Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
How to prove universally-quantified formula is true by contraposition?
For all natural numbers $x$ and $y$, if $x+y$ is odd, then $x$ is odd or $y$ is odd.
How do I prove the following statement is true by contraposition without using a truth table and theorems?
| The contrapositive of "if $x+y$ is odd then either $x$ is odd or $y$ is odd" would be "if $x$ and $y$ are both even or both odd, then $x+y$ is even".
The sum of two even numbers, $2a$ and $2b$, is $2(a+b)$, which is even. The sum of two odd numbers, $2a+1$ and $2b+1$, is $2(a+b+1)$, which is even too. This is sufficien... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3134362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Harmonic series in probability mass function problem Suppose $X$ is a discrete random variable with possible values $\{1, 2, 3,\dots\}.$ Further, suppose the p.m.f is $$c\left(\frac{1}{x}-\frac{1}{x+1}\right)\enspace\text{s.t. $c > 0$}$$
Find c and $E[X].$
Idea:
We have $$1=\sum_{x=1}^{\infty}c\left(\frac{1}{x}-\frac{1... | Hint: Evaluate the series up to a finite $N$ first, then take the limit as $N\to \infty$. The series up to a finite $N$ will be a telescoping series, i.e. most of the terms will cancel, making it easy to evaluate.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3134508",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Motivate why we may calculate $P[XGiven two independent random variables $X $ and $Y $ with continuous densities we know that there exists a regular conditional distribution $P[X<Y|Y=y] $, and futher that the regular conditional distribution has density $f_{X,Y}(x,y)/f_Y(y) $.
How do we motivate that $P[X<Y]=\int_\inft... | This is a special case of the very general "double expectation theorem" of conditional expectations:
Theorem.
Let $(\Omega,\mathcal{F},P)$ be a probability space, $\mathcal{G} \subseteq \mathcal{F}$ a $\sigma$-algebra of events, and $Z$ a $\mathcal{F}$-measurable random variable such that either $Z$ is non-negative ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3134595",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all n for which z is a purely imaginary number
Find every $n\in \Bbb{N}$ for which $z=\dfrac{(1-i)^{3n+4}}{{2^n}(1+i)}$ is a purely imaginary number.
I know $1-i = \sqrt 2\cdot e^{\frac{7\pi}{4}i}$
Then argument of the numerator is
$(3n+4)$ $\cdot \frac{7\pi}{4} +2k\pi \quad k \in \Bbb{Z}$
And I know $1+i= \s... | Since $2^n\in\mathbb R$ and since $\dfrac{1-i}{1+i}=-i$, the number $z$ is purely imaginary if and only if $(1-i)^{3n+3}$ is real. Can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3134705",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Verifying a $K$-basis for a primitive extension $K\subset K(\alpha)$ Let $K\subset L:=K(\alpha)$ be a primitive field extension of degree $n$ and we define $c_i\in L$ as
\begin{align*}
\sum_{i=0}^{n-1}c_i x^i=\frac{f^{\alpha}_K}{(x-\alpha)}\in L[x]\quad(1)
\end{align*}
where $f^{\alpha}_K$ is the minimal polynomial... | Yes, your definition of basis is correct. Here, "basis" is just the usual linear algebra definition of basis, which applies here because $L$ is indeed a $K$-vector space.
Your long division idea is the right direction to head towards. Setting $f^\alpha_K(x) = x^n + d_{n-1}x^{n-1} + \ldots + d_0$, the first few terms of... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3134838",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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A proof strategy for $L_XY=[X,Y]$ I'm trying to prove that $$L_XY=[X,Y]$$ I do realize that there are other proofs given of this assertion on stackexchange. However, I'm looking for ways to prove it using my strategy, as given below:
Let $\phi(t)$ be the flow of the vector $X$ at the point $p$. Then $$L_XY=\lim\limits... | Let's just act on a function:
$$ L_X(Y) (f) = X (Y(f)) -Y(X(f)),$$
by the Leibnitz rule and the property $L_X(f)=X(f)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3134981",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Proving a multi variable function bijective I understand the theory of how to prove a mutli variable function bijective, however I somehow can neither prove this function injective or surjective:
$$f: \mathbb N\times \mathbb N \rightarrow \mathbb N, (a, b) \mapsto {(a+b)}^{2} + a$$
I tried to start with $f(a, b) = f(a... | The map is certainly not surjective, as it has been pointed out in the comments.
Let us try to prove injectivity. I will rewrite the function $(a+b)^2+a$ as $c^2+a$ where $c=a+b$, and in particular note $c\geq a$ because $b\in\mathbb{N}$. Argue by contradiction, suppose there exist $a, c, \hat{a}, \hat{c}$ with $c\neq ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3135072",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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If $f$ is closed and convex then $f = f^{**}$ proof question.
If $f$ is closed and convex then $f = f^{**}$
Let $f$ be closed and convex. Then $f^* = \sup_x(y^Tx - f(x))$.
Since $$\{h(x) = ax + b | h(x) \le f(x) \text{ for all $x$ }\} = \{h(x)
= y^Tx + c | y \in \text{dom}(f^*), c \le - f^*(y)\}$$ we have $$f(x) = \... | For the affine minorant part, notice that we may write
$$
f^{**}(x) = \sup_{y} x^\top y - f^*(y) =
\sup_{y, \beta} \left\{ x^\top y - \beta \ \middle|\ \beta \geq f^*(y) \right\} \\
= \sup_{y, \beta} \left\{ x^\top y - \beta \ \middle|\ \beta \geq \sup_{z}
z^\top y - f(z) \right\} = \sup_{y, \beta}
\left\{ x^\top y - \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3135378",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Is it possible to prove that the limit, as n goes to infinity, of the sequence $a_n = 3^n/7^n$ is $0$? Intuitively this seems to be true, but is there a way to actually prove it instead of just saying "the denominator gets bigger faster"? I'm not sure that the sequence can be modeled by a function, but even in doing th... | This is a pretty standard and basic result related to sequences.
Theorem: Let $x\in\mathbb {R} $ then the sequence $\{a_n\} $ defined by $a_n=x^n$
*
*diverges to $\infty$ if $x>1$.
*converges to $1$ if $x=1$.
*converges to $0$ if $|x|<1$.
*oscillates finitely with values $1,-1$ if $x=-1$.
*oscilla... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find when two sin graphs hit $x=0$ at the same time Lets say I have 2 sine functions. They could be any sine functions but I'm more interested in the ones that get bigger/smaller as they go.
So we have:
$$
y_1 = \sin{(x^2)}
$$
$$
y_2=\sin{\left(\sqrt{x}\right)}
$$
I'm interested in when they both hit $y=0$ at the same ... | You have $x^2 = \pi n$ and $\sqrt{x} = \pi m$. Eliminating $x$ gives $n = \pi^3 m^4$. As $\pi$ is irrational $m$ must be 0, and hence $x = 0$ is the only solution.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3135783",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Two players throw a die until the sequence $1,2,3$ appears, and the winner is the one who roll $3$. What is the probability the second player wins?
Alameda and Belisario alternate turns throwing a fair die. Alameda plays first and they continue throwing, one at a time, until the sequence $1$-$2$-$3$ appears. Whoever t... | Let's use states.
We'll label a state according to how much of the $1,2,3$ chain has been completed and according to who's turn it is. Thus you start from $(A,\emptyset)$, and the other states are $(B,\emptyset),(X,1),(X,1,2)$ Win and Loss (Where $X\in \{A,B\}$. In a given state $S$ we let $p_S$ denote the probabilit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3135987",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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$d_p$ and $d_\infty$ in $\mathbb{C}^n$ are uniformly equivalent I need to prove this
Show
with the metrics $d_p$ and $d_{\infty}$ in $\mathbb{C}^n$ are uniformly equivalents, with $p \in [1, \infty)$.
So, I have in my book two definitions about equivalence metrics in a metric space $(X,d)$.
1) Two metrics $d_1$ and $d... | For any $p\ge 1$ and any $u=(u_1,u_2,\dots,u_n)\in\mathbb{R}^n$,
$$\|u\|_p = \left(\sum_{k=1}^n |u_k|^p\right)^{1/p} \le \left(\sum_{k=1}^n (\sup_k|u_k|)^p\right)^{1/p} = \left(n\|u\|_{\infty}\right)^{1/p}=n^{1/p}\|u\|_{\infty}$$
Estimate each term in the sum by the largest one. It's that simple. For the distance, appl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3136177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
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Evaluate the limit: $\lim\limits_{n\to\infty} \frac{4^nn!}{(3n)^n}$
Evaluate the following limit
$$
\lim_{n\to\infty} \frac{4^nn!}{(3n)^n}
$$
I've shown the limit is equal to $0$. One may for example use the ration test by which:
$$
\begin{align}
\frac{x_{n+1}}{x_n} &= \left({4\over 3}\right)^{n+1}\frac{(n+1)!}{(n... | Here is another approach.
Using Stirling's approximation $n!\sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$ you get
$$\frac{4^nn!}{(3n)^n}\sim \sqrt{2\pi n}\frac{4^n(n/e)^n}{3^n n^n}=\sqrt{2\pi n}(\frac{4}{3e})^n$$
which clearly tends to zero as $n\to\infty$, because $4<3e$.
(The symbol $\sim$ means that the ratio of the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3136272",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Movement of $1/z$ in complex plane How is moving $\frac{1}{z}$ in complex plane if $z$ is described by a circle which has radius $r$ and center $a+b*i$
I've just started complex algebra and still having some trouble imagining it.
How one does even solve this kind of problems, I don't want the complete solution I just w... | Hint: Compute $\dfrac1{r\bigl(\cos(\theta)+i\sin(\theta)\bigr)}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3136522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Polar decomposition of an operator and the partial isometry Let $A \in L(H)$, the space of bounded operators on the Hilbert space $H.$ I need to show that there is a partial isometry $U$ (i.e. $U^{*}U$ and $UU^{*}$ are projections), such that: $$A=U|A|$$ is the polar decomposition of $A,$ whereby $|A|=(A^{*}A)^{1/2}.$
... | Let $\sqrt{A^*A}$ denote the unique positive square root of $A^*A$. Then
\begin{align}
\|Ax\|^2 &=\langle A^*Ax,x\rangle \\
& = \langle \sqrt{A^*A}x,\sqrt{A^*A}x\rangle \\ & =\|\sqrt{A^*A}x\|^2
\end{align}
Let $U$ be defined so that $U=0$ on $\mathcal{N}(\sqrt{A^*A})$ and be defined on $\mathcal{N}(\sqrt{A^*A}... | {
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"timestamp": "2023-03-29T00:00:00",
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Example of two homotopically equivalent manifolds such that one admits a symplectic structure and the other does not A smooth manifold $M$ admits a symplectic structure if there is an alternating
non degenerate $2$-form $\omega \in \Lambda^2(M)$ that is also closed i.e. $d\omega = 0.$
Usually we can express obstruction... | There are many examples in four dimensions, in fact there are infinitely many examples where $M_1$ and $M_2$ are actually homeomorphic.
Let $M_1$ be a simply connected Kähler surface which is not spin. Then $M_1$ is symplectic and is homeomorphic to the smooth manifold $M_2 = b^+\mathbb{CP}^2\# b^-\overline{\mathbb{CP}... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Continuous function and limit $ f(x,y)=\left\{\begin{matrix} (x-y)\sin\frac{1}{x}\sin\frac{1}{y} & , xy\neq 0 \\ 0 &, x=y=0 \end{matrix}\right. $ I have this function:$$ f(x,y)=\left\{\begin{matrix}
(x-y)\sin\frac{1}{x}\sin\frac{1}{y} & , xy\neq 0 \\ 0
&, x=y=0
\end{matrix}\right. $$
a) Show that $ \lim_{x\rightarrow... | $sin{(1/x)sin(1/y)}$ is an interval and $x-y$ under the limit of $(x,y)\to{(0,0)}$ is an interval too. if a function in a close interval is continuous, its valued field is an interval, but its converse proposition may be not true. for example, your function is a counterexample. so you should separate your problem into ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3136903",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Is there any neat way to calculate this contour integral $PV\int_{\mathbb{R}^n}\frac{e^{i(b_1z_1+\cdots+b_nz_n)}}{\prod_{j$$\mathrm {PV}\int_{\mathbb{R}^n} \frac{e^{i(b_1z_1+\cdots+b_nz_n)}}{\prod_{j<k}(z_k-z_j)}dz=?$$
Other than integrate this term by term? It is in fact the Fourier transform of the inverse Vandermond... | So finally you meant the Fourier transform of the distribution $$F(x) = PV(\frac{1}{\prod_{j < k} (x_k-x_j)})$$
that is $$\hat{F}(\omega) = \int_{\mathbb{R}^n} F(x) e^{-i \langle \omega,x \rangle}d^n x$$
Then
$$1=F(x)\prod_{j < k} (x_k-x_j)$$
implies $$(2\pi)^n i^{n(n-1)/2} \delta(\omega)=\prod_{j < k}(\partial_j-\pa... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3137018",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to approach this question on subspaces?
Determine which of the following are subspaces of $3 \times 3$ matrix $M$
all $3 \times 3$ matrices $A$ such that the trace of $A$ is $\mbox{tr}(A) = 0$.
What does trace mean?
| Let $V$ be a vector space.
Then $W \subset V$ is a subspace of $V$ if
a) $0 \in W$;
b) $u, v \in W$ then $u + v \in W$;
c) $u \in W$ e $\lambda \in \mathbb{R}$, then $\lambda u \in W$.
The trace of a matrix $A$ of order 3 is given by $\mbox{tr} (A) = a_{11} + a_{22} + a_{33}$ (it is the sum of the principal diagonal).
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3137160",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove $ u \cdot (u \times (\nabla \times u)) = 0 $ Prove $$ u \cdot (u \times (\nabla \times u)) = 0 $$
Where '$u$' is a 3D-velocity vector.
I came across this for a proof for converting Euler's Equation to the Bernoulli expression for a steady-state, in compressible fluid.
Anyone know why this is the case?
| For simplicity, call ${\bf v} = \nabla \times {\bf u}$. The idea is that ${\bf u} \times {\bf v}$ is perpendicular to both ${\bf u}$ and ${\bf v}$, so that
$$
{\bf u} \cdot({\bf u} \times {\bf v}) = 0
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3137381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Find all positive integers $a, b, c$ such that $21^a+ 28^b= 35^c$ .
Find all positive integers $a, b, c$ such that $21^a+ 28^b= 35^c$.
It is clear that the equation can be rewritten as follows:
$$
(3 \times 7)^a+(4 \times 7)^b=(5 \times 7)^c
$$
If $a=b=c=2$ then this is the first possible answer to this issue. It i... | Note that $21=3\times7$, $28=4\times7$ and $35=5\times7$, and so by unique factorization the numbers $21^a$, $28^b$ and $35^c$ are all distinct for all positive integers $a$, $b$ and $c$.
By unique factorization we see that the left hand side of
$$21^a+28^b=35^c,$$
is divisible by $7^{\min\{a,b\}}$ and hence $c\geq\min... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3137485",
"timestamp": "2023-03-29T00:00:00",
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"question_score": "10",
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Find the number of elements $a^{x}=x$ depending on $ a> 0 $ My tryLet $$f(x)=a^{x}-x=e^{x\cdot ln a}-x$$So $$f'(x)=a^{x-1}\cdot x-1$$Then I should examine the $ f $ monotonicity.But when I can do it I have:$$f'(x)>0$$$$a^{x-1}\cdot x>1$$However I don't know what can I do in this moment. I tried:for $x>0$: $(x-1)lna>-ln... | You have differentiated $a^{x}$ wrongly. It should be $a^{x}\ln(a)$. But I suggest you sketch the graphs of $y=x$ and $y=a^{x}$ on the same axes and consider the number of intersections, and the conditions on $a$ for there to be any intersections at all.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3137612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Integrate $\int \frac{\sqrt{x^2-1}}{x^4}dx$ I am trying to integrate $\int \frac{\sqrt{x^2-1}}{x^4}dx$ via trig substitution. I decided to substitute $x = \sec\theta$ into the square root and $dx = \sec\theta \tan\theta\,d\theta$.
$$\int \frac{\sqrt{\sec^2 \theta-1^2}}{\sec^4\theta} \,dx
= \int \frac{\sqrt{\tan^2\theta... | $$
\begin{aligned}
& \int \frac{\sqrt{x^{2}-1}}{x^{4}} d x \\\stackrel{y=\frac{1}{x}}{=} &\int\frac{\sqrt{\frac{1}{y^{2}}-1}}{\frac{1}{y^{4}}}\left(-\frac{1}{y^{2}} d y\right)\\
=& -\int y \sqrt{1-y^{2}}d y \\
=&\frac{\left(1-y^{2}\right)^{\frac{3}{2}}}{3}+C
\\=&\frac{\left(x^{2}-1\right)^{\frac{3}{2}}}{3 x^{3}}+C
\en... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3137719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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why set $S$ is not closed under addition? some doubts :
Consider $\mathbb{R}^2$ over $ \mathbb{R}. $
Then
Given set $ S = \{(x,0) : x \in \mathbb{R}\} \cup \{ (0,y) : y \in \mathbb{R}\}$
My question is that why set $S$ is not closed under addition?
My attempt : I think set $S$ is closed under the addition... | First of all, do you agree that $(1,1) \not\in \{(x,0) : x \in \mathbb{R}\}$ and $(1,1) \not\in \{(0,y) : y \in \mathbb{R}\}$? If yes, then it is obviously not in their union.
You are confusing two different concepts, I think.
Consider the plane ($\mathbb{R}^2$) and then let $l_1$ be the line $\{(x,y)\in\mathbb{R}^2: y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3137844",
"timestamp": "2023-03-29T00:00:00",
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If X, Y independent then X/Y and Y are independent? if two random Variables $X, Y$ are independent
Does that mean that $\left(\frac{\ X }{Y}\right)$ and $Y$ are independent?
for example is it true that E[$\left(\frac{\ X }{Y}\right)$|$Y$]= E[$\left(\frac{\ X}{Y}\right)$] because of the independence of the random var... | Suppose that $X,Y$, satisfy $\mathbb{P}( X=-1 ) =\mathbb{P}( X=1 )=1/2 $ and
$\mathbb{P}( Y=-1 ) =\mathbb{P}( Y=2 )=1/2 $.
Set $Z=X/Y$, then $\mathbb{P}( Z=1/2, Y=-1)=0$ however
$\mathbb{P}(Z=1/2)\neq 0$ and $\mathbb{P}( Y=-1)\neq 0$.
So, they are not independent.
For the conditional what we have is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3138053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to calculate $\int_{0}^{4} \sqrt{\frac{t+4}{t}}dt$? I need help with this problem. I'm calculating the length of the smooth simple arc of the portion of the parabola $y^2=16x$ which lies between the lines $x=0$ and $x=4$.
So I first parametrized the function like this: $t=x\Rightarrow y=\pm\sqrt{16t}$, thus $f(t)=... | You can also set $u = 1+\frac4t$ to obtain
$$\int_0^4 \sqrt{1+\frac4t}\,dt = \begin{bmatrix} u = 1+\frac4t \\ du = -4(u-1)^2\,dt \end{bmatrix} = \int_{2}^\infty \frac{\sqrt{u}}{4(u-1)^2}\,du$$
You can decompose the latter function as
$$\frac{\sqrt{u}}{4(u-1)^2} = \frac1{16}\left(\frac1{(\sqrt{u}-1)^2}-\frac1{(\sqrt{u}+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3138197",
"timestamp": "2023-03-29T00:00:00",
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} |
Question about Fouriers Inversion Theorem. This is my professors definition of the FIT.
Fourier Inversion Theorem: Assume that $f\in L^2(\mathbb{R}).$ Define the Fourier transform to be
$$\widehat{f}(\xi)=\int\limits_{\mathbb{R}}f(y)e^{-iy\xi} \ dy.\tag 1$$
Then, as an equality in $L^2(\mathbb{R})$ we have the inv... | The choice of variables that were used is confusing. Note the $x$ and $y$ in your second equation of
$$f(x)=\frac{1}{2\pi}\int\limits_{\mathbb{R}}\widehat{f}(y)e^{ixy} \ dy. \tag2$$
are basically "dummy" variables, with $x$ being a placeholder for the variable of the function $f$ and $y$ specifying the variable being i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3138321",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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Evaluate :$\int_{-1}^{1} 2\sqrt{1-x^2} dx $
Evaluate: $$\int_{-1}^{1} 2\sqrt{1-x^2} dx $$
The answer is $\pi$
My attempt
$x = \sin(u), dx = \cos(u)du$
$$\int_{-1}^{1} 2 \sqrt{1-\sin^2(u)}\cos(u)du = \int_{-1}^{1} 2 \cos^2(u)du =\int_{-1}^{1} \frac{1}{2}(1+\cos(2u))du = \bigg(\frac{u}{2} + \frac{1}{2}\sin(2u) \bigg)\B... | Geometrically , the unit circle can be represented as $$x^2+y^2=1$$
so $$y=\pm \sqrt{1-x^2}$$ and your case $y=+ \sqrt{1-x^2}$
So $\int_{-1}^1\sqrt{1-x^2} dx $ is the area of a (upper )semi circle, which is $\frac{\pi}{2}$. So $$2 \int_{-1}^1 \sqrt{1-x^2}dx =2 \frac{\pi}{2}=\pi$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3138443",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
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How do I find/estimate the unknown given that the equation has exactly $2$ solutions? My problem:
The equation $x(x^2-7) = 2x+c$ has exactly $2$ solutions. Estimate the value(s) of $c$, writing you answer(s) in the form $c_1< c <c_2$, where $c_1$ and $c_2$ are integers.
My question: How do "$2$ solutions" contribute... | Consider $f(x)=x^3-9x$.
Now, $$f'(x)=3x^2-9=3(x-\sqrt3)(x+\sqrt3).$$
For $f'(x)=0$ we obtain $x=\sqrt3$ or $x=-\sqrt3$.
For $x=\sqrt3$ we obtain $c=-6\sqrt3$ and
$$x(x^2-7)-2x-c=0$$ it's
$$(x-\sqrt3)^2(x+2\sqrt3)=0,$$ which has two different roots.
For $x=-\sqrt3$ we obtain $c=6\sqrt3$ and the following equation.
$$(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3138621",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How did Gauss conjecture there were nine Heegner numbers? Coming from someone not very knowledgable in algebraic number theory it seems odd. At the time they didn't have the computing power to determine whether very high values (>>163) were Heegner numbers; so, why even assume there was a finite amount rather than infi... | Gauss had plenty of computing power. He calculated class numbers up to 2000, and found they got scarcer as he climbed higher, with none at all after 163. That seemed enough to conjecture there weren't any more.
Gauss was working with quadratic forms, rather than quadratic fields, and the bigger the discriminant, the e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3138747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
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How many monoids, groups, etc are there defined on a finite set of size $n$? Suppose we have a set $X$. We can define a binary relation $\cdot :X\times X\to X$, and get an algebraic structure $(X,\cdot)$.
There are $|X|^{|X|^2}$ such binary relations that can be defined. I.e. let $N=n^{n^2}$ where $n=|X|$.
Now, if we... | For groups this is a well studied problem with no easy answer. See finite simple groups classification for reference. If you take enough axioms though then this computation can become quite simple, for instance number of finite abelian groups of order $n$ can be easily described look at this question for example.
In su... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3138829",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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A decreasing transfinite sequence of subsets of a countable set. Let $X$ be a countable set and $(S_\alpha)_{\alpha< \rho}$ is a decreasing transfinite sequence of subsets of $X$ in the sense that
$$
S_\alpha \supset S_\beta
$$
whenever $\alpha<\beta$. Here $\rho$ is some fixed ordinal.
Suppose that $(S_\alpha)_{\alph... | If it is strictly decreasing, then $A_\alpha=S_\alpha\setminus S_{\alpha+1}$ is non-empty, and $A_\alpha\cap A_\beta=\varnothing$ whenever $\alpha\neq\beta$.
How many pairwise disjoint subsets a countable set can have?
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Conserving algebraic quantities I have the relations
$$r_0^{\pm} = \frac{1}{2}\left (\frac{\hat \rho_0\zeta_1^{1/2}}{\hat \rho_1\zeta_0^{1/2}} \pm \frac{\zeta_0^{1/2}}{\zeta_1^{1/2}} \right )$$
$$r_1^{\pm} = \frac{1}{2}\left (\frac{\hat \rho_1\zeta_2^{1/2}}{\hat \rho_2\zeta_1^{1/2}} \pm \frac{\zeta_1^{1/2}}{\zeta_2^{1... | For $i\in\{0,1\}$ you want to show that $R_i^2+T_i^2=1$, or equivalently
$$(r_i^+)^2-(r_i^-)^2=1.$$
For the sake of legibility let $a_k:=\hat{\rho}_k$ and $b_k:=\zeta_k^{1/2}$. Then for $i=0$ we have
$$r_0^{\pm}=\frac12\left(\frac{a_0b_1}{a_1b_0}\pm \frac{b_0}{b_1}\right)
=\frac{a_0b_1^2\pm a_1b_0^2}{2a_1b_0b_1},$$
and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3139131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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In a normal linear model (with intercept), show that if the residuals satisfy $e_i = a + \beta x_i$, then each residual is equal to zero. In a normal linear model (with intercept), show that if the residuals satisfy $e_i = a + \beta x_i$, for $i = 1\dots n$, where $x$ is a predictor in the model, then each residual is ... | Since your regression model has intercept, we can assume, the X matrix for the regression has the form
$$
X =
\begin{pmatrix}
1 & x_1^T \\
1 & x_2^T \\
\vdots & \vdots \\
1 & x_n^T \\
\end{pmatrix}
$$
Note that the residual must be orthogonal to every vector in column space of $X$.
This is because the predicted value,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3139297",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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What does it mean for a number to be independent of ZFC?
Since the definition of the Busy Beaver function by Radó in 1962, an interesting open question has been what [is] the smallest value of $n$ for which $BB(n)$ is independent of ZFC set theory.
Source: the first sentence of the abstract of the paper A Relatively ... | Sure, $BB(7918)$ is some number. But it is provably beyond the capabilities of ZFC to figure out which number that is, or even an upper bound for that number.
Specifically, given any (very large, but constructively described) integer $D$, the statement $BB(7918)<D$ can never be proven with ZFC.
As to the bonus question... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3139456",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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"answer_id": 0
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Proving a function $p:V→[0,\infty)$ forms a norm on V if and only if the unit ball $\{{x∈V|ρ(x)<1}\}$ is convex I have a function $p:V→[0,\infty)$ where V is a vector space and we know $p$ to be positive, absolutely homogeneous and non-degenerate (i.e.: we know $p$ satisfies all norm conditions other than the triangle ... | Some hints: If you knew that the closed unit ball $\{ x \in V \mid \rho(x) \le 1 \}$ was convex, then for nonzero vectors $x$ and $y$ you could use the convexity to conclude:
$$ \rho \left( \frac{\rho(x)}{\rho(x) + \rho(y)} \cdot \frac{x}{\rho(x)} + \frac{\rho(y)}{\rho(x) + \rho(y)} \cdot \frac{y}{\rho(y)} \right) \le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3139569",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Show that n(n + 2)(n + 4) is either divisible by 16, or is an odd number. I understand more or less how to do this problem, however, I am having trouble actually showing that n can be divisible by 16.
Here's what I have done so far
If n is an odd integer, then n = 2k + 1, where k is any integer.
2k + 1 * (2k + 3) * (2k... | Your approach is fine and, as noted by Randall, will lead to a solution. The other way to handle the problem is to note that if $n$ is even, either $n \equiv 2 \pmod{4}$ or $n \equiv 0 \pmod {4}$, and in either case the result will follow.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3139659",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Sum of cross terms vs sum of squares? What do we know about sum of squares vs sum of cross terms? Does one always dominate the other? Any theorems on that?
e.g
for
$a^2 + b^2 + c^2 \ < ? > \ ab + ac + bc $
for any number of terms.
Thank you
| for $a,b,c\in R(a\neq b\neq c)$
$(a-b)^2+(b-c)^2+(c-a)^2>0$
$a^2+b^2+c^2>ab+bc+ca$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3139752",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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Cartesian Product over a list of objects in MAGMA I'm currently trying to create the Cartesian Product of certain objects (namely: Character Tables of different finite groups).
However, it seems like I don't really understand the car<$...$> constructor.
In the documentation it says, that car<$...$> expects a list of se... | You can store the character tables in a list using [* ... *],
for example
F := FiniteField(3);
P<x> := PolynomialRing(F);
S := [ x^2 + 1, x^3 +x^2 +x +2];
CT := [* CharacterTable(UnitGroup(ext<F|f>)) : f in S *];
Then if you want to create a cartesian product space, you can use the CartesianProduct command as so:
CTP ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3139943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Probability of matching 5 cards from a deck of 40 Suppose we have a deck of 40 cards which has, 5 Aces, 6 Kings, 9 Queens, 20 Jacks. A game is played where a contestant will continuously draw cards until they have 5 matching cards (not necessarily in order). The cards are drawn without replacement.
I'm trying to find t... | Suppose a win occurs when the fifth ace is drawn on draw $r+1$, for $0 \le r \le 16$. On the previous draw, the hand must have contained four aces and no more than four of any of the other ranks, and then the contestant must draw an ace on draw $r+1$.
Let's try to find the probability of being in a favorable state on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3140034",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Problem similar to Gambler's Ruin I am working on a problem that is similar to the gambler's ruin problem.
We start with a bet of $\$1$.
*
*With probability $p$ the game is won and we win the amount of the bet, i.e., if we bet $\$1$, we win $\$2$.
*With probability $1-p =: q$ the game is lost.
If we win the game, ... | You can split your sum into two separate sums:
$$E(p)=\sum \limits_{k=1}^{\infty}(2^k-1)p(1-p)^{(k-1)}=p\sum \limits_{k=1}^{\infty}2^k(1-p)^{k-1} - p\sum \limits_{k=1}^{\infty}(1-p)^{k-1} \\ = 2p\sum \limits_{k=0}^{\infty}2^{k}(1-p)^{k} - p \sum \limits_{k=0}^{\infty}(1-p)^{k}$$
The sum $\sum\limits_{k=0}^{\infty}2^{k... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3140164",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Finding integer solution to a quadratic equation in two unknowns
We have an equation:
$$m^2 = n^2 + m + n + 2018.$$
Find all integer pairs $(m,n)$ satisfying this equation.
| Simpler start: separating variables to either side gives:
$$m^2-m=n^2+n+2018$$
which then factors roughly for the variables as:
$$m(m-1)=n(n+1)+2018$$
which since both pairs(m,m-1) and (n,n+1) are consecutive integers, you can divide both sides by two giving:
$$\frac{m(m-1)}{2}=\frac{n(n+1)}{2}+1009$$
But, $\frac{y(y+1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3140268",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Euler's totient $\phi$ function as a product of primes Many web pages say that Euler's totient function $\phi(n)$ can be given as
$$\phi(n)=n \prod_{p|n} \biggl(1- \frac{1}{p} \biggr)$$
But $\phi(1)=1$, and no primes divide $1$. Surely this gives
$$\phi(1)=\prod_{p|1} \biggl(1- \frac{1}{p} \biggr)=0n=0$$
Is $\phi(1)$ a... | There's no problem here.
$\phi(1)=1$ even according to the product definition;
there are no primes dividing $1$, so it's an empty product, which is $1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3140411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Show that there exists a $t \in [0, 1]$ such that $\alpha(f) =f(t)$ for all $f \in C_\mathbb{R}[0, 1]$.
Suppose $\alpha$ is a nonzero multiplicative linear functional on $C_\mathbb{R}[0, 1]$. Show that there exists a $t \in [0, 1]$ such that $\alpha(f) =f(t)$ for all $f \in C_\mathbb{R}[0, 1]$.
I approached this by ... | Nice!
Here's a more direct approach.
*
*$\alpha (f) = \alpha(1 f ) = \alpha(1)\alpha(f)$. Therefore
$$ (*)\quad \alpha(1) =1,$$
because $\alpha$ is not identically zero.
*
*If $f\ge 0$
$$ (**) \quad 0 \le \alpha (\sqrt{f})^2 = \alpha(f).$$
*
*Since $0\le \|f\|_\infty-f$, it follows that
$$ (***) \q... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3140669",
"timestamp": "2023-03-29T00:00:00",
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Let $k$ be a non-algebraically closed field and $I\subset k[x_1,\dots, x_n]$ be maximal ideal. Is $V_{\bar{k}}(I)$ necessarily finite? Let $k$ be a non-algebraically closed field and $I\subset k[x_1,\dots, x_n]$ be a maximal ideal.
$\textbf{Q:}$ Is $V_{\bar{k}}(I)=\{x\in\bar{k}^n\vert \forall f\in I, f(x)=0\}$ necessa... | Yes. Let $R=k[x_1,\cdots,x_n]$ and $\overline{R}=\overline{k}[x_1,\cdots,x_n]$. Then $\overline{R}$ is an integral extension of $R$ and both rings are normal domains, and we may apply going up and going down to see that $V_{\overline{k}}(I)$ also has dimension zero, and the dimension zero closed subsets of affine space... | {
"language": "en",
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Construct a set of points having some properties (colouring related)
Construct a finite set of points $S$, all in the same plane, such
that:
*
*Every line in the plane intersects $S$ in no more than $4$ points.
*If the points of $S$ are arbitrarily coloured with two colours, there are 3 collinear points ... | I'll number the points from top to bottom and on each line from left to right. We will try to construct a coloring that fails to satisfy condition $2$ and show that we can't do it. I'll think of the colors as red and green.
The corners of the triangle are points $1, 10$, and $13$. If all three corners are red, then ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3140905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Prove that this is a surjection and find the kernel We have a map $\alpha: G \rightarrow S(G)$, where $S(G)$ is the group of all bijections from $G$ to $G$. And $\alpha(g) = f_g$, where $f_g(a) = gag^{-1}$. It's easy to prove that this is a homomorphism and its kernel is the set of $g\in G$ such that $f_g = \operatorna... | In general, $\alpha$ is not surjective. This follows already from the fact that $|S(G)|=n!$ when $|G|=n$ (and $n!>n$ for $n>2$). Also, we have $\alpha(g)(1)=1$ for all $g\in G$, but (unless $n=1$) there exist bijections $\in S(G)$ that map $1$ elsewhere.
In summary, $\alpha$ is surjective iff $G$ is trivial.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3141101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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How to show that $|D_{2n}| = 2n$ via the presentation? Consider the dihedral group
$$D_{2n}= \langle a,b \mid a^n = 1 = b^2, b^{-1}ab = a^{-1}\rangle$$
How can I show that $|D_{2n}| = 2n$?
I'm trying to show that we can write every element in the form
$$x = a^i b^j$$ where $i= 0, 1, \dots, n-1$; $b = 0,1$.
I managed ... | Assume $D_{2n}= \langle a,b \mid a^n = 1 = b^2, b^{-1}ab = a^{-1}\rangle$ has this presentation. $D_{2n}\neq\emptyset$ since $1\in D_{2n}$. Let $a,b\in D_{2n}$ such that $a\neq b$, $a,b\neq1$ else this presentation is futile.
$\textit{Claim}$: $|a|=n$ and $|b|=2$.
$b^2=1.$ Then $ |b|\:\Bigg|\:2$. Since $b\neq1\implies... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3141226",
"timestamp": "2023-03-29T00:00:00",
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How does this work? $| [f(x)−g(x)] − (L−M) | \leq | f(x)−L | + | g(x)−M |$ My teacher was showing my class a proof for the limit difference rule using the epsilon-delta definition, and nearing the end, he showed us this:
|[f(x)−g(x)] − (L−M)| ≤ |f(x)−L| + |g(x)−M| . I know what the triangular inequality is, and how it... | The trick is that
$$|M-g(x)|=|g(x)-M|$$ because $$|M-g(x)|=|(-1)(g(x)-M)|=|-1|\cdot|g(x)-M|=|g(x)-M|$$
So that $$|(f(x)-g(x))-(L-M)|=|(f(x)-L)+(M-g(x))|\leq |f(x)-L|+|g(x)-M|$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3141310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Incomplete elliptic integral and Jacobi's form The incomplete elliptic integral of the first kind is written (using trigonometric form) : $ F(\varphi,k)=\int_{0}^{\varphi} \frac{1}{\sqrt{1-k^2 \sin^2(\theta)}} \mathrm{d}\theta $.
Then, it is noted everywhere that if we make the change of variable $t=\sin(\theta)$, then... | Because the original integral makes sense for $\theta$ in a neighborhood of $0$. There, $\cos$ is positive.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3141411",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Are these two graphs isomorphic? Why/Why not? Are these two graphs isomorphic?
According to Bruce Schneier:
"A graph is a network of lines connecting different points. If two graphs are identical except for the names of the points, they are called isomorphic."
Schneier, B. "Graph Isomorphism"
From Applied Cryptogr... | Both claims are correct.
Mapping $$e_1 \to c_1, \qquad e_2 \to c_3, \qquad e_3 \to c_5, \qquad e_4 \to c_2, \qquad e_5 \to c_4$$ maps the edges of the left graph precisely to those of the right graph, so that map defines an isomorphism of graphs.
The right graph has cycles of length $3$ (e.g., $aefa$) but he left gra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3141500",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 6,
"answer_id": 3
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Other inner products for $\mathbb{R}^n$ For $\mathbb{R}^n$, the standard inner product is the dot product. It is defined as $ \langle v,\,w\rangle = \sum_i v_i \cdot w_i $. I am aware that any scaled version, namely
$ \langle v,\,w\rangle = \sum_i\lambda_i\cdot v_i \cdot w_i $ will still satisfy the 4 inner product r... | I agree with SmileyCraft. In finite dimensional vector spaces, bilinear transformations, as linear transformations, can be written in terms of the values that they adopt in a given base:$$\left \langle x,y \right \rangle=\sum_{i,j=1}^{n}x_iy_j\left \langle e_i,e_j \right \rangle.$$
I believe you can arrive in this repr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3141609",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 4
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What is the "determinant" of two vectors? I came across the notation $\det(v,w)$ where $v$ and $w$ are vectors. Specifically, it was about the curvature of a plane curve:
$$\kappa (t) = \frac{\det(\gamma'(t), \gamma''(t)) }{\|\gamma'(t)\|^3}$$
What is it supposed to mean?
| They formed a matrix by stacking $\gamma'(t)$ and $\gamma''(t)$ next to each other as column vectors. You can also regard it as the cross product of the two vectors if you extend both with a $z=0$ coordinate and take the z component of the resulting vector (that way you can relate it to the 3d formula in a way).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3141770",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
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orthogonal chebyshev polyhomials This Theorem says
statement:
$\int_{-1}^{1} \frac{T_n(x)T_m(x)}{ \sqrt{1+x^2} } dx = 0 ;$ when $n\ne m $
proof:
"substitute $x= cos \theta$ "
and that's it.
So I am wondering should I start with this
$\int_{-1}^{1} \frac{cos(\theta n)cos(\theta m)}{\sqrt{1+x^2}} dx $
or with this
$\i... | When you do this, you apply the substitution rule for integrals, fully. Everything with $x$ in it, including the denominator, the $dx$, and the limits, transform.
On the other hand, the statement that you're trying to prove is incorrect. The correct form has $\sqrt{1-x^2}$ in the denominator instead.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3141913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Arithmetic Progression Time and Work Problem
A group of men working at the same rate can finish a job in $45$ hours. However, the men report to work, one at a time, at equal intervals over a period of time. Once on the job, each man stays until the job is finished.
If the first man works five times as many hours as ... | Suppose that there are $M$ men in the group. We are told that it takes $45M$ man-hours to finish the job. Now suppose the first man comes to work at time $0$ and works until time $T,$ and that the other men arrive at intervals of $h$ hours. The second man arrives at time $h$ and works until time $T$ so he works $T-h... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3142054",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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} |
Find value of $|z_2+z_3|$ Given that complex numbers $z_1,z_2,z_3$ lie on unit circle and
$$|z_1-z_2|^2+|z_1-z_3|^2=4$$ Then find value of $|z_2+z_3|$
My try:
We can take $z_1=e^{i\alpha}$, $z_2=e^{i\beta}$ and $z_3=e^{i\gamma}$
So we have
$$|z_1-z_2|=2\sin\left(\frac{\alpha-\beta}{2}\right)$$
$$|z_1-z_3|=2\sin\left(\f... | Hint: Writing $$\frac{\beta-\gamma}{2}=\frac{\beta-\alpha}{2}+\frac{\alpha-\gamma}{2}$$ and combing this with trigonometric identity for the sum of cosine $$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$$ we get
\begin{align}\cos\left(\frac{\beta-\gamma}{2}\right)&=\cos\left(\frac{\beta-\alpha}{2}+\frac{\alpha-\gamma}{2}\rig... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3142187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Alice and Bob take turns to remove numbers from a list Alice and Bob play the following game. In the beginning there is list of numbers
$$\{0, 1, 2,\dotsc, 1024\}.$$
Alice starts, and removes 512 numbers of her choice. Bob continues and removes 256 numbers of his choice. Alice continues and removes 128 numbers of her... | Given a set $X$, let $A(X)$ denote the largest distance between any two points in $X$ and $B(X)$ denote the smallest distance between any two different points in $X$. For a two-element set, $A(X)=B(X)$. We start with $A(X)=1024$ and $B(X)=1$. I will show Alice can halve $A(X)$ at each of her turns, and Bob can double $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3142291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Simple Displacement of Parametric Equations Dispute A parametric equation with $\frac{dx}{dt}$ = something, $\frac{dx}{dt}$ = something, has a resultant velocity vector by pythagorean theorem to be $\sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2}$. Calculus theorem dictates that integral of velocity over interval equals d... | I understand this actually now. The "paradox" I was referring to with infinite x steps and y steps is called the staircase paradox and can be used to falsely claim pi is 4. By doing what I did with the diagonal of the x coordinate and y coordinate you are finding the distance (length) of a path that follows straight fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3142410",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Find the coefficient of $x^{13}$ in the convolution of two generating functions
Four thiefs have stolen a collection of 13 identical diamonds. After the
theft, they decided how to distribute them.
3 of them have special requests:
*
*One of them doesn't want more than 2 diamonds ($\leq2$).
*The other one... | It is a laborious work.
$$\frac{d^n}{dx^n}\frac{1}{(1-x)^2(1-x^2)}=-\frac{1}{2}(-2-n)_n(-1+x)^{-3-n}+\frac{1}{4}(-1-n)_n(-1+x)^{-2-n}+\frac{1}{8}(-n)_n(1+x)^{-1-n}-\frac{1}{8}(-n)_n(-1+x)^{-1-n}$$
The bracket symbol is the Pochhammer symbol.
$$f^{(10)}(x)=\frac{d^{10}}{dx^{10}}\frac{1}{(1-x)^2(1-x^2)}=$$
$$\frac{-72576... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3142542",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
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if $M \otimes_A (A_m/mA_m)=0$ for every maximal ideal $m \subset A$, then $M=0$, $M$ finitely generated
Suppose $M$ is a finitely generated $A$-module. Prove that if $M \otimes_A (A_m/mA_m)=0$ for every maximal ideal $m \subset A$, then $M=0$. Subscrpit $_m$ means localization at $m$.
First consider the exact sequenc... | Hint:
$$M \otimes_A A_{\mathfrak m}/\mathfrak mA_{\mathfrak m}\simeq M_{\mathfrak m}/\mathfrak mM_{\mathfrak m}.$$
Then use Nakayama's lemma.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3142696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Let $B \subseteq \biguplus^{\infty}_{n=1} A_n$, show that $\mathbb{P}(B)=\sum^{\infty}_{n=1} \mathbb{P}(A_n) \mathbb{P}(B|A_n)$
Question:
Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space with events
$A,B\in\mathcal{A}$. Now, let $B \subseteq \biguplus^{\infty}_{n=1} A_n$, where $A_n \in \mathcal{A}$ for ... | Hint: if $\biguplus$ means disjoint union,
$$B = B\cap\left(\biguplus_{n=1}^\infty A_n\right) =
\biguplus_{n=1}^\infty(B\cap A_n).
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3142803",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Interpretation of Einstein notation for matrix multiplication Consider the matrix product $C = AB$ where $A \in \mathbb{R}^{m \times n}, B \in \mathbb{R}^{n \times p}$. The Einstein summation notation for this is
$$
c_{ik} = a_{ij}b_{jk}.
$$
Is there any example from math, physics, engineering, statistics etc. where e... | Define a third order tensor whose components are equal to zero unless all three indices are equal
$${\cal H}_{ijk} = \begin{cases}
1 \quad{\rm if}\; i\!=\!j=\!k \\
0 \quad{\rm otherwise} \\
\end{cases}
$$
Then you can use Einstein notation to write
$${\cal D}_{ijk} = A_{ip}{\cal H}_{pjs}B_{sk}$$
This tensor is a usefu... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3142957",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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If $\sqrt{a}-\sqrt{b}$ is a root of a polynomial with integer coefficients, then so is $\sqrt{a}+\sqrt{b}$.
If $\sqrt{a} - \sqrt{b}$, where $a$ and $b$ are positive integers and non-perfect squares, is a root of a polynomial with integer coefficients, then $\sqrt{a} + \sqrt{b}$ also is.
It seems to hold some relation... | Returning to this question to complete my work in my previous answer - the crucial insight to the following argument was provided by the answer of @robjohn.
Supposing $a$, $b$ are positive integers, $a \neq b$ such that none of $a$, $b$, $ab$ are perfect squares, then for any polynomial with integer coefficients $p(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3143101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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About branch points of a holomorphic map Let $F:X \to Y$ be a holomorphic map between Riemann surfaces. $q \in Y$ is a branch point if it is the image of a ramification point. How to prove that the set of branch points is a discrete subset of $Y$. This is from Rick Miranda's Algebraic curves and Riemann surfaces. Thank... | You forgot to say $F$ is non-constant. Then again, I guess $Ram(F)$ is not defined for $F$ non-constant.
In general for any map $F: X \to Y$ of any topological spaces $X$ and $Y$ with $X$ compact and $Y$ Fréchet/T1 and for any closed discrete subspace $A$ of $X$, we have $F(A)$ discrete.
Proof: Closed discrete subspace... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Definition of convergent sequence using only closure operator I have interest in Kuratowski closure axioms for topology. I would like to know how to define convergent sequence using only closure operator such that it is the same to definition of convergent sequence of equivalent axiomatic framework of topological space... | If $s: \mathbb{N} \to X$ is a sequence in $X$, we can define that $s$ converges to $x \in X$ by
$$x \in \bigcap \{\operatorname{cl}(s[A]) : A \subseteq \mathbb{N} \text{ infinite }\}$$
which can be shown by considerations as William Elliott gave as well.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3143356",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Transforming Quadrics in Characteristic 2 I’m trying to solve the following problem given in a textbook:
Let $k$ be an algebraically closed field and $Q=V(F)$ a quadric in $\mathbb{P}^3(k)$, where $F$ is an irreducible polynomial in $X,Y,Z,T$, and hence gives rise to a quadratic form on $k^4$ which we assume is non-de... | Having spent more time on this, I think I have solved the problem, and the result is in fact true.
Arf defines non-singularity for quadratic forms of characteristic $2$ here, explained in English here. From these papers, we see that if $Q$ is non-singular over a field of characteristic $2$, we can write $$Q=(aX^2+XT+bT... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3143476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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If $f(x+y)=f(x)f(y)-g(x)g(y)$ and $g(x+y)=f(x)g(y)+f(y)g(x)$, with $f'(0)=0$, determine $[f(x)]^2+[g(x)]^2$ Given the expressions:
$f(x+y)=f(x)f(y)-g(x)g(y)$
$g(x+y)=f(x)g(y)+f(y)g(x)$
the exercise is to show that $[f(x)]^2+[g(x)]^2$ is constant for all real $x$ and determine its value, knowing that $f$ and $g$ are rea... | If we let $\phi(x)=f(x)+ig(x)$, then $\phi$ satisfies a functional equation
$$
\phi(x+y)=\phi(x)\phi(y).
$$ This gives $\phi(x)=\phi(x)\phi(0)$, so either $\phi \equiv 0$ or $\phi(0)=1$. Since $\phi \equiv 0$ is a trivial solution, we assume $\phi(0)=1$. Differentiating with $y$ and plugging $y=0$, we obtain $\phi'(x)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3143564",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
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If the median AM of a triangle ABC bisects the angle $\hat{A}$, then the triangle is an isosceles. Can we solve the above problem using only the criteria for congruent triangles (i.e., without using the fact that the sum of the angles of a triangle is $180^\circ$)?
| This is the most basic approach, I believe. Consider the triangle $\triangle ABC$, in which $AD$ is both median and bisector.
*
*Extend $CD$ to a segment $DC'\cong CD$.
*Then $\triangle ACD \cong \triangle BDC'$ by SAS criterion.
*Consequently you have $\angle BC'D \cong \angle ACD$ and $AC \cong BC'$.
*For trans... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3143797",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 0
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Expected number of colors in a sampling of colored balls without replacement, Suppose there is a box containing differently colored balls. There are $G$ colors, each color having $n$ balls of this color, i.e. $G \times n$ balls.
What is the expected number of different colors in a sample of size $s$ without replacement... | We have from first principles that the PGF in $u$ with the coefficient
on $[u^q]$ representing the probability of $q$ different colors /
coupons not being seen in a sample of size $s$ is given by
$$\frac{1}{s!} {nG\choose s}^{-1} s! [z^s]
\left(u + \sum_{k=1}^n \frac{n!}{(n-k)!} \frac{z^k}{k!}\right)^G.$$
This sim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3143895",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simple integral with $e^x$ - how to decompose it? How to calculate this integral?
$$\int \frac{e^{2x}+2}{e^x+1}dx$$
I have tried various substitions such as: $t = e^x, t = e^x + 1, t = e^x +2, t = e^{2x}$
and none seem to work.
According to wolframalpha I can simplify this expression into:
$$\frac{e^{2x}+2}{e^x +1} = ... | \begin{align}
\frac{e^{2x}+2}{e^x +1}&=\frac{(e^{x})^2+2e^x+1-2e^x+1}{e^x +1}\\
&=\frac{(e^x+1)^2-2e^x+1}{e^x +1}\\
&=e^x+1+\frac {-2e^x-2+3}{e^x +1}\\
&=e^x+1-2+\frac {3}{e^x +1}\\
&=e^x + \frac{3}{e^x+1} - 1
\end{align}
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3144032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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How can we construct an 8x8 grid with minimal squares Links to similar questions:
What is the minimum number of squares needed to produce an $ n \times n $ grid?
How can we draw $14$ squares to obtain an $8 \times 8$ table divided into $64$ unit squares?
The second link is a similar question, but at the bottom, someone... | For all $n\ge 4$, the optimal number of squares is $2(n-1)$.
A construction, taken from Jorik's answer, is as follows.
If $n$ is even,
*
*$n-2$ squares have lower left corner $(0,0)$, whose widths comprise all integers between $1$ and $n-1$ except for $n/2$.
*$n-2$ squares have upper right corner $(n,n)$, with the... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3144226",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Missing constraint While reading this paper, I stumbled over the statement (1, 8) about linear independence in trees.
Just to make sure, that I understand it correctly: He means conical independence and forces $a_{i, k} \geq 0$?.
Linear independence as soon as the number of vertices $n$ is larger than the dimension sho... | No, this is linear independence.
A tree always has $n$ vertices and $n-1$ edges, so here we are talking about the linear independence of $n-1$ elements of an $n$-dimensional space. This should always be possible; there is no dimension argument against it.
Another way to phrase the argument for proving this linear inde... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3144376",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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System of equations can be interpreted as intersection of $3$ planes in $3$-dimensional space I'm struggling with this question.
I have worked out that attempting to solve for the $x_i$ leads to a contradiction, and that
$$\begin{vmatrix}1&4&6\\1&-2&1\\2&14&17\end{vmatrix}=0$$
So there is no solution for $x$. But wha... | Labelling the coefficient matrix's rows as $R_1,R_2,R_3$, we have $R_3=3R_1-R_2$ but $R_1$ and $R_2$ independent, but $3\cdot18-(-6)\ne-6$ so the third equation is not a linear combination of the first two. This means that
*
*the planes corresponding to $R_1$ and $R_2$ intersect in a line
*$R_3$'s plane is parallel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3144524",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Simplifying $\prod\limits_{k\neq j=0}^{n-1}\frac1{\lambda_{n,k}-\lambda_{n,j}}$ for $\lambda_{n,k}=\exp\frac{i\pi(2k+1)}{n}$ I have been able to show that for $n\in\Bbb N_{\geq2}$ $$\phi(n)=\int_0^1\frac{dx}{x^n+1}=\sum_{k=0}^{n-1}\Gamma_{n,k}\log\frac{\lambda_{n,k}-1}{\lambda_{n,k}}$$
Where $$\lambda_{n,k}=\exp\frac{i... | Defining the polynomial
\begin{align}
P(x)&=x^n+1\\
&=\prod_{j=0}^{n-1}\left( x- \lambda_{n,j}\right)
\end{align}
we can express its derivative at $x=\lambda_{n,k}$ as:
\begin{align}
P'(\lambda_{n,k})&=\prod_{k\neq j=0}^{n-1}\left( \lambda_{n,k}-\lambda_{n,j} \right)\\
&=\frac{1}{\Gamma_{n,k}}
\end{align}
But we ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3144672",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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$\| X'(t) \| = (\| X(t) \|)'$
Let $X : \mathbb{R} \to \mathbb{R}^n$ be a $C^1$ function. Let $\| .\|$ be the norm : $\| v \| = \max_{1 \leq i \leq N} \mid v_i \mid$. Then is it true that :
$$\| X'(t) \| = (\| X(t) \|)'$$ ?
I am wondering if in general if I have any function $f : \mathbb{R}^n \to \mathbb{R}^p$ and... | For $n=1$ the identity function $X(x)=x$ is a $C^{1}$ function. In this case $|X(x)|$ is not even differentiable at $0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3144813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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What is the matrix of $A$? We know the following about the linear map $A$: $\mathbb{R}$$^3$ -> $\mathbb{R}$$^3$:
$A$ is orthogonal
$A$(1,2,2) = (1,2,2)
The vector (2,0,-1) is eigenvector for eigenvalue -1
dim $E_1$ = 1
Determine the matrix of $A$
I'm not quite sure which properties to use, such that i can create a ma... | From the given conditions you have two equations $Av_1=v_1$ and $Av_2=-v_2$.
Notice that here additionally $v_1^Tv_2=0$ what means that both vectors are orthogonal.
You can find also transformed the third vector $v_3$ using as input vector cross product of $v_1$ and $v_2$, the result vector is orthogonal to both ... | {
"language": "en",
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"source": "stackexchange",
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Limit of a function, given the recurrence relation
Let $f(n)$ be a function defined for $n\ge 2$ and $n\in N$ which follows the recurrence(for $n\ge 3$) $$\displaystyle f(n)=f(n-1) +\displaystyle \frac {4\cdot (-1)^{(n-1)} \cdot \left(\displaystyle \sum_{d \vert (n-1)} (\chi (d))\right) }{n-1}$$ where $d\vert (n-1)$ m... | A preliminary lemma relates your $\chi$ function with the Gaussian integers:
$$ 4\sum_{d\mid n}\chi(d) = r_2(n) = \left|\{(a,b)\in\mathbb{Z}^2:a^2+b^2=n\}\right| $$
hence your question is equivalent to the determination of the Dirichlet L-series
$$ L=\sum_{n\geq 1}\frac{(-1)^{n+1} r_2(n)}{n} $$
which is conditionally c... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3145071",
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"source": "stackexchange",
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How can I prove that $(a_1+a_2+\dotsb+a_n)(\frac{1}{a_1}+\frac{1}{a_2}+\dotsb+\frac{1}{a_n})\geq n^2$ I've been struggling for several hours, trying to prove this horrible inequality:
$(a_1+a_2+\dotsb+a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dotsb+\frac{1}{a_n}\right)\geq n^2$.
Where each $a_i$'s are positive and $n$ is ... | Here is the proof by induction
that you wanted.
I added a more exact
version of
the identity used
in the proof
at the end.
Let
$s_n
=u_nv_n
$
where
$u_n=\sum_{k=1}^n a_k,
v_n= \sum_{k=1}^n \dfrac1{a_k}
$.
Then,
assuming
$s_n \ge n^2$,
$\begin{array}\\
s_{n+1}
&=u_{n+1}v_{n+1}\\
&=(u_n+a_{n+1}) (v_n+\dfrac1{a_{n+1}})\\
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3145187",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
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Problem of probability distribution A box contains $N$ tickets numbered $1, 2, 3,...., N$. If $m$ tickets are drawn one by one from the box without replacement, then find the mean of the sum of the numbers obtained on the tickets drawn.
I have approached the sum as below.
Let $X_i$ denote the number on the $i$th ticket... | Your calculation is actually correct, though it is not trivial to see it.
On one hand your argument is right, after the first number is drawn, the second number ($X_2$) only has one of $N-1$ possible values. Even worse, if $X_1=1$, then the expected value of $X_2$ will be $\frac{2+3+\ldots+N}{N-1}=\frac{N^2+N-2}{2(N-1)... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
Example of series where $\sum a_n$ convergent but $\sum n {a_n}^2$ divergent? Can anyone suggest an example of a sequence $\{a_n\}_n\subset (0,\infty)$ such that $\sum a_n$ is convergent but $\sum n {a_n}^2$ is divergent?
| Since we must have $a_n = \Omega(\frac{1}{n})$ we think about sparsness, so something like: $a_{2^m}=\frac{1}{m^2+1}$, all other a's being very small, say $a_n = 2^{-n}$ if $n$ is not a power of two, will do since then for $n=2^m$, $na_n^2 = \frac{2^m}{(m^2+1)^2}$ which obviously goes to infinty, so the series $\sum n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3145479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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} |
Proving there are infinitely many rational numbers in $[x,y]$
Prove if $x$ and $y$ are real numbers with $x \lt y$, then there are infinitely many rational numbers in the interval $[x,y]$.
What I got so far:
Let $x,y \in \Bbb R$ with $x \lt y$
Let $S = [x,y]$
By the density of $\Bbb Q$ in $\Bbb R$, $\exists r \in \B... | Okay, so I'll give it another shot given the feedback.
Proof:
Let $x,y \in \Bbb R$ with $x \lt y$ and $S = [x,y]$
Suppose there are only $n$ rational numbers between $x$ and $y$ such that:$$x \lt r_1 \lt \cdot \cdot \cdot \lt r_n \lt y$$
But since $\Bbb Q$ is dense in $\Bbb R$, there exists $r_{n+1} \in \Bbb Q$ such th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3145550",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 2
} |
Probability of Prime within radius around number Here is my question: do we have any kind of estimate about $p_{k, d}(n)$ the probability that there are at least $k$ prime numbers in a radius of $d$ around $n$?
Do you have any suggestions regarding related work?
For instance, we know that for $n$ there is a prime $p : ... | Which other unsolved problems, have necessary restrictions on the prime gaps? a related question I just got answered. As the comments on your question talk about though, there's not really a restriction. Primorials (products of all primes up to a number) have potentially massive gaps nearby, You can gaurantee all numbe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3145655",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Examples of non-unitary isometries on finite dimensional Hilbert spaces? I was reading the question A Finite Dimensional non-Unitary Isometry?, which gives an example of a non unitary isometry which is a map $T: R \rightarrow R^2 $. This question is based on a previous question Difference between an isometric operator ... | Very generally, if $X$ is a finite-dimensional vector space and $A,B:X\to X$ are linear maps such that $AB=1$, then $BA=1$. Indeed, if $AB=1$, then $A$ and $B$ must be invertible (consider their determinants), and so $$BA=BA(BB^{-1})=B(AB)B^{-1}=BB^{-1}=1.$$
From a different perspective, a linear isometry $T:X\to Y$ b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3145778",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Find all solutions, if any, to the equation $[28][x]-[22] = [33]$ in $\mathbb{Z}_{51}$. Find all solutions, if any, to the equation
$[28][x]-[22] = [33]$ in $\mathbb{Z}_{51}$.
I know this simplifies to
$[28][x] = [55]$,
which can be rewritten as
$28x \equiv 55 \bmod{51}$.
From here do I use SMT and split it, then pr... | You can certainly solve the congruence $\ 28x \equiv 55 \bmod{51}\ $ (which you can rewrite as $\ 7x \equiv 1 \bmod{51}\ $) by using the same procedure used in the proof of the Chinese remainder theorem. That is, if $\ x_1\ $ satisfies the congruence $\ x_1 \equiv 1 \bmod{3}\ $, and $\ x_2\ $ the congruence $\ 7x_2 \e... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3146002",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is this proof of the uniqueness of prime factorizations unnecessarily long? I am learning textbook Analysis I by Herbert Amann and Joachim Escher. The authors present Prime Factorization Theorem
and a proof of uniqueness
On the basis of authors' proof, I have found a shorter way to fulfill the task as follows:
L... | *
*Your proof uses the following fact: if $p_0$ is prime and $p_0\nmid a$, $p_0\nmid b$, then $p_0\nmid ab$. While this is not a very advanced number theory fact, it does require proof given the standard (in my experience) definition of a prime as a number having exactly two positive divisors. It's quite possible that... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3146090",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Is there a way to find $\min\{m(|X-c|), \:c \in \mathbb{R}\}$? Suppose X is a random variable, such that $F(t) := P(X < t) \in C(\mathbb{R})$. Is there a way to find $\min\{m(|X-c|), c \in \mathbb{R}\}$? Here $m$ stands for median.
I know the solutions for two particular cases: (and they both use a similar method):
If ... | Not a full solution but an idea that gives useful shortcuts (sometimes).
Imagine the PDF of $X$ as a picture. The the PDF of $X-c$ is of course just shifting, and $|X-c|$ is then folding around $c$.
What is $median(|X-c|)$? Since $|X-c|$ has a definite lower bound at $0$, the median is just the point $m>0$ where th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3146231",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why is there unique plane which passes through given point and is parallel to given line I was trying to solve one question which is asking to find a plane which passes through given point and is parallel to given line.
The given point is $M(2,-5,3)$ and the given line is given as an interesection of the planes $2x-y+3... | You are right: there are infinitely mane planes passing through a point and parallel to a given line.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3146342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Prove, that for every real numbers $ x \ge y \ge z > 0 $, and $x+y+z=\frac{9}{2}, xyz=1$, the following inequality takes place Prove, that for every real numbers $ x \ge y \ge z > 0 $, and $x+y+z=\frac{9}{2}, xyz=1$, the following inequality takes place:
$$ \frac{x}{y^3(1+y^2x)} + \frac{y}{z^3(1+z^2y) } + \frac{z}{x^... | Note: I have found a solution.
Shall we observe that each term of the LHS sum is of the form $\frac{x^4z^4}{y+z}$, the inequality is equivalent to
$$\sum_{cyc}{\frac{x^4z^4}{y+z}} > \frac{1}{3}{(xy+yz+zx)} $$
But from Titu's Lemma, we have
$$ \sum_{cyc}{\frac{x^4z^4}{y+z}} = \sum_{cyc}{\frac{(x^2z^2)^2}{y+z}} \ge \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3146462",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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If $A \in \mathcal{L}(H)$ and $\langle A(u),u \rangle \geq \langle u, u \rangle$, then $A$ is invertible. Exercise :
Let $H$ be a Hilbert space and $A \in \mathcal{L}(H)$ such that :
$$\langle A(u),u \rangle \geq \langle u, u \rangle \; \forall u \in H$$
Show that $A$ is invertible.
Attempt/Thoughts :
The inequal... | If $Au = Av$ then $$\|u-v\| \le \|Au - Av\| = 0$$ so that $u=v$ too.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3146658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
} |
How to find limit of sum $\lim\limits_{n\to\infty}\sum_{k=1}^{100n}\frac{k^p}{n^{p+1}}$ How can I find this limit? I've tried to use Stolz theorem, but have not succeed. I have heard smth about Riemann sums, but have not found good algorithm how to use it. Can you help me to solve it with the help of riemann sums or sh... |
I thought it might be instructive to present an approach that does not use Riemann sums. To that end, we proceed.
Note that
$$\sum_{k=1}^N \underbrace{\left(k^{p+1}-(k-1)^{p+1}\right)}_{=(p+1)k^p+O(k^{p-1})}=N^{p+1}$$
which by induction reveals that
$$\sum_{k=1}^N k^p=\frac{N^{p+1}}{p+1}+O(N^p)\tag1$$
Hence, we ha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3146805",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Are all values of $x$ solutions for $e^{2\ln(\sin(x))} = 1 - e^{2\ln(\cos(x))}$ in $\mathbb R$?
Does all values of $x$ in $\mathbb R$ satisfy equation: $$e^{2\ln(\sin(x))} = 1 - e^{2\ln(\cos(x))}$$
I am asking this, because by checking WolframAlpha solution there is an answer: (all values for $x$ are solutions over r... | Note that :
$$e^{2\ln \sin x} + e^{2 \ln \cos x} = 1 \Rightarrow e^{\ln (\sin x)^2} + e^{\ln (\cos x)^2} = 1 \Leftrightarrow \sin^2x + \cos^2x = 1 \rightarrow \text{true} \; \forall x \in \mathbb R$$
Restrictions apply so as the initial expression holds, so that narrows down the solution set. Note the usage of $\Righta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3146923",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Range of radical functions. Suppose we have $f(x)=\sqrt{x-1}+\sqrt{5-x}$; how do we find range for this function? Single radicals are easy , but two of them are in this particular function.I have the domain of the function and I can only think of differentiation to get the maximum of the function in the valid domain t... | By the AM-GM inequality, $$\sqrt{2+t} \sqrt{2-t} \le \frac {(2+t)+(2-t)} 2 = 2,$$
so $$(\sqrt{2+t}+\sqrt{2-t})^2 = (2+t)+(2-t)+2\sqrt{2+t}\sqrt{2-t}=4+2\sqrt{2+t}\sqrt{2-t}\le 8$$
so
$$\sqrt{2+t}+\sqrt{2-t}\le\sqrt{8}.$$
Now let $x=t+3.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3147086",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Let $x_1$ and $x_2$ be independent uniform variables from [0, 2]. What is the probability that $|x_1-x_2| \leq 1$? What I have so far for the solution
Since they are both continuous uniform variables. And because they are independent, we can say that
$$f(x_1, x_2)=\frac{1}{4}$$
$$P(|x_1-x_2| \leq 1) = P(x_1-1 \leq x_2 ... | In addition to the condition $x_1-1 \leq x_2 \leq x_1+1$ you have to remember that $x_2$ has to lie between $0$ and $2$. For example, if $x_1 <1$ then $x_1-1 <0$ so the integral w.r.t. $x_2$ cannot start from the negative number $x_1-1$.
If $x_1 <1$ then the integral w.r.t. $x_2$ starts from $0$ and if $x_1 >1$ then t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3147369",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Matrices Inequality Proof Recently, I read a paper and there is a step which turns out not obvious to me. The statement is as follows:
All matrices here are real matrices. $F$ is an arbitrary square matrix. $\Psi$ is a symmetric positive definite matrix. Let $$\lambda_{\max}(A)\equiv\text{The maximum eigenvalue of symm... | Let $A=\Psi^{-1/2}(I-F)^T\Psi(I-F)\Psi^{-1/2}$ and $y=\Psi^{1/2}x$. Then $\Psi^{-1}(I-F)^T\Psi(I-F)=\Psi^{-1/2}A\Psi^{1/2}$ is similar to $A$ and hence the inequality in question can be rewritten as
$$
y^TAy\le\lambda_\max(A)y^Ty.
$$
Now the inequality holds because $A$ is positive semidefinite.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3147491",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Two forms related by an automorphism are in the same cohomology class? Let $f: M \to M$ define an automorphism on the smooth manifold M.
Given a differential form $\omega \in \Omega^k$ is it true that the de Rham cohomology class of $\omega$ and $f^*\omega$ are the same? That is, does $[\omega]=[f^*\omega]$.
| No. One example: take the torus $X = \mathbb{R}^2/\mathbb{Z}^2$. The flip-flop on the factors interchanges the closed forms $dx$ and $dy$ which are linearly independent in $H^1(X)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3147645",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Formula (how to calculate) Y axis cross-point of two intersecting lines i.e. I have two lines:
A) Orange (Y axis starts at: 6, end at: -3)
B) Green (Y axis starts at: 5, end at: -2)
the start/end X axis values are same. Please note, I don't have SLOPE(angle) information, we only know what I've mentioned.
How to cal... | I've also successfully used this formula:
xCoef = (orangeStartY - greenStartY)/(greenEndY-greenStartY-orangeEndY+orangeStartY)
CrossPointY = xCoef * (greenEndY-greenStartY) + greenStartY
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3147734",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Ratio of $\frac{\zeta(2n)}{\zeta(n)}$ from infinite product involving primes Given that
$$\zeta(n)=\sum_{k=1}^\infty \frac{1}{k^n}=\prod_{k=1}^\infty \frac{1}{1-\frac{1}{(p_k)^n}}\tag{1}$$
where $n>1$ and $p_k$ is the $k^{th}$ prime.
Proof of the Euler product formula for the Riemann zeta function
It immediately follow... | Yes, for all $s\in \Bbb C$ with $\Re(s)>1$ we have
$$
\frac{\zeta(2s)}{\zeta(s)}=\sum_{n=1}^{\infty}\lambda(n)n^{-s}.
$$
Here $\lambda(n)$ is the Liouville function, defined by $\lambda(1)=1$ and
$$
\lambda(n)=\lambda(p_1^{e_1}\cdots p_r^{e_r})=(-1)^{\sum_{i=1}^re_i}.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3147884",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Showing that the unit sphere is a surface I was going through Andrew Pressley's book and on the place where they have discussed surfaces, there is one example which deals with the unit sphere. I have understood to the point where they have taken the surface patch and the fact that it will not be able to cover the whole... | Lots of ways to do this. One way would be to consider the function $f: \mathbb R^3\rightarrow\mathbb R$ defined by $x \mapsto ||x||^2 $. Check that $1$ is a regular value of this smooth map and $S^2=f^{-1}(1)$ (Use Implicit Function Theorem). The other way would be to look at the stereographic projection $S^2-N\rightar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3148035",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is it true that $(a^2-ab+b^2)(c^2-cd+d^2)=h^2-hk+k^2$ for some coprime $h$ and $k$?
Let us consider two numbers of the form $a^2 - ab + b^2$ and $c^2 - cd + d^2$ which are not both divisible by $3$ and such that $(a, b) = 1$ and $(c,d) = 1$. Running some computations it seems that the product $$(a^2 -ab + b^2)(c^2 - c... | There is this Identity:
$[(ac+bd)^2-(ab(c^2+d^2)-(abcd)+cd(a^2+b^2))+(bc-ad)^2]=(a^2-ab+b^2)(c^2-cd-d^2)$
Hence for:
$(a^2-ab+b^2)(c^2-cd-d^2)=(h^2-hk+k^2)$
$h=(ac+bd)$
$k=(bc-ad)$
$hk=(ac+bd)(bc-ad)$
Condition (c,d)=(2b,b-2a)
For $(a,b,c,d)=(3,7,14,1)$ we get:
$(49^2-49*95+95^2)=(37)*(183)=6771$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/3148152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 1
} |
Vector Field Exponential Map I've got ${\bf v} = x^2\partial_x$, and I'm trying to find $\exp(\varepsilon{\bf v})$, but I'm having some trouble.
If I define ${\bf v}^{n+1} = {\bf v}{\bf v}^n$ then I get a different outcome to ${\bf v}^{n+1} = {\bf v}^n{\bf v}$.
For example:
$${\bf v}^2 = {\bf vv} = (x^2\partial_x)(x^2\... | You are looking at the simplest Lie advective flow in perturbation theory (as applied in physics: QFT) and the workhorse example in the 19th century book of Georg Sheffer cited.
v generates a shift operator, and it pays to define suitable canonical coordinates,
$$
y=-1/x, \qquad \Longrightarrow \qquad x^2 \partial_x=\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/3148273",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
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