Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Show the following: $\displaystyle\sum_{n=1}^\infty\dfrac1{n(n+k)}=\dfrac{H_k}k$.
For each $n\in\Bbb N$ with $n\geq1$ is $\displaystyle H_n:=\sum_{k=1} ^n\dfrac 1k$ the $n$*-th partial sum of the harmonic series.* $k\in\Bbb N$ with $k\geq1$. Show that $\displaystyle\sum_{n=1}^\infty\dfrac1{n(n+k)}=\dfrac{H_k}k$
How w... | Note that $$\frac{1}{n(n+k)} = \frac{1}{k}\left(\frac{1}{n} - \frac{1}{n+k}\right).$$
| {
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"url": "https://math.stackexchange.com/questions/607897",
"timestamp": "2023-03-29T00:00:00",
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How is it possible calculate Volumes from Adding together Areas? You can't find the volume for a $2D$ object, but by finding the area of say a square $A$ and multiply by any height $H$ you get the volume of cube with base $A$ and height $H$.
Here's my question. Is the volume of an object made of a very large amount of... | I think this is just a matter of conceptualisation. But Calculus is better because it gives us the concept and the language to do the calculations.
Another way is instead of thinking that a 2D area has zero height, I would like to think that height is undefined in the 3rd dimension. Then when you multiply by the height... | {
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Why is $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$ Why is $\sin(\alpha+\beta)=\sin\alpha\cos\beta+\sin\beta\cos\alpha$? How can we find the RHS if we don't know what it is? (instead of proving the identity itself)
I could find a geometric solution in Wikipedia, but is there any solution that doesnt req... | The differential equation
$$f''=-f$$
has a unique solution for given initial conditions $f(0)=x_0$, $f'(0)=y_0$. To show uniqueness assume that $g$ is another solution for the same initial values. Then $h:=f-g$ also satisfies $h''=-h$, and $h(0)=h'(0)=0$. But then
$$\frac\partial{\partial t}(h(t)^2+h'(t)^2)=2h(t)h'(t)+... | {
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Prove the limit $\mathop {\lim }\limits_{x \to {0^ + }} {x^{\sin (x)}} = 1$ Prove the following limit:
$$\mathop {\lim }\limits_{x \to {0^ + }} {x^{\sin (x)}} = 1$$
I can use limits arithmatic, squeezing principle, "well-known" limits etc..
We didn't learn Lopital law so I can't use it.
I tried to use the above tools, ... | EDIT: Note that on $(0,1)$, $x^a < x^b$ whenever $a>b$. The adjustment below reflects this. I give a shout out to Baranovskiy for point out this error.
I have an alternative proof. Note that $1 = x^0 \geq x^{\sin(x)} \geq x^x$ in some neighbourhood $(0,\delta)$. What does this tell us? Then squeeze.
| {
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Real life examples of commutative but non-associative operations I've been trying to find ways to explain to people why associativity is important.
Subtraction is a good example of something that isn't associative, but it is not commutative.
So the best I could come up with is paper-rock-scissors; the operation takes ... | The averaging operation, defined by $$a\oplus b= \frac{a+b}2$$ is commutative but not associative.
| {
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Simple Proof question Image : http://postimg.org/image/dkn0d5uen/
I'm studying Spivak's calculus and I have a really simple question :
I'm only in the first chapter on "The basic properties of numbers"
So far, we have the following propostion
P1 : (a+b)+c=a+(b+c)
P2 : a+0=0+a=a
P3 : a+(-a)=(-a)+a=0
Now, he tries to pro... | Spivak wants to show that zero is the unique additive identity on $\mathbb{R}$. That is, he want to prove that if we have $a+x=a$ then $x$ must identical to zero. He assumes P1, P2 and P3 to prove this. In particular, he uses P2 in the last step. If $0+x=0$ then using P2 we can conclude that $x=0$ without P2 we can not... | {
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RSA cryptosystem with special prime
Let $p < 2^{1000}$ and $q=3 \cdot 2^n - 1$ for $500 < n < 1000$ be
primes and set $n=pq$ to be the modulus of the RSA cryptosystem. Find
an attack on this system and how many operations that are required to
succeed.
My attempt at a solution: Set $m=pq$ and compute $d_n = \gcd... | Your approach is correct; if the set of possible values of $q$ is so much restricted, the easiest solution is to search it exhaustively.
The tricky solution is to check the candidates for $q$ for primality first (this can be done once, even if we're trying to break multiple instances of the cryptosystem with different ... | {
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Help on basic set theory question. Prove or Disprove: For every two sets $A$ and $B$, $(A\cup B)-B=A$.
I believed it was true, so first I showed that $(A\cup B)-B$ is a subset of $A$. My question is how do I prove that $A$ is a subset of $(A\cup B)-B$?
What I have first is what follows:
Suppose there exists an arbitra... | HINT: You are implicitly assuming that $A\cap B=\varnothing$.
| {
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How does $\sin(x-2\pi) = \sin(x)$? How does $\sin(x-2\pi) = \sin(x)$? Is it so that you can split $\sin(x-2\pi)$ into $\sin(x) - \sin(2\pi)$ and that equals $\sin(x) - 0 = \sin(x)$? Please help. Thank you
| Sine is a periodic function with period 2$\pi$. Perhaps the easiest way to see this is to use the formula $\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)$
| {
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Volume of Solid Bounded by the cylinders $y=x^2$ and $y=z^2$ and the plane $y=1$ Find the volume of the solid bounded by the cylinders $y=x^2, y=z^2$ and the plane $y=1$
I think the integral should be: $$\int_0^1\int_{-\sqrt y}^\sqrt y\int_{-\sqrt y}^\sqrt y\ dx\,dz\,dy$$
Could someone tell me if this is correct?
| This is not quite right. The region of integration there is not curved but straight like this:
Instead, maybe to
use cylindrical coordinates.
Also please look at this question here: Find the volume common to two circular cylinders, each with radius r, if the axes of the cylinders intersect at right angles. (using d... | {
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work rate problem. GMAT -related I'm looking for a good formula/system to use for these problems. Too often I'm just relying on raw intuition and it takes me too much time to solve these questions. Is there a good starting place to solve these problems? What's like a good step 1 and step 2?
Six machines, each working a... | Any time you have multiple machines working together, you add their rate to get the total rate. In this case, the machines all have the same rate, so their total rate for the 6 machines is just 6r. We know that rate*time=work, so we know that 6r*12days=W (W can represent the job they complete)
Now we want to know ho... | {
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How to calculate the interest amount per day I need to implement this calculation in my project...
Its a simple calculation But I dont know... I googled about that but can't to find the solution....
I have the following values (note:its a dynamic value)
Interest rate (per day) => 0.17%
Amount => 1500
Days => 15
How ca... | If the amount is compounded every day then the total with interest after 15 days worth of interest is $$1500\times\underset{\text{15 times}}{\underbrace{1.0017\times1.0017\times\dots\times1.0017}}=1500\times1.0017^{15}
$$
(hopefully if that's not quite right you can change it so that it's the right number, i.e. compou... | {
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Infinite Series $\sum\limits_{n=1}^\infty\frac{H_{2n+1}}{n^2}$ How can I prove that
$$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$$
I think this post can help me, but I'm not sure.
| Different approach:
$$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\sum_{n=1}^\infty\frac{H_{2n}+\frac{1}{2n+1}}{n^2}$$
$$=\sum_{n=1}^\infty\frac{H_{2n}}{n^2}+\sum_{n=1}^\infty\frac{1}{n^2(2n+1)}$$
where
$$\sum_{n=1}^\infty\frac{H_{2n}}{n^2}=4\sum_{n=1}^\infty\frac{H_{2n}}{(2n)^2}$$
$$=4\sum_{n=1}^\infty\frac{1}{2n}\left(-\in... | {
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"timestamp": "2023-03-29T00:00:00",
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Uniqueness of Unitary operator i saw the post "Polar decomposition normal operator" (Polar decomposition normal operator). There was that such a $U$ is unique iff the image of $T$ is dense. Some lines later by the comments there is that we also can say that $||T||>\delta||x||$, but this means that $T$ is invertible. Ca... | For a bounded operator $A$ with polar decomposition $A = UP$, $U$ is the (canonical, if you'd like) partial isometry
$$
(A^*A)^{\frac{1}{2}}h \stackrel{U}{\mapsto} Ah
$$
from $Ran(A^*A) = Ran(A^*A)^{\frac{1}{2}}$ to $Ran(A)$. If $A$ is invertible, so is $A^*A$. This make $U$ unique.
Assuming $U$ admits unitary extens... | {
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Another question about integrable functions with a transform I am an engineering student, and taking a real analysis course at demand of my advisor, my inexperience in proofs is giving me hard time. I stumbled upon this example, whose proof left as an exercise. Could you please give me some clues? Here is the example:
... | Here is a lacunary solution.
*
*Approximate $f$ in $\mathbb L^1$ by a simple function (a linear combination of characteristic functions of measurable sets: this reduced the proof where $f$ is such a function.
*By linearity, do it when $f=\chi_B$, where $B$ is a Borel subset of $\mathbb R^n$, with finite Lebesgue m... | {
"language": "en",
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Stuck on simple partial integration $$\int_0^3\frac{|x-y|}9dy=\frac19\left(x-\frac32\right)^2+\frac14$$
Could someone enlighten me regarding this partial integration?
I feel like i'm missing something but I dont know what I am doing wrong.
| Hint
$$\int_0^3\frac{|x-y|}{9}dy=\int_0^x\frac{x-y}{9}dy+\int_x^3\frac{y-x}{9}dy$$
| {
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Two different M/G/Infinity queues I need some help with this problem. We have a service system where customers arrive randomly, following a Poisson process (intensity λ). The times that the customers spend in service are independent, following a Weibull distribution with parameters α > 0 and β > 0. The times in service... | The analytic result you're discussing for the mean of $L$ is
$$ \lambda \alpha \Gamma\left(1+\frac{1}{\beta}\right)$$
which I guess you've obtained, your question definitely suggests this. It's the product of $\lambda$ and the mean waiting time a single customer experiences which is the mean of the Weibull distribution... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to work out this easy fraction? I need help working out this fraction, I know it seems quite easy but I'm a bit stuck.
The question is:
$$\frac{\frac {5}{2}}{\frac{5}{9}}$$
My attempt was changing the denominators by multiplying by $2$ to make $18$ and then changing the same way the nominators:
$$\frac{\frac {45}... | 5/2 * 9/5
Multiply by what you're dividing by's reciprocal.
| {
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"question_score": "3",
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Solutions of $f(x)\cdot f(y)=f(x\cdot y)$ Can anyone give me a classification of the real functions of one variable such that $f(x)f(y)=f(xy)$? I have searched the web, but I haven't found any text that discusses my question. Answers and/or references to answers would be appreciated.
| There is a classification of the functions $f:\mathbb R\to\mathbb R$ satisfying
$$
f(x+y)=f(x)+f(y), \quad\text{for all $x,y\in\mathbb R$}. \qquad (\star)
$$
These are the linear transformations of the linear space $\mathbb R$ over the field $\mathbb Q$ to itself. They are fully determined once known on a Hamel basis o... | {
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In how many ways can we select n objects from a collection of size 2n that consists of n distinct and n identical objects? In how many ways can we select $n$ objects from a collection of size $2n$ that consists of $n$ distinct and $n$ identical objects?
The answer is $2^n$ and I really don't see how they get this. Sele... | Let $A=\{a_1,\ldots,a_n\}$, where the $a_k$ are mutually distinguishable, and let $S$ be $A$ together with $n$ indistinguishable objects. An $n$-element subset $X$ of $S$ is completely determined when you know $X\cap A$: if $|X\cap A|=k$, the remainder of $X$ is just $n-k$ of the indistinguishable objects. $A$ has $2^n... | {
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Cups of water from a bucket We have an empty container and $n$ cups of water and $m$ empty cups. Suppose we want to find out how many ways we can add the cups of water to the bucket and remove them with the empty cups. You can use each cup once but the cups are unique.
The question: In how many ways can you perform t... | Let $L=n+m$ and $D=n-m \ge 0$
Hint 1: consider a sequence $X=(x_1,x_2, ... x_L)$, associate a filled cup with $x_i=1$ and an empty with $x_i=-1$. Let $C(n,m)$ count all such binary sequences of length $L$ with the two restrictions: $\sum_{k=1}^j x_k\ge 0$, $\forall j$ and $\sum_{k=1}^L x_k =D$
Then, the total number of... | {
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How to find $f(x)$ and $g(x)$ when only given $f(g(x))$ I've learned how to find $f(g(x))$ when given the two $f(x)$ and $g(x)$ functions fairly easily, but I haven't found anywhere online showing how to do the opposite. For this question I'm working on I'm asked to find $f(x)$ and $g(x)$ if $\cos^2(x) = f(g(x))$.
Can... | This isn't possible to do uniquely, since for example
$$
f\left(x\right)=x
$$
and
$$
g\left(x\right)=\cos^{2}x
$$
gives you the desired result. However, I think the answer they are looking for is
$$
f\left(x\right)=x^{2}
$$
and
$$
g\left(x\right)=\cos x
$$
so that
$$
f\left(g\left(x\right)\right)=\left(\cos x\right)^{2... | {
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If D is an Integral Domain and has finite characteristic p, prove p is prime. So the question is simply.
If $D$ is an integral domain and has finite characteristic prove that the characteristic of $D$ is a prime number.
This is my proof.
Assume $p$ is the characteristic of $D$. Let $a$ be a non zero element of $D$.... | Hint $\ $ The finite characteristic $\,n\,$ is just the size of the natural image of $\Bbb Z$ in $D\,$, via $\,1_\Bbb Z \mapsto1_D.$ This image is a subring of $D$ isomorphic to $\,\Bbb Z/n,\,$ which is a domain $\iff n\,$ is prime.
| {
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Linear combination and Basis
*
*Consider a set of five arbitrary 2x2 matrices. Can you always write one as a linear combination of the others? Explain. Repeat for five arbitrary 3x3 matrices.
*For each of the following sets explain whether or not the set is/could be a basis for the space mentioned.
a) Four vectors i... | The first part of the first question is asking whether every set of five $2\times 2$ contains one that is a linear combination of the other four; the second part is asking the same question about sets of five $3\times 3$ matrices. The first part can be paraphrased as follows: is every set of five $2\times 2$ matrices l... | {
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"timestamp": "2023-03-29T00:00:00",
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Find The Eigenvalues and Eigenvectors of the Hermitian Matrix Find the eigenvalues and eigenvectors of the $2\times2$ hermitian matrix.
$$\pmatrix{\epsilon_1&|V|e^{i\alpha}\\
|V|e^{-i\alpha}&\epsilon_2}$$
I know to find eigenvalues, you use $|A-\lambda I|$, but this is giving me difficult results to find an exact ... | We can start off by solving the more general case system in order to simplify matters:
$$\begin{bmatrix}a & b\\c & d\end{bmatrix}$$
This produces the eigenvalue / eigenvector pairs:
*
*$\lambda_1 = \dfrac{1}{2} \left(-\sqrt{a^2-2 a d+4 b c+d^2}+a+d\right)$
*$v_1 = \left(\dfrac{-(-a+d+\sqrt{a^2+4 b c-2 a d+d^2})}{2... | {
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Method of characteristics: $xu_t -tu_x = u$ I am just learning the method of characteristics. Suppose I want to solve
$$xu_t - tu_x = u \quad u(x,0) = h(x)$$
I write
$$\dot{t}(s) = x\\
\dot{x}(s) = -t \\
\dot{z}(s) = z$$
If $t(0) = 0$ and $x(0) = x_0$, we have (I think)
$$t(s) = x_0\sin(s)\\
x(s) = x_0\cos(s)\\
z(s) = ... | Clearly, the domain of a solution of $xu_t-tu_x=u$, can not contain any circle centered at the origin. But this is the only restriction!
Take $\Omega=\mathbb R^2\smallsetminus C$, where $C$ is a curve starting from the origin and going to infinity, assume also that $C$ does not intersect to positive $x$-axis. Let $h=h(... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Find $c_1,c_2,c_3\in\mathbb{Q}$ such that $(1+\alpha^4)^{-1}=c_1+c_2\alpha+c_3\alpha^2$ in $\Bbb Q(\alpha)$. Let $\alpha\in \overline{\mathbb{Q}}$ be a root of $X^3+X+1\in\mathbb{Q}[X]$. So this is the minimal polynomial of $\alpha$ because it's irreducible in $\mathbb{Q}[X]$. I had to find the minimal polynomials of $... | Hint
Apply the extended Euclidean algorithm to find polynomals $g,h$ with
$$
g(X^4 + 1) + h(X^3 + X + 1) = 1.
$$
Then consider this equation mod $X^3 + X + 1$ and plug in $\alpha$.
Note: It's not strictly neccesary, but you may want to reduce $X^4 + 1$ mod $X^3 + X + 1$ first.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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solving an expression based on sin $\theta$ If $\sin^2 \theta = \frac{x^2 + y^2 + 1}{2x}$, then $x$ must be equal to what?
What does the following solution mean?
$0 \le \sin^2 \le 1$
This implies $0 \le \frac{x^2 + y^2 + 1 }{ 2x } \le 1$
This implies $\frac{(x - 1)^2 + y^2 }{2x} \le 0 $
This implies $x = 1$.
Can you ... | We have $$(x-1)^2=x^2+1-2x\ge0\iff x^2+1\ge 2x$$
and since $\sin^2\theta\ge0$ hence we have $x>0$ and then
$$1\ge\sin^2\theta =\frac{x^2+y^2+1}{2x}\ge1\quad (>1\;\text{if}\; y\ne 0\;\text{which gives a contradiction})$$
hence $y=0$ and
$$\sin^2\theta=\frac{x^2+1}{2x}=1\iff x=1$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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LCR Odds - 6 'C's in a row. Played the game LCR. One unlucky guy rolled 3 - 'C's and then to make matters worse rolled 3 - 'C's again on his next turn. So what are the odds of that happening?
| While @Ross's answer is entirely correct, there is a broader question here. The question asked "what are the odds of that happening". What is "that"? The specific event of rolling six C's in a row does indeed have probability $\frac{1}{46656}$.
However, if instead your unlucky friend had rolled six R's in a row, you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/610453",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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Converting Context Free Grammar to Chomsky Normal Form This is an exercise that I had to complete in my class and I struggled a lot with it
$$\begin{align*}
&S\to 0A0\mid 1B1\mid BB\\
&A\to C\\
&B\to S\mid A\\
&C\to S\mid\epsilon
\end{align*}$$
Added from comments: This is what I have but I am sure it is wrong: replac... | (a) Eliminate -productions.
Since C can produce , A can also produce , so B can produce , so S can produce .
therefore, we need to change the productions for every symbol.
S → 00 | 0A0 | 11 | 1B1 | B | BB
A → C
B → S | A
C → S
(b) Eliminate any unit productions in the resulting grammar.
We note that A, B, and C will al... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Show that $\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$ In this question we are asked to show that
$\sum_{k=2012}^{n} 2^k\binom{n}{k} = \Theta(3^n)$
What I did:
$\sum_{k=2012}^{n} 2^k\binom{n}{k} = \sum_{k=2012}^{n} 2^k*1^{n-k}\binom{n}{k} \leq \sum_{k=0}^{n} 2^k*1^{n-k}\binom{n}{k} = (2+1)^n = 3^n$, using newton's... | You note that the term you are omitting from the Newton binomial are all smaller than $\binom{n}{k}$ in growth (the $2^k$ are smaller than $2^{2012},$ so constant. And the binomial is at most a polynomial of degree $2012,$ so subexponential.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/610691",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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A fair dice is tossed until a number greater than $4$ appears. The probability that an even number of tosses will be required is A fair dice is tossed until a number greater than $4$ appears. The probability that an even number of tosses will be required is:
$A. 1/2$
$B. 3/5$
$C. 1/5$
$D. 2/3$
What I did: The probabili... | You didn't go wrong. The probability of "success" is 1/3, so the probability of succeding in $2k$ tosses is
$$\left(\frac{2}{3}\right)^{2k-1} \frac{1}{3}$$
And
$$\sum_{k=1}^\infty\left(\frac{2}{3}\right)^{2k-1} \frac{1}{3}=\frac{1}{3} \frac{3}{2}\sum_{k=1}^\infty\left(\frac{4}{9}\right)^k =\frac{1}{2} \frac{4/9}{1-4/9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/610731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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find the limit of $\mathop {\lim }\limits_{x \to 0 } \frac{a}{x}\left[ {\frac{x}{b}} \right]$ $$a,b \gt 0$$
$$\mathop {\lim }\limits_{x \to 0 } \frac{a}{x}\left[ {\frac{x}{b}} \right]$$
So, I know that if x is $x \in \mathbb{Z}$ then the limit is $a\over [b]$
I couldn't figure out the solution for $x \notin \mathbb{Z}$... | Hint
Assume that $b>0$, prove that
$$\mathop {\lim }\limits_{x \to 0^+ } \frac{a}{x}\left[ {\frac{x}{b}} \right]=0$$
and
$$\mathop {\lim }\limits_{x \to 0^{-} } \frac{a}{x}\left[ {\frac{x}{b}} \right]=\infty$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/610794",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Why is true? $\mathop {\lim }\limits_{x \to 0} \frac{x}{a}\left[ {\frac{b}{x}} \right] = \frac{b}{a}$ $$\begin{array}{l}a,b > 0\\\mathop {\lim }\limits_{x \to 0} \frac{x}{a}\left[ {\frac{b}{x}} \right] = \frac{b}{a}\\\end{array}
$$
I asked already a similar question, but I'm still not sure what makes it true.
As $x$ is... | We have,
$\displaystyle\mathtt{\lim_{x\to\,0}\dfrac{x}{a}\left[\dfrac{b}{x}\right]}$
$\displaystyle=\mathtt{\lim_{x\to\,0}\dfrac{x}{a}\left(\dfrac{b}{x}-\left\{\dfrac{b}{x}\right\}\right)}$
$\displaystyle=\mathtt{\lim_{x\to\,0}\dfrac{x}{a}\cdot\dfrac{b}{x}-\dfrac{x}{a}\cdot\left\{\dfrac{b}{x}\right\}}$
$\displaystyle=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/610882",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Derive the Frenet equations I was looking for a derivation of the Frenet equations. I've been following this reference but I've been having problems in understanding this statement (found before Eq.$(2.19)$):
When r'($s+\Delta s$) is moved from $Q$ to $P$, then r'($s$), r'($s+\Delta s$) and r'($s+\Delta s$)-r'($s$) fo... | You've mis-typed things a little -- in each place where you have $|r'(s + \Delta s) - r'(s + \Delta s)|$, you should have $|r'(s + \Delta s) - r'(s)|$; I suspect this is a cut-and-paste error.
The claim the authors make -- that the difference vector EQUALS $\Delta \theta \cdot 1$ -- is false. But what's true is that ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/610983",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
probability of drawing balls from an urn An urn contains four balls: one red, one green, one yellow, and one white.
Two balls are drawn without replacement from the urn. What is the probability of getting a red ball and a white ball? Assume that the balls are
equally likely to be drawn
Here's what I've tried:
Probabili... | Probability of getting a red or white ball on first draw:$\frac{1}{2}$
probability of getting the remaining ball that is either red or white $\frac{1}{3}$
probability both of these happen: $\frac{1}{2}\cdot \frac{1}{3}\approx0.167$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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In differential equations, do transient state and steady state always go hand in hand? Do transient state and steady state go hand in hand in differential equations? Meaning, if there is a transient state, is there always going to be a steady state?
Also, if there are neither of them, then is it always pure resonance?... | Lets say we have a spring with an external force.
We solve the second order ODE and arrive at:
$$x(t) = \dfrac{e^{-4 t}}{9} [3 \cos(4 \sqrt{3} t) - 11 \sqrt{3}) \sin(4 \sqrt{3} t)] + 2 \sin(8 t) \\ y(t) = 2 \sin(8t)$$
A plot of these functions is:
You can see that the $e^{-4t}$ becomes negligible when $t$ is large. Th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/611162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Proof about GCD's Prove that if $a, b$ and $c$ are integers with $b \neq 0$ and $a=bx+cy$ for some integers $x$ and $y$, then $\text{gcd}(b,c) \le \text{gcd}(a,b).$
I don't understand how to show (b,c) is less than (a,b)?
| $\ \begin{eqnarray}\color{}{(b,c)}&&\mid\, \color{sienna}b,\ \ \ \color{sienna}c \\ \Rightarrow\ (b,c)&&\mid \,\color{sienna}b\,x\!\!+\!\!\color{sienna}c\,y=\color{#c00}a\!\end{eqnarray}\bigg\rbrace\ $ so $\ \bigg\lbrace\begin{eqnarray}(b,c)&\mid&\ \color{#c00}a,\,b\\ \,\Rightarrow (b,c) &\color{#0a0}\le& (a,b) \end{e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/611232",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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For every $n \in \mathbb{N}$, find a function $f$ which is differentiable $n$ times at $0$, but not $n+1$ times. For every $n \in \mathbb{N}$, (in this problem $\mathbb{N}$ starts at $n = 1$) find a function $f$ which is differentiable $n$ times at $0$, but not $n+1$ times.
The function I chose is the following:
$$
f_n... | Consider $x^\alpha\sin(x^{-1})$ for a suitable $\alpha$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/611296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 2
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find $a_n$, $a_n=\sqrt{\frac{a_{n-1}^2+a_{n+1}^2}{2}}, a_1=10(a_n\in \mathbb{N},n=2,3,4,\cdots)$ I would appreciate if somebody could help me with the following problem
Q: find $a_n$
$$a_n=\sqrt{\frac{a_{n-1}^2+a_{n+1}^2}{2}}, a_1=10(a_n\in \mathbb{N},n=2,3,4,\cdots)$$
| Follow Ewan's answer.
Obviously, $c_2\ge0$, otherwise $b_n=a_n^2=c_1+c_2n<0$ for large $n$.
If $c_2>0$, then $a_{n+1}-a_{n}>0$.
However, $a_{n+1}-a_{n}=\sqrt{c_1 + c_2 n+c_2}-\sqrt{c_1 + c_2 n}=\frac{c_2}{\sqrt{c_1 + c_2 n}+\sqrt{c_1 + c_2 n+c_2}}<\frac{c_2}{2\sqrt{c_1+c_2n}}$
So for $n>c_2-c_1/c_2$, $0<a_{n+1}-a_n<\f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/611396",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
how to show $E[|X|]= \sigma$ where $X \sim N(0, \sigma^2)$ let $X \sim N(0, \sigma^2)$ I want to show $$E[|X|]= \sigma$$
thanks for help
| Let $Z\sim N(0,1)$ so that $X=\mu+\sigma Z\sim N(\mu,\sigma^2)$.
Then $Y=|X|$ has the Folded Normal Distribution if $\mu\ne 0$ and the Half-Normal Distribution when $\mu=0$.
In your case $Y$ has the Half-Normal Distribution with cdf $F$ and pdf $f$
$$
\begin{align}
F(y) & = 2 \Phi\left(\frac{y}{\sigma}\right) - 1 = \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/611490",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Relation between Supp and sheaves Let $\mathcal{F}$ be a cohorent sheaf on projective scheme $X$.
My question is simple... If $\operatorname{dim}\operatorname{Supp}\mathcal{F}$ is zero, then $\mathcal{F}(n) =\mathcal{F}$ for any integer $n$??
| One should read the comments by Matt E before reading the following. I am keeping this just as a record.
Let $X = \hbox{Proj} \, (K [x,y])$ and $Y = \hbox{Proj} \, (K[x,y]/(x^2))$. Write $i: Y \hookrightarrow X$ and $F = i_* \mathcal{O}_Y$. Then $\dim \hbox{Supp} \, F = 0$. One has $\Gamma(X, F) = 1$, but $\Gamma(X... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/611573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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$A\in\mathrm{M}_{n\times n}(\mathbb{C})\implies A^TA$ is diagonalisable? I have been set some work to do over the holidays, and one of the questions gives a hint that is as follows:
$A\in\mathrm{M}_{n\times n}(\mathbb{C})\implies A^TA\text{ is diagonalisable}$.
I know that
*
*$A\in\mathrm{M}_{n\times n}(\mathbb{R})\... | This is false.
$$
A=\pmatrix{\frac{1}{2}+i&1\\-1-\frac{i}{2}&i}\qquad A^TA=\pmatrix{2i&1\\1&0}\qquad \mathrm{Spectrum}(A^TA)=\{i\}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/611703",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
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Examples where derivatives are used (outside of math classes) I want to know what is the use of derivatives in our daily life.
I have searched it on google but i haven't find any accurate answer. I think it is mostly used in Maths but I want to know its use in other departments i.e physics, chemistry, biology and econ... | All of the above.
It is actually easier to explain physics, chemistry, economonics, etc with calculus than without it.
For example:
Velocity is derivative of position with time.
Derivative of momentum (by time) is force.
Derivative of Gibbs free energy with number of atoms is chemical potential.
Etc.
I do honestly use ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/611749",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Let T = {1000,1001...,9999}. How many numbers have at least one digit that is 0... Let T = {1000,1001...,9999}. How many numbers have at least one digit that is 0, at least one that is 1 and at least one that is 2. For example, 1072 and 2101 are two such numbers.
I have no idea to solve such probability question, I sho... | Let's break up the different cases, shall we?
*
*If the leading digit $d$ is not one of the special digits, that leaves us with $d \in \left\{3,4,5,6,7,8,9\right\}$. So we have $7$ different options for the leading digit. After that, there are $3! = 6$ different options for our special digits — $\left\{(d,0,1,2),(d,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/611830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Lemma 1.3.4(b) in Bruns and Herzog My question refers to the proof of the second of the following lemma given in Cohen-Macaulay rings by Bruns and Herzog.
Lemma 1.3.4 (Bruns and Herzog): Let $(R,\mathfrak m,k)$ be a local ring, and $\phi:F \rightarrow G$ a homomorphism of finite $R$-modules. Suppose that $F$ is free, ... |
Let $(R,\mathfrak m,k)$ be a local ring, and $\phi:F \rightarrow G$ a homomorphism of finite free $R$-modules. Then $\phi \otimes k$ is injective iff $\phi$ is injective, and $\phi(F)$ is a free direct summand of $G$.
"$\Leftarrow$" This is not difficult and I leave it to you.
"$\Rightarrow$" Let $\{e_1,\dots,e_m\}\s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/611900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Why does $\left(1+ 1/k\right)^k$ converge to $e$ as $k \to\infty$? I came across this when learning about sequences and series. No one proved it to me tho! Is there a link anywhere? Or would it be beyond what I know? (First term of analysis)
| Of course, this depends on how you define $e$. Here is something that would convince a calculus student.
Taking logs we have:
$\log((1 + \frac{1}{k})^k) = k \log(1 + \frac{1}{k})$
So we can take this limit as $k\rightarrow\infty$ using l'hospitals rule.
$$ \lim_{k\rightarrow\infty} k \log(1 + \frac{1}{k}) = \lim_{k\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/611995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Initial Value Problem
We have initial value problem $y''=x^{1/3}y$ with $y(0)=y'(0)=0$.
Does it have a unique solution?
I have tried using the Picard-Lindelöf theorem, but I cannot reduce it to a order 1 ODE. Thank you.
| Check this
Theorem: Let $p(t), q(t)$, and $g(t)$ be continuous on $[a,b]$, then the differential equation
$$y'' + p(t)y' + q(t)y = g(t),\quad y(t_0) = y_0,\, y'(t_0) = y'_0 $$
has a unique solution defined for all $t$ in $[a,b]$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $x^2+x+3=0$ mod $27$ I was preparing for my Number Theory class for next semester and one of the questions that I came upon is to solve $x^2+x+3=0$ mod $27$. I have seen modular arithmetic before but never one that involved using mod with an equation. I plugged it into WolframAlpha and it gave me $x=11,15$. My qu... | There is no harm in completing the square. We have that $2^{-1}=-13$, so $x^2+x+3=x^2-26x+3=0$ gives $(x-13)^2+3-13^2=0$, or $(x-13)^2=4=2^2$. This means that $(x-13)^2-2^2=0$ so that $(x-15)(x-11)=0$. Note you cannot really conclude $x-15=0$ or $x-11=0$ right away, since we're not working over an integral domain.
ADD... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/612103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$ In the following thread
I arrived at the following result
$$\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$$
Defining
$$H_k^{(p)}=\sum_{n=1}^k \frac{1}{n^p},\,\,\, H_k^{(1)}\equiv H_k $$
But, it was after lo... | I think it is reasonable to start with:
$$\sum_{k=1}^{+\infty}\frac{H_k^{(2)}H_k}{k^2}=\sum_{k=1}^{+\infty}\frac{H_k}{k^4}+\sum_{k=1}^{+\infty}\frac{H_k}{k^2}\sum_{1\leq j< k}\frac{1}{j^2},\tag{1}$$
that leads to:
$$\sum_{k=1}^{+\infty}\frac{H_k^{(2)}H_k}{k^2}=\left(\sum_{k=1}^{+\infty}\frac{H_k}{k^2}\right)\left(\sum_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/612181",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "44",
"answer_count": 6,
"answer_id": 4
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Book Recommendations for Linear Algebra Proofs I'm taking a graduate Linear Algebra course and have limited experience writing proofs (mostly from a discrete math class). Can anyone recommend good books to teach you how to write proofs for linear algebra? My impression is that they're very different from other kinds of... | Another book you might consider is Curtis' Abstract Linear Algebra. It claims that it could be used for a "first course," but is quite sophisticated (but I have to agree with the claim). It takes some care to introduce the more abstract topics not usually covered in undergrad classes. For instance, exact sequences of m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/612242",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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For what values of $b\in \mathbb R$ is $\pi-b$ rational? Just a simple short question.
I'm looking for values $b$ such that $\pi-b$ is a rational number.
Obviously $\pi$ is such a number, but are there more?
Edit: $b$ is in $\mathbb R$
| Clearly, and almost vacuously $b=\pi+q$ for some rational number $q$. You can easily show that these are the only ones.
| {
"language": "en",
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Finding the area of a region? I sketched out the graph to get this figure but I can't seem to find the area of the shaded region... would one Y = 4 and the other Y = 8?
I, in all honesty, am so flabbergasted with this question, any guidance will be much appreciated. I got 28/3 for the first integral (0 to 4) and 292/3... | HINT Find the point of intersection, set the limits $a = 0$ and $b =$ the point you found, set up the integral with those limits and the function $f(x) - g(x)$, and set up another integral from $b$ to $16$ with $g(x) - f(x)$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Is it true or not : if $u(z)$ is harmonic, $u(\overline{z})$ is also harmonic. Is it true or not : if $u(z)$ is harmonic, $u(\overline{z})$ is also harmonic.
My try :
$u(z)=u(x,y)$ is harmonic
Define $s=-y$
Let $U := u(\overline{z})=u(x,-y)=u(x,s)$ : $$\frac{\partial U}{\partial x}=\frac{\partial u}{\partial x} \Right... | A funciton $\phi$is harmonic if
$$ \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} = 0$$
It can be shown that $$ \frac{\partial^2 }{\partial x^2} + \frac{\partial^2 }{\partial y^2} = \frac{\partial}{\partial z}\frac{\partial}{\partial \bar{z}}$$
This means for a function of a complex variab... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/612481",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Sum of difference of numbers in an arrangement of the numbers $0,1,2,\cdots, n$ A seemingly interesting (easy?) problem came to mind and I thought it would be nice to ask your opinion about it.
Suppose we are going to arrange numbers $0$ to $n$ in a row in such a way that the sum of the difference between adjacent num... | I have an idea how to obtain an upper bound which seems to be good. We have to estimate from above the sum $S=\sum_{i=1}^n |a_i-a_{i-1}|$ where the sequence $a_0,\dots, a_n$ is a permutation of the sequence $0,1,\dots,n$. When we open the moduli in the expression for $S$, we obtain $S=\sum_{i=0}^n\varepsilon_i a_i$, wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/612553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is $SO_n({\mathbb R})$ a divisible group? The title says it all ... Formally, if
$SO_n(\mathbb R)=\lbrace A\in M_n({\mathbb R}) |AA^{T}=I_n, {\sf det}(A)=1 \rbrace$
and $W\in SO_n(\mathbb R)$, is it true that for every integer $p$, there is a $V\in SO_n(\mathbb R)$ satisfying $V^p=W$ ?
This is obvious when $n=2$, be... | Every compact connected Lie group $G$ (and $SO(n)$ is both compact and connected) is divisible since its exponential map $\exp$ is surjective (see here). By surjectivity of $\exp$, every element $g$ of $G$ has the form $\exp(v)$ (with $v\in {\mathfrak g}$, the Lie algebra of $G$) and, thus, for $h=\exp(v/p)$ we obtain ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/612649",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "19",
"answer_count": 2,
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how do you read: "$\lim (n-1)/(n-2) = 1$" The limit of the fraction $(n-1)/(n-2)$ with $n$ approaching $+\infty$ is $1$. But how do you read that, concisely? (say during chalking the formula on a blackboard). Is it acceptable to say "lim", for instance?
I would read "the limit of n minus one over n minus two, with n ap... | I tend to read $\lim\limits_{x\to a}f(x)$ as
'The limit as $x$ approaches $a$ of the function $f(x)$ is . . .'
As for dealing with the ambiguity when verbalising a quotient like this, I find that the use of the word 'all' is helpful to distinguish between the possible numerators: $n - 1$ and $1$. I would say
'$n$ m... | {
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"source": "stackexchange",
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Applying Fermat's Little Theorem: $6^{1987}$ divided by $37$ Find the remainder when $6^{1987}$ is divided by $37$.
Because 37 is prime we have: $6^{36}$ mod $37 = 1$. I tried to get a nice combination like: $1987 = 36 * 55 + 7$, so we would have $(6^{36})^{55}6^{7}=6^{1987}$.
Then, I've taken mod $37$, which is: $6^{1... | Hint $\rm\ mod\ n^2+1\!:\,\ \color{#c00}{n^2}\equiv -1 \ \Rightarrow\, n^{4k+3}\! = n(\color{#c00}{n^2})^{2k+1} \equiv - n.\ $ Yours is special case $\,\rm n=6.$
| {
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How can a negative multiplied by a negative give positive? On first look this can seem weird. But I can explain what I am looking for.
We all know from elementary maths that $(-\times-)=+$.
Now, lets say there are 3 cows and I say they will become doubled after one year so $3*2=6$.
And lets say I have $-3$ cows(which ... | Multiplication by a positive number is rather intuitive, so $(+ \times +)$ and $(+\times -)$ are relatively easy (for the latter, think you multiply a debt, for example). So the hard one is $(- \times -)$. I may be uneasy to understand this directly, but if you want to keep laws of arithmetic as they are with positive ... | {
"language": "en",
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Properties of matrix exponential I know that the solution to system $x' = Ax$ is $e^{At}$, and I'm aware of various methods to calculate the exponential numerically. However I wonder if there are some analytical results.
Namely, I'm interested in matrices of type $A_{i,i} = -A_{i+1,i}$; $A_{n,n}=0$. These matrices are ... | This seems to transform the original problem into a more complicated one... A direct approach is as follows.
For every $1\leqslant i\leqslant n-1$, let $a_i=-A_{i,i}$. The hitting time $T$ of $n$ starting from $1$ is the sum of $n-1$ independent random variables, each exponential with parameter $a_i$. In particular,
$$... | {
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"source": "stackexchange",
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find the point which has shortest sum of distance from all points? I want to find a point in the Cartesian plane so that sum of distances from this point to all points in the plane be minimum.
For example we have the points: $(x_1,y_1),(x_2,y_2),(x_3,y_3), . . .(x_n,y_n)$. Now find a point - we call this $(X,Y)$ - so t... | I think what you are looking for is the Geometric median
I'd recommend having a look at this question at stackoverflow
| {
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Integration of a rotated triangle: determining the slope of a side I'm working through MIT's OCW single-variable calculus course. The problem is taken from Problem set 7: Question 4B-5.1
Q: Find the volume of an equilateral triangle with side length a which is rotated around one of its sides.
I am actually doing the ca... | You were close, but height needs to be $a\sin\left(\frac \pi 3\right)$. Then slope is given by $\dfrac {\frac {a\sqrt 3}2}{\frac a2} = \sqrt 3$.
| {
"language": "en",
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"source": "stackexchange",
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Finding the value of a logarithmic expression involving an infinite GP Find the value of $(0.16)^{\displaystyle\log_{2.5}(\frac13+\frac1{3^2}+\frac1{3^3}+\cdots)}$.
I could solve the series. It gave $$(0.16)^{\log_{2.5}0.5}$$
Unable to proceed from here.
| HINT:
$$0.16=\frac4{25}=\left(2.5\right)^{-2}$$
$$\text{Now, }\displaystyle a^{\log_am }=m$$
| {
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How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter? This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and proof wi... | There is really an elementary proof of a more general result: if one polygon fits within the other (boundaries may touch) then its perimeter is smaller than that of the bigger polygon
http://www.cut-the-knot.org/m/Geometry/PerimetersOfTwoConvexPolygons.shtml
The idea is that the sides of the small polygon may be extend... | {
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Hunt for a function. I am looking for any nontrivial function $f(z): \mathbb{C}\to\mathbb{C}$ such that:
*
*$f(z)$ is an entire function.
*A $z_p\in\mathbb{C}$ exists for which $\Re(f(z))\geq\Re(f(z_p))~\forall z\in\mathbb{C}$ with $\infty>\Re(f(z_p))>-\infty$.
If you do not have an example, but can give me some st... | Such a function must be constant.
Consider $g(z) = \exp(-f(z))$. We have
$$
\left|g(z)\right| = \exp(-\Re(f(z)) \le \exp(-\Re(f(z_p)).
$$
Since $g$ is entire and bounded, it must be constant. Hence $f$ is constant too.
| {
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Limit with missing variables
Find the values for $a$ and $b$ such that
$$\lim_{x \to 0} \frac{\sqrt{a + bx} - \sqrt{3}}{x} = 3$$
Basically what I did so far was I started by multiplying by the conjugate. and obtained $$\frac{a+bx-3}{x(\sqrt{a+bx}+\sqrt 3)}$$ I don't know what to do after this.
| Being finite, numerator $\to 0\,\Rightarrow\,a = 3,$ so it is $\,\displaystyle\lim_{x\to0}\dfrac{f(x)-f(0)}x = f'(0),\ \ \ f(x) = \sqrt{3+bx}$
| {
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Question about differentiation of series Suppose $$f(x)=\sum_{n=1}^\infty (-1)^{n+1} \ln (1+\frac{x}{n}), \quad x\in[0,\infty).$$ I need to show that $f$ is differentiable on $(0,\infty).$
proof: I try to show differentiability using the classical defintion of limit.
$$\lim _{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ $$... | $$ \lim_{h\to 0} \sum_{n=1}^\infty (-1)^{n+1}\frac{\left(\ln(\frac {n+x+h}n) - \ln(\frac {n+x}n)\right)}{h} \\
= \lim_{h\to 0} \sum_{n=1}^\infty (-1)^{n+1}\frac{\ln(\frac {n+x+h}{n+x})}{h} \\
= \lim_{h\to 0} \sum_{n=1}^\infty (-1)^{n+1}\frac{\ln(1+\frac {h}{n+x})}{h} $$
Now use the formula $\ln(1+a) = a + O(a^2)$ as $a... | {
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Finding the derivative using quotient rule... $$\frac{\text{d}}{\text{dt}}\dfrac{2(t+2)^2}{(t-2)^2}$$
I applied the quotient rule:
$$\dfrac{[2(t+2)^2]'(t-2)^2-2(t+2)^2[(t-2)^2]'}{(t-2)^4}$$
$$\dfrac{4(t+2)(t-2)^2-2(t+2)^22(t-2)}{(t-2)^4}$$
$$\dfrac{4(t+2)(t-2)-4(t+2)^2}{(t-2)^3}$$
This was part of a problem where I nee... | Your answer is correct, Wolfram is just simplified.
$$ \frac{(t-2)^2 \cdot 4(t+2) - 2(t+2)^2 \cdot 2(t-2)}{(t-2)^4}$$
$$=\frac{(t-2)(t+2) \cdot 4(t-2) - 4(t+2)}{(t-2)^4}$$
$$= \frac{(t+2) \cdot (4t - 8 - 4t - 8)}{(t-2)^3}$$
$$= \frac{-16(t+2)}{(t-2)^3}$$
| {
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Why $\int{\log^2(2\sin(\pi x))}dx\neq\frac{\log^3(2\sin(\pi x))\sin(\pi x)}{3\pi \cos(\pi x)}$? Apparently I completely forgot the basics of calculus after 13 years of not studying it. Why won't:
$$\frac{d \log^2(2\sin(\pi x))}{dx} = \frac{\log^3(2\sin(\pi x))\sin(\pi x)}{3\pi \cos(\pi x)}$$
UPDATE:
Sorry everybody, I... | Using Chain rule:$$\frac{d }{dx}\log^2(2\sin(\pi x))=2\log(2\sin(\pi x))\times[2\times\pi\times\cos(\pi x)]\times\frac{1}{2\sin(\pi x)}$$
| {
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Example of a non locally-compact space with a dense locally compact subspace. Let $X$ be a topological space such that $A \subset X$ is a dense subspace which is locally compact and $B \subset X$ is a dense subspace which is not locally compact (at all of its points).
Is it possible to find such $X$? if it is, an examp... | 1)If $X$ must be Hausdorff, take $X=\mathbb{R}$, $A=X-\{0\}$, $B=\mathbb{Q}$.
2)If $X$ must be a non-locally compact Hausdorff space, let $Y=\{1/n|n\in \mathbb{N}^*\}$ and $X=(\{0\}\times \mathbb{Q}) \cup (Y\times \mathbb{R})$, $X$ is not locally compact at the point $(0,0)$.
Let $A=Y\times \mathbb{R}$, $A$ is dense in... | {
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Proof that $\mathbb G_n \bigcap \mathbb G_m = \mathbb G_{(m:n)}$ Being $\mathbb G_n$ the roots of unity for $n \in \mathbb N$, prove that $\mathbb G_n \bigcap \mathbb G_m = \mathbb G_{(m:n)}.$
| Hint
Recall
$$z\in\mathbb G_n\iff z^n=1$$
By double inclusion
*
*if $d|n$ prove that
$$z\in\mathbb G_d\Rightarrow z\in\mathbb G_n$$
and hence which inclusion we can deduce?
*Use the Bézout's theorem:
$$d=\gcd(m,n)\iff \exists u,v\in\mathbb Z,\; um+vn=d$$
to get the other inclusion.
| {
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find a the following series this is just somthing i thought about (i dont know if there is an answer)
Let $ \sum_{n=1}^{\infty}a_n \to L_1$
($a_n $ is a positive sequence)
find a sequence $b_n$ such that:
$\lim_{n\to\infty} \dfrac{b_n}{a_n} = \infty$
and $\sum_{n=0}^{\infty}b_n = L_2$ (the series converges)
if there ... | NOTICE
This is an answer to the original question. While I was writing the answer the OP changed the condition $a_n/b_n\to\infty$ to $b_n/a_n\to\infty$.
Take $b_n=a_n^2$. Since $\sum a_n$ converges, $a_n\to0$, and $a_n/b_n=1/a_n\to\infty$. Also, $a_n$ is bounded. Let $A$ be an upper bound. Then $0\le b_b\le A\,a_n$, so... | {
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Show that $A= ([0,\sqrt2] \cap \Bbb Q ) \subset \Bbb Q $ is not compact. We have $\Bbb Q$ equipped with the Euclidean Metric.
Show that $A=([0,\sqrt2] \cap \Bbb Q ) \subset \Bbb Q $ is not compact.
How would you go about showing this?
You can make on open cover $\{O_n\}= ((-\infty, \sqrt2 -\frac1n) \cap \Bbb Q)$ for ... | If a subset $A$ of $\mathbb R$ is compact, then it is also closed. In particular, if a ssequence $\{x_n\}_{n\in}\subset A$ converges to $x$, then $x\in A$. Let
\begin{align}
x_1=&1\\
x_2=&1.4 \\
x_3=&1.41 \\
x_4=&1.414 \\
x_5=&1.4142 \\
etc.
\end{align}
I.e.,
$$
x_n=\frac{\lfloor10^n\sqrt{2}\rfloor}{10^n},
$$
Clearly ... | {
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Inductive proof of $\,9 \mathrel| 4^n+6n-1\,$ for all $\,n\in\Bbb N$ Prove that for all $n\in\mathbb N$, $9 \mathrel| (4^n+6n-1)$.
I have shown the base case if $n=0$ (which is a natural number for my course).
I'm now trying to prove that $9k=4^n+6n-1$.
I substituted $n+1$ for $n$, and have $4^{(n+1)}+6(n+1)-1$, which ... | It's really simple this is how I'd approach it:
$4^{n+1}+6(n+1)-1=4(4^n+6n-1)-18n+9=4(4^n+6n-1)-9(2n-1)$
and since you've already proved that $4^n+6n-1=9k$ then you will have: $4(4^n+6n-1)-9(2n-1)=9k*4-9(2n-1)$
$=9(4k-(2n-1))$ and since $k$ and $2n-1$ are both integers then you can let $k'=4k-(2n-1)$ .
Now you'll have ... | {
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Is a set that is an abelian group under addition and a group under multipliation a field? I suspect the answer to my question is yes, but I'm just checking my understanding. If we have a set which is an Abelian group under addition and a group under multiplication is it then defined as a field?
Cheers Matt
| In general, no, because the distributive laws may not hold.
With that and a minor modification, yes. The multiplicative group has to leave out $0$, because $0$ won't be invertible.
So if you have a ring (with identity) such that the nonzero elements are an abelian group, then yes, you have a field.
If the nonzero eleme... | {
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positive harmonic function has a zero limit at a point on the boundary Let $u$ be a positive harmonic function in $\{ \Re{z} > 0\}$ such that
$\lim_{r \rightarrow 0^+} u(r) = 0 $.
Prove that then $\lim_{r \rightarrow 0^+} u(re^{i\theta}) = 0 $ for any $\theta \in (-\pi/2, \pi/2)$.
What I have tried is to consider the ... | The assumption that $u$ is positive should bring to mind Harnack's inequality. Let's state its special case for the disk: if $u$ is positive and harmonic in $\{z:|z-z_0|<R\}$, then
$$\frac{R-|z-z_0|}{R+|z-z_0|}u(z_0)\le u(z) \le \frac{R+|z-z_0|}{R-|z-z_0|}u(z_0)$$
for all $z$ in the disk.
The precise form of constants... | {
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What is the big picture behind AKS algorithm? Despite a number of question on AKS algorithm here, there does not seems to anything related to the idea behind it (for those who don't know, AKS primality testing is found in PRIMES is in P).
I read through the paper, check all the step for correctness (and also fix a few ... | There is an expository article by Granville titled"It is easy to determine whether a given integer is prime," which answers exactly the question you are asking. The article is worth the read, and it won the 2008 Chauvenet prize for its exposition.
Edit: This article was also referenced by Will Jagy in the comments.
| {
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Simple functional equation: $\frac 1 2 [\alpha(x - 1) + \alpha(x + 1)]$, $\alpha(0) = 1$, $\alpha(m) = 0$ I have a simple functional equation:
$$
\alpha(x)
=
\frac 1 2 [\alpha(x - 1) + \alpha(x + 1)]\,,
\qquad \alpha(0) = 1\,,\quad\alpha(m) = 0
$$
I know it has a linear solution $\alpha(x) = ax + b$ but I don't have an... | In your last comment to date, you rightly observe that for $b(x)=a(x)-a(x-1)$ you get the equation
$$b(x+1)=b(x)$$
Its solution are all functions with period 1.
Now
$$a(x+n)=a(x+n-1)+b(x)=a(x+n-2)+2\,b(x)=...=a(x)+nb(x)$$
using the periodicity of $b$. This leads to the idea to consider
$$c(x)=a(x)-x\,b(x).$$
It sati... | {
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"url": "https://math.stackexchange.com/questions/614610",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Dirichlet series for inverse of Eta function We know that $$ \frac{1}{\zeta (s)} = \sum_{n=1}^{\infty} \frac{\mu(n)}{n^{s}} $$
but what happens with $$ \frac{1}{\eta (s)} = \sum_{n=1}^{\infty} \frac{b(n)}{n^{s}} $$
with $ \eta (s) = (1-2^{1-s})\zeta (s) ?$
Can I evaluate the coefficients $ b(n) $ ? Perhaps I should a... | $$\frac{1}{\eta(s)}=\frac1{(1-2^{1-s})\cdot\zeta(s)}=\frac1{1-2^{1-s}}\cdot\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=\sum_{n=1}^{\infty}\frac{b(n)}{n^{s}}\iff b(n)=\frac{\mu(n)}{1-2^{1-s}}$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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$\cos^2\frac{1}{2}(\alpha-\beta)=\frac{3}{4}$ if........... Help please: If $\sin\alpha+\sin\beta= \sqrt{3} (\cos\beta-\cos\alpha)$ then show that $\cos^2\frac{1}{2}(\alpha-\beta)=\frac{3}{4}$ please tell me how can... | Apart from the Prosthaphaeresis Formulas already mentioned with the unmentioned assumption that $\displaystyle \sin\frac{\alpha+\beta}2\ne0$
we can try as follows :
Rearranging we have $\displaystyle\sin\alpha+\sqrt3\cos\alpha=\sqrt3\cos\beta-\sin\beta$
As $\displaystyle 1^2+(\sqrt3)^2=4,$ we write $1=2\sin30^\circ,\s... | {
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Find $\displaystyle \lim_{x \to 0} \frac{x \cdot \operatorname{cosec}(2x)}{\cos(5x)}$ I am having difficulties to find the limit for
$$\lim_{x \to 0} \frac{x \cdot \operatorname{cosec}(2x)}{\cos(5x)}$$
I tried to get rid of $ \operatorname{cosec} $ fist
$$\lim_{x \to 0} \frac{\dfrac{x}{\sin(2x)}}{\cos(5x)}$$
Probably I... | HINT : You can use $\lim_{x\to 0}\frac{\sin(2x)}{2x}=1.$
| {
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Solving special boolean equation set I have boolean equation sets that look like this (where ^ means xor):
eq 1: x1^x3^x5^x6^x9^x10^x11^x13^x17^x18 = 0
eq 2: 1^x1^x3^x10^x12^x17 = 0
eq 3: 1^x2^x3^x5^x8^x10^x14^x16 = 0
eq 4: 1^x4^x5^x6^x7^x8^x10^x16^x17 = 0
eq 5: x2^x5^x8^x11^x13^x14^x17^x18 = 0
(This example has 18 v... | We can phrase the problem as follows:
Given $M \in \mathbb{F}_2^{m\times n}$ and $b \in \mathbb{F}_2^m$ with $n>m$, find the solution $x \in \mathbb{F}_2^n$ to $Mx=b$ for which $\sum_{i=1}^n x_i$ is minimized.
That is, this may be regarded as a "linear programming" problem, of sorts. Below is my initial attempt at a... | {
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How do you prove that $\Bbb{Z}_p$ is an integral domain? Let $\Bbb{Z}_p$ be the $p$-adic integers given by formal series $\sum_{i\geq 0} a_i p^i$. I'm having trouble proving that it's an integral domain.
| As $\Bbb Z_p/p^n\Bbb Z_p\cong\Bbb Z/p^n\Bbb Z$, to show $a,b\ne0\Rightarrow ab\ne0$ it suffices to show that $ab\not\equiv0$ mod $p^n$ for a high enough power $p^n$ given $a,b\ne0$ (see vadim's hint in the comments).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/615102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 4,
"answer_id": 2
} |
Queueing model - expected outflow Can anybody please help me how to tackle this question?
We have one server.
The service time is random with mean 1 minute
The arrival rate is constant with 3 customers/minute, but they leave if the server is occupied.
A)
Assume the server is empty. How long would it take before one ent... | The arrival rate is $3$ per minute, so $1$ every $20$ seconds. That means you have to wait on average $10$ seconds for the next customer. Each customer takes $60$ seconds of time, so in total, thats $1$ customer every $70$ seconds. Per hour, you get $\frac 1{70}\cdot3600=\frac{360}7$.
Not that I am assuming that arriva... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/615186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Doubts in Trigonometrical Inequalities I'm now studying Trigonometrical Inequalities, and I've just got struck when I have modified arguments to my trigonometrical functions, for example:
$\sqrt{2} - 2\sin\left(x - \dfrac{\pi}{3} \right) < 0$ when $-\pi < x < \pi$
With some work I've got: $\sin\left(x - \dfrac{\pi}{3} ... | The inequality
$$
\sin \left( x - \frac{\pi}{3} \right) > \frac{\sqrt{2}}{2}
$$
implies that
$$
\frac{\pi}{4} < x - \frac{\pi}{3} < \frac{3\pi}{4}.
$$
Adding $\frac{\pi}{3}$ to all three expressions yields
$$
\frac{7\pi}{12} < x < \frac{13\pi}{12}.
$$
If you impose the initial restriction, then the upper bound is $\pi$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/615272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Is the power of 1/2 same thing as principal square root? $\sqrt{9} = 3$
9 has 2 square roots: 3 and -3.
What is $9^\frac12$? Is $9^\frac12 = \sqrt{9} = 3$ or is $9^\frac12 = \pm3$?
| $a^b$ is always positive for $a\in\Bbb R^+$ and $b\in\Bbb R$. This is because $f(x)=a^x$ satisfies the functional equation $f(x+y)=f(x)f(y)$ (i.e. $a^{x+y}=a^xa^y$). It follows that $f(2x)=f(x)^2,$ or $$a^x=f(x)=f\left(\frac x2\right)^2\ge 0$$
Consequently, $9^\frac12\ge 0\implies 9^\frac12=3$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/615435",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Help finding the $\lim\limits_{x \to \infty} \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}}$ I need help finding the $$\lim_{x \to \infty} \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}}$$
I did the following:
$$\begin{align*}
\lim_{x \to \infty} \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \... | $$\lim_{x \to \infty} \frac{\sqrt[3]{x} - \sqrt[5]{x}}{\sqrt[3]{x} + \sqrt[5]{x}}=\lim_{x \to \infty} \frac{x^{1/3}-x^{1/5}}{x^{1/3} + x^{1/5}}=$$
$$=\lim_{x \to \infty} \frac{1-x^{1/5-1/3}}{1 + x^{1/5-1/3}}=\lim_{x \to \infty} \frac{1-x^{-2/15}}{1 + x^{-2/15}}=\lim_{x \to \infty} \frac{1-\frac{1}{x^{2/15}}}{1 + \frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/615505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
How to find the graph of $e^{2x}$ and $\ln y$ I'd like to know how find the graph I just only know how to draw the common function graph as circle or ellipse but I don't know how to draw.
For example
$e^{2x}$ and $\ln y$.
Note:the interval for $e^{2x}, \;[0,\ln 2].$
How can I determine $\ln 2$ on the graph?
It's bett... | You should know the graph of $y=e^x$.
Then, the graph of $y=e^{2x}$ is the 'shorten' graph of $y=e^x$ in the direction of the $x$ axis . You'll see the both graphs here.
Also, note that $e^0=1, e^{2\ln 2}=4$, which means the graph of $y=e^{2x}$ passes the two points $(0,1),(\ln 2, 4).$ However, we don't know the exac... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/615593",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Diagonalization and eigenvalues Let $A$ $\in M_3(\mathbb R)$ which is not a diagonal matrix. Pick out the cases when
$A$ is diagonalizable over $\mathbb R$:
a. when $A^2 = A$;
b. when $(A - 3I)^2 = 0$;
c. when $A^2 + I = 0$.
I could eliminate c. by using the equation $\lambda^2+1$ and showing if a matrix has to satisfy... | Not sure what the question is, so I will take a stab at the answer. I am assuming you want to know what additional condition is needed to ensure that it is diagonalizable.
a) $A^2=A$
In this case the eigenvalues of $A$ have to be zero or one. Since we need three eigenvalues,
at least one of them is repeated. Now Jordon... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/615666",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
What values can $v-e+f$ attain if $G$ is a planar (non connected) graph? Let $G=(V,E)$ be a planar graph and choose planar representation.
If $G$ is connected, then according to Euler's formula, we have $$v − e + f = 2,$$ were $v$ is the number of vertices, $e$ the number of edges and $f$ the number of regions bounded ... | note that one of the faces of a planar graph is the unbounded region "outside" the graph. this is in common to the separate connected components, so you can then sum over components and obtain:
$$ v-e+f = 1 + C
$$
where $C$ is the number of connected components.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/615753",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
Map $\{x+iy \mid x^2+y^2<1 \text{ and } x^2 + (y-1)^2<2\}$ conformally to UHP From an old qualifying exam:
Let $D$ be the domain $$D :=\{x+iy \mid x^2+y^2<1 \text{ and } x^2 +
(y-1)^2<2\}.$$
*
*Map the domain onto the upper half-plane.
*Obtain a function $f(z)$ analytic in the domain $D' := D \cap \{x+iy ... | First from the definition it seems $D := \{|x^2+y^2|<1\}$ or the unit disc itself. If that is not a mistake, then the map $z \rightarrow i \dfrac{1+z}{1-z}$ sends the unit disc to the upper half plane conformally.
However, a more interesting problem would be to map $D:= \{|x^2+y^2| >1 \text{ and } x^2 + (y-1)^2 < 2\}$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/615842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis? I've seen a lot of definitions of notions like boundary points, accumulation points, continuity, etc, and axioms for the set of the real numbers. But I have a hard time accepting these as "true" definitions or acceptable axioms and... | Contrary to a common misconception, one will not find an epsilon, delta definition of continuity in Cauchy even if you look with a microscope. On the other hand, you will find his definition of continuity in terms of infinitesimals: every infinitesimal increment $\alpha$ necessarily produces an infinitesimal change $f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/615925",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "22",
"answer_count": 3,
"answer_id": 0
} |
Sequential sums $1+2+\cdots+N$ that are squares While playing with sums $S_n = 1+\cdots+n$ of integers,
I have just come across some "mathematical magic"
I have no explanation and no proof for.
Maybe you can give me some comments on this:
I had the computer calculating which Sn are squares,
and it came up with the fol... | I see. Nobody answered this the way I would have... Taking $u = 2 n+1,$ we are solving $$ u^2 - 8 m^2 = 1. $$ A beginning solution is $(3,1).$ Given a solution $(u,m),$ we get a new one $$ (3 u + 8 m, u + 3 m). $$ Then $n = (u-1)/2$ for each pair.
So, with $n^2 + n = 2 m^2$ and $u = 2 n + 1,$ we get triples
$$ (n,u,m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/616072",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
algebra help in AoCP intro to induction In the introduction of mathematical induction, section 1.2.1 of Knuth's Art of Computer Programming, I'm struggling with (4) especially this relation:
$\phi^{n-2} +\phi^{n-1} = \phi^{n-2}(1+\phi)$
Looks like my algebra skills have gone all to rot, if I ever had any, so I am hopin... | Distributing and using properties of exponents,
\begin{align*}
\phi^{n - 2} (1 + \phi) &= \phi^{n - 2} \cdot 1 + \phi^{n - 2} \cdot \phi^1 \\
&= \phi^{n - 2} + \phi^{n - 1}
\end{align*}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/616120",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Inequality from a Sequence I am working on a problem and I am lead to prove the following inequality which is true based on writing out the sequence and on the fact that it should be true based on what I am trying to prove.
Let $s_0,s_1,s_2,....$ be a sequence of positive numbers satisfying $s_0=s_1=1$ and $s_n = s_{n... | Letting $A\lt B$ be the roots of the equation $x^2-x+\alpha=0,$ we have
$$s_{n+1}-As_{n}=B(s_{n}-As_{n-1})=\cdots=B^n(s_1-As_0)=B^n(1-A).$$
$$s_{n+1}-Bs_{n}=A(s_{n}-Bs_{n-1})=\cdots=A^n(s_1-Bs_0)=A^n(1-B).$$
Hence, we have
$$s_n=\frac{B^n(1-A)-A^n(1-B)}{B-A}.$$
Hence,
$$2s_{n+1}-s_n\ge 0$$
$$\iff 2\cdot B^{n+1}(1-A)-2\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/616225",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
largest fraction less than 1 What is the mathematically rigorous way to answer the question: "what is the largest fraction less than 1"? (or to explain why it cannot be answered in the manner worded).
| There is no such fraction.
To see that this is the correct answer, note that for any fraction $\frac{p}{q}$ less than one, there is a slightly bigger fraction which is still less than one; in particular, $\frac{p+q}{2q}$ is a fraction such that $\frac{p}{q} < \frac{p+q}{2q} < 1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/616294",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 1
} |
Dimension of space spanned by row vectors Question is to find dimension of spaces spanned by vectors :
$$\alpha_1=(1,1,0,1,0,0),\\
\alpha_2=(1,1,0,0,1,0),\\
\alpha_3=(1,1,0,0,0,1),\\
\alpha_4=(1,0,1,1,0,0),\\
\alpha_5=(1,0,1,0,1,0),\\
\alpha_6=(1,0,1,0,0,1).$$
I tried to make it down to row echelon form but it is not g... | When we write the given vectors in a matrix form and reduce it to a row echelon thus obtaining the rank as 4 where the 1 St four columns are linearly independent implies that the first four scalars are zero..so out of the 6 vectors given when only 4 are linearly independent..we can say that the vector space is spanned ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/616354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Visualizing a complex valued function of one real parameter I'm looking for a way to capture/graph or visualize it in my head, but I can't find how..
a 2-dimensional path won't do, because it doesn't reveal the rate-of-change..
2 1-dimensional graphs on top of each other doesn't help much either..
3-dimensional space... | [> with(plots):
f := z-> exp(I*z):
complexplot3d(f, -2-2*I .. 2+2*I);
Or
[> complexplot(exp(I*x), x = -Pi .. Pi);
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/616440",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Pick's theorem application In a game let us say that we have $3$ ways to score points, getting $1,2 \text{ or } 3$ points. I make a total of $30$ points. What are the various ways I can make $30$ points?
If we plug in values, we have $a + 2b + 3c = 30$ and we have to find ordered pairs $a,b,c$ where $a,b,c$ are whole ... | I don't know why you would want to use Pick's theorem to solve this, but you are trying to find the number of lattice points satisfying your equation. These lie in a triangle, so if you project it into the $x,y$ plane the number of lattice points stays the same, and you can use use pick's theorem (assuming you can comp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/616518",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find all solutions of ${\frac {1} {x} } + {\frac {1} {y} } +{\frac {1} {z}}=1$, where $x$, $y$ and $z$ are positive integers Find all solutions of ${\frac {1} {x} } + {\frac {1} {y} } +{\frac {1} {z}}=1$ , where $x,y,z$ are positive integers.
Found ten solutions $(x,y,z)$ as ${(3,3,3),(2,4,4),(4,2,4),(4,4,2),(2,3,6),(... | We may as well assume $x\le y\le z$ (and then count rearrangements of the variables as appropriate). The smallest variable, $x$, cannot be greater than $3$ (or else $1/x+1/y+1/z\lt1/3+1/3+1/3=1$), nor can it be equal to $1$ (or else $1/x+1/y+1/z=1+1/y+1/z\gt1$). So either $x=2$ or $x=3$.
If $x=3$, then $y=z=3$ as wel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/616639",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 1
} |
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