Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Solving quadratic equations in modular arithmetic Is there a general way to solve quadratic equations modulo n? I know how to use Legendre and Jacobi symbols to tell me if there's a solution, but I don't know how to get a solution without resorting to guesswork. Some examples of my problems:
$x^2 = 8$ mod 2009
$x^2 + 3... | The answer depends on the value of the modulus $n$.
*
*in general, if $n$ is composite, then solving modulo $n = \prod p_i^{e_i}$ is equivalent to solve modulo each $p_i^{e_i}$. However, this requires knowing the factorization of $n$, which is hard in general (in a computational way): there are cryptosystems based o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1257648",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Show that $(S^\perp)^\perp=\overline {\operatorname{span}(S)}$ . Let $H$ be a Hilbert Space. $S\subseteq H$ be a finite set .Show that $(S^\perp)^\perp=\overline {\operatorname{span} (S)}$ .
Now $\operatorname{span}(S)$ is the smallest set which contains $S$ and $\overline{\operatorname{span}(S)}$ is the smallest close... | You don't need $S$ to be a finite set. This is true for any subset $S\subseteq \mathscr{H}$.
First of all notice that $S^{\perp}$ is a closed subspace of $\mathscr{H}$ (using continuity of inner product) and $S \subseteq (S^{\perp})^{\perp}$ for any subset $S \subseteq \mathscr{H}$. Thus $Span(S) \subseteq (S^{\perp})... | {
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"url": "https://math.stackexchange.com/questions/1257748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Finding coefficients, Legendre polynomials. Say I have a function
$f(\theta) = 1 + \cos^2(\theta)$
that can be expressed terms of the Legendre polynomials. When calculating coefficients should I change the Legendre polynomials from $x$ variables to theta variables? e.g. The third Legendre usually written:
$(0.5(3x^2-... | The answer depends on what you're seeking. Legendre polynomials $P(x)$ form an orthonormal basis on $[-1,1]$, so any nice function $f(x)$ on $[-1,1]$ can be written as a linear combination of them. Your $f(\theta)=1+\cos(\theta)^2$, while nice, will have a pretty awful expansion in terms of $P(\theta)$, but a much nice... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1257843",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$R/I^n$ is a local ring
Let $I$ be a two sided ideal of a ring $R$ such that $I$ is maximal as a right ideal. I need to show that $R/I^n$ is a local ring, for every $n \geqslant 1$.
For $n=1$ I was able to show that the quotient $R/I$ is a division ring and so it is a local ring (because the non-invertible elements f... | You need to show that for each $n$, $R/I^n$ has a unique maximal ideal (just the definition). The ideals in $R/I^n$ are in bijective, inclusion-preserving correspondence to the ideals of $R$ which contain $I^n$ (the bijection is induced by the quotient map). Therefore, the image of $I$ in $R/I^n$ is a unique maximal id... | {
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Statistics Problems, I don't understand what this means.. P(A)=0.46 and P(B)=0.42
If P(B∣A)= 0.174
what is P(A∩B)?
| One may recall that
$$
P(B|A)=\frac{P(A\cap B)}{P(A)}
$$ giving
$$
P(A\cap B)=P(A)\times P(B|A)
$$Here you then have
$$
P(A\cap B)=0.46\times 0.174=0.08004.
$$
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show a random walk is transient I was going through some problems related to Markov chains and I got stuck on this bit:
We are given a random walk on $Z$, defined by the transition matrix $p_{i,i+1}=p$ and $p_{i,i-1}=1-p$. How to show that if $p\neq 0.5$ the walk is transient?
| Since the process is irreducible, we can assume without loss of generality that $X_0=0$, and it suffices to show that $\mathbb P(N_0<\infty)<1$, where
$$N_0=\inf\{n>0: X_n=0\}$$
is the time until the first return to $0$. Let
$$F(s) = \mathbb E\left[s^{N_0}\right]$$
be the generating function of $N_0$. It can be shown ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1258103",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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formula for the $n$th derivative of $e^{-1/x^2}$
$f(x) = \begin{cases} e^{-1/x^2} & \text{ if } x \ne 0 \\ 0 & \text{ if } x = 0 \end{cases}$
so
$\displaystyle f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0} = \lim_{x \to 0} \frac {e^{-1/x^2}}x = \lim_{x \to 0} \frac {1/x}{e^{1/x^2}} = \lim_{x \to 0} \frac x {2e^{1/x... | Show by induction that $f^{(n)}(x)=P_n(\frac 1 x) \mathbb{e} ^{-\frac 1 {x^2}}$ with $P_n$ a polynomial function of degree $3n$, and then compute (again by induction if you want) $\lim \limits _{x \to 0^+} \space f^{(n)}(x)$. You'll have to use l'Hospital's thorem.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1258219",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
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Arithmetic progression - terms divisible by a prime.
If $p$ is a prime and $p \nmid b$, prove that in the arithmetic progression $a, a+b, a +2b, $ $a+3b, \ldots$, every $p^{th}$ term is divisible by $p$.
I am given the hint that because $\gcd(p,b)=1$, there exist integers $r$ and $s$ satisfying $pr+bs=1$. Put $n_k$ ... | Apply the extended Euclidean algorithm to find $r$ and $s$.
Suppose $p=31$, $b=23$
$$\begin{array}{c|c|c}
pr+bs & r & s \\
\hline p=31 & 1 & 0 \\
b=23 & 0 & 1 \\ \hline
8 & 1 & -1 \\
7 & -2 & 3 \\
1 & 3 & -4 \\
\end{array}$$
$31\cdot 3 + 23 \cdot -4 = 93-92 = 1 \quad \checkmark$
The process works by, at each step, subt... | {
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"timestamp": "2023-03-29T00:00:00",
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Is the multiplicative order of a number always equal to the order of its multiplicative inverse? Is it true that $ord_{n}(a)=ord_{n}(\bar{a})$ $\forall n$?
Here, $\bar{a}$ refers to the multiplicative inverse of $a$ modulo $n$ and $ord_{n}(a)$ refers to the multiplicative order of $a$ modulo $n$.
| Yes, it is. Since $a \bar{a}=1$, it follows that for any positive integer $k$ we have $a^k (\bar{a})^k=1$. It follows that $a^k=1$ if and only if $(\bar{a})^k=1$. In particular, if $k$ is the smallest positive integer such that $a^k=1$, then $k$ is the smallest positive integer such that $(\bar{a})^k=1$.
| {
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Area between $ 2 y = 4 \sqrt{x}$, $y = 4$, and $2 y + 4 x = 8 $ Sketch the region enclosed by the curves given below. Then find the area of the region.
$ 2 y = 4 \sqrt{x}$, $y = 4$, and $2 y + 4 x = 8 $
Attempt at solution:
I guess I'm supposed to divide the areas into several parts, and then sum up the areas of those ... | Simplify your boundary equations:
$$y = 2 \sqrt{x}$$
$$y = 4$$
$$y = -2x +4$$
Sketch the area. You ought to try hand-sketching it to verify.
Split into two double integrals.
$$Area = \int_{x=0}^1\int_{y=lower curve}^{higher curve} dydx+ \int_{x=1}^4\int_{y=lower curve}^{higher curve}dydx$$.
UPDATE/EDIT: You ought to ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Steinhaus theorem for topological groups $G$ is a locally compact Hausdorff topological group, $m$ is a (left) Haar measure on $X$, $A$ and $B$ are two finite positive measure in $G$, that is $m(A)>0$, $m(B)>0$.
My question is:
Can we conclude that $AB= \{ab, a\in A, b\in B\}$ contains some non-empty open set of G?
Is ... | Here is another proof, using regularity of the measure instead of convolution.
Claim: The result holds when $B=A^{-1}$.
Proof: By regularity there is a compact set $K$ and an open set $U$ such that $K\subset A\subset U$ and such that $m(U)<2m(K)$. The multiplication map sends $\{1\}\times K$ into $U$, so by continuit... | {
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What is the intuition/motivation behind compact linear operator. Compact Linear Operator is defined such that the operator will map any bounded set into a relatively compact set. Why is this property so special that it can be named as "compact"? Does it share some similar properties as compact sets? What is the motivat... | The set of compact operators (in a Hilbert space) is exactly the set of norm limits of finite rank operators. This is perhaps a more natural definition than the one you indicate. Many of the nice properties of finite rank operators have analogues for compact operators. You can view compactness has a slight generalizati... | {
"language": "en",
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Poisson power series We have a Poisson power series of
$$Y=\sum\limits_{k=0}^{\infty}e^{-\pi\lambda v^2}\frac{(\pi\lambda v^2)^k}{k!}(A)^k $$
If we have a disk with radius $v$
where A is defined as the density of a distance of some node from the origin placed randomly inside the disk, $A=\frac{2x}{v^2}$.
If i try to p... | You need to learn a little about convergence of infinite series
before you tackle this.
First step: Consider the geometric series:
$$A = 1/2 + 1/4 + 1/8 + \cdots = \sum_{k=1}^\infty 1/2^k.$$
It can be evaluated as follows:
$$(1/2)A = 1/4 + 1/8 + \cdots.$$
So $A - (1/2)A = 1/2$ and $A = 1.$
You can get very close to th... | {
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Given $L = L_1 \cap L_2$ where $L_1 \in NP$ and $L_2 \in coNP$, how do I express L as a symmetric difference of 2 sets in NP? My ultimate goal is to show that $L \in PP$, but I need to figure out the title question first as an intermediary step. Any help is appreciated, thanks in advance.
| Recall that NP is closed under intersection. Hence $L_1\cap\overline{L_2}$ is in NP. Finally we can realize that the symmetric difference of this set and $L_1$ is exactly $L_1\cap L_2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Product of rings: If $K$ is an ideal of $R\times S$, then there exists $I$ ideal of $R$, $J$ ideal of $S$ such that $K=I\times J$. Let $R$ and $S$ be two rings. We consider the product $R\times S$.
It is a ring with operations of sum and product defined coordinate by coordinate, i.e.
$$(r_1, s_1) + (r_2, s_2) = (r_1 + ... | For (a) you should also note that $(a,b)(i,j) \in I \times J$ (you want $I \times J$ to be two-sided).
For (b): Let $I := \{i \in R \mid \exists s \in S : (i,s) \in K\}$ and $J := \{j \in S \mid \exists r \in R: (r,j) \in K\}$. Then $I$ is an ideal: $I$ is non-empty, if $i,i' \in I$, say $(i,s), (i',s') \in K$, then $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1258998",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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Image of point of codimension one has codimension one? I'm working on the following exercise (7.2.3) from Liu's Algebraic Geometry and Arithmetic Curves:
Let $f: X \rightarrow Y$ be a morphism of Noetherian schemes. We suppose that either $f$ is flat or $X,Y$ are integral and $f$ is finite surjective.
Let $x \in X$ ... | No, this is not true. There is a famous example of Nagata of a Noetherian local domain $A$ of dimension $2$ which has a finite overring $A \subset B$ with two maximal ideals $\mathfrak m, \mathfrak n \subset B$ with $\dim(B_{\mathfrak m}) = 2$ and $\dim(B_{\mathfrak n}) = 1$. Let $X$ be the spectrum of $B$ and $Y$ be t... | {
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"timestamp": "2023-03-29T00:00:00",
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Computing Ancestors of # for Stern-Brocot Tree Reading about the Stern-Brocot tree, the article gives this example:
using 7/5 as an example, its closest smaller ancestor is 4/3, so its left child is (4 + 7)/(3 + 5) = 11/8, and its closest larger ancestor is 3/2, so its right child is (7 + 3)/(5 + 2) = 10/7.
Getting t... | "But, how can I figure out the closest smaller and larger ancestors of 7/5?"
Here is a method using a version of the subtractive Euclidean algorithm:
A : 7 (1) - 5 (0) = 7
B : 7 (0) - 5 (1) = -5
C : 7 (1) - 5 (1) = 2 A + B
D : 7 (1) - 5 (2) = -3 B + C Adding smallest positive to 'smallest' (lowest absolute ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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range of $m$ such that the equation $|x^2-3x+2|=mx$ has 4 real answers. Find range of $m$ such that the equation $|x^2-3x+2|=mx$ has 4 distinct real solutions $\alpha,\beta,\gamma,\delta$
To show how I got the wrong answers.
From $|x^2-3x+2|=mx$
I got the two case $x^2-3x+2=mx$ when $x>2 $ or $ x<1$
and $x^2-3x+2=-mx$ ... | There is some positive value $m$ such that $y=mx$ is tangent to $y=-(x^2-3x+2)$.
This value must make $0$ the discriminant of the equation
$$x^2-3x+2=-mx$$
That is, $$m^2-6m+1=0$$
The least root of this equation is $$3-2\sqrt2$$
So $0<m< 3-2\sqrt 2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Show that $l^2$ is a Hilbert space
Let $l^2$ be the space of square summable sequences with the inner product $\langle x,y\rangle=\sum_\limits{i=1}^\infty x_iy_i$.
(a) show that $l^2$ is H Hilbert space.
To show that it's a Hilbert space I need to show that the space is complete. For that I need to construct a Cau... | A typical proof of the completeness of $\ell^2$ consists of two parts.
Reduction to series
Claim: Suppose $ X$ is a normed space in which every absolutely convergent series converges; that is, $ \sum_{n=1}^{\infty} y_n$ converges whenever $ y_n\in X$ are such that $ \sum_{n=1}^{\infty} \|y_n\|$ converges. Then the spac... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "25",
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Prove that a continuous function of compact support defined on $R^n$ is bounded. I am working through a few sets of notes I found on the internet and I came across this exercise. How do I prove that a continuous function $f$ of compact support defined on $R^n$ is bounded?
It seems believable that it is true for $f$ in ... | Put $K = {\rm supp}(|f|)$. Since $K$ is compact and $|f|$ is continuous, then $\sup_{K} |f| < \infty$ and the supremum is attained on $K$. Since $f$ is zero off $K$, we are done.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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probability of chess clubs problem The chess clubs of two schools consists of, respectively, 8 and 9 players. Four members from each club are randomly chosen to participate in a contest between the two schools. The chosen players from one team are then randomly paired with those from the other team, and each pairing pl... | There are 24 or (4!)ways to pair the 4 members of the 2 school teams, and 6 or (3!) Ways to pair the members ensuring that rebecca and elise are paired. Hence the probability that rebecca and elise are paired given that they were chosen is 3!/4!. Now we must multiply this probability that both rebecca and elise are cho... | {
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"timestamp": "2023-03-29T00:00:00",
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Orthogonality lemma sine and cosine I want to know how much is the integral $\int_{0}^{L}\sin(nx)\cos(mx)dx$ when $m=n$ and in the case when $m\neq n$. I know the orthogonality lemma for the other cases, but not for this one.
| If $n=m$, then we have
$$\intop_{x=0}^{L}\sin\left(nx\right)\cos\left(nx\right)\mathrm{d}x
=\left[\frac{\sin^2\left(nx\right)}{2n}\right]_{x=0}^{x=L}
=\frac{\sin^2\left(nL\right)}{2n},$$
and for $n\neq m$ we have
$$\intop_{x=0}^{L}\sin\left(nx\right)\cos\left(mx\right)\mathrm{d}x
=\frac{1}{2}\left(\intop_{x=0}^{L}\sin\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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What's wrong with my permutation logic? The given question:
In how many ways the letters of the word RAINBOW be arranged, such that A is always before I and I is always before O.
I gave it a try and thought below:
Letters A, I and O should appear in that order. Then there are four places in which all the remaining f... | You would be entirely correct if there really were 16 places ( 19 in all ) and the places not filled with letters were filled with spaces so that
R***A****I*N*BO**W*
was truly distinct from
*R**A****I*N*BO***W ( I has used * because spaces are hard to count )
what $ ^{16} P _4$ counts is the number of four-tuples ... | {
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Find a nontrivial unit polynomial in $\mathbb Z_4[x]$ Find a unit $p(x)$ in $\mathbb{Z}_4[x]$ such that $\deg p(x)>1$
What I know:
A unit has an inverse that when the unit is multiplied by the inverse we get the identity element.
But I am confused by the concept of degree
"If $n$ is the largest nonnegative number
for w... | Take $p(x) = 2x^2+1$. Observe that $p(x)^2 = (2x^2+1)^2 = 4x^4+4x^2+1 = 1$, and $p(x)$ has degree $2$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Four 6-sided dice are rolled. What is the probability that at least two dice show the same number? Am I doing this right? I split the problem up into the cases of 2 same, 3 same, 4 same, but I feel like something special has to be done for 2 of the same, because what if there are 2 pairs (like two 3's and two 4's)?
Thi... |
I split the problem up into the cases of 2 same, 3 same, 4 same, but I feel like something special has to be done for 2 of the same, because what if there are 2 pairs (like two 3's and two 4's)?
Yes. Your cases are
*
*1 quadruplet: $\binom{4}{4}\times \binom{6}{1}$ arrangements.
*1 triplet, 1 singleton: $\binom{... | {
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Are derivatives linear maps? I am reading Rudin and I am very confused what a derivative is now. I used to think a derivative was just the process of taking the limit like this
$$\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$
But between Apostol and Rudin, I am confused in what sense total derivatives are derivati... | If $f:M\to N$ is some (possibly nonlinear) function (here I have in mind a diffeomorphism), then the Jacobian $J_f$ can be viewed as a linear map taking tangent vectors at some point $p \in M$ and returning a tangent vector at $f(p) \in N$.
Let's consider the case where $M$ and $N$ are both $\mathbb R^3$. $f$ is some ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Help with factorial inequality induction proof: $3^n + n! \le (n+3)!$ So I'm asked to prove $3^n + n! \le (n+3)!$, $\forall \ n \in \mathbb N$ by induction. However I'm getting stuck in the induction step.
What I have is:
(n=1)
$3^{(1)} + (1)! = 4$ and $((1) + 3)! = (4)! = 24$
So $4 \le 24$ and the statement holds for... | The proof for n = 1 is correct.
Now, let's go on to the inductive step. Let us consider that $\color{blue}{3^n + n!} \le \color{green}{(n+3)!}$ holds true for some integer n. Then the next step is to check what happens to the inequality for some integer n+1:
$3^{n+1}+(n+1)!=$
$3 \times 3^n + (n+1) \times n!=$
$(3^n ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260126",
"timestamp": "2023-03-29T00:00:00",
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Compute $\lim_\limits{n\to\infty}a_n$ where $a_{n+2}=\sqrt{a_n.a_{n+1}}$ I managed to show that the limit exists, but I don't know how to compute it.
EDIT:
There are initial terms: $a_1=1$ and $a_2=2$.
| Note that $$ a_{n+2}\sqrt{a_{n+1}}=a_{n+1}\sqrt{a_n} =\cdots =a_2\sqrt{a_1}=2$$
Hence limit is $2^\frac{2}{3}$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260341",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 0
} |
A valid method of finding limits in two variables functions? I was wondering if in finding the limit of a two variables function (say, $F(x,y)$), I can choose the path by let $y=f(x)$, then find the limit in the same way of that in one variable functions.
For example,
$$
\lim_{(x,y) \to (0,0)} \frac{xy}{x^2+xy+y^2}
$$
... | If the limit does not exist, you can find two paths that disagree. However, in case the limit does exit, your failure to find such paths is not a proof of anything. You will need to show that the limits are the same for all paths.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
How to show real analyticity without extending to complex plane Suppose we have some $f \in C^\infty(\mathbb{R},\mathbb{R}).$ For example, $$f(x)=(1+x^2)^{-1}.$$ Using complex analysis, we can easily show $f$ is real analytic. Is there an easy, general method to show this which doesn't use complex analysis?
| The most popular general method is to calculate the general term $f^{(n)}(a)$, and if that's possible, and for every $a$, find an interval $[a-h,a+h]$, such that
if
$$
M_n=\max_{x\in[a-h,a+h]}\lvert \,f^{(n)}(x)\rvert,
$$
then
$$
\limsup_{n\to\infty} \left(\frac{k_n}{n!}\right)^{\!1/n}=L<\infty.
$$
The $M_n$ can be ap... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260511",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How many ordered pairs are there in order for $\frac{n^2+1}{mn-1}$ to be an integer? For how many ordered pairs of positive integers like $(m,n)$ the fraction
$\frac{n^2+1}{mn-1}$
is a positive integer?
| We have:
$$n^2+1=kmn-k$$
so we have $n$ divides $k+1$ we can write $k+1=nt$ so that $$n^2+1=(nt-1)(mn-1)$$
but if $m,t,n>1$ we have $(nt-1)(mt-1)\geq (2n-1)^2>n^2+1$impossible
if either $t=1$ or $m=1$ in the two cases $n-1$ divides $n^2+1$ but we know that $n-1=\gcd(n^2+1,n-1)=1$ or $2$, so that $n=1$, $n=2$ or $n=3$ a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260617",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible
Prove that $f=x^4-4x^2+16\in\mathbb{Q}[x]$ is irreducible.
I am trying to prove it with Eisenstein's criterion but without success: for p=2, it divides -4 and the constant coefficient 16, don't divide the leading coeficient 1, but its square 4 divides the cons... | $$x^4-4x^2+16=(x^2-(2+\sqrt{-12}))(x^2-(2-\sqrt{-12}))$$
No rational roots and no factorization into quadratics over the rationals. The polynomial is irreducible over the rationals
edit for those who commented that this is not enough. I factorized over $\mathbb{C}[X]$ and thus proved that there are no rational solution... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 9,
"answer_id": 8
} |
Analytical solution for $\max{x_1}$ in $(x_n)_{n\in\mathbb{N}}$ Let be $x_1,x_2,x_3,\ldots,$ a sequence of positive integers. Suposse the folowing conditions are true for all $n\in\mathbb{N}$
*
*$n|x_n$
*$|x_n-x_{n+1}|\leq 4$
Find the maximun value of $x_1$
I can't solve this analytically, I've start with $x_1=50$ ... | If $n\ge 9$ then there is at most $1$ multiple of $n+1$ within distance $4$ of $x_n$, and there is at most $1$ multiple of $n$ within distance $4$ of $x_{n+1}$, so a term determines and is determined by the next term. Hence the terms of the sequence appearing for $n\ge 9$ are injectively determined by $x_9$
We clearly ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260837",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
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How to factor the polynomial $x^4-x^2 + 1$ How do I factor this polynomial: $$x^4-x^2+1$$
The solution is: $$(x^2-x\sqrt{3}+1)(x^2+x\sqrt{3}+1)$$
Can you please explain what method is used there and what methods can I use generally for 4th or 5th degree polynomials?
| Actually you have:
$$x^4-x^2+1=x^4+2x^2+1-3x^2=(x^2+1)^2-(\sqrt3 x)^2 $$
and use the identity $a^2-b^2=(a-b)(a+b)$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260918",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 4,
"answer_id": 1
} |
Lindebaum's Lemma seemingly inconsistent with Gödel's incompleteness theorem? Lindenbaum's Lemma: Any consistent first order theory $K$ has a consistent complete extension.
First Incompleteness Theorem: Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.
... | What you've shown, as Asaf points out, is that Goedel's incompleteness theorem implies that $PA$ has no computable completion.
This addresses your question completely. However, at this point it's reasonable to ask, "How complicated must a completion of $PA$ be?"
It turns out the answer to this question is extremely wel... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1260994",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Dimension of $m\times n$ matrices I'm a bit confused on the notion of the dimension of a matrix, say $\mathbb{M}_{mn}$. I know how this applies to vector spaces but can't quite relate it to matrices.
For example take this matrix:
$$ \left[
\begin{array}{ccc}
a_{11}&\cdots&a_{1n}\\
\vdots & \vdots & \vdo... | The term ''dimension'' can be used for a matrix to indicate the number of rows and columns, and in this case we say that a $m\times n$ matrix has ''dimension'' $m\times n$.
But, if we think to the set of $m\times n$ matrices with entries in a field $K$ as a vector space over $K$, than the matrices with exacly one $1$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1261057",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Corollary of Gauss's Lemma (polynomials) I am trying to prove the following result. I have outlined my attempt at a proof but I get stuck.
Any help would be welcome!
Theorem:
Let $R$ be a UFD and let $K$ be its field of fractions.
Suppose that $f \in R[X]$ is a monic polynomial.
If $f=gh$ where $g,h \in K[X]$ and $g$ i... |
why does $g$ and $h$ being monic imply that $k$ and $t$ are in $R$?
Because $kg$ and $th$ are primitive. In particular, they belong to $R[x]$. Since the highest coefficient of $kg$ is $k$, and the highest coefficient of $th$ is $h$, both $t$ and $h$ belong to $R$.
why does $k$ and $t$ being invertible in $R$ imply ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1261154",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Generalized way to solve $x_1 + x_2 + x_3 = c$ with the constraint $x_1 > x_2 > x_3$? On my example final exam, we are given the following problem:
How many ways can we pick seven balls of three colors red, blue, yellow given
also that the number of red balls must be strictly greater than blue and
blue strictly grea... | For the case $n = 3$, since $x_2 > x_3 \geq 0 \Rightarrow x_2 \geq x_3+1 \Rightarrow x_2=x_3+1+r, r \geq 0$, and similarly, $x_1 > x_2 \Rightarrow x_1 \geq x_2+1 \Rightarrow x_1=x_2+1+s = (x_3+1+r)+1+s = x_3+2+r+s$. Substituting these into the equation: $x_3+2+r+s + x_3 + 1 + r + x_3 = c \Rightarrow 3x_3+2r+s = c-3$. F... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1261258",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
almost sure convergence for non-measurable functions Let $(\Omega,\mathscr{F},P)$ be a probability space. Assume for each $n$, $Y_n:\Omega\rightarrow\mathbb{R}$ is a function but $Y_n$ is not necessarily $\mathscr{F}$-measurable. In this case, is it still meaningful to talk about almost sure convergence of $Y_n$? Conce... | Not only a.s. convergence but pointwise convergence, as well, can be defined in the case of sequences of non measurable functions. Let, for instance, $$([0,1],\mathscr A=\left\{\emptyset,[0,1/2],(1/2,1],[0,1]\right\},\mathbb P((0,1/2]))=\mathbb P((1/2,1])=1/2)$$
be a probability space, and let $$X_n(\omega)=\frac{\omeg... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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user friendly proof of fundamental theorem of calculus Silly question. Can someone show me a nice easy to follow proof on the fundamental theorem of calculus. More specifically,
$\displaystyle\int_{a}^{b}f(x)dx = F(b) - F(a)$
I know that by just googling fundamental theorem of calculus, one can get all sorts of answers... | The key fact is that, if $f$ is continuous, the function $G(x)=\int_a^xf(t)\,dt$ is an antiderivative for $f$. For this,
$$
\frac1h\,\left(\int_a^{x+h}f(t)\,dt-\int_a^xf(t)\,dt\right)
=\frac1h\,\int_x^{x+h}f(t)\,dt\xrightarrow[h\to0]{}f(x).
$$
The justification of the limit basically plays on the fact that $f$ is conti... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1261432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
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Is it possible to prove this? $\ln(\frac{x}{x-1}) < \frac{100}{x} $ for $ x > 1$ $-\ln(1-(\frac{1}{x})) < \frac{100}{x} $ for $ x > 1$ is what I want to prove. I pulled a negative sign out and I got $\ln(\frac{x}{(x-1)}) < \frac{100}{x} $ for $ x > 1$.
How do I continue with this proof?
Or is it actually possible to p... | It is not true for $x=1+e^{-100}$. We then have
$$ \log\frac{x}{x-1} = \log(x)+100 > 100 $$
but
$$ \frac{100}{x} < 100 $$
because $x>1$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1261527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 5,
"answer_id": 0
} |
Can you help me subtract intervals? I was reading my abstract math textbook and they subtracted
$[3, 6] - [4, 8) = [3, 4)$. I was wondering if someone could write out how they got to $[3, 4$). I looked at wikipedia and it said I should go
$[a, b] - [c, d] = [a-d, b-c]$. When I did this, I got $[-5, 2)$. I would be tha... | You are looking at two different definitions of $A-B$:
Set difference: $A - B = \{ x\in A \, \mid \, x \notin B \}$ which in this case gives $$[3,6]−[4,8) = [3,4)$$
Interval arithmetic: $A - B = \{ x-y \in \mathbb{R} \, \mid \, x\in A, \,y \in B \}$ which in this case gives $$[3,6]−[4,8) = (-5,2]$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1261613",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 0
} |
Explain this inequality, related to logarithms I am trying to understand a proof of Stirling's formula.
One part of the proof states that, 'Since the log function is increasing on the interval $(0,\infty)$, we get
$$\int_{n-1}^{n} \log(x) dx < \log(n) < \int_{n}^{n+1} \log(x) dx$$ for $n\geq 1$.'
Please could you exp... | $$\int_{n-1}^n\log(x) dx<\int_{n-1}^{n}\log( n) dx=\log(n)$$
using $\log(n)>\log(x)$ for $n-1\leq x<n$.
Similarly:
$$\int_n^{n+1}\log(x)dx>\int_n^{n+1}\log(n)dx=\log(n)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1261723",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
ADMM formalization I found lots of examples of ADMM formalization of equality constraint problems (all with single constraint). I am wondering how to generalize it for multiple constraints with mix of equality and inequality constraints.
I have a problem
Minimize $||A_x||_1 + \lambda ||A_y||_2 $, such that:
$$A_x X ... | One way to formulate the problem using ADMM is to let the ADMM-variable $X$ contain $A_x$ and $A_y$, i.e. $X = [A_x; A_y]$ (semi-colon denotes stacking, as in Matlab etc.), and let $Z=[Z_1; Z_2; Z_3; Z_4]$ contain four blocks corresponding to $A_x$, $A_x$, $A_x$ and $A_y$ respectively. (I will write $Q$ for $X-I$, wher... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Would this theorem also work for any integer $n$, not necessarily a prime ?
Would this theorem also work for any integer $n$, not necessarily a prime ?
I don't see why it should not, can you verify it or do you have an counterexample for a nonprime integer ?
| In short, it depends on the notion of irreducibility.
In a commutative rings that are not domains, there are problems with divisibility - or, the situation is simply a bit more complicated: one gets several different notions of associated elements, thus, several notions of irreducible elements, etc.
Just to demonstrat... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Find the Fourier Transform of $2x/(1+x^2)$ I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.
| Hint: Taking the derivative with respect to $k$ of $$F(k)=\int_{-\infty}^{\infty}\frac{1}{1+x^2}e^{ikx}dx$$
yields
$$F'(k)=i\int_{-\infty}^{\infty}\frac{x}{1+x^2}e^{ikx}dx$$
Thus, the Fourier Transform of $\frac{2x}{1+x^2}$ is $-2i$ times the derivative with respect to $k$ of the Fourier Transform of $\frac{1}{1+x^2}$.... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1261977",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Derivatives - optimization (minimum of a function) For which points of $x^2 + y^2 = 25$ the sum of the distances to $(2, 0)$ and $(-2, 0)$ is minimum?
Initially, I did $d = \sqrt{(x-2)^2 + y^2} + \sqrt{(x+2)^2 + y^2}$, and, by replacing $y^2 = 25 - x^2$,
I got $d = \sqrt{-4x + 29} + \sqrt{4x + 29}$, which derivative d... | For better readability, $$S=\sqrt{29+4x}+\sqrt{29-4x}$$
$$\dfrac{dS}{dx}=\dfrac2{\sqrt{29+4x}}\cdot4-\dfrac2{\sqrt{29-4x}}\cdot4$$
For the extreme values of $S,$ we need $\dfrac{dS}{dx}=0\implies29+4x=29-4x\iff x=0$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1262073",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Area enclosed by an equipotential curve for an electric dipole on the plane I am currently teaching Physics in an Italian junior high school. Today, while talking about the electric dipole generated by two equal charges in the plane, I was wondering about the following problem:
Assume that two equal charges are placed... | Here is another method based on the curve-linear coordinates introduced by Achille Hui. He introduced the following change of variables
$$\begin{align}
\sqrt{(x+1)^2+y^2} &= u+v\\
\sqrt{(x-1)^2+y^2} &= u-v
\end{align} \tag{1}$$
Then solving for $x$ and $y$ we shall get
$$\begin{align}
x &= u v\\
y &= \pm \sqrt{-(u^2-1)... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1262174",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "20",
"answer_count": 4,
"answer_id": 2
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How can I solve the integral $ \int {1 \over {x(x+1)(x-2)}}dx$ using partial fractions? $$ \int {1 \over {x(x+1)(x-2)}}dx$$
$$ \int {A \over x}+{B \over x+1}+{C \over x-2}dx $$
I then simplified out and got:
$$1= x^2(A+B+C) +x(C-2B-A) -2A$$
$$A+B+C=0$$
$$C-2B-A=0$$
$$A=-{1 \over 2}$$
However, I'm stuck because I don't ... | Generally you want to avoid simultaneous equations. So rather than collect coefficients of powers of $x$ as you have done, write it as $1=A(x+1)(x-1)+Bx(x-1) +Cx(x+1)$. Since this is an identity, you can substitute any value of $x$ into this. Therefore substitute values which will make brackets disappear. For example, ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1262263",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
How to find a basis of a linear space, defined by a set of equations Problem
Find a basis of the intersection $P\cap Q$ of subspaces $P$ and $Q$ given by:
$$ P:
\begin{cases}
x_1 - 2 x_2 + 2 x_4=0,\\
x_2 - x_3 + 2 x_4 = 0
\end{cases}
\qquad Q:
\begin{cases}
-2 x_1 + 3 x_2 + x_3 -6 x_4=0,\\
x_1 - x_2 - x_3 + 4 x_4 = 0
\... | Having:
$$
\begin{cases}
x_1 = 2 x_3 - 6 x_4,\\
x_2 = x_3 - 2 x_4\\
\end{cases}
$$
We could set
1) For the first element of basis:
\begin{split}
x_3=1,\quad x_4=0:\\
x_1 = 2\cdot 1 - 6\cdot 0 = 2,\\
x2 = 1 - 2 \cdot 0 = 1
\end{split}
So, we get: (2, 1, 1, 0).
2) Second element of basis
\begin{split}
x_3=0,\quad x_4=1:... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1262342",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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The residue of $9^{56}\pmod{100}$ How can I complete the following problem using modular arithmetic?
Find the last two digits of $9^{56}$.
I get to the point where I have $729^{18} \times 9^2 \pmod{100}$. What should I do from here?
| By Carmichael's function, the order of any residue of $100$ divides, and is a maximum of, $20$ (the same as for $25$). We see that $9$ is a square, so the maximum order of $9 \bmod 100$ is $10$ (or divides $10$ if it is less). This gives
$$9^{56}\equiv 9^6 \bmod 100$$
Then we can simply multiply a few small powers:
$$9... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1262409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
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Set theory (containing Power Set) Need Help in a proof I am confirming whether my proof is correct or not and need help.
If $ A \subseteq 2^A , $ then $ 2^A \subseteq 2^{2^A} $
Proof:
Given: $ \forall x ($ $ x\in A \rightarrow \exists S $ where $ S \in 2^A \wedge x \in S )$ --($0$)
Goal: $ \forall S \forall ... | Here is how I would write down this proof, in a way which makes clear the inherent symmetry. (Ignore the red coloring for now, I will use that below.)$
\newcommand{\calc}{\begin{align} \quad &}
\newcommand{\op}[1]{\\ #1 \quad & \quad \unicode{x201c}}
\newcommand{\hints}[1]{\mbox{#1} \\ \quad & \quad \phantom{\unicode{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1263483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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How to calculate the integral $I=\int\limits_0^1\frac{x^n-1}{\ln(x)} \,\mathrm dx$ How can we calculate this integral:
$$I=\int\limits_0^1\frac{x^n-1}{\ln(x)}\,\mathrm dx$$
I believe that integral is equal to $\ln(n+1)$, but I don't lnow how to prove it.
| Let: $$I(n)=\int_0^1\dfrac{x^n-1}{\ln x}dx$$
Then: \begin{align}I'(n)&=\dfrac{d}{dn}\int_0^1\dfrac{x^n-1}{\ln x}dx=\int_0^1\dfrac{\partial}{\partial n}\left[\dfrac{x^n-1}{\ln x}\right]dx\\
&=\int_0^1 x^n dx=\left.\dfrac{x^n+1}{n+1}\right\vert_0^1\\
&=\dfrac{1}{n+1} \end{align}
Therefore: $$I(n)=\int I'(n)=\ln(n+1)$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1263568",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
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Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets? Let $G$ be a finite group, and $g$ be a an element of $G$. How could we go about proving $g^{|G|}=e$ without using cosets? I would admit Lagrange's theorem if a proof without talking about cosets can be found.
I have a proof for abel... | Let me take a shot-
Let $o(g)=n$ for some arbitrary $g \in G$, then $g^n=e$ (and $n$ is least such positive integer), now if suppose $g^{|G|}\neq e$, then there exist there exist $t \in \mathbb{Z}$ which is also greater than $1$ such that $g^{|G|t} = e$, but then by division algorithm $\exists \ $ $q,r \in \mathbb{Z}$... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 2
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Why must $n$ be even if $2^n+1$ is prime? This is a necessary step in a problem I am working on.
| $n$ has to be a power of $2$.
suppose $n$ is not a power of $2$, then let $n=dk$ with $d$ odd.
Then $2^n+1=(2^k)^d+1=(2^k)^d+1^d$ ,now use the high school factorization for $x^d +y^d$ which is true when $d$ is odd that says $x^d+y^d=(x+y)(x^{d-1}-x^{d-2}y+x^{d-3}y^2\dots +y^{d-1})$.
In this case $x=2^k$ and $y=1$.
We ... | {
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Invertible skew-symmetric matrix I'm working on a proof right now, and the question asks about an invertible skew-symmetric matrix. How is that possible? Isn't the diagonal of a skew-symmetric matrix always $0$, making the determinant $0$ and therefore the matrix is not invertible?
| No, the diagonal being zero does not mean the matrix must be non-invertible. Consider $\begin{pmatrix} 0 & 1 \\ -1 & 0 \\ \end{pmatrix}$. This matrix is skew-symmetric with determinant $1$. Edit: as a brilliant comment pointed out, it is the case that if the matrix is of odd order, then skew-symmetric will imply singu... | {
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Evaluating trigonometric limit: $\lim_{x \to 0} \frac{ x\tan 2x - 2x \tan x}{(1-\cos 2x)^2}$
Evaluate $\lim_{x \to 0} \cfrac{ x\tan 2x - 2x \tan x}{(1-\cos 2x)^2} $
This is what I've tried yet:
$$\begin{align} & \cfrac{x(\tan 2x - 2\tan x)}{4\sin^4 x} \\
=&\cfrac{x\left\{\left(\frac{2\tan x}{1-\tan^2 x} \right) - 2\... | Well, let's try something different from using power series expansions. Here, we simplify using trigonometric identities to reveal that
$$\frac{x\tan 2x-2x \tan x}{(1-\cos 2x)^2}=\frac{2x}{\sin 4x }=\frac{1}{2\text{sinc}(4x)}$$
where the sinc function is defined as $\text{sinc}(x)=\frac{\sin x}{x}$.
The limit as $x \... | {
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The definition of span In Linear Algebra by Friedberg, Insel and Spence, the definition of span (pg-$30$) is given as:
Let $S$ be a nonempty subset of a vector space $V$. The span of $S$,
denoted by span$(S)$, is the set containing of all linear
combinations of vectors in $S$. For convenience, we define
span$(\e... | Let $S$ be a non-empty subset of a vector space $V$. The the set of all linear combinations of finite sets of elements of $S$ is called the linear span of $S$ and is denoted by $L(S)$.
| {
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Defining Equivalence relations So I am not really comfortable with equivalence relations, so this example from Wikipedia gives me trouble. Here is what it says:
Let the set $\{a,b,c\}$ have the equivalence relation $\{(a,a),(b,b),(c,c),(b,c),(c,b)\}$.
So it can also be $\lbrace(a,a), (b,b), (c,c) \rbrace$ if I so choos... | Your relation is indeed an equivalence relation:
Reflexivity clearly holds, as symmetry does. Transitivity does hold since
$$\forall x,y,z\in X: xRy \wedge yRz \implies xRz $$
is a true statement: $xRy \wedge yRz$ is true in this case if and only if $x=y=z$, in which case clearly (reflexivity) $xRz$.
EDIT: Simpler argu... | {
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Is it true that $H = H_1 \times H_2\times \dots \times H_r$? Suppose that $G= G_1 \times \dots\times G_r$ be a decomposition of group $G$ into its normal subgroups. Let $H_i \leq G_i$ for every $i$. We know that for every $i \neq j$, we have $[G_i, G_j]=1$ and so $[H_i, H_j]=1$.
a) Is it true that $H := H_1 H_2 \cdots... | Yes, note that that set is closed under the group operation and inverses since all the subgroups commute with each other, so it is a subgroup.
More explicitly consider $(h_1\cdots h_r)(k_1\cdots k_r)= (h_1k_1)(h_2k_2)\cdots(h_rk_r)$.
| {
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Can we have a one-one function from [0,1] to the set of irrational numbers? Since both of them are uncountable sets, we should be able to construct such a map. Am I correct?
If so, then what is the map?
| Both sets $[0,1]$ and $[0,1]\setminus\mathbb Q$ have the same cardinality $\mathfrak c=2^{\aleph_0}$, so there is a bijection between them.
If you want write down some explicit bijection, you can use basically the standard Hilbert's hotel argument which shows that if $|A|\ge\aleph_0$ then $|A|+\aleph_0=|A|$.
So let us... | {
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Adding rows to calculate the determinant.
Evaluate the determinants given that $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}=-6.$
*
*$\begin{vmatrix} a+d & b+e & c+f \\ -d & -e & -f \\ g & h & i \end{vmatrix}$
*$\begin{vmatrix} a & b & c \\ 2d & 2e & 2f \\ g+3a & h+3b & i+3c \end{vmatrix}$
... | Geometrically, the fact that you can add multiples of rows to each other while keeping the determinant the same, is a reflection of the fact that the determinant can be seen as the volume of the parallelepiped with the rows or columns as it's vectors.
Adding a row to a row has the effect of simply skewing the parallele... | {
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Finding particular solution to $y'' + 2y' - 8y = e^{2x} $ $$y'' + 2y' - 8y = e^{2x} $$
How do I find the particular solution?
I tried setting: $y = Ae^{2x} => y' = 2Ae^{2x} => y''= 4Ae^{2x}$
If I substitute I get: $4Ae^{2x} + 4Ae^{2x} - 8Ae^{2x} = e^{2x} => 0 = e^{2x}$
What am I doing wrong?
| You already received answers.
When you have a doubt (and when your work leads to something as $0=e^{2x}$ as you honestly pointed out), just do what you did but considering now that $A$ is a function of $x$. So, $$y=A\,e^{2x}$$ $$y'=A'\,e^{2x}+2A\,e^{2x}$$ $$y''=A''\,e^{2x}+4A'\,e^{2x}+4A\,e^{2x}$$ Plug all of that in t... | {
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Find the value of the given limit.
The value of $\lim_{x\to \infty} (x+2) \tan^{-1} (x+2) - x\tan^{-1} x $ is $\dots$
a) $\frac{\pi}{2} $ $\qquad \qquad \qquad$ b) Doesn't exist $\qquad \qquad \qquad$ c) $\frac{\pi}{4}$ $\qquad \qquad$ d)None of the above.
Now, this is an objective question and thus, I expect that th... | Set $1/x=h$ to get $$\lim_{h\to0^+}\dfrac{(1+2h)\tan^{-1}\dfrac{1+2h}h-\tan^{-1}\dfrac1h}h$$
$$=\lim_{h\to0^+}\dfrac{\tan^{-1}\dfrac{1+2h}h-\tan^{-1}\dfrac1h}h+2\lim_{h\to0^+}\tan^{-1}\dfrac{1+2h}h$$
Now,
$$\tan^{-1}\dfrac{1+2h}h-\tan^{-1}\dfrac1h=\tan^{-1}\left[\dfrac{\dfrac{1+2h}h-\dfrac1h}{1+\dfrac{1+2h}h\cdot\dfrac... | {
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Inverse Rule for Formal Power Series I am just really starting to get into formal power series and understanding them. I'm particularly interested in looking at the coefficients generated by the inverse of a formal power series:
$$\left(\sum_{n\ge 0}a_nx^n\right)^{-1}=\sum_{n\ge 0}b_nx^n$$
I first thought that my appr... | As closed a form as I could get I posted here. It looks pretty ugly... but I'm not sure how much prettier it can get.
| {
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What does $\sim$ in $X\sim \mathcal{N}(\mu,\sigma^{2})$ really mean? This is a bit of a silly question, but I can't seem to find the answer anywhere.
I feel like $X\sim \mathcal{N}(\mu,\sigma^{2})$ means that $\sim$ is a relation, but if it is a relation, what precisely is this relation?
If this is a relation, could I ... | It means "distributed like". It's just a shorthand way of saying "$X$ assumes a normal distribution with these parameters."
| {
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Let $a_n$ be defined inductively by $a_1 = 1, a_2 = 2, a_3 = 3$, and $a_n = a_{n−1} + a_{n−2} + a_{n−3}$ for all $n \ge 4$. Show that $a_n < 2^n$.
Suppose that the numbers $a_n$ are defined inductively by $a_1 = 1, a_2 = 2, a_3 = 3$, and $a_n = a_{n−1} + a_{n−2} + a_{n−3}$ for all $n \geq 4$. Use the Second Principle... | Hint: Consider instead the sequence defined by the same initial values and $\displaystyle a_n = \sum_{k=0}^{n-1} a_k$ for $n \geq 4$.
| {
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Prove that $f_n(x)=\frac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$ Title says it all; I have to prove that the function sequence $f_n(x)=\dfrac{1-(x/b)^n}{1+(a/x)^n}$ is uniformly convergent on the interval $[a+\epsilon ; b - \epsilon]$, with $0<\epsilon<(b-a)/2$ - I'v... | For all $x \in [a - \epsilon, b + \epsilon]$,
$$|f_n(x) - 1| = \frac{(a/x)^n + (x/b)^n}{1 + (a/x)^n} \le (a/x)^n + (x/b)^n \le \left(\frac{a}{a + \epsilon}\right)^n + \left(\frac{b - \epsilon}{b}\right)^n.$$
Thus
$$\sup_{x\in [a-\epsilon,b+\epsilon]} |f_n(x) - 1| \le \left(\frac{a}{a+\epsilon}\right)^n + \left(\frac{b... | {
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I cannot solve this limit $$
\lim_{n\to\infty}\frac{(\frac{1}{n}+1)^{bn+c+n^2}}{e^n}=e^{b-\frac{1}{2}}
$$
I am doing it like this, and I cannot find the mistake:
$$
\lim_{n\to\infty}\frac{1}{e^n}e^{n+b+c/n}=
\lim_{n\to\infty}e^{n+b-n+c/n}=e^b
$$
| hint: $a_n=\left(\dfrac{\left(1+\frac{1}{n}\right)^n}{e}\right)^n \Rightarrow \ln (a_n) = \dfrac{\ln(1+\frac{1}{n})-\frac{1}{n}}{\frac{1}{n^2}}\to 0$ by L'hospitale rule with $x = \frac{1}{n} \to 0 \Rightarrow a_n \to 1$. Given that the answer is $e^{b-1/2}$, can you figure the other part ?
| {
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Entropy Calculation and derivation of logarithm I have probabilities as
$$p_1 = 0.4,\ p_2 = 0.3,\ p_3=0.2,\ p_4=0.1$$
hence entropy is given by:
$$H(x) = -\big(0.4\cdot \log_2(0.4) + 0.3\cdot \log_2(0.3) + 0.2\cdot \log_2(0.2) + 0.1\cdot \log_2(0.1)\big)$$
I derive this to
$$H(x) = -\big(1 - \log_2(10) + 0.3\cdot \log... | You could take $$\begin{align}-\big(1 - \log_2(10) + 0.3\cdot \log_2(3)\big) = -\big(1 - \log_2(2\cdot 5) + 0.3\cdot \log_2(3)\big) \\ = -\big(1 - 1-\log_2(5) + 0.3\cdot \log_2(3)\big) \\ =\log_2(5)-0.3\cdot \log_2(3)\end{align}$$ I don't think there is much left you can do with this besides stick it into a calculator... | {
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Can the range of a variable be inclusive infinity? Can a range be $[0, \infty]$ or must it be $[0, \infty)$ because you can never quite reach infinity?
Clarification:
$[0, 1]$ means $0 \leqslant x \leqslant 1 $, while $(0, 1)$ means $0 < x < 1 $. My question is whether infinity can be written as inclusive when statin... | An example of where this is actually used are measure functions. For instance, the Lebesgue measure, $\lambda$ on (certain subsets of) $\Bbb R$ has the property that $$\lambda((a,b)) = b-a $$
when $a,b$ are finite, while, for example $$\lambda(\Bbb R) = \infty$$.
| {
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Why represent a complex number $a+ib$ as $[\begin{smallmatrix}a & -b\\ b & \hphantom{-}a\end{smallmatrix}]$? I am reading through John Stillwell's Naive Lie Algebra and it is claimed that all complex numbers can be represented by a $2\times 2$ matrix $\begin{bmatrix}a & -b\\ b & \hphantom{-}a\end{bmatrix}$.
But obvious... | Matrix representation of complex numbers is useful and advantageous because we can discover and explore new concepts, like this:
$\begin{bmatrix}\hphantom{-}a & b\\ -b & a\end{bmatrix}$ ---> complex numbers
$\begin{bmatrix}\hphantom{-}a & b\\ \hphantom{-}0 & a\end{bmatrix}$ ---> dual numbers
$\begin{bmatrix}\hphantom{-... | {
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Write 100 as the sum of two positive integers
Write $100$ as the sum of two positive integers, one of them being a multiple of $7$, while the other is a multiple of $11$.
Since $100$ is not a big number, I followed the straightforward reasoning of writing all multiples up to $100$ of either $11$ or $7$, and then find... | While certainly not the ideal solution, this problem is certainly in the realm of Integer Programming. As plenty of others have pointed out, there are more direct approaches. However, I suspect ILP solvers would operate quite efficiently in your case, and requires less 'thought capital'.
| {
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Using trigonometry to predict future position Intro
I'm currently creating an AI for a robot whose aim is to shoot another robot.
All I want to do is to be able to calculate at what angle to shoot my bullet, so that it hits my enemy, with the assumption that the enemy continues moving at the same bearing and velocity.
... | Name the point where you robot shoots from $P$, the point of departure of the enemy robot $Q$, and the point where the bullet hits the enemy $R$. In triangle $PQR$ we then have
$$
\angle Q=90^o+(180^o-90^o-a)+b=180^o+b-a
$$
and from the law of sines we know that
$$
\frac{q}{\sin Q}=\frac{p}{\sin\theta}\iff \frac{\sin\t... | {
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Result of solving an unsolved problem? I know that when some of the previously unsolved problems were solved they created new fields in mathematics. May someone explain to me what would be the result of a major problem like the Hodge Conjecture being solved vs a "smaller" problem like "Do quasiperfect numbers exist?" i... | Most of the time, the actual result isn't important as the theory. The reason why problems are unsolved is because either the math doesn't exist yet, or some connection between current fields has not been established yet.
Either way, creating new math and connecting existing math are the real reasons why solving open ... | {
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Chromatic number $\chi(G)=600$, $P(\chi(G|_S)\leq 200) \leq 2^{-10}$ I am learning martingale and Hoeffding-Azuma inequality recently but do not how to apply the those inequality or theorem here.
Let $G=(V,E)$ be a graph with chromatic number 600,i.e. $\chi(G)=600$. Let $S$ be a random subset uniformly chosen from $V$.... | Hints:
*
*If you choose a set at random, then with probability at least 1/2 the chromatic number is at least 300 (why?).
*Azuma's inequality shows that the chromatic number of $G|_S$ is concentrated around its mean.
*Since the chromatic number is always between 0 and 600, and its median is above 300, its mean can'... | {
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A formal proof that the function $ x \mapsto x^{2} $ is continuous at $ x = 4 $. Problem: Show $f(x)=x^2 $ is continuous at $ x = 4$.
That is to say, find delta such that:
$ ∀ε>0$ $ ∃δ>0 $ such that $ |x-a|<δ ⇒ |f(x)-f(a)|<ε$
Where $a=4$, $f(x)=x^2$,and $f(a)=16$.
So in order to do to do these delta/epsilon proofs, I... | $|x^2|<16+ε$ does not imply $|x^2-16|<ε$
For example, take $x= 0 , \epsilon = 1$.
Then $|x^2| = 0 < 17 = 16+ε$
But $|x^2-16| = |0^2-16| = 16 \geq 1 = ε $.
So in the current form, your proof is invalid.
| {
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Can some explain very quickly what $ |5 x + 20| = 5 $ actually means? I don’t mean to bother the community with something so easy, but for the life of me, I can’t remember how to do these. I even forgot what they were called, and I’m referring to the use of the “$ | $” symbol in math. I think I may have been studying t... | $$|5x+20|=5$$
We want to consider both $5$ and $-5$ for this absolute value function. So we solve for both $5$ and $-5$.
$$5x+20=5$$
$$5x=-15$$
$$\boxed{x=-3}$$
$$5x+20=-5$$
$$5x=-25$$
$$\boxed{x=-5}$$
If we plug $-3$ and $-5$ back into the original equation, we should get an equivalent statement. So the final answer i... | {
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Showing that $\sin(x) + x = 1$ has one, and only one, solution Problem:
Prove that the equation $$\sin(x) + x = 1$$ has one, and only one solution. Additionally, show that this solution exists on the interval $[0, \frac\pi2$]. Then solve the equation for x with an accuracy of 4 digits.
My progress:
I have no problems v... | Derivative of the LHS is always greater than or equal to zero and it is zero only at points that $\cos(x) = -1$ which is $x = 2n\pi+\pi$. So, the function is always increasing which crosses level $1$ at only one point. Since $\sin (x) + x$ is more than $1$ at point $\frac{\pi}{2}$ and $0$ at point $0$, it should have o... | {
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Where is the mistake in proving 1+2+3+4+… = -1/12? https://www.youtube.com/watch?v=w-I6XTVZXww#t=30
As I watched the video on YouTube of proving sum of $$1+2+3+4+\cdots= \frac{-1}{12}$$
Even we know that the series does not converge.
First I still can't prove what wrong in infinite sum of $$1-1+1-1+1+\cdots= \frac{1}{2... | In proving that 1-1+1-1+... = 1/2, you add two divergent series, which can sometimes produce a convergent series. However, here, (I'll mark our sum I = 1-1+1-1+...) I+I = 2*I = 2-2+2-2+... and that still diverges, although we shifted one of those to get that 2*I = 1. That's a contradiction.
That is the main problem wit... | {
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Famous Problems the Experts Could not Solve After Yitang Zhang stunned the mathematics world by establishing the first finite bound on gaps between prime numbers, it got me thinking about the following question:
$\underline{\text{Question}}:$ What are other examples of proofs provided by younger, less accomplished math... | Kurt Heegner showed the Stark-Heegner theorem. At that time, he wasn't connected to any university, in fact, no one looked to his proof until Stark showed the same result. He wasn't young, but Yitang Zhang wasn't either.
| {
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Finding the inverse of a number under a certain modulus How does one get the inverse of 7 modulo 11?
I know the answer is supposed to be 8, but have no idea how to reach or calculate that figure.
Likewise, I have the same problem finding the inverse of 3 modulo 13, which is 9.
| ${\rm mod}\ 11\!:\,\ \dfrac{1}7\equiv \dfrac{12}{-4}\equiv -3\ $ (see Gauss's algorithm. for an algorithmic version of this).
Or, compute the Bezout identity $\,\gcd(11,7) = 2(11)-3(7) = 1\,$ by the Extended Euclidean Algorithm $ $ (see here for a convenient version). $ $ Thus $\ {-}3(7)\equiv 1\pmod{11}$
${\rm mod}\ 1... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1266282",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Proving Convergence and Absolute Convergence of Power Series How do you prove the following claim?
If a power series $\sum_{n=0}^{\infty} a_n (x-a)^n$ converges at some point $b ≠ a$, then this power series converges absolutely at every point closer to $a$ than $b$ is.
Here's what I tried so far. Does this proof make s... | There's something a bit strange about your proof. Look at the line after "Rearranging, we get:".. That inequality you say is because of (*) but if you cancel $\frac{|(x-a)^n|}{|(b-a)^n|}$ it's simply equivalent to $|a_n(b-a)^n|<1$. So you don't need to go to all that work to get to this inequality.
Also, how do you ... | {
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"url": "https://math.stackexchange.com/questions/1266382",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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} |
How to prove that $I+A^{T}A$ is invertible
Let $A$ be any $m\times n$ matrix and $I$ be the $n\times n$ identity.
Prove that $I+A^{T}A$ is invertible.
| The matrix $I+A^TA$ is invertible because it is positive definite: if $v\in\mathbb{R}^n$ is nonzero, then
$$
v^T(I+A^TA)v=v^Tv+v^TA^TAv=|v|^2+|Av|^2\geq |v|^2>0.
$$
In particular, all the eigenvalues are positive and their product must too be positive (i.e. nonzero).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1266495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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} |
What are "tan" and "atan"? As the title says, I'm confused on what tan and atan are. I'm writing a program in Java and I came across these two mathematical functions. I know tan stands for tangent but if possible could someone please explain this to me. I have not taken triginomotry yet (I've taken up to Algebra 1) so ... | A quick google search of "java atan" would tell you that it stands for "arctangent", which is the inverse of tangent. Tangent is first understood as a ratio of non-hypotenuse sides of a right triangle. Given a non-right angle $x$ of a right triangle, $\tan(x)$ is the ratio $\frac{o}{a}$ where $o$ is the length of the... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Is a polynomial ring over a UFD in countably many variables a UFD? Let $R$ be a UFD. It is well know that $R[x]$ is also a UFD, and so then is $R[x_1,x_2,\cdots,x_n]$ is a UFD for any finite number of variables. Is $R[x_1,x_2,\cdots,x_n,\cdots]$ in countably many variables also a UFD? If not, what about if we take $\ti... | Hint: first show that for any $n\in\mathbb N$ and any $f\in R[x_1,x_2,\cdots,x_n]\subset R[x_1,x_2,\cdots]$, $f$ is irreducible in $R[x_1,x_2,\cdots]$ iff it's so in $R[x_1,x_2,\cdots,x_n]$; then notice any element of $R[x_1,x_2,\cdots]$ lies in $R[x_1,x_2,\cdots,x_n]$ for some $n$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1266784",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Finding the eigenvectors of a matrix that has one eigenvalue of multiplicity three This is a simple question, which hopefully has a quick answer. I have a given matrix A, such that
\begin{equation} A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix} \end{equation}
Since it's fairly straightforward, I'... | Your answer is correct, and Matlab is being problematic. Here's some discussion of why this isn't so surprising:
Defective eigenvalue problems are numerically problematic. This is because diagonalizable matrices are dense in the space of all matrices. Consequently an arbitrarily small arithmetic error can make a nondia... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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$\langle A,B\rangle = \operatorname{tr}(B^*A)$ "define the inner product of two matrices $A$ and $B$ in $M_{n\times n}(F)$ by $$\langle A,B \rangle = \operatorname{tr}(B^*A), $$ where the {conjugate transpose} (or {adjoint}) $B^*$ of a matrix $B$ is defined by $B^*_{ij} = \overline{B_{ji}}.$
Prove that $\langle B,A \ra... | Hint: $\operatorname{tr}(AB)=\operatorname{tr}(BA)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1266928",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Something screwy going on in $\mathbb Z[\sqrt{51}]$ In $\mathbb Z[\sqrt{6}]$, I can readily find that $(-1)(2 - \sqrt{6})(2 + \sqrt{6}) = 2$ and $(3 - \sqrt{6})(3 + \sqrt{6}) = 3$. It looks strange but it checks out.
But when I try the same thing for $3$ and $17$ in $\mathbb Z[\sqrt{51}]$ I seem to run into a wall. I c... | There is one crucial difference between $\mathbb{Z}[\sqrt{6}]$ and $\mathbb{Z}[\sqrt{51}]$: one is a unique factorization domain, the other is not. You have to accept that some of the tools that come in so handy in UFDs are just not as useful in non-UFDs.
One of those tools is the Legendre symbol. Given distinct primes... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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I can't find the critical points for this function. I showed my work :) So, I have to find Critical Points of $y=\frac{1}{(x^3-x)}$
I know the derivative.
Derivative = $(3x^2-1)/(x^3-x)^2$
To find Critical Points I equal to $0$.
$x=1/\sqrt3$ and $x=-1/\sqrt3 $
But Critical points are the Max and Min value of your graph... | The points you found are just local extrema. Regarding the + and - $\infty$, you can simply note that those are local maxima of the function.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267338",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Differential Equations: Recursive Functions Functions I have some familiarity with look so, $y^\prime(x) = \tan(x+2)$: straightforward expression of the first derivative of y as a function of x.
But say I have a function, $y^\prime(x) = \cos{(y)}$? I'm not sure what 'y' is supposed to signify when it's being called re... | You have: $\dfrac{dy}{dx} = \cos y \to \dfrac{dy}{\cos y} = dx \to x = \displaystyle \int \sec ydy$. Can you continue? more generally, you have a separable ODE: $y' = f(y)$, then the way to solve it is "separate" the $x$ and the $y$. like the one I did.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
How to approximate Heaviside function by polynomial I have a Heaviside smooth function that defined as
$$H_{\epsilon}=\frac {1}{2} [1+\frac {2}{\pi} \arctan(\frac {x}{\epsilon})]$$
I want to use polynominal to approximate the Heaviside function. Could you suggest to me a solution? Thanks
UPDATE: This is Bombyx mori re... | It is not possible to use polynomial as Heaviside step function with a good average precision, because any polynomial is infinite at both positive and negative infinity and Heaviside is not. Second, you would need to have zero value for all negative values. Even if you take that the value is only approximately zero you... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
prime numbered currency The unit of currency is the Tao(t) the value of each coin is a prime number of Taos. The coin with the smallest value is 2T there are coins of every prime number Value under 50.
Help! I don't under stand if the question means that 1coin equals 2T.
Some pointers in the right direction would be ve... | The T after the number 2 isn't a variable in this case - it's the unit. Is that what the problem was?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267591",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Find $P + Q + R$ I was doing questions from previous year's exam paper and I'm stuck on this question.
Suppose $P, Q, R$ are positive integers such that $$PQR + PQ + QR + RP + P + Q + R = 1000$$ Find $P + Q + R$.
It seems easy but I am not getting the point from where I should start. Thank you.
| Note that
$(1 + P) (1 + Q) (1 + R) = 1 + P + Q + R$
$+ PQ+ PR + QR + PQR = 1 + 1000 = 1001, \tag{1}$
by the hypothesis on $P$, $Q$, $R$; also,
$1001 = 7 \cdot 11 \cdot 13, \tag{2}$
all primes; thus we may take
$P = 6; \;\; R = 10; \;\; Q = 12, \tag{3}$
or some permutation thereof; in any event, we have
$P + Q + R = 28... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
} |
Universal quadratic formula? Is there any way to write the quadratic formula such that it works for $ac= 0$ without having to make it piecewise?
The traditional solution of $x = (-b \pm \sqrt{b^2 - 4ac}) / 2a$ breaks when $a = 0$, and the less-traditional solution of $x = 2c / (-b \pm \sqrt{b^2 - 4ac})$ breaks when $c ... | Answering my own question, but I just realized this algorithm on Wikipedia works if we cheat a little and don't consider $\operatorname{sgn}(x) = |x| \div x$ a "piecewise" function:
$${\begin{aligned}x_{1}&={\frac {-b-\operatorname{sgn}(b)\,{\sqrt {b^{2}-4ac}}}{2a}}\\x_{2}&={\frac {2c}{-b-\operatorname{sgn}(b)\,{\sqrt ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 1
} |
Are sections of vector bundles $E\to X$ over schemes necessarily closed embeddings? A brief question, if $\pi\colon E\to X$ is a vector bundle over a scheme $X$, is it automatic that any section $s\colon X\to E$ always a closed embedding?
| First, a general observation:
If $p : Y \to X$ is a separated morphism of schemes and $s : X \to Y$ satisfies $p \circ s = \mathrm{id}_X$, then $s$ is a closed embedding.
Indeed, under the hypotheses, we have the following pullback diagram,
$$\require{AMScd}
\begin{CD}
X @>{s}>> Y \\
@V{s}VV @VV{\Delta}V \\
Y @>>{\la... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267827",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Galois group of the simple extension Let $K=Q(\sqrt3,\sqrt5)$. Show that the extension is $K/Q$ is simple and also Galois extension. Determine its Galois group.
I showed that extension is simple because $K=Q(\sqrt{3}+\sqrt{5})$
But I can not find its Galois group?
| Hint: Notice that any $\sigma \in \mathrm {Aut} L$ is determined by the images of $\sigma (\sqrt{3})$ and $\sigma (\sqrt 5)$ then the possibilities are $\sigma (\sqrt 3) \in \{-\sqrt 3, \sqrt 3 \}$ and $\sigma (\sqrt 5) \in \{-\sqrt 5, \sqrt 5\}$.
Can you see how this is related to $\mathbb Z_2 \times \mathbb Z_2$?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/1267913",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Gravitational force inside a uniform solid ball - evaluation of the integral in spherical coordinates - mistake I have been reading this PDF document: www.math.udel.edu/~lazebnik/BallPoint.pdf
While trying Case A I found a small error (a $2$ was missing) but I was able to follow the argumentation and got to the solutio... | By symmetry,
$$
\int_{0}^{\pi} \sqrt{a^{2}\cos^{2} \phi + R^{2} - a^{2}} \cos\phi \sin\phi\, d\phi = 0.
$$
Loosely, the radical and $\sin\phi$ are "even-symmetric" on $[0, \pi]$, while $\cos\phi$ is "odd-symmetric". That is, substituting $\psi = \phi - \frac{\pi}{2}$ gives
$$
\int_{0}^{\pi} \sqrt{a^{2}\cos^{2} \phi + R... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268052",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Differentiate $\sin^{-1}\left(\frac {\sin x + \cos x}{\sqrt{2}}\right)$ with respect to $x$
Differentiate $$\sin^{-1}\left(\frac {\sin x + \cos x}{\sqrt{2}}\right)$$ with respect to $x$.
I started like this: Consider $$\frac {\sin x + \cos x}{\sqrt{2}}$$, substitute $\cos x$ as $\sin \left(\frac {\pi}{2} - x\right)$,... | An alternative approach is to use Implicit Differentiation:
\begin{equation}
y = \arcsin\left(\frac{\sin(x) + \cos(x)}{\sqrt{2}} \right) \rightarrow \sin(y) = \frac{\sin(x) + \cos(x)}{\sqrt{2}}
\end{equation}
Now differentiate with respect to '$x$':
\begin{align}
\frac{d}{dx}\left[\sin(y) \right] &= \frac{d}{dx}\left[... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268153",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 2
} |
How to integrate $\int \cos^2(3x)dx$ $$\int \cos^2(3x)dx$$
The answer according to my instructor is:
$${1 + \cos(6x) \over 2} + C$$
But my book says that:
$$\int \cos^2(ax)dx = {x \over 2} + {\sin(2ax) \over 4a} + C$$
I'm not really sure which one is correct.
| There are two methods you can use. Integration by parts and solving for the integral, or the half angle formula.
Remember that the half angle formula is given by $\cos^2(x) = \frac12 (1+\cos(2x))$ and also $\sin^2(x) = \frac12(1-\cos(2x))$.
Thus $$\int \cos^2(3x) dx = \frac12 \int (1+\cos(6x)) dx = \frac12 \left(x + \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/1268278",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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