Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Ask for different ways to solve $(x+y)dy+(y+1)dx=0$ $(x+y)dy+(y+1)dx=0$
Rewrite the equation to
$\frac{dx}{dy}+\frac{x}{y+1}=\frac{-y}{y+1}$
I can use used the integrating factor $μ(y)=e^{-\int\frac{1}{y+1}dy}$ to solve it.
The answer is $x=\frac{-y^2}{2(y+1)}+\frac{C}{y+1}$
I'm curious that are there any other metho... | See that we can write as
$$(x+y)dy+(y+1)dx=0$$
$$xdy+ydx+ydy+dx=0$$
$$d(xy)+ydy+dx=0$$ now just integrate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1548761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to show $1 +x + x^2/2! + \dots+ x^{2n}/(2n)!$ is positive for $x\in\Bbb{R}$?
How to show $1 + x + \frac{x^2}{2!} + \dots+ \frac{x^{2n}}{(2n)!}$ is positive for $x\in\Bbb{R}$?
I realize that it's a part of the Taylor Series expansion of $e^x$ but can't proceed with this knowledge? Also, I can't figure out the sign... | If we use Taylor's theorem with an integral remainder we get:
$$ 1+x+\frac{x}{2}+\ldots+\frac{x^{2n}}{(2n)!} = e^x-\frac{1}{(2n)!}\int_{0}^{x}t^{2n}e^{x-t}\,dt \tag{1}$$
hence in order to prove the non-negativity of the LHS it is enough to show that:
$$ \int_{0}^{x} t^{2n}e^{-t}\,dt \leq (2n)! \tag{2}$$
holds for any $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1548882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Proving there is no negative real number greater that all negative rational numbers How would I use the Archimedean property to prove that there is no negative real number greater that all negative rational numbers?
| Let $x$ be any negative real number. Then $-x$ is positive, and by the Archimedean property there is a natural number $n$ such that $0<-x<n$.
Then $-\frac 1n$ is a negative rational number, and
$$x<-\frac 1n<0$$
Thus $x$ is not greater than all negative rational numbers. Since $x$ are arbitrary, there is no negative re... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1548994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Odds of rolling a $6$ in three chances I'm working on a dice game to play with the kids I work with. I'm trying to recall calculating probability but its been a while. Its for a table top DND type game and they are rolling for the properties of their character.
What are the odds of rolling a $6$ with a D6 given $3$ cha... | You can most easily calculate this with the counter probability.
What is the probability of rolling no 6.
$$P(\text{no } 6) = \frac 56 \frac 56 \frac 56=\frac{125}{216}$$
Then the probability of rolling at least one 6 (which I think is the outcome you're interested) is
$$P(\text{ at least one 6}) = 1 -\left(\frac 56\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1549096",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Does the Dirac belt trick work in higher dimensions? If the Dirac belt is in 4-space, is it still true that when the belt is initially given a 360 degree twist then it cannot be untwisted?
I assume this is so because SO(n) is not simply connected, but I am still a bit fuzzy on how this all works. (For instance, even i... | One can show the following: Given a noncompact connected surface $S$, any two smooth embeddings $f_0, f_1: S\to R^4$ are isotopic. (In particular, there are no twisted bands in $R^4$.) Proofs are a bit involved, but the key fact that for a generic smooth homotopy $F: S\times [0,1]\to R^4$ between $f_0, f_1$, for each $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1549164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Prove that $\inf{U(f,P)}\geq 1/2$
Define $f(x)=\begin{cases} x\;\;\;\text{if the point $x\in[0,1]$ is rational}\\ 0\;\;\;\text{if the point $x\in[0,1]$ is irrational} \end{cases}$
Prove that $\inf{U(f,P)}\geq 1/2$.
Let $P=\{x_1,x_2,\cdots,x_i\}$ be a partition of $[0,1]$ with $x_i=\frac{i}{n}$ where $i\geq 1$ and $n... | Let $\mathcal P_0$ be a partition of $[0,1]$. Refine this partition if necessary to a partition $\mathcal P$ containing $n+1$ points so that $x_{i}-x_{i-1}=1/n.\ $Note that $x_0=0$, $x_{n}=1$ and in general $x_i=i/n$.
We have $U(\mathcal P_0)\geq U(\mathcal P)=\frac{1}{n}\sum_{i=1}^{n}f(x_i^{*})$.
Now let $\epsilon >0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1549291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
Show that the curve has two tangents I'm a little stuck on a math problem that reads as follows:
Show that the curve $x = 5\cos(t), y = 3\sin(t)\, \cos(t)$ has two tangents at $(0, 0)$ and find their equations
What I've Tried
*
*$ \frac{dx}{dt} = -5\sin(t) $
*$ \frac{dy}{dt} = 3\cos^2(t) - 3\sin^2(t) $ because of... | You computed the slopes as $\frac{dy}{dx}=\frac{3}{5}$ and $\frac{dy}{dx}=-\frac{3}{5}$, but for some reason when you wrote the equations of the tangent lines you took the reciprocal of these slopes and wrote $y=\frac{5}{3}x$ and $y=-\frac{5}{3}x$. They should be $y=\frac{3}{5}x$ and $y=-\frac{3}{5}x$. Other than that,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1549432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
How many 4 digit numbers are divisible by 29 such that their digit sum is also 29? How many $4$ digit numbers are divisible by $29$ such that their digit sum is also $29$?
Well, answer is $5$ but what is the working and how did they get it?
| Just to give a different take on things, in order for a number to be a multiple of $29$ and have digit sum congruent to $2$ mod $9$, the number must be of the form
$$29(9k+1)=261k+29$$
For the number to have four digits, we need $1000\le261k+29\le9999$, which requires $4\le k\le38$.
Now for the digit sum to be not j... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1549552",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
does uncorrelated imply independence for 2 valued random variables? Let $(X_n)_{n \geq 1}$ be identically distributed random variables such that $\mathbb{P}[X_1 = 1] = \mathbb{P}[X_1 = -1] = \frac{1}{2}$ and for any $i_1< \cdots < i_p, \mathbb{E}[X_{i_1} \cdots X_{i_p}] = 0$ for any $p \in \mathbb{N}.$ Can I conclude ... | I think I solved it. First of all we shift and rescale the random variables, so we denote by $Y_i = \frac{X_i + 1}{2}.$ It is not difficult to check that $\mathbb{E}[Y_{i_1} \cdots Y_{i_p}] = \mathbb{E}[Y_{i_1}] \cdots \mathbb{E}[Y_{i_p}] $ and $\mathbb{P}[Y_i = 0] = \mathbb{P}[Y_i = 1] = \frac{1}{2}.$
But on the oth... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1549655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Give me hints for evaluating this limit Evaluate this limit:
$$\lim_{n\to\infty}{\left\{\left(1+\frac{1}{n}\right)^n-\left(1+\frac{1}{n}\right)\right\}}^{-n}$$
Please give me some hints. If you provide a complete answer instead, please include a spoiler tag.
| Hint.
$$\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n=e^x=\exp(x)$$
From here on you can easily conclude the result as you mentioned in the comments of your post.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1549779",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Prove the following limits without using l'Hospital and Sandwich theorem Prove the following limits:
$$\lim_{x \rightarrow 0^+} x^x = 1$$
$$\lim_{x \rightarrow 0^+} x^{\frac{1}{x}}=0$$
$$\lim_{x \rightarrow \infty} x^{\frac{1}{x}}=1$$
They are not that hard using l'Hospital or the Sandwich theorem. But I curious if the... | Using $a^b=\exp(b\ln a)$ as definition of exponentiation with irrational exponents, it is natural to take logarithms; then the claims are equivalent to
$$\lim_{x\to 0^+}x\ln x=0,\qquad \lim_{x\to 0^+}\frac 1x\ln x=-\infty, \qquad \lim_{x\to \infty}\frac 1x\ln x=0.$$
Substituting $x=e^{-y}$ for the first two and $x=e^y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1549843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Compute $\lim_{n\to\infty}n^{-(n+\nu)}\sum_{k=1}^nk^k$ for $\nu\in\Bbb R$
Let $\nu\in\Bbb R$ and
$$S_{\nu}:=\lim_{n\to\infty}\frac{1}{n^{n+\nu}}\sum_{k=1}^nk^k$$
What is the value of $S_{\nu}$?
Disclaimer: This is not a Homework question. Indeed, I thought at this problem when reading this question.
We have
$$ n... | As pointed out in a comment, we have
\begin{align*}n^n\ \leq\ \sum_{k=1}^nk^k \ &\ \leq n^n+(n-1)^{n-1}+\sum_{k=1}^{n-2}k^k\leq n^n+(n-1)^{n-1}+\overbrace{(n-2)(n-2)^{n-2}}^{=(n-2)^{n-1}} \\ &\leq n^n+2(n-1)^{n-1}\leq n^n+2n^{n-1}=n^n\Big(1+\frac{2}{n}\Big) \qquad \forall n\in\Bbb N.
\end{align*}
So that
$$n^{-\nu}\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1549944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
5 geometric shapes, all touching each other I was playing aroud with shapes, which all connected. I managed to get 3 and 4 shapes all connected to each other, but I can't get 5 to work in 2D.
Does anyone have an idea what these shapes are called and also how to get 5 shapes connected? It would be the best, if all shap... | It is impossible to make five 2D shapes (on a flat plane) which all touch each other. This is a consequence of the four color map theorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1550054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Every field has an algebraic closure proof Below is the proof in Fraleigh.
The part that I don't understand is the construction of set $A$. What does "every possible zero of any $f(x)$" mean? When the textbook talked about "zeros", they are always defined in a larger field, now there is no such single extension fields... | The set $A$ is not a field it is just a sufficiently large set. For example one could define it slightly more formally as $A = \bigcup \{(f,i)\,|\, f \in F[x];i \in \{0,...,\text{degree}(f)-1\}\}$. Once again for example one could define $\Omega = P(A) \cup F$ (where $P(A)$ denotes the power set of $A$). The idea is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1550127",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
proving the validity I need to prove the validity of the formula:
$Q= \forall x \forall y \forall v \ F(x,y,f(x,y),v, g(x,y,v)) \rightarrow \forall x \forall y \exists z \forall v \exists u \ F(x,y,z,v,u)$
I thought the best way to do this is by proving that $\neg Q$ is unsatisfiable and I tried the Tableaux Method but... | If I remember my tableaus, you get $\forall x,y,zF(\ldots)$ (the antecedent) and $\neg\forall x,y\exists z\ldots$ (the negation the of consequent).
You should have a rule that allows you to conclude
$$¬∃z∀v∃uF(a,b,z,v,u)$$
for some $a,b$. To reduce $¬∃z$ we need something to apply it to. The terms we have lying aroun... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1550345",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Integrate $\int \frac{\arctan\sqrt{\frac{x}{2}}dx}{\sqrt{x+2}}$ $$\int \frac{\arctan\sqrt{\frac{x}{2}} \, dx}{\sqrt{x+2}}$$
I've tried substituting $x=2\tan^2y$, and I've got:
$$\frac{1}{\sqrt2}\int\frac{y\sin y}{\cos^4 y} \, dy$$
But I'm not entirely sure this is a good thing as I've been unable to proceed any further... | the taylor series of $$\frac{\sin y}{\cos^4y}=\sum_{n=1}^{\infty }\frac{(2(2n-1)!-1)y^{2n-1}}{(2n-1)!}$$
so
$$\frac{1}{\sqrt2}\int\frac{y\sin y}{cos^4 y}dy=\frac{1}{\sqrt2}\int\sum_{n=1}^{\infty }\frac{(2(2n-1)!-1)y^{2n}}{(2n-1)!}dy$$
$$=\frac{1}{\sqrt2}\sum_{n=1}^{\infty }\frac{(2(2n-1)!-1)y^{2n+1}}{(2n+1)(2n-1)!}+C$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1550405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
Is Peirce's law valid? Could anyone help me see how Peirce's law ((P→Q)→P)→P is valid?
It seems to me that from (P→Q), P need not follow.
e.g: assume p=pigs can fly, q=1+1=2
Then if (if pigs can fly → 1+1=2) then pigs can fly
since the first part is a valid inference (1+1=2) is always true how can P follow?
| $$((P \implies Q) \implies P) \implies P$$
$$((\text{Pigs can fly} \implies 1+1=2) \implies \text{Pigs can fly}) \implies \text{Pigs can fly}$$
$$((\text{False} \implies \text{True}) \implies \text{False}) \implies \text{False}$$
$$(\text{True} \implies \text{False}) \implies \text{False}$$
$$\text{False} \implies \tex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1550479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Map from $\mathbb{A}^1 \rightarrow \mathbb{A}^2$ Let the map $\varphi_n:\mathbb{A}^1 \rightarrow \mathbb{A}^2$ be defined by $t\rightarrow(t^2,t^n)$.
-Show that if n is even, the image of $\varphi_n$ is isomorphic to $\mathbb{A}^1$ and $\varphi_n$ is 2:1 away from 0.
-Show that if n is odd, $\varphi_n$ is bijective, a... | For the even case, you are showing that the image is isomorphic to $\mathbb{A}^1$. It is not necessary (and not true) that $\varphi_n$ is the isomorphism. In fact, your answer almost contains the map from $\mathbb{A}^1$ to $\varphi_n(\mathbb{A}^1)$ and the map in the other direction. (You will need to check that they a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1550605",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
$\sum_{n=1} ^\infty \frac{8}{n15^n}$ without calculator I just can't figure out which function to use to get the sum. I tried with ln, but that gives me an alternating series.
| The exchange between integral and series is justified because the convergence is uniform when far from $1$.
$$
\sum_{n=1} ^\infty \frac{8}{n15^n}
=8\,\sum_{n=1}^\infty \int_0^{1/15}t^{n-1}\,dt
=8\,\int_0^{1/15}\left(\sum_{n=1}^\infty t^{n-1}\right)\,dt\\
=8\,\int_0^{1/15}\frac{dt}{1-t}=-8\,\log(1-t)\left.\vphantom{\in... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1550738",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluate the Integral: $\int e^{2\theta}\ \sin 3\theta\ d\theta$ $\int e^{2\theta}\ \sin 3\theta\ d\theta$
After Integrating by parts a second time, It seems that the problem will repeat for ever. Am I doing something wrong. I would love for someone to show me using the method I am using in a clean and clear fashion. T... | When you do it with integration by parts, you have to go in the "same direction" both times. For instance, if you initially differentiate $e^{2 \theta}$, then you need to differentiate $e^{2 \theta}$ again; if you integrate it, you will wind up back where you started. If you do this, you should find something of the fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1550841",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Evaluating $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ with a rectangular contour I need to try to evaluate $\int_{0}^{\infty}\frac{\sin(ax)}{\sinh(x)}dx$ and it seems like this is supposed to be done using some sort of rectangular contour based on looking at other questions.
My main issue is that I am unsure how t... | To answer your third question, you can do whatever you want, which we will see later. Because we want to evaluate the integral over the real axis, we will include $\mathbb{R}$ (except $z=0$) in the contour. For this integral, this is possible, maybe not in other integrals. Because $\sinh(z)=0$ for all integer multiples... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1550932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Higher homotopy groups meaning I am developing intuition for higher homotopy groups but it's very hard for me to visualize what $\pi_2$ represents (and $\pi_n$ for that matter). I know that $\pi_2(S^2) \cong \mathbb{Z}$ and can kind of see "wrapping" a sphere around itself an integer number of times. But, something l... | The question in its current form is too general. However, the case $\pi_n(S^{n-1})$ is relatively easy and was discovered first.
In Milnor's Topology from differential viewpoint, you can find a quite intuitive explanation of the fact that elements of $\pi_3(S^2)$ correspond to framed cobordism classes of $1$-dimension... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1551050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Understanding Vacuously True (Truth Table) I don't know very much about formal logic, and I'm trying to understand the concept of vacuously true statements. Consider the truth table below:
$$\begin{array} {|c|}
\hline
P & Q & P\implies Q & Q\implies P & P\iff Q \\ \hline
T & T & T & T & T \\ \hline
T & F & F & \color... | Here is why we say $P \implies Q$ is true if $P$ is false and $Q$ is true:
Let $P$ be the statement "it's raining outside" and $Q$ be the statement "the car is wet".
In order for $P \implies Q$ to be true, what needs to happen is: every time it rains outside, it better follow that the car is wet. That's all you need t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1551320",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 0
} |
Simplifying $2\cos(t)\cos(2t)-\sin(t)\sin(2t)$ How do I simplify $2\cos(t)\cos(2t)-\sin(t)\sin(2t)$? I know this should be possible, but I don't know how.
I have tried the $\cos(t)\cos(u)-\sin(t)\sin(u)=\cos(t+u)$, but I don't know what to do with the $2$ in front of $\cos(t)$.
| $$2\cos(t)\cos(2t)-\sin(t)\sin(2t)=\frac{\cos(t)+3\cos(3t)}{2}$$
Proof:
$$2\cos(t)\cos(2t)-\sin(t)\sin(2t)=\frac{\cos(t)+3\cos(3t)}{2}\Longleftrightarrow$$
$$2\left(2\cos(t)\cos(2t)-\sin(t)\sin(2t)\right)=\cos(t)+3\cos(3t)\Longleftrightarrow$$
$$2\left(\cos(-t)+\cos(3t)-\sin(t)\sin(2t)\right)=\cos(t)+3\cos(3t)\Longlef... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1551410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Complex equation: $z^8 = (1+z^2)^4$ What's up with this complex equation?
$ z^8 = (1+z^2)^4 $
To start with, there seems to be a problem when we try to apply root of four to both sides of the equation:
$ z^8 = (1+z^2)^4 $
$ z^2 = 1 + z^2 $
which very clearly doesn't have any solutions, but we know there are solutions: ... | The solutions of $z^8 = (1+z^2)^4$, and the problem you see, are clearly seen from a factorization process:
$$\begin{align}
z^8 - (1+z^2)^4 &= 0 \,, \\
[z^4 + (1+z^2)^2] [z^4 - (1+z^2)^2] &= 0 \,, \\
[z^2 + i(1+z^2)] [z^2 - i (1+z^2)] [z^2 - (1+z^2)] [z^2 + (1+z^2)] &= 0 \,.
\end{align}$$
From the last line, it is cle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1551522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 3
} |
Prove that functional equation doesn't have range $\Bbb R.$ Prove that any solution $f: \mathbb{R} \to \mathbb{R}$ of the functional equation
$$ f(x + 1)f(x) + f(x + 1) + 1 = 0 $$
cannot have range $\mathbb{R}$.
I transformed it into
$$ f(x) = \frac {-1} {f(x + 1)} - 1 = \frac {-1 - f(x + 1)} {f(x + 1)} $$
I tried to... | Assume that the range of $f$ is all of $\Bbb R$. Then there are $a,b\in\Bbb R$ with $f(a)=0$ and $f(b)=-1$. But then
$$ 1=f(a)f(a-1)+f(a)+1=0$$
and
$$ 1=f(b+1)f(b)+f(b+1)+1=0,$$
both of which are absurd.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1551600",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Is This Matrix Diagonalizable? Consider the matrix A below:
$$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 & 0 & -2 \\ 1 & 2 & 4\end{pmatrix}$$
Is the matrix A diagonalizable? If so, then find a diagonal matrix D and an invertible matrix P such that $P^{-1}AP=D$.
I know it is supposed to be diagonalizable but I have tried to so... | Assuming $A$ is an $n\times n$ matrix, $A$ diagonalizable if $P_A(\lambda)$ has $n$ distinct real roots or if, for each eigenvalue, the algebraic multiplicity equals the geometric multiplicity. So let's find out:
$$P_A(\lambda) = det(A - \lambda I) = det \begin{pmatrix} 3-\lambda & 2 & 2 \\ -1 & -\lambda & -2 \\ 1 & 2 ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1551703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Regarding ranks of row echelon form matrices How do I prove that the rank of a matrix in reduced row echelon form is equal to the number of non-zero rows it has?
| By definition of row echelon form, all the non-zero rows are linearly independent. (Think on all the non-zero leading coefficients which are "aligned" ).
So the non-zero rows form a basis for the row space (i.e the subspace spanned by all the rows). Therefore their number equals to the dimension of this space, as requ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1551819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Negative binomial distribution.Calculate the probability that exactly 12 packages are inspected. In a shipment of 20 packages, 7 packages are damaged. The packages are randomly inspected, one at a time, without replacement, until the fourth damaged package is discovered.
Calculate the probability that exactly 12 packag... | The distribution is not negative binomial, for in the negative binomial the trials are independent. Here we are not replacing after inspecting. The resulting distribution is sometimes called negative hypergeometric, but the term is little used.
We can use an analysis close in spirit to the one that leads to the negativ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1551940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Divisivility and Sylow p-subgroups Hello I have been working in this problem for two days but i can't get the answer, I would appreciate any help or hint.
Let $p$ be prime, $m\ge 1$, $r \ge 2$ and $(p,r) = 1$.
If there is a simple group $G$ such that $ |G| = p^ {m}r $, then $ p^ {m}|(r-1)! $
| Pick a Sylow $p$-subgroup $P$ and let $G$ act on $G/P$ by left multiplication: $p\cdot gP = pgP$. This gives a homomorphism $G\to S(G/P)=S_r$, which is clearly not identically equal to $1$. Since $G$ is simple, the kernel is thus trivial and hence $S_r$ contains an isomorphic copy of $G$ as a subgroup. So $p^m r| r!$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1552051",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to prove $\sum_p p^{-2} < \frac{1}{2}$? I am trying to prove $\sum_p p^{-2} < \frac{1}{2}$, where $p$ ranges over all primes. I think this should be doable by elementary methods but a proof evades me.
Questions already asked here (eg. What is the value of $\sum_{p\le x} 1/p^2$? and Rate of convergence of series of... | Here's a solution that exploits a comment of Oscar Lanzi under my other answer (using an observation that I learned from a note of Noam Elkies [pdf]). In particular, it avoids both the identity $\sum_{n \in \Bbb N} \frac{1}{n^2} = \frac{\pi^2}{6}$ and using integration.
Let $\Bbb P$ denote the set of prime numbers and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1552136",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 5
} |
Primes on the form $p^2-1$ Prove that there exists a unique prime number of the form $p^2 − 1$ where $p\geq 2$ is an integer.
I have no idea how to approach the question. any hints will be greatly appreciated
| $n(x) = n^2-1$
$n(x) = n^2-1^2$
$n(x) = (n+1)(n-1)$
For it to be prime it has to be only be divisible by itself. What to do from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1552283",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
} |
Prove $x_n$ converges IFF $x_n$ is bounded and has at most one limit point I'm not entirely sure how to go about proving this so hopefully someone can point me in the right direction. The definition I have for a limit point is "$a$ will be a limit point if for a sequence $x_n$ there exists a subsequence $(x_{n_k})$ suc... | First show: a point $x_0$ is a limit point of a sequence $\{x_n\}$ if and only if for every $\varepsilon>0$, there exists an open ball $B(x_0,\varepsilon)$ centered at $x_0$ with radius of $\varepsilon$ that contains infinite number of $x_n$.
In the leftward direction of the original problem, let $x_0$ be the unique li... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1552411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
The probability of two spies on the same team, and the probability after switching? I was watching a TV show, and the setting is pretty simple:
There are 10 players in total, within them there are 2 "spies", players initially form 2 teams of 5 people, and there are 3 rounds of game, where at the end of each round, we ... | If a player is switched at random (i.e., disregarding the element of politics and trickery), then the probability should certainly remain the same.
Consider shuffling a deck of cards, and suppose you "win" if the ace of spades is on top. Before looking at the top card, you're allowed to make any two cards switch place... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1552498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Finding extrema of $\frac{\sin (x) \sin (y)}{x y}$ I need to find the extrema of the following function in the range $-2\pi$ and $2\pi$ for both $x$ and $y$, but I don't know how to go about doing it since it's a bit weird and not similar to other functions I've seen:
$$f(x,y)=\frac{\sin (x) \sin (y)}{xy}$$
I've evalua... | As said in comments, if $$F=\frac{\sin (x) \sin (y)}{xy}$$ $$F'_x=\frac{\cos (x) \sin (y)}{x y}-\frac{\sin (x) \sin (y)}{x^2 y}=\frac{\sin (y) (x \cos (x)-\sin (x))}{x^2 y}$$ $$F'_y=\frac{\sin (x) \cos (y)}{x y}-\frac{\sin (x) \sin (y)}{x y^2}=\frac{\sin (x) (y \cos (y)-\sin (y))}{x y^2}$$ where we see appearing the s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1552622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Winning Strategy with Addition to X=0 Problem:
Two players play the following game. Initially, X=0. The players take turns adding any number between 1 and 10 (inclusive) to X. The game ends when X reaches 100. The player who reaches 100 wins. Find a winning strategy for one of the players.
This is my solution, which ho... | Community wiki answer so the question can be marked as answered:
Yes, you are correct. Also note the links in the comments for further reading about similar games and a systematic way of solving them.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1552729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Unique solution of quadratic expression I have a quadratic expression: $x^HAx - B$ in which I have to solve for $x$. $A$ is a 4 x 4 positive definite hermitian matrix and $x$ is a 4 x 1 vector. I have solved it by considering Singular Value Decomposition of A as well as Cholesky decomposition of A but each time I have ... | If I understand correctly, you want to find $x$ such that $x^HAx = B$, where $B \geq 0$ is a real number. Let $v_1, \ldots, v_4$ be an orthonormal basis of eigenvectors of $A$ with $Av_i = \lambda_i v_i$. Write $x = a_1 v_1 + \ldots + a_4 v_4$. Then you have
$$ x^HAx = \left< Ax, x \right> = \sum_{i=1}^4 \lambda_i |a_i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1552904",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability Deck of Cards In a hand of 13 playing cards from a deck of 52 whats the probability of drawing exactly one king.
My approach would be $${4 \choose 1}*{48 \choose 12}/{52 \choose 13}*{2}$$
I divided by 2 because I felt I had ordered the king and the other 12 cards chosen but this is wrong. Can someone please... | you have two disjoints subsets: 'kings' and 'everything else'. $\binom{4}{1}\binom{48}{12}$ simply means 'any 1 out of 4' AND 'any 12 out of 48'. There's no order involved. Any form of order would be if, for example, you would need to get 3 kings. Then you would need to divide by $3!$ because the order doesn't matter. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Are Morrey spaces reflexive? Since $L^{p,0}=L^p$ and $L^1$ is not reflexive, thus in general Morrey space is not reflexive, but how about for $L^{p,\lambda}$ with $1<p<+\infty$ and $0<\lambda<n$, where $n$ is the dimension of domain.
What's more, it seems that the dual space for Morrey spaces are not clear so far?
| For any function space defined as "some supremum is finite" without further conditions like continuity or vanishing, you should expect that:
(a) the space is nonreflexive and nonseparable;
(b) its dual is too large to be described in concrete terms.
The reason for both things is that $\ell_\infty$ embeds into the space... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553083",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
limit of $n \left(e-\sum_{k=0}^{n-1} \frac{1}{k!}\right) = ?$ As in the title:
$$
\lim_{n\to\infty} n \left(e-\sum_{k=0}^{n-1} \frac{1}{k!}\right) = ?
$$
Numerically, it seems 0, but how to prove/disprove it?
I tried to show that the speed of convergence of the sum to e is faster than $1/n$ but with no success.
|
I tried to show that the speed of convergence of the sum to e is faster than $1/n,$ but with no success.
Have you tried Stirling's approximation ?
In my opinion, a far more interesting question would have been trying to prove that
$$\lim_{n\to\infty}(n+a)\bigg[~e^b-\bigg(1+\dfrac b{n+c}\bigg)^{n+d}~\bigg]~=~b~e^b~... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553173",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 4
} |
Is there a sequence such that $a_n\to0$, $na_n\to\infty$ and $\left(na_{n+1}-\sum_{k=1}^n a_k\right)$ is convergent? Is there a positive decreasing sequence $(a_n)_{n\ge 0}$ such that
${\it i.}$ $\lim\limits_{n\to\infty} a_n=0$.
${\it ii.}$ $\lim\limits_{n\to\infty} na_n=\infty$.
${\it iii.}$ there is a real $\ell$ ... | Assume $\{a_n\}$ exists.
Let $a_n = f (n)$. By i and ii, we have $f (n) = o (1)$ and $f (n) = \Omega (1/n)$. By Euler-McLaurin summation formula, we have $$\sum_{k = 1}^{n} f (k) = c_1 F (n) + o (F (n))$$ for a constant $c_1$, where $F$ is anti-derivative of $f$, and $F (n) = o (n)$ and $F (n) = \Omega (1)$. Then, by i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
"answer_count": 3,
"answer_id": 2
} |
Show that linear transformation is not surjective? Given the matrix, $$M =
\begin{bmatrix}1&7&9&3\\2&15&19&8\\7&52&66&27\\3&4&10&-24\end{bmatrix}$$
Show that the linear transformation $T_m: \mathbb R^4 \to \mathbb R^4$ defined by the multiplication of column vectors of the left by $M$ is not surjective by exhibiting a ... | A very important identity that you may have seen in your course so far is $$\text{Im}(M)^{\perp}=\text{Ker}\left(M^{T}\right).$$
This is true because if $\vec{y}\in\text{Im}(A)^\perp$ then by definition it
satisfies $\langle\vec{y}\cdot M\vec{x}\rangle=0$
for all $\vec{x}$ and so $$\langle\vec{y}\cdot M\vec{x}\ra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How can we choose $z$ to make this series absolute convergent? I have the following series:
$\sum_{n=0}^\infty \frac{n^{\ln (n)}}{3^n} (z+2)^n$
The question is, how can we choose $z \in \mathbb{C}$ to make this absolute convergent?
For the first sight it seemed impossible to have such $z$, since $n^{ln n}$ should be wa... | The reciprocal of the radius of convergence, as given by the Cauchy-Hadamard Theorem is
$$
\begin{align}
\limsup_{n\to\infty}\left(\frac{n^{\log(n)}}{3^n}\right)^{1/n}
&=\frac13\limsup_{n\to\infty}n^{\log(n)/n}\\
&=\frac13\limsup_{n\to\infty}e^{\log(n)^2/n}\\
&=\frac13e^{\limsup\limits_{n\to\infty}\log(n)^2/n}\\[3pt]
&... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Does this type of "cyclic" matrix have a name? Let $\{a_1, a_2, ..., a_n\} \subset \mathbb{C}$ and consider the matrix of the form
$$
\begin{bmatrix}
a_1 & a_2 & ... & a_{n-1} & a_n\\
a_2 & a_3 & ... & a_n & a_1\\
.\\
.\\
.\\
a_n & a_1 & ... & a_{n-2} & a_{n-1}
\end{bmatrix}
$$
Does this type of matrix have... | Yes, it is a type of Circulant Matrix.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553662",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Determine the sign of the integral $\int_0^\pi x\cos x\,dx$ without calculating it
Without explicitly evaluating, determine the sign of the integral $$\int_0^{\pi} x\cos(x)dx$$
I know $x\cos(x) > 0$ when $0 < x < {\pi}/2$ and $x\cos(x) < 0$ when $\pi/2 < x < \pi$, and, in fact, the end result is negative, but I'm uns... | $$\int_0^{\pi} x\cos(x)dx=\int_0^{\frac \pi 2} x\cos(x)dx+\int_{\frac \pi 2} ^{\pi} x\cos(x)dx$$
let $u=\pi-x$ in the second integral
$$\int_0^{\pi} x\cos(x)dx=\int_0^{\frac \pi 2} x\cos(x)dx - \int_0^{\frac \pi 2} (\pi-u)\cos(u)du$$
$$=\int_0^{\frac \pi 2} (2u-\pi)\cos(u)du$$
which is negative because $\cos(u)\ge 0$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 1
} |
Recurrence relation for the determinant of a tridiagonal matrix Let
$$f_n := \begin{vmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
& & \ddots & \ddots & b_{n-1} \\
& & & c_{n-1} & a_n
\end{vmatrix}$$
Apparently, the determinant of the tridiagional matrix above is given by the recurrence relation
$... | For $n \ge 2$ using Laplace expansion on the last row gives
\begin{align}
f_n &=
\begin{vmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
& & \ddots & \ddots & b_{n-3} \\
& & & c_{n-3} & a_{n-2} & b_{n-2} \\
& & & & c_{n-2} & a_{n-1} & b_{n-1} \\
& & & & & c_{n-1} & a_n
\end{vmatrix}
\\
&=
(-1)^{2n-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
Evaluate $\int \frac{x+1}{(x^2-x+8)^3}\, dx$ Could you give me a hint on how to find $$\int \frac{x+1}{(x^2-x+8)^3}\, dx$$
It doesn't seem like partial fractions are the way to go with here and using the integration by parts method seems to be tedious.
I have also tried substituting $(x^2-x+8)$ but it gets even more c... | We don't need to use any integration by parts.
We start by completing the square
$$\int \frac{x+1}{(x^2-x+8)^3} dx = \int \frac{x-\frac{1}{2}+\frac{3}{2}}{((x-\frac{1}{2})^2+\frac{31}{4})^3} dx =$$
$$\underbrace{ \int \frac{x-\frac{1}{2}}{((x-\frac{1}{2})^2+\frac{31}{4})^3} dx}_{=:I_1}+ \frac{3}{2}\underbrace{\int\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553924",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
} |
Easy way of memorizing values of sine, cosine, and tangent My math professor recently told us that she wanted us to be able to answer $\sin\left(\frac{\pi }{2}\right)$ in our head on the snap. I know I can simply memorize the table for the test by this Friday, but I may likely forget them after the test. So is there a ... | First of all, you know the square diagonal is $\sqrt 2$ of the side, so sine and cosine of $45^\circ$ is $1/\sqrt 2 = \sqrt 2/2$ and tangent is $1$.
Recall also the isosceles right triangle and find the described ratios in it
(see https://commons.wikimedia.org/wiki/File:Reglas.svg)
For the rest of the first quarter... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1553990",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "62",
"answer_count": 12,
"answer_id": 8
} |
Showing the integral of a function is finite almost everywhere Suppose$\ E \subset \mathbb R$ is closed. Let$\ d(y) = \inf \{|x-y| : x \in E \} $ and let $\ M(x) = \int_0^1\frac{d^a(y)}{|x-y|^{(1+a)}} dy $ , for some arbitrary constant $a$.
Show that $\ M(x)$ is finite everywhere in $E$ except on a set of measure zero.... | First of all, observe that it is enough to extend the $y$ integration over $(0,1)\cap E^c$ only because $d(y)=0$ on $E$. Also, since this set is open, I can write it as a disjoint union of open intervals $I_n$, and $M(x)=\sum \int_{I_n} d^a(y)/|x-y|^{1+a}\, dy$.
Now let's look at $\int_E M(x)\, dx$ (as you already did ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1554122",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Let S be a cubic spline that has knots t0 < t1 < · · · < tn. Let S be a cubic spline that has knots $t_0 < t_1 < · · · < t_n$. Suppose that
on the two intervals $[t_0, t_1]$ and $[t_2, t_3]$, S reduces to linear polynomials. What does
the polynomial S look like on the interval $[t_1, t_2]$ (linear, quadratic, or cubic)... | The shape of S within interval $[t_1,t_2]$ could be linear, quadratic or cubic. Each "segment" of this cubic spline in between two knots is a cubic polynomial by its own. Therefore, the shape could look linear, quadratic or cubic depending on the control points configuration. If your cubic spline is required to be $C^1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1554192",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Conditional expectation of Y given X equals expectation of X? Let $Y\vert X \sim N(X,X^2)$ We can write
$$E[Y] = E[E[Y\vert X]] =E[X]$$
Where the first equality follows from the law of iterated expectations, and the second equality confuses me. Where does the second equality come from? $X$ is distribution uniformly ove... | If $$Y \mid X \sim \operatorname{Normal}(\mu = X, \sigma^2 = X^2),$$ then what is $$\operatorname{E}[Y \mid X]?$$ This is obviously simply $X$: Given the value of $X$, $Y$ is normal with mean $\mu = X$, thus the expected value of $Y$ given $X$ is $X$. So $$\operatorname{E}[Y \mid X] = X.$$
Next, just take the expect... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1554301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Maximum likelihood estimator for family of uniform distributions Question:
Based on the random sample $Y_1 = 6.3$ , $Y_2 = 1.8$, $Y_3 = 14.2$, and $Y_4 = 7.6$, use the method of maximum likelihood to estimate the parameter $\theta$ in the uniform pdf
$f_Y(y;\theta) = \frac{1}{\theta}$ , $0 \leq y \leq \theta$
My atte... | If you take the minimum $Y$ (or something smaller than the maximum) as your estimate for $\theta$, the probability of observing a $Y>\theta$ is zero. But since you have observations in that area this cannot be the MLE.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1554416",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How do I prove that among any $5$ integers, you are able to find $3$ such that their sum is divisible by $3$? How do I prove that among any $5$ integers, you are able to find $3$ such that their sum is divisible by $3?$
I realize that this is a number theory question and we use modular arithmetic, but I'm unsure of whe... | Consider the following seven numbers:
$a_1\\a_2\\a_3\\a_4\\a_5\\a_1+a_2+a_3+a_4\\a_2+a_3+a_4+a_5$
Since there are seven numbers three of them must have same remainder modulo $3$. If all are from the first five then their sum is multiple of $3$ otherwise one of the four-element-sum and one of its summand must be include... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1554573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 5,
"answer_id": 4
} |
Volume bounded by elliptic paraboloids Find the volume bounded by the elliptic paraboloids given by $z=x^2 + 9 y^2$ and $z= 18- x^2 - 9 y^2$.
First I found the intersection region, then I got $x^2+ 9 y^2 =1$. I think this will be area of integration now what will be the integrand. Please help me.
| You are wrong about the intersection region. Equating the two expressions for $z$ gives
$$x^2+9y^2=18-(x^2+9y^2)$$
$$2(x^2+9y^2)=18$$
$$x^2+9y^2=9$$
and thus here $z=9$.
The volume of your intersection can be divided into to parts: $0\le z\le 9$ where the restrictions on $x$ and $y$ are $x^2+9y^2\le z$, and $9\le z\le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1554685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Differentiability of $f(x+y) = f(x)f(y)$ Let $f$: $\mathbb R$ $\to$ $\mathbb R$ be a function such that $f(x+y)$ = $f(x)f(y)$ for all $x,y$ $\in$ $\mathbb R$. Suppose that $f'(0)$ exists. Prove that $f$ is a differentiable function.
This is what I've tried:
Using the definition of differentiability and taking arbitrary... | By assumption
$$
\lim_{h \to 0}\frac{f(h) - f(0)}{h} = f(0)\lim_{h \to 0}\frac{f(h) - 1}{h}
$$
exists,
implying
$$
l := \lim_{h \to 0}\frac{f(h)-1}{h}
$$
exists;
hence,
if $x \in \Bbb{R}$, then
$$
\frac{f(x+h) - f(x)}{h} = \frac{f(x)[f(h) - 1]}{h} \to f(x)l
$$
as $h \to 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1554765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Interior and exterior of a subset in the relative topology of the ambient space Let $X = (0, 4] \cup \{ 6 \} \cup [10, 11] \subset \Bbb R$. How do I find the interior and exterior of $A = (0, 2] \cup \{ 6 \} \cup (10, 11]$ in $X$?
Any help would be very welcome! Thanks in advance.
| $A^o=(0,2)\cup\{6\}\cup(10, 11]$. ($B_1(6)\cap X=\{6\}\subset A$...)
$ext A=(2,4]\cup \{10\}$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1554856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Zero divided by zero must be equal to zero What is wrong with the following argument (if you don't involve ring theory)?
Proposition 1: $\frac{0}{0} = 0$
Proof: Suppose that $\frac{0}{0}$ is not equal to $0$
$\frac{0}{0}$ is not equal to $0 \Rightarrow \frac{0}{0} = x$ , some $x$ not equal to $0$ $\Rightarrow$ $2(\frac... | 0/0 is indeterminate. Which means it has infinite number of values.
(0)(1) = 0 => 0/0 = 1
(0)(2) = 0 => 0/0 = 2
(0)(4) = 0 => 0/0 = 3 .....
From above it can be shown that 0/0 has infinite number of values which make it meaningless. Hence the term.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1554929",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "52",
"answer_count": 16,
"answer_id": 15
} |
Show that rank of skew-symetric is even number $$A = -A^T$$
I assume that $A$ is not singular.
So $$\det{A} \neq 0$$ Then $$ \det(A) = \det(-A^T) = \det(-I_{n} A^T) = (-1)^n\det(A^T) = (-1)^n\det(A)$$
So I get that $n$ must be even.
But what about odd $n$? I know it has to be singular matrix. Hints?
| Note: This answer is essentially based on this one by Jason DeVito. I have merely added some details.
I assume $A$ is real matrix. Note that it's rank as a real matrix equals its rank when considered as a complex matrix.
So from now on we consider $A$ as a complex matrix.
It is proved here that all the eigenvalues of $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Show that $\lim_{\epsilon\to0^{+}}\frac{1}{2\pi i}\int_{\gamma_\epsilon}\frac{f(z)}{z-a} \, dz=f(a)$
Let $a\in\Bbb C$ and $r>0$ and denote by $B(a,r)\subseteq \Bbb C$ the open ball of center $a$ and radius $r$. Assume that $f:B(a,r)\to\Bbb C$ is a continuous function and for each $\epsilon>0$ let $\gamma_\epsilon:[0,2... | $f$ is continuous in $a$, therefore for all $\eta > 0$ there exists
a $\delta > 0$ such that
$$
|f(z) - f(a)| < \eta \text{ for all } z \in B(a, \delta) \, .
$$
Then for $0 < \epsilon < \delta$
$$
\left| \frac{1}{2\pi i}\int_{\gamma_\epsilon}\frac{f(z)}{z-a} \, dz - f(a) \right| = \left | \frac{1}{2\pi}\int_0^{2\pi}(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555195",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
What is the integral of $\frac{x-1}{(x+3)(x^2+1)}$? I've worked with partial fractions to get the integral in the form $$\int\frac{A}{x+3} + \frac{Bx + C}{x^2+1}\,dx$$ Is there a quicker way?
| Notice, for $(Bx+C)$ part, you should use separation as follows $$\int \frac{x-1}{(x+3)(x^2+1)}\ dx=\int \frac{-2}{5(x+3)}+\frac{2x-1}{5(x^2+1)}\ dx$$
$$=\frac{1}{5}\int \left(-\frac{2}{x+3}+\frac{2x}{x^2+1}-\frac{1}{x^2+1}\right)\ dx$$
$$=\frac{1}{5} \left(-2\int \frac{1}{x+3}\ dx+\int \frac{d(x^2)}{x^2+1}-\int\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Finding the sum of the infinite series Hi I am trying to solve the sum of the series of this problem:
$$
11 + 2 + \frac 4 {11} + \frac 8 {121} + \cdots
$$
I know its a geometric series, but I cannot find the pattern around this.
| Your series is a geometric series with a common ration of $q =\dfrac{2}{11}$ and a first term $a_0$ equal to $11$.
$-1 < \dfrac{2}{11}< 1$ so the series is convergent.
Its sum is equal to
$$a_0 \cdot \dfrac{1}{1-q} = 11 \cdot \dfrac{1}{1-\dfrac{2}{11}} = \dfrac{121}{9}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555429",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Subsequences and convergence. Do all subsequences have to converge to the same limit for the sequence to be convergent? I was asked this question:
Prove that $a_n$ converges if and only if:
$a_{2n},a_{2n+1},a_{3n}$ all converge
I thought this was an easy generic question until I read the hint which said:
Note: It is no... | I think the word "required" in the hint is confusing. Focus instead on the last sentence:
Note: It is not required that the three sub-sequences have the same limit. This needs to be shown
In other words, you need to show that as a result of the given hypotheses, the three sub-sequences do have the same limit. You nee... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
When is a quasiprojective variety Kobayashi hyperbolic? I am looking for some (simple and well-known) sufficient conditions on a quasiprojective variety to be Kobayashi hyperbolic.
I realize that in this generality it may be a complicated (maybe even hopelessly naive) question, so less generality is perfectly OK.
A bi... | I'm certainly far from knowing the subject well. However, what follows quite easily is the following:
Suppose $X$ is Kobayashi hyperbolic. Then $X$ cannot contain any rational curves, and any rational map $f:A\to X$ from an abelian variety (or more generally any compex torus) must be constant.
The reason is simply that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555636",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
How do you solve 5th degree polynomials? I looked on Wikipedia for a formula for roots of a 5th degree polynomial, but it said that by Abel's theorem it isn't possible. The Abel's theorem states that you can't solve specific polynomials of the 5th degree using basic operations and root extractions.
Can you find the roo... | As mentioned above, no general formula to find all the roots of any 5th degree equation exists, but various special solution techniques do exist. My own favourite:
- By inspection, see if the polynomial has any simple real solutions such as x = 0 or x = 1 or -1 or 2 or -2. If so, divide the poly by (x-a), where a is th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555743",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 3,
"answer_id": 0
} |
If $f$ is Riemann integrable and $g$ is continuous, what is a condition on $g$ such that $g \circ f$ has the same discontinuity set as $f$? I know that if $f$ is Riemann integrable and $g$ is continuous, then the discontinuity set of $g \circ f$ is contained in the discontinuity set of $f$. How would I go about finding... | I don't know if Riemann integrability has much influence here. What you are aiming at is that where $f$ is discontinuous you would have to make $g$ map values taken by $f$ near the discontinuity to different values.
For example if we take $f$ to be the Heaviside step function and we see that $f$ takes values $0$ and $1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555932",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $f(x)=\sin^{23}x-\cos^{22}x$ and $g(x)=1+\frac{1}{2}\arctan|x|$,then the number of values of $x$ in the interval $[-10\pi,20\pi]$ Let $f(x)=\sin^{23}x-\cos^{22}x$ and $g(x)=1+\frac{1}{2}\arctan|x|$,then the number of values of $x$ in the interval $[-10\pi,20\pi]$ satisfying the equation $f(x)=\text{sgn}(g(x)),$ is ... | $$f(x)=sin^{23}x-cos^{22}x$$ then
$$f'(x)=sinxcosx\left(23sin^{21}x+22cos^{20}x\right)=sinxcosx \times g(x)$$
In First Quadrant obviously $f'(x) >0$, so $f$ is Increasing in $(0 \: \frac{\pi}{2})$
In Second Quadrant, $sinx \gt 0$ and $cosx \lt 0$ and $g(x) \gt 0$ so $f'(x) \lt 0$ in $(\frac{\pi}{2} \: \pi)$
and in Thir... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1555998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A child sits at a computer keyboard and types each of the 26 letters of the alphabet exactly once, in a random order. A child sits at a computer keyboard and types each of the 26 letters of the alphabet
exactly once, in a random order.
How many independent children typists would you need such that the probability
that ... | You're right. The probability of getting "EXAM" is $p=\frac{23!}{26!}$. Now, suppose, at least $n$ children are needed. If the total number of occurings we get is $X$, then $P(X=r)={n \choose r} \cdot p^r \cdot (1-p)^{(n-r)}$, according to the theory of Barnoulli Trials. Now, $P(X \geq 1) \geq 0.9 \implies P(X=0) \leq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
What is the chance of rolling a die and getting the number six three times at exactly 10 rolls?
What is the chance of rolling a die and getting the number six three times at exactly 10 rolls?
I was asked this question in my statistic class. I thought the way to do this was $(1/6)^3 \times (5/6)^7$, because that is ge... | You’ve computed the probability of rolling six three times and some other number seven times in that order. You also need to account for all of the other ways in which three sixes can come up in ten rolls. The binomial distribution is called for here. If we let $X$ be the number of times a six is rolled, then $$
P(X=3)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556197",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
When is implication true? If we have $p\implies q$, then the only case the logical value of this implication is false is when $p$ is true, but $q$ false.
So suppose I have a broken soda machine - it will never give me any can of coke, no matter if I insert some coins in it or not.
Let $p$ be 'I insert a coin', and $q$ ... | The fact some specific values satisfy the formula doesn't mean the formula is true in general. "It is noon now and it rains" is true right now and right here, but at other place or in other time this will appear false.
Your implication will turn out true if you check it keeps satisfied by any possible combination of it... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556298",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 4
} |
Prove that $m-n$ divides $p$ Let $a_1,a_2,a_3$ be a non constant arithmetic Progression of integers with common difference $p$ and $b_1,b_2,b_3$ be a geometric Progression with common ratio $r$. Consider $3$ polynomials $P_1(x)=x^2+a_1x+b_1, P_2(x)=x^2+a_2x+b_2,P_3(x)=x^2+a_3x+b_3$. Suppose there exist integers $m$ and... | (Too long for a comment)
The claim is not true. A counterexample is
$$m=5,\quad n=3,\quad p=3,\quad r=\frac 35,\quad a_1=-11,\quad b_1=\frac{75}{2}.$$
In the following, I'm going to write about how I figured out these values and about that the claim is true if $b_1$ is an integer.
We have
$$P_1(m)=P_2(n)\iff m^2+a_1m+b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556387",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Delta Method corollary Consider the Delta Method as stated in van der Vaart Theorem 3.1 at page 26 (you can find the page here https://books.google.co.uk/books?id=UEuQEM5RjWgC&pg=PA32&lpg=PA32&dq=van+der+vaart+theorem+3.1+because+the+sequence+converges+in+distribution&source=bl&ots=mnRJLD8XLC&sig=inIMmSPvWDfrPc6r4U7dnu... | If $T$ follows a Gaussian distribution on $\mathbb{R}^{n}$ with mean $0$ and covariance matrix $\Sigma$, denoted by $T \, \sim \, \mathcal{N}(0,\Sigma)$, and if $A$ is a $n \times n$ real matrix, we have :
$$ AT \, \sim \, \mathcal{N}(0,A \Sigma A^{\top}) $$
If $f \, : \, \mathbb{R}^{n} \, \rightarrow \, \mathbb{R}^{n}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Expectation of inverse of sum of iid random variables - approximation I am working with the set of i.i.d discrete random variables $\{\zeta_1, \ldots, \zeta_n\}$. Each one of them can take either of the $m$ values $\{z_1, \ldots, z_m\}$ with corresponding probabilities $\{p_1, \ldots , p_m\}$.
I am trying to understan... | Let $p=P(\zeta\ge z)$. Then as $n\to\infty$ $$LHS\approx E(\frac 1 {B(n,p)};B(n,p)\ge 1)\approx \frac 1 {np}+\frac {np(1-p)}{(np)^3}=\frac 1 {np}+\frac {1-p}{(np)^2}=\frac 1{np}(1+\frac{1-p}{np})$$
while $RHS=\frac 1 {np}$. Hence $LHS\sim RHS$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556586",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Showing that a cubic extension of an imaginary quadratic number field is unramified. Let $\alpha^3-\alpha-1=0$, $K=\mathbb Q(\sqrt{-23})$, $K'=\mathbb Q(\alpha)$, and $L=\mathbb Q(\sqrt{-23},\alpha)$.
Then I am asked to show that the field extension $L/K$ is unramified.
I know that if $\mathfrak p\in\operatorname{Max}... | Here’s an approach quite different from what you had in mind, purely local and not ideal-theoretic.
The field $K$ is ramified only at $23$, and since the $\Bbb Q$-discriminant of $k=\Bbb Q(\alpha)$ is of absolute value $23$ as well, the only possibility for ramification of $K'=Kk$ over $K$ is above the prime $23$, in o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Complex numbers problem I'm trying to use the relevant rules and definitions but making little progress.
| Simplify it in $x,y$ you get two parts i.e. two equations and it's done. $(x+iy)(1+x-iy)+\frac{5.(x).(1+2i)}{5}-2x-4=0+0i$ now we have two equations. The imaginary part and real part ie $(x+x^2+y^2)+x-2x-4=0$ and $ y+xy+2x=0$ two equations two unknowns solve for $x,y$ and you are done with it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Find Least Squares Regression Line I have a problem where I need to find the least squares regression line. I have found $\beta_0$ and $\beta_1$ in the following equation
$$y = \beta_0 + \beta_1 \cdot x + \epsilon$$
So I have both the vectors $y$ and $x$.
I know that $\hat{y}$ the vector predictor of $y$ is $x \cdot \b... | First define
X = [ones(size(x)) x];
then type
regress(y,X)
Observations:
*
*the first step is to include a constant in the regression (otherwise you would be imposing $a=0$).
*the output will be a vector with the OLS estimates $(a,b)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Why is $e$ the number that it is? Why is $e$ the number that it is? Most of the irrational number that we learn about in school have something to do with geometry, like $\pi$ is the ratio of a circle's diameter to its circumference. Those numbers can also be derived using maths ($\pi = \lim_{n\to\infty}4\sum_{k = 1}^... | The identity (which can be seen as a definition) $e^{i\theta} = \cos\theta + i\sin \theta$ is geometrically significant. If you delve into the subject of complex analysis, you'll understand that $e^{i\theta}$ can be seen as a "rotation".
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1556943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 11,
"answer_id": 7
} |
A more "complete picture" of the relationship between various modes of convergence I am trying to generate/looking for a more comprehensive/complete list/diagram of how the 4 major modes (as listed on wikipedia) of convergence of random variables relate to each other:
*
*Distribution (law)
*Probability
*Almost sur... | Gearoid de Barra's 2003 book Measure Theory and Integration contains an entire section called "Convergence Diagrams" (7.3, pp.128-131) concerning six modes of convergence in three different settings: almost everywhere, in mean, uniform, in $L^p$, almost uniform, and in measure. The settings are: the general case, the c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1557137",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to solve $2^x=2x$ analytically. $$2^x=2x$$
I am able to find the solutions for this equation by looking at a graph and guessing. I found them to be $x_1=1$ and $x_2=2$.
I also can also find them by guess and check, but is there anyway algebraically to solve this problem?
| Not without using the Lambert $W$ function, which is really just a fancy reformulation of the problem, and not actually a solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1557285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Why Radians oppose to degrees? Degrees seem to be so much easier to work with and more useful than something with a $\pi $ in it. If I say $33\:^\circ $ everyone will be able to immediately approximate the angle, because its easy to visualize 30 degrees from a right angle(split right angle into three equal parts), but ... | The radian is the standard unit of angular measure, and is often used in many areas of mathematics. Recall that $C = 2\pi r$. If we let $r =1$, then we get $C = 2\pi$. We are essentially expressing the angle in terms of the length of a corresponding arc of a unit circle, instead of arbitrarily dividing it into $360$ de... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1557372",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Of what is the Hopf map the boundary? Consider a generator $x$ of the singular homology group $H_3(S^3)$. I think of this (perhaps wrongly?) as something like the identity on $S^3$, cut up into simplices. Now we have the Hopf fibration $\eta: S^3 \to S^2$, which gives us $\eta_*x \in H_3(S^2) = 0$. Thus $\eta_*x$ is a ... | $\newcommand{\Cpx}{\mathbf{C}}\newcommand{\Proj}{\mathbf{P}}$Let $M$ be the complex surface obtained by blowing up the unit ball in $\Cpx^{2}$ at the origin. The exceptional curve at the origin is a complex projective line, i.e., an $S^{2}$. The boundary of $M$ is $S^{3}$, and radial projection from the unit ball to th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1557466",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
With a 45-gallon and a 124-gallon jug with no marketings (but known capacities), how do you measure exactly 1 gallon of water? I could imagine filling up the $124$, then filling the $45$ from it, so there is $124-45=79$ left in the $124$, then again so there is $37$. Then I'm lost!
| You could use the 124-gallon jug four times to get 496 gallons, and subtract the 45-gallons 11 times to get 1 gallon.
How to solve:
Let $x$ be the number of 124-gallons and $y$ the number of 45-gallons. We want solutions to
$$ 124x + 45y = 1$$
Obviously, one will be negative. Taking modulo 45, we get
$$ 34x = 1 \mod 45... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1557535",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proof of trignometric identity Could you help me prove this? I've gotten stuck, need some help..
$$\sin^2\Theta + \tan^2\Theta = \sec^2\Theta - \cos^2\Theta$$
Here's what I've done so far:
Left Side:
$$\sin^2\Theta + \frac{\sin^2\Theta}{\cos^2\Theta}=\frac{\sin^2\Theta\cos^2\Theta+\sin^2\Theta}{\cos^2\Theta}$$
Right ... | Hint: $\sin^2\theta+\cos^2\theta=1.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1557632",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
If $T: V \to V $ is an operator with $\mathrm{rank}(T)=n-1$, then there exists a vector $v \in V$ such that $T^{n-1}(v) \ne 0$
Let $V$ be an $n$-dimensional vector space over a field $F$ and $T: V \to V$ an operator. If $\mathrm{rank}(T)=n-1,$ prove that $T^{n-1}$ is not equal to the $0$ operator on $V$.
We need to s... | Question 1: No, it's not correct. You can't conclude that $T^{n-1}(v)\neq \alpha T^{n-1}(w)$ just because $T(v)\neq \alpha T(w)$. If you're still stuck on the problem, I suggest trying to prove that $\dim\ker(T^k)\leq k$ for all $k$ by induction on $k$.
Question 2: When $n=1$, the hypotheses do imply $T=0$. However,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1557732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Existence of integer sequence for any positive real number I am stuck on the following problem:
Given any $\lambda > 0$ there exists strictly increasing sequence of positive integers $$ 1 \leq n_1 < n_2 < ... $$ such that
$$\sum _{i=1}^{\infty} \frac{1}{n_i} = \lambda$$
Any help will be appreciated.
Thank you.
| You can use the following lemma : for any two sequence $u_{n}$ and $v_{n}$ from
$\mathbb{R}^{\mathbb{N}}$ that tend to $+\infty$ , and $u_{n+1}-u_{n} \to 0$
then the set $\{u_{n}-v_{m}|(n,m)\in \mathbb{N}^{2}\}$ is dense in $\mathbb{R}$ .
Proof :
let $e>0$ there is $k$ such that $\forall n \geq k$ we have $|u_{n+1}-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1557854",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Condition that all 3 roots of $az^3+bz^2+cz+d=0$ have negative real part The problem is - 'find the condition that all 3 roots of $f(z)=az^3+bz^2+cz+d=0$ have negative real part, where $z$ is a complex number'.
The answer - '$a,b$, and $d$ have the same sign.'
Honestly, I have no clue about how to proceed. Here is wha... | Ok, I think I have it.
Since $z$ is a complex number, $f(z)$ must have a complex root. As they occur in conjugate pairs, 2 out of 3 roots of f(z) must be imaginary. The remaining one shall be purely real (and obviously rational). Now, all three have negative real parts. So, lets name the roots-$\alpha=-x+iy, \beta=-x-i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1557977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
Is $xRy \iff x+y = 0$ an equivalence relation? $R$ is a relation on real numbers. $xRy \iff x+y = 0 $.
Is it an equivalence relation?
My answer is no
proof:
-(Reflexive) let $x = a$ , $aRa \iff 2a=0$.
Since $2a = 0$ doesn't hold for every real number $a$, $R$ is not reflexive.
Since $R$ isn't reflexive, $R$ is not an ... | It is not only correct, it is almost the most straightforward approach you could take. Nicely done!
The only improvement I would suggest is that you demonstrate a particular real number $a$ for which $a\:R\:a$ fails to hold. For example, you could just say: "Since $1+1=2\ne 0,$ then $1\:R\:1$ fails to hold, and so $R$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1558066",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
"Exploding Dice" vs. "Non-Exploding Dice" For a roleplaying game I'm designing, I'm using a core resolution mechanic that comes down to counting "successes" on rolled dice. Characters roll a handful of dice (six-sided) and count each die that generated a 4, a 5, or a 6. Dice that generated a 1, a 2, or a 3 are discarde... | Let's call $x$ the average of successes per dice. First, let's examine the case in which they can not explode 6s. 3 possible successes, 3 possible fails, so $x=1/2$
Now the case where they explode twice. There are 3 possible successes and 3 possible fails, but also 1 case where 2 additional dices are rolled (so and add... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1558183",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Inverse image of a vector subspace I've a question on the inverse image in linear functions.
Let $f:V\rightarrow W$ be a linear function between two vector spaces.
I know that if I have to find the inverse image of a particular vector $\vec{w} \in W$ I can find it as
$f^{-1}(\vec{w})= \Big\{ { \vec{x_0}+\vec{y}} \Big... | your guess about the inverse image of the subspace is right.
Take $\vec{x} \in f^{-1}(\mathscr{K})$, so that $f(\vec{x}) \in \mathscr{K}$.
Hence we will have,
\begin{align}
f(\vec{x})&= \sum \limits_{i=1}^{k}c_i \vec{v_i}\;\;\;\text{for some scalar} c_i\\
&= \sum \limits_{i=1}^{k}c_if(\vec{x_i})\\
&= f(\sum \limits_{i=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1558276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Finding XOR of all even numbers from n to m Problem : Given a range $[n,m]$, find the xor of all even numbers in this range. Constraints : $1 \le n \le m \le 10^{18}$
How do we solve this problem?
P.S. I asked this question at stackoverflow but that was not the appropriate place to ask this question. So I re-asked it h... | Something like this:
For bit $k$,
where $k=0$ is the
units bit,
blocks of length
$2^{k+1}$
xor to zero,
since there are
$2^k$ with a $0$
and
$2^k$ with a $1$.
Therefore,
we can xor the $k$-th bit
from
$i=
n+\lfloor \frac{m-n}{2^{k+1}} \rfloor
$
to
$m$
to get that bit,
since the blocks of
length $2^{k+1}$
xor to zero.
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1558404",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Show that $f$ is uniformly continuous given that $f'$ is bounded. Let $D\subset \mathbb{R}$ be an interval and $f:D\rightarrow \mathbb{R}$ a continuously differentiable function such that its derivative $f':D\rightarrow \mathbb{R}$ is bounded. Show that $f$ is uniformly continuous.
I will admit before I begin; I am sta... | Let $f:[-X,X]\rightarrow \mathbb{R}$ has the conditions that you want. Let $\epsilon>0$, then we want to find $\delta>0$ such that:
$$\forall x,y\in [-X,X](|x-y|<\delta \rightarrow |f(x)-f(y)|<\epsilon)$$
By Mean value theorem for every $x,y\in [-X,X],x<y$, there exists $c\in (-X,X)$ such that
$$\frac{f(y)-f(x)}{y-x}=f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1558493",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Can you always decrease the variance by removing outliers? Consider a finite set of real numbers. Let its variance be $V$. Let the highest number be $h$ and the lowest number be $l$, and let's assume $l < h$.
Let $x$ be an arbitrary number with $l < x < h$.
Now create a new set by removing one element equal to $h$ and... | I think I may have found the proof.
Number the points $p_1,p_2,...,p_N$ from lowest to highest.
Replace one of the extreme points, $p_E$ where $E \in \{1,N\}$, by some $x$ with $p_1<x<p_N$. Call the new set ${q_1,...,q_N}$.
Now
$$
Var(q) = \frac{1}{N} \sum (q_i - \bar{q})^2
\le \frac{1}{N} \sum (q_i - \bar{p})^2
$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1558622",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Limit of $n\ln\left(\frac{n-1}{n+1}\right)$ I'm having trouble with this limit:
$$\lim_{n\to ∞}n\ln\left(\frac{n-1}{n+1}\right)$$
It's supposed to be solvable without using l'Hospital's rule. I'm guessing it's a case for the squeeze theorem, but I'm not exactly sure. Any advice?
| Notice, let $\frac{1}{n}=t$, hence $$\lim_{n\to \infty}n\ln\left(\frac{n-1}{n+1}\right)$$ $$=\lim_{t\to 0}\frac{1}{t}\ln\left(\frac{\frac{1}{t}-1}{\frac{1}{t}+1}\right)$$ $$=\lim_{t\to 0}\frac{\ln\left(\frac{1-t}{1+t}\right)}{t}$$
$$=\lim_{t\to 0}\frac{\ln\left(1-t\right)-\ln\left(1+t\right)}{t}$$
Now, using Taylor's s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1558833",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 4
} |
Condition for Sums in a Set Let $$ S = \{ a_1,a_2, \cdots , a_k \}$$ be comprised of divisors of $n \in \mathbb{N}, n>1$ and $n$ not prime. Suppose we select $p$ elements from the set $S$ (possibly more than once for each divisor), and the $p$ chosen elements have a total sum of $q$. Prove that it is always possible t... | The restriction on $n$ isn't necessary: it's true for $n$ prime as well as $n$ composite (and even, trivially, for $n=1$). Indeed, I came up with the following solution by considering the case where $n$ is prime.
Suppose that $a_1,\dots,a_p$ are (not necessarily distinct) divisors of $n$, and set $q=a_1+\cdots+a_p$. De... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1558921",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Why does $p$ have to be moderate in the Poisson approximation to binomial random variable? So the proof that a binomial rv with large $n$ approximates a poisson rv with $\lambda = np$ (given below) doesn't seem to use the fact that $p$ is moderate/small, so why does wikipedia and my textbook (Ross) state this as a cond... | Suppose for example that $p=1/2$ and $n$ is say $100$. Let $i=50$. Then $\frac{n(n-1)(n-2)\cdots(n-i+1)}{n^i}$ is quite far from $1$. So the argument quoted no longer works.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1559023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Orthogonal complement of the column space of a matrix Let $H =\operatorname{Col}(A)$, where $$A =\begin{pmatrix}
1&2\\
2&4\\
3& 1\end{pmatrix}$$
Find $H^\perp$, the orthogonal complement
of $H$.
$H$ is the same thing as $A$, and as I understand it, Orthogonal complement means the span of vectors that are orthogonal to ... | In general, for any matrix $A \in \mathbb{C}^{m\times n}$, the answer may be obtained, using those relations:
$$ \text{im}(A^*) =\ker(A)^{\perp} ~~\text{and}~ ~\ker(A^*) = \text{im}(A)^{\perp}$$ furthermore
$$\mathbb{C}^{m} = \ker(A^*) \oplus \text{im}(A) ~~\text{and}~ ~ \mathbb{C}^{n} = \ker(A) \oplus \text{im}(A^*)$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1559151",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Double Angle identity??? The question asks to fully solve for $$\left(\sin{\pi \over 8}+\cos{\pi \over 8}\right)^2$$
My question is, is this a double angle formula? And if so, how would I go about to solve it?
I interpreted it this way; $$\left(\sin{\pi \over 8}+\cos{\pi \over 8}\right)^2$$
$$=2\sin{\pi \over 4}+\le... | It is related to a double-angle identity. The relevant identities you need are:
$$\sin^2\theta + \cos^2\theta = 1$$
and
$$\sin 2\theta = 2\sin\theta\cos\theta$$
You will also need to expand your binomial, using the basic algebraic identity
$$(A+B)^2 = A^2 + 2AB + B^2$$.
So, start by expanding the binomial; then use th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1559257",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 0
} |
How to show that det(A)≤1?
Let $A = (a_{ij})_n$ where $a_{ij} \ge 0$ for $i,j=1,2,\ldots,n$ and $\sum_{j=1}^n a_{ij} \le 1$ for $i = 1,2,\ldots,n$.
Show that $|\det(A)| \le 1$.
Should I use the definition of matrix:
$$\det(A)=\sum \textrm{sgn}(\sigma)a_{1,\sigma(1)}a_{2,\sigma(2)}...a_{n,\sigma(n)}$$
I don't unde... | Just take modulus the determinant expression you wrote. Note each term on the right under sum can be bounded by using G.M. $\le$ A.M.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1559353",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
} |
How to 'get rid of' limit so I can finish proof? Suppose $\sup_{x \in \mathbb{R}} f'(x) \le M$.
I am trying to show that this is true if and only if $$\frac{f(x) - f(y)}{x - y} \le M$$
for all $x, y \in \mathbb{R}$.
Proof
$\text{sup}_{x \in \mathbb{R}} f'(x) \le M$
$f'(x) \le M$ for all $x \in \mathbb{R}$
$\lim_{y \to ... | well, first of all, we have to presume f is continuous and differentiable on R. This statement isn't true otherwise.
1) Suppose $\frac{f(x) - f(y)}{x - y} > M$ for some $x, y \in \mathbb R$.
By the mean value theorem, there exist a $c; x <c < y$ where $f'(c) = \frac{f(x) - f(y)}{x - y}$.
So $f'(c) > M$.
So $\sup f'(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1559485",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How to solve the quadratic form I am a physicist and I have a problem solving this
\begin{equation}
Q(x)=\frac{1}{2}(x,Ax)+(b,x)+c
\end{equation}
In a book it says that:
"The minimum of Q lies at $\bar{x}=-A^{-1}b$ and
\begin{equation}
Q(x)=Q(\bar{x})+\frac{1}{2}((x-\bar{x}),A(x-\bar{x}))
\end{equation}
How do I go to ... | You have
$$
\frac{1}{2}((x-\bar{x}),A(x-\bar{x}))=\frac{1}{2}(x,Ax)-(x,A\bar{x})+\frac{1}{2}(\bar{x},A\bar{x})
$$
so that
$$
\tag{1}
\frac{1}{2}((x-\bar{x}),A(x-\bar{x}))+Q(\bar x)=\frac{1}{2}(x,Ax)-(x,A\bar{x})+(\bar{x},A\bar{x})+(b,\bar x)+c.
$$
On the other hand:
$$
(\bar x,A\bar x)+(b,\bar x)=(A^{-1}b,b)-(b,A^{-1}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1559610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Augmented Algebras Recently I started to study Operads. My reference is Algebraic Operads of Jean-Louis Loday and Bruno Vallette. In this book they define augmented algebra of the following form:
an $\mathbb{K}$-algebra $A$ is augmented when there is a morphism of algebras $\epsilon: A\rightarrow \mathbb{K}$ called aug... | Define $\phi \colon A \to \mathbf K1_A \oplus \ker \epsilon$ by
$$ \phi(a) = \bigl(\epsilon(a)1_A, a - \epsilon(a)1_A\bigr) $$
then $\phi$ is obviously linear and one-to-one. And if $(r1_A, a) \in \mathbf K 1_A \oplus \ker \epsilon$ is given, then
$$ \epsilon(r1_A + a) = r + \epsilon(a) = r $$
hence $$ \phi(r1_A + a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1559719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.