Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Good sources on studying fractals (the mathematical, and not just the pretty pictures version)? Particularly, I'm interested in learning about the dimensions (whether it's always possible to find them, and if so, a concrete way of calculating them) of different types of fractals (given by the Hausdorff dimension, accor... | The author who would be a great start for looking at fractals constructed by iterated function systems and then studying their features like Hausdorff dimension is Kenneth Falconer. His books from the 90's are a great base to start from. If you want to see what is an amazing use of linear algebra to study functions and... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Solving for $(x,y): 2+\frac1{x+\frac1{y+\frac15}}=\frac{478}{221}$ Solving for $x,y\in\mathbb{N}$:
$$2+\dfrac1{x+\dfrac1{y+\dfrac15}}=\frac{478}{221}$$
This doesn't make any sense; I made $y+\frac15=\frac{5y+1}5$, and so on, but it turns out to be a very complicated fraction on the left hand side, and I don't even know... | Yes, there is a very large and important mathematical theory, called the theory of Continued Fractions. These have many uses, both in number theory and in analysis (approximation of functions).
Let's go backwards.
The number $\frac{478}{221}$ is $2+\frac{36}{221}$, which is $2+\frac{1}{\frac{221}{36}}$.
But $\frac{22... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/447512",
"timestamp": "2023-03-29T00:00:00",
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Covariant derivative of $Ricc^2$ How to derivate (covariant derivative) the expressions $R\cdot Ric$ and $Ric^2$ where $Ric^2$ means $Ric \circ Ric$? Here, $Ric$ is the Ricci tensor seen as a operator and $R$ is the scalar curvature of a Riemannian manifold.
| We want to compute
$$(\nabla(R\cdot Rc))(X)$$
Connections commute with contractions, so we start by considering the expression (where we contract $X$ with $Rc$, obtaining $Rc(X)$)
$$\nabla(R\cdot Rc\otimes X) = (\nabla (R\cdot Rc))\otimes X + R\cdot Rc\otimes (\nabla X)$$
Taking the contraction and moving terms around... | {
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Calculating derivative by definition vs not by definition I'm not entirely sure I understand when I need to calculate a derivative using the definition and when I can do it normally.
The following two examples confused me:
$$ g(x) = \begin{cases} x^2\cdot \sin(\frac {1}{x}) & x \neq 0 \\ 0 & x=0 \end{cases} $$
$$ f(x)... | Given a function $f(x)$ and a point $x_0$, it's possible to take $f'(x_0)$ by differentiating normally only if $\lim\limits_{x \rightarrow x_0} f'(x)$ exists. However, this is not the case for the $g(x)$ that you gave.
As $x \rightarrow 0$, $-\cos\left(\frac{1}{x}\right)$ oscillates, and, in fact, in any small neighbou... | {
"language": "en",
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"question_score": "3",
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Solve a set of non linear Equations on Galois Field I have the following set of equations:
$$M_{1}=\frac{y_1-y_0}{x_1-x_0}$$
$$M_{2}=\frac{y_2-y_0}{x_2-x_0}$$
$M_1, M_2, x_1, y_1, x_2, y_2,$ are known and they are chosen from a $GF(2^m).$ I want to find $x_0,y_0$
I ll restate my question.
Someone chose three distinct... | The first system can be solved in the usual way, provided the "slopes" $M_i$ are distinct. Solve each for the knowns $y_k$, $k=1,2$ and subtract. You can then get to
$$x_0=\frac{M_2x_2-M_1x_1-y_2+y_1}{M_2-M_1},$$
and then use one of the equations you already formed with this $x_0$ plugged in to get $y_0.$ Since this me... | {
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Cauchy theorem for a rectangle. Here $\delta R$ will give the boundary of a rectangle taking positively.
This is a theorem of the book Complex Analysis An Introduction to The Theory of Analytic Function on One Variable by L. V. Ahlfors, chapter4: Complex Integration
Let $f(z)$ be analytic is the set $R'$ obtained from ... | A favorite trick for estimating a line integral $\int_\gamma f$ is that it is bounded by $M\cdot L(\gamma)$, where $|f(z)|\leq M$ for all $z$ in the trace of $\gamma$, and $L(\gamma)$ the length of the curve.
Here we can use for $M$ the reciprocal of the minimum distance of $z$ from $\zeta$, that is the minimum distan... | {
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Differentiability of $xy^{\alpha}$ I was asked to prove that
$|xy|^{\alpha}$ is differentiable at $(0,0)$ if $\alpha > \frac{1}{2}$.
Since both the partial derivatives are zero, I concluded that this function is differentiable if and only if the following holds:
$$ \lim\limits_{(x,y)\to (0,0)} \frac{|xy|^{\alpha}}{\sqr... | Polar co-ordinates.
$\displaystyle \lim_{(x,y)\to(0,0)}\frac{|xy|^{\alpha}}{\sqrt{x^2+y^2}}=\lim_{r\to 0}r^{2\alpha-1}|\sin \theta \cos \theta|$ which $\to 0$ if $\alpha>\frac{1}2$
| {
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Number of trees with a fixed edge Consider a vertex set $[n]$. By Cayley's theorem there there are $n^{(n-2)}$ trees on $[n]$, but how can one count the following slightly modified version:
What is the number of trees on $[n]$ vertices where the edge $\{1,2\}$ is definitely contained in the trees?
| It seems that you can adjust this proof.
First, we denote by $S_n$ the number of trees with one of your edges fixed, and then we follow, as to count the number of ways the directed edges can be added to form rooted trees where your edge is added first. As in original proof, we can pick the root in $n$ ways, and add the... | {
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Let $R, P , Q$ be relations, prove that the following statement is tautology
Let $P, R, Q$ be relations. Prove that:
$\exists x(R(x) \vee P(x)) \to (\forall y \neg R(y) \to (\exists xQ(x)
\to \forall x \neg P(x)))$ is a tautology.
How do I do so? please help.
| Relations of arity 1 (that is the relations that take only one argument) are just subsets of the universe. Therefore, your formula can be translated into
$$(R \cup P \neq \varnothing) \to (R = \varnothing \to (Q \neq \varnothing \to P = \varnothing))$$
and then simplified into
$$(R \neq \varnothing \lor P \neq \varnoth... | {
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Evaluating $\lim\limits_{x \to 0} \frac1{1-\cos (x^2)}\sum\limits_{n=4}^{\infty} n^5x^n$ I'm trying to solve this limit but I'm not sure how to do it.
$$\lim_{x \to 0} \frac1{1-\cos(x^2)}\sum_{n=4}^{\infty} n^5x^n$$
I thought of finding the function that represents the sum but I had a hard time finding it.
I'd apprecia... | Note that the first term is $$\frac 1 {1-(1-x^4/2+O(x^8))} = \frac 2 {x^4}(1+O(x^4))$$ and that the second term is $$4^5 x^4(1+O(x))$$
(The question keeps changing; this is for the $$\cos x^2$$ denominator.)
| {
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What rational numbers have rational square roots? All rational numbers have the fraction form $$\frac a b,$$ where a and b are integers($b\neq0$).
My question is: for what $a$ and $b$ does the fraction have rational square root? The simple answer would be when both are perfect squares, but if two perfect squares are mu... | We give a fairly formal statement and proof of the result described in the post.
Theorem: Let $a$ and $b$ be integers, with $b\ne 0$. Suppose that $\frac{a}{b}$ has a rational square root. Then there exists an integer $e$, and integers $m$ and $n$, such that $a=em^2$ and $b=en^2$,
Proof: It is enough to prove the resul... | {
"language": "en",
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How to prove that $\lim\limits_{x\to0}\frac{\tan x}x=1$? How to prove that
$$\lim\limits_{x\to0}\frac{\tan x}x=1?$$
I'm looking for a method besides L'Hospital's rule.
| One way to look at it is to consider an angle subtended by two finite lines, both of magnitude r, where the angle between them is x (we take x to be small). If you draw this out, you can see there are "3 areas" you can consider. One is the area enclosed with a straight line joining the two end points, an arc and lastly... | {
"language": "en",
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convert ceil to floor Mathematically, why is this true?
$$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a+b-1}{b}\right\rfloor$$
Assume $a$ and $b$ are positive integers.
Is this also true if $a$ and $b$ are real numbers?
| Suppose, $b\nmid a$. Let $\lfloor\frac{a+b-1}{b}\rfloor=n \implies nb+r=a+b-1$ where $0\leq r<b$. So $\frac{a}{b}+1=n+\frac{r+1}{b}$. So $\lfloor\frac{a}{b}+1\rfloor=\lceil\frac{a}{b}\rceil=n$.
This is not true in general as shown by Lyj.
| {
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I don't see how Cauchy's proof of AM $\ge$ GM holds for all cases? I am reading Maxima and Minima Without Calculus by Ivan Niven and on pages $24-26$ he gives Cauchy's proof for the $AM-GM$ . The general idea of the proof is that $P_{n}$ is the proposition $$(a_{1}+a_{2}+\cdot \cdot \cdot a_{n}) \ge n(a_{1} a_{2} \cdot... | We wish to show that the case of the inequality with $n$ non-negative numbers proves the case with $n-1$ numbers. Take any $n-1$ non-negative numbers. We may put any non-negative number for $a_n$ since we already know it to be true in the case of $n$ numbers. We simply choose $a_n=g$ where $g$ is as you defined it.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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helix and covering space of the unit circle Does a bounded helix; for instance $\{(\cos 2\pi t, \sin 2\pi t, t); -5\leq t\leq5\}$ in $\mathbb R^3$ with the projection map $(x,y,z)\mapsto (x,y)$ form a covering space for the unit circle $\mathbb S^1$?
If yes, does it disprove: There is a bijection between the fundamen... | Yes it's a covering map. Since the fundamental group of $S^1$ is $\mathbb{Z}$, and since each fiber is the set of $(\cos(t_0),\sin(t_0),t_0+2\pi k)$, $k \in \mathbb{Z}$, there is no conflict to the bijection you mention.
Note the question has been reformulated and this answer is no longer relevant; I'm awaiting further... | {
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Applications of logic What are some applications of symbolic logic? I tried using Google and Bing but just got a bunch of book recommendations, and links to articles I did not understand.
| It aids mathematicians to explain or describe something i.e. a specific statement in such a way that other mathematicians will understand, also it makes the statement open to manipulation, furthermore it is less time consuming to write an equation than the whole statement. Also it is a universal language which is under... | {
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How to show that there is no condition that can meet two inequalities? Here's an excerpt from Spivak's Calculus, 4th Edition, page 96:
If we consider the function
$$
f(x)= \left\{ \matrix{0, x \text{ irrational} \\
1, x \text{ rational}} \right.
$$
then, no matter what $a$ is, $f$ does not approach an number $l$ ne... | When dealing with equations including absolute values, it's often convenient to rewrite them as equations without absolute values.
In your case,
$$|0-l| < \frac{1}{4}$$
gives:
$$-\frac{1}{4} < l < \frac{1}{4}$$
And
$$|1-l| < \frac{1}{4}$$
gives:
$$\frac{3}{4} < l < \frac{5}{4}$$
And now it's easy to spot the contradi... | {
"language": "en",
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Relation between integral and distribution Let $\mu$ be a (positive) measure on $\mathbb{R}^d$ and $f$ be a $\mu$-measurable function on $\mathbb{R}^d$. How to prove that
\begin{equation}
\int_{\mathbb{R}^d} |f(x)|^p d\mu(x)=p\int_{0}^{\infty} \gamma^{p-1} \mu(\{x\in \mathbb{R}^d:|f(x)|>\gamma\}) d\gamma
\end{equation... | You can do it using the Indicator-Fubini trick. Here's how it goes:
$$\int_{\mathbb{R}^d}|f(x)|^pd\mu(x) = \int_{\mathbb{R}^d}\int_0^{|f(x)|}pu^{p-1} dud\mu(x)$$
$$ = \int_{\mathbb{R}^d}\int_0^{\infty}1_{\{u < |f(x)|\}}pu^{p-1} dud\mu(x)$$
By Fubini Theorem,
$$ = \int_0^{\infty}pu^{p-1}\int_{\mathbb{R}^d}1_{\{u < |f(x)... | {
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Number Theory divisibilty How can I check if $$12^{2013} + 7^{2013}$$ is divisible by $19$?
Also, how can I format my questions to allow for squares instead of doing the ^ symbol.
| Proposition : $a+b$ divides $a^m+b^m$ if $m$ is odd
Some proofs :
$1:$ Let $a+b=c,$
$a^m+b^m=a^m+(c-a)^m\equiv a^m+(-a)^m\pmod c\equiv \begin{cases} 2a^m &\mbox{if } m \text{ is even } \\
0 & \mbox{if } m \text{ is odd } \end{cases}\pmod c $
$2:$ If $m$ is odd, $a^m+b^m=a^m-(-b)^m$ is divisible by $a-(-b)=a+b$
as $\fr... | {
"language": "en",
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characteristic prime or zero Let $R$ be a ring with $1$ and without zero-divisors. I have to show that the characteristic of $R$ is a prime or zero. This is my attempt:
This is equivalent to finding the kernel of the homomorphism $f\colon \mathbb{Z}\rightarrow R$ which has the form $n\mathbb{Z}$. There are two cases. S... | The claim being proved here is not true; it fails to hold for the zero ring. It doesn't have nontrivial zero divisors (because it doesn't have nontrivial elements at all), yet its characteristic $1$ is neither zero nor prime.
The flaw in the proof is that it assumes that just because $m-n$ is positive and nonprime, it ... | {
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Is $x^2-y^2=1$ Merely $\frac 1x$ Rotated -$45^\circ$? Comparing their graphs and definitions of hyperbolic angles seems to suggest so aside from the $\sqrt{2}$ factor:
and:
| Almost. $$x^2-y^2=1\iff (x+y)(x-y)=1$$
By rotating by $-45^\circ$ you move the point $(x,y)$ to $(\hat x,\hat y) = \frac1{\sqrt 2}(x-y,x+y)$, so what you really get is $\hat y = \frac1{\hat x\sqrt 2}$
| {
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Prove that the angle $\theta=\arccos(-12/17)$ is constructible using ruler and compass. Could I just do this?
Proof: if we want to show $\arccos\left(\frac{-12}{15}\right)$ is constructible, can't I just say, take $x_0=\cos(\theta)=-\frac{12}{17}$ implies $17x_0+12=0$ which says that $f(x_0)=17x+0+12$ is a polynomial w... | The way I would do it is with a triangle whose three sides are $51$, $92$, and $133$. Then Law of Cosines.
| {
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Do groups, rings and fields have practical applications in CS? If so, what are some? This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be applied (directly or indirectly) or has g... |
I always thought they were useful (instinctively) but failed to see where/how. Are they just theoretical concepts without practical applications?
I'm 100% sure you've written programs $p_1,p_2,p_3$ before, which take data $\mathrm{in}$ in and after the code did what it should it you get computed data $\mathrm{out}$ a... | {
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Geometric series sum $\sum_2^\infty e^{3-2n}$ $$\sum_2^\infty e^{3-2n}$$
The formulas for these things are so ambiguous I really have no clue on how to use them.
$$\frac {cr^M}{1-r}$$
$$\frac {1e^2}{1-\frac{1}{e}}$$
Is that a wrong application of the formula and why?
| The sum of a geometric series:
$$\sum_{k=0}^{\infty} ar^k = \frac{a}{1-r}$$
In your case:
$$r = e^{-2},\ a = e^3$$
Just remember to subtract the first two elements..
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Coloring sides of a polygon In how many ways we can color the sides of a $n$-agon with two colors?
(rotation is indistinguishable!)
| We can answer the following equivalent and more general question: how many necklaces with $n$ beads can be formed from an unlimited supply of $k$ distinct beads? Here, two necklaces are considered the same if one can be transformed into the other by shifting beads circularly. To formalize this, define a string $S=s_1s_... | {
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Rational number to the power of irrational number = irrational number. True? I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question.
Now, his inital solution was like this: let's take a ... | Consider $2^{\log_2 3}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Integration $\int \frac{\sqrt{x^2-4}}{x^4}$ Problem :
Integrate $\int \frac{\sqrt{x^2-4}}{x^4}$
I tried : Let $x^2-4 =t^2 \Rightarrow 2xdx = 2tdt$
$\int \frac{\sqrt{x^2-4}}{x^4} \Rightarrow \frac{t^3 dt}{\sqrt{t^2+4}(t^4-8t+16)}$
But I think this made the integral too complicated... please suggest how to proceed.. Tha... | Besides to @Ron's answer, you can see that the form Differential binomial is ruling the integral here. Let to write integrand as follows:
$$\int(x^2-4)^{1/2}x^{-4}~dx$$
So, $m=-4,~~p=1/2,~~n=2$ and so $\frac{m+1}{n}+p=-1\in\mathbb Z$ and so the method says that you can use the following nice substitution:
$$x^2-4=t^{2}... | {
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Zeroes of a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ for $n\geq 2$ Why can the zeros of a holomorphic map $f:\mathbb{C}^n\to \mathbb{C}$ with $n\geq 2$ have no isolated zeros (or poles if we write it as meromorphic)?
Someone says the $n$-times Cauchy Integral formula is enough, but how does it work?
| Ted is of course perfectly correct. :)
Alternatively the claim follows from Hartogs' theorem, https://en.wikipedia.org/wiki/Hartogs%27_extension_theorem, whose proof may be the origin of the Cauchy integral formula hint you received.
| {
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$6$ people, $3$ rooms, $1$ opening door $6$ people spread in $3$ distinguishable rooms, every room needs one person who opens the door.
There are ${6 \choose 3}\cdot 3 \cdot 2$ options to choose the three door opener persons and let them open one certain room, so that every room is opened by one person.
Further, there ... | If you choose an order for the six people and put the first two into room $1$ the next two into room $2$ and the final pair into room $3$, and then you put the first named of each pair as the door-opener, every one of the $6!$ orders of people gives you a unique arrangement of people in accordance with the criteria in ... | {
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Why "integralis" over "summatorius"? It is written that Johann Bernoulli suggested to Leibniz that he (Leibniz) change the name of his calculus from "calculus summatorius" to "calculus integralis", but I cannot find their correspondence wherein Bernoulli explains why he thinks "integralis" is preferable to "summatorius... | If you wanted the correspondence. Maybe, this would be sure, the initial quotation states the exact point. As for the reason, some say that it was to rival Newton on proving that he had invented calculus first. The Bernoulli's too were involved in the controversy.This page explains the controversy.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/449686",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 3,
"answer_id": 1
} |
Compute $\lim_{n\to\infty} nx_n$ Let $(x_n)_{n\ge2}$, $x_2>0$, that satisfies recurrence $x_{n+1}=\sqrt[n]{1+n x_n}-1, n\ge 2$. Compute $\lim_{n\to\infty} nx_n$.
It's clear that $x_n\to 0$, and probably Stolz theorem would be helpful. Is it really necessary to use this theorem?
| Since $(1+x)^n\geqslant nx+1$ we obtain that $x_n\geqslant x_{n+1}$. As the sequence is positive and decreasing, it must converge. Call this limit $\ell$. Consider the non-negative functions $$f_n(x)=\frac{\log(1+nx)}n$$
They have the property that $$\log(1+x_{n+1})=f_n(x_n)$$
Since $x_n$ is decreasing, and since the $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/449748",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
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A problem about the convergence of an improper integral Let $f:\mathbb R\longrightarrow\mathbb R$ be a function with
$$f(x)=\frac{1}{3}\int_x^{x+3} e^{-t^2}dt$$
and consider $g(x)=x^nf(x)$ where $n\in\mathbb Z$.
I have to discuss the convergence of the integral
$$\int_{-\infty}^{+\infty}g(x)dx$$ at the varying o... | Basically, you are considering the integral
$$ I := \frac{1}{3}\int_{-\infty}^{\infty}x^n f(x) dx= \frac{1}{3}\int_{-\infty}^{\infty}x^n \int_{x}^{x+3}e^{-t^2}dt \,dx.$$
Changing the order of integration yields
$$ I = \frac{1}{3}\int_{-\infty}^{\infty}e^{-t^2} \int_{t-3}^{t}x^ndx \,dt $$
$$ = \frac{1}{3}\int_{-\infty}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/449813",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 6,
"answer_id": 4
} |
Why is $1/n^{1/3}$ convergent? I thought because $p<1$ it would be divergent, but apparently not. Why is that?
| You, certainly, know the series $\sum (1/n)$ and know that it is divergent. There is a nice approach in which we can test the divergence or convergence. That is the Quotient Test or Limit comparison test. According to it, if $$\lim_{n\to\infty}\frac{u_n}{v_n}=A\neq0, ~~\text{or}~~ A=\infty$$ then $\sum u_n$ and $\sum ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/449868",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 5,
"answer_id": 2
} |
Men and Women: Committee Selection There is a club consisting of six distinct men and seven distinct women. How many ways can we select a committee of three men and four women?
| There are "$6$-choose-$3$" $=\binom{6}{3}$ ways to select the men, and "$7$-choose-$4$" =$\binom{7}{4}$ ways to select the women. That's because
$(1)$ We have a group of $6$ men, from which we need to choose $3$ for the committee.
$(2)$ We have $7$ women from which we need to choose $4$ to sit in on the committee.
Si... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/449940",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Every bounded function has an inflection point? Hello from a first time user! I'm working through a problem set that's mostly about using the first and second derivatives to sketch curves, and a question occurred to me:
Let $f(x)$ be a function that is twice differentiable everywhere and whose domain is $ \Bbb R$. If ... | If $f''>0$ identically and $f$ is bounded, then $f'\leq 0$ identically (for otherwise, $f(\infty)=\infty$). Likewise, $f'\geq 0$ since otherwise $f(-\infty)=-\infty$. Hence $$f'=0$$ identically, so $f$ is constant. The result follows similarly in the case $f''<0$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/450001",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Simplifying compound fraction: $\frac{3}{\sqrt{5}/5}$ I'm trying to simplify the following:
$$\frac{3}{\ \frac{\sqrt{5}}{5} \ }.$$
I know it is a very simple question but I am stuck. I followed through some instructions on Wolfram which suggests that I multiply the numerator by the reciprocal of the denominator.
The pr... | This means
$$
3\cdot \frac{5}{\sqrt{5}}=3\cdot\frac{(\sqrt{5})^2}{\sqrt{5}}
=3\sqrt{5}
$$
You're multiplying twice for the reciprocal of the denominator.
Another way to see it is multiplying numerator and denominator by the same number:
$$
\frac{3}{\frac{\sqrt{5}}{5}}=\frac{3\sqrt{5}}{\frac{\sqrt{5}}{5}\cdot\sqrt{5}}
=... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450158",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 8,
"answer_id": 2
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Question on monotonicity and differentiability Let $f:[0,1]\rightarrow \Re$ be continuous. Assume $f$ is differentiable almost everywhere and $f(0)>f(1)$.
Does this imply that there exists an $x\in(0,1)$ such that $f$ is differentiable at $x$ and $f'(x)<0$?
My gut feeling is yes but I do not see a way to prove it. Any ... | As stated in another answer the Cantor function is a counterexample.
You would need to assume differentiability for all $x \in (0,1)$. "Almost everywhere" is not good enough.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/450230",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
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Differentiating $\tan\left(\frac{1}{ x^2 +1}\right)$ Differentiate: $\displaystyle \tan \left(\frac{1}{x^2 +1}\right)$
Do I use the quotient rule for this question? If so how do I start it of?
| We use the chain rule to evaluate $$ \dfrac{d}{dx}\left(\tan \frac{1}{x^2 +1}\right)$$
Since we have a function which is a composition of functions: $\tan(f(x))$, where $f(x) = \dfrac 1{1+x^2}$, this screams out chain-rule!
Now, recall that $$\dfrac{d}{dx}(\tan x) = \sec^2 x$$
and to evaluate $f'(x) = \dfrac d{dx}\le... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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$5^m = 2 + 3^n$ help what to do how to solve this for natural numbers $5^m = 2 + 3^n$ i did this
$5^m = 2 + 3^n \Rightarrow 5^m \equiv 2 \pmod 3 \Rightarrow m \equiv 1 \pmod 2$
now if i put it like this
$ 5^{2k+1} = 2 + 3^n $
what to do ??
now is this right another try :
$ m = n \Rightarrow 5^n - 3^n = 2 = 5 -... | I like to reformulate the problem a bit to adapt it to my usances.
$$5^m = 2+3^n \\5^m = 5-3+3^n \\
5(5^{m-1}-1) = 3(3^{n-1}-1) $$
$$ \tag 1 {5^a-1 \over 3} = {3^b-1 \over 5} \\
\small \text{ we let a=m-1 and b=n-1 for shortness}$$
Now we get concurring conditions when looking at powers of the primefactor decomp... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450365",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
"answer_count": 2,
"answer_id": 0
} |
Sum of a geometric series $\sum_0^\infty \frac{1}{2^{1+2n}}$ $$\sum_0^\infty \frac{1}{2^{1+3n}}$$
So maybe I have written the sequence incorrectly, but how do I apply the $\frac{1}{1 - r}$ formula for summing a geometric sequence to this? When I do it I get something over one which is wrong because this is suppose to m... | This is
$$
\begin{align}
\frac12\sum_{n=0}^\infty\frac1{4^n}
&=\frac12\frac1{1-\frac14}\\
\end{align}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/450589",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
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$\lim_{(x,y)\to (0,0)} \frac{x^m y^n}{x^2 + y^2}$ exists iff $m+ n > 2$ I would like to prove, given $m,n \in \mathbb{Z}^+$, $$\lim_{(x,y)\to (0,0)}\frac{x^ny^m}{x^2 + y^2} \iff m+n>2.$$
(My gut tells me this should hold for $m,n \in \mathbb{R}^{>0}$ as well.) The ($\Rightarrow$) direction is pretty easy to show by con... | If $m+n>2$, you can divide into two cases, by observing that you can't have $m<2$ and $n<2$.
First case: $m\ge2$.
$$
\lim_{(x,y)\to (0,0)}\frac{x^ny^m}{x^2 + y^2}
=
\lim_{(x,y)\to (0,0)}\frac{y^2}{x^2 + y^2}x^ny^{m-2}
$$
where $n\ge1$ or $m-2\ge1$. The fraction is bounded, while the other factor tends to zero.
Similarl... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450669",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Ratio test: Finding $\lim \frac{2^n}{n^{100}}$ $$\lim \frac{2^n}{n^{100}}$$ as n goes to infinity of course.
I know that the form os $\frac{a_{n+1}}{a_n}$
$$\frac {\frac{2^{n+1}}{(n+1)^{100}}}{\frac{2^n}{n^{100}}}$$
$$\frac{2n^{100}}{(n+1)^{100}}$$
I am not clever enough to evaluate that limit. To me it looks like it s... | In
Prove that $n^k < 2^n$ for all large enough $n$
I showed that
if $n$ and $k$ are integers
and $k \ge 2$
and
$n \ge k^2+1$,
then
$2^n > n^k$.
Set $k = 101$.
Then, for $n \ge 101^2+1 = 10202$,
$2^n > n^{101}$
or
$\dfrac{2^n}{n^{100}} > n$
so $\lim_{n \to \infty} \dfrac{2^n}{n^{100}} = \infty$.
This obviously shows th... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450733",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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divisibility test let $n=a_m 10^m+a_{m-1}10^{m-1}+\dots+a_2 10^2+a_1 10+a_0$, where $a_k$'s are integer and $0\le a_k\le 9$, $k=0,1,2,\dots,m$, be the decimal representation of $n$
let $S=a_0+\dots+a_m$, $T=a_0-a_1\dots+(-1)^ma_m$
then could any one tell me how and why on the basis of divisibility of $S$ and $T$ by $2,... | If $m=1$ then $S$ and $T$ uniquely determine the number. In any case $S$ determines the number mod $3$ and $9$, and $T$ determines the number mod $11$, as André pointed out. Unfortunately, as André also pointed out, even the combination of $S$ and $T$ does not generally determine divisibility by any of the numbers $2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450818",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 2
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Power series for $(1+x^3)^{-4}$ I am trying to find the power series for the sum $(1+x^3)^{-4}$ but I am not sure how to find it. Here is some work:
$$(1+x^3)^{-4} = \frac{1}{(1+x^3)^{4}} = \left(\frac{1}{1+x^3}\right)^4 = \left(\left(\frac{1}{1+x}\right)\left(\frac{1}{x^2-x+1}\right)\right)^4$$
I can now use
$$\frac... | You could use the binomial expansion as noted in the previous answer but just for fun here's an alternative
Note that:
$$ \dfrac{1}{\left( 1+y \right) ^{4}}=-\dfrac{1}{6}\,{\frac {d^{3}}{d{y}^{3}}}
\dfrac{1}{ 1+y } $$
and that the geometric series gives:
$$ \dfrac{1}{ 1+y }=\sum _{n=0}^{\infty } \left( -y \right) ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450900",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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How to evaluate a zero of the Riemann zeta function? Here is a super naive question from a physicist:
Given the zeros of the Riemann zeta function,
$\zeta(s)
= \sum_{n=1}^\infty n^{-s}$,
how do I actually evaluate them?
On this web page I found a list of zeros. Well I guess if the values, call one ${a_\text{zer... | The Riemann Zeta function has zeros as follows:
(1) From the Euler Product follows $\zeta(s)\not=0$ for $\mathrm{Re}\,s>1$.
Taking the functional equation into account results that the only zeros outside the critical strip $\{ s\in\mathbb C\mid 0\leq\mathrm{Re}\,s\leq1\}$ are the trivial zeros $-2,-4,-6,\ldots$
(2) Bey... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/450972",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
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What is the meaning of $(2n)!$ I came across something that confused me $$(2n)!=?$$ What does this mean: $$2!n!, \quad 2(n!)$$ or $$(2n)!=(2n)(2n-1)(2n-2)...n...(n-1)(n-2)...1$$
Which one is right?
The exercise is to show that $$(n+1)\bigg|\left(\begin{array}{c}2n\\n\end{array}\right)$$Then I thought of using the co... | Hint: You can verify by a computation that
$$\frac{1}{n+1}\binom{2n}{n}=\binom{2n}{n}-\binom{2n}{n+1}.$$
Detail: We have
$$\frac{1}{n+1}\binom{2n}{n}=\frac{1}{n(n+1)}\frac{(2n)!}{(n-1)!n!}=\left(\frac{1}{n}-\frac{1}{n+1}\right) \frac{(2n)!}{(n-1)!n!} .$$
Now
$$\frac{1}{n}\frac{(2n)!}{(n-1)!n!} =\binom{2n}{n}\qquad\t... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 0
} |
How to find minimum of sum of mod functions? How to find minimum value of $$|x-1| + |x-2| + |x-31| + |x-24| + |x-5| + |x-6| + |x-17| + |x-8| + \\|x-9| + |x-10| + |x-11| + |x-12|$$ and also where it occurs ?
I know the procedure for find answer for small problems like the following
what is the minimum value of |x-2| ... | Let's make it generic, you want to minimise
$$f(x) = \sum_{k = 1}^n \lvert x - p_k\rvert,$$
where, without loss of generality, $p_1 \leqslant p_2 \leqslant \ldots \leqslant p_n$.
How does the value of $f(x)$ change if you move $x$
*
*left of $p_1$,
*between $p_k$ and $p_{k+1}$,
*right of $p_n$?
A simple counting... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451201",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Solving for radius of a combined shape of a cone and a cylinder where the cone is base is concentric with the cylinder? I have a solid that is a combined shape of a cylinder and a concentric cone
(a round sharpened pencil would be a good example)
Know values are:
Total Volume = 46,000
Height to Base Ratio = 2/1
(Heigh... | Cone height = $ \sqrt 3 R$.
Cylinder height = Total height - Cone heght
$$= (4 -\sqrt3 R ) $$
Total Volume
$$ V_{total} = V_{cone}+ V_{cyl } = \pi R^2 [ (4- \sqrt 3) R ] + \frac\pi3 R^2 \cdot \sqrt3 R = 46000. $$
from which only unknown $R^3, R $ can be calculated.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/451277",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
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Double integrals over general regions Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order and explain why it's easier. $$\int\int_{D}{ydA}, \text{$D$ is bounded by $y=x-2, x=y^2$}$$
I'm having trouble with setting up the integral for both type I and type 2 ... | Plotting both $y=x-2$ and $x=y^2$ in wolfram alpha:
Based upon the points of intersection, you can see that the region D exists for $(x,y)$ with $x\in[1,4]$ and $y\in[-1,2]$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/451336",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
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Lower bound on the probability of maximum of $n$ i.i.d. chi-square random variables exceeding a value close to their number of degrees of freedom I am wondering if there is a tight lower bound on the probability of a maximum of $n$ i.i.d. chi-square random variables, each with degree of freedom $d$ exceeding a value cl... | I'm not sure if this is helpful, but taking a look at what Wikipedia has to say about the cdf of the Chi-Square distribution, it appears that
\begin{align*}
P[X < d+\delta]^n
&= \left( \left( \frac{d+\delta}{2}\right)^{d/2} e^{-d/2}
\sum_{k=0}^\infty \frac{\left(\frac{d+\delta}{2}\right)^k}{\Gamma(d/2 + k + 1)}... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451413",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Help with $\lim_{x \to -1} x^{3}-2x=1$ I believe that I completed this problem correctly but I could use a second set of eye's to verify that I used the right methods. Also if you have a suggestion for a better method of how to solve this I would appreciate any advice.
Prove $$\lim_{x \to -1} x^{3}-2x=1$$
Given $\epsi... | Your method looks like taken "out of the blue": why did you choose $\;\delta\,$ as you did? Why did you do that odd-looking calculations in the third line (which I didn't understand right away, btw)?.
I propose the following: for an arbitrary $\,\epsilon>0\,$:
$$|x^3-2x-1|=|(x+1)(x^2-x-1)|<\epsilon\iff |x+1|<\frac\epsi... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451479",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Probability of getting a certain sum of two dice; confusion about order If you roll two six-sided dice, the probability of obtaining a $7$ (as a sum) is $6/36$.
Here is what is confusing me. Aren't $(5,2)$ and $(2,5)$ the same thing? So we shouldn't really double count?
Thus by that logic, wouldn't the actual answer be... | We can perfectly well decide that the outcomes are double $1$, a $1$ and a $2$, double $2$, and so on, as in your proposal. That would give us $21$ different outcomes, not $36$.
However, these $21$ outcomes are not all equally likely. So although they are a legitimate collection of outcomes, they are not easy to work w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451579",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Is there a name for this type of logical fallacy? Consider a statement of the form: $A$ implies $B$, where $A$ and $B$ are true, but $B$ is not implied by $A$.
Example: As $3$ is odd, $3$ is prime.
In this case, it is true that $3$ is odd, and that $3$ is prime, but the implication is false. If $9$ had been used inst... | This is not strictly a fallacy; this is a result from the truth-table definition of implication, in which a statement with a true conclusion is necessarily given the truth-value true, i.e., for any formula $A\rightarrow B$ , if B is given the truth-value T, then, by the truth-table definition of $ \rightarrow $, it fol... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451646",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 7,
"answer_id": 5
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Find the projection of the point on the plane I want to find the projection of the point $M(10,-12,12)$ on the plane $2x-3y+4z-17=0$. The normal of the plane is $N(2,-3,4)$.
Do I need to use Gram–Schmidt process? If yes, is this the right formula?
$$\frac{N\cdot M}{|N\cdot N|} \cdot N$$
What will the result be, vector... | You can use calculus to minimize the distance (easier: the square of the distance) of M from the generic point of the plane.
Use the equation of the plane to drop a variable, obtaining a function of two independent variables: compute the partial derivatives and find the stationary point. That's all.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/451722",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
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"answer_id": 1
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How to show $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$? I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule.
The problem is I don't quite like the solution, as it depends ... | The given limit is $$\lim_{x\rightarrow 1}\frac{\sum_{k=1}^nx^k-n}{x-1}\\
=\lim_{x\rightarrow 1}\frac{\sum_{k=1}^n(x^k-1)}{x-1}\\
=\sum_{k=1}^n \lim_{x\rightarrow 1} \frac{(x^k-1)}{x-1}$$
Now, $$\lim_{x\rightarrow 1} \frac{(x^k-1)}{x-1}\\
=\lim_{x\rightarrow 1} (\sum_{j=0}^{k-1}x^j)=k$$
Hence the given limit becomes $$... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451799",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "49",
"answer_count": 8,
"answer_id": 5
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Is taking cokernels coproduct-preserving? Let $\mathcal{A}$ be an abelian category, $A\,A',B$ three objects of $\mathcal{A}$ and $s: A\to B$, $t: A' \to B$ morphisms.
Is the cokernel of $(s\amalg t): A\coprod A'\to B$ the coproduct of the cokernels of $s$ and $t$?
In case it's wrong: Is it true if we restrict $s$ and $... | Colimits preserve colimits, so colimits do preserve coproducts, and cokernels are colimits. However, this means something different than what you suggest.
Usually, $s \coprod t$ is used to mean the morphism $A \coprod A' \to B \coprod B$; with this meaning, we do have
$$ \text{coker}(s \coprod t) = \text{coker}(s) \cop... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451891",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Angles between two vertices on a dodecahedron Say $20$ points are placed across a spherical planet, and they are all spaced evenly, forming the vertices of a dodecahedron. I would like to calculate the distances between the points, but that requires me to find out the angles between the vertices.
From the origin of the... | It is maybe not a good habit to use short-cut formulas/constructions from Wikipedia, since a real mathematician should construct his geometric or analytic geometric figures herself/himself, however, I believe that it is necessary when you have not enough time.
Wikipedia already constructed a dodec with 20 vertices: $(... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/451943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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how to draw graphs of ODE's In order to solve this question How to calculate $\omega$-limits I'm trying to learn how to draw graphs of ODE's. For example, let $p\in \mathbb R^2$ in the case of the field $Y=(Y_1, Y_2)$, given by:
$Y_1=-y_2+y_1(y_1^2+y_2^2)\sin\left(\dfrac{\pi}{\sqrt{y_1^2+y_2^2}}\right)$
$Y_2=y_1+y_2(y_... | This is a nice example of what a nonlinear term can do to a stable, but not asymptotically stable, equilibrium. It helps to introduce the polar radius $\rho=\sqrt{y_1^2+y_2^2}$, because this function satisfies the ODE
$$\frac{d\rho }{dt} = \frac{y_1}{\rho}\frac{dy_1}{dt}+\frac{y_2}{\rho}\frac{dy_2}{dt} = \rho^3\sin \f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452007",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
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Prove that $\sqrt 2 + \sqrt 3$ is irrational I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. My first question is, is this reasoning correct?
Secondly, the book wants me... | If $\sqrt 3 +\sqrt 2$ is rational/irrational, then so is $\sqrt 3 -\sqrt 2$ because $\sqrt 3 +\sqrt 2=\large \frac {1}{\sqrt 3- \sqrt 2}$ . Now assume $\sqrt 3 +\sqrt 2$ is rational. If we add $(\sqrt 3 +\sqrt 2)+(\sqrt 3 -\sqrt 2)$ we get $2\sqrt 3$ which is irrational. But the sum of two rationals can never be irrati... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452078",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "43",
"answer_count": 9,
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Understanding "Divides" aka "|" as used logic What are the rules for using the divides operator aka "$\mid$"? Is it false to say $2\mid5$ since $5/2$ = $2.5$ and $2.5\notin\mathbb{Z}$? Or does my question imply a misunderstanding?
I am seeing this for the first time in the 6.042J, Lecture 2: Induction on MIT OCW.
Thank... | That's correct. We use the divide operator to denote the following:
We say for two integers $a$, $b$, that $a$ divides $b$ (or $a|b$) if $b = ka$ for some integer $k$.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "9",
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Given a rational number $p/q$, show that the equation $\frac{1}{x} + \frac{1}{y} = \frac{p}{q}$ has only finite many positive integer solutions. How can i solve this,
Given a rational number $p/q$, show that the equation $\frac{1}{x} + \frac{1}{y} = \frac{p}{q}$ has only finite many positive integer solutions.
I thoug... | Hint: A solution always satisfies
$$ \frac{x + y}{xy} = \frac{p}{q} $$.
Conclude that $xy = r q$ for some positive integer $r$. Now what?
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/452213",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Closure of a certain subset in a compact topological group Suppose that $G$ is a compact Hausdorff topological group and that $g\in G$. Consider the set $A=\{g^n : n=0,1,2,\ldots\}$ and let $\bar{A}$ denote the closure of $A$ in $G$.
Is it true that $\mathbf{\bar{A}}$ is a subgroup of $\mathbf{G}$?
From continuity of m... | I really like Fischer's solution. I would like to post an answer based on Fischer's answer in the "filling the blank" spirit - certainly helpful for beginners like me.
We need to get help from the two theorems below (refer to section 1.15 in Bredon's Topology and Geometry):
*
*In a topological group $G$ with unity e... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Solving for $x$: $3^x + 3^{x+2} = 5^{2x-1}$ $3^x + 3^{x+2} = 5^{2x-1}$
Pretty lost on this one. I tried to take the natural log of both sides but did not get the result that I desire.
I have the answer but I would like to be pointed in the right direction. Appreciated if you can give me some hints to this question, th... | Just take $$3^x$$ as common factor on right side then
$$3^x(1+3^2)=5^{2x-1}\Rightarrow ln({3^x(1+3^2)}) = ln(5^{2x-1})\Rightarrow xln(3) + ln(10) = (2x-1)ln(5)\Rightarrow ln(10)+ln(5)=2xln(5)-xln(3)\Rightarrow x=\frac{ln(10)+ln(5)}{2ln(5)-ln(3)}$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/452346",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 3
} |
show numbers (mod $p$) are distict and nonzero Let's start with the nonzero numbers, mod $p$,
$1$, $2$, $\cdots$, $(p-1)$,
and multiply them all by a nonzero $a$ (mod $p$).
Notice that if we multiply again by the inverse of $a$ (mod $p$) we get back the numbers
$1$, $2$, $\cdots$, $(p-1)$.
But my question is how the a... | It is important to point out that $\Bbb Z_p$ is a field if (and only if) $p$ is prime. This means in particular that every element that is not $0$ has an inverse. Since $p\not\mid a$ is equivalent to $a\not\equiv 0\mod p$, $a$ has an inverse $a^{-1}$. But then $$x\equiv y\mod p\iff ax\equiv ay\mod p$$
since we can rev... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452432",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 1
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we need to show gcd is $1$ I need to show if $(a,b)=1,n$ is an odd positive integer then $\displaystyle \left(a+b,{a^n+b^n\over a+b}\right)\mid n.$
let $\displaystyle \left(a+b,{a^n+b^n\over a+b}\right)=d$
$\displaystyle d\mid {a^n+b^n\over a+b}=(a+b)^2(a^{n-3}-a^{n-4}b\dots+b^{n-3})-2ab(a^{n-3}\dots+b^{n-3})-ab^{n-2}... | Modulo $a+b$, you have $a^kb^{n-k}\equiv (-1)^kb^n$, so
$$ \frac{a^n+b^n}{a+b}=a^{n-1}-a^{n-2}b+a^{n-3}b^2\mp\cdots -ab^{n.2}+b^{n-1}\equiv n$$
so $n=\frac{a^n+b^n}{a+b}+(\ldots)\cdot (a+b)$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/452510",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How to calculate weight of positive and negative values. We have used formula formula to calculate weight as,
$$ w_1 = \frac{s_1}{s_1 + s_2 + s_3};$$
$$ w_2 = \frac{s_2}{s_1 + s_2 + s_3};$$
$$ w_3 = \frac{s_3}{s_1 + s_2 + s_3};$$
However, their is possibility of negative and positive numbers. Even all can be negative o... | I'm a little confused by your question. When you say the weight can be positive or negative, do you mean just the value can be negative or positive because of the measurement technique, or is it actually a negative weight? I would assume the first (for example, if you slow down really fast in an elevator, and measure y... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452566",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Which of these numbers is greater: $\sqrt[5]{5}$ or $\sqrt[4]{4}$? I know that this is the question of elementary mathematics but how to logically check which of these numbers is greater: $\sqrt[5]{5}$ or $\sqrt[4]{4}$?
It seems to me that since number $5$ is greater than $4$ and we denote $\sqrt[5]{5}$ as $x$ and $\sq... | The function $$n^{1/n}=e^{(1/n)\log n}$$ goes to $1$ as $n$ becomes infinite. Also, taking derivatives shows that it is monotonically decreasing whenever $n>e.$ If we consider only integers now, we have that for $n=4,5,6,\ldots,$ the sequence decreases.
It follows that $$4^{1/4}>5^{1/5}.$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/452635",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
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Closed representation of this integral I was wondering whether there is some easy closed representation for $\int_R^{\infty} e^{-k(r-R)}r^{l+1} dr$, where $l\in \mathbb{N_0}$ $k>0$ and $R>0$.
| $$ I=\int_R^{\infty} e^{-k(r-R)}r^{l+1} dr= e^{kR}\int_R^{\infty} e^{-kr}r^{l+1} dr. $$
Now, using the change of variables $kr=t$ gives
$$ = e^{kR}\int_R^{\infty} e^{-kr}r^{l+1} dr = \frac{e^{kR}}{k^{l+2}}\int_{Rk}^{\infty} e^{-t}t^{l+1} dr $$
$$I = \frac{e^{kR}}{k^{l+2}}\Gamma( l+2, Rk ),$$
where $ \Gamma(s,x) $ is ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452702",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Prove that every closed ball in $\Bbb R^n$ is sequentially compact. Question:
Prove that every closed ball in $\Bbb R^n$ is sequentially compact.
A subset $E$ of $\Bbb R^n$ is said to be squentially compact $\iff$ every sequence $x_k\in E$ has convergent subsequence whose limit belongs to $E$
Solution:
Let $B_R(a)... | First, note that a closed ball is a closed set.
To prove this, let $E\subset \mathbb{R}$ is a closed ball of radius $R$ around $y$. Let $x\in E^{C}$. Now, let $r=d(x,y)-R>0$ and $z\in B_{r}(x)$. Then $$d(y,x)\le d(y,z)+d(z,x)<d(y,z)+r\\
\Rightarrow d(y,z)>R$$ Hence $z\notin E$ and hence the open ball $B_r(x)$ is total... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452789",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
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Finding a point within a 2D triangle I'm not sure how to approach the following problem and would love some help, thanks!
I have a two-dimensional triangle ABC for which I know the cartesian coordinates of points $A$, $B$ and $C$.
I am trying to find the cartesian coordinates of a point $P$. I know the lengths (distanc... | Let $(x,y)$ be the coordinate of $P$ and $(x_i,y_i)$ be the coordinates of $A,B,C$ respec. $(i=1,2,3)$ Since you know the lengths of $PA, PB,PC$, you'll get three equations like $$(x-x_i)^2+(y-y_i)^2=d_{i}^2,\ i=1,2,3$$ Then, subtract the equation $i$ from equation $j$ to get something like $$x(x_i-x_j)+y(y_i-y_j)+x_i^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452845",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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"answer_id": 0
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Testing for convergence of this function For the integral
$$\int_2^\infty \dfrac{x+1}{(x-1)(x^2+x+1)}dx .$$
Can I know if it's convergent or not? If it does can I know how to evaluate it?
I tried to use $u$ substitution but it didn't work.
| Hint: Apply partial fractions.
$$\frac{x+1}{(x-1)(x^2+x+1)}=\frac A{x-1}+\frac{Bx+C}{x^2+x+1}$$
$$x+1=(A+B)x^2+(A-B+C)x+(A-C)$$
$$\therefore A=\frac23,B=-\frac23,C=-\frac13$$
Now we know that: $$\int \frac{x+1}{(x-1)(x^2+x+1)}dx=\frac23\int\frac1{x-1}dx-\frac13\int\frac{2x-1}{x^2+x+1}dx\\ =\frac23\ln|x-1|-\frac13\ln(x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452930",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 4,
"answer_id": 0
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The Effect of Perspective on Probability My friend and I are tearing each other to bits over this, hope someone can help.
Coin flip experiment:
Define a single trial as 10 coin flips of a fair coin. Perform an arbitrarily large number of trials. At some number of trials n, you notice that your distribution is extremely... | If you want to consider something really interesting, consider that as you add more flips, the probability of getting exactly half heads and tails goes down and this isn't that hard to compute mathematically as the sequence is the fraction of $2n \choose n$ divided by $(2n)^2$ for 2n flips. The denominator will grow a... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/452980",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 4,
"answer_id": 2
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Fourier Transform of short pulse I'm trying to take the fourier transform of a short laser pulse, represented by $E(t) = E_oe^{-(t/\Delta T)^2}\times e^{-i \omega t}$
E is the electric field of the laser pulse. $E_o$ and $\Delta T$ are both constants. Specifically I want to know if there are beats in the fourier transf... | The $e^{-i\omega t}$ factor in your pulse means only a phase delay by $\omega t$, otherwise the pulse is a Gaussian pulse which means that the Fourier transform will also be Gaussian and it will be $$\large \mathcal{E}(f)=E_0\Delta T\sqrt{\pi}e^{-\Delta T^2 \pi^2\left(f+\frac{\omega}{2\pi}\right)^2}$$
So, the power w... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/453062",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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Limit of $n^2\sqrt{1-\cos(1/n)+\sqrt{1-\cos(1/n)+\ldots}}$ when $n \to \infty$ Compute the limit:
$$\lim_{n \to \infty} n^2\sqrt{1-\cos(1/n)+\sqrt{1-\cos(1/n)+\sqrt{1-\cos(1/n)+\ldots}}}$$
| Hints:
*
*For every $a\gt0$, $b=\sqrt{a+\sqrt{a+\sqrt{a+\cdots}}}$ is such that $b^2+a=b$ and $b\gt0$, thus $b=\frac12+\frac12\sqrt{1+4a}$.
*When $n\to\infty$, $1-\cos(1/n)\to0$.
*When $a\to0$, $\frac12+\frac12\sqrt{1+4a}\to1$.
*Hence the limit you are after is $\lim\limits_{n\to\infty}n^2\cdot1=+\infty$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/453139",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$ show that
$$\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$$
using different ways
thanks for all
| Using the formula found in my answer,
$$
\begin{aligned}
\int_{0}^{\infty} \frac{\sin ^{3} x}{x^{3}} &=\frac{\pi}{2^{3} \cdot 2 !}\left[\left(\begin{array}{l}
3 \\
0
\end{array}\right) 3^{2}-\left(\begin{array}{l}
3 \\
1
\end{array}\right) 1^{2}\right] \\
&=\frac{3 \pi}{8}
\end{aligned}
$$
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/453198",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 8,
"answer_id": 7
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On the Space of Continuous Linear Operators on LCTVS Suppose that $X$ is a locally convex topological vector space (LCTVS) and that $L(X)$ denotes the space of all continuous linear operators on $X$.
Question. How can we construct a topology on $L(X)$ which compatible with the vector space structure of $L(X)$?
I need ... | There is no natural topology on the dual space of a locally convex space unless the original space has a norm. In this case, if the original space is complete, then the $\sup_{B}$ norm is a norm on the dual space. Here $B$ is the unit ball in the original space.
I think the conventional way to endow a topology on the ... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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"answer_id": 1
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One-to-one functions between vectors of integers and integers, with easily computable inverses I'm trying to find functions that fit certain criteria. I'm not sure if such functions even exist. The function I'm trying to find would take vectors of arbitrary integers for the input and would output an integer. It is one-... | What you need is a bijection $\alpha : \mathbb{N}^2 \to \mathbb{N}$ which is easy to compute in both forward and backward directions. For example,
$$\begin{align}
\mathbb{N}^2 \ni (m,n)
& \quad\stackrel{\alpha}{\longrightarrow}\quad \frac{(m+n)(m+n+1)}{2} + m \in \mathbb{N}\\
\mathbb{N} \ni N
& \quad\stackrel{\alpha... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/453436",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically $0$. Let $I$ be a generalized rectangle in $\Bbb R^n$
Suppose that the function $f\colon I\to \Bbb R$ is continuous. Assume that $f(x)\ge 0$, $\forall x \in I$
Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically... | This may be only a minor variation on an earlier answer, but maybe it adds something.
Suppose there's some point $x_0$ where $f(x_0)>0$. Let $\varepsilon=f(x_0)/2$. Then by continuity, there is some $\delta>0$ such that for $x$ in the open interval with endpoints $x_0\pm\delta$, the distance between $f(x)$ and $f(x_0... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Orientation of manifold given by external normal field
Consider the unit sphere $S^1$ of $\mathbb{R}^2$. This is a 1-dimensional manifold. And an orientation $\sigma$ of $S^1$ is given by the orientated atlas $\left\{\phi_1,\phi_2\right\}$ with the maps
$$
\phi_1\colon (-\pi,\pi)\to S^1\setminus (-1,0), t\mapsto ... | I think you want to show that, if you pick a point $p\in S^{1}\subset\mathbb{R}^{2}$, then $\nu(p),v(p)$ is positively oriented in $T_{p}\mathbb{R}^2$, where $v(p)$ is the oriented basis for $T_{p}S^{1}$ you get from the atlas $\{\phi_1,\phi_2\}$. Is this what you meant by "the external normal field $\nu$ is positive ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/453549",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
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Why do I disagree with my calculator? I followed the order of opperations but my answer disagrees with my calulator's.
Problem: $331.91 - 1.03 - 19.90 + 150.00$
Calculator answer: $460.98$
My answer: $162.98$
Why the discrepancy?
| The calculator is correct of course (at least in magnitude of the answer, I haven't actually calculate it, and I won't). I bet you are having problems with those negative terms.
For this particular exercise and infinite more like this one, you can take another approach, more intuitive than just following rules like a ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/453631",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 6,
"answer_id": 1
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Formal (series/sum/derivative...) I have come across a lot of cases where terms such as formal sum rather than simply sum is used, similarly in case of derivatives/infinite series/power series.
As I understand in case of series/sum, the term formal is used when the notion of convergence is not clear.
I would appreciate... | A formal sum is where we write something using a $+$ symbol, or other way normally used for sums, even when there may be no actual operation defined.
An example. Someone may have defined a quaternion as a formal sum of a scalar and a (3-dimensional) vector, for example. Before this definition, at least in that book... | {
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"url": "https://math.stackexchange.com/questions/453690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "12",
"answer_count": 2,
"answer_id": 0
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Determinant of PSD matrix and PSD submatrix inequality I'm reading this paper and in the appendix I see the following statement:
For $A \in R^{m\times m}, B \in R^{n\times m}, C \in R^{n\times n}$,
if $D = \begin{bmatrix}A & B\\B^T & C\end{bmatrix}$ is positive semi-definite then,
$det(D) \leq det(A)det(C)$
This is giv... | Lemma: If $A$ and $B$ are symmetric positive-definite, then $\det(A+B) \geq \det(A)$.
This follows from Sylvester's determinant theorem: if $L$ is a Cholesky factor of $B$,
$$\det(A+B) = \det(A)\det(I + L^TA^{-1}L) \geq \det(A)$$
since $L^TA^{-1}L$ is symmetric positive-definite, and adding $I$ shifts the spectrum by o... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/453840",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
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How is the Inverse Fourier Transform derived from the Fourier Transform? Where does the $\frac{1}{2 \pi}$ come from in this pair?
Please try to explain the Plancherel's theorem and the Parseval's theorem!
$ X(j \omega)=\int_{-\infty}^\infty x(t) e^{-j \omega t}d t$
$ x(t)=\frac{1}{2 \pi} \int_{-\infty}^{\infty} X(j \om... | I am going to "derive" this heuristically, because the question is concerned with the origin of the $1/(2 \pi)$ factor. Questions about integrability, order of integration, limts, etc., are to be smoothed over here (unless, of course, I have erred somewhere in the derivation - then all bets are of course off).
Conside... | {
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"url": "https://math.stackexchange.com/questions/453946",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Is this function convex or not? Is this function convex ?
$$
f(\mathbf y) = { \left| \sum_{i=1}^{K} y_i^2e^{-j\frac{2\pi}Np_il} \right| \over\sum_{i=1}^{K}y_i^2}
$$
where : $ P = \{p_1,p_2,\cdots,p_K\} \subset\{1,2,\cdots,N\} $
I tried to plot it and see whether it is convex or not, but since we are able to plot for t... | No. The function is homogeneous of order zero, i.e., $f(ty)=f(y)$ for any nonzero $t\in\mathbb{R}$, $y\in\mathbb{R}^n$. If such a function is convex, it is necessarily constant. To see this, pick any two linearly independent $x$, $y\in\mathbb{R}^n$, and let $z=x+y$, so that any two of $x$, $y$, $z$ are linearly indepen... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
solve differential equation $x^2\frac{dy}{dx}=2y^2+yx$ How to solve this equation?
$x^2\frac{dy}{dx}=2y^2+yx$
I tried to separate variables, but I always have both $x$ and $y$ on one side of equation.
| HINT:
Divide either sides by $x^2$ to get $$\frac{dy}{dx}=2\left(\frac yx\right)^2+\frac yx\text{ which is a function of } \frac yx$$
So, we can put $\frac yx=v\iff y=vx $ to separate the variables
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/454150",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
What Are the Relations for the Polar Hypercomplex form $a + bi + cj + dk$? Olariu in "Complex Numbers in $N$ Dimensions" has polar hypercomplex numbers described by its generators as
\begin{gather}
\alpha^2 = \beta, \\
\beta^2 = 1, \\
\gamma^2 = \beta, \\
\alpha\beta =\beta\alpha = \gamma, \\
\alpha\gamma =\gamma\alpha... | I will attempt an answer at this point. First, I think you would like us to read section 3.4 which is found at pages 113-137. I make no claim to understand all those results, clearly the author has spent some time developing exponentials and trigonometric functions as well as studying the structure of zero-divisors and... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454204",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
What is $S^3/\Gamma$? Let G is a group and H is a subgroup of G. I know $G/H$ is the quotient space but I have no idea about what $S^3/\Gamma$ is, where $S^3$ is the sphere and $\Gamma$ is a finite subgroup of $SO(4)$. In this case $S^3$ has not structure of a group and $\Gamma$ is not subgroup of the sphere. So, what ... | $\mathrm{SO}(4)$ acts on $S^3$ in the obvious way if we think of $S^3$ as the set of unit vectors in $\Bbb R^4$. Then any subgroup $\Gamma \leq G$ acts on $S^3$ as well. $S^3/\Gamma$ is the the orbit space of this $\Gamma$-action, i.e.
$$S^3/\Gamma = S^3/\!\sim$$
where
$$x \sim y \quad \iff \quad x = g \cdot y \text{ f... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454340",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 2,
"answer_id": 0
} |
GRE test prep question [LCM and divisors]
Let $S$ be the set of all positive integers $n$ such that $n^2$ is a
multiple of both $24$ and $108$. Which of the following integers are
divisors of every integer $n$ in $S$ ?
Indicate all such integers:
$A:12$
$B:24$
$C:36$
$D:72$
The answers are $A$ and $C$
First I took t... | Let $n=p_1^{a_1}\cdots p_r^{a_r}$ where the $p_i$ are primes, so $n^2=p_1^{2a_1}\cdots p_r^{2a_r}$. As you observed, $n^2$ must be a multiple of $LCM(24, 108)=2^{3} 3^{3}$, so $2a_1\ge 3$ and $2a_2\ge 3$ with $p_1=2$ and $p_2=3$.
Therefore $a_1\ge 2$ and $a_2\ge 2$, so n is a multiple of $2^{2} 3^{2}=36$. Thus S con... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454409",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
show that $\int_{0}^{\pi/2}\tan^ax \, dx=\frac {\pi}{2\cos(\frac{\pi a}{2})}$ show that $$\int_{0}^{\pi/2}\tan^ax \, dx=\frac {\pi}{2\cos(\frac{\pi a}{2})}$$
I think we can solve it by contour integration but I dont know how.
If someone can solve it by two way using complex and real analysis its better for me.
thanks... | Sorry for being late
$$
\begin{aligned}
\int_0^{\frac{\pi}{2}} \tan ^a x d x&=\int_0^{\frac{\pi}{2}} \sin ^a x \cos ^{-a} x d x \\
& =\int_0^{\frac{\pi}{2}} \sin ^{2\left(\frac{a+1}{2}\right)-1} x \cos ^{2\left(\frac{-a+1}{2}\right)-1} x d x \\
& =\frac{1}{2} B\left(\frac{a+1}{2}, \frac{-a+1}{2}\right) \\
& =\frac{1}{2... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 2
} |
Double integral in polar coordinates Use polar coordinates to find the volume of the given solid inside the sphere $$x^2+y^2+z^2=16$$ and outside the cylinder $$x^2+y^2=4$$
When I try to solve the problem, I keep getting the wrong answer, so I don't know if it's an arithmetic error or if I'm setting it up incorrectly. ... | It's almost correct. Recall that the integrand is usually of the form $z_\text{upper}−z_\text{lower}$, where each $z$ defines the lower and upper boundaries of the solid. As it is currently set up, you are treating the sphere as a hemisphere, where your lower boundary is the $xy$-plane. Hence, you need to multiply by $... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454546",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
In plane geometry is it possible to represent the product of two line segments (p and q) as a line segment? In plane geometry the product of two line segments p and q can be represented as the area of a rectangle with sides p and q. Or at least that is the premise assumed here. Assuming that is correct, the question i... | Yes it is possible.
We are given two line segments, of lengths $a$ and $b$. Draw two say perpendicular lines (it doesn't really matter) meeting at some point $O$. On one of the lines, which we call the $x$-axis, make a point $A$ such that $OA=a$. (Straightedge and compass can do this.) On the other line, which we ca... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454620",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Why is there no explicit formula for the factorial? I am somewhat new to summations and products, but I know that the sum of the first positive n integers is given by:
$$\sum_{k=1}^n k = \frac{n(n+1)}{2} = \frac{n^2+n}{2}$$
However, I know that no such formula is known for the most simple product (in my opinion) - the ... |
These two formulas give n! This was discovered by Euler. Reference this link for further reading. http://eulerarchive.maa.org/hedi/HEDI-2007-09.pdf
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/454689",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "27",
"answer_count": 5,
"answer_id": 3
} |
Why can't $x^k+5x^{k-1}+3$ be factored? I have a polynomial $P(x)=x^k+5x^{k-1}+3$, where $k\in\mathbb{Z}$ and $k>1$. Now I have to show that you can't factor $P(x)$ into two polynomials with degree $\ge1$ and only integer coefficients. How can I show this?
| Assume $k\ge2$.
By the rational root theorem, only $\pm1$ and $\pm3$ are candidates for rational roots - and by inspection are not roots. Therefore,
if $P(x)=Q(x)R(x)$ with $q:=\deg Q>0$, $r:=\deg R>0$, we conclude that $q\ge2$ and $r\ge 2$. Modulo $3$ we have $(x-1)x^{k-1}=x^k-x^{k-1}$, hence wlog. $Q(x)\equiv (x-1)x^... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454756",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Searching sum of constants in a polynomial Consider the polynomial $f(x)=x^3+ax+b$, where $a$ and $b$ are constants. If $f(x+1004)$ leaves a remainder of $36$ upon division by $x+1005$, and $f(x+1005)$ leaves a remainder of $42$ upon division by $x+1004$, what is the value of $a+b$?
| The general knowledge to be used is that if
$$P(x)=(x-a)Q(x)+R$$
where $P$ and $Q$ are polynomials and $R$ a constant. Then, evaluating on $x=a$ we get $P(a)=R$. This is called the Polynomial remainder theorem.
You are looking for $f(1)-1=a+b$.
We have that $f(x+1005)$ gives remainder $42$ after division by $x+1004=x... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454830",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
Is there a closed form solution for partial sums of $1/(2^{2^0}) + 1/(2^{2^1}) + 1/(2^{2^2}) + \ldots$ Title says it all, this is such a classical looking series,
$$\frac1{2^{2^0}} + \frac1{2^{2^1}} + \frac1{2^{2^2}} + \ldots.$$
So, I was just wondering, is there a closed form solution known for the partial sums? If s... | I doubt there's a closed expression for it, but the partial sums arise here:
http://oeis.org/A085010
and in particular the decimal expansion of the infinite series is here:
http://oeis.org/A007404
There is a (paywall) article called "Simple continued fractions for some numbers" where the problem being considered is to ... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454910",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Prove that $P(X)$ has exactly $\binom nk$ subsets of $X$ of $k$ elements each. Let set $X$ consist of $n$ members. $P(X)$ is power set of $X$.
Prove that set $P(X)$ has exactly $$\binom nk = \frac{n!}{k!(n-k)!}$$ subsets of $X$ of $k$ elements each. Hence, show that $P(X)$ contains $2^n$ members.
Hint: use the binomial... | I think some hints might be more appropriate here than actually answering the question:
1) If you take $k$ out of $n$ elements, you automatically also take $n-k$ out of $k$ elements - the other ones. The order of these elements doesn't matter.
2) All subsets of a finite set with $n$ elements have a certain amount of el... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/454974",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 1
} |
$24\mid n(n^{2}-1)(3n+2)$ for all $n$ natural problems in the statement. "Prove that for every $ n $ natural, $24\mid n(n^2-1)(3n+2)$"
Resolution:
$$24\mid n(n^2-1)(3n+2)$$if$$3\cdot8\mid n(n^2-1)(3n+2)$$since$$n(n^2-1)(3n+2)=(n-1)n(n+1)(3n+2)\Rightarrow3\mid n(n^{2}-1)(3n+2)$$and$$8\mid n(n^{2}-1)(3n+2)?$$$$$$
Would n... | Hint:
$(n-1)n(n+1)$ is product of three consecutive numbers. So they are always divisible by $8$ and $3$ if $n$ is odd. You don't even have to bother about $3n+2$. Now think, what happens to $3n+2$ if $n$ is even.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/455043",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Sum of $\sum_2^\infty e^{3-2n}$ $$\sum_2^\infty e^{3-2n}$$
I only have memorized the sum of at index zero. So I reindex
$$\sum_0^\infty e^{-2n-1}$$
This gives me the sum as
$$\frac{1}{1- \frac{1}{e} }$$
This is wrong. Why?
| There is an $e^3$ factor that we can bring to the front. For the rest, we want
$$\sum_{2}^\infty e^{-2n}.\tag{1}$$
It is nice to write out (1) at greater length. It is
$$\frac{1}{e^4}+\frac{1}{e^6}+\frac{1}{e^8}+\frac{1}{e^{10}}+\cdots.\tag(2)$$
Now if you know that when $|r|\lt 1$ then $a+ar+ar^2+ar^3+\cdots=\frac{... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/455094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 5,
"answer_id": 4
} |
An example of a simple module which does not occur in the regular module? Let $K$ be a field and $A$ be a $K$-algebra.
I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem $A$ , as a left regular module, can be written as a direct sum of indecomposable $A$-modules, that is
$A=\oplus_{i=1}^n S_i$ where ea... | Consider the algebra $K[T]$. The simple $K[T]$-modules are of the form $K[T]/P(T)$ for some irreducible polynomial $P$. These do not occur as submodules of $K[T]$, since every such submodule contains a free module, and hence is infinite dimensional over $K$.
| {
"language": "en",
"url": "https://math.stackexchange.com/questions/455223",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 4,
"answer_id": 1
} |
Can a function with just one point in its domain be continuous? For example if my function is $f:\{1\}\longrightarrow \mathbb{R}$ such that $f(1)=1$.
I have the next context:
1) According to the definition given in Spivak's book and also in wikipedia, since $\lim_{x\to1}f$ doesn't exist because $1$ is not an accumulati... | Based on the definitions Spivak gave, I suspect that (as discussed in comments) his definition of continuity is based on the assumption that we're dealing with functions defined everywhere, or at very least having domains with no isolated points. His definition does indeed break down (badly) for functions such as yours... | {
"language": "en",
"url": "https://math.stackexchange.com/questions/455296",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "18",
"answer_count": 6,
"answer_id": 4
} |
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