Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Number of solutions to $x^2\equiv b \mod p^n$ For an odd prime $p$, and some integer $b,n$. I'm interested in finding the number of solutions to $$x^2 \equiv b \mod p^n$$
Researching this led me into Hensel's lemma but I want to verify I understood correctly.
By Hensel's lemma, a solution $x_i$ to $f(x)\equiv 0 \text{ ... | Since $p$ is odd and $p\nmid b$, the proof given by @André Nicolas can be easily formalized as follows : the cyclic group $W_n := (Z/p^nZ)^* $ admits exactly one cyclic subgroup of order $(p-1)$ and one of order $p^{n-1}$, and since these orders are co-prime, $W_n \cong F_p^* \times C_{p^{n-1}}$. Because $p$ is odd, $C... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769291",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Prove that $\sum_{k=1}^n \frac{1}{n+k} = \sum_{k=1}^{2n} \frac{1}{k}(-1)^{k-1}$ using induction I'm trying to prove (using induction) that:
$$\sum_{k=1}^n \frac{1}{n+k} = \sum_{k=1}^{2n} \frac{1}{k}(-1)^{k-1}.$$
I have found problems when I tried to establish an induction hypothesis and solving this because I've learne... | This is kind of a cute induction proof. The key in this case will be a careful rearrangement of the terms. Let's get started. I'll assume the base case has been checked, so let's move on to the induction step. That is, we assume then that
$$
\sum_{k=1}^{2n} (-1)^{k-1}\frac{1}{k} = \sum_{k=1}^n \frac{1}{n+k}.
$$
It seem... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769421",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Find $\lim_{n \to \infty} \left( \frac{3^{3n}(n!)^3}{(3n)!}\right)^{1/n}$ Find $$\lim_{n \to \infty} \left( \frac{3^{3n}(n!)^3}{(3n)!}\right)^{1/n}$$
I don't know what method to use, if we divide numerator and denominator with $3^{3n}$, I don't see that we win something. I can't find two sequences, to use than the squ... | Use equivalents and Stirling's formula: $\;n!\sim_\infty \sqrt{2\pi n}\Bigl(\dfrac n{\mathrm e}\Bigr)^{\!n}$:
$$\biggl(\!\frac{3^{3n}(n!)^3}{(3n)!}\!\biggr)^{1/n}\!\sim_\infty\left(\frac{3^{3n}\sqrt{(2\pi n)^3\strut}\Bigl(\dfrac{n}{\mathrm e}\Bigr)^{\!3n}}{\sqrt{6\pi n}\Bigl(\dfrac{3n}{\mathrm e}\Bigr)^{\!3n}}\right)^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769570",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 2
} |
Prove that if $x$ is conjugate to $x^{-1}$ and $y$, that $y$ is conjugate to $y^{-1}$ The full question: Prove the following:
a) If $x$ is conjugate to $x^{-1}$ and $y$, then $y$ is conjugate to $y^{-1}$
b) If $x$ is conjugate to $x^{-1}$ in a finite group, $G$, and $x \neq x^{-1}$, then the conjugacy class of $x$ has ... | From $h^{-1}yh=x=g^{-1}x^{-1}g$ we obtain by inverting $h^{-1}y^{-1}h=g^{-1}xg=g^{-1}(h^{-1}yh)g$, which finally gives
$$
y^{-1}=hg^{-1}h^{-1}yhgh^{-1}=(hgh^{-1})^{-1}y(hgh^{-1}).
$$
Part c) has been shown already in this MSE quation.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769683",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Fourier decomposition of solutions of the wave equation with respect to the spatial variable Say I have a wave equation of the form $$\nabla^{2}f(t,\mathbf{x})=\frac{1}{v^{2}}\frac{\partial^{2}f(t,\mathbf{x})}{\partial t^{2}}$$ which is clearly a partial differential equation (PDE) in $\mathbf{x}$ and $t$.
Is it valid... | When considering a function $f(t,\mathbf x)$ on $[0,\infty)\times \mathbb{R}^n$, we can focus on one time slice at a time, fixing $t$ and dealing with a function of $\mathbf x$ only. The Fourier transform can be applied to this slice, since it's just a function on $\mathbb{R}^n$. The result can be denoted $\tilde f(t,\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1769930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove algebraically that, if $x^2 \leq x$ then $0 \leq x \leq 1$ It's easy to just look at the graphs and see that $0 \leq x \leq 1$ satisfies $x^2 \leq x$, but how do I prove it using only the axioms from inequalities? (I mean: trichotomy and given two positive numbers, their sum and product is also positive).
| $x \geq x^2$
If $x = x^2$,
$x - x^2$ = $0$ (Subtract x^2 from both sides)
$x^2 - x$ = $0$
$x(−x+1$) = $0$ (Factor left side of equation)
$x = 0$ or $−x+1 = 0$ (Set factors equal to 0)
$x = 0$ or $x = 1$
Check intervals in between critical points. (Test values in the intervals to see if they work.)
$x ≤ 0$ (Doesn't work... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1770047",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 6,
"answer_id": 5
} |
Counter example for uniqueness of second order differential equation I have a second order differential equation,
\begin{eqnarray}
\dfrac{d^2 y}{d x^2} = H\left(x\right) \hspace{0.05ex}y \label{*}\tag{*}
\end{eqnarray}
where, $\,H\left(x\right) = \dfrac{\mathop{\rm sech}\nolimits\left(x\right) \mathop{\rm sech}\nolimi... | Consider the constant coefficient first order linear dynamical system of dimension $n$ \begin{equation}{\bf{x}}'=A{\bf{x}},\end{equation} where $A$ is an $n\times n$ constant matrix and ${\bf{x}}$ is an $n$-dimensional vector. If $A$ has $k$ eigenvalues with positive real parts, then system above has a $k$ dimensional... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1770171",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 1,
"answer_id": 0
} |
If we have a real line and $X$ is any subset of it, let $Y$ be a set such that $X\subseteq Y \subseteq \bar{X}$ If we have a real line and $X$ is any subset of it, let $Y$ be a set such that $X\subseteq Y \subseteq \bar{X}$. Prove that $\bar{X}=\bar{Y}$
My Attempt
We can use definition and show that $\bar{X} $ is in $\... | Another answer, more general (this property is true in every topological space) :
$\overline{X}$ is the smallest (for the inclusion) closed set that contain $X$
$Y$ is contained in $\overline{X}$ that is closed, so $\overline{Y}\subset \overline{X}$ because $\overline{Y}$ is the smallest closed set that contain $Y$ (he... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1770292",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proving no vector potential for gravitation field defined on all of $\mathbb{R}^3 -$ origin Let:
$$F=\frac{x,y,z}{(x^2+y^2+z^2)^{3/2}}$$ Show that there is no vector potential for F which is defined on all of $\mathbb{R}^3 - \text{origin}$
I can find a vector potential which is not well-behaved on the z-axis quite easi... | Hint: If there were a vector field $G$ defined in the complement of the origin such that $F = \nabla \times G$, then Stokes' theorem would imply
$$
\iint_{S} F \cdot n\, dS = 0
$$
because the sphere has empty boundary. On the other hand, you can calculate
$$
\iint_{S} F \cdot n\, dS
$$
explicitly (without calculus, eve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1770381",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Finding the volume of the region bounded by $z=\sqrt{\frac{x^2}{4}+y^2}$and $x+4z=a$. Cylindrical coordinates. I would like the answer to preferably be done using either using a surface integral, or an integral with substitutions. But anything other than this is alright, if nothing else exists.
I have to find the volum... | $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \ov... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1770498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
} |
Bijection from $\mathbb {Z}^3$ to $\mathbb {Z}$ I am not a mathematician. Let $\mathbb {Z}$ be a positive integer set. I need to know whether there exist a bijection from $\mathbb {Z}^3$ to $\mathbb {Z}$, what might be a possible mapping?
I know that bijection exists from $\mathbb {R}^3$ to $\mathbb {R}$.
| You have an injective map $Z\rightarrow N$ defined by $f(n)=2^n, n>0$ and $f(n)=3^{-n}, n\leq 0$, this induces an injective map $g:Z^3\rightarrow N$ defined by $g(a,b,c)=2^{f(a)}3^{f(b)}5^{f(c)}$. The image of $g$ is in bijection with $N$, now take a bijection between $N$ and $Z$ and compose with $g$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1770640",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Solving Trig Equation with Unknown Inside and Outside of Function In my physics course, we're covering physical pendulums, and we are to essentially analyze the range of angles within the interval $\left[0, \frac{\pi}{6}\right]$ to show that $\sin\theta \approx \theta$. (I completed my analysis using Desmos.)
After cre... | Likely, your instructor wants you to find the approximate value where the error is 1%. Your equation is transcendental and won't be solved algebraically.
You could use a numerical method such as Newton's Method to approximate the roots to desired accuracy. That presupposes your familiarity with calculus (or at least w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1770780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How is the null space related to singular value decomposition? It is said that a matrix's null space can be derived from QR or SVD. I tried an example:
$$A= \begin{bmatrix}
1&3\\
1&2\\
1&-1\\
2&1\\
\end{bmatrix}
$$
I'm convinced that QR (more precisely, the last two columns of Q) gives the null space:
$$Q= \begin{bm... | For an $m \times n$ matrix, where $m >= n$,
the "full" SVD is given by
$$
A = U\Sigma V^t
$$
where $U$ is an $m \times m$ matrix, $\Sigma$ is an
$m \times n$ matrix and $V$ is an $n \times n$ matrix.
You have calculated the "economical" version of the SVD
where $U$ is an $m \times n$ and $S$ is $n \times n$.
Thus, you ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1771013",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "29",
"answer_count": 1,
"answer_id": 0
} |
What does it mean the notation $\int{R\left( \cos{x}, \sin{x} \right)\mathrm{d}x} $ Sometimes I find this notation and I get confused:
$$\int{R\left( \cos{x}, \sin{x} \right)\mathrm{d}x} $$
Does it mean a rational function or taking rational operations between $\cos{x}$ and $\sin{x}$ ?
Can you explain please?
Update: ... | Here $R$ is a function of two variables $s$ and $t$. For instance, if $$R(s,t) = \frac{s}{1+t}$$ then $$R(\cos x , \sin x) = \frac{\cos x}{1 + \sin x}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1771125",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
bijective function from [a,b] to [c,d] Im trying to think about bijective function from the closed interval [a,b] to the closed interval [c,d]. When $a,b,c,d \in \mathbb{R}$ and $a < b,\;c < d$.
Is there such a function?
| The idea is to construct a line whose domain is $[a, b]$ and whose range is $[c, d]$
Hence, two points on the line will be
$$
p_0 = (a, c) \\
p_1 = (b, d)
$$
Since we want $a \to c$, $b \to d$, and a straight line between them.
The slope of such a line will be
$$ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{d - c}{b - a}$$
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1771247",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
if $p\mid a$ and $p\mid b$ then $p\mid \gcd(a,b)$
I would like to prove the following property :
$$\forall (p,a,b)\in\mathbb{Z}^{3} \quad p\mid a \mbox{ and } p\mid b \implies p\mid \gcd(a,b)$$
Knowing that :
Definition
Given two natural numbers $a$ and $b$, not both zero, their greatest common divisor is the larges... | It depends on what definition of greatest common divisor you use. You probably use the second one.
Definition 1
Given natural numbers $a$ and $b$, the natural number $d$ is their greatest common divisor if
*
*$d\mid a$ and $d\mid b$
*for all $c$, if $c\mid a$ and $c\mid b$, then $c\mid d$
Theorem. The ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1771373",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 0
} |
Hyperbola equation proof I've been trying to prove the canonical form of the hyperbola by myself.
$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $
I started from the statement that
A hyperbola may be defined as the curve of intersection between a right circular conical surface and a plane that cuts through both halves of the ... | If $a \neq 0,$ the cutting plane is parallel to the $x$ axis but not parallel to the $y$ axis. Starting at the point $(x,y,z) = (0,0,b)$ on this plane,
and traveling along the line of intersection of the cutting plane and the $y,z$ plane (that is, the line that simultaneously satisfies $z = ay+b$ and $x = 0$),
we can m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1771498",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
$T$ can be $\infty$ with positive probability From Williams' Probability with Martingales
How exactly do we know $T$ can be $\infty$ with positive probability
or
$$P(T = \infty) > 0 \text{ ?}$$
I'm guessing that that means there is a positive probability that for some $c$, the sum, $X_1 + \cdots + X_r$, will never... | Indicating the dependence of $T$ on $c$ explicitly, you have
$$\{T_c=\infty\}=\{\sup_n|M_n|\le c\}$$
Therefore,
$$\bigcup_{c>0,c\in\Bbb Q}\{T_c=\infty\}=\{\sup_n|M_n|<\infty\}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1771606",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Diophantine Equation $x^2+y^2+z^2=c$
$x^2+y^2+z^2=c$
Find the smallest integer $c$ that gives this equation one solution in
natural numbers.
Find the smallest integer $c$ that gives this equation two distinct
solutions in natural numbers.
Find the smallest integer $c$ that gives this equation three distinct
solu... | Since the equation is symmetric: if $(x,y,z)$ is solution than every permutation of$(x,y,z)$ is the solution too. So it's impossible for this equation to have 2 solutions (because if for example $x\neq y$ than we have at least three distinct permutations).
The smallest $c$ for three distinct solutions is $6$: $(2,1,1),... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1771724",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
Projectivised tangent bundle of 2 sphere I'm trying to understand how rotations act on the "projectivised" tangent bundle of the sphere.
Let $S^2$ be the two sphere and denote by $P(TS^2)$ the tangent bundle where each tangent space $T_xS^2$ is taken to be a projective vector space. I'm trying to show that given any $... | You can visualize this action very explicitly: a tangent vector to a point $p \in S^2$ is literally a little vector tangent to $S^2$ inside of $\mathbb{R}^3$, and rotation acts in the obvious way. The rotations around the axis through $p$ act transitively on unit tangent vectors at $p$ (and so act transitively on the p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1771842",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
If it has an emergency locator, what is the probability that it will be discovered? Okay so here's the question
Seventy percent of the light aircraft that disappear while in flight
in a certain country are subsequently discovered. Of the aircraft that
are discovered, 60% have an emergency locator, whereas 90% of ... | $P(D \mid E)$
$= P(D \cap E)/P(E)$
$= P(D \cap E)/P((E \cap D) \cup (E \cap \overline{D}))$
$= P(D \cap E)/\{P(E \cap D) + P(E \cap \overline{D})\}$
$= \{P(D) \cdot P(E \mid D)\}/\{(P(D) \cdot P(E \mid D)) + P(\overline{D}) \cdot P(E \mid \overline{D})\}$
$= (0.70 \cdot 0.60)/((0.70 \cdot 0.60) + (0.30 \cdot 0.10))$
$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1771920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Still not clear why longest path problem is NP-hard but shortest path is not? I've heard/read many times that shortest path problem is P, but longest path problem is NP-hard. But I have a problem with this: we say longest path problem is NP-hard because of graphs with positive cycles. But if you think about graphs with... | The longest path problem is commonly understood as follows: given a graph, find the longest simple path. Simple means that no vertex is visited more than once. Only in graphs with cycles can a vertex be visited more than once. The shortest path problem, however, is commonly defined for simple paths in acyclic graphs.
T... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772053",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Why is the domain of these two functions different? I tried graphing the functions $y = x^{.42}$ and $y = (x^{42})^{\frac{1}{100}}$. From my understanding of the laws of exponentiation these two expressions should be equivalent. What I found out after graphing these two functions was that they were equivalent for all n... | For $b > 0$ then $b^{n/m}=\sqrt [m]{b^n} $ is well defined whereas the same definition for $b < 0$ is not. If $b < 0$ for example, then $b^{3/2} = \sqrt {b^3} $ is not defined as $b^3$ is negative. But $3/2 = 6/4$ makes $b^6$ positive so a 4th root is possible.
So note: $.42=42/100 =21/50$ but $42/100$ is not in lo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772133",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
Can the sum be simplified? $\sum_\limits{x=y}^{\infty} {x \choose y}\left(\frac{1}{3}\right)^{x+1}$ Let:
$$f(y) = \sum_{x=y}^{\infty} {x \choose y} \left(\frac{1}{3}\right)^{x+1}$$
Can this be simplified somehow? Is it a standard probability distribution?
I can only get as far as:
$$f(y) = \frac{1}{3} \sum_{x=y}^{\inft... | $$f(y)=\sum_{x=y}^∞ \binom xy(\tfrac 13 )^{x+1} = {\tfrac 12}^{1+x}$$
This arose in the context of a probability problem in which X is a geometrically distributed random variable (p=1/3) and Y is a binomially distributed random variable (p=1/2, n=x)
In effect you have a sequence of trials each with three equally prob... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772229",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
DTFT of the unit step function If i apply the DTFT on unit step function, then i get follow:
$$DTFT\{u[n]\}=\sum_{n=-\infty}^{\infty}u[n]e^{-j\omega n}=\sum_{n=0}^{\infty}e^{-j\omega n} = \frac{1}{1-e^{-j\omega}}$$.
Now i have the problem, if $|e^{-j\omega}|$ = 1, the sum diverges.To handle this case, i know that $e^{-... | Hi guys I think this way is better than others,What's Ur idea?
$u[n]=f[n] + g[n] $
Where:
$f[n]= {1\over2}$ for $-\infty<n<\infty $
and
$g[n]=\left\{
\begin{array}{c}
{1\over2} \text{ for } n\ge 0 \\
{-1\over2} \text{ for } n<0
\end{array}
\right.$
do:
$ \delta [n] = g[n] - g[n-1]$
U Know DTFT of $\delta[n]$ is $1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772285",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Prove $\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$ I've been given a proof that shows the following:
If $f:[a,b]\to \mathbb C$ is a continuous function and $f(t)=u(t)+iv(t)$ then $$\left| \int_a^b f(t) dt \right| \leq \int_a^b \left| f(t) \right| dt$$ The proof begins by letting $\theta$ be th... | If $A = \int_a^b f(t)dt$, notice that, since $\theta$ is the principal argument of $A$,
$$A = |A| e^{i \theta}$$
Which gives:
$$|A| = Ae^{-i\theta} = \left(\int_a^b f(t)dt \right) e^{-i\theta} = \int_a^b e^{-i\theta} f(t)dt$$
Hence $\int_a^b e^{-i\theta} f(t) dt = |A| \in \Bbb R$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772406",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
$\bigcup X$ finite implies $\mathcal P(X)$ is finite. Can anyone help with this past paper question from a Set Theory exam.
Prove that, for all sets $X$, $\bigcup X$ finite implies $\mathcal P(X)$ finite.
I am using the Kuratowski definition of finiteness, ie A is finite if every Kuratowski inductive set for A contai... | The key facts are:
*
*If $B$ is finite and $A\subseteq B$, then $A$ is finite; $\quad(1)$
*If $A$ is finite, then $\mathcal{P}A$ is finite. $\quad(2)$
Notice that $A\subseteq\mathcal{P}(\bigcup A)$. If this is not immediately obvious: fix $a\in A$, then every $\gamma\in a$ must also be in $\bigcup A$ by definitio... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772537",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 2
} |
How to write the real projective plane as a pushout of a disk and the mobius strip? I heard in topology class that the real projective plane is obtained by gluing a disk along the boundary of the mobius strip. I was wondering - how can I write this as a pushout?
Also, how can I write a mobius strip itself as a pushout?... | We have two copies of $S^1$ here: the $S^1$ that is the boundary of $D^2$, and the $S^1$ that forms the boundary of the Möbius band.
$$\require{AMScd}
\begin{CD}S^1 @>>> D^2 \\
@VVV @VVV \\
M @>>> \mathbb{RP^2}\end{CD}$$
Actually showing that these are the same might take a little more work, depending on how familiar ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772630",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Equations of Custom Curves How would one find a equation or a function to describe a custom curve in a coordinate system?
Like for example, when someone presents you with a graph of a repeating curve, how would one find its equation to describe it?
Here is my sketch of the example:
What would be the equation to descr... | There's no general process for finding a parametrisation of a given curve. It is a question of having some experience and then trying things until you find something that looks good.
If we look at one copy of your curve, we can se that $y$ increases, then decreases about the double of the increase and then increases to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772776",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Third Order Differential Equations I am having trouble solving the third order differential equation
$y'''+y'=0$
It was given to me in a quiz (which I got wrong) with boundary conditions
$y(0) = 0$
$y'(0)=2$
$y(\pi)=6$
I know that the obvious trial solution is $y=Ae^{rx}$ but I kind of get stuck after here. I have see... | Let's see:
If $y=e^{rx}$, then the Chain Rule tells us that $y' = re^{rx}$ and $y''' = r^3e^{rx}$.
If $y$ is a solution, that means $y'''+y'=0$.
In other words, $r^3e^{rx} + re^{rx}=0$
If you factor, you find that $(r^3+r)e^{rx}=0$. This happens when the (characteristic) polynomial $p(r) = r^3+r = r(r^2+1)$ has roots: ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1772908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Minimizing quadratic objective function on the unit $\ell_1$ sphere I would like to solve the following optimization problem using a quadratic programming solver
$$\begin{array}{ll} \text{minimize} & \dfrac{1}{2} x^T Q x + f^T x\\ \text{subject to} & \displaystyle\sum_{i=1}^{n} |x| = 1\end{array}$$
How can I re-write t... | In one of the other answers (unfortunately now deleted) it is incorrectly assumed that we can apply a standard variable splitting technique, without worrying that both the positive and negative parts can become nonzero. I posted a comment already that this probably needs some additional binary variables to fix this. L... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1773038",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Can geometric programs be solved more efficiently than general convex optimization problems? I want to solve an optimization problem for which I have already proven that it is feasible and convex. Introducing further variables and considering a special case of the problem, I can formulate it as geomtric programming pro... | Depends on your alternative; I'm one of the gpkit developers, and in comparisons we've run GPs solve much faster than naive gradient descent (it's worth noting that not all GPs are convex without the transformation).
However, if your problem is can be solved by another convex solver (e.g. it's also a valid LP) then th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1773155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do I calculate the number of trials required to verify that a failed intermittent test is fixed? Say I've got a software test that fails randomly one out of ten times. I make a change to the code which I hope will fix it. I know ten trials is not sufficient to verify the fix. How many trials do I need so I can be X... | Let $ 0 < p < 1$ be the probability that the test fails and $X$ the random variable that represents the number of tests needed to get a (first) fail.
Suppose you tried $N$ tests and got no fails. The probability that you get this result under the hypothesis
$$ H_0 : \text{the software still has a problem} $$
is
$$\begi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1773246",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Range of a P-name I am working on a set theory problem from Kunen's Set Theory book, and it involves knowing $\text{ran}(\tau)$ where $\tau$ is a $\mathbb{P}$-name. The entire section loves to talk about the domain of things like $\tau$, but not the range. Is
$$\text{ran}(\tau) = \{p \in \mathbb{P} : \exists \sigma ... | The name $\tau\in M^\mathbb{P}$ is an element of the set $M$, and in particular, it is a relation. So you can compute its range
$$
\mathrm{ran}(\tau) = \{p\in \mathbb{P} : \exists \sigma\,(\langle\sigma,p\rangle\in\tau)\}.
$$
But in this particular problem, $\tau$ is the name of a function and you have the expression
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1773346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Show a set is open using open balls The set is $ \{ (x_1 , x_2) : x_1 + x_2 > 0 \}$
I wanted to solve this using open balls, so I said let $y = (y_1, y_2)$ be in the stated set. Then create an open ball $ B_r (y)$ around this point with radius $r = \frac {y_1 + y_2}{2} $. (My professor suggested this radius, but I u... | Actually there is a general result:
If function $f:\mathbb{R}^n\to \mathbb{R}$ is continuous on $\mathbb{R}^n,$ then for every real number $a,$ the set $\{x\in\mathbb{R}^n\mid f(x)>a\}$ is open in $\mathbb{R}^n$ (with respect to the natural Euclidean topology).
The proof of this result is elementary. For convenience, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1773486",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
How can I show that there cannot exist a homotopy from $\mathcal{C_1}$ to $\mathcal{C_2}$? Consider the diagram below with an annulus $\mathcal{A}$ and two circles in the annulus $\mathcal{C}_1$ and $\mathcal{C}_2$.
In $\mathbb{R^n}$, there clearly exists a homotopy between any two circles. But if we have this scenari... | I think you mean "homotopy" instead of "homeomorphism".
If that's the case, then you can use the fact that "homotopic" is an equivalence relation, in particular it is transitive.
The circle $\mathcal{C}_2$ is obviously homotopic to the trivial circle. So if you prove that $\mathcal{C}_1$ is not homotopic to the trivial... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1773572",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
In a loop, if $(xy)x^r = y$ then $x(yx^r)=y$ Consider a loop $L$, that is, a quasigroup with an identity, and recall that a quasigroup $L$ is a set together with a binary operation such that, for every $a$ and $b$ in $L$, the equations $ax=b$ and $ya=b$ have unique solutions $x$ and $y$ in $L$. Further denote $x^r$ the... | Substituting $yx^r$ for $y$ in the premise yields $
\left(x\left(yx^r\right)\right)x^r=yx^r$. The conclusion follows since the solution of $ux^r=b$ is unique.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1773698",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Radius of convergence of two power series I am trying to find the radius of convergence and trying to figure out the behaviour on the frontier of the disk of convergence of the following power series:
a) $\sum_{n=1}^{\infty} \dfrac{n!}{(2-i)n^2}z^n$
b) $\sum_{n=1}^{\infty} \dfrac{1}{1+(1+i)^n}z^n$
I know that the radiu... | One would rather use the ratio test.
For a), one obtains, as $n \to \infty$,
$$
\left|\dfrac{(n+1)!}{(2-i)(n+1)^2}\times \dfrac{(2-i)n^2}{n!}\right|=\frac{n^2}{(n+1)} \to \infty
$$ thus $R=0$.
For b), one obtains, as $n \to \infty$,
$$
\left|\dfrac{1}{1+(1+i)^{(n+1)}}\times \dfrac{1+(1+i)^n}{1}\right|=\left|\dfrac{1}{1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1773765",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Is the general equation for a straight line not considered a linear function in linear algebra? Is the general equation for a straight line, which we called a linear function in highschool, i.e.
$$f(x)=mx+c \tag{1}$$
not considered to be a linear function according to the linear algebra definition that it need to satis... | You are correct, the term linear is sligtly misused in high school, when affine functions from $\mathbb R$ to $\mathbb R$ are also called linear.
That said, two points that make this a little less annoying:
*
*"linear" in this term simply means "forming a line", and you can't argue with the fact that $f(x)=kx+n$ for... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1773857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Cardinality of equivalence relations on N I asked a similar question yesterday about well ordered sets, now I am having troubles with equivalence relations.
Could someone suggest an injection from a well known set of cardinality
$2^{\aleph_{0}}$ to the set of all equivalence relations of $\mathbb{N}$?
Many thanks in a... | For each subset $S$ of $\mathbb{N}$, declare all elements of $S$ to be equivalent, and all elements outside $S$ to only be equivalent to themselves. It's not quite an injection, but if you ignore singleton and empty $S$, then it is.
And because there are uncountably many subsets of $\mathbb{N}$, removing countably many... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1774003",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Problem in Linear Algebra about dimension of vector space Let U and V be finite dimensional vector spaces Over $\mathbb R$, Let $L(U,V)$ be the vector space of linear transformations from $U$ to $V$, and Let $W$ be a vector subspace of $U$.
If $Z$= {$T$ $\in L(U,V)$: $T(w)=0$ for all w $\in W$}, then What is the vec... | Hint: Choose a basis of $W$ and extend it to a basis of $U$.
Express $T$ in that basis as a matrix.
How many degrees of freedom are left?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1774075",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What is the relationship between the width, height, and radius of an arc? What is the relationship between the width ($w$), height ($h$), and radius ($r$) of an arc?
Specifically, the relationship in terms of $h$.
I know this is a simple question - I'm a hobbyist engineer, and I'm having one of those moments, where you... | Another take.
Let $(0,-r)$ and $(x, y)$ be the endpoints of the arc. Then
$h = |r - y|$ and $w = |x|$.
We know $x^2 + y^2 = r^2$ so $h = |r \pm \sqrt{r^2 - w^2}| = r \pm \sqrt{r^2 - w^2}$. And $w = |x| = \sqrt{r^2 - y^2} = \sqrt{r^2 - (h-r)^2} = \sqrt{2hr - h^2}$
=====
However not your interpretation of width and he... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1774177",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Evaluate the double integral $\iint_D\sqrt{4-x^2-y^2}$ bounded by semi-circle I would appreciate it if someone can help me solve this question, as I'm struggling to get its answer.
Q: Evaluate the double integral $$\iint_D\sqrt{4-x^2-y^2}dxdy$$ bounded by semi-circle $$x^2+y^2=2x$$ and lying in first quadrant
Thanks
| We have: $$x^2+y^2=2x$$
Moving everything to the left: $$x^2-2x+y^2=0$$
Completing the square: $$(x-1)^2+y^2=1$$
Which is a circle with centre $(1,0)$, hence: $$0\le x\le2$$
Solving $y$: $$y=\sqrt{1-(x-1)^2}$$
So the integral becomes: $$\int_{x=0}^{x=2}\int_{y=0}^{y=\sqrt{1-(x-1)^2}}\sqrt{4-x^2-y^2}\ \mathrm dy\ \mathr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1774331",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A union of n sets with diferent order, how can I prove? Let$k_1,k_2,...,k_n$ be any ordination of the indexes $1,2,...,n$then $\bigcup_{i=1}^{n}A_{k_i}=\bigcup_{i=1}^{n}A_i$
I tried the double contention technique with no good results, is there a handy theorem I can use?
| Let $x\in \bigcup_{i=1}^{n}A_{k_i}$. Then for some m, $x\in A_{k_m}$. But $ A_{k_m}$ is one of $A_1, \dots A_n$. So $x$ is in on of $A_1, \dots A_n. $ Hence $x \in\bigcup_{i=1}^{n}A_i$. We proved so far: $\bigcup_{i=1}^{n}A_{k_i}\subseteq\bigcup_{i=1}^{n}A_i$.
Similarly: $\bigcup_{i=1}^{n}A_i\subseteq \bigcup_{i=1}^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1774457",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$
Determine the coefficient of $x^{18}$ in $\left(x+\frac{1}{x}\right)^{50}$.
I know he Binomial Theorem will be useful here, but I am struggling to use it with any certainty.
| Hint The general term of the binomial expansion is $${50 \choose k} x^k \left(\frac{1}{x}\right)^{50 - k} = {50 \choose k} x^{2k - 50} .$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1774573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Why do authors make a point of $C^1$ functions being continuous? I've just got a little question on why authors specify things the way they do. Is their some subtlety I'm missing or are they just being pedantic?
I've encountered the function spaces $C^k[a,b]$ a few times this year and usually the author will make a po... | As Dave Renfro commented, this may be useful for pedagogical reasons, even if it's logically unnecessary. One difficulty that people often have is putting too much trust in formulas. Of course if $f'$ is to exist, $f$ must be continuous, but a formula for $f'$ might sometimes exist and be continuous without $f$ being... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1774648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Continuity of the Box-Cox transform at λ = 0: Why is it the log? The Box-Cox power transform frequently used in statistical analysis takes the value (x^λ -1) /λ for λ not equal to zero, and ln(x) for λ=0. I would like to see a demonstration, that need not be a full formal proof, that this family of transformations is ... | Since $\lim_{\lambda \rightarrow 0} x^\lambda = x^0 = 1$ we can use L'hopitals rule to get
$$
\lim_{\lambda \rightarrow 0} \frac{x^\lambda -1}{\lambda} = \lim_{\lambda \rightarrow 0} \frac{\ln x e^{\lambda \ln x}}{1} = \ln x e^0 = \ln x
$$
showing the continuity.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1774747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Continuous deformation of loop to point. Suppose I have a homotopy from a loop around the origin to a constant loop which is not the origin.
Prove that the origin is in the image of the homotopy.
Basically prove that if I deform a loop to a point, at some point in time it has to cross the origin.
I have tried to prove ... | If the origin is not in the image, then consider your original loop as a loop in space $\mathbf{R}^2 \backslash(0,0)$.
Since your deformation avoids the origin, it can be viewed, again, as a deformation in the "punctured" space that deforms the loop encompassing the origin to the constant map.
But the loop in the const... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1774919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove $(x_n)$ defined by $x_n= \frac{x_{n-1}}{2} + \frac{1}{x_{n-1}}$ converges when $x_0>1$ $x_n= \dfrac{x_{n-1}}{2} + \dfrac{1}{x_{n-1}}$
I know it converges to $\sqrt2$ and I do not want the answer. I just want a prod in the right direction.
I have tried the following and none have worked: $x_n-x_{n-1}$ and this g... | We have: By AM-GM inequality: $x_n > 2\sqrt{\dfrac{x_{n-1}}{2}\cdot \dfrac{1}{x_{n-1}}}=\sqrt{2}, \forall n \geq 1$. Thus: $x_n-x_{n-1} = \dfrac{1}{x_{n-1}} - \dfrac{x_{n-1}}{2}= \dfrac{2-x_{n-1}^2}{2x_{n-1}} < 0$. Hence $x_n$ is a decreasing sequence,and is bounded below by $\sqrt{2}$. So it converges, and you showed... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775041",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
How can you check if there exists a valid magic square with given initial conditions? For example, if I have a $4\times4$ magic square that looks like so:
\begin{pmatrix}
\hspace{0.1ex}2 & 3 & \cdot & \cdot\hspace{1ex} \\
\hspace{0.1ex}4 & \cdot & \cdot & \cdot\hspace{1ex} \\
\hspace{0.1ex}\cdot & \cdot & \cdot & \cdot... | In the present case, you can determine it by checking this list. There seems to be no such magic square.
In the general case, you can treat the emtpy squares as variables, introduce the $2n+2$ constraints and solve the corresponding system of linear equations. If you prescribe $k$ squares, this will leave $n^2-k-(2n+2)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Number of distinct values Question:
How many possible values of (a, b, c, d), with a, b, c, d real, are there
such that abc = d, bcd = a, cda = b and dab = c?
I tried multiplying all the four equations to get:
$$(abcd)^2 = 1$$
Not sure how to proceed on from here. Won't there be infinite values satisfying this equation... | 1) If $abcd = 0$, i.e., if one at least among $a,b,c,d$ is zero, all are clearly zero. Thus there is a solution:
$$a=b=c=d=0$$
2) If $abcd \neq 0$, let $A=\ln(|a|), B=\ln(|b|), C=\ln(|c]), D=\ln(|d|).$
Taking absolute values and then logarithms of the 4 equations, we obtain the following linear homogeneous system
$$\b... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775266",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Question involving functions and permutations If $A=\{1,2,3,4\},B=\{a,b,c\}$, how many functions $A\to B$ are not onto?
My Try: so one element in $B$ shouldn't have a preimage in A so one element is excluded(for convenience) so for $4$ elements in $A$ there are $2$ in B hence total ways are $16$ then $2$ elements in $... | You overcounted the "one element excluded" mappings
(assuming they were supposed to be "exactly one element excluded").
Yes, there are $16$ ways to map $A$ onto the two remaining elements of $B$
after one element of $B$ is excluded, but if you exclude $a$ (for example),
then two of the mappings you produce in this way ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
} |
Finding the Exponential of a Matrix that is not Diagonalizable Consider the $3 \times 3$ matrix
$$A =
\begin{pmatrix}
1 & 1 & 2 \\
0 & 1 & -4 \\
0 & 0 & 1
\end{pmatrix}.$$
I am trying to find $e^{At}$.
The only tool I have to find the exponential of a matrix is to diagonalize it. $A$'s eigenvalue is 1. Therefore, ... | this is my first answer on this site so if anyone can help to improve the quality of this answer, thanks in advance.
That said, let us get to business.
*
*Compute the Jordan form of this matrix, you can do it by hand or check this link. (or both).
*Now, we have the following case: $$ A = S J S^{-1}.$$ You will fin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775469",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 7,
"answer_id": 3
} |
Find the value of P such that the seris converge $$\sum_{n=2}^{\infty} \frac{1}{\log^p(n)} \tag{1.}$$
$$\sum_{n=2}^{\infty} e^{n(p+1)} \tag{2.}$$
In 1 if $p=0$ then the whole series is 1.
in 2 I can look at some individual results.
The question is what is the correct way to answer this.
| This can be shown by using Ratio Test.
For (1), $$a_n=\ln(n)^{-p}$$
The ratio is, $$\frac{a_{n+1}}{a_n}=(\frac{\ln(n)}{\ln(n+1)})^p$$
Thus for any $p\gt 0$, the ratio is less than 1 and the series absolutely converges.
For (2), $$a_n=e^{n(p+1)}$$
The ratio is, $$\frac{a_{n+1}}{a_n}=(\frac{e^{(n+1)}}{e^n})^{(p+1)}=e^{p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775557",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A weaker characterization of convergence in distribution $X_n$ and $X$ are real-valued random variables. If $X_n \to X$ in distribution, then we know that
$$P\{X_n \leq a\} \to P\{X \leq a\}$$
for every $a$ at which $x\mapsto P\{X \leq x\}$ is continuous. For a certain problem I have at hand this condition is too stron... | If a sequence $(d_k)$ from $D$ decreases to $x$, then $F_n(x)\le F_n(d_k)$, so $\limsup_nF_n(x)\le F(d_k)$ for all $k$, so $\limsup_nF_n(x)\le F(x)$. If $(d_k)\subset D$ increases to $x$, and $x$ is a continuity point of $F$, then in like fashion $\liminf_nF_n(x)\ge F(x)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775608",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to solve this algorithmic math olympiad problem? So, today we had a local contest in my state to find eligible people for the international math olympiad "IMO" ...
I was stuck with this very interesting algorithmic problem:
Let $n$ be a natural number ≥ 2, we take the biggest divisor of $n$ but it must be differen... | Firstly, note that: $$n=2^{775}3^{310}7^{155}$$
Let the number of steps to get from $x$ to $1$ be $f(x)$.
Then, note that the biggest divisor of $2x$ is always $x$.
Therefore: $$f(2x)=f(x)+1$$
For example: $$f(30)=f(15)+1$$
Applying to here: $$f(n)=f(3^{310}7^{155})+775$$
Now, when $x$ is not divisible by $2$, the bi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775719",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
"answer_count": 3,
"answer_id": 1
} |
Prove that the line integral of a vector valued function does not depend on the particular path Let C denote the path from $\alpha$ to $\beta$.
If $\textbf{F}$ is a gradient vector, that is, there exists a differentiable function $f$ such that
$$\nabla f=F,$$ then
\begin{eqnarray*}
\int_{C}\textbf{F}\; ds &=& \int_{\... | In general, for a smooth vector field $\vec F(\vec r)$, Helmholtz's Theorem guarantess that there exists a scalar field $\Phi(\vec r)$ and a vector field $\vec A(\vec r)$ such that
$$\vec F(\vec r)=\nabla \Phi(\vec r)+\nabla \times \vec A(\vec r)$$
Then, forming the line integral along a path $C$, from $\vec r_1$ to $\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Linear exponential sum over $(\mathbb{Z}/(p^t))^*$ Let $p$ be a prime and $t$ a natural number. Let us denote $(\mathbb{Z}/(p^t))^*$ to be the group of units of $\mathbb{Z}/(p^t)$.
I have the following exponential sum
$$
S = \sum_{w \in (\mathbb{Z}/(p^t))^*} e^{2 \pi i a w/ p^t},
$$
which I am convinced is $0$ if $a \... | Write that sum as $S(a,t)$. Here are some thoughts:
*
*If $\gcd(a,p)=1$, then $S(a,t)=S(1,t)$ because raising a primitive $p^t$-root of unity to $a$ still gives you a primitive $p^t$-root of unity.
*If $\gcd(a,p)>1$, then $S(a,t)=m S(a',t')$ for some $m\in \mathbb N$ and $a'<a$ with $\gcd(a',p)=1$ and $t'<t$.
So, t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1775936",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Can three vectors have dot product less than $0$?
Can three vectors in the $xy$ plane have $uv<0$ and $vw<0$ and $uw<0$?
If we take $u=(1,0)$ and $v=(-1,2)$ and $w=(-1,-2)$
$$uv=1\times(-1)=-1$$
$$uw=1\times(-1)+0\times(-2)=-1$$
$$vw=-1\times(-1)+2\times(-2)=-3$$
is there anyway to show that without examples, just wo... | The dot product between two vectors is negative when the angle $\theta$ between them is greater than a right angle, since
$$
u \cdot v = |u||v|\cos(\theta).
$$
So when three vectors point more or less to the vertices of an equilateral triangle all three dot products will be negative.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1776069",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Question on Indefinite Integration: $\int\frac{2x^{12}+5x^9}{\left(x^5+x^3+1\right)^3}\,\mathrm{d}x$ Give me some hints to start with this problem: $${\displaystyle\int}\dfrac{2x^{12}+5x^9}{\left(x^5+x^3+1\right)^3}\,\mathrm{d}x$$
| Hint
Take $x^5$ common from denominator so that it's cube it will come out as $15$ and take $x^{15}$ common from the numerator. Put the remaining denominator as $t$ numerator becomes $dt$. Hope you can do the rest.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1776317",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Calculation of total number of real ordered pairs $(x,y)$
Calculation of total number of real ordered pairs $(x,y)$
in $x^2-4x+2=\sin^2 y$ and $x^2+y^2\leq 3$
$\bf{My\; Try::}$ Given $x^2-4x+2=x^2-4x+4-2=\sin^2 y\Rightarrow (x-2)^2-2=\sin^2 y$
Now Using $0 \leq \sin^2 y\leq 1$. So we get $0\leq (x-2)^2-2\leq 1\Righta... |
Now How can I solve it after that
I don't know how to continue from that. So, let us take another approach.
Solving $x^2-4x+2-\sin^2 y=0$ for $x$ gives
$$x=2\pm\sqrt{2+\sin^2y}$$
Now $x=2+\sqrt{2+\sin^2y}$ does not satisfy $x^2\le 3$. So, we have $x=2-\sqrt{2+\sin^2y}$.
Now
$$\left(2-\sqrt{2+\sin^2y}\right)^2+y^2\le ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1776444",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How can I show the uniqueness of homomorphism? Let $R$ be a commutative ring and let $k(x)$ be a fixed polynomial in $R[x]$.
Prove that there exists a unique homomorphism $\varphi:R[x]\rightarrow R[x]$ such that
$\varphi(r)=r\;\mathrm{for\; all\;}r\in R\quad\mathrm{and}\qquad\varphi(x)=k(x)$
and I found such ring homom... | Let's say we have a homomorphism $\phi$ that satisfies $\phi(r)=r$ for $r \in R$ and $\phi(x)=k(x)$. We need to prove that this homomorphism equals your homomorphism above, which we can do simply by using the definition of homomorphisms and the values of $\phi$ that are given.
Consider $\phi(f(x))$:
$$\phi\left(\sum_{i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1776579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Are these sets Borel-measurable? Are the following sets Borel-measurable and if so, what is the value of the measure?
1) A = {(x,y) ∈ $[0,1]^2$| x and y rational}
2) B = {(x,y) ∈ $[0,1]^2$ | x or y rational}
3) C = {(x,y) ∈ $[0,1]^2$ | x and y irrational}
4) D = {(x,y) ∈ $[0,1]^2$ | x=y}
| Some hints to help you further in this.
1) Singletons are Borel-measurable and countable unions of Borel-measurable sets are Borel-measurable. Conclusion: countable sets (which are countable unions of singletons) are measurable. The Lebesgue measure of a singleton is $0$. What can you conclude then about the Lebesgue m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1776708",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Factor $6x^2 −7x−5=0$ I'm trying to factor
$$6x^2 −7x−5=0$$
but I have no clue about how to do it. I would be able to factor this:
$$x^2-14x+40=0$$
$$a+b=-14$$
$$ab=40$$
But $6x^2 −7x−5=0$ looks like it's not following the rules because of the coefficient of $x$. Any hints?
| When in doubt, you always have the quadratic equation. But if you really want to do it this way, you have to consider how the dominant term can factor. Here, you have either $6=6\times 1$ or $6=3\times 2$, so $(6x+a)(x+b)$ or $(3x+a)(2x+b)$. Try both. In both cases, you get the Viete-like formulas, just like you wrote ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1776775",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 10,
"answer_id": 5
} |
Curious inequality: $(1+a)(1+a+b)\geq\sqrt{27ab}$ I was recently trying to play with mean inequalities and Jensen inequality. My question is, if the following relation holds for any positive real numbers $a$ and $b$
$$(1+a)(1+a+b)\geq\sqrt{27ab}$$
and if it does, then how to prove it. By AM-GM inequality we could obtai... | Use AM-GM:
$$\frac{1}{2}+\frac{1}{2}+a\ge3\sqrt[3]{\frac{a}{4}}\\
\frac{1}{2}+\frac{1}{2}+a+\frac{b}{3}+\frac{b}{3}+\frac{b}{3}\ge6\sqrt[6]{\frac{ab^3}{108}}\\\therefore(1+a)(1+a+b)\ge\left(3\sqrt[3]{\frac{a}{4}}\right)\left(6\sqrt[6]{\frac{ab^3}{108}}\right)=\sqrt{27ab}$$
Equality holds iff $\frac{1}{2}=a=\frac{b}{3}$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1776857",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 3
} |
What is the sum of this alternating series: $\sum_{n=1}^\infty \frac{(-1)^n}{4^nn!}$? I need to find the sum of an alternating series correct to 4 decimal places. The series I am working with is: $$\sum_{n=1}^\infty \frac{(-1)^n}{4^nn!}$$ So far I have started by setting up the inequality: $$\frac{1}{4^nn!}<.0001$$Even... | The sum of this series is known, since it is the expansion of $\;\mathrm e^{-x}-1\;$ for $x=\frac14$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1776931",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 1
} |
Bayes Theorem probability Past Exam Paper Question -
Prof. Smith is crossing the Pacific Ocean on a plane, on her way to a conference.
The Captain has just announced that an unusual engine fault has been
signalled by the plane’s computer; this indicates a fault that only occurs once in
10,000 flights. If the fault repo... | When confused, it is useful to break the problem into parts, and work through a fictitious scenario.
The prof will survive all false alarms and $30\%$ of correct alarms
Suppose the prof takes $1,000,000$ trips (to avoid decimals)
A fault is likely to occur $100$ times, of which $99\% \;\; or\;\;99$ would sound the alar... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1777064",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 7,
"answer_id": 1
} |
Infinite Product $\prod_n^\infty \frac{1}{1-\frac{1}{n^s}} \rightarrow$? Can this $$P_n(s)=\prod_{m=2}^n \frac{1}{1-\frac{1}{m^s}}$$ for s>1 and $\lim_{n\rightarrow\infty}$ be written any simpler (does it converge)? When $m$ runs here only over the primes this is the famous Euler-Product form of the Riemann zeta functi... | Hint:
Note that $1-\frac1x = \frac{x-1}{x}$. Then you can simplify
$$\begin{align*}\log P_m(s) &= -\sum_n \log(1-\frac{1}{m^s}) \\
&= -\sum \log(\frac{m^s-1}{m^s}) \\
&= \sum\log(m^s)-\log(m^s-1) \end{align*}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1777129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Fermat's little theorem question: why isn't $a^p \equiv 1$? Fermat's little theorem says that $a^p \equiv a \pmod p$.
I have kind of a stupid question. Since $p \equiv 0\pmod p $, why isn't $a^p \equiv a^0 \equiv 1 \pmod p$ ?
| While $x\equiv y\pmod p$ implies $a+x\equiv a+y\pmod p$ as well as $a\cdot x\equiv b\cdot y\pmod p$ (and even $x^a\equiv y^a\pmod p$), it does not imply $a^x\equiv a^y\pmod p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1777279",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 1
} |
For $R$-modules M, is $M\cong R^{\oplus n}\otimes_RM\cong M^{\oplus n}$? I wanted to explicitly give a bilinear map $R^{\oplus n}\times M\longrightarrow M^{\oplus n}$ when trying to prove that $R^{\oplus n}\otimes_RM\cong M^{\oplus n}$ and ended up with $(r_1,...,r_n,m)\mapsto ((\prod_{i=1}^nr_i)m,...,(\prod_{i=1}^nr_i... | The map you wrote is not an isomorphism. It is instead a map that arguably parametrizes (though not uniquely) all multiples of the diagonal map $M\rightarrow M^{\oplus n}$ (where the multiple, depending on an element $(r_1,\ldots,r_n)$ of $R^{\oplus n}$, is $\prod_ir_i$). This is why, as you observe, you could make a s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1777358",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Sum of squares of integers divisible by 3 Suppose that $n$ is a sum of squares of three integers divisible by $3$. Prove that it is also a sum of squares of three integers not divisible by $3$.
From the condition, $n=(3a)^2+(3b)^2+(3c)^2=9(a^2+b^2+c^2)$. As long as the three numbers inside are divisible by $3$, we can ... | Induction on the power of $9$ dividing the number. Begin with any number not divisible by $9,$ although it is allowed to be divisible by $3.$ The hypothesis at this stage is just that this number is the sum of three squares, say $n = a^2 + b^2 + c^2.$ Since this $n$ is not divisible by $9,$ it follows that at least one... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1777521",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How can I Show that, $\arcsin\left(\frac{b}{c}\right)-\arcsin\left(\frac{a}{c}\right)=2\arcsin\left(\frac{b-a}{c\sqrt2}\right)$ Where $c^2=a^2+b^2$ is Pythagoras theorem. Sides a,b and c are of a right angle triangle.
Show that,
$$\arcsin\left(\frac{b}{c}\right)-\arcsin\left(\frac{a}{c}\right)=2\arcsin\left(\frac{b-a}{... | HINT:
$$\dfrac a{\sin A}=\dfrac b{\sin B}=\dfrac c1$$
$\implies\arcsin\dfrac ac=\arcsin(\sin A)=A$ as $0<A<\dfrac\pi2$
Now $A=\dfrac\pi2-B$
and $\dfrac{b-a}{\sqrt2c}=\dfrac{\sin B-\sin A}{\sqrt2}=\dfrac{\sin B-\cos B}{\sqrt2}=\sin\left(B-\dfrac\pi4\right)$
Can you take it from here?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1777635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Discreet weighted mean inequality
Let ${p_{1}},{p_{2}},\ldots,{p_{n}}$ and
${a_{1}},{a_{2}},\ldots,{a_{n}}$ be positive real numbers and let
$r$ be a real number. Then for $r\ne0$ , we define
${M_{r}}(a,p)={\left({\frac{{{p_{1}}a_{1}^{r}+{p_{2}}a_{2}^{r}+\cdots+{p_{n}}a_{n}^{r}}}{{{p_{1}}+{p_{2}}+\cdot... | Let $$w_{i} = \frac{p_{i}}{\sum_{i=1}^np_{i}}\implies\sum_{i=1}^nw_{i}=1 $$
we recall that by Holder inequality one has
$$\sum_{i=1}^nw_{i}A_i B_i
\le \left(\sum_{i=1}^nw_{i}A_i^{\color{red}{q}}\right)^{\color{red}{1/q}}\left(\sum_{i=1}^nw_{i}B_i^{\color{red}{q'}}\right)^{\color{red}{1/q'}}~~~~~~{\color{red}{1/q}}+{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1777754",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Evaluation of $\int_{0}^{2}\frac{(2x-2)dx}{2x-x^2}$ Evaluate $$I=\int_{0}^{2}\frac{(2x-2)dx}{2x-x^2}$$
I used two different methods to solve this.
Method $1.$ Using the property that if $f(2a-x)=-f(x)$ Then $$\int_{0}^{2a}f(x)dx=0$$
Now $$f(x)=\frac{(2x-2)}{2x-x^2}$$ So
$$f(2-x)=\frac{2(2-x)-2}{2(2-x)-(2-x)^2}=\frac{... | Hint: $u=2x-x^2$ and $du=(2-2x)dx$.
You will get:
$$I=-\ln(2x-x^2)|_{x=0}^{x=2}$$
Lets treat both boundary values as variables $a$ and $b$
$$I_{a,b}=-\ln(2x-x^2)|_{x=a}^{x=b}=-\ln(2b-b^2)+\ln(2a-a^2)=\ln\left(\frac{2a-a^2}{2b-b^2}\right)=\ln\left(\frac{a(2-a)}{b(2-b)}\right)$$
Now take the limit $(a,b)\to(0,2)$:
$$\lim... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1777926",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Are these facts about the Poisson process correct? Before studying theorems one by one, I want to check whether it is right what I know about Poisson process.
Let $\left\{N(t)\right\}$ be a Poisson process. Then
*
*the number of the event occur during time $t\sim{}Poisson(\lambda{}t)$
*Each time interval between ad... | Yes, those are correct. Here is more useful information:
*
*The interarrival times are iid's.
*The conditional distribution of arrival time $T_1$, $\:P[T_1 \leq \tau \mid N(t)=1]$
with $\tau \leq t$ is uniformly distributed over $(0,t)$, $\:P[T_1 \leq \tau \mid N(t)=1] = \frac{\tau}{t}$. And this generalizes to lat... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1778059",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How can I prove that $S=\{x_k:k\in \Bbb N\}\cup\{x_0\}$ is closed in $\Bbb R^n$? How can I prove that if $\{x_k\}$ is a convergent sequence in $\Bbb R^n$ with limit $x_0$, then $S=\{x_k:k\in \Bbb N\}\cup\{x_0\}$ is closed in $\Bbb R^n$?
This is my attempt:
It suffices to show that $x_0$ is the only limit point of $S$.
... | Proceed this way :
Let $x=\lim_\infty x_k$ and $A=\{x_k\vert k\in\mathbb N\}\cup\{x\}$.
You want to prove that for any sequence $(y_k)_{k\in\mathbb N}\in A^{\mathbb N}$, if $(y_k)$ has a limit $y\in \mathbb R^n$, then $y\in A$.
Now there are two cases :
*
*either $(y_k)$ is stationary, in which case the result is t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1778190",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Banach Contraction mapping of $\Phi(f)(x)=\int_0^x \frac{1}{1+f(t)^2}dt$ , find a fixed point.
Let $(X,d)$ be metric space with $d(f,g)=\sup |f(x)-g(x)|$ where $X$ is the set of continuous function on $[0,1/2]$. Show $\Phi:X\rightarrow X$ $$\Phi(f)(x)=\int_0^x \frac{1}{1+f(t)^2}dt$$ has a unique fixed point $f(0)=0$.
... | A useful approach for these sorts of estimates is to use the mean value theorem.
(a) Let $g(x) = {1 \over 1+x^2}$, then $g'(x) = -{2x \over (1+x^2)^2 }$ and
$g''(x) = 2 { 3 x^2 -1 \over (1+x^2)^3 }$. It is not hard to show that
$|g'|$ has a maximum of $L={3 \sqrt{3} \over 8} <1 $ at $x = \pm { 1\over \sqrt{3}}$.
Hence ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1778310",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
How do I find the smallest positive integer $a$ for which $a^n \equiv x \pmod{2^w}$? $x$ is fixed odd positive integer value. $n$ and $w$ are fixed positive integer values. $a$ is positive integer value.
I am interested for $n=41$ and $w=160$, but would appreciate a general algorithm.
I know how to find any $a$ for whi... | Note that here, the exponent $n$ is odd, while the order of every element modulo $2^w$ divides $\phi(2^w)=2^{w-1}$. Since $n$ and $2^{w-1}$ are relatively prime, the solution to $a^n=x\pmod{2^w}$ exists and is unique modulo $2^w$. So if you have a solution $a<2^w$, then it is indeed the smallest positive solution.
Anot... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1778579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
A pair of continued fractions that are algebraic numbers and related to $a^2+b^2=c^m$ Similar to the cfracs in this post, define the two complementary continued fractions,
$$x=\cfrac{-(m+1)}{km\color{blue}+\cfrac{(-1)(2m+1)} {3km\color{blue}+\cfrac{(m-1)(3m+1)}{5km\color{blue} +\cfrac{(2m-1)(4m+1)}{7km\color{blue}+\cfr... | Too long for a comment.
If you let $a=-1$ and $b=2m+1$ of the general continued fraction in this post, it reduces to the first continued fraction in this post (with $k=1$) and is expressible as a quotient of gamma functions,
$$x=-\tan\Big(\frac{\pi(m+1)}{4m}\Big)=\frac{\tan\Big(\frac{\pi}{4m}\Big)+1}{\tan\Big(\frac{\p... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1778673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Is a function of $\mathbb N$ known producing only prime numbers? It is well known that a polynomial $$f(n)=a_0+a_1n+a_2n^2+\cdots+a_kn^k$$ is composite for some number $n$.
What about the function $f(n)=a^n+b$ ?
Do positive integers $a$ and $b$ exists such that $a^n+b$ is prime for every natural number $n\ge 1$ ?
I ... | The answer to your question is no. For any $a>1,b\geq 1$ there will always exists $n$ such that $a^n+b$ is composite.
Suppose that $a+b$ is prime, (otherwise we are finished) and consider $n=a+b$ and look at $a^{a+b}+b$ modulo $a+b$. Then by Fermat's little theorem, which states that $$a^p\equiv a\pmod{p}$$ for any pr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1778736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 2,
"answer_id": 1
} |
Proving divisibility for $256 \mid 7^{2n} + 208n - 1$ I can't come up with a way of proving this:
$$256 \mid 7^{2n} + 208n - 1\\ \forall n \in \Bbb N$$
I've tried by induction but couldn't see when to apply the inductive hypothesis...
$$P(n+1) = 7^{2n+2}+208n+207$$
How can I continue?
Thank you
| We have transforming step by step suitably,
$$7^{2n}+208n-1=$$
$$(48+1)^n+(256-48)n-1=$$
$$(2^4\cdot3+1)^n+(256-2^4\cdot3)n-1 =$$
$$\sum_{k=0}^{k=n-1}\binom nk (2^4\cdot3)^{n-k}+1+256n-2^4\cdot3n-1=$$
$$=\sum_{k=0}^{k=n-2}\binom nk (2^4\cdot3)^{n-k}+256n+2^4\cdot3n-2^4\cdot3n=$$
$$=\sum_{k=0}^{k=n-2}\binom nk (2^4\cdo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1778846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
} |
counting probability with multiple cases There are four different colors of paint one can use for four different houses. If one color can be used up to three times, how many total possibilities are there?
I approached the problem by splitting it into cases:
1) each color used once
2) one color used twice
3) two colors... | Choose colours, then choose houses for each colour choice.
1) each color used once
$$\binom{4}{4}\cdot\binom{4}{1}\binom{3}{1}\binom{2}{1}$$
2) one color used twice
(plus two colours used once)
$$\binom{4}{1}\binom{3}{2}\cdot\binom{4}{2}\binom{2}{1}$$
3) two colors used twice
$$\binom{4}{2}\cdot\binom{4}{2}$$
4)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779045",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 5,
"answer_id": 3
} |
Prove or disprove that $\sqrt[3]{2}+\sqrt{1+\sqrt2}$ is a root of a polynomial Prove or disprove that there is a polynomial with integer coefficients such that the number $\sqrt[3]{2}+\sqrt{1+\sqrt2}$ is a root.
My work so far:
Let $P(x)=x^3-2$. Then $\sqrt[3]{2}$ is a root of $P(x)$
Let $Q(x)=x^4-2x^2-1$. Then $\sqrt... | $$\mathbb{Q}[\alpha,\beta]/(\alpha^3-2,\beta^4-2\beta^2-1)$$
is a vector space over $\mathbb{Q}$ with dimension $12$: a base is given by $\alpha^n \beta^m$ for $0\leq n\leq 2$ and $0\leq m \leq 3$. If follows that if we represent $(\alpha+\beta)^k$ for $k=0,1,\ldots,11,12$ with respect to such a base, we get $13$ vecto... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
} |
A combinatorial task I just can't solve Suppose you have $7$ apples, $3$ banana, $5$ lemons. How many options to form $3$ equal in size baskets ($5$ fruits in each) are exist?
At first I wrote:
$\displaystyle \frac{15!}{7!3!5!} $
But its definitely not true because I do not care about permutations of apples in one $5$... | We need to be very careful about what we are assuming is distinguishable and what is not distinguishable. Both @almagest and @windircursed have correct answers with different sets of assumptions:
@almagest assumed that both the baskets and the fruit are indistinguishable while
@windircursed assumed that only the bask... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779299",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Calculate $ \lim_{x\to0}\frac{\ln(1+x+x^2+\dots +x^n)}{nx}$ My attempt:
\begin{align*}
\lim_{x\to0}\frac{\ln(1+x+x^2+\dots +x^n)}{nx} &= (\frac{\ln1}{0}) \text{ (we apply L'Hopital's rule)} \\
&= \lim_{x \to0}\frac{\frac{nx^{n-1}+(n-1)x^{n-2}+\dots+2x+1}{x^n+x^{n-1}+\dots+1}}{n} \\
&= \lim_{x \to0}\frac{nx^{n-1}+(n-1)x... | Besides using L'Hospital's Rule,
By the definition of derivative,
$\displaystyle\lim \limits_{x\to0}\frac{\ln(1+x+x^2+...+x^n)}{nx}=\lim \limits_{x\to0} \frac{\ln(1+x+x^2+...+x^n)-\ln1}{n(x-0)}=\left.\frac{1}{n}\frac{d}{dx}(ln(1+x+x^2+...+x^n))\right|_{x=0}$
$=\displaystyle\left.\frac{1}{n}\frac{1+2x+...+nx^{n-1}}{1+x+... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779394",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 0
} |
integration using change of variables
find $$\iint_{R}x^2-xy+y^2 dA$$ where $R: x^2-xy^+y^2=2$ using $x=\sqrt{2}u-\sqrt{\frac{2}{3}}v$ and $y=\sqrt{2}u+\sqrt{\frac{2}{3}}v$
To calculate the jacobian I take $$\begin{vmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v}\\
\frac{\partial y}{\partial u}... | You're sloppy with notation, you're just gluing differentials next to the Jacobian after you get the determinant. Let's settle this question once and for all:
The Jacobian is used in place of the chain rule, so
$$\left|\frac{\partial (x,y)}{\partial (u,v)}\right|=\frac{4}{\sqrt3}$$
Now, just like you can write $dx=\fra... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779468",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Proposition 13 in Royden's Real Analysis According to the Proposition 13 Ch.1 in the book Real Analysis by Royden:
Let $F$ be a collection of subsets of a set $X$. Then the intersection $A$ of all σ-algebras of subsets of $X$ that contain $F$ is a σ-algebra that contains $F$. Moreover, it is the smallest σ-algebra of ... | Mathematical "translation":
Let $F\subset2^{X}$. Let $\mathcal{B}$ be the set of all $\sigma$-algebras $B$ on $X$ such that $F\subset B$. Then, $A=\cap_{B\in\mathcal{B}}B$ is a $\sigma$-algebra such that $F\subset A$. Moreover, for any $\sigma$-algebra $A^{\prime}$ on $X$ such that $F\subset A^{\prime}$, $A\subset A^{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779584",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Can multiplication and division be treated as logical operations? A few of my friends and I were playing around with math (more specifically, why (-1)(-1)=1) and we figured out that multiplication (with regards to signs) was an "nxor" operation (I.E. If we treat "1" as "true" and "-1" as "false," than the values of mul... | " My questions are these: is this line of thought similar to any current area of mathematical research?"
Yes, absolutely. The area you rediscovered is called algebraic logic. I think it is not a very active area of research any more, but in 50's and 60's it was rather active. Especially Tarski school of logic did many ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779680",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Prove that $11^{2n}+5^{2n+1}-6$ is divisible with $24$ for $n∈ℤ^+$ Prove that $11^{2n}+5^{2n+1}-6$ is divisible with $24$ for $n∈ℤ^+$
I've been trying to solve it by using modulo;
$11^{2n}+5^{2n+1}-6≡ (11^2 mod24)^n + 5*(5^2mod24)^n-6 = 1^n + (5*1^n)-6 = 0$
Is this the right way to tackle the problem? I am not certain ... | $$11^{2n}+5^{2n+1}-6=(5\cdot24+1)^n+5(24+1)^n-6\equiv 1+5-6\equiv0\pmod{24}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779793",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
$2^n=n$ and similar equations Is it possible to solve equations in the form $k^n=n$ for n and if so, How? I am new to logarithms and so would be glad if someone could explain even if there is an obvious answer.
Also What about $k^{a+b}=a$ for a? Or $k^{ab}=a$?
| All these equations can be standardized to the form
$$xe^x=y$$ for which the general solution has been studied in depth and is denoted as the function $x=W(y)$ known as Lambert's.
*
*$k^n=n$
Write $k^n=e^{-x}$, i.e. $x=-n\log(k)$, and $e^{-x}=-\dfrac x{\log(k)}$ or $xe^x=-\log(k)$.
*
*$k^{a+b}=a=k^ak^b$
Like ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1779919",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Generalised Gauss sums Let $\chi$ be a non-trivial Dirichlet character modulo an odd prime $p$
and let $f(x) \in \mathbb{Z}[x]$ be a polynomial. We define the generalised Gauss sum $$ G(\chi, f):=\sum_{y \in \mathbb{F}_p^*}
\chi(y) \left(\frac{f(y)}{p}\right).$$ Under which conditions for $f$ can we prove that
$$ G(\ch... | Let $\eta$ be a generator of the character group $\widehat{\Bbb{Z}_p^*}$. Therefore $\chi=\eta^k$ for some integer $k$, $0<k<p-1$ and the Legendre symbol is equal to $\eta^{(p-1)/2}$. Therefore
$$
G(\chi,f)=\sum_{y\in\Bbb{F}_p}\eta(y^kf(y)^{(p-1)/2}).
$$
The general Weil bound for multiplicative character sums says tha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1780202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to prove that if $f$ is continuous a.e., then it is measurable. Definition of simple function
*
*$f$ is said to be a simple function if $f$ can be written as
$$f(\mathbf{x}) = \sum_{k=1}^{N} a_{_k} \chi_{_{E_k}}(\mathbf{x})$$
where $\{a_{_k}\}$ are distinct values for $k=1, 2, \cdots, N$ and
$\chi_{_{E_k}}(\math... | Suppose $f: [a,b]\to \mathbb R$ is continuous at a.e. point of $[a,b].$ Let $D$ be the set of points of discontinuity of $f$ in $[a,b].$ Then $m(D)=0.$ We therefore have $E = [a,b]\setminus D$ measurable, and $f$ is continuous on $E.$
Let $c\in \mathbb R.$ Then
$$\tag 1 f^{-1}((c,\infty)) = [f^{-1}((c,\infty))\cap E] \... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1780336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 1,
"answer_id": 0
} |
How to integrate $\int_{0}^{1}\text{log}(\text{sin}(\pi x))\text{d}x$ using complex analysis Here is exercise 9, chapter 3 from Stein & Shakarchi's Complex Analysis II:
Show that:
$$\int_{0}^{1}\text{log}(\text{sin}(\pi x))\text{d}x=-\text{log(2)}$$
[Hint: use a contour through the set $\{ri\,|\,\,r\geq0\} \cup\{r\,\,... | Let $I$ be the integral given by
$$I=\int_0^1 \log(\sin(\pi x))\,dx \tag 1$$
Enforcing the substitution $ x \to x/\pi$ in $(1)$ reveals
$$\begin{align}
I&=\frac{1}{\pi}\int_0^\pi \log(\sin(x))\,dx \\\\
&=\frac{1}{2\pi}\int_0^\pi \log(\sin^2(x))\,dx\\\\
&=\frac{1}{2\pi}\int_0^\pi \log\left(\frac{1+\cos(2x)}{2}\right)\,d... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1780449",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Weak convergence and strong convergence on $B(H)$ Let $\mathcal{A} \subset B(H)$ be a weak closed convex bounded set of self-adjoint operators. If $A_n \rightarrow_{wo} A\in \mathcal{A}$, do we have $A_n \rightarrow A$ strongly?($A_n$ is a sequence in $\mathcal{A}$)
| Let $H=\ell^2(\mathbb{N})$ and let $\mathcal{A}$ be the set of all self-adjoint elements in the unit ball of $B(H)$. Define $S_n\colon H\to H, S_n \xi(k)=\xi(k+n)$. Then $\|S_n\|=1$, hence $A_n:=\frac 1 2(S_n+S_n^\ast)\in \mathcal{A}$.
The adjoint of $S_n$ is given by
$$
S_n^\ast\xi(k)=\begin{cases}\xi(k-n)&\colon k\ge... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1780565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Special Gamma function integral I'm trying to evaluate this integral $$\int_{0}^{1} \sin (\pi x)\ln (\Gamma (x)) dx$$
and I got to the point, when I need to find
$\displaystyle \int_{0}^{\pi } \sin (x)\ln (\sin (x)) dx$
but everything I tried just failed,or either I was not able to put in the borders . Could you pleas... | Let we put everything together.
$$ I = \int_{0}^{1}\sin(\pi x)\log\Gamma(x)\,dx = \int_{0}^{1}\sin(\pi z)\log\Gamma(1-z)\,dz \tag{1}$$
leads to:
$$ I = \frac{1}{2}\int_{0}^{1}\sin(\pi x)\log\left(\Gamma(x)\,\Gamma(1-x)\right)\,dx \tag{2}$$
but $\Gamma(x)\,\Gamma(1-x) = \frac{\pi}{\sin(\pi x)}$, hence:
$$ I = \frac{\log... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1780677",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 2
} |
Relative entropy (KL divergence) of sum of random variables Suppose we have two independent random variables, $X$ and $Y$, with different probability distributions.
What is the relative entropy between pdf of $X$ and $X+Y$, i.e.
$$D(P_X||P_{X+Y})$$
assume all support conditions are met.
I know in general pdf of $X+Y$ i... | Let $f(t)$ be the PDF of $X$ and $g(t)$ be the PDF of $Y$.
$$D_{KL}(P_X\parallel P_{X+Y}) = \int_{-\infty}^{+\infty}f(x)\log\frac{f(x)}{(f*g)(x)}\,dx$$
does not admit any obvious simplification, but the term
$$\log\frac{f(x)}{(f*g)(x)}=\log\frac{\int_{-\infty}^{+\infty} f(t)\,\delta(x-t)\,dt}{\int_{-\infty}^{+\infty} ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1780790",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Find all integral solutions of the equation $x^n+y^n+z^n=2016$
Find all integral solutions of equation
$$x^n+y^n+z^n=2016,$$
where $x,y,z,n -$ integers and $n\ge 2$
My work so far:
1) $n=2$ $$x^2+y^2+z^2=2016$$
I used wolframalpha n=2 and I received the answer to the problem (Number of integer solutions: 144)
2) ... | An approach that can sometimes help to find solutions to this type of equation is to consider the prime factors of the given integer. In this case:
$$2016 = 2^5.3^2.7 = 2^5(2^6-1) = 2^5((2.2^5)-1)$$
Hence:
$$\mathbf{2016 = 4^5 + 4^5 + (-2)^5}$$
And also (finding a solution of $x^3+y^3+z^3=252$ by trial and error or fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1780881",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
What is the 'meaning' of nowhere dense set? In some books, nowhere dense set is defined to be $int(\bar A)=\emptyset$ but meanwhile is defined to be $int(A)=\emptyset$ in some books(e.g. Munkres).
So what is the 'meaning' (i.e motivation, intuitive/geometric meaning etc.) of nowhere dense set? Thank you.
| A set $A$ is nowhere dense if every nonempty open set contains a nonempty open set which is disjoint from $A.$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1780973",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
Counterexamples of Cumulative Distribution Function ( multidimensional ) For simplicity, we consider 2-dimensional.
We consider the function $F:\mathbb{R}^2\to\mathbb{R}$.
Let $F$ satisfies:
*
*$0\leq F(x_1,x_2)\leq1$ for $(x_1,x_2)\in\mathbb{R}^2$
*$\left\{\displaystyle\begin{matrix}
\displaystyle\lim_{x_1,x_2\upa... | The function
$$
F(x, y):=\begin{cases}
1& \text{if $x+y\ge0$}\\
0& \text{otherwise}
\end{cases}
$$
satisfies conditions (1) and (2). It also meets condition (3), since for any $(x, y)$ there is $h>0$ such that $F(x+h, y+h)=F(x, y)$. However, condition (4) is not satisfied since
$$
F(1,1) + F(0,0) = 1 + 0 < 1 + 1 = F(1,... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1781101",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Eigen values of matrix formed by column vector multiplied by row vector. Let $u$ and $v$ be column vectors in $\mathbb{R}^n$. Let $A = uv^T$ that is matrix formed by column vector multiplied by row vector.
What are all the eigen values and eigen vectors of $A$? What is the rank of $A$?
| If one of $u$ and $v$ is zero, then $A=0$ and the case is trivial.
Suppose then $u\ne0$ and $v\ne0$. Then $A\ne0$ and has rank at most $1$ (the rank of a product can't be greater than the rank of the factors). So the rank is $1$.
Consider $x=v$; then $uv^Tv=(v^Tv)u$ by direct computation.
The other eigenvalue is $0$. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1781189",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.