Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Let $x=\frac{1}{3}$ or $x=-15$ satisfies the equation,$\log_8(kx^2+wx+f)=2.$ Let $x=\frac{1}{3}$ or $x=-15$ satisfies the equation,$\log_8(kx^2+wx+f)=2.$If $k,w,f$ are relatively prime positive integers,then find the value of $k+w+f.$
The given equation is $\log_8(kx^2+wx+f)=2$ i.e. $kx^2+wx+f=64$
Since $x=\frac{1}{3}... | Eliminating $f$ gives $44k=3w$. Take $w=44,k=3$. Then we get $f=49$. Note that the general solution is $k=3h,w=44h,f=64-15h$, but the requirement that the numbers are relatively prime positive integers forces $h=1$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simple asymptotic analysis problem I came across a problem that I tried to formalize as follows:
Let say i have two functions $x(t)$ and $y(t)$ such that for $t \rightarrow t_0$
$$
\left\{
\begin{array}
\;y(t) \rightarrow -\infty \\
x(t) \rightarrow -\infty \\
y(t) = o(x(t))
\end{array}
\right.
$$
Shouldn't be in s... | No, $ 2^{y(t)} = O(2^{x(t)})$ is false.
Consider the example
$$ \left\{ \begin{array} y(t) = -\frac1{|t-t_0|} \\ x(t) = -\frac2{|t-t_0|} \\ y(t) = \frac12 x(t) = O(x(t)) \end{array} \right. $$
But if we let $u = \frac1{|t-t_0|}$, then
$2^{-u}$ is not $O(2^{-2u})$, although it is $O(\sqrt{2^{-2u}})$: as $u\to\infty... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Does every topology have a subbasis? I know that every topology is generated by a basis.
Is it true that every topology has a subbasis which generates it? If not, what makes it not possible to "synthesize" a subbasis out of a basis?
Having studied linear algebra, I am intuitively comfortable with the idea of basis, but... |
Thm:
Any class $\mathcal{A}$ of subsets of a non-empty set $X$ is the subbase for a unique topology on $X$. That is, finite intersections of members of $\mathcal{A}$ form a base for a topology $\mathcal{T}$ on $X$.
Proof:
We show that the class $\mathcal{B}$ of finite intersections of members of $\mathcal{A}$ satisf... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1720524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Limit of a spiraling sequence in $\mathbb{R}^2$ I did some serious mistakes while typing a question few minutes ago.
Let $z_0=(0,0)$ and $z_1=(1,0)$ and define the sequence $z_n$, for any $ n \ge 2 $, as the endpoint of a line drawn perpendicular from $z_{n-1}$ and which is half the distance of the line joining $z_{n-2... | HINT
You can consider each coordinate separately. Look at the vectors of changes every 2 steps, so you have for $x$, for example:
$1, -1/4, 1/16, -1/64, \ldots$
which you must sum. It is a geometric series with common factor $-1/4$ and initial value of $1$.
Similar picture will be in the $y$ direction.
| {
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"question_score": "2",
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Restriction of an isomorphism to an invariant subspace may fail to be surjective
I'm wondering whether the restriction of a vector space automorphism $f : V \to V$ to an invariant subspace $W \subset V$ can fail to be surjective, i.e. $f\vert_W : W \to W$ is not an automorphism.
Clearly, this can only happen if $W$ (... | Try a shift operator on $\ell^1(\mathbb Z)$.
| {
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Prove that $a-b=b-a\Rightarrow a=b$ without using properties of multiplication. Yesterday my Honors Calculus professor introduced four basic postulates regarding (real) numbers and the operation $+$:
(P1) $(a+b)+c=a+(b+c), \forall a,b,c.$
(P2) $\exists 0:a+0=0+a=a, \forall a.$
(P3) $\forall a,\exists (-a): a+(-a)=(-a)... | You are right, you can't only from those axioms, and here is why.
Consider $A=\mathbb{Z}/2\mathbb{Z}$. Then $1-0=1=0-1$. But $0 \neq 1$.
But you can show this is true if $2 \neq 0$ and your ring is a integral domain, where $2:=1+1$. In fact,
$a-b=b-a \implies a=b-a+b \implies 0=b-a+b-a \implies 0=2b-2a \implies 0=2(b-... | {
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Determine the quadratic character of $293 \bmod 379$. Determine the quadratic character of 293 mod 379.
Did several other problems like this with 3, 5, 60, -1 and 307 all mod 379 but still having a tough time with this problem. I can post up work from these examples if helpful. Any help is appreciated.
So far I have...... | $\newcommand{kron}[2]{\left( \frac{#1}{#2} \right)}$
You are trying to determine the Legendre symbol $\kron{293}{379}$. There is a pretty general recipe when trying to compute Legendre symbol $\kron{a}{p}$ when $p$ doesn't divide $a$ and $p$ is an odd prime. The tools available are
*
*Explicitly find all the squares... | {
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Solving inequality in complex plane I have to graphically represent the following subset in the complex plane being z a complex number:
$A={1<|z|<2}$
However after trying to do it on WolframAlpha it says that "inequalities are not well difined in the complex plane".
What I did previously was solve it like it was a reg... | The geometric interpretation of this would be an open annulus centred at $0$ with inner radius $1$ and outer radius $2$.
To see this, observe that $|z|$ represents the Euclidean distance from $z$ to $0$. The condition $1 < |z|$ restricts $z$ to be outside of the unit disk, and similarly for $|z| < 2$.
| {
"language": "en",
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"question_score": "1",
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Uniformly continuous independent of metrics? Let $(X,d)$ and $(Y,e)$ be metric spaces. A map $f:X\to Y$ is uniformly continuous if for each $\epsilon>0$ there exists $\delta >0$ such that whenever $d(x,y)<\delta$ we have $e(f(x),f(y))<\epsilon$.
Suppose $f:X\to Y$ is uniformly continuous w.r.t. $d$ and $e$. My question... | No. Let $d$ be the usual metric on $\mathbb R$, and let $d'$ be the metric $d'(x,y)=|\arctan(x)-\arctan(y)|$. It's not hard to check that $d'$ induces the same topology on $\mathbb R$ (use the continuity of $\arctan$). Then the identity function on $\mathbb R$ is uniormly continuous when considered as a function $(\mat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that a set of homotopy classes has a single element This is from Munkres section 51 problem 2b
Given spaces $X$ and $Y$ let $[X,Y]$ denote the set of homotopy classes of maps of $X$ into $Y$. Show that if $Y$ is path connected, the set $[I,Y]$ has a single element. Here $I=[0,1]$.
My approach to the problem is... | Your proof is correct!
Just one comment, though: perhaps you can mention why the maps $F$ and $G$ are continuous, for the sake of more clarity.
| {
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A net in $\mathbb{R}$ Let $\{x_j\}_{j\in J}\subset \mathbb{R}$ be a net, $J$ is a directed set.
If $\{x_j\}_{j\in J}$ does not converge to 0, then there is a subnet$\{x_b\}_{b\in B}$, $B$ is a directed set, that $x_b\rightarrow x,$ where $x$ is either $\infty,-\infty,$ or a nonzero real number.
I see this in a proof o... | Classical fact: if every subnet (in any space $X$) of a net has itself a subnet that converges to some fixed $p$, then the original net converges to $p$. (This can be proved using the cluster point fact, if you like.)
Suppose we have net $(x_j)_{j \in J}$ that does not converge to $0$. As $[-\infty, +\infty]$ (the two ... | {
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finding recursive formula and show it converges to a limit Suppose we are playing cards and we start with $1000$ dollars. Every hour we lose $\frac{1}{2}$ of our money and then we buy another $100$ dollars. I am trying to find $x_n$ for the amount of money the player has after $n$ hours.
I think we can just take $x_n =... | What you did is not wrong, but it's not complete.
What you did prove:
If the sequence $x_n$ has a limit, then the limit is equal to $200$.
What you did not prove:
The sequence $x_n$ has a limit.
Also, that's not what the question is asking you. The question says you need to find a formula for $x_n$, not the limit ... | {
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Probability for "drawing balls from urn" I'm afraid I need a little help with the following:
In an urn there are $N$ balls, of which $N-2$ are red and the remaining are blue. Person $A$ draws $k$ balls, so that the first $k-1$ are red and the $k$ ball is blue. Now Person B draws m balls:
What's the probability for Per... | The hypergeometric distribution accounts for the change in probability of success, but counts number of success in finite predetermined sample, which is not what you want. The geometric distribution, which counts number of trials until the first success works in a different setting: constant probability of success and ... | {
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"question_score": "4",
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Limit evaluation for oscillating function I want to evaluate the following oscillating limit x tends to infinity $$\sin(\sqrt{x+1})-\sin(\sqrt{x})$$
I tried evaluating this limit using trigonometric transformations but didn't arrive at the answer
| $$\lim_{x\to\infty}\left(\sqrt{x+1}-\sqrt{x}\right)=\lim_{x\to\infty}\left(\sqrt{x+1}-\sqrt{x}\right)\left(\frac{\sqrt{x+1}+\sqrt{x}}{\sqrt{x+1}+\sqrt{x}}\right)=$$
$$\lim_{x\to\infty}\frac{\left(\sqrt{x+1}-\sqrt{x}\right)\left(\sqrt{x+1}+\sqrt{x}\right)}{\sqrt{x+1}+\sqrt{x}}=$$
$$\lim_{x\to\infty}\frac{1}{\sqrt{x+1}+\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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If $x_{n+1}=f(x_n)$ and $x_{n+1}-x_n\to 0$, then $\{x_n\}$ converges Let $f:[0,1] \rightarrow [0,1]$be a continuous function. Choose any point $x_0 \in [0,1]$ and define a sequence recursively by $x_{n+1}=f(x_n)$. Suppose $\lim_{n \rightarrow \infty}x_{n+1}-x_n =0$, does this sequence converge?
| Yes. Let $I = \lim_{k \rightarrow \infty} I_k$ where $I_k$ is the closure of $\{x_k,x_{k+1},...\}$. Note that because $x_{k+1}-x_k \rightarrow 0$, $I$ is connected, thus either a singleton or an interval. If $I$ is a singleton, we are done. If $I$ is an interval $(x_-,x_+)$, then $f(x)=x$ on this interval. It then foll... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Definite integral of sign function I need to calculte the integral of $F(x)=\text{sign}(x)$ (A partial function)
between $x=-1$ and $x=2$.
Of course we need to seperate the integral between $x>0$ and $x<0$
but is it a case of improper integral ? or just seperate and calculate?
|
Notice:
*
*When $x<0$:
$$\text{sign}(x)=-1$$
*When $x>0$:
$$\text{sign}(x)=1$$
*When $x=0$:
$$\lim_{x\to0^+}\text{sign}(x)=1$$
$$\lim_{x\to0^-}\text{sign}(x)=-1$$
\begin{align}
\int_{-1}^{2}\text{F}(x)\space\text{d}x &= \int_{-1}^{2}\text{sign}(x)\space\text{d}x \\
&= \int_{-1}^{0}\text{sign}(x)\space\text{d}x+\... | {
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"timestamp": "2023-03-29T00:00:00",
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How do I factorize this numerator? $$\lim_{x\to -3} \frac 1{x+3} + \frac 4{x^2+2x-3}$$
I have the solution I just need to know how i turn that into:
$$\frac {(x-1)+4}{(x+3)(x-1)}$$
I know this might be really simple but I'm not sure how to factorise the numerator.
Thanks in advance!
| You may use common denominator technique:
$$
\frac 1{x+3} + \frac 4{x^2+2x-3} = \frac {1}{x+3} + \frac {4}{(x+3)(x-1)} = \frac {1 \times (x-1) + 4 \times 1}{(x+3)(x-1)} = \frac{x-1+4}{(x+3)(x-1)}
$$
Take a denominator which divisible to both denominators as the common denominator and divide that by each denominator se... | {
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Solving the following trigonometric equation: $\sin x + \cos x = \frac{1}{3} $ I have to solve the following equation:
$$\sin x + \cos x = \dfrac{1}{3} $$
I use the following substitution:
$$\sin^2 x + \cos^2 x = 1 \longrightarrow \sin x = \sqrt{1-\cos^2 x}$$
And by operating, I obtain:
$$ \sqrt{(1-\cos^2 x)} = \dfrac... | An even simpler way is to substitute $x \rightarrow y - \frac{\pi}{4}$
You equation is now $$\sqrt{2} \sin(y) = \frac{1}{3}$$
The solutions are
$$\begin{aligned} x & = \arcsin \left( \frac{\sqrt{2}}{6} \right) - \frac{\pi}{4} + 2\pi n \\ x & = \arccos \left( \frac{\sqrt{2}}{6} \right) - \frac{\pi}{4} + 2 \pi n
\end{al... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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"answer_id": 7
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$\exists\text{ set }X:X=X^X$? Given sets A and B, define the set $B^A$ to be the set of all functions A $\to$ B.
My question is: Is there a set X such that X = $X^X$?
Has this something to do with the axiom of regularity?
| Hint:
*
*By a cardinality argument it follows $|X|=1$.
*Now, can such a $X$ satisfy $X = X^X$?
| {
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Number of ways to color paired shoes so that each pair will have different color There are 4 pairs of shoes of different sizes. Each of the 8 shoes can be colored with one of the four colors: Black, Brown, White & Red. In how many ways can one color shoes so that in at least three pairs, the left and the right shoes do... | The number of ways to color the shoes so that three pairs are odd-colored is
$$
N = \text{(number of ways to pick three pairs)} * \text{(number of ways to color each odd pair)}^3 * \text{(number of ways to color the one remaining pair)}
$$
which is
$$
N = C(4,3) * (12)^3 * 4 = 4 * 12^3 * 4.
$$
It looks like you forgot... | {
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Finding ALL solutions of the modular arithmetic equation $25x \equiv 10 \pmod{40}$ I am unsure how to solve the following problem. I was able to find similar questions, but had trouble understanding them since they did not show full solutions.
The question:
Find ALL solutions (between $1$ & $40$) to the equation $25x \... | Let's use the definition of congruence. $a \equiv b \pmod{n} \iff a = b + kn$ for some integer $k$. Hence, $25x \equiv 10 \pmod{40}$ means $$25x = 10 + 40k$$ for some integer $k$. Dividing each side of the equation $25x = 10 + 40k$ by $5$ yields $$5x = 2 + 8k$$
for some integer $k$. Thus,
$$5x \equiv 2 \pmod{8}$$... | {
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"timestamp": "2023-03-29T00:00:00",
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Confidence interval for the Proportion Question on my Homework:
The Pew Research Center has conducted extensive research on the young adult population (Pew Research website, November 6, 2012). One finding was that 93% of adults aged 18 to 29 use the Internet. Another finding was that 21% of those aged 18 to 28 are marr... | The aim of a confidence interval is to estimate an interval for a population parameter using sample information. In this case we are interested in estimating a confidence interval for the population proportion $p$ of adults aged $18$ to $29$ that use the internet. To that end, we draw a sample of size $n=500$ adults wi... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Does the sequence $\frac{n!}{1\cdot 3\cdot 5\cdot ... \cdot (2n-1)}$ converge? I'm trying to determine if this sequence converges as part of answering whether it's monotonic:
$$
\left\{\frac{n!}{1\cdot 3\cdot 5\cdot ... \cdot (2n-1)}\right\}
$$
First, I tried expanding it a bit to see if I could remove common factors i... | The reciprocal of the term of interest is
$$\begin{align}
\frac{(2n-1)!!}{n!}&=\left(\frac{(2n-1)}{n}\right)\left(\frac{(2(n-1)-1)}{(n-1)}\right)\left(\frac{(2(n-2)-1)}{(n-2)}\right) \cdots \left(\frac{5}{3}\right)\left(\frac{3}{2}\right)\\\\
&=\left(2-\frac{1}{n}\right)\left(2-\frac{1}{n-1}\right)\left(2-\frac{1}{n-2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1722673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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A convex subset of normed vector space is path-connected Let $(N, \|\;\|)$ be a normed vector space and $(X,\tau)$ a convex subset of $(N,\|\;\|)$ with its induced topology. Show that $(X,\tau)$ is path-connected, and hence also connected.
What I have done so far is Let $a, b \in N$, then construct $X = \{x: x = at+ (1... | Let $x\in X$ and $r>0$, and consider the open ball $B(x;r)=\{y\in X:\|x-y\|<r\}$. For any $y,z\in B(x;r)$ and $t\in(0,1)$, we have
\begin{align}
\|ty + (1-t)z - x\| &= \|t(y-x) + (1-t)(z-x)\|\\
&\leqslant t\|y-x\| (1-t)\|z-x\|\\
&<r,
\end{align}
so $B(x;r)$ is convex. Write $f(t) = t(a-b) + t$, then it is clear that $f... | {
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If $dF_p$ is nonsingular, then $F(p)\in$ Int$N$ Here is the problem 4-2 in John Lee's introduction to smooth manifolds:
Suppose $M$ is a smooth manifold (without boundary), $N$ is a smooth manifold
with boundary, and $F: M \to N$ is smooth. Show that if $p \in M$ is a
point such that $dF_p$ is non-singular, t... | $dF_p$ is nonsingular then there exists open set $U\subset M$ s.t. $F:U\rightarrow F(U)$ is a diffeomorphism Hence $F(p)$ is inerior point in $F(U)$ where $F(U)$ is open in $N$.
(Reference : differential forms and applications - do Carmo 60p.
If $H =\{ x|x_n\leq 0\}$ assume that $V$ is open in $H$. And $f :
V\subset H... | {
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"timestamp": "2023-03-29T00:00:00",
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Where does the basis of row space matrix comes from? So what I know: According to Strang's book the row space basis comes from where the pivots are in the upper triangular. We take those rows and therefore we have our basis. My problem lies here: "The row space of A has the same basis with the U because the two row spa... | "It is true that $A$ and $U$ have different rows, but the combinations of the rows are identical: same space!" with
$$A= \left(\begin{matrix}2 & 4&5 \\ 8& 0& 3\end{matrix} \right)$$ and $$U= \left(\begin{matrix}1 & 0&0.375 \\ 0& 1& 1.0625\end{matrix} \right)$$
Answer: Yes: $(8,0,3) = 8 \cdot (1,0,0.375)$ and
$(2,4,5)=... | {
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Calculate sum of infinite series by solving a differential equation Calculate the sum of the infinite series
$$\sum_{n=0}^{\infty}\frac{1}{(3n)!}$$ by solving an aptly chosen differential equation.
I know that one can solve a differential equation by assuming that we can write the solution as a power series in the for... | From a polynomial or power series $f(x)$ you can "punch out" the odd degree terms by taking the even part $\frac12(f(x)+f(-x))$. In a similar fashion you can produce "holes" with period $3$ in the coefficient sequence by combining $f(e^{ik\frac{2\pi}3}x)$, $k=-1,0,1$.
As the remaining coefficients in the given series c... | {
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"answer_id": 1
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probability of divisibility by $5$ Let $m,n$ be $2$ numbers between $1-100$ . what is the probability that if we select any two random numbers then $5|(7^m+7^n)$ . My attempt last digit should be $5$ or $0$ so $7$ powers follow the pattern $7,9,3,1,7...$ so $m,n$ should be such that if one gives $7$ as last digit other... | As you would have noticed by now, $7^k\equiv7^{k\bmod4}$.
And as you noted, $5|(7^m+7^n)\iff|m-n|\equiv2\pmod4$.
The probability for that in the case of $m,n\in\mathbb{N}$ is obviously $\frac14$.
Since $4|100$, the probability in the case of $m,n\in[1,100]$ is also $\frac14$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1723177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Determining the limits of integration in multiple integrals over delta-functions In using Feynman parametrisation, I have noticed different expressions given in the literature that seem to imply
$$
\int_0^1dx\int_0^1dy\int_0^1dz\delta(1-x-y-z)f(x,y,z)=\int_0^1dx\int_0^{1-x}dyf(x,y,z)|_{z=1-x-y}.
$$
However I have been ... | I have figured out the answer to this. The result is not totally general but may be found in each case by writing the integrals as integrals over an infinite range and putting Heaviside step functions in the integrand to restore the finite range of integration.
The Dirac-deltas change the arguments of the step function... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Find $a^{100}+b^{100}+ab$ $a$ and $b$ are the roots of the equation $x^2+x+1=0$.
Then what is the value of $a^{100}+b^{100}+ab$?
Here's what I found out:
$$a+b=-1$$
$$ab=1$$
but how to use this to find that I don't know! Someone please answer my query.
| The roots of your equation are $$x = -\frac{1}{2} \pm i \frac{\sqrt{3}}{2}$$
We have $e^{\frac{2\pi i}{3}}$ and $e^{\frac{4\pi i}{3}}$, both which remain unchanged in magnitude and direction upon exponentiating 100 times as $e^{\frac{2\pi i}{3}} = e^{\frac{200 \pi i}{3}}$ and $e^{\frac{4\pi i}{3}} = e^{\frac{400 \pi i}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1723413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
How many 20 digit numbers have 10 even and 10 odd digits? How can I perform operations so as to get this value? Number should not have leading zeros.
| Case 1: First digit is odd
*
*How many choices do we have for this first digit?
There are $5$ ways of choosing a digit for this place from $1,3,5,7,9$
*Now in how many ways can we build the rest of the number?
For the remaining 19 digits, we need to choose 9 positions to be odd and the rest to be even.
Number ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1723525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Determine the point of intersection between $f(x) = x^2$ and its normal in the point $(a, a^2)$ Determine the point of intersection between $f(x) = x^2$ and its normal in the point $(a, a^2)$
Answer:
This should be easy enough...
$f'(x) = 2x$
The tangent line in the point $(a, a^2)$ is $y - a^2 = 2 (x - a) \rightarrow ... | The slope of the normal line is going to be: $-\frac{1}{2a}$.
set $g(x)=-\frac{1}{2a}x+(a^2+\frac{1}{2})$
You want to solve $g(x)=f(x)$.
$-\frac{1}{2a}x+(a^2+\frac{1}{2})=x^2$
$x^2+\frac{1}{2a}x+\frac{1}{16a^2}=(a^2+\frac{1}{2})+\frac{1}{16a^2}$
$x=\pm\sqrt{(a^2+\frac{1}{2})+\frac{1}{16a^2}}-\frac{1}{4a}$
Personally, I... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1723628",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Calculating Edge points of a rectangle in 2D I'm building a computer game and I got stuck during a math calculation:
The game is a 2D game and is based on a Cartesian coordinate system.
I know the coordinates of E and F. From there I know the angle of EF (Also the angle of AB and CD). I also know the length of AB and ... | Let E be $(0, 0)$:
F = $(EF\cos\theta, EF\sin\theta)$
A = $(-\frac{AB}{2}\cos\theta, \frac{AB}{2}\sin\theta)$
B = $(\frac{AB}{2}\cos\theta, -\frac{AB}{2}\sin\theta)$
C = $(EF\cos\theta + \frac{CD}{2}\cos\theta, EF\sin\theta - \frac{CD}{2}\cos\theta)$
D = $(EF\cos\theta - \frac{CD}{2}\cos\theta, EF\sin\theta + \frac{CD}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1723755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
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Can the system of equations be extracted from its solution? While I was solving the secondary school exam of 2014 I came across a question that states:
After solving those equations: $a_{1}x + b_{1}y = c_{1}$ and $a_{2}x + b_{2}y = c_{2}$, we found that x = $\frac{-7}{\begin{vmatrix}
3 & 1 \\
1 & -2
\end{vmatrix}}... | In view of the main determinant:
$$\begin{vmatrix}3 & 1\\1 & -2\end{vmatrix}$$
one may infer that the system is
$$\begin{cases}3x_1+x_2&=&c_1\\1x_1-2x_2&=&c_2\end{cases}$$
Using Cramer's rule (https://en.wikipedia.org/wiki/Cramer's_rule), the two numerators can be written under the form:
$$\begin{vmatrix}c_1 & 1\\c_2 &... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1723888",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Dumb question: what is $\sum\limits_{n = 1}^k 1$ Let $\{I_n\}$ be a collection of intervals on $\Bbb {R}$, whose length denoted by $I_n = |I_n|$
Then what is $\sum\limits_{n = 1}^k (I_n + \alpha)$, where $\alpha$ is some real number?
The notation confuses me:
$$\sum_{n = 1}^k (I_n + \alpha) = \sum_{n = 1}^k I_n + \sum... | If you translate an interval by $\alpha$, it's length stays the same. Suppose $I=[3,4]$ and $\alpha=5$, $I+\alpha=[8,9]\implies |I+\alpha|=9-8=1=|I|$. Note that $|I+\alpha|\neq |I|+\alpha$.
So
$$
\sum_{n=1}^k |I_n+\alpha|=\sum_{n=1}^k|I_n|=\sum_{n=1}^k|I_1|=|I_1|\sum_{n=1}^k1=k|I
_1|
$$
| {
"language": "en",
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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 to prove the number of poles minus the number of zeros is $2-2g$? I want to show that, for all differentials on the same Riemann surface S the
number of poles minus the number of zeros, counting multiplicities, always equals $2-2g$. It says this can be deduced from the following result:
By Riemann-Roch theorem we ... | The number you want to compute is $$\deg K_S,$$ the degree of the canonical divisor. This appears in the Riemann-Roch formula: $$h^0(K_S)=\deg K_S+1-g+h^0(K_S-K_S).$$ Now you need to use:
*
*your knowledge of $h^0(K_S)=g$, and
*$h^0(K_S-K_S)=h^0(\mathcal O_S)=1$.
Then you get $\deg K_S=2g-2$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1724171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find last two digits of $33^{100} $ Find last two digits of $33^{100}$.
My try:
So I have to compute $33^{100}\mod 100$
Now by Euler's Function $a^{\phi(n)}\equiv 1\pmod{n}$
So we have $33^{40}\equiv 1 \pmod{100}$
Again by Carmichael Function : $33^{20}\equiv 1 \pmod{100}$
Since $100=2\cdot40+20$ so we have $33^{100}=1... | 33^4 = 21 mod 100 ;33^20 = 01 mod 100 ;33^100 = 01 mod 100 ;Yes the
last 02 digits are 01
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1724246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
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What is $\arctan(x) + \arctan(y)$ I know $$g(x) = \arctan(x)+\arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right)$$
which follows from the formula for $\tan(x+y)$. But my question is that my book defines it to be domain specific, by which I mean, has different definitions for different domains:
$$g(x) = \begin{cases}\arct... | Here is a straightforward (though long) derivation of the piece wise function description of $\arctan(x)+\arctan(y)$.
We will show that:
$\arctan(x)+\arctan(y)=\begin{cases}-\frac{\pi}{2} \quad &\text{if $xy=1$, $x\lt 0$, and $y \lt 0$} \\\frac{\pi}{2} \quad &\text{if $xy=1$, $x\gt 0$, and $y \gt 0$} \\ \arctan\left(\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1724348",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
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Find the sum $\sum _{n=1}^{\infty }\left(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n}\right)$ $$\sum _{n=1}^{\infty }\left(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n}\right)$$
On their own, all three are divergent, so I thought the best way would be to rewrite it as:
$$\frac{\sqrt{n+2}-2\sqrt{n+1}-\sqrt{n}}{\sqrt{n+2}-2\sqrt{n+1}-\sqrt{n}}\cdo... | Render the summand as $(\sqrt{n+2}-\sqrt{n+1})-(\sqrt{n+1}-\sqrt{n})$ and use telescoping. The difference between two consecutive square roots goes to zero as the arguments go to infinity. You should be able to see the answer immediately.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1724446",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Probability - determine the probability for an event I have this probability question from homework
A system consists of $N$ chips in a parallel way, such that if at least one of the chips are working the system fully operates.
The probability that throughout a work day, a chip will get broken is $\frac{1}{3}$
Note tha... | You are looking to calculate $\mathsf P(X_1\mid X\neq 0)$ when $X$ is the count of chips that work at the end of the day, given a rate of failure, $q=1/3$ (i.i.d. for each chip) and $X_1$ is the event that chip one works. It is the probability that a specific chip works when given that at least one chip does.
Your ap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1724548",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Using Fourier Transform to solve an ODE Consider the differential equation
$$f^{iv}+3f^{''}-f=g$$
I have read that taking the Fourier Transform of both sides gives
$$\left(i\lambda\right)^{4}F\left(\lambda\right)+3\left(i\lambda\right)^2F\left(\lambda\right)=G\left(\lambda\right)-F\left(\lambda\right)=G\left(\lambda\r... | You can combine
*
*Linearity of the Fourier transform:
$$\mathcal{F}\left[\sum_{\forall k} g_k\right] = \sum_{\forall k} \mathcal{F}[g_k]$$
*Differentiation becomes multiplication with the frequency:
$$\mathcal{F}[g'] = i\lambda\mathcal{F}[g]$$
First use 1) to separate each term.
Then use 2) as many times you need... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to verify that one of equations in a polynomial system is redundant? I know that system of polynomial equations
$$
p_1(x_1,\dots,x_n)=0,..., p_N(x_1,\dots,x_n)=0
$$
has infinitely many solutions.
I computed some of them numerically and notices that they always satisfy one more polynomial equation
$$
q(x_1,\dots,x... | Hint: If you compute the row echelon form of the matrix given by the polynomial system, then the all-zero rows will be the redundant ones.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1724747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 1
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Find the sum of the series $\sum_{n=1}^{\infty}(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n})$ I need to test the convergence and find the sum of the following series:
$\sum_{n=1}^{\infty}(\sqrt{n+2}-2\sqrt{n+1}+\sqrt{n})$
But i am not really sure what kind of series is this?
Since
$2\sqrt{n+1}>\sqrt{n}+\sqrt{n+2} \forall n \in \m... | Just write
$$\sqrt{n+2}-2\sqrt {n+1}+\sqrt n=(\sqrt{n+2}-\sqrt {n+1})+(\sqrt n-\sqrt{n+1})$$
and then telescope.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1724923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Integrate $t^2y'=-5ty$ I'm reading the notes and it says:
$t^2y'=-5ty$ where $y=y(t)$
Using direct integration we can conclude that $y=kt^{-5}$
I don't understand how this can be integrated directly. I can see that the solution is correct, but how did they achieve it?
| Cancel $t$ at first sight. Conveniently separate the $x,y$ variables on either side of the equation.
$$\frac{dy}{dt}=-\frac{5 y }{t} $$
$$\frac{dy}{y} + \frac{5 \,dt }{t} = 0 $$
Both are logs on integration.
$$ y t ^5 = const. $$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1725006",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Even and odd functions in Laurent decomposition I have the following problem and have no idea how to approach it, could anyone give me any hint about it? Thanks!
Suppose that $f(z) = f_0(z) + f_1(z)$ is the Laurent decomposition of an analytic function $f(z)$ on the annulus $\{A < |z| < B \}$. Show that if $f(z) $ is a... | Continuing with Martin R's answer, we have that $\lim_{z \to \infty} f_1(z) = 0$, so $f_1$ is bounded.
Also, since $f_0$ is analytic, and hence continuous, on the compact set $|z| < B$, $f_0$ is bounded. Therefore, $h(z)$ is bounded and analytic on the entire plane. Liouville's theorem implies that $h(z)$ is identical... | {
"language": "en",
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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 to solve $\int \dfrac{x^5\ln\left(\frac{x+1}{1-x}\right)}{\sqrt{1-x^2}} dx$ Consider the integral
$$\int \dfrac{x^5\ln\left(\frac{x+1}{1-x}\right)}{\sqrt{1-x^2}}dx$$
How to start integrating?
Any hint would be appreciated.
| Using the substitution $ x = \sin(t) $
The integral simplifies to
$$\int \sin^5t \cdot \ln \left[\frac{1+ \sin(t)}{1-\sin(t)}\right] dt$$
Further simplifications yields:
$$ 2\int \sin^5t \cdot \ln[\sec(t)+\tan(t)] dt $$
we recognize $\ln[\sec(t)+\tan(t)]$ as the antiderivative of $\sec(t)$ which suggests integrating by... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1725465",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Give an example that the following condition does not imply WARP I know how to prove that Weak Axiom of Revealed Preference (WARP) implies the following condition: if $a\in B_1, B_1 \subseteq B_2, a\in C(B_2)$, then $a\in C(B_1)$. $C$ here is a notation for choice correspondence. Could someone provide an example that t... | Let $B_1 = \{w,x,y\}$ and $B_2 =\{x,y,z\}$. Let $C(B_1)=\{x\}$ and $C(B_2)=\{y\}$. This example satisfies your property (it holds vacuously), but fails WARP.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Is $\cot(x)=\frac{1}{\tan(x)}$? I just found in some place say that $\cot(x)=\frac{1}{\tan(x)}$.
If you think about it as a function they are definitely not same. but if you think about as a relation between angle in a right angle triangle and its side they will be equal.
so the question is its right to say they are ... | The tangent is defined as $$\tan\alpha =\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}=\frac{\sin~\alpha}{\cos~\alpha}$$
and the cotangent as
$$\cot\alpha =\frac{\text{adjacent}}{\text{opposite}}=\frac{b}{a}=\frac{\cos\alpha}{\sin\alpha} $$
what leads us to
$$\frac{\cos\alpha}{\sin\alpha} = \tan\left(\frac{\pi}{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1725619",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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"answer_id": 2
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What is the new probability density function by generating a random number by taking the reciprocal of a uniformly random number between 0 and 1? I have a random number generator which can generate a random number between $0$ and $1$.
I attempt to generate a random number between 1 and infinity, by using that random nu... | Let the old probability density function be $f_1(x)$, and the new one be $f_2(x)$.
We have:$$
\int_1^af_2(x)\mathrm dx=\int_\frac1a^1f_1(x)\mathrm dx
$$where $a>1$.
We also know that $f_1(x)$ is uniform, and spans from $0$ to $1$. Therefore, $f_1(x)=1$ in that interval.
Therefore:$$
\int_1^af_2(x)\mathrm dx=\int_\frac1... | {
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"timestamp": "2023-03-29T00:00:00",
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Simplify $AC'+A'C+BCD'=AC'+A'C+ABD'$ How to prove that $$AC'+A'C+BCD'=AC'+A'C+ABD'$$
approch: a way to demonstrate is expressed in its canonical form.
Any hint would be appreciated.
| In
$$AC'+A'C+BCD'=AC'+A'C+ABD'$$
If any term out of $AC'$ and $A'C$ becomes $1$, then, both the LHS and RHS will become $1$ and the equality holds.
So, the case left out is when both $AC'$ and $A'C$ are $0$. Then, suppose $A=1$. Then, $C'=0\Rightarrow C=1$. Similarly, if $A=0$, we have $A'=1$ which implies $C=0$.
Th... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Estimate the bound of the sum of the roots of $1/x+\ln x=a$ where $a>1$ If $a>1$ then $\frac{1}{x}+\ln x=a$ has two distinct roots($x_1$ and $x_2$, Assume $x_1<x_2$). Show that $$x_1+x_2+1<3\exp(a-1)$$
First I tried to estimate the place of the roots separately. I have got that $x_1\leq \frac{1}{a}$ and $\exp(a-1)<x_... | I can show the inequality when $a$ is close enough to $1$ (namely, $a\leq 1+ln(5/4)\approx 1.22$) or when $a$ is big enough (namely, $a \geq 1+\ln(5) \approx 2.6$). In the sequel $f(x)$ denotes $x+\ln(\frac{1}{x})$.
When $a$ is close to $1$. Let us put $w=\sqrt{e^{a-1}-1}$. The inequality then becomes $x_1+x_2\leq 2+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1725944",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
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Prove the solution of $f''(x)-4f(x)=0$ is $f(x)=\sum_{p=0}^{\infty} \frac{4^{p+1}}{(2p)!}x^{2p}$ I'm wondering about this question :
We have the differential equation $f''(x)-4f(x)=0$ and we want to find $f$ as a power serie with $f(0)=4$ and $f'(0)=0$. I would like to prove the only solution is $f(x)=\sum_{p=0}^{\inft... | From
$$
(n+2)(n+1)a_{n+2}-4a_n=0 \tag1
$$ you deduce, for $n=0,1,2,\ldots$,
$$
a_{n+2}=\frac4{(n+2)(n+1)}a_n \tag2
$$ giving, with $n:=2p$,
$$
\begin{align}
a_{2(p+1)}&=\frac4{(2p+2)(2p+1)}\:a_{2p}
\\\\&=\frac4{(2p+2)(2p+1)}\cdot \frac4{2p(2p-1)}\:a_{2(p-1)}
\\\\&= \cdots
\\\\&=\frac4{(2p+2)(2p+1)}\cdot \frac4{2p(2p-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1726041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Given a $3m\times m$ table, how many ways can it be filled by $x$'s & $o$'s such that each column has at least $2$ $x$'s? Looking at this treat I thought it's a pretty easy one. Apparently not :]. Well,
Given a $ 3m \times m $ table, how many ways can we fill it with $x$'s and $o$'s, such that each column has at least... | Given the solution (and working backwards) we understand that the solution considers each column separately:
*
*$2^{3m}$ ways to fill it (without restrictions).
*$3m$ ways with one $x$,
*$1$ way with no $x$
So, there are $(2^{3m}-3m-1)$ ways to fill one column with the given restriction. Since there are $m$ colu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1726128",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Taking inverse Fourier transform of $\frac{\sin^2(\pi s)}{(\pi s)^2}$ How do I show that
$$\int_{-\infty}^\infty \frac{\sin^2(\pi s)}{(\pi s)^2} e^{2\pi isx} \, ds = \begin{cases} 1+x & \text{if }-1 \le x \le 0 \\ 1-x & \text{if }0 \le x \le 1 \\ 0 & \text{otherwise} \end{cases}$$
I know that $\sin^2(\pi s)=\frac{1-\co... | Note that we can write
$$\begin{align}
\int_{-\infty}^\infty\frac{\sin^2(\pi s)}{(\pi s)^2}e^{i2\pi sx}\,ds&=\frac12\int_{-\infty}^\infty\frac{1-\cos(2\pi s)}{(\pi s)^2}e^{i2\pi sx}\,ds\\\\
&=\int_0^\infty \frac{1-\cos(2\pi s)}{(\pi s)^2}\,\cos(2\pi sx)\,ds\\\\
&=\int_0^\infty \frac{\cos(2\pi sx)-\frac12\left(\cos(2\pi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1726241",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
How many fixed points are there for $f:[0,4]\to [1,3]$ Let , $f:[0,4]\to [1,3]$ be a differentiable function such that $f'(x)\not=1$ for all $x\in [0,4]$. Then which is correct ?
(A) $f$ has at most one fixed point.
(B) $f$ has unique fixed point.
(C) $f$ has more than one fixed point.
Here, $f:[0,4]\to [1,3]\subset [0... | Note that a derivative has the intermediate value property. Therefore if $f'(x)$ is never $1$, then $f'(x)>1$ for all $x$ or $f'(x)<1$ for all $x$. In the first case, the function $g(x)=f(x)-x$ is increasing, so it can have at most one root; in the second case it is decreasing and the same applies.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1726329",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
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How to rewrite $i^3$ I learned that: $\sqrt[3]i=(e^{\frac{\pi}{2}i+2k\pi i})^\frac{1}{3}=e^{\frac{\pi i}{6}+\frac{2}{3}k\pi i}$ for k={0, 1, 2}
Now how about this case:
$i^3=(e^{\frac{\pi}{2}i+2k\pi i})^3=e^{\frac{3\pi i}{2}+6k\pi i}$ for k={???}
Why would it be $+6k\pi i$? Why not $+2k\pi i$? It seems that for exampl... | When we say $z = re^{ti + 2k\pi i}$ the $2k\pi i$ is just the period and is always there much as "plus a constant"
So really $\sqrt[3]i=(e^{\frac{\pi}{2}i+2k\pi i})^\frac{1}{3}=e^{\frac{\pi i}{6}+\frac{2}{3}k\pi i + 2k'\pi i}$ But $\frac{2}{3}k\pi i$ is now significant and means more that simply "give or take a few ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1726436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Reindexing Exponential Generating Function I have an exponential generating function, and I need to double check what the teacher said, because I'm having trouble coming to the same result. Also, I need to verify what I am coming up with, and the reasoning behind it.
My generating function is: $$xe^{2x} = x\sum_{n=0... | Here is a slightly different answer, which might also be helpful for this and similar tasks.
Situation: The generating function for the exponential series is already known.
\begin{align*}
e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}
\end{align*}
We want to find the exponential series of $xe^{2x}$, i.e. the coefficients $a_n$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1726570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is there a sequence with an uncountable number of accumulation points? Let $(x_{n})_{n \geq 1}$ sequence in $\mathbb{R}$. Is there a sequence with an uncountable number of accumulation points?
Thank you!
| Since the rational numbers are countable, we know there is a bijection $f:\mathbb N\to\mathbb Q$. Let $x_n=f(n)$. Then this is a sequence which contains every rational number, and its set of accumulation points is $\mathbb R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1726710",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
"answer_count": 2,
"answer_id": 1
} |
How to find the value of a variable in a probability distribution function? I encountered a mathematics problem that I don't know how to solve.
$P(X=x) = a(\frac{5}{6})^x$ is a probability distribution function for the probability distribution of the discrete random variable $X$ for $x = 0,1,2,3\dots$
I know that the s... | As you said in your question, we must have
$$ 1=\sum_{x=0}^{\infty}\mathbb{P}(X=x)=a\sum_{x=0}^{\infty}\Big(\frac{5}{6}\Big)^x$$
Using the formula for the sum of a geometric series, we obtain
$$ 1=\frac{a}{1-\frac{5}{6}}=\frac{a}{\frac{1}{6}}$$
so $a=\frac{1}{6}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1726817",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Show that if G is a connected simple k-regular graph with k ≥ 2 and χ'(G) = k, then G is Hamiltonian Hi I am really lost on this problem. The notation χ'(G) = k means that the graph has a proper edge coloring of size k. I am only to the point where I know our graph G is comprised of cycles, and has an even number of ve... | After a bit of searching, I've found this graph:
which is a cubic, connected, non-Hamiltonian graph with chromatic index $3$.
Another counterexample could be also the Barnette-Bosák-Lederberg Graph which is also planar.
No wonder you can't prove it, the claim is false.
I hope this helps $\ddot\smile$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1726933",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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How do I show that $\sum_{k = 0}^n \binom nk^2 = \binom {2n}n$? $$\sum_{k = 0}^n \binom nk^2 = \binom {2n}n$$
I know how to "prove" it by interpretation (using the definition of binomial coefficients), but how do I actually prove it?
| To prove binomial identities it is often convenient to use the coefficient of operator $[x^k]$ to denote the coefficient of $x^k$. So, we can write e.g.
\begin{align*}
\binom{n}{k} = [x^k](1+x)^n\tag{1}
\end{align*}
Note, this is essentially the same approach as that of @Batominovski but with a somewhat more algebrai... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1727088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 3
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What is $2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty$ equal to? I came across this question while doing my homework:
$$\Large 2^{\frac{1}{4}}\cdot4^{\frac{1}{8}}\cdot8^{\frac{1}{16}}\cdot16^{\frac{1}{32}}\cdots\infty=?$$
$$\small\text{OR}$$
$$\large\prod\limits_{x=1}^{\... | For every $x\in \mathbb R$ which $|x|\lt 1$, we have: $$\sum_{n=0}^\infty x^n=\frac{1}{1-x}$$ From here: $$\sum_{n=1}^\infty nx^{n-1}=\frac{1}{(1-x)^2}$$ Now, multiplying $x^2$ in both side we get that: $$\sum_{n=1}^\infty nx^{n+1}=\frac{x^2}{(1-x)^2}$$ And so: $$\sum_{n=1}^\infty n\left(\frac{1}{2}\right)^{n+1}=1$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1727174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Parametrization of $a^2+b^2+c^2=d^2+e^2+f^2$ Is there an existing parametrization of the equation above that is similar to Brahmagupta's identity for $a^2+b^2=c^2+d^2$? I need either a reference to look it up or a hint to solve it. Thanks.
| Above equation (a^2+b^2+c^2)=(d^2+e^2+f^2) has parametric solution.
Refer to Tito piezas on line book "collection of algebraic Identities".
Section sum of squares. The answer is given below;
(a,b,c)=[(p+q),(r+s),(t+u)] and
(d,e,f)=[(p-q),(r-s),(t-u)]
Condition is (pq+rs+tu)=0
After parametrization of (pq+rs+tu)=0, th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1727303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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is there a ring homomorphism $f:H^*(T^n;R)\to H^*(T^n;R)$ of graded rings which is not induced by a continuous map $T^n\to T^n$? Let $R$ be a commutative ring with unit $1_R$ and let $\xi\in H^1(S^1;R)$ be a generator ($H^1(S^1;R)$ is the first singular cohomology group of $S^1$). Let $p_i:(S^1)^n\to S^1$ be the projec... | When $R=\Bbb Z$ the answer to your question is no:
Since $T^n$ is a $K(\pi_1(T^n),1)$, every group endomorphism of $\pi_1(T^n)$ is induced by a map $T^n\to T^n$. (See for example Prop 1B.9 in Hatcher). Therefore, any endomorphism of $\pi_1(T^n)^{ab}=H_1(T^n)=H^1(T^n)$ is also induced by a map of spaces.
Since $H^*(T^n)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1727381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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General sum of $n$th roots of unity raised to power $m$ comprime with $n$ I am trying to find a reference for the following proposition:
Let $m$ and $n$ be coprime. Then,
$$ \sum_{k=0}^{r-1} \exp\left( i \frac{2\pi}{n} k m \right) = 0 $$
if and only if $r$ is an integer multiple of $n$.
Can anyone point a basic textb... | It can be proven easily using geometric summation. Observe that
$$ \sum_{k=0}^{r-1} \exp\left(i \frac{2\pi k m}{n} \right) = \sum_{k=0}^{r-1} \exp(2\pi i m/n)^k = \frac{1-\exp(2\pi i m/n)^r}{1-\exp(2\pi i m/n)}
$$
The sum is $0$ if and only if $\exp(2\pi i m/n)^r = 1$, which corresponds to $n|rm$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1727503",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Covariance of X and Y on a Quadrilateral How should I go about determining the covariance? Also, how can I use intuition to determine if it should be positive or negative?
| If $Y$ on average gets bigger as $X$ gets bigger, then the covariance is positive; if $Y$ on average gets smaller as $X$ gets bigger, then the covariance is negative. You're just picking a random point in that quadrilateral and asking that question about the horizontal and vertical coordinates of that point.
The covar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1727608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Poisson Distribution: What's the probability of getting a first week without any events when you are told that 5 events occurred within a month? I'm told that the probability of getting $n$ murders per month in London can be modelled as a Poisson distribution with rate $\lambda$. I'd like to calculate the probability t... | First, you have to convert the monthly event rate to a weekly event rate (assuming that there are 4 weeks in a month). So if the monthly event rate is $\lambda$, the weekly event rate is $\lambda/4$. Then you want a conditional probability: if the random number of events per week is $$X \sim \operatorname{Poisson}(\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1727722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Bell number as summation of Stirling numbers of the second kind
A different summation formula represents each Bell number as a sum of
Stirling numbers of the second kind
$ B_n=\sum_{k=0}^n \left\{{n\atop k}\right\} $
The Stirling number $\left\{{n\atop k}\right\} $ is the number of ways
to partition a set of cardi... | The Stirling number $n\brace k$ is actually defined for all pairs of non-negative integers. Of course ${n\brace k}=0$ when $k>n$. Since a non-empty set cannot have a partition into $0$ parts, it’s also clear that we want ${n\brace 0}=0$ for $n>0$. It turns out, however, to be convenient to set ${0\brace 0}=1$, as it ma... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1727835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
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Minimizing a sum of exponential functions I want to minimize this function:
$$ g_0(\psi)= \sum_{m=0}^{M-1}e^{j(am^2+bm)}e^{jm\psi} $$
where $a$ and $b$ are constants for which I want to minimize the function.
Can anyone help me regarding this. Will some optimization technique work?
| In your expression:
$$g_0(\psi)= \sum_{m=0}^{M-1}e^{j(am^2+bm)}e^{jm\psi}$$
why dont you express the exponent (or more precisely the coefficient of $j$) in this way:
$$am^2+m(b+\psi)=a\left(m+\dfrac{b+\psi}{2a}\right)^2-\dfrac{(b+\psi)^2}{4a}$$
In this way, you could factor out $e^{-j\frac{(b+\psi)^2}{4a}}$ and concent... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1727982",
"timestamp": "2023-03-29T00:00:00",
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Sum of a series $\frac {1}{n^2 - m^2}$ m and n odd, $m \ne n$ I was working on a physics problem, where I encountered the following summation problem:
$$ \sum_{m = 1}^\infty \frac{1}{n^2 - m^2}$$ where m doesn't equal n, and both are odd. n is a fixed constant
We could alternatively write:
$$ \sum_{j = 1}^\infty \frac{... | Assume $N>i\geq1$. Starting by a partial fraction decomposition, one may write
$$
\begin{align}
&\sum_{j=1,\,j\neq i}^N\frac1{(2i-1)^2-(2j-1)^2}
\\\\&=\frac1{4(2i-1)}\sum_{j=1,\,j\neq i}^N\left(\frac1{j+i-1}-\frac1{j-i}\right)
\\\\&=\frac1{4(2i-1)}\left(\sum_{j=1}^N\frac1{j+i-1}-\frac1{2i-1}-\sum_{j=1,\,j\neq i}^N\frac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1728090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Conjecturing concerning $\mathrm{cl}(\mathrm{int}S) = S$ Let $S$ be a subset of $\mathbb{R}$, $\mathrm{int}S$ denote the set of interior points of $S$, $\mathrm{bd}S$ denote the set of boundary points of $S$, $S'$ denote the set of accumulation points of $S$, and $\mathrm{cl}S$ denote the closure of $S$.
The problem st... | You cannot assume that $S$ is an interval. There are infinite closed sets with empty interior; one interesting example is the Cantor set.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1728181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Limiting question:$\displaystyle \lim_{x\to\ 0} \frac{a^{\tan\ x} - a^{\sin\ x}}{\tan\ x -\sin\ x}$ How do I find the value of $$\lim_{x\to\ 0} \frac{a^{\tan\ x} - a^{\sin\ x}}{\tan\ x - \sin\ x}$$
in easy way.
| As Henry W; commented, Taylor series make thigs quite simple.
$$A=a \tan(x) \implies \log(A)=\tan(x)\log(a)=\Big(x+\frac{x^3}{3}+\frac{2 x^5}{15}+O\left(x^6\right)\Big)\log(a)$$ $$A=e^{\log(a)}\implies A=1+x \log (a)+\frac{1}{2} x^2 \log ^2(a)+\frac{1}{6} x^3 \left(\log ^3(a)+2 \log
(a)\right)+O\left(x^4\right)$$ So... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1728390",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Density of smooth positive functions Let $\Omega$ be an open bounded set of $R^n$. For $f\in L^2(\Omega)$ such that $f>0$, a.e. in $\Omega, $ there is $(f_k)\subset W^{2,\infty}(\Omega)$ such that $f_k\to f$ in $L^2(\Omega)$. My question is:
Is it possible to chose $f_k>0,\; a.e. \; \Omega, \forall k?$
| Yes. This can be obtained by the typical approach via mollification:
*
*Extend $f$ to $\mathbb{R}^n \setminus \Omega$ by $1$.
*Let $f_k$ be the convolution of $f$ with a smooth convolution kernel.
*This directly yields $f_k > 0$ a.e. and $f_k \to f$ in $L^2(\Omega)$ if the convolution kernels are appropriately cho... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1728561",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Intuitive explanation for why $\left(1-\frac1n\right)^n \to \frac1e$ I am aware that $e$, the base of natural logarithms, can be defined as:
$$e = \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n$$
Recently, I found out that
$$\lim_{n\to\infty}\left(1-\frac{1}{n}\right)^n = e^{-1}$$
How does that work? Surely the minus si... | If you know that $$\lim_{n \to \infty}\left(1 + \frac{1}{n}\right)^{n} = e\tag{1}$$ (and some books / authors prefer to define symbol $e$ via above equation) then it is a matter of simple algebra of limits to show that $$\lim_{n \to \infty}\left(1 - \frac{1}{n}\right)^{n} = \frac{1}{e}\tag{2}$$ Clearly we have
\begin{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1728752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "36",
"answer_count": 17,
"answer_id": 8
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Solve: $\int\frac{\sin 2x}{\sqrt{3-(\cos x)^4}}$ (and a question about $t=\tan \frac{x}{2}$) I tried substituting $$t=\tan \frac{x}{2}$$ but the nominator is $\sin {2x}$, so is there a way to get from $$\sin x=\frac{2x}{1+x^2}$$ to an expression with $\sin 2x$?
| $$\int\frac{\sin(2x)}{\sqrt{3-\cos^4(x)}}\space\text{d}x=$$
Use $\sin(2x)=2\sin(x)\cos(x)$:
$$2\int\frac{\sin(x)\cos(x)}{\sqrt{3-\cos^4(x)}}\space\text{d}x=$$
Substitute $u=\cos(x)$ and $\text{d}u=-\sin(x)\space\text{d}x$:
$$-2\int\frac{u}{\sqrt{3-u^4}}\space\text{d}u=$$
Substitute $s=u^2$ and $\text{d}s=2u\space\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1728941",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Mapping The Unit Disc To The Hemisphere?
Question: Can a disc drawn in the Euclidean plane be mapped to the surface of a hemisphere in Euclidean space ?
If $U$ is the unit disc drawn in the Euclidean plane is there a map, $\pi$, which sends the points of $U$ to the surface of a hemisphere, $H,$ in Euclidean space ... | If by a "circle" you mean the set of all points inside a circle (e.g., points whose distance from some center $C$ is less than or equal to 1), then the answer is "yes" and one solution is called "stereographic projection;" another is "vertical projection".
If you have a point $(x, y)$ in the unit disk (the "filled in ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1729012",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
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Some elements of the function field of the Fermat curve For $n>0$, consider the Fermat curve:
$$C(n): \{X^n+Y^n=Z^n\}\subset\mathbb P^2(\mathbb C)$$
the function field of $\mathbb C(n)$ can be explicitly described in the following way. It is the set of all fractions $\frac{f}{g}$ satisfying the following conditions:
... | Let's say you want to write
$$ (*) \qquad \frac{f(\alpha,\beta,1)}{g(\alpha,\beta,1)} = \sum_{j=0}^{n-1}\frac{p_j(\alpha)}{q_j(\alpha)}\beta^j,$$
for all points $(\alpha:\beta:1)\in C(n)$. Clearing denominators, we can suppose that $f$ and $g$ have integer coefficients. Since $\beta^n = 1-\alpha^n$, we can write
$$ f... | {
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"url": "https://math.stackexchange.com/questions/1729145",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Counterexample to switching limit and integral on compact domain If $f:[a,b] \times [c,d] \to \mathbb{R}$ is continuous on the compact rectangle (and, hence, uniformly continuous) then it holds for $y_o \in [c,d]$ that
$$\lim_{y \to y_0}\int_a^b f(x,y) \,dx = \int_a^b \lim_{y \to y_0}f(x,y) \,dx =\int_a^b f(x,y_0) \,dx... | Try $$ f(x,y) = \cases{ \dfrac{x}{y^2} \exp(-x/y) & if $y \ne 0$\cr
0 & if $y = 0$ }$$
on $[0,1] \times [0,1]$, with $y_0 = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1729271",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Distribute 5 distinguished balls to 3 undistinguished cell On a first glimpse it looks a very easy one, but I find that a bit odd.
Assuming we have 5 distinguished balls, and 3 undistinguished cells, in how many ways can we distribute the balls in the cells, when we have at least 1 ball in each?
My first guess was to ... | The numbers are small, so one can use cases: (i) $1,1,3$, and (ii) $1,2,2$.
Case (i) The "team of $3$" can be chosen in $\binom{5}{3}$ ways, and now we have no further choices.
Case (ii) The lonely one can be chosen in $\binom{5}{1}$ ways. For every such choice, the person from the remaining $4$ with the lowest student... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1729396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Alternative injection from $(0,1)$ to $p(\mathbb{N} )$ I know the standard injection is to consider the binary expansion of a number in the interval, but I was wondering if it is possible to create an injection using a decimal expansion.
To that effect first denote the decimal expansion of a $r\in (0,1)$ as $r=0.d_1d_2... | This almost works, but if you have more than one $0$ digit, you’ll get multiple copies of $1$ in the description of $f(x)$. To avoid this problem, define
$$f(x)=\left\{p_i^{d_i+1}:i\in\Bbb Z^+\right\}\;.$$
Added: As Henning Makholm notes below in the comments, the original idea of setting
$$f(x)=\left\{p_i^{d_i}:i\in\B... | {
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"answer_count": 1,
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} |
Is there a ring homomorphism between $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$? Is there a ring homomorphism between $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$?
Also for any rings $R_{1}$ and $R_{2}$ does there exist a ring homomorphism $\phi$ : $R_{1} \rightarrow$ $R_{1} \times R_{2}$?
Note: I am allowin... | Suppose $R$, $S$ and $T$ are rings. Giving a ring homomorphism $f\colon R\to S\times T$ is the same as giving homomorphisms $g\colon R\to S$ and $h\colon R\to T$.
Let's see why. First, the projection maps $p\colon S\times T\to S$ and $q\colon S\times T\to T$ are ring homomorphisms, so if we are given $f\colon R\to S\ti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1729637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
How can I prove that a linear recurrence $x_{n+1} = αx_n - β$ will contain a composite number in the sequence? I'm working on a homework problem about finite automata and I got stuck trying to prove a fact about prime numbers that I think should be true.
Given a prime $p$ and integers $α$ and $β$, can I show that the s... | The solution of the recurrence is
$$
x_n = \alpha^n x_0 -\beta(1+\alpha+\cdots+\alpha^{n-1})
$$
Since $x_0=p$, we get
$$
x_n \equiv -\beta(1+\alpha+\cdots+\alpha^{n-1}) \bmod p
$$
If $\alpha \equiv 1 \bmod p$, then
$$
x_n \equiv -\beta n \bmod p
$$
and so $x_n \equiv 0 \bmod p$ for all $n$ that are multiples of $p$.
If... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1729755",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Suppose that a sequence is Cesaro summable. Prove.... Suppose that a sequence $a_{n}$ is Cesaro summable. Prove that
$$\lim_{n \to \infty }\frac{a_{n}}{n}=0$$
| Suppose $(a_1 + \cdots +a_n)/n \to L \in \mathbb R.$ Then
$$\frac{a_n}{n} = \frac{a_1 + \cdots + a_n}{n} - \frac{n-1}{n}\frac{a_1 + \cdots + a_{n-1}}{n-1} \to L-L = 0.$$
Let's apply this to the Cesaro sums: Suppose $(S_1 + \cdots +S_n)/n \to L\in \mathbb R,$ where $S_n = a_1 + \cdots +a_n.$ By the above, $S_n/n \to 0.$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1729855",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove $f$ is not differentiable at $(0,0)$ For
$$f(x,y)=\begin{cases}
\frac{x|y|}{\sqrt{x^2+y^2}} & \text{ for }(x,y)\neq (0,0)\\
0 & \text{ for } (x,y)=(0,0)
\end{cases}$$
I'm trying to prove $f$ is not differentiable at $(0,0)$. I showed if $f$ is differentiable at $(0,0),$... | It's easy. You only have to see that if $f$ is differentiable in $(0,0)$, then $f'(0,0)$ is a linear transformation. So:
$$f'(0,0)\cdot (1,1)=f'(0,0)\cdot((1,0)+(0,1))=f'(0,0)\cdot(1,0)+f'(0,0)\cdot(0,1).$$
And that is a contradiction, because $f'(0,0)\cdot (1,1)\not = f'(0,0)\cdot(1,0)+f'(0,0)\cdot(0,1).$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1729978",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 1
} |
Vector field on manifold I've only seen a vector field $V$ on a manifold $M$ as a mapping $V:M\to TM$. Is it true that they can also be seen as a mapping $V:C^{\infty}\left(M\right)\to C^{\infty}\left(M\right)$? How would $V$ work in the second case?
| There are a couple of things to point out here. Let $M$ be a smooth manifold.
*
*Every tangent vector at $p$ may be thought of as a derivation at $p$: indeed, if $v_p$ is a tangent vector at $p$ and $f: U \to \mathbf{R}$ is a smooth function in a neighbourhood $U \subseteq M$ of $p$, then, $v_p(f)$ can be thought of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1730074",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Is there a regular hexagon with integral corners? I'm looking for a regular hexagon in $\mathbb{R}^2$, whose corners are integral, i.e. the coordinates are integers.
The hexagon cannot lie "flat" (with upper and lower line segments horizontal), since then $h = \frac{\sqrt{3}}{2} w$ with $w$ width and $h$ height of the ... | Yes we can have a regular hexagon with integral vettex coordinates ... in three dimensions. Think of a face-centered cubic lattice. If we set the length unit to half the edge of a cubic unit cell then all lattice points have integer coordinates -- including those forming a hexagonal close-packed plane perpendicular t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1730204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$.
If $n = a^2 + b^2 + c^2$ for positive integers $a$, $b$,$c$, show that there exist positive integers $x$, $y$, $z$ such that $n^2 = x^2 + y^2 + z^2$.
I feel that the pr... | This can be generalized a bit. it happens that every positive integer $n$ is the sum of four squares, $n = A^2 + B^2 + C^2 + D^2.$ The we get, from manipulating quaternions with integer coefficients,
$$ n^2 = (A^2 + B^2 - C^2 - D^2)^2 + (-2AC +2BD)^2 + (-2AD-2BC)^2 $$
https://en.wikipedia.org/wiki/Lagrange%27s_four-squ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1730327",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Can a null graph be considered k-regular, for any k ? By null graph I mean a graph without any vertices. I did not find any mention of this anywhere.
| Looking at the definition of a k-regular graph it's obvious taht you can consider the empty graph k-regular for any k. Why? Because since it has no vertices the statement: "every vertex has degree k" is true for any k.
But I am not sure how that helps and in what scenarios.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1730405",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Probability of triangle to be obtuse
Two points $A,B$ fixed on a plane(distance = 2).
C - random choosen point inside circle with radius $R$ with center at the center of $AB$
Find probability of triangle $ABC$ to be obtuse
My thoughts:
*
*If $C$ lies in the circle - $ABC$ will be rectangular
*Opposite the larger... | Any such triangle $ABC$ with $AB$ diameter and $C$ any point within the circle will be obtuse.
Proof:-
(Note:-$AO=OB=r$)
Let,C be any point in the circle.$AC$ is extended to meet the circumference at $X$.So,$\angle AXB=\alpha=90^0$ and $\angle \beta>0^0$.So,$\angle ACB=\gamma=\alpha+\beta=90$+something $>0.$So,$\angle... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1730551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Understanding One of Fermat's little theorem's proof Fermat's Little theorem :
Let $p$ be a prime which does not divide the integer $a$, then $a^{p-1} \equiv 1 (\mod p)$.
Leibniz's proof
*
*suppose that $ra$ and $sa$ are the same modulo $p$, then $r \equiv s (\mod p)$. {So the first p-1 multiples of a are distinct a... | Because if $ak \equiv 0 \pmod{p}$, for some $1 \leq k \leq p-1$, then $p \mid ak$. But, by Euclid's Theorem, as $\gcd(p,k) = 1, p \mid a$, a contradiction.
Therefore, $ak \not\equiv 0 \pmod{p}$, for all $1 \leq k \leq p-1$. Then we've got $p-1$ distinct possible values of $k$. Since that, for every $k$ there must be a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1730637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
suppose a sample is taken from a symmetric distribution whose tails decrease more slowly than those of a normal distribution I was wondering how to go about this question about Probability QQ Plots, the question is,
suppose a sample is taken from a symmetric distribution whose tails decrease more slowly than those of a... | Maybe it helps to have an example. The Laplace distribution has 'fatter' tails than a normal. We can easily generate some data
from a Laplace distribution. The difference of two exponential
distributions with the same rate is Laplace. (See Wikipedia on
'Laplace distribution', third bullet under Related Distributions. i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1730792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Why weren't continuous functions defined as Darboux functions? When we were in primary school, teachers showed us graphs of 'continuous' functions and said something like
"Continuous functions are those you can draw without lifting your pen"
With this in mind I remember thinking (something along the lines of)
"Oh, t... | The Darboux definition does not correspond very well with our intuition about continuity. For example, the Conway function takes on every value in every interval, and is therefore Darboux. However it is not continuous, and I don't think we want it to be continuous, because it certainly doesn't agree with your teacher... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1730911",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "38",
"answer_count": 5,
"answer_id": 0
} |
Why does addition not make sense on infinite vectors? I was reading http://www.math.lsa.umich.edu/~kesmith/infinite.pdf to learn more about infinite dimensional vector spaces, and the author argues that the standard basis ($e_i$ is the sequence of all zeroes except in the i-th position, where there appears a 1), does n... | The real problem is that infinite sums should be understood as some sort of limit of a converging sequence. But while that operation may make sense, it isn't a vector space operation.
If you add a topology, then you can introduce infinite sums and things will make perfect sense (as long as the sum converges). In this... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1731030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
How do I prove that if A is a $n × n$ matrix with integer entries, then $det(A)$ is an integer? My proof:
if A is an n x n matrix with integer entries, then this means that every entry of A contains an integer. Therefore, any integer operated on by another integer is also an integer. A determinant is calculated by a s... | Your approach is correct. You could extend it by giving a definition for the determinant and pointing out the closedness of the involved operations.
One such definition defines the determinant of a matrix as alternating multi-linear form of the column vectors of that matrix. This definition involves only additions and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1731132",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Find he local maximum and minimum value and saddle points of the function? Find he local maximum and minimum value and saddle points of the function: $$f(x,y)=x^2-xy+y^2-9x+6y+10$$
The answer is a min of $(-4,1), f(-4,1)=73$
I got a min of $(12/5,-21/5)$
my
$$
f_x=2x-y-9\\
f_y=-x+2y+6
$$
set $f_x = 0 = f_y$ and we get ... | You're both wrong (or maybe you made a mistake in copying $f$): the minimum is at $(4,-1)$ where the value is $-11$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1731227",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
behavior of the Linear system of an ODE model I am working on a predator-prey model and the linearization about and equilibrium point $(0,e_2)$ has Jacobian matrix as follows
$$\mathcal{J} = \begin{pmatrix}
0 & 0\\
b& -b
\end{pmatrix},$$
where the parameters $e_2$ and $b$ are positive. I never have dealt with this ki... | Such a Jacobian matrix does not suffice to determine the stability of the fixed point. Compare the behaviour of the differential systems $$x'=ax^3\qquad y'=x-y+c$$ around their fixed point $(0,c)$ for some positive $a$ (unstable) and for some negative $a$ (stable).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1731358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to compute this double integral I'm trying to show that $\int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^2}dxdy \neq \int_0^1\int_0^1\frac{x^2-y^2}{(x^2+y^2)^2}dydx$ by computing these integrals directly.
I tried using polar coordinates with no success as the bounds of integration caused problems.
I also tried the subst... | The integral is not absolutely convergent, so we cannot change the order of integration, or the coordinates used, and expect to get the same answer.
Polar Coordinates
If we try to convert to polar coordinates,
$$
\begin{align}
\int_0^1\int_0^1\frac{x^2-y^2}{\left(x^2+y^2\right)^2}\,\mathrm{d}x\,\mathrm{d}y
&=\int_0^1\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1731441",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
$3$ children riddle, compute the ages based on information given A man has $3$ children such that their ages add up to some number $x$, and whose ages multiply to some number $y$, such that $xy = 756$. What are the ages of the $3$ children?
Letting the ages be $a$, $b$, and $c$ of the three children, what we know is t... | There is also another "reasonable" solution which is the ages are $3.5$, $4$, and $4.5$. Some children's ages are expressed in "halves" although it is not as common as "wholes". ($3.5 + 4 + 4.5$) * ($3.5 * 4 * 4.5$) does indeed equal $756$. It is interesting to note that this is $12 * 63$ and if you were to guess th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1731520",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
Solution of a riccati equation. We're given a self-adjoint equation as follows : $$ \dfrac{d}{dt}[t \frac{dx}{dt}]+(1-t)x = 0$$ We first convert this into a riccati equation , and hence we get : $$ \dfrac{du}{dt} + (\dfrac{1}{t})u^{2} +(1-t)=0$$ Now we want the solution of this riccati equation of the form $ct^{n}$. Ho... | As you said you must have a solution to get the other one. by a simple guess you can make the guess that solution is $u=-t$. and then find the other root. the first root must be given in problem or else you must guess it.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1731685",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
An inequality to find the range of an unknown coefficient Find the range of $a$ such that
$$a(x_1^2+x_2^2+x_3^2)+2x_1x_2+2x_2x_3+2x_1x_3 \geq 0, x_i\in \mathbb{R}$$
I tried to use Cauchy Inequality but it seems not...
| Answer: $a \ge 1$
Let $a=1:$ $$x_1^2+x_2^2+x_3^2+2x_1x_2+2x_2x_3+2x_1x_3 =(x_1+x_2+x_3)^2\ge 0$$
If $a>1$ then $$(a-1)(x_1^2+x_2^2+x_3^2)+x_1^2+x_2^2+x_3^2+2x_1x_2+2x_2x_3+2x_1x_3 \ge 0$$
If $a<1, a=1- E$ then
$(x_1+x_2+x_3)^2-E(x_1^2+x_2^2+x_3^2)<0$ at $x_1=-x_2-x_3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1731807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
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