Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Probability of winning a game
$A$ and $B$ play a game.
*
*The probability of $A$ winning is $0.55$.
*The probability of $B$ winning is $0.35$.
*The probability of a tie is $0.10$.
The winner of the game is the person who first wins two rounds. What is the probability that $A$ wins?
The answ... | Ties don't count, don't record them. So in effect we are playing a game in which A has probability $p=\frac{0.55}{0.90}$ of winning a game, and B has probability $1-p$ of winning a game.
Now there are several ways to finish. The least thinking one is that A wins with the pattern AA, or the patterns ABA, or BAA.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/377301",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Finding the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$
Find the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$.
Please brief about the concept behind this to solve such problems. Thanks.
| Using Euler-Fermat's theorem.
$\phi(7)=6$
$2^{6} \equiv 1 (\mod 7) \implies2^4.2^{96} \equiv 2(\mod7)$
$3^{6} \equiv 1 (\mod 7) \implies3^4.3^{96} \equiv 4(\mod7)$
$4^{6} \equiv 1 (\mod 7) \implies4^4.4^{96} \equiv 4(\mod7)$
$5^{6} \equiv 1 (\mod 7) \implies5^4.5^{96} \equiv 2(\mod7)$
$2^{100}+3^{100}+4^{100}+5^{100} \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377378",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 7,
"answer_id": 1
} |
Differential Equation - $y'=|y|+1, y(0)=0$ The equation is $y'=|y|+1, y(0)=0$.
Suppose $y$ is a solution on an interval $I$. Let $x\in I$.
If $y(x)\ge 0$ then $$y'(x)=|y(x)|+1\iff y'(x)=y(x)+1\iff \frac{y'(x)}{y(x)+1}=1\\ \iff \ln (y(x)+1)=x+C\iff y(x)+1=e^{x+C}\\ \iff y(x)=e^{x+C}-1$$
Then $y(0)=0\implies C=0$. So $y(... | I think my solution above is correct.
there are a few details missing:
it is necessary to show that $y(x)\leq 0\iff x\leq0$ and $y(x)\ge 0\iff x\ge 0$ which allows me to define $y$ the way I do.
also it is necessary to check that $y$ is differentiable at $x=0$ and it is because:
$$\lim _{x\to 0^+}\frac{e^x-1}{x-0}=1=\l... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Determine number of squares in progressively decreasing size that can be carved out of a rectangle How many squares in progressively decreasing size can be created from a rectangle of dimension $a\;X\;b$
For example, consider a rectangle of dimension $3\;X\;8$
As you can see, the biggest square that you can carve out... | Following the algorithm you seem to be doing, cutting the largest possible square off a rectangle, it is a simple recursive algorithm. If you start with an $n \times m$ rectangle with $n \ge m$, you will cut off $\lfloor \frac nm \rfloor m \times m$ squares and be left with a $(n-\lfloor \frac nm \rfloor m) \times m$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Length of period of decimal expansion of a fraction Each rational number (fraction) can be written as a decimal periodic number. Is there a method or hint to derive the length of the period of an arbitrary fraction? For example $1/3=0.3333...=0.(3)$ has a period of length 1.
For example: how to determine the length of... | Assuming there are no factors of $2,5$ in the denominator, one way is just to raise $10$ to powers modulo the denominator. If you find $-1$ you are halfway done. Taking your example: $10^2\equiv 9, 10^3\equiv -1, 10^6 \equiv 1 \pmod {13}$ so the repeat of $\frac 1{13}$ is $6$ long. It will always be a factor of Eul... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "33",
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$11$ divides $10a + b$ $\Leftrightarrow$ $11$ divides $a − b$ Problem
So, I am to show that $11$ divides $10a + b$ $\Leftrightarrow$ $11$ divides $a − b$.
Attempt
This is a useful proposition given by the book:
Proposition 12. $11$ divides a $\Leftrightarrow$ $11$ divides the alternating sum of the digits of $a$.
Pro... | Since $11$ divides $10a+b$, then
$$
10a+b=11k
$$
or
$$
b = 11k-10a
$$
so
$$
a-b=a-11k+10a=11(a-k)
$$
which means that $11$ divides $a-b$ as well, since $a,b,k$ are integers.
Update
To prove in opposite direction you can do the same
$$
a-b=11k\\
b=a-11k\\
10a+b=10a+a-11k=11(a-k)
$$
or in other words, if $11$ divides $a-... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377729",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 0
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Approximating measurable functions on $[0,1]$ by smooth functions. Let $f$ be a measurable function on $[0,1]$. Is there a sequence infinitely differentiable $f_n$ such that one of
*
*$f_n\rightarrow f$ pointwise
*$f_n\rightarrow f$ uniformly
*$\int_0^1|f_n-f|\rightarrow 0$
is true?
| Uniform convergence is surely too much to ask for. As Wikipedia suggests, uniform convergence theorem assures that the uniform limit of continuous functions is again continuous. Hence, as soon as $f$ is discontinuous, all hope of finding smooth $f_n$ uniformly convergent to $f$ is gone.
The statement involving the inte... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377808",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b} \overset{?}{\iff}\gcd(a,b)=1$ $$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b} \overset{?}{\iff} \gcd(a,b)=1$$
This seems almost painfully obvious because it is just $ar+bs=1$ in another form. This second form is the definition of coprimality, so what else is my professor looking for?
| If $\gcd(a,b)=1$ then since the greatest common divisor is the smallest positive integer that can be represented as a linear combination of a and b then we have that there are integers r and s such that
$1=ra+sb$
By dividing by ab we have that
$\frac{1}{ab}=\frac{s}{a}+\frac{r}{b}$.
Now if we suppose that $\frac{1}{a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/377896",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Definition of $b|a \implies 0|0$? The definition I'm using for $b|a$ (taken from Elementary Numbery Theory by Jones & Jones):
If $a,b \in \mathbb{Z}$ then $b$ divides $a$ if for some $q \in \mathbb{Z}$ $a = qb$.
However, I have $0 = q\cdot0$ for any $q$ I choose. So this seems to imply that $0$ divides $0$ which I k... | The statement $0$ divides $0$ and the "quantity" $0/0$ are different things. The first is exactly the statement that there exists some $a$ such that $0a=0$ and the second is not a number
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/377996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that $X\times Y$, with the product topology is connected I was given this proof but I don't clearly understand it. Would someone be able to dumb it down for me so I can maybe process it better?
Since a topological space is connected if and only if every function from it to $\lbrace 0,1\rbrace$ is constant. Let $F... | The basic fact we use is the one you start out with (which I won't prove, as I assume it's already known; it's not hard anyway):
(1) $X$ is connected iff every continuous function $f: X \rightarrow \{0,1\}$ (the latter space in the discrete topology) is constant.
So, given connected spaces $X$ and $Y$, we start with an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Big-O notation in division Let $r(x)=\frac{p(x)}{q(x)}$. Expending $p$ and $q$ around 0 gives
$$
\frac{p_0+p'_0x+\mathcal{O}(x^2)}{q_0+q'_0x+\mathcal{O}(x^2)}.
$$
Now the claim is that the above expression is equal to
$$
\frac{p_0+p'_0x}{q_0+q'_0x}+\mathcal{O}(x^2).
$$
| Try to evaluating the difference:
$\displaystyle \frac{p_0+p'_0x+\mathcal{O}(x^2)}{q_0+q'_0x+\mathcal{O}(x^2)}-\frac{p_0+p'_0x}{q_0+q'_0x}$
and recall that $p\mathcal{O}(x^2)=\mathcal{O}(x^2)$ and $x\mathcal{O}(x^2)=\mathcal{O}(x^2)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/378115",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Solve $\dfrac{1}{1+\frac{1}{1+\ddots}}$ I'm currently a high school junior enrolling in AP Calculus, I found this website that's full of "math geeks" and I hope you can give me some clues on how to solve this problem. I'm pretty desperate for this since I'm only about $0.4%$ to an A- and I can't really afford a B now.... | This is the Golden ratio (also known as $\varphi$) expressed using countinued fraction. This number is solution of the $x^2-x-1=0$ quadratic equotation.
This quadratic equotation you can wrote as $x=1+\frac{1}{x}$ and this form is used to construct cotinued faction used in your question.
See wikipedia for "Golden Ratio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378162",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 4,
"answer_id": 3
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whether or not there exist a non-constant entire function $f(z)$ satisfying the following conditions In each of the case below, determine whether or not there exist a non-constant entire function $f(z)$ satisfying the following conditions.
($1$) $f(0)=e^{i\alpha}$ and $|f(z)|=1/2$ for all $z \in Bdr \Delta$.
($2$) $f(e... | Looks good to me. Surely you can find an example for (4)? A first degree polynomial should do the job.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/378253",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
} |
construction set of natural number logic I identify the natural number $0$ with the empty set $\emptyset$, $1$ with $S(0)$, $2$ with $S(1)$, etc, etc.
The axiom of infinity says $\exists x (\emptyset\in x\wedge \forall z\in x\space z\cup\{z\}\in x)$ and the Axiom schema of specification says $\forall y_0,...,y_n\exis... | Let $y$ be an inductive set whose existence follows from the axiom of infinity, then consider $\{x\subseteq y\mid x\text{ is inductive}\}$. This is a definable collection of members of the power set of $y$, so it is a set, and $y$ is there so it's not an empty set.
Now take the intersection of all those sets. This is a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378325",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Prove that if $A - A^2 = I$ then $A$ has no real eigenvalues Given:
$$ A \in M_{n\times n}(\mathbb R) \; , \; A - A^2 = I $$
Then we have to prove that $A$ does not have real eigenvalues.
How do we prove such a thing?
| By using index notation, $A-A^2=I$ can be written as $A_{ij}-A_{ik}A_{kj}=\delta_{ij}$. By definition: $A_{ij}n_i=\lambda n_j$. So that, $A_{ij}n_i-A_{ik}A_{kj}n_i=\delta_{ij}n_i$, hence $\lambda n_j -\lambda n_k A_{kj}=n_j$, whence $\lambda n_j -\lambda^2 n_j=n_j$, or $(\lambda^2-\lambda+1)n_j=0$, $n_j\neq 0$ an eigen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378386",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 6,
"answer_id": 5
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If $S_n = 1+ 2 +3 + \cdots + n$, then prove that the last digit of $S_n$ is not 2,4 7,9. If $S_n = 1 + 2 + 3 + \cdots + n,$ then prove that the last digit of $S_n$ cannot be 2, 4, 7, or 9 for any whole number n.
What I have done:
*I have determined that it is supposed to be done with mathematical induction.
*The formul... | First show that for $n^2$ the last digit will always be from the set $M=1,4,5,6,9,0$ (I don't know how to create those brackets with your version of TeX, \left{ doesn't seem to work).
Then consider all cases for the last digit of $n$ (last digit is a $1$, I get a $2$ as last digit for $n(n+1)= n^2 + n$ and so on). If y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378478",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
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Is the linear dependence test also valid for matrices? I have the set of matrices
$
\begin{pmatrix}
1 & 0 \\
0 & 0 \\
\end{pmatrix}
$
$
\begin{pmatrix}
0 & 1 \\
0 & 0 \\
\end{pmatrix}
$
$
\begin{pmatrix}
0 & 0 \\
1 & 0 \\
\e... | $\mathbb{R}^{N\times N}$ as a linear space (with addition between elements and multiplication by scalars) is no different than $\mathbb{R}^{2N}$ endowed with these same operations. There is a bijective mapping between elements and operations in these two spaces, so any tricks you know about vectors is $\mathbb{R}^4$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378541",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Stieltjes Integral meaning. Can anybody give a geometrical interpretation of the Stieltjes integral:
$$\int_a^bf(\xi)\,d\alpha(\xi)$$
How would we calculate?
$$\int_a^b \xi^3\,d\alpha(\xi)$$
for example.
| $$\int_a^b \xi^3\,d\alpha(\xi)=\int_a^b\xi^3a'(\xi)d\mu(\xi) $$
in case that $a$ is differentiable. Also $\mu(\xi) $ indicates the Legesgue measure.
In order to find the above formula, you have to use Radon-Nikodym derivatives. (http://en.wikipedia.org/wiki/Radon%E2%80%93Nikodym_theorem).
Generally the function $a$ gi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
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Existence of a certain functor $F:\mathrm{Grpd}\rightarrow\mathrm{Grp}$ Let $\mathrm{Grpd}$ denote the category of all groupoids. Let $\mathrm{Grp}$ denote the category of all groups. Are there functors $F\colon\mathrm{Grpd}\rightarrow \mathrm{Grp}, G\colon\mathrm{Grp}\rightarrow \mathrm{Grpd}$ such that $GF=1_{\mathrm... | Such a pair of functors do not exist.
Reason 1 (if you accept the empty groupoid)
In the category of groups every pair of objects have a morphism between them. While in the the category of groupoids there is no morphism from the terminal object to the initial object. It follows that the initial object can't be in the i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378757",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "23",
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Meaning and types of geometry I heard that there's several kind of geometries for instance projective geometry and non euclidean geometry besides the euclidean geometry. So the question is what do you mean by a geometry, do you need truly many geometries and if yes what kind of results we can find in one geometry and... | Different geometries denote different sets of axioms, which in turn result in different sets of conclusions. I'll concentrate on the planar cases.
*
*Projective geometry is pure incidence geometry. The basic relation expresses whether or not a point lies on a line or not. One of its axioms requires that two differen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Graph of $\quad\frac{x^3-8}{x^2-4}$. I was using google graphs to find the graph of $$\frac{x^3-8}{x^2-4}$$ and it gave me:
Why is $x=2$ defined as $3$? I know that it is supposed to tend to 3. But where is the asymptote???
| Because there is a removable singularity at $x = 2$, there will be no asymptote.
You're correct that the function is not defined at $x = 2$. Consider the point $(2, 3)$ to be a hole in the graph.
Note that in the numerator, $$(x-2)(x^2 + 2x + 4) = x^3 - 8,$$ and in the denominator $$(x-2)(x+ 2) = x^2 - 4$$
When we si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Testing for convergence in Infinite series with factorial in numerator I have the following infinite series that I need to test for convergence/divergence:
$$\sum_{n=1}^{\infty} \frac{n!}{1 \times 3 \times 5 \times \cdots \times (2n-1)}$$
I can see that the denominator will eventually blow up and surpass the numerator,... | We have
$$a_n = \dfrac{n!}{(2n-1)!!} = \dfrac{n!}{(2n)!} \times 2^n n! = \dfrac{2^n}{\dbinom{2n}n}$$
Use ratio test now to get that
$$\dfrac{a_{n+1}}{a_n} = \dfrac{2^{n+1}}{\dbinom{2n+2}{n+1}} \cdot \dfrac{\dbinom{2n}n}{2^n} = \dfrac{2(n+1)(n+1)}{(2n+2)(2n+1)} = \dfrac{n+1}{2n+1}$$
We can also use Stirling. From Stirli... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/378991",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Find all matrices $A$ of order $2 \times 2$ that satisfy the equation $A^2-5A+6I = O$
Find all matrices $A$ of order $2 \times 2$ that satisfy the equation
$$
A^2-5A+6I = O
$$
My Attempt:
We can separate the $A$ term of the given equality:
$$
\begin{align}
A^2-5A+6I &= O\\
A^2-3A-2A+6I^2 &= O
\end{align}
$$
This impl... | The Cayley-Hamilton theorem states that every matrix $A$ satisfies its own characteristic polynomial; that is the polynomial for which the roots are the eigenvalues of the matrix:
$p(\lambda)=\det[A-\lambda\mathbb{I}]$.
If you view the polynomial:
$a^2-5a+6=0$,
as a characteristic polynomial with roots $a=2,3$, then an... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/379076",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 5,
"answer_id": 0
} |
Summing series with factorials in How do you sum this series?
$$\sum _{y=1}^m \frac{y}{(m-y)!(m+y)!}$$
My attempt:
$$\frac{y}{(m-y)!(m+y)!}=\frac{y}{(2m)!}{2m\choose m+y}$$
My thoughts were, sum this from zero, get a trivial answer, take away the first term. But actually I don't think this will work very well.
This que... | For example, one can write
\begin{align}
\sum_{y=0}^m\frac{y}{(m-y)!(m+y)!}
&=
\sum_{k=0}^m\frac{m-k}{k!(2m-k)!}
\\
&=\frac{m}{(2m)!}\sum_{k=0}^m{2m \choose k}-\frac{1}{(2m-1)!}\sum_{k=1}^{m}{2m-1\choose k-1}
\\
&=
\frac{m}{2(2m)!}\left[{2m\choose m}+\sum_{k=0}^{2m}{2m \choose k}\right]-\frac{1}{(2m-1)!}\sum_{k=0}^{m-1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/379149",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
closest pair in N-Dimensional I have to find the closest pair in n-dimension, and I have problem in the combine steps.
I use the divide and conquer.I first choose the median x, and split it into left and right part, and then find the smallest distance in left and right part respectively, dr, dl.
And then dm=min(dr,dl)... | The closest pair was either already found, or is in the 2-d-thick slab which can only include a low number of points. No need to reduce the dimension, just apply the algorithm recursively left, right and on the slab (cycling the direction the separating hyperplane is perpendicular to), optimality is implicit. Here are ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/379243",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx$ Prove that: $\frac{d}{dt} \int_{-\infty}^{\infty} e^{-x^2} \cos(2tx) dx=\int_{-\infty}^{\infty} -2x e^{-x^2} \sin(2tx) dx$
This is my proof:
$\forall \ t \in \mathbb{R}$ (the improper integral coverge absolutely $\forall \ t \in \mathbb{R}$) I consider: $g(t)... | You way looks good. Here's an alternate way: evaluate both integrals, and see that the derivative of one equals the other.
For example,
$$\begin{align}\int_{-\infty}^{\infty} dx \, e^{-x^2} \, \cos{2 t x} &= \Re{\left [\int_{-\infty}^{\infty} dx \, e^{-x^2} e^{i 2 t x} \right ]}\\ &= e^{-t^2}\Re{\left [\int_{-\infty}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/379282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Square and reverse reading of an integer For all $n=\overline{a_k a_{k-1}\ldots a_1 a_0} := \sum_{i=0}^k a_i 10^i\in \mathbb{N}$, where $a_i \in \{0,...,9\}$ and $a_k \neq 0$,
we define $f(n)=\overline{a_0 a_1 \ldots a_{k-1} a_k}= \sum_{i=0}^k a_{k-i}10^i$.
Is it true that, for all $m=\overline{a_k a_{k-1}\ldots a_1 a_... | If $m=...4$, then $m^2=...6$, but $f(m)=4...$ and $f(m)^2=1...$ or $2...$ (because $4^2=16$ and $5^2=25$).
The same can be calculated explicitly for $m$ ending in $5, \ldots, 8$, and only a little bit different for $9$.
If $m=...9$, then $m^2=...1$, but $f(m)=9...$ and $f(m)^2=8...$ or $9...$ not $1...$ (as $9^2=81$, $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/379369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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How to enumerate the solutions of a quadratic equation When we solve a quadratic equation, and let's assume that the solutions are $x=2$, $x=3$, should I say
*
*$x=2$ and $x=3$
*$x=2$ or $x=3$.
What is the correct way to say it?
| You should say $$x=2 \color{red}{\textbf{ or }}x=3.$$ $x=2$ and $x=3$ is wrong since $x$ cannot be equal to $2$ and $3$ simultaneously, since $2 \neq 3$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/379437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 2
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Solve for $x$: question on logarithms. The question:
$$\log_3 x \cdot \log_4 x \cdot \log_5 x = \log_3 x \cdot \log_4 x \cdot \log_5 x \cdot \log_5 x \cdot \log_4 x \cdot \log_3 x$$
My mother who's a math teacher was asked this by one of her students, and she can't quite figure it out. Anyone got any ideas?
| Following up on Jaeyong Chung's answer, and working it out:
$$ 1 =\log_3x\log_4x\log_5x$$
$$1=\frac{(\ln x)^3}{\ln3\ln4\ln5}$$
$$(\ln x)^3 = \ln3\ln4\ln5$$
$$(\ln x) = \sqrt[3]{\ln3\ln4\ln5}$$
$$x = \exp\left(\sqrt[3]{\ln3\ln4\ln5}\right) \approx 3.85093$$
EDIT: And, of course, the obvious answer that everyone will ove... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Uncountability of the equivalence classes of $\mathbb{R}/\mathbb{Q}$ Let $a,b\in[0,1]$ and define the equivalence relation $\sim$ by $a\sim b\iff a-b\in\mathbb{Q}$. This relation partitions $[0,1]$ into equivalence classes where every class consists of a set of numbers which are equivalent under $\sim$,
My textbook sta... | If you know $\Bbb Q$ is countable, that covers the second half. Then use the fact that a countable union of countable sets is again countable to show that there must be uncountably many classes.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/379591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Learning trigonometry on my own. I have been self teaching myself math beginning with a grade 10 level for a while now and need learn trigonometry from near scratch.
I am seeking both books and perhaps lectures on trigonometry and possibly geometry as some overlap does exist. I am not looking for algebra/precalc textbo... | The Indian mathematician Ramanujan learned his trigonometry from Sidney Luxton Loney's "Plane Trigonometry". Since it's a free Google book, what have you got to lose?
http://books.google.com/books?id=Mtw2AAAAMAAJ&printsec=frontcover&dq=editions:ix4vRrrEehgC&hl=en&sa=X&ei=Qu2CUeznBaO-yQH2tYCwDA&ved=0CDQQ6AEwAQ#v=onepag... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
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"answer_id": 0
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Another Birthday Problem (Probability/Combinatorics) What is the smallest number of people in a room to assure that the probability that at least two were born on the same day of the week is at least 40%?
I understand when approaching this type of problem, you simplify it so there's only 365 days. Also, I thought you ... | Here are the first several results for the same day of the week:
$$
\begin{align}
1-\frac77&=0&\text{$1$ person}&(0\%)\\
1-\frac77\frac67&=\frac17&\text{$2$ people}&(14.29\%)\\
1-\frac77\frac67\frac57&=\frac{19}{49}&\text{$3$ people}&(38.78\%)\\
1-\frac77\frac67\frac57\frac47&=\frac{223}{343}&\text{$4$ people}&(65.01\%... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/379705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solve a written problem with matrix I have the following problem described here:
The government attributes an allocation to the children who benefits child-care services. The children are splitted inside 3 groups: preschool, first cycle and second cycle. The allocation is different for each group, 2$ for the first cycl... | There's an easier solution, I think. Take the Rainbow and Nimbus schools alone. This yields a system of equations:
$$
43x+320+140y=589\\
100x+176+80y=556.
$$
Two variables and two equations means you can find the solutions for $x$ and $y$. Following that, you can plug those values into your formula for Cumulus and fin... | {
"language": "en",
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Is there a correct order to learning maths properly? I am a high school student but I would like to self-learn higher level maths so is there a correct order to do that?
I have learnt pre-calculus, calculus, algebra, series and sequences, combinatorics, complex numbers, polynomials and geometry all at high school level... | Quite often the transition to higher, pure math is real analysis. Here proofs really become relevant. I would suggest this free set of down-loadable notes from a class given at Berkeley by Fields medal winner (math analog of Noble Prize) Vaughan Jones.
https://sites.google.com/site/math104sp2011/lecture-notes
They are ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Stokes' Theorem Let $C$ be the following, let $C$ be the curve of intersection of the cylinder $x^2 + y^2 = 1$ and the given surface $z = f(x,y)$, oriented counterclockwise around the cylinder. Use Stokes' theorem to compute the line integral by first converting it to a surface integral.
(a) $\int_C (y \, \mathrm{d}x +... | Just to get you started, here's the details for (a). Start by finding the curl of the vector field $\mathbf{F}=\langle y,z,x\rangle$. You get
$$\nabla\times\mathbf{F}=\det\begin{pmatrix} \mathbf{i} &\mathbf{j}&\mathbf{k}\\\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&\frac{\partial}{\partial z}\\y&z&x\end{pma... | {
"language": "en",
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How long will it take Marie to saw another board into 3 pieces? So this is supposed to be really simple, and it's taken from the following picture:
Text-only:
It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long will it take for her to saw another board into
$3$ pieces?
I ... | The student is absolutely correct (as Twiceler has correctly shown).
The time taken to cut a board into $2$ pieces (that is $1$ cut) : $10$ minutes
Therefore, The time taken to cut a board into $3$ pieces (that is $2$ cuts) : $20$ minutes
The question may have different weird interpretations as I am happy commented:<br... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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I roll 6-sided dice until the sum exceeds 50. What is the expected value of the final roll? I roll 6-sided dice until the sum exceeds 50. What is the expected value of the final roll?
I am not sure how to set this one up. This one is not homework, by the way, but a question I am making up that is inspired by one. I'm ... | The purpose of this answer here is to convince readers that the distribution of the roll 'to get me over $50$' is not necessarily that of the standard 6=sided die roll.
Recall that a stopping time is a positive integer valued random variable $\tau$ for which $\{\tau \leq n \} \in \mathcal{F}_n$, where $\mathcal{F}_n = ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Bounded integrable function Let $f : \mathbb{R} \to \overline{\mathbb{R}}$ be an integrable funtion. Given $\varepsilon > 0$ show that there is a bounded integrable function $g$ such that $\int |f - g| < \varepsilon$.
I was wondering if I could get a hint.
| First, as $f$ is integrable, it takes infinite values on a negligible set, so we can assume that $f$ take its values on $\Bbb R$.
Writing $f=\max\{f,0\}+(f-\max\{f,0\})$, we can write $f$ as the difference of two measurable integrable non-negative functions. So we are reduced to the case $f\geqslant 0$ is integrable a... | {
"language": "en",
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Inf and sup for Lebesgue integrable functions Let $D \subset \mathbb{R}$ be a measurable set of finite measure. Suppose that $f : D \to \mathbb{R}$ is a bounded function. Prove that
$$\sup\left\{\int_D \varphi \mid \varphi \leq f \text{ and } \varphi \text{ simple}\right\} = \inf\left\{\int_D \psi \mid f \leq \psi \tex... | Suppose the inf and the sup are equal. Then for any $\epsilon > 0$ there exist simple functions $\varphi \le f \le \psi$ with the property that $\displaystyle \int_D \psi < \int_D \varphi + \epsilon$. Use this fact to construct sequences $\varphi_n$ and $\psi_n$ of simple functions with the property that $\varphi_n \l... | {
"language": "en",
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Is $f(x,y)$ continuous? I want to find out if this function is continuous:
$$(x,y)\mapsto \begin{cases}\frac{y\sin(x)}{(x-\pi)^2+y^2}&\text{for $(x,y)\not = (\pi, 0)$}\\0&\text{for $(x,y)=(\pi,0)$}\end{cases}$$
My first idea is that
$$\lim_{(x,y)\to(\pi,0)} |f(x,y)-f(\pi,0)|=\lim_{(x,y)\to(\pi,0)}\left|\frac{y\sin(x)}{... | Let $u=x-\pi$. Then you're wondering about $$\lim_{(u,y)\to(0,0)}\frac{y\sin(\pi+u)}{u^2+y^2}$$
$$\lim_{(u,y)\to(0,0)}\frac{-y\sin u}{u^2+y^2}=$$
$$\lim_{(u,y)\to(0,0)}\frac{-yu}{u^2+y^2}\frac{\sin u}{u}$$
Can you show it doesn't go to zero? Look at $y=u$ and $y=-u$, for example.
| {
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"timestamp": "2023-03-29T00:00:00",
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Are all vectors straight lines? Is there a math field that deals with quadratic, cubic etc. vectors? Or a non-linear equivalent of a vector? If so, why are they so much less common than linear vectors?
| From your response to my comment in your OP, I'll talk about why vectors are always considered "rays" geometrically.
The most important interpretation of a vector is as a list of numbers (we'll say from $\Bbb R^n$.) The list of numbers determines a point in $\Bbb R^n$, and if you imagine connecting this to the origin w... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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a totally ordered set with small well ordered set has to be small? doing something quite different the following question came to me:
1)If you have a totally ordered set A such that all the well ordered subset are at most countable, is it true that A has at most the cardinality of continuos?
2)More in general is it tru... | If $\kappa>2^\omega$, then $\kappa^*$ is a counterexample: all of its well-ordered subsets are finite. (The star indicates the reverse order.) However, if you require all well-ordered and reverse well-ordered subsets to be countable, the answer is yes to the more general question.
Let $\langle A,\le\rangle$ be a linear... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
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Understanding bicomplex numbers I found by chance, the set of Bicomplex numbers. These numbers took particularly my attention because of their similarity to my previous personal research and question. I should say that I can't really understand the fact that $j^2=+1$ (and must of other abstract algebra) without using m... | The algebra you discribed in the question is tessarines.
So, there are two matrical representations of tessarines.
*
*For a tessarine $z=w_1+w_2i+w_3j+w_4ij$ the matrix representation is as follows:
$$\left(
\begin{array}{cccc}
{w_0} & -{w_1} & {w_2} & -{w_3} \\
{w_1} & {w_0} & {w_3} & {w_2} \\
{w_2} & -{w_3} & {... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/380488",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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Bayesian learning Imagine we assume there are two different types of coins:
*
*Coin A: a fair coin, p(heads) = 0.5.
*Coin B: biased to heads at p(heads)=0.7.
We then want to learn from samples which coin we are flipping. Assume a naive prior over the two coins, so we have a Beta distribution, $\beta_0(1,1)$.
You ... | I don't understand how a beta distribution enters into it. A beta distribution is usually used in the context of an unknown probability lying anywhere in $[0,1]$. We have no such parameter here; all we have is an unknown binary choice between coins $A$ and $B$. The most natural prior in this case is one that assigns pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/380567",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Finding a continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f(\mathbb{R})$ is neither open nor closed Find a bounded, continuous function $f: \mathbb{R} \to \mathbb{R}$ such that $f(\mathbb{R})$ is neither open nor closed?
| Take $f(x) = \arctan(x^2)$. Then, $f(\mathbb{R}) = [0, \pi/2)$.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Is there something faulty about this statement? Show any prime of the form $3k+1$ is of the form $6k+1$.
I came up with my own solution that made perfect sense to me, but when I read the text's solution, it argued that for the primes that are of the particular form are $6k+1 = 3(2k)+1$. But doesn't that really say the... | Another way to phrase it is
"If k is a positive integer such that 3k+1 is prime
then k is even".
The proof, of course, is easy:
If k is odd, then k=2h+1 for some integer h.
But 3k+1 = 3(2h+1)+1 = 6h+4 is even
and therefore not prime.
| {
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"source": "stackexchange",
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Probability of draws at random with replacement of five tickets
$400$ draws are made at random with replacement from $5$ tickets that are marked $-2, -1, 0, 1,$ and $2$ respectively. Find the expected value of: the number of times positive numbers appear?
Expected value of $X$ number times positive number appear $= ... | The idea is right, but note that $0$ is not positive. So the probability we get a positive on any draw is $\frac{2}{5}$.
So if $X_i=1$ if we get a positive on the $i$-th draw, with $X_i=0$ otherwise, then $E(X_i)=\frac{2}{5}$. Now use the linearity of expectation to conclude that the expected number of positives in $4... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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An NFA with $\Sigma = \{1\}$ with $x^2$ accepting runs on strings $1^x$ for all $x \geq 0$ - how to construct? One of my homework assignments requires us to construct an NFA over the alphabet $\{1\}$ which has exactly $x^2 + 3$ accepting runs over the input string 1^x for all $x \in \mathbb{N}$. Now, the +3 part is sim... | There's no such NFA.
Assume there was an NFA over the alphabet ${1}$ which accepts all strings whose length is $n^2 + 3$ for some $n \in \mathbb{N}$. I.e., the NFS is supposed to accept strings of lengths $3,4,7,12,19,28,39,\ldots$.
If there was such an NFS, the language $$
\Omega = \left\{\underline{1}^{n^2+3} \,:\,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/380812",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "2",
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"answer_id": 1
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Is G isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$? If $ G=\{3^{m}6^{n}|m,n \in \mathbb{Z}\}$ under multiplication then i want prove that this G
is isomorphic to $\mathbb{Z} \oplus \mathbb{Z}$.Can any one help me to solve this example?
please help me. thanks in advance.
Can i define $\phi:\mathbb{Z} \oplus \mathbb{Z} ... | Hint: $2^k=3^{-k}6^k$.${}{}{}{}{}{}{}{}$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Prove that $\log X < X$ for all $X > 0$ I'm working through Data Structures and Algorithm Analysis in C++, 2nd Ed, and problem 1.7 asks us to prove that $\log X < X$ for all $X > 0$.
However, unless I'm missing something, this can't actually be proven. The spirit of the problem only holds true if you define several ext... | One way to approach this question is to consider the minimum of $x - \log_a x$ on the interval $(0,\infty)$. For this we can compute the derivative, which is
$1 - 1/(\log_e a )\cdot x$. Thus the derivative is zero at a single point, namely $x = 1/\log_e a,$ and is negative to the left of that point and positive to the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/380963",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Normal subgroup of a normal subgroup Let $F,G,H$ be groups such that $F\trianglelefteq G \trianglelefteq H$.
I am asked whether we necessarily have $F\trianglelefteq H$. I think the answer is no but I cannot find any counterexample with usual groups. Is there a simple case where this property is not true?
| Let $p$ be a prime, and let $G$ be a $p$-group of order $p^3$. Let $H \leq G$ be a non-normal subgroup of order $p$ (equivalently, $H$ is of order $p$ and not central). Then $H$ is contained in a subgroup $K \leq G$ of order $p^2$. In this case $H \trianglelefteq K \trianglelefteq G$, but $H$ is not normal in $G$.
For ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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"answer_id": 3
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Complex roots of polynomial equations with real coefficients Consider the polynomial $x^5 +ax^4 +bx^3 +cx^2 +dx+4$ where $a, b, c, d$ are real
numbers. If $(1 + 2i)$ and $(3 - 2i)$ are two roots of this polynomial then what is the value of $a$ ?
| Adding to lab bhattacharjee's answer, Vieta's Formulas basically tell you that the negative fraction of the last term (the constant) divided by the coefficient of the first term is equal to the product of the roots. Letting r be the 5th root of the polynomial (since we know 4), $$-\frac{4}{1} = (1−2i)(1+2i)(3+2i)(3-2i)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/381114",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simple/Concise proof of Muir's Identity I am not a Math student and I am having trouble finding some small proof for the Muir's identity.
Even a slightly lengthy but easy to understand proof would be helpful.
Muir's Identity
$$\det(A)= (\operatorname{pf}(A))^2;$$
the identity is given in the first paragraph of the foll... | This answer does not show the explicit form of $\textrm{pf}(A)$ but it proves that such a form must exist as a polynomial in the entries of $A$.
Let $A$ be a generic skew-symmetric $n \times n$ matrix with indeterminate entries $A_{i j}$ on row $i$ column $j$ for $0 \leq i < j \leq n$. I will prove by induction in $n$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Find a polar representation for a curve. I have the following curve:
$(x^2 + y^2)^2 - 4x(x^2 + y^2) = 4y^2$ and I have to find its polar representation.
I don't know how. I'd like to get help .. thanks in advance.
| Just as the Cartesian has two variables, we will have two variables in polar form:
$$x = r\cos \theta,\;\;y = r \sin \theta$$
We can also use the fact that $x^2 + y^2 = (r\cos \theta)^2 + (r\sin\theta)^2 = r^2 \cos^2\theta + r^2\sin^2 \theta = r^2\underbrace{(\sin^2 \theta + \cos^2 \theta)}_{= 1} =r^2$
This gives us $$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/381353",
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"question_score": "1",
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Explicit Functions on $\mathbb{C}$ The following question on last years Complex Analysis exam paper, and Im a little stuck on it..
$(i)f(z)=e^{z^2}$ find the explicit formulas for $u(x,y)$ and $v(x,y)$ such that:
$f(x+iy)=u(x,y)+iv(x,y)$
(ii) Find all functions $v: \mathbb{R}^2\rightarrow\mathbb{R}^2$ such that
$f(x+iy... | From Euler formula $e^{iz}=\cos z+i\sin z$,
$e^{z^2}=e^{x^2-y^2+2ixy}=e^{x^2-y^2}\cos(2xy)+ie^{x^2-y^2}\sin(2xy)$
$f$ $\mathbb{C}$-differentiable implies $u,v$ $\mathbb{R}$-differentiable plus Cauchy-Riemann equations
$$u_x=v_y$$
$$u_y=-v_x$$
Then
$$v_y(x,y)=3x^2-3y^2\longrightarrow v=3x^2y-y^3+H(x)$$
$$v_x(x,y)=6xy\l... | {
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"url": "https://math.stackexchange.com/questions/381450",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$k(tx,ty)=tk(x,y)$ then $k(x,y)=Ax+By$ A friend asked me today the following question:
Let $k(x,y)$ be differentiable in all $\mathbb{R}^{2}$ s.t for every
$(x,y)$ and for every $t$ it holds that $$k(tx,ty)=tk(x,y)$$ Prove that
there exist $A,B\in\mathbb{R}$ s.t $$k(x,y)=Ax+By$$
I want to use the chain rule someh... | First $k(0,0)=0$. $k(x, y)=\lim_{t\to 0}\frac{k(tx, ty)}{t}=xk_x+yk_y$ where $k_x=\partial_xk|_{(x,y)=(0,0)}, k_y=\partial_yk|_{(x,y)=(0,0)}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/381512",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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What is the Fourier transform of $f(x)=e^{-x^2}$? I remember there is a special rule for this kind of function, but I can't remember what it was.
Does anyone know?
| Caveat: I'm using the normalization $\hat f(\omega) = \int_{-\infty}^\infty f(t)e^{-it\omega}\,dt$.
A cute way to to derive the Fourier transform of $f(t) = e^{-t^2}$ is the following trick: Since $$f'(t) = -2te^{-t^2} = -2tf(t),$$
taking the Fourier transfom of both sides will give us
$$i\omega \hat f(\omega) = -2i\ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/381597",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "15",
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Expected number of edges: does $\sum\limits_{k=1}^m k \binom{m}{k} p^k (1-p)^{m-k} = mp$
Find the expected number of edges in $G \in \mathcal G(n,p)$.
Method $1$: Let $\binom{n}{2} = m$. The probability that any set of edges $|X| = k$ is the set of edges in $G$ is $p^k (1-p)^{m-k}$. So the probability that $G$ has ... | Related problems:(I), (II). Consider the function
$$ f(x)=( xp+(1-p) )^m = \sum_{k=0}^{m} {m\choose k} p^k(1-p)^{m-k}x^k $$
Differentiating the above equation with respect to $x$ yields
$$ \implies mp( xp+(1-p) )^{m-1} = \sum_{k=1}^{m}{m\choose k} k p^k(1-p)^{m-k}x^{k-1}. $$
Subs $x=1$ in the above equation gives the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/381716",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Need help solving - $ \int (\sin 101x) \cdot\sin^{99}x\,dx $ I have a complicated integral to solve.
I tried to split ($101 x$) and proceed but I am getting a pretty nasty answer while evaluating using parts.
are there any simpler methods to evaluate this integral?
$$
\int\!\sin (101x)\cdot\sin^{99}(x)\, dx
$$
| Let's use the identity $$\sin(101x)=\sin(x)\cos(100x)+\cos(x)\sin(100x)$$
Then the integral becomes
$$\int\sin^{100}(x)\cos(100x)dx+\int\sin^{99}(x)\sin(100x)\cos(x)dx$$
Integrating the first term by parts gives
$$\int\sin^{100}(x)\cos(100x)dx=\frac{1}{100}\sin^{100}(x)\sin(100x)-\int\sin^{99}(x)\sin(100x)\cos(x)dx$$
P... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/381867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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A parameterized elliptical integral (Legendre Elliptical Integral) $$
K(a,\theta)=\int_{0}^{\infty}\frac{t^{-a}}{1+2t\cos(\theta)+t^{2}}dt
$$
For $$ -1<a<1;$$ $$-\pi<\theta<\pi$$
I know this integral to be a known tabulated Legendre elliptic integral, however the very fact that the numerator is parameterized completel... | This is not elliptic integral, this can be expressed in terms of elementary functions:
\begin{align}
K(a,\theta)=\int_0^{\infty}\frac{t^{-a}dt}{t^2+2\cos\theta\, t+1}=\frac{1}{2i\sin\theta}\int_0^{\infty}\left(\frac{t^{-a}}{t+e^{-i\theta}}-\frac{t^{-a}}{t+e^{i\theta}}\right)dt=\\
=\frac{1}{2i\sin\theta}\left(\frac{\pi ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Let $\ \varphi \, : \, V\rightarrow V\ $ be a linear transformation. Prove that $\ Im(\varphi \, \circ \varphi) \subseteq Im \,\varphi\ $ Let V be a vector space and $\ \varphi \, : \, V\rightarrow V\ $ be a linear transformation. Prove that:
$$\ Im(\varphi \, \circ \varphi) \subseteq Im \,\varphi\ $$
I am struggling t... | Recall that
$$\mathrm{Im}(\varphi)=\{\varphi(x)\quad|\quad x\in V\}$$
Now take $y\in \mathrm{Im}(\varphi\circ \varphi)$ then there's $x\in V$ such that
$$y=\varphi\circ \varphi(x)= \varphi( \underbrace{\varphi(x)}_{z\in V})=\varphi(z)\in \mathrm{Im}(\varphi)$$
so we have
$$\ Im(\varphi \, \circ \varphi) \subseteq Im \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/382013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Integration by parts, Reduction
I was able to complete part (a) easily by using integration by parts. I ended up getting:
$$I(n) = -\frac{1}{n} \cos x\cdot \sin^{n-1}x + \frac{n-1}{n}· I(n-2)$$
For question (b), When I integrated $1/\sin^4x$ and subbed in $n = -4$, I get the following equation:
$$\frac{1}{4}·\cos x·\... | Putting $n=-2,$ in $$I_n=-\frac1n\cos x\sin^{n-1}x+\frac{n-1}nI_{n-2}$$
we get $$I_{-2}=-\frac1{(-2)}\cos x\sin^{-2-1}x+\frac{(-2-1)}{(-2)}I_{-2-2}$$
$$\implies \frac32I_{-4}=I_{-2}-\frac{\cos x}{2\sin^3x}$$
Now, $$I_{-2}=\int\sin^{-2}xdx=\int \csc^2xdx=-\cot x+C$$
Can you finish it from here?
| {
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"timestamp": "2023-03-29T00:00:00",
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Ways of merging two incomparable sorted lists of elements keeping their relative ordering Suppose that, for a real application, I have ended up with a sorted list A = {$a_1, a_2, ..., a_{|A|}$} of elements of a certain kind (say, Type-A), and another sorted list B = {$b_1, b_2, ..., b_{|B|}$} of elements of a different... | For merging N sorted lists, here is a good way to see that the solution is $$\frac{(|A_1|+\dots|+|A_N|)!}{|A_1|!\dots |A_N|!}$$
All the $(|A_1|+\dots|+|A_N|)$ elements can be permuted in $(|A_1|+\dots|+|A_N|)!$ ways. Among these, any solution which has the ordering of the $A_1$ elements different from the given order h... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 3,
"answer_id": 2
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Binary Decision Diagram of $(A\Rightarrow C)\wedge (B\Rightarrow C)$? I made a Binary Decision Diagram for $(A\vee B)\Rightarrow C$, which i think is correct.
Know i want o make a Binary Decision Diagram for $(A\Rightarrow C) \wedge (B\Rightarrow C)$ but i can't. I can make 2 BDD's, one for $(A\Rightarrow C)$ and one... | $(A\Rightarrow C)\lor(B\Rightarrow C) \equiv (\lnot A\lor C)\lor(\lnot B\lor C)\equiv \lnot A\lor\lnot B\lor C$.
So, this is true iff either $A$ is false or $B$ is false, or $C$ is true.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Prove by mathematical induction that $1 + 1/4 +\ldots + 1/4^n \to 4/3$ Please help. I haven't found any text on how to prove by induction this sort of problem:
$$
\lim_{n\to +\infty}1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3}
$$
I can't quite get how one can prove such. I can prove basic di... | If you want to use proof by induction, you have to prove the stronger statement that
$$
1 + \frac{1}{4} + \frac{1}{4^2} + \cdots+ \frac{1}{4^n} = \frac{4}{3} - \frac{1}{3}\frac{1}{4^n}
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 10,
"answer_id": 5
} |
Spanned divisors and Base Points Let $X$ be a smooth algebraic variety.
We say that a line bundle $\xi\in H^1(X,\mathcal{O}^\ast)$ is spanned is for each $x\in X$ there is a global section $s\in H^0(X,\mathcal{O}(\xi))$ with $s(x)\neq 0$.
Let $\xi=[D]$ be the line bundle associated to a divisor $D$. If $\xi$ is spanne... | Assume that the line bundle $\mathscr{O}(D)$ is spanned. So given $x \in X$, there exists a section $s \in H^0(X, \mathscr{O}(D))$ such that $s(x) \neq 0$. Let $D'$ denote the divisor of zeroes of the section $s$; then $x$ is not contained in $\mathrm{Supp}(D')$.
The remaining issue is to show that $D'$ is linearly eq... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Roots of cubic polynomial lying inside the circle Show that all roots of $a+bz+cz^2+z^3=0$ lie inside the circle
$|z|=max{\{1,|a|+|b|+|c| \}}$
Now this problem is given in Beardon's Algebra and Geometry third chapter on complex numbers.
What might be relevant for this problem:
*
*author previously discussed roots ... | I'd simply be looking at showing that the $z^3$ term was dominant, so there could be no roots beyond the bound. I don't think it is at all sophisticated.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How to write this conic equation in standard form? $$x^2+y^2-16x-20y+100=0$$
Standard form? Circle or ellipse?
| Recall that one of the usual standard forms is:
$(x - a)^{2} + (y - b)^{2} = r^{2}$ where...
*
*(a,b) is the center of the circle
*r is the radius of the circle
Rearrange the terms to obtain:
$x^{2} - 16x + y^{2} - 20y + 100 = 0$
Then, by completing the squares, we have:
$(x^{2} - 16x + 64) + (y^{2} - 20y + 100) ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Number of Invariant Subspaces of a Jordan Block I'm asking this question on behalf of a person I'm supposed to be tutoring who has this problem as part of eir homework.
The problem is "How many invariant subspaces are there of a transformation $T$ that sends $v\mapsto J_{\lambda,n}v$" where $J_{\lambda,n}$ is a Jordan ... | You are right. Let $J_{\lambda,n}=\lambda\,I+N$ where $N$ is the nilpotent part, which maps $e_i\mapsto e_{i-1}$ if $i>1$ and $e_1\mapsto 0$.
*
*Observe that the invariant subspaces of $\lambda\,I+N$ coincide with those of $N$.
*Assume that $v=(v_1,..,v_k,0,..,0)$ with $v_k\ne 0$, and consider its generated $N$-inv... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Limit $\frac{\tan^{-1}x - \tan^{-1}\sqrt{3}}{x-\sqrt{3}}$ without L'Hopital's rule. Please solve this without L'Hopital's rule? $$\lim_{x\rightarrow\sqrt{3}} \frac{\tan^{-1} x - \frac{\pi}{3}}{x-\sqrt{3}}$$
All I figured out how to do is to rewrite this as $$\frac{\tan^{-1} x - \tan^{-1}\sqrt{3}}{x-\sqrt{3}}$$
Any help... | We want
$$L = \lim_{x \to \sqrt{3}} \dfrac{\arctan(x) - \pi/3}{x - \sqrt3}$$
Let $\arctan(x) = t$. We then have
$$L = \lim_{t \to \pi/3} \dfrac{t-\pi/3}{\tan(t) - \sqrt{3}} = \lim_{t \to \pi/3} \dfrac{t-\pi/3}{\tan(t) - \tan(\pi/3)} = \dfrac1{\left.\dfrac{d \tan(t)}{dt} \right\vert_{t=\pi/3}} = \dfrac1{\sec^2(t) \vert_... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Induction proof: $\dbinom{2n}{n}=\dfrac{(2n)!}{n!n!}$ is an integer. Prove using induction: $\dbinom{2n}{n}=\dfrac{(2n)!}{n!n!}$ is an integer.
I tried but I can't do it.
| This is one instance of a strange phenomenon: proving something seemingly more complicated makes things simpler.
Show that $\binom{n}{k}$ is an integer for all $0\leq k\leq n$, and that will show what you want. To do so, show by induction on $n$ that $$\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}.$$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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compare of eigenvalues $\lambda_1(D_a)$ and $\lambda_1(D_c)$. Let $f(x)$ be a smooth function on $[-1,1]$, such that $f(x)>0$ for all $x\in(-1,1)$,$f(-1)=f(1)=0$. consider $\gamma\subset\Bbb{R}^2$ the graph of the $f(x)$. Let $T_a$ the symmetry with respect to axis $x$ and $T_c$ the central symmetry with respect to ori... | Fact 1. In the special case
$$f(-x)\le f(x),\qquad 0\le x\le 1\tag0$$ the inequality $\lambda_1(D_a)\le \lambda_1(D_c)$ holds.
Proof. Let $u$ be the first eigenfunction for $D_c$. Extend it to $\mathbb R^2$ by zero outside of $D_c$. For $(x,y)\in\mathbb R^2$ define
$$v(x,y)=\begin{cases}
\max(u(x,y),u(-x,y))\quad &... | {
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Can every infinite set be divided into pairwise disjoint subsets of size $n\in\mathbb{N}$?
Let $S$ be an infinite set and $n$ be a natural number. Does there exist partition of $S$ in which each subset has size $n$?
*
*This is pretty easy to do for countable sets. Is it true for uncountable sets?
*If (1) is tru... | A geometric answer, for cardinality $c$. For $x, y \in S^1$ (the unit circle), let $x \sim y$ iff $x$ and $y$ are vertices of the same regular $n$-gon centered at the origin. Now the title of this question speaks of infinite sets generally, which is a different question. The approach by Hagen von Eitzen is probably abo... | {
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Prove by mathematical induction for any prime number$ p > 3, p^2 - 1$ is divisible by $3$? Prove by mathematical induction for any prime number $p > 3, p^2 - 1$ is divisible by $3$?
Actually the above expression is divisible by $3,4,6,8,12$ and $24$.
I have proved the divisibility by $4$ like:
$$
\begin{align}
p^2 -1 ... | Hint:
$p \equiv 1$ or $-1 (\mod3) \implies p^2 \equiv 1 (\mod 3)$ for every $p>3$
| {
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"timestamp": "2023-03-29T00:00:00",
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Solving systems of linear equations using matrices, 3 equations, 4 variables I understand how to solve systems of linear equations when they have the same number of variables as equations. But what about when there are only three equations and 4 variables?
For example, when i was looking through an exam paper, i came ... | There are a couple of things you have to pay attention to when solving a system of equations.
The first thing you want to pay attention to is the rank of the corresponding matrix, defined as the number of pivot rows in the Reduced Row Echelon form of your matrix (that you get at via Gaussian elimination). You can think... | {
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How to solve equations of algebra? Let $a_i>0, b_i>0$ ($i=1,2,\ldots,N$). How to prove that there exist unique $x_i>0$ ($i=1,2,\ldots,N$) such that
$$a_ix_i^{b_i}+x_1+x_2+\cdots+x_N=1,\;\;i=1,2,\ldots,N.$$
Thank you.
| Replace $x_1+\ldots+x_N$ in the equation by $S$. A solution to the problem must have $0<S<1$, since $0<a_ix_i^{b_i}=1-S$. Then
$$
x_i=(\frac{1-S}{a_i})^{1/b_i}=f_i(S).
$$
$f_i(S)$ are continuous functions from $[0,1]$ to $\mathbb{R^{+}}$, so the same is true for the sum $F(S)=\sum_{i=1}^N f_i(S)$. $F(S)$ is strictly m... | {
"language": "en",
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Rotating x,y points 45 degrees I have a two dimensional data set that I would like to rotate 45 degrees such that a 45 degree line from the points (0,0 and 10,10) becomes the x-axis. For example, the x,y points (1,1), (2,2), and (3,3) would be transformed to the points (0,1), (0,2), and (0,3), respectively, such that t... | Here is some good background about this topic in wikipedia:
Rotation Matrix
| {
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"timestamp": "2023-03-29T00:00:00",
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An intuitive idea about fundamental group of $\mathbb{RP}^2$ Someone can explain me with an example, what is the meaning of $\pi(\mathbb{RP}^2,x_0) \cong \mathbb{Z}_2$?
We consider the real projective plane as a quotient of the disk.
I didn't receive an exhaustive answer to this question from my teacher, in fact he sa... | You can see another set of related pictures here, which gives the script for this video
Pivoted Lines and the Mobius Band (1.47MB).
The term "Pivoted Lines" is intended to be a non technical reference to the fact that we are discussing rotations, and their representations. The video shows the "identification" of the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "39",
"answer_count": 4,
"answer_id": 2
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Singular asymptotics of Gaussian integrals with periodic perturbations At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$,
$$
\int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} \int_0^1 f(x)\,dx + O(1)
\tag{1}
$$
as $A \to 0^+$.
How... | Your estimate of error term is correct. The following are just some supplementary details to make your argument more rigorous.
Let $(f_n)_{n\ge 1}$ be the sequence of partial sums of the Fourier series of $f$, i.e.
$$
f_n(x) = \int_0^1 f(t)\,dt + \sum_{k=1}^n \big(a_k \cos(2\pi kx) + b_k \sin(2\pi kx)\big).
$$
Note th... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "10",
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Dirichlet series generating function I am stuck on how to do this question:
Let d(n) denote the number of divisors of n. Show that the dirichlet series generating function of the sequence {(d(n))^2} equals C^4 (s)/ C(2s).
C(s) represents the riemann zeta function, I apologize, I am not very accustomed with LaTex. Any... | There are many different ways to approach this, depending on what you are permitted to use. One simple way is to use Euler products.
The Euler product for $$Q(s) = \sum_{n\ge 1} \frac{d(n)^2}{n^s}$$ is given by
$$ Q(s) = \prod_p \left( 1 + \frac{2^2}{p^s} + \frac{3^2}{p^{2s}}
+ \frac{4^2}{p^{3s}} + \cdots \right).$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/383733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How big is the size of all infinities? "Not only infinite - it's "so big" that there is no infinite set so large as the collection of all types of infinity..."
What does exactly mean? How many infinities are there? I've heard there are more than infinite infinities? What does that mean? Is that true? Will anyone ever b... | In the world of natural numbers it is known that
$2 ^ a \neq 3 ^ b$ for any pair of positive integers a and b.
This is true for any pair of primes.
So if we believe there is only one infinitely large natural number ($\infty$)
From the above statement: $2 ^ \infty \neq 3 ^ \infty$
Let $2 ^ \infty $ be $\infty_{2}$
Let... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/383811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Evaluating Complex Integral. I am trying to evaluate the following integrals:
$$\int\limits_{-\infty}^\infty \frac{x^2}{1+x^2+x^4}dx $$
$$\int\limits_{0}^\pi \frac{d\theta}{a\cos\theta+ b} \text{ where }0<a<b$$
My very limited text has the following substitution:
$$\int\limits_0^\infty \frac{\sin x}{x}dx = \frac{1}{2i}... | For the first one, write $\dfrac{x^2}{1+x^2+x^4}$ as $\dfrac{x}{2(1-x+x^2)} - \dfrac{x}{2(1+x+x^2)}$. Now
$$\dfrac{x}{(1-x+x^2)} = \dfrac{x-1/2}{\left(x-\dfrac12\right)^2 + \left(\dfrac{\sqrt3}2 \right)^2} + \dfrac{1/2}{\left(x-\dfrac12\right)^2 + \left(\dfrac{\sqrt3}2 \right)^2}$$
and
$$\dfrac{x}{(1-x+x^2)} = \dfrac{x... | {
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"answer_id": 0
} |
Practice Preliminary exam - evaluate the limit This is from a practice prelim exam and I know I should be able to get this one.
$$
\lim_{n\to\infty} n^{1/2}\int_0^\infty \left( \frac{2x}{1+x^2} \right)^n
$$
I have tried many different $u-$substitions but to no avail. I have tried
$$
u = \log(1+x^2)
$$
$$
du = \frac{2x... | Let $x = \tan(t)$. We then get that
\begin{align}
I(n) & = \int_0^{\infty} \left(\dfrac{2x}{1+x^2} \right)^n dx = \int_0^{\pi/2} \sin^n(2t) \sec^2(t) dt = 2^n \int_0^{\pi/2} \sin^n(t) \cos^{n-2}(t) dt\\
& = 4\int_0^{\pi/2}\sin^2(t) \sin^{n-2}(2t)dt \tag{$\star$}
\end{align}
Replacing $t$ by $\pi/2-t$, we get
$$I(n) = 4... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/383953",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 0
} |
prove that : $ \sum_{n=0}^\infty |x_n|^2 = +\infty \Rightarrow \sum_{n=0}^\infty |x_n| = +\infty $ i wanted to prove initially that a function is well defined and i concluded that it's enough to prove this statement for $x_n$ a sequence :
$ \sum_{n=0}^\infty |x_n|^2 = +\infty \Rightarrow \sum_{n=0}^\infty |x_n| = +\in... | Hint: If $\sum a_n < \infty$, then $\displaystyle \lim_{n \to \infty} a_n = 0$. In particular for large $n$, we have $|a_n| \leq 1$. Then
$$
|a_n|^2 = |a_n| |a_n| \leq \dots
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/384015",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Which probability law? It may be a basic probability law in another form, but I cannot figure it out. Why can we say the following:
$P(A∩B|C) = $$P(A|B∩C)P(B|C)$
Thank you.
| You should simply expand upon the definition:
$P(A \wedge B | C) = \frac{P(A \wedge B \wedge C)}{P(C)}$.
$P(A | B \wedge C) = \frac {P( A \wedge B \wedge C)}{P( B \wedge C )}$.
$P(B|C)=\frac{P( B \wedge C)}{P(C)}$.
Now, we solve for $P(A \wedge B \wedge C)$ in the first two above and equate them:
$P(A \wedge B | C) \c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384088",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Can't prove this elementary algebra problem $x^2 + 8x + 16 - y^2$
First proof:
$(x^2 + 8x + 16) – y^2$
$(x + 4)^2 – y^2$
$[(x + 4) + y][(x + 4) – y]$
2nd proof where I mess up:
$(x^2 + 8x) + (16 - y^2)$
$x(x + 8) + (4 + y)(4 - y)$
$x + 1(x + 8)(4 + y)(4 - y)$ ????
I think I'm breaking one of algebra's golden rules, but... | Let us fix your second approach.
$$
\begin{align}
&x^2+8x+16-y^2\\
=&x^2+8x+(4-y)(4+y)\\
=&x^2+x\big[(4-y)+(4+y)\big]+(4-y)(4+y)\\
=&(x+4-y)(x+4+y).
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/384129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Longest antichain of divisors I Need to find a way to calculate the length of the longest antichain of divisors of a number N (example 720 - 6, or 1450 - 4), with divisibility as operation.
Is there a universally applicable way to approach this problem for a given N?
| If $N=\prod_p p^{e_p}$ is the prime factorization of $N$, then the longest such antichain has length $1+\sum_p e_p$ (if we count $N$ and $1$ as part of the chain, otherwise subtract $2$) and can be realized by dividing by a prime in each step.
Thus with $N=720=2^4\cdot 3^2\cdot 5^1$ we find $720\stackrel{\color{red}2},... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384212",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
If $S\times\mathbb{R}$ is homeomorphic to $T\times\mathbb{R}$, and $S$ and $T$ are compact, can we conclude that $S$ and $T$ are homeomorphic? If $S \times \mathbb{R}$ is homeomorphic to $T \times \mathbb{R}$ and $S$ and $T$ are compact, connected manifolds (according to an earlier question if one of them is compact th... | For closed 3-manifolds, taking the product with $\mathbb{R}$ doesn't change the fundamental group, so if the two products are homemorphic, the original spaces have the same fundamental group, and closed 3-manifolds are uniquely determined by their fundamental group, if they are irreducible and non-spherical.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/384288",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 1,
"answer_id": 0
} |
Group of invertible elements of a ring has never order $5$
Let $R$ be a ring with unity. How can I prove that group of invertible elements of $R$ is never of order $5$?
My teacher told me and my colleagues that problem is very hard to solve. I would be glad if someone can provide me even a small hint because, at this... | Here are a couple of ideas:
*
*$-1 \in R$ is always invertible. If $-1 \neq 1$, then it follows that $R^*$ should have even order, a contradiction. Therefore, $1=-1$ in ring $R$, so in fact $R$ contains a subfield isomorphic to $\mathbb{F}_2$.
*Let $a$ be the generator of $R^*$. Consider the subring $N \subseteq R$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "21",
"answer_count": 2,
"answer_id": 0
} |
Integrating a school homework question. Show that $$\int_0^1\frac{4x-5}{\sqrt{3+2x-x^2}}dx = \frac{a\sqrt{3}+b-\pi}{6},$$ where $a$ and $b$ are constants to be found.
Answer is: $$\frac{24\sqrt3-48-\pi}{6}$$
Thank you in advance!
| On solving we will find that it is equal to -$$-4\sqrt{3+2x-x^{2}}-\sin^{-1}(\frac{x-1}{2})$$
Now if you put the appropriate limits I guess you'll get your answer.
First of all write $$4x-5 = \mu \frac{d(3+2x-x^{2})}{dx}+\tau(3+2x-x^{2})$$
We will find that $\mu=-2$ and $\tau=-1$.
$$\int\frac{4x-5}{\sqrt{3+2x-x^{2}}}=\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384530",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Finding Markov chain transition matrix using mathematical induction Let the transition matrix of a two-state Markov chain be
$$P = \begin{bmatrix}p& 1-p\\
1-p& p\end{bmatrix}$$
Questions:
a. Use mathematical induction to find $P^n$.
b. When n goes to infinity, what happens to $P^n$?
Attempt: i'm able to find
$$P^n... | Your initial $P^1$ matrix is has first row $[p,1-p]$ and second row the reverse of that. Your goal matrix for $P^n$ also has its entries in the same form, with first row say $[a_n,b_n]$ and second row the reverse of that. So an approach would be to multiply the matrix for $P^n$ by the matrix $P$, and its top row will b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384592",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
If $\{w^k|w\in L\}$ regular implies L regular? If L is a language and the language
$$\tilde{L}:=\{x^k,x\in L, k\in\mathbb{N}\}$$
is regular, does that imply that L is regular? ($|L|<\infty$ gives equivalence)
We came across this question when trying to prove an explicit example, namely
Prove that $L=\{a^{p^k}\mid p\t... | No. Consider $L = \{a^{n^2} \colon n \ge 1\}$. As $a \in L$, your $\tilde{L} = \mathcal{L}(a^*)$, which is regular, but $L$ isn't.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/384644",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Find the value of $\int_{-\infty}^\infty \int_{-\infty}^\infty e^{-(x^2+xy+y^2)} \, dx\,dy$
Given that $\int_{-\infty}^\infty e^{-x^2} \, dx=\sqrt{\pi}$. Find the value of $$\int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+xy+y^2)} \, dx\,dy$$
I don't understand how I find this double integral by using the given d... | For fun, I want to point out that much more can be said. The spectral theorem for real symmetric matrices tells us that real symmetric matrices are orthogonally diagonalizable. Thus, if $A$ is some symmetric matrix, then there exists an orthogonal matrix $U$ and diagonal matrix $D$ such that $A$ can be written as $A=U^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384732",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 0
} |
Showing $(v - \hat{v})\,\bot\,v$ $\fbox{Setting}$
Let $V$ be an inner-product space with $v \in V$.
Suppose that $\mathcal{O} = \{u_1, \ldots, u_n\}$ forms an orthonormal basis of $V$.
Let $\hat{v} = \left\langle u_1, v\right\rangle u_1 + \ldots + \left\langle u_n,v\right\rangle u_n$ denote the Fourier Expansion of $v$... | $$
\begin{align}
\langle v-\hat{v},\hat{v}\rangle
&=\left\langle\color{#C00000}{v}-\color{#00A000}{\sum_{k=1}^n\langle v,u_k\rangle u_k},\color{#0000FF}{\sum_{k=1}^n\langle v,u_k\rangle u_k}\right\rangle\\
&=\left\langle\color{#C00000}{v},\color{#0000FF}{\sum_{k=1}^n\langle v,u_k\rangle u_k}\right\rangle-\left\langle\c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
The set of numbers whose decimal expansions contain only 4 and 7 Let $S$ be the set of numbers in $X=[0,1]$ that when expanded as a decimal form, the numbers are 4 or 7 only.
The following are the problems.
a), Is S countable ?
b), Is it dense in $X$ ?
c), Is it compact ?
d), Is it perfect ?
For a), I want to say that ... | Hints: For b, can you get close to $0.2?$
For c, you are correct that it is bounded, so you need to investigate closed. Let $y \in [0,1]$, but $y \not \in S$. Then there is some digit of $y$ that is not $4$ or $7$....
For d, you need to show that any point of $S$ is a limit of a sequence of other elements of $S$. L... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384856",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Calculating time to 0? Quick question format:
Let $a_n$ be the sequence given by the rule: $$a_0=k,a_{n+1}=\alpha a_n−\beta$$
Find a closed form for $a_n$.
Long question format:
If I have a starting value $x=100000$ then first multiply $x$ by $i=1.05$, then subtract $e=9000$. Let's say $y$ is how many times you do it.... | Let's $u_0 = x$, and $u_{n+1} = iu_n - e$.
To solve this equation, let's $l= il-e$, and $v_n = u_n - l$.
Then $v_{n+1}= i v_n$ ans $v_n = i^n v_0$, so $u_n = l + i^n v_0 = l + i^n (x - l)$.
But $l = e/(i-1) = 180 000$. So $u_n = 180 000 - 80000*1.05^n$. So $u_n \leq 0$ is equivalent to
$$180 000 - 80000*1.05^n \leq 0$$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384867",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Is Vector Calculus useful for pure math? I have the option to take a vector calculus class at my uni but I have received conflicting opinions from various professors about this class's use in pure math (my major emphasis). I was wondering what others thought about the issue. I appreciate any advice.
| The answer depends on your interests, and on the place you continue your education. In some areas in the world, PhD's are very specialized so any course that is not directly related to the subject matter is not necessary. One could complete a pure math PhD and not know vector or multivariate calculus.
However, this i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/384940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Homogenous measure on the positive real halfline
Define a measure $\mu\not=0$ on positive real number $\Bbb R_{>0}$ such that for any measurable set $E\subset\Bbb R_{>0}$ and $a\in \Bbb R_{>0} $, we have $\mu(aE)= \mu(E)$, where $aE=[ax;x\in E]$.
I am totally blank about this problem. I ponder on it several times bu... | I'll answer in greater generality.
A common way to construct a measure is to take a nonnegative locally integrable function $w$ and define $\mu(E)=\int_E w(x)\,dx$. This does not give all measures (only those that are absolutely continuous with respect to $dx$) but for many examples that's enough.
In terms of $w$, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/385008",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
} |
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$ Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
| $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}=\frac{28}{243}+\frac{10}{81} \log \left(\frac{2}{3}\right).$$
Hint: Change the order of summation:
$$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}=\sum_{n=1}^\infty\sum_{k=1}^n\frac{(-1)^n n^4}{2^n k}=\sum_{k=1}^\infty\sum_{n=k}^\infty\frac{(-1)^n n^4}{2^n k}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/385067",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 3,
"answer_id": 0
} |
Why isn't there a continuously differentiable surjection from $I \to I \times I$? I was asked this question recently in an interview. Why can't there be a a continuously differentiable map $f \colon I \to I \times I$, which is also surjective? In contrast to just continuous, where we have examples of space filling curv... | The image of any continuously differentiable function $f$ from $I$ to $I\times I$ has measure zero; in particular, it cannot be the whole square.
To see this, note that $\|f'\|$ has a maximum value $M$ on $I$. This implies that the image of any subinterval of $I$ of length $\epsilon$ lies inside a circle of diameter $M... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/385165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
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