Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Why do we eventually end up with $0$ in Euclidean Algorithm? I'm new to number theory, I just understood the proof of Euclidean Algorithm and how it cleverly uses the fact that $\mathrm{gcd}(a,b) = \mathrm{gcd}(b,r)$ repeatedly, where $a$ is the dividend, $b$ is the divisor and $r$ is the remainder.
Although one thing,... | Assuming $a$ and $b$ are positive and $a>b$, by definition, $r$ is less than $b$. Then $b<a$ and $r<b$, so you have smaller numbers than you started with. If $a<b$, then just switch $a$ and $b$. If $a=b$, then $\gcd(a,\,b)=a=b$. Since $b$ and $r$ are still non-negative (also by definition), the only possibility is to g... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2954010",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 0
} |
Prove or disprove $f(n) = O(f(2n))$ I wonder how to to prove or disprove that $f(n) = O(f(2n))$
I have tried many function, and think it is right, but still don't have any idea how to prove.
Could anyone give me a hint about it?
| If
\begin{equation}
f(n)=\begin{cases}1 & n\text{ even,}\\ n& n\text{ odd,}\end{cases}
\end{equation}
then
\begin{equation}
\mathcal{O}(f(2n))=\mathcal{O}(1)\text{,}
\end{equation}
but $f(n)$ is not asymptotically constant.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2954085",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Probability of having a $4$-digit PIN number with strictly increasing digits What is the probability of having a PIN number (digits $0$-$9$, starting with consecutive zeros allowed) with strictly increasing digits?
We easily deduce that, if $a_1, a_2, a_3, a_4$ are the respective digits, then $a_1<7, a_2<8$ and $a_3<9... | From a set of $\{0, 1, \cdots, 9\}$, choose any subset of 4 numbers. Such subset is in 1-to-1 correspondence with a 4-digit pin with increasing digits.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2954213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Some notation for vector space $\mathbb{R}^\mathbb{R}$, $\mathbb{R}^{X}$, $C(X)$ I am reading some slides for functional analysis, and it mentioned that $\mathbb{R}^\mathbb{R}$, $\mathbb{R}^{X}$, and $C(X)$ are all vector spaces. Since the slides are so brief and it doesn't provide an further details. Is there anyone c... | For sets $X$ and $Y$, the set of maps $X \to Y$ is sometimes denoted $Y^X$. (Compare the power set $2^X$, which can be thought of as the set of maps $X \to \{0, 1\}$ by associating to a set $U\subset X$ its indicator function.) If $Y$ has some additional structure, then $Y^X$ generally does as well; in particular, if $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2954348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Number repositioning You are given $2^n$ numbers, in one step you move the numbers in odd positions to the beginning of the list and the numbers in even positions to the end of the list, keeping the initial order among them. Prove that after $n$ such steps, you will get the initial list.
How do I do this? Someone told ... | Index the $2^n$ list positions of elements as $\underbrace{00\ldots00}_n, \underbrace{00\ldots01}_n, \ldots,\underbrace{11\ldots10}_n, \underbrace{11\ldots11}_n$. (These are indices given to the $2^n$ positions, not to elements that may move around the list)
In each reposition / shuffle step, for the element at positio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2954446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Total waiting time of exponential distribution is less than the sum of each waiting time, how so? I am reading my textbook and find a weird phenomenon.
The example says that Anne and Betty enter a beauty parlor simultaneously. Anne to get a manicure and Betty to get a haircut. Suppose the time for a manicure (haircut) ... | This was an interesting problem... I was also initially surprised by the answer of $12$ minutes. The solution given by the text is very clever and concise, but because I have nothing better to do I also solved it using the longer way.
Let $X$ be the time to get a manicure, $X \sim Exp(\lambda_1 = \frac{1}{20})$
Let $Y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2954618",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\frac{3}{5} + \frac{4}{5}i$ is not a root of unity I want to prove that $z=\frac{3}{5} + \frac{4}{5}i$ is not a root of unity, although its absolute value is 1.
When transformed to the geometric representation: $$z=\cos{\left(\arctan{\frac{4}{3}}\right)} + i\sin{\left(\arctan{\frac{4}{3}}\right)}$$ Accordin... | As is shown in this answer, Niven's theorem says that $\sin(\pi p/q)$ is rational only when $\sin(\pi p/q)\in\left\{-1,-\frac12,0,\frac12,1\right\}$. However, $\sin\left(\arg\left(\frac35+\frac45i\right)\right)=\frac45$, so we know that $\arg\left(\frac35+\frac45i\right)$ is not a rational multiple of $\pi$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2954737",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 7,
"answer_id": 4
} |
Number of possible permutations of three pairs of socks, one blue, one black, and one white? There are 3 colours of socks; blue, black and white and each colour has two pairs of socks and we need to distribute them to six people and the left leg socks are distinguishable from right leg socks. What are the possible ways... |
There are three pairs of socks, one black, one blue, and one white. In how many ways can the socks be permuted if left socks are distinguishable from right socks?
Since there are six different socks, they can be arranged in $6!$ orders.
There are three pairs of socks, one black, one blue, and one white. In how man... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2954840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Given $f:[0,+\infty]\to\Bbb [1,+\infty)$, if $1/f\in\mathcal{R}[0,+\infty)$ then is $f$ not Lipschtiz? I thought of proceeding by contrapositive.
Assume $f(x)\ge1$ and $f'(x)\le K\in\Bbb R$ for all $x\ge0$. Then for any $t>0$ it follows that \begin{align} \int_0^tf'(x)\ dx=f(t)-f(0)\le Kt\end{align}i.e. $f(t)\le Kt+f(0... | You don't need to use derivatives.
By the assumption of Lipschitzianity, there exists a constant $K>0$ such that
$|f(x) - f(y)| \leq K |x-y|$ for every $x,y$.
In particular,
$$
f(t) - f(0) \leq K t
\qquad \forall t\geq 0,
$$
and then you can proceed as in your proof.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2954933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $\lim\limits_{x \to \pm\infty}\dfrac{x^3+1}{x^2+1}=\infty$ by the definition. Problem
Prove $\lim\limits_{x \to \pm\infty}\dfrac{x^3+1}{x^2+1}=\infty$ by the definition.
Note:
The problem asks us to prove that, no matter $x \to +\infty$ or $x \to -\infty$, the limit is $\infty$,which may be $+\infty$ or $-\infty.... | Possibly correct but unreadable.
Consider
$$
\frac{x^3+1}{x^2+1}=x-\frac{x-1}{x^2+1}>x
$$
whenever $x>1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2955103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
My question is about Independent events of the following question in which tree diagram is said to make. How to do this? Suppose identical tags are placed on both the left ear and the right
ear of a fox. The fox is then let loose for a period of time.
Consider the two events $C_1 = \{\text{left ear tag is lost}\}$ and ... | Let $A$ be the event where exactly one tag is lost and $B$ be the event where at most one tag is lost. Then we are looking for
$$
P(A\mid B).
$$
By the definition of conditional probability,
$$
P(A\mid B) = \frac{P(A\cap B)}{P(B)}.
$$
If exactly one tag is lost, then it's true that at most one tag is lost, so $A\subset... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2955409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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What is the determinant of this linear map? Consider the map $f:M_{n}(K)\to M_n(K)$
$$f(A) = A^t,A\in M_n(K).$$
What is the determinant of this map?
After working on some examples I recognized that the matric associated with this map is a permutation matrix and so its determinant is the sign of the permutation. Exclud... | Yes, it is correct. Alternatively, since $f(H)=H$ for every symmetric matrix $H$ and $f(K)=-K$ for every skew-symmetric matrix $K$, the matrix space $M_n(K)$ is the sum of two eigenspaces of $f$, one of dimesion $n(n+1)/2$ for the eigenvalue $1$ and the other of dimension $n(n-1)/2$ for the eigenvalue $-1$. Hence $\det... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2955524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$2(a^8+b^8+c^8)=(a^4+b^4+c^4)^2$ if and only if $a,b,c$ are the lengths of a right angled triangle Here is a problem from the book: Everything connected to Pithagoras
It is well known that in every right angled triangle $ABC$, $a^2+b^2=c^2$. Howevee, there are some more complicated equations as well. Here is one:
Prove... | Bear with me as I ramble a bit ...
The relation in question resembles a way of writing Heron's formula for the area of a triangle with sides $x$, $y$, $z$:
$$16\;|\triangle xyz|^2 = \left(x^2+y^2+z^2\right)^2-2\left(x^4+y^4+z^4\right) \tag{1}$$ Now, some (most?) people think of Heron's formula as more like
$$|\triangle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2955666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Injection from cardinal $\lambda$ to cardinal $\kappa$ implies $\lambda\leq\kappa$ I'm trying to prove that if there is an injection $f:\lambda\to\kappa$ (for $\lambda$,$\kappa$ cardinal numbers) then $\lambda\leq\kappa$. This is not true if they are just ordinal numbers, for example it is easy to build an injection fr... | Hint: The statement you are trying to prove is more or less just a disguised version of the Schroder-Bernstein theorem.
A full proof is hidden below.
Suppose there is an injection $f:\lambda\to\kappa$ but $\kappa<\lambda$. Then the inclusion map is an injection $i:\kappa\to\lambda$. Since there are injections in bo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2955822",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why is $|x|$ defined as $\sqrt{x^2}$ instead of $(\sqrt{x})^2$? I can't seem to understand this even though it might be utterly simple for some people. For me, saying $|x|=\sqrt{x^2}$ is a bit weird since $\sqrt{x^2}$ doesn't force positivity as there are always two possible square roots of a number,
$\sqrt{x}=+\sqrt... | You can see that $\sqrt{x^2}$ is defined for all $x\in\Bbb R$, while $(\sqrt{x})^2$ is defined only for $x\ge 0$. And that's a huge difference.
Furthermore, $|x|$ is not defined as $(\sqrt x)^2$. Instead, $|x|$ is defined as $x$ if $x\ge 0$ and as $-x$ if $x<0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2955934",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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$Z$ is a random variable that follows a standard normal distribution. Is $X = Z^2$ independent from $Y = Z^3$? I think the title is pretty clear about the problem. Should I try to find the joint probability of $X$ and $Y$ and decide if $\, f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)$? If so, how am I going to find a joint distr... | $EXY^{2} \neq EXEY^{2}$ so $X$ and $Y^{2}$ are not independent. This implies that $X$ and $Y$ are not independent. [Use a table of moments of $Z$ to see that $EXY^{2} \neq EXEY^{2}$. You can also argue analytically using that fact that 6-th moment is strictly smaller than the 8th moment]. Alternative proof: if $X$ and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2956050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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Given a positive sequence $\{a_n\}$ where $a_{n+1}=a_n+\frac{n}{a_n}$, can one find an asymptotic expansion of $a_n-n$?
Given a positive sequence $\{a_n\}$ such that
$$a_{n+1}=a_n+\frac{n}{a_n}$$
Can one find an symptotic expannsion of $a_n-n$?
I want one which has the term $O(1/n)$, or a stronger one.
| Let $b_n=a_n-n$.
Then
$$b_{n+1}=\left(1-\frac 1{a_n}\right)b_n$$
Thus if $a_0>1$ then $a_n>1$ and $b_n>0$ for all $n\in\Bbb N$.
Consequently, $0<b_{n+1}<b_n$ hence the sequence $b_n$ has limit, say $b\geq 0$, and $a_n\sim n$ as $n\to\infty$.
\begin{align}
\log\left(\frac{a_{n}-n}{a_2}\right)
&=\log\left(\frac{b_{n}}{b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2956144",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Partial integration CDF I am reading a textbook which claims that we can obtain by partial integration, for CDF $F(x)$:$$\int_{t}^{\infty} 1-F(x) \frac{dx}{x}=\int_{t}^{\infty} (\log u -\log t) dF(u) $$
I am aware that the latter integral is a Riemann-Stieltjes integral, but I am not sure how to go from the first to t... | Hope this helps:
\begin{align}
\int_t^\infty(1-F(x))\frac{dx}{x}
& = (1-F(x))\log x|_t^\infty-\int_t^\infty (-dF(x)) \log x\\
& = -(1-F(t))\log t+\int_t^\infty\log u dF(u)\\
& = -\int_t^\infty dF(u)\log t+\int_t^\infty\log u dF(u)\\
& = \int_t^\infty(\log u-\log t)dF(u).
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2956305",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Piecewise function and differentiation quotient We have function $y = f(x) = |2x + 1|$
Can we simply say that $|2x + 1|' = 2$?
No because if we use the chain rule then we get $\left(\left|2x+1\right|\right)'\:=\frac{2\left(2x+1\right)}{\left|2x+1\right|}$
Redefine the same function piecewise without using the absolute ... | Note that for the limit of a difference quotient to exist, you must have
$$\lim_{x\rightarrow x_0^{\color{red}{-}}}\frac{f(x)-f(x_0)}{x-x_0}=\lim_{x\rightarrow x_0^{\color{red}{+}}}\frac{f(x)-f(x_0)}{x-x_0}$$
You do not have that. You have
$$\lim_{x\rightarrow x_0^{\color{red}{-}}}\frac{f(x)-f(x_0)}{x-x_0}=-2\neq 2=\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2956512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Span of sine and cosine equals span of positive and negative complex exponentials proof I need to prove that
$$
\newcommand{\C}{\mathbb{C}}
\newcommand{\span}{\text{span}_\C}
\span\{\cos(kx), \sin(kx)\}=\span\{e^{ikx}, e^{-ikx}\}
$$
This is my proof but I am not 100 % sure it's correct, mainly because I'm not sure I c... | I think your proof is basically correct though, perhaps, too cumbersome. I'd go as follows:$\newcommand{\C}{\mathbb{C}}\newcommand{\span}{\text{span}_\C}$
$$
\begin{cases}\cos(kx)=\frac{e^{ikx}+e^{-ikx}}{2} \\{}\\ \sin(kx)=\frac{e^{ikx}-e^{-ikx}}{2i}\end{cases}\;\;\implies
\span\{\cos(kx), \sin(kx)\}\subset\span\{e^{ik... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2956601",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Validate the proof that the sequence $x_n = \sum_{k=1}^{n} {1\over n+k}$ is bounded.
Let $n \in \mathbb N$ and:
$$
x_n = \sum_{k=1}^{n} {1\over n+k}
$$
Prove that $x_n$ is a bounded sequence.
I'm wondering whether the proof below is valid.
Since $n \in \mathbb N$ we have that $x_n$ is strictly greater than $0$. ... | Yes, you have $(\forall n\in\mathbb{N}):0<x_n<y_n$, but asserting that the sequence $(x_n)_{n\in\mathbb N}$ is bounded means that there are constants $a$ and $b$ such that$$(\forall n\in\mathbb{N}):a<x_n<b.$$That's easy, though, after what you did. Just take $a=0$ (of course) and $b=1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2956674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Find count of this functions roots:$\sqrt{x+1}-x^2+1=0$ There is an equation here:
$$\sqrt{x+1}-x^2+1=0$$
Now we want to write the equation $f(x)$ like $h(x)=g(x)$ in a way that we know how to draw h and g functions diagram.
Then we draw the h and g function diagrams and find the common points of them. So it will be nu... | Guide:
*
*First draw $\sqrt{x}$.
*Now think of having drawn $h(x)$, how would you draw $h(x\color{red}+1)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2956791",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Difference real and complex fourier series I'm working on fourier series and I'm trying to compute the fourier transformation for the $2\pi$-periodic function of $f(x)=x^2$ with $x \in [-\pi,\pi]$.
Now with the real way, that is $$f(x) \sim \frac{a_{0}}{2}+\sum\limits_{n=1}^{\infty}a_{n}\cos(nx)+b_{n}\sin(nx)$$ and I ... | With $f(x) = x^2$, the complex Fourier series should be indexed by the integers. That is,
$$ f(x) \sim c_0 + \sum_{n=-\infty}^{\infty} c_n \mathrm{e}^{inx}, $$
where the Fourier coefficients are given by
$$ \frac{1}{2\pi} \int_{-\pi}^{\pi} x^2 \mathrm{e}^{-inx}\, \mathrm{d} x = \begin{cases}
\frac{2}{n^2}(-1)^n & \tex... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2956922",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Evaluate $\lim_{x \to 4} \frac{x^4-4^x}{x-4}$, where is my mistake? Once again, I am not interested in the answer. But rather, where is/are my mistake(s)? Perhaps the solution route is hopeless:
Question is: evaluate $\lim_{x \to 4} \frac{x^4 -4^x}{x-4}$.
My workings are:
Let $y=x-4$. Then when $x \to 4$, we have that ... | The following step is not allowed
$$\lim_{x \to 4} \frac{x^4-4^x}{x-4}=\lim_{y \to 0} \frac{(y+4)^4-4^{y+4}}{y}\color{red}{=\lim_{y \to 0} \frac{(y+4)^4}{y}-\lim_{y \to 0} \frac{4^{y+4}}{y}}$$
Refer also to the related
*
*Analyzing limits problem Calculus (tell me where I'm wrong).
*Evaluate $ \lim_{x \to 0} \left(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2957072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Question about metric function Suppose that $(X,d)$ is a metric space. Prove that $d:X\times X\to \mathbb{R}$ is a continuous.
Remark: I know that there a lot of similar topics such as this question. Please do not duplicate because the question which I am going to ask I did not meet in other topics.
Let $(x_0,y_0)$ som... | You are quite right to question what metrics or topologies you should be using when you judge whether $d : X \times X \to \Bbb{R}$ is continuous. The product topology on $X \times X$ (where $X$ is given the metric topology induced by $d$) and the standard topology on $\Bbb{R}$ are the ones that make sense.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2957205",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Combinatorics rolling a die six times How many ways are there to roll a die six times such that there are more ones than twos?
I broke this up into six cases:
$\textbf{EDITED!!!!!}$
$\textbf{Case 1:}$ One 1 and NO 2s --> 1x4x4x4x4x4 = $4^5$. This can be arranged in six ways: $\dfrac{6!}{5!}$. So there are $\dfrac{6!}{... | If the question is "where is my error", than plese ignore this answer, which is giving an other way to count. The idea is that there are either more $1$'s, case (1), or more $2$'s, case (2), or they occur equaly often, case $(=)$. Of course, the count of possibilities for case (1) is the same as for case (2), so we sim... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
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Understanding $E[x^2]$ in a variance question I know variance is equal to $V[x]=E[x^2] - (E[x])^2 $, but how do you expand $E[x^2]$ for some x if your given the necessary information...
What I mean for example is if you suppose $x = ys+(1-y)r$ then would $E[x^2] = E[(ys+(1-y)r)^2] = E[(ys+(1-y)r) * (ys+(1-y)r)]$?
I'm t... | If $x$ is your random variable, then $x^2$ is just a 'transformation' of that random variable.
Remember that the expectation operator $\text{E}[ \cdot]$, when applied to a random variable $x$, just gives you back the 'weighted average' of your support's values i.e. $\text{E}[ x] = \sum_{i \in S}\mathbb{P}(x = i)\times... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2957607",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Is it true that the number is divisible by $p$?
Question: Let $a, b, c$ be positive integers and $p>3$ be a prime ($ a$ isn't divisible by $p$).
Consider a quadratic polynomial $P(x) = ax^2+bx+c$, and assume that there exists $2 p-1$ consecutive positive integers: $$x+1, x+2, ..., x+2p-1$$
satisfying that $P(x+i)... | This is just a question of arithmetic mod $p$. Writing $\bar n$ for $n$ mod $p$, the reduction mod $p$ of the given binomial yields an $f(X)=\bar aX^2+\bar bX+\bar c \in \mathbf F_p [X]$, with $\bar a \neq \bar0$. The OP hypothesis says that $f(\bar x)$ is a square in $\mathbf F_p$ for $p$ distinct values between $\bar... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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A Special Property of e? I was debating whether I should post this question over at MathEducatorsStackExchange or here, sorry if this was the wrong forum for this question.
I am teaching Calculus I this semester and I found the following problem, which I gave to a group for one of their group challenge problems:
"Assu... | I. Why $b^{\frac{1}{\ln(b)}}=e$- note 2 logarithm properties:
*
*$\frac{1}{log_p(q)}=log_q(p)$
*$B^{log_B(C)}=C$
II. The history and (quick) winning of $e$:
Find the derivative of the function $f(x)=a^x$ by definition:
$$\displaystyle \lim_{h\rightarrow0}\frac{a^{x+h}-a^x}{h}=\displaystyle \lim_{h\rightarrow0}\frac... | {
"language": "en",
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"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 3
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Prove that $7^n+2$ is divisible by $3$ for all $n ∈ \mathbb{N}$
Use mathematical induction to prove that $7^{n} +2$
is divisible by $3$ for all $n ∈ \mathbb{N}$.
I've tried to do it as follow.
If $n = 1$ then $9/3 = 3$.
Assume it is true when $n = p$. Therefore $7^{p} +2= 3k $ where $k ∈ \mathbb{N} $. Consider no... | $7^{n} +2$ is divisible by $3$ iff $7^{n} +2 - 3 = 7^{n} -1$ is divisible by $3$.
Now, $7^{n} -1 = (7-1)(7^{n-1}+7^{n-2}+\cdots+1)$ is even divisible by $6$.
| {
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"source": "stackexchange",
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"answer_id": 5
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Does $\sqrt{i^4} = i^2$? I'm assuming it doesn't, because if it did, then $1 = \sqrt{1} = \sqrt{i^4} = i^2 = -1$. In general, does $\sqrt{x^4} = x^2$?
| In general,
$$\sqrt{x^2} = \pm x$$
so technically,
$$\sqrt{i^4} = \pm i^2$$
so what you gave is one of the possible solutions.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Given $F(x,y)$, what does it mean to compute $dF(X)$ for $X=x \frac{d}{dx}+y\frac{d}{dy}$? Given $F(x,y)$, what does it mean to compute $dF(X)$ for $X=x \frac{d}{dx}+y\frac{d}{dy}$?
My idea:
$dF(x,y)$ is the same as the Jacobian of $F(x,y)$. But in order to plug in $X$, then what should I do?
| By definition,
$$
dF(X) = \frac{\partial F}{\partial x} dx(X) + \frac{\partial F}{\partial y} dy(X).
$$
Here,
$$
dx(X)
= dx(x \partial_x + y \partial_y)
= \{ \text{ linearity } \}
= x \, dx(\partial_x) + y \, dx(\partial_y)
= x \cdot 1 + y \cdot 0 = x
$$
Likewise, $dy(X) = y.$
Thus,
$$
dF(X) = \frac{\partial F}{\pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2958318",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$\iint 1/(x+y)$ in region bounded by $x=0, y=0, x+y =1, x+y = 4$ $\iint 1/(x+y)$ in region bounded by $x=0, y=0, x+y =1, x+y = 4$ using following transformation: $T(u,v) = (u - uv, uv)$.
I want to make sure that my method is correct
Calculating the jacobian, I get $u$
since $x+y = u$, i get that $1 < u < 4$
Additional... | Yes the method is correct, indeed we are considering the following change of coordinates
*
*$u=x+y \implies 1\le u \le 4$
*$v=\frac{y}{x+y}\implies 0\le v \le 1$
indeed for any fixed value for $u=x+y\,$ we have that $y$ varies form $0$ to $u$ and therefore $v$ varies from $0$ to $1$
the jacobian is
$$du\,dv=|J|dx\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Expressiong $\frac{t+2}{t^3+3}$ in the form $a_o+a_1t+...+a_4t^4$, where $t$ is a root of $x^5+2x+2$ Expressing $\frac{t+2}{t^3+3}$ in the form $a_o+a_1t+...+a_4t^4$, where $t$ is a root of $x^5+2x+2$.
So i can deal with the numerator, but how do I get rid of the denomiator to get it into the correct form? Thanks in ad... | Using the Euclidean algorithm for computing $\gcd(x^3+3,x^5+2x+2)$, we get
$$
367=(10 x^4 - 31 x^3 - 14 x^2 - 30 x + 113)(x^3+3)+(-10 x^2 + 31 x + 14)(x^5+2x+2)
$$
and so
$$
367=(10 t^4 - 31 t^3 - 14 t^2 - 30 t + 113)(t^3+3)
$$
Thus,
$$
\begin{align}
367\frac{t+2}{t^3+3}
&=(t+2)(10 t^4 - 31 t^3 - 14 t^2 - 30 t + 113)\\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2958566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Does this equation have a complex number solution? Does this equation have any solutions:
$$\sqrt{z^2+z-7}=\sqrt{z-3}?$$
I know it does not have any real number solutions, but how about complex number solutions?
I understand that when you solve this problem algebraically, you get $z=\pm 2$ as solutions. But when you in... | You must consider what is meant by $\sqrt\;$. There are several possible square root functions here: one has domain $\mathbb R^+$ and codomain $\mathbb R$, one has domain $\mathbb R$ and codomain $\mathbb C$, and one has domain $\mathbb C$ and codomain $\mathbb C$.
If you use the first function, then it is undefined wh... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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Proving $x^2 - 4y^2 = 7$ has no natural numbers Ok so I needed to prove this by contradiction.
Let $P:~x^2 - 4y^2 = 7$ and $Q:~x,y$ are not natural numbers
Note that $N$ does not include $0$
OK to begin to prove by contradiction we are given $P$ and I would assume $\text{not}(Q)$ so the equation would have natural num... | Suppose there is an integer solution. since $y>0$, we have $x-2y < x+2y$.
Hence we have $x-2y =1$
and $x+2y=7$.
if we subtract them, we have $4y = 6$ and $y = \frac32$ which is a contradiction.
Alternatively, think of what values can $x^2 \pmod{4}$ take. Just take modulo $4$ and you can see the contradiction.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Does a real number with this decimal expansion for $r$ and $r^2$ exist? Does there exist a real number $0< x <1$, such that the decimal expansions of $x$ and $x^2$
are the same, starting from the
millionth term, and neither expansion has an infinite tail of zeroes?
I was thinking $x=0.\overline{999}$, but... | We can concoct an example quite easily. Suppose we want the difference between $x$ and $x^2$ to be 0.1:
$$x-x^2=0.1$$
where the order $x-x^2$ is mandated by $0<x<1$, so $x^2<x$. Solving this, we get two admissible values $x=\frac{1\pm\sqrt{0.6}}2$.
Thus (taking $x=\frac{1+\sqrt{0.6}}2$) we have
$$x=0.88729833\dots$$
$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2958853",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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An ellipse intrinsically bound to any triangle Given any triangle $\triangle ABC$, we build the hyperbole with foci in $A$ and $B$ and passing through $C$. The hyperbole always intersects the side of the triangle that is opposite to the vertex through which it pass in two points $D$ and $E$.
Similarly, we can build ot... | Showing that the points in question lie on a common conic is straightforward.
I've renamed the points thusly: $D_B$ and $D_C$ are the points where the hyperbola through $A$ meets $\overline{BC}$; the subscripts indicate the closer vertex. Likewise for $E_C$, $E_A$, $F_A$, $F_B$.
Now, simply note that
$$|BD_B| = |CD_C... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Complex Anti-derivative of tan(z)
Show $tan(z)$ has a complex anti-derivative on $S=\mathbb{C}\backslash((-\infty,-\pi/2]\cup[\pi/2,\infty))$
If F(z) is the complex antiderivative of $tan(z)$ on $S$, find $F(i)$, if $F(0)=0$
I know that for $D\subset \mathbb{C}$ open and starshaped, and if $f:D\rightarrow \mathbb{C}$... | Answer to second part: the antiderivative vanishing at $0$ is given by the integral $\int_{[0,z]} tan (\zeta)\, d\zeta$. so $F(i)=\int_{[0,i]} tan (\zeta)\, d\zeta =i\int_0^{1} tan (it)\, dt$. Now $\tan (it)=\frac {i\sinh t} {\cosh (t)}$. Make the substitution $u=\cosh t$ to evaluate the integral.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How many positive divisors of 2000? I thought that the number of divisors of a number was the product of the indices in its factorisation, plus $2$ (for 1 and the number itself). For instance, $2000=2^{4} \cdot 5^{3}$, so it would have $4 \cdot 3 + 2 = 14$ divisors.
Apparently, however, $2000$ actually has $20$ diviso... | The factors are of the form of $2^x\cdot 5^y$ where $x$ takes value from $0$ to $4$ and $y$ takes values from $0$ to $3$.
Hence there is a total of $(4+1)(3+1)=20$ factors.
Your method miss out number such as $5^y, y>0; 4^x, x>0$ and double counted $2000$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 3
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What are relations? I understand that a function may be considered as a set of ordered pairs which relate the elements between two sets. I understand that a function is a subset of the cartesian product between the two sets and it can be defined by an equation like $y=x+1$ or $f(x)=x+1$, on a specified domain for $x$.
... | A function is a "special" relation where there are no two pairs $(x,y_1)$ and $(x,y_2)$.
Example : the relation "father-to-child" is not a function, because a father may have more than one children.
The relation "child-to-father" is a function because a child has exactly one father.
"Less than" is a relation but not ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2959414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Intro to Classical Number Theory I am having trouble understanding page 53, of A Classical Introduction to Modern Number Theory, by Kenneth Ireland and Michael Rosen.
Corollary 3. $$(-1)^{(p-1)/2} = \left(\frac{-1}{p}\right)$$ Where the right-hand side is the Legendre symbol.
The part I'm tripping over is the next... | The congruence has a solution iff the Legendre symbol is equal to $1$ which is the case iff the power of $-1$ in the explicit formula for the Legendre symbol is even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2959541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How to solve $(x \mod 7) - (x \mod 8) = 5$? I'm trying to solve $(x \mod 7) - (x \mod 8) = 5$ but no idea where to start. Help appreciated!
| If CRT = Chinese Remainder Theorem is known then we can reformulate it as
$$\begin{align}
x&\equiv a\!+\!5\!\!\pmod{7},\ \ \ \overbrace{ 0\le a\!+\!5 \le 6}^{\Large \iff a\ =\ 0,1}\\
x&\equiv a\quad\pmod{8},\ \ \ 0\le a\le 7\end{align}\qquad \qquad\qquad $$
Solve the $\,a=0\,$ case $\,x\equiv (5,0)\bmod (7,8),\,$ th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that the exact values of the square roots of $z=1+i$ are... Show that the exact values of the square roots of $z=1+i$ are...
$w_0=\sqrt{\frac{1+\sqrt{2}}{2}}+i\sqrt{\frac{-1+\sqrt{2}}{2}}$
$w_1=-\sqrt{\frac{1+\sqrt{2}}{2}}-\sqrt{\frac{-1+\sqrt{2}}{2}}$
My attempt
Let $z=1+i\in \mathbb{C}$. Then
$r=|z|=\sqrt{2}$
... | Setting $$\sqrt{1+i}=A+Bi$$ then $$1+i=A^2-B^2+2ABi$$ and we have to solve
$$A^2-B^2=1$$
$$2AB=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2959857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Maximise $(x+1)\sqrt{1-x^2}$ without calculus Problem
Maximise $f:[-1,1]\rightarrow \mathbb{R}$, with $f(x)=(1+x)\sqrt{1-x^2}$
With calculus, this problem would be easily solved by setting $f'(x)=0$ and obtaining $x=\frac{1}{2}$, then checking that $f''(\frac{1}{2})<0$ to obtain the final answer of $f(\frac{1}{2})=\fra... | You can solve the problem using geometry.
$(x+1)\sqrt{1-x^2}$ is the area of triangle
$$(-1, 0), (x, \sqrt{1-x^2}), (x, -\sqrt{1-x^2}).$$
This unit inscribed triangle has maximal area if and only if it is a equilateral triangle.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
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Is this a correct interpretation of the fundamental theorem of algebra? I tried reading the Wikipedia page but it's stated in terms of complex roots, and I don't really understand how that relates to the following proposition:
if a real valued polynomial:
$$\sum_{i=0}^n a_i x^i = 0$$
for all $x \in \mathbb{R}$, is it r... | There are two questions in your post. The answers to them are different.
Is this a correction interpretation of the fundamental theorem of algebra?
No, according to the standard meaning of "fundamental theorem of algebra," which says every polynomial with real coefficients has at least one complex root. That theorem ... | {
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"timestamp": "2023-03-29T00:00:00",
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Is this the correct interpretation of the differential? I am going through Tenenbaum and Pollard's book on differential equations and they define the differential $dy$ of a function $y = f(x)$ to be the function $$
(dy)(x,\Delta x) = f'(x) \cdot (d\hat{x})(x, \Delta x)
$$
where
*
*$\Delta x$ is a variable denoting a... | Yes, differentials and derivatives are 2 ways of doing the same thing to a function.
The derivative operator $\frac{d}{dx}$ hits functions and $\frac{d}{dx}f$ is itself a function. The differential operator $d$ hits functions and $df$ is itself a function. Therefore with this definition of $df$, you can think of deriv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Binomial Distribution: Stochastic Dominance Suppose
*
*$X_1 \sim \operatorname{Bin}(N_1,p)$ and $X_2 \sim \operatorname{Bin}(N_2,p)$
*$N_2>N_1$
Does $X_2$ first-order stochastically dominate $X_1$?
| Yes! One proof I know is via coupling. Let $X_{1},\ldots,X_{n}$ be an IID sequence of Bernoulli random variables
with $P\left[X_{i}=1\right]=p$. Also, we have $S_{n}:=\sum_{i=1}^{n}X_{i}\sim\text{Bin}\left(n,p\right)$.
Then $S_{n+1}\geq S_{n}$. Hence, $P\left[S_{n+1}>s\right]=P\left[\sum_{i=1}^{n+1}X_{i}>s\right]\geq P... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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What is the probability that some random event won't happen in the next 10 minutes given it happened exactly twice in the last 120 minutes? The title says pretty much all. The event is fully random, it has the same chance any minute, it can happen multiple times at the same minute. How is it possible to calculate that?... |
I'm not really familiar with random distribution types
You need to get a grasp of probability distributions to understand the problem and solution.
You can think of the time between the events i.e. some one passing through the path that you are looking at as a random variable. It could be 1 minute or 8 minutes or 1... | {
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Is it allowed to solve this inequality $x|x-1|>-3$ by dividing each member with $x$? Is it allowed to solve this inequality $x|x-1|>-3$ by dividing each member with $x$? What if $x$ is negative?
My textbook provides the following solution:
Divide both sides by $x: $ $\frac { x | x - 1 | } { x } > \frac { - 3
} { x }... | For $x\geq0$ this inequality is always true.
Assume that $x<0$, so $x=-y$ for some positive $y$ and we get $$y|\;\underbrace{y+1}_{>0}\;|<3\implies y(y+1)<3 ...$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2961023",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Number of positive integer solutions of inequality I was preparing a class on polynomials (high school level). The handbook I use always contains some questions from math olympiads. The following question is asked:
What is the number of positive integer solutions of the following inequality:
$$(x-\frac{1}{2})^1(x-\frac... | This may look shorter, but then again it is not really different from your argument:
The polynomial changes its sign at $\frac12, \frac32,\ldots, \frac{4021}2$ and is positive as $x\to+\infty$, hence positive for $x>\frac{4021}2$. In the $2010$ intervals determined this way, the polynomial is alternatingly positive and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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GRE question: maximize $\int_{0}^{3}f(x)dx$ where $f$ is a differentiable function such that $f(3)=7$ and $f'(x) \geq x$ for all $x>0$ Let $f$ be a differentiable real valued function such that $f(3)=7$ and $f'(x) \geq x$ for all positive $x$. What is the maximum possible value of $$\int_{0}^{3}f(x)dx$$
The answer is ... | Between $x=0$ and $x=3$, $f'(x)>0$ so $f$ increases. So we've got an increasing function on $[0,3]$ and we want to maximize its integral. Since the function is fixed to go through $(3,7)$, the steeper it is the lower the value of the integral. (As an example, see $f_1$ and $f_2$ below, $f_2$ is steeper so it has a sma... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How to determine if a surjective homomorphism exists between two groups? My question sheet asks about whether a surjective homomorphism exists between various symmetric groups and various $Z_n$ groups, for example between $S_3$ and $Z_3$, or $A_4$ and $Z_3$. To be honest, I don't really know where to start at all - I'v... | A good place to start would be the first isomorphism theorem: if $ f: G\rightarrow H$ is a homomorphism, then $G/\ker f\simeq \operatorname{Im} f$. So if $f : G \rightarrow H$ is a surjective homomorphism, then $G / \ker f \simeq H$ since surjectivity implies $\operatorname{Im} f = H$. So to determine whether such a ho... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving that $\sum\limits_{n=1}^∞\frac{a_n}{s_n^2}$ converges
Let $a_n>0\;$ ($n=1,2,...\,$) with $\sum\limits_{n=1}^\infty a_n$ divergent and $s_n =\sum\limits_{k=1}^na_k$. For all $n \ge 2$, prove that $\sum\limits_{n=1}^∞\dfrac{a_n}{s_n^2}$ converges.
Proof: For all $n \ge 2$, we have $\dfrac{a_n}{s_n^2} \le \df... | Your proof is correct if $a_n$ is nonnegative since it then follows that
$$\frac{a_n}{S_n^2} \leqslant \frac{a_n}{S_nS_{n-1}}= \frac{S_n - S_{n-1}}{S_nS_{n-1}}= \frac{1}{S_{n-1}}- \frac{1}{S_n}$$
and the RHS has a telescoping sum.
You also need to make it clear that $1/S_n \to 0$ as $n \to \infty$ to argue that the tel... | {
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"question_score": "3",
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Why is Monte Carlo integration randomly sampled? As I understand, Monte Carlo integration uses stochastic sampling to sample points.
Obviously, you would need many samples to reach an accurate result, but why does this process have to be random?
Would using a symmetrical grid of very dense samples (e.g. a 1 million by... | I just wanted to revisit this question to provide what I feel is a more intuitive explanation of why a randomly sampled algorithm is faster at providing a result.
Look at the two image below. Both do not show the complete, full quality image. They are both approximations of the actual image (which is analogous to the g... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Convergence in distribution of the sum of two dependent random variables I have the following question about the limiting distribution of the sum of two random variables say $Z_n = X_n+Y_n.$ I know the following:
*
*Conditioned on $X_n,$ $Y_n$ has a CLT i.e.,
$$\mathbb P (Y_n \le z | X_n) \to \phi(z)$$
where $\phi... | Use characteristic functions. $Ee^{it(X_n+Y_n)} =E e^{itX_n}E(e^{itY_n}|X_n)$. Note that $E(e^{itY_n}|X_n) \to \phi (t)$ uniformly and $E e^{it(X_n)} \to \phi (t)$. It follows easily from these that $Ee^{it(X_n+Y_n)} \to \phi (t)^{2}$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Clarification of definition of quotient group - Does $Na, a \in G$ mean "cosets of $N$ in $G$"? The definition of quotient group from mathworld.wolfram.com is:
For a group $G$ and a normal subgroup $N$ of $G$, the quotient group of $N$ in $G$, written $G/N$ and read "G modulo N", is the set of cosets of $N$ in $G$.
... | The notation "$Na$" means exactly "the coset of $a$ in $G/N$."
So $G/N$ is exactly the collection of cosets $\{Na\mid a\in G\}$. Note that this set notation does not irredundantly list the cosets.
I don't think "the set of cosets which have elements $Na, a \in G$" is a valid way of saying it. The cosets don't have $Na$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Distance of a plane I have posted this question in Stack Overflow programming forum. Someone there feels it might be more suited to Mathematics.
I have to warn you I am rusty in math and was terrible in Algebra.
Using Accord Framework plane class. The help on this page specifies a constructor that creates a plane a c... | Either the API or the documentation are wrong.
Regardless the other arguments, Offset is deemed to specify the distance from the plane to the origin, and DistanceToPoint should return the same value.
I strongly suspect that Offset is in reality the parameter d of the equation, and the normal vector is never normalized... | {
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Is this a proof of "$\log(a^x) = x\log(a)$"? I'm not sure if it's right, but could this be proof of "$\log(a^x) = x\log(a)$"?
For example $\log(1000) = \log(10*100)=\log(10) + \log(100) = \log(10) + \log(10*10) =
\log(10) + \log(10) + \log(10) = 3\log(10)$
| This is not a proof, your statement is an example. In order to prove a relationship like
\begin{equation}
\tag{1}
\label{1}
\log(a^x) = x\log(a)
\end{equation}
you have to take into account all possible values of $x$ and $a$ for which \ref{1} should hold. A simple version of the proof you are looking for can be done by... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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A question about how to show two polynomials are not equal Let $f\in k[x_i,i=0,\ldots ,n]$ be a a polynomial which contains $x_0$ (i.e. polynomial like $f=x_1$ is not allowed), where $k$ is a field with characteristic $0$. We define a map
$$Q_a: k[x_i,i=0,\ldots ,n] \to k[x_i,i=1,\ldots ,n]$$
which sends $f(x_0,\ldots,... | Start from the Leading coefficient,(If two were equal) the coefficient of the highest power of the polynomial must be equal, then the second highest must be equal ...
As for the proof, you may consider use induction on the degree of polynomials.
| {
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Create a model from the given text (linear programming/optimization) I'm practicing for a linear programming test and here is a task I like to see if I did it correct and if not maybe how to do it correctly? Need to create a mathematical model whose requirements are represented by linear relationships.
A company wants... | The company wants to maximize the profit: $4000x_1 +3000x_2 \rightarrow \color{red}{\max}$
The company has (at most) 720 hours time for the installation:
$4x_1+6x_2 \color{red}{\leq} 720$
... and (at most) 480 hours for the finishing within a
production-cycle:
$6x_1+3x_2 \color{red}{\leq} 480$
The other two const... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Are Baire class functions closed under pointwise limits? I am confused about the notion of Baire functions (real or complex valued) on a compact space $X$.
The set of Borel functions on $X$, $Bo(X)$ is defined to be the set of those functions $f$ for which $f^{-1}(U)$ is a Borel set, when $U$ is open.
On the other h... | It's quite easy, really: let $f_n$ be a sequence of Baire functions (so from $\operatorname{Ba}(X)$) tending pointwise to $f$. By definition for each $n$, $f_n \in \operatorname{Ba}_{\alpha_n}(X)$ for some $\alpha_n < \omega_1$. In $\omega_1$, every countable set has an upper bound so there is some $\beta_0 < \omega_1$... | {
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"source": "stackexchange",
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Does anybody know the name of the discrete distribution with these properties? I'm looking for a distribution which has the following properties. I don't know what it's called so I'm having a hard time finding references to it.
Properties:
*
*Domain is over a finite range of integers (distribution is discrete and t... | The solution is again a discrete, truncated Gaussian. With Lagrange multipliers, the objective function is
$$
\sum_n p_n\log p_n+\lambda\sum_np_n+\mu\sum_nnp_n+\nu\sum_nn^2p_n\;.
$$
Setting the derivative with respect to $p_i$ to zero yields
$$
\log p_i+1+\lambda+\mu i+\nu i^2=0\;,
$$
which yields a Gaussian, with the ... | {
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Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$
Let $a, b, c \in \mathbb{R^+}$ and $abc=8$ Prove that $$\frac {ab+4}{a+2} + \frac {bc+4}{b+2} + \frac {ca+4}{c+2} \ge 6$$
I have attempted multiple times in this question and the only method that I ... | Let $a=\frac{2y}{x}$ and $b=\frac{2z}{y}$, where $x$, $y$ and $z$ are positives.
Thus, $c=\frac{2x}{z}$ and by AM-GM we obtain:
$$\sum_{cyc}\frac{ab+4}{a+2}=\sum_{cyc}\frac{\frac{4z}{x}+4}{\frac{2y}{x}+2}=2\sum_{cyc}\frac{z+x}{x+y}\geq6\sqrt[3]{\prod_{cyc}\frac{z+x}{x+y}}=6.$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 3
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Question about commutator of normal subgroups Let $\mathbf G$ be a group and $H, K \trianglelefteq \mathbf G$ ($H$ and $K$ are normal subgroups of $\mathbf G$).
It follows that $[H,K]$, the subgroup of $\mathbf G$ generated by elements of the form $h^{-1}k^{-1}hk$, with $h \in H$ and $k \in K$ is also a normal subgroup... | Let $G=H=K=\langle t \rangle$ where $t$ is of order $4$.
Let $a=b=1$, $c=t^{-1}$, $d=t$.
Then $a^{-1}b=1\in [H,K]$.
Also $a^{-1}c=t^{-1}\in H$, and $b^{-1}d=t\in H$.
Then $c^{-1}d=t^2\in K$. But $c^{-1}d\not=1$, so $c^{-1}d\not\in [H,K]$.
| {
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Projection definition in Linear Algebra Why is a mathematical projection defined as $P^2 = P$? I understand that it might be because once a vector has been projected onto a space, if projected again, it should give the same thing. Is there anything more to it?
| There's little more. If $P$ is a projection and $w\in\operatorname{Im}P$, we want to have $P(w)=w$. This is equivalent to the assertion:$$(\forall v\in V):P\bigl(P(v)\bigr)=P(v).$$And this, in turn, is equivalent to $P^2=P$. $\phantom{}$
| {
"language": "en",
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Implied meaning of lower case latin letters $p$, $q$, $u$ etc. in probability? Are there any generalizations that should be assumed for (lower case) $p$, $q$ and $u$ in probability theory? For example, $q$ is often assumed to signify, in relation to $p$, that
$$
q=1-p.
$$
Is there a standard implied meaning/relation th... | In general, one should not rely on notation to be common across the literature, but nonetheless conventions exist. Here are a few that I am aware of:
General
*
*$i,j,n,m,k,l \in \mathbb{N}$ indices
Probabilistic Setting
*
*$\mu = \mathbb{E}[X]$: mean of a random variable $X$
*$\sigma = \text{Var}[X]$: variance... | {
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Strange succession
An ant starts from the origin of a cartesian plane and makes $n \ge 2$ steps of lengh $d_1, d_2, \cdots, d_n$. Is there a condition (necessary and sufficient) on $d_i$'s for which the ant can come back to the origin after the $n$ steps? (The ant can move in any direction.)
I think the claim is $d_... | Your condition is clearly necessary and there is no sufficient condition. At each step, even if the distance is right you only reach the origin if the angle you choose is exactly right, representing a point out of $[0,2\pi)$. As there are only a countable number of steps, there are only (at most) a countable number o... | {
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Find $n$ such that $\int_1^n \lfloor{x}\rfloor\lfloor{\sqrt x}\rfloor dx>60$
Find the smallest positive integer $n$ such that $$\int_1^n \lfloor{x}\rfloor\lfloor{\sqrt x}\rfloor dx>60$$
where $\lfloor.\rfloor$ is the GIF. I couldn't find any decent method rather than explicitly evaluating it to a few terms by brute f... | The function $f(x):=\lfloor x\rfloor\cdot\lfloor\sqrt{x}\rfloor$ satisfies $$f(x)=\left\{\eqalign{\lfloor x\rfloor\qquad&(1\leq x<4)\cr 2\lfloor x\rfloor\qquad&(4\leq x<9)\ .\cr}\right.$$
It follows that
$$\int_1^8 f(x)\>dx=1+2+3+8+10+12+14=50\>, $$
and $\int_1^9f(x)\>dx=50+\int_8^9f(x)\>dx=66$.
The answer to your ques... | {
"language": "en",
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"source": "stackexchange",
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Convergence of negative geometric series in the p-adic integers In real analysis, I am learning about convergence of series in metric spaces. My professor states that in the metric space of $\mathbb Z$ with the $2$-adic metric, one of the series converges and the other does not:
$$ \sum_{n=0}^\infty 2^n$$
$$ \sum_{n=0}... | Both of these series definitely converge in $\mathbb{Z}_2$. As you say, the sequences of partial sums are both Cauchy.
However, the second series converges to an element of $\mathbb{Z}_2$ which is not an element of $\mathbb{Z}$; namely, $1/3$. To see this, use the old geometric series trick:
$$
\frac{1}{3} = \frac{1}{1... | {
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Show that if G is a planar, simple and 3-connected graph, then the dual graph of G is simple and 3-connected I've been thinking about this question for several days now and I haven't come up with a satisfactory answer yet. The part of proving that the dual graph is simple (under the assumption that the original graph i... | If You can make the dual graph disconnected by removing two edges, this means that after removing one edge, there is a bridge, i.e., an edge with the same face on both sides. This means that before removing the first edge, there were two faces sharing two edges. In the original graph this means two vertices joined by t... | {
"language": "en",
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Closed form for a sum Please, i need help with this example, step by step.
Calculate the value of the next summation, i.e. express simple formula without the sum:
$$\sum_{n_1 + n_2 + n_3 + n_4 = 5} \frac{6^{n_2-n_4} (-7)^{n_1}}{n_1! n_2! n_3! n_4!}$$
I think the formula is
$\sum_{n1 + n2 + n3 + n4 = n} = \frac{n!* x^... | Hint. The multinomial theorem is
$$(x_1 + x_2 + \ldots + x_k)^n = \sum_{n_1 + \cdots + n_k = n} {n \choose n_1, \ldots, n_k} x_1^{n_1} x_2^{n_2} \cdots x_k^{n_k}$$
Now let $k=4$, $n=5$ and find values of $x_1,x_2,x_3,x_4$ to make the right-hand side summand look a bit like the summand in your expression. Note that $$... | {
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How to show that $\lim_{x \rightarrow \infty } \frac{p(x)}{2^{\sqrt x}} = 0$ I want to show that the below limit is 0 for any polynomial $p(x)$ with degree $n$
$$\lim_{x \rightarrow \infty } \frac{p(x)}{2^{\sqrt x}} = 0$$
If I apply the l'Hopital's Rule, the numerator, eventually, will be zero. What about the denumerat... | It suffices to show
$\lim_{x \rightarrow \infty}\dfrac{x^n}{2^{√x}} =0$
(why?).
1) Let $y=√x, y >0.$
Then $F(y)= \dfrac{y^{2n}}{2^y}.$
2) Let $e^a=2,$ $a >0.$
$F(y)= \dfrac{y^{2n}}{e^{ay}}.$
$e^{ay} \gt \dfrac{(ay)^{2n+1}}{(2n+1)!}$
(Series expansion).
$F(y) \lt \dfrac{(2n+1)! y^{2n}}{(ay)^{2n+1}}=$
$\dfrac{(2n+1)!}{a... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Deriving $\sqrt2 \approx 1 + \frac13 + \frac1{3 \cdot 4} - \frac1{3 \cdot 4 \cdot34}$ Here is a wierd expansion for $\sqrt2$ found in the ancient Indian mathematical literature.
$$1 + \frac13 + \frac1{3 \cdot 4} - \frac1{3 \cdot 4 \cdot34} = \frac {577}{408}$$
Today we know that the resulting fraction can be obtained u... | EDIT: This is an answer for an algorithm to generate the unit fractions of the expansion :
$$\sqrt2 \approx 1 + \frac12 - \frac1{3 \cdot 4} - \frac1{3 \cdot 4 \cdot34}$$
(an answer to the actual question is provided by Gerry Myerson in his first comment)
$$-$$
This (signed) Egyptian fraction may be obtained by starting... | {
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Prove that $T_n = 3n^2 -60n + 301$ is positive for every $n$ I recently did a Mathematics exam from a previous year, and I stumbled across a question's answer I struggled to fully understand.
It is given: The quadratic pattern $244 ;~ 193 ;~ 148 ;~ 109;~ \ldots$
I've determined the $n$-$\textrm{th}$ term as $T_n = 3n^2... | Completing the square enable you to see the vertex clearly. For $f(x)=ax^2+bx+c$ where $a>0$, we can see the minimal value that it can attain.
Alternatively, just see that the discriminant is negative, that is the function doesn't intersect the $x$-axis and it doesn't change sign. Since one of the term is positive, eve... | {
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If $f$ is a polynomial, how does $f(\frac{d}{dt})$ act on $y$? If $f\left(\frac{d}{dt}\right)=a_n\frac{d^n}{dt^n}+\dots+a_1\frac{d}{dt}+a_0$, then whether
$$f\left(\frac{d}{dt}\right)(y)=a_n\frac{d^n}{dt^n}y+\dots+a_1\frac{d}{dt}y+a_0y$$
or
$$f\left(\frac{d}{dt}\right)(y)=a_n\frac{d^n}{dt^n}y+\dots+a_1\frac{d}{dt}y+a_0... | The correct answer is $$(\frac{d}{dt})(y)=a_n\frac{d^n}{dt^n}y+\dots+a_1\frac{d}{dt}y+a_0y$$
Otherwise you lose the linearity of your operator.
| {
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Proof by contradiction. Statement negation This should be an easy question. Yet, the provided solution confuses me.
The question comes from "Understanding analysis" by S. Abbot, 2nd edition (Exercise 1.2.11).
Negate the statement. Make an intuitive guess as to whether the claim or its negation is the true statement.
... |
There exists a real number $x > 0$ such that $x < 1/n\;\;\forall n \in \mathbb{N}$.
Is this statement really valid? Let's check.
$nx<1 \;\;\forall n\in\mathbb{N}$ is valid only for $x\leq 0$.
(Because $n$ becomes very large, and if $x\gt 0$ then $nx$ diverges to infinity)
So there is no such $x$ exist.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2964449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Forward Euler Method Given Two Step Sizes I am attempting to compute an approximation of the solution with the forward Euler method in $[0,1]$ with step lengths $h_{1}= 0.2$, $h_{2}= 0.1$ given the initial value problem below
$$\frac{dy}{dz}=\frac{1}{1+z}-y(z)\quad y(0)=1$$
I am not sure what to do when I am given two ... | The problem asks for solving the differential equation twice. Once for the step size of $h=.1 $ and once for the step size of $h= .2$ and compare the results.
As you know different step sizes give you different results with the smaller step size smaller error is made .
| {
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Proving $\int_0^\pi \frac{\log(1+x\cos (y))}{\cos y}dy=\pi \arcsin x$ How would I go about proving that, $$\int_0^\pi \frac{\log(1+x\cos (y))}{\cos (y)}\,dy=\pi \arcsin (x)$$
I have tried to do this by computing the integral directly, but it appears to be too difficult. Maybe there is a better approach to this that I d... | HINT:
Using Feynman's Trick (differentiating under the integral), we have for $|x|<1$
$$\begin{align}
\frac{d}{dx}\int_0^\pi \frac{\log(1+x\cos(y))}{\cos(y)}\,dy&=\int_0^\pi \frac{1}{1+x\cos(y)}\,dy\tag1
\end{align}$$
Use contour integration or apply the Weierstrass substitution to evaluate the integral on the right-ha... | {
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Differentiation Operator is not a bounded operator for polynomials If you consider the space of all polynomials on [0,1] (defined as $P_{[0,1]}$ as subspace of $C_{[0,1]}$) then the differentiation operator is not a linear bounded operator on this space. Why is that? this doesn'
t make any sense to me.
| There's no reason to assume that the operator is bounded here. The norm on $C_{[0, 1]}$ measures the size of a function in terms of its values, but the derivative is really about steepness. A function can be arbitrarily steep even though it takes only small values.
To be explicit, the norm of $x^n$ is clearly $1$, whil... | {
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"source": "stackexchange",
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Linear-algebra first course problem about orthogonal matrices I am trying to demonstrate next assert about matrices:
$A$ is a matrix of $n$ order, with $n$ odd, that obeys $A A^T =I$ and $\det\, A=1$. Then $\det\,(A-I)=0$.
I have tried a number of things but none of them work. That $n$ is odd seems to indicate to the t... | We have $\left(-1\right)^n = -1$ (since $n$ is odd). But
\begin{equation}
\left(A-I\right)A^T = \underbrace{AA^T}_{=I} - A^T = I-A^T = \left(I-A\right)^T .
\end{equation}
Taking determinants of both sides of this equality, we obtain
\begin{align}
\det\left(\left(A-I\right)A^T\right)
&= \det\left(\left(I-A\right)^T\righ... | {
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How to show $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ given by $f(x,y)=x^3$ is an open map? I know for instance that $g: \mathbb{R} \rightarrow \mathbb{R}$ given by $g(x)=x^3$ is open. Since $g$ is a homeomorphism, then its inverse $g^{-1}$ is continuous. Let $V \subset \mathbb{R}$ be a open set, then $(g^{-1})^{-1}(V)=... | Hint:
$f$ is the composition of the $1$st projection, which is an open map, by the cube function, which is a homeomorphism, hence is an open map.
| {
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"url": "https://math.stackexchange.com/questions/2965102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If rank$(A) = 2$, then $A^2 \neq 0_3$
Let $A$ be a real $3 \times 3 $ matrix such that rank$(A) = 2$.
Prove that $A^2 \neq 0_3$. where $0_3$ represents the null matrix of order $3$.
I am looking for a solution involving only basic manipulation using matrices. I already have a better solution using the range and the n... | Let us write $A=(C_1,C_2,C_3)$ where $C_i$ is the column $i$ of $A.$
Assume $A^2=0.$ That is, we have that $C_1,C_2,C_3$ are two linearly independent solutions of the system $Ax=0.$ But since $A$ is of order $3$ and has rank $2$ this is not possible.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2965228",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to prove $f(x) = 4x^{3} + 4x - 6$ has exactly one real root?
How can I show that $f(x) = 4x^{3} + 4x - 6$ has exactly one real root?
I think the best way is to show $f'(x) = 12x^2 + 4 > 0$ for all $x \in \mathbb{R}$. Thus, $f'(x)$ has zero real roots. Thus, $f(x)$ has at most one real root.
I thought about tryi... | Your intuition for the first one is correct!
$f(0)<0$ and $f(1)>0$, so by IVT $f$ has a root say $x_0$
Suppose $f$ has another root $x_1 \neq x_0$ with $x_0<x_1$ .Then $f(x_0)=f(x_1)=0$ and by Rolles theorem $\exists$ $c \in (x_0,x_1)$ such that $f'(c)=0$, contradicting to the fact $f'(x)>0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2965499",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 4,
"answer_id": 0
} |
Category theory - Prove that $\operatorname{Hom}$ preserves representations for quasi-inverse functors
Let $F: \mathcal C \to \mathcal D$ and $G: \mathcal D \to \mathcal C$ be quasi-inverse functors, and let $H : \mathcal C \to Set$ be a representable (contravariant) functor with representative $X \in \mathcal C$. Pro... |
$\newcommand\cat\mathscr\DeclareMathOperator\id{id}$Let $F:\cat C\rightleftarrows\cat D:G$ be quasi-inverse functors.
Then $F,G$ are fully faithful and there exists natural isomorphisms $\varepsilon:F\circ G\to\id_{\cat C}$ and $\eta:\id_{\cat D}\to G\circ F$ such that
\begin{align}
&\eta_GG(\varepsilon)=1_G&
&F(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2965615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Example of a metric space where diameter of a ball is not equal twice the radius My question is regarding the notion of balls in metric spaces, and specifically about their diameters. If $(X,d)$ is a metric space and $A \subset X$, then the diameter of $A$ is defined by
$$
d(A) = \sup \{ d(a_1,a_2) : a_1 \text{ and } a... | Consider the discrete metric $d$ on a set $X$:
$$d(x,y)=\begin{cases}
0,&\text{if }x=y\\
1,&\text{if }x\ne y\;.
\end{cases}$$
Consider the ball of radius $r=1/2$ centered at $x$
Then $B(x,r)=\{x\}$
Now by definition, $\operatorname{diam} A = \sup\{ d(a,b) : a, b \in A \}$
Applying it to our case where $A=B(x,r)$, we h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2965697",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 7,
"answer_id": 4
} |
How would you integrate $\frac{Si(x)}{x}$? The function $Si(x)$ can be obtained when we integrate
$\frac{\sin(x)}{x}$. But how would we go about integrating $\frac{Si(x)}{x}$?
More information about the function $Si(x)$ can be found here https://en.wikipedia.org/wiki/Trigonometric_integral
Edit: Just checked wolframal... | There is no reason for suspecting that the antiderivative of $\operatorname{Si}(x)/x$ can be expressed in terms of “known” functions.
The power series expansion of $\operatorname{Si}(x)$ is
$$
\operatorname{Si}(x)=\sum_{n\ge0}\frac{(-1)^nx^{2n+1}}{(2n+1)^2\cdot(2n)!}
$$
Therefore the power series expansion of $\operato... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2965902",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
How to prove that $3^{x^2+x} (x+1)^{-x} \Gamma (x+1)\ge 1$ for $x>0$? Let
$$f(x)=3^{x^2+x} (x+1)^{-x} \Gamma (x+1).$$
Drawing a picture with any computer algebra system,
it is obviously that $f(x) \ge 1$ on $[0,\infty)$.
But How can we prove this? If we take derivative, then we get
$$
\frac{\mathrm d}{\mathrm dx}\lo... | Here's a proof for $x > 1$.
If $c > 1$,
since
$x! > \sqrt{2\pi x}(x/e)^x$
for $x > 1$,
if $x > 1 $ then
$\begin{array}\\
f(x)
&=c^{x^2+x} (x+1)^{-x} \Gamma (x+1)\\
&=c^{x^2+x} (x+1)^{-x} x!\\
&>c^{x^2+x} (x+1)^{-x} \sqrt{2\pi x}(x/e)^x\\
&=\sqrt{2\pi x}\left(c^{x+1} \dfrac{x}{e(x+1)}\right)^x\\
&>\sqrt{2\pi x}\left( \d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2966021",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Formula for the least element on the spectrum Let $A$ be a self-adjoint operator defined on a dense subset of an Hilbert space $\mathcal{H}$. Assume that $A$ is bounded below in the sense there is $m \in \mathbb{R}$ such that
$$\langle Ax,x\rangle \geq m,~\forall x : \|x\| = 1.$$
I want to show that:
$$ m = \inf\{\lamb... | Let $m=\inf\;\{ \lambda : \lambda\in\sigma(A) \}$. Then, for every positive integer $n$, $E_{A}[m,m+1/n] \ne 0$. So there exists a unit vector $x_n\in\mathcal{D}(A)$ such that $E_{A}[m,m+1/n]x_n = x_n$, which gives
\begin{align}
0 & \le \langle (A-mI)x_n,x_n\rangle \\
& = \int_{m}^{m+1/n}(\lambda-m) d\langle E(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2966123",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
} |
Stirling Numbers of second kind defined in terms of coefficients Prove that $t^n = \sum_{k=1}^n S(n,k)t^{\underline k}$ where $t^{\underline k}$ denotes the $k$-th falling power $t(t-1)(t-2)\ldots(t-k+1)$ of$~t$.
I know that we'll have to use the recurrence relation for $S(n,k)$ here but since the summation itself invo... | For the induction proof as per the comment we use
$${n+1\brace k} = k{n\brace k} + {n\brace k-1}$$
which says that we put $n+1$ into one of $k$ sets of a set partition
of $n$ into $k$ sets or we join it as a singleton to a partition of
$n$ into $k-1$ sets. The base case is
$$t^1 = \sum_{k=1}^1 {1\brace k} t^\under... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2966247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Odd or even function Show whether the function f is odd , even or other wise
Where $$f(x) = 2 , x\in ]0,\infty[ , f(x) =- 2 , x\in ]-\infty , 0]$$
I think that the function is odd because it is symmetric around the origin point , for the value 0 in the domain since -0=0 , f(0) and f(-0) can not be the additive invers... | It is not odd due to the reason that $f(0) \neq - f(0)$ (By definition of an odd function $f(-x) = -f(x) $ is satisfied in every $x$ and $-x$ in the domain) . It is obviously not even either.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2966428",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Comparision asymptotic notation I can get the result of an asymptotic two expressions by using limit or definitions of Big-Oh. However, I cannot express the following one in terms of $n$.
$$\sum_{i=1}^{n} i^k$$
I want to compare it with $n^{k+1}$.
| HINT
Recall that by Faulhaber's formula
$$\sum_{i=1}^n i^{k} = \frac{n^{k+1}}{k+1}+O(n^k)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2966500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
Position of 2D Brownian motion exiting quarter plane Let $X_t = (X_t^1,X_t^2)$ a planar brownian motion without drift with independent components startet at $X_0 = (1,1)$ and $\tau := \inf \lbrace t\ge 0: X_t \notin (0,\infty)^2 \rbrace$ the first time the process leaves the positive quadrant. So one component of $X_\t... | Starting at any $(x_0,y_0)$, the exit distribution is the Cauchy distribution proportional to $1/((x-x_0)^2/y_0^2+1)$. You'll recognize this as the fundamental solution for $(x,0)\in \partial\mathbb{H}$ of the Laplace equation of the upper half plane $\mathbb{H}$, evaluated at $(x_0, y_0)$. Read any textbook on stochas... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2966647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Writing $\frac{1}{2}+i$ as $1+i+\frac{i^2}{2}$ in Needhams Complex Analysis text This question is from the first chapter of Needham's "Visual Complex Analysis Text"
My question is in regards to the sequence they get in the solution below. Getting the first term in the sequence to be $1$ and the second being $1+i$, fo... | The author used the following equivalences:
$$\begin{align}\text{East} &: i^0 =i^4=i^{8\ }=...=1\\
\text{North} &: i^1 =i^5=i^{9\ }=...=i\\
\text{West} &: i^2 =i^6=i^{10}=...=-1\\
\text{South} &: i^3 =i^7=i^{11}=...=-i\end{align}$$
And in general $i^k = i^{k\ \ (\text{ mod } 4)}$.
It's worth mentioning also that multi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2966806",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What methods can be used to solve $ \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $ I'm seeking methods to solve the following definite integral:
$$ I = \int_{0}^{\frac{\pi}{2}} \frac{x}{\tan(x)} \:dx $$
| The method I took was:
First make the substitution $t = \tan(x)$
$$ I = \int_{0}^{\infty} \frac{\arctan(t)}{t\left(1 + t^2\right)} \:dt $$
Now, let
$$ I\left(\omega\right) = \int_{0}^{\infty} \frac{\arctan(\omega t)}{t\left(1 + t^2\right)} \:dt $$
Thus,
\begin{align}
\frac{dI}{d\omega} &= \int_{0}^{\infty} \frac{t}{t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2966938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How can I prove that $\mathbb{Q}(\sqrt[3]{2},i)=\mathbb{Q}(\sqrt[3]{2}+i)$? It is easy to show that $\mathbb{Q}(\sqrt[3]{2}+i)\subseteq \mathbb{Q}(\sqrt[3]{2},i)$.
But how I can show that $\mathbb{Q}(\sqrt[3]{2},i)\subseteq\mathbb{Q}(\sqrt[3]{2}+i)$?
I can't find a way to express $\sqrt[3]{2}$ in terms of $\sqrt[3]{2}... | Consider numbers of the form
$$x_0+x_1a+x_2a^2+x_3i+x_4ai+x_5a^2i$$
where $\sqrt[3]2=a$ and $x_i\in\mathbb Q$. It is easily shown that these numbers form a vector space $V$ under addition and are closed under multiplication.
Now consider the powers of $\sqrt[3]2+i=a+i=z$ from $z^0=1$ to $z^5$. Certainly all these numbe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2967060",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
On the notion of 'winding numbers' of maps $\mathbb{C} \setminus \{0\} \to \mathbb{C} \setminus \{0\}$ In complex analysis, the winding number (around the origin) of a continuous loop $\gamma: [0,1] \to \mathbb{C} \setminus \{0\}$ is the number of times the loops "winds" around zero, which is given by the integral
$$\... | Kenny Wong's answer is instructive because it does it "by hand", but here's a much more condensed version:
A very basic result is that the winding number of $\gamma$ only depends on the path homotopy class of $\gamma$.
Let $r:\mathbb{C}\setminus\{0\} \to S^1, z\mapsto \frac{z}{|z|}$, and $i:S^1\to \mathbb{C}\setminus\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2967207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
show 1 is not a linear combination of the polynomials 2 and x in Z[x] I want to prove that 1 is not a linear combination of the polynomials 2 and x in Z[x], can you give any hints for this?
What I have done is construct $f(x), g(x) \in Z[x]$ such that
\begin{align}
2f(x) + x g(x)=1
\end{align}
Let
\begin{align}
f(x)... | Indeed, there are no polynomials $f,g$ with integer coefficients such that $xf(x)+2g(x)=1$,
since the constant coefficient on the left-hand side is divisible by 2.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2967331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
How can I prove that eventually one function will overtake another, and find when? Given for example 2 functions,$\ n^{100} $ and$\ 2^n$. I know that$\ 2^n$ grows faster and that therefore there is some$\ n$ where it will eventually overtake $\ n^{100} $ but how can I prove this, and also maybe find that$\ n$?
| Solve $2^n > n^{100} \iff n\ln 2>100\ln n\iff \dfrac{n}{\ln n} > \dfrac{100}{\ln 2}$. Let $n = 2^k$, then $ \dfrac{n}{\ln n} = \dfrac{2^k}{k}$,and you solve $2^k > 100k$. Observe the first integer solution $k$ for this is $k = 10$. Thus $n = 2^{10} = 1,024$ .
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2967615",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 0
} |
Create a set of N numbers with no common rational factor Question:
So I want to create a set of real numbers $\{a\}_{N} = \{a_{1}, a_{2}, \ldots, a_{N}\}$ such that if there exists a common factor between all of the elements, it must be irrational.
In this way, the fraction $\frac{a_{i}}{a_{j}} \notin \mathbb{Q} \; \f... | Well, your solution attempt also works. It's not that much different or more difficult to prove that $\frac{\sqrt{p}}{\sqrt{q}}$ is irrational for different primes $p,q$ than proving $\sqrt{2}$ is irrational.
Your 'tripping me up' fear is not without reason, of course, but you circumvented it by chosing roots of prime... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2967724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
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