text{ann}_l(S)$, resp. $\text{ann}_r(S)$, for the left (resp. right) annihilator of a left ideal $S$ of $R$. Any tips on how to show that $\text{ann}_l(\text{ann}_r(I)) \subseteq I$?
(the reverse inclusion is st... | Since $R$ is semisimple, $I=Re$ for some idempotent $e$.
It's not hard to show that $ann_r(Re)=(1-e)R$ and $ann_l(eR)=R(1-e)$.
Then $ann_l(ann_r(Re))=ann_l((1-e)R)=R(1-(1-e))=Re=I$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Geometric sequence problem with the last term (2018) undetermined.
$x_0 = 1$
$x_{n+1}=2x_{n}+1$
$S_n=x_0+x_1+\ldots+x_n$
$\text {Find } S_{2018}$
How can I solve it?
I tried to understand the sum of sequence but I couldn't and this is what I got:
$$1+3+7+15+31+\ldots$$
I really don't know how to calculate the $2018^... | Hint:
Write the sequence $x_0+1$, $x_1+1$, $x_2+1$, $x_3+1$, ...
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Question regarding the proof of the Hartogs cardinal theorem. I am reading through the text The Foundations of Mathematics by Kenneth Kunen. On pages 54 and 55 he gives the following proof of the Hartogs cardinal theorem.
Theorem For every set $A$, there is a cardinal $\kappa$ such that $\kappa \npreceq A$.
Proof: Let... | This is just a typo. In the sentence
Observe that $\alpha \preceq X$ iff $\alpha = \text{type}(X,R)$ for some $(X,R) \in W$ (See Exercise I.$11.19$).
it should say $\alpha\preceq A$ instead of $\alpha\preceq X$. (It does not even make sense to say $\alpha\preceq X$, since no specific $X$ has been defined and the $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969560",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is every measure $0$ set a set of discontinuities of a Riemann integrable function? Let $f:[a,b]\rightarrow\mathbb{R}$ be bounded, and let $D$ be its set of discontinuities. Then Lebesgue's criterion states that $f$ is Riemann-integrable if and only if $D$ has Lebesgue measure $0$.
My question is, for any subset $D$ o... | It is well known that any $F_{\sigma}$ set is the set of discontinuities of some function. We can also make this function bounded. So any $F_{\sigma}$ set of measure $0$ is the set of discontinuities of a Riemann integrable function. As pointed out in the comments not every of measure $0$ is the set of discontinuities ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969665",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Given initial positions and velocities of two boats, do they collide? This is a homework question from a precalculus class that I'm a TA for.
Boat $A$ is initially at position $(1,4)$ and moves at a constant velocity $\langle 3,5 \rangle$. Boat $B$ is at position $(7,2)$ and moves at a constant velocity of $\langle 1,... | Let $P_A(a)$ denote the position of the boat $A$ at time $a$, and let $P_B(b)$ denote the position of boat $B$ at time $b$. From the initial positions and velocities given, we have:
$$
\begin{align}
P_A(a) &= (1,4) + a\langle 3,5 \rangle
&\qquad
P_B(b) &= (7,2) + b\langle 1,10 \rangle
\\
&= (3a+1, 5a+4)
&\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Continuity of $\begin{cases}(xy+y^2)/(x^4+y^2)&\text{if }(x,y)\neq(0,0),\\0&\text{if }(x,y)=(0,0)\end{cases}$ at origin using polar coordinates Study the continuity of $$f(x,y)=\begin{cases}\dfrac{xy+y^2}{x^4+y^2}&\text{if }(x,y)\neq(0,0),\\0&\text{if }(x,y)=(0,0),\end{cases}$$ at $(x,y)=(0,0)$ using polar coordinates.... | Although your argument contains a grain of truth, it is not quite correct as it is written, since you wrote that the limit $\lim_{(x,y)\to (0,0)} f(x,y)$, which doesn't exist, is equal to the limit $\lim_{r \to 0} (\cdots)$, which does exist (for $\sin \theta \neq 0$) if you just view it as an ordinary single-variable ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2969960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Which grows at a faster rate $\sqrt {n!}$ vs $(\sqrt {n})!$ when $n \rightarrow \infty$? Which grows at a faster rate $\sqrt {n!}$ vs $(\sqrt {n})!$ ? How to solve such type of questions considering $n \rightarrow \infty$?
| As alluded to in the comments, $(\sqrt{n})!$ doesn't make sense, so I'm going to compare the growth of $n!$ to the growth of $\sqrt{(n^2)!}$. Or, equivalently, compare $(n!)^2$ to $(n^2)!$.
Let $a_n = \frac{(n!)^2}{(n^2)!}$. Then
\begin{align*}
\frac{a_{n+1}}{a_n} &= \frac{((n+1)!)^2 \div (n!)^2}{((n+1)^2)! \div(n^2)!}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2970061",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
A matrix calculus problem in backpropagation encountered when studying Deep Learning I am studying the Algorithm 6.4 in the textbook Deep Learning, which is about backpropagation.
I am confused by this line:
$$\nabla_{W^{(k)}}J = gh^{(k-1)T}+\lambda\nabla_{W^{(k)}}{\Omega(\theta)}$$
This equation is derived by calculat... | I'm going to use subscripts because they're easier to type, and retains the use of superscripts to indicate things like transposes and conjugates.
Algorithm 6.4 tells you how to calculate the vector $g$. It's a chain of derivatives extending from the output layer back to the $k^{th}$ layer
$$\eqalign{
g &=
\frac{\part... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2970202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is it true that if the set $A=\{x:f(x) =c\}$ is measurable for every $c$ in $\Bbb R$, then $f$ is measurable?
Let $f: [0;1] \to \Bbb R$. Is it true that if the set $A=\{x:f(x) =c\}$ is measurable for every $c$ in $\Bbb R$, then $f$ is measurable?
I have this counter example
$f(x) =x$ if $x$ belong to $P$, $f(x) =-x$ ... | As you think, the answer to the question is no. However, I am not sure how your example will work. Maybe it does, but I am unable to show that there exists a measurable set $U$ such that $f^{-1}(U)$ is not measurable in your example. I had a lapse of intelligence a little bit. The comments both under the question a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2970305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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We have $\triangle ABC$ and we must find $PM$
Here we have $\triangle ABC$ and some information related:
$$\angle A=60^\circ$$
$$\angle B=45^\circ$$
$$AC=8$$
$$CM=MB$$
The vertical line $PM$ is perpendicular to $BC$.
So now we want to calculate $PB$.
$$PB=?$$
I have tried some different ways to calculate it. But it wa... | Guide:
I would construct line $PC$ and show that $\angle APC$ is $90^\circ$.
After that I can use trigonometry to obtain $PC$ and I can use trigonometry one more time to obtain the length of $PM$.
Edit:
$PC$ is the line connecting $P$ to $C$. Notice that we have $PM = CM$. Then, we can compute $\angle ACP = 75^\circ -... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2970459",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to calculate the envelope of the trajectory of a double pendulum? Consider a double pendulum:
Background
For the angles $\varphi_i$ and the momenta $p_i$ we have (with equal lengths $l=1$, masses $m=1$ and gravitational constant $g=1$):
$\dot{\varphi_1} = 6\frac{2p_1 - 3p_2\cos(\varphi_1 - \varphi_2)}{16 - 9\cos^2... | This is not a complete or rigorous answer, but it should point you in the right direction.
The double pendulum is a Hamiltonian system; energy is conserved. The most extreme points of the trajectory occur when both legs are at standstill ($p_1=p_2=0$) as all the energy is positional. The points where this is the case c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2970573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
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Why is my translation of $\exists{x}\,(C(x) \rightarrow F(x))$ into an English sentence wrong? Let $\text{C(x): x is a comedian}$ and $\text{F(x): x is funny}$
Let $$\alpha:\quad\exists{x}\,(C(x) \rightarrow F(x))$$ and the domain consists of all people.
I needed to translate $\alpha$ into English so what I did was I l... | The domain is of people, not commedians.
$\exists x~(\lnot C(x)\lor F(x))$ would read as "There is some person who is not a commedian or is funny."
Although equivalent, $\exists x~(C(x)\to F(x))$ more directly translates as "There is some person who would be a commedian only if they were funny."
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2970732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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When is the exponential law in topology discontinuous? By $Y^X$ I mean the space of continuous functions $X \to Y$ with the compact-open topology (by compact I don't require Hausdorff. Consider the exponential law $Z^{X \times Y} \to (Z^Y)^X$ where the map is well-defined a a set map (this is not hard to see). However ... | There is a paper by P. Booth and J. Tillotson, Pacific J. Math.
Vol. 88. No. 1, 198 (downloadable) which discusses exponential laws $X ^{Z \times Y} \cong (X^Y)^Z$ for various function space topologies, and topologies on $X \times Y$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2970857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Integrate $\int x\sec^2(x)\tan(x)\,dx$ $$\int x\sec^2(x)\tan(x)\,dx$$
I just want to know what trigonometric function I need to use. I'm trying to integrate by parts. My book says that the integral equals
$${x\over2\cos^2(x)}-{\sin(x)\over2\cos(x)}+C$$
| Let $u = x, dv = \sec^2 x\tan xdx\implies v = \displaystyle \int\sec^2 x\tan xdx= \displaystyle \int \tan xd(\tan x)= \displaystyle \int wdw, w = \tan x$. Can you put it together ?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2970958",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
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Derivation of the bivariate normal distribution I am reading through a derivation of the bivariate normal distribution, (from the US Defence Department!) and came across an expression that I can't understand.
The derivation starts off with the observation that the total area, $A$, under the curve of the distribution i... | $C$ is part of $f(x)$ as can be seen in step $(3)$ if one would just put it in front of the integral sign, and is therefore part of the variance, $σ_X^2$. $A$ is just $1$. The point of this step is to prove that $k={1\over σ_X^2}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2971070",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Limit with factorial (Stolz & Stirling) We have this limit:
$\lim\limits_{n \to \infty} (\ln(n!))/n \to \infty$
I know that it should be answered with Stolz or Stirling, but I don't know how.
| Stirling's approximation tells us that the following is true:
$$ln(n!) = n \cdot ln(n) - n + O\left(ln(n)\right)$$
If we divide this by $n$, we have:
$$\frac{ln(n!)}{n} = ln(n) - 1 + O\left(\frac{ln(n)}{n}\right)$$
As $n \rightarrow \infty, \frac{ln(n!)}{n} \rightarrow \infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2971172",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Prove that every even degree polynomial function has a maximum or minimum in $\mathbb{R}$
Prove that every even degree polynomial function $f$ has maximum or minimum in $\mathbb{R}$. (without direct using of derivative and making $f'$)
The problem seems very easy and obvious but I don't know how to write it in a math... | Let $f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$. Let us assume that $a_n>0$ (the case in which $a_n<0$ is similar). Then\begin{align}\lim_{x\to\pm\infty}f(x)&=\lim_{x\to\pm\infty}a_nx^n\left(1+\frac{a_{n-1}}{a_nx}+\frac{a_{n-2}}{a_nx^2}+\cdots+\frac{a_0}{a_nx^n}\right)\\&=+\infty\times1\\&=+\infty.\end{align}Therefore... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2971313",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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is there are any short cut method to find the determinant of A? Find the determinant of A
$$A=\left(\begin{matrix}
x^1 & x^2 & x^3 \\
x^8 & x^9 & x^4 \\
x^7 & x^6 & x^5 \\
\end{matrix}\right)$$
My attempts : By doing a laplace expansion along the first column i can calculate,but it is a long process My qu... | \begin{align*}
\det(A) =\begin{vmatrix}
x^1 & x^2 & x^3 \\
x^8 & x^9 & x^4 \\
x^7 & x^6 & x^5 \\
\end{vmatrix}
&=\color{red}{x^5}\begin{vmatrix}
x^1 & x^2 & x^3 \\
x^8 & x^9 & x^4 \\
\color{red}{x^2} & \color{red}{x^1} & \color{red}{1}
\end{vmatrix}\\
&=x^5\cdot x^4 \cdot x^1\begin{vmatrix}
1 & x & x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2971406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Elementary contour integral I have an integral
$$
\int_{-\infty}^{\infty}\frac{1}{(\omega^{2}-4)(\omega-2-i)(\omega+2-i)}d\omega
$$
And I wish to evaluate this using Cauchy's Integral Theorem. I understand that with a simple pole on the real axis like
$$
\frac{sin(x)}{x}
$$
We can break the contour around $x=0$ and us... | Hint: All the poles are simple, so you could break the integrand into a sum of four simple fractions of the form $\frac{c_k}{w-p_k}$, right?
Then just deal with each integral separately.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2971521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Orthogonality of a matrix where inner product is not the dot product Here are the definitions I am using:
*
*Orthogonality: Two vectors $x$ and $y$ are orthogonal iff $\langle x,y \rangle=0$.
*Orthonormal: If two vectors $x$ and $y$ are orthogonal and $||x|| = 1 = ||y||$ then $x$ and $y$ are orthonormal.
*Orthog... | Yes, the matrix is orthogonal. All orthogonal matrices have columns with orthonormal vectors with respect to the dot product, regardless of your choice of inner product.
There is something called an orthogonal linear transformation, that is, $TT^* = T^*T = I$. The matrix representation of these linear transformations w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2971690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
General solution of $y''-2xy'+2y=0$ I have to find the general solution of the differential equation $$y''-2xy'+2y=0.$$ An obvious solution is $y(x)=ax$ but I am unable to find another solution. Any hint as to how to proceed or which kind of method to apply is greatly appreciated.
| Making $z = x y$ we obtain
$$
z''-\frac{2(x^2-1)}{x}z' = 0
$$
now making $v = z'$
$$
v'-\frac{2(x^2-1)}{x}v = 0
$$
which is separable.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2971813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Poisson probability with Bayes? The number of goals scored every month is a Poisson with lambda 5 :
$$
P(X=x) = \frac{e^{-5}5^x}{x!} (x=0,1,2,3,4....)
$$
What is the probability of at least 4 goals scored next month given two goals scored next month.
I need to compute the following :
P(x >= 4 | X=2)
$$
P(x \ge 4)=1 -... | The problem statement isn't quite clear but it looks like it wants you to find $\mathsf P(X\geq4\mid X\geq 2)$
It's clear then from Bayes' Theorem that
$$\mathsf P(X\geq4\mid X\geq 2)=\frac{\mathsf P(X\geq4,X\geq 2)}{\mathsf P(X\geq2)}=\frac{\mathsf P(X\geq4)}{\mathsf P(X\geq2)}=\frac{1-\mathsf P(X\leq3)}{1-\mathsf P(X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2971939",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Proof of Heine-Borel Theorem; Bartle I'm reading through the proof for the Heine-Borel Theorem in Bartle's Elements of Real Analysis and getting caught on one point:
We assume that $K$ is a compact subset of $\mathbb{R}^p$ and let $x$ be an element in the complement of $K$. Then we let $G_m=\{y\in\mathbb{R}^p:|x-y|>\fr... | Note that $\{G_m : m \in \Bbb{N}\}$ is an open cover of $K$. By compactness of $K$, there must be a maximum $M \in \Bbb{N}$ such that only finite number of $G_m$ with $m\leq M$ cover $K$. But since those are nested subsets, then $G_M$ alone must cover $K$ and does not contain $x$. Because $G_M$ and $ \{z : |z-x| \leq 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do we know if a third vector is on the plane of the first and second vector? I have to add its components and if it gives me zero,then is it on the plane?
Suppose that the vectors have three components.
Thank you.
| Use the fact that a plane is a two dimensional object. That means that the size of the base is two (you can describe any vector in that plane as a linear combination of the vectors in the base). Now create a matrix with the components of those three vectors as rows (or columns). If the determinant is not zero, then the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Study the converge $\sum_{n=1}^{\infty}(\sqrt[n]{n}-1)$ I need to study the convergence of the series
$\sum_{n=1}^{\infty}(\sqrt[n]{n}-1)$
Now, I know that if we have a series $\sum_{n=1}^{\infty}a_n$ with positives elements and we can find a series $\sum_{n=1}^{\infty}b_n$ so that $0<a_n<b_n$ then if $\sum_{n=1}^{\in... | The series diverges by comparison with $\sum \frac{1}{n}\log n$.
Let $n \ge 2$. Since $\sqrt[n]{n} = e^{\frac{1}{n}\log n}$, the mean value theorem gives $\sqrt[n]{n} - 1 = e^{c_n}\cdot \frac{1}{n}\log n$ for some $c_n\in \left(0, \frac{1}{n}\log n\right)$. Now $e^{c_n} > 1$, so that $$\sqrt[n]{n} - 1 > \frac{1}{n}\log... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972418",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Numerical differentiation with Binomial Theorem In George Shilov's Elementary Real and Complex Analysis, there is a problem which asks us prove
If $f$ is twice differentiable on some open interval and the second derivative is continuous at $x$, then prove that
$$f''(x)=\lim_{h\rightarrow 0}\frac{f(x)-2f(x+h)+f(x+2h)... | I'm not sure I'm contributing anything, maybe I misunderstood since this feels like essentially a duplicate of this link. $\newcommand{\fd}{\Delta}$ Its not so hard (via e.g. repeated applications of l'Hopital, as Paramanand shows in that link) to show that you are actually interested in iterated forward finite differ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 3
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Find the value of an integer $a$ such that $ a^2 +6a +1 $ is a perfect square. I was able to solve this but it required me using hit and trial at one step. I was wondering if i could find a more solid method to solve it.
p.s. this is the first time im asking a question here so sorry if i couldn't construct the question... | hint
$$a^2+6a+1=b^2$$
$$\iff a^2+6a+1-b^2=0$$
the reduced discriminant is
$$\Delta'=9-1+b^2=b^2+8$$
$$a=-3\pm \sqrt{b^2+8}$$
thus
$$b^2+8=c^2$$
and
$$(c+b)(c-b)=8$$
$$=4×2=-4×(-2)$$
$$c+b=\pm 4,\;\; c-b=\pm 2$$
gives $$\;\; b=\pm 1$$
and in all cases,
$$a=0 \text{ or } a=-6$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Inequality of real numbers with exponent For $a,b>0$ are two real numbers and $p\geq 1$. Is the following inequality true
$$|a^p-b^p|\leq|a-b|^p\;\;?$$
| Let $b\geq a$ then $\>b=a+\delta\>$ and $\>\delta>0$
$$
\\|a^p-b^p|\leq|a-b|^p
\\(a+\delta)^p-a^p\leq\delta^p
\\(a+\delta)^p\leq a^p+\delta^p
$$
but $a>0$ and $\delta\geq0 => (a+\delta)^p=a^p+\delta^p+\delta*x\>(x\geq0)($
Binomial theorem$)=>$
$$
\\a^p+\delta^p\geq(a+\delta)^p=a^p+\delta^p+\delta*x\geq a^p+\delta^p
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972714",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
Curve of possible locations of a post office between two buildings One of my students had to answer this question, and it is actually stumping me.
Let $A,B$ be two buildings 10 miles apart. Suppose $P$ is a post office such that the distance between $A$ and $P$ is 2 miles more than the distance between $P$ and $B$. Th... | Your desired locus is one branch of a hyperbola.
You can see this by placing A and B at convenient points on the Cartesian plane--say A at (-5, 0) and B at (5, 0). Place P at (x, y). Use the distance formula to get an equation for your locus:
$$\sqrt{(x--5)^2+(y-0)^2} = 2 + \sqrt{(x-5)^2+(y-0)^2}$$
Simplify that equati... | {
"language": "en",
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Prove $L = F(β)$, where $β^p = F$ via Hilbert's Theorem 90 Let $F$ be a field that contains a primitive $p$-th root of unity, where $p$ is a prime. I wish to prove that if $L$ is Galois over $F$ and $[L : F] = p$, then $L = F(β)$, where $β^p = F$. Does anyone know of a proof via Hilbert's Theorem 90?
That is, making u... | Let $\zeta$ be a primitive $p$-th root of unity in $F$. Then $N_{L/F}(\zeta)=\zeta^p=1$. By Hilbert 90, there is $\alpha\in L^*$ with
$\sigma(\alpha)/\alpha=\zeta$, that
is
$$\sigma(\alpha)=\zeta\alpha.$$
But then
$$\sigma(\alpha^p)=\sigma(\alpha)^p=\zeta^p\alpha^p$$
so that $\alpha^p=a\in F$. But $\sigma(\alpha)\ne\al... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2972880",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Bezout's Identity: Finding A Pair of Integers
There is only one integer $x$, between 100 and 200 such that the integer pair $(x, y)$ satisfies the equation $42x + 55y = 1$. What's the value of $x$ in this integer pair?
We know that
$$\begin{align} x &= x_0 + 5t \\ y &= y_0 - 4t \end{align}$$
But we need to know what ... | First you need to find all integer solutions of $42x+55y=1$.[See this post]
Here $\gcd(42,55)=1$, so by Euclidean algorithm, $$1=42(-17)+55(13)\;[\text{check!}]$$
So integer solutions are $$x=-17+55r$$ $$y=13-42r$$ where $r \in \Bbb{Z}$
For your task, you need to find one $r$ so that $100 \leq x \leq 200$ and $r$ sa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2973025",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Independence of random products mod $p$ Let $p$ be a large prime and $1 \leq q < p$ be chosen uniformly at random. Let $0 \leq r_1 \neq r_2 < p$ be arbitrary but fixed.
Question: for sufficiently large $p$, are $qr_1 \bmod p$ and $qr_2 \bmod p$ statistically independent? (i.e. if I take $p$ sufficiently large, can I m... | I have two questions to you:
*
*Why do you allow $r_1=0$ or $r_2=0$? I think you should not.
*Do you mean that $r_1$ and $r_2$ remain fixed when you change (said, increase) $p$?
Anyway... For $0 < r_1, r_2 < p$, the expected value of each of these two random variables is $\dfrac{p}{2}$, the variance is $\dfrac{p(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2973118",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Empty Forest in Graph Theory I was reading the Kruskal's Algorithm application and there was a statement about starting with an empty forest. WHat should I understand from the term 'empty forest'? Does it mean an edge that is not connected or should I think of it as a forest with 1-component?
| Usually an empty object simply doesn't contain any of whatever its highest-level member happens to be. You can talk about forests in terms of nodes and edges, but the conceptual vantage point of a forest is that it contains trees. If it's empty, it just doesn't contain any trees.
By extension then, it wouldn't contain ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2973222",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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What is a "singular system" mean? Currenly I'm working on a problem as an apprentice. I'm currently reading the book "An introduction to the Mathematical Theory of Inverse Problems" by Andreas Kirshch.
In that book, on page 32, in Theorem 2.6, there is a sentence like this
Let $K: X \to Y$ be compact with singular sys... | In the Appendix, theorem $A.53$ clarifies your doubt!
Theorem $A.53$ (Singular Value Decomposition).
Let $K : X \rightarrow Y$ be a linear
compact operator, $K^∗ : Y \rightarrow X$ its adjoint operator, and $μ_1 ≥ μ_2 ≥ μ_3 . . . > 0$
the ordered sequence of the positive singular values of $K$, counted relative to its... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Proving that $GL_n$ ($ \mathbb{R}$) $: = \{A \in \mathbb{R}^{n \times n}: \det A \neq 0\}$ is open in $\mathbb{R}^{n \times n}$ How can one prove that the set of invertible matrices $GL_n := \mathbb{R} : = \{A \in \mathbb{R}^{n \times n}: \det A \neq 0\}$ is open in $\mathbb{R}^{n \times n}$?
${\mathbb{R}^{n \times n}... | The determinant function $$\det: \mathbb{R^{n^2} \to \mathbb{R}}\\A \mapsto \det A$$ is continuous because it is a polynomial one.
The set $GL_n(\mathbb{R})=\{A \in M_n(\mathbb{R}) | \det A \ne 0\}$ is the preimage of the open set $\mathbb{R} \setminus \{0\}$, so, for the continuity of the determinant function, it is o... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Partial derivative of cross-entropy I am trying to make sense of this question.
$$E(t,o)=-\sum_j t_j \log o_j$$
How did he derive the following?
$$\frac{\partial E} {\partial o_j} = \frac{-t_j}{o_j}$$
| You are missing a reformulation Christopher Bishop (1995) took for the cross-entropy, because the formulation
\begin{equation}
E=-\sum_j t_j \log (y_j)
\end{equation}
does not have a minimum value of zero.
However, his reformulation below for cross-entropy error has a minimum value of zero:
\begin{equation}
E=-\sum_j t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2973580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Proving that a sequence $a_n: n\in\mathbb{N}$ is (not) monotonic, bounded and converging $$a_n = \left(\dfrac{n^2+3}{(n+1)^2}\right)\text{ with } \forall n\in \mathbb{N}$$
$(0\in\mathbb{N})$
Monotonicity:
To prove, that a sequence is monotonic, I can use the following inequalities:
\begin{align}
a_n \leq a_{n+1}; a_n <... | Hint: $$a_{n+1}-a_n=2\,{\frac {{n}^{2}-n-4}{ \left( n+2 \right) ^{2} \left( n+1 \right) ^{
2}}}
$$
Second hint: $$a_n=\frac{n^2(1+\frac{3}{n^2})}{n^2(1+\frac{2}{n}+\frac{1}{n^2})}$$ this tends to $1$ for $n$ tends to infinity
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
Expanding the supremum metric on $C[0,1]$ to $C[a,b]$
Let $X=C[0,1]$ be the set of all continuous functions on the interval
$[0,1]$. Define:
$$d_1(f,g)= \sup {\{ \left \lvert {f(t)-g(t)} \right \rvert : \ t \in [0,1]} \}$$
I want to expand this supremum metric of continuous functions defined on $[0,1]$ to the supr... | The map you have given is in fact not a bijection; remark that if $0\leq b \leq 1$, $g$ is not even defined at $x=b$. Instead, for a function $f \in C[a,b]$, we define $g \in C[0,1]$ to be:
\begin{align*}
g(x) = f \left( (b-a)x + a \right)
\end{align*}
You should check for yourself that this is indeed a bijection. How... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2973887",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Sum the first $n$ terms of the series $1 \cdot 3 \cdot 2^2 + 2 \cdot 4 \cdot 3^2 + 3 \cdot 5 \cdot 4^2 + \cdots$ The question
Sum the first $n$ terms of the series:
$$ 1 \cdot 3 \cdot 2^2 + 2 \cdot 4 \cdot 3^2 + 3 \cdot 5 \cdot 4^2 + \cdots. $$
This was asked under the heading using method of difference and the answer ... | \begin{align*}
\sum_{k=1}^n k(k+2)(k+1)^2&=\sum_{k=1}^n (k+3)(k+2)(k+1)k-2\sum_{k=1}^n (k+2)(k+1)k\\
&=24 \sum_{k=1}^n \binom{k+3}{4}-12 \sum_{k=1}^n \binom{k+2}{3}\\
&=24\binom{n+4}{5}-12\binom{n+3}{4}
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2973979",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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How to prove that if the integral of a nonnegative function is zero, then the function is zero This is required to prove that the expression $d(f,g)=\int|f-g|dt$ is a metric. We need to show that if $\int |f-g|dt=0$ then $f=g$. It seems obvious but how to prove it? This relation is not given in Paul's Online Math Notes... | Let's forget continuity for a minute.
Let $u$ be a measurable function that's also non-negative almost everywhere on $I=[a, b]$, such that $$\int_a^bu=0$$
Then for $\epsilon>0$, let $$S_\epsilon=\{x\in[a,b] \textrm{ such that } u(x)\geq\epsilon\}$$
Then by definition of $S_\epsilon$, $$\int_a^bu\geq\int_{S_\epsilon}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Covariance intuition to formula How we prove below transition mathematically?
*
*Double Summation from
$$
\dfrac{1}{N^2}\sum_{i=1}^{N}\sum_{j=i+1}^{N}(x_i - x_j)(y_i - y_j) \tag{1}
$$
to
$$
\dfrac{1}{2N^2}\sum_{i=1}^{N}\sum_{j=1}^{N}(x_i - x_j)(y_i - y_j) \tag{2}
$$
I understand this visually though (imagining a... | The paper you have linked indeed answers your question. The problem that it is done "in reverse" is not an issue since an equality is proven. The equality holds both ways. Doing it in reverse would be an issue when proving a theorem which is only a necessary, but not a necessary and sufficient condition.
$\overline{x} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2974227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Covariant derivative on the dual bundle The covariant derivative on the dual bundle is defined as follows:
$\nabla^{*}: \Gamma(TM) \times \Gamma(E^*) \ni (X, t) \mapsto \nabla_X^{*} t \in \Gamma(E^*)$
, where for any section $s \in \Gamma(E)$,
$(\nabla_X^{*} t)(s) = L_X(t(s)) - t(\nabla_X s)$.
Remark: $\Gamma(E^... | As written in the post,
$$\nabla^*_Xft(s)=X(ft(s))-ft\nabla_Xs.$$
Now, using the Leibniz rule for the first summand on the right,
$$=X(f)t(s)+fX(t(s))-ft\nabla_Xs=X(f)t(s)+f\nabla^*_Xt(s).$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2974614",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can any integer, not a multiple of three, be represented as $n = \sum_{i=0}^{a-1} 3^i \times 2^{b_i}$? This question has some relevance to the Collatz conjecture. It was originally based on trying to represent a number like this: Finding whether $\dfrac{2^k - (2\cdot3^{n-1} + 2^{t_0}3^{n-2} + 2^{t_0+t_1}3^{n-3} .... + ... | I deduced the same thing when study the Collatz conjecture, here is the proof without the restrictions and some stuff related.
Let $G_n = \{m \in \mathbb{N} \,|\,\gcd(m,n) = 1\}$
We can do this for all $G_n$, for example $n = 2$, but we only interested in $n = 3$.
Lemma:
For all $n \in G_3$, exists $a \in \mathbb{Z} : ... | {
"language": "en",
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Stirling numbers of the second kind - proof For a fixed integer k, how would I prove that
$$\sum_{n\ge k} \left\{n \atop k\right\}x^n= \frac{x^k}{(1-x)(1-2x)...(1-kx)}.$$
where $\left\{n \atop k\right\}=k\left\{n-1 \atop k\right\}+\left\{n-1 \atop k-1 \right\}$
| The definition of ${n\brace k}$ through the recursion is easily seen to be equivalent to the combinatorial definition "the number of ways for partitioning a set with $n$ elements into $k$ non-empty subsets". $m^n$ can be interpreted as the number of functions from $[1,n]$ to $[1,m]$: if we classify them according to th... | {
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Partial derivative of an integral from zero to infinity How would one go about taking the derivative of this integral?
$$\frac{\partial}{\partial C_T} \int_{0}^{\infty} U(C_T)e^{-\delta t}dt$$
| $$
\int_{0}^{\infty} U(C_T)e^{-\delta t}dt=U(C_T)\int_{0}^{\infty} e^{-\delta t}dt=\frac{U(C_T)}{\delta}
$$
and
$$\frac{\partial}{\partial C_T} \int_{0}^{\infty} U(C_T)e^{-\delta t}dt=\frac{U'(C_T)}{\delta}.
$$
| {
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Inverse of a primitive recursive bijection Is it true or false that the inverse of a primitive recursive bijection $f: \mathbb{N} \to \mathbb{N}$ is also primitive recursive (pr)?
| No this is not true!
Let's $A$ the Ackermann-Peter function (defined by $A(x)=A(x,x)$ with this definition). Note that $A$ is fast growing, recursive but not primitive recursive.
But $A^{-1}$ defined by $A^{-1}(x)=1+\max\{y\;|\;A(y)\le x\}$ ($0$ if the set is empty) is recursive primitive (and very slow growing).
Now ... | {
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Prove (¬(∀(¬()))) ⊢ (∃ ()) by Natural Deduction I want to prove (¬(∀(¬()))) ⊢ (∃()) using only the basic rules of the Natural Deduction system for propositional logic and predicate logic.
I am not sure how to get rid of the negation before the universal quantifier.
How should I prove this?
| If in doubt, try reductio ....
So after the initial premiss, assume $\neg\exists xPx$
Now what?
You'll have to make another assumption to get anywhere ....
So suppose $Pa$ (the obvious thing ...why??)
Then you can infer $\exists xPx$, contradiction!
So you can infer $\neg Pa$
And that gives you $\forall x\neg Px$
And n... | {
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What is the Euler characteristic of the hyperboloid of one sheet I would like to know what's the Euler characteristic of the hyperboloid of one sheet. I know that $2-2g$ is the Euler characteristic where g is the number of "holes". Using this fact, Euler characteristic of the hyperboloid is -2. Am I right?
| No. The "fact" you mention is not stated in a rigorous way; there is no definition of 'hole'. (I really dislike this phrasing because of confusions like this.) The precise statement is that if $\Sigma_g$ is the compact surface without boundary of genus $g$, then $\chi(\Sigma_g) = 2-2g$.
The hyperboloid of one sheet is... | {
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Limit Evaluation - $\lim_{x\to \infty} \frac{1-e^x}{e^{2x}}$ $\lim_{x\to \infty} \frac{1-e^x}{e^{2x}}$
My guess is to evaluate by dividing all terms by $e^x$, which works and gives me Eulers identity.
But why should that be right? I thought we are only supposed to divide by the highest exponent term in the denominator... | Your tought is correct but there is not needing of Euler'e identity, indeed dividing both numerator and denominator by $e^x$, we obtain
$$\dfrac{1-e^x}{e^{2x}}=\dfrac{\frac1{e^x}-\frac{e^x}{e^x}}{\frac{e^{2x}}{e^x}}=\dfrac{\frac1{e^x}-1}{e^x}$$
and then it suffices to observe that the numerator tends to $-1$ (that is b... | {
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Prove that two sequences of integers that have the same sum and product must be the same.
Given two sequences of nondecreasing distinct positive integers such that $$x_1 + x_2 + ... + x_i = y_1 + y_2 + ... + y_i , i>0$$ and that $$x_1x_2 ... x_i = y_1y_2 ... y_i$$
Prove/disprove that the sequences are equal i.e. $$... | Counterexample:
$12+4+3 \ =\ 9+8+2$
$12\cdot4\cdot3 \ = \ 9\cdot8\cdot2$
Moreover, for $\ i>2\ ,\ $you can always find infinitely many counterexamples.
| {
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What does '$N × Q$' represent in this relation? In relation $\{(x, y) ∈ N × Q | y = \sqrt x\}$
what does '$N × Q$' represent?
| $\mathbb N$ is the set of natural numbers and $\mathbb Q$ is the set of rationals.
$\mathbb N \times \mathbb Q$ is the Cartesian product of the two sets : $\mathbb N$ and $\mathbb Q$.
Thus, $(x, y) \in \mathbb N \times \mathbb Q$ means that $(x, y)$ is an ordered pair where $x$ is a natural and $y$ is a rational.
| {
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If a seimigroup's left identity is unique, can it be two-side identity?
If a seimigroup's left identity is unique, can it be two-side identity?
The answer is true if we talk it in a ring. Like construct $(be-b+e)a=a$. But in a semigroup I can't image how to construct a equation so we can use the condition uniquenes... | That is an interesting question.
Let $S=${$x,y$} (where $x$ is not equal with $y$). $S$ with the following operation is a semigroup:
$xy=y ,xx=y, yx=y, yy=y$.
We see that $y$ is the unique left identity of $S$, but $y$ is not an identity, other wise we have $yx=x$ then we have $x=y$.
| {
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What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power?
What is the smallest integer greater than 1 such that $\frac12$ of it is a perfect square and $\frac15$ of it is a perfect fifth power?
I have tried multiplying every perfect square ... | Hint: Let the required number be x:
$\frac{1}{2}x= A^2$
$\frac{1}{5}x= B^5$
$\frac{1}{2}x+\frac{1}{5}x =A^2+B^5$
$\frac{5x+2x}{10}=A^2+B^5$
$7x=10(A^2+B^5)$
⇒ $x=10k$; $k ∈ N $.
So x is a power of 10.
The smallest 5th power of 10 is $10^5$ so the number must be $5\times 10^5=500000$.
| {
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Group of matrices form a manifold or euclidean space There is a very interesting question How can a group of matrices form a manifold. From the answers it looks more like group of matrices form euclidean space than a general manifold. I understand that euclidean space is a manifold, but manifold is very general and ha... | Take $SO(2,\mathbb{R})$, for instance. This is the group of the $2\times2$ orthogonal matrices whose determinant is $1$. But then$$SO(2,\mathbb{R})=\left\{\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}\,\middle|\,\theta\in\mathbb R\right\}.$$This can be seen as a circle in $\mathbb{R}^2$. The... | {
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Why is $\pi/2$ omitted from the solution of $\cot x = 3 \sin 2x$?
Why is it that the solution of
$$\cot x = 3 \sin 2x \quad(\text{for the interval}\; -\pi < x < \pi)$$
does not include $\pi/2$, even though if this is graphed, it shows intersections at $x = \pm\pi/2$?
Please see graph below. (The solutions menti... | You are right the values $x=\pm \frac{\pi}2$ are solutions of the equation
$$\cot x=3 \sin 3x \implies \cot \pm\frac{\pi}2=3 \sin \pm\pi=0$$
maybe it was not included since it is considered a trivial solution.
added after editing
The values $x=\pm \frac{\pi}2$ seem to be indeed included among the solutions.
| {
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Inner Space Projection Using Matrices In my math class today, we proved that the ratio of the area of an inner space to that of the inner space projected by some matrix $A$ is equal to $|det(A)|$. In other words, if the area of an inner space is $a$, the area of that inner space projected by a matrix $M$ $= |det(M)|*a$... | If the transformation is linear then the matrix $M$ is determined if you know the images of two general points (i.e. two points that do not lie on the same line through the origin). Essentially, the entries in each row of $M$ are the solution to a pair of simultaneous linear equations.
Of course, in many cases the tran... | {
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What is the value of k if equation $x^3-3x^2+2=k$ has three real roots and if one real root? I'd like help understanding this question graphically actually. My book finds the derivative of f(x)=$x^3-3x^2+2$ and gets 3x(x-2), thus finding maxima and minima of f(x) as 2 and -2. Then it just says that for three real roots... | Consider $g(x)=x^{3}-3x^{2}=x^{2}(x-3)$, so that $f(x)=g(x)+(2-k)$. The plot of $g(x)$ is as follows;
So we see that $g(x)$ has $2$ roots, one at $x=0$ and the other at $x=3$. It also has a local minimum at $x=2$, with value $f(2)=-4$ and a local maximum of at $x=0$.
Now, $f(x)$ is just a vertical translation of $g(x)... | {
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The prime index of subgroup and some investigations I have a task: "Let $p$ be a prime number and let $G$ be a group with a subgroup $H$ of index $p$ in $G$. Let $S$ be a subgroup of $G$ such that $H\subset S\subset G$. Prove that $S = H$ or $S = G$."
So $[G : H] = p$ $\Rightarrow$ $|G| = |H|p$. Supposably, that $|H| ... | You don't need all those variables.
Moreover, the groups in question need not be finite.
Suppose $H$ is a subgroup of $G$ with $[G:H]=p$, for some prime $p$.
If $S$ is a subgroup of $G$ such that $H\subseteq S\subseteq G$, then
$$p=[G:H]=[G:S]\cdot [S:H]$$
https://en.wikipedia.org/wiki/Index_of_a_subgroup#Propertie... | {
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How to maximize returns in this scenario You have a machine. You can put money into it. You have $s$ initial budget. $p$ percent of the time the machine will double your investment. $(100-p)$ percent of the time it will just swallow your money and not return anything. You can choose a ratio $a$ of your money to reinves... | Let $p' = p/100$ denote the probability of winning at a turn with the machine. Your initial capital is $W_0 = s$. After the first turn where you bet $aW_0$, your wealth is
$$W_1 = W_0 + aW_0X_1 = W_0(1 +aX_1),$$
where $X_1$ is a binary random variable such that $P(X_1 = 1) = p'$ and $P(X_1 = -1) = 1-p'$.
Assume tha... | {
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Proving Regularly Closed If $U$ is open and $A=\overline{U}$, then $A$ is regularly closed.
Note that: $A=\overline{U}=U \cup U' \Rightarrow U \subset A \Rightarrow U \subset Int(A)$, since $U$ is open.
A set $A$ is regularly closed iff $A=\overline{Int(A)}.$
$\Rightarrow$
Let $x\in A=\overline{U}$. Let $V$ be an open ... | Since U is open U = int U.
So A = cl U = cl int U. Thus
cl int A = cl int cl int U = cl int U = A.
Exercise. Show by set inclusions cl int cl int U = cl int U.
| {
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$\alpha \nmid n \Rightarrow \alpha$ is a generator of group $G$ Let $G=\left \langle x \right \rangle$ a cyclic group s.t. $ord(x)=n$. Let $a=x^\alpha$ s.t. $\alpha \nmid n$ and $0<\alpha<n$ . Prove that $x^\alpha$ is a generator of $G$.
My partial answer: Let $A:=\left \langle x^\alpha \right \rangle$. We'll assume in... | This is not true. For example, suppose $\text{ord}(x)=10$, and take $\alpha=6$. Then
$$<x^\alpha>=\{e, x^6, x^{12}=x^2, x^8, x^{14}=x^4\}$$
which clearly is not the whole group.
However, if $\gcd(\alpha,n)=1$ this is true, since then by the euclidean algorithm there exist $k,l \in \mathbb{Z}$ such that $k\alpha+ln=1$... | {
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Find $\cos(\alpha+\beta)$ if $\alpha$, $\beta$ are the roots of the equation $a\cos x+b\sin x=c$
If $\alpha$, $\beta$ are the roots of the equation $a\cos x+b\sin x=c$, then prove that $\cos(\alpha+\beta)=\dfrac{a^2-b^2}{a^2+b^2}$
My Attempt
$$
b\sin x=c-a\cos x\implies b^2(1-\cos^2x)=c^2+a^2\cos^2x-2ac\cos x\\
(a^2+... | Guide: $c= a\cos \alpha +b\sin \alpha = a\cos \beta + b\sin \beta \implies a(\cos \alpha -\cos \beta) = b(\sin \beta-\sin \alpha)\implies \dfrac{a^2}{b^2}=\dfrac{(\sin \alpha - \sin \beta)^2}{(\cos \alpha - \cos \beta)^2}=m\implies RHS = \dfrac{m-1}{m+1}=...LHS$
| {
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An exercise about bifunctors from Riehl's "Category Theory in Context" This is an exercise from E.Riehl's book "Category Theory in Context" (p.48, ex.1.7.vii)
Prove that a bifunctors $F\colon\mathsf{C}\times\mathsf{D}\to\mathsf{E}$ determines and is uniquely determined by:
*
*A functor $F(c,-)\colon\mathsf{D}\to\mat... | Yes, to your first question, or more precisely to $F(1_c,g)$
where $1_c$ is the identity morphism on $c$.
The morphisms you seek in your second are the $F(f,1_d)$.
| {
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Properties of conditional variance Full statement of problem:
Let $(\Omega,\mathcal{F},P)$ be a probability space and $\mathcal{G}\subset \mathcal{F}$ a $\sigma$-algebra. Let $X\in L^2$. Define $$\text{var}(X \mid \mathcal{G})=E[|X-E[X \mid \mathcal{G}]|^2\mid \mathcal{G}].$$ Prove that:
(a) $\text{var}(X \mid \mathca... | Hints for part (b):
*
*Show that $$\mathbb{E} \big[ (X-\mathbb{E}(X \mid \mathcal{G}) \mathbb{E}(X \mid \mathcal{G}) \big]=0 \tag{1}$$ and $$\mathbb{E}(X- \mathbb{E}(X \mid \mathcal{G}))=0. \tag{2}$$
*Conclude from $$X-\mathbb{E}(X) = (X-\mathbb{E}(X \mid \mathcal{G})) + (\mathbb{E}(X \mid \mathcal{G})-\mathbb{E}(X... | {
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solution of a system of equations in algebraic closure of GF2 I have a set of equations and I want to know whether there exist solutions of these equations in an extension of Galois field $\mathbb{GF}_2$ and what are they? Is there a procedure to check this?
For example, the following set of equations in variables ${a,... | By Hilbert's Nullstellensatz the set of common zeros of those polynomials in the algebraic closure is empty if and only if the constant polynomial 1 is in the ideal $I$ they generate in the ring of polynomials $R$. Actually this is exactly what the weak Nullstellensatz states, and that suffices here. In your example ca... | {
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Conditional Expectation on the first outcome Assume Independent trials, resulting in one of the outcomes 1, 2, 3, 4, 5 with respective probabilities $p_i$ for $i=1,2,3,4,5$ and $\sum_i p_i = 1$
Let $Z$ be the number of trials needed until the initial outcome has occurred exactly $5$ times. example: if we get $1,3,3,4,1... | Indeed, it is $1+$ something. You are using the Linearity of Expectation.
$\mathsf E[Z\mid O_i]$, is the expected time until that first outcome ($i$) has its fourth subsequent occurance.
What type of distribution is the count of Bernoulli trials until the next success?
What type of distribution is the count of Ber... | {
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Matrix with no negative elements = Positive Semi Definite? A matrix $A$ is positive semi-definite IFF $x^TAx\geq 0$ for all non-zero $x\in\mathbb{R}^d$. If all elements of $A$ are non-negative, does this guarantee that $A$ is positive semi-definite?
| In general no. One way of defining positive definiteness is through the leading principal minors of a matrix. The $k^{th}$ leading minor if found by computing the deturminant of the matrix after deleting the last $n-k$ colomns and rows in an $n \times n $ matrix. It is quite common to see a matrix with all positive ent... | {
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Let $I$ be an ideal in a Noetherian ring. Show that either $I$ contains an $R$-regular element or else $aI=0$ for some $0\neq a\in R$. Let $I$ be an ideal in a Noetherian ring. Show that either $I$ contains an $R$-regular element or else $aI=0$ for some $0\neq a\in R$.
How would I prove this? Also what does $aI=0$ mean... | Yes this is an important property of Noetherian rings. It is Theorem 82 in Kaplansky's Commutative Rings, which he prefaces as "a result that is among
the most useful in the theory of commutative rings."
In the literature you often encounter this property as Property (A)
A ring is said to have Property (A) if every ... | {
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Combinatorics: How many persons like apples and pears and strawberries? Out of $32$ persons, every person likes to eat at least one of the following type of fruits: Strawberries, Apples and Pears. (Which means that there does not exist any person, who does not like to eat any type of fruit). Furthermore, we know that $... | Part (a) is an ill-posed question (almost in the spirit of this other question). By inclusion/exclusion, using the given values the number of people liking all three fruits is
$$32-(20+18+28-10-16-12)=32-28=4$$
Then the number of people liking apples and some other fruit(s) is $16+10-4=22$, which is greater than the 2... | {
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How to determine the gcd of a set I'm stuck at a question.
The question states that $K$ is a field like $\mathbb Q, \mathbb R, \mathbb C$ or $\mathbb Z/p\mathbb Z$ with $p$ a prime. $R$ is used to give the ring $K[X]$. A subset $I$ of R is called an ideal if:
• $0 \in I$;
• $a,b \in I \to a−b \in I$;
• $a \in I$ $r ... | The proof is essentially the same as in $\Bbb Z$ via Euclidean descent, i.e. by division with remainder.
It suffices to show $I = (a_1,\ldots,a_n) = (d)$ is principal, since then $\,a_i \in (d)$ implies $\,d\,$ is a common divisor of the $a_i,\,$ necessarily greatest since $\,d\in I\,\Rightarrow\, d = r_1 a_1 +\cdots +... | {
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Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Show that $\ell(M)<\infty$ if and only if $\text{Supp}(M)\subset m\text{-Spec}(R)$ Let $R$ be a Noetherian ring and $M$ a finite $R$--module. Show that $\ell(M)<\infty$ if and only if $\operatorname{Supp}(M)\subset m\operatorname{-Spec}(R)$.
What does $\ell(M)<\... | $l(M)<\infty$ means that $M$ has a composition series.we always assume $R$ is Noetherian in following:
definition1:let $N$ be an aribitray module,a prime ideal $P$ is called the associated prime ideal of $N$ if $p=ann(x)$ for some $X\in N$.and denote $Ass(N)$ be the set of all the associted prime ideals.
it is clear ... | {
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Proof of : $\cap_{f\in E'}Ker(f)= \left\{0 \right\}$ Let $(E,N)$ be a normed vector space.
The dimension of $E$ could be infinite.
Let $E'= \left\{f:E \rightarrow \mathbb{K} ,\quad f \quad linear \quad and \quad continuous \right\}$.
do we have $\cap_{f\in E'}Ker(f)= \left\{0 \right\}$ ?
| If $\mathbb K = \mathbb R$ or $\mathbb C$, then this is True by Hahn-Banach Theorem.
I am not certain but you have more general version of Hahn-Banach for other fields such as p-adic fields.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2979257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Group structure of rotating-square puzzle Suppose we arrange the numbers $1$ through $6$ at the "vertices" of the shape formed by aligning the sides of two squares, as shown below:
In this "puzzle," the only moves allowed are rotating the vertices of either square counterclockwise.
I would like to find the group $G$ t... | GAP shows that the group is in fact isomorphic to $S_5$. A geometric interpretation was requested for what $5$ things are being permuted. Consider the following $5$ sets of edges between the vertices.
$(1,2,5,4)\leftrightarrow(orange,blue,purple,green)$
$(2,3,6,5)\leftrightarrow(red,blue,purple,green)$
To see that thi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2979561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Ring of order $p^2$ must be commutative. See this relevant post
Ring of order $p^2$ is commutative.
Either I don't understand the above or it isn't quite complete (presumably the former).
Let $R$ be a ring of order $p^2$. Then $char(R)=p$ and $Z(x)=p,p^2$ for any $x \in R$. In the latter case there is nothing to show. ... | We recall that $R$ has an additive subgroup of order $\text{char}(R)$; thus, since $\vert R \vert = p^2$, either $\text{char}(R) = p$ or $\text{char}(R) = p^2$.
If
$\text{char}(R) = \vert R \vert = p^2, \tag 1$
we are done, since every element of $r$ is a sum of $1$s; so suppose
$\text{char}(R) = p; \tag 2$
then
$\Bbb... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2979716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Combinatoric meaning of $a_n=5a_{n-1} - 6a_{n-2}$ I've solved the following recurrence relation: $a_n=5a_{n-1} - 6a_{n-2}$ using generating functions, to be: $a_n=3^n-2^n$. It is possible to give a meaning to $3^n-2^n$, and that is:
Consider the following set: $S=\{a,b,c\}$.
$3^n$ is the number of sequences of length n... | Let $a_n$ be the number of sequences of length $n$ from the set $S$ with at least one $a$
Let $b_n$ be the number of sequences of length $n$ from the set $S$ with no $a$
Then using the combinatorial analogy we can easily say
*
*$a_n=3 a_{n-1} + b_{n-1}$ since we can append any of the three to a satisfactory sequenc... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2979816",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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A continuous Function from $R$ to a Banach Space is Borel-Measurable I think the notation of my book is a bit odd, so I'm having trouble finding any other sources to help me with this proof. The thing I'm trying to prove is that a continuous function, $f$, from $R$ to a Banach Space, $B$, is Borel-Measurable. The defin... | Fix $n$. Then $f([-n,n])$ is a compact set. hence it can be covered by a finite number of open balls of radius $\frac 1 n$, say $A_{n,1},A_{n,2},\cdots, A_{n,k}n$. Let $B_{n,1}=A_{n,1},B_{n,2}=A_{n,2}\setminus A_{n,1}$, $\cdots$, $B_{n,k_n}=A_{n,k_n}\setminus \cup_{j=1}^{k_n-1} A_{n,j}$. Then the sets $f^{-1}(B_{n,j})$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2979945",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How Isolate y from $a=\sqrt{(y^2+(a+x)^2)^3}$ Just that!
I'm asking for a method to isolate $y$ from such expressions:
$$a=\sqrt{(y^2+(a+x)^2)^3}$$
or even easier:
$$a=\sqrt{(y^2+x^2)^3}.$$
EDIT:
I'm just trying to revisit some of the basics in algebraic equation manipulation.
Even it's sooo easy to find the school ex... | It might help to see how $\sqrt{(y^2+(a+x)^2)^3}$ is built starting from y
$\begin{align}
& y \\
& {{()}^{2}}\to {{y}^{2}} \\
& +{{(a+x)}^{2}}\to {{y}^{2}}+{{(a+x)}^{2}} \\
& {{()}^{3}}\to {{({{y}^{2}}+{{(a+x)}^{2}})}^{3}} \\
& \sqrt{{}}\to \sqrt{{{({{y}^{2}}+{{(a+x)}^{2}})}^{3}}} \\
& =a \\
\end{align}$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2980246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Geometric intuition of the dimension of Grassmannians and flag manfolds I wish to understand geometrically (not just algebraically) why the dimension of the Grassmanian $G(k,n)$ is $k(n-k)$ and the dimension of a flag manifold $F(k_{1},k_{2},...,k_{n},N)$ is $\sum_{i=1}^{n}k_{i}(k_{i-1}-k_{i})+Nk_{n}$ (in fact with und... | The idea is to use the standard affine charts of $G(k,n)$. Start with the $k$-plane $P \subset \mathbb{R}^n$ (say, the one spanned by $e_1, \dots, e_k$) and
a $(n-k)$-plane $P^\perp$ transverse to $P$ (say, the one spanned by $e_{k+1}, \dots, e_n$).
The set of all $k$-planes transverse to $P^\perp$ is an open neighborh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2980359",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Proving the division theorem with strong induction The exercise goes like this:
Prove the division theorem using strong induction. That is, prove that for $a \in \mathbb{N}$, $b \in \mathbb{Z}^+$ there always exists $q, r \in \mathbb{N}$ such that $a = qb + r$ and $r < b$. In particular, give a proof that does not use... | You can prove it by strong induction on $a$.
For $a=0$, it is trivial.
Now, consider an arbitrary $a\in\mathbb N$ and assume that each $a'<a$ can be written as $qb+r$, with $r<b$. Now, if $a<b$, you can write $a$ as $0\times b+a$. Otherwise, consider $a-b$. By the induction hypothesis, it can be written as $bq+r$, with... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2980504",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
If $z = f(x,y)$ is continuous on $\mathbb{R}^2$, then is $f(g_1(x), g_2(x))$ measurable, where $g_1$ and $g_2$ are measurable? Let $z = f(x,y)$ be a continuous function on $\mathbb{R}^2$, and $g_1(x), g_2(x)$ be real valued functions on $\mathbb{R}^1$. Prove $F(x) = f(g_1(x), g_2(x))$ is a measurable function on $[a,b]... | Recall the composition of two measurable functions is measurable. Since $f$ is continuous, it is measurable. Let $g:\mathbb{R} \rightarrow \mathbb{R}^2$ map $x$ to $g(x) = (g_1(x),g_2(x))$. If $A = I_1 \times I_2 \subset \mathbb{R}^2$ is a rectangle (Cartesian product of intervals $I_1 \subset \mathbb{R}$ and $I_2 \sub... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2980653",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$t \in \mathbb{R}$ so that $f(t)=\int_{0}^{+\infty}e^{-tx}\frac{\sin x}{x}dx, t\in\mathbb{R}$ exists
$$f(t)=\int_{0}^{+\infty}e^{-tx}\frac{\sin x}{x}dx, t\in\mathbb{R}$$
I need to find out for which $t \in \mathbb{R}$ this integrals exists (meaning it doesn't diverge) as Riemann-integral at first and then as Lebesgue... | The Cauchy criterion can be helpful. The improper integral $\displaystyle \int_0^\infty f(x) \, dx$ exists and is finite if and only if $$\displaystyle \lim_{n,m \to \infty} \int_n^m f(x) \, dx = 0.$$
Let $t < 0$. Integrate over a half-period of the $\sin$ function to estimate
$$\int_{2k\pi}^{2k\pi + \pi} e^{-tx} \frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2980797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing an orthogonalisation process Can anyone show that:
$\mathbf{a_{\perp}}=\mathbf{a}-\frac{\mathbf{x}\mathbf{x}^T}{\mathbf{x}^T\mathbf{x}}\mathbf{a}$,
$\mathbf{a}\in\mathbb{R}^N$, $\mathbf{x}=(1,1,\dots,1)^T\in\mathbb{R}^N$
results in $\mathbf{a}$ becoming orthogonal to $\mathbf{x}$?
Thank you for your help.
| We have that
$$\left(\mathbf{a}-\frac{\mathbf{x}\mathbf{x}^T}{\mathbf{x}^T\mathbf{x}}\mathbf{a}\right)\cdot \mathbf{x}=\mathbf{a}^T\mathbf{x}-\mathbf{a}^T\frac{\mathbf{x}\mathbf{x}^T}{\mathbf{x}^T\mathbf{x}}\mathbf{x}=\mathbf{a}^T\mathbf{x}-\mathbf{a}^T\mathbf{x}=0$$
Refer also to the related
*
*Writing projection i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2980906",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Relationship between $\langle \nabla f(\overline{x}), x - \overline{x}\rangle > 0$ and the the minimality of $\overline{x}$. Say $f$ is a differentiable function over a convex set $X$ with $\langle \nabla f(\overline{x}), x - \overline{x}\rangle > 0$ for all $x, \overline{x} \in X$ such that $x \neq \overline{x}$. Can ... | Assuming you meant that $\langle \nabla f(\bar{x}), x - \bar{x}\rangle > 0, \forall x \neq \bar{x}$, this should imply a local minimum. From the differentiability of $f$, you know that near $\bar{x}$:
$$
f(x) = f(\bar{x}) + \langle \nabla f(\bar{x}), x - \bar{x} \rangle + o(\| x - \bar{x}\|),
$$
where $o(\| x - \bar{x}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2981014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is the optimal way to take $16$ question true/false test with four attempts? Question:
Suppose a student is taking a $16$-question true/false test with four attempts. They must keep the store that they obtain after the fourth trial.
He or she does not know the answer to any question. After the test is completed, t... | Here is one strategy that beats yours:
Attempt 1: Give random answers to questions 1-5 and leave the rest blank.
Attempt 2: Give random answers to questions 6-10 and leave the rest blank.
Attempt 3: Give random answers to questions 11-15 and leave the rest blank.
Final test: For questions 1-5 give the same answers as i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2981136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Seeking methods to solve $ I = \int_{0}^{\infty} \frac{\sin(kx)}{x\left(x^2 + 1\right)} \:dx$ I am currently working on an definite integral that requires the following definite integral to be evaluated.
$$ I = \int_{0}^{\infty} \frac{\sin(kx)}{x\left(x^2 + 1\right)} \:dx$$
Where $k \in \mathbb{R}^{+}$
I was wondering... | The method I took was:
Let
$$ I(t) = \int_{0}^{\infty} \frac{\sin(kxt)}{x\left(x^2 + 1\right)} \:dx$$
Take the Laplace Transform
\begin{align}
\mathscr{L} \left[I(t) \right]&= \int_{0}^{\infty} \frac{\mathscr{L} \left[\sin(kxt)\right]}{x\left(x^2 + 1\right)} \:dx \\
&= \int_{0}^{\infty} \frac{kx}{\left(k^2x^2 + s^2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2981245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
How to rectify "time lost " by pendulum clock by alteration in its length? question:
If a clock loses $5$ seconds per day ,what is the alteration required in the length
of pendulum in order that the clock keeps correct time
$(a)\dfrac{4}{86400} $times its original length be shortened
$(b)\dfrac{1}{86400}$ times its o... | $$1-\left(\dfrac{86395}{86400}\right)^2 = 1-\left(1-\dfrac{5}{86400}\right)^2 = 1-\left(1-\dfrac{10}{86400}+\dfrac{5^2}{86400^2}\right) = \dfrac{1}{8640}-\dfrac{1}{298598400}$$ which suggests to me that you might be expected to give answer $(b)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2981438",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
about the p-adic logarithm It is well known that we can define the logarithm in the p-adic setup by the usual power series that converges in $B(1,1^-)$. In Schickoff "Ultrametric calculus" there is an extension of the $log$ from the unit ball to $C_p^{\times}$ (called $LOG$) and it is proved that it is locally analyt... | Sure. Local analyticity can be judged solely from the behavior at the identity, $1$. And there, the function is given by the series that you know.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2981602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a $4$-component link such that upon removing any one of them you get the Borromean link? Is there a $4$-component link such that upon removing any one of them you get the Borromean link?
I've managed to get close but not quite. What I have gets me something similar to the Borromean link but two of the componen... | You can sort of think of the Borromean rings as lying on three faces of a tetrahedron, with the center triangle of the usual presentation at a vertex. By adding a component corresponding to the fourth face in a way so that the link has tetrahedral symmetry, you get the following:
While the outer component looks funny... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2981717",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
$\infty\cdot 0$ Indetermination without L'Hopital? Evaluate $\lim_{n\to\infty}n\cdot r^n$ , being $0<r<1$.
I dont know if I took the proper steps, but I get to this point:
$$\lim_{n\to\infty}n\cdot r^n=\lim_{n\to\infty}n \lim_{n\to\infty}r^n = \infty \cdot 0 $$
I dont know how to solve this indetermination without L'... | You want $\lim_{n\to\infty}n\exp -cn$ with $c:=-\ln r>0$. Since $\int_0^\infty n\exp -cn\operatorname{d}n=c^{-2}$ is finite, the limit is $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2981843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 8,
"answer_id": 3
} |
About the C*-algebra of the Schrodinger representation of the Weyl C*-algebra We start with the Weyl C*-algebra $\mathcal{W}$ for a finite dimensional symplectic space and we consider the irreducible Schrodinger representation $\pi:\mathcal{W}\rightarrow \mathcal{B}(\mathcal{H})$ where $\mathcal{H}=L^2(\mathbb{R}^n)$. ... | $\pi(\mathcal{W})$ is strictly smaller than $B(H)$.
Since the Weyl algebra $\mathcal{W}$ is simple, every nontrivial representation is faithful. Therefore $\pi:\mathcal{W}\rightarrow\pi(\mathcal{W})\subseteq B(H)$ is a bijective *-homomorphism (surjective to its image), hence $\pi(\mathcal{W})\cong\mathcal{W}$ as C${}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2981977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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