Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
About lower upper bound of a subset of all topologies for a non empty set X
How i may prove what is stated? That the topology generated using the theorem D as in the image, is equal to the lower upper bound of a non-empty family of topologies for a non empty set $X$.
Definitions:
The lower upper bound of a non-em... | If you want to prove it in detail, all you have to show is that last statement:
the class of all unions of finite intersections forms a topology
which is to say, this class is closed under the formation of finite intersection and arbitrary unions.
Doesn't it follow by construction?
(Notice that $\bigcup \varnothing... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2072807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Calculating the limit of $\frac{x^2\sin(\frac{1}{x})}{\sin x}$ $$ \lim _{x\to 0} \frac{x^2\sin(\frac{1}{x})}{\sin x}$$
Okay since sine is bounded ${x^2\sin(\frac{1}{x})} \ \ \to 0$
$\sin x\to 0$ Thus we can apply l'hospitals to it .
Applying l'Hospital's we get :
$$\lim_{x\to 0}\frac{\sin(\frac{1}{x})(1-2x)}{\cos x}$$... | Recall that
$$\lim_{x\to0}\frac{\sin(x)}x=1$$
and
$$-x<x\sin(1/x)<x$$
Thus,
$$\lim_{x\to0}\frac{x^2\sin(1/x)}{\sin(x)}=\lim_{x\to0}\frac x{\sin(x)}x\sin(1/x)=0$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2072914",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 3
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Make a $2$ unit bottle from $4$ and $3$ unit bottles Assume that we have only two bottles. The first one's volume is $3$ units and the second one's volume is $4$ units. We can perform $3$ kinds of operations :
*
*Fill an empty bottle with water.
*Empty a full bottle.
*Pouring the contents of one bo... | The way to solve this kind of problem is generally similar.
Given $a$ and $b$ with $gcd(a,b)=1$ we can apply Bezout to find $(u,v)$ such that $au+bv=1$.
In fact here $1$ (1 liter) is not interesting, but similarly there exists $(u,v)$ such that $au+bv=2$.
Here $4*(+2)+3*(-2) = 8-6=2$, the signs are also of interest:
... | {
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Two mappings with constant composition Can you tell me if my answer is correct? Thank you so much!!!
Here is the problem:
If $f,g$ are mappings of $S$ into $S$ and $f\circ g$ is a constant function, then
(a) What can you say about $f$ if $g$ is onto?
(b) What can you say about $g$ if $f$ is 1-1.
Original Image
As $f\ci... | An informal, yet correct, answer provided you know what all the terms mean can be:
$g$ is onto means $g(S) = \{g(x)|x \in S\}$ can, and will be anything. So $f(g(x))$ can, and will, have $g(x)$ being any value yet $f(g(x))$ is constant so if $g(x)=y$ is .... anything....$f(y)$ is constant. So $f$ is constant.
$f$ is ... | {
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Finding the $k^{th}$ root modulo m We know that method of finding $k^{th}$ root modulo $m$, i.e. if $$x^k\equiv b\pmod m,\tag {$\clubsuit$}$$ with $\gcd (b,m)=1$, and $\gcd(k,\varphi(m))=1$, then $x\equiv b^u\pmod m$ is a solution to $(\clubsuit)$, where $ku-v\varphi(m)=1$. Because
$$\begin{array}
{}x^k &\equiv \left(... | $b^{\large ku}\!\equiv b\pmod{\!pq}\,$ is case $\,i,j,k=1\,$ of this generalization of the Fermat Euler $\color{blue}{\rm (E)}$ theorem.
${\bf Theorem}\,\ \ n^{\large k+\phi}\equiv n^{\large k}\pmod{\!p^i q^j}\ \ $ if $\,p\ne q\,$ are prime, $ \ \color{#0a0}{\phi(p^i),\phi(q^j)\mid \phi},\, $ $\, i,j \le k\ \ \ $
$... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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the set of linear transformations. Let S={T:$\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$:T is a linear
transformation with T(1,0,1)=(1,2,3), T(1,2,3)=(1,0,1)} .Then S is
(a) A Singleton Set
(b) A finite set containing more than one element
(c) A countable set
(d) An uncountable set
MY APPROACH:
T(1,0,1)=(1,2,3)$\Longri... | Hint/Solution: A linear transformation is determined by its values on a basis,hence $S$ is uncountable.(Of course you need to fill some details)
| {
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"timestamp": "2023-03-29T00:00:00",
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Horospherical ham sandwich Let $B_1, \ldots B_n$ be Borel sets in $\mathbb{H}^n$, and $\mu$ (absolutely continuous w.r.t) the hyperbolic volume. Is there a horosphere $H$ that cuts each of the $B_i$'s into two parts with equal $\mu$-measure ?
(For Euclidean space with hyperplanes in place of horosphere this is the folk... | Nice question.This fails however already in the case $n=2$ though, taking $B_1, B_2$ two concentric balls of suitable radii, $r_1, r_2$ with $r_1$ is sufficiently small. To prove this consider first the case when $r_1=0$ and the measure is positive concentrated at one point, the center. Once you understood this case, a... | {
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"timestamp": "2023-03-29T00:00:00",
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A basis to a orthogonal set Let $(E, \langle \cdot, \cdot \rangle)$ be an $n$-dimensional Hilbert space and $A,B \colon E \to E$ linear isomorphisms.
Does there exist a basis $\{e_{1},...,e_{n}\}$ of $E$ such that $\mathcal{B}=\{A(e_{1}),...,A(e_{n}),B(e_{1}),...,B(e_{n})\}$ are a orthogonal set?
Hints or solutions a... | No, it doesn't (for $n\neq 0$). Because a orthogonal set of non-zero vectors is linearly independend. As your space is $n$ dimensional there are at most $n$ linearly independend vectors.
Indeed, let $\{v_1, \dots, v_m \}\subseteq E$ be a orthogonal set of non-zero vectors. Assume they are linearly dependend
$$ \sum_{j=... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Prove that $f$ is Riemann-integrable on $[a,b]$ Assume that $\{r_n\}$ is an index of rational numbers in the interval $[a,b]$ and $\{v_n\}$ is a sequence of non-zero real numbers which converges to $0$.
Define $f:[a,b] \to \mathbb R$ this way :
If $x=r_n$ , $f(x)=v_n$
If $x \notin \mathbb Q \cap [a,b]$ , $f(x)=0$ ... | I think in this way it should work:
Consider for the sake of simplicity $v_n >0$ for each $n \in \mathbb{N}$. Let $ 1 > \epsilon >0$ By definition there exists $n_m \in \mathbb{N}$ such that $v_n < \epsilon$ for each $n \ge n_m$.
Now observe that one can assume that $ a = r_1 < r_2 < . . . < r_{n_m} = b$. Take $\delta$... | {
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Prove $10^{n+1}+3\cdot 10^n+5$ is divisible by $9$? How do I prove that an integer of the form $10^{n+1}+3\cdot 10^{n}+5$ is divisible by $9$ for $n\geq 1$?I tried proving it by induction and could prove it for the Base case n=1. But got stuck while proving the general case. Any help on this ? Thanks.
| $10^{n+1}+3\cdot 10^{n}+5=10^{n}(10+3)+5=1300\cdots05$ has digit sum equal to $9$ and so is a multiple of $9$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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$\lim \limits_{n \to \infty} \frac{\sin x_n}{x_n}$ if $\lim \limits_{n \to \infty} x_n =0$ How to easily prove that
$$\lim \limits_{n \to \infty} \frac{\sin x_n}{x_n}=1,$$
if $\lim \limits_{n \to \infty} x_n =0$?
I proved it using inequality
$$ 1-\frac{x^2}{2}<\frac{\sin x}{x}<1$$
therefore,
$$1\xleftarrow[\text{$x_... | $$
\lim \limits_{x \to 0} \frac{\sin x}{x}=\lim \limits_{x\to0} \frac{\sin x-\sin 0}{x-0}=(\sin x)'|_{x=0}=\cos 0=1.
$$
| {
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"timestamp": "2023-03-29T00:00:00",
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Stuck at evaluating $\sum\limits_{d \mid n} \tau(d)$ It seems the number of nonnegative integer solutions to the equation $xyz=n$ is given by
$$\sum\limits_{d \mid n} \tau(d)$$
$\tau$ is the number of divisors function. I'm wondering if there is a way to simplify this sum. Really appreciate any kind of help. Thank you... | Factor $n$ as $p_1^{\alpha_1}\cdot\ldots\cdot p_k^{\alpha_k}$. Then every solution is associated with three vectors (the exponents in the factorizations of $x,y,z$) with non-negative integer components and sum given by $(\alpha_1,\ldots,\alpha_k)$. By stars and bars, it follows that the number of solutions is given by ... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Maximum number of components in a Graph Containing $n$ vertices and $k$ edges I know that in a Graph $G$ Containing $n$ vertices and $k$ edges,$G$ will contain atleast $n-k$ components.
Explanation-:
Graph containing $n$ vertices and $0$ edges will contain $n$ components.Each time adding an edge will reduce the compo... | The maximum number of components comes from using complete graphs. For instance, with two edges, you lose two components. With three edges, you can make a complete subgraph and lose no edges. The result is $n-l$ such that $l$ is the lowest integer satisfying $k\le \frac{l(l+1)}2$, where $k$ is the number of edges, $n$ ... | {
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"timestamp": "2023-03-29T00:00:00",
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How many ways can 8 teachers be distributed among $4 $ schools? There are several ways that the teachers can be divided amongst $4$ schools, namely here are the possible choices I came up with:
$1) 1 1 1 5$
$2) 1 1 2 4$
$3) 1 1 3 3$
$4) 1 2 2 3$
$5) 2 2 2 2$
now given the fact that say $2213$ is the same as $1 2 2 3$ i... | Assuming distinct teachers, distinct schools, and having identified the $5$ patterns, a foolproof mechanical way is to sum up the product of two multinomial coefficients for each case, one for the pattern, the other for the frequencies of singletons, doubles, triples, etc, viz.
$\binom{8}{1,1,1,5}\binom{4
}{3,1} + \bin... | {
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"url": "https://math.stackexchange.com/questions/2074092",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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The valid interval of the maclaurin series for $\frac{1}{1+x^2}$ The Maclaurin series for $\frac{1}{1-x}$ is $1 + x + x^2 + \ldots$ for $-1 < x < 1$.
To find the Maclaurin series for $\frac{1}{1+x^2}$, I replace $x$ by $-x^2$.
The Maclaurin series for $\frac{1}{1+x^2} = 1 - x^2 + x^4 - \ldots$.
This is valid for $-1 <... | The MacLaurin series for $\frac{1}{1-x}$ converges when $|x| < 1$. If we substitute $-x^2$ for $x$, then our radius of convergence is when $|-x^2| < 1 \implies |x^2| < 1 \implies -1 < x < 1$.
Hope this helps!
| {
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the New Year will be 2017: how many pairs of integer solutions to $x^2 + y^2 = (2017)^3$? We're almost in 2017. I wonder how many pairs of integer solutions has the following diophantine equation:
$$x^2 + y^2 = (2017)^3$$
Thanks in advance.
| Borrowing from this answer here
The answer is just $\frac{3+1}{2}=2$. ( or $4$ if the order matters).
I checked it with this code:
#include <bits/stdc++.h>
using namespace std;
typedef long long lli;
lli N=2017;
int isq(lli N){
lli s=sqrt(N);
if(s*s==N) return(1);
return(0);
}
int main(){
N=N*N*N;
... | {
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"timestamp": "2023-03-29T00:00:00",
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Converse Truth Table I do not understand how the converse ($B \Rightarrow A$) truth table is logical.
For instance, take the statement, "If I am in Paris, then I am in France".
If I am in Paris, then I am in France. Therefore, $A \Rightarrow B$, since if I am in Paris, then I must also be in France. However, ($B \not \... |
Let's assume you have a number,for example 5. When you are asked 5 is in which circle you will say it is in B. Then by this statement one can say your number is also in A.
But lets assume you have number 3. And you are asked in which circle number is? You day it is in circle A. So one can't say the number is in B!
Thi... | {
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What is the Fourier transformation of H(-t)? I want to start by the definition of Fourier transformation. It is
$$\mathcal{F}[H(-t)]=\int_{-\infty}^0 e^{-j{\omega}t}\,dt =\left.\frac{1}{-j\omega}e^{-j{\omega}t}\right|_{-\infty}^0 $$
But when $t \rightarrow{-\infty}$, the result goes to the infinity, right?
| As stochasticboy321 mentioned, taking the "Fourier transform" of $H(t)$ requires some slightly more sophisticated machinery. Details are given in this post: Heaviside step function fourier transform and principal values.
The idea is that one considers generalized functions which are not defined by pointwise values, but... | {
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Find the value of $\theta$, which satisfy $3 − 2 \cos\theta − 4 \sin\theta − \cos 2\theta + \sin2\theta = 0$. We have to find the value of $\theta$, which satisfy $3 − 2 \cos\theta − 4 \sin\theta − \cos 2\theta + \sin2\theta = 0$.
I could not get any start how to solve it .
| HINT:
$$4-2\cos x+2\sin x\cos x-4\sin x=1+\cos2x$$
$$-(\cos x-2)+\sin x(\cos x-2)=\cos^2x$$
$$-(\cos x-2)(1-\sin x)=1-\sin^2x$$
| {
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"timestamp": "2023-03-29T00:00:00",
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$\text{Prove That,}\;f(n) = \prod\limits_{i=1}^{n}(4i - 2) = \frac{(2n)!}{n!}$
$$\text{Prove That,}\;f(n) = \prod_{i=1}^{n}(4i - 2) = \frac{(2n)!}{n!}$$
This is a problem from Elementary Number theory.
My Work:
The statement is true for $i=1$. So, if it is true for $i=k$ it must be true for $ i = k+1$ . I am ... | $$\prod_{i = 1}^n (4i - 2) = \prod_{i = 1}^n 2 \, (2i - 1) = 2^n \, \prod_{i = 1}^n (2i - 1) \frac{\prod\limits_{i = 1}^n 2i}{\prod\limits_{i = 1}^n 2i} = 2^n \frac{\prod\limits_{i = 1}^{2n} i}{2^n \, \prod\limits_{i = 1}^n i} = \frac{(2n)!}{n!}$$
| {
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"timestamp": "2023-03-29T00:00:00",
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"question_score": "1",
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Solve $\cos x-\sin(2x)=0$ Solve $\cos x-\sin(2x)=0$
I did:
$$\cos x=\color{blue}{\sin(\pi /2-x)}$$
therefore:
$$\color{blue}{\sin(\pi /2-x)}=\sin(2x)$$
Can I do that:??
now to solve only for $\pi/2-x=2x$
so $x=\pi/6+2\pi k$
| The first step is ok, but for the second we have that
$$\sin A=\sin B\iff A+B=(2k+1)\pi\text{ or }A-B=2k\pi$$
Then, from the equality $\sin(\pi/2-x)=\sin 2x$ we get two sets of solutions:
*
*$\pi/2-x+2x=(2k+1)\pi$
*$2x-(\pi/2-x)=2k\pi$
| {
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"timestamp": "2023-03-29T00:00:00",
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Do we assume $f_n$'s map into $\Bbb{R}$ or $\Bbb{C}$ in Theorem 7.8 Rudin's *Principles of Mathematical Analysis*?
Theorem 7.8 The sequence of functions $\{f_n\}$ defined on $E$ converges uniformly on $E$ if and only if for every $\epsilon > 0$ there exists an integer $N$ such that $m \geq N, n \geq N, x \in E$ implie... | For each $x \in E$, the sequence $(f_n(x))_{n \in \mathbb N}$ is a Cauchy-sequence in $ \mathbb R$ or ($\mathbb C$).
| {
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"timestamp": "2023-03-29T00:00:00",
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How to differentiate $\frac{x}{1-\ln(x-1)}$? I'm working on the following problem found in James Stewart's Calculus Early Transcendentals, 7th Ed., Page 223, Exercise 27. I'd just like to know where my work had gone wrong?
Please differentiate: $f(x)=\frac{x}{1-\ln(x-1)}$
My work is below. First I apply quotient rule ... | The following is correct:
$$f'(x)=\frac{\left(1-\ln(x-1)\right)(1)-\left((x)(-\left(\frac{1}{x-1}\right)(1)\right)}{\left(1-\ln(x-1)\right)^2}$$
The following is incorrect:
$$f'(x)=\frac{\left(1-\ln(x-1)\right)-(x-\left(\frac{1}{x-1}\right))}{\left(1-\ln(x-1)\right)^2}$$
On the right term in the numerator, you added $x... | {
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"timestamp": "2023-03-29T00:00:00",
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Differential equation in polar coordinates I have the following system:
$\frac{dx}{dt} = 3x + y - x(x^2+y^2)$
$\frac{dy}{dt} = -x +3y -y(x^2+y^2)$
Converting this to polar coordinates gives us:
$\frac{dr}{dt} = r(3-r^2)$
$\frac{d\theta}{dt} = -1$
This gives us a solution $\theta(t) = -t + \theta_0$. What would the solu... | You have
$$\frac{1}{r (\sqrt{3}-r)(\sqrt{3}+r)} dr = dt.$$ Use partial fractions on the left hand side and integrate.
| {
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Applications of complex numbers to solve non-complex problems Recently I asked a question regarding the diophantine equation $x^2+y^2=z^n$ for $x, y, z, n \in \mathbb{N}$, which to my surprise was answered with the help complex numbers. I find it fascinating that for a question which only concerns integers, and whose a... | All of the trigonometric identities easily follow from $e^{bi} = \cos b + i \sin b$. Sine and cosine sum from $e^{(a+b)i} = e^{ai}e^{bi}$, De Moivre's theorem, and so on.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2075039",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "58",
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"answer_id": 10
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Example of stronger programming formulation. So I was revising the course combinatorial optimization and I find the statement $$P_1 \text{ is a stronger formulation than } P_2 \text{ if } P_1 \subset P_2. $$ Can someone give me an example of two formulations for which one is stronger than the other? And why is it usefu... | Usually in combinatorial optimization, for example in integer linear programming, we want to relax the feasible set such that the problem become easier, such as relax it to a linear programming problem.
The feasible set become larger that way. If the relaxed feasible set is too large, the approximation become crude. H... | {
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If $A(z_1)$ and $(z_2)$ are two points in argand plane, find $\angle ABO$ If $A(z_1)$ and $(z_2)$ are two points in argand(complex) plane such that $$\frac{z_1}{z_2}+\frac{\overline{z_1}}{\overline{z_2}}=2$$. Find the value of $\angle ABO$ where $O$ is origin.
Using given condition, I found that Real part of $\frac{z_1... | Let $z_1 / z_2=a+bi$ with $a,b \in \mathbb{R}$. Then $\bar z_1 / \bar z_2=a-bi$ and the given condition gives $(a+bi)+(a-bi) = 2 a = 2 \iff a = 1 \iff z_1/z_2 = 1 + bi$.
The angle $\angle ABO = \arg((z_1-z_2) / z_2)=\arg(z_1/z_2-1)=\arg(1+bi-1)=\arg(bi) = \pm \pi / 2$. Ignoring orientation, $\angle ABO = \pi / 2\,$.
| {
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"timestamp": "2023-03-29T00:00:00",
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Prove $\sum\limits_{k=1}^n a_k\sum\limits_{k=1}^n \frac1{a_k}\le\left(n+\frac12\right)^2$ then $\max a_k \le 4\min a_k$
Prove that
if
\begin{align}
&0<a_1,a_2,\dots,a_n \in \mathbb R,
&\left(\sum_{k=1}^n a_k\right)\left(\sum_{k=1}^n \frac1{a_k}\right)\le\left(n+\frac12\right)^2\\
\end{align}
then
$$\max_k \space a... | Hint:
\begin{align*}
\min_j a_j &\leq a_k \quad \text{ for every } 1 \leq k \leq n\\
\sum_{k = 1}^n \min_j a_j &\leq \sum_{k = 1}^n a_k\\
n \, \min_j a_j &\leq \sum_{k = 1}^n a_k
\end{align*}
Similarly
$$
n \frac{1}{\max\limits_j a_j} \leq \sum_{k = 1}^n \frac{1}{a_k}
$$
Hence
$$
n^2 \frac{\min\limits_j a_j}{\max\limit... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2075303",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 3
} |
Given $\lim_{n\to\infty} a_n = a$ what is the limit $\lim \limits_{n \to \infty}\frac{a_n}{3^1}+\frac{a_{n-1}}{3^2}+\ldots+\frac{a_1}{3^n}$?
Given $\lim \limits_{n \to \infty}a_n = a$ then I need to find the limit $\lim \limits_{n \to \infty} \frac{a_n}{3} + \frac{a_{n-1}}{3^2} + \frac{a_{n-2}}{3^3} + \dotso + \frac{a... | Fix $\epsilon > 0$. Since $\displaystyle\lim_{n \to \infty}a_n = a$, there exists an $N \in \mathbb{N}$ such that $|a_n-a| < \epsilon$ for all $n \ge N$.
Let $S_n := \dfrac{a_n}{3}+\dfrac{a_{n-1}}{3^2}+\cdots+\dfrac{a_2}{3^{n-1}}+\dfrac{a_1}{3^n}$. Then, $S_{n+1} = \dfrac{1}{3}S_n+\dfrac{1}{3}a_{n+1}$ for all $n \in \m... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2075424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Find $\lim\limits_{x\to\pi/4} \frac{1-\tan(x)^2}{\sqrt{2}*\cos(x)-1}$ without using L'Hôpital's rule. Find $$\lim_{x\to\pi/4} \frac{1-\tan(x)^2}{\sqrt{2}\times \cos(x)-1}$$ without using L'Hôpital's rule.
I can solve it using L'Hôpital's rule, but is it possible to solve it without using L'Hôpital's rule?
| Just another way to do it.
Let $x=y+\frac \pi 4$, expand and simplify. You should get $$\lim_{x\to\pi/4} \frac{1-\tan^2(x)}{\sqrt{2}\times \cos(x)-1}=\lim_{y\to 0}\frac{2 (\cos(y)-\sin (y)+1)}{(\cos (y)-\sin (y))^2}$$ which looks very simple.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2075487",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
proving $t^6-t^5+t^4-t^3+t^2-t+0.4>0$ for all real $t$ proving $t^6-t^5+t^4-t^3+t^2-t+0.4>0$ for all real $t$
for $t\leq 1,$ left side expression is $>0$
for $t\geq 1,$ left side expression $t^5(t-1)+t^3(t-1)+t(t-1)+0.4$ is $>0$
i wan,t be able to prove for $0<t<1,$ could some help me with this
| Let $p(t) = t^6 - t^5 + t^4 - t^3 + t^2 - t +2/5$. Observe that
$$ p(t) = \begin{bmatrix} 1\\t\\t^2\\t^3\end{bmatrix}^\intercal \begin{bmatrix}2/5&-1/2&0&0\\-1/2&1&-1/2&0\\0&-1/2&1&-1/2\\0&0&-1/2&1\end{bmatrix}\begin{bmatrix} 1\\t\\t^2\\t^3\end{bmatrix}
$$
The matrix in the middle is positive definite, from which it fo... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2075580",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Number of zeroes of solution to $y''(x)+e^{x^2}y(x)=0$ in $[0,3π]$ The question is to investigate the number of zeroes of $y''(x)+e^{x^2}y(x)=0$ in $[0,3π]$.
Solving this ODE would not be an easy task as one has to use the power series solution and then investigating the zeroes of the solution will require more analysi... | (Moved from a deleted duplicate question, answered Feb 18 '17 at 8:44, since it contains a more elementary approach)
See the Sturm-Picone comparison theorem which tells you that you have at least as many roots as $\cos x$ on $[0,3π]$.
You could apply it to the segments $[0,π]$, $[π,2π]$ and $[2π,3π]$ separately to get... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2075647",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 1,
"answer_id": 0
} |
Why is the solution to $x-\sqrt 4=0$ not $x=\pm 2$? If the equation is $x-\sqrt 4=0$, then $x=2$.
If the equation is $x^2-4=0$, then $x=\pm 2$.
Why is it not $x=\pm 2$ in the first equation?
| Actually while solving a quadratic equation we drop one step to short the answer or to save time whatever it may be.
THE STEP IS: $$x^2-4=0\tag{Step $1$}$$ $$x^2=4\tag{Step $2$}$$ $$x=\pm\sqrt4\tag{Step $3$}$$ $$x=\pm2\tag{Step $4$}$$ In our solution we drop the $3^{rd}$ step. Actually the thing is that as the $(+)$ s... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2075745",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 13,
"answer_id": 4
} |
$P$ is projection matrix iff $A$ is reflection matrix? I have the following definition
An $n \times n$ matrix $A$ is a reflection matrix if and only if $A^2 = I$ and $A^T= A$. A projection matrix is $P = 1/2(A+I)$.
I was wondering if I can conclude that $P$ is a projection matrix if and only if $A$ is a reflection ma... | Since you’re restricting $P$ to an orthogonal projection, consider one of the standard ways to construct the (orthogonal) reflection of a vector relative to some subspace of $\mathbb R^n$: find the orthogonal rejection of the vector from that subspace and reverse it. That is, if $W\subset\mathbb R^n$ is a subspace and ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2075813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
A graph problem. Two friends $A$ & $B$ are initially at points $(0,0)$ & $(12,7)$ respectively on the infinite grid plane. $A$ takes steps of size $4$ units and $B$ takes steps of size $6$ unit along the grid lines. Show that it is not possible for them to meet at a point.
It is my problem.I can find the number of ways... | I assume that the definition of "meet" by the OP is that both $A$ and $B$ must end a move at the same point.
Consider a move that $A$ makes from $(x_1,y_1)$ to $(x_2,y_2)$
Notice that the sum of the "net movement" of $A$ in the $x$ and $y$ directions (i.e. $|x_1 - x_2| + |y_1 - y_2|$ where $|x|$ is the absolute value o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2075915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to solve this using definite integral? I already have asked $2$ questions on similar topics. In that questions I have equations of parabola and line. And equations have $y$ and $x$ variable.
But in this question I have functions.
$$f(x) = |x| - 1 \quad\mbox{and}\quad g(x) = 1 - |x|.$$
a) Sketch their graphs.
b) Usi... | I presumed you know how to sketch the region. Then
$$A=2\int_{0}^1[(1-x)-(x-1)]dx=\int_0^1(4-4x)dx=2.$$
In set notation, the region $R$ is given by
$$R=\{(x,y):-1\leq x\leq 1,f(x)\leq y\leq g(x)\}.$$
The line $x=0$ or known as the $y$-axis serves the line of symmetry.
Okay,this is the graph of the region $R$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2076020",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Show that the fractional power of a linear operator is closed Let $H$ be a $\mathbb R$-Hilbert space and $(\mathcal D(A),A)$ be a linear operator.
Assume $(e_n)_{n\in\mathbb N}\subseteq\mathcal D(A)$ is an orthonormal basis of $H$ with $$Ae_n=\lambda_ne_n\;\;\;\text{for all }n\in\mathbb N\tag 1$$ for some $(\lambda_n)_... | The projection of $x_n -x$ onto the $e_k$ component must converge to zero. Multiply this projection by $\lambda_k^\alpha$ to get that
$$\lambda_k^\alpha \langle x_n,e_k\rangle\to\lambda_k^\alpha \langle x,e_k\rangle .\tag{1}$$
The left-hand side of $(1)$ is the same as $\langle A^\alpha x_n,e_k\rangle$ which must conve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2076107",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is it proper form to write $f'(x)$ in terms of $a$ when using the $x \to a$ method? For first principles derivatives, I solve for $f'(x)$ by doing the following: $$f'(x) = \lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
Since $x$ approaches $a,$ the final answer will be in terms of $a.$ Is this considered proper form? I had alway... |
Since $x$ approaches $a$, the final answer will be in terms of $a$. Is this considered proper form?
No, this is not proper. The left-hand side is looking for an answer in terms of $x$ while the right-hand side is in terms of $a$. This really doesn't make any sense. In order to fix this, you should make $a$ approach $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2076461",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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What were some major mathematical breakthroughs in 2016? As the year is slowly coming to an end, I was wondering which great advances have there been in mathematics in the past 12 months. As researchers usually work in only a limited number of fields in mathematics, one often does not hear a lot of news about advances ... | The Non Existent Complex 6 Sphere by Michael Atiyah
"The possible existence of a complex structure on the 6-sphere has been a famous unsolved problem for over 60 years. In that time many "solutions" have been put forward, in both directions. Mistakes have always been found. In this paper I present a short proof of the ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2076565",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "186",
"answer_count": 6,
"answer_id": 0
} |
An equation with two variables is unsolvable for either one, but how can I know if it's unsolvable as an expression for both? Weird title perhaps, so let me illustrate with the question that got me thinking about this problem:
You are buying a laptop and have two to choose from. What is the
difference between the or... | I believe its because the difference between $a$ and $b$ is dependent on their values in every case and it just so happens the case where the difference did not depend on them is when they both drop by the same percentage. Let me illustrate using the example you've given.
$$0.65a - 0.55b = 50$$
$$0.55a - 0.55b = 50 - 0... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2076650",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 3,
"answer_id": 1
} |
Given a graph with distinct edge weights and a not-minimum ST, there always exist another ST of lesser total weight that differs only by one edge I have to show that, if all the edge weights of a graph are distinct, given a spanning tree $T$ that is not a MST, there always exist a spanning tree $T'$ of lesser total wei... | This is true even if not all the edge weights are distinct.
Define
$$ w'(e) = \begin{cases} 2 \cdot w(e) &\text{ if } e \text{ belongs to } T \\ 2 \cdot w(e) + 1 & \text{otherwise}\end{cases} $$
then sort all the edges according to $w'$ (rather than $w$). Observe that it produces the same ordering as sorting accordin... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2076735",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Prove that $e^x, xe^x,$ and $x^2e^x$ are linearly independent over $\mathbb{R}$
Question: Prove that $e^x, xe^x,$ and $x^2e^x$ are linearly independent over $\mathbb{R}$.
Generally we proceed by setting up the equation
$$a_1e^x + a_2xe^x+a_3x^2e^x=0_f,$$
which simplifies to $$e^x(a_1+a_2x+a_3x^2)=0_f,$$ and furtherm... | Suppose $a_1e^x + a_2xe^x+a_3x^2e^x=0 $ for all $x$.
Setting $x=0$ shows that $a_1 = 0$.
Now note that $a_2xe^x+a_3x^2e^x=0 $ for all $x$ and hence
$a_2e^x+a_3xe^x=0 $ for all $x \neq 0$. Taking limits as $x \to 0$ shows
that $a_2 = 0$, and setting $x=1$ shows that $a_3 = 0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2076908",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 1
} |
How to calculate Limit of $(1-\sin x)^{(\tan \frac{x}{2} -1)}$ when $x\to \frac{\pi}{2}$.
How to calculate Limit of $(1-\sin x)^{(\tan \frac{x}{2} -1)}$ when $x\to \frac{\pi}{2}$.
We can write our limit as $\lim_{x\to \frac{\pi}{2}}e^{(\tan \frac{x}{2} -1) \log(1-\sin x)}~ $ but I can not use L'Hopital rule.
Is ther... | Using (elementary) Taylor series, to low order.
As you noticed, $$
(1-\sin x)^{(\tan \frac{x}{2} -1)}=
\exp\left( (\tan \frac{x}{2} -1) \ln (1-\sin x)\right)
$$
Now, since I am much more comfortable with limits at $0$ than at other points, let us write $x = \frac{\pi}{2}+h$ and look at the limit of the exponent when $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077014",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Determinant of odd matrix Given a matrix $A = \{{a_{i,j}}\} \in M_{7\times7}(\Bbb R)$
It is said that
$a_{i,j} = 0$ if $i$,$j$ are both odd.
Show that $det(A) = 0$
Any hints?
| Let $e_1,\dots,e_7$ denote the standard basis of $\Bbb R^7$. Let $P$ be the permutation matrix
$$
P = \pmatrix{e_1&e_3&e_5&e_7&e_2&e_4&e_6}
$$
Then $PAP^T$ can be written in the form
$$
M = P^TAP = \pmatrix{0_{4 \times 4} & M_{12}\\M_{21}&M_{22}}
$$
$M_{21}$ is $3 \times 4$, so it has linearly dependent columns.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077167",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
How to prove $\sqrt{1000} < x < 1000$? I have been given that
$$x = \frac 21 \times \frac 43 \times \frac 65 \times \frac 87 \times \cdots \times \frac {996}{995} \times \frac{998}{997} \times \frac {1000}{999}$$
How can I prove that $\sqrt{1000} < x < 1000$?
| \begin{align}
x^2 &= \left(\frac 21 \times \frac 21\right) \times \left(\frac 43 \times \frac 43\right) \times \cdots \times \left(\frac{1000}{999} \times \frac {1000}{999}\right) \\
&\ge \left(\frac 21 \times \frac 32\right) \times \left(\frac 43 \times \frac 54\right) \times \cdots \times \left(\frac{1000}{999} \time... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077269",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Finding the Tangent Line to a Surface
What is the equation of the tangent line to the intersection of the surface $z = \arctan (xy)$ with the plane $x=2$, at the point $(2,\frac{1}{2}, \frac{\pi}{4})$
The intersection of $x=2$ and $z= \arctan (xy)$ produces the curve $z = \arctan (2y)$ in the $yz$-plane. Thus, the pa... | For your first question:
is there a conceptual/logical error in plugging the $x=2$ before taking the (partial) derivative.
No, and the reason is that you're plugging in a value for $x$ and you're taking the partial derivative with respect to (wrt) $y$. When you take the partial wrt $y$, you treat $x$ as a constant a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077363",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Prove this relation for areas: $2[\triangle BOD] = [\square COME]$ In the figure, $D$ and $M$ are the midpoints of $AB$ and $AC$, respectively. Then prove that $2\left[\triangle BOD\right] = \left[\square COME\right]$
My Attempt
*
*$\left[\triangle COM\right]=\left[\triangle CME\right]$
*$\left[\triangle BCD\righ... | Extend $AO$ to meet $BC$ at $N$.
$AN$ is median since $O$ is centroid. Therefore, $\Delta ANB=\Delta ANC$ and $\Delta ONB=\Delta ONC$.
$$\Rightarrow \left[\Delta ANB\right]-\left[\Delta ONB\right] = \left[\Delta ANC\right]-\left[\Delta ONC\right]$$
$$\Rightarrow \frac{\left[\Delta AOB\right]}{2} = \frac{\left[\Delta A... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077476",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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If $|z^2-1|=|z|^2+1$, show that $z$ lies on imaginary axis If $|z^2-1|=|z|^2+1$, how do we show that $z$ lies on imaginary axis ?
I understand that I can easily do this if I substitute $z=a+ib$. How do we solve it using algebra of complex numbers without the above substitution ?
My Attempt:
$$
|z|^2+|1|^2=|z-1|^2+2\mat... | Use the fact that $|z|^2 = z\bar{z}$.
Squaring both sides of the given equality yields
\begin{align}
|z^2-1|^2 &= (z\bar{z} + 1)^2\\
(z^2 - 1)(\bar{z}^2 - 1) &= (z\bar{z} + 1)(z\bar{z}+1)\\
z^2 + 2z\bar{z} + \bar{z}^2 &= 0\\
(z + \bar{z})^2 &= 0\\
z = -\bar{z}
\end{align}
from which it follows that the real part of $z$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077551",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 0
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How to simplify an expression that does not have a common factor I am trying to simplify this expression :
$$9a^4 + 12a^2b^2 + 4b^4$$
So I ended up having this :
$$(3a^2)^2 + 2(3a^2)(2b^2) + (2b^2)^2$$
However, after that I don't know how to keep on simplifying the equation, it is explained that the answer is $(3a^2 + ... | after the binomial formula $$(x+y)^2=x^2+2xy+y^2$$ we get: $$(3a^2+2b^2)^2=9a^4+12a^2b^2+4b^4$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 5,
"answer_id": 4
} |
Beta distribution, as $\epsilon \to 0$, $(-\epsilon\,\log(B_\epsilon), -\epsilon\,\log(1 - B_\epsilon)) \implies (\xi E_a, (1 - \xi)E_b)$. Fix $a$, $b > 0$. For $\epsilon > 0$, let $B_\epsilon$ be distributed according to a Beta distribution with parameters $\epsilon a$ and $\epsilon b$. Now, I wish to show that as $\e... | To show convergence in distribution, we must show for $x, y \in \mathbb{R}$:
$$\lim_{\epsilon \to 0}\mathbb{P}(-\epsilon \text{log}(B_{\epsilon}) \leq x, -\epsilon \text{log}(1 - B_{\epsilon}) \leq y) = \mathbb{P}(\zeta E_{a} \leq x, (1 - \zeta)E_{b} \leq y)$$
PART I (Left-hand side)
We start be rewriting some expressi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077712",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
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Algorithm for Random irregular polygon in between two shapes This is not a homework problem. It is meant as a challenge for people who really enjoy math and have time to spare.
Background Info
Suppose you have a 2D Cartesian coordinate system. There are three shapes: R, C, and P.
R is a large rectangle. Its left side i... | Without some geometric condition on what it means for $R$ to be "large" and $C$ to be "small," there may not be a solution for $n=3$, i.e., $P$ a triangle:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077831",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
what are the different applications of group theory in CS? What are some applications of abstract algebra in computer science an undergraduate could begin exploring after a first course?
Gallian's text goes into Hamming distance, coding theory, etc., I vaguely recall seeing discussions of abstract algebra in theory of ... | The theory of Grobner Basis is a way to solve simultaneous multivariable polynomial equations. The key component underlying this is something called Buchberger's Algorithm (which given an ideal $I$ of some ring $R = k[x_1,\dots,x_n]$) computes the Grobner basis, an especially nice way to represent the ideal that makes... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077915",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "41",
"answer_count": 11,
"answer_id": 1
} |
Replacing floor operation with modulus I have this array of characters
aaaaaaaaaabbbbbbbbbbccccccccccddddddddddeeeeeeeeeeffffffffffgggggggggghhhhhhhhhhiiiiiiiiiijjjjjjjjjjkkkkkkkkkkllllllllllmmmmmmmmmmnnnnnnnnnnooooooooooppppppppppqqqqqqqqqqrrrrrrrrrrssssssssssttttttttttuuuuuuuuuuvvvvvvvvvvwwwwwwwwwwxxxxxxxxxxyyyyyyyy... | Sure. For positive $x, k$ integers, and % being the usual mod operator
floor(x/k) = (x - x % k)/k
As dxiv notes, this might be redundant (depending on the language you're using) - often, for integer types, $x/y$ is already a $floor$ operation.
One issue here is that various languages will disagree about the results wh... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2077996",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
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} |
A property similar to paracompacness Definitaion1: A family $\{A_t\}_{t\in S}$ of subsets of a
topological space $X$ is locally finite if for every point $x\in
X$ there exists a neighborhood $U$ of $x$ such that the set
$\{s\in S : U \cap A\neq \emptyset\}$ is finite.
Dfinition2: A topological space $X$ is called a *-s... | Even more can be said, suppose $X$ has the property that every open cover $\mathcal{U}$ has a point-finite subcover. Then $X$ is compact. It's clear that $\ast$-spaces have this property (as locally finite implies point-finite).
Proof: let $\mathcal{U}$ be any open cover of $X$. Let $U_0$ be any non-empty open set from... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Proof without induction of the inequalities related to product $\prod\limits_{k=1}^n \frac{2k-1}{2k}$ How do you prove the following without induction:
1)$\prod\limits_{k=1}^n\left(\frac{2k-1}{2k}\right)^{\frac{1}{n}}>\frac{1}{2}$
2)$\prod\limits_{k=1}^n \frac{2k-1}{2k}<\frac{1}{\sqrt{2n+1}}$
3)$\prod\limits_{k=1}^n2k-... | Notice that in problem #$1$ if you raise each side to the $n$th power, then it is equivalent to showing that the product of the $n$ factors of the form
$$\left(1-\frac{1}{2k}\right)\tag{1}$$
is greater than $\left(\frac{1}{2}\right)^n$. But that is clearly true since each factor in equation $(1)$ is greater than or equ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078166",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Distribute balls to cells problem I couldn`t understand how did they get to (b) solution.
Can someone please give an explanation?
Thanks!
| An explanation of the expression has already been given, I'd only like to add that since you are likely to encounter many problems of the balls in cells type,(and worked out differently in different books !), you might like to standardize the method.
The one I use for counting the arrangements is a multiplication of tw... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
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limit using Taylor's theorem i need to find the limit of $\lim _{x \to 0}f(x)$
$f(x)=\frac{\left (\sinh \left (x \right ) \right )^{n}-x^{n}}{\left (\sin \left (x \right ) \right )^{n}-x^{n}}$
i tried this $f(x)=\frac{(\frac{\sinh (x)}{x} )^{n}-1}{(\frac{(\sin (x)}{x})^{n}-1}$
and with Taylor's theorem $\lim _{x \t... | Using the binomial theorem at this point, you get
$$
\lim_{x\rightarrow 0}f(x)=\lim_{x\rightarrow 0}\frac{1+\frac{nx^2}{6}+\epsilon(x^3)-1}{1-\frac{nx^2}{6}+\tilde\epsilon(x^3)-1}
=\lim_{x\rightarrow 0}\frac{\frac{nx^2}{6}+\epsilon(x^3)}{-\frac{nx^2}{6}+\tilde\epsilon(x^3)}.
$$
(Here $\epsilon$ and $\tilde\epsilon$ rep... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078377",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
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Studying convergence of recursively defined sequence $a_1=2\text{,}\; a_{n+1}=2\sin(a_n)$ Firstly sorry for duplicate if this was asked before, i couldnt find.
This sequence is not monotone, but it seems convergent, i have plotted with Maple. Any hints to prove this is a Cauchy sequence? Or another method?
By the way, ... | Hints:
*
*Show that $|a_n|\le2$ for all $n$
*Show that if $a_n\to L, L=2\sin L$
*Determine the stability of the fixed points of the map $x\to 2\sin x$
*Deduce the convergence behaviour of $a_n$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078463",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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How many factors of $N$ are a multiple of $K$? How many factors of $N = 12^{12} \times 14^{14} \times 15^{15}$ are a multiple of $K = 12^{10} \times 14^{10} \times 15^{10}$ ?
Any approach to attempt such questions ?
| Look at it this way:
$$N=2^{38}3^{27} 5^{15} 7^{14}$$
Now ask yourself how many factors are a multiple of:
$$K=2^{30}3^{20} 5^{10} 7^{10}$$$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078573",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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Question involving Householder Matrix This is an exam question that I'm having trouble solving.
Given a unit vector $v^Tv=1$, the Householder matrix is defined as $H=I-2vv^T$.
The first question is: given column vector $x$, if $Hx=c\cdot e_1$ where $c$ is constant and $e_1$ is the first vector of the canonical basis, f... | It might be a little simpler to look at what $H$ does. If $x || v$ then
$Hx = -x$ and if $x \bot v$, we have $Hx = x$. So to invert, we just apply $H$ again.
To confirm, check that $H^2 = I$, in fact, $H$ is orthogonal, which gives it
desirable numerical properties.
I suspect the purpose of the question was not to have... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078652",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
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Complex Functions using polar form Given that Z1.Z2 $\ne$ 0
use the polar form to prove that
Re($Z_1\bar Z_2$)= |$Z_1$||$Z_2$| iif $\theta_1 - \theta_2 = 2n\pi$
, n = ±1,±2,...,±n
and $\theta_1 = Arg(Z_1) , \theta_2 = Arg(Z_2)$
| Hint: given that $\operatorname{Re} z = \frac{1}{2}(z+\bar z)$, $\operatorname{Im} z = \frac{1}{2i}(z-\bar z)$ and $|z|^2=z \bar z\,$:
$$
\begin{align}
\operatorname{Re}(z_1\bar z_2) = |z_1| |z_2| \;\;& \iff\;\;(z_1 \bar z_2+\bar z_1 z_2)^2 = 4\, z_1 \bar z_1 z_2 \bar z_2 \\
& \iff\;\; (z_1 \bar z_2-\bar z_1 z_2)^2 = ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078740",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Taking the derivative inside the integral (Liebniz Rule for differentiation under the integral sign) I have a function I would like to differentiate but am wondering if I my method is allowable:
If $\displaystyle f(x)= \int_{a}^{b} h(t) \:\mathrm{d}t$, what is the derivative of $f$ with respect to $x$, if $x$ occurs i... | To me the interesting result is the measure theory statement which intuitively requires:
*
*$f_x$ exists (derivative of $f_x$ wrt to x)
*$f_x$ remains measurable.
Then:
$$ \frac{d}{dx} \int_{\Omega} f(x, \omega) d\omega = \int_{\Omega} f_x(x, \omega) d\omega$$
is true.
Formally according to wikipedia:
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078844",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Verify the solution of the wave equation with Heaviside initial condition. I am interested in solving the following wave equation in three dimensional space:
$$ \begin{cases}
u_{tt} & = c^2\Delta u\\
u(x,0) & = 0\\
u_t(x,0) & = h(|x|),
\end{cases} $$
where $h(r) = H(1-r)$ for $r>0$, $H(\cdot)$ being the Heaviside funct... | Since the initial condition is spherically symmetric we have $u=u(r)$ where $r$ is the radial coordinate. The wave-equation in (3D) spherical coordinates can be written $v_{tt} = c^2v_{rr}$ where $v(r,t) = ru(r,t)$ so $v(r,0) = 0$ and $v_t(r,0) = rH(1-r)$. Since this is just the standard one-dimensional wave equation w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Inverse of a factorial I'm trying to solve hard combinatorics that involve complicated factorials with large values.
In a simple case such as $8Pr = 336$, find the value of $r$, it is easy to say it equals to this: $$\frac{8!}{(8-r)!} = 336.$$
Then $(8-r)! = 336$ and by inspection, clearly $8-r = 5$ and $r = 3$.
Now th... | I just wrote this answer to an old question. Using $a=1$, we get a close inverse for the factorial function:
$$
n\sim e\exp\left(\operatorname{W}\left(\frac1{e}\log\left(\frac{n!}{\sqrt{2\pi}}\right)\right)\right)-\frac12\tag{1}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2078997",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "37",
"answer_count": 5,
"answer_id": 2
} |
If $x(b-c)+y(c-a)+z(a-b)=0$ then show that .. I am stuck with the following problem that says:
If $x(b-c)+y(c-a)+z(a-b)=0$ then show that
$$\frac{bz-cy}{b-c}=\frac{cx-az}{c-a}=\frac{ay-bx}{a-b}$$ where $a \neq b \neq c.$
Can someone point me in the right direction? Thanks in advance for your time .
| HINT:
$$-(a-b)z=x(b-c)+y(c-a)$$
$$\frac{bz-cy}{b-c}=\dfrac{-b\{x(b-c)+y(c-a)\}-c(a-b)y}{(b-c)(a-b)}=?$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2079068",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Permutations : A person picks 4 numbers from a set of 9, what are the total ways he can Win? In a casino, a person picks 4 numbers from a set of 9 numbers (3 Even, 6 Odd). A person wins if at least one of those 4 numbers is Even. What are the total ways he can Win ?
There are two approches I believe.
1) One is to find... | The second method is wrong because you are over counting.
Suppose the even numbers are $e_1,e_2,e_3$ and the odd numbers are $o_1,o_2,\cdots,o_6$.
When you choose a number from the set of three evens, suppose you get $e_1$. Out of the remaining $8$, you choose $3$, let them be $e_2,o_1,o_2$.
So you have $e_1,e_2,o_1,o_... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2079134",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Why Is Inverse of a Polynomial Still a Polynomial? The fourth paragraph of the wikipedia article algebraic extension states that $K[a]$ is a field, which means every polynomial has an inverse. The inverse has to be a polynomial over $K$ as well. It seems it requires $K$ to be a non-integral domain. How do we resolve th... | Polynomials over a field $K$, that is, elements of $K[x]$, don't generally (unless they have degree zero) have an inverse.
$K[a]$ is a different object, it is isomorphic to the quotient ring $K[x]/(m)$ where $m$ is the minimal polynomial of $a$ (you can see this via the first isomorphism theorem using the morphism $\va... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2079202",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
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A category is $\mathbf{J}$-complete? Let $\mathbf{C}$ be a category and $\mathbf{J}$ be an index category. What does it mean to say that $\mathbf{C}$ is $\mathbf{J}$-complete? Is it just saying that all $\mathbf{J}$ shaped diagrams in $\mathbf{C}$ have a limit?
| Yes, this is exactly what it means. A category is $\mathbf{J}$-complete when it has all $\mathbf{J}$-shaped limits.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2079306",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can't recognize the series. Can somebody take a look? I am solving a question as part of which I got the below mentioned series. I tried a lot but couldn't recognize this.
$$
^m C_m m^n - {^m C_{m-1} (m - 1)^n} + \cdots \pm {^mC_1} 1^n
$$
I am sure it's expansion of some famous series.
Can somebody help?
| We use the notation $\binom{m}{j}$ instead of $^mC_j$ and we also use the coefficient extraction operator $[z^n]$ to denote the coefficient of $z^n$ in a series. This way we can write e.g.
\begin{align*}
n![z^n]e^{jz}=j^n
\end{align*}
We obtain for $m\geq 1$
\begin{align*}
\sum_{j=1}^m\binom{m}{j}(-1)^{m-j}j... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Limit question related to integration Find the limit,
$$L=\lim_{n\to \infty}\int_{0}^{1}(x^n+(1-x)^n)^{\frac{1}{n}}dx$$
My try:
$$ \int_{0}^{\frac{1}{2}}2^{\frac{1}{n}}xdx+ \int_{\frac{1}{2}}^{1}2^{\frac{1}{n}}(1-x)dx< \int_{0}^{1}(x^n+(1-x)^n)^{\frac{1}{n}}dx< \int_{0}^{\frac{1}{2}}2^{\frac{1}{n}}(1-x)dx+ \int_{\frac{... | Substitute 1/2+y for x, y going from -1/2 to 1/2. This a symmetric integral so the positve and negative intervals contribute equally and it suffices to consider only the positve interval. As n goes to infinity (1/2-y)^n is neglible compared to (1/2+y)^n. Hence the root approaches 1/2+ y and the integral approaches 2*In... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2079437",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 4
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Solve the inequality and show solution sets on the real line $$-\frac{x+5}{2} \le \frac {12+3x}{4}$$
I always have issues with problems like this. I chose to ignore the negative sign at the beginning and got the answer. Is that a good method? When solving this inequality what is the best method for no mistakes?
My answ... | $$-\frac{x+5}{2} \le \frac {12+3x}{4}$$
Multiply $4$ both sides:
$$-2(x+5) \le 12+3x$$
$$-2x-10 \le 12 +3x$$
$$-22 \le 5x$$
$$x \geq \frac{-22}{5}$$
Your mistakes:
$-2(12+3x)=-24-6x$ rather than $-24+6x$. If this is just a typo, the next line is fine.
After this mistake, surprisingly in the next line, you corrected y... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Can't come even near to the solution.. Can somebody take a look? Fermat's little theorem states that if $p$ is a prime number, then for any integer $a$, the number $a^p − a$ is an integer multiple of $p$. In the notation of modular arithmetic, this is expressed as $$a^p \equiv a \mod p.$$
Use Fermat's little theorem to... | We have that $p \mid x^a - 1$, or $x^a \equiv 1 \pmod{p}$. So first of all $x \not\equiv 0 \pmod{p}$, so that $x^ p - x \equiv 0 \pmod{p}$ implies $x^{p-1} \equiv 1 \pmod{p}$. (As the prime $p$ divides $x^ p - x = x (x^{p-1} - 1)$, and $p \nmid x$.)
And then, $a$ being prime, $x$ has order either $1$ or $a$ modulo $p$.... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
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Is it possible to find the sum of the infinite series $1/p + 2/p^2 + 3/p^3 + \cdots + n/(p^n)+\cdots$, where $p>1$? Is it possible to find the sum of the series:
$$\frac{1}{p} + \frac{2}{p^2} +\frac{3}{p^3} +\dots+\frac{n}{p^n}\dots$$
Does this series converge? ($p$ is finite number greater than $1$)
| Hint:
$$\frac{1}{1-x}=1+x+x^2+\dots$$
Differentiate this formula (it is uniformly convergent for $|x|<1$) and multiply with x. Finally set $x=1/p$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2079726",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Closed subset of quasi-separated space is quasi-separated? Is every closed subset of quasi-separated space quasi-separated?
A topological space is called quasi-separated if the intersection of two quasi-compact opens is quasi-compact. Here quasi-compact means every open cover has a finite subcover.
| Let $X$ be a space with the following property: each point has a n.h. basis consisting of quasi-compact opens.
Then (since any closed subset of a quasi-compact set is quasi-compact), we deduce that any closed subset $Z$ of $X$ inherits this property.
Furthermore, one easily sees that any quasi-compact open subset $V$ o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2079807",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
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Compute a canonical divisor Consider $C,C'$ two cubics in the plane with $C$ smooth. There are $9$ basepoint on the linear system generated by $C$ and $C'$ so if we blow them we get a map $X \to\mathbb P^1$, where $X$ is $\mathbb P^2$ blown-up at 9 points. Now is my question : how to compute $K_X$ ? I saw that $K_X = -... | Here is a formula you should learn (say from Hartshorne). If $Y$ is a smooth surface, $p\in Y$ is a point and $\pi:X\to Y$ the blow up of $p$ and $E$ the exceptional divisor, then $K_X=\pi^*K_Y+E$.
In your case, $K_X=f^*K_{\mathbb{P}^2}+\sum E_i$. Since $K_{\mathbb{P}^2}=-C$, we get $K_X=-f^*C+\sum E_i$ and thus $K_X$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2079898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 1,
"answer_id": 0
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Compute the $n$-th power of triangular $3\times3$ matrix I have the following matrix
$$
\begin{bmatrix}
1 & 2 & 3\\
0 & 1 & 2\\
0 & 0 & 1
\end{bmatrix}
$$
and I am asked to compute its $n$-th power (to express each element as a function of $n$). I don't know at all what to do. I tried to compute some values manually to... | Define
$$J = \begin{bmatrix}
0 & 2 & 3\\
0 & 0 & 2\\
0 & 0 & 0
\end{bmatrix} $$
so that the problem is to compute $(I+J)^n$. The big, important things to note here are
*
*$I$ and $J$ commute
*$J^3 = 0$
which enables the following powerful tricks: the first point lets us expand it with the binomial theorem, and th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2079950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "26",
"answer_count": 6,
"answer_id": 3
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Linearity of expectations - Why does it hold intuitively even when the r.v.s are correlated? An experiment - say rolling a die, is performed a large number of times, $n$. Let $X$ and $Y$ be two random variables that summarize this experiment.
Intuitively(by the law of large numbers), if I observe the values of $X$, ov... | For intuition, suppose the sample space consists of a finite number of equally probable outcomes (this is of course not true for all probability spaces, but many situations can be approximated by something of this form). Then
$$ E(X+Y) = \frac{(x_1+y_1)+(x_2+y_2)+\cdots+(x_n+y_n)}n $$
and
$$ E(X)+E(Y) = \frac{x_1+x_2+\... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080030",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 5,
"answer_id": 3
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How can an isolated point be an open set? I have the following definition:
In a metric space $(X,d)$ an element $x \in X$ is called isolated if $\{x\}\subset$ X is an open subset
But how can $\{x\}$ be an open subset? There has to exist an open ball with positive radius centered at $x$ and at the same time this open ... | Let $X$ be any non-empty set, and define a function $d:X\times X\to\Bbb R$ as follows: for $x,y\in X$,
$$d(x,y)=\begin{cases}
0,&\text{if }x=y\\
1,&\text{if }x\ne y\;.
\end{cases}$$
You can easily check that this function $d$ is a metric on $X$; it is commonly called the discrete metric on $X$. Now observe that for if ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080165",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 3,
"answer_id": 0
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Constructing two matrices that do not commute I need to construct square matrices $A$ and $B$ such that $AB=0$ but $BA \neq 0$.
I know matrix multiplication is not commutative, but I don't know how to construct such matrices. Thanks in advance.
Edit: looking for some simple way
| Pick $\mathrm u, \mathrm v, \mathrm w \in \mathbb R^n \setminus \{0_n\}$ such that $\neg (\mathrm u \perp \mathrm v)$ and $\mathrm v \perp \mathrm w$. Define
$$\mathrm A := \mathrm u \mathrm v^{\top} \qquad \qquad \qquad \mathrm B := \mathrm w \mathrm v^{\top}$$
whose traces are
$$\mbox{tr} (\mathrm A) = \mathrm v^{\to... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080245",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 8,
"answer_id": 5
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Why is $-\log(x)$ integrable over the interval $[0, 1]$ but $\frac{1}{x}$ not integrable? I don't understand why some functions that contain a singularity in the domain of integration are integrable but others are not.
For example, consider $f(x) = -\log(x)$ and $g(x) = \frac{1}{x}$ on the interval $[0, 1]$. These func... | Think about it this way - what's the inverse?
$$y = \frac{1}{x}; x = \frac{1}{y}$$
$$y = -\log x; x = e^{-y}$$
Looking at it this way, it's clear that as $y$ shoots off to infinity, $x$ approaches zero much faster in one case than in the other.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080334",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 4,
"answer_id": 1
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Derivative of nuclear norm of $xx^T-V$ The function is
$$f(x) = \| x x^T - V \|_*$$
where $\| \cdot \|_*$ denotes the nuclear norm and $V$ is a given matrix. $x$ is a vector. Please tell me how to differentiate $f(x)$. And, if it is possible, please show me how to compute the 2nd derivative of $f(x)$.
| Define a new matrix variable $$M=xx^T-V$$Then find the differential of the function in terms of this new variable
$$\eqalign{
f &= \operatorname{tr}\sqrt{M^TM} \cr
\cr
df &= \frac{1}{2}(M^TM)^{-1/2}:d(M^TM) \cr
&= \frac{1}{2}(M^TM)^{-1/2}:(dM^TM+M^TdM) \cr
&= (M^TM)^{-1/2}:M^TdM \cr
&= M(M^TM)^{-1/2}:dM \cr
... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Counting the total number of possible passwords I'm working from Kenneth Rosen's book "Discrete Mathematics and its applications (7th edition)". One of the topics is counting, he gives an example of how to count the total number of possible passwords.
Question: Each user on a computer system has a password, which is si... | The number of passwords with at least one digit plus the number of passwords with no digit equals the total number of passwords.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080563",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 1
} |
Integrate $\int x e^{x} \sin x dx$
Evaluate:
$$\int x e^{x} \sin x dx$$
Have you ever come across such an integral? I have no idea how to start with the calculation.
| This can also be solved another way (a little longer but correct nevertheless and could be more basic and readable for people who just started learning calculus):
We will use the results of following(easy - just substitute):
$$
∫ e^x \sin (x) dx = \frac{e^x \sin (x) - e^x \cos (x)}{2} + C
$$
$$
∫ e^{\ln (x)+x}dx = ∫ xe... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080664",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 7,
"answer_id": 3
} |
A function that is not a derivative of any derivable function That's basically it. I need to find a function on the interval $[0,1]$ that isn't a derivative of any derivable function.
I've found one possible solution which sets $0$ for every $x \in \mathbb{Q}$, and $1$ for every $x$ from $\mathbb{R}\setminus \mathbb{Q}... | yes, to expand on MathematicsStudent1122's reply, derivatives have the Darboux property (see Darboux's theorem on wiki), that is, if $f:I\to\mathbb{R}$ is differentiable in the interval $I$ and $f'$ takes two values, then it takes all the values in between. So the function $g$ which is $0$ on the rationals and $1$ on ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080747",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Coincidence set closed equivalent to diagonal closed Prove
For any topological space $Y$ and any continuous maps $f, g : Y → X$, the set
$\{y ∈ Y : f(y) = g(y)\}$ is closed in $Y$
is equivalent to
The diagonal $∆ = \{(x, x) : x ∈ X\}$ is a closed subset of $X × X$, in the product
topology.
I've proved that th... | Note that $\Delta$ is the coincidence set of the two projections from $X\times X$ to $X$. The projections are $p_1 : X\times X\to X,\;(x,x')\mapsto x$ and $p_2 : X\times X\to X,\;(x,x')\mapsto x'$.
Indeed, it's the set of points $(x,x')$ s.t. $x=x'$...
Thus the diagonal is closed.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2080863",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
Where does this proof that the abs. value of the determinant of a 2x2 matrix A is the area of the image of the unit square under A go wrong? If we have a matrix $A\in M_2(\mathbb{R})$, $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, then $det(A)=ad-bc$. $A$ sends the unit square to the parallelogram whose sides are... | The area of a parallelogram isn't the product of the two side lengths, it's the product of its base and height. If $\theta$ is the angle between adjacient sides, then the area becomes $||(a,c)||\, ||(b,d)||\sin(\theta)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2081032",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Other ways to evaluate $\lim_{\theta\to 0} \frac{\sin2\theta}{2\theta}$? What steps should be taken to find the limit:
$$\lim_{\theta\to 0} \frac{\sin2\theta}{2\theta}$$?
I went about evaluating the limit using the fundamental rules of limits. I noticed that $\lim_{\theta\to 0}$ $\sin\theta\over\theta$ $=1$ and that th... | Well, since very fortunately L'Hopital rule is not prohibited here, why don't we use it:
$$
\lim_{x\to 0 }\frac{\sin (2x)}{2x}=\lim_{x\to 0}\frac{2\cos 2x}{2}=1.
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2081150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
A question about neighboring fractions. I have purchased I.M. Gelfand's Algebra for my soon-to-be high school student son, but I am embarrassed to admit that I am unable to answer seemingly simple questions myself.
For example, this one:
Problem 42. Fractions $\dfrac{a}{b}$ and $\dfrac{c}{d}$ are called neighbor fract... | We have to prove that for positive integers $a,b,c,d$, we either have
$$\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}$$
or
$$\frac{c}{d}<\frac{a+c}{b+d}<\frac{a}{b}$$
First of all, $\frac{a}{b}=\frac{a+c}{b+d}$ is equivalent to $ab+ad=ab+bc$, hence $ad=bc$, which contradicts the assumption $ad-bc=\pm 1$. We can
disprove $\fr... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2081252",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 1
} |
Integrating $\int\frac{5x^4+4x^5}{(x^5+x+1)^2}dx $ In the following integral:
$$I = \int\frac{5x^4+4x^5}{(x^5+x+1)^2}dx $$
I thought of making partial fractions , then solve it .
But I am not able to make partial fractions.
| $\displaystyle \int\frac{5x^4+4x^5}{(x^5+x+1)^2}dx = \int\frac{5x^4+4x^5}{x^{10}(1+x^{-4}+x^{-5})^2}dx = \int\frac{5x^{-6}+4x^{-5}}{(1+x^{-4}+x^{-5})^2}dx$
put denominator is $=t$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2081525",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Evaluating $\int\frac{\cos^2x}{1+\tan x}\,dx$ I'd like to evaluate the following integral:
$$\int \frac{\cos^2 x}{1+\tan x}dx$$
I tried integration by substitution, but I was not able to proceed.
| This one always works for rational functions of $\sin x$ and $\cos x$ but can be a bit tedious. Set:
$$ z = \tan x / 2$$
so that
$$ \mathrm{d}x = \frac{2\,\mathrm{d} z}{1 + z^2}$$
$$ \cos x = \frac{1 - z^2}{1 + z^2}$$
$$\sin x = \frac{2z}{1 + z^2}$$
Now, you have a rational fraction in $z$ that you can integrate by sta... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2081612",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 4
} |
Greens function with non-zero boundary condition When solving a differential equation using a Greens function, is it possible to solve a problem with non-zero boundarys directly using a Greens function? For example, when solving a problem with non-zero boundarys I've broken the problem into multiple pieces, using the G... | It's possible to recombine the pieces into a single formula for solution of the equation $\Delta u=f$ in $\Omega$, $u=g$ on $\partial\Omega$. Namely,
$$
u(x) = \int_\Omega G(x,y)f(y)\,dy + \int_{\partial\Omega} \frac{\partial G(x,y)}{\partial n}g(y)\,dy
$$
(With multiplicative constants subject to the normalization of ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2081696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
How do we write the derivation of distance formula of two points from two different quadrants in a cartesian plane? I learnt the derivation of the distance formula of two points in first quadrant I.e., $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ where it is easy to find the legs of the hypotenuse (distance between two p... | Consider the following diagram:
Now to find the wanted distance,By the Pythagorean theorem,we need to know the size of the Edges BC and AC.
Suppose A and B to be:$$A(x_1,y_1),B(x_2,y_2)$$
And to find AC ,We need to subtract the length of AD from CD. Observe that C has the y component equal to the point B. So:
$$AC =\ve... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2081792",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Simple Polynomial Algebra question to complete the square $4y^2+32y = 0$
$4(y^2+8y+64-64) = 0$
$4(y+4)^2 = 64$
Is that correct?
| I made a comment, and I would do things slightly differently, so $$4y^2+32y=0$$
divide by $4$$$y^2+8y=0$$ add $\left(\frac 82\right)^2=4^2=16$ to both sides $$y^2+8y+16=16$$Rewrite the left-hand side as a square $$(y+4)^2=16$$If you want the $4$ back you can multiply through by $4$ at the end. If you want a pure square... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2081864",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 3
} |
Show that the line through $P$ and $Q$ is perpendicular to the surface at $P$. Consider a smooth surface given by the function $z = f(x,y)$, such that the partial
derivatives of $f(x,y)$ exist. Suppose $Q$ is a point that does not lie on the surface,
and $P$ is the nearest point on the surface to $Q$. Show that the lin... | The question is incorrect I am afraid. $Q$ also must be at a minimum distance to the surface.
The required normal direction comes from cross-product $ \partial z / \partial x$ X $ \partial z / \partial y $
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Prove $ \frac{1}{2} (\arccos(x) - \arccos(-x)) = -\arcsin(x)$ The identity I need to prove is this, and I am very close but I am missing a negative sign which I cannot find.
$$ \frac{1}{2} (\arccos(x) - \arccos(-x)) = -\arcsin(x)$$
I started off by using the $\cos(A-B)$ double angle formula, using $\arccos(x)$ and $\ar... | Maybe using the following identities will help: $$\arccos a-\arccos b= \arccos (ab +\sqrt {(1-a^2)(1-b^2)}) \tag {1}$$ $$\arcsin a +\arcsin b= \arcsin (a\sqrt {1-b^2} +b\sqrt {1-a^2}) \tag{2}$$ After applying $(1)$, use the triangle formula to convert $\arccos $ to $\arcsin $ component and thus LHS=RHS. Hope it helps. ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to solve this series: $\sum_{k=0}^{2n+1} (-1)^kk $ $$\sum_{k=0}^{2n+1} (-1)^kk $$
The answer given is $-n-1$. I have searched for how to do it, but I have problems simplifying the sum and solving it. How do you go about solving this?
| Another approach is to exploit the sum $$\sum_{k=0}^{2n+1}x^{k}=\frac{1-x^{2n+2}}{1-x}.
$$ Taking the derivative we have $$\sum_{k=0}^{2n+1}kx^{k}=\sum_{k=1}^{2n+1}kx^{k}=x\frac{-\left(2n+2\right)x^{2n+1}\left(1-x\right)+1-x^{2n+2}}{\left(1-x\right)^{2}}
$$ hence taking $x=-1
$ we get $$\sum_{k=0}^{2n+1}k\left(-1\ri... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082210",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 5,
"answer_id": 4
} |
Solving First Order ODE using the integrating factor approach
I am trying to solve the differential equation, but I do not understand the method. Here is my working:
| The step from line 3 to line 4 isn't correct: $\frac{di}{dt} e^t + ie^t = \frac{d}{dt}(ie^t)$, so line 4 should read $\frac{d}{dt}(ie^t) = 10t$ which can then be integrated directly.
In general, the ideas behind an integrating factor are outlined here as well as I could explain (if not better).
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082350",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Commutative ring $R\neq \{0\}$ in which every subring is finite is a field question on proof
Assume we are given a commutative ring $R \neq \{0\}$ with no zero
divisors (not necessarily with a unit element) in which every proper subring only has finitely many elements. Show
that $R$ is a field.
I am aware of the ... | Assuming you're talking about possibly nonunital rings and that the statement is about proper subrings being finite, we can observe that each ideal is a subring. Also I assume $R\ne\{0\}$.
Suppose $xR$ is proper for every $x\in R$, $x\ne0$. Being a finite commutative ring with no nonzero divisors, $xR$ is a field, so i... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082505",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Limit problem involving bijective function Let $f: \Bbb N^{\star}\to \Bbb N^{\star}$ a bijective functions such that exists $$\lim _{n \to \infty} {\frac {f(n)} {n}}.$$ Find the value of this limit.
I noticed that, if $f(n)=n$, then the limit is $1$. I couldn't make more progress. Can you help me?
| Because this limit is finite, exist $M$ and $n_0$ such that $f(n)/n<M$ for all $n>n_0$ ie is bounded ae.
Now, because $f(n)<Mn $ for all $n>n_0$ then $f=O(n)$ and by definition the limit request is 1
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082602",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Determine two changing variables only knowing the result So, about a decade ago my company came up with pricing for some banners that we sell. the prices are as follows.
$43.68 for a 3x4 banner
$44.52 for a 3x6 banner
$46.36 for a 3x8 banner
$50.00 for a 3x10 banner
$52.54 for a 3x12 banner
and I can not figure out wh... | Each time you go up a size you add 2 feet to the length. The added cost for these 2 feet varies from 0.84 to 3.64, which is quite a variation. This shows you will not be able to generate a formula of $A + B(length)$ that fits the old data, as $B/2$ should be the added cost per square foot. Clearly labor is not $63$ ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/2082696",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
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